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ot
0 +
p ><
>) - 0
(2)
9Momentum equation : 0 0 0~ < p > < U~ > +0-~zk < p > < Uk > < U~ > =
o o~ < p > < uku~ > O
-I" 0 :c~l}s~
where the strain tensor
<
Sik > -
(~
Oxk
<
>)
(3)
2 o
For closing this open system, the Jones and Launder's [8] k - e model is chosen as the closure. The equations for the new Reynolds stresses < Ur~U,.k > are : < p > < Ur, U,.k > -
2
--zany: < p > < k > +#t < &k > O
The turbulent viscosity is: # t - <
(4)
p > Ct, ft,
Equations for the turbulent kinetic energy < k > and the dissipation rate < e > are identical to classical ones. It is worth pointing out that Jones and Launder's k - c model was originally developed for the classic approach. However, the incoherent Reynolds stresses < Ur~Ur~ > to be modelled are only a part of the classical Reynolds stresses u~u~, since u~u~ - < u,.,u,.~ > + ~]i~]j. For this reason it is necessary to modify the model. In this work, we simply adjust the constant C, which is directly linked to the turbulent viscosity to 0, 06, a value obtained by numerical tests [7]. The other constants in the equations are kept the same as their original values. This kind of modification is the simplest one. It must be understood as
494 the first stage of fitting classical models to this new decomposition. Nevertheless, we will check that coherent values obtained with this slight modification are close to experimental results. 3. N U M E R I C A L
METHOD
The computation code used is based on the explicit MacCormack scheme which is second order accurate in time and in space. The grid contains 1 2 0 . 100 points, and is non-uniform both in x and y directions. Since the development of the Kelvin-Helmholtz type instabilities in the mixing layer depends directly on the boundary layer in the inlet of the backward facing step, it has to be refined in this region. The grid is also stretched near the solid wall in order to allow the application of the near-wall damping functions in the turbulence model. In the inlet section (x = - 1 H ) , the velocity profile < U(y) > is obtained by interpolating the experimental results. However, From the wall (y = 0) to the first measurement point it is obtained from the simulation of a flow over the flat plate by a parabolic code. The profiles for < k > and < c > are also constructed in the same way. An example of space map of coherent vorticity deduced from semi-deterministic modelling is given in figure 1. 4. E X P E R I M E N T A L
APPROACH
4.1. E x p e r i m e n t a l a p p a r a t u s The backward facing step model is fixed in the middle of the wind-tunnel $10 of C.E.A.Toulouse. The test volume is 1 m wide by 2.2 m height by 2 m long. The plate where initial boundary layer is developing is 0.45 m, the reattachment plate is 1.2 m and the step height is 65 mm. The aspect ratio is more than 10 and so, ensures that we simulate a flow in an infinitely large tunnel. Reference velocity is 40 m/s (Reynolds number based on step height is about 170000, Mach number 0.12). Some preliminary Laser Doppler Velocimetry measurements gave access to mean velocity fields and help to check that the flow is two dimensional as well as to find reattachment length (Xr = 5.7 H) and
Figure 1. Example of space map of numerical coherent vorticity
495 adequate position of the rakes in the flow. Then, an experimental study was developed using 8 X hot wires rakes in the mean-gradient direction behind the backward facing-step. The rake is located in the upper part of the separated layer, where the longitudinal velocity is positive for two reasons : the former, some visualisations (in initial laminar flows) show that CSs are confined in this upper part (Zaffalon [9]) and the latter, hot wires do not allow to determine the velocity sign in the separated bubble. The measurements of instantaneous streamwise and transverse velocity with a sampling frequency of 12 kHz during about 40 seconds allow a good definition of the CS's transit and provide a number of detected events high enough for statistical treatments. The Table 1 presents features of measurements: location and extent of rake (AY), intervals between wires (Ay) and vorticity thickness (5~): Table 1 Characteristics of measurements
AY(mm)
Ay(mm)
5~(mm)
25 25 50 75
3.2 3.2 6.4 9.6
15 20 35 ?
x/H-1.2 x/H=2.2 x/H=4.2 x/H=6.2
4.2. T h e v o r t i c i t y - b a s e d conditional m e t h o d This procedure is based on the instantaneous spanwise vorticity (using the Taylor hypothesis in the flow direction and finite difference schemes for evaluating derivative terms). It gives us a time display of the vorticity along the transverse length of the rake. We apply a numerical filter (LP without phase difference) to smooth out high frequencies and improve the definition of high vorticity areas. Practically, we use a method very close to the vorticity-based conditional sampling technique developed by Hayakawa [6] : from a discrete time series of simultaneous U and V signals at N locations separated by Ay, during T with a time step At, we define the instantaneous vorticity :
1 z(t, y) -
= E
Vi+l,j -- Vi-l,j
2At
Ui,j+l
-
-- Ui,j-1
2/ y
(5)
U~ is the average convection velocity of CS's in the longitudinal direction (U~ - 0.5 x Um~• in our study). We obtain spatio-temporal maps of vorticity fields. To isolate CS features from the complete motion, we impose a threshold Th on vorticity to select areas where their magnitude is strong. Positions (t~, y~) of maximum amplitude in these areas are supposed to be the centers of CS's:
~z(tc, yc) > Th
and
wz(t~, y~) - Max,o~a, Wz(t, y)
(6)
Then, we obtain a spatio-temporal matrix I that indicate center positions:
if
Ii,j - 1,
t ~ - i x At
and
yc-j
x Ay
else
I i , j - O,
(7)
496 The application of phase average on instantaneous signals allow to extract the coherent motion (f) and the pure random motion ft. Furthermore, the coherent motion is assumed to be the CS's motion. We choose not to class as the same event structures centered on different transverse positions. These CS's indeed have distinct signatures (Aubrun [4]). So, to determine the mean CS centered on a fixed transverse location, we respectively align original unfiltered realizations of all transverse positions with respect to each center of reference position and then, we apply ensemble-average on U and V. For instance, to educe the mean CS centered on the probe 4, the detection signal is reduced to vector Ii,4. For 7- time delay with respect to the center of structure, the phase-average operator is: 1
tN
f~(r, y ) = (f(t, y ) ) - (fi,j) - -~ ~ fi+~,j
(8)
t--t1
Two criteria are also implemented : - the shape criteria which consists on using the first-step phase-average vorticity wcz as reference scheme to eliminate all events too different of the expected shape. - the size criteria which eliminates too small events. Too large events are naturally smoothed out by filtering. 5. R E S U L T S 5.1. p r e l i m i n a r y r e m a r k s It is important to mention that even though coherent structures inferred from Semideterministic modelling and from space-time conditional sampling are comparable, they are not completely identical: - Solving ensemble-averaged Navier-Stokes equations assumes that the unsteadiness of the motion is periodic and enables us to access to the global 2-D spatial field of coherent motion. - Experimentally, we have to build a detection criteria to detect the spatio-temporal center of each structure (reference time) and apply, a posteriori, the phase average operator. Because, in reality, the unsteadiness is pseudo-periodic, the coherent motion obtained possesses a bias as soon as one moves away from the reference instant. Futhermore, to correctly compare with numerical results, we convert the time direction into a longitudinal direction with the Taylor hypothesis: y) --
v)
(9)
5.2. C o m p a r i s o n of t i m e - a v e r a g e d r e s u l t s To validate the mean velocity, turbulent intensities and Reynolds stresses fields deduced from numerical approach, we compare some profiles (in sections multiple of the step height H) with the ones obtained by Laser doppler velocimetry technique (figure 2). The figure (a) shows the velocity < U >/Uo from X / H = 1 to X / H = 6.
It is obvious that the numerical results are similar to the experimental ones in the first two sections which are still far from the reattachment zone. From X / H = 3 to X / H = 6 the profiles reveal that the shear layer develops too slowly due to the reduction of the
497
a)
b)
2
1.5 -< 0.5
0
c)
d)
2
I
1.5 "r>...
--
1 0.5 0 0
2
4
X/H
6
8
0
2
4
6
8
Figure 2. comparison between numerical (solid line) and experimental (dotted line) results about a) mean velocity ~/Uo, b) and c) turbulent intensities u"---2/U~and -V-~/U~,d) Reynolds stresses u'v'/U~
constant C u. And for this reason the reattachment length predicted numerically(6.15H) is larger than the experimental result(5.7H). The figures (b)-(d) represent the -~/Uo 2, -~/Uo 2 and u'v'/Uo 2 profiles. The underestimation of these quantities comes also from the diminution of Cu. In fact, when Cu is reduced, the quantity < Ur~U,.~> reduces, too. In contrary, the quantity ~ j which represents the coherent motion should increase. In this work, the coherent motion indeed increases due to the diminution of the turbulent viscosity but not sufficiently. This explains that the simulation results are much smaller than the experimental ones.
5.3. Comparison of unsteady results The first stage is to compare characteristic frequencies of the unsteady motion which are associated with the transit of coherent structures. Experimental and numerical results are given in Table 2. Frequencies are very close to experimental data. It means that the semi-deterministic modelling enables to educe the real unsteady motion. We now look more precisely at the respective distribution of coherent motions in both cases (numerical and experimental). We observe that, whatever the studied section, the qualitative distribution of coherent motion for an isolated (not in pairing process) coherent structure is similar. However, the growth of coherent structures is more important in the numerical case. Consequently, we choose to describe the coherent motion only in the fixed section X/H = 2.2, where the sizes of the numerical and experimental structure are
498 Table 2 Comparison of characteristic frequencies X / H - 1.2 X / H = 2.2 experimental 372 285 numerical 390 230
X / H = 4.2
X / H - 6.2
164 170
78 79
comparable (figure 3). We compare numerical and experimental coherent vorticity 9 • h/Uo, unsteady coherent longitudinal and transversal velocities (t/Uo and ~/Uo , the incoherent kinetic energy restricted to 2D 1/2(u'--~ + -v--~)/U~ and the production of incoherent motion from coherent motion [-(Ur~)~ Ox - ( v ~ ) ~Oy - (u~v~)(~ Oy + ~ Ox) ] • h/U 3 " Reader has to keep in mind that the experimental longitudinal direction is obtained by the Taylor hypothesis. At first, it is important to mention that these distributions are typical of vortical structures in shear layer (even in plane mixing layer [5]). So, as expected, in each case, the concentration of coherent vorticity (definition of a coherent structure in a shear layer) is well defined and is associated with a distribution of coherent velocities characterized by over and under-velocity areas. These ones are indicative of vortical events and assert that the peak of vorticity is not only due to monotonous shear effect. Quantitatively, numerical magnitudes are noticeably lower experimental ones. The incoherent kinetic energy and the production are both characterized by maximum of magnitude at saddle-points, at each sides of coherent structure and have a minimum value at the center. Similar distributions have been validated in [4]. Contrary to vorticity and velocities, their experimental and numerical magnitudes are identical. It seems that the semi-deterministic modelling is well adapted to predict the unsteady motion (see characteristic frequencies), the qualitative distribution of coherent and incoherent motions and that it correctly verifies magnitudes of incoherent energy distribution and coherent to incoherent energy transfers. Only the magnitudes of unsteady coherent velocity and thus of vorticity are under-estimated. This fact entails that in time-averaged results, when numerical turbulence intensities and Reynolds stresses(u~u~) are lower than experimental ones, only the coherent contribution (uiuj) is really under-estimated. 6. C O N C L U S I O N Our study had two goals: Using a conditional method to educe experimental coherent motion and separately, developing the semi-deterministic modelling that directly provides the numerical coherent motion in the same configuration of the backward facing step flow. - Comparing precisely numerical and experimental results. Concerning the differences with plane mixing layer, despite an ambiant turbulent level much higher, we have checked, that similar vortical coherent structures can be identified
-
499 and that the higher turbulence level in the separated flow does not alter their coherence. The semi-deterministic approach is a rather rough approach based on a simple generalization of k - c from steady to unsteady case, but it correctly predicts the coherent motion in this flow. Characteristic frequencies and morphologies of coherent structures are in agreement with experimental ones. In contrary, two differences exist about the evolution of the size of coherent events when they go towards the reattachment area (the growth of numerical coherent structures is higher) and the magnitudes of coherent vorticity and velocities (experimental structures are more amplified). Nevertheless, the comparison between numerical and experimental coherent motions has not been made previously for this situation and it shows that the semi-deterministic approach is able to predict the coherent motion, even though some modelling improvements are necessary to improve the introduction of the characteristics of incoherent turbulence. In the work of Kourta [3] modelling coefficients are not constant values but explicit timedependant functions of both ensemble-averaged strain and rotation. First results are promising. REFERENCES 1. H. Ha Minh, Order and disorder in turbulent flows:their impact on turbulence modelling. Osborne Reynolds Centenary symposium. UMIST- Manchester, May, 24, 1994. 2. A.K.M.F Hussain, Coherent structures, reality and myth!, Phys. Fluids 26 (10), october 1983. 3. A. Kourta, Computation of vortex shedding in solid rocket motors using timedependent turbulence model. Journal of Propulsion and Power, Vol. 15, No. 1, januaryfebruary 1999. 4. S. Aubrun, Etude exp~rimentale des structures coh~rentes dans un ~coulement turbulent d~coll~ et comparaison avec une couche de m~lange. Th~se de I'INPT. Toulouse, France, january, 28, 1998. 5. E. Vincendeau, Analyse conditionnelle et estimation stochastique appliqu~es k l'~tude des structures coh~rentes dans la couche de m~lange. Th~se soutenue s l'universit~ de Poitiers, France. 1995. 6. M. Hayakawa, Vorticity-based conditional sampling for identification of large-scale vortical structures in turbulent shear flows. Dans Bonnet J.P. and Glauser M.N. Eds., 1993, "Eddy structure identification in free turbulent shear flows", Proceedings of the IUTAM Symposium- POITIERS Octobre 1992, Kluwer Academic Publishers. 7. P.L. Kao, Etude num~rique des instabilit~s convectives et des structures coh~rentes dans des couches de m~lange libres ou d~coll~es. Th~se de I'INPT. Toulouse, France, may, 29, 1998. 8. W.P. Jones, B.E. Launder, The prediction of laminarization with a two-equation model of turbulence. Int. Heat and Mass Transferts, No. 5, pp 301-314, 1973. 9. P. Zaffalon, Contribution l'~tude a~rodynamique d'un ~coulement de marche. M~moire de diplSme d'ing~nieur C.N.A.M. Centre R~gional associ~ de Toulouse. 1993
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Figure 3. comparison between experimental (at left) and numerical (at right) coherent results about a) and b) coherent vorticity -{wz} x h/Uo; c) and d) unsteady coherent longitudinal velocity s x 100; e) and f) unsteady coherent transversal velocity ~/Uo x 100; g) and h) incoherent kinetic energy 1/2(u '2 + v'2)/U 2 x 100; i) and j) production Ox
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
501
Experimental investigation of the coolant flow in a simplified reciprocating engine cylinder head D. CHARTRAIN, A.-M. DOISY, M. GUILBAUD and J.- P. BONNET CEAT- Universit6 de Poitiers Laboratoire d'Etudes A&odynamiques-UMR CNRS n~ 43 rue de rA&odrome, 86036 Poitiers Cedex, France The objective of this study is to develop an experimental study of the coolant flow in a simplified transparent model of an internal combustion engine cylinder head in order to build a data bank to validate CFD codes. Simple laws of engineering relevant for pressure losses are provided. A set of two-dimensional cylindrical tubes, perpendicular to the flow models the cooling circuit of a 16 valves cylinder head. The test fig-set-up consists in a closed circuit with a pump. Two measurement techniques, Laser Doppler Anemometer (LDV) and Particle Image Velocimeter (PIV) have been used. Mean values, moments and vorticity have been determined. Pressure on the various cylinders and head pressure losses have also been measured. Moreover, the effect of separation on downstream tubes have been investigated. As a simple test, comparison with a conventional k-e code is provided. 1.INTRODUCTION For car builders, the major research topics for reciprocating engines concern the improvement of the performances and the decrease of the pollution due to exhaust emissions from motors. These improvements require the optimisation of every domain involved in the engine conception such as internal aerodynamics, acoustics, strength of material, hydraulics, aeraulics, ...with different constraints in each domain. For most of the design divisions of the factories, the constraints concerning the engine cooling are in general not considered of major interest and are only partially taken into account, in general only after having def'mitely chosen the engine geometry. Consequently, engineers in charge of the cooling circuit design generally use only a cut and try approach based on the knowledge acquired during previous conceptions and on defects revealed during endurance tests, as for example observation of cracks. Thus, the modifications proposed are only of weak importance and involve small change of the initial concept: for example modifications of the size of the pipes in the cylinder head, of the cylinder head gasket or by locating some pipe restrictions in the remaining space. The objective of these improvements can be to force the flow in region where the tests have shown the thermal exchanges to be too low. This methodology is no longer satisfactory for the engineers facing more and more severe conflicting economical and performance constraints. Recent advances in experimental methods and numerical predictions make more rational approaches possible. Several studies deal with engine coolant, Hoag[1], Finlay[2], French[3], Shalev[4], Aoyagi[5], Davis[6], Bederaux-Cayne[7], and Priede[8]. The conclusions of most of the authors point out that it is necessary to develop a new and rationale methodology for a better integration of the cooling problems in the engine design. Due to the progress of computational fluid dynamics, new approaches can be now used to optimise the coolant system, giving accurate informations about the flow structure in the cylinder head/block. These computations can be associated with the more advanced experimental techniques. As the geometry is 3D
502 and quite complex, numerical codes have to be checked on simplified models. This has been done, for example by comparing computations and measurements, Sandford [9] using LDV and PIV, or Coll6oc[10], [ 1]. As usual for the computation of such internal flows, the results are very sensitive to the turbulence model and the wall function chosen. Some papers give results on real models, [1], Liu[11] or Arcoumanis[12] using transparent engine and as cooling fluid, a fluid having same index of refraction that the model walls. These authors use the refractive index matching technique for optical methods. The study presented here is aimed to define a specific experimental methodology. For the cooling purpose, the objective is, indeed, to insure an optimal circulation of the cooling fluid in order to have an efficient heat rejection of the heat developed in the cylinder block and the cylinder head in order to improve the thermal fatigue fife. The purpose of this work is to analyse the flow characteristics, the heat exchange being directly deduced from the knowledge and predictions of the hydrodynamic field. In order to simplify the experiments, we choose a quasi 2D configuration; however, the design of the tubing is chosen in such a way that the essential geometrical characteristics of a realistic configuration are kept. We first present the model and the experimental set-up associated. Then, we present a global flow analysis and a freer description of the mean and turbulent velocity fields. Velocity measurements have been performed using Particle Image Velocity and been associated with some Laser Doppler Anemometry to check the accuracy of the measurements, particularly concerning the flow statistics. These results will enable a physical interpretation of the phenomena leading to a better description of the heat exchange. This flow appear to be more complex that the flow observed by Watterson et al. [13] in the tube bundle of a heat exchanger. The flow is not established in the downstream cylinders and varies from one cylinder to another. In order to test the ability of CFD codes to simulate such a complex flow, even in a simplified geometry, computations both in 2D and 3D configurations have been achieved with the FLUENT software. 2. TEST RIG SET-UP DESCRIPTION f
Y
J
___~x z
,
i ]
Tank
300 litres
~Filter r Isolating \ 7 Valve /X
4 ,~
Cylinderaxis i --
] By-Pass [
i ..........
Model
RegulationValve
PressureTaos
Figure lb Details of the experimental model
~
7
0
mm
X=250mm
X=334m m
9 Flowmeter
Figure la Test-rig experimental circuit
Figure lc Distribution of pipes and cylinders
503 The test fig-set-up consists in a closed circuit with a centrifugal pump driving the water into the model. A schematic of the test circuit is presented on figure l a. The flow is driven to the model, figure lb, via a long pipe followed by a small divergent inlet pipe to limit the pump effects; the exit is a simple divergent. A by-pass system, associated with two valves, is used to adjust the flow rate. Temperatures and pressures are measured in several locations of the circuit. A 3001 tank is used as plenum chamber. The flow rates used during the tests are measured with a turbine flowmeter, and are respectively 130 l/mn and 240 l/mn. The model is 340mm long, 150mm high and 100 mm wide. It is a simplified representation of the cooling circuit of 16-valve cylinder head of a four-cylinder engine. A two-dimensional bundle of staggered cylindrical tubes, as presented on figure lc, has been machined in acrylic. The characteristic dimensions (tube diameter, size of the coolant passage) have realistic values. The heights have been enlarged to obtain a quasi two-dimensional flow in the horizontal plane of symmetry of the model. Each simplified cylinder is composed of 5 cylindrical tubes: 4 of each (0ffi26mm) representing the two inlet and the two exhaust pipes surrounding the fifth tube (0ffi20mm) of smaller diameter, representing of the spark plug. The distance between the two adjacent large tube axes of one cylinder is 40mm and the minimum distance between small and large tubes (the so-called port bridge) is 5.3mm. Cylinder 1 corresponds to pipes 1 to 4, cyl.2 to pipes 5 to 8, cyl.3 to pipes 9 and 12 and cyl. 4 to pipes 13 to 16. The walls of the model are made of Altuglass in order to make use of optical methods. The configuration of the cylinders and numbers attributed to the cylinders are given on figure lc. The frame of reference XYZ is def'med on the figure. The X axis lies in the direction of the flow. The Y axis is vertical; Z is perpendicular to the cylinder axis of the model. The flow restrictions imposed by the cylinder head gasket and the influence of the cylinder block are not included in this model. Most of the measurements have been performed for a flow rate of 2401/min (=14m3/h). The flow enters and exits from the model by two rectangular sections with area 26mm*127mm. A Reynolds number based on the hydraulic diameter Dh of these sections can be def'med as ReDh=(Dh.U)/v (U is average velocity in a section computed from the flow rate and v the kinematic viscosity). With Dhffi4.3cm for 2401/min, we obtained ReDh=50800. Some tests have been also performed at the reduced flow rate of 1301/min, then leading to ReDh=25400. 3. GLOBAL CHARACTERISATION OF THE F L O W 0 3.
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Figure 2 Head pressure loss coefficient versus the Reynolds number
504 After a first qualitative description of the flow by visualisations using laser sheets (video and photographs), input/output pressure measurements have been performed to study the variation of the head pressure loss in the model giving two types of information about the cooling circuit. First, this is an important parameter for the design of the thermo-hydraulic circuit (radiator, pump .... ). Secondly, it can be also considered that there is an analogy between head losses and global heat exchanges between engine walls and the fluid. Then, the head losses can be considered as a global indicator of the thermal exchanges. The head pressure losses have been measured for flow rates ranging between 30 and 2401/min. Two pressures taps are located in the middle section of the model in plane Y---0, see figure 2, in the inlet and outlet sections. Figure 2 shows the variation of the head pressure loss coefficient Cp-Ap/(0.5pU 2) versus the Reynolds number (p is the fluid density). Inaccuracies are estimated less than +_5%. The different curves obtained have similar shapes. These measurements can be interpolated with a simple power law: ACp--a(ReDh)b. A value of-1/5 (.192) for the exponent b is obtained with an accuracy of 3.3%. The head pressure loss coefficient can be considered as decreasing as ReDh-1/5. This result is quite close to the power law obtained for the friction coefficient in several classical configurations such as turbulent boundary layer, pipe flow .... Indeed, for a pipe flow, the experimental Blasius law is i x C r . - - - ~ I C f d x ~ Reyx 115, and for a turbulent boundary layer, the total friction coefficient /k
-0
follows a law proposed by Falkner C F , , ~ ReYx1/7 . Even if it is difficult to assert formal links between these relations and the results observed in our experiments, we can therefore assume that the variation of the head pressure loss coefficient is a dissipative phenomena linked to viscous friction of turbulent type. 4. DETAILED FLOW ANALYSIS Measurements have been performed by Particle Image Velocimetry giving an instant vector map of the flow. The laser sheet is produced by a double cavity 200mJ Nd-YAG laser working at reduced power (about 15mJ). Two images are recorded on a CCD-camera with 768*484 pixels. The flow is seeded with Rilsan particle (diameter ~_25~rn after filtering). Each picture is divided into interrogation areas in which a vector is computed. An overlapping window technique with a 50% ratio has been used; giving about 1363 vectors (2 to 5 vectors per mm2). Each field is the average of 500 or 50 instant fields. These fields have the typical dimensions of 100mm*70mm. A FFF based on cross-correlation is performed between two successive images: once the location of the correlation peak is determined, the velocity vector is computed giving the time step between these pictures. A Dantec Flowmap software is used. Some complementary measurements have been performed by Laser Doppler Anemometry to check the accuracy of the measurements. Agreement is quite good except some alteration of the results in PIV due to shadow regions and to an increase of the laser light intensity produced by the various tubes acting as lenses. Furthermore, LDV is able to give accurate measurements of the velocity fluctuations. The flow is almost homogeneous as soon it enters the model. In the last plane (close to the exit), a converging effect can be observed. The measurements show that there is an area with approximately 100mm width in the middle of the model where the flow pattern can be considered as two-dimensional. In the following sections, only the results corresponding to the central OXZ planes, located in this zone, will be given. Figure 3 presents the mean velocity magnitude (from the X and Z components
505
FIG. 3 -
Velocity iso-values in the Y -
0 plan (L.D. V).
FIG. 4 - U rms ( x/~u'2) iso-values in the Y - 0 plan.
FIG. 5 -
Velocity iso-values in the Y -
0 plan ( R N G , k - e).
506 measured in planes Y=0). The observation of these plots shows that the flow is almost established in the middle of the first cylinder. The velocity is quite low between the large pipes (~0.15m/s) and show weak variation outside of the tubes (between larger tubes and outer walls), from 0.45 and 0.7m/s. Finally, except in the last cylinder, the velocity inside the cylinder (in the port bridges between the larger and small ones) is always greater than 0.7m/s, with larger values in the direction of the cylinder exit. For example, for the first cylinder, the maximum velocity is 1.9m/s (1.5*V, where V is the entrance mean velocity) but only 0.7m/s (0.55"V) for the last one. The flow is deviated toward the lateral walls between the entrance and the first cylinder. The flow becomes then more uniform in the Z direction before being accelerated again at the exit. In the first cylinder, high velocity magnitude is observed for ct----.45~ on pipe 1 and 315 ~ on pipe 2 and in the port bridge: the angle {x is defined on figure 1c. Figures 6a, b,c show the streamlines in Y=0 plane deduced from the previous velocity measurements respectively for cylinders 1-2, 2-3 and 3-4. The large deviation of the walls at the entrance is responsible for the creation of the lateral eddies located between X=0 and 20mm and for a strong deviation of the streamlines at the entrance section. The first cylinder is responsible for a strong obstruction of the flow. Large size eddy systems can be also observed between the cylinders, and, with a weaker intensity, inside the larger pipes of one cylinder. The fluid coming from the outer part of the model is directed into the eddies but does not seem to pass inside the port bridges. The flow is quite similar in the various port bridge cylinders. Figures 3 or 6 show separation for {x-~90~ (~, angular position defined in figure l c) on the large pipes (with non-symmetrical flow with respect to OX) and for et=135 and 225 ~ for the small ones. Other measurements (not presented here) show that the mean flow structure is nearly independent of the flow rate.
Figure 6a Streamlines deduced from LDV measurements for cylinder 1-2
Figure 6b Streamlines deduced from LDV measurements for cylinders 2-3
Figure 6c Streamlines deduced from LDV measurements for cylinder 3-4
Figure 7 Streamlines computed for cylinders 1-2
507
Uniformity o f the flow rate distribution is often used as criteria to evaluate the quality o f cooling in various sections of the cylinder head. F r o m the measurements, the flow rate has been calculated for various sections normal to X axis (shown on figure 1): 8 central sections and 8 lateral ones corresponding to X - 3 0 - 7 0 - 1 1 0 - 1 5 0 - 1 9 0 - 2 3 0 - 2 7 0 and 300mm. The c o m p u t e d flow rates correspond to a unit height of the model. Figure 8 show the axial distribution o f the inner flow rates deduced from measurements in the central sections. 0.006
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The variation with X is quite weak inside the cylinders but the decrease is about 50% between cylinders 1 and 2 and weaker for the last ones. An increase o f the flow rate appears in the last cylinder, due to the converging effect near the exit. Figure 9 shows the corresponding values for the outer flow, between a row of pipes and the outer walls. The flow rate increases slightly with X, due to fluid transfer from the outer part to the inner part o f the model. The transverse flow rate (through sections parallel to OX) has been computed between the cylinders. A negative flow rate is observed in cylinder 1 (from outer to inner part) becoming almost null in the following ones. This implies that the eddies present between the pipes induced a very small fluid transfer. By adding the inner and outer flow rates (multiplied by a factor o f two to take into account the symmetry), the value of the total flow rate is lower than the entrance flow rate. This means that the flow is not perfectly two-dimensional and that a part o f the flow is deviated toward the upper and lower parts of the model (along the tubes). The deviation begins just at the model entrance. Thanks to the numerous measurements obtained, a large amount o f qualitative and quantitative data on the flow structure are available. It is thus possible to discuss the flow topology: location of the stagnation points, areas with separation or recirculating zones. Wall pressure measurements around the larger circular pipes will complete the results obtained on the velocity field. A pipe element has been instrumented with twelve pressure taps, at midheight, every 30 ~. By rotating this instrumented pipe and locating it at the place of the various tubes, pressure measurements have been performed every 10 ~ on the 16 larger pipes. Examples o f the results are presented on figures 10-a and b, where the pressure difference between pressure on the cylinder and the static pressure in the entrance section has been plotted versus the angular position (x.
508 Plots relative to pipes 1 and 2 (figure 10a) give the location of the two stagnation points on each pipe: the first one for ~-326 ~ for pipe 1 and for ~t-34 ~ for pipe 2; the second ones are located for a - 1 8 0 ~ (pipe 1) and tx-178 ~ (pipe 2). If we consider that the measured pressure is equivalent to a dynamic pressure (-1/2pU2), the corresponding velocity is U-1.34m/s for a pressure difference of 900Pa. This value is in agreement with the values obtained from the corresponding local external velocity measurements (see figure l c, for example).
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a) Pipes 1 and 2 b) Pipes 15 and 16 Figure 10 Wall pressure distribution It can also be observed that there is a zone with very low pressure on pipe 1, between ot-355 ~ and or-150 ~ This interval corresponds to the part of the flow which is accelerated along the pipe being deviated on the model sides. Downstream, along the pipe, the flow suffers separation generating a small recirculation zone connected to the second stagnation point (ct=180~ On figure 10-b, for pipes 15 and 16, the pressure values are systematically shifted toward the negative pressures. This can be attributed to an increase of the head pressure loss when the distance to the entrance section increases. To help the interpretation of the graphs presented on figure 10, the pressure values have been added to the velocity measurements presented in the previous sections: on the figure 6 presenting the streamlines, the wall pressure distributions on the 16 pipes have been plotted as colour part of the tube walls. For pipes 3 and 4, in the port bridges, it is easy to recognise the zone with acceleration of the fluid at the cylinder exit associated with the low-pressure zone (in blue). The pressure peaks due to the presence of a recirculating zone between the pipes is clearly obvious on the figure. Similarly to pipes 1 and 2, stagnation points for ct-310 ~ (respectively 50 ~ are located on pipes 5 and 6, corresponding to the impact of the central jet. This time, the dynamic pressure (160mbar) corresponds to the velocity U-~0.57m/s (result here also in good agreement with the velocity measurements). A zone with a small high pressure due to the recirculating zone for ~ - 4 0 ~ (respectively 325 ~ can also be observed. The similarity of the pressure distribution on pipes 3/4 and 7/8 (for instance location of the peak and minimum values) has to be noticed. The turbulent RMS velocity in X direction obtained by LDV has been plotted on figure 4 as a contour plot. The turbulent kinetic energy is very high at the first cylinder exit. It appears that the jet issuing from the first cylinder is highly turbulent, therefore the jets issuing
509 from the others cylinders seem to be less turbulent. This high turbulence level is of prime interest for the thermal exchange ratios. Turbulence intensities can be very high, RMS values can reach 20% of the local velocity. For the prediction of such an internal flow, a simple gradient model will be probably poorly efficient. Then, we perform, as a preliminary study, a simple computation with a closure k-e model. 5. TEST OF A k-g MODEL In order to check the ability of a k-e model to predict such a flow, we have performed computations both for a 3D configuration on half a configuration (using symmetry property of the flow). The purpose is not to improve predictions of models and codes, but is to analyse the behaviour of a conventional closure model in such a relatively simple geometry. Then no attempt for optimisation have been tried. We use a computational domain including both the divergent and convergent pipes located at the inlet and outlet of the model. The inlet boundary conditions have been determined from the measured corresponding conditions (turbulence intensity of 17%, turbulent characteristic length of 10mm and mean velocity of 2m/s). A conventional RNG k-e model with a non-equilibrium function close to the walls and a second order discretisation proposed by the Fluent code, UNS Version 4.1.9 have been used. The mesh consists of 72230 nodes. The smaller size mesh size (close to the cylinder walls) is 0.7mm along the pipe and lmm radially. In the core of the flow, in one cylinder, the meshes are 0.Smm by 0.5mm. Hexahedral grids typically 10mm is used in the z direction. The convergence have been reached for 594 iterations, the residual being decrease from l i f E to 10-5 during the calculations. The results are presented on figure 5 which corresponds to the experimental configuration given on figure 3. In the first cylinder, the predicted boundary layer separation is close to the one observed in the tests. However, the transverse flow rate is not correctly predicted. To assert this conclusion, figure 7 presents the calculated streamlines in the first cylinder and is to be compared with the experimental ones of figure 6a. The computations show more intense vortex motions, particularly between the pipes corresponding to the same cylinder. Between two cylinders, the predicted motion is more correct, with no transverse flow-rate, corresponding to the observation during the tests. Into the first cylinder, the recirculation area computed behind the first pipe has a too large extent and interacts with the second; this last one has not been observed in the experiments. So the calculations predict a positive flow-rate (from the central part to the sides) contrary to the negative one measured. This inversion of the sign of the flow between experiments and computations is still observed into the second cylinder. 6. CONCLUSIONS AND PERSPECTIVES We have developed a simplified model representing the essential hydrodynamic features of cooling circuit which is more complex than a regular tube arrangement. The model can be considered as two-dimensional and easy to study by computational fluid dynamics. The measurements methods used are complementary and a relatively complete database has been obtained. A head pressure loss coefficient varying as the -1/5 power of the Reynolds number has been proposed. The sensitivity of this law to the geometric location of the pipes is yet to be analysed. Due to the asymmetric repartition of the tubing, the internal flow in the model is quite complex with eddy systems specific to each cylinder. Between the cylinders 2 to 4, the flow seems to be relatively established and independent of the X position. The first cylinder
510 plays a particular role because it corresponds to the impact of the entrance jet but also is responsible for the major part of the pressure loss. The exit is also associated to the strong turbulent intensity. The intricate nature of the problem, which strong asymmetric interactions such as the coupling ~tween separations and walls, strong accelerations and sudden expansions, etc.., represents a challenge for optimisation and computation. A crude test of a commercial code has been performed. It shows that a RNG-k-e model predicts with some details most of the overall features of the flow. However, a detailed analysis shows, as it was expected, that the separation zones and their interactions with inner obstacles are not predicted with a sufficient accuracy. For the purpose of heat transfer evaluation, this drawback can be dramatic. With the present data base, higher order codes, such as ASM or RSM, can be tested with better chances of success. ACKNOWLEDGEMENTS The authors thank gratefully the Renault Company (Direction de la strat6gie et des Avants Projets) for the financial support of the experimental part of the work presented and for an efficient collaboration during this study. REFERENCES 1. K.L.Hoag and S.Brasmer, SAE Technical Paper 891897, (1989). 2. I.C.Finlay, D.Harris, D.J.Boam and B.I.Parks, Proc. Instn. Engrs., 199, 3, (1985) 207. 3. C.C.J.French and K.A.Atkins, Proc. Instn. Engrs., 187, 3, 1973. 4. M.Shalev, Y.Zvirin and A.Stotter, Int. J. Mech. Sci., 25, 7, (1983) 471. 5. Y.Aoyagi, Y.Takenaka, S.Niino, A.Watanabe and I.Joko, SAE Technical Paper 880109, 1988. 6. G.D.Davis and R.J.Christ, SAE Technical Paper 960883, 1996. 7. W.S.Bederaux-Cayne, SAE Technical Paper 960881, 1996. 8. T.Priede and D.Anderton, Proc. Instn., Mech. Engrs., 198D, 7, 1984. 9. M.H.Sandford and I.Postlethwaite I., SAE Technical Paper 930068, 1993. 10. A.Coll6oc, Mellat and J.M.Boyer, Note technique 1096/93/1847, 4 th AACHEN Colloquium Automobile and Engine Technology, 1993. 11. C.H.Liu, C.Valfidis and J.H.Whitelaw, Experiments in Fluid, 10 (1990) 50. 12. C.Arcoumanis, J.M.Nouri, J.H.Whitelaw, G.Cook and D.M.Foulkes, SAE Technical paper 910300, 1991. 13. J.K.Watterson, W.N.Dawes, A.M.Savill and A.J.White, 3rd ERCOFFAC-IAHR Workshop on Refined Flow Modelling, Lison, Portugal, 1995.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
511
S e c o n d a r y flow in c o m p o u n d sinuous/meandering channels Y. Muto and T. Ishigaki Ujigawa Hydraulics Laboratory, DPRI, Kyoto University Yoko-Oji Shimomisu, Fushimi, Kyoto 612-8235, Japan
Secondary flow and its related problems in compound sinuous/meandering channels are discussed. The structure of secondary flow in sinuous/meandering channels during floods is illustrated by detailed velocity measurements using a fibre optic laser Doppler anemometer (FLDA) and an advanced flow visualisation technique using a small submergible video camera. Some data analyses were carried out in order to estimate energy expenditure by secondary flow and its effect on the macro flow structure. The results clearly shows relatively large effect of secondary flow on estimating conveyance of the channel system.
1. INTRODUCTION Rivers during floods often inundate their adjacent plains and show the behaviour of socalled compound channel flow. Compound channel flow is known to have a distorted 3dimensional nature mainly due to the velocity difference between the main channel and the flood plain. A shear layer is formed at the junction of the main channel and the flood plain, and fluid exchange takes place through this junction region (see e.g. Knight and Shiono, 1990; Tominaga and Nezu, 1991). In case of meandering channel such flow structure is more complex. Velocity distributions in those channels are much distorted because directions of the main stream are different between the main channel (lower layer) and the flood plain (upper layer) (see e.g. Toebes and Sooky, 1967; Stein, 1990). Detailed velocity measurements in a meandering channel with flood plains using a laser Doppler anemometer were carried out by Schr6der et al. (1991) and Shiono and Muto (1993). They could successfully illustrate growth and decay processes of secondary flow under overbank conditions, but in a rather limited meandering geometry condition. Muto et al. (1997) also discussed based on velocity measurements unique features of compound meandering flow as to secondary flow and the shear layer instability. They pointed out that these features can take quite important roles when considering conveyance capacity of this sort of channel. This paper deals with secondary flow and its related problems in compound sinuous/meandering channels. Detailed velocity data measured by a fibre optic laser Doppler anemometer (FLDA) were mainly used. Direct visualisation for secondary flow cell in a compound sinuous channel was also carried out by a small submergible video camera. Contribution of secondary flow in energy expenditure mechanism, mainly that for the channel conveyance, was estimated by spectrum analysis for fluctuating velocity data and 1-D type loss coefficient analysis.
512
2. E X P E R I M E N T A L SETUP Figure 1 shows a plan view of the experimental flume and meandering channel. The flume was made of perspex with a rectangular cross section, 10.8m long, 1.2m wide and 0.35m deep. The valley slope of the flume was set at 0.001. The main channels and the flood plains were formed of polystyrene boards. The channel had a rectangular cross-section of 0.15m wide and 0.053m deep. The channel sinuosity s were 1.09, 1.37 and 1.57 corresponding the arc of channel bend q9 of 60 ~ 120 ~ and 180 ~ respectively with the bend central radius rc of 0.425m. s=1.37 .
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Water depth H (m) 0.0525 0.0633 0.1078 0.0519 0.0630 0.1059 0.0532 0.0631 0.1087
Mean velocity Us (m/s) 0.23"7 0.157 0.352 0.197 0.129 0.282 0.170 0.113 0.268
Friction velocity u, (m/s) 0.0166 0.0121 0.0225 0.0148 0.0120 0.0221 0.0140 0.0120 0.0226
Reynolds number Re (• 2.63 0.82 6.26 2.19 0.66 4.92 1.95 0.62 5.16
Froude number Fr 0.431 0.412 0.495 0.359 0.340 0.401 0.307 0.299 0.374
The other relevant information on the experimental setup can be found elsewhere (Muto, 1997) together with the possible error factors and their minimising efforts.
513
3. FLOW STRUCTURE 3.1. Depth Dependency Figure 2 shows secondary flow behaviour in a vector form watching from the upstream for the bankfull and Dr=0.15 flows in the s=1.37 channel. Figure 2(left), for the bankfull flow, shows that the dominant secondary flow cell is developed through a bend section. The fully developed clockwise cell can be observed at the bend exit (Section 5). On the other hand, for the overbank flow, Figure 2(right), an anticlockwise cell recognised at Section 1 near the inner wall suddenly collapses in the latter half of the bend, synchronous with the appearance of a new clockwise cell along the inner wall from Section 3. This new cell immediately grows and occupies most of the cross section in the crossover sections. No. 13
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514 These results come from the difference of the originating and developing processes of the cell. That is, it is the centrifugal force that governs the secondary flow structure in the inbank case. Whereas for the overbank flow the structure is controlled by the flow interaction in the crossover region. This shear layer effect in overbank cases is so strong as to nullify the centrifugal effect. Figure 3 supports this explanation. The figure shows a result of flow visualisation in the crossover region. The visuali~;ation was carried out using a small submergible video camera and plastic beads whose specific gravity is Section =.o
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Figure 4. Secondary flow vectors in s=1.37 channel for Dr=0.50.
515 1.05 as a tracer. Secondary flow cell developed through the crossover region is clearly captured in the figure. The figure also shows that the intruding flow from the upstream flood plain is strong enough to reach the main channel bed and is partly involved into the cell. Figure 4 shows secondary flow vectors in Dr=0.50. As the flooding depth further increases, the cell becomes larger, especially in the bend section. However the inducing mechanism of the cell seems to be the same as Dr=0.15. Flow expansion and contraction behaviour in the latter half of bend to the crossover is more clearly seen. The structure of secondary flow described here is similar to that illustrated by Schr6der et al. (1991). In their results the cell induced through the crossover section in overbank cases attains its maximal size at the apex section. Such a cell however grows more rapidly in our results, being maximum at the end of crossover, and is already in the decaying process after entering into the consecutive bend sections. This difference could result from the vegetation on the flood plain. The vegetated flood plain used in Schr6der's study retards the upper layer flow, thus the inductive force of secondary flow cell is weaker than in non-vegetated cases. 3.2. Sinuosity Effect
Figure 5 shows secondary flow vectors for Dr=0.15 flow in various channel sinuosities. The effect of channel sinuosity on the secondary flow structure can be summarised as follows.
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516 (1) Inducing and developing mechanisms of secondary flow is basically the same irrespective of the channel sinuosity. That is, secondary flow cell develops through the crossover region for all sinuosities and this dominant cell is consistently observed near the water surface along the inner wall. These facts indicates that the upper layer flow induces secondary flow in the main channel. The intruding upper layer flow runs over the main channel with some angle, whether large or small, due to the meander channel geometry and entrains the fluid near the junction (boundary) of the upper and lower layers. (2) Generally speaking, as the channel sinuosity increases, the mechanism developing secondary flow becomes enhanced. This is mainly due to a larger crossover angle of a more sinuous channel, which results in stronger entrainment in the lateral direction near the layer boundary in the main channel. In s=l.09 case the secondary flow cell, different from the other two cases, doesn't develop up to occupying the whole channel cross section. On the other hand, in s=1.57 case developing process of the cell in size is quite similar to that in s=1.37 case. Vectors within the cell in s=1.57 case seems slightly larger, which means that stronger secondary flow is induced, than those in s=1.37 case. In order to examine the effect of sinuosity on the developing process of secondary flow more closely, the strength of secondary flow %Syz defined by Equation (1) is calculated:
=
+w:,u / L u aA
(1)
12 Here the integration A is applied over the area below the interracial r ~_ s=1.09 - i - s:1.37 -A- s:1.57 10 boundary. The result is shown in Figure 6. The figure clearly shows 8 enhancing effect of a larger channel sinuosity on producing secondary co 6 flow. As the sinuosity increases, the peak of %Syz appears at more 4 downstream section and its peak value becomes larger. It is 2 considered that for the tested cases a larger sinuosity contributes to set 0 a longer crossover zone where the 0.00 0.25 0.50 0.75 1.00 Streamwise distance from an apex to the next x / ( L / 2 ) upper layer flow gives energy to secondary flow. As a result, secondary flow in a more sinuous Figure 6. Strength of secondary flow ~ z for Dr=0.15. channel can receive more energy through the longer supplying zone and this leads to a larger and later-appearing peak of ~ z.
4. ENERGY ESTIMATION
4.1. Spectrum Analysis In order to estimate energy expenditure by secondary flow, spectrum analysis was carried out. The wavenumber spectrum should be used in the analysis. However direct measurement
517
for the spectrum is in practice extremely difficult. Moreover the transformation from the frequency spectrum into the wavenumber spectrum applying Taylor's frozen turbulence hypothesis is questionable in such a complex flow case. Thus the following discussion is made based on the frequency spectrum. The spectrum was calculated using a FFT technique. The number of data points w a s 212 = 4096. Figure 7 shows spatial distributions of spectra for the streamwise component at the bankfull level in the crossover region for the s=1.37, Dr=0.50 flow. The figure shows that the positions 1,0~
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Figure.7 Normalised spectrum at the interface in the crossover for the streamwise component, s=1.37, Dr=0.50. of a clear peak in the productive subrange of spectra change from one section to the other, as indicated by arrows in the figure. This change of the peak appearance can be closely related to the position of the secondary flow cell shown in Figure 4. The spectral peak mainly appears in the frequency range from 0.3Hz to 1.0Hz. If Taylor's hypothesis of Eq. (2) can be applied,
E(k) = Uc ~- F(f),
f k - 2~ U c
(2)
518 (where E(k) = wavenumber spactrum, F(f) = frequency spectrum, Uc = convection velocity), the corresponding length scale (the reciprocal of wavenumber) is from 2cm to 8cm, which is about the order of the main channel depth. This is an evidence that secondary flow of the channel depth scale takes an important role in energy expenditure. To see the contribution of the spectral distribution to the total energy, the cumulative spectrum is considered. The cumulative spectrum K(f) defined for the streamwise component, for example, is written as follows:
K.(f)
= u- -fo
(3)
Figure 8 shows an example of the cumulative spectra together with their corresponding frequency spectra. The frequency range which iswithin the secondary flow scale is assessed as follows. According to Imamoto et al. (1989), the diameter of cell generally distributes around its mean d m from 0.4d m to 1.6d m. If this distribution can also be applied to the case being considered, using Eq. (2), the frequency range governed by the secondary flow scale can be estimated as 0.625fp to 2.5fp, where fp is the peak frequency. The length scale which corresponds to the peak frequency can be considered as the mean size of the secondary flow cell, as was examined above. Figure 8 also shows the estimated range of secondary flow. The estimated contribution to energy expenditure due to secondary flow is 35% to 50%. On the other hand, the turbulence contribution is mostly over 50%. It can be said that the effect of secondary flow on energy expenditure is quite large and it dissipates as much energy as turbulence does. : E
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4.2. Loss Coefficient Analysis According to Ervine and Ellis (1987), conveyance of compound meandering channels can be estimated in an 1-D form similar to that for pipe flow by introducing several energy loss coefficients relevant to the flow mechanisms taken place in the channel. Secondary flow is the case for the lower layer, where it works as one of the main energy loss factors. The equation for the lower layer proposed by Ervine and Ellis is as follows:
519 V2 SO K-, 2g s
(4)
K = K b f + K s f + Kts
where V = section averaged velocity, So = valley slope and K = loss coefficients and subscripts bf, sf and ts are boundary friction, secondary flow and turbulence respectively. Kbe is expressed by Darcy-Weisbach type friction factor. K,s is derived by Muto (1997) using interfacial shear data working on the boundary to the upper layer.
f
(s)
Kbf - 4R
Kts
f" = ~
Ra '
f"=
22"a 2
(6)
pV
where f = friction factor, R = hydraulic radius, f" = turbulent friction factor, R a = imaginary radius defined at the boundary and "ta = interfacial shear stress averaged over the boundary. On the other hand, rational derivation for Ksf in overbank cases is at the current stage not established yet. Thus in this study secondary flow contribution was evaluated by Eqs. (4), (5) and (6) together with adopting measured velocity V and shear stress 1:a. The results are shown in Table 2. The table clearly shows large contribution of secondary flow in energy expenditure, especially for the shallow flooding case. Table 2. Energy loss balance in the lower layer by the loss coefficient method. Energy loss coefficient s Dr Total Friction Turbulence Secondary flow K K~f I~ Ksf 1.09 0.15 0.668 0.244 0.068 0.356 (37%) (10%) (53%) 0.50 0.271 0.245 -0.030 0.053 1.37
1.57
0.15
1.111
0.50
0.475
0.15
1.244
0.50
0.542
(91%)
(-10%)
(2o%)
0.257 (23%) 0.257 (54%) 0.259 (21%) 0.260 (48%)
0.220 (20%) 0.115 (24%) 0.361 (29%) 0.243 (45%)
0.635 (57%) 0.103 (22%) 0.623 (50%) 0.039 (7%)
5. CONCLUSION Secondary flow behaviour under overbank conditions in meandering channels for various experimental conditions is successfully illustrated by detailed velocity measurements and an advanced flow visualisation technique. Its effect to the channel system as to channel
520 conveyance is evaluated by spectrum analysis and loss coefficient analysis. The spectrum analysis shows that a clear peak seen in the productive range of the spectra has close relation to a scale of secondary flow. This relation, together with careful examination of visualisation results, enables to estimate secondary flow range and its contribution to energy balance. The results shows its quite large contribution in energy expenditure, 35% to over 50 % and sometimes being dominant over that of turbulence. The loss coefficient analysis also supports this large contribution of secondary flow.
ACKNOWLEDGEMENT
The authors would like to thank Prof. H. Imamoto, Kyoto University and Dr K. Shiono, Loughborough University, for their advice and comments. This research programme is partly supported by the Japanese Ministry of Education, Science, Sports and Culture (Monbusho) on the Grant-in-Aid for Encouragement of Young Scientists (A) (No. 10750389) in which the first author is the principal investigator.
REFERENCES
Ervine , DA and Ellis, J (1987), Experimental and computational aspects of overbank floodpalin flow, Trans. Royal Society of Edinburgh, Earth Science, Vol.78, pp.315-325. Imamoto, H, Ishigaki, T and Nishida, M (1989), Experimental study on the turbulent flow in a trapezoidal open channel, Annuals, DPRI, Kyoto Univ., No.32B-2, pp.935-949. Knight, DW and Shiono, K (1990), Turbulence measurements in a shear region of a compound channel, J. Hydr. Res., Vol.28, No.2, pp.175-196. Muto, Y (1997), Turbulent Flow in Two-Stage Meandering Channels, PhD thesis, Bradford University, UK. Muto, Y, Shiono, K, Imamoto H and Ishigaki, T (1997), Three dimensional flow structure for overbank flow in meandering channels, J. Hydroscience and Hydraulic Engineering, Vo1.16, No.l, pp.97-108. Schr6der, M, Stein, CJ and Rouv6, G (1991), Application of the 3D-LDV-Technique on physical model of meandering channel with vegetated flood plain, Proc. 4th Int. Conf. on Laser Anemometry-Advances and Applications, Cleveland, Ohio, USA. Shiono, K and Muto, Y (1993), Secondary flow structure for in-bank and over-bank flows in trapezoidal meandering compound channel, Proc. 5th Int. Symp. on Refined Flow Modelling and Turbulence Measurements, Paris, France, pp.645-652. Stein, CJ (1990), M~iandrierende Flie[Sgew~isser mit tiberstr6mten Vorl~indern experimentelle Untersuchung und numerische Simulation, Mitteilung des Instituts ftir Wasserbau und Wasserwirtschaft, RWTH Aachen, Nr. 76. Toebes, GH and Sooky, AA (1967), Hydraulics of meandering rivers with floodplains., J. Waterways and Harbors Div., Proc. ASCE, Vol.93, No.WW2, pp.213-236. Tominaga, A and Nezu, I (1991), Turbulent structure in compound open-channel flows, J. Hydr. Eng., Vo1.117, No.l, pp.21-41.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 1999 Elsevier Science Ltd.
521
Diffused turbulence distortion by a free surface M. A. Atmane and J. George Institut de MScanique des Fluides-INP Avenue du Professeur Camille Soula, 31400 Toulouse, France
1. I n t r o d u c t i o n Interracial gas-liquid mass/heat transfer modelling requires a good understanding of the effects of turbulence and the way it is generated. Indeed, the case where turbulence is produced at the interface by the wind-shear, for example, is basically different from that one where turbulence is produced in the bulk flow and transported towards the interface. It seems, from recent studies (Kumar et a/.[1]), that the way turbulence is produced can induce dramatic changes in the global mass/heat transfer factor formulation. It is hence interesting to show the origine of the difference between different cases. It has been demonstrated by Hunt [2], that due to the presence of a thicker viscous boundary layer, the small structures developing near a shear-free interface are greater than the ones present near a sheared interface. Mc Cready et al. [3] confirmed numerically this result and Gulliver ~ Tamburrino [4] did show it experimentally. However, the turbulence transport problem and its deformation by a free surface remains open. Basically, such a configuration has to be studied keeping in mind two kinds of problems:
The importance of the turbulence generation process: turbulent structures can be produced either at a wall, at sheared interface, or in the bulk by a mixing process (in the agitated vessels for example). The main difference between these configurations is that a mean current is present and generates a mean interracial shear in the first two cases while there is no mean shear in the last one. Turbulence can keep, however, the same intensity magnitude order in all cases.
The interactions between the internally generated turbulence and a free surface can be viewed following different approaches. Hirsa & Willmarth [5] choosed to study a vortex pair interacting with a free surface when Brumley & Jirka [6] came up with new experimental results concerning the grid turbulence deformation by free surfaces. In spite of a difference between the two configurations studied by these authors, some characteristic results can be drawn from Hirsa & Willmarth's study: depending on the free surface contamination, the vortex pair rolls up, generates an opposite sign tangential vorticity and rebound from the surface. This gives rise to horizontal motions assimilated in mass transfer studies to the 'surface renewal'.
522 2. E x p e r i m e n t a l
procedures
Our experiments are performed in a tank agitated by microjets as described by George et al.[7]. One hundred microjets with 0.7ram diameter are disposed on a plate in the bottom of a tank (0.45x0.45x0.8m3). Jets are disposed in rows, the distance between jets is 40ram. The distance of merging can be estimated around 4 times the distance between jets, that is 160mm. The Reynolds number based on the jets exit velocity and the jets diameter is greater than 4000, hence, large scale instabilities described by Villermaux & Hopfinger [8] should be avoided. Holes are regularly arranged between jets in order to ensure the water recirculation. Water is supplied to the microjets by a pump and a second tank permits to maintain the interface level constant. A two components LDA system is used to measure simultaneously horizontal (U) and vertical (W) velocities. A second LDA device is associated to the first one to measure simultaneous velocities at two locations. Integral length scales are derived computing the correlation functions of vertical velocity measured at two different locations separated by ~0"
R (7 + 70)=
w2 +
(1)
For a vertical separation r0 - (0, 0, z0), the integral of this correlation function gives a longitudinal length scale estimation L z. If Y0~0= (x0, 0, 0), the integral will give an estimation of the lateral length scale L~. Integral time scales are directly estimated from the velocity signals following the same procedure used to compute the length scales. The temporal auto-correlation is estimated, for different time lags ~-, by:
w(t)w(t + T) 3.
(2)
Diffused turbulence characteristics
The measurements to be presented in this section were carried out in the 'homogeneous' part of the flow. The homogeneity we are talking about concerns horizontal planes. Fernando & De Silva [9]) showed that agitated vessels experiments are not able to generate a 'pure' turbulence in that sense that mean motions are always present. Nevertheless, the presence of such motions seems to be necessary to permit the transport of turbulence from the bottom towards the interface and to maintain an energy level high enough in the immediate proximity of the free surface. Turbulence intensity is evaluated comparing the vertical turbulent intensity u~ to the mean U~ profile. It does show an approximative value of ui/Ui = 0.5 - 0.6 in the homogeneous planes (in the intermediate region located between the forcing plane and the free surface). 3.1. K i n e t i c e n e r g y b a l a n c e The objective when using the agitated tank is to carry out experiments in the part of the flow where mean motions, and therefore turbulence production, are avoided. In the absence of production and transport, the only mechanism able to maintain the turbulence level is the diffusion process. Let us examine what happens in our case.
523 The transport equation of the turbulent kinetic energy writes:
Ok
Ok
OUi
0 ( v Ok
o--i + u~-5~ + ~J-~x~ + ~ ' -
1
~x~ + -v~-;p + k ~ ) + ~ - o
(a)
we recognize the time dependant T K E variation, the transport term, the production term, the diffusive term (molecular diffusion, diffusion by pressure, turbulent diffusion) and the dissipation term. Assuming a steady flow with homogeneity in horizontal planes and horizontal symmetry (U = V, u = v and On/Oxn= On/Oyn), equation (3)can be expressed as follow:
P r + Ad + Dt + R = 0
(4)
where
2~ou -~ow+ 2~--~(0U P~=
-~x +
Ad - 2U Ok
-~z
OW
-~z +-~x )
W Ok
+
Oz
D t = 2 ff--~-~-k+ ff---~w k and R is the sum of the remaining terms (dissipation and pressure-velocity correlation). All terms in equation (4) (except the term R) are evaluated from the velocity signals recorded in the region where the flow is well developed, that is where its structure does not depend from the flow near the injection plane. Figure (1) shows the different terms participation to the total equilibrium in the equation (4). Keeping in mind that the flow is dominated by the shear just above the jets, we can state that a complete change occurs in the structure of turbulence when it reaches the interracial region. From a production-advection-dissipation equilibrium it moves towards a diffusion-dissipation one. Here, we are in presence of a turbulence in which diffusive term dominates but where production, despite its weak value, still participates in the T K E budget because of the presence of motions at large scales.
3.2. Energy decay law In an ideal case, in the absence of mean-shear and thereby of turbulent production, the only source known to provide turbulence far from the bottom is the turbulent diffusion. Furthermore, dissipation causes a loss of energy when moving from a horizontal plane to another above it. The vertical diffusion-dissipation T K E equilibrium acting in the homogeneous part of the flow then writes:
a (-~-~+ ~__~) + c = o
Oz
(5)
where c represents the dissipation rate. According to Magnaudet's [10] analysis, the diffusive term can be made propGrtionnal to the third order vertical velocity correlation:
524
Figure 1. Terms in T K E equilibrium equation in the diffusive layer.
Figure 2. Normalized vertical and horizontal fluctuating velocity behaviour. The forcing plane is at 0.
O(wk + l~-pp)/Oz ~ O-w-g/Oz. On the other hand, Batchelor [11] states, from his work on turbulence governed by a production-dissipation equilibrium, that the dissipative term can be assumed proportionnal to w----ff3/2/L,where L is the integral length scale. Equilibrium in equation (5) can then be expressed:
O~
Oz (wa) ~
~---~3/2 L
It is obvious that a variation in the size of turbulent structures affects the TKE decay mode. Thus, a linear behavior of L(z) gives rise to the energy decay law following z -n. Figure(2) shows both horizontal and vertical rms velocity profiles. These profiles are normalized by those obtained in the 'homogeneous' flow. This can be achieved measuring the fluctuations (uh, wh) when the interface is removed. We observe that the vertical component does not decrease the way the horizontal one does. It induces a weak anisotropy in the bulk flow. Actually, the mean motions appear to be active through their gradients as found by Risso & Fabre [12]. Indeed, these authors, studying a confined jet flow, show that the mean motion gradient can act together with the pressure in the energy redistribution process between the horizontal and the vertical components velocity. Thus, the flow can reach a quasi-isotropic state impossible to get in our case. In presence of the confinement, the flow structures itself to supress differences initialy created by the vertical forcing at the bottom. Concerning the power law exponent n in the decay law of the i th velocity component u~ ~ z -n, it is necessary to point out that it does differ depending on the hydrodynamical conditions. The most frequently observed values are n ~ 1.5 for the vertical velocity component and n ~ 1 for the horizontal one.
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.
.
.
.
.
:
. . . .
150
1
2
3
4
5
6
7
8
z (,r~)
Figure 3. Vertical time scale evolution in the diffusive layer.
.... 0,01
', . . . . 0,015
', . . . . 0,02
I .... 0,025
I .... 0,03
', . . . . 0,035
Hm) 0,04
Figure 4. Vertical length scale evolution: o- evaluated from a temporal time scale, *- evaluated from two pointcorrelations.
3.3. Integral time and length scales Length scales are usually estimated in agitated tank using rotating probes. We have been trying to perform such an estimation by measuring the velocities at two different locations. These measured length scales will also be compared with those based on the time scale computation. Computing L z and L~ (longitudinal and lateral length scales) from the correlation function defined in relation (1), we can get another isotropy indicator. Indeed, it is known that L z = 2L~, in an ideal isotropic turbulence. In our experiments, the longitudinal length scale value is not exactly twice the lateral one. Actually, the L Z / L ~ ratio ranges between 1.5 and 2.2. Then, our flow is not far from the perfect isotropic state. In order to put light on the turbulence structure, we try to estimate the length scales by a second method using the integral time scale. Let us first have a look at the vertical time scale variation. Figure (3) shows a vertical profile of Tw. It does confirm the expected power law growth Tw = az nt with nt ~ 2 - 2 . 2 . A new length scale estimation is then built based on T~ and the vertical fluctuating velocity: LT = Tww. We compare in Figure(4) the length scales estimations given by the eulerian and the lagrangian methods. The comparison reveals a relatively good agreement between both estimations. The discrepancy might come from the origine (L = 0) of the linear approximation chosen for each method. This origine, called virtual origine, corresponds to the horizontal plane where turbulent eddies are generated. In our experiments, the virtual origine is located 5 - 6 c m above the forcing plane.
3.4. High order correlations High order correlations, normalized by the relevant quantities, can describe some features of the turbulence structure. Figures(5, 6) show vertical profiles of both the skewness
526
350
350 ~0
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~
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000o
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Figure 5. Vertical variation of the skewness factor.
i
i
i
t
t
J
i
J
i
i
J
10
t
15
Flatness
Figure 6. Vertical variation of the flatness factor.
S~ i and flatness F~i factors. These factors are defined as follows:
3
Ui
--~3/2 Ui ~Tui
=
---~4 Ui
In figure (5), we observe that the skewness factor reaches an equilibrium value in the region located between the bottom and the bulk volume, for both velocity components. Sw decreases from zero to reach a minimum value of about - 0 . 6 at the location where the interactions between two neighbour jets is supposed to occur. This is a consequence of the existence of a maximum of w 3 which results from the interactions of two turbulent layers. Veeravali & Warhaft [13] studying such a configuration did find that two mechanisms dominate the interactions: a continuous turbulent diffusion action and more intermittent penetration effects. Above that region, we observe an increase in the Sw values. Concerning S~, it starts from 0 at the bottom, increases to reach a constant value of about 1.4 and keeps it until the interface. 4. T u r b u l e n c e f e a t u r e s near t h e interface 4.1. S o m e t h e o r i c a l p r e d i c t i o n s The most important theorical contribution to the turbulence-boundary layer interactions is the rapid distortion theory calculation done by Hunt ~ Graham [14]. Solving a linear form of the momentum transport, they end with the fluctuating velocities behaviour. The normal component decreases following z -1/3 whereas the tangential one growths as z 1/3 in the so-called irrotationnal layer characterised by a thickness of about
527 m
one turbulence integral length scale. The dissipation rate (and then the rms vorticity a; 2) and the TKE are constant in this layer and, following this analysis, these quantities do not change except within the viscous layer. This theory seems to work nicely at high Re numbers. At weak Re, some corrections are necessary as stated by Hunt [2]. This is due to the fact that the theory does not take into account the vorticity production by baroclinic effects or by the distortion of small eddies by the largest ones when approaching the interface. These predictions are supported by exprimental works of Brumley & Jirka [6] and Kit et al. [15].
0,5
1
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,
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.
.
.
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.
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.
.
.
.
.
.
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.-o-- Kit et al.
-1,3 !
,
.......
"1
,1
,,~-1,3
Thomas & Hancock
-1,5
Figure 7. Horizontal fluctuating velocity normalized by the fluctuating velocity at depth equal to one turbulence integral scale.
.......
omas & Hancock
;i ::
- . o - Kit et al. ....... B r u m l e y &
Jirka
-- - Hunt & Graham -1,5
Figure 8. Vertical fluctuating velocity normalized by the fluctuating velocity at depth equal to one turbulence integral scale.
4.2. E n e r g y r e d i s t r i b u t i o n below t h e interface At a free surface, the normal velocity fluctuations must vanish to satisfy the boundary condition at the interface. Since energy is maintained following the R D T (Hunt ~ Graham [14]), the loss observed on the vertical component of the TKE is recovered by the tangential ones. Figures (7, 8) show the vertical behavior of both vertical and horizontal velocity fluctuations. These quantities are normalized by the homogeneous ones measured at z = - L , where L is the turbulence integral length scale in the vicinity of the interface. As can be seen, the vertical velocity fluctuations are in good agreement with R D T predictions. The horizontal component shows a more complex behavior. The u-ff maximum, predicted at the interface by the theory, can be reached either at the interface or below it, depending on the role played by viscous effects. Let us consider the Reynolds number based on L and x/~, Re - x/~L/v. For a weak agitation level, a thick viscous layer appears and affects the tangential velocity growth. Hunt [2] estimates the viscous layer thickness to be roughly equal to 2L Re -1/2. In the case presented here, the Reynolds
528 0,001
0,01 ........
1,00E-01
L~
I
0,1 ........
I
1 ........
A []
1,00E-03<~
o z=-L/20 1,00E-07
I
f(hz)
100
........
[] .................. A [] ....................... ..... A A 9 ~ ................... - 5 / 3 [] A 9 ........................
~
Injected Energy
1,00E-05-
........
A
-
.,..
~
10
I
[] z = - L / 8
~
~ 1 7 6~ . ~ o
'i
.9..............
%g
Injection s c a l e
,, z = - L
o
1,00E-09
Figure 9. Vertical velocity spectra beneath the interface
number is equal to 937. We do consider, according to the values given by Hunt's estimation of the viscous layer thickness, that the u 2 maximum is drawn further down from the interface because of viscous effects, the estimated thickness and the measured one being of the same order. At greater Re, Thomas & Hancock [16] indicate that the viscous layer is not thick enough to affect the tangential fluctuating velocity behavior. The energy redistribution is mainly attributed to the role of the pressure-velocity correlation term appearing in the T K E balance. Walker et al. [17] emphasize the fact that this term plays either as a factor of a return to isotropy or of anisotropy in the bulk flow depending on which kind of interactions exist between the flow and the boundary. 4.3.
Length
scales
and spectra
Frequency spectra are computed from the inverse Fourier transform of the auto-correlation function of the fluctuating vertical velocity. They give two kinds of informations. The first one concerns the way turbulence scales interact through the energy cascade and the second one is relative to the scales at which energy is provided and dissipated. Figure (9) represents spectra of the vertical velocity signals recorded below the free surface at three different depths. The 'blocking' effect predicted by R D T is obvious in this figure, the deeper the location point the greater the injection temporal scale (and therefore the length scale). The 'blocking' effect expresses in fact the 2D trend of the turbulent eddies emerging from the diffused turbulence with a strong 3D character. Due to this effect, the inertial equilibrium subrange is more and more narrow when approaching the interface. It is clear then that the lower frequencies, representing the large turbulence scales, are the first ones affected by inhomogeneity. Furthermore, it is shown that the integral time scale based on the depth and the vertical rms velocity fluctuation provides us in the spectra with a good injection scale approximation. We can emphasize here the fact that
529 this statement is true within the so-called 'source layer' in RDT. This result seems to be interesting because it explains the blocking as a direct role the interface plays on the largest scales development. In other terms, at a first stage, the interface does not affect transfers between the different scales of the turbulence, it only changes the size of the biggest eddies and gives them the possibilty to approach it.
4.4. Interfacial t u r b u l e n c e i n t e r m i t t e n c y A useful information can be extracted from the high order correlations shown in figures (5,6). Just beneath the interface, the skewness and flatness factors of the vertical velocity give us an interesting diagnosis. Sw has a value roughly equal to - 0 . 7 and a high Fw is observed (~ 10, 15). Following Townsend's [18] classification, these values can be synonym of a strong turbulent eddies intermittency. He states that near a plane of symmetry, say a flat interface, the energy flux should have a weak value, and the velocity near this plane should fluctuate in time between an intermittent high intensity, rapidly vanishing, and a lower one more often observed. In Townsend's terminology, the turbulence structure is spotty. This is confirmed by the Perot & Moin's [19] DNS of a homogeneous turbulence submitted to a free surface. These authors did show that turbulence below the interface is dominated by impingement motions responsible for 'splatting' 2-D eddies. Further diagnosis were obtained by Atmane [20] measuring both velocity and scalar (oxygen concentration) fields simultaneously in order to characterize the local mass transfer. The overall picture emerging from this study was that these fields are highly related through the interfacial intermittency. Actually, examining simultaneously scalar and velocity fields, it is shown that motions at two separated scales can be observed giving rise to a slow process regularly broken by rapid motions. This is what is called here intermittency.
5. C o n c l u s i o n An experimental setup is used to generate a turbulence without mean circulation. This kind of turbulence is dominated by diffusive transport. Some new diagnosis are presented in this study. They concern mainly the turbulence structure changes along the vertical direction. The exact time and length scales are determined, they show an increase with respect to z. It is worth noting that mean motions are playing a significant role in maintaining the turbulence intensity. When submitting such a turbulence to a free surface, turbulence structure changes and does adapt itself in order to satisfy the boundary conditions. It is shown that the RDT, despite its pure kinematic approach and the fact that it does not take into account the internal interactions and non-linearities inherent to turbulence, restores the main features of experimental results: eddies are flattened near the interface while turbulence intensity remains important in the horizontal planes. Velocity spectra in the interface proximity indicate its blocking effect. This effect starts at a depth roughly equal to one turbulence integrale scale. Furthermore, a weak intermittency associated to a normal vorticity generation is observed near the free surface through the skewness factor. This already has been demonstrated by Veeravalli & Warhaft [13] studying two turbulence mixing layers interactions.
530 REFERENCES
S. Kumar, R. Gupta, and S Banerjee. An experimental investigation of the characteristics of the free-surface turbulence in channel flow. Phys. of Fluids, 10-2:437-456, 1998. J. C. R. Hunt. rlhrbulence structure in thermal convection and shear-free boundary layers. J. Fluid Mech., 138:161-184, 1984. M. J. McCready, E. Vassiliadou, and T. J. Hanratty. Computer simulation of turbulent mass transfer at a mobile interface. AIChE Journal, 32-7:1108-1115, 1986. J. S. Gulliver and A. Tamburrino. Turbulent surface deformation and their relationship to mass transfer in an open-channel flow. In B. Jahne, editor, Air-Water Mass Transfer: Selected Papers 3thd Int. Syrup. Gas Transfer at Water Surfaces, 1995. A. Hirsa and W. W. Willmarth. Measurements of vortex pair interaction with a clean or contaminated free surface. J. Fluid Mech., 259:25-45, 1994. B. H. Brumley and G. H. Jirka. Near-surface turbulence in a grid-stirred tank. J. Fluid Mech., 183:235-263, 1987. J. George, F. Minel, and M. Grisenti. Physical and hydrodynamical parameters controlling gas-liquid mass transfer. Int. J. Heat Mass Transfer, 37-11:1569-1578, 1994. E. Villermaux and E. J. Hopfinger. Periodically arranged co-flowing jets. J. Fluid Mech., 263:63-92, 1994. H. J. S. Fernando and I. P. D. De Silva. Note on secondary flows in oscillating-grid, mixing-box experiments. Phys. Fluids, 5-7:1849-1851, 1993. 10. J. Magnaudet. Modelling of inhomogeneous turbulence in the absence of mean velocity gradients. Advances in Turbulence IV, 51:525-531, 1993. 11. G. K. Batchelor. The theory of homogeneous turbulence. Cambridge University Press, sixth edition, 1993. 12. F. Risso and J. Fabre. Diffusive turbulence in a confined jet experiment. J. Fluid Mech., 337:233-261, 1997. 13. S. Veeravalli and Z. Warhaft. The shearless turbulence mixing layer. J. Fluid Mech., 207:191-229, 1989. 14. J. C. R. Hunt and J. M. R. Graham. Free-stream turbulence near plane boundaries. J. Fluid Mech., 84:209-235, 1978. 15. E. L. G. Kit, E. J. Strang, and H.J.S. Fernando. Measurements of turbulence near shear-free density interfaces. Jr. Fluid Mech., 334:293-314, 1997. 16. N. H. Thomas and P. E. Hancock. Grid turbulence near a moving wall. J. Fluid Mech., 83-3:481-496, 1977. 17. D. T. Walker, R. I. Leighton, and L. O. Garza-Rios. Shear-free turbulence near a flat free surface. J. Fluid Mech., 320:19-51, 1996. 18. A. A. Townsend. The structure of turbulent shear flow. Cambridge University Press, second edition, 1976. 19. B. Perot and P. Moin. Shear-free turbulent boundary layers, part 1. physical insights into near-wall turbulence. J. Fluid Mech., 293:199-227, 1995. 20. M. A. Atmane. Th~se de Doctorat, INP-Toulouse, 1998. .
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Transition
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
533
Surface curvature and pressure gradient effects on boundary layer transition
B. J. Abu-Ghannam a, H. H. Nigim a and P. Kavanagh b
a Department of Mechanical Engineering, Birzeit University, Birzeit, Palestine b Department of Mechanical, Iowa State University, Ames, IA 50010, USA
Abstract The effects of convex curvature on boundary layer transition were examined experimentally in low and high free-stream turbulence, and in zero and favorable pressure gradients. For high turbulence with zero pressure gradient, transition along convex surfaces was found to be similar to that on flat surfaces, however at low turbulence levels, transition was significantly delayed. The effect of favorable pressure gradients in delaying transition was observed even in high turbulence flows. The usually held conclusion that in high turbulence flows, favorable pressure gradients have little effect on transition was shown to be inappropriate. A new pressure gradient parameter was introduced to express flow acceleration. When transition results were presented in terms of this new parameter, the effect of favorable pressure gradients in delaying transition was demonstrated more clearly.
1. I N T R O D U C T I O N Boundary layer transition from the laminar to the turbulent state is known to be affected by many parameters. Among these parameters, the flee-stream turbulence and pressure gradient are thought to be the most influential. Previous experimental investigations have been conducted mainly on flat, smooth surfaces at ambient temperature for steady, incompressible flows. Flows of this type can now be predicted with some confidence using either empirical relationships or turbulence modells that rely to some degree on empiricism [1]. Unfortunately, boundary layer flow over turbomachine blades and many other bodies is markedly different from the simple flat surface case. For turbomachine blades the surfaces are generally concave, convex, or concave-convex with constant or varying curvature. The effect of surface curvature on boundary layer transition was realized early by Liepmann [2]. Up to 80% of the total profile loss for atypical turbine airfoil (depending on the loading level) might be attributed to the suction (convex) surface boundary layer [3]. A recent survey on the role of transition in gas turbines indicated that the effect of convex surfaces has not been given sufficient attention [4]. It was thus appropriate that further experimental study be
534
made in investigate the effect of convex surface curvature on the location of the onset of transition, the length of the transition zone and the development of boundary layer parameters through the transition zone. Accelerated flows usually exist over much of the curvature portion of the suction side of turbine blades. The pressure gradient is customarily expressed in terms of Thwaite's pressure gradient parameter, )~0. However, it was found that the acceleration parameter K t is more appropriate for flows with a favorable pressure gradients where transition takes place via the bypass mode [4]. In terms of )~0 or Kt, the effect of favorable pressure gradients on the momentum thickness Reynolds number at the start of transition, Re0s, was shown to be small, and, furthermore to decrease as turbulence level increased. Nevertheless, acceleration of the flow has a significant retarding effect on transition that cannot be demonstrated when presented in terms of Re0s and )~0 or K t . In recent years more attention has been given to the physics of the transition process following the principles of spot formation theory [5]. Several transition modells and theories have been proposed for the development of the transitional boundary layer [1,4,6,7]. A considerable progress has been made in the modelling of transition on the basis of spot generation rate and intermittency. All of these methods assume that the onset of transition is known and hence require Re0s to proceed with the calculations. However, the prediction of the onset of transition remains an unsolved issue, and empirical correlations are the only available tools to deal with the onset of transition, under the high free-stream turbulence levels characteristic of turbomachines.
2. EXPERIMENTAL SETUP AND PROCEDURE As shown in Figure 1, an experimental setup consisting of a blowing wind tunnel with a (250 x 840 mm) working section was used to study the effects of convex surface curvature on transition. Air velocities could be obtained in the range of 5-4m/s, and three turbulence grids of different sizes and a fine screen were used to generate different free-stream turbulence levels. The turbulence generating grids were installed one at a time in the plenum box upstream of the test section and turbulence level was measured near the test surface leading edge. Boundary layer measurements were made on three convex test surfaces. The first surface was formed from a stainless steel (20-gage) plate rolled to a radius of curvature of 450 mm. This plate was 250 mm wide and 710 mm long in the streamwise direction. At the plate leading edge an 8 mm stainless steel bar was machined and fitted producing a smooth rounded inlet. The second and third surfaces were formed from a Lexan (carbonated plastic) plate which was 840 mm wide, 1220 mm long and 3 mm thick. The Lexan was flexible enough to fit into circular grooves cut in the wooden sides of the test section. The grooves were long enough to allow the sliding of the test plate to form an inlet incidence angle of -2.5 degrees. This negative incidence angle together with the rounded plate leading edge ensured no boundary layer separation at the inlet. The same Lexan plate was tested in two configurations, one with a radius of curvature of 760 mm, and the other with a radius of curvature of 1520 mm. To check the experimental procedure and results with previous boundary layer measurements, experiments were also carried out on a flat surface. An adjustable Lexan control wall was also installed in the working section along with the curved
535
AirFlow
TostSoction i3,'Contractio ii e
Plenum
Round-to-Square Transition j Ducta ~ / ~ ' ~
/___~,~ulenceGeneratiTng "1 ~
--. ___2
Plate
Figure 1. Experimental set-up.
or flat surfaces to generate the required pressure distribution along the test surface. More information about the wind tunnel construction can be found in reference [8]. The boundary layer measurements on either the curved or flat surfaces were made using a small flattened mouth Pitot tube of 1 mm outside diameter in combination with a static tube of similar diameter. These two tubes were installed 20 mm apart (transverse to the flow) on a thin mylar sheet and traversed in the streamwise direction at a constant height above the test surface. The mylar sheet was 100 mm wide and wrapped around the test surface. The start of transition was located at the point where the ratio of the measured near-wall velocity to free stream velocity reached a minimum, and the end of transition where this ratio reached a maximum. Boundary layer velocity profiles were measured using another miniature flattened Pitot tube traversed across the boundary layer at several streamwise locations. These measurements were corrected for both wall proximity [9] and turbulence effects [10]. Static pressure was also measured by a set of wall static pressure taps installed along the test plate center line. The streamwise component of the free stream turbulence level was measured and checked using a hot wire anemometer [ 11 ]. Values for turbulence level quoted in this work are those measured near the test plate leading edge. Because the turbulence grids were installed at about 1250 mm upstream of the test surface leading edge, measurements reported in [11] showed a very small decay of the turbulence level over any transition zone detected in this work.
536
25.0
e. Convex surface, r =450 Q Flat surface, 1st pressure gradient ~, Flat surface, 2nd pressure gradient
1.0
0.0
f I ~-.--Lx..._.,~__- 250.0
I
f 500.0
9
-4
9
~L
End of
z 750.0
-2
S t a r~t o ~
.
.
~ansition
-1.0
-2.0
-3.0
\
-4.0
l 250
r
I 500
Distance from leaomg edge X. mm
Figure 2. Pressure distribution along the centerlineof curved and flat surfaces.
Figure 3. Movement of the Transition zone with reference velocity; Tu =6.4%, r =450 mm.
3. E X P E R I M E N T A L RESULTS The first set of experiments was conducted on the test convex surface of 460 mm radius of curvature. The flow was accelerated as shown by the pressure coefficient distribution C D in Figure 2. The pressure coefficient was calculated based on the air dynamic head as measured by a Pitot-static tube installed 150 mm upstream of the test surface leading edge. The pressure coefficient distribution varied only slightly with the reference velocity. Figure 3 demonstrates the movement of the transition zone with the reference velocity for different turbulence levels. The first set of measurements were made on a flat surface under the test pressure distributions 1 and 2 shown in Figure 2, while the second set of experiments was conducted using the Lexan sheet with radii of curvature of 760 and 1520 mm in zero pressure gradient flows. In order to check the curvature effects, the transition results collected on the curved test surfaces were compared with the flat surface empirical modell, first, reported in AbuGhannam and Shaw model [12]. Their modell gives the momentum thickness Reynolds number at the start of transition as; Re0s - 163 + Exp [F(X0 ) - F(X0 ) Yu/6.91 ], where F (X0) for accelerated flow is given by, F()~0 ) = 6.91 + 2.84 )~0 - 12.27 ()~0)2. It was found that for the same turbulence level (Tu) and pressure gradient parameter ()~0), Re0s, for the curved surface increased an average of 20 percent over those values for the flat surface. The turbulence level in this comparison was rather high (T u > 3.2 percent). At the lower turbulence level of 0.8 percent produced by the fine screen, measurements failed to
537
X 9 r = L80 cm 90 c ~
o
L| W
~1Si 9 /~
~.
ang and mon [13]
90 and 180 cm .
r = ~ and r = 30 in r = 30
Abu-Ghannam and Shaw [ 12]
el
0
1
2
3
4
5
6
Turbulance level, Tu%
Figure 4. Effect of surface curvature and turbulence level on Rexs in zero pressure gradient
, Reee=Z667Re0s
,
Re• = Re• + 16.8 (Re• 0"8 ~ , / _
"
n 9
9
_
Cl
O
/
O0 O/
/
/
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~
~__~
3.2
O
9.0
0
l
~
i00
L
I
200
~
ci
6.4
[
I
300
Je / /~(
Curved
9
{
400
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Figure 5. Relationship between Rees and Re0e for curved and flat surface
0
I I
A TuT.
.3 . 2
O
6 .4 9.0
~(~
9
9 I
/
~
I 2
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.
rl .
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~ O
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9 9
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Pressure Gradient
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.
I 5
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~o
Zero I
6
I 7
10-5 Rexs
Figure 6. Relationship between Rexs and Rexe in different flow conditions
538
6007oot
6.4.2 3Tu% Flat O 52 surface
curved 99surface
600 ~[. I ~-
1.2 + Abu-Ghannam [14 __ Abu-Ghannamand Shaw[12]
/ ~"
500
~3.2 6.4
Flat 0_surface
curved 99surface
1.2 + Abu-Ghannam [ 14 .Asshown Abu-Ghannamand Shaw [12]
500
Tu=l.2
~
400
Tu=l.2
.
~00
300
300
a 200
o
~,a,
eo
"lu = 3.2
o
200
Tu = 6.4
I00
!
o
I 0
1
Tu = 3.2
I
I 2
:
,I 3
I
, I
.,
I
I 5
z
tOO
o 6
100 X0
Figure 7. Effects of pressure gradient parameter X0 at Transition on Rees.
9
,3.2
3. -
). 5
o. s
".0
106 Kt
Figure 8. Effects of acceleration parameter K t at transition Re0s.
detect any transitional flow anywhere over the entire length of the curved surfaces tested under both zero and favorable pressure gradient conditions. In fact transition did not occur at T u = 0.8 and r = 1520 mm when the pressure gradient was made slightly adverse. These transition results suggested that curvature effects are more evident in flows at low turbulence level. In Figure 4, the Abu-Ghannam and Shaw zero pressure gradient transition modell is shown in terms of Rexs (rather than Re0s ) calculated using the Blasius laminar boundary layer relation, Re 0 = 0.644 (Rex)0.5. In addition to present results, Figure 4 shows results obtained by others [12, 13]. From this figure and Figures 5-6, which relate start of transition momentum thickness and length Reynolds numbers to their counterpart at end of transition, it is clear that for zero pressure gradient, turbulence level dominates the transition process such that the effect of curvature is evident only in flows with T u < 2 percent. However, the curvature effect increases quickly as the turbulence level decreases below 2%. As the start of transition is delayed by convex curvature and favorable pressure gradients, the end of transition is also delayed and the length of the transition zone is stretched. However, it can be seen from Figures 5 and 6 that even with these changes in the start and length of transition, the relationships between momentum thickness and length Reynolds numbers at the start and end of transition remain basically the same as for the flat surface case with zero pressure gradient. The transition results obtained in a favorable pressure gradient and high turbulence plotted in Figure 7 show that for the same value of X0 and T u, the momentum thickness Reynolds number Re0s is on average, 20 percent greater for the curved surface than for the flat surface, while Figure 10 shows that the length Reynolds number Rexs is almost doubled. This
539
Tu%
Flat s u r f a c e
curved surface
T
[]
3.2
6.4 9 9 1.2 + A b u - G h a n n a m [ 14 As s h o w n A b u - G h a n n a m and S h a w [12]
4------"4"--"-+
Tu%
Flat surface
c u r v e d surface
3.2 [] 6.4 O 9 1.2 + A b u - G h a n n a m [ 14 As s h o w n A b u - G h a n n a m and S h a w [12]
-4-
T u = 1.2
Tu = 3.2
Tu = 1.2 6
|-
/
200 ~
o
~
0
2
z,
Tu = 6.4
0 ~-00
o
L
0
1,
1
',
J
,
5
i
r
I
~0
lO -4 LX
Figure 9. Effect of pressure gradient parameter ~.X at transition on Re0s.
o I 0
.,
!
L
1
,
!
:
I
~
1
i0
5
r I~
10-4 ZX
Figure 10. Effect of pressure gradient parameter )~X at transition on Rexs.
contradict previous conclusion that curvature effects are only significant for low turbulence flows. To resolve this paradox, it seems that the pressure gradient parameter 2~0may not be the appropriate parameter to describe the acceleration of the flow. Furthermore, Figure 8 shows curved and flat surface results in accelerated flow plotted against the acceleration parameter K t. From Figures 7 and 8, it can be seen that the present results agree with the Abu-Ghannam and Shaw modell, but the question of both Re0s and Rexs being larger for the curved surface cases (for the same ~.0 and Tu) remains unanswered. However, a possible explanation can be drawn from examining figure 2 which shows the pressure distribution in the three test cases. It is clear from figure 2 that the pressure gradient dP/dX in the curved surface case is larger than that in the two flat surface cases, especially between the plate leading edge and the start of transition point. On the other hand, ~.0 and K t are not always larger at the point of transition on the curved surface. It follows that neither 2~0nor K t is a suitable parameter to express the flow acceleration at transition since either may increase or decrease as dU/dX changes. The fact remains that flow acceleration should be expressed in terms of X, U, dU/dX, and v . A suitable non-dimensional pressure gradient parameter was sought and defined as Z,x = (X2/v) dU/dX. This parameter calculated at the start of transition always increases as dU/dX increases, since X s increases with dU/dX. The advantage of using ~.x is demonstrated in Figures 9 and 10 where points of higher acceleration have larger values of ~.x and Re0s, thus clearly demonstrating the effect of a favorable pressure gradient in delaying transition. One also notices the improvement of the distribution of the data points in Figures 9 and 10. This improvement is also evident when
540 additional transition data from Abu-Ghannam [ 14] are plotted in terms of)~x. Figures 9 and 10 also show that using )~x provides a smooth and logical departure from the widely accepted values for Re0s measured in constant pressure flows. With this in mind, it is possible to explain some of the conflicting results reported in the literature. For example, Vijayaraghavan and Kavanagh [15], working on the suction side of the turbine airfoil cascade, detected transition at a streamwise distance significantly larger than that predicted by the Abu-Ghannam and Shaw flat plate modell.
4. C O N C L U S I O N S The results presented and discussed in this paper led to the following conclusions about the surface curvature and pressure gradient effects on boundary layer transition. These are:
,
Transition is delayed and stretched by surface convex curvature. This effect is only significant in low turbulence (< 2%) flows and increases as a turbulence level decreases. Transition is significantly delayed by fluid acceleration even in high turbulence flows. To demonstrate this effect, it is important to use the correct acceleration parameter which was found to be )~x = (X2/v) dU/dX . Using )~x improved the relationships between data in constant pressure flows and in accelerated flows. The relationships between the relative location of the start and end of natural transition are not affected by turbulence level, pressure gradient or surface convex curvature.
NOMENCLATURE CD
H Kt r
Rex Re0 Tu U UR X 0 )~x V
P
Pressure Coefficient Shape factor Acceleration parameter, (v / U 2) dU/dX Radius of surface curvature Reynolds number based on streamwise distance Reynolds number based on momentum thickness Turbulence intensity Local free-stream velocity Reference velocity Surface coordinate in streamwise direction Boundary layer momentum thickness Pressure gradient parameter (x2/v) dU/dX Pressure gradient parameter, (02/v) dU/dX Kinematic viscosity Density
Subscripts e s
End of transition start of transition
541
REFERENCES 1 Narasimha, R., The Dynamics of Transition Zone and its Modelling, Lecture series 1991-06, Von Karman Institute of Fluid Dynamics, 1991. 2 Liepmann, H. W., Investigations on Laminar Boundary-Layer Stability and Transition on curved boundaries, NACA ACR 3H30 (NACA-WR-W-107), 1943. 3 Sharma, O. P., Wells, R. A., Schlinker, R. H., and Baily, D. A.," Boundary Layer Development on Turbine Airfoil Suction Surfaces", J. Eng. Power, Vol. 104, (1982), pp. 698-706. 4 Mayle, R. E., The Role of Laminar-Turbulent Transition in Gas Turbine Engines, ASME paper 91-GT-261, 1991. 5 Emmons, H. W., The Laminar-Turbulent Transition in the Boundary Layer-Part 1, J. Aero. Sci., Vol. 18, (1951), pp. 490-498. 6 Volino, R. J., and Simon, T. W., Bypass Transition in Boundary Layers Including Curvature and Favorable Pressure Gradient Effects, NASA Report 17187, (1991). 7 Walker, G. J., The Role of Laminar-Turbulent Transition in Gas Turbine Engines: A Discussion, paper submitted for 1992 ASME Turbo Expo, Cologne, 1992. 8 Halstead, D. E., The Use of Surface-Mounted Hot-Film Sensors to Detect Turbine-Blade Boundary-Layer Transition and Separation, M.Sc. Thesis, Iowa State University, 1989. 9 Young, A. D., and Maas, J. N., The Behaviour of a Pitot Tube in Transverse Total Pressure Gradient, ARC R&M No. 1770 (1937). 10 Goldstein, S., A Note on the Measurement of Total Head and Static Pressure in a turbulent stream, Proc. Roy. Soc. A, Vol. 155, (1936). 11 Vijayaraghavan, s. B., Effects of Free Stream Turbulence, Reynolds Number, and Incidence Angle on Axial Turbine Cascade Performance, Ph.D. Dissertation, Iowa State University, 1987. 12 Abu-Ghannam, B. J., and Shaw, R., Natural Transition of Boundary Layers: the Effects of Turbulence, Pressure Gradient and Flow History, J. Mech. Eng. Sci., 22, (1980), pp. 213-228. 13 Wang, T., and Simon, T. W., Heat Transfer and Fluid Mechanics Measurements in Transitional Boundary Layers on Convex-Curved Surfaces, ASME paper 85-HT-60, (1985). 14 Abu-Ghannam, B. J., Boundary Layer Transition in relation to Turbomachinery blades, Ph.D. Thesis, University of Liverpool, England, 1979. 15 Vijayaraghavan, S. B., and Kavanagh, P., Effect of Free Stream Turbulence, Reynolds Number, and Incidence on Axial Turbine Cascade Performance, ASME paper 88-FT-152, (1988).
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
543
Calculating Turbulent and Transitional B o u n d a r y - L a y e r s with T w o - L a y e r Models of Turbulence R. Schiele, F. Kaufmann, A. Schulz and S. Wittig Institut ftir Thermische Str6mungsmaschinen, Universit~it Karlsruhe (TH), 76128 Karlsruhe, Germany
A two-layer model of turbulence combining a two-equation model in the fully turbulent region with a one-equation turbulence model near the wall is used to calculate turbulent and transitional boundary-layers. By simulating turbulent boundary-layers the validity of the model and the correct coupling of the two layers are reviewed. A new intermittency function is introduced into the one-equation model enabling the application of the so called TRANSIC-T model ('TRANSitional Intermittency Controlled Two-Layer' model) to transitional boundary-layers. The model is verified against transitional boundary-layer flows under varying conditions. Its performance is discussed by comparison to experimental data, a revised version of the two-layer model by Fujisawa et al. [ 1] and four low-Reynolds number k-e turbulence models capable of simulating transitional flows reasonably well.
1.
INTRODUCTION
Two-layer models of turbulence have recently gained popularity (e.g. [2]). Most of the models of this type proposed in the literature use the same standard k-e turbulence model of Launder and Spalding [3] in the bulk of the flow, but different one-equation turbulence models in the viscosity affected near-wall region. The reduced number of required grid points and the associated lower computing time and storage demand, a higher numerical stability in comparison with low-Reynolds number k-E models as well as their potential capability to cope with transitional boundary-layer flows make their application very attractive. Due to their favourable properties two-layer models have been used by various authors for the simulation of attached and separated fully turbulent flows. Combining the one-equation model by Wolfshtein [4] with the standard k-E model successful flow simulations were reported by Chen and Patel [5,6], Patel et al.[7], Iacovides and Launder [8], Abou Haidar et al. [9] and Arman and Rabas [10]. Fujisawa et al. [1], Franke [11] and Cordes [12] were able to predict separated and unsteady flowfields in good agreement to experimental data by applying the one-equation model of Norris and Reynolds [13] in the near-wall region while using the standard k-e model in the core flow. Lakehal et al. [ 14] used the same model combination to simulate the flow- and temperature field behind a film cooling injection. The two-layer model presented here employs a slightly modified version of the oneequation k-1 model developed by Rodi et al. [ 15] in the viscosity-affected near-wall region and
544 the standard k-e model in the core flow. The model by Rodi et al. [ 15] was developed with the aid of direct numerical simulation data (DNS), taking advantage of information not being available from experiments. It therefore shows an improved performance when compared to older one-equation models. When two-layer models are applied coupling of the two domains is of major concern. Correct matching implies that neither the one- nor the two-equation model are used outside their limits of validity. Therefore, a criterion which takes the local properties of turbulence into account is the best means to locate the matching region. Accordingly, in the present model, the two layers are connected with reference to the ratio of eddy viscosity l.tt and molecular viscosity kt. Following a suggestion in [15] the standard k-e model is used when this ratio is greater than 16, while for l.tt/l.t less than 16 the one-equation model is employed. Even by considering this physically sound criterion discontinuities may occur in the matching region due to the different equations used for the calculation of the turbulent length scales. In the first part of the paper the possible impact of these discontinuities on the simulation is assessed by comparing calculated turbulent boundary-layer profiles with data from measurements and direct numerical simulations. Calculating transitional boundary-layers with two-layer turbulence models is possible by introducing an intermittency function T into the eddy viscosity relation near the wall. Through transition 3, is increased from 0 for laminar flow to 1 in fully turbulent flow. At the same time the standard k-e model is used in its original version describing the turbulent core flow. This approach reflects the typical flow situation under which transitional boundary-layers occur i.e. a turbulent main flow and a boundary-layer which changes from a laminar to a turbulent state. Fujisawa et al. [ 1] employed a similar concept to simulate transitional boundary-layers by altering the eddy viscosity relation of the one-equation model of Norris and Reynolds [13] while keeping the standard k-e model unchanged. Using the same model combination Cho et al. [16] performed promising calculations of the two-dimensional unsteady flow field in a linear turbine cascade including transition on the blade surfaces. In a former work [ 17] the authors of the present paper used a new intermittency function to model transition for the first time with a two-layer model employing the one-equation model by Rodi et al. in the near-wall region. In continuing this successful work a new set of correlations for describing transition progress is presented in the second part of the current paper. These correlations improve the model's performance even further. This is verified by calculating different transitional boundary-layer flows with and without pressure gradients. All calculations were performed with an implicit and forward marching finite-volume boundary-layer procedure. See [ 18] for a detailed decription of the code. 2.
TURBULENCE MODELS
2.1
TRANSIC-T
The TRANSIC-T model employs the one-equation model by Rodi et al. [ 15] in the nearwall region and the standard k-e model by Launder and Spalding [3] in the core flow. Using the standard values for the empirical constants appearing in the transport equations for the turbulence parameters k and e the two-equation model is utilized in its form widely applied in the literature.
545 The one-equation model proposed by Rodi et al. uses the normal fluctuation ~/v -~- as velocity scale of the turbulent motion instead of ~ employed in most other models e.g. Norris and Reynolds [ 13]. Therefore, the eddy viscosity relation reads
~l,t = p 4V-72 l~t,v
(1)
where l~v is an appropriate length scale described by l~v = Cl,/~ y. Rodi et al. [15] quote a value of 0.33 for Cl,u. This value has been revised to 0.30 during the present investigation resulting in an improved model's performance. The one-equation model solves the same semiempirical transport equation for the turbulent kinetic energy k as the standard k-E model, but uses an algebraically prescribed distribution for the turbulent length scale. The relevant relations are deduced from DNS data simulating boundary-layer flows fairly well up to y+--50. For a more detailed description of the model the reader is referred to [ 15].
2.2
TLK-T
Fujisawa et al.[1] described a two-layer model for the prediction of fully turbulent and transitional boundary-layers by combining the standard k-e model with the one-equation model of Norris and Reynolds [ 13]. It is denoted as TLK-T as it is a Two-Layer model using k as velocity scale being capable to simulate Transition. To the authors' knowledge besides [ 17] this work has been the only investigation in which a two-layer model is used to simulate transitional, steady state boundary-layer flows. It is therefore interesting to compare the so called TLK-T model to the TRANSIC-T model, giving a good impression of the relative peformance of the models and the validity of the two-layer approach in general. In the TLK-T model coupling of the two calculation domains is realized with reference to the damping function f~t in the eddy viscosity relation of the one-equation model. For values of fu less than 0.95 the one-equation model is applied while the two-equation model is used when f~t becomes greater than 0.95. In turbulent equilibrium boundary-layers this matching criterion is equivalent to a ratio of lxt/lx of approximately 36 which in tern corresponds to a dimensionless wall distance y+ between 80 and 90 [ 12]. Details of the model may be found in [ 1].
3.
CALCULATING TURBULENT BOUNDARY-LAYERS
As stated before when two-layer models of turbulence are used correct coupling of the calculation domains is of major concern. In the following, boundary-layer profiles will be looked at in detail being the best means to quantify the possible impact discontinuities may have on the simulation. To assess the overall quality of the solutions results are compared to data from experiments and direct numerical simulations. Additionally, profiles obtained by applying typical lowReynolds number k-(~ turbulence models are shown in the figures, namely the models proposed by Launder and Sharma [ 19] and the model by Lam and Bremhorst [20], abbreviated subsequently as LS and LB model. Fig. 1 shows velocity profiles in a zero pressure gradient fully turbulent boundary-layer at a momentum thickness Reynolds number Re0 of 15000. Besides measured and calculated values the law of the wall is presented using standard values for )c and C being 0.41 and 5.2,
546 respectively. All models lead to results which are in good agreement with the experimental data with the two-layer models being slightly advantageous. Coupling of the two calculation domains is performed in the TRANSIC-T model at y+ of 43.2 and in the TLK-T model at y+ of 74.4. Discontinuities in the velocity profiles could not be detected, even when looking at the profiles in strong enlargement. Considering nonisothermal boundary-layers, similar results were obtained showing no visible effect of the matching of the models on the temperature profiles.
30
30 LsTRANSIC-T (9
20
TLK-T LB
/~ .-""
exp. resul
(9
20
_~...'"
I i
i
+
10
U+=y + ' 10 . / ~
U+= y+ ,,'./;" ~ 7
Iny§ + C
U+=I/~: Iny+ + C
1/ '
I0 ~
Figure 1.
'
' ' , ' " 1
,
,
101
,,
....
I
10 2
'
y+ [ - ]
'
' ' " " 1
I0 s
'
. . . . . . . .
10 4
100
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....
'1
101
. . . . . . . .
i
102
. . . . . . . .
y+ [ - ]
I
. . . . . . .
10 ~
10 4
Velocity profiles in a zero pressure gradient turbulent boundary-layer at Re0=15000: experimental data from Coles and Hirst [21]
Discrepancies due to the use of different turbulence models should be most obvious when boundary-layer profiles of the turbulence parameters k and e and the Reynolds stresses - u ' v ' are examined. Analyzing the profiles of the turbulence parameters no visible effect on k and only a minor effect on e were found. The largest discontinuities were observed when Reynolds stresses were considered. Therefore, Fig. 2 shows Reynolds stress profiles at a momentum thickness Reynolds number of 1410. Besides the current calculations results from direct numerical simulations [22] are presented. The Reynolds stresses are given in dimensionless form as - u ' v '+ = - u ' v ' / u 2 Both two-layer models describe the Reynolds stress profiles in ,C.
excellent accordance to the DNS data outperforming the low-Reynolds number models especially close to the wall. The TRANSIC-T model leads to the best results of the models considered for y+ less than 10. The models are matched at y+=46 in the TRANSIC-T model and at y+=82.7 in the TLK-T model. Around these coordinates discontinuities of the Reynolds stress profiles are visible. Still, discontinuities remain small in size as well as in extension bearing no detectable influence outside of the matching region. Looking at the results it becomes clear that by applying the two-layer models profiles in fully turbulent boundary-layers are excellently predicted and coupling of the models leads to neglectable discontinuities of the relevant quantities.
547
1.00
t
-~
TRANSIC-T LS
.
I I 1.00
TLK-T LB
0 0.75
0.75
0.50
0.50
0.25
0.25
+>
?
0.00
0.00
0~
Figure 2.
101
y+ [ - ]
10 2
10 3
, ~
10 ~
jy
l,
. . . . . . . . . . . . . . . .
101
y+ [ - ]
10 2
10 3
Reynolds stress profiles in a zero pressure gradient turbulent boundary-layer at Re0=1410: DNS-data from Spalart [22]
4.
M O D E L I N G OF T R A N S I T I O N
4.1
TRANSIC-T
4.1.1
Intermittency Function
Transition cannot directly be simulated by applying the two-layer turbulence model, since it is originally intended for use in fully turbulent boundary layers. In order to extend the model for the calculation of transitional flows, an intermittency function T is therefore introduced into the eddy viscosity relation of the one-equation model. Eq. (1) then becomes =
(2)
By describing the relative fraction of time for which the flow is turbulent T is equal to zero in laminar and attains a value of one in turbulent Hows. Through transition T gradually increases from zero to one. Being part of the eddy viscosity relation (eq. 2) the intermittency controls the increase of IXt from zero to its value in fully turbulent flow. It should be emphasized that intermittency is always set to one in the domain of the core flow thus reflecting the situation under realistic conditions e.g. in gas turbine engines. The intermittency f T=
0.0 1-exp(-4.65r15)+q(1-rl)(r12-rl +0.5)
; Re 0 < A- Re0,s ; A. Ree, s < R e e
1.0
; Re e > Reo,E
(3)
is given as a function of the dimensionless coordinate 11 =
Re e - A. Reo,s
(4)
Reo, E - A. Reo,s In this Reo is the local momentum thickness Reynolds number and Re0,s and Reo,E denote its values at start and end of transition, respectively.
548 1.00
r-----I
I
0.75
r-
0.50
New I n t e r m i t t e n c y Narasimho (1957)
function
i
//
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/I
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Surface
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Distance
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!
Figure 3.
///
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I
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~
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Reynoldsnumber
'
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1.00
Ree
Intermittency function
Fig. 3 shows this new intermittency function together with a typical dependence from the literature (Narasimha [23]). The comparison clearly indicates that the new function differs from conventional approaches by incorporation of a pretransitional zone in which intermittency grows from 0.0 to 0.1. This allows for the first time for the consideration of the unstable laminar region upstream of the actual transition zone. The pretransitional domain starts at the location at which the local momentum thickness Reynolds number reaches the value of A'Re0,s. For predicting transitional boundary-layer flows correlations are needed to determine start and end of transition as well as the location of the beginning of the pretransitional region. 4.1.2
Pretransitional Flow
Strong negative pressure gradients in laminar boundary layers hinder the production of turbulent spots. The dimensionless acceleration parameter K = v / U ~ .dUe/dx is used to describe this effect. If K is greater than its critical value of 3x10 -6 [24], no new turbulent spots form and existing fluctuations are damped. Accordingly, the intermittency in this region is set to zero and the calculation is made under laminar conditions. If K is lower than this critical value, however, the possibility of turbulent spots in the flow can no longer be excluded and the beginning of transition is in principle possible. Due to this consideration the pretransitional flow has to start at the point where the acceleration parameter K drops below its critical value: T~
0.0 ]13.0;1.0]
9 K > Kkrit -- 3.0x 10-6 9 K < Kkrit = 3.0X10 -6
4.1.3
S t a r t of T r a n s i t i o n
and A calculated by" A = Re~ Re0,s K-K~,,
i@
(5)
The best way to characterize the location at which the laminar boundary-layer starts transitioning to its turbulent state is by specifying the value of the momentum thickness Reynolds number Reo,s at start of transition.
549 A detailed examination of a representative database consisting of almost 50 different test cases shows in agreement with the results of earlier investigations (e.g. [25], [26]) a dependence of Re0,s upon the free stream turbulence level in the core flow. An increase of the free stream turbulence causes the transition zone to begin at a lower value of Re0,s. In accordance with Dunham [27], to correlate this effect a turbulence parameter Xs is used which is defined as the average value of inlet and local free stream turbulence level. By inserting this value in percent the start of transition may acceptably predicted by the relation Re0,s = 500 Xs0"65 4.1.4
(6)
End of Transition
With the begin of the pretransitional zone and the start of transition described above, the only additional requirement for the model is a correlation for the end of transition. Again, the momentum thickness Reynolds number serves as the decisive parameter. As Re0,E appears to be dependent on Xs in a similar manner as Re0,s, the correlation may be expected to be of the form Re0,F, = D. Re0,s (7) with D being a constant. However, assuming a constant value for D leads to unsatisfactory predictions and is therefore replaced by a more sophisticated relation which takes the influence of the pressure gradient on transition progress into account. Positive pressure gradients shorten the transition length while negative gradients lead to a stretched transition zone. The Pohlhausen parameter ~,0, which is a function of local pressure gradient and momentum thickness turned out to be the best measure for the correlation of this effect. Careful examination then revealed that D is best obtained by: D=
0.04 ~s +2.2
" ~s > 0.0
0.055~s +2.2
9 ~s <0.0
(8)
with the new transition parameter ~ts described by:
_ro0 + us _.
- e x p (- 50~o, s)]. Tus ~,
15 )
"~o,s > 0
~s =
(9) [1- exp (50~,0,s)]. Tus t,
4.2
,5 )
"g0,s < 0
TLK-T
In the TLK-T model a similar approach as described above is employed to allow for a simulation of transitional boundary-layers. Neglecting the pretransitional flow zone the intermittency function depends on the parameters at start and end of transition only. To predict start of transition Fujisawa et al. [1] apply the correlation of Abu-Ghannam and Shaw [25] while end of transition is obviously taken from Re0,E = 2"Re0,s. In their paper the authors state to employ Abu-Ghannam and Shaw's correlation also for end of transtition. In contrast, recalculating the test cases shown in their paper reveals that they most probably used the
550
relation stated above which leads to improved results. This relation will be used also for the calculations in the current paper.
5.
C O M P U T A T I O N OF T R A N S I T I O N A L F L O W S
The performance of the two-layer models of turbulence for the calculation of transitional flows is evaluated against experimental data. Moreover, a comparison to various lowReynolds number models is provided by parallel application of the Launder and Sharma [ 19] and the Lam and Bremhorst [20] model. Additionally, improved versions of theses models denoted LS-PTM and LB-PTM are employed [28]. In an extensive study evaluating a large number of low-Reynolds number k-e models it was found that these models produced the best results for the simulation of transitional boundary-layers (Sieger et al. [29,30]). The test cases are based upon measurements made at Rolls-Royce plc., which were made publicly available by Coupland [31] and the ERCOFTAC Special Interest Group on Transition (e.g. Savill [32]). These experimental data were selected since they are well documented and allow for an accurate specification of all inlet and boundary conditions needed for the simulation. The test case denoted as T3A is a zero pressure gradient boundary-layer flow, whereas test cases T3C2 and T3C5 obey a favourable-to-adverse pressure gradient. The latter flows are qualitatively representative for the suction side of state of the art turbine blades (see e.g. Schiele et al. [33]), along which the velocity reaches its maximum value after approximately two thirds of the surface length and then decelerates. For the T3C2 test case transition starts far downstream in the decelerated region of the flow. Due to a higher Reynolds number the T3C5 flow is prone to an earlier transition starting already in the area of acceleration. All three test cases have a constant inlet turbulence level of 3 %. Figs. 4 through 6 show the friction coefficient cf, the shape factor H~z and the momentum thickness 0 versus the surface distance x for all test cases and turbulence models considered. It may clearly be seen that the new TRANSIC-T model provides the best simulation of the transitional boundary-layer flows, especially when the start and the end of the transition zone are looked at. Besides transition, laminar and turbulent boundary-layer quantities are predicted well leading to an excellent agreement between experimental data and simulation along the entire surface. When applying the TLK-T model very reasonable results are obtained, too. While the measured data of the T3C2 test case are reproduced very well, the calculated onset of transition takes place slightly too late for the two other cases (T3A and T3C5). This leads to an underestimation of the skin friction coefficient cf and an overestimation of the shape factor H~2 during transition. The low-Reynolds number models all lead to simulations of inferior quality as compared to the results obtained with the two-layer models. Using the LS-model transition starts too early in all cases whereas the LB-model results in a too late start of transition for the T3A, a flow separation for the T3C2 case and a correct start of transition for the T3C5 case. In any case transition proceeds too fast when these two models are used. Applying the PTM models improves the quality of the calculations in some cases. Nevertheless, agreement to experimental data is unsatisfactory with underestimating turbulent skin friction with the LSPTM model and predicting a too slow transition progress with both models.
551 10.0
10.0 T3A
TRANSIC-T LS LS--PTM
, - - - , 7 . 5 t ,, "
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Figure 4. Measurements and predictions in a zero pressure gradient boundary-layer flow: Testcase T3A 6.
CONCLUSIONS
Two-layer models of turbulence were shown to be capable to simulate fully turbulent boundary-layers without any discontinuity of significance present in the matching region of the calculation domains. By incorporating a new intermittency function into the one-equation model by Rodi et al. [15], the so called TRANSIC-T model has been developed being capable of simulating transitional boundary-layer flows. Pretransitional flow is considered by an increase of intermittency up to a value of 0.1. A new set of correlations to describe progress of transition
552
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Figure 5. Measurements and predictions in a favourable-to-adverse pressure gradient boundary-layer flow: Testcase T3C2 has been presented. The model is verified against experimental data showing excellent agreement between measurements and simulations. By comparing the TRANSIC-T model to another two-layer model of turbulence and four low-Reynolds number k-e models the superior quality of the solutions by the new model is demonstrated. Besides the TRANSIC-T model only the TLK-T model leads to acceptable results, substantiating the potential of two-layer models of turbulence in general.
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Figure 6. Measurements and predictions in a favourable-to-adverse pressure gradient boundary-layer flow" Testcase T3C5 REFERENCES 1
2 3 4
Fujisawa, N., Rodi, W., Sch6nung, B., In: Rotating Machinery: Transport Phenomena. Proc. 3 rd Int. Symp. on Transport Phenomena and Dynamics of Rotating Machinery (1990), Kim, J.H., Yang, W.-J. (eds.), pp. 651-663, Hemisphere Publishing Corporation (1992). Rodi, W., AIAA-Paper 91-0216 (1991). Launder, B.E., Spalding, D.B., Computer Methods in Applied Mechanics and Engineering, Vol. 3, pp. 269-289 (1974). Wolfshtein, M., Int. J. of Heat and Mass Transfer, Vol. 12, pp. 301-318 (1969).
554 5 6
7 8 9 10 11 12 13 14 15 16 17 18
Chen, H.C., Patel, V.C., AIAA Journal, Vol. 26, pp. 641-648 (1988). Chen, H.C., Patel, V.C., In: Turbulent Shear Flows 6, Andr6, J.-C., Cousteix, J., Durst, F., Launder, B.E., Schmidt, F.W. Whitelaw, J.H. (eds.), pp. 215-231. Springer-Verlag (1989). Patel, V.C., Chon, J.T., Yoon, J.Y., ASME - J. of Engineering for Power, Vol. 113, pp. 579-586 (1991). Iacovides, H., Launder, B.E., ASME-Paper 90-GT-24 (1990). Abou Haidar, N.I., Iacovides, H., Launder, B.E., AGARD-CP-510 (1992). Arman, B., Rabas, T.J., Numerical Heat Transfer, Part A, Vol.25, pp. 721-741 (1994). Franke, R., Dissertation, Institut fur Hydromechanik, Universit~it Karlsruhe (TH) (1991) Cordes, J., Fortschrittsbericht VDI Reihe 7, Nr. 214, VDI-Verlag (1992). Norris, L.H., Reynolds, W.C., Report No. FM-IO, Department of Mechanical Engineering, Stanford University, USA (1975). Lakehal, D., Theodoridis, G.S., Rodi, W., In: 11 th Symp. on Turbulent Shear Flows, Vol. 1, pp. 3.13-3.18 (1997). Rodi, W., Mansour, N.N., Michelassi, V., ASME - J. of Fluids Engineering, Vol. 116, pp. 196-205 (1993). Cho, N.-H., Liu, X., Rodi, W., Sch6nung, B., ASME - J. of Turbomachinery, Vol. 115, pp. 675-686 (1993) Sieger, K., Schiele, R., Kaufmann, F., Wittig, S., Rodi, W., ERCOFTAC Bulletin, Vol. 24, pp. 21-25 (1995). Wittig, S., Rodi, W., Sill, K.H., Rtid, K., Eriksen, S., Scheuerer, G., Schulz, A., FVVVorhaben Nr. 241 (1983).
19 Launder, B.E., Sharma, B.I.,Letters Heat Mass Transfer, Vol. 1, pp. 131-138 (1974). 20 Lam, C.K.G.; Bremhorst, K., A S M E - J. of Fluids Engineering, Vol. 103, pp. 456-460 (1981). 21 Coles, D.E., Hirst, W.A., Proc. Computation of Turbulent Boundary-Layers - 1968 AFOSR-IFP-Stanford Conference, Vol. II (1968). 22 Spalart, P.R., J. Fluid Mechanics,Vol. 187, pp. 61-98 (1988). 23 Narasimha. R., J. of the Aeronautical Sciences, pp. 711-712 (1957). 24 Jones, W.P, Launder, B.E., Int. J. Heat and Mass Transfer, Vol. 15, pp. 301-314 (1972). 25 Abu-Ghannam, B.J., Shaw, R., J. Mechanical Engineering Sciences, Vol. 22, No. 5, pp. 213-228 (1980). 26 Mayle, R.E., ASME-Paper 91-GT-261 (1991). 27 Dunham, J.,AGARD-AG-164 (1972). 28 Schmidt, R.C., Patankar, S.V., NASA Contractor Report 4145 (1988). 29 Sieger, K., Schulz, A., Crawford, M.E., Wittig, S., In: Engineering Turbulence Modeling and Measurements 2, Rodi,W., Martelli, F. (eds), Elsevier Science Publishers, pp. 593602 (1993). 30 Sieger, K., Schiele, R., Schulz, A., Wittig, S., In: 5 th Int. Symp. on Transport Phenomena and Dynamics of Rotating Machinery, Vol. A, pp. 454-471, (1994). 31 Coupland, J., Personal Communications (1993). 32 Savill, A.M., In: Engineering Turbulence Modeling and Measurements 2, Rodi, W. Martelli, F. (eds), Elsevier Science Publishers, pp. 583-592 (1993). 33 Schiele, R., Sieger, K., Schulz, A., Wittig, S., In: Proc. 12 th Internatl. Symposium on Air Breathing Engines, Billig, F.S. (eds), pp. 1091-1101 (1995).
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
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Turbulence Modeling and Computation of Viscous Transitional Flows for Low Pressure Turbines A. Chernobrovkin and B. Lakshminarayana Center for Gas Turbines and Power, The Pennsylvania State University, 153 Hammond Building, University Park, PA 16802 ABSTRACT Variation of the flow Reynolds number between the take off and cruise conditions significantly affects the boundary layer development on a low-pressure turbine blading. A decreased Reynolds number leads to flow separation on the suction surface of the blading and increased losses. A numerical simulation has been carried out to assess the ability of the NavierStokes solver to predict the transitional flows in a wide range of Reynolds numbers and inlet turbulence intensities. A number of turbulence (including the Algebraic Reynolds Stress Model) and transition models have been employed to analyze the reliability and the accuracy of the numerical simulation. A comparison between the prediction and the experimental data reveals good correlation. However, the analysis shows that the artificial dissipation in the numerical solver may have a profound effect on the transition in a separated flow. NOMENCLATURE Cp Cx K k2ke
k4 m m
S R R Re Rey Tu X Y u',v', w' u'
Pressure coefficient, Cp=(p01-p)/pol-p2) Axial chord length Turbulent kinetic energy Coefficient of the second-order artificial dissipation, turbulence equations Coefficient of the fourth-order artificial dissipation Strain rate tensor Viscous residual Rotation rate tensor Reynolds number, based on chord and outlet velocity Turbulent Re number, ~ y/v Turbulence intensity Axial length measured from leading edge Distance in normal to surface x, y, and z components of fluctuating velocity Total fluctuating velocity (RMS value)
U V y+ e 7
Total velocity y-component of velocity Inner variable, u~y/v Turbulence dissipation rate Intermittency
Subscripts 0,inl 1 2 ARSM e k-E ref s t tr vis w
Total, inlet, free stream Inlet Outlet Based on Algebraic Reynolds Stress Model Values at the edge of the boundary layer Based on k-E model Reference value Separation inception Turbulent Transition inception Viscous Quantity at the wall
556 1.
INTRODUCTION The transition from laminar-to-turbulent flow on the blade surface is a common, yet complex, phenomenon in turbomachinery. The boundary layer development, losses, efficiency, and heat transfer are greatly affected by the transition. The ability to accurately predict the onset and length of the transition are very important in the design of efficient machines. The transition in a low-pressure turbine may occur in either bypass form or through the development of a separation bubble, depending on the Reynolds number between the take-off and the cruise condition. At cruise condition, the flow Reynolds number may be less than half of the value of the take-off condition. This may result in separated flow and efficiency degradation. The development of a reliable prediction technique may lead to an improved efficiency and weight/thrust characteristics. Considerable effort has been spent in the investigation of the ability of different turbulence models to predict various types of transitional flows. A systematic approach undertaken by the ERCOFTAC group is summarized by Savill (1997). Nevertheless, very few investigators were focused on turbomachinery flows with the transition over a laminar separation bubble, especially at a high level of freestream turbulence. The objective of the research is to gain a detailed understanding of the unsteady transitional flows in low-pressure turbines, with emphasis on separation-induced steady transition. The test case chosen for this study is the simulation of separation and transition of the flow over the suction surface of low-pressure turbine cascade blade investigated experimentally by Qiu and Simon (1997). The influence of the free stream turbulence and pressure gradient is investigated. An existing Navier-Stokes unsteady flow solver is used. Three low-Reynolds number forms of two-equation turbulence models have been incorporated and tested for accuracy. In order to overcome the over-prediction of turbulence kinetic energy and the dissipation rate near the leading edge, several modifications of the production terms have been incorporated in the code as well as the Algebraic Reynolds Stress Model. 2. DESCRIPTION OF THE TEST CASE The experimental data in a simulated LP turbine cascade have been used to assess the ability of the numerical solver. A schematic of the facility is shown in Fig. 1 (Qiu and Simon, 1997). The cascade flow was simulated using a channel with a convex and concave wall profiled as suction and pressure surfaces of a turbine blade. A flow suction device was utilized to simulate periodic flow near the leading edge. Experiments were carried out with the inlet flow velocity ranging from 3 to 12.5 m/s, which corresponds to Re number from 50, 000 to 200, 000. A number of turbulence generators were utilized to generate the flow with 0.5, 2.5 and 10% inlet turbulence intensities. Boundary layer characteristics were measured using a hot-wire probe. Coordinates of measurement locations are given in Table 1. According to Qui and Simon (1997), the uncertainty in the mean velocity is 3.6%, and the fluctuating velocity is 4%.
Table 1. Location of the experimental data points on the suction blade surface N x/Cx N x/Cx N x/Cx N x/Cx P2 0.0398 P5 0.4667 P8 0.6889 P11 0.8593 P3 0.2111 P6 0.5506 P9 0.7457 P12 0.9111 P4 0.3778 P7 0.6247 P10 0.8173 P13 0.9728 3. NUMERICAL PROCEDURES AND MODELS The flow solver is based on full, Favre averaged, Navier-Stokes equations. An explicit fourstage Runge-Kutta scheme is used for the time integration of both mean-flow and turbulence
557
equations. A compact second-order accurate central difference flux evaluation scheme is employed for convection terms. Diffusion terms are discretized using second-order accurate central differences. A detailed description of the numerical procedure for the steady solver can be found in Kunz and Lakshminarayana, 1992. The solver was intensively verified and validated for different flows. A modified solver was successfully utilized for the numerical simulation of unsteady transitional flows in compressor and turbine cascade (Chernobrovkin and Lakshminarayana, 1998). A number of turbulence models ranging from two-equation low Re k-e models to full Reynolds stress models are employed for the turbulence closure. In current research a set of low Re k-e turbulence models (Chien, 1982, denoted as CH; Lam-Bremhorst, 1981, denoted as LB; and Fan-Lakshminarayana-Barnett, 1993, denoted as FLB), and hybrid ARS/ k-e models are employed for the numerical simulation. ARS component of the model represents an implicit ARSM (Rodi, 1972). To avoid the problem with the low Re ARS model, the near wall region is calculated using a low Re k-e model. Turbulence quantities based on ARS model are interfaced with those based on the low Re k-e model using the matching function: Re y 1 tanh(fl 1 - ~ / y + m,,ch -- 1) f m -- ~ ( tanh(fl) + 1) Rvi~ = f mRaRsM + (1- f m )Rk-, where: y§match--inner variable at matching point , 13- slope constant, R - residual Numerical simulation shows that for the LP turbine flow, the utilization of the explicit ARSM does not modify the solution outside the boundary layer, but may cause a stability problem. To avoid this instability and minimize the CPU time, the ARSM is used only up to twice the boundary layer thickness from the blade surface. The numerical investigation requires a verification to ensure the grid independence. A number of grids (121 x 71, 141 x 91, and 241 x 181) have been utilized to study the grid dependency on the solution. The maximum distance between the surface and the first grid point varies between y+=0.8 for the course grid to y+=0.12 for the finest grid. At X/Cx=0.65, coarse grid has 20 grid points within the boundary layer and 18 grid points within the laminar sublayer. The fine grid has 35 and 15 points correspondingly. Numerical predictions based on coarse and fine grid are very close to each other. In some cases, the fine grid solution was not stable in the transition region over a separation bubble. The difference between the solutions based on fine and coarse grid is minimal for the converged solution. All the reported simulation data in this paper is based on 141 x 91 grid. Additional discussion on the code verification is presented later. The numerical simulation of the transitional flow in turbomachinery cascades reveals the deficiency of the standard k-e model in predicting the flow with a high free stream turbulence. Non-physical increase in the turbulence intensity near the stagnation point may "contaminate" the boundary layer turbulence and trigger an earlier transition. An elevated level of the turbulence at the mid-passage, in the zone of maximum flow acceleration, leads to 2-3% higher level of freestream turbulence at the boundary layer edge. The replacement of the turbulence production term based on S- S with those based R is used to improve the prediction of the turbulence field near the stagnation point. The standard form of the prediction term is utilized for the flow calculation in the free stream. After the incorporation of the production term modification, the inlet distribution of the dissipation rate is set to provide the measured kinetic energy along the boundary layer edge.
558
4. PREDICTION USING k-E MODEL
4.1 Case Re=200000, Tu=10% The distribution of the surface pressure predicted by different turbulence models is compared with the experimental data in Fig. 2. Since the flow is fully attached at this Reynolds number, the pressure increases monotonically along the rear part of the suction surface. There is no difference between the blade pressure distribution predicted by various turbulence models. Flow with Re = 200,000 and Tu=10% is fully attached on the suction surface (Fig. 3). The flow with Re=50,000 and Tu= 10% has transition over laminar separation bubble as shown in Fig. 4. In the laminar part of the boundary layer (experimental locations P2-P7, Table 1), the predicted velocity field exactly match the measured values. For brevity, this comparison is not shown. Velocity and turbulence intensity profiles at locations P8-P13 are shown in Fig. 5. The beginning and the end of the transition, as well as the separation location predicted by various turbulence models, are compared with the data in Table 2.
Table 2 Inception and length of the transition, separation and reattachment points, Re=200000, Tu=10%. ~::
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Prediction Prediction Prediction FLB model CH model LB model Transition inception, x/Cx 71% -65% 57-59% 61% End of transition, x/Cx 81% 82% 94% 80-82% Separation, x/Cx attached ~ attached 75% attached Reattachment, X/Cx no reattachment i Flow visualization indicates the presence of a very small separation bubble at x/Cx=0.7 ,,,,
,
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,
The simulation based on the CH model predicts the transition at about 58% of the chord. The best agreement between the measured and the predicted velocity profile is achieved in simulations based on FLB and LB models. At locations P8 and P 11 the predicted velocity profiles are identical to the experimental data. Between P8 and P11 the measured profile is less steep in comparison with the numerical solution. The transition inception is predicted at about 4% of the chord upstream of the measured location. The numerical simulation predicts a shorter transition and hence the flow field is near the end of the transition at the location P10. The turbulence profile is very close to 'fully developed' profiles. As a consequence of this, the predicted velocity profiles at P9-P10 are closer to turbulent profile when compared to the data. The main drawback of the CH model is the existence of a very thin separation zone near the 75% of the chord. The CH model, based on y+, is known for its poor performance in separated flows. The CH model overpredicts the turbulence intensity in the turbulent boundary layer by 5070%.
4.2 Case Re=50000, Tu=10% Reduced Re number results in the development of a medium size separation bubble (Fig. 4). The separation zone is characterized by the presence of a flat zone in the pressure distribution at x/Cx-0.75 on the suction surface of the blade (Fig. 2b). A solution based on the FLB model correctly predicts this trend, which shows an earlier return to an adverse pressure gradient. As confirmed by the velocity profiles, this is due to a smaller separation bubble and earlier reattachment of the flow. In spite of the presence of the separation bubble in the simulation based on FLB and CH models, the region of constant pressure is not clearly predicted by these two models.
559 Velocity profiles are plotted in Fig. 6. Experimental data indicates that the flow separates at about 70% of the chord and the transition to turbulence occurs over a laminar separation bubble at about 81% of the chord. All three models predict a laminar separation at the same location; x/Cx=71-72% of the chord. The length of the laminar region inside the separation bubble varies for different models. The CH model gives the transition immediately after the inception of the separation at 71% of the chord. A simulation based on the FLB model predicts the transition inception at 78% of the chord, with distance between the separation point and inception of transition equal to 7%, which is close to the measured value. The transition to turbulence over the separation bubble is characterized by the inception of the transition in the shear layer with further penetration through the separation zone. The maximum turbulence intensity is located farther from the wall in comparison with high Re cases (Fig. 6). Contrary to the measured data, the numerical simulation predicts strong backward flow inside the separation flow, (point P10). An additional turbulence production due to the higher shear stresses in the zone of separated flow increases the turbulence intensities in the separation bubble. The predicted turbulence profile has a smoother distribution, and its maximum is located closer to the wall in comparison with the experimental data. In the case of FLB and LB models, an increased level of turbulence in the separation zone leads to a smaller thickness of the separation bubble and an earlier reattachment. A comparison between the numerical simulation and the data is given in Table 3.
Table 3 Inception and length of the transition, separation and reattachment points, Re=50000, Tu=10%. Transition inception, x/Cx End of transition, x/Cx Separation, x/Cx Reattachment, X/Cx
Expe_riment ............~Bi__mgd__e__!_i...................._CH mode! ....... LB mode! 76% 78% 72% 75% >p13, 97.28% 84% 85% 86% 70% 71% 72% 72% 91% 93% no reattachment no reattachment ,,
4.3 Case Re=50000, Tu=2.5 % This case is the most difficult to compute. A low level of turbulence at a low Re number leads to an inherently unsteady flow with an unsteady separation bubble and transitional zone. Even though the results presented in this paper is based on the steady solution, the analysis of the convergence and unsteady flow simulation indicate the need for an implementation of the time accurate simulation to achieve better resolution of the flow physics. No results for the LB model is presented, because attempts to stabilize the solution using increased artificial dissipation resulted in total damping of the separated flow. For the CH and FLB models, the flow has moderate fluctuation in the size and extent of the separation bubble. The data presented is calculated as an average of these fluctuations. This variation affects the rear part of the separation region and does not influence the location of the separation point and the transition inception point. A comparison between the predicted and the measured surface pressure distribution is shown in Fig. 2c. Both models underpredict the extent of the separation bubble, which results in a shorter zone of constant pressure. A comparison of the measured and the predicted velocity and turbulence intensity profiles (Fig. 7) shows similar trends to the cases described earlier. The numerical solver overpredicts the turbulence intensity in the transition zone; while for the flow with high freestream turbulence level, the maximum turbulence intensity is underpredicted (similar to Re=200000 case with Tu=2.5% and Tu=10%). There is no peak in turbulence fluctuations above the separation bubble in the transition region. The prediction based on the FLB
560 model has a smaller separation bubble thickness and an earlier reattachment. The size of the separation bubble is equal to the experimental data at point P9, about 1/2 of the experimental value at point P10 and 1/3 at point P11. For the FLB model, the variation in the thickness of the separation bubble was 50%. A low level of the freestream turbulence delays reattachment from the 85% to 99% of the chord. A comparison between the numerical simulation and the data is given in Table 4. Table. 4 Inception and length of the transition, separation and reattachment points, Re=50000, Tu=2.5 %. ............................................... Experiment ........ Transition inception, x/Cx End of transition, x/Cx Separation, x/Cx Reattachment, X/Cx
81% 96% 69% no reattachment
Prediction . . . . . . Prediction ...........Prediction ..... FLB model CH model LB model -80% -78% 69% 86% 88% 97% 71% 69% 72% 99% no reattachment 90%
5. PREDICTION USING HYBRID ARSM/k-e MODEL
Experimental data for the transition over a laminar separation bubble (e.g., Wang et al., 1998) shows a strong redistribution of the turbulent kinetic energy between components in the transition zone. Peaks of u' and v' are equal to each other, while in the attached turbulent boundary layer v' is equal to 40% of the u' component. The k-e model is unable to capture this redistribution as well as the overall anisotropy of the turbulence field associated with the transition process. A numerical simulation based on the hybrid k-e/ARSM has been carried out to investigate the ability of this model to improve the prediction of the transition flow over the LP turbine blading. Results of the current research as well as previously reported simulations (e.g., Abid et al., 1995) indicate that a numerical solution strongly depends on the k-e model used. A comparison of the prediction based on hybrid models with different low Re k-e (CH, FLB, LB) led to the conclusion that transition inception is controlled by the k-e model and is close to those predicted by a corresponding k-e model. Therefore, the FLB model has been chosen as the model with the best results based on the previous computations. Results of the numerical simulation based on hybrid k-e/ARSM are identical to the prediction based on k-e approach for the high Re case. High Re case is characterized by the transition in an attached boundary. Maximum shear stresses are located close to the blade surface in the region where viscous residual is calculated using k-e approach. ARSM part is used only for the outer layer. Therefore, the influence of the hybrid approach is minimal. In contrast, the shear layer above the separation zone is located in the zone where viscous residual is calculated using ARSM. Thus, the hybrid approach has a more profound effect. For the high Re case (Re--50 000, Tu=10%), this effect is minimal and can be seen only in the turbulence field. The utilization of a hybrid model moves the peak of the fluctuation velocity further from the wall, and closer to the measured location. There is only a minor change in the predicted velocity field. A comparison between the predicted and the experimental data for the case with Re=50, 000 and Tu=2.5% (Fig. 8) reveals an improvement in both predicted velocity and turbulence. Current numerical simulations have been carried out without the pressure strain terms. As a result, the w' component is equal to the v' component. An analysis of the turbulent case indicates that the streamwise component has about 50% of the total turbulent kinetic energy, while v' and w' have 25% each. No significant change in the balance between the different components is
561 found in the transition region (no more than 5% variation). Hybrid turbulence model overpredicts maximum amplitude of the fluctuation velocity similar to k-e model. However, redistribution of the turbulence energy between turbulence components plays a major role in improving the velocity prediction. 6. PREDICTION USING k-e MODEL IN CONJUNCTION WITH THE TRANSITION MODEL A numerical simulation of the transitional flows based on the turbulence model generally does not provide an adequate level of accuracy and robustness. Incorporation of transition models is a potential way to improve the transition prediction. Transition models use an empirical or a semi-empirical correlation to calculate the inception and end of transition, as well as the intermittency distribution in the transition region zone. A number of models were developed to calculate the inception and end of transition in an attached flow. The most common approach is the calculation of the transition inception using an empirical correlation and the calculation of the intermittency distribution using the approach suggested by Dhawan and Narasimha (1958), in conjunction with the correlation for the non-dimensional spots breakdown parameter (e.g., Gostelow and Walker, 1991; Mayle, 1991). Recently Mayle and Schulz (1997) proposed a method to calculate pretransition laminar fluctuatuations. In the current research, a model by Abu-Ghannam and Shaw (1980) is utilized for the transition prediction in an attached flow. Intermittency distribution is based on Dhawan and Narasimha (1958) Transition models for separated flows correlate the distance between the separation and the transition inception. In the current paper the model due to Davis et al. (1985) is used. As in the case of the attached flow, an intermittency distribution is based on Dhawan and Narasimha (1958) formula. Even though this relation was suggested for the attached flow, a comparison presented in Qui and Simon (1997) indicates that it can be applied to the separated flow. Six combinations of transition model incorporations have been analyzed. Two approaches have been used to incorporate intermittency distribution into solver; through an additional dumping function in eddy viscosity calculation, denoted as Var.1, and as a dumping function in an expression for the turbulence production, denoted as Var. 2. Each case was combined with three different distributions of the intermittency factor: 1. Step distribution: y=0, for x<Xtrand ~/=1, for x>Xtr 2. One-dimensional: y(x) = Maxy('f(x,y)) (i.e. maximum intermittency across boundary layer at current location) or :y(x) based on transition model 3. Two-dimensional: ~y(x,y) All cases are calculated using the FLB turbulence model, which gave the best prediction among the CH, LB, and FLB models. In attached flow transition, the implementation of the transition model does not have any significant effect on the velocity and turbulence distribution. No significant influence of the method of the implementation of the transition model (Var. 1, Var. 2, or type of ~) is found. As stated above, the utilization of Abu-Ghannam and Shaw correlation predicts an earlier transition in comparison with both the experiment and the prediction based on "pure" turbulence model. The current approach may only postpone the transition inception. Therefore, the prediction based on the "pure" turbulence model and the transition model produce practically identical flow fields. Simulation with the experimental distribution of the intermittency factor improves the prediction the turbulence kinetic energy distribution in the vicinity of the transition inception and an earlier end of the transition. Nevertheless, there is no improvement in the velocity distribution at location P10.
562 In contrast to the high Re cases, the way the transition model is incorporated and the type of intermittency distribution used has a profound effect on the prediction of low Re flow (Fig. 9). The incorporation of the transition model with the direct effect on the eddy viscosity (Var. 1) resulted in the development of the larger separation bubble in comparison with the experimental data (Fig. 9). The separation zone extends beyond the location of the trailing edge. Utilization of the two-dimensional distribution of the intermittency factor led to a further increase in the separation bubble size. The flow prediction based on the application of the intermittency distribution to the calculation of the production term (Var.2) leads to the prediction of a much smaller separation bubble and reattachment near the trailing edge. The predicted height and extent of the separation zone are closer to the measured values, which is a consequence of the delayed inception of transition. However, an overall deviation of the predicted velocity profile from the experimental data is greater in comparison with the "pure" turbulence model for all cases except the "step" transition model. This is due to the double damping of the eddy viscosity in a transition zone. Even though Y distribution indicates that the transition zone should extend beyond the trailing edge, all but one (one-dimensional model, Var. 1) has the end of the transition upstream of the trailing edge. A numerical prediction based on the one dimensional distribution and "step" distribution in conjunction with Var. 2 gave the most accurate prediction of the separation bubble size and location, even though it does not improve the turbulence intensity distribution in comparison with the simulation without the transition model. 7. E F F E C T OF ARTIFICIAL DISSIPATION ON THE TRANSITION P R E D I C T I O N The effect of utilizing only the fourth-order artificial dissipation term has been also investigated. This approach does not alter the result of the analysis presented below (beyond absolute values of the artificial dissipation coefficient). However, the employment of only the fourth-order artificial dissipation leads to a significantly increased sensitivity of the code to the turbulence field development near the leading edge. A moderate flow disturbance generates a significant increase in the turbulence kinetic energy, which decreases rapidly downstream. Numerical modeling shows that this increase can not be explained as a transition with relaminarization further downstream, because it may be reproduced at any location within the first 30% of the chord by placing the disturbance source (e.g., locally skewed grid). The predicted location of the separation inception, beginning and end of transition and reattachment point as a function of the artificial dissipation is shown in Fig. 10. Both the insignificant as well as excessive levels of artificial dissipation result in an earlier transition. The values of k4 and k2k~vary from the level below the stability limit to a level at which the artificial dissipation causes a significant non-physical diffusion. It should be noted, based on the previous experience with the solver, that the recommended variation of the k2k~ was 0.01 - 0.02. Within this range, the variation of the predicted and measured location of the transition inception is within 2.5% of the chord. An earlier transition inception results in a smaller separation bubble (Fig. 10) The primary source of the early transition in the case of a small k2k~ is a slight numerical instability of the scheme. For k2k~< 0.075, a moderate odd-even decoupling generates a premature transformation from the laminar to the turbulent boundary layer. An increase in the artificial dissipation also results in an earlier transition inception. It is possible to identify zones with a different behavior of the scheme. For simulations with k2ke < 0.02, the variation of the artificial dissipation term affects only the transition inception, but the transition length is essentially constant. This fact indicates that, within this range, the artificial dissipation acts as a destabilizing factor. A comparison of the streamwise distribution of the turbulent kinetic energy based on
563
differing values of k2k~ shows that slope of k is constant; i.e., the transition zone is shifted upstream without diffusion of the k field. Therefore, in this zone, the artificial dissipation is similar to the physical disturbances (freestream turbulence, noise etc.). For k2k~>0.02, the artificial dissipation leads to both an earlier transition and an increased transition length. This is the consequence of the streamwise/stream diffusion of the turbulent kinetic energy. 8. C O N C L U D I N G REMARKS A numerical simulation of the flow in a low-pressure turbine was carried out to assess the ability of different turbulence models to predict transitional flow at different Re and Tu levels. Best results were obtained with the FLB model (without the transition model). Implementation of the hybrid k-e/ARSM improves the prediction for Re = 50,000, Tu = 2.5%. While having minimum impact in all other cases, this modification contributed to the redistribution of the turbulent kinetic energy between various components in the transition region. Utilization of the transition model does not result in an improved flow simulation. Analysis of the turbulence characteristics in the transition zone shows that the lack of improvement is due to interference between the transition model and the low Re turbulence model. In the current prediction, the transition inception from the "pure" k-e model is located only about 2% of the chord upstream of the measured location. An enforcement of the transition through the intermittency function leads to a double damping of turbulence in the transition zone. A number of factors have been found to be essential for an accurate prediction of the transition. The first factor is the need to limit the turbulence production near the leading edge to ensure an accurate development of the laminar boundary layer. A second factor is the need to modify the freestream turbulence equation. This problem, as well as the first one, is due to the poor performance of a standard k-e model in the case of strong normal stresses. Without the adjustment of the freestream turbulence, the turbulence intensity may be over-predicted by 2-3%. The elevated level of the turbulent kinetic energy may affect the transition inception prediction. 9. A C K N O W L E D G M E N T This work was supported by the NASA Grant NAG 3-2025, with Dr. David Ashpis as the technical monitor. The authors wish to acknowledge the availability of the Supercomputer time at NASA Ames and the San Diego NSF Supercomputing Center.
10. REFERENCES Abid, R., Rumsey C., and Gatski, T., 1995, "Prediction of Non-Equilibrium Turbulent Flows with Explicit ARSM". AIAAJournal, Vol. 33, No.11, November. Abu-Gannam, B.J. and Shaw, R., 1980, "Natural Transition of Boundary Layers - the Effect of Turbulence, Pressure Gradient, and Flow History", J. of Mechanical Engineering Science, Vol.22, No.5, pp. 213-228. Chernobrovkin, A. and Lakshminarayana, B., 1998, "Experimental and Numerical Study of Unsteady Viscous Flow due to Rotor-Stator Interaction in Turbines; Part II: Simulation, Integrated Flow Field & Interpretation", AIAA 98-3596, AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Cleveland, Ohio, July 13-15, pp. 1-16. Chien, K.-Y., 1982, "Predictions of Channel and Boundary-Layer Flows With a Low-Reynolds-Number Turbulence Model," AIAAJournal, Vol. 20, No. 1, pp. 33- 38. Davis, R. L., Carter, J. E., and Reshotko, E., 1987, "Analysis of Transitional Separation Bubbles on Infinite Swept Wings," AIAAJournal, Vol. 25, No. 3, March, pp. 421- 428. Dhawan, S. and Narasimha, R., 1958, "Some Properties of Boundary Layer During the Transition from Laminar to Turbulent Flow Motion," Journal of Fluid Mechanics, Vol.3, pp.418-436. Fan, S., Lakshminarayana, B., and Barnett, M., 1993, "Low-Reynolds-Number k-e Model for Unsteady Turbulent Boundary-Layer Flows," AIAAJournal, Vol. 31, No. 10, pp. 1777 - 1784, October 1993.
564
Gostelow, J. P. and Walker, G.J., 1991, "Similarity Behavior in Transitional Boundary Layer Over a Range of Pressure Gradient and Turbulence Levels," J. ofTurbomachinery, Vol. 112, pp. 198-205. Kunz, R. and Lakshminarayana, B., 1992, "Three-dimensional Navier-Stokes Computations of Turbomachinery Flows Using an Explicit Numerical Procedure and a Coupled k-e Turbulence Model", Journal of Turbomachinery Vol.114, pp. 627-642, July. Lam, C. K. G. and Bremhost, K., 1981 "A Modified Form of the k -e Model for Predicting Wall Turbulence," Journal of Fluids Engineering, Vol. 103, September, pp. 456 - 460. Mayle, R. E., 1991, "The Role of Laminar-Turbulent Transition in Gas Turbine Engines," J. of Turbomachinery, Vol. 113, pp. 509-537. Mayle, R. E., and Schulz A, 1997, " The path to predicting bypass transition" J.of Turbomachinery, Vol 119, No3, pp 405-411 Qiu, S. and Simon, T. W., 1997, "An Experimental Investigation of Transition as Applied to Low Pressure Turbine Suction Surface Flows," ASME 97-GT-455, ASME Turbo Expo 97, Orlando, Florida, June 1997. Rodi, W 1976, "New Algebraic Relation for Calculation of the Reynolds Stresses," ZAMM, Vol. 56, p.219. Savill, A. M., 1997, "COST-ERCOFTAC Transition SIG. Evaluation of Turbulence Models for Predicting Transition in Turbomachinery Flows," Minnowbrook II, Workshop on Boundary Layer Transition in Turbomachines, September 7 - 10. Wang, T. and Hatman, A., 1998, "A Prediction Model for Separated-Flow Transition," ASME Paper 98-GT-237, Gas Turbine Congress and Exhibition, Stockholm, Sweden, June 2-5, 1998.
Fig. 1 Schematic of the experiment
Fig. 2 Surface pressure distribution
Fig. 3 Transition in attached boundary layer
Fig. 4 Transition over a laminar separation bubble
565
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
567
A General Model for Transition in Wall-Bounded Flows D. C. Weatherly ~* ~Department of Mechanical Engineering 521 CRMS Building University of Kentucky Lexington, Kentucky 40506-0108, USA A simple ODE analog for the near-wall streak-vortex mechanism is used to construct a transition model that responds to classical instability modes, local velocity and pressure disturbances, and wall roughness. The ODE analog governs the evolution of two scalar fields, which represent the local amplitudes of near-wall quasi-streamwise vortices and streaky velocity disturbances, respectively. A local Reynolds number characterises the response to vortical disturbances. At low values, all disturbances decay. At high Reynolds numbers, finite disturbances induce transition to a self-sustaining high-amplitude state. The model is intended for application to Reynolds-averaged Navier-Stokes calculations or large-eddy simulations in which the near-wall features cannot be resolved. The model also estimates turbulent wall stress using outer layer velocities, by assuming logarithmic behavior for 0 _< y+ _< 65" it may thus have potential as a general wall model. 1. I N T R O D U C T I O N Accurate prediction of transition to turbulence is important for the successful design of modern aircraft and turbomachinery. Existing numerical transition models lack universality, and for application to a given unknown flow, they must be calibrated to match experimental measurements of some closely related flow. Despite decades of study, the dynamics that govern transition (and developed turbulence also) have remained veiled, and the strategy of existing models is still to correlate data rather than to mimic those dynamics. Studies by a number of authors over the last ten years have focussed on transient algebraic growth of disturbances in shear flows. This linear process converts weak vortical initiating disturbances into strong streaky disturbances. The studies suggest that complete transition occurs when transient growth triggers strong enough new disturbances (via nonlinearity) that the process becomes self-sustaining. This theory establishes a link between transition processes and the onset of self-sustaining turbulent motions; it also successfully predicts the forms of those motions. Neither of these results can be achieved through classical normal-mode stability theory. *Current Address: Lexmark International, Inc., 740 New Circle Road, Lexington, Kentucky 40511, USA; [email protected]
568 If transition does occur this way, then low-dimensional analogs for the relevant physical mechanisms might yield quasi-universal transition models. The analyses by Trefethen and coworkers provide a theoretical framework and candidate low-dimensional analogs. The objective of this work is to adapt one such low-dimensional analog to predict transition and wall friction in wall-bounded flows. The vortical initiating disturbances are small near-wall features of a quite general form. They may be driven by several phenomena, including free stream turbulence, free stream sound, flow over wall roughness, and classical instability waves. To the extent that these phenomena produce similar vortical initiating disturbances, they are dynamically equivalent for the transition process. A model constructed to respond to all these inputs might be applied to highly disturbed turbomachinery flows. The scheme uses a low-dimensional analog to model the near-wall streak-vortex mechanism. It is applicable to Reynolds-averaged Navier-Stokes calculations or large-eddy simulations in which the near-wall features cannot be resolved economically. 2. R A T I O N A L E
The impulse of this work is to build a model that actually mimics turbulence transition dynamics. Since the relevant dynamics are still in some doubt, the proposed model depends on a plausible conception of how transition and self-sustaining turbulence occur in boundary layers. This conception is based on the recent analyses by Trefethen and co-workers [1,2,14] of the fundamental near-wall turbulence mechanism. Consider the Navier-Stokes equations linearized around a linear velocity profile. The evolution operator for this velocity distribution is non-normal. Even when no disturbance growth is predicted by classical normal-mode stability theory, transient linear growth is possible for disturbances that have the form of streamwise vortices. The output due to such vortical input disturbances takes the form of streamwise streaks of low- and highvelocity fluid. The growth rate and ultimate maximum amplitude of the streaky disturbances is proportional to the input amplitude and to a Reynolds number composed from the vortex diameter, the mean velocity difference across the vortex diameter, and the kinematic viscosity" Re~
=
Vh u
(1)
At high Reynolds numbers, even quite weak quasi-streamwise vortex motions are strongly amplified into long-lived streaks. In the nonlinear case, energy can be returned from the output streaks back to the input vortices under fairly general conditions. If enough energy is recycled in this way, then a feedback loop is established and the total disturbance energy jumps to a dramatically larger sustained value. This phenomenon has been named nonlinear boot-strapping; it may occur for all Reynolds numbers larger than a critical value, in the presence of sufficiently strong initiating disturbances. This condition is associated with the completion of transition. The threshold disturbance amplitude ecrit apparently satisfies
o(R<),
(2)
569 where the exponent a likely falls in the range -1.75 < c~ < - 1 . 2 5 .
(3)
Notice that the theory explains both the transition to turbulence and the self-regeneration of the turbulence by the same streak-vortex mechanism. It is consistent with physical observations and direct numerical simulations of both incompressible [5-10,15] and compressible [11-13] boundary layers. While Trefethen et al. develop the theory for canonical Couette and Poiseuille flows, it likely applies much more widely. These canonical flows represent the mean velocity profiles in near-wall layers where the streak-vortex mechanism is observed: y+ < 100. They are reasonable approximations to the mean near-wall velocity profiles in even highly disturbed flows, such as turbomachinery flows. This explanation of the development of turbulence has several interesting implications. If a near-wall layer of high shear can be identified, the velocity, layer thickness, and kinematic viscosity determine the amplification factor of the local non-normal disturbance operator. Any disturbance that has some quasi-streamwise vortex component may initiate the events described above. Disturbances might be due to free stream turbulence, free stream sound, near-wall flow over wall roughness elements, fluid jets exiting the wall, wall-normal velocity caused by strongly amplified instability waves at large scales, or quasi-streamwise vortices convected from their places of origin upstream. All such disturbances, both largeand small-scale, can be described in terms of their tendency to induce quasi-streamwise vortices and thereby stimulate the non-normal boot-strapping effect. Because the Reynolds number Re~ is a measure of the amplification, the boot-strapping effect can take place at any Reynolds number larger than some minimum value, Re~,~,.it. If transition has occurred, then relatively strong quasi-streamwise vortices are convected to cells downstream. We suppose that under this concentrated stimulation, the bootstrapping mechanism occurs at the same minimum self-sustaining Reynolds number Re~, ~,.it. This minimum Reynolds number in effect fixes the scale of the near-wall features in terms of wall units. For example, in a standard incompressible boundary layer, the quasistreamwise vortices form and re-form between the wall and y+ - 65. The Reynolds number Re,., for this layer is about 1000. This may be taken to be the minimum Reynolds number Rew,~,.it at which the physical process can be self-sustaining. If the outer velocity distribution is known from calculations, then the minimum Reynolds number rule allows us to estimate the thickness of the near-wall high shear layer. This yields the inner length scale, and assuming logarithmic behavior within the buffer layer, leads directly to an estimate of the turbulent wall shear stress. The studies by Trefethen and coworkers contain several simple analogs for the streakvortex mechanism. In the next section, a transition model is developed from one such analog, using the ideas just described. 3. N U M E R I C A L
MODEL
This section begins with an overview of the proposed transition model. Details are discussed in the following subsections. The analog for the streak-vortex mechanism is a 2 x 2 system of ordinary differential equations (ODE). The system governs the evolution of scalars u and v that represent the
570
amplitudes of streaky velocity disturbances and quasi-streamwise vortices, respectively, at each surface grid cell. The quantity u is local; it grows or decays strictly in accordance with the governing ODE. However, the quantity v propagates: it is taken to be the maximum value contributed locally from several sources, including free stream turbulence, wall-normal velocity, intense sound, flow over wall roughness, and quantity v convected from the cell immediately upstream. The ODE system contains an amplification factor R analogous to the near-wall Reynolds number Re,,. The factor R is determined locally at each time step using the large-scale velocity and kinematic viscosity, and some estimate of the near-wall high-shear layer thickness. The time advance for each local model system depends on the viscous time scale and the time step of the large-scale computation in which the transition model is embedded. The magnitude of the vector (u, v) T is a measure of the degree of completion of transition. Wall shear stress is estimated as a linear combination of the corresponding laminar and fully turbulent wall stresses, using the degree of transition as a weighting function.
3.1. O D E Analog for Streak-Vortex M e c h a n i s m Baggett & Trefethen have analyzed the transition behavior of several simple analogs for the near-wall disturbance evolution operator [2]. These analogs may evidently be adapted for engineering use in light of the above considerations. The analog used here is a modified version of a 2 x 2 system of O DE's devised by the Trefethen group:
o(.) ( . 1
Ot
v
1 --l~ -s
(,) V
V
V
,4, "
In the two-component solution vector, component u is the analog for the streamwise streak disturbances, and component v represents the quasi-streamwise vortices. As in the physical disturbance evolution operator, the Jacobian m~trix of the analog has a linear non-normal part that causes transient amplification, and an energy-neutral nonlinear part that redistributes energy from one component to another. The exponent s controls the threshold disturbance decay rate c~ (Eqn. 2). The Trefethen group has determined that a probably takes values near-1.25 for plane Couette flow and -1.75 for plane Poiseuille flow. In the experiments reported here, s was chosen to give c~ ,,~ -1.5. Table 1 Experimentally determined parameters for the model system.
0.60 0.70 0.80
-1.302 -1.505 -1.680
2.4 2.3 2.2
0.3292 0.3264 0.3144
0.812 0.809 0.795
571 1.60 1.40 1.20 1.00
v_O= 0.20 /
0.80
/ 0.60 0.40
/
/,
....
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:/
..
................... . ........v 0 = 0 . 1 7 .............................
v 0=0.16
0.20 0.00 5
10
15
20
25
30
35
40
45
50
t__viscous/t__convective
Figure 1. Behavior of analog system for R - 5.0 and s - 0.7; initial disturbances all have u - 0.0.
Experiments give rough estimates of the decay rate exponents a for three values of s, and the corresponding variation of the minimum sustainable value of amplification factor R. These results are presented in Table 1. Table 1 also shows the product I ( u , v)l in boot-strapping mode at the minimum sustainable value of R, and the vector magnitude I[(~, v)TII 9 This product is an analog for the production of new vortices from streaky disturbances via nonlinear effects. When normalized by its maximum boot-strapping value, it indicates the local degree of transition. The minimum amplification factor -Re,it which allows this system to achieve transition is about 2.3, depending on the value of exponent s. The system response to several initiating disturbances is shown in Fig. 1. For this demonstration, R = 5.0, s = 0.7, and component u was set to zero initially in all cases. The threshold disturbance amplitude in this case is about v = 0.165. The model ODE system is chosen to give relatively smooth response when bootstrapping occurs. In spite of this, experiment shows that the system is best integrated using an implicit scheme; the backward Euler scheme is chosen. The time scale for the transition of the analog system is just the Reynolds number R. Proper resolution of the transient behavior of the system requires relatively short time steps. In this case, 1000 steps per characteristic time are used to minimize inaccuracy that would change the effective threshold disturbance value.
3.2. Amplification Factor R The Reynolds number Rew indicates the tendency of the local near-wall shear layer to amplify and transform vortical initiating disturbances into streaky output responses (Eqn. 1). To estimate this quantity, appropriate values must be determined for the shear layer thickness h, velocity V, and kinematic vicosity u. Appropriate procedures are discussed in the next subsection. To match the behavior of the physical and analog systems, the physical amplification
572 factor Rew is simply scaled by the ratio of the critical values (R~,~t/Re~,~,~t) to get the analogous quantity R.
3.3. Shear Layer Thickness, Velocity, and Wall Friction The local high-shear layer thickness and velocity must be estimated to compute the amplification factor R for the analog system. It is important to remember that the layer thickness h is actually the diameter of the typical active vortical disturbances at a given location, and the velocity V is the velocity difference in the mean shear environment across such vortices. One of three rules determines these quantities, depending on the local state of transition. In laminar regions, the thickness h is equivalent to the local ~9~ boundary layer thickness. As transition begins, these quantities diverge. In all cases, conditions at a given location strongly affect circumstances immediately downstream. For regions in which local disturbances exhibit decay during a typical transit time across the cell, an integral method appropriate for laminar flow is used to estimate the boundary layer thickness as it evolves downstream. In these regions, the 89~ thickness of the boundary layer is an appropriate value of h, and 0.95 V~ is used for the layer velocity. For regions in which completion of transition is indicated by full boot-strapping amplitudes, an alternate procedure is used. In these fully turbulent near-wall regions, we assume that the layer thickness and its velocity adjust themselves such that the minimum sustainable Reynolds number Re~,c,.it is achieved. Assuming a logarithmic mean velocity near the wall, Re~,c,.it ~ 1000 if we take the layer thickness h to be y+ = 65. We further assume that the velocity profile is piecewise linear near the wall. The outer portion of the profile is given by two velocities known from resolved-scale calculations. The inner portion satisfies the no-slip condition at the wall. The maximum inner profile velocity V and corresponding wall-normal distance h must simultaneously yield the prescribed value of Re~ and also fall on the outer profile. Solving these simultaneous relations,
-
11} '
(5)
where known values are identified by subscripts. The velocity V then follows by substitution into the Re~ definition. Assuming the standard logarithmic velocity distribution from the wall out to y+ - 65, the velocity is V = 15.5 V,. The friction velocity V~ then gives the turbulent wall shear stress directly. For intermediate regions in which disturbances grow during a typical cell residence time but do not result in full boot-strapping amplitudes for the analog system, transition may be in progress. The shear layer thickness h is passed unchanged to the cell immediately downstream, while the boundary layer thickness evolves in accordance with the local wall shear stress. For downstream cells, the shear layer thickness is no longer the entire boundary layer thickness. The effective wall shear stress is the weighted sum of the stress of an equilibrium laminar layer with the local (~99 thickness, and the stress of a fully turbulent layer with the local shear layer thickness h. The latter stress is computed as in the fully turbulent case just described, using the thickness h given from the upstream cell and the velocity V estimated at height h by extrapolation of outer layer velocities
573 toward the wall. The weighting factor is the ratio of the local model quantity (u 9v) to that product in full boot-strapping mode. This product is the analog for the physical regeneration of quasi-streamwise vortices from streaks, a friction indicator. 3.4. Local Values of (u, v) w Component u represents the amplitude of the local streamwise streaks. Physically, the streaks result from the action of quasi-streamwise vortices in the mean shear environment. They are relatively long-lived and quiescent; therefore, the value for the streak field is set to zero initially, and non-zero values develop naturally as the local model is advanced in time. This is analogous to the localized production and decay of the streaks in the streak-vortex mechanism. Component v represents the nondimensional amplitude of quasi-streamwise vortices in the streak-vortex mechanism. No diameter is associated with the amplitude value here, although this is an important factor physically. For simplicity, we assume that the diameter of the vortices is the same as the presumed high-shear layer thickness. Physically, the vortices are relatively long-lived, and they convect downstream, decaying and re-forming as they move. Within established turbulent boundary layers, most locations receive strong vortices by convection from upstream. However, disturbances of almost any type may induce some quasi-streamwise vortical motion. The value of component v at a given location is proportional to the strongest vortical velocity present due to any source, and normalized by the local shear layer velocity V. Calibration experiments establish scaling factors that allow several types of disturbances to be considered on an equivalent basis. 3.5. T i m e Scales The analog system must be advanced in time an appropriate amount at each step. The physical amplification process is governed by the viscous time scale T~ = h 2/z/. Its nondimensional time scale is the ratio of viscous and convective time scales; this is just the Reynolds number [14]: =
h2/v h/v
=
T~
(6)
We are interested in the change experienced by a disturbance as it convects through the cell. The transit time of a disturbance convecting across the cell of length dx at 80% of the shear layer velocity V is dx = 0.S
(7)
This transit time must be expressed in terms of the characteristic viscous time scale. The disturbance experiences about Tt/Tv - (dx/0.8 9 V ) / ( R ~ 9 h / V ) - (1.25/R~), (dx/h)
(8)
characteristic time periods during cell transit, where (dx/h) is the ratio of the cell length to the shear layer thickness. For adequate resolution of the 2x2 system behavior, 1000 time steps are used for each characteristic time period of the streak-vortex cycle. This leads to (1250/R~) 9 (dx/h)
574 steps of h2/(1000 9v) units of physical time for the 2x2 system at each grid cell at each large-scale time step. 4. N U M E R I C A L
EXPERIMENTS
The numerical experiments concern transition on an adiabatic flat plate in a zero streamwise pressure gradient in incompressible air flow. The boundary layers are described by expressions developed by Chapman & Rubesin [3] for flat plates with arbitrary wall temperature distributions and free stream Math numbers. Their velocity expressions give the Blasius solution in the limit of zero Mach number and adiabatic wall. The experiments demonstrate the model behavior for flows at several Reynolds numbers, each with a particular distribution of free stream turbulence levels. The cases are the ERCOFTAC T3A, T3B, and T3AM datasets [4]. The friction coefficient cf is the comparison variable, since it simultaneously shows the transition behavior and tests the proposed wall shear stress scheme. To simulate the availability of outer flow quantities, velocity values were taken from the T3 test cases at wall-normal distances of about 1.5 mm and 4.5 mm at each grid cell. The outer locations fell within y+ distances of about 80, 150, and 130 for cases T3A, T3AM, and T3B, respectively. Turbulence levels in the datasets have been normalized by free stream velocities. The scaling factors were calibrated to give the best possible fit for case T3B, and the other two cases were run with the same constants for comparison. In each case, twelve large-scale time steps were taken; the solution magnitudes built up during an initial transient period, and a steady state was reached within about six time steps. The results are shown in Fig. 2a-c. It is important to observe first that the proposed model does indeed sustain transition in the demonstrations: the analog system shares the generic behavior of the physical system. With the scaling factors for model quantities R and v properly adjusted, the model gives an adequate representation of the T3B data (Fig. 2a). Transition occurs in the correct location and over an appropriate streamwise distance. Large overshoots in cf occur at the beginning and end of the transition length, but the computed values match experimental values well over most of the range. While the computation of cf in the laminar region is strictly a product of the Chapman & Rubesin expressions, the values in the transitional and turbulent regions are bona fide predictions of the proposed model. In the T3A and T3AM cases (Fig. 2b,c), the model predicts complete transition well ahead of the experimental locations, and there is very little transition length in either case. However, the computed friction factor matches the experimental values well in the fully turbulent region downstream. The transition behavior of the global system based on the chosen ODE system appears to be inappropriately abrupt. Experiments with a number of implementation details showed that this trait could be adjusted very little. 5. C O N C L U S I O N S The experience so far accumulated with the proposed transition model suggests the following conclusions: The proposed model can, in principle, account for the effects of all classes of disturbances
575 0.008
i
i
computed measured
0.007
,
0.006 0.005 o
0.004 0.003 0.002 (a)
0.001 0.000
!
I
1
500
1000
1500
0.008
|
|
|
computed measured
0.007
,
0.006 0.005 0.004 0.003
r r
0.002
o
(b)
0.001 t 0.000
i
i
i
500
1000
1500
0.008
|
computed measured
0.007
,
0.006 0.005 0.004 0.003 0.002 0.001
~~Jo 9
0.000
o
o
r
9
9
o
o
o
o
~176(c)
i
i
i
500
1000
1500
x (ram)
Figure 2. Comparison of computed and measured skin friction coefficients cf for fiat plates in air with zero pressure gradient: (a) Case T3B: unit Reynolds number R e ,,~ 618 mm -1, free stream velocity ~ ~ 9.4 m/s, and turbulence intensity lu'l/uo+ ~ 6.0%; (b) Case T3A: R e ,~ 361 mm -1, Voo ~ 5.4 m/s, ]u'l/uo+ ,~ 3.0%; (c) Case T3AM: R e ~ 1307 rnrn -a, V++ ~ 19.8 m/s, lu'l/uo+ ,~ 0.9%. Model exponent s - 0 . 7 for all cases.
576 that cause classical and by-pass transition. The construction is valid also for supersonic flows in which turbulence originates adjacent to the wall. The model shares the generic behavior of the corresponding physical system over the small range of conditions tested. However, the model does not reproduce measured transition locations very well at conditions other than the calibration conditions. Another ODE system with more gradual transition behavior should replace the current system. This might better represent observed transition lengths, and would also allow larger time steps for integration. The proposed method of estimating turbulent wall shear stress using known velocity values and the minimum Rew rule is successful here. It has potential as a wall model, and should be tested with a RANS or LES scheme that computes the outer velocities at the same time. These tests would be especially interesting in computations of high Reynolds number flows, in which the available velocity values are only known at locations far outside of the buffer layer.
REFERENCES 1. Jeffrey S. Baggett, Tobin A. Driscoll, and Lloyd N. Trefethen. A mostly linear model of transition to turbulence. Physics of Fluids A, 7, 1995. 2. Jeffrey S. Baggett and Lloyd N. Trefethen. Low-dimensional models of subcritical transition to turbulence. Physics of Fluids, 9:1043-53, 1997. 3. D.R. Chapman and M. W. Rubesin. Temperature and velocity profiles in the compressible laminar boundary layer with arbitrary distribution of surface temperature. Journal of the Aeronautical Sciences, 16:547-565, 1949. 4. ERCOFTAC World Wide Web page: http://fluindigo.mech.surrey.ac.uk/ 5. Javier Jimenez and Parviz Moin. The minimal flow unit in near-wall turbulence. Journal of Fluid Mechanics, 225:213-40, 1991. 6. Ugo Piomelli, Yunfang Yu, and Ronald J. Adrian. Subgrid-scale energy transfer and nearwall turbulence structure. Physics of Fluids, 8:215-24, 1996. 7. Stephen K. Robinson. The Kinematics of Turbulent Boundary Layer Structure. PhD thesis, Stanford University, April 1991. 8. Stephen K. Robinson. The kinematics of turbulent beundary layer structure. Technical Memorandum 103859, NASA, April 1991. 9. Stephen K. Robinson. Coherent motions in the turbulent boundary layer. Annual Review of Fluid Mechanics, 23:601-639, 1991. 10. P. Spalart. Direct numerical simulation of a turbulent boundary layer. Journal of Fluid Mechanics, 187:61-98, 1988. DNS. 11. E. F. Spina, J. F. Donovan, and A. J. Smits. On the structure of high-Reynolds-number supersonic turbulent boundary layers. Journal of Fluid Mechanics, 222:293-327, 1991. 12. E. F. Spina and A. J. Smits. Organized structures in a compressible, turbulent boundary layer. Journal of Fluid Mechanics, 182:85-109, 1987. 13. E. F. Spina, A. J. Smits, and S. K. Robinson. The physics of supersonic turbulent boundary layers. Annual Reviews of Fluid Mechanics, 26:287-319, 1994. 14. Lloyd N. Trefethen, Anne E. Trefethen, Satish C. Reddy, and Tobin A. Driscoll. Hydrodynamic stability without eigenvalues. Science, 261:578-584, 1993. 15. Jigen Zhou, Ronald J. Adrian, and S. Balachandar. Autogeneration of near-wall vortical structures in channel flow. Physics of Fluids, 8:288-90, 1996.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
577
The Use of a Turbulence Weighting Factor to Model By-Pass Transition. J. Steelant ~ 9 ~Section of Aerothermodynamics MPA, ESTEC/ESA, Keplerlaan 1, P.O. Box 299, 2200 AG Noordwijk, The Netherlands.
To simulate transitional heat transfer in turbine cascades, the conditionally averaged Navier-Stokes equations are used. To cover both the physics of freestream turbulence and the intermittent flow behaviour during transition, a turbulence weighting factor 7- is used. A transport equation is presented for this T-factor including convection, diffusion, production and sink terms. In combination with the conditioned Navier-Stokes equations, this leads to improvements in the calculation of flow characteristics in both the transitional layer and the freestream. The method is validated on transitional heat transfer measurements in a linear turbine cascade at typical operational conditions. 1. I N T R O D U C T I O N By-pass transition emanates mainly from high free-stream turbulence and leads to a transition far further upstream than what would be expected for natural transition. In contrast with natural transition which emanates from the breakdown of amplified disturbances within the boundary layer, the by-pass transition is mainly determined by the free-stream turbulence which affects the pre-transitional (pseudo-laminar) layer directly by diffusion and indirectly by pressure fluctuations. This diffusion of turbulent eddies has an intermittent character and is first localized in the outer layer of the laminar boundary layer. A similar but different kind of intermittent behaviour is also seen during the transition where the flow in the boundary layer is characterized by distinct turbulent and laminar phases alternating in function of time. In the latter, the intermittent behaviour has been quantified by the intermittency factor 7. This factor is the relative fraction of time during which the flow is turbulent at a certain position. It evolves from 0% before the transition point up to 100% at the end of transition. The same relative fraction of time can be taken to quantify the intermittent behaviour of the turbulent eddies on the underlying pseudo-laminar boundary layer. This parameter, in the sequel referred to as freestream factor w, is 0% near the wall and tends to 100% in the freestream. *The author wishes to thank Prof. R. Mayle and Prof. T. Arts for their helpful discussions during the course of this work. The work reported here is related to research activities, undertaken in the European Space Agency's Facilities, in the framework of a fellowship granted through the Training and Mobility of Researchers Programme financed by the European Community.
578 2. T R A N S I T I O N
MODEL
When modelling transition, it is essential to take these factors into account. Global time averaging used for classical turbulence modelling is not valid, in intermittently changing flows. To describe the transitional zone and the outer layer zone, it is necessary to use conditional time averaging. These averages are taken during the fraction of time the flow is laminar or turbulent respectively. As we are only interested in the state of the flow, i.e. laminar or turbulent, at a certain position, it is sufficient to make use of a turbulence weighting factor T, which is the sum of the intermittency factor 3' and the freestream factor w: T - 3'+aJ.
(1)
As a consequence, this factor ~- incorporates two effects: firstly the diffusion of freestream turbulent eddies into the boundary layer and secondly the transport and growth of the turbulent spots during transition. Hence, this factor is 0% in the vicinity of the wall within the pretransitional boundary layer, and 100 % in the freestream and inside a fully turbulent boundary layer. Near the wall, the T-factor is identical to the intermittency factor 3' as ~ = 0 close to the wall. Applying the conditionally averaging technique onto the Navier-Stokes equations leads to a set of laminar and turbulent equations for mass, momentum and energy [1,2]. These conditioned equations differ from the original Navier-Stokes equations by the presence of source terms which are function of the turbulence weighting factor T:
8U1 ---=--8Fl 8Gl 8Fvl ~- 8Gvl S~. . . . . . . -F l, Ot ~ Ox -~ Oy Ox Oy 0~7t OFt O-Gt _ OFv~ OGv~ .
a-Y-
ay
-
+
oy
+s[
Writing out the first equation of each set results respectively in a conditioned laminar and turbulent mass equation: cg~t vl _
Ot
Ox
1
~t
0y
0t 0~-
with
~ t Ut
OPt Vt __ 1
T
0y 0~
0~-
Similar expressions can be written for the momentum and energy equations. In order to close the equation, ~- has to be determined. The factor 7- has a spatial distribution which largely depends on the conditions of the mean flow and the boundary layer. For simple cases, one could set up an algebraic expression. For more general flows, a transport equation is more appropriate. This has been done for the evaluation of the intermittency factor 3' in previous work [2]. To evaluate the turbulence weighting factor ~-, the transport equation is extended with a diffusion and dissipation term and is given by 0~ ~-
Ox
t
0~ ~-
Oy
=
D~-+P~.-E~-.
(2)
579 The derivation of equation (2) can be found in [3]. The terms DT, P7 and E7 represent respectively diffusion, production and dissipation of 7-" 0 oxi
-
0T oxi
-
2f
(1 -
_--
-)V-ln(1
-
-
2.5,,
~
0~ 0T On On"
The production term P7 is however switched off on the wall prior to the transition point
-g
7
given by" Rex8 - 400094Tu~ l3s - 105254Tu1~ wich is derived in [3]. The value of ~- is set at 1 at the inlet boundary condition. The quantity ~ - v/~ 2 + ~2 is the local velocity amplitude and #7, f7 and/3 are the following functions"
#7 -
f~.#33Tu~~
f7
--
{ T < 0.45" 7->0.45"
/32
-
125• .
T)] -5(1-7),
A- -- 1 - exp [--256
1 -- exp[--1.735tg(5.45T--.95375) -- 2.2] 1
ru~ fufM
(PocUc~) 2 #~
The subscript oo refers to the value in the freestream. The coefficient V7 quantifies the diffusion of the turbulent freestream eddies into the boundary layer with fur being a damping function. The function f7 models the distributed breakdown near the onset. The/32-function is related to the growth of the turbulent spots in combination with Mach number effects (fM) and pressure gradient effects (fn):
fM -fu
[1 + 0"58M~
with P R C -
-2'
-- 1 + f T ( P R C - 1),
{ (474Tu~29) [1-exp(2ç176 10 -3227K~59s~
KM < 0 KM > 0
with M the local isentropic Mach number, Tuoo and Tut~ the turbulence intensity respectively in the free stream and at the leading edge. As the evaluation of Tuoo within the cascade is not evident, it is evaluated algebraically. The local freestream turbulence level is determined by Tuoo = Tut~(Ul~/U~o) 3/2 In the presence of a shock, this level is increased to a value of 15% [4]. In the wake-region the Tuo~-level is set at a physical value of 20%. The local compressible pressure gradient parameter KM is given by: K M =
#~ pooUoo
ll-
M2
I
dp dx
9
3. T E S T C A S E S
The capability of the present transition model is demonstrated on a turbine guide vane operating at typical design conditions. This guide vane has been experimentally investigated by Arts et al. [5] in order to understand the influence of Mach number, turbulence intensity and Reynolds number on the transitional heat transfer distribution. The blade profile has a chord of c = 67.647mm and a pitch to chord ratio of g/c = 0.85. The total inlet temperature is set at approximately T01=420K. The wall temperature is considered to be at a nearly constant level of 300K during the measurement. To validate
580 Table 1 Description of the different test cases. C~s~ MUR245 MUR241
II M2,,~
Tu,(%)
0.924 1.089
4 6
(%) 4.52 3.34 4.52
Rec,2 2.10 6 2.10 6 2.10 6
Figure 1. Geometry and block location
the present transition model, three different test cases are considered which are described in table 1. The incoming turbulence level Tui is measured 55mm upstream of the leading edge plane (x/cax = -1.487). As no length scale or dissipation were measured, the turbulence level at the leading edge Tule is estimated by the correlation of Roach [6] 5
Tu - 0.80 (~) 7, where the rod diameter is d - 3mm. The computational domain consists of two blocks. The first is a O-grid with 433 nodes along the blade and 73 nodes normal to the blade. The first point is situated at a location y+ < 1. The second block is a H-grid of 217 x 49 nodes and is placed in line with the outlet angle. Figure 1 shows the geometry and the location of the different blocks. 4. T U R B U L E N T
RESULTS
Before moving to the results of the present transition model, it is important to recognize how the used turbulence model behaves in a turbine cascade flow. It is generally known that, in the vicinity of stagnation regions, the classical turbulence models generate a too
581 2000 Exp Exp Exp
M_2,is M_2,is M_2,is
= 0.922 = 0.922 = 0.922
Num H Tul = 6% Tul = 4% Tul = 1%
x []
1500
.................................................
E
i ..................................................
!
.........................................
i 1000
'
.............................................................................................................................
x 500
x. ~ :.
.............................................................................. ~( ~
E:I E:] C] ~ X
•
i......... x . ~ ................................... ix x
X
[]
~.................. m - x ..... x - X ............ ---.:.................................................. x ~ ( ~
-x----x---•
: i [ ] [ ] IZI I=1 [ ] iZ]
0
! ................................................
i I
-0.1
-0.05
F.I[][]
i
[]
::
I:'1 [ ] r l IZI i~ F1 . IZl[]
::
,,
I
i
0
0.05
O. 1
s [m]
Figure 2. Fully turbulent heat transfer distribution for MUR239 (Tui =6 %) obtained with modified k-e model; full line: calculated, symbols: experiments (other Tui also shown).
high turbulence kinetic energy. As a consequence the corresponding heat transfer rate is far too high in comparison to experimentally measured data. A remedy exists in reformulating the production term according to Kato and Launder [7]. Classically this term is written as Pk = #tS 2 where S is the magnitude of the shear tensor: S - V/SijSij. Using Pk = #tS~ instead reduces the excessive raise near stagnation regions but falls back to the same analytical expression for boundary layer flows. The value ~ is the magnitude of the vorticity tensor ~ - V/~ij~ij. Applying this modification to the k-c model of Yang and Shih [8] results in a much better heat transfer distribution with a lower stagnation peak (see fig. 2). The heat transfer coefficients on the pressure side are placed on the left hand side, while those on the suction side are placed on the right hand side. The parameter s is the distance along the blade surface in meter. Although the obtained values are much lower, they still remain too high in comparison to the experiments both in the stagnation region and at the end of the suction surface. Similar overpredictions have been regularly reported by other researchers, e.g. see [9] and [10]. The necessary grid refinement was verified as nearly identical results were obtained on a mesh with a coarser refinement near the wall. This holds both for the laminar, turbulent and transitional results. In the sequel, only the k-c turbulence model of Yang and Shih with the Kato-Launder modification is used. To interprete the transitional results, one should keep in mind the possible overprediction of the turbulent heat transfer due to the underlying turbulence model.
582
2000
Exp Exp Exp 1500
Num H Num G* 1000 = 0.922 T u l = 6 % = 0.922 Tul = 4% = 0.922 Tul = 1%
M_2,is M_2,is M_2,is
.................................................
....... x m
~.................................................................................................................................................... i
E
.
E3 E3
1000 9
"v-
J.
500
i
x
-.
i
%
x
-...;. \ x ................................................ k ; . : ..........~ ......... x . x . . • ~
[]E3E3
[] j
0 -0.1
i ~(x x
mi -0.05
" I : : dt r , ~
Din[]
x
~ ) ~ "",o D
..
/
!
."
t" ............. ~ . ~ . . . . . . . . . . . . y _ . _ _ _ ! . . ~
,.9
,.
x
~."
x i
"" . . . . . i . . . . . . . . . . . . . . . 0 s Ira]
,-
t '
................................
~ ~ )~ ~g.~ ~( ~
.,."
!
i 0.05
'-'
O. 1
Figure 3. Heat transfer distribution for MUR239 (Tui - 6 %); full line: calculated heat transfer, dashed line: intermittency (x 1000), symbols: experiments (other Tui also shown).
5. T R A N S I T I O N
RESULTS
From the turbulent results, one can clearly notice that the obtained transition happens too early and too rapid. This defect is generally seen by all kinds of two-equation turbulence models. Applying the present transition model on test case MUR239 results in the heat transfer distribution given in figure 3. The overall distribution agrees very well with the experimental results which is mainly due to the correct prediction of the evolution of the turbulence weighting factor near the wall. The predicted heat transfer rate in the pretransitional boundary layer is generally lower than the experimental values both on the suction and pressure side. At high freestream turbulence, velocity fluctuations are induced in the boundary layer prior to transition. This inherently results in higher heat transfer rates [11]. This mechanism is however not incorporated in the present model. A possible method dealing with freestream turbulence effects is the model proposed by Volino [12]. Hence, pure laminar values are obtained which are almost identical to the low Tu-level date (Tu - 1%). As mentioned above, the fully turbulent level is still higher than the experimental level and can be attributed to the underlying turbulence model. The near-wall distribution of the turbulence weighting factor 7-, which corresponds with the intermittency factor 3', indicates that the transition is almost completed on the suction side (dashed line in fig. 3). At the pressure side, the transition is not completed as the intermittency reaches the 70% value near the trailing edge. The Mach number distribution is given in figure 4 together with experimental results taken at slightly different
583
1.4
I
' Num Exp M_2,is = 0.875 E x p M_2,is. = 1.02
12 I ...............................
.............................................................................................
...................................................
~
.
.
.
.
~
~
;' ............... i .....................
il..................... ~
1
._~
x ~
.
.
....................
: ....................................... i ~
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
-~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.8
x
0.6
...........................................................................
0.4
.............................................................
0.2
.......................................................................................
0
0.02
0.04
s [m]
0.06
0.08
0.1
Figure 4. Isentropic Mach number distribution for MUR239; full line: calculated, symbols: experiment.
Mach numbers. Lowering only the turbulence level to 4% (MUR245) reduces the spot production parameter such that a shock wave appears within the transition zone (fig. 5). The sudden raise in heat transfer is very well predicted but it ends with a too large overshoot. On the pressure side, the intermittency evolution is lower that in the previous case due to the lower turbulence level. On the suction side, however, the intermittency undergoes a steep increase downstream of the shock. This leads to a shorter transition zone ending before the trailing edge. The corresponding Mach number distribution is given in fig. 6 and differs slightly on the suction side from the result in fig. 4. The presence of a thinner, laminar layer allows a longer persistence of the low pressure region than in the MUR239 case ending with a shock wave. Increasing the outlet isentropic Mach number from M2,is - 0.922 in MUR239 to M2,is = 1.089 in MUR241, results in a lower pressure distribution on the suction side but has little or no effect on the pressure side (fig. 7). The higher acceleration in the region s - 3 to 4 cm compared to MUR239 decreases the spot growth resulting in 7 - 15% at position s - 4cm instead of the 26% value in the MUR239 case (fig. 8). The deceleration from s - 4 to 6 cm increases the spot growth again with a sharp raise in heat transfer as a consequence. The slightly accelerated region further downstream attenuates this raise. Near the aft, the heat transfer rate ends with a peak due to the presence of a shock at the trailing edge. On the pressure side the intermittency distribution is almost identical to MUR239 due to the identical pressure distribution.
584 2000 Exp Exp Exp
Num H Num G* 1000 = 0.922 Tul = 6% = 0.922 Tul = 4% = 0.922 Tul = 1%
M_2,is M_2,is M_2,is
1 oo
.................................................
1000
.................................. i•
....... x []
i ...................................................
•
i .....:.................................................
. ,\ ~
Xx
i...
, ; , , ."-
Xx
[]
X
X
./ .'
~i
x
x
:
~
'..
mi -0.05
~x x
x
............... ~ x ..... x - • ............... : : - / % "
~
[][]
i......... x.
~<
xt-%x
x x x
.................. " P !
-.. rn rn rq" ~Jxa
i 0 -0.1
i
x
xx
~
................................... I ........... ~ ......... ~ " - ~ ~'-.
::
J~x•
x
rn rn rn
................................
..~.:...................................................
9
500
..................................................
noD
i
~
!
'],.,..,"
!
.....................................
O m[],_tn
.--"
" .......
i. . . . . . . . . . . . . . . . . . . . . . . 0 s [m]
0.05
0.1
Figure 5. Heat transfer distribution for MUR245 (Tui =4 %); full line: calculated heat transfer, dashed line: intermittency (x 1000), symbols: experiments (other Tui also shown).
1.4
' Num Exp M_2,is = 0.875 Exp M_2,is = 1.02
1.2
.......................................................
x ~
............................................................................................ i i ~ i
~
,
,,!
,,
i i ........................................
..........................................
1
"-~,
.........
0.8
...................../
.......
0.6
0.4
..........................
0.2
0
0.02
0.04
0.06
0.08
0.1
s [ml
Figure 6. Isentropic Mach number distribution for MUR245; full line: calculated, symbols" experiment.
585 Num 1.4
-
Exp M_2,is = 0.875 Exp M_2,is = 1.02
x ~
........................................................................................ ! :
1.2
...............................................................................
! ........................................ i i
................. ~ " ~ :
..............................
1
"-I
0.8
0.6
0.4
0.2
0
0
0.02
0.04
0.06
s [m]
0.08
0.1
Figure 7. Isentropic Mach number distribution for MUR241; full line: calculated, symbols: experiment.
I
=
Exp Exp
M..2,is M..2,is
= =
-..........................................................................................................
1200
.......................................................................................................................................................................................................
,ooo
.................................. : ~ i . . . .
M. 2,is
....... x
-
800
Exp
' Num H Num G* 1000 .1 T u l = 6 %
1400
2 0 0
= 4% = 1%
[]
~x
~9i ~ x
: k x i ~x
'-.,
..........-~:........~
~ ~A ,~ " i~,v~x
~:.
x
x/
x~~D~x ~
...............................................................................
x'
:i ~ ' ~/ X X x x ~ , . d
x
.... ~
~
x/
M:
.~-~
'
,"
.,"
~ ................................................. / : ................
- / ~
x/i
XxxX,~
................................................. J ~ _ . ' : , . : . . . . . X ~ . . ~ . ~ . ~ [] [] [] []:D [] ",.
-0.05
::
~t~-xx F.:" \~,~,x
.................................................. i............... x . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0 -0.1
1
................................................ ~: .................................................. ~................................. I,: ...........
i-'...
400
Tul Tul
i
,
"-..
600
.1 .1
|
.
~
.
m
"
m
~
.,.'
6D''
.................. _. .............................
'x ................... : ........................................... :" .... i ............................................ 9 : /
" ..... ~ .............. 0
s [m]
" 0.05
0.1
Figure 8. Heat transfer distribution for MUR241 (Tui=4%); full line: calculated heat transfer, dashed line" intermittency (x 1000), symbols: experiments (other Tui also shown).
586 6. C O N C L U S I O N The use of the conditioned Navier-Stokes equations in combination with a transport equation for the turbulence weighting factor allowed the calculation of transitional heat transfer distributions in a turbine cascade operating at typical design conditions. The numerically obtained results are in good correspondance with the experimental data. Besides pressure gradient and turbulence level, the effect of compressibility needs to be taken into account to assure the correct prediction of both the transition onset and length. The present model, however, does not allow the prediction of the higher heat transfer rates in the pretransitional boundary layer. This pleads for a further extension of the model to incorporate the physics of the freestream pressure fluctuations affecting the underlying boundary layer.
REFERENCES 1.
2. 3.
4. 5.
6. 7.
8. 9. 10.
11. 12.
Steelant J. and Dick E. , 'Modelling of Bypass Transition with Conditioned NavierStokes Equations coupled to an Intermittency Equation'. Int. J. for Numerical Methods in Fluids, 23:193-220, 1996. Steelant J. and Dick E. , 'Calculation of Transition in Adverse Pressure Gradient Flow by Conditioned Equations'. ASME 96-GT-160, 1996. Steelant J. and Dick E. , 'A Transport Equation of a Turbulence Weighting Factor for Modelling By-Pass Transition'. In K.D. Papailiou et al., editor, 'Proc. of 4th European CFD Conference, Athens', pages 535-540. J. Wiley, 1998. ISBN 0-471-98579-1. Lee S., Lele S.K. and Moin P. , 'Isotropic Turbulence Interacting with a Weak Shock Wave'. J. of Fluid Mechanics, 251:533-562, 1993 and corrigendum 264:373-374, 1994. Arts T., Lambert de Rouvroit M. and Rutherford A.W. , 'Aero-Thermal Investigation of a Highly Loaded TranSonic Linear Turbine Guide Vane Cascade'. Technical Report TN-174, Von Karman Institute, 1990. Roach P.E. , 'The Generation of Nearly Isotropic Turbulence by means of Grids'. Heat and Fluid Flow, 8:2:82-92, 1989. Kato M. and Launder B.E. , 'Modified Form of Turbulent Flow around Stationary and Vibrating Square Cylinders'. In Proc. Turbulent Shear Flows, Kyoto, pages 10-4, 1993. Yang Z. and Shih T.H. , 'New Time Scale Based k - e Model for Near-Wall Turbulence'. AIAA J., 31(7), 1993. Zhang L.J. and Glezer B. , 'Turbine Airfoil External Heat Transfer Measurement in a Hot-Cascade'. ASME-Paper 97-GT-327, 1997. Biswas D. and Fukuyama Y. , 'Calculation of Transitional Boundary Layers with an Improved Low-Reynolds-Number Version of k-~ Turbulence Model'. J. of Turbomachinery, 116:765-773, 1994. Mayle R.E., Dullenkopf K. and Schulz A. , 'The Turbulence that Matters'. J. of Turbomachinery, 120:402-409, 1998. Volino R.J. , 'A New Model for Free-Stream Turbulence Effects on Boundary Layers'. J. of Turbomachinery, 120:613-620, 1998.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
587
Modelling of separation-induced transition to turbulence with a second-moment closure I. Had~i~ and K. Hanjali~
Department of Applied Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
The paper presents some results of application of a low-Re-number second-moment closure to the modelling of laminar-to-turbulent transition induced by a separation bubble. The same model, tested earlier in a number of low- and high Re-number flows, was found also to reproduce reasonably well several cases of by-pass transition, as well as cyclic sequence of laminarization and turbulence revival in oscillating flows at transitional Re numbers, without any artificial transition triggering. The present paper focusses on separation-induced transition in a flow over a flat plate with circular leading edge, and in a flow on a plane surface on which a laminar separation bubble was generated by imposed suction and blowing on the wall-opposite side. The results show reasonable agreement with available experimental and DNS data. The importance of applying higher order discretisation schemes for reproducing both the bubble and the transition is also discussed.
1. Introduction Transition from laminar flow to turbulence can be classified in three major groups (Mayle 1991), depending on the mechanism by which the turbulence is created: natural-, bypass- and separation-induced transition. The phenomenon of natural transition due to growth of perturbations of the laminar flow into Tollmien-Schlichting waves, which break into turbulence by non-linear effects, is beyond the reach of Reynolds-averaged single-point closure. However, the transition in situations where most of development stages in the natural transition are bypassed by some strong forced disturbances can, in principle, be reproduced by statistical models. Such are the cases of bypass transition which is maintained by diffusion of turbulence into a boundary layer from the free-stream or from some other turbulence source. It is, however, possible that a boundary layer separates while still laminar, forming a bubble and generating turbulence around reattachment, which then penetrates into the bubble and incoming laminar flow. This kind of separation-induced transition is often associated with adverse pressure-gradients and occurs e.g. at leading edge of an airfoil or gas turbine blades. Despite inability to capture the essence of the transition mechanism, it is generally accepted that the single-point turbulence closures offer more flexibility and better prospects for predicting
588 real flows with transition than classical linear stability theory. Progress reports on the activity of the ERCOFTAC Transition Special Interest Group (SIG), (Savill 1993" Savill 1996) provide a good overview of the current achievements of single-point closures in modelling the diffusioncontrolled bypass transition. Most results reported were obtained with low-Re-number eddy viscosity models (EVM), often modified to reproduce the class of flows considered. Although the outcome is still inconclusive it seems that the models which use the invariant turbulence parameters such as the turbulence Re number, perform better than those which use the local wall distance. Satisfying the correct wall limits of the shear stress and of energy dissipation rate was also found to be important for a successful reproduction of the transition. Several contributions using the Reynolds-stress models with low-Re-number modifications seemed all to perform better than two-equation models. Major advantages of the second moment closure (SMC) are located in the provision to account for anisotropy of the free stream- and of the near-wall stress field, particularly in the ability to reproduce the normal-to-the-wall velocity fluctuations. Another merit is in the exact treatment of the turbulence production and of effects of streamline curvature. These features help also in handling other forms of non-equilibrium phenomena, such as separation and reattachment, which are frequently encountered with different forms of transition. A major controversy is the artificial triggering of transition by a pointwise input of turbulence energy and "switch on" of the turbulence model at either a preset location, or one calculated from empirical correlations. While such an approach may be appealing and realizable for simple geometries, it is impractical in more complex flows simply because such information are not known in advance. Introducing intermittency as additional parameter, computed from empirical correlation (or even by a separate empirical transport equation, Savill 1996), was found helpful in handling very low free-stream turbulence (FST). However, the level of empiricism involved exceeds by far that of common single-point modelling. Besides, performances in more complex flows are still unknown: to authors knowledge, none of such models was tested in flows other than by-pass transition on a flat plate. In this work we apply the model which was tuned to reproduce reasonably well a number of non-equilibrium attached and separated flows at both high and low Re numbers, including bypass transition for FST_>3% and cyclic transition and relaminarization in oscillating flows. The model is based on standard high-Re-number second-moment closure with linear pressure strain in which the coefficients are variables in terms of turbulent Re-number Ret and invariants of stress and dissipation tensor anisotropy. More details can be found in earlier publications (e.g. Hanjali6 et al. 1997) and it will suffice to give a summary of the equations and coefficients used in Appendix. It should be noted that the same model and coefficients have been used consistently in computing all flows reported earlier and the present ones. Also, no artificial triggering, nor other empirical input were used to trigger the onset of transition.
2. Transition on a flat-plate with semi-circular leading edge Transitional flow over a flat-plate with a semi-circular leading edge was investigated experimentally at Rolls-Royce ASL, Coupland (1995). Although controlled by free-stream turbulence, the experiments indicate that the transition is preceded and enhanced by a thin laminar-like separation bubble at the plate just behind the circular edge. This very thin bubble is the major cause of enhancement of turbulence near the wall, as illustrated by a significant increase in the streamwise normal Reynolds stress in that region. Predicting the correct shape and size of
589 the separation bubble, which is crucial for predicting correctly the transition, has been a major challenge in which most conventional low-Re-number models failed. Several computations were reported within the ERCOFTAC Transition SIG, and in the literature, all based on eddy-viscosity models. Chen et al. (1994) reported a success by modifying the production of kinetic energy as suggested by Kato and Launder (1993) Pk = C~f, eSf2, (where S and f2 are modules of the mean rate of strain and mean vorticity respectively. With an addhoc modification of the production of k as Pk = C , f , cS (0.15S + 0.85f2)) they obtained even closer agreement with experiments. Papanicolau and Rodi (1997) reported good predictions using a two-layer model, but with empirically defined intermittency and the transition onset determined from integral correlation in terms of momentum thickness and free stream turbulence. Such and similar ad hoc modifications can be found in the literature. In this work we applied the low-Re SMC model and, for comparison, the Launder-Sharma eddy viscosity model. Several cases with different free-stream conditions were considered, which vary in the mean free-stream velocity and the FST level. Major attention was given to the T3L4 case with the mean free-stream velocity Uo = 5m/s and FST about 5.5%. The computations were performed with finite-volume method using block-structured, non-orthogonal grid. The computational domain extends 100 mm upstream and 500 mm downstream from the plateedge, and 150 mm away from the plate in the free-stream. The domain was divided into two blocks. The outer block, covering the free-stream, contains 184 x 15 CVs. The inner block covers a region of 25 mm next to the wall with locally refined grid with 308 x 60 CVs. In direction normal to the wall the grid distribution was highly non-uniform so that y+ at the first point to the wall never exceeded 0.3. The optimised grid covered the region of separating bubble with up to 28 CVs in the width (negative streamwise mean velocity) and about 55 CVs in the length. The stagnation region was covered with 60 CVs. A grid with the same number of CVs, but distributed differently so that it is coarser at the front of the plate (bubble region) with only about 25 CVs in axial direction, was used to assess the sensitivity of the solution to the grid density. The length of the separation bubble, mean velocity and streamwise Re-stress profiles, computed on both grids differed by less than 2%.. Uniform profiles of all variables were specified at the inlet. The values are chosen in accord with experimental data (Coupland 1995). Only streamwise normal stress was measured and in lack of other components, the isotropic turbulence was represented with v/-~/U~ - 0.0627 and L~ = ka/2/e = 27.3mm. Although not fully adequate, these initial conditions provide appropriate decay of turbulence in the free-stream as can be seen in Fig. 2 from profiles of u/Uo. Fig. 1 shows calculated stream lines for the eddy-viscosity and the SMC model. The separation bubble was predicted only with the SMC model and is in close agreement with the experiments. Fig. 3 shows evolution of maximum value of streamwise normal Reynolds stress. The experiments show that the stress increases in the bubble region reaching the maximum near reattachment. The experiments do not give the length of the bubble, but from the mean-velocity profiles it can be recognised that the reattachment point lies between 19 and 21 mm). The contours of turbulence kinetic energy given in Fig. 1 illustrate the development of k around leading edge and in the separation bubble. The much too high value of k in stagnation region predicted with EVM does not allow formation of separation bubble and therefore the real transition cannot be captured. In contrast to this result the SMC model compute proper development of turbulence. This is basically due to the correct prediction of the production and the dissipation of turbulence kinetic energy in the stagnation region. It is known that the wall
590
EVM
SMC
SMC "
~
Figure 1" Computed stream lines and contours of turbulence kinetic energy k for T3L4 case obtained with EVM and SMC, R e - U o 2 R / u - 3333. For the value of k see Fig. 3.
0.006
.02O
0.015
0.026
0.075
0.3
x,~zr~
Z
SMC
s .015
_ _ _ _
.010
Z _
~ (,
,
Z _ .005 _ _ _ lilt
IIll]ltl~lltll[lllllllllll~ll
IIl[Itlll
IIII
?l
,,,
1
0
"7,,
......
o u/u~
.020
SMC .015 ,o .010
',1
o
h% .005
O.
0
.1
0
.1
0
0
.1
0
.I
.020
Ii .
.015
.
.
.
.
ii
.
EVM ,co.s/" U
.010
~o .005
O.
_
t
i
)" 0
.1
0
1
r
.1
0
jj
1
kL,, 0
,,,, .1
~ 0
,, .1
U/Uo
Figure 2:T3L4 case. SMC results for the mean velocity U, and all components of the Reynolds-stress tensor and EVM results for turbulence kinetic energy. reflection model of Gibson and Launder (1978) which is used here in the SMC does not work correctly in the impingement flows, Craft (1991). However, in the case here considered the stagnation region is rather small and the boundary layer develops fast around the leading edge giving rise to the shear production and normal-to the flow pressure reflection, which is well
591
Exp. T3L4 u/.U~, Comp. RSM ~ o Comp. EVM Comp. EVM k )PUo
O|
o
|
/ I I/i
- I I I I
0
~ I
I
I
Sep. bubble: Comp. RSM i
I t
I\l
.0E
i
I
I
[ I
I
t
I
I
.04
I
I
I
I
x,
I
I
i
I
7"~ .06
Figure 3:T3L4 case. Evolution of the maximum of u 2 in streamwise direction. SMC and EVM results. Separation bubble indicated as predicted with SMC. reproduced by the conventional wall-echo model. Except the narrow region around stagnation point, the curvature of the streamlines are smaller than e.g. in an impinging jet. Fig. 2 shows the turbulent stress components at several positions computed with k - c and SMC model. The positions at which the results are compared are located near the point of separation (x=0.006 m), in the middle of the bubble (x=0.015 m), just behind the point of the reattachment (x=0.026 m), in the recovery region (x=0.075 m) and in the fully developed boundary layer (x=0.300 m). As seen, the results obtained with k - c model show poor agreement with experimental data which is a direct consequence of failure to predict the separation bubble. The maximum of turbulence kinetic energy is computed with Launder-Sharma model in front of the plate. The evolution of the turbulence kinetic energy downstream is shown in Fig. 3. Because of much too large value of k the model cannot predict the very thin bubble and therefore there is no transition in the bubble region. In contrast to this results the Reynolds-stress model reproduces correct value and almost correct shape of maximum of u 2 but the peak of this is about 4 mm (20% of the bubble length) behind the experimental one. Fig. 3 shows a drastic difference in predicting turbulence kinetic energy using EVM and SMC. The SMC calculates the maximum in the middle of the bubble as indicated by experiments. The SMC results can be regarded as good. Fig. 2 shows the profiles of the mean velocity and all components of the Reynolds-stress tensor at five selected positions, as said above. The show a good development through the flow. It is known that a possible stress anisotropy in the free-stream can influence the evolution of the flow near the wall as well as the transition. The v component of Reynolds-stress exceeds the u component at first three positions shown in Fig. 2 which is most likely unrealistic and is a consequence of the assumed isotropic turbulence at the inlet and in the free-stream. Nevertheless, the stress anisotropy at position x - 0.3m is consistent with the model results of a boundary-layer with zero pressure gradient. Both results are obtained with the second-order discretisation of solved equations (the convective terms in momentum equations were approximated with the QUICK scheme and in the
592
model equations with the TVD scheme and UMIST limiter was used). It should be pointed out that with the UDS scheme the SMC model predicted much too short bubble (about 6mm only) and the mean velocity and stresses in the recirculation region are far from experimental results. We computed two other cases, T3L3 and T3L6, both with about 2.5% FST but T3L3 is with Uo = 5m/s and T3L6 is with U0 = 1Orals. Because of substantially smaller FST, these results were less successful. In both cases the SCM predicts the separation bubble and the transition to turbulence induced by this bubble. However the predicted separation bubble is in both cases too long as compared with experiments. These results demonstrate the weakness of the singlepoint closures to predict the transition at lower level of free-stream turbulence (smaller than 3%). Nevertheless these results are better than any in the literature of which is the authors are aware of.
3. Transition to turbulence in a laminar separation bubble In this case, for which DNS were performed recently by Spalart and Strelets (1997), the separation of an incoming laminar boundary layer was created by imposed suction along the computational domain boundary opposite to the wall. The flow separates due to the adverse pressure gradient induced by suction. Transition to turbulence is sudden and occurs at a short distance (within first half of the bubble). Since the suction is limited to a small part of the domain, the pressure gradient drops to almost zero at the end of the suction region so that the fluid reattaches the wall around that point. The length of the separation bubble is about two flow-domain widths. In this way created bubble becomes highly turbulent although the incoming boundary layer is laminar. The turbulence generated by the bubble penetrates into the laminar part of the flow and is transported downstream by convection. A great challenge for turbulence models is the prediction of the coexistence of the virtually laminar flow portion and a highly turbulent ones without imposing any artificial measures (triggering) from outside. Besides, since the incoming flow is laminar, there is no ambiguity nor uncertainty regarding the inflow conditions, what is often a major problem in reproducing computationally transitional flows. The DNS yield the peak of the streamwise normal component of the Reynolds-stress tensor within the separation bubble close to the reattachment point, what was also observed in experiments by Coupland (1995) for T3L cases. According to Spalart and Strelets (1997) the flow is defined by three non-dimensional parameters: deceleration factor S = Qs/(YUo) which is the ratio between the suction flow rate Qs and the incoming flow rate YUo; the bulk Reynolds-number Rey -- YUo/u and the Reynoldsnumber based on the boundary layer length X (measured from the origin of the boundary-layer) at which the suction occurs Rex = XUo/u. In this particular case S = 0.3, Rex = 105 and Rey = Rex/3 were chosen (Spalart and Strelets 1997) to make the separation bubble large enough and at fairly large Reynolds number. The boundary conditions used for DNS were also applied in the RANS computations. A Blasius laminar boundary layer with 5 -- 6.8639 10 -3 Y was specified at the inlet. For computational convenience a very small turbulence level is assumed on the inflow boundary of the solution domain, k/U 2 = 10 -8. The top boundary was defined by zero vorticity and prescribed suction velocity respectively by
V(x ) -
5'
_e
_1/2(x/~)2
(1)
so that the suction flow rate is given by Q~ - f~o V(x)dx" Spalart and Strelets (1997) argued that cr does not influence the solution if small enough and we used here o- = 0.24.
593 The computations were performed on the numerical meshes of 200x50 CVs. Fig. 4a shows comparison of the wall friction factor obtained with the low-Re-number SMC to DNS data. The length of the separation bubble is well reproduced by the model. However, the model cannot predict the strong acceleration of the reversal flow near the end of the separation bubble. The DNS data show that the bubble has a tendency to "split" into two parts around X = 3.4, where the wall shear stress has a small but positive value. Similar underprediction of is observed in the computation of the backward-facing step flow at lower Re-numbers with the same model by Hanjali6, Jakirli6, and Had~i6 (1997), although in that case the flow is fully turbulent and the separation occurs from a sharp edge. Nevertheless, the results of the SMC are consistent. Fig. 4b shows development of the maximum values of the Re-stress components along the flow predicted with SMC and compared with DNS data. Although only the DNS data for two components are available, the figure indicates that the position of maximum Reynolds-stress obtained with SMC is in good agreement with the DNS data. The model does not capture fully the peak value, and the increase in Reynolds-stresses in the transitional region is too sharp. In the recovery region beyond the bubble, where the zero pressure boundary layer develops, the
Cy
~/-~/Uo
is predicted values of stresses are in good agreement with DNS. The peak value of about 0.25 which is equal to the value measured by Coupland (1995) for T3L4 cases (with 5%
i-J/Uo
in the recovery region is about 0.1 which is about 50% lower than measured FST). The by Coupland (1995) which is a consequence of high level of free-stream turbulence. In view of the simultaneous presence of a laminar and a fully turbulent region in the flow domain considered, the results can be regarded as pretty good. Fig. 5 shows profiles of the mean velocity U and all components of the Reynolds-stress tensor at several positions.
9
9
DNS Spalart and Strelets(1997) Comp. Low-Re Models
.010
0.30 0.25 .005
o U/Uo
F
o,V/Uo
0.20
......
0.15
....
0.10
o.
! L
---
!
u/U o
o
v/U ~
~ %
/Uo
uvlu:
0.05 0.00 -0.05
-.005 0
l
2
3
4
5
6X
7
I
0.0
2.0
4.0
6.0
Figure 4: a) Friction factor along the wall in the flow with separation bubble, b) Evolution of the maximum of the Reynolds-stress components along the flow.
4. Conclusions A low-Reynolds number second-moment closure model was shown to predict a transition from laminar to turbulent regime in two flows where the transition is induced by a laminarseparation bubble. The bubble was created in one case by low-turbulence free flow impinging on the round leading edge of a finite thickness plate, and in the other by imposed adverse pres-
594
~,~
Velocity Vectors = 1.16 2 m/s
u/~ =
1.000
=
0.200
=
0.200
=
O.OfO
1 v/u. .
.
.
.
.
.
.
.
.
.
.
.
uv/u~"
<~ I
I
I
I
I
I
2
3
4
5
6
7
Figure 5: Mean velocity vectors, streamlines and contours of ble flow computed with the low-Re-number SMC model.
V/-~/Uoin the transitional separation bub-
595 sure gradient on the incoming laminar boundary layer. In both cases the model reproduced well the transition location and transition-induced adjustment of the mean and turbulence flow parameters. These predictions, as well as the earlier reported ones of by-pass transition on a flat plate with zero and variable pressure gradient, and cyclic transition and laminarization in oscillating flows, were all achieved without having to introduce any artificial triggering of the onset of transition, nor other empirical input. Although the model shows some difficulties in reproducing by-pass transition at lower free stream turbulence (< 3%), it is demonstrated that the model can predict the coexistence of virtually laminar and fully turbulent flow regimes in a single separated flow.
References Chen, W. L., E S. Lien, and M. A. Leschziner (1994). Computational modelling of turbulent flow in turbomachine passage with low-Re two-equation models. In Computational Fluid Dynamics '94, pp. 517-524. Coupland, J. (1995). Personal Communication. Craft, T. J. (1991). Second-Moment Modelling of Turbulent Scalar Transport. Ph.D. thesis, UMIST Manchester, UK. Gibson, M. M. and B. E. Launder (1978). Ground effects on pressure fluctuations in the atmospheric boundary layer. J. Fluid Mech. 86, 491-511. Hanjali6, K., S. Jakirli6, and I. HadZi6 (1997). Expanding the limits of "equilibrium" secondmoment turbulence closures. Fluid Dynamics Research 20, 25-41. Launder, B. E. and B. I. Sharma (1974). Application of energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transfer 1,131-138. Mayle, R. E. (1991). The role of laminar-turbulent transition in gas turbine engines. Journal of Turbomachinery 113, 509-537. Papanicolau, E. L. and W. Rodi (1997). Computation of separated-flow transition using a two-layer model of turbulence. In ASME, Int. Gas Turbine & Aeroengine Congress & Exhibition. Savill, A. M. (1993). Some recent progress in the turbulence modelling of by-pass transition. In C. G. Speziale, R. M. C. So and B. E. Launder (Eds.), Near-Wall Turbulent Flows. Elsevier Science Publishers B.V. Savill, A. M. (1996). One-point closure applied to transition. In M. Hallb/~ck et al. (Eds.), Turbulence and Transition Modelling. Kluwer Academic Publishers. Spalart, E R. and M. K. Strelets (1997). Direct and Reynolds-averaged numerical simulations of a transitional separation bubble. In Proc. of 11 th Syrup. on Turbulent Shear Flows.
596
Appendix: The low-Re-number Second-Moment Closure Duiuj
Dt
Oxk
+ a2ij -- cij
Oxl
Dc
Oxk
Dt
~
k +C~u-ujuk
~j
02Ui O~Ui
c
- C~4f4kl2k~k + Sl
OxkOxl
OxjOxl
k
The pressure-strain model" dPij,1 -- - C l C a i j
(~ij,2 -
(~ i j ,1 -- C ~ f w -k
w (~ij,2
_
2 PkSij) ( P i j - -~
-62
U k Um n k Ttm (~i j -- -~ U i U k Ttk Ttj -- -~ U k U j Tbk Tti
C ~ f w (\(~km,2nkftm(~ij -- ~3 o(~ik,2ftkTtj --
~rbkj,2nkni) ~g2
C = 2.5AF1/nf;
C1 -- C -~- v / ' A E 2 ;
-
C2 - 0.8A1/2; A-I-~
}
"1 ; '
\ 150
F - min{0.6; A2};
1
f~=min
..~xn;1.4 ;
C~' - m a x ( 1 - 0.7C; 0.3);
9(A2-Aa)
l-
Ret -
e
C~ - min(A; 0.3)
A2 - a i j a j i
A3 -- a i j a j k a k i
aij =
]E2 -- eijeji
E3 -- e i j e j k e k i
eij --
9 E-I-~(E2-E3)
pc
UiUj k
2 (~ij 3
cij
2
c
3
(~ij
The model of the stress dissipation rate:
2
I~-7~+(~k~k + ~j~kn,~k + ~k~Ink~l~nj)f~ 1 k (1 + -~--~UpUq -npnqfd3 )
cij -- (1-- fs)-~e6ij + f~e L - 1 - v~E2;
f a - (1 + 0.1Ret)-l;
f~-
1-
C~2 - 1.4 exp
C~
[(
--~-)
Other variables are deffined as"
{[(1 0'2
-1
-~l~
,'0 ~A,
g - c - 2u OklOxl/z
The basic coefficients take the following values" Cs =0.22
C~=0.18
C~1= 1.44
C~2= 1.92
C~3=0.25
C/=2.5
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
597
The Effect of a Single Roughness Element on a Flat Plate Boundary Layer Transition M. Ichimiya Department of Mechanical Engineering, University of Tokushima 2-1 Minami-Josanjima, Tokushima 770-8506, Japan
A turbulence wedge that develops downstream from a single roughness element placed in a laminar boundary layer on a flat plate is experimentally investigated. Mean and fluctuating velocities and the Reynolds stress components were measured. Mean streamwise vorticity was estimated and velocity vectors were plotted. Many streamwise vortices appeared within the turbulence wedge just behind the roughness. Farther downstream, on the other hand, many of these streamwise vortices disappeared, and a pair of streamwise vortices were found on both interfaces between the wedge and the outer laminar region. The wedge expanded outwardly since the turbulent fluid within the wedge overflowed the wedge. At both sections just behind the roughness and farther downstream, the fluctuating velocities and the Reynolds stresses were larger than the values in the fully developed turbulent boundary layer over the whole height. The mixing length was also obtained.
1. INTRODUCTION The laminar-turbulent transition is important in engineering studies. Although we are close to fully understanding it, there remain many unresolved problems. DNS calculations and experiments were recently done [ 1,2], and detailed aspects were made clear. One of the topics of interest is the turbulent transition due to roughness elements. The present author performed measurements of a turbulence wedge by a single hot-wire [3 ]. The turbulence wedge developed downstream of a single roughness element in a laminar boundary layer. The streamwise velocity distribution and the shape of the wedge were shown. Many streamwise vortices were supposed to exist just behind the roughness. On the other hand, farther downstream from the velocity distributions a pair of streamwise vortices were estimated to exist on the interfaces between the wedge and laminar region outside the wedge. To identify the streamwise vortices and to clarify the relationship between them and the wedge, measurements of the velocities in the wall-normal and spanwise directions are desirable. In this report, the velocities of those two components were measured by V- and X-shaped hot-wires. Moreover, we obtained the vorticity of the streamwise vortices and conditional averages of the velocities based on whether the flow is turbulent or non-turbulent. The turbulence wedge develops not only in the wall-normal direction (such as in a general twodimensional boundary layer), but also in the spanwise direction. Measurements of the spanwise transverse velocity component can only be seen in the wedge on a rotating cylinder [4-6]. The wallnormal component of the velocity in the wedge has never been measured.
598 2. E X P E R I M E N T A L A P P A R A T U S A N D M E A S U R E M E N T M E T H O D S
The wind runnel employed in this experiment is a blowing type. Its total length is approximately 8 m, and the test section is 400 • 150 mm in cross section and 2 m in length. A flat Bakelite plate was used in the experiment. A wall opposite the working side of the plate permitted adjustments of the pressure gradient. A zero pressure gradient was obtained along the plate. The velocity profile near the leading edge was of the Blasius type. The experiment was conducted for constant unit Reynolds number Um/V= 5 • 105 rn-~ (The reference main flow velocity at the leading edge U~ was about 7.5 m/s). Then, the turbulence intensity of the free stream in the whole inlet cross section was kept below 0.2%. Tani et al. [7] showed that the turbulence intensity of the main flow does not affect the condition of the transition induced by the roughness element. Figure 1 shows the coordinate system and turbulence wedge. The roughness element is a cylinder 2 mm in both diameter d and height k, and is located 100 mm downstream from the leading edge on the center line (z = 0 mm) of the flat plate. Since the boundary layer at this position without the roughness element is laminar with a thickness of about 2.2 mm, the height of the roughness element k is nearly equal to the boundary layer thickness. The roughness Reynolds number based on k and velocity at y = k is 996, thus satisfying the condition under which the turbulence wedge develops from the roughness [7,8]. Here we employ normalized coordinates, X(= (x-xk)/ k), Y(=y / k) and Z (= z / d) (xk is the x position of the roughness, 100 mm). V- and X-shaped hot-wire probes with two tungsten sensing elements each 5 gm in diameter and 1 mm wide were used in the measurements. Two sensing elements of the V-shaped hot-wire are on the x-z plane and can measure streamwise and spanwise velocities, whereas, those of the X-shaped hot-wire are on the x-y plane and can measure streamwise and wall-normal velocities. The output voltage from the hot-wire has been digitized at a 5-kHz sampling frequency during about a 13-second sampling period. To obtain a conditional averaged velocity, a detector function, which discriminates whether an instantaneous flow is turbulent or non-turbulent, was first obtained. A first-order time derivative of an instantaneous streamwise velocity was adopted as the detector function. From the above detector function alone, a perfect discrimination could not be expected. Thus, the threshold level was adjusted to obtain a reasonable value, s o that an intermittency distribution matched the familiar characteristic flow field ofKovasznay et al. [9].
•
/ t=====t-= / / I I ~:'[ ]
Turbulence
..
Wodgo II1~ r 2->~1!1 Probe
10
,
/
Um Single Flat Plate Roughness Element
Layer
Figure 1. Coordinate system and turbulence wedge
599 3. RESULTS AND DISCUSSION
3.1. Mean Velocity Figure 2 show the wall-normal distributions of spanwise mean velocity components W normalized by the free stream velocity U,. The height from the wally is normalized by the boundary layer thickness 6. Within the turbulence wedge, the velocities take large values near the wall and smaller values farther from the wall at both X = 50 and 200. At X = 50, velocities take various signs with Z near the wall; thus indicating that the spanwise velocity changes its direction frequently, although symmetry with respect to the center of the wedge is fairly well maintained. This suggests that many streamwise vortices exist, and that they are symmetric with respect to the wedge center. This will be considered again later with the aid of the vorticity. At X = 200 and Z > 0, most points near the wall take positive values, although velocities take negative values at Z =-12.5. Likewise, in this streamwise position, symmetry with respect to the center is well established, but unlike at X = 50 the existence of a pair of strearnwise vortices is suggested. In Fig. 2 distributions of the normal average W, turbulent average Wr and non-turbulent average WNat the interfaces of the wedge are also shown. Turbulent and non-turbulent averages are shown in the region of intermittency factors V > 0.1 and V < 0.9, respectively. Absolute values of the turbulent average are larger than those of the non-turbulent average except for the region near the wall. From this distribution, the turbulent region tends to overflow the wedge, causing it to expand in a lateral direction.
x&_ o 1
- ~
X = 50 5
-4-3.5-3-2.5-2-1.5-1-0.5
0.8 ~0.6 >0.4 0.2 0 -0.10 0 0 0 0 0 0 0 0 0 0 0 0 0.1 W/Ue X= 50 1 Z=0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 10 15
0.8 ~0.6 >0.4 0.2 0 -0.10 0 0 0 0 0 0 0 0 0 0 0 0 0.1 W/Ue X : 200 0 5 10 517.52~ 30 1 Z=-12.5 2.5 7.5 .2.51 25 35 I I I I 0.8 ~0.6 >0.4 0.2 0 -0.020 0 0 0 0 0 0 0 0 0 0 0 00.02 W/Ue
Z=-5
0.8
-4
LR
1 I
i ~
Io ~
o~
-3.5
, II
o/A
i
i
i
" -W/U e J
I
o~WT/Ue N/Ue t
~..~0.6 t~Oo~z& (z~ >0.4
0
i
F
-0.04 0 0 0 0.04 0.08 -0.02 0.02 0.02 0.02 0.06 0.1 W/Ue WT/Ue WN/U e
X = 50
Z= 4 ' ~8'
1 0.8 20.6 >0.4 02
4.5 ' &~'
~
5 '~'
/
%0
o~ ~oO
9
0
9
,~o
~'
-0.04
_0 0.04 0.02 0 0.04 O.O:Z- 0.02 0 0.04 0.02 W/U e WT/U e WN/U e
X=200 1 0.8 ~o0.6
>0.4 0.2 0
Z=15
17.5 OOI I I I
20 I I 0 o
I
I
I
I
-0.04 0 0.04 0.02 0 0.04 -0.02 0.02 0 0.04 0.02 W/U e WT/U e WN/U e
Figure 2. Distribution of the mean velocity component in the streamwise direction
600 0.041
,
0.04
o og
,
,
,
,
I
,
,
,
,
0.02 0.01
, I
f
|
I
I
I
I
I
I
I
I
I
I
I
O0
I
I
Y-2
0.02 0.01
0
D 0.02
. oolo o.oo o. o.o4 0.02 -0.04 -0"02
0.03 0.02 0.01
_ -
i
i
i
i
i
i
i
i
i
i
i
i
-0.01 -0.02
I
I
I
I
I
I
I
I
I
I
I
I
I
0 5 10 15 20 25 30 35 5 10 15 Z Z Figure 3. Distribution of the mean velocity component in the wall-normal direction -15-10-5
0
Figure 3 show the distributions of wall-normal mean velocity components V in the spanwise direction. At X - 50 the distribution near the wall at -5 < Z < 5 varies markedly with Z. A comparison of this distribution with the streamwise mean velocity U/Ue in the previous paper [2] helps to understand the behavior of the fluid motion. At the position where U/Ue is minimal, V/U~ takes positive values, whereas at the position where U/Ue is maximal, V/U~ takes negative values, namely, where fluid rises, low-velocity fluid near the wall is elevated and U / U~ is minimal, whereas where fluid falls, high-velocity fluid far from the wall is suppressed and U/U~ is maximal. This suggests that in this streamwise position, many streamwise vortices exist, and that they are litted up or pushed down at their respective positions. Outside the wedge, the velocities are almost zero. Far from the wall, at Y= 2, the maxima or minima cannot be seen (unlike distributions near the wall). At X = 200, like U/Ue and W/U~, extreme change are not seen, so it may be said that the many streamwise vortices at X = 50 disappeared. At Z = 10, values are slightly smaller than the surrounding values, whereas at 16.5
601
3.0
2.0
X=50 I
,/,/
~~
~
I
~
I
= o k /0.04
,j ).04 ( " _
/
>-1.5
i
'
~.o8 /
o.os
4
0.12
_~
,~.'
i//~ ~
~xk/U~ /=-0.008
-,~,',
, ./
~
,.o
-a
~
,
"~,---,// ,/,A/
,
/ / /
-o.oo4
-
, 0 / , 0.004
-
,
Z
Z
Figure 4. Isopleths of the mean vorticity component in the streamwise direction V/U'=O'05 T__.> x - 50
5
w/u.=o.05
V/U,--O.05 ' ~ X = 200
W/U,,=O.05
1
i
'L[ i'
3 2
0
......~...........~ ........ .....~ ....... ..........~,iil
-6
-4
-2
0
2
4
6
0
0
5
10
15
Z
...........J.............t ......................... r......................."
20
25
30
35
Figure 5. Velocity vectors In this position, which corresponds to the interface between the wedge and the outer laminar region, a streamwise vortex exists which rotates from the outside to the inside of the wedge in this position. At X - 200, as Figs. 2 and 3 suggest, a positive vorticity exists in the region of Z < 0, and this vorticity makes a pair with the negative vorticity in Z > 0. Thus, at X - 200, a pair of streamwise vortices exists at both interfaces, although the vorticity is only about one-tenth of that at X - 50. Figure 5 show the velocity vectors on the y-z plane. These vectors were obtained from the mean velocities W / Ue and V/Ue in Figs. 2 and 3, respectively. At X - 50, the direction of the vector coincides well with the sense of vortex rotation in Fig. 4. The vectors also confirm the existence of the many streamwise vortices.
3.2. Fluctuating Velocity Figure 6 show rms values of fluctuating velocity in the streamwise direction u' / Ue with Y as a
602
0.040"06I-)( -'5'0 ' ~ ~ '
' ' '-I
o
0.04 0.02 :z)o 0.04o
0.08 0.04006
OO2o
0.04 ,,>0.02 ~o.o o
Y=l-
0.04
0.04
-
0.04 0.02
0.04 0.020
o o~
!
-~ o o~
o
OO2o7f
, -15-10-5
0 Z
5
1015
, , , , , , , , 0
5
Y~O.7[
101520253035 Z
Figure 6. Distribution of the fluctuating velocity component in the streamwise direction
0.04 0.03 0.02 0.01 0 0.03 0.02 221 <'>O.O~l
.05 04
~io3 o2
,_
_ , / ~ . . . ~ ~
I
I
I
I
I
I
I
I
o;ol ~
0 .03 02
DO:01
-> 0.03 0.02 0.01 i i i Ly~0,5i 0 0.03 0.02 0.01 0 -15 -10 -5 0 5 1 0 1 5 Z
-,,, ,,A
~:o.~;
-
o ~~.0 O2
oiol
i
0
0.03 0.02 0.01 0
~,, ,,,,, 0
5
~
v:o.?s,
10 15 20 25 30 35 Z
Figure 7. Distribution of the fluctuating velocity component in the wall-normal direction parameter. At X = 50, values fluctuate markedly as in V/Ue.The positions of maxima and minima in the distributions do not correspond well to the positions of the suppression and elevation of fluid considered in Sec. 3.1. The mean velocity U increases monotonously with the distance from the wall. However the fluctuating velocity u increases at first but then decreases with the distance from the wall. Thus, for the mean velocity, it is clear that the fluid which comes down from above always has a higher velocity, but for the fluctuating velocity it does not necessarily follow that the fluid which comes down from above has a lower fluctuating velocity. Figure 7 show the fluctuating velocity in the wall-normal direction v' / Ue. This component also fluctuates markedly at X = 50 because of the many streamwise vortices. At X = 200, the distributions at Y = 0.25, 0.5 and 1 take almost constant values in the region of Z < 10 and reach their maxima at Z ~ 12.5. They suddenly decrease in the region of 15 ~_Z<22.5 and fall to almost zero at 22.5 _~Z.
603
0.06 0.04 0.02 0 0.04 0.02 ,,, 0 D 0.04 -~ 0.0
0.081 , , , , m , , , , , , , , 0.06 ~ X = 200 0.04 ~ "%. m
~
m
0 . 0 4 0 " 0 2 ~ -
Y = I
-
D
~%--~
Y=2:
,, 0.02 ~ " ~ , ~ _ ~ _ . Y = I 0 0.04
:
0.04
0.04 0.02 0 0.04 0.02 0
o.O2ol0.02 0 5-10-5
0 Z
5
;
y
1015
0
5
.~
101520253035 Z
Figure 8. Distribution of the fluctuating velocity component in the spanwise direction 0.002
.
0.002 0.001 0 0.002
_l_ ' ' ' ' X =
w''''
=
,m
o.oo I=>,o.ool
u
u
m
n
i
m
i
m
o.ool Y=I -
0
Y=0.5 0
o 1015
m
O
o 5
m
Y=2
o.ool
0 Z
m
- 200 -
0
o
o 0.002 0.001 o -o.ool -15-10-5
u
0.001 ,.,
o.ool
a
m
0
5
m
Y=0.25 m
~
m
101520253035 Z
Figure 9. Distribution of the Reynolds stress component -uv/U 2 Figure 8 show the fluctuating velocity in the spanwise direction w ' / tendency is the same as for u' and v'. Figures 9 and 10 show the Reynolds stress component-uv/U2 a n d - u w / U fluctuates markedly at X = 50 because of the many streamwise vortices.
Ue. 2 .
On the whole, the
This distribution also
3.3. Downstream Development of Turbulence Wedge Figure 11 show distributions of the fluctuating velocities and Reynolds stress components at the wedge center (Z = 0). For all velocities and Reynolds stresses, the distributions are larger than the Klebanoffprofile [ 10] for ally/fiboth at X - 50 and 200; although values at X - 200 are smaller than at X = 50, at X = 900, they almost coincide with the Klebanoff profile. The height y / f i w l ~ c h takes a maximum value proceeds to the wall at the streamwise distance X~ finally, at X = 900, it almost coincides with the position in which the Klebanoff profile takes the maximum. It is considered that
604 0.002 0.001 0.002 0.001
i
i
i
i
i
I
i
-
,
,
,
Y-1
i
-
0.003 ,,, ,~0.002 0.001 0 l= 0.003 '0.002 0.001
("4
E3
Y:0.sJ ~ Y : 0 ~ 2 5
0.002 0.001 -o.ool
t
0.001
~
:oO:t
-15-10-5
0 5 10 15 0 5 10 15 20 25 30 35 Z Z Figure 10. Distribution of the Reynolds stress component - u w / U 2
0.12 0.11 0.10 :~0.09 :~ 0.08 ,~0.07 ~0.06 >0.05 ::~0.04 -=0.03 0.02 0.01 0 0.01
3.0
x 10 -3
X-~e2-~/Ue2 '
2.5
c)
2.~ 1.5
C)
~
'
I''"1
'
50
A
z~
900
"
[]
200
s (D
'
~
o
2
I'"'1
A--A ,tL ~
Z--
,a, A
A
-
1.0
o I
1
'
A"
,,~
4'
Klebanoff;-Q-q- ,~
0.5
0.1 y/6
'
0.01
'EI'~/"'~,, ,el
0.1 y/6
e
A
_
, [] , I - ' ~
1
2
Figure 11. Streamwise distribution of the fluctuating velocities and the Reynolds stresses on the wedge center the many streamwise vortices within the wedge might make fluctuating velocities and stresses larger than the value in the ordinal flat-plate boundary layer. After the many vortices disappear, the fluctuating velocities and the stresses decrease gradually and become values near the Klebanoff profiles. The mixing length is a well-known classical concept [11,12]. In this study, a streamwise variation in the mixing length distribution is examined, and the downstream development of the wedge will be considered. This study estimates the mixing length 1using the well-known calculation in equation (1).
605 0.12
I o.o7O.Eo8O./' O, I
i
i
j
i
i
i
i
i
t
i
/
1
o.o6 "" 0.05 0.04 I- ~g~ 0.03 d 0.02 0.01 0 0 0.2
X --~-- 50 '~
0.4 0.6 y/6
0.8
1
1.0
Figure 12. Distribution of the mixing length
l -
-uv
(1)
Figure 12 shows the distributions of the mixing length. The chain line in Fig. 12 shows an equation of I = 0.4y ~ W (K is the Karman constant). At X = 50 and 200, the gradients of the mixing length in y-direction almost coincide with the Karman constant K near the wall, whereas away from the wall 1 ~ 6 t a k e s values around 0.1. It is interesting that although at X = 50 and 200, where the turbulence wedge has not attained fully developed turbulence [3], and the fluctuating velocities and the Reynolds stresses are larger than the values in the fully developed turbulence, the gradient of the mixing length near the wall is almost 0.4, as it is at X= 900.
4. CONCLUSIONS The mean and fluctuating velocities and the Reynolds stress components in the turbulence wedge developing downstream from a single roughness element in a laminar boundary layer on a flat plate were examined. The following conclusions can be drawn. (1) Just behind the roughness, the mean and fluctuating velocity vary markedly in the spanwise direction, and from the isopleths of the vorticity many streamwise vortices exist within the turbulence wedge. The direction of rotation of vortices next to each other is opposite. The distribution of the velocities farther downstream, on the other hand, do not vary markedly, and many streamwise vortices disappear to be replaced by a pair of streamwise vortices on both interfaces between the wedge and the outer laminar region. (2) From the conditional averages of the mean velocity in the spanwise direction, it is found that the wedge expands outwardly since the turbulent fluid within the wedge overflows the wedge. (3) At both sections just behind the roughness and farther downstream, the fluctuating velocities
606 and the Reynolds stresses are larger than the values in the fully developed turbulent boundary layer over the whole height. It may be considered that the effects of the many streamwise vortices still remain even in the section in which a pair of streamwise vortices exist. (4) The gradient of the mixing length near the wall almost corresponds to the Karman constant.
5. ACKNOWLEDGEMENTS
The author wishes to express his thanks to Prof. I. Nakamura of Nagoya University, Prof. S. Yamashita of Gifu University and Profs. Y. Nakase and J. Fukutomi of the University of Tokushima for their kind guidance and constant encouragement throughout the course of this investigation, as well as to Mr. M. Fukunaga and S. Kondoh of the University of Tokushima for their generous cooperation.
REFERENCES
1.
D.S. Henningson, A. Lundbladh and A. V. Johansson, A Mechanism for Bypass Transition from Localized Disturbances in Wall Bounded Shear Flows, J. Fluid Mech., 250(1993), 169207. 2. B.G.B. Klingmann, On Transition due to Three-Dimensional Disturbances in Plane Poiseuille Flow, J. Fluid Mech., 240(1992), 167-195. 3. M. Ichimiya, Y. Nakase and J. Fukutomi, Structure of a Turbulence Wedge Developed from a Single Roughness Element on a Flat Plate, in Engineering Turbulence Modelling and Experiments 2, W. Rodi and F. Martelli(eds.), Elsevier Science, (1993)613-622. 4. S. Yamashita, M. Ichimiya and I. Nakamura, Experiments on the Effects of a Single Protrusion on the Boundary Layer Mound a Cylinder Spinning in an Axial Flow - Mean and Fluctuating Velocities in a Turbulence-Wedge Region, AIAA paper, No. 88-3762-CP (1988). 5. S. Yamashita, M. Ichimiya, I. Nakamura and K. Ogiwara, Effects of a Single Protrusion on the Boundary Layer Mound a Cylinder Rotating in an Axial Flow (Change in Flow Properties in the Turbulence Wedge with the Speed Ratio), Proc. 2nd KSME-JSME Fluids Eng. Cons (1990) 1-190-1-195. 6. S. Yamashita, M. Ichimiya and I. Nakamura, The Effect of a Single Protrusion on the Boundary Layer Around a Cylinder Rotating in an Axial Flow - Conditional Measurement in the Intermittent Region of a Turbulence Wedge -, Proc. 1lth Australasian Fluid Mech. Conf., Hobart (1992) 231-234. 7. I. Tani, H. Komoda, Y. Komatsu and M. Iuchi, Boundary-Layer Transition by Isolated Roughness, Aeron. Res. Inst. Univ. Tokyo Rep. No. 375 (1962). 8. M. Mochizuki, Smoke Observation on Boundary Layer Transition Caused by a Spherical Roughness Element, J. Phys. Soc. Jpn, 16-5 (1961) 995-1008. 9. L.S.G. Kovasznay, V. Kibens and R.F. Blackwelder, Large-Scale Motion in the Intermittent Region of a Turbulent Boundary Layer, J. Fluid Mech., 41-2 (1970) 283-325. 10. P.S. Klebanoff, Characteristics of Turbulence in a Boundary Layer with Zero Pressure Gradient, NACA Tech. Rep., No. 1247 (1955). 11. L. Prandtl, Bericht fiber Untersuchungen zur ausgebildeten Turbulenz, ZAMM, 5 (1925) 136139. 12.L. Prandtl, Bemerkungen zur Theorie der freien Turbulenz, ZAMM, 22 (1942) 241-243.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
607
FEATURES OF LAMINAR-TUBULENT TRANSITION IN A FREE CONVECTION BOUNDARY LAYER NEAR A VERTICAL HEATED SURFACE Yu. Chumakov a and S. Nikolskaja b aDepartment of Hydroaerodynamics, State Technical University, St.-Petersburg, Russia bDepartment of Hydroaerodynamics, State Technical University, St.-Petersburg, Russia
This article presents the results of an experimental study of the free convection boundary layer formed on a vertical isothermally heated plane surface. Three regimes of the flow were studied. They are laminar, transition and turbulent. A method to measure the surface shear stress and heat flux on the plate is described. New empirical dependencies of these values within a wide range of the Grashof number are obtained. Some features of the flow in the transition area which are appropriate only to the free convection flow are described.
1. E X P E R I M E N T A L UNIT The generator of the free convection flow is a vertical aluminium plate 90 cm wide and 4.95 m high. On the back side of the plate there are 25 heaters which are controlled by the electronic system which is able to keep the set heat regime for a long time (6-8 hours). When setting any regime to each of 25 sections the different laws of the surface heating, in particular the constant surface temperature, can be simulated by the surface height. Due to the considerable height of the plate one can obtain all the three flow regimes such as laminar, transitional and developed turbulent up to a Grashof number of 4.5 x 10 ll . The sensor was traversed in the air flow with a traverse gear. The precision of the movement along the vertical coordinate is about 1 cm and along the normal coordinate with regard to the surface (i.e. across the boundary layer) about 1 micrometer. The movement along the normal coordinate is remote controlled. In addition, the measurement of the flow parameters in one section of the boundary layer is totally automated thanks to the use of the specially designed equipment and computer software. In the course of the experiment the average and fluctuation temperature components were measured. All the measurements were made by using a resistance thermometer and a hot wire anemometer. As sensor a tungsten wire of 5 micrometers diameter and 3-4 mm length was used. It is known that when using the hot wire anemometer to measure the velocity in nonisothermal flow the influence of the temperature should be taken into account to process the hot wire anemometer readings. In our opinion, almost all existing methods of thermal compensation cannot be used for this free convection flow which has
608
low average velocities and high turbulence degree. Therefore, a new method of thermal compensation by the actual temperature was developed. Without describing this method in detail it should be mentioned that unlike other methods of the thermal compensation by the average temperature, in this method the readings of the hot wire anemometer which correspond to the actual temperature in the given point of the area are processed taking into account the actual temperature in this point. All the measurements were made at constant surface temperature which is 70~
The
air temperature at the exterior border of the boundary layer was constant and equal to 25-26~ up to the height 2 meters and then it was increased and reached 27-28~ reached 5 meters.
when the height
2. C A L I B R A T I O N UNIT To carry out a new method of the velocity determination a special calibration unit was designed. The main principle of the unit operation is a regular sensor movement at the set velocity in the stationary non-isothermal air. The sensor goes about a horizontal tube axis. The carriage speed is determined by the time during which the carriage passes the basis distance. To create non-isothermal flow, the air in the tube is heated with the use of infrared emission This unit can calibrate the sensors at velocities from 1 to 50 cm/s and for temperatures from 20 to 80~ For the simultaneous measurement of velocity and temperature the two wire sensor was used: the hot wire measures the velocity and the cold one measures the temperature. These two tungsten wires were located 3.5 mm from each other and perpendicular to the axis of the tube. Such a design of the sensor eliminates the influence of the hot wire on the cold one. The calibration consisted in obtaining the measurements sufficient for the statistic treatment at different sensor movement velocities by the unevenly heated stationary air. The sensor calibration time is about 2 hours. After recording all the required measurements the data processing by computer is carried out. For the further use of the obtained data the calibration results are provided as a ratio of the hot wire anemometer voltage to the flow velocity. The parameter is the air temperature. It is very important to note that when using a measurement procedure like this, there is a thermal compensation of the readings of the hot wire anemometer by the actual temperature value. This is one of the key features of the measurement in this work.
3. ANALYSIS OF E X P E R E M E N T A L DATA The transition region is still seldom studied, mainly because of the measuring difficulties which are due to big intermittency and intensive fluctuations. A method of the thermal compensation of the hot wire anemometer by the actual temperature used in this work does not depend on the intensity of the fluctuations. This made it possible to make measurements in the transition region.
609 3.1. Profiles of average and fluctuation values of longitudinal velocity and temperature within the transition region It is commonly known that the profiles of average velocities and temperature depends a lot on the flow regime in the boundary layer. For example, based on our data, in the case of the turbulent flow regime the profiles of the longitudinal velocity component become more filled compared to such profiles in the laminar flow regime, and the exterior region of the boundary layer (i.e. the region from the maximum of the average velocity to the exterior layer boundary) is more than 90% of the total layer thickness. The thickness of the boundary layer at the unit was varied within the range 2-3 cm in the lower part of the plate (Grashof number Gr x
was about 105 + 108 ) to 20 cm in the upper part of the plate (Grx ~ 10l~ + 1011). Here
GrX = g~ATx
3 / v 2 is the Grashof number where g - gravitational acceleration, 13 - volume
expansion coefficient, AT =
T w -T~
-
difference of surface temperature
Tw
and air
temperature T~ beyond the boundary layer, x - longitudinal coordinate along the surface. The analysis of the obtained data shows that the average velocity and temperature profiles are very close to the laminar profiles even at Grashof numbers 3.109 . At the same time, based on other flow characteristics, the transition processes have already started developing. Especially, the increase of the fluctuation intensity is evident. I.e. if the start of the profile characteristics change is used as a criterion of the transition start, then the transition slows down. More intensive fluctuations do still not result in a change of the average characteristics. In the middle of the transition region the boundary layer becomes thicker very fast, the velocity maximum is decreased and the slope of the temperature profile near the surface is quickly increased. The value of Grashof number corresponding to the end of the transition region when the profiles become like for the turbulent regime turns out to be lower than the value of Grashof number determined based on other characteristics. In other words, at the beginning of the transition region the average velocity and temperature profiles start changing slightly with significant conservatism, but coming down to the end of this region the profiles become
0.4-
I IIIII11
m
!
]U
0.3-
i l llilll
~
,~.
~, .,
0.2-
i
z:~
~ l li~ll
Grx*10"10 0
%. ~%/,,,,
I--
3.7
I
IT
i
i l llllli
Grx*10-10
1 I I IIllll
I
J l lilll
I
I II IIII
,~
02
/X 4.0
/x
4.2
~ '
"'5<>~v'~~,'
0.10.0-
' Y" 1
'
10
,, ,,ll 100
Figure 1. Distribution of the intensity of longitudinal velocity fluctuation.
0.0
I
I ,, i,,, i
1
,Y'I II Iiii I
10
Figure 2. Distribution of the intensity of temperature fluctuation..
100
610 very fast like those for the developed turbulent flow regime. In this case the fluctuation characteristics are relaxed at some distance along the plate to the constant value corresponding to the developed turbulent regime. The results of the measurement of the intensity profiles of longitudinal velocity I U (y)
(I U =(uZ)l/Z/Um ) and temperature IT(Y) (I T =(t-Z)l/Z/AT ) component fluctuation in the transition region of the boundary layer are shown on Fig. 1,2. These figures show that at the beginning of the transition region the intensity profiles have two maxima: one is near the external boundary of the viscous sublayer and the other one is in the average velocity maximum zone. Then, while the boundary later develops downstream, two maxima become one big maximum, and the profiles of fluctuation intensity I U (y) 6 I T (y) becomes like for the turbulent regime. Unfortunately the available literature provides only two references about work when similar measurements in the transition region were made. In [1] when measuring in air the authors also noted the formation of two maxima on the profile I U (y) in the transition region. The authors of [2] observed the similar phenomena when making the experiments in water. However, the information in [1,2] is not sufficient to make any conclusions on the location of these maxima with regard to the boundary layer regions. 3.2. Results of The Measurement of Wall Heat Flux and Shear Stress
The results of the measurement of the average temperature T and velocity U profiles are used to determine the wall heat flux qw and shear stress ~w. Also the temperature and velocity ratios obtained by integrating the equations of the boundary layer are used. These ratios are represented with the use of Boussinesq approximation. The equations are integrated in the near wall region but the turbulent shear stress -9(uv) and heat flux -OCp(Vt) as well as the left parts of the initial equations are neglected. Hence, as a result of the integration of simplified boundary layer equations the followings ratios of average temperature T and average velocity U are obtained under respective boundary conditions. Linear temperature profile:
T =Tw--~-~y
(1)
and cubic velocity profile:
U=
xw g[3(Tw - Too) y2 + 9gqw y3 yI ~t 2o 6~o
(2)
where X - heat conductivity coefficient, ~t- dynamic viscosity coefficient, 9- gas density. It should be noted that in case of laminar flow regime the ratios of the temperature and the velocity in the near wall area will be the same. That's why in this work a term "viscous sublayer" which is usually applied to the turbulent flow, will denote that part of the boundary layer where the equations (1) and (2) are valid irrespective of the flow regime. For the use of the equations (1) and (2) to obtain qw and Xw the values of the derivatives dU/dy and dT/dy on the surface must be known. These values are determined by extrapolating the equations (1 and 2) to the wall. The extrapolation will result inevitably to a
611 number of difficulties such as a determination of the coordinate of the first measuring point, the determination of the wall 1000 _'ll'll ,llllllll l li,i,lll l lltlllll l llllllll l li,lli,l t lllll zone influence on the readings of the neighboring hot wire Nu~ anemometer sensor. In the _ course of this work an _ extrapolation method was developed. This method was checked many times and 100 carefully by comparing to the results which were available in the literature. The results of the determination of the heat flux on the surface are shown in Gr~ 10~ l" Fig.3 as a relation of Nusselt number N u x ( N u x = h x / ~ , where h - local heat exchange 1E+6 1E+7 1E+8 1E+9 1E+10 1 E + l l coefficient) to Grashof number G r x. The approximation of Figure 3. Dependence of Nusselt number on Grashof number. the experimental points in the laminar flow area gives the following relation (curve # 1)" -
m
m
Nu x
= 0.279 x G r x 0.262
at
Gr x
=5x105 +2.8x109,
in the developed turbulence area (curve #3)" Nu x
1000
= 0.0547 x G r x o.361 _ ,,,,,I
, ,,,,J.I
, ,,,,.d
at
, ,,,~.,I
, ,,~,J,,I
NUx
_ _
_
_
Gr x =
, ,,,,,.I
f
1.4 x 10 lO +5x1011
, ill
~
_
Change of N u x in the transition region can be presented by the following relation (curve # 2)"
_
"~"
100--
present work
.
~
Nu x
at
/ llili]
/
Grx I l lillii I
I l lillilJ
Gr x
=3.5x109+6x109.
This figure shows that there is a local maximum Nux in the
/
10--
=3.75x10 -ll xGrx 1.3041
i l llllil]
I l llilli]
I l lllillJ
I Iiii
IE+6 IE+7 IE+8 IE+9IE+I0IE+11
Figure 4. Comparison of the data of the present work with other data.
transition area. This phenomenon is explained below. Fig.4 shows the comparison of the data of the present work with the data of other authors. It can be noted that the results are very close in the laminar
612
and turbulent regime. The agreement is less good in the transition regime. In the present tests there were more than 100 "~w profiles of the average ouo 2 velocity. In Fig.5 the results 3 * 10-2_ . of these tests are shown as a relation of Grashof
to
Z w / ( p U 2)
number,
U b = ( g [ 3 A T u ) 1/3 .
where In the
laminar flow area the change of the shear stress can be expressed by the following equation:
**
1;___E_w _
, ~ 0 0954
pU~ - 0.743 x ~ r x"
Gr _
_
iiiii I
i i iiiill I
1E+6
i i iii1111
1E+7
i i 1111111
1E+8
i i iiiiii I
1E+9
x
i i illiii I
1E+10
at
I i
1E+11
Gr x = 5 . 9 x 1 0 5 + 2 •
.
Let's note that the value of the initial Grashof number is very low. In the turbulent flow area the Grashof number is considerably higher than in other references. The variation of the wall shear stress can be expressed by Figure 5. Dependence of wall shear stress on Grashof number.
~w
- 0.0752 x GrO183
at
G r x = 7.9x 109 +4.5 x 1011 .
,,,,.,I ,t,,,,,,.l ,l,,,,,,l ,,,,,,,,I ,,,,,,.I ,,,,,,,,I ,,, present work
"cw
0--
A,
[5]
-9
t+]
@ +
[31 [
The transition area turns out to be very short. In addition, we obtained a few experimental points. However, the following equation may be given for the transition area:
+
Xw - 8.45 x 103 x G r ; ~
at
Gr x - 2 . 5 x 1 0 9 + 5 . 5 x 1 0 9 . Ra I IIIIII I
I I IIIII11
1E+6
1E+7
I IIIIII11
I I IIIII11
1E+8
I I IIIII11
1E+9
I I IIIII11
1E+10
X II
1E+11
Figure 6. Comparison of the data of the present work with the other data.
For the free convection flow the wall shear stress in the transition area is decreased. For the forced convection flow the wall shear stress in the transition area is increased.
613 Hence, there is a radical difference between free convection flow and forced convection one. Fig.6 shows the comparison of the present data with the results of other authors. The Rayleigh number Ra x is used as abscissa in order to compare the results obtained in diverse physical media. A quite wide spread in the experimental data of different authors is noted: different values of ~w/(Pf U~) and a different distribution, particularly a different inclination of curves as well as a diverse location and extent of the transition area can be noted. The curve inclination Tw/(PfU 2) = f(Rax)
obtained in the current work in the
laminar area is close to the corresponding theoretical value, and the curve inclination in the turbulent zone is the intermediate value between the data of [3,6]. To our mind, the results of the current work are close to work [3]. In both works the velocity was measured by hot-wire anemometer, and special attention was given to the investigation of the near-wall region. In conclusion, let us get back to the analysis of the Z IT m a x characteristics, given in Fig.3 and bring up one of the 0.2-possible explanations of the A local maximum appearance in ~x A 0.1 number Nux dispersal at the Z~ end of the transition area. Gr X Such a phenomenon was noted A Z~ Z& Z~ Z~ A Z~& 0.0 I I I I I I l I I,,,,, I I lllllll I I llll~llI I lllJ~llI I~lll,ll I I J in some other references, but 1E+6 1E+7 1E+8 1E+9 1E+10 1E+11 without explaining the reasons of its appearance. In our work, Figure 7. Distribution of the maximum intensity of the the careful and detailed temperature fluctuations along the surface. measuring of diverse characteristics in the transition regime zone enabled us to explain this interesting and unusual phenomena, typical only for the free convection flow. Figs. 7, 8 show the i l l , , I , i J,,,,,I, ,,llJlll i ~lllllll i , , , , , , , I , ,,,lllll II 0.6 _ dependencies of the A Z maximum intensity of the temperature and velocity 04-Z fluctuations along the A Z ZX Z section. Fig.9 shows the Z Z _----0.2maximum of the average Z /k Z Z velocity (Um) from Grashof Z Z ~ ~ ~ ~ Grx Z number along the section. It 0.0 IIIII] I I I IIIII I I I I IIII1[ I I IIIIII I I I I IIIll I I I I IIIII I I II is perfectly shown that in the 1E+6 1E+7 1E+8 1E+9 1E+10 1E+11 transition area (from 0.3
_
Figure 8. Distribution of the maximum intensity of the velocity fluctuations along the surface.
Grx= (2 + 3)
• 109
(0.8 + 1.3) x 10 l~
to the
maximum velocity decreases from (0.48 + 0.52) m/s to 0.4 m/s, which is 20%. The alteration of the maximum velocity
614 along the section occurs practically simultaneously with falling of the wall shear stress, i.e. with decreasing of the derivative d U / d y y = 0 . This process causes a sharp increase in the fluctuation level and a fast growth of the boundary layer thickness. It can be suggested that the sharp increase of ,,,,,I , ,,,,,,,I , ,,,,,,,I ,,,,,,,,I , ,,,,,,,I , I,,,,,ll I IIm Z Z the fluctuation velocity m 0.6 t ~ , m/s is the main reason of A A these phenomena: A A A "i~ "-'A A A Lt~il~i'-" firstly, the average flow O.4 mA A~-~t~~ A energy, spent up to the A AA A A beginning of the A A transition only to 0.2 accelerate the laminar flow longitudinal Grx 0.0 (growing of x w a n d IIIII I I I I III111 I I IIIIII I I I I IIIII I 1E+6
1E+7
1E+8
1E+9
'"'""1
''"'"'1
1E+10
"
1E+11
the laminar area), begins "to transfer" to the Figure 9. Distribution of the maximum of the average velocity fluctuation flow; at the along the surface, same time there occurs capturing of new portions of the cold air from the external boundary layer (increasing of the layer thickness, intermittency). Therefore, there is an increase of the air mass, participating in the longitudinal flow. Because of the terminal velocity of heating, the air mass temperature decreases and, as a result, a buoyancy force ,which is the unique source of the flow, decreases likewise. This process causes the decrease of the maximum value of the velocity, and also the decrease of the inclination of the velocity profile on the surface (i.e. decrease of x w). On the other hand, it promotes increasing the intensive heat exchange, i.e. the local maximum in equation N u x ( G r x ) is formed (see Fig.3). The recovering of the previous maximum value of Um
in
the velocity occurs in the large space within the range of Grashof numbers (1 + 5) x 10l~ .
CONCLUSIONS 1. The method of the measurement of the velocity with the use of hot wire anemometer is modified to measure the velocities in non-isothermal flow of high turbulence degree. While studying the processes of the transition from the laminar regime to the turbulent one a number of features were discovered. These features distinguish this flow from the forced convection flow. 2. At the beginning of the transition area the profiles of the velocity and temperature fluctuation have two maxima. While the flow develops downwards, these maxima are combined together, and the value of this maximum is higher than in the developed turbulent flow area. 3. The transition area shows the decrease of the wall shear stress value and average velocity which is maximum by the section. At the end of the transition area the intensity of the heat transfer from the wall is non-monotonously changed..
615 4. The shear stress and the heat flux on the surface for the three flow regimes were measured. New approximation dependencies on Grasshof numbers are proposed. This work is conducted thanks to the financially assistance by the Russian Funds of the Fundamental Research Study (project: 96-02-19461).
REFERENCES
1. Miyamoto M., Katoh Y., Kurima J., Taguchi Y., Trans. JSME, Ser.B, V.60, N 571, (1994) 971. 2. Jaluria Y., Gebhart B., J. Fluid Mech., V.66, N 2, (1974) 309. 3. Tsuji T., Nagano Y., Int. J. Heat Mass Transfer, V.31, N 8, (1988) 1723. 4. Cheesewright R., Mirzai M.H., Proc. 2nd U.K. National Conf. Heat Transfer, Glasgow, C 140/88, (1988) 79. 5. Smith R.R., Ph.D. Thesis, Queen Mary College, Univ. of London, (1972). 6. Cheesewright R., Ierokipiotis E.G., Proc. 7th Int. Heat Transfer Conf., Munich, FRG, V.2, NC31, (1982) 305.
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0
Turbulence Control
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 1999 Elsevier Science Ltd.
619
On active control of high-lift flow F. Tinapp, W. Nitsche Department of Aeronautics and Astronautics, Technische Universit~it Berlin, Marchstr. 14, 10587 Berlin, Germany The active control of airfoil flows is of outstanding importance in future technological applications. For example, maintaining a boundary layer on a wing as much as possible in the laminar state reduces the skin-friction drag. A turbulent boundary layer however, is more resistant to separation than a laminar one. By shifting the onset of separation towards higher angles of attack by means of active flow control, lift is further enhanced and the drag is reduced. Recent investigations have shown that it is highly effective to excite the flow through a narrow spanwise suction/blowing slot, either to damp the turbulent structures in the boundary layer, or to enhance them. Combining the excitation mechanism with a control device, enables the system to operate automatically and to achieve optimal effects under varying flow conditions. In this paper an example of flow control on a simple high-lift airfoil will be presented. Periodic blowing and suction trirough a narrow slot is applied to the separated flow in order to achieve reattachment and thus to increase lift for high angles of attack.
1.
INTRODUCTION
Modern transport aircraft wings have to provide very high lift-coefficients in low speed flight during take-off and landing. This leads to good payload/range capabilities for a given field length and a reduction of the noise footprint in the airport area. Therefore high-lift systems are of complex mechanics, generally consisting of a combination of leading-edge slats and multiple trailing-edge flaps. At high angles of attack the flow over high-lift wings may separate, resulting in a lift reduction and in an increase of drag. If the onset of separation could be delayed towards a higher angle of attack, it will either be possible to achieve a higher lift or to reduce the complexity of the high-lift system (Figure 1). Recent investigations showed clearly that periodic excitation of the separated shear layer results in a partial reattachment and therefore in an increase of lift and decrease of drag (Hsiao et a.1990, Dovgal 1993 and Seifert et a.1996). The present investigation treats this problem and deals with experiments aimed on the separation control via excitation of the separating boundary layer on the flap.
620
2
E X P E R I M E N T A L APPARATUS
2.1 Test model Two different test models for use in wind tunnel and water tunnel experiments were necessary. They are geometrically similar but different in scale. The first test model for wind tunnel experiments, called model A is a generic two element high-lift configuration, consisting of a 180 mm chord length NACA 4412 main airfoil and a NACA 4415 flap with 72 mm chord length, both of 400 mm span. The flap is mounted at a fixed position underneath the trailing edge of the main airfoil, thus forming a gap of 6.3 mm height with an overlap of 4.9 mm (Figure 2). The tested configuration was chosen in accordance with the experiments made by Adair & Home (1989). The angle of attack of the whole configuration o~ can be varied between 3 ~ and 20 ~ while the flap-angle q can be adjusted between 3 ~ and 50 ~ To ensure turbulent separation of the flow, turbulator strips are placed close to the leading edge of the main airfoil and of the flap. For dynamic excitation of the separated shear layer, the flap was equipped with a 0.3 mm wide spanwise slot at 3.5% chord length (figure 3). Static pressure taps were distributed along the midsection of upper and lower surface of the test model and the configuration was mounted on a 3-component balance to measure the aerodynamic forces on the test model. The second test model for water tunnel experiments (model B) is identical to model A but differs in scale. Due to the smaller testsection of the water tunnel, the dimension of model B is reduced on 55% of model A, thus resulting in a chord length of the main airfoil of 100 mm and 40 mm of flap chord length. This test model is also equipped with a excitation slot at the same relative position as model A.
2.2 Excitation system The periodic oscillating pressure pulses are generated externally by a electrodynamic shaker driving a small piston. The excitation signal generated this way is brought into the flap resulting in an oscillating jet emanating perpendicular to the chord from the narrow slot near the flap leading edge (see figure 3). To present the two excitation parameters, frequency and intensity, the Strouhal number ,nd a nondimensional impulse coefficient can be calculated: F St
" l h,t,.
with the flap chord as characteristical length lchar.
--
b/oo
c~, = 2.
9
, with H=slot width, c=chord of main airfoil, v'=velocity fluctuation at the slot exit
621
2.3 Test bed Pilot tests were carried out in a closed water tunnel with a cross-section of 330 x 255 mm. Therefor the smaller test model B was used and a rotating valve, producing periodic pressure pulses, was applied as excitation mechanism. LDV-measurements around the whole test configuration, using a Polytech LDV-580 were conducted at a free stream velocity of uo~=1.6 m/s resulting in a Reynoldsnumber based on the chord length of the main airfoil of 160000. The wind tunnel, used for the main experiments, is a closed one, with a 1.4 x 2 m 2 crosssection. To achieve again a Reynolds number of 160000, the free stream velocity is fixed at u==14 m/s. Due to the short span of the test model A, two side walls reaching from top to bottom of the testsection are installed. All external installations for the test configuration are hidden inside these walls to minimize the influence on the measurement results.
3
EXPERIMENTS
3.1 Pilot tests in the water tuanel To gain an insight into the flow behaviour in the case of natural flow and excited flow, LDV-measurements are carried out around the small test model in the water tunnel. In figure 4 the results are plotted for the case of ~=8 ~ and r1=35 ~ In the upper graphic (figure 4a) the streamlines, calculated from the LDV results are plotted around the whole test configuration. At these conditions, the flow separates over the flap, forming a big, closed recirculation area, while the flow over the main airfoil remains nearly completely attached. In the vector plot (figure 4b) it can be recognized that there is just a small recirculation area very close to the trailing edge of the main airfoil. When the excitation is activated (figure 4c) the recirculation over the flap disappears and the flow reattaches almost entirely. The changed flow field over the flap also has an effect on the flow behaviour over the main airfoil: the small separated region close to the main airfoil trailing edge vanish in case of excitation. The influence of excitation frequency and intensity on reattachment behaviour was also investigated (Tinapp & Nitsche 1998). As an example for the influence of excitation intensity on the reattachment behaviour, three velocity profiles at a fixed position for three different excitation intensities are plotted in figure 5. The dotted line represents the velocity profile of the separated flow for the case of no excitation. It is recognizable, that a weak excitation (a) is not able to suppress the recirculation area over the flap completely, but clearly reduces it in size. A stronger excitation (b) eliminates the backflow, resulting in a much better reattaching of the flow to the flap. The reattached jet is more pronounced and closer to the wall than in the case of weak excitation, nevertheless the velocity profile in case (b) still has a clear tendency towards separation. Increasing the excitation intensity further more, does not change the flow behaviour remarkably but still enhances the velocity profile very close to the flap surface. To better understand the mechanism of reattachment, phase averaged measurements in the flap region were undertaken. In figure 6a the vertical velocity of the excitation jet at the slot outlet is plotted against the positions of the rotating valve used for excitation. Three points of interest are marked in the graphic: In case of a closed valve (b) the velocity of the excitation jet is zero, when the valve opens (c) the jet starts to emanate from the excitation slot and reaches a maximum when the valve is completely open (d).
622 The lower graphics of figure 6 show the flowfields in the flap domain at different moments of excitation that correspond to the three points marked in figure 6a. The positions of the reattached jets are marked in the graphics by the thick arrows. It can be seen, that the flow is attached during the whole excitation cycle (even for the case of a closed valve), but does oscillate in phase with the periodic excitation, resulting in a slight up and down movement of the attached jet. In the moment of starting excitation (figure 6c) a small separation bubble in the vicinity of the excitation slot is formed by the onset of the excitation jet and the flow over the flap moves slightly upwards. The separation bubble disappears immediately afterwards, when the excitation pulse reaches a maximum and the flow reattaches completely to the flap surface.
3.2 Wind tunnel experiments To determine the lift coefficient CL, the wind tunnel test model was mounted on a 3component balance that measures the aerodynamic forces. In figure 7a the lift behaviour of the test model is plotted versus the angle of attack for different flap angles in the absence of excitation. The results show a typical behaviour of highlift configurations. For a fixed flap angle, the lift raises with increasing angle of attack of the whole configuration t~. At a certain angle of attack flow separation on the flap and on the main airfoil inhibits further lift increase and CL drops down to low values. Raising the flap angle shifts the whole lift curve towards higher values due to the enhanced camber of the test model, while flow separation now occurred at lower angles of attack. Maximum lift is obtained in the case of 1"1=35~ with a very peaked lift characteristic at ct=4 ~ For flap angles higher than q=35 ~ the flow over the flap is always separated resulting in a completely changed lift characteristic, as shown in figure 7a for the case of 11=39~ The lift behaviour of the high-lift configuration can be changed by introducing periodic disturbances through the slot on the flap. In figure 7b two exemplary cases (r1=35 ~ and r1=39 ~ are plotted without excitation (dotted line) and with excitation (solid line). The excitation parameters were chosen as F=80 Hz = St=0.4 and %=4010 -5. It can clearly be seen, that the lift is enhanced by flow excitation. Especially in the case of q=39 ~ the recovered flow over the flap yields a strong increase i~, lift up to a value higher than the maximum lift that could be achieved without flow excitation. To depict the enhancement of lift due to flow excitation, the increase of lift as a percentage of the lift coefficient without excitation(cL,exi, ' --CL.b,,.,.ic)/CL,b,.,i, is plotted in figure 8. The achievable lift enhancement is best (about 30%) at a flap angle of q=39 ~ and ix=4 ~ for other configurations the lift improvement obtained by flow excitation is less. It can be seen, that the excitation works best at post-stall conditions. The former separated flow over the flap is reattached, resulting in a recovering of lift. To investigate the influence of excitation frequency and intensity, further measurements are carried out. The test model is set at high angles of attack (ix=7 ~ q=41 ~ to cause flow separation over the flap. In figure 9 the required excitation intensity to achieve reattachment is plotted against excitation frequency marked by the solid line. It can be seen that at low frequencies around F=30 Hz (St = 0.15) minimum excitation intensity is necessary to achieve reattachment of the flow. If reattachment of the flow occurs the excitation intensity can be reduced down to a certain value at which the flow re-separates again (marked by the dashed
623 line). The hysteresis is very small at low frequencies but grows with rising excitation frequency. This indicates a different mechanism between achieving reattachment of the former separated flow and remaining the reattached flow in it's attached state. In the first case, a big recirculation area has to be influenced, which requires big vortices to be introduced into the flow field to achieve a reduction of the separated area. If the flow is attached over the flap, relatively small turbulent structures (St = 1) have to be produced to enhance the mixing process between the shear layer over the flap and the outer flow. This explains, why high frequencies are more effective to remain the flow attached. 3.3 Conclusions Improvement of the lift behaviour of a simple high-lift configuration can be achieved by periodic blowing and suction through a narrow slot at the flap leading edge. This method works best at post-stall conditions, that means, if the flow over the flap is yet separated. The wind tunnel experiment yields, that by introducing periodic disturbances of a certain frequency and intensity into the separated flow, reattachment of the flow can be achieved, resulting in a enhancement of the lift up to 30% of the lift. Low frequencies around St -- 0.15 are best to achieve reattachment of the separated flow, while higher frequencies (St --- 1) work better to remain the flow attached over the flap. LDV measurements of the flow field in a water tunnel around a test model with periodic blowing have shown, that the excited flow over the flap is constantly attached and does oscillate strongly in phase with the excitation frequency.
REFERENCES Adair, D., Horne, W.C. 1989, Turbulent Separating Flow Over and Downstream of a TwoElement Airfoil, Experiments in Fluids, Vol. 7, pp. 531 Dovgal, A. 1993, Control of Leading-Edge Separation on an Airfoil by Localized Excitation, DLR-Forschungsbericht DLR-FB-93-16 Hsiao, F.B., Liu, C.F., Shyu, L.Y. 1990, Control of Wall-Separated Flow by Internal Acoustic Excitation, AIAA Journal, Vol. 28, No.8, pp. 1440 Seifert, A., Darabi, A., Wygnanski, I. 1996, On the Delay of Airfoil Stall by Periodic Excitation, Journal of Aircraft 33, No.4, pp. 691 Tinapp, F., Nitsche, W. 1998, LDV-measurements on a high-lift configuration with separation control, Ninth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Conference Proceedings Vol. 1, pp. 19.1
624
CL
////////////////
airfoil with flap
I
-....
.
Y / ~
~
0.67 m ~_
air:~ ~:];cflap ~cparat, on control
~
(~6,3
mm
1.4 m
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Figure 3: Sketch of the investigated test model with a spanwise blowing slot to excite the separated flow
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Figure 4: Results of LDVflowfield measurements in a watertunnel with geometrical reduced test model (Cmain---100 mm, Cflap=40mm) The conditions of the experiment were: u~= 1.6 m/s, Re=150000, or=8~ 11=35~ a) flowfield around the whole test configuration, streamlines calculated from LDV-results, no excitation, flow separated over the flap b) vectorplot of the separated flowfield in the flap region in the case of no excitation c) reattachment of the flow due to flow excitation through the slot near the flap leading edge (F=40 Hz --" St = 1)
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30 25 9 39~ ,..., 'E
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~ 10 F i g u r e 9- Hysteresis of excitation intensity between reattachment and reseparation.
5
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10
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
627
A Demonstration of MEMS-based Active Turbulence Transitioning W a y n e P. Liu and G e o r g e H. Brodie Naval Surface Warfare C e n t e r - Carderock, MD U S A 1.0
INTRODUCTION The majority of turbulence research has been targeted at reducing turbulence for o b v i o u s drag and noise benefits. H o w e v e r , there are a few situations where turbulence or an accelerated transition to turbulence would be desirable. Laminar separation and the resulting stall about the leading edge of an airfoil can sometimes be prevented with a transition to turbulence which can provide the needed m o m e n t u m to delay separation. Such an application could be useful for sailplane wings, wind turbine blades, Remotely Piloted Vehicles, and other airfoils in low Reynolds number flows. Model testing is another application where turbulent flow fields are desirable. If the flow field about a full-scale submarine is nearly turbulent, then it w o u l d be a more faithful simulation to provide turbulent or at least transitional flows about a scale model rather than laminar flows. For that reason, straight-ahead, steady drag testing has traditionally e m p l o y e d trip strips, studs or grit to produce a turbulent b o u n d a r y layer about the model. H o w e v e r , how could this be accomplished for a f r e e - s w i m m i n g model such as a r a d i o - c o n t r o l l e d submarine model? Slow speed m a n e u v e r s conducted by a cylindrical model could suffer a laminar cross flow separation which w o u l d p r o d u c e drastically higher drag values than a separation achieved with a turbulent b o u n d a r y layer. One immediate and significant p r o b l e m is that the flow field about m a n e u v e r i n g body is unsteady and not fully u n d e r s t o o d . A second p r o b l e m is that even with a predictable flow field, it would still be impractical to cover the body with studs or grit without irnpacting the -'~0.3 m total drag characteristics of the model. s~bwoofer A future approach to this O.050m --problem could be to cover the model hull with an array of tiny MicroElectro-Mechanical-System (MEMS) sensors and actuators. O n-body 0.2m I sensors could identify critical flow i field characteristics such as velocity, 0.9m p r e s s u r e and shear stress fields to 1.2m reveal stagnation and separation Flow P regions. This flow field information, when combined with tiny hull splitter alate mounted MEMS actuators, could be ! used to excite the natural flow instabilities about the hull to achieve Fig. 1. Wind tunnel set-up of cylinder with an accelerated transition to MEMS sensors, span-wise slit and sub-woofer. turbulence. This s e n s o r - a c t u a t i o n
J N
~
628 effect could also be used on low Reynolds number airfoils to prevent laminar separation on an active basis. It is the goal of this investigation to demonstrate the use of an on-body sensor-actuation effect in achieving active turbulence transition about a cylindrical body. As shown in figs. 1 and 2, state of the art MEMS sensors mounted on the cylinder perimeter will be used to evaluate classic flow phenomena such as stagnation, separation and the vortex shedding. This data will provide the necessary information to properly tune and locate internal acoustic forcing. Acoustic disturbances aimed at the separation region from a thin span-wise slit on the cylinder, will then be amplified by the shear layer instability to accelerate the transition to turbulence. 2.0
BACKGROUND The effect of flow perturbations matched to instability frequencies has been studied since the 1960's. Klebanoff, Tidstrom and Sargent [1] induced 3-D flow perturbations amidst a field of Tollmien-Schlicting (TS) waves on a flat plate with a vibrating ribbon. Inherent or natural instabilities of the flow then amplified these disturbances to achieve a rapid growth in span-wise velocity irregularities, eventually developing turbulent flow. Later, Bloors [2], Gerrard [3], and Peterka and Richardson [4] investigated the effects of a uniform acoustic field imposed externally upon a cylinder. Their research showed a strong correlation between changes in cylindrical flow characteristics and perturbations which matched instability and vortex shedding frequencies. They concluded that small levels of acoustic forcing enhanced the flow entrainment at strategic points about a cylinder and achieved variable lift and separation characteristics. The amplification of velocity fluctuations through instability frequencies has been called a natural tripping device by Mueller [5]. For low Reynolds number airfoils with a leading edge separation bubble, Mueller showed how the instability frequencies of the bubble's shear layer amplified natural incoming Fig. 2. Cross-sectional view of flush mounted disturbances to achieve turbulent MEMSsensors and acoustic slit on cylinder. mixing and subsequently gain the additional momentum needed for reattachment. Other evidence of initiating turbulence with acoustic forcing was demonstrated by Ahuja [6] and Hsiao [7]. Ahuja transitioned flows to turbulence using an external source of sound, while Hsiao used internal acoustic excitation from a span-wise slit on the cylinder. Again, it was shown that external or internal acoustic excitations, performed at instability and
629 vortex s h e d d i n g frequencies, could amplify velocity fluctuations to transition flows to turbulence. Hsiao and other a e r o d y n a m i c i s t s such as N i s h i o k a [8], B a r - S e v e r [9], and W y g n a n s k i [10], then s h o w e d how local forcing about the leading edge separation point of an airfoil could prevent stall at high angles of attack. They found that the p r o p e r l y tuned and located flow p e r t u r b a t i o n s , w h e t h e r acoustic, mechanical or pneumatic based, were amplified by the separated shear layer to increase flow fluctuations, thereby reattaching the separated flow. Forcing frequency, amplitude and location were found to control the effectiveness of such perturbation a r r a n g e m e n t s . Unlike many previous experiments which used o f f - b o d y sensors to track flow fluctuations, this study will use an o n - b o d y array of MEMS sensors to determine stagnation, separation and vortex s h e d d i n g characteristics about a cylinder. MEMS t e c h n o l o g y has recently surged to the forefront of s e n s o r research and offers a sub-millimeter m e a s u r e m e n t resolution which is critical for s t u d y i n g the d e v e l o p m e n t of flow phenomena. Research by Ho and Tai [1 1,12, 13] has d e m o n s t r a t e d the success of MEMS devices in controlling and evaluating b o u n d a r y layer phenomena. 3.0
EXPERIMENTAL SETUP This i n v e s t i g a t i o n was conducted at the Low T u r b u l e n c e Wind Tunnel of the Naval Surface Warfare Center in Carderock, MD (USA). The tunnel is 1.2 m high by 0.6 m wide and can reach m a x i m u m wind velocities of a p p r o x i m a t e l y 43 m/s. As s h o w n in fig. 1, a 0.9 m long test cylinder of 50 mm diameter was mounted vertically in the test tunnel; a splitter plate was held 0.3 m off the floor of the wind tunnel to allow for sensor cable runs and cameras. The middle 0.2 m section of the e x p o s e d test cylinder was specially c o n s t r u c t e d to house 4 MEMS s e n s o r skins which were divided by thin 0.25 mm s p a n - w i s e slit. The four s e n s o r skins were staggered in the s p a n - w i s e direction to minimize each o t h e r ' s d o w n s t r e a m disturbance and p r o v i d e d a c o v e r a g e of 180 deg about the cylinder perimeter. The cylinder could be rotated by a stepping motor to aim the slit at any angle from the stagnation point. F i g u r e 2 s h o w s that the MEMS s e n s o r skins are flush mounted onto the cylinder over a range of 180 deg, with the slit dividing the c o v e r a g e into 64 s e n s o r s per 9 0 d e g . The s e n s o r skins each measured 2 cm long by 1 cm wide by 0.1 mm thick. On each skin, 32 tiny sensors were arrayed in the streamwise direction over a length of 22 mm--a spatial resolution of almost 0.5 mm or 1.4 deg per sensor. Although a total of 128 s e n s o r s on a span of 180 deg were available for the experiment, only a limited n u m b e r of amplifier channels were p u r c h a s e d to due to funding constraints. As a result only 33 s e n s o r s were on line for the experiment; the sensors were picked to maximize r e s o l u t i o n about the separation region. S e n s o r outputs r e p r e s e n t raw voltages as calibration was not required to determine the flow diagnostics of stagnation, separation and vortex shedding.
4.0
DISCUSSION OF RESULTS If active turbulence transitioning is to be a c c o m p l i s h e d on an u n s t e a d y p l a t f o r m equipped with a skin of sensors and actuators, then the platform s e n s o r s must be able to detect stagnation, separation, and vortex shedding characteristics. Plots will presented to 1) d e m o n s t r a t e how these critical flow characteristics could be detected by s e n s o r outputs and 2) how this information
630 could be applied to actively transition the flow to turbulence. Unless o t h e r w i s e noted, all results were achieved at a Reynolds number of 2 5 , 0 0 0 . 4.1
DETECTION OF C R I T I C A L C H A R A C T E R I S T I C S Stagnation The high spatial resolution and temporal r e s p o n s e of the MEMS s e n s o r s allows the identification of the stagnation point to a high degree of accuracy. The stagnation point is required as a reference and check on the separation region. Figure 3 shows the time traces of 3 c o n s e c u t i v e s e n s o r s located over a 2.9 deg (1.2 mm) span at the stagnation point. As s h o w n by Mangalam [14], the two outer sensors displayed strong oscillations which are 180 deg out of phase with each other while the middle s e n s o r has an oscillation of precisely twice the frequency of the two outer sensors. The two outer s e n s o r s are on o p p o s i n g sides of the cylinder diameter and therefore show out of phase oscillations due to the alternating vortex shedding. The middle sensor is located precisely at the stagnation point and thus s h o w s twice the frequency of the outer sensors. Stagnation can thus be detected by comparing the outputs of consecutive sensors until an out-ofphase or double frequency situation is detected. Separation. Many investigations of the separation Fig. 3. Shear stress sensor outputs about region on cylinders have been based the stagnation point of a cylinder, upon the average and rms (standard deviation) values of hot films located on the surface. For laminar b o u n d a r y layers, separation is typically identified by a m i n i m u m in the average shear stress s e n s o r outputs, along with a c o r r e s p o n d i n g jump in rms s e n s o r outputs ( B e l l h o u s e [15], Coder [16]). Figures 4 and 5 presents such average and rms shear stress s e n s o r outputs for 33 MEMS sensors placed about the perimeter of a 50 mm diam. cylinder in air flow at Re = 2 5 , 0 0 0 . Figure 4 s h o w s average sensor outputs for baseline conditions as the s p a n - w i s e slit is rotated to 82 and 102 deg from stagnation. Slit positions of 82 and 102 deg are s h o w n to produce nearly the same peak and m i n i m u m and compare favorably against data from Bellhouse [15], which s h o w s a m i n i m u m of 88 deg for laminar separation. Comparable data from the 82 and 102 deg slit position show that premature separation is not caused when the slit is p o s i t i o n e d near the typical separation region. Figure 5 presents rms sensor outputs for baseline conditions at slit p o s i t i o n s of 82 and 102 deg from stagnation. Again, the d o w n s t r e a m position of 102 deg compares well with the 82 deg position and proves that the peak and m i n i m u m rms values at 82 deg are naturally occurring and are not induced by the slit.
631 1.2
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20 40 60 80 100 120 140 160 180 Angle from stagnation (deg)
Fig. 4. Average MEMS shear stress sensor outputs about the cylinder perimeter.
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Fig. 5. RMS (Std. Dev.) of MEMS shear stress sensor outputs about the cylinder perimeter.
F r o m figs. 4 and 5 it can be seen that the separation zone for this test cylinder occurs in the same general vicinity, 80 to 90 deg, as identified in a t h o r o u g h compilation of cylinder separation data by Coder [16]. Figure 6 compares time traces for 7 individual sensors located about the separation region for baseline conditions at Re - 2 5 , 0 0 0 . The individual traces s u p p o r t the trends in fig. 5 - - s t r o n g and regular oscillations at 76 and 79 deg from stagnation are followed by a rapid drop and gradual rise in r a n d o m fluctuations. The sudden change in strength and regularity of fluctuations about the separation region is better s h o w n in fig. 7, which plots a y-direction time history mapping of the shear stress voltages as mapped out by 14 individual sensors clustered about a 9 mm region. Moving from left to Fig. 6. Time histories of MEMS shear stress right, it can be seen that regular, sensor outputs about the cylinder separation high energy fluctuations upstream region, of separation are quickly diluted into more r a n d o m and lower energy fluctuations as separation is encountered. Such a rapid change in frequency strength and regularity can be quickly determined with a spectrum analyzer. Results from such a spectrum analysis are s h o w n in fig. 8, which plots spectral analyses from the individual time traces s h o w n on fig. 6. It is seen
632 that the solitary vortex shedding frequency peak (30Hz) is immediately diluted at the separation region with more energy distributed to higher frequencies. Separation can thus be detected through the minimum of mean and rms shear stress sensor outputs, as well as through the change in spectral energies of the sensor outputs. Instability Frequencies. The instability frequencies which amplify incoming disturbances are predictable and typically constitute a brand band of frequencies. As shown by Hsiao, Bloors, Peterka and Richardson, there exists a ratio between the instability and vortex shedding frequencies which varies with Reynolds number. For Fig. 7. Time history mapping of 14 shear Reynolds numbers of 20,000, a stress sensor outputs over separation region. ratio of about 10 exists between instability and shedding frequencies. At higher Reynolds numbers of about 50,000, the ratio becomes 20. This ratio of instability and shedding frequencies varies as a function of Reynolds number and was shown to be a linear-log relationship. Hsiao also showed that the range of instability frequencies which can provide disturbance amplification was broad; effective frequencies were typically found at Strouhal numbers (St) of 1.0 to 3.0 for Re = 20,000. S trouhal number is defined as" St = [frequency (f)*characteristic length(D)]/freestream velocity (U)
Fig. 8. Spectral analysis of shear stress sensor outputs about cylinder perimeter with no
forcing.
633 An estimate can therefore be made of the instability frequencies by measuring the vortex shedding frequencies just upstream of separation. Shedding frequencies on a cylinder with a known flow velocity can also be estimated with a Strouhal number of 0.21. 4.2
APPLICATION OF FLOW CONTROL
Fig. 9. Spectral analysis of shear stress sensor outputs about cylinder perimeter with forcing at St=2 (275 Hz).
As shown in figs 3 to 8, sensor data were used to identify: 1) the stagnation point, 2) the separation region and 3) vortex shedding frequencies. These inputs were then used to tune acoustic disturbances at 275 Hz (St=2) and to aim the slit at the separation region, 82 deg beyond stagnation. With properly tuned and located acoustic forcing, dramatic changes are achieved in the spectral energies as shown in fig. 9. Prominent energy peaks are observed at the disturbance frequency of 275 Hz, its first harmonic at 550 Hz, and the vortex shedding frequency of 30 H z . It appears that flow forcing at the instability frequency has re-energized the spectral energies at the vortex shedding frequency to extend the life of the peak from 85 to 127 deg beyond stagnation. By contrast, the baseline spectral analysis of fig. 8 displays a permanent drop in the vortex shedding energies for all points beyond 85 deg. The emergence of the forcing frequency and extended vortex shedding peaks are shown in the time traces of the corresponding sensors on fig. 10. Downstream of the forcing at 85 deg, the regular large scale disturbances due to vortex shedding are immediately replaced by rapid fluctuations tuned to the input disturbance of 275 Hz. Between 90 and 120 deg beyond stagnation, a wave form matching the vortex shedding frequency is eventually superimposed upon the forcing disturbances. Figure 11 compares the average shear stress values about the cylinder for Re = 25,000 with and without forcing at 275 Hz. It is seen that the average values are approximately equivalent for the baseline and forcing conditions up to the forcing location (82 deg). Downstream of the forcing between 80 and 127 deg, average shear stress increases significantly for the forcing situation. At 140 deg, average values for both the forcing and baseline conditions re-
634
Fig. 10. Time history traces of shear stress sensors on cylinder perimeter with forcing at St=2 (275 Hz). 1.2
~ I::1~Forcing ~ at 82 ~ ' i 9No forcing
1
e g 0
'~0.8
converge. The trends seen with forcing are c o m p a r a b l e to the traces seen in Bellhouse [ 15] for transition flows. F i g u r e 12 plots the rms shear stress values about the c y l i n d e r for Re = 2 5 , 0 0 0 with and w i t h o u t forcing at 275 Hz. Equivalent values for both the baseline and forcing situations are seen up to the forcing point at 82 deg. H o w e v e r , between 80 and 127 deg, m a x i m u m rms values for the forcing situation are almost three times larger than those of the baseline condition. At 140 deg, rms values for both the forcing and baseline conditions reconverge. As predicted by Hsiao, Bloors, Peterka and R i c h a r d s o n , the most effective forcing frequency matched the instability frequencies, which were found to be a p p r o x i m a t e l y ten times greater than the m e a s u r e d vortex s h e d d i n g frequency at Re=25,000. Other testing at Re = 5 0 , 0 0 0 s h o w e d that an effective f r e q u e n c y of about 950 Hz, or a p p r o x i m a t e l y 20 times the vortex shedding f r e q u e n c y , was required to yield increased fluctuations. Effective forcing frequencies were found at St = 1 to 3.5. 0.25
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Fig. 12. RMS (Std. Dev.) of MEMS shear stress sensor outputs about the cylinder perimeter with and without forcing.
635 Figure 13 shows a wake profile taken 5 diameters 9No forcing downstream of the cylinder for [] Forcing at St=2 "l 2.5 both baseline and forcing rag .i AForcing at St-1.27 situations. It is shown that disturbances at St=1.27 and 2 9 produced reductions in the wake velocity deficit. While it is understood that the wake is 9 & Q asymmetric with forcing from only in & tV 1 one side, this should represent 9rrl A | adequate evidence to prove that m & m--i "ql flow control has been achieved 0.5 ,," >. with the acoustic forcing at the nn instability frequencies. It would appear that 6 7 8 9 properly tuned flow disturbances Velocity (m/s) were amplified by the separated Fig. 13. Wake profile of cylinder with and shear layer to entrain higher without flow forcing. momentum flows; this entrainment of higher momentum flows re-energized the boundary layer to delay separation. A delay of separation could be inferred from the reduced cylinder wake and the extended spectral energies at the vortex shedding frequency--with flow forcing, the vortex shedding peak is extended from 85 to 127 deg beyond stagnation. r -
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CONCLUSIONS An array of on-body MEMS shear stress sensors and internal acoustic actuation was used to actively transition the flows to turbulence about a cylinder. On-body sensors identified stagnation, separation and vortex shedding data; this information was then used to direct an internal acoustic disturbance at the separation region with a disturbance frequency of St=2. Acoustic disturbances tuned to the instability frequencies were aimed at the separation region to produce dramatically higher shear stress fluctuations downstream of the forcing and a reduced cylinder wake. It would appear that properly tuned flow disturbances were amplified by the separated shear layer to entrain higher momentum flows; this entrainment of higher momentum flows re-energized the boundary layer to delay separation. Other characteristics of forcing at the instability frequency included spectral energy peaks at the forcing frequency and re-energized spectral energies at the vortex shedding frequency from 85 to 127 deg. Without forcing, spectral energies at the vortex shedding frequency were typically dissipated at the separation point. 6.0 o
,
REFERENCES
Klebanoff, K. D. Tidstrom and L.M. Sargent, "The three-dimensional nature of boundary layer instability," Journal of Fluid Mechanics, Vol. 12, 1962, pp. 1-35. Bloors, M. Susan, "The transition to turbulence in the wake of a circular cylinder," Journal of Fluid Mechanics, Vol. 19, June 1964, pp. 290303.
636 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
15. 16.
7.0
Gerrard, J. H., "A Disturbance-Sensitive Reynolds Number Range of the Flow Past a Circular Cylinder," Journal of Fluid Mechanics, Vol. 22, May 1965, pp. 187-196. Peterka, J. A. and Richardson, P. D., "Effects of Sound on Separated Flows," Journal of Fluid Mechanics, Vol. 37, June 1969, pp. 265-287. Mueller, T. J., "Low Reynolds Number Vehicles," AGAR-Dograph No. 288, Neuilly sur Seine, France, 1985. Ahuja, K. K., Whipkey, R.R. and Jones, G.S. "Control of Turbulent Boundary Layer Flows by Sound," AIAA Paper 83-0726, 1983. Hsiao, F.-B., Shyu, R.-N., and Chang, R.C., "High Angle of Attack Airfoil Performance Improvement by Internal Acoustic Excitation," AIAA Journal, Vol. 32, No. 3, March 1994, pp. 6 5 5 - 657. Nishioka, M., Asai, M., Yoshida, S., "Control of Flow Separation by Acoustic Excitation," AIAA Journal, Vol. 28, No. 11, November 1990, pp. 1909- 1915. Bar-Sever, A., "Separation Control on an Airfoil by Periodic Forcing," AIAA Journal, Vol. 27, 1989, pp. 8 2 0 - 821. Katz, Y., Nishri, B., and Wygnanski, I., "The Delay of Turbulent Boundary Layer Separation by Oscillatory Active Control," AIAA Paper 89-1027, March 1989. Jiang, F., Y.-C. Tai, K. Walsh, T. Tsao, G.-B. Lee, C.-M. Ho, "A Flexible MEMS Technology and its First Application to Shear Stress Sensor Skin," IEEE-MEMS, Jan 1996. Jiang, F., Y.-C. Tai, B. Gupta, R. Goodman, S. Tung, J. B. Huang, an C.-M. Ho, "A Micromachined Shear Stress Sensor Array," Proc. IEEE MEMS-96 Workshop, San Diego, pp. 110-115, 1996. Ho, C.-M., S. Tung, G.-B. Lee, Y.-C. Tai, F. Jiang, T. Tsao, "MEMS - A Technology for Advancements in Aerospace Engineering," AIAA paper 07-0545, Reno, 1997. Mangalam, S. M., and Kubendran, L.R., Experimental Observations on the Relationship between Stagnation Region Flow Oscillations and Eddy Shedding for Circular Cylinder. Instability and Transition, ICASE NASA LaRC Series, Hussaini and Voigt, eds., Springer-Verlag, 1990. Coder, D. W., "Location of Separation on a Circular Cylinder in Crossflow as a Function of Reynolds Number," NSRDC Report 3647, November 1971. Bellhouse, B. J., D.L. Schultz, "Determination of mean and dynamic skin friction, separation and transition in low-speed flow with a thinfilm heated element," Journal of Fluid Mechanics, 1966, Vol. 24, part 2, pp. 379-400.
ACKNOWLEDGEMENTS The authors would like to acknowledge the funding and encouragement provided by Dr. Bruce Douglas, Director of Research at NSWC. Dr. Douglas manages the In House Laboratory Independent Research (ILIR) program at NSWC which provides funding for a wide variety of critical basic research. The authors would also like to thank Mr. Jack Gordon for his insight and technical assistance during the experiment. As he has so generously done during his 33 year career with NSWC, Mr. Gordon provided critical mentoring, advice and humor during the experiment. Mr. Gordon is now retiring from NSWC, but the results of his mentoring will continue to produce long after his retirement.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
637
E v o l u t i o n o f instabilities in an a x i s y m m e t r i c i m p i n g i n g jet S.V. Alekseenko, A.V.Bilsky, D.M. Markovich and V.I. Semenov Institute of Thermophysics, Siberian Branch of RAS, Lavrentyev Ave., 1, Novosibirsk, 630090, Russia
The experimental study of the evolution of axisymmetrical and spiral instabilities has been performed for the round impinging jet flow. In order to create strictly periodical coherent vortices in the jet shear layer the external low amplitude excitation was applied. Digital analysis of flow visualisation patterns at low Reynolds numbers allowed to obtain the values of phase velocity of the large scale structures along their trajectory. For the large Reynolds numbers the electrodiffusion method and Fourier analysis were applied to determine the flow characteristics. It is stated that attenuation in structure's velocity occurs nonmonotonically along the impingement surface. The manifestations of subharmonics resonance were found. The conditions for transition from axisymmetric mode of the large scale instabilities to the spiral one were determined.
1. INTRODUCTION It is well known that the development of flow instabilities in shear layers influences essentially the mixing processes. The initial instability waves roll up into the discrete vortices which can further merge with the creation of more large structures. In free axisymmetric jet or wake flow the initially axisymmetric vortex structures lose their symmetry downstream and the spiral instability modes appear. An active control of the flow can be provided by the external excitation which can lead to resonant amplification of large-scale vortex structures. First comprehensive study of free forced axisymmetric jet was done by Crow & Champagne (1971). In the recent work, Paschereit et al. (1995) generated resonant subharmonic interaction between the two axisymmetric travelling waves in the jet shear layer and they observed the strong influence of the initial phase difference between the subharmonic and the fundamental harmonic during external excitation of the flow. On the other hand, as it was stated by Gutmark & Ho (1983), the spatial disturbances of individual facilities can change the characteristics of jet flow and so the scatter of obtained characteristics reaches 100% in literature. Thus it is necessary to control as more parameters of experimental facility as possible. The spiral instabilities in free shear layers have been studied by many authors. Browand and Laufer (1975) discovered the growth of energy of spiral disturbancies further downstream in free axisymmetric jet, Cohen et al. (1983) found that with increasing jet's velocity the transition to spiral modes occurs to be located more close to the nozzle exit. The impinging jets have been studied extensively for the last three decades. The turbulent structure of impinging round jet was studied in detail by Donaldson et al. (1971). The
638 characteristics of plane impinging jet were presented in the work by Gutmark et al. (1978). The instability analysis for confined jet flows was done only by Ho & Nosseir (1981) who supposed that the decrease of vortices frequency occurs as a result of collective interaction of large-scale structures in the near-wall region of high speed impinging air jet. A structural image of the impingement region was statistically derived by means of analysis of the surfacepressure fluctuations in the work by Kataoka et al. (1985). Kataoka et al. (1987) associated the heat transfer intensification near the stagnation point with the periodical renewal of the surface by large-scale vortex structures penetrating into this region. The unsteady separations of radial wall jet flow in the vicinity of an impingement region were observed by Didden & Ho (1985) and Ozdemir & Whitelaw (1992). The first measurements of the instant velocity field in the round impinging jet were made by Landreth & Adrian (1990) with the use of PIV technique which, however, have not been provided with high spatial resolution. The recent work by Meola et al. (1995) shows the presence of equidistant azimutal structures which appear in impinging nonisothermal round jet. The present work is devoted to the study of instabilities evolution during the round jet impingement under the action of periodical forcing.
2. EXPERIMENTAL TECHNIQUE AND PROCEDURE
The sketch of impinging jet flow is shown in Figure 1. The experimental set-up consisted of a test section representing the rectangular channel made of Plexiglas, with the dimensions of 86xl 62xl 600 mm 3, the system of pumps and flow meters, a reservoir, connecting tubes and apparatus for measurements. A well-profiled round nozzle was inserted through the side wall of a channel. The submerged round jet issuing from the nozzle impinged normally on the opposite wall (measuring plate) of the channel. The skin friction probes were placed at the measuring plate which could be shifted and this allowed to change the radial position of each probe with an accuracy of 0.1 mm. To measure the wall shear stress and liquid velocities the electrodiffusion method was applied. The details of this technique are described in the author's work (Alekseenko and Markovich, 1994). The electrical signals from the probes passed to the a.d. transformer through the d.c. amplifiers. A complete data processing was accomplished by a personal IBM computer. The computer program allowed us to determine the mean values of the velocity and skin friction, its rms. pulsations, spectral density of the velocity and friction pulsations and mutual spectra also. For the spectral density Figure 1. Sketch of the measuring sell of the impinging jet flow. Re = 1000; H / d = 2
639
Figure 2. The visualisation of impinging jet flow. Re = 1000; Sh = 0.5; (a) - H/d = 2; (b) - 3
estimations the Fast Fourier Transform technique was applied. Each array of the experimental data, which was processed by FFT, consisted of about 100 segments of 2048 points. By the analysis of the signals phase difference between two skin friction probes which were placed close to each other ( Ar = 0.5 mm), it was possible to measure the phase velocity of the largescale structures. The visualisation of flow was performed with the aid of hydrogen bubbles which were produced on the surface of platinum wire during water hydrolysis. The excitation of the jet was provided by a standard electrodynamic vibration exciter connected by the instrumentality of the silphone with the plenum chamber. The sinusoidal excitant signals conveyed from the generator through the power amplifier to the exciter. The initial oscillations of flow embodied the axisymmetric mode (m = 0) and their rms. value changed from "ff/U0 = ~ / ~ 2 / U o = 0.0001 to 0.001 depending on the experimental conditions. The forcing frequency fl , was characterized by the Strouhal number, Sh d = f f . d / U 0 . The velocity measurements near the nozzle exit have shown that the imposed oscillations of level mentioned above do not influence the initial flow characteristics. The level of natural turbulence measured in the vicinity of nozzle was in the range of u ' / U o = ~ - ~ / U o = 0.005 § 0.008 at the nozzle axis and 0.05 + 0.06 at the centre of shear layer. The value of momentum thickness 0 at the nozzle exit, obtained from the velocity profile, equals to 0 ~ 0.1 mm. During the experiments the three values of Reynolds number were tested: Re = 1000 (lowest one), 12700 and 25200. Here Re = U o 9d / v , Uo is the mean flow rate velocity at the nozzle exit, d is the nozzle diameter equal to 10 mm and v is the kinematic viscosity of electrochemical solution equal to 1.04.10 -6
m 2 / s . The distance H between the edge of a
nozzle and the plate could be changed in a wide range - from 10 to 60 mm (H/d = 1 + 6).
640 EXPERIMENTAL RESULTS AND DISCUSSION
The flow visualisation pattems are shown in Figures 2 a, b for different nozzle-to-plate distances which were equal correspondingly to H/d = 2 and 3. It can be observed from the plots that the large scale vortex structures are developing in the mixing layer and interacting downstream with the barrier. In order to analyse the dynamics of the vortices along their trajectory the measurements of phase velocities of the large scale structures were fulfilled. With the aid of digital processing of the instantaneous flow patterns the values of structure's velocities were calculated beginning from the instant of originating the nonlinear wave formations in the free jet shear layer. These data are shown in Figure 3, a, b for the lowest Reynolds number studied, Re = 1000, and for H/d equal correspondingly to 2 and 3. In both cases the phase velocity of structures natively falls up to a certain minimum value in the location of the maximum structure's deceleration. Further downstream the large-scale vortex insignificantly accelerates under the action of negative pressure gradient and then, with increasing the radial distance, phase velocity for corresponding frequency decreases again. Phase velocity dependencies for free jet flow are shown also in Figure 3. The attenuation of velocity of the structures occurs by another way in this case. Similar consistent patterns of large scale instabilities persist also in impinging jet for more high Reynolds numbers. The measurements of local wall shear stress values were performed for Re - 12700 and 25200 for the conditions of external excitation of flow in order to provide the information both for the development of large scale vortex formations and the broad-band turbulence structure. In Figures 4 a, b the radial distributions of mean wall shear stress and rms. pulsation's level are shown versus the forcing frequency fi . Here 1:'= X/'~'2/'Cmax, "Cmax is the maximum value of mean friction. The amplitude and frequency of initial sinusoidal perturbations were determined by measuring the local velocities with electrochemical probe "blunt nose" at the centre of the nozzle exit. For presented data the dimensionless amplitude
1.0 4~ k O ~
0.8
0 9 0
1.0 ,i }
freejet impingingjet, H/d=2 0
Vy
O freejet 9 impinging jet, H/d=-3
0.8
~ r
0.6
0.6
(a)
0.4 0
(b)
0.4 |
I
2
x/d
|
4
0
I
'
2
x/d
I
4
Figure 3. Phase velocity of large scale structures in impinging jet. Re = 1000, Sh = 0.5, (a) H/d = 2, (b)- 3.
641 50 fr (Hz) Sh
0.40. 40 0.30
,
~
~,
90 150 210 260
0.34 0.57 0.8 1.0
k
0.20
!
0
9 t [] a
30
20
O
(b)
Ca) 0.10
P
0.00 0
2
4
r/d
6
0
2
r/d
4
6
Figure 4. Distributions of mean wall shear stress (a) and its turbulent intensity (b) for the different frequencies of excitation. Impinging round jet, Re = 25200, H/d = 2 1E+2
lIE+2
i
i
!
1 I
11~+1
1E+I
I
I:
I
'
~
;
i1
I J
1
lIE+0
1E+0 #L~..,,
1
J . . . . i,,
,,
1
1E-1
.~
,J
....
.f~
~Ii
I
I
1E-1
-" ,ii
,
~.~_i
1E-2
i ! i
~ " 1E-2 ~u
i. 1E-3
iJ
i
1E-3 |m
II
II 1E-4 1E-5 0.01
i
I
J . ~
I
l
i
1E-4 1E-5 0.10
Sh (a)
1.00
0.01
O.lO
Sh
1.oo
(b)
Figure 5. Spectral distributions of wall shear stress pulsations at different locations on the impingement plate. Re = 25200; dashed lines - unexcited jet, solid lines excited jet. (a) - r / d - 1.1, (b) - 2.2.
of velocity pulsations was constam and equal to e = u*/u o = 0.001. At forcing frequency J) = 90 Hz, lying outside the range of the natural frequency of coherent eddies, see below, (let's call the centre of this range as the most probable frequency fm), the mean friction distribution changes weakly, whereas the pulsation level grows more significantly with simultaneous displacement of its maximum towards the stagnation point. In the range of forcing frequency J) = 130 - 180 Hz (Sh - 0.5 - 0.68) close to fm, the mean skin friction at any point on the surface decreases, the second maximum of mean friction
642
disappears and the pulsation's level attains the greatest amplification (up to 1E+O % 42%) at r/d ~ 1. Further increase in J) OA ~OA ..... ~O~j sDv vCl..~ leads to reverse changes. In our previous ) o O0 0 1 0 work (Alekseenko et al., 1996) we ) 9 2 b qualified such an effect as | 3 D 0 0 "quasilaminarisation" of the flow, 1E-1 ~ ~ ~_~-J~ because similar distributions for the wall shear stress and its pulsations can be 1E+2 .,-r q-. observed in the low-Re impinging jet --; (b) flows. One more confirmation of the ---O---O 1E+I " o O --~ effect of turbulence suppression can be O _,,w" I found in Figure 5, a, where the spectral 1E+O -~ ' ~ - ---distributions of the wall shear rate pulsations are presented for the unforced O| 9 and forced conditions. This spectrum 1E-2 ~ | C corresponds to the point at the C IE-3 impingement surface where coherent 0.0 1.0 2.0 3.0 4.0 5.0 vortices penetrate from the free jet shear r/d layer with highest intensity. In the absence of forcing (grey line) one can Figure 6. Distributions of phase velocity (a) and observe the pronounced hump in the intensity (b) of fundamental and subharmonic spectral distribution. This hump, along the impingement surface. Re = 25200, however, is not sharp enough and thus H/d = 2. Excitation is carried out at frequency comprises the contributions of the f/ = 150 Hz (Sh = 0.57). 1 - fundamental natural coherent structures with certain harmonic, 2 - subharmonic, 3 - most probable frequency range. When the excitation is frequency in unforced case. applied by the frequency corresponding to the centre of natural hump or its neighbourhood (region of sensitivity of the jet), the sharp amplification of the coherent structures appears in such a way that they become much more powerful and strictly periodical (see black line in Figure 5, a). At the same time the vortices of neighbouring scales are suppressed strongly in the wide range of frequencies. In Figure 5, b the spectra are presented for certain location in the far field of radial wall jet flow (r/d = 2.2). In this point the vortex merging occurs with high degree of probability, i.e. the two sequential vortices are pairing with the creation of more powerful and slow structure. The maximum on the spectral distribution in Figure 5, b shows the subharmonic's intensity. In order to analyse the dynamics of fundamental harmonic and subharmonic during jet impingement let's consider the evolution of corresponding structures. Figure 6, b demonstrates the development of instability waves along the impingement surface. The data were obtained from the measured spectra of the wall shear stress pulsations. Both the intensity of harmonic with most probable frequency for the unforced jet and the intensity of main harmonic for the forced jet develop similar to the rms. friction pulsations. The most probable (preferable) frequency fmp corresponds to the maximum of power spectrum of skin friction pulsations in the absence of forcing (Figure 5, a). If the forcing frequences lie in a range mentioned above, fl. = 130 + 180 Hz (Sh = 0.5 + 0.68), one can observe a sharp amplification of pulsations at response frequency f~ (which is equal to the forcing frequency f (a)
643
2.5
~
]
',
...........
1
2.0
~
....
2
1.5
". -',
,,
1.0
"{,i
71:
-
r/d
3
II 1 0 2 II~ 3
,,;.....,,........, _ ~
""
~a-~_
I ,i
iz<~ 7I/2
/,
0.5
~,
0.0 0
200
400
600
800
10 0
f Hz
ll--O---q
00
Figure 7. Dispersive curves for different points on impingement surface. Re = 40400, H/d = 2, 1 - r/d
=
1.0, 2 - 1.3, 3 - 1.9.
2
4
6
H/d
Figure 8. Phase difference between the coherent pulsations in two symmetrical points on the measurement plate. Re = 25200, Sh = 0.57. Centre of symmetry stagnation point.
= f l = 150 Hz, see Figure 5, a) and its harmonics, 2f~, 3fl, etc. The growth of higher harmonics occurs similar to the main one but their intensity is lower. The subharmonic fr//2 develops by the other way. It achieves its maximum value at r/d ~ 2, and this fact demonstrates that at this point the first vortex merging occurs. At distance r/d ~ 1 the intensity of subharmonic has a small plateau, and, as it is seen from Figure 6, b, its position coincides with the coordinate of the main harmonic's maximum. With increasing radial distance from r/d ~ 1 to r/d ~ 2 the sharp growth of subharmonic's intensity takes place. According to Ho and Huerre (1984), this is manifestation of subharmonic resonance phenomena. One of the conditions of its existence is the local equilibrium of the fundamental harmonic. The phase velocity of subharmonic begins to grow simultaneously with its intensity (see Figure 6, a) and with falling of the main harmonic's velocity. At r/d ~ 2 it also reaches the maximum value and further downstream decreases slowly. Main harmonic's velocity in Figure 6, a develops similarly to the case of low Reynolds number (Figure 3). The results obtained for another Reynolds number (Re = 12700) show the similar behaviour of the main flow characteristics. In Figure 7 the dispersive curves are shown for the three characteristic points at the impingement surface. The phase velocity of discriminate frequencies corresponding to the multiple harmonics is much more higher to compare with broad-band turbulent fluctuations. It manifests the organised structure of these instabilities. In order to study the transition from axisymmetric mode (m = 0) of large scale instabilities to the spiral one (m = 1) the phase difference between the coherent pulsations (as a gauge of spirality) was measured for different points at the impingement surface. The corresponding wall shear stress probes were placed symmetrically in relation to the stagnation point of the flow. Several radial distances were tested beginning from r/d = 1 and up to r/d = 3 where the
644 coherent component of pulsations is still substantial. In Figure 8 the phase difference distributions are shown for three values of r/d. The point r/d = 1 can be considered to test the appearance of the spiral instability in free jet at least up to H/d = 4 + 5. The corresponding curve shows that spiral modes appear in the free jet (Aq~ becomes substantially non-zero) after the structure propagates the pass of approximately 4.5 nozzle diameters. This fact is in accordance with Browand & Laufer (1975) and theoretical predictions of Michalke (1971). At the same time in the far field of the radial flow the substantial phase difference was found beginning from the smallest nozzle-to-plate distances so the structures lose their axial symmetry more early in the case when they propagate along the solid surface.
4. S U M M A R Y
The experimental study of the axisymmetrical and spiral instabilities evolution has been performed for the round impinging jet under different conditions. In order to create strictly periodical coherent vortices in the jet shear layer the external low amplitude excitation was applied. Digital analysis of flow visualisation patterns at low Reynolds numbers allowed to obtain the values of phase velocity of the large scale structures along their trajectory. For the large Reynolds numbers the electrodiffusion method and Fourier analysis were applied for flow characteristics determination. It is stated that attenuation in structure's velocity occurs nonmonotonically along the impingement surface. The manifestations of subharmonics resonance were found. The conditions for transition from the axisymmetric mode (m = 0) of the large scale instabilities to the spiral one (m = 1) were determined. Measurements in the vicinity of near field of radial wall jet after impingement show that the spiral modes appear after the structure propagates the pass of approximately 4.5 nozzle diameters. In the far field of the radial jet flow the substantial phase difference was found beginning from the smallest nozzle-to-plate distances so the structures loss their axial symmetry more early in the case when they propagate along the solid surface.
REFERENCES Alekseenko S.V. and Markovich D.M., 1994, Electrodiffusion diagnostic of wall shear stresses in impinging jets. J. Appl. Electrochemistry, 24, 626-631. Alekseenko S.V., Markovich D.M. and Semenov V.I., 1996, Resonance effects in an impinging round jet, in C.-J.Chen, C. Shih, J. Lienau and R.J. Kung (eds), Flow Modelling and Turbulence Measurements VI, 109-116, A.A.Balkema Publishers, Rotterdam. Browand F.K. and Laufer J., 1975, The role of large scale structures in the initial development of circular jets. In: Turbulence in liquids, eds. J.L. Zakin, G.K. Patterson.- Princeton, N. J.: Science, 33-44. Cohen J., Gutmark E. and Wygnanski I., 1983, On model distribution of coherent structures in a jet, AIAA Journal Crow S.C. and Champagne F.H., 1971, Orderly structure in jet turbulence, J.Fluid Mech, 48, 547-591. Donaldson C.D., Snedeker R.S. and Margolis D.P., 1971, A study of free jet impingement. Part II. Free jet turbulent structure and impingement heat transfer, J. Fluid Mech., 45, 477.
645 Gutmark E. and Ho C.M., 1983, Prefered modes and the spreading rates of jets, Physics of Fluids, 26 (10), 2932-2938. Gutmark E., Wolfshtein M. and Wygnanski I., 1978, The plane turbulent impinging jet, J. Fluid Mech, 88, 737-756. Ho C.M. and Huang L.S., 1982, Subharmonics and vortex merging in mixing layers, J.Fluid Mech, 119, 443-473. Ho C.M. and Huerre P., 1984, Perturbed free shear layers, Ann. Rev. Fluid Mech., 16, 365424. Ho C.M. and Nosseir N.S., 1981, Dynamics of an impinging jet. Part I. The Feedback Phenomenon, J. Fluid Mech, 105, 119-142. Kataoka K., Kamiyama Y., Hashimoto S. and Komai T., 1985, Mass transfer between a plane surface and an impinging turbulent jet: the influence of surface-pressure fluctuations, J. Fluid Mech, 119, 91-105. Kataoka K., Suguro M., Degawa H., Maruo K. and Mihata I., 1987, The effect of surface renewal due to large-scale eddies on jet impingement heat transfer, Int. J. Heat Mass Transfer, 30, 559-567. Landreth C.C. and Adrian R.J., 1990, Impingement of a low reynolds number turbulent circular jet onto a flat plate at normal incidence, Experiments in Fluids, 9, 74-84. Meola C., de Luca L. and Carlomagno G.M., 1995, Azimutal instability in an impinging jet: adiabatic wall temperature distribution, Experiments in Fluids, 18, 303-310. Michalke A., 1971, Instabilitat eines kompressiblen runden Freistrahls unter Berucksichtigung des Einflusses der Strahlgrenzschichtdicke. Z. Flugwiss., 19, 319-328. Ozdemir I.B. and Whitelaw J.H., 1992, Impingement of an axisymmetric jet on unheated and heated flat plates, J. Fluid Mech, 240, 503-532. Paschereit C.O., Wygnanski I. and Fiedler H.E., 1995, Experimental investigation of subharmonic resonance in an axisymmetric jet, J. Fluid Mech, 283, 365-407.
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Aerodynamic Flows
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
649
A s s e s s m e n t o f E d d y V i s c o s i t y M o d e l s in 2 D a n d 3 D S h o c k / B o u n d a r y - L a y e r Interactions T. Coratekin a* , A. Schubert t and J. Ballmann t aLehr- und Forschungsgebiet ftir Mechanik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany The two-dimensional 24 ~ compression corner and the three-dimensional asymmetric 7~ 11 ~ double fin configuration are simulated as examples of shock wave/turbulent-boundary-layer interactions. Incoming Mach numbers were chosen 2.84 and 7.39 for the 2D case and 3.95 for the 3D interaction. The Favre-averaged Navier-Stokes equations with Fourier's law for heat transfer are solved approximately using a 3D finite volume scheme. Turbulent effects are incorporated through Wilcox's k - ca-model and the explicit algebraic Reynolds stress model of Johansson and Wallin. Convective terms are discretized with either Roe or AUSM flux splitting. The computed surface pressure and skin friction for the compression corner are in good agreement with the experiment, especially with the k - a~-model. In the 3D case, the computed surface pressure distribution on the bottom wall is in reasonable agreement with the experiment until the two shocks interact. Some discrepancies are observed beyond the crossing point. The computed heat transfer coefficient shows significant deviations from the experiment. 1. I N T R O D U C T I O N Many crucial problems still remain unsolved in the field of supersonic and hypersonic aerodynamics. The interaction of turbulent boundary-layers with shock waves is one of them and therefore a subject of widespread interest in hypersonic aircraft design. If strong enough, the interacting shock can cause the boundary-layer to separate. The strong pressure gradients involved produce an increase in skin friction and heat flux. In the worst case, the heat flux can be so high that the local temperature reaches beyond the melting point of the surface material and hence, causes irreversible damage to the fuselage. To prevent this type of structural failure in future hypersonic aircrafts, it is necessary to gain detailed information on the phenomenon itself on one hand, and profound understanding of the physics involved on the other. Shock/boundarylayer interactions occur preferably at concave supersonic flow deflections like rudder actuations or inlet configurations. The two-dimensional compression corner and the three-dimensional single'or double fin configuration are well known examples of flows involving shock/boundary-layer interaction. For the 2D case, the numerical investigation of a 24 ~ compression corner flow using a two-equation *PhD Student, DFG Graduiertenkolleg "Transportvorg~ingein Hyperschallstr6mungen" tphD Student, DFG Graduiertenkolleg "Transportvorg~ingein Hyperschallstr6mungen" ~;Professor
650
turbulence model is a common test case and hence, almost serves as validation. Settles [ 1] conducted the corresponding experiment at Ma~ = 2.84 in full detail. A similar experiment was performed by Schulte-R6dding et al. [2] at Ma~ = 7.39. In recent years however, attention has focused on more complex three-dimensional problems, like the symmetric crossing-shock interaction [3]. Extensive experimental and computational research on this configuration has elucidated the wave and streamline structure. Good agreement has been observed between computed and experimental surface pressure and flowfield profiles. Most of the published work on shock wave/turbulent boundary-layer interaction has incorporated the turbulent effects through either two-equation or full Reynolds stress models. Without any doubt, full Reynolds stress closures definitely contain more physics than two-equation models but they present an inherent difficulty with wall-bounded flows, where near-wall models that typically depend on the unit normal to the wall must be introduced - a feature that makes it virtually impossible to systematically integrate second order closures in complex geometries [4]. Moreover, compressible Reynolds stress models are not as well developed as their incompressible counterparts. Therefore, until new methods are fully developed, it is preferable to use two-equation models - with an anisotropic eddy viscosity systematically obtained from an algebraic second order closure - in complex, compressible wall-bounded turbulent flows. The purpose of this paper is to study the two-dimensional 24 ~ compression corner on one hand, and the three-dimensional asymmetric 7~215 11 ~ crossing-shock interaction on the other. Two different turbulence models are tested to account for Reynolds stress effects. One is Wilcox's [5] k - ~ - m o d e l model including low-Reynolds and compressibility corrections. The other is the explicit algebraic Reynolds stress model of Johansson and Wallin [6] based on the former k - w-model. Results are compared with the experiments mentioned above for the 2D case and with the experimental studies of Zheltovodov et al. [7] for the three-dimensional interaction.
2. PHYSICAL MODEL 2.1. Governing Equations Continuum mechanics provide the basis for describing unsteady turbulent hypersonic flows of compressible viscous fluids, as they would occur on the fuselage of future aerospace planes flying above 90 kms altitude. The corresponding balance equations are the Favre-averaged Navier-Stokes equations with Fourier's law for heat transfer. Additional equations are necessary for the Reynolds stresses, the turbulent dissipation and other heat flux correlations. All balance equations are used in their integral form but will hereafter be presented in differential form for the sake of brevity. Conservation of mass, momentum and energy then take the form
Ot
Ot
§
Ox~ +
+
-0
Ox~ 0
(1)
. . . .
-I
OXj
+
_
(2)
651
where Einstein summation convention is applied to repetitive Greek indexes. The overbar denotes a Reynolds averaged quantity whereas a tilde i stands for a Favre average. Turbulent fluctuations are provided with a double prime o". The framed expressions in Equations 2 and 3 show the major difference with ordinary Navier-Stokes equations for laminar flow. These correlations are time-averaged products of fluctuating quantities and must be considered as additional unknowns. Air is modeled as a fluid obeying to the perfect gas equation of state and the perfect gas caloric equation. Assuming that the pressure viscosity can be neglected, the mean molecular viscous stress for a newtonian fluid becomes
(4)
where the mass-averaged dynamic viscosity/~ - #(T) is determined by Sutherland's law. Similarly, the molecular heat flux is
qi -
cp/207~
(5)
P r Oxi
with the molecular Prandtl number P r - 0.72. To close the above system of partial differential equations, a turbulence model needs to be v j , the turbulent heat flux cppv i"~" ~ , introduced to determine the Reynolds stress tensor - p v~i"~" the molecular diffusion tiav~ and the turbulent transport pvav~v i -"~` "~" (also known as triple velocity correlation). 2.2. The k - w - T u r b u l e n c e Model The first possibility to model the Reynolds stresses is to use the Boussinesq hypothesis. Two additional transport equations for the turbulence kinetic energy k and the specific dissipation rate ~ are introduced to determine k and the turbulent viscosity #T -- -pc~*k/w"
O-fk ot
O-fw ot
O-fkS,~ +
OfJ),
0
:
O~w~,y
( # + O ' * # T ) O~~xk]
(6)
+
w
0~),
(7)
-
For compressible shear flows, both Equations 6 and 7 can be provided with a pressure dilatation term [4]. This term is skipped in the present paper. The turbulence model coefficients/3, /3", c~, c~*, cr and cr* are computed as proposed by Wilcox [5] with low-Reynolds and compressibility corrections. The value of w at the wall is determined by introducing effects of surface
652 roughness as proposed in [5]. A specific sand-grain roughness k + < < 1 then provides for a hydraulically smooth surface. Modelling effort for the remaining unknown correlations cppvi-,,,-r,,x, ti,xv~ and pvxvAv i ' ' ''~ " is substantially less. The turbulent heat flux is modelled in a similar way to its molecular counterpart by introducing a turbulent Prandtl-number: tt cppr"v i =
pTCp OT
Prt Oxi
(8)
with Prt usually set to 0.89. Finally, the molecular diffusion and the turbulence transport can be neglected for flows with Mach numbers up to the supersonic range. For hypersonic flows however, the most commonly used approximation is [5]:
Ok
tix v~ -- -~ pv~ v~ v i
(9)
2.3. The Explicit Algebraic Reynolds Stress Model In the case of an EARSM, the Reynolds stresses are given by
(10)
-
where aij is the Reynolds stress anisotropy. Neglecting the advection and diffusion terms in the exact transport equation for aij yields an algebraic expression that can be written as [8]:
vivj (p _ e) _ Pij - ~ij + YIij k
(11)
Depending on the choice of the models for r and IIij, general tensor theory can provide an explicit algebraic equation for aii. The complete detail of EARSM theory lies beyond the scope of this study and is therefore skipped for the sake of brevity. The model presented hereafter was originally proposed by Johansson and Wallin [6] and has been found to be well suited for wall bounded compressible flows. It contains the general linear form of the rapid pressure strain rate model [9], the Rotta model for the slow pressure strain and an isotropic dissipation rate tensor. In its 2D form, it reads
a~j = f l/~l Sij + +
( 1 - f ~ ) i - ~ 3s (B2 - ~ )4( S i k S k j - -
(s 4-(l-S1
~IIsSiy ) (12)
with the mean strain tensor Sij and the mean rotation t e n s o r ~ij, both normalized by the turbulent time scale T = k/c. The coefficients fll and/74 appearing in Equation 12 are given by
653
/~1 = N]~4 - -
6
N
with the two invariants IIs =
N3_C,1N2 -
(13)
5 N 2 - 2II~
S~,~,S~,../andIIa
= f2~.~f2~.y. N is the physical solution to
(27 ~-dIIs + 2IIa ) N + 2ClIIa - 0
(14)
which can be solved analytically [6] with
C'1 - ~
C~--I--sT0x,~)
(15)
where the Rotta coefficient C1 is set to 1.8. Low-Reynolds corrections are incorporated through a simple wall damping function fl -1 - e x p ( - B l y +) of"van Driest" type with B1 = 0.038 and B2 = 1.8. The near wall correction is strictly valid only for 2D parallel mean flow but the very near wall flow is, however, near parallel and two-dimensional also for quite complex flow fields so that it can be used as a first approximation. Since this EARSM is used in combination with the k - ~o-model, Equations 6 and 7 still remain necessary to determine k and w. In this case, of course, the Reynolds stresses rij are given by Equation 10 and not by the Boussinesq hypothesis. The models for turbulent heat flux, turbulent transport and molecular diffusion remain unchanged.
3. NUMERICAL METHOD Full details of the numerical method can be found in [ 10]. The complete system of partial differential equations with corresponding boundary conditions is solved numerically with the DLR FLOWer Code [11 ] using a block-structured, explicit finite volume upwind scheme according to the modifications published in [ 10]. The flowfield is therefore divided into non-overlapping quadrilateral (2D) or hexahedral (3D) cells in general curvilinear coordinates. All conservation laws and turbulence equations in integral form are then applied to each cell. This leads to a system of ordinary, time dependent differential equations that can be solved with a Runge-Kutta method. In order to account for the direction dependent transport of information in the inviscid part of the equations, the convective part is discretized with an upwind scheme. Two different upwinding schemes are used in the present study and are briefly outlined below. Roe's scheme is part of the Godunov-type flux difference splitting methods. Due to the matrix-vector multiplications involved, it is quite CPU-time consuming but has the potential to provide a sharp resolution of discontinuities, like for example shock waves. Unfortunately, it can become unstable in the vicinity of strong shocks and it displays an uncertain amount of numerical viscosity, depending on the value of the stabilizing "entropy-fix"-paramter 3. A scheme with too much numerical viscosity may cause non-physical separation behaviour.
654 An alternative can be the AUSM-scheme (Advection Upstream Splitting Method), which is part of the flux-vector-splitting methods. The major advantage compared to Roe's scheme is its computational cost. Conversely, a well known deficiency of the AUSM-scheme are the pressure oscillations near solid walls. Consistency with the central differences used for the diffusive terms is achieved by MUSCLExtrapolation (Monotonic Upstream Scheme for Conservation Laws). Applied to the primitive and turbulent variables, it improves the formal accuracy in space of the upwind discretization to second order. The minmod limiter ensures that variable do not overshoot in the vicinity of strong gradients. Time integration is performed by a 5-stage higher order Runge-Kutta scheme. Local time stepping is used as a convergence acceleration technique to reach the asymptotical steady state solution of the considered flow problems. In the 3D case, full-multigrid provides an efficient initialization of the flow field. 4. RESULTS AND DISCUSSION
Complete flow conditions for all three test configurations are shown in Table 1.
Table 1 Flow conditions.
Maoo Reoo(m-1) poo(Pa) Too(K) 50 (m)
2D supersonic ramp 2.84 6.5-107 2.4-104 103.3 0.026
2D hypersonic ramp 7.39 1.7-107 8.7.103 255.7 0.0027
3D double fin 3.95 8.6.107 1.1-104 63.2 0.0023
4.1. The 2D compression corner The 24 ~ 2D compression corner was used to validate the k - w turbulence model of the FLOWer code, to test the proposed EARSM and to demonstrate grid-independence. The corresponding grid details are therefore shown in Table 2.
Table 2 Grid details for the 2D compression corner at Maoo - 2.84.
ZXxmn/60 ZXXma/60 Coarse Grid Baseline Grid Fine Grid
50 100 200
30 60 120
0.177 0.088 0.044
0.185 0.092 0.046
ZXy , /60 6.15- 10 -5 3.08.10 -5 1.54- 10 -5
ZXymo/60 0.538 0.269 0.134
0.708 0.343 0.177
A boundary-layer equation code [5] with the same k - a~-model was used to generate the upstream variable profiles. The boundary-layer thickness of the experimental and the computed
655
variable profiles matched reasonably well. For all computations, the wall was supposed to be adiabatic. Figure 1 shows the computed skin friction coefficient at M a ~ = 2.84 for all three grids. It can be seen that the results for the baseline case and the fine grid are almost identical. The coarse grid however, is obviously below the required level of refinement to ensure grid-independence. These computations were performed using the AUSM-scheme for the discretization of the convective terms.
k-o~ Computation E A R S M Computation
Baseline case Fine grid ....... Coarse grid
0.002
q
zx
a. 8
V
} Experiment
0.001 n
r I
I
I
-o.~
I
I
'-O?o's'
I
I
6
i
,
X [m]
,
,
I
. . . .
0.05
I
o.1
i
i
o ' :o'.1 ' '-6~o5' ' '~'
,
Figure 1. Skin friction coefficient for 24 ~ compression ramp at M a ~ - 2.84.
30 25
v } Experiment A
20
/
o ~
o . o o ~~
'
'o'.~' ' '
Figure 2. Surface pressure for 24 ~ compression ramp at Mao~ - 2.84.
k-o~ Computation E A R S M Computation
0.002
' '6.~5'
X [m]
--
15
I~.
10
A v
k-e Computation EARSM Computation
} Exp
5 II
-o.o5
o 0.05 x [m]
o.1
Figure 3. Skin friction coefficient for 24 ~ compression ramp at M a ~ - 2.84.
0:~,
~ , , , ,r "o.~ -o.o5
o x [m]
0.05
o.1 ,,
Figure 4. Surface pressure for 24 ~ compression ramp at Mao~ - 7.39.
Figures 2 and 3 show the computed surface pressure distribution and skin friction coefficient at M a ~ - 2.84 for both the k - ~ - m o d e l and the EARSM in comparison with experimental
656
data [1 ]. These computations used the less "viscous" Roe discretization with 6 < 10 -5. The agreement is quite satisfactory with the k - a2-model which accurately predicts the position of separation and reattachment as well as the intensity of the separation. The EARSM gives acceptable results for the position of the separation but obviously overpredicts its size. It also shows a higher skin friction coefficient after reattachment. A similar behaviour can be observed in Figure 4 which shows the surface pressure distribution at Ma~ - 7.39. Results are compared with pressure measurements of Schulte-R6dding [2] which have an experimental uncertainty of 4-2%. For stability reasons, the computation was performed using the AUSM-scheme. The remarkable prediction quality shown here by the k - a;-model is in contrast with earlier results of Bardina [12], where its performance was judged to be rather poor compared to the Spalart-Almaras and SST models. However, the Mach-number domain covered in [ 12], ranging from subsonic to transonic, is different. At higher Mach-numbers, compressibility effects are expected to play a significant role and the behaviour of turbulence models must be re-examined. Furthermore, the authors could not realize whether low-Reynolds corrections have been used in [12] or not. In the case of the k - a;-model presented herein, these corrections considerably improved the results for supersonic flows.
4.2. The 3D asymmetric crossing shock interaction A general view of the double fin configuration can be seen in Figure 5 and the corresponding grid details are given in Table 3. The corresponding experiment was conducted by Zheltovodov et al. [7] with an experimental uncertainty of +0.5% in the pressure measurements and 4-10 to 15% in heat transfer measurements. A block structure was used to compute the incoming flat plate turbulent boundary layer and the crossing shock interaction in two separate blocks. Fullmultigrid with was employed to obtain a favorable initialization of the flowfield. To further minimize CPU-time, the convective terms were discretized according to the AUSM-scheme. In Figure 6 the computed pressure distribution along the throat middle line (z - 0.03675mm) agrees well with the experiment except at the end where a strong shock-wave reflection from the side walls takes places. Figure 7 shows surface pressure distributions at three different sections in flow direction. The sections at x -- 46 and 79 mm are located upstream of the crossing shock interaction and show close agreement with the experiment. At x = 112 mm, which is located after the shock-wave intersection, the measured pressure valley corresponding to the vortex generated by the 11 ~ fin, seems underpredicted by the k - a;-model. Similarly, the pressure plateau due to the 7 ~ fin is obviously overpredicted.
Table 3 Grid details for the 3D double fin with 128 x 80 • 80 cells. 7.41.10 -4
1.26
3.70.10 -4
0.618
3.70.10 -4
0.882
20.37
0.301
Figure 8 is the computed heat transfer coefficient compared with experimental results. The heat transfer coefficient is defined as
CA =
qw(x,z) -
(16)
657
Ma> 1
[]
7 ~
/
Experiment k-co Computation
J
0.1
I
~
.075''~ 0
Figure 5. Grid for the 3D double fin (every fourth point is shown).
7[ 6[-
[] • v --
I~.~ 5 f I1.
Experiment ( x = 4 6 m m ) Experiment (x=79 mm) Experiment (x = 112 mm) k-coComputation
3
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
0
i
i
,
,
I
. . . .
0.01
Z " ZTML [rn]
I
....
o.b5 ....
x [m]
0.0041
. . . .
o'.l ....
o. 5'
Figure 6. Surface pressure for the 3D double fin along the throat middle line.
0.005 F
k-co Computation ~
Experiment
F
Lx
-0.03-0.02-0.01
[]
I
0.02 0.03
Figure 7. Bottom-wall pressure distribution at x = 46, 79 and 112 mm.
X [m]
Figure 8. Heat transfer coefficient for the 3D double fin along the throat middle line.
where qw(x, z) = -nOT~On is the total wall heat flux computed in an isothermal case where the fluid temperature Tw at the wall is set to 265 K. In the adiabatic case, the wall boundary condition is OT/On = 0 and Taw(z, z) is the local temperature of the fluid at the wall. The agreement with the experiment is quite poor. Upstream of the shock-wave intersection, the deviation from the experiment is about -t-30%. After the interaction, it becomes more than 100%. There may be two major explanations for this discrepancy. One is perhaps the strong unsteady character of the flowfield and the other, the determination of Ch itself. In fact, the temperature difference in Equation 16 represents a difference in fluid temperature whereas in the experiment, the temperature is the real temperature of the wall. The same applies to the heat transfer, which is measured through the fluid/wall boundary. Hence, to obtain a proper computation of the heat transfer coefficient, it might be necessary to simulate at least a small part of the solid boundary with adequate material properties.
658 5. CONCLUSION Two and three-dimensional shock-wave/turbulent boundary layer interactions have been investigated for the 24 ~ compression corner and the double fin crossing shock interaction. The flow over the compression corner was simulated at Ma~ = 2.84 and Ma~ = 7.39 using Wilcox's k - ~ - m o d e l [5] with compressibility and low-Reynolds corrections and the EARSM of Johansson and Wallin [6]. Three-dimensional simulations at Ma~ -- 3.95 were performed with the k - ~ - m o d e l only. The numerical code uses a block-structured, explicit finite volume scheme with a second order upwind discretization for the convective terms. Roe's and the AUSM flux splitting methods can be applied optionally. In the 2D case, results obtained with the k - ~ - m o d e l are in good agreement with the experiment. The EARSM tends to overpredict the size of the separation region. The simulation of the 3D crossing shock interaction gives acceptable results for the bottom wall surface pressure distribution. Heat transfer predictions however remain quite poor. All 3D computations were performed on Fujitsu VPP300 vector supercomputer and needed approximately 100 hours CPU-time with local-time-stepping and full-multigrid techniques.
REFERENCES 1.
Settles G. S., Fitzpatrick T. J., Bogdonoff S. M., "Detailed Study of Attached and Separated Compression Corner Flowfields in High Reynolds Number Supersonic Flow", AIAA Journal, Vol. 17, pp. 579-585, (1979). Schulte-R6dding J. H., Olivier H., "Experimental Investigations on Hypersonic Inlet Flows", AIAA-Paper 98-1528, (1998). Gaitonde D., Shang J., Visbal M., "Structure of a Double-Fin Turbulent Interaction at High Speed", AIAA Journal, Vol. 33, No. 2, pp. 193-200, (1995). Speziale C. G., "Simulation and Modeling of Turbulent Flows", ICASE/LaRC Series in Computational Science and Engineering, Edited by T. B. Gatski, M. B. Hussaini, J. L. Lumley, Oxford University Press, pp. 185-236, (1996). Wilcox D. C., "Turbulence Modeling for CFD", DCW Industries, (1994). 6. Wallin S., Johansson, A. V., "A New Explicit Algebraic Reynolds Stress Turbulence Model", Proc. Sixth European Turbulence Conference, Lausanne, (1996). Zheltovodov A., Maksimov A., Shevchenko A., Vorontsov A., Knight D., "Experimental Study and Computational Comparison of Crossing-Shock Wave-Turbulent Boundary Layer Interaction", Proceedings of the International Conference on Method of Aerophysical Research-Part 1, Russian Siberian Div., Academy of Sciences, pp. 221-230, (1994). Rodi W."A New Algebraic Relation for Calculating the Reynolds stresses", Z. angew. Math. Mech. 56, T219-221, (1976). Launder B. E., Reece G. J., Rodi W., "Progress in the development of a Reynolds-stress closure", J. Fluid. Mech., Vol. 41, (1975). 10. van Keuk J., Ballmann J., Schneider A., Koschel W., "Numerical Simulation of Hypersonic Inlet Flows", AIAA-Paper 98-1526, (1998). 11. Deutsches Zentrum ftir Luft- und Raumfahrt, "FLOWer Version 114.1", Project ,
.
.
.
,
o
.
MEGAFLOW, (1997). 12. Bardina J.E., Huang EG., Coakley T.J., "Turbulence Modeling Validation", AIAA-Paper 97-2121, (1997).
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
659
A s s e s s m e n t of E x p l i c i t A l g e b r a i c S t r e s s M o d e l s in T r a n s o n i c F l o w s T. Rung, H. Liibcke, M. Franke, L. Xue, F. Thiele ~ and S. Fu b ~Hermann-F5ttinger-Institut fiir Str5mungsmechanik, Technische Universit/it Berlin, Miiller-Breslau-Strasse 8, D-10623 Berlin, Germany bDepartment of Engineering Mechanics, Tsinghua University, Beijing 100084, PRC A class of recently developed explicit algebraic stress models (EASM) [2] is subjected to a critical assessment in transonic engineering flows. It is found that the performance and robustness of these models is hampered with inaccuracies due to an inappropriate regularization practice inherent to the stress-strain relation. The origin of the regularisation is attributed to an inadequate closure assumption for the production-to-dissipation ratio in non-equilibrium conditions. To remedy the defect, a simple approximation for an explicit representation of P/c within the stress-strain relation is proposed. The emerging unregularized EASM satisfies the positivity requirement for the eddy-viscosity and obeys to the realizability principle. Several exemplary 2D and 3D testcases, employing different numerical algorithms, are investigated to assess the predictive capabilities of the present proposal including a linear truncation. Supplementary, the mathematical rigour of the approach is used to combine the non-linear stress-strain relation with a multiscale methodology and pressure-dilatation models. The paper aims to argue the potential advantages of an unregularized EASM methodology in transonic engineering flows. 1. I N T R O D U C T I O N The accurate prediction of compressible turbulent flows is still predominantly restricted to design conditions, in which the flow field is governed by inviscid phenomena while viscous or turbulent effects are of minor importance. As the industrial need rises for reliable simulations at off-design conditions, more accurate - and therefore more elaborate - approaches towards modelling turbulence are required. The present research focusses on transonic aerodynamic flows at low turbulence Mach numbers (Mr <0.3) invoking mild to strong shock/boundary-layer interaction. Although compressibility effects are primarily felt through the changes in mean density and turbulence behaves almost as if it were incompressible, these flows are far from being trivial. Various studies have demonstrated the inability of the most prominent standard Boussinesq-viscosity models to render the fundamental physics of turbulence in flows exposed to non-equilibrium high-load conditions. In contrast to this, the second-moment closure framework provides an increased predictive accuracy, however at the expense of significant numerical disadvantages and an enhanced computational cost. In view of an efficient modelling practice featuring an enhanced range of validity, recent
660 efforts indicate that non-linear two-parameter models are a viable approximation of the full second-moment closure in engineering flows. Based on integrity basis methods, Gatski and Speziale (GS) [2] proposed a non-linear eddy-viscosity model for incompressible flows, which constitutes an explicit solution to the algebraic stress model (ASM) of Rodi [5] in the limit of stationary turbulence. The advantage of the GS explicit algebraic stress model (EASM) over other non-linear modelling strategies is the mathematically sound transfer of desirable model properties inherent to the elaborate second-moment closure models to a less elaborate, application-oriented modelling practice. Thus, the GS EASM methodology in the context of compressible flows is in the focal point of this study. 2. T U R B U L E N C E
MODELLING
The derivation of the compressible EASM starts from the implicit algebraic stress model [5] which is based on the equilibrium turbulence assumption. For stationary, compressible flows in an inertial frame of reference the ASM reads: D
~p.uiuj) = u~u} D (pk) = Pij + Oij - peij Dt k Dt
with
D (pk) = Pkk -- pekk + Okk Dt 2 "
(1)
In (1)Pij---pu~u'kOUj/OXk-PU'ku}OUi/OXk represents the generation rate of the Reynolds stresses (with P=Pkk/2), ~ij and eij are, respectively, the pressure-strain correlations representing the mechanism of turbulence redistribution and relaxation, and the dissipationrate tensor. Due to the low turbulence Mach number density fluctuations are neglected (Morkovin's hypothesis). For modelling purposes, it is convenient to distinguish between a conventional trace-free and a dilatational part of the redistribution tensor, viz. q~J _ p'
-g~x~ + Ou~
= p
(r
with
+ 5 Cd
Cd -- ~kk = p, 0 U'm 1 p OXm p
(2)
The dissipation-rate tensor is usually split into isotropic (2//35ijc) and deviatoric parts (eijD) with the latter being absorbed in the pressure-strain model. The present study is confined to solenoidal contributions to the isotropic part. In the following, any trace-free pressure-strain model that is a linear function of the Reynolds stresses can be adopted:
r
- c~j.
=
-2Clcb~j + C~k(S~j - ~5~jSmm) + C4k(b~kWjk + bjkW~)
2
+C3k(bikSkj + bjkSki -- -~bijblkSkl)
2 5~je + e~j.
(3)
Cij ~--- 5 with,
bij = u~u} 2k
15i j 3
Sij
l ( OUi OUj ) - -2 ~ + ~
l ( OUi Wij - -~ Oxj
OUj ) Ox~
(4)
representing the anisotropy tensor of Reynolds stresses, mean strain-rate and vorticity tensors, respectively. Substituting equation (2)-(4)into the ASM (1) yields:
+(c~ - 2) (b~Sj~ + b j ~ . - ~
(5)
661 The latter equation employs the following abbreviations: 1
35ijSkk
~ j -- Sij --
and
C1-
g --
+
1 +
9
J
gic
incompressible part:
- -r
3
Skk ,
%
(6)
9
compressible part:
gc
where the second term of the coefficient g accounts for compressibility effects. Provided that the turbulent time scale k/E and the coefficient g are known, the implicit ASM (5) can be rearranged as an explicit algebraic expression for bij [2] by the use of integrity basis methods [12]. Invoking 2D mean flow, the relation can be further simplified based on the Caley-Hamilton theorem, which finally results in a quadratic stress-strain relation"
where, o6/~_
6oik~kj' Sm m-2 _
* k 2
*
__
Yt - - Ctt -~" ~
C# - -
s-
a-
fll --
(2/3
-
C2/2)/g,
oOmkOOkmetc. and
~2 -
~1
1-2/37/2 +2~ 2
(s) (1
-
C4/~)/g,
~3 -
(2 - C 3 ) / g .
Expressions (7) and (8) indicate that the difference between the compressible and the incompressible formulation is confined to the coefficient g and the definition of the irreducible invariants of the mean strain-rate tensor. Essentially, the compressible contribution is absorbed into the definiton of Sij, which facilitates the same structure of the ASM for compressible and incompressible flows. Strictly speaking, the assumption of twodimensionality is violated in compressible flows, as the trace eliminating contribution to 6oij is 3D by nature. Accordingly, the assumed 2D simplifications of the Caley-Hamilton theorem are partially invalid. The stress-strain relation, however, becomes significantly more complicated if the compressibility effects are not absorbed, but explicitly considered via a sixth invariant (kSkk/c) [1]. In particular, the individual coefficients are thereby afflicted by numerous possibilities for a singular behaviour through non-linear invariant interactions . The model can be extended to 3D mean-flow situations by an additional cubic term, e.g.
bij-h{~'q'(7)}~*'-
23/3a(L't/e)r/2 + ~2 [-~2S2 -
4rl2,aWij~/3~ 2
fl2ke ( 1 - ~ ) ( 6 o 2 k W k j +
6o2kWki)].
(9)
The Caley-Hamilton theory proves that a five term basis is generally sufficient for a complete representation of 3D mean-flow situations, disregarding the specific non-linear modelling practice adopted [1,12]. The coefficients resulting from the formal reduction process, however, are somewhat cumbersome. Thus, the above given cubic extension (9) is a viable approximation to the rigorous approach, preserving the model's simplicity. Pressure-dilatation model In general, the coefficient g defined in eq.(6) should always remain positive. The incompressible part of g, which will be discussed hereafter, is unlikely to cause a violation of the
662 positivity constraint. The compressible part of the coefficient, however, has the potential to drive g towards zero. The present approach absorbs all compressible effects in the pressure-dilatation term which needs to be modelled: g-
C 1 - 1 + p---~ + - - c
with
Cd(Mt, P/(pc), S)
and
Mt2 - 2k/a 2 .
(10)
Although q~d does generally not conform to Cd, the adopted pressure-dilatation model might serve as an approximation for Cd. In the following, two different models suggested by Sarkar (r s) [8] and aistorcelli (r [6] are investigated, viz. cR _ 0.5C (ozMtS) 2 [P/(pc) - 1]
and
Cds -- -0.15cMt (P/(pe) - 4Mt/3) .
(11)
In order to evade possible singularities of g, the adopted versions of the pressures-dilatation models explicitly refer to the homogeneous shear flow, where further contributions related to k Skk/e vanish. Ristorcellis approach clearly indicates that pressure dilatation is a nonequilibrium phenomenon, which can either stabilze or destabilize turbulence. In contrast to Sarkars model, the physics is not modelled by the turbulent Mach number but the gradient Mach number (MRS). The parameter a is actually described by the Kolmogorov relationship e = a (2k/3) 15 L -1. For the sake of simplicity, c~ is assumed to take a constant value of a = l . 0 throughout the present study, which is in line with recommendations of Ristorcelli. DNS data reveals that dilatation dissipation scales with M 4 [6], thus the effect is negligible in transonic aerodynamic flows. Coefficients a n d R e g u l a r i s a t i o n A major obstacle of the EASM approach (7) is the dependency of the coefficients ~1,2,a on the production-to-dissipation ratio P/e, which effectively gives rise to a non-linear, thus implicit formulation of the Reynolds stresses. To circumvent this undesirable feature, Gatski and Speziale [2] recommended to assign P/c to the asymptotic equilibrium value (~2) as a universal constant. This approach is associated with severe drawbacks for non-equilibrium conditions, e.g. large strain rates, and necessitates a subsequent regularization of cu to avoid negative or singular values. The regularization is afflicted by inferior predictive capabilities for appreciable, but not necessarily large strain rates, since it features a wrong asymptotic behaviour. Moreover, the considered pressure-dilatation models demand an anccurate representation of P/c. In an effort to remove any ad-hoc assumption for P/e, Wallin & Johansson [9] and Girimaji [3] presented a self-consistent explicit formulation by reference to 2D mean flow situations. The self-consistent approach is more elaborate and involves a selection criterion for g, an extension to 3D flows is unfeasible. The procedure can be simplified by neglecting a specific term of the integrity basis pertaining to S~ via (73=2 [9], however a renouncement of this term is highly debatable [2]. The present effort aims to adopt a less elaborate approach, which refers to a simple approximation of P/c from a linear truncation. The coefficients employed in this study are based on a slightly modified version the linearized SSG pressure-strain model [2]: $
C1 ~- 2 . 5 ,
C2 - -
0.39 (73 = 1.25, (74 = 0.45,
P/(pe) from eq.(13).
(12)
The choice of C~ depends on the approximation of P/c actually used in the model, which stems from the corresponding SSG expression for the incompressible coefficient g/~, viz. =1 P 0.8+r gic f l [ C 1 - 1] + 0(S, t2) ~ [(1.7 + 0.9P/(pc)) - 1] + P/(pE) .fl } _ _ _ _ pe 1.9 -
-
663 Table 1 Performance of the present EASM in incompressible homogeneous shear flow. S--3.3 S--5.7 Present DNS Present DNS bll 0.180 0.179 0.215 0.215 b22 -0.130 -0.127 -0.155 -0.153 b12 -0.158 -0.143 -0.157 -0.158 c~ 0.096 0.055
s2
present approximation of g 12
I
I
2
4
,
4
I
6
8
8
12
2D-analytical evolution of g //// 246810
.-S
,
,
4
8
,._S
12
Figure 1. Evolution of approximated and analytical modelling coefficient g
[
4. + 1.83x/0.2S 2 + 0.8~'-~ 2
~2
fl -- 1 + 0.95 1 - t a n h
(s)2] ~-~
.
(13)
A rigorous analysis reveals a minor dependency of (7/from the considered Cd which is negligible for low Mt. The function fl provides an upper bound for c~ in the limit of vanishing strain rates. A detailed outline of the derivation of r is beyond the scope of this paper. It should be mentioned that a number of constraints, e.g. stress realizability or positivity of c~, suggest lims-+oo r ~ S. Figure (1) depicts the evolution of the approximated g in the (S-~) map. The model has been successfully validated for a variety of benchmark flows [1] and provides a fairly reasonable predictive accuracy as indicated in Table 1. As opposed to the original GS model, the anisotropy coefficient c~ and the normal-stress components remain positive under any circumstances. Accordingly, the model does not require a regularization, which displays a major advantage of the approach. Background model The approach can be coupled to any two-equation background model. The present study is confined to modelled transport equations for the turbulence kinetic energy k and the specific dissipation rate w (c.f. Wilcox [11]), which is frequently employed in present industrial aerodynamic simulations:
Dpk = p + Dt ~
--2 -~xi
+ p(r
'
= a----if- +
Dt
~
-2 -~xi
-~pc~,w 2 (14) "
The coefficients take their standard values c~=5/9, cl,=0.09 , ~=0.83 and the effective viscosity is defined by # e - # t + 2 # . A fixed relation (c, wk = c) between the energy dissipation and w is assumed. All calculations have been obtained in conjunction with low-Re boundary conditions. Multi-scale methodology Recently, the use of two-scale models in compressible flow simulations has received renewed attention The idea of the multiscale methodology is to separate the total energy spectrum into large scale (kv) and small scale (e) contributions. Accordingly, the stresstransport equation (1) is subdivided into two parts. Details of the modelling frame are given by [10]. In the framework of the present modelling strategy attention is drawn to the contraction of the large-scale stress-transport equation, which is solved as an additional passive scalar-transport equation:
D(pk,)Dt
- P 1-
+--~-
- -~
pe
with
k - k, + e .
(15)
664 Obviously, the energy transfer and thereby the cross-section between the two scales is aligned with the coefficient Ca. In the limitting case of Ca=2, the multi-scale methodology reduces to the standard single-scale approach (k~--+0). The considered pressure-strain model (3) consists of rapid and slow parts. The rapid parts are traditionally associated with large scales, whereas the slow part is affected by both scales. Hence, the only change of the stress-strain relation is the modification of the slow term coefficient C1, where the individual scales contribute separately:
C1-Clp
+Cle
[
1
1-
with
Cle-1,
Clp=
1-0.5Ca
. (16)
The coefficient C~ is in line with the realizability principle, C~p forces the the resulting C1 to conform to the value given by eq.(12) in incompressible local equilibrium conditions. 3. C O M P U T A T I O N A L
APPROACH
Two different Finite-Volume Codes, viz. FLOWer and ELAN have been employed as testbeds for the evaluation of turbulence models under consideration. FLOWer is being devised by DLR, DASA Airbus and several universities under the aegis of the MEGAFLOW project. It is a density-based solver using a cell-vertex scheme on block-structured grids. Spatial discretization is based on central differences with added artificial dissipation, an explicit Runge-Kutta scheme is employed to integrate in time. Turbulence models available besides algebraic mixing-length models include several linear one- and two-equation eddy-viscosity models. ELAN consists of a pressure-based semi-structured, multi-block method. The algorithm employs a cell-centered, fully co-located storrage arrangement for all transport properties. Diffusion terms are approximated using second-order central differences, whereas advective fluxes are approximated using higher-order monotonicity preserving schemes. The odd-even decoupling problem of the cell-centered scheme is supressed with a fourth-order artificial dissipation pressure term in the continuity equation. The solution is iterated to convergence using a compressible pressure-correction approach. Various turbulence-transport models up to cubic second-moment closure models are implemented based on the apparent pressure/viscosity principle. 4. R E S U L T S A N D D I S C U S S I O N The first examples focus on a generic 2D transonic bump flow (Delery's CASE C [4]) and the transonic flow over an RAE 2822 airfoil [4]. These testcases are of particular interest when attention is drawn to shock/boundary-layer interaction. The results should display the ability of the considered turbulence closures to capture the onset and strength of shock-induced separation, as well as to mimic the recovery process aft of reattachment. The geometry of the bump flow and the shock-structure predicted by the quadratic EASM based on a semi-structured 30000 nodes mesh are shown in upper part of Fig. (2). The testcase involves subsonic inlet (T0=300K) and outlet conditions (p,,~60kPa) at Re=l.1 106. The outlet pressure is adjusted to match the experimentally determined location of the shock root at the lower wall. In order to assess the performance of the various
665 turbulence models Figure (2) compares the predicted surface pressure distributions and streamwise velocity profiles obtained with two quadratic (NLEA) and four linear truncations (LEA) with experiments and predictions of a Wilcox k - w model. In contrast to the standard model, all truncations of the present suggestion return a fairly accurate representation of the shock structure and adequately capture the recirculation regime indicated by the lower wall pressure plateau downstream of the shock. Comparing to the Wilcox
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model, the linear truncation already benefits a great deal from the rationale of the EASM modelling practice, which is a first central result of the investigations. Thus, the subsequent validation of the multi-scale approach and the pressure-dilatation models mainly refers to the linear truncation. Ristorcelli's pressure-dilatation model (LEA+r which seems to provide more sound turbulence physics, yields a stronger stabilization of turbulence, followed by a weaker recovery downstream of the reattachment point. Moreover, it was experienced that cR is sensitive to subtle variations of the Kolmogorov parameter a. In conjunction with an a priori estimate of a, the occurence of the gradient Mach number might inhibit the numerical stability, thus Sarkar's suggestion (LEA+r s) is favourized for computational reasons. A suprising outcome of this study is that the multi-scale approach did not significantly affect the results for this testcase, as indicated by a comparison of LEA and LEA+MS. The superior performance of the non-linear approach in the recovery regime is caused by the models' ability to resolve the normal stress anisotropy in the
666 Table 2 Predicted aerodynamic performance for the RAE 2822 airfoil CASE 10 [4]. Wilcox k - w LEA+r S NLEA+r Exp. (7/ 0.811 0.787 0.767 0.743 Cd 0.024 0.023 0.022 0.024
vicinity of the shock. Aft of the shock, a significant return to isotropy is induced by the non-linear model which enhances the momentum recovery process. It should be mentioned that the calculations aborted employing the regularized GS formulation. This is due to the occurrence of severe negative component energies in the vicinity of the shock. A thorough assessment of a particular model's performance should generally include testcases featuring both weak and strong shock/boundary-layer interaction. Therefore, the second example refers to two distinct RAE-2822 airfoil flows [4], where turbulence modelling aspects have a significant impact on the predictive accuracy of the solution. Calculations are performed on various structured and semi-structured grids of approximately 22000 nodes with different numerical procedures. Fig. (3) depicts the performance of three turbulence closures for the incipiently separated CASE 9 (M=0.73, Re=6.5 106, c~=2.79 ~ in comparison to CASE 10 (M=0.75, Re=6.2 106, c~=2.79 ~ in which shock induced separation occurs. Results obtained from the linear (LEA+r and the non-linear approach (NLEA+r s) using Sarkars pressure-dilatation model are seen to return a better overall prediction of the shock location when compared to the Wilcox model. Table 2 summarizes the experimental and computed lift and drag coefficients for the more challenging CASE 10. The comparison reveals encouraging predictive improvements of the EASM methodology.
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As a final example the flow around the ONERA M6 wing at M=0.84, Re=11.7 106 and c~=5.06 ~ is presented. As is the case with basically all realistic aerodynamic configurations, only pressure measurements are available [7]. Computations are performed on a grid consisting of approximately 1.5 million grid nodes and 48 blocks. Fig. (4) shows
667 the computed surface pressure coefficient on the upper surface of the wing employing the NLEA model. A A-structure of two shocks merging at 63% half span can be identified. Shock-induced separation occurs in the outboard region. A smaller A-structure is present where the merged shock reaches the wing tip. In oder to assess the predictive accuracy of the various models, Fig. (4) compares the computed pressure coefficient to the experimental data in the wing sections marked in Fig. (4). The suction peak is met correctly in all calculations. However, despite the shock location being too far downstream for all models, the prediction quality rises from Wilcox k - w to LEA to NLEA, the latter already leading to quite satisfactory results.
Figure 4. Results for the ONERA M6 wing flow [7].
5. C O N C L U D I N G R E M A R K S The paper aims to convey the merits of a simple, unregularized EASM. The suggested formulation approximately preserves the models' consistency in non-equilibrium conditions and satisfies the positivity requirement for the eddy-viscosity and the turbulenceenergy components. In conclusion, it offers enhanced numerical stability and significantly improves the predictive accuracy compared to the original regularized formulation. The results indicate, that even linear truncations of the present EASM achieve a more realistic predictive response than simple isotropic-viscosity models. The merits of the EASM, however, often rely on higher grid resolution and quality. The 2D assumption rooted into the derivation of the compressible quadratic stress-strain relation is a viable approach towards an all-speed engineering turbulence modelling frame. The principal benefits of deriving a non-linear stress-strain relation from second-moment
668 closures are elucidated by the mathematically sound extension to more complex flow phenomena and modelling practices, e.g. pressure-dilatation or multi-scale models. The multi-scale technique did not significantly affect the performance of the EASM in this study, thus the computational surplus is hardly defensible. On the other hand, the use of recently suggested pressure-dilatation models is recommendable in conjunction with the EASM approach. Sarkar's pressure-dilatation model was preferred to Ristorcelli's suggestion for numerical reasons. The adopted pressure-dilatation models only account for integral effects of compressibility through the change of turbulence energy. In order to increase the predictive performance for the turbulence structure, an improved modelling of the deviatoric part of the pressure-strain correlation model is required. ACKNOWLEDGMENTS The authors greatly acknowledge the financial support by German Ministry of Education and Research (BMBF) under the umbrella of the MEGAFLOW project (Grant. No. 20A9505H). The last author (SF) appreciated the support of TU Berlin's foreign exchange unit. REFERENCES
1. S. Fu, T. Rung, W. Chen, H. Liibcke. Collaborative research on flow physics modelling. Internal Report, TU Berlin-Tsinghua University Beijing (1998). 2. T. B. Gatski and C. G. Speziale. On explicit algebraic stress models for complex turbulent flows. J. Fluid Mech., 254 (1993) 59. 3. S.S. Girimaji. Fully-explicit and self-consistent Reynolds stress model. ICASE Report 95-82, NASA Langley Research Center (1995). 4. W. Haase, F. Bradsma, E. Elsholz, M.A. Leschziner, D. Schwamborn. EUROVALA European initiative on validation of CFD Codes. Notes on Numerical Fluid Mech. (42), Vieweg, 1993. 5. W. Rodi. A new algebraic relation for calculating the Reynolds stresses. ZAMM, 56 (1976) T219. 6. J . R . Ristorcelli. Some results relevant to statistical closures for compressible turbulence. ICASE Report 98-1, NASA Langley Research Center (1998). 7. Schmitt, V.; Charpin, F.: Pressure Distributions on the ONERA M6 Wing at Transonic Mach Numbers. AGARD AR-138 (1979). 8. S. Sarkar. The pressure-dilatation correlation in compressible flows. Phys. Fluids A, 4 (1992) 2674. 9. S. Wallin and A. V. Johansson. A new explicit algebraic Reynolds stress turbulence model including an improved near-wall treatment, Proc. of the 6th Symp. on Flow Modelling and Turbulence Measurements A.A. Balkema, 1996. 10. D. C. Wilcox. Multiscale Model for Turbulent Flows. AIAA Journal, 26 (1988) 1311. 11. D. C. Wilcox. Turbulence Modeling for CFD. DCW Industries, Inc., La Cafiada, 1993. 12. Q.-S. Zheng. Theory of representation for tensor functions- A untied invariant approach to constitutive relations. Appl. Mech. Rev. 47 (1994) 545.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
669
D e t a c h e d - e d d y s i m u l a t i o n of a n airfoil a t h i g h a n g l e of a t t a c k M. Shur ~, P. R. Spalart b, M. Strelets ~, and A. Travin ~ ~Federal Scientific Center "Applied Chemistry", St. Petersburg 197198, Russia UBoeing Commercial Airplane Group, P.O. Box 3707, Seattle, WA 98124, USA We present the first true applications of Detached-Eddy Simulation (DES), in the sense of being three-dimensional. DES was defined in 1997 with hopes of combining the strengths of Reynolds-averaged methods and of Large-Eddy Simulations, in a non-zonal manner, to treat separated flows at high Reynolds numbers. We first simulate isotropic turbulence, to check the concept in LES mode and set its adjustable constant. Smooth inertial ranges are obtained up to the cutoff in the spectra. We then treat an airfoil in the challenging r~gime of massive separation and do so very successfully, in that lift and drag are within 10% of the experimental results at all angles of attack, to 90 ~. Such an accuracy is not achieved with traditional modelling, even unsteady, which gives up to 40% error. Cost puts a pure LES of the same flow (at Reynolds number 105 and beyond) out of reach on any computer, yet we use personal computers for the DES, and about 200,000 grid points. On the other hand, grid refinement, domain-size and Reynolds-number studies have not been completed yet. Hysteresis in the 1 5 - 25 ~ range has not been addressed. 1. B R I E F D E S C R I P T I O N
OF T H E D E S A P P R O A C H
Additional discussion and, we hope, better context are provided in a companion paper [1]. In [2] we presented DES, giving an operational description that was complete except for the value of a constant, and two-dimensional (2D) preliminary exercises. The DES activity so far has been restricted to the S-A one-equation turbulence model [3], but we see no essential obstacles to using other models. Here, we present in w a basic study that sets the constant, and in w an applied study which we hope will reveal much of the potential of the method. w gives our outlook. DES was born after we found to our surprise that, on wings, traditional LES will be simply unaffordable for several decades [2]. This is patent even with very favorable assumptions such as allowing only 8,000 grid points for each cube of size ~, where (~ is the boundary-layer thickness, at any Reynolds number. This assumption is much more favorable than the state of the art can justify: today's proven LES methods resolve the near-wall streaks, which would bring an immense penalty at industrial Reynolds numbers. The number of &sized cubes needed to fill the boundary layer on a wing, even without flaps, is in excess of 107 [2], thus demanding 1011 grid points under the best assumptions (8000 points per cube, unstructured grid adapted to the boundary layer). This will be impossible for many years. The flaw in optimistic past announcements of LES for
670 aerodynamics [4] was to average 5 over the wing surface, when the correct procedure for LES purposes is to account for the cubes, and therefore to average 1/52 instead. This reduces the "representative" 5, roughly from 1/100 to 1/1000 of the airfoil chord. The 1011 estimate makes it clear that in the foreseeable future large regions of thin boundary layer on a wing will be treated with Reynolds-Averaged Navier-Stokes (RANS) methods. Not only do these regions place unmanageable demands on LES, but they are not very challenging for RANS, at least until separation is approached. Cars place lesser demands on LES than wings, but the feasibility gap is many orders of magnitude wide. It is also clear that regions with massive separation such as that behind a spoiler or a car present a tremendous challenge for RANS turbulence models, especially the simple ones. The models are built and calibrated in thin shear flows; many aim only at the primary Reynolds shear stress. A case of immediate failure outside thin shear flows is the free vortex: all models predict a very excessive diffusion in that flow [5]. In contrast, LES is generally successful in free shear flows, given a sensible number of grid points, and has no limit to its physical content in the sense that finer grids will resolve smaller eddies which in turn improves the accuracy of the larger stress-carrying eddies. This needs to be qualified for wall-bounded flows [1]. Quite a few colleagues have talked informally of combining RANS and LES, but we do not know of actual attempts. Most people envision zonal methods, in which this region would be explicitly "tagged RANS" and that region "tagged LES" by the user. DES differs in that the following single equation is applied; it is the S-A model with a new length scale d - min(d, CDESA) instead of the raw d, which is the distance to the wall and plays a strong role in the standard S-A model, as it controls the destruction term [3]. A is the local grid spacing, taking the maximum over the three directions (not the cube root). The intention is that in boundary layers, A far exceeds d and the standard S-A model rules since d - d; the model comes with all its experience base and fair accuracy. Away from walls, we have d - CDEsA and the model turns into a simple one-equation Sub-Grid-Scale (SGS) model, close to Smagorinsky's in the sense that both make the "mixing length" proportional to A. We are counting on the weak sensitivity of LES to its SGS model, away from boundaries, and have only CDES to adjust. On the other hand, the approach retains the full sensitivity to the RANS model's predictions of boundary-layer separation [1]. Still, the field of responsibility of the model is here much narrower than when it also controls a variety of thick, distorted, three-dimensional (3D) vortical regions. The simplicity of the approach, which has only one new constant, is attractive. The price is of course that 3D time-dependent calculations are obligatory if any region is to "go into" LES. If the boundary layers remain attached and the steady 2D solution is stable, DES finds that solution. In general, the user applies fine resolution in the regions of interest, which is similar to "tagging" these regions for LES treatment, but is implicit. It is possible to activate LES only in a region of particular interest. For instance, a RANS solution could be continued with a refined grid in a landing-gear well or at a rearview mirror, if we needed fine unsteady physics only there. Noise sources may also be determined much better than they could empirically from a RANS solution. Although DES extends the reasoning behind "Unsteady RANS" (URANS, which has been 2D [6-8] and sometimes called VLES) and was encouraged by the accuracy trend of URANS, it is more ambitious and expensive [1]. It is closer to VLES as in [9].
671 2. H O M O G E N E O U S
TURBULENCE
Here we exercise DES in pure LES mode into the Kolmogorov inertial range, hoping to show that the modified S-A model makes a fair SGS model, and to find the optimal value of CDES. Undoubtedly, DES will be applied even with grids too coarse for the spectrum to reach into the inertial range; we predict that a value that performs well here will also be efficient then. By "efficient" we mean keeping the solution smooth enough to prevent numerical pollution of the larger scales, without suppressing the physically valuable eddies close to the cutoff. Note that the optimal CDES weakly depends on the differencing scheme, as each scheme has an "effective" grid size that is not in the same ratio to the raw A. Schemes with higher accuracy will perform best with slightly smaller values of CDES, as they can accurately follow smaller eddies. We use the experiment of Comte-Bellot and Corrsin [10] and create initial conditions from their spectrum with random directions and phases. The differencing scheme is centered and fourth-order accurate. A fifth-order upwind scheme was found to be too dissipative to yield the ideal spectral slope near the cutoff. The time integration is by an implicit three-layer second-order-accurate scheme, with a CFL number below 0.1 everywhere. The centered scheme works well, in the sense of allowing a spectral slope of - 5 / 3 close to the cutoff wavenumber, with the Smagorinsky model and Cs ,~ 0.2. The spectral slope is also our criterion for setting CDES. With both models, the average eddy viscosity scales closely with A 4/3, as expected in the inertial range. Figure 1 shows spectra at t = 2, late in the experiment; the energy has decayed by 75%. The value of CDES is 0.65. The low end of the spectrum is quite good. Both 643 simulations return spectra a little steeper than the experiment, but very close to a - 5 / 3 slope. The slope is quite sensitive to CDES near the cut-off; altering it by +0.05 causes a very noticeable change. The 323 simulation displays only a moderate "ripple" effect in the smaller wavenumbers (1 to 10), compared with 643. The test is satisfactory. Again, LES results depend more on the resolution than on the SGS model. 10 -1
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672
Figure 2. DES grid over NACA 0012 airfoil
3. AIRFOIL
Our airfoil study is very promising in comparison with experimental results, but is not a complete validation due to time and computing constraints. We do not present systematic grid improvements beyond the plausible grid shown in figure 2, either in terms of the grid spacing or of the size of the spanwise period. The Reynolds-number dependence is weak after stall, which makes a value of 105 acceptable, but higher values will be studied. Before stall, in view of the sensitivity to transition, we cross-check the code with another code and high-Reynolds-number measurements. All cases here use the turbulence model in fully turbulent mode. Although the S-A model can simulate laminar boundary layers, we do not yet have a method to predict natural transition, and do not wish to arbitrarily adjust transition points. The NACA 0012 airfoil has been very widely studied up to its stalling angle of attack, roughly a = 15 ~ [11-13]. In the post-stall r~gime, we rely on [14] for forces on 0012 itself and on [15] for local pressure information that carries over from thin plates. The numerical method uses the upwind scheme. It is more accurate in stretched grids than the centered scheme we used in w which furthermore seems unable to provide stable solutions in the RANS regions, where the cell Reynolds numbers are very high. Mittal & Moin give an encouraging discussion of upwind-biased schemes for LES, while observing the same severe steepening of the spectrum (energy removal at high wave-numbers) we did [16]. The spanwise boundary conditions are periodic, with a length equal to the chord c of the airfoil. This value is much smaller than that of Najjar & Vanka for their thin plate [17], namely, 27~ x c. However, their value appears conservative to us, and 1 x c is
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acceptable in a first a t t e m p t . Runs in progress with period 2 x c both at 45 ~ and at 90 ~ which we speculate will be the most demanding, show a very weak effect. Visualisations shown below suggest that the spanwise flow structures are not severely confined, at least up to c~ = 45 ~ Similarly, the 25 spanwise grid points offer a plausible resolution of the larger flow structures. Roughly, the model operates as RANS within 0.026 x c of the surface, and as LES outside. In the other two directions, free-air conditions are applied at a distance of about 15 x c for most cases. The near-wall grid spacing is 10 -4 x c, following normal practice for RANS at a Reynolds number of l0 s. The number of points in the normal direction is 61, with slower stretching than for an attached flow. Finally, 141 points are on the airfoil in the chordwise direction, making an "O" grid. Our aim was to have fairly balanced grid cells in the most active region, say 1/3 c over the airfoil. The time step is At = 0.025 x c/Uoo where Uoo is the freestream velocity, and we apply dual time-stepping. A complete separated case includes 100 to 200 chords of travel and consumes 4 to 6 weeks on a Pentium II/266MHz computer. We begin with visualisations. Figure 3a-c is a side view with three simultaneous crosssections of the DES. Eddies with sizes rather smaller than c are resolved, especially near the upper airfoil surface, but not very fine with this grid. URANS, meaning an unsteady
674
Figure 4. End views of the DES at c~ - 45 ~ 9wx contours; cross-sections at 0.5, and 0.7.
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simulation with the standard S-A model (d - d), suppresses the smaller of these eddies and sheds large smooth vortices as shown by figure 3d, which resembles other URANS studies [6,7]. The three DES frames are noticeably different, indicating that the simulation has sustained the random initial three-dimensionality. In our experience, 3D URANS on this and similar flows actually damps out the three-dimensionality, at least with a spanwise period equal to c (it is very likely that URANS could sustain oblique vortex shedding with a long enough period). Figure 4 presents end views of the streamwise vorticity wx in different planes, over the upper surface of the airfoil (the lower surface carries no appreciable streamwise vorticity). Some areas appear not to obey periodicity, from z - 0 to z - 1; this is due to the postprocessing program using one-sided differences when computing the vorticity along the edges. The spanwise domain equal to one chord c seems to contain a few "typical" flow structures, indicating that the period is at least acceptable. Figure 5 presents the average forces over a wide range of angle of attack c~. Two branches denote chord Reynolds numbers Rc - 105 and 3 x 106 up to 8 ~ . We also crudely render the experimental scatter, in the 13 to 25 ~ range (for Rc > 106), by drawing two curves (beyond 25 ~, we have only one source and are unable to estimate scatter). The post-stall NACA measurements are at R~ - 2 x 106 [14]. The DES are at only 105, which helps contain the computational cost, while having a small impact on the flow beyond stall. At lower angles, and with the simulations being fully turbulent, we cannot expect to
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obtain the exact stall angle and m a x i m u m lift coefficient C/max. McCroskey [12] shows a strong trend for Clm~x in the range from 3 x 105 to 107, and the dependence on transition is strong. E x p e r i m e n t s suggest a value in the 0.8 to 0.95 range for Rc = 105 with natural transition [12]. Our fully turbulent simulations return a CZm~x near 1 with a near 13 ~ At a = 8 ~ DES produces steady a t t a c h e d flow, and is equivalent to RANS. We compare our code with the established NASA code INS2D, fitted with a Boeing module for transition prediction [18]. For these tests, the best results were obtained with a s o m e w h a t finer grid t h a n the one used for the stalled cases (figure 2) and of the C type, which better fits attached flows. W h e n run with n a t u r a l transition, INS2D agrees very well with N A C A results [11]. W h e n run fully turbulent the codes agree within a few % on both lift and drag, which gives us confidence. The experimental results at 105 with n a t u r a l transition (estimated from [12]) show a few % more lift t h a n we find. Beyond stall, the agreement between DES and experiment is most gratifying. This is
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Figure 6. Pressure coefficient. Left, DES (---, a = 8~ - - - - , a = 20~ - - , a = 45~ ---, a=90~ Center, a = 20 ~ ( - - - - , DES; o, Expt a = 20 ~ ~; [:3, Expt a = 20 ~ $). Right, various methods at 45 ~ (--, 3D D E S ; - - -, 2D U R A N S ; - - - , "2D DES"; o, Expt. [15]).
consistent with results of Najjar & Vanka [17], who performed 3D DNS but at Rc = 1000 and a = 90 ~ only. They treated a flat plate, but that makes little difference at such high a. Also in line with their findings, 2D URANS shows a large drag excess at a = 90 ~ The spanwise mixing caused by 3D eddies appears to greatly modify the primary 2D shed vortices, and in turn the mean flow. In contrast, the 2D URANS results are fairly good up to 45~ we have not identified any clear difference in flow pattern that would explain why. Finally, we conducted a "2D DES" at 45 ~ although it is unnatural. The results are very poor, as both lift and drag are over-predicted by 70%, compared with true DES. It is possible that in 2D URANS, the turbulence model gives a first approximation of the cross-talk between spanwise motions and in-plane mixing, at some angles of attack. The 60 ~ and 90 ~ results indicate that this hypothesis is not reliable, however. Our final figure, 6, shows the pressure coefficient Cp. On the left, four DES cases are shown. At a = 8 ~ we see the usual suction peak and the flatter spot of a separation bubble around x / c - 0.08 (which is dependent on details of the turbulence-model implementation, at this low Re). At 20 ~ and beyond, the suction has collapsed and the upper-surface Cp is nearly flat; this is correct behavior. The experimental data provided by Dr. W. J. McCroskey show a strong hysteresis in the 13~ ~ range [13]. We have yet to perform a sweeps with DES. In the center graph at a = 20 ~ the DES agrees well with measurements in a stalled state, obtained when a is decreasing (denoted by a = 20 ~ $). This is consistent with the Reynolds-number difference: the experiment at 4 x 106 and the DES at 105, giving a thicker boundary layer and a tendency for early separation (furthermore, the a = 20 ~ 1" measurement was not typical even for an increasing-a condition). Both curves have a spike for x very close to 0. The pitching-moment coefficient Cm is-0.12, when the experimental band i s - 0 . 0 6 t o - 0 . 1 1 [13].
677 On the right of figure 6 we compare three methods at 45 ~ All agree well on the lower surface. Experiments show that the pressure is flat on the upper surface, and the level was estimated from flat plates [15]. The DES results are very satisfactory, whereas 2D URANS and especially "2D DES" are too high and fail to produce the flat character. Again this is very consistent with DNS [17].
4. O U T L O O K Based on this first application, DES appears highly promising. Granted, this is exactly the kind of flow it was conceived for. No surprises were encountered, and no changes were made relative to the initial publication [2]. The CDES constant was set based on one flow, and used on the other (the upwind differencing scheme probably makes the airfoil cases insensitive to GOES,within a range). Running DES in 2D gave poor results, as others have found for 2D DNS; this is a fair finding since DES is fundamentally a 3D method. The forces are close enough to experimental results to be considered as within the error band. That band is fairly wide for separated flows in 2D geometries; the experiment has side walls with troublesome boundary layers, while the simulation has spanwise periodicity which introduces its own uncertainty. Future DES work will take at least two directions. The first is to explore the sensitivity of the present airfoil simulations to numerical and physical parameters, including: differencing scheme and grid spacing in all directions including time; spanwise period; GOES value; Reynolds number; direction of c~ variation; and mode of transition. This will raise the cost by over an order of magnitude, but is essential to rule out a fortuitous agreement and to establish DES as having a clearly superior physical basis, compared with steady and unsteady RANS. In view of figure 5, we are optimistic. We are less positive about the desire to harmonize the spatial differencing schemes between isotropic-turbulence and airfoil studies. The other direction is to a three-dimensional geometry with massive separation and challenging physics. Delta wings come to mind rapidly, possibly with vortex breakdown, as do spoilers and other obstacles on a wall. A third research direction would be an extensive study of the circular cylinder up to high Reynolds numbers. We can now hope for better results than from 2D URANS [7], but the question of transition prediction could make it difficult to separate sources of error. We also need to simulate a channel or boundary layer with an LES grid. This is not a natural DES application and there is no reason to expect a very good intercept for the log law, but this flow is the model for the flow far downstream of an obstacle.
Acknowledgements We enjoyed the active help of Drs. Kusunose, Allmaras and McCroskey. Dr. Venkatakrishnan reviewed the manuscript. CPU time was donated by SGI on an Origin 2000, and Dr. Kremenetsky guided the adaptation of the code. Authors listed alphabetically.
678
REFERENCES 1. P. R. Spalart, 1999. Strategies for turbulence modelling and simulations. 4th Int. Symp. Eng. Turb. Modelling and Measurements, May 24-26, Corsica. 2. P. R. Spalart, W-H. Jou, M. Strelets, and S. R. Allmaras, 1997. Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. 1st AFOSR Int. Conf. on DNS/LES, Aug. 4-8, 1997, Ruston, LA. In Advances in DNS/LES, C. Liu & Z. Liu Eds., Greyden Press, Columbus, OH. 3. P. R. Spalart and S. R. Allmaras, 1994. A one-equation turbulence model for aerodynamic flows. La Rech. A drospatiale, 1, 5-21. 4. D. R. Chapman, 1979. Computational aerodynamics development and outlook. AIAA J. 17, 12, 1293-1313. 5. O. Zeman, 1995. The persistence of trailing vortices: a modeling study. Phys. Fluids A 7, 135-143. 6. P.A. Durbin, 1995. Separated flow computations using the k - c - v 2 model. AIAA J. 33, 4, 659-664. 7. M. L. Shur, P. R. Spalart, M. Kh. Strelets and A. K. Travin, 1996. Navier-Stokes simulation of shedding turbulent flow past a circular cylinder and a cylinder with a backward splitter plate. Third Eur. CFD Conf, Sept. 1996, Paris. 8. S. A. Orszag, V. Borue, W. S. Flannery and A. G. Tomboulides, 1997. Recent successes, current problems, and future prospects of CFD. AIAA 97-0431. 9. R. E. Childs and D. Nixon, 1987. Turbulence and fluid/acoustic interaction in impinging jets. SAE 87-2345. 10. G. Comte-Bellot and S. Corrsin, 1971. Simple Eulerian time correlation of full- and narrow-band velocity signals in grid-generated 'isotropic' turbulence. J. Fluid Mech. 48, 273-337. 11. I. H. Abbott and A. E. von Doenhoff, 1959. Theory of wing sections, including summary of airfoil data. Dover, New York. 12. W. J. McCroskey, 1987. A Critical Assessment of Wind Tunnel Results for the NACA 0012 Airfoil, AGARD-CP-429. 13. W. J. McCroskey, K. W. McAlister, L. W. Carr and S. L. Pucci, 1982. An experimental study of dynamic stall on advanced airfoil sections. NASA TM 84245. 14. S. F. Hoerner, 1958. Fluid-Dynamic Drag. http://members.aol.com/hfdy/home.htm. 15. F. H. Abernathy, 1962. Flow over an inclined plate. ASME J. Basic Eng. 61,380-388. 16. R. Mittal and P. Moin, 1997. Suitability of upwind-biased finite difference schemes for large-eddy simulation of turbulent flows. AIAA J. 35, 8, 1415-1417. 17. F. M. Najjar and S. P. Vanka, 1995. Effects of intrinsic three-dimensionality on the drag characteristics of a normal flat plate. Phys. Fluids 7 (10), 2516-2518. 18. K. Kusunose and H. V. Cao, 1994. Prediction of transition location for a 2-D NavierStokes solver for multi-element airfoil configurations. AIAA 94-2376.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
679
T R A N S I T I O N AND T U R B U L E N C E M O D E L L I N G F O R D Y N A M I C STALL AND B U F F E T
W. Geissler a and L.P. Ruiz-Calavera b a DLR, Institute of Fluid Mechanics, Bunsenstrasse 10, 37073 G6ttingen, Germany.
b INTA, Aerodynamics Division, Carretera de Ajalvir km 4.5, 28850 Torrejon de Ardoz, Spain
The influence of various turbulence models (Baldwin-Lomax, Spalart-Allmaras, k-o), SST k-o)) and transition models (Chen-Thyson with various transition onset criteria) in calculations involving unsteady separated flows (dynamic stall and shock buffet) is tested. The results are compared against experimental data and other published calculations. For these very sensitive flow problems different turbulence models produce very different results, which also depend on the details of the implementation of a particular turbulence model, even if the same Navier-Stokes solver scheme is used.
1. INTRODUCTION The proper computation of unsteady viscous flows around airfoils remains an outstanding and important problem in fluid dynamics. This is becoming an specially pressing issue because as the design of aircrafts, helicopters and jet engines is improved, a large number of unsteady phenomena appear that have serious implications in terms of achievable performance or safety, and that need to be predicted accurately as soon as possible in the design cycle. This paper addresses two of them, namely dynamic stall and shock buffet. Dynamic Stall refers to the unsteady separation and stall phenomena of aerodynamic bodies forced to execute time-dependent large amplitude motions. It is a complex fluid dynamic problem of practical importance that occurs on manoeuvring aircraft, retreating blades of helicopter rotors, compressor cascades, and wind turbine blades. It often leads to the initiation of stall flutter. The flow is characterized by massive unsteady separation and formation of large scale vortical structures which create extended hysteresis effects. As a result, the maximum values of lift, drag, and pitching moment can highly exceed their static counterparts. Shock buffet refers to the self-induced periodic shock oscillations at a fairly
680 constant frequency that appear when an airfoil moving in the transonic regime exceeds a certain angle of attack (the so-called buffet onset boundary). This instability is not produced by airfoil motion but by shock-boundary layer interactions that induce alternating separation and re-attachment. Periodic aerodynamic forces occur which can excite the structure in a phenomenon known as buffeting and may lead to fatigue failures. Recently investigations have been started to favourably influence dynamic stall and/or buffet by means of dynamic deformations of the airfoil (nose droop, bumps, etc.) at flow separation and/or shock location positions. For this work to progress it is absolutely necessary to have the capability to reliably predict the effect of such dynamic geometry modifications on the airflow. Experimental investigations are difficult and costly, not only because of the intrinsic difficulty of designing a deforming wind-tunnel model, but also because both problems depend on a large number of parameters (airfoil shape, Mach number, Reynolds number, reduced frequency, type of motion, wind tunnel noise, wind tunnel wall interferences, threedimensional effects, etc.). The use of Computational Fluid Dynamics is thus an attractive alternative as long as it can assure accurate results. Turbulence and transition modelling remain the weak link in the simulation of this type of flows with either massive separation or shock-boundary layer interaction. This is specially true for the unsteady case. The objective of this study is to identify reasonably accurate and robust transition and turbulence models for these kinds of applications. Other authors have performed similar exercises, also using Navier-Stokes solvers, both for the dynamic stall case [1-4] and to a lesser extent for buffet [5,6]. In the following, four turbulence models are used: the Baldwin-Lomax algebraic model [7], the oneequation Spalart-Allmaras model [8], the two-equation Wilcox k-co model [9], and the two-equation Menter's SST k-c0 model [10]. The performance of these models for dynamic stall calculations is evaluated against two different cases: the NACA 0015 airfoil experiment of [11] and the NACA 0012 experiment of [12]. The former has the advantage that the boundary layer was tripped in the leading edge region to ensure a fully turbulent flow. It is thus an ideal case to test turbulence models decoupled from the transition model, which is of primary importance to the overall development of the suction side flowfield if it is dominated by leading-edge separation. The influence of the transition model is tested for the NACA 0012 airfoil measured in [12] for both tripped and untripped flow cases. These tests may be assumed as preparations for dynamically deforming the leading edge of the airfoil (DDLE concept, [12]) where the influence of transition is of even greater importance with respect to separation control as in rigid airfoil cases. The development of corresponding transition models suitable for deforming airfoils remains a challenging task for future investigations. For buffet, the well known 18% circular-arc airfoil frequently investigated experimentally by McDevitt, [13], is used. A further reason why these test cases have been selected is because they have been already calculated in the past by other authors [2,3,5], either successfully (like the circular-arc airfoil) or unsuccessfully (like the NACA 0015 deep stall case),
681 so that comparison with their results can also be made. 2. N U M E R I C A L M E T H O D
In the present study, the full time-dependent compressible Reynolds-Averaged Navier-Stokes equations in strong conservation-law form are solved using an implicit finite-difference method on a body fitted C-type grid which moves with the airfoil. The numerical algorithm adopted is the well known Beam and Warming spatially factored scheme [14], which is first order time-accurate in the dissipation and second order time-accurate for the convection terms. Spatial derivatives are calculated centrally and kept second order accurate. The viscous terms are retained in both the ~ and TI-directions. Artificial dissipation is added both explicitly and implicitly according to Jameson's type blending of second and forth order terms based on the computed pressure field. More details of the particular implementation of the scheme can be found in [15]. The Baldwin-Lomax (B-L) algebraic turbulence model [7], the Spalart-Allmaras (S-A) one-equation model [8], Wilcox's k-c0 model [9], and Menter's Shear Stress Transport (SST) modification of the k-c0 model [10] are used. Standard versions of these models have been implemented. Unless otherwise stated, the BaldwinLomax calculations have been performed with the original wake constant value of Cwk=0.25, [7] to enhance flow separation, and not with the often used value of Cwk=l.0 which improves numerical convergency but yields considerably stronger interaction and thus underpredicts the separated domain. 3. D Y N A M I C S T A L L O N N A C A 0015 A I R F O I L
The tests [11] were performed at a freestream Mach number of M=0.29 and a Reynolds number (based on chord length) of Re=l.95x106. The airfoil oscillates harmonically in pitch about its quarter-chord point with a reduced frequency (based on half chord) of k=0.1. The oscillation amplitude remains fixed at a1=4.2 ~ and variations of the mean angle of attack a 0 lead to different flow regimes. The light stall regime is obtained for a 0 = l l ~ and it is characterized by moderate trailing edge separation which develops at the peak of the cycle. The flow remains separated for a large portion of the downstroke and a recirculating region of about half a chord length is observed. The deep stall regime is obtained for a0=15 ~ and it is characterized by massive flow separation which develops before the peak angle of incidence. At peak incidence and before the downstroke, the dynamic stall vortex is shed and a trailing edge vortex forms. The flow remains separated for a large part of the downstroke and significant hysteresis effects are obtained. The calculations were performed using the same values for the different numerical parameters reported in [2] and [4], where the same numerical scheme to integrate the Navier-Stokes equations is used. This is done to demonstrate that due to the sensibility of this type of flows, the use of nominally identical numerical scheme and turbulence model does not necessarily assure identical results, caused
682 1.60
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Fig.1 (Left) Light Dynamic Stall, Fig.2 (Right) Deep Dynamic Stall on NACA 0015 Airfoil Influence of Turbulence Modelling, Comparisons with Experiments [11]. by inevitable small differences in the implementation of the codes. In particular a 361x71 grid with 271 points on the airfoil was used. The spacing of the first grid point at the surface in the normal direction is 0.00002 chords, and the grid boundaries are located at 15 chords in all directions. A nondimensional time step of At=0.0108 (based on chord length and freestream speed of sound) corresponding to 10.000 constant time-steps per period were used. For the SST k-co calculations a finer grid had to be used to obtain converged results, namely 361x81 with a distance from the surface to the first grid point of 0.00005. The number of timesteps per period was increased to 15.000. In all cases the calculations proceed for a total of three cycles, the results being repeatable beyond the second cycle. The flow is considered to be fully turbulent Figures 1 and 2 present the unsteady airloads (lift, drag, and pitching moment about quarter-chord point) during the third cycle calculated with the different turbulence models and compared with the experimental data for the light and deep stall cases respectively. For the light stall case (Figure 1) the B-L (either with Cwk=0.25 or =1.0) and ko~ models predict almost no separation. The S-A model gets fairly close to the
683 experimental lift hysteresis loop although shows earlier flow r e a t t a c h m e n t during the downstroke. On the other hand, and for unknown reasons it has poor results in terms of pitching moment. The SST k-c0 has excellent agreement with the experimental data during the upstroke, but then excessive separation develops during the downstroke. These results have not been plotted to improve the visibility of the figures. This is surprising as in [4] this model proved to be one of the best. Recent calculations on the RAE 2822 airfoil under steady transonic flow conditions have shown t h a t the SST-k-o) model shows the best match compared to experimental data. In the deep stall case only the S-A model does a good job in reproducing the physics of the flow. The lift hysteresis is predicted reasonably well, although the drag and pitching moment hysteresis loops indicate that it delays the onset of separation. The smaller extreme values of drag and pitching moments point to an underprediction of the extent of separation. The SST k-o3 model was not used in this case. 4. DYNAMIC
STALL
ON NACA
0012 AIRFOIL
It is well known from both numerical [15] and experimental [12] investigations t h a t stall on a symmetric NACA 0012 airfoil may be of both leading and trailing edge type. At moderate Reynolds numbers compressibility effects as well as the bursting of a laminar separation bubble lead to dynamic stall onset and proper computational modelling of the transition process is necessary to match real flow features. In a comprehensive numerical study [16],[17] the transition effects have been modelled by a combination of the algebraic Baldwin-Lomax turbulence model with the Chen-Thyson transition model. Special emphasis was placed on the application of transition onset criteria: fixed prescribed transition onset location (experimental tripping position), instantaneous position of the upper surface pressure minimum, - Michel's criterion to simulate free transition conditions. -
In the latter case the match of the two functions Re 0 (x) and f(x) = 1.174. (1.0 + 22400/(Re~
(1)
with Re x and Re 0 as Reynolds numbers based on freestream velocity Uoo rather t h a n on the boundary layer edge velocity U e. The Chen-Thyson model computes the transitional flow regime with an effective eddy viscosity applying the intermittency function
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Fig.3: Pressure Minimum during Upstroke Motion of NACA0012 Airfoil. Influence of Different Transition Onset Positions, Comparison with Experiments (NASA Ames), [12]. Calculation on a 157x59 C-grid. with G~tr= 213 [log(Rextr) - 4.7323]/3 as the transition constant. The effect of transition modelling with the Chen-Tyson transition model combined with the different transition onset criteria have been studied intensively for deep dynamic stall conditions. Figure 3 shows as a typical result the development of minimum upper surface pressures versus incidence. The computed results have been compared with experimental data, [12], for both fixed and free transition conditions respectively. As can be seen in Fig.3 a considerable improvement on the dynamic stall characteristics has been achieved with the application of transition modelling. F u r t h e r improvements of transition modelling are necessary if the time dependent separating flow is influenced by dynamic airfoil deformation. In [17] the effect of transition on a NACA 0012 airfoil with deforming leading edge is investigated, in [18] corresponding investigations of a dynamic nose-droop device have been discussed. It is to be expected that with the rather simple Chen-Thyson transition model this kind of unsteady flow control cannot be handled with success. More sophisticated models are necessary to improve the results.
685 0.4
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The tests of McDevitt, l / / / /\ [13],were performed at [ / [ / /\ ~" anangle~176176 ~ ~176 / f / / \ / ~ ~ and at a Reynolds [ / ~ ]\ / number (based on chord) [ / ~ / / / of Re=lX.0xX0 6, in a o2~ / ~ / ~ wind tunnel with " I / J~ / ~. contoured upper and [ / / ~ vlg.a lower walls to approximate the airfoil 0.4010 20.0 ' 40.0 ' 60.0 ' 80.0 streamlines in free air at T* a nominal Mach number of 0.775. The actual test Mach number was changed slowly at an approximate rate of dM/dt=0.001 up to a value of 0.76, at which point the airfoil experienced selfinduced periodic shock oscillations with a frequency of about 189 Hz (Buffet Onset). After the oscillatory flow was established, it persisted as the Mach number was lowered to approximately 0.73, at which point the flow became steady again (Buffet Decay). The calculations were performed using a 353x71 grid with 257 points on the airfoil. The spacing of the first grid point at the surface in the normal direction is Aql=0.000005 chords, and the grid boundaries are located at 10 chords in all directions. A nondimensional time step of At=0.0005 (based on chord length and freestream speed of sound)was used. The hysteresis effect is tested by starting the calculations at a Mach number of
686 Buffet Established at M=0.76
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from the very beginning, in [5] steady-state mode calculations are used up to the point of buffet onset. At M=0.75 the trailing edge separation point reaches the shock position and it develops a large-scale motion displaying limit cycle behaviour, Figure 4. This oscillation is maintained at M=0.76 with a frequency of 183 Hz very
~ 70
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close to the experimental result. Starting with the solution at M=0.76, the code is then run at decreasing Mach numbers. The flow remains unsteady at every Mach number down to M=0.72, Figure 5, at which time the oscillations damp out. It must be pointed out t h a t the exact points of initiation and finish of the hysteresis region show certain dependency with the grid and time step 80 used. For example, a coarser
~
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0.73 and progressively restarting the calculation at successively higher Mach number increments of 0.01. At Mach numbers up to 0.74 the flow shows a very small high-frequency oscillation which is not associated with shock motion. Interestingly this oscillation has not been reported in [5], probably because contrary to w h a t is done in the present study where the calculations are performed time-accurately
40 T*
687 grid and/or a longer time-step move them to 0.76 and 0.73 in accordance with the experimental results and the numerical results reported in [5]. Figure 6 shows the well established lift oscillation versus time at M=0.76 with both amplitude and frequency very close to the experimental data reported in [13]. The enlarged scale of Fig.6 shows t h a t the time dependency of the lift is not sinusoidal but has a repeating unsymmetrical behaviour also in close correspondence with experiments. Figure 7 shows selected pressure distributions over a cycle of shock oscillations on the upper surface of the airfoil. It can be seen t h a t the shock is moving over a quite large region between x/c=0.5 and x/c=0.82 with considerable increase in s t r e n g t h during its downstream movement. The most u p s t r e a m location does not exceed x/c=0.5 in contrast to experiments where the shock movement was observed even further upstream. In [5] it has been found t h a t wind tunnel wall interference effects may be responsible for this discrepancy. The above described results were obtained with the S-A turbulence model. Similar calculations performed with the B-L model resulted in buffet t h a t started at M=0.76 with a frequency of 189 Hz. This seems to agree even better with the experimental d a t a t h a n the S-A results, but caution has to be exercised, because the ragged appearance of the lift signal, Figure 8, indicates the presence of too much instability in the flow. The use of the B-L model with a wake constant of Cwk=l.0 recovers the smooth signal, but buffet onset is delayed until M=0.78 and it happens at a frequency of 218 Hz. In [5] no differences were found between the S-A and the B-L models except for the amplitude of oscillation which was about four times smaller with the B-L model. This is not seen here, with both models predicting amplitudes of the same magnitude and corresponding to t h a t of the S-A model in [5]. Finally, the SST k-o) model predicts buffet start already at M=0.74 in correspondence with its tendency to overpredict separation as already seen in the dynamic stall case. 6. C O N C L U S I O N S An evaluation of the ability of different turbulence and transition models to predict u n s t e a d y flow in dynamic stall and shock buffet cases has been conducted. None of the models considered is capable to fully accurately predict the dynamic stall cases. The buffet case resulted in good agreement with the experiments although it m u s t be reminded t h a t the configuration used (circular-arc airfoil) is somehow "easier" t h a n a normal supercritical airfoil. The Spalart-Allmaras model has consistently proved to be the best among those considered for these kinds of application. Even if the same numerical scheme for the Navier-Stokes equations and the same turbulence model is used the results of other authors are not necessarily reproduced, probably due to the fact t h a t the details of the implementation have an important influence on these very sensitive type of flows.
REFERENCES [ 1] Dindar, M.; Kaynak, U.; "Effect of Turbulence Modelling on Dynamic Stall of a NACA 0012 Airfoil"; AIAA Paper 92-0027; January 1992
688 [2] Srinivasan, G.R.; Ekaterinaris, J.A.; McCroskey, W.J.; "Dynamic Stall of an Oscillating Wing, Part 1: Evaluation of Turbulence Models"; AIAA Paper 93-3403, August 1993 [3] Ekaterinaris, J.A.; Menter, ER.; "Computation of Separated and Unsteady Flows with One and Two Equation Turbulence Models"; AIAA Paper 94-0190; January 1994 [4] Ekaterinaris, J.A.; Srinivasan, G.R.; McCroskey, W.J.; "Present Capabilities of Predicting Two-Dimensional Dynamic Stall"; AGARD CP-552; 1994 [5] Rumsey, C.L.; Sanetrik, M.D.; Biedron, R.T.; Melson, N.D.; Parlette, E.B.; "Efficiency and Accuracy of Time-Accurate Turbulent Navier-Stokes Computations"; Computers and Fluids, Vol. 25, pp. 217-236; 1996 [6] Bartels, R.E.; "Flow and Turbulence Modelling and Computation of Shock Buffet Onset for Conventional and Supercritical Airfoils"; NASA TP- 1998-206908; 1998 [7] Baldwin, B.S.; Lomax, H.; "Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows"; AIAA Paper 78-257; January 1978 [8] Spalart, ER.; Allmaras, S.R.; "A One-Equation Turbulence Model for Aerodynamic Flows"; AIAA Paper 92-0439; January 1992 [9] Wilcox, D.C.; "Reassessment of the Scale-Determining Equation for Advanced Turbulence Models"; AIAA Journal Vol. 26, N. 11, pp. 1299-1310; November 1988 [ 10] Menter, F.R.; "Improved Two-Equation k-co Turbulence Models for Aerodynamic Flows"; NASA TM-103975; October 1992 [ 11] Piziali, R.A.; "2D and 3D Oscillating Wing Aerodynamics for a Range of Angle of Attack Including Stall"; NASA TM 4632; September 1994 [ 12] Chandrasekhara,M.S.,Wilder, M.C.,Carr,L.W.,"Reynolds Number Influence on 2-D Compressible Dynamic Stall", AIAA 34th Aerospace Sciences Meeting & Exhibit, Jan 1518,1996,Reno,NV [13] McDevitt, J.B.; "Supercritical Flow About a Thick Circular-Arc Airfoil"; NASA TM78549; 1979 [ 14] Steger, J.L.; "Implicit Finite-Difference Simulation of Flow about Arbitrary TwoDimensional Geometries"; AIAA Journal Vol. 16, N. 7, pp. 679-686; July 1978 [ 15] Geissler, W.; "Instation~ires Navier-Stokes-Verfahren ftir beschleunigt bewegte Profile mit Abl6sung"; DLR-FB 92-03; 1992. [16] Geissler,W.,Carr,L.W.,Chandrasekhara,M.S.,Wilder,M.C.,Sobieezky,H., "Compressible Dynamic Stall Calculations Incorporating Transition Modelling for Variable Geometry Airfoils",AIAA 36th Aerospace Sciences Meeting & Exhibit, Jan 1215,1998,Reno,NV. [ 17] Geissler,W.,Chandrasekhara,M.S.,Platzer,M.F.,Carr,L.W.,"The Effect of Transition Modelling on the Prediction of Compressible Deep Dynamic Stall", The Seventh Asian Congress of Fluid Mechanics,Dec 8-12,1997,Chennai (Madras), India. [18] Geissler, W.,Sobieczky,H.,"Dynamic Stall Control by Variable Airfoil Camber", AGARD-CP-552,Oct 1994,pp 6.1-6.10.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
689
Scrutinizing Flow Field Pattern around Thick Cambered Trailing Edges: Experiments and Computations E. Coustols ~, G. Pailhas ~ & P. Sauvage b
a ONERA/DMAE, Department of Modelling in Aerodynamics and Energetics, 2 avenue Edouard Belin, 31 055 Toulouse cedex, France. b presently, AEROSPATIALE A/SW/SF, 316 Route de Bayonne, 31 060 Toulouse cedex, France.
1. I N T R O D U C T I O N Research has been conducted rather recently at ONERA, in the field of supercritical airfoils equipped with thick cambered trailing edges. For transonic applications, the aerodynamic performance of such airfoils, previously referred to as D.T.E. "Divergent Trailing Edge" [1], could be put forward mainly from computational tools. Indeed, when considering relatively high values of the lift; coefficient, total drag reduction could be evidenced though base drag increase, because of increased airfoil loading and subsequent wave drag decrease. As the physical aspects induced by such non-conventional trailing edge geometries are identical, whatever the flow regime, a rather detailed experimental study has undergone at the hydrodynamic tunnel of ONERA. The aim is to scrutinize the flow field pattern downstream of an airfoil equipped with three non-zero-thickness trailing edges: (i) a reference trailing edge (RTE) having a base thickness of 0.5% airfoil chord length (c); (ii) two increased cambered trailing edges with the same trailing edge angle, but different incremental base thickness, either 0.2% c (CTE 1) or 0.5% c (CTE 2). Apart from static pressure measurements, a two-dimensional Laser Doppler Velocimetry (L.D.V.) system, operating in the backward-scattering mode, has provided very detailed velocity measurements in the close vicinity of each trailing edge. Experiments have been conducted at turbulent conditions, for a free stream Reynolds number (based upon the airfoil chord length) less than 1 million. The hypothesised flow field pattern has been carefully characterised: a very complete database has been generated for several trailing edge shapes [2,3]. This database is used for validating the computational tools that will be considered for future optimisation of specific wing design with non-zero thickness trailing edges. Consequently, a numerical approach has been conducted, aiming at testing the
690 behaviour of different turbulence models under such aerodynamic and geometric conditions. The Reynolds-Averaged Navier-Stokes solver, developed by ONERA [4], has been applied to the airfoil with the tested trailing edge geometries. Grid generation has led to a "C-H" type mesh topology, with specific refinement in the vicinity of the cambered trailing edges. Several one- and two-transport equation models, rather recently implemented in this code, have been considered. Detailed comparisons between experiments and computations are discussed with a special emphasis being placed on pressure distributions, boundary layer and near wake surveys in the trailing edge vicinity. 2. E X P E R I M E N T A L STUDY 2.1
Water tunnel
Experiments have been carried out in the circulatory-type water tunnel of ONERA]DMAE [2]. The horizontal test section is 0.5 m large, 0.3 m high and 3 m long; the lateral walls allow perfect optical access for L.D.V. measurements. The velocity variations at the test section entrance are lower than 1% of the mean velocity. The main parameters governing the flow are constantly controlled and adjusted by a microcomputer via an automaton. 2.2
Model OAT15A
~RTE --
CTE I
....
CTE2
I
i
I
I
I
0.96
0.97
0.98
0.99
1.0
x/c
Figure 1. Trailing Edge Definitions.
The reference model for the present investigation is an OAT15A airfoil having a maximum thickness ratio of 12.3% and a chord length (c or C) of 400 mm; the upper surface contour is constrained to be the same as the original OAT15A whereas the airfoil geometry is modified on the lower surface only. The airfoil is manufactured in such a way that the rear part (last 19% and 15% chord length of the upper and lower sides, respectively) is removable, allowing to mount rather easily the different shapes of trailing edges to be tested.
The three following two-dimensional trailing edges have been considered (Fig. 1): - the reference trailing edge("RTE"), having a base thickness of 0.5% c; - t w o increased cambered trailing edges with the same trailing edge angle, but different incremental base thickness: 0.2% c ("CTE 1~) and 0.5% c ("CTE 2").
691 The geometry of these trailing edges results from a numerical optimisation performed at ONERA/DAAP using a compressible Euler method coupled with an integral boundary layer method (referred to as "ISES") [5]. The model, with the reference trailing edge "RTE", is fitted with 64 static pressure taps located on both suction and pressure sides, with one on the base. The "CTE l"and "CTE 2" shapes comprise one and two supplementary pressure taps in their base, respectively. 2.3 T e s t C o n d i t i o n s
The experiments are conducted for a tunnel water speed of 2 ms a (+1%) and a water temperature of 293K (+_2K). This leads to a free stream Reynolds number based on the model chord length of about 800,000. Transition is tripped on the lower and upper sides of the model by a 0.2 mm diameter trip wire, stuck parallel to the leading edge, at about 3% of chord length downstream of the leading edge. 2.4 M e a s u r e m e n t s
A two-component L.D.V. is used in a backward scattering mode; it allows to obtain the mean and fluctuating quantities of the velocity in a plane aligned with the upstream flow direction. Step by step motors under computer control do positioning of the measuring point in the X and Y directions. Data acquisition is performed over about 2000 samples. Time needed for such an acquisition is used for calculating the mean value, the standard deviation and the correlation of the two velocity components for the point measured just before. According to the size of the measurement volume, no data could be obtained at a distance less than 0.2 mm to any model side or base. Wake and boundary layer surveys have been performed in a single plane located at one third of the span airfoil. Details about the measurement technique can be found in [2,3]. 3. NAVIER-STOKES SOLVER AND T U R B U L E N C E M O D E L L I N G 3.1 G o v e r n i n g E q u a t i o n s .
The "CANARI" code developed by ONERA, solves the three-dimensional compressible Reynolds-Averaged Navier-Stokes (RANS) system of equations [4]. The perfect gas law is used to compute the static pressure. The diffusive fluxes of the RANS equations system are evaluated following the Stokes hypothesis and the Boussinesq assumption. The molecular viscosity is given by the Sutherland law; the eddy viscosity being derived from the considered turbulence model. The viscous stress tensor and the heat flux vector are then related to the velocity gradient and temperature gradient, the standard and turbulent Prandtl numbers being assumed constant and equal to 0.72 and 0.9, respectively.
692
3.2 T u r b u l e n c e Modelling. Momentum and turbulence equations are decoupled; such a procedure should lead to the correct steady state. The code had originally an algebraic mixing length model [6] and a two-transport equation model [7]. Improvements were brought by Sauvage [3] and Houdeville [8] by extending the code with several one- and two-transport equation models, knowing that some of them have provided correct flow predictions for rather thick trailing edges [9,10,11]. The following turbulence models have been selected for the present study: - the Baldwin-Barth [BB] [12] and Spalart-Allmaras [SA] [13] models which require the solution of a transport equation for a quantity, derived either from the turbulent Reynolds number R t ([BB]) or from the eddy viscosity ([SA]). - the two-transport equation k-s type models: Jones-Launder [JL] [7], Chien [Ch] [14] and Nagano-Tagawa [NT] [15]; these models differ essentially from i) wall proximity treatment with damping functions expressed as a function of either y+ or R~; ii) isotropic or non-isotropic dissipation rate e. - the two-transport equation model k-l from Smith [Sm] [16], derived from a k-k/ model, uses an equation for the turbulent length scale, l. Near wall treatment is eliminated, resulting in an equation which is easier to handle numerically. 3.3 N u m e r i c a l Method. The solver is based upon a finite volume method; the discretization is cellcentered (grid points and flow variables are no longer collocated). In order to keep the modular feature of the solver, the time integration procedure requires a uncoupling of the RANS system and of the two-transport equation system. The former is solved with a frozen eddy viscosity; then, the latter is solved with fixed mean flow quantities. The algorithm involves a second-order accurate four stage Runge-Kutta scheme with an implicit residual smoothing; the local time stepping has been specified in order to satisfy a CFL stability criterion [4]. Each fractional step can involve the succession of three stages: explicit calculation, artificial numerical dissipation correction and implicit resolution. Boundary and matching conditions are treated by applying characteristics relations. 3.4 Grid g e n e r a t i o n
Figure 2. Mesh Refinement (Zoom)
This code admits multi-domain structured meshes with possible noncoincidence of the nodes or block overlapping. For our specific purpose, grid generation leads to a C-H mesh topology, satisfying the following requirements: outer boundaries as far as 10 chord lengths away from the airfoil (using the far-field vortex condition [17]) and a value of y § close to 0.3 in the trailing edge vicinity. Mesh convergence has been also verified [3].
693
Appropriate adaptation of the mesh refinement is necessary in each trailing edge vicinity. An example of mesh geometry is given in Figure 2, for the OAT15ACTE 1 airfoil with 216"100 ("C") and 130"247 ("H') mesh points, in the longitudinal * normal to the wall directions including 49 points in the base. The presence of any wind tunnel wall was not simulated ("infinite" atmosphere).
4. RESULTS AND D I S C U S S I O N 4.1 P r e s s u r e d i s t r i b u t i o n a l o n g the airfoil. The measured longitudinal pressure distributions, obtained for 0.9the RTE, CTE 1 and CTE 2 trailing O.6i edges, are plotted on Figure 3, for : almost the same value of the lift; 0.38 es ~ a a a 9 I 8 8..= 9 9 9 coefficient (0.47). The Cp value has ~o.~ o been corrected in order to take into m * , o O 0 { -0.~ 9 RTEa= 1.25- Cl= 0.464 account the wind tunnel "blockage" -0.6" a C T E 1 - a = O. 75 - C l = 0 . 4 6 8 effect. As expected, an important 9 CTE2 - a-0.00-C1-0.465 -0.9 pressure gradient modification is -X2 ~.~ observed in the last 10% of chord 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x/C length of both upper and lower Figure 3. Measured Pressure Distributions sides of the airfoil. 1.2
-. ~g
~p
. . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
I
. . . .
I
' " '
'
!
. . . .
I
'
'
'
The combined effects of cambered lower side and increased divergence angle of the trailing edge result in an increase of rear airfoil loading, leading to smaller values of the angle of attack, for the same global lift; coefficient. Indeed, Aa is equal to -0.5" and-1.25" for CTE 1 and CTE 2, respectively (cf. Fig. 3). o.s:
o.
a~
11
.... . . . . . . . .
9
._
9
m
.
~ao9 Fa'p.
.o.42
- C! = 0.47
a=O.O0
-- [SA]
or~O.O0 - C l = 0 . 4 7 9.. [ B B I w:O.O0 - Cl = 0.46 -- (ISES) a=0.30CI = 0.47 -i.2
, . ,
0.0
*l
. . . .
0.1
. . . .
0.2
9
0.8
I
,
u . . . .
i
G3
!
0.~
.
. . . .
0.4
.
.
.
u . . . .
I
115
!
0.9
x/C
.
i ~s)~o.3o.ct-o.47
-O.&] . . . .
0.6
.
.
.
u . . . .
0.7
.
u . . . .
u . . . .
0.8
! 119J
.
0.9
.
.
.
I
1.0
-I.2
.... 110
~
6
I .... 111
I .... 112
I ....
0.3
t ....
0.4
I ....
O.J
.
.
-- [~_1 i ....
0.6
.
.
.
a=O.O0I ....
0.7
CI = 0.47
I ....
0.$
~*'~'q 0.9
1.0
u
I.O
G8
G~U
O.9
x/C
0.95
Figure 4. Computed Pressure Distributions - OAT15A-CTE 2 Airfoil.
;.6
694 Computations with the above-mentioned turbulence models (cf. w are performed using the same grid. The value of the angle of attack used for such computations corresponds to the geometrical value corrected from wall effects, allowing to get approximately the right value of the lift coefficient [2,3]. Whatever the turbulence model is, the agreement between experiments and computations is rather good, except maybe in the leading edge region where the correction due to the "blockage" might not be sufficient. An illustration is given in Figure 4, for the thickest base OAT15A-CTE 2. The pressure gradient variations in the rear part of the airfoil are very well reproduced by all models, including the decrease of the adverse pressure gradient on the upper side due to rear airfoil loading. On the last 5% of chord length of the lower side, the flow smoothly decelerates and then undergoes a very great acceleration due to camber. [Ch] and [NT] models are the only ones to slightly overpredict both the intensity of the adverse pressure gradient in that rear pressure side and the velocity peak in the leading edge vicinity. Modification of the overall circulation may explain it. It should be pointed out that the method used to define these trailing edge shapes [5] does not provide the correct pressure gradient behaviour in the rear part of the airfoil, by underestimating the increased rear loading. Computations were performed at a slightly higher value of the angle of attack. 4.2 P r e s s u r e d i s t r i b u t i o n in t h e b a s e . From experiments, it looks as if the lower and upper surfaces appear to be uncoupled, since the pressure level on the upper side differs tremendously from that on the lower side. Therefore, the pressure coefficient along the base, CPb, is very close to the value recorded from the last pressure taps of the upper surface downstream of x/C=99 % [3]. When looking at the computed CPb value, some differences exist depending upon the turbulence model (Fig. 5); anyhow, these computations allow to confirm the rather constant pressure level in the ydirection, hypothesised from 2 or 3 pressure taps, yet. Changes at the trailing edge between CTE1 and CTE2 are directly related to the variation in the angle of attack, necessary for having a constant lift. 0.2
-
9 0.1
c'-...-._-_-*-_---_-_"'_7.Z 9
o.o
Experiment
--
[SA]
9"
[BB]
--.
ISra]
-..
[hrU
--"
[Ch]
---
[JLI
o
.4).I-
CTE 1 '''
-0.2
.0.8
I ' ' '
.0.6
I ' ' '
-0.4 y/C
t ' ' '
-0.2
I *lff2
0.0
o.1
o.oI --"~ .
"
.
.
.
.
...........
.
. .
~, .....
. .
.
.
.
~.1~ -0.2
I
-I.0
.
.,q q . - ~ - t - " . - ' "
.
=~.=:.-_2" .
.
.
.
.
.
.........
.
"~'--.'.
w,
9
CTE 2 ,
,
,
i
-0.8
,
,
,
i
-0.6
,
,
ylC
,
i
-0.4
,
,
,
i
.0.2
,
,
,
~
,/if2
0.0
Figure 5. Pressure distribution along the base of the OAT15A-CTE 1 and CTE 2.
695
Integration along the base provides an average computed Cpb value. Generally speaking the models overestimate the pressure level along the base. The [Ch], [JL] and [NT] models predict a base pressure in better agreement with the experimental values than the [BB], [SA] and [Sin] ones. However, if one thinks to apply such a code for future optimisation, it is very important to guess correctly the base drag increase between the CTE 1 (or CTE 2) and the RTE, rather than the absolute value. [SA] and [Sm] provide the correct drag increase for the CTE 1 trailing edge; however, for the CTE 2 trailing edge, the [Ch] model over-estimates the base pressure increase while the other models behave rather well. When comparing to the ISES estimates, improvements brought by using the "CANARI" code are noticeable.
CTE 1 CTE 2
Exp. 0.050 0.146
ISES 0.041 0.094
ACp~ = (-Cp b)rEs- (-Cp ~) cry. [BB] [SA] [Sm] [Ch] 0.080 0.050 0.040 0.090 0.140 0.124 0.117 0.160
[JL] 0.058 (*)
[NTI 0.073 0.141
(*) No convergence (see following page).
4.3 N e a r - w a k e s u r v e y s
These experiments were mainly aimed at scrutinizing the flow in the vicinity of the trailing edge. Downstream of the CTE 1 & 2 bases, a recirculation area corresponding to negative longitudinal velocities is captured. When plotting the streamline pattern, two contra-rotating vortices are recorded in a confined area, the dimensions of which are tightly related to the base thickness (e): indeed, it is about (2e * e) in the longitudinal * transverse directions. The computed topological structure is strongly dependent upon the mesh refinement downstream of the base. Illustration is provided on Figure 6 for computations performed with the [Sm] model and two grids: M1 (and M2) "C'type: 216"70 (216"100) "H"-type: 110"177 (130"247) with 39 and 49 grid points in the base, respectively. Indeed, the strength of the lower vortex can be more or less foreseen; such a refinement affects essentially the transverse component of the mean velocity and not the longitudinal one [3],
Figure 6. Effect of grid refinement on topological structure (OAT15A-CTE 1).
696
Figure 7. Near Wake surveys downstream of the OAT15A-CTE2 airfoil (zoom)
697
Computations of the viscous flow downstream of the OAT15A-CTE 2 airfoil are performed with different turbulence models, using the same refined mesh. In some cases, convergence might not be obtained because of strong (numerical) oscillations of the flow in the near wake: this is the case with the [JL] model for the thickest cambered trailing edges. The other turbulence models allow to capture rather precisely these two vortex-type structures in the near wake (Fig. 7). However, the computed vortices are located too close to the base, which is consistent with a too upstream location of the re-attachment point. Furthermore, the less extent of computed separated flow is in agreement with weaker pressure computed base values, compared to experimental values. It could be pointed out that the camber of the lower side does not induce a "diving" motion of the flow. The suction effect due to the low base pressure counteracts the guideline of the flow by the strong variation of the geometry. The level of the computed velocity modulus in the centre of the separation area is not in perfect agreement with the estimated one from wake surveys. Generally speaking, the turbulence models under-estimate the maximum defect of the longitudinal velocity component in the wake, though its spreading along the longitudinal direction seems correct [3]; illustration is given on Figure 8 for few two-transport equation models. The plot of the shear stress profile in the near wake reveals that no turbulence model is able to faithfully reproduce the recorded asymmetry between the upper and the lower parts of the wake (Fig. 8); the [BB] and [SA] models generate similar results to that given by the [Sin] one.
1.0= as:0.6:~ o.4:-
a....F
, x eriment
P B ~
I
~ ~] ---tSml !~
0.2-_
- -
I
'
-0.05
I
0.0 y/C
'
I
0.05
Experiment --- [Sml
0.02:
9
-.- [NT1
".~ 0.0
[Chl
- - [Ch]
-0.02-:
x/c=1.05
0.0-: -0.2-0.1
0.05
":'i '
I
0.1
-o.o.~
. . . .
-20
I
x/c= l. 05 '
I
-10 0 <.u'v'>/U2(,11~
'
'
'
'
I
2O
Figure 8. U- and u'v' profiles (OAT15A-CTE 2 airfoil). 5. DISCUSSIONS Experiments carried out with an airfoil equipped with couple of increased cambered trailing edges have allowed to obtain a very precise description of the turbulent flow. The pressure gradient has been tremendously modified, especially in the last 10% of chord length on both sides of the airfoil, leading to an increase of the rear airfoil loading; the thicker the base, the larger the rear loading is. Thus, to reach a given lift coefficient, an airfoil with a cambered trailing edge has
698 to be set at a lower incidence than that of the same airfoil with a "standard" base. Such a mechanism would allow to lessen wave drag for transonic conditions. A rather complete database, issued from boundary layer and wake surveys performed in each trailing edge vicinity, has then been generated. A RANS solver, developed by ONERA, including several one- and two-transport equation turbulence models, has been applied to such aerodynamic configurations. All the tested models predict rather correctly the physical observations and topological modifications induced by such cambered trailing edges. The [BB] and k-e type models require a specific treatment in the separated regions because of correctortype functions of either y§ or R~, while the [SA] and [Sin] models require only an appropriate estimate of the wall distance. Nevertheless, the Spalart-Allmaras and Smith models are producing results in better agreement with experiments than the other selected turbulence models; these former notably guess the consistent total drag increases, essentially due to base drag increase, even though not discussed in the present paper. The [SA] model is less sensitive to grid refinement than the [Sm] model, yet. Applications of these models at transonic conditions will remain the following challenge. These two models have shown a great numerical robustness, which may be worth testing in RANS solvers devoted to compute possible complex industrial flow configurations.
REFERENCES [1] P.A. Henne P.A. and R.D. Greg III, J. Aircraft, 28 (5): 333-345 (1991). [2] G. Pailhas, P. Sauvage, Y. Touvet and E. Coustols, In Proc. 9th Int. Symp. on App. of Laser Tech. to Fluid Mech., Lisbonne, Vol. 1:19-3-1 19-3-8 (1998). [3] P. Sauvage, Ph. D. Thesis SUPAERO Toulouse (1998). [4] A.M. Vuillot, V. Couailler and N. Lamis, AIAA Paper 93-2576 (1993). [5] M. Drela and M.B. Giles, A A Paper 87-0424 (1987). [6] R. Michel, C. Quemard and R. Durand, ONERA Technical Note 154 (1969). [7] W.P. Jones and B.E. Launder, Int. J. of Heat and Mass Transfer, 15(2) (1972). [8] R. Houdeville, ONERA Technical Report 72/5025.61 (1997). [9] S.K. Stanaway and W.J. McCroskey, AIAA Paper 92-0024 (1992). [10] J.A. Ekatenaris and F.R. Menter, AIAA Paper 94-0190 (1994). [11] E. Monsen and R. Rudnik, A A Paper 95-0089 (1995). [12] B.S. Baldwin and T.J. Barth, A A Paper 91-0610 (1991). [13] P.R. Spalart and S.R. Allmaras, La Recherche Aerospatiale, 1:5-21 (1994). [14] K.Y. Chien, AIAA Journal, 20(1): 33-38 (1982). [15] Y. Nagano and M. Tagawa, Journal of Fluids Engineering, 112:33-39 (1990). [16] B.R. Smith, AIAA Paper 94-2386 (1994). [17] J.L. Thomas and M.D. Salas, AIAA Journal, 24(7): 1074-1080 (1986).
Acknowledgements. The authors gratefully acknowledge the support that Airbus Industrie and the Service Technique des Programmes A~ronautiques have provided for the research reported in this paper.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
699
Turbulent Structure in the Three-Dimensional Boundary Layer on a Swept Wing Motoyuki ITOH a and Minoru KOBAYASHI b a Department of Mechanical Engineering, Nagoya Institute of Technology, Nagoya, Japan b Nippon Sharyo, LTD, 1-1, Sanbonmatsu-cho, Atsuta-ku, Nagoya, Japan
The effects of mean flow three-dimensionality on the turbulent structure of the boundary layer on a swept wing are investigated experimentally. The configuration of the test model used to simulate the infinite swept wing condition is close to that used by Berg et al. [1]. Mean flow measurements show that the location of the maximum cross-flow velocity is within the log-law region for the streamwise velocity profiles. In the near-wall region, the streamwise turbulence intensity, I~u ---~/U r , decreases with increasing three-dimensionality. A four quadrant analysis is applied to the fluctuating velocity components in the local mean flow and wall-normal directions. At downstream stations, the contributions to the turbulent shear stress from quadrant 4 (sweeps) are predominant in the region near the wall. It is revealed that strong ejections are suppressed by the effects of increased three-dimensionality.
1. INTRODUCTION Many engineering boundary layer flows, such as those over swept wings, within turbomachinery and over hulls of ships, are turbulent and three-dimensional. There have been many experimental and theoretical works on the three-dimensional turbulent boundary layers (see reviews by Bradshaw [2]; Johnston and Flack [3]). In those investigations, several basic differences were found between two- and three-dimensional turbulent boundary layers. One of them is that the direction of the shear-stress does not aligned with the direction of the velocity-gradient vector. Another important difference is that the structure parameter A1 is smaller for three-dimensional boundary layers than for two-dimensional ones. However, causal structural changes have not been fully understood. Eaton [4] reviewed several experiments and simulations examining the near-wall structure of three-dimensional turbulent boundary layers and concluded that it is appropriate to interpret the structure of the three-dimensional turbulent boundary layer as a distorted version of the two-dimensional one, i.e. low- and high speed streaks and quasistreamwise vortices are dominating the flow near the wall (Robinson [5]). Chiang and Eaton [6] and Flack [7] studied the near-wall structures of three-dimensional turbulent boundary layers using flow visualization technique. Their conclusions, however, contradict each other in that asymmetry of the near-wall structure found by Chiang and Eaton [6] was not observed in Flack's experiments [7].
700
In the present work, the three-dimensional turbulent boundary layer was produced in a model similar to that used by Berg et al. [1]. Measurements of the mean velocity profiles and the Reynolds stresses were obtained by the use of hot-wire anemometry. The mean velocity distributions obtained were discussed in view of the theoretical prediction by Degani et al. [8]. To study the effects of mean flow three-dimensionality on the structure of turbulence, a four-quadrant analysis was applied to the u * - and v-components, where u* and v represent the fluctuating velocity components in the local mean flow and wall-normal directions, respectively.
2. E X P E R I M E N T A L APPARATUS AND P R O C E D U R E Experiments were performed in a low-speed wind tunnel at Nagoya Institute of Technology. The model (Fig. 1) used to simulate the boundary layer on an infinite swept wing is similar to that used by Berg et al. at NAL in Netherlands [1]. The model was installed in the working section o (800 x 800 x 3000 ram) of the wind tunnel. The leading edge of the flat test plate is swept at 35 with respect to the direction of oncoming free stream. A 1.6 mm trip wire was glued on the test plate at 50 mm downstream of the leading edge. In order to obtain the required pressure distribution on the test plate (i.e. the pressure is constant along the generator on the test plate) as accurately as possible, ingenious contrivances used by Berg et al. [1] were adopted here also. The only novel design in the present model was the flat roof and the side walls extending upstream of the flexible roof and the curved side walls. With these extended walls, the mean flow two-dimensionality was more closely approximated at the upstream measuring stations (x = -108 mm) than for the configuration without those walls. The infinite-swept condition was obtained approximately by adjusting the pressure inducing roof, the curved side walls and a blockage attached to the downstream end of the model (see Fig. 1). The upstream ends (x = 0) of the flexible roof and the curved side walls are situated at 400 mm downstream of the leading edge of the flat test plate. Velocity measurements were carried out in the boundary layer on the flat test plate at 5 stations (x = -108, 108, 252, 396, 540 mm) along a line parallel to the tunnel axis. Two types of single-wire (hot-wire) probes were used. One had a wire normal to that rotation axis and was ---'g -""-7 used for the mean flow measurements as well as Reynolds stress u and w ,where u and w represent the fluctuating velocity components in the local free-stream and cross-flow directions, respectively. The other had a wire inclined at an angle of about 45 ~ to its rotation axis and was used for measurements of Reynolds stresses v , uv and ~ . The hot-wire was 3.1 /z m in diameter made of tungsten and copper plated at both ends, having an active length l = 0.5 mm (see l + in Table 1). To obtain the two instantaneous velocity components in the local mean flow and wall-normal directions, an X-wire probe was used. The two wires of the X-wire probe was separated by 0.3 mm from each other. Wall shear stress "C w was evaluated from the mean velocity gradient in the viscous sublayer. To this end, the measurements of mean velocity distribution were made down to the wall-normal distance y ~" 0.05 mm. Since heat conduction to the wall caused erroneous hot-wire readings near the wall itself, a similar procedure to that used in our previous work [9] was adopted to correct the near-wall data. The free stream velocity flowing into the model was kept constant (15 m/s) throughout the present work. The uncertainty for the velocity data is estimated to be -----0.01 for the nondimensional mean velocity components Us/U~ and Un/Ue, +- 0.03 for /u--Z/U v , ~,---z w /U v , + -- 0.06 for v--r/U r 2 and ~*'V/U r
2
. The uncertainty in "C w is estimated at + 8 ,%o of its absolute value.
701 Curved Side Walls
Measurement Station Test Plate
--....
k -----v~/>-~L, ~, J,T-
,/
I/./-/_..Y..t..7_...
; 'I "- o I
///-/;7r
,, /I/-/-..7-./..7
;I
~" /-
o:/o? / /
c Blockagc
Static Pressure Hole L.
400
1..
aoo
/ I
]
Guide Wall
"'~ "I
~Fl.at Roof/Flexible Roof
Figure 1. Sketch of the model employed for boundary layer measurements
Table 1 External-stream and boundary layer parameter x[mm]
Ue[m/s]
J U,[m/s]
~,.-
ReO
-!08
15.4
0.739
I.~
4 . 6 0 x i 0 -s
- 1 . 9 7 x 1 0 -s
1240
108
14.3
0.622
3.2
3 . 6 2 x 1 0 -s
3 . 5 5 x 1 0 -s
1620
252
13.5
0.533
6.4
3 . 1 0 x l O -s
7 . 1 1 x l O -s
2330
396
12.6
0.432
14.2
2 . 3 6 x 1 0 -s
1 . 2 2 x 1 0 -2
3520
28.6
2 . 0 2 x 1 0 -s
1 . 1 6 x l O -=
5830
540
11.9 .
x[mm]
.
.
0.379
.
.
.
.
.
.
.
.
.
.
.
6' [ram]
~m[mm]
.
.
.
.
0 I mm
o.*
-!08
12.6
1.69
1.24
1.32
24.0
108
16.4
2.39
1.72
1.39
20.5
252
21.6
3.78
2.63
1.44
17.5
396
31.8
6.65
4.29
1.55
14.1
540
47.1
7.33
1.58
12.6
p+
--
11.6
v
~P
U~: Q,
pU~
~x
1)
702
3. R E S U L T S AND DISCUSSION 3.1. Surface pressure distributions The local coefficient of pressure Cp was defined as Cp = 2(P - P0)/ 0 U 2 (1) where P is the local wall static pressure, P0 is P at the reference station (x = 0) and U is the velocity of oncoming uniform free stream. The static pressure distributions along the generators are shown in Fig. 2. The approximation to the infinite swept wing condition is comparable to that of Berg et al. [1]. Figure 3 shows the static pressure distribution parallel to the tunnel axis. For comparison,the corresponding pressure distributions in the experiments of Berg et al. (NLR) and Bradshaw and Pontikos [10] are shown in the same figure. The velocity measurement stations are indicated by arrows. Note that the favorable pressure gradient is obtained at the most upstream velocity measurement station (x = -108 mm) and the values of nondimensional (adverse) pressure gradient P + at two downstream stations (x = 396 mm and x = 540 mm) are nearly equal to each other (see Tble 1). The latter situation seems to be useful, at least for the near-wall region, to discern the effects of three-dimensionality from those of adverse pressure gradient.
r
I
9
I
I
'
'
I
'
I
'
I
0 x=576mm [3 x=432mm
0.4
x=288mm
A x=144mm x=-72mm
~------<~-----~,o._..._~.o____._~ ~______cr-------~
a, 0.2 r,.)
0.4
A
i
-0.3
,
I
-0.2
,
I
t
-0.1
l
0
,
I
0.1
I
I
I
a, 0.2 L)
A
-~ ~
I
I
,
I
0.2
-z'/1 Figure 2. Surface-pressure distributions parallel to leading edge of test plate
0 -400
I
---NLR
, , _~ ~ e ~ / / I
0
-
- - - Oradshaw , i
I
400
l
800
x[mm] Figure 3. Surface-pressure distributions parallel to tunnel axis
3.2. Mean flow field Figure 4 shows the mean velocity distributions, where Us and Un are the mean velocity components in the free-stream and cross-flow directions, respectively. Both velocity components are normalized by the free-stream velocity at the outer edge of the boundary layer Ue. It is noticed that the peaks of the Un-profiles are very close to the wall. Due to the adverse pressure gradient, as seen in Fig. 3, the shape of the Us profile becomes more and more decelerated with going downstream. Flow angles measured relative to the local free-stream direction, ~ - q5 e, are shown in Fig. 5. The maximum skew angle is around 30 ~ , which is between those of Berg et al.
703
[1] and Bradshaw and Pontikos [10]. The turning angle of the free-stream, 4) e, was less than 10 ~ (referred to the tunnel axis) at the most downstream velocity measurement station (x = 540 mm). Values of the traditional integral parameters are shown in Table 1, together with other boundary layer parameters, where Cf is the skin friction coefficient (Cf = 2 z" w/ 0 Ue 2 ). The boundary layer thickness (~ 99, the displacement thickness ~ * and the momentum thickness 0 are
1
0.8
301~
u
. . . . o x---108mm [] x= 108mm O x= 252mm A x= 396mm V x= 540ram
[ '~ I ~7
I ~
'
~.
2
%., 06 o x=-lO8mm
[] x= 108mm
04 R b '
O x= 252mm
" ~
"0. --
A x= 396mm V x= 540ram
1
o
o
0.5 Y / ~ 99
Y/(~ 99 Figure 4. Mean velocity distributions
1
Figure 5. Directions of velocity in boundary layer
89 ,
"~" 40 13
v
9
v
vl,,,
I
'
'
'
' ' ' " 1
'
'
'
' ' ' " 1
u,~ T cos(r w- 4' O"
O Un/U I sin( ~ w - ~ e~~
f -
30 .S
o.2L
o o zs v
VV
oO +
+
U =5.621o~ v +5.0 a ~ 20
x=lO8mm x=252mm x=396mm x=540mm
10
~7 o
~o.1
% AA
o
i
0
'~
o
V o
O
~DFIFI %% o o vzx glD 1-113 o
Us/Ue Figure 6. Polar plots of velocity distributions
010 0
101 10 2 y+=U 1- y /
10 3
Figure 7. Log-law plot of mean velocity components
704
defined based on the Us-profiles. The shape factor H and the Reynolds number R o are defined a s H = (3 */ 0 a n d R 0 = U e 0 / V ,respectively. Figure 6 shows the polar plots of the mean velocity distributions. The straight lines radiating from the origin represent the directions of wall shear stress at respective stations. The collateral flow in the near-wall region, where the data are on the straight lines, was found to extend only up toy = 5 "~ 10. The log-law plots for the mean velocity components Us and Un are shown in Fig. 7, where ( 0 w - 4) e) represents the angle between the directions of the wall shear stress and the local free-stream. It is evident in the figure that, with going downstream, the extent of collateral flow in the near-wall region increases and the location of the maximum Un shifts outward from the wall. A log-law region is apparent for Us-component but not for Un-component, probably because the Reynolds number is not large enough. The location of the maximum Un is within the log-law region for Us-component. All these results are in accordance with the theoretical prediction by Degani et al. [8]. Furthermore, it is noticeable that the log-law region for the Us-profiles can be described by the expression established for the flat plate boundary layer. +
3.3. Turbulence field Figure 8 shows the turbulence intensities /--U--~ and /-w-~. In the outer region (y+ > 200), both / u ~ / U r and / ~ / U r increase with going downstream. This may be ascribable to the effect of adverse pressure gradient (Nagano et al. [11]). It should be remarked that, in the near-wall region, / ~ - z / U r decreases appreciably with going downstream from x = 396 mm to x = 540 mm. This seems to be due to the effects of increased three-dimensionality. On the other hand, / ~ / U r increases even in the near-wall region with going downstream from x = 396 mm to x = 540 ram, as seen in Fig. 8(b). These results are consistent with the numerical prediction (Moin et al. [12]) studying the effects of three-dimensionality on the turbulence statistics. The wall-normal turbulence intensity /v-Z is shown in Fig. 9. As we used an X-wire probe to obtain / v -values, the data in the near-wall region could not be obtained. The increase of /U r with going downstream seems to be due to the dominant effect of the adverse pressure gradient [11]. Figure 10 shows the turbulent shear stress - ~ V , where u* is the turbulent velocity component
,
w
. . . . .
'I
'
'
'
' ' ' " 1
'
'
41
9 ' ' ' ' ' 1
i
o x=-108mrn t) x= 1 0 8 m m
I- o / o [._o / a 2 ~v
o x= 2 5 2 m m -~ x - 3 9 6 m m
i.
~0~
*
*
,|ll,I
i
10~
i
i
i|**|l
I
!
i
it,,,l
102 y+=U ~ y/~'
(a) streamwise component
10~
'I'0~
........ ,
. . , ..... ,
........ ,
x---10Smm x= 108ram x= 2 5 2 m m x - 396mm x= 5 4 0 m m
lOt
10 y+=U r Y/~'
(b) cross-flow component
Figure 8. Distributions of turbulence intensities
103
705
in the local mean flow direction. It should be noted that, in the region y/ 6 99 < 0.4, the value of --6"*"~/U r z decreases with going downstream from x = 396 mm to x = 540 mm. This seems to be caused by the effect of increased three-dimensionality. Although all the components of Reynolds shear stress were obtained by the technique of a rotated hot-wire, they will not be disussed here any more because the main concern in the present work is to clarify the causes for decreasing structure parameter, A1, with increasing three-dimensionality. The Reynolds stresses data obtained in the present work are available in machine-readable form for use by interested workers. The distributions of the turbulent kinetic energy, q z/2, are shown in Fig. 11. The values of q z/U r near the wall decrease with going downstream from x = 396 mm to x = 540 mm, which also seems to be due to the effect of increased three-dimensionality. Figure 12 shows the distributions of structure parameter A1 (ratio of shear-stress magnitude in the plane parallel to the wall to twice the turbulent kinetic energy). As has been well known, the value of A1 decreases with increasing three-dimensionality. In Fig. 12, A1 decreases from about 0.15 (at x = -108 mm) to 0.11 (at x = 540 mm) in the central region of the boundary layer. Figures 13 (a) and (b) show the turbulent transport of turbulence energy component u*--Tr and the turbulent shear stress u*v. It is seen in Fig. 13(a) that with going downstream the values of vu* 2/U r 3 decrease near the wall and become negative at the stations x = 396 mm and X = 540 mm. The negativevalues of vu* z/U r 3 indicate that the turbulence energy u* z is transported
2
....
o x=-lO8mm o x= 1 0 8 m m o x= 2 5 2 m m x- 396mm v x-- 5 4 0 r a m
r
,
,
,
i
,
,
,
o o ,o a
~----~.
,
x---108mm x= 1 0 8 r a m x= 2 5 2 m m x: 396mm
73
1 -.>,.
1.
!
!
!
I
!
!
!
Y/(~
i
|
0.5
..
i
i
i
"
!
[
y/6
99
i
I
99
Figure 10. Turbulent shear stress
Figure 9. Wall-normal component of turbulence intensity 20
i
0.5
1
0.2 o o o ,a v
x=-lO8mm x= 1 0 8 m m x= 2 5 2 m m x= 3 9 6 m m x= 5 4 0 m m
'
'
'
'
!
'
'
'
9
1
o
~10
-
-
,
i
|
,
.I
i
|
!
O.5
y / 6 99 Figure 11. Distributions of turbulent kinetic energy
o x=-lO8mm o x= 1 0 8 m m o x= 2 5 2 m m |
|
a x= 3 9 6 m m v x= 5 4 0 r a m i
|
I
--I
t
.
|
i
0.5
y/6 99 Figure 12. Distributions of structure parameter A1
I
t 1
706
'IL ~
1.5
.
0.5
~,~
.
1 0.5
-
.
.
.
w
.
.
. . o x=-lO8mm o x= 1 0 8 m m o x= 2 5 2 m m t, x= 3 9 6 m m
-
" 7 "
[
o
I -o.s
ox:
-1'--
-0.5
o x= 2 5 2 m m a x= 3 9 6 m m v x= 540ram
v
;
-1 .~
-
--|
-1.5
1
0.5 y/ 6
-
I
|
|
i
99
[
,
0.5 y/ 6
(a) ~
,
|
i
1
99
(b) W~V Figure 13. Distributions of turbulent transport
toward the wall. It will be known from Fig. 13(b) that the turbulent transport of - v u * is also toward the wall in the same region as for u* 2. The wallward turbulent transport has also be found in the turbulent boundary layers with adverse pressure gradient ([11][13]). 3.4. A four quadrant analysis To investigate the effects of mean flow three-dimensionality on turbulence producing events near the wall, a four quadrant analysis (Lu and Willmarth [14]) was applied to the u* and v components. The fractional contributions to the Reynolds stress -W*--~from respective quadrants (i) in the u*-v plane are shown in Fig. 14 for x = 540 mm. The parameter H is a threshold defined by H = [ u * v ] / u ~ ' ~ ~ commonly called "hole size". For evaluating strong events, the data with H = 2 are also shown as well as those with H = 0. It was observed that, at the upstream stations (x _<- 252 mm), the contribution from quadrant 2 (Q2) is the largest of all across the boundary layer. However, at the downstream stations (x >= 396 ram), the fractional contribution from Q2 reduces markedly near the wall and the contribution from Q4 predominates instead, as seen in Fig.14. Similar results were reported for the boundary layer with a strong adverse pressure gradient [12]. The ratio of contributions to -u--~ from Q2 and Q4 are shown in Fig. 15. It is noted that the Q4 events (sweeps) are predominant in the region where wallward turbulent transport is observed in Fig. 13. It should be remarkable that a plateau region exists in the central part (0.35 < y/ (~ 99 < 0.55) of the curve for the most downstream station (x = 540 mm). Figure 16 shows the intermittency factor "( i for x = 540 mm. It was found, by comparing
.,/
9/-][-
ok i=f
white:H--'O '
o 9 i=2 0 9 i=3
black:H=2
. . . .
2u,n
]
~:!
4
"*
.
l
~7
1
~
....
-0"4t
.
o
.
.
.
.
.
I
.
o x=-lO8mm o x= 108mm o x= 2 5 2 m m A x-- 3 9 6 m m
[
X=
' /// //,,(" ///-
~ / d/
"x,, "-"
f
5
-
l
0.5 y/6
.
.
.
.
1 99
Figure 14. Fractional contributions to -u*v from respective quadrants in u*-v plane at x = 540 mm
0 ~
"o
~
,
,
,
I
0.5
y/6
.
.
.
.
99
Figure 15. Ratio of the contributions to -u*v from Q2 and Q4 with H = 2
"x
707
0.6
. o i=l o 9 i=2 o * i=3
t
1',
.,..,,
9
. . wh'ite: H--.'0 blaclcH=2
20
. '
'
'
o x=-108mm o x= 108mm o x= 252mm x= 396mm v x= 540mm
10
i=4
0.4
b-
b,,
5
~ /'//P" /// f
I q " X ~ " ~
/
\
/
V
~"'~.,. 1 '~x~
0.2
1 0
i
0.50
0.5
|
i
i
0.5
y/6' 99
y/6
,02f '
'
151
'
[ !/
Iflack : i=4
o, o.
1010
!
i
t 1
99
,
,
white : i--2 black : i=4
,
,
,
.../
. . . .
o 9 x=-108mm o ,, x= 108mm o . x= 252mm
tx 9 x= 396mm v 9 x= 540mm
[ t
o-
---~
x] ~
J
x=-108mm o , x= 2 5 2 m m v 9 x= 540mm x= 108mmZX 9 x= 396mm
. . . .
i
Figure 17. Ratio of the intermittency factors of Q2 and Q4 with H = 2
Figure 16. Intermittency factor of respective events at x = 540 mm
103[~1:w'hite:i=2
,
~ 0.5
. . . .
y / 6 99 Figure 18. Time intervals of Q2 and Q4 events with H = 2
e_~----~-----~-_
.
x]
.
.
.
.
5
1
/
1
0
0.5
y / 6 99 Figure 19. Mean duration of respective events with H = 2
these data with those for the upstream stations (not shown here), that 3" i-values for i = 2 with H = 2 (or i = 4 with H = 0) decrease with going downstream near the wall. The ratio of 3" i for Q2 and Q4 events with H = 2 are shown in Fig. 17. Note that the qualitative agreement is obtained between Figs. 15 and 17. Figure 18 shows the mean time intervals between ejections (T § 2) or sweeps ~'q:'4) with H = 2 2, where ~ = TU ~ / v . It is noticeable that ~ 2 decreases with going downstream at the upstream stations but it increases from x = 396 mm to x = 540 mm. This remarkable evidence suggests that strong ejections are suppressed by the effects of increased mean flow three-dimensionality. As we have not visualized the near-wall structure, it is not known whether one sign of vortex produces ejections that are considerably weaker than the other sign (namely, asymmetry of the near-wall structure [6]) in the region of increased three-dimensionality. The mean durations of Q2 and Q4 events with H = 2 are shown in Fig. 19. As seen in the figure, the normalized mean durations of both events (strong ejections and sweeps) decrease with going downstream, and the difference between those of ejections and sweeps becomes smaller. Figure 20 shows the conditional averages of u * , v and - u * v normalized with their rms values. Although the u*-pattern changes little, the width at halfheight of the v - and u ' v - p a t t e r n reduce as the mean flow three-dimensionality increases. It is noted that the (-uv)'-values at far from the detection time (T + = 0) are rather small at x = 540 mm. These results may interpret some aspects for the reduction in (-fi'~)' with increased three-dimensionality.
1
708
[
.....
x=-lOSmm x= 108mm
.... .....
x= 2 5 2 m m x= 3 9 6 m m
-2 ,;:.~.~ Z" "~
=
v
| ["
7'--,
7.-.,
'
'
!
'
{~\ ~
~
.... ....
'
[
x= 2 5 2 m m [ x= 3 9 6 m m -[
2
I
~o
-10
0
10
~,0
-20
-10
T+ (a) ejections
0
l0
20
T+ (b) sweeps
Figure 20. Conditional averages of u*, v and -u*v with H = 2 at y
+
~ 50
4. CONCLUSIONS An experimental study has been made of the turbulent structure of the three-dimensional boundary layer on an infinite swept wing. The results obtained are summarized as follows. 1. The location of the maximum Un is within the log-law region of Us-profiles, where Un and Us represent the mean velocity components in the cross-flow and local free-stream directions, respectively. 2. In the near-wall region, the streamwise turbulence intensity, [ u-'~/U r , decreases with increasing mean flow three-dimensionality. On the other hand, the cross-flow component, / w '~'2' AJ r , increases. 3. At downstream stations, the turbulent transport to the wall was observed in the near-wall region where sweeps were dominating. 4. A four quadrant analysis suggests that strong ejections are suppressed by the effects of increased three-dimensionality.
REFERENCES o
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
B. V. D. Berg, et al., J. Fluid Mech., 70(1975), 127. P. Bradshaw, Ann. Rev. Fluid Mech., 19(1987), 53. J. P. Johnston and K. A. Flack, J. Fluid Eng., 118(1996), 219. J. K. Eaton, AIAA J., 33(1995), 2020. S. K. Robinson, Ann. Rev. Fluid Mech., 23(1991), 601. C. Chiang and J. K. Eaton, Exp. Fluids, 20(1996), 266. K. A. Flack, Exp. Fluids, 23(1997),335. A. T. Degani, et al., J. Fluid Mech., 250(1993), 43. M. Itoh, et al., Exp. Therm. Fluid Sci., 5(1992), 359. P. Bradshaw and N. S. Pontikos, J. Fluid Mech., 159(1985), 105. Y. Nagano, et al., Turbulent Shear Flows 8, (1992), 7. P. Moin, et al., Phys. Fluids A, 2(1992), 1846. P-A. Krogstad and P. E. Skate, Phys. Fluids, 7(1995), 2014. S. S. Lu and W. W. Willmarth, J. Fluid Mech., 60(1973), 481.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
709
Flowfield characteristics of swept struts in supersonic annular flow K.E. Williams " and F.B. Gessner b "Adroit Systems Inc., Bellevue, WA 98004, USA bDepartment of Mechanical Engineering, Box 352600, Seattle, WA 98195, USA 1. INTRODUCTION Supersonic flow past a strut mounted on an endwall generates a complex vortical flow pattern near the strut/endwall intersection. Recent studies by Williams et al. [ 1,2] based on supersonic flow past diamond-shaped struts mounted between concentric cylinders have shown that four distinct vortices are formed in the vicinity of each strut/endwall intersection. These vortices consist of two horseshoe vortices generated at the leading and trailing edges of the strut and two, eontra-rotating corner vortices formed along the compression and expansion faces of the strut which propagate as shown in Fig. 1. In reference to applications, diamond-shaped struts are one means of supporting the outer cowl of the supersonic through-flow fan engine [3,4]. This type of strut configuration with lentieular (curved) surfaces to minimize shock losses has also been proposed as a means of supporting swirling jet fuel injectors in some combustor applications [5]. Local flow behavior in these geometries is influenced to some extent by compression face corner vortices that have propagated into the wake region behind each strut, as shown in Fig. 1. More specifically, these vortices can distort flow at the compressor face of the supersonic through-flow fan engine and can lead to non-uniform spray dispersal downstream of a strut-mounted, swirling jet fuel injector. These propagating vortices can also be a complicating factor when simulating oblique shock wave/tip vortex interactions of the type that occur in high-speed flight [6,7], especially if the height-to-chord ratio of the vortex generator (strut) in the model simulation is relatively small, which could lead to the paths of tip and corner induced vortices being in relatively close proximity downstream of the strut trailing edge. These examples are intended to illustrate that vortices generated by a strut/endwall intersection may have an important bearing on the downstream flow within a particular geometry and affect overall performance. The purpose of this paper is to present the results of a study concerned with the effect of strut sweep on local flow behavior in the vicinity of a strut/endwall intersection. This work represents an extension of an earlier computational and experimental study in which supersonic flow about unswept, endwall-mounted struts was investigated [8]. 2. E X P E R I M E N T A L PROGRAM Experiments were conducted in a continuous, open-loop supersonic flow facility whose design and operation are discussed by Williams et al. [9]. The test section is composed of a rotatable centerbody (83.0 mm OD) positioned concentrically within a stationary cowl (118.6 mm OD) to yield an annular gap width of 17.8 mm. Four symmetric, equally spaced, diamond-shaped struts were mounted on the centerbody as shown in Fig 2. The struts were made from stainless steel with faces polished to a near-mirror finish. The strut height (h) and
710
Compression face
Approaching a endwall ~ boundarYlayer
~ 1
Expansion face [CompressionfaceJ [ r I [ cornervortex J
Strut
~
I
I
k.- ......
.,~l"L~~~Expans,~.onl Trailingedge I i! *'" ~ horseshoe vortex 1 I I I I 1 ~
~'~,=,.._ ~
II
aceJ I corner vortex J
I
J~J'J~
I I
j l l i m l l i i i l ii l i J
i
Leading edge horseshoe vortex
i i
I
i j
i j
j i
. . . . . Clockwise rotation as viewed from downstream 9-9 9 9Counter clockwise rotationas viewedfrom downstream Fig. 1
Observed vortices as a strut/endwall intersection in supersonic flow.
chord width (c) are 17.8 mm and 25.7 mm, respectively, yielding a strut height-to-chord ratio h/c = 0.7 and a dimensional gap width Ar/c = 0.7, where Ar is a radial coordinate measured from the centerbody surface, as shown in Fig. 2. The maximum strut thickness (t) is 3.18 mm, yielding a 7.1 o half-wedge angle for the compression and expansion faces of the strut. Sweep was accomplished by holding the cross section of the swept strut the same as that of the unswept strut (dimensions c and t the same for both struts) and, in effect, rotating the unswept strut about its midchord intersection point with the centerbody (at Ar/c = O, x/c = 0.5) to a sweep back angle of 45 ~ and then adjusting its length to intersect the cowl and centerbody surfaces as shown in Fig. 3.
Pitot pressure data were taken with two pitot tubes made from nested stainless steel tubing (0.30 mm tip diameter) designed to facilitate measurements near the cowl and centerbody. Wall static pressures were measured by means of wall taps having an orifice diameter of 0.34 mm. A cone-cylinder probe, 0.64 mm in diameter with a cone half-angle of 5~ was used for static pressure measurements in the flow. A family of two-tube Conrad probes was used to measure pitch and yaw angles in cross planes normal to the primary flow. Details of the probe configurations and data reduction procedures are given by Williams [10]. Experimental
711 results in this paper are based on data taken at a location downstream of the strut trailing edge (x/e = 4). At this location pitot pressure and flow angle data were taken in radial increments of 0.5 mm over the annular gap width for a total of 35 measurements at each circumferential (0) location. Static pressure data were also taken in radial increments of 0.5 mm, except near each endwall. Between the strut plane of symmetry (0 = 0 ~ and the plane of symmetrybetween adjacent struts (e = 45~ pressure probe data were taken at 30 circumferential positions from 1~ to 3 ~ apart, depending in the amount of clustering needed to resolve the local shock structure. Data were also taken in 1~ increments on either side of the wake centerline (at 0 = 0 ~ to check flow symmetry. Experimental data taken at x/c = -0.5 (0.5 chord lengths ahead of the strut leading edge) were used to initialize the computations. At this streamwise location the boundary layers on the cowl and centerbody surfaces were partially developed and fully turbulent, with a core flow Mach number of 2.9 and a Reynolds number based on chord width of3xl05, and the displacement and momentum thicknesses were nominally the same on each endwall (5"~ 1.0 mm, 0 ~ 0.22 ram). 3. C O M P U T A T I O N A L M O D E L The equations of motion in strong conservation form were solved using NPARC [11 ], which is an updated version of the PARC code [12]. Closure to this system of equations was effeeted by specifying the turbulent Prandtl number as 0.9 and employing a combined version of the Baldwin-Lomax [13] and P.D. Thomas [14] turbulence models, which is one of the options in NPARC. The grids for the calculations were created using GRIDGEN [15]. Symmetry was invoked to limit the circumferential extent of the computational domain to a region between 0 = 0 ~ (strut plane of symmetry) and 0 - 45 o (plane of symmetry between adjacent struts). The grids for both the unswept and swept struts had 147 x 75 x 75 nodes in the x, r, and 0 directions, respectively, and were clustered near anticipated strut-induced leading and trailing shock locations, as shown in Fig. 4, where grid lines through every other nodal point are drawn. The grids extend to the strut, centerbody, and cowl surfaces, where no-slip, adiabatic wall conditions were specified. The first grid plane adjacent to each of these surfaces was located well within the viscous sublayer of the initial boundary layer on each surface (y+ < 1). Details of the computational procedure are given by Williams [ 10]. Plane of strut
(~0 deg)
and strut surface
Exit plane
Cemerbody surface
a) Unswept strut
Fig. 4
Plane of symmetry between struts (0=-45deg)
Computational grids.
b) Swept strut
712 4. RESULTS AND DISCUSSION 4.1 Shock Structure It is instructive to examine first the secondary shock structure induced at the strut/centerbody and strut/cowl intersections, inasmuch as compression and expansion waves generated at these locations impinge on the endwall boundary layers within which the leading and trailing edge horseshoe vortices are propagating. Figure 5 shows computed shock and expansion wave behavior in the vicinity of the strut for both the unswept (Fig. 5a) and swept (Fig. 5b) strut configurations. As the leading edge shock wave generated by each strut configuration intersects the endwall boundary layers, expansion fans are formed which propagate radially away from the intersections. These expansion fans affect the flow by turning streamlines that have just crossed the leading edge shock wave away from the duct midheight toward the endwalls. Similarly, the expansion fan generated at the strut midchord interacts with the endwall boundary layers to generate shock waves that propagate away from the endwalls. These "intersection shocks" create ashock diamond pattern on the strut plane of symmetry behind the strut as shown in Fig. 5. It will be shown shortly that when these weak shock waves interact with an endwall boundary layer downstream of the strut trailing edge, vorticity is reoriented and vortical flow patterns may develop in the interaction region. 4.2 Limiting Streamlines Computed limiting streamlines on the cowl and centerbody surfaces are shown in Fig. 6 for both the unswept (Fig. 6a) and swept (Fig. 6b) strut configurations. The lines C1 and C2 are lines of coalescence (separation) which respectively bound the leading and trailing edge
Fig. 5
Computed shock waves and expansion fans generated by strut/endwall intersections.
713
Fig. 6
Computed limiting streamlines on the cowl and centerbody surfaces.
horseshoe vortices generated by the strut. The lines D1 and D2, where they appear, are lines of divergence (attachment) which also bound these vortices. The finite extent of the C and D lines in this figure should be interpreted only as a subjective measure of actual separation and attachment phenomena in the flow, inasmuch as these lines are not wholly representative of the actual flow because of inadequate grid resolution. For example, the computations show that the trailing edge horseshoe vortices on the cowl and centerbody surfaces maintain their integrity well downstream of the strut trailing edge (at least to x/c = 5), so that lines C2 in Figs. 6a and 6b should extend over this distance for each strut configuration. Flow visualization results which depict this behavior are given in Ref. 10, but are not included in this paper, because black and white reproduction of the color photographs tends to obscure observed coalescence of the limiting streamlines. One interesting feature of Fig. 6a is that lines C3 and D3 observed on the cowl for the unswept strut originate near x/c = 3, 0 = +_ 15~ which is at a position where the strut/centerbody intersection shock shown in Fig. 5a first impinges on the cowl. Fluorescent dye/oil flow visualization of limiting streamlines on the centerbody surface, in fact, shows that a crescent-shaped line of coalescence (C3) is formed downstream of each strut configuration that extends outward from the C2 line of coalescence which bounds the trailing edge horseshoe vortex [10]. This behavior is not evident in computed limiting streamline behavior on the centerbody for the swept strut (Fig. 6b), again because of inadequate grid resolution. In contrast to this behavior, experimentally observed lines at coalescence which bound the compression corner face vortices that are propagating toward the center of the duct along the expansion face of each strut (refer to Fig. 1) are simulated well by the computations. More specifically, the computed lines of coalescence shown in Fig. 7 which bound these vortices (C4) are in good agreement with flow visualization results reported by Williams [10].
714
Fig. 7
Computed limiting streamlines on the strut.
4.3 Corner Vortical Flow Structure Cross flow vectors computed in the corner regions of the compression face/endwall intersections at 75 percent of the streamwise distance along the compression face of the strut are shown in Fig. 8. The vortex observed in each corner is bounded by lines of coalescence and divergence on the strut and endwall, as denoted by points C4 and D4 respectively, at this streamwise location. Inasmuch as the views in this figure are perspective views along computational grid lines which sweep downstream away from the strut, some of the bounding surfaces are canted as a result of three-dimensional effects. Although Fig. 8 shows that strut sweep has little effect on the size of the corner vortex bounded by the eenterbody, the corner vortex bounded by the cowl for the swept strut is roughly twice as large as that for the unswept strut. This behavior is apparently the result of a relatively strong interaction induced by the intersection of the backward swept strut with the cowl (refer to Fig. 7b), which also induces a much stronger leading edge horseshoe vortex at the strut/cowl intersection compared to that observed for the unswept strut [ 10]. Figure 9 shows cross flow vectors and vortices computed in the corners of the expansion face/endwall intersections at 75 percent of the streamwise distance along the expansion face. The vortex in each corner is bounded by lines of coalescence and divergence, as denoted at
Cowl ~ o.oc , . , . . . . . . .~- _ ~'li
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Strut~\\~'~,~,Q :
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Computed cross flow vectors near the expansion face/endwall intersections.
715 this location by points C5 and D5, respectively. Note that the rotational sense of each expansion face comer vortex is opposite to that of its compression face counterpart (compare Figs. 8 and 9). For the unswept strut, a comparison of Figs. 8a and 9a shows that the magnitude of cross flow in the expansion face comer region is much larger than that observed in the compression face corner region at the s aae relative streamwise distance. For the swept strut, a comparison of Figs. 8b and 9b shows that the expansion face/cowl vortex is also much smaller than the compression face/cowl vortex. The expansion face comer vortices in Fig. 9 continue to migrate along the comers formed by the strut/endwall intersections until they eventually merge with horseshoe vortices generated at the strut trailing edge, as shown in Fig. 1. In contrast, the compression face comer vortices shown in Fig. 8 migrate toward the duct midheight along the expansion faces of the strut, as shown schematically in Fig. 1, and as implied by the lines of convergence C4 shown in Fig. 7. 4.4 Cross-Planar Distributions Pitot pressure contours, static pressure contours, and Mach number contours downstream of the swept strut trailing edge at x/c = 4 are shown in Figs. 10, 11 and 12, respectively.The measured results within the circumferential interval - 6 ~ _<0 _<6 o are indicative of the level of symmetry observed about 0 = 0 o in the experimental flow. In general, Figs. 10-12 show that there is close correspondence between measured and computed distributions for each variable, except in the wake region centered about 0 = 0 ~ The concentration of steep gradients in the central region of the flow near 0 = 22 ~ and 36 o correspond, respectively, to the circumferential positions of the trailing edge shock and of the crossing leading edge shock from the neighboring strut at 0 = 90 o.
Computed and measured cross-flow vectors at x/c = 4 are shown in Fig. 13. Again, computed and measured distributions outside the wake region agree very well. The relatively strong, inwardly directed cross flow at 0 = 30 o near each endwall is due primarily to the crossing shock from the neighboring strut which has entered the flow domain and has a relatively strong influence on the endwall boundary layers at this location. The inwardly directed vectors near the cowl and centerbody are compatible with the limiting streamlines for the swept strut shown in Fig. 6b, which are canted on each surface toward the strut plane of symmetry at, and near, x/c = 4, 0 = 30 o. Examination of Fig. 10 shows that discrepancies between computed and measured pitot pressures exist in the wake region behind the strut. This can be seen more clearly in Fig. 14, which is a circumferentially expanded view at the results in Fig. 10 centered about 0 = 0 o. Figure 14 shows that both the width of the wake and the level of distortion in the wake are underestimated by the computations. The measured contour variations between A r/c = 0.2 and 0.3 near 0 = 0 ~ are the direct result of a vortex pair corresponding to two, contra-rotating vortices generated at the compression face/centerbody intersection that are propagating in the wake region. Analysis of cross flow data taken in this region has shown that these vortices are roughly centered at Ar/c = 0.27, 0 = + 2 ~ [10]. The measurements did not reveal, however, whether vortices generated at the compression face/cowl intersection actually exist in the flow at x/c = 4, although their presence at this location was confirmed experimentally for the unswept strut [10]. The point to note here is that the computed pitot pressure contours in Fig. 14b are relatively undistorted in the same region where vortex-induced distortion is observed experimentally, because the prescribed isotropic eddy viscosity turbulence model does not predict the presence of comer induced vortices in the wake region behind a strut. This shortcoming should be addressed in future work by developing a more representative turbulence model for the wake region.
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This paper has presented the results of a study related to the effect of strut sweep on the nature of supersonic flow past diamond-shaped struts mounted on a circular centerbody positioned concentrically within a circular cowl. Computed results which compare shock structure, limiting streamline patterns, and cross flow vector behavior for both unswept and swept struts show that the local flow structure near, and downstream of, each strut/endwall intersection is basically similar, with some observed differences between the size and strength of comerinduced vortices along the intersection. Comparisons between measured and computed distributions downstream of the swept strut-trailing edge show that the prescribed turbulence model is performing properly, except in the wake region behind the strut. Further work on model development is needed in order to improve predictive capabilities in this region.
718 REFERENCES 1 Williams, K. E., Harloff, G. J. and Gessner, F. B., Investigation of Supersonic Flow About Strut/Endwall Intersections, AIAAJ33 (4), 586-594, 1995. 2 Williams, K. E. and Gessner, F. B., On the Evolution of Strut-Induced Vortices in Supersonic Annular Flow, AIAA Paper 96-0324, 1996. 3 Franciscus, L. C., The Supersonic Through-Flow Turbofan for High Mach Propulsion, AIAA Paper 87-2050, 1987. 4 Barnhart, P. J., A Preliminary Design Study of Supersonic Through-Flow Fan Inlets, AIAA Paper 88-3075, 1988. Heiser, W. H. and Pratt, D. T., Hypersonic Airbreathing Propulsion, AIAA Publications, New York, 1993. Rizzetta, D. P., Numerical Simulation of Oblique Shock-Wave/Vortex Interaction, AIAAJ33 (8), 1441-1446, 1995. Smart, M. K. and Kalkhoran, I. M., Effect of Shock Strength on Oblique Shock Wave/Vortex Interaction, AIAAJ33 (11), 2137-2143, 1995. Williams, K. E. and Gessner, F. B., Flowfield Characteristics of Struts in Supersonic Annular Flow, Exp. Therm. FhtidSci. 17 (1 & 2), 156-164, 1998. Williams, K. E., Gessner, F. B. and Harloff, G. J., Design and Operation of a Supersonic Annular Flow Facility, AIAAJ32 (7), 1528-1531, 1994. 10. Williams, K. E., Investigation of Supersonic Flow About Strut/Endwall Intersections in an Annular Duct, Ph.D. Thesis, Dept. of Mech. Engrg., University of Washington, Seattle, WA, 1995. 11. NPARC 1.2a User Notes, Arnold Engineering Development Center, 1994. 12. Cooper, G. K. and Sirbaugh, J. R., PARC Code: Theory and Usage, AEDC-TR-89-15, Arnold Engineering Development Center, 1989. 13. Baldwin, B. S. and Lomax, H., Thin Layer Approximation and Algebraic Model for Separated Turbulent Flows, AIAA Paper 78-257, 1978. 14. Thomas, P. D., Numerical Method for Predicting Flow Characteristics and Performance of Nonaxisymmetirc Nozzles-Theory, NASA CR 3147, 1979. 15. Steinbrenner, J. P., Chawner, J. R. and Fouts, C. L., The Gridgen 3D Multiple Block Grid Generation System, WRDC-TR-90-3022, Vols. 1 and 2, Wright-Patterson Development Center, 1990. .
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10. Turbomachinery Flows
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
721
Flow In a Radial Outflow Impeller Rear Cavity of Aeroengines X. Liu Pratt & Whitney Canada, 1801 Courtneypark Drive Mississauga, ONT. L5T 1J3, Canada Tel. (905) 564-7500 Ext. 5421, Fax (905) 564-3899 xiaoliu.liu @pwc.ca ABSTRACT In order to guide the engine air system design, a comprehensive parametric CFD analysis has been carried out on a radial outflow impeller rear cavity. Eight different cases have been analyzed, with the inlet flow swirl factor ranging from 0 to 2.1, the through flow number ranging from 3.5x103 to 3.5x104, and the rotational Reynolds number ranging from 1.2x107 to 2.9x107. The influence of the above three flow parameters are studied in detail. The CFD results provide a comprehensive data base for engine air system design optimization.
NOMENCLATURES Moment = 1 o)2R 5
Cm "-" ~
--2f') m
Cw-
~tRtip
moment coefficient exerted to the flow by the rotating wall
tip
- through flow number
K = swirl factor = air tangential velocity / (Rco) m = mass flow rate Ps = static pressure of flow Ptip = static pressure at cavity tip R = radius from the engine centre line
(oRt2ip Re 0 -
V
- rotational Reynolds number
T = absolute total temperature of flow AT = temperature change due to windage co = rotational speed of the impeller (rad/s) 9 = air density at the cavity hub ~t = air viscosity at the cavity hub v = air kinematic viscosity at the cavity hub subscript: tip = at cavity tip hub - at cavity hub
722 1. I N T R O D U C T I O N The cavity at the rear side of the centrifugal compressor impeller in a gas turbine engine is called Impeller Rear Cavity or IRC. The IRC is a rotor-stator cavity between the rotating impeller rear surface and the stationary combustor case and bearing house surfaces. In a conventional design, all air flow enters the IRC from the impeller tip with the compressor exit temperature and flows radially inward through the IRC, exiting the cavity at the bottom. Air at the bottom of the IRC is hotter than that at the tip, due to the windage (or friction work) from the rotating impeller rear surface. This temperature profile is unfavorable to the impeller life, because the impeller stress is more critical at the bottom than at the tip due to the centrifugal load. In order to address this problem, a new design is investigated here. As is shown in Fig.l, most of the flow is introduced to the IRC from inlet A at the bottom and flows
Fig. 1 The velocity field in meridian plane for case 1
723 radially outward through the IRC. The majority of the flow exits from exit D at the IRC top. The impeller tip bleed flow into the IRC is reduced to a small amount, but it is still positive to maintain the compressor stability. Since the flow is radially outward, the windage will accumulate radially outward. Therefore, the flow will be cold at the bottom of the IRC where the impeller stress is more critical, and hot at the IRC tip where the impeller stress is less critical. In this way, the impeller life will be increased significantly, so that better engine durability can be achieved. A good understanding of the flow in such a radial outflow impeller rear cavity, in particular the pressure and temperature variation, is essential for the engine air system design. The temperature distribution determines the impeller life and the pressure distribution determines the thrust bearing load. In order to guide the design, CFD analyses have been carried out on the flow in such a radial outflow impeller rear cavity.
Flow in rotating cavities is a major research subject for gas turbine air system technology, and has attracted more and more researchers. Recently, the co-rotation cavities were studied by several researchers. Karabay et al. (1997) investigated the pre-swirl system for turbine cooling and confirms that free-vortex flow occurs throughout most of the rotating cavity. Pilbrow et al. (1998) continued this work and investigated the heat transfer in this pre-swirl system. Mirzaee et a1.(1997) studied the flow and heat transfer in a co-rotating cavity with peripheral inflow and outflow. All of the above investigations are a combined experimental and numerical study. Liu (1997) conducted a comprehensive CFD study on the flow in a corotation radial inflow cavity between turbine disk and coverplate, and found that the flow behaves more and more like a free vortex with increasing mass flow, but more and more like a forced vortex with increasing rotational speed. The closed (i.e. no through flow) rotating cavity is studied by Bohn and Gier (1997) using CFD, and they demonstrated that turbulence has a considerable influence on the overall heat transfer as well as on the local heat transfer distribution. CFD has been proven to be the most powerful tool to study the rotating cavity flows. The research on a simplified cylindrical rotor-stator cavity was pioneered by Daily and Nece (1960) and later it was systematically studied by Owen (1988) and his co-workers. Recently, Gaetner (1997) correlated experimental data from various sources and developed a new correlation which improved the analytical prediction of friction torque for a simple cylindrical cavity. For a complicated rotor-stator turbine cavity, Tekriwal (1997) conducted a CFD analysis, and successfully predicted the flow, heat transfer, windage and temperature. However, the complicated rotor-stator impeller rear cavity with multiple inlets and outlets is the subject of the present research.
2. THE C A L C U L A T I O N M E T H O D The NS3D CFD code developed in Pratt & Whitney Canada has been used for this analysis. This code solves the 3-D compressible Navier-Stokes equations using a finite element method. It employs a k-t0 turbulence model (Wilcox, 1988) and a wall function. Details about this code is described by Peeters et al. (1992). This code has been well validated with a lot of experimental data, but most of the validation work is published only within Pratt & Whitney Canada. Following are just a few examples. Hall (1996) calculated the flow in the Krain impeller using this code and compared with the experimental results of Krain (1987). Good agreement with experiments is found in terms of pressure ratio, temperature ratio, overall
724
performance and detailed flow features. Peeters (1997) calculated the PW206 impeller using this code and compared with the measurements conducted in PWC. Again good agreement was obtained in terms of pressure and temperature distribution. The flow situation in an impeller is in close similarity to the flow situation in the impeller rear cavity in the sense that in both cases the flow is between a rotor and a stator wall. Zhou et al. (1995) calculated the flow in a 180 ~ turn coolant passage and compared with the measurement done in Pratt & Whitney. Excellent agreement was obtained in pressure distribution and Nusselt number distribution. The current calculation was done on a multi-block grid which consists of 21 blocks and 35022 nodes. Since the flow is axisymmetric, three meridian planes were used. 3-D calculation was carried out with periodic boundary condition applied to the front and the back meridian plane. The calculation was performed on a HP work station. Each iteration takes about 170 seconds. Good convergence is achieved typically after 500 iterations. The impeller rear cavity geometry is shown on Fig.1. The impeller rotates at a high speed, typically ranging from 10000 to 30000 rpm. Rotating wall boundary condition is applied to the impeller rear surface as shown in Fig.1. Stationary wall boundary condition is applied at the combustor case wall and at the bearing house wall as shown in Fig.1. Since the exact thermal boundary condition is not known, adiabatic wall boundary condition is applied to all walls. Therefore, the temperature change is purely caused by the windage. The air flow enters the cavity at the inlet A. Some of this flow exits at exit B, and the rest of the air flows radially outward. This outward flow joins with the small amount of impeller tip bleed flow from inlet C, and then exits the cavity at exit D, as shown on Fig. 1. At inlet A and C, air mass flow rate and the swirl factor K are specified. At inlet C, the swirl factor K is 0.82 for all cases. At exit B static pressure is specified, and at exit D mass flow rate is specified. For this cavity, the ratio of Rhub/Rtip is 0.38. Eight different cases were calculated which covers the whole envelope of engine operating conditions. The parameters for each case are listed in Tablel. The main parameters are the swirl factor K at inlet A, C w, and Re 0. C w is calculated according to the net radial outward through flow, i.e. the mass flow at inlet A minus the mass flow at exit B. Table 1. The main parameters for the calculation cases
C a s e No.
K at inlet A
Cw
Reo
1
0.0
1.7x10 4
2.0x10 7
2
0.7
1.7x10 4
2.0x10 7
3
1.5
1.7x 10 4
2.0x 10 7
4
2.1
1.7x10 4
2.0x10 7
5
1.6
3.5x10 3
2.0x10 7
6
1.4
3.5x10 4
2.0x10 7
7
1.4
1.7x 10 4
1.2x 10 7
8
1.6
1.7x10 4
2.9x10 7
725
3. RESULTS AND DISCUSSIONS Fig.1 shows the calculated velocity field in the meridian plane for case 1. For this case, flow enters the cavity at inlet A with a swirl factor K of zero. Due to the viscous shear force, the rotating impeller rear surface imparts the tangential momentum onto the flow, so that the flow in the cavity rotates with but slower than the impeller rear surface. The centrifugal force is radially outward while the pressure gradient force is radially inward. Near the rotating wall the flow velocity is close to the wall surface velocity. Therefore, the centrifugal force is stronger than the pressure gradient force, so that the flow is pumped radially outward along the rotating wall as is seen in Fig. 1. Near the stationary wall, the centrifugal force is almost zero because the flow velocity is zero on the stationary wall. Therefore, near the stationary wall the flow is forced radially inward by the pressure gradient force, as is seen on Fig.1. This forms the complicated secondary flow feature in the cavity. However, the net through flow is radially outward, and the majority of the flow exits the cavity at exit D. Fig. 2 shows the various contour plots in the meridian plane for case 3. For this case, the flow enters the cavity at inlet A with a swirl factor of 1.5. The static pressure contours are shown on Fig.2a. As is seen, the contour lines are all horizontal and almost straight lines. This shows that the pressure gradient is predominantly in the radial direction, because the flow field is dominated by the tangential velocity component. Due to the swirl, the pressure increases monotonically from hub to tip. The contours of the swirl factor K is shown in Fig.2b. The flow coming in at inlet A with K of 1.5, but the swirl dissipates very quickly to a value of about 0.8 not far away from inlet A. Except near the inlets and the exits, the K contour lines are almost horizontal lines. This indicates that the swirl factor K is mainly a function of the radius R. Fig.2c shows the contours of the absolute total temperature. Since the flow is radially outward, the windage accumulates radially outward. Therefore, the temperature increases from hub to tip, as shown in Fig.2c. This is in contrast with the conventional radially inflow IRC. As a result, the temperature at the hub is reduced considerably compared to the conventional radial inflow IRC, and the impeller life will be improved significantly because the impeller is stress critical at the hub. Therefore, the basic objective of this radial outflow IRC is confirmed by Fig.2c. It is interesting to note that at the bottom of the cavity the temperature is higher close to the rotating wall, while at the top of the cavity the temperature gradient is mainly in the radial direction and the gradient is quite large because most of the windage is generated at the top of the cavity where the surface velocity is high. v
tip
0.849 0.855
~, 0.868 o.861 EF 0.874 0.880 HG 0.893 0.887
I 0.899 KJ 0.912 0.906 ML 0.924 0.918 0.931
~i
~
0.950
,~ 0.962 0.956 uT 0,969 0.97,5 v 0.981 0.988 X 0.994
; 0.000 0.082 C 0.163 0.245 s 0.326 0.408 H G I J
0.489 0.571 0.6,52 0.734 0.816 L 0.897 0.979 1.060 1.1 ; 1.223 1.305
R !.387 1.468
1.00o
Fig.2 Contours of various flow parameters for case 3
9
0.747 0.761 C 0.774 0.788 EF 0.801 0.814 HG 0.828 0.841 J 0.855 0.868 K 0.881 0.895 0.908 0.921 0.935 0.948 QR 0.962 0.975 S 0.988
726 1.00
1.10
K=0 0.95
__ /
/
.~
0.90
a.
0.85
__:
/
~
/
_-
.-'"
.-
0.80 , 0.5
0.3
.
, 0.7
R/Rtip
a) Pressure
O
\
,.l,..-
...- .....
.... ..
0.70
K=0
,.
".
\
i....s...-"'""
...
0.75
0.90
.....
- -
K=0.7
-'-
K=1.5
,,e
0.50
...... K = 2 . 1
0.30
.
0.10
, 0.9
\
.........
\.
"~
0.3
0.5
- -
K=0.7
-'-
K=1.5
".....
K=2.1
..................
..4
............................. // I
"--
0.7
0.9
R/Rtip
b) K factor in the middle of the cavity
Fig.3 Radial distribution of flow parameters with different K at inlet A Cw = 1.7x 104, Re0=2x 107
The radial distribution of static pressure and K factor are shown in Figs. 3 to 5. The pressure distribution determines the bearing load of the engine, while the K factor distribution plays an important role in impeller life calculation. The distributions with different K factor at inlet A are shown on Fig.3. As can be seen, with increasing K factor at inlet A, the pressure gradient increases (Fig.3a). This is because as the flow coming in at inlet A has higher and higher swirl, the K factor throughout the cavity is increased as shown on Fig.3b. This produces higher centrifugal force which results in larger radial pressure gradient. It is interesting to note that in the top region corresponding to the narrow neck of the cavity, the distribution of pressure and K factor are almost independent of K at inlet A. The pressure increases monotonically from hub to tip, while the K factor in the mid portion of the cavity is lower than that in the hub and tip region. Fig.4 shows the distribution with different Cw. Generally speaking, within the range of Cw, the distributions do not change very much. With increasing Cw, the K factor at the lower portion of the cavity increases slightly but the K factor at the upper portion reduces slightly. As a result, the overall pressure variation does not change much. The swirl at the lower portion increases with mass flow because there it is mainly imported from inlet A, while the swirl at the upper portion reduces with mass flow because there it is mainly imparted from the rotating wall. Fig.5 shows the distribution with different Re 0. As is seen, with increasing Re 0, the pressure gradient increases significantly throughout the cavity. The K factor reduces with increasing Re 0, but the shape of the distribution is quite similar. In fact, as Re 0 increases from 2.0x 107 to 2.9x 107, there is not much difference in K factor distribution.
727
1.00
1.00
L._ O .,,-, O
~C I .
12.
0.90
-9
--
Cw=3.5e3
0.80
\,\'"
- -
Cw=1.7e4
;,,
0.95
0.90
0.70 -
,.i--,
,,e,
13.
-0.85
Cw=l.7e4
0.60 0.50
...... C w = 3 . 5 e 4
0.40 --..,.....,.....->"
0.80
,
0.3
.
0.5
,
,
0.7
0.9
0.30
0.3
I
I
0.5
0.7
R/Rtip
1
'
0.9
R/Rtip
a) Pressure
b) K factor in the middle of the cavity
Fig.4 Radial distribution of flow parameters with different C w Re0=2x 107, K at inlet A= 1.5
1.00
1.00 .
-
.-
0.80
0.90 CI. "-
ix. 13.
O
.- -"
........... - - R e 0 = 1.2x 107 ......... - - Re0=2.0x 107 ....'~ .. "..... Re0=2.9xl0 7
q')
0.80
*i, \
~ Re0= 1.2x107 - - Re0=2.0xl 0 7
':~.{,..,..,..~...:
..... 9
0.90
O
,,,I
0.70 0.60 0.50
= "
'i
0.40 '~..~ ...... :.'"
0.70 0.3
i
I
0 5
0.7 R/Rtip
a) Pressure
'
1
0.9
0.30 0.3
I
I
I
0.5
0.7
0.9
R/Rtip
b) K factor in the middle of the cavity
Fig.5 Radial distribution of flow parameters with different Re 0 Cw= 1.7x 104, K at inlet A-- 1.5
The overall flow characteristics are summarized by the pressure ratio across the cavity, the moment coefficient and the temperature change due to windage. The windage work generated by the rotating surface is absorbed by the mass flow exiting at exit D. This produces a AT, i.e. the temperature change due to windage. Figs. 6 to 8 show the influence of K factor at inlet A, C w, and Re 0 on the above three flow parameters.
728 1.00
9e-04
.+
|
~.
0.90 -
'-
8e-04
8
0.80 0.70-t 0.0
. . . . . I 0.5 1.0 1.5 2.0 2.5 K factor at inlet A
5e-04
i
0.0
9
!
i
I
0.5 1.0 1.5 2.0 K factor at inlet A
2.5
b) Moment coefficient
a) Pressure ratio across the cavity
0.13
6e-04
-
0.12 "tO
.c_
0.11
Fig.6 The influence of K factor at inlet A Cw= 1.7x104, Re0=2x 107
.~ o.lo <~
0.09 0.08 'i 0.0
i
i
i
'
i
0.5 1.0 1.5 2.0 K factor at inlet A
2.5
c) Temperature change due to windage
Fig.6 shows the influence of K factor at inlet A. As can be seen in Fig.6a, the pressure ratio reduces with increasing K factor at inlet A. This is obvious considering the discussions of Fig.3. With increasing K factor at inlet A, the swirl is increased throughout the cavity as shown earlier. This reduces the shear force on the rotating surface and thus reduces the moment coefficient as shown in Fig.6b. The reduced moment generates less windage, and results in the reduced AT due to windage as shown in Fig.6c. It is should be noted that the effect of K factor at inlet A on all the three parameters are all close to linear. The influence of C w is shown on Fig.7. The pressure ratio is fairly insensitive to C w as shown in Fig.7a. Although the moment coefficient (Fig.7b) increases slightly with Cw, the temperature change due to windage (Fig.7c) decreases significantly with Cw, because with increased C w there is more mass flow to absorb the windage, which results in a smaller temperature change. Fig. 8 shows the influence of Re 0. As can be seen, Re 0 has a strong influence on every parameter. The pressure ratio (Fig.8a) reduces with Re 0, because the flow swirl in the cavity increases with the rotational speed of the impeller. The moment coefficient (Fig.8b) increases with Re 0. This indicates that the moment increases faster than o)2 according to the definition of Cm. Because of the increased moment and rotational speed, the temperature change due to windage increases with Re 0 as shown in Fig.8c.
729 0.900
7e-04
"= 0.875
o~ D.
0.850
oE 6e-04
o~ 0.825
13_
0.800 3.0e+03
i
1.9e+04 Cw
3.5e+04
a) Pressure ratio across the cavity 0.20 r
5e-04 3.0e+03
!
1.9e+04
Cw
3.5e+04
b) Moment coefficient
m
m
"~ 0.15 ._=
"~
0.10
Fig.7 The influence of C w Re0=2x 107, K at inlet A= 1.5
~3 0.05 3.0e+03
!
1.9e+04 Cw
3.5e+04
c) Temperature change due to windage
0.95
8e-04
ca.
~. 0.90
E 7e'04
~= 0.85 r
~_
6e-04
0.80
0.75 1e+07
, 2e+07
m 3e+07
Re0
5e-04 I e+07
i
2e+07
3e+07
Ree
a) Pressure ratio across the cavity
b) Moment coefficient
0.3 E
~ <1
0.2
0.1 0.0 le+07
Fig.8 The influence of Re 0 Cw= 1.7x 104, K at inlet A= 1.5
i
2e+07 Re e
3e+07
c) Temperature change due to windage
730 4. SUMMARY AND CONCLUSIONS Comprehensive CFD analyses have been conducted on a newly proposed radial outflow impeller rear cavity of an aeroengine. Eight different flow cases are studied with different flow inlet swirl, different mass flow rate and different rotational speed, which covers the whole envelope of engine operation. The CFD results revealed the complicated secondary flow features in the cavity. The result confirms that with the radial outflow design the temperature at the hub is lower than that at the tip. Such a temperature profile is favorable for the impeller life. It is shown that increasing the K factor at inlet A results in a larger pressure change and a smaller temperature change due to windage. Increasing the rotational Reynolds number will increase both the pressure change and the temperature change due to windage. However, the pressure ratio is not too sensitive to the through flow number, while the temperature change due to windage reduces with increasing through flow number. The effect of K factor at inlet A on various flow parameters is quite close to linear. The CFD results have provided detailed distribution of pressure, temperature and swirl factor, etc., which serve as a comprehensive data base for the engine air system design optimization. REFERENCES: Bohn, D. and Gier, J. (1997), "The effect of turbulence on the heat transfer in closed gasfilled rotating annuli", ASME paper 97-GT-242. Daily, J. W. and Nece, R.E. (1960), "Chamber dimension effects on induced flow and frictional resistance of enclosed rotating disks", ASME J. of Basic Engineering, March, 1960,pp.217-232. Gaetner,W. (1997), "A prediction method for the frictional torque of a rotating disk in a stationary housing with superimposed radial outflow", ASME paper 97-GT-204. Hall, D. (1996), "Validation of NS3D for the Krain Impeller", Pratt & Whitney Canada Memo, Jan. 29, 1996. Karabay, H., Chen, J.X., Pilbrow, R., Wilson, M. and Owen, J.M. (1997), "Flow in a cover-plate pre-swirl rotor-stator system", ASME paper 97-GT-243. Krain, H. (1987), "Swirling impeller flow", ASME paper 87-GT-19. Liu, X. (1997), "Flow in a co-rotation radial inflow cavity between turbine disk and coverplate", ASME paper 97-GT- 137. Mirzaee, I., Gan, X., Wilson, M. and Owen, J. M. (1997), "Heat transfer in a rotating cavity with a peripheral inflow and outflow of cooling air", ASME paper 97-GT-136. Owen, J.M. (1988), "An approximate solution for the flow between a rotating and stationary disc", ASME paper 88-GT-293. Peeters, M. (1992), "Finite element solutions of the Navier-Stokes equations for compressible internal flows", J. Propulsion, vol.8, No. 1, Jan.-Feb., 1992, pp. 192-198. Peeters, M. (1997), "NS3D analysis of PW206 impellers", Pratt & Whimey Canada Memo, Feb.3, 1997. Pilbrow, R., Karabay, H., Wilson, M. and Owen, J.M. (1998), "Heat transfer in a coverplate pre-swirl rotating-disc system", ASME paper 98-GT-113. Tekriwal, P. (1997), "Flow and heat transfer computations in a rotating cavity with tangential seal air flow", ASME paper 97-GT-216. Wilcox, D.C. (1988), "Reassessment of the scale-determining equation for advanced turbulence models", AIAA Journal, vol. 26, No.11, Nov., 1988, pp.1299-1310. Zhou, J.M., Robichaud, M.P., Ghaly, W.S., Habashi, W.G. and Riahi, A. (1995), "CFD predictions of heat transfer with applications to turbine blade cooling", presented at the 5th CASI Aerodynamics Symposium, May 9-10, 1995, Montreal, Canada.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
Experimental
731
I n v e s t i g a t i o n of T u r b u l e n t W a k e - B l a d e I n t e r a c t i o n in
Axial Compressors A. Sentker and W. Riess a ~Institute for Turbomachinery, University of Hannover, Appelstr.9, 30167 Hannover, Germany The fundamentally unsteady character of flow in multi-stage turbomachines is traditionally neglected in most of the flow calculation methods. The rapid development of hard- and soft-ware for numerical flow simulation has opened possibilities for investigation of the unsteady flow in turbomachines. The still existing methodical problems in the simulation especially of turbulent flow makes the experimental validation of numerical results indispensable. In a low-speed research axial compressor with incompressible flow conditions and relatively large geometric dimensions a flow measuring system based on split-film probes has been installed and extensively operated. Measurements of the unsteady flow field and the turbulence properties concerning the blade wakes have been made and reported. In the subsequent investigation the passage of the IGV wakes and the rotor blade wakes of the first rotor through the adjacent stator blade row have been explored in one measuring plane in front of the stator, four measuring planes within the stator and a measuring plane behind the stator at the mean radius of the blading. Measurements of the unsteady absolute velocity field in the Sl-plane and the turbulent fluctuations have been made. The transportation of the IGV wake through the stator is one result of the measurements. The decay of the fluctuation components inside the stator blade row has been investigated in detail. 1. E X P E R I M E N T A L
FACILITY AND INSTRUMENTATION
The experimental investigations of the unsteady flow field have been conducted in the first stage of a two-stage low speed axial compressor (NGAV).The compressor consists of an inlet guide vane and two repeating stages. It is operated in a closed system with atmospheric pressure because of its large dimensions. In addition to the advantage of less contamination in the compressor the construction with an integrated watercooler assures a constant fluid temperature during the experiments. The main design parameters are listed below in table 1. For more information see [1] [2]. Split-film probes with a radially oriented sensor have been attached to a mechanical traverse system and mounted at different axial positions on the low-speed axial compressor. With this traversing system an automatical positioning of the probes in radial and circumferential direction is possible. The split-film probes (Typ R57, Dantec) are operated in constant temperature mode with an overheat ratio of 1.7. They consist of two 3#m nickel films deposited on a quartz fibre with 200#m
732 Table 1 NGAV Geometry outer diameter chord length rotor chord length stator blade height number of rotor blades number of stator blades
Design Parameters Da
lr 18 h
Zr z8
760ram
rotational speed 75mm mass flow rate 90mm total pressure ratio 140mm Reynolds number rotor 30 Reynolds number stator 26
nnenn rh 1-I
ReLA1 ReLE1
3000r~v 16.5~-~8ec 1.07 4.5. 105 4. 105
diameter and an active length of 1.25 mm. For the measurement of the unsteady velocity field in a turbomachine this probe type has the advantage of relatively small dimensions in circumferential direction and simultaneous acquisition of two components of the unsteady velocity vector in the 'S1-Plane'. Frequencies up to 175 kHz can be measured.
2. M E A S U R E M E N T
PROCEDURE
AND EVALUATION METHOD
For the measurements of the unsteady flow field in the first stage of the NGA V a rotational speed of 2900 rev/min is choosen. The operating point is close to the design point with a total mass flow rate of 16.8 k%~ and a pressure ratio of I] - 1.07.
Figure 1. Measuring planes in first stage All investigations are conducted at 50% blade height. The probes have been traversed circumferentially in the axial gap in front of the first stator (E2), in four planes inside the stator blade row (E2a, E2b, E2c, E2d) and in one measuring plane behind the first stator
733 (E3) which is indicated in fig.1. The stepping varied from 0.5 degrees inside the stator blade row to 0 . 5 - 1 degrees in the planes E2 and E3. During the experiments a sampling rate of 200kHz and a total measuring time of 2.62sec is choosen. This leads to a total number of 219 single measured values at each measuring point. A continuously recorded trigger, which is fixed to the rotor, enables a correct collection of the measured values into 4200 equidistant time windows - one window has the size of 4.926#s- per revolution. The samples in each time window are ensemble averaged, which leads to 4200 values of 5' and ci,2 for the 30 rotor pitches. An arithmetic averaging of the 30 rotor pitches in a phase locked mode delivers 140 values of -~' and ci~2 for an average rotor pitch [3],[4]. In this paper only data for the average pitch are presented. The absolute frame of reference for the evaluation of the mean square fluctuation components and an 'unresolved unsteadiness' calculated according to a formula normally referred to the turbulence intensity in macroscopically steady flows is the mainflow (subscript m]) and the crossflow (subscript c]) plane [5].
3. E X P E R I M E N T A L
RESULTS AND INTERPRETATION
3.1. Unsteady absolute velocity field The unsteady velocity field in E2, E2b, E2d and E3 is presented in figures 2 to 5. In the three dimensional plots the axis 'measuring position' means the probe position in the circumferential direction (see fig.l) and the axis 'rotor position' is the time axis related to the actual rotor position. A step of one degree rotor position corresponds to a time
Figure 2. Unsteady absolute velocity field in E2
step of 57.47#s. The unsteady velocity field behind the rotor in E2, presented in fig.2, reveals three characteristics of the flow. First the rotor wakes moving from the right to the left side diagonally through the flow field. They are characterised by a small velocity defect and a higher absolute velocity on the suction side of the rotor wakes. The downstream stator blades, too, exert an influence on the unsteady flow field behind the rotor. In the range of-9 t o - 4 and 6 to 11 degrees measuring position the absolute velocity is about 5~ lower due to the potential effect exerted on the
unsteady velocity distribution by the stator blades. Third the wake of the IGV shows at 1.5 ~ measuring position. It is responsible for the higher absolute velocity here (see [3]).
734
Figure 3. Unsteady absolute velocity field in E2b
Figure 4. Unsteady absolute velocity field in E2d
In E2b (fig.3) the characteristic velocity distribution inside a stator channel shows a higher velocity near the suction side of the blade compared to the pressure side. The drop is
Figure 5. Unsteady absolute velocity field in E3
about 13~ or 9.4% of the maximum velocity. In this measuring plane the diagonally moving rotor wake has a lower absolute velocity inside the wake compared to E2 but the increased velocity on the suction side has nearly disappeared. The velocity gradient inside the stator channel from pressure side to suction side has decreased in E2d, where it is only about 3.5% (see fig.4). The velocity defect due to the rotor wake can still be measured at this axial measuring position, but the minimum inside the rotor wake is lower.
Near to the suction side of the stator channel at 7.5 ~ measuring position the absolute velocity increases suddenly. This abrupt increase indicates that not the pressure gradient inside the stator channel but the influence of the inlet guide vane, which was characterised by an increase of the absolute velocity in E2, is the reason for the higher velocity at - 7 . 5 ~ measuring position.
735 Behind the stator in measuring plane E3 (fig.5) the unsteady velocity field is more complex with the additional disturbation caused by the wakes of the stator blades and the potential effect of the downstream second rotor. Spatially fixed at 3~ and -10.5 ~ measuring position the wakes of the stator blades cause a strong decrease of the unsteady absolute velocity. Additionally regions with a lower velocity moving diagonally through the diagram with a distance of 12 ~ that is one rotor pitch, can be seen. They are due to the potential effect of the downstream rotor and not the result of the upstream rotor wakes, because the spacing is very precise. The velocity defect due to the rotor wakes should be shifted by the velocity gradient inside the stator channel. The potential influence of downstream rotor prevails. This result will be proven in the following chapter by the presentation of the unsteady fluctuation components. 3.2. T u r b u l e n c e i n t e n s i t y d i s t r i b u t i o n first s t a g e The turbulence intensity, or unresolved unsteadiness, which is calculated from the mean square fluctuation components divided by the mean absolute velocity [3], gives a good insight into the behaviour of the unsteady flow field travelling through the stator. Considering the turbulence intensity in E2 (fig.6) most striking is the high turbulence in the wakes of the rotor blades with more than 5%. With a distance of 12~ rotorposition they move from the right to the left side through the three dimensional diagram. The wake of the IGV is also characterised by an increased turbulence intensity (3%). It is situated spatially fixed at 0.5 ~ to 3~ measuring position. At the position where the rotor and IGV wakes intersect the IGV wake is shifted, with the part on the suction side of the roFigure 6. Turbulence intensity distribution in E2 tor wake moving faster than that on the pressure side. Inside the stator channel in E2b presented in fig.7 the rotor wake is still visible due to its high turbulence intensity of 4.5%. The shape of the rotor wake is changing inside the stator channel. The absolute velocity near the suction side of the channel is higher so that the part of the rotor wake near to the suction side of the channel is transported faster downstream. This results in a twisting of the rotor wake in the three dimensional diagram. Near the pressure side of the stator channel occurs a spatially fixed region with a higher turbulence intensity. It can be assumed that the increased turbulence at this measuring position results from an IGV wake. Changing the operating point of the compressor leads to other positions of this region of increased turbulence inside the stator channel. This fact underlines, that the higher turbulence near the suction side of the stator channel is not generated by the velocity or pressure distribution inside the stator channel but originates from the IGV wake. Considering fig.8, where the turbulence intensity in E2d is shown, supports the observation
736 that an influence of the IGV wake on the unsteady flow field is still detectable there. In a measuring range of - 5 . 5 ~ to - 7 . 5 ~ a spatially fixed region with increased turbulence
Figure 7. Turbulence intensity distribution in E2b
Figure 8. Turbulence intensity distribution in E2d
intensity is visible, which can be identified as the IGV wake. Compared to the absolute velocity distribution the position of the IGV wake has changed, because the highest turbulence occurs inside the wake and the max. velocity in fig.4 comes from the suction side of the IGV wake. The maximum turbulence inside the rotor wake in E2d is less than
Figure 9. Turbulence intensity distribution in E3
in E2b with 4%. In the measuring plane behind the stator, presented in fig.9, the wakes of the stator with a turbulence intensity of about 10% are dominant. They are spatially fixed at 3.5 ~ and -10.5 ~ measuring position. In contrast to fig.5 the diagonal regions with an increased turbulence level are the wakes of the upstream rotor, transported through the stator. The time distance between two adjacent rotor wakes is no longer constant at 12 ~ but has diminished. The wakes are shifted clockwise due to the velocity gradient inside the stator pitch.
737 Considering the position of the interaction between stator and rotor wake, a difference of 6~ rotor position between the part of the rotor wake on the suction side compared to the part on the pressure side of the stator wake can be determined. Near the pressure side of the stator wake the curvature of the rotor wakes is strong. The IGV wake, too, is weakly visible at - 4 . 5 ~ to - 7 . 5 ~ measuring position. The increase of the turbulence intensity due to the IGV wake is not very strong behind the stator. It has been nearly mixed out. In fig.10 the way of the IGV wake through the first stage is reconstructed schematically from the measured unsteady flow data. The extension in circumferential direction in E2a and E2b could not be measured because the IGV wake is situated at the border of the max. possible probe traverse. The IGV wake follows an imaginary streamline on its way through the stator. Its circumferential extension grows from E2 to E3, it follows the curvature of the stator blade.
Figure 10. IGV wake in first stage
3.3. F l u c t u a t i o n c o m p o n e n t s in first stage The decay of the undisturbed rotor wake during the passage through the stator can be shown by selecting sections through the three dimensional diagrams at times (i.e. rotor positions), when the rotor wake is in the middle of the stator channel. The dotted line in fig.10 indicates the actual rotor wake positions choosen for the detailed examination of the unsteady flow components within the stator in fig.ll and fig 12. These positions are choosen because there neither the side walls nor the IGV wake exert an influence on the rotor wake. The evaluation of the absolute fluctuation components avoids the reference to - and thus the influence o f - the varying velocity level through the stator, as this per definition is the case for the turbulence intensity.
738 In fig.11 the mean square fluctuation components in mainflow direction c'~f ~ of the selected sections are plotted over the measuring position for all measuring planes. In E2 the m 2 fluctuations in mainflow direction in the rotor wake reach the highest value with 12 ~=. m 2
The fluctuations in the rotor wake decrease continuously down to E2d with 6~- and remain constant then down to E3. In the undisturbed regions outside of the rotor wake the minimum value of the fluctuations in mainflow direction is nearly constant at 2 ~-~ in all m 2 measuring planes from E2 to E2d, it increases to 3~- in E3 behind the stator.
cm~ [w]
12 12-
I0 10
84
~i
E2.c i~ E2b
~,
E3 . . \ j \ "',,
~
~/
o
,
/ ~
/
~
I
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~
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4
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2 E2b ~/
~:.
I
-5 -10 Circumferential Position [o]
Figure 11. Mean square fluctuation components in mainflow direction
\
J~L:.:
2
.... s
I
1I
8
! \
~ \ I
"-
I
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-5 -lo Circumferential Position [o]
Figure 12. Mean square fluctuation components in crossflow direction
In the measuring planes E2b and E2c the mean square fluctuations in mainflow direction c'~f 2 increase near the suction side of the channel due to the wake of the IGV. An increase near the pressure side of the channel is not fully understood yet. Considering the development of the mean square fluctuations in crossflow direction c'ci2 presented in fig.12 they are of the same level as the longitudinal fluctuations within the rotor wake in plane E2. Remarkably they decrease in the rotor wake from E2 to E2b by /rt 2 about 50% to 5.5 s~-~. The decrease continues to E2d, where 3 7 are reached. Additionally the shape of the rotor wake changes. The decrease of c'c/2 on the suction side of the rotor wake is smoother, the wake becomes broad. The fluctuation components in crossflow direction in E2b and E2c are higher near the suction side of the stator channel, too, due to the IGV wake, but no increase towards the pressure side is detectable. Figure 13 shows the maximum values of the fluctuation components inside the rotor wake over a nondimensional length scale lstage/laziaZ. Behind the rotor the unsteady flow field inside the rotor wake is nearly isotropic, with c'/2ma x = C'~I2max" Entering the stator channel the fluctuations in crossflow direction decrease exponentially whereas the fluctuations in mainflow direction decrease linearly. The plot of the ratio c'i2max/ Ccf ' 2max underlines this phenomenon. It increases from 1 to 2.5. The description presented here of the decay of the fluctuation components in the rotor wake through the stator by measurements at different axial positions and time instants
739
m f Ctcf 2
Ct
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mf m2
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I
I
50
60
70
80
90
i00
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l~o~/l~o~ [%] Figure 13. Maximum of fluctuation components
is open to a number of criticisms in detail. For example it cannot be ensured, that the maximum of the wake has always been captured. Therefore the stepping of the probe in circumferential direction has to be diminished. Additionally a comparison with measurements at other operating points of the compressor would be helpful in order to give a more general impression of the wake decay. But, as other investigations have shown, the comparison with the results at other time steps is consistent with the results presented here, only the absolute range of the values is slightly changing.
4. C O N C L U S I O N S This paper gives details of the unsteady absolute velocity field and the turbulence field mid plane at 50% blade height of the first stage of a two stage low- speed axial compressor. At six different axial positions in front of, inside and behind the first stator the unsteady flow field has been examined in space and time. In detail it is shown:
1. the IGV wake moving through the stator blade row. It is situated at spatially fixed positions, following the curvature of the stator blade. The turbulence intensity in the IGV wake decreases slightly and the wake becomes broader, but is not fully mixed out behind the stator. 2. the characteristics of the rotor wakes at different axial positions within and behind the stator blade row. The rotor wake is shifted clockwise inside the stator blade row due to the velocity gradient between the suction and the pressure side of the channel. The turbulence intensity in the rotor wake decreases on the way through the stator, the longitudinal and the lateral fluctuations show a distinctly different decay mode. REFERENCES Traulsen, D., Axialverdichterpriifstand zur Untersuchung yon RandzonenstrSmungen, Fortschrittberichte VDI, Reihe 7, Nr. 161 (1989)
740 2. Sentker, A., Riess, W., Turbulence Intensity Measurements in a Low Speed Axial Compressor, Engineering Turbulence Modelling and Experiments 3, W.Rodi and G.Bergeles, Eds., pp. 773-783, Elsevier Science B.V., 1996 3. $entker, A., Riess, W., Unsteady Flow and Turbulence in a Low Speed Axial Compressor, 8th International Symposium on Unsteady Aerodynamics and Aeroelasticity of Turbomachines, Stockholm, 15.-18. Sept. 1997 4. Sentker, A., Riess, W., Measurement of unsteady flow and turbulence in a low speed axial compressor, Experimental Thermal and Fluid Science 17, p. 124-131, 1998 5. Zaccaria, M.A., Lakshminarayana, B., Unsteady Flow Field Due to Nozzle Wake Interaction With the Rotor in an Axial Flow Turbine: Part I - Rotor Passage Flow Field, Journal of Turbomachinery, Vo1.119, pp.201-213, 1997 Acknowledgment The Deutsche Forschungsgemeinschaft (DFG) supported the construction of the Low Speed Axial Compressor (NGAV) in the program SFB 211. The authors want to express their sincere gratitude.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
741
A n e x p e r i m e n t a l study of the u n s t e a d y characteristics of the turbulent w a k e o f a turbine b l a d e Marina Ubaldi and Pietro Zunino Istituto di Macchine e Sistemi Energetici, Universit?a di Genova Via Montallegro 1, 16145 Genova, Italy An experimental investigation on the time-varying characteristics of the flow in the turbulent wake of a large scale turbine cascade is described. An ensemble averaging technique has been applied to LDV instantaneous data in order to separate coherent contributions due to the vortex structures from random contributions due to turbulence. Organised periodic structures have been identified in the flow and their evolution in time has been documented. 1. I N T R O D U C T I O N Wakes from turbine blades are unsteady in character because of the presence of large organised vortex structures, known as von Karman vortex streets. Vortex shedding represents an important cause of energy losses, blade dynamic loadings, mechanical vibrations and noise. In spite of the importance of this phenomenon the flow in the wake behind turbine blades is generally viewed as the superposition on a time averaged flow of an unresolved turbulence comprehensive of all the unsteady fluctuations regardless of their scale. The technical literature is still scarce of contributions concerning the time-varying characteristics of the blade wakes [ 1], because of the experimental difficulties associated with the large gas velocities and small blade trailing edge dimensions resulting in very large shedding frequencies and scarce spatial resolution of measurements. Therefore our present knowledge on the mechanism of vortex shedding from blade trailing edges is still mainly based on the results of detailed experimental investigations behind cylinders (e. g. Cantwell and Coles [2], Lyn et al. [3]). The present contribution is part of a research activity carried out within the framework of a Brite Euram project and aimed at the improvement of the knowledge on the time-varying turbine wake flow characteristics. The general program included systematic investigations on large scale models carried out in different laboratories making use of complementary experimental techniques. The present paper reports results of a LDV detailed investigation in the turbulent near wake of a turbine blade in cascade. A phase-locked data acquisition and ensemble average technique suitable for LDV measurements has been developed and used together with the triple decomposition scheme for the instantaneous velocity to identify the periodic organised structures embedded in the turbulent wake flow. 2. E X P E R I M E N T A L DETAILS
2.1. Test facility and instrumentation The experiment was conducted in the IMSE low speed wind tunnel. The facility is a blowdown continuously operating variable speed cascade tunnel with an open test section of 500x300 mm.
742 The flow was surveyed downstream of the central blade of a three-blade large scale turbine linear cascade. The blade profde, designed at VKI (Cicatelli and Sieverding [4]), is representative of a coolable hp gas turbine nozzle blade with a large trailing edge (D/c = 0.05). A three-blade configuration has been selected allowing larger blade dimensions within the given test section with the purpose of lowering the vortex shedding frequency and increasing the spatial resolution of the measurements. The cascade is shown in Fig. 1 The main geometrical parameters of the cascade and the test conditions are summarised in Table 1. The coordinates of the blade are given in [4]. The three blades are instrumented at mid-span with a total of 110 pressure tappings. Periodicity condition was monitored by comparing the pressure distributions on the central blade and the two adjacent ones and achieved by modifying the curvature of the adjustable tailboards and throttling two lateral apertures at the cascade inlet [5]. In the present experiment a four-beam two-colour laser Doppler velocimeter with backscatter collection optics (Dantec Fiber Flow) has been used. The fight source is a 300 mW argon ion laser tuned to 488 nm (blue) and 514.5 nm (green). The probe consists of an optical transducer head of 60 mm diameter connected to the emitting optics and to the photomultipliers by means of optic fibres. With a front lens of 400 mm focal length and a beam separation of 38 rnm, the optical probe volume is 0.12 mm of diameter and 2.4 mm of length. The probe volume was oriented with the larger dimension along the spanwise direction, in order to have better spatial resolution in the blade-to-blade plane. A Bragg cell is used to apply a frequency shift (40 MHz) to one of each pair of beams, allowing to solve directional ambiguity and to reduce angle bias. The signals from the photomultipliers were processed by two Burst Spectrum Analysers. The probe was traversed using a three-axis computer controlled probe traversing system with a minimum linear translation step of 8 gin. The flow was seeded with a 0.5-2 gm atomised spray of mineral oil injected in the flow at about 2 chord upstream of the cascade leading edge.
Table 1 Cascade conditions.
geometry
and
test
Cascade Geometry
Chord length c = 300 mm Pitch to chord ratio g/c = 0.7 Aspect ratio h/c - 1.0 Inlet blade angle [3 '1 = 0~ Gauging angle [3'2 = 19.1 ~ Test Conditions Relative inlet total pressure P,1 = 3060 Pa Inlet total temperature T~ = 293 K Inlet turbulence intensity Tu = 1% Outlet isentropic Mach number M2is = 0.24 Outlet Reynolds number Re2is = 1.6.106
Fig. 1 Test section with the three-blade cascade installed.
743
2.2. Measuring procedure According to the triple decomposition scheme proposed by Reynolds and Hussain [6] for the study of coherent structures in shear flows, a generic velocity component v in a generic position P can be represented as the sum of the time averaged contribution V, the fluctuating component due to the periodic motion (~" - V) and the random fluctuation v'. v-V+(~-V)+v' (1) The time-varying mean velocity component ~" can be obtained by ensemble averaging the samples. 1 K, ~'(i) = - - ~ ~ v(i,k) (2) /~-/" k _-1
i = 1 ... I is the index of the phases into which a vortex shedding period is subdivided and k = 1... K i is the index of the samples for each window associated with a particular phase i. The time-averaged velocity component V can be simply determined as
where
I
V= -12; ~'(i) I
(3)
i=1
Ensemble averaged variances which are proportional to the Reynolds stress normal components are obtained as v'2 (i) :
(K i
[v(i,k)-~(i)]
(4)
In the present experiment an ensemble averaging technique suitable for LDV data processing has been implemented. The technique makes use of a single-sensor hot-wire probe which is sensitive to the passage of the organised structures to generate a reference signal. The hot-wire output is band-pass filtered, amplified and processed by means of an electronic device capable of producing a suitable TTL signal of 5 ~ts duration in correspondence of the maximum of the hot-wire signal. This phase information is provided to the synchronisation input port of the BSA processors operating in "encoder enabled mode". The arrival time of each LDV velocity realisation is recorded together with the phase reference signal. To allow the ensemble average, the data are sorted into 50 phase bins each representing a particular phase of the vortex shedding cycle. To obtain statistically accurate ensemble averages of the LDV velocities, 50000 validated data for each component have been sampled at each measuring point. For a vortex shedding frequency of about 1300 Hz the bin width is 15 ~ts and the mean number of samples for each bin is 1000.
2.3. Measurements accuracy A specific evaluation of the errors for frequency domain processors is given by Modarress et al. [7]. For the present experiment, the resolution of the BSA processor, depending on the record length of the FVI" and on the background noise, was below 1 per cent of the mean velocity, even in the worst cases. Ensemble averaging procedure, where the samples in each bin are statistically independent because collected during different cycles, should be exempt from statistical bias [3]. Angle bias was minimised by moving the fringe pattern in the probe volume by means of the Bragg cell. Statistical uncertainty in mean and rms velocities depends on the number of sampled data, turbulence intensity and confidence level [8]. Considering a typical value of 1000 sampled data, a confidence level of 95 and 20 per cent of local turbulence intensity, uncertainties of 1.3 and 5 per cent are expected for the mean and rms velocities, respectively.
744
3. RESULTS AND DISCUSSION The time-varying characteristics of the blade wake flow are strongly influenced by the boundary layer state at the blade trailing edge (Sieverding and Heinemann [9]). The blade boundary layer development mainly depends on the surface velocity distribution and free stream turbulence. The mid-span surface velocity distribution is typical of a front-loaded profile with a moderate suction side diffusion rate [5]. Boundary layer measurements along the blade profile described in details in [5] show that, in case of natural development, the boundary layer at the trailing edge is turbulent on the suction side and laminar-transitional on the pressure side. The mixed boundary layer condition at the trailing edge increases the Strouhal number and results in a broader vortex shedding frequency spectrum [9]. The necessity to have well defined and reproducible boundary layer conditions at the trailing edge has led to the decision to enforce turbulent conditions also on the pressure side by installing a 0.4 mm diameter trip wire. Instantaneous velocities were measured in several points in the wake by means of a single sensor hot-wire probe, in order to get a preliminary overview of the time-varying characteristics of the flow [10]. A peak of energy, corresponding to the vortex shedding frequency has been observed at about 1300 Hz. The corresponding Strouhal number S = f D / U 2 is 0.26.
3.1. Mean flow properties Before analysing the periodic structure of the unsteady flow in the wake, a survey of the mean flow field (Fig. 2) can be appropriate, as it is an important term of comparison with numerical prediction and represents the base flow upon which the motion of organised structures is superimposed.
,.,.
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Fig. 3 Instantaneous patterns of the periodic velocity components. Streamwise velocity distribution U,./U 2 shows the extension and the rapid decay of the wake velocity defect. Because of the continuity condition, large transverse mean velocity components U, / U 2 of opposite sign are present on the two sides of the wake. This effect extends in cross-stream direction well beyond the wake defect region and free-stream flow is drawn towards the wake centre. Both streamwise velocity distribution and vector plot show the extension of the mean separated flow. The mean closure point is about 0.7 D. For circular cylinders this quantity is
746 sensitive to the Reynolds number based on diameter and freestream velocity Re v. Cantwell and Coles [2] estimated x,/D = 1 for Re D = 140000, Mc Killop and Durst [11] x/D = 1.65 for Re D = 15000. Both experiments were performed in the so called "shear layer transition" region [ 12], characterised by laminar separation from the surface and turbulent transition in the shear layer. In the present investigation the Re D value (Re D = 85 000) is intermediate. In spite of that, the boundary layer state at the trailing edge is turbulent on both sides, depending on the large blade chord Reynolds number (Re2/s = 1.6-106), the streamwise velocity gradients and the use of a trip wire on the pressure side. Boundary layer conditions at the separation points explain the reduced length of the mean recirculating region found in the present investigation as well as the large value of the Strouhal number (S = 0.26) which is characteristic also of the shedding from circular cylinders with Re D > 106 [ 13].
3.2. Ensemble averaged velocity components Figures 3 to 5 show the instantaneous images of the periodic characteristics of the wake flow for four instants in a period (t/T = 0, 0.25, 0.50, 0.75). The periodic part of the streamwise velocity components U s (left of Fig. 3) shows an antisymmetric periodic structure with peaks and valleys which alternate from the pressure to the suction side of the wake, proceeding in streamwise direction. The periodic part of the cross-stream component U, (right of Fig. 3) is structured in cores of positive and negative values, approximately centred in the wake, which alternate. The nuclei of U, appear displaced in respect of those of U,, of about half wavelength in streamwise direction. The kinematic structure of the periodic flow, as comes out from the present results, looks qualitatively similar to that presented by Cantwell and Coles [2] for the wake downstream of a cylinder, but the magnitude of the periodic velocity components, made non-dimensional with the downstream velocity U 2, is much lower. This difference may be attributed to various factors, like a less defined vortex shedding frequency, the boundary layer state at separation, the ratio of the boundary layer thicknesses to the distance between the separation points. The combination of the two velocity patterns (U,.-U~) and ( U , - U , ) gives rise to the rolling up of the periodic flow in a row of vortices as shown by the vector plots in Fig. 4.
3.3. Vorticity Identification of coherent structures and analysis of theft evolution in time have been carried out with the aid of the vorticity of the periodic flow ( ~ - ~ ) . In Fig. 4 the vortices are highlighted by zones of positive and negative intense spanwise vorticity alternating in streamwise direction. The vorticity patterns are structured into individual nuclei of opposite vorticity located on opposite sides, at a small distance from the wake centerline, in close agreement with the finding of the Cantwell and Coles investigation [2]. These nuclei extend backward by means of ribs embracing on the two sides the incoming nucleus of opposite vorticity. The ribs on the two sides of each vortex are characterised by a different level of vorticity with larger values on the side from which the vortex has been shed off. Local peaks of vorticity in the region x/D < 3 are very high. Peak values of the order of twenty for the quantity (o3 - ~ ) c ! U 2 indicate that the local periodic vorticity is as large as twenty times the mean streamwise acceleration (U2 / c) that the flow undergoes through the cascade. The vortex formation model of Gerrard [14] is very useful for interpreting instantaneous pictures of unsteady flow in the vortex formation region. This model postulates that at the end
747
Fig. 4 Instantaneous vector plots and vorticity patterns of the periodic flow. of the formation process of one vortex, fluid from the opposite side bearing vorticity of opposite sign is drawn across the axis. When this vorticity equals and cancels out the local vorticity, the vortex is shed off. In Fig. 4 the vortices shed from the suction side are clockwise and the associated vorticity is negative (black). At t/T = 0.25 (second flames from the top), a clockwise vortex extending from x/D = 0 on the suction side to x/D= 1.7 on the centreline, characterised by negative (black) vorticity, is shown just at the shedding instant. Fluid entrained from the pressure side is rolling up in a counterclockwise loop filling the base region. The positive (white) vorticity of
748
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0I)~:::~3~ -1 L t/T = 0.75
1 o
1
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. . . .
,
1
. . . .
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,
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,
5
. . . .
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Fig. 5 Instantaneous patterns of the normal Reynolds stresses. this fluid tends to cancel out the local negative vorticity on the suction side at x/D = 0.0, allowing the shedding of the clockwise vortex. At t/T = 0.50 (third frames), the clockwise suction side vortex has been shed off, while the core of negative vorticity associated with the pressure side vortex in formation is growing, being fed by the flow from the pressure side. At t/T = 0.75 (fourth frames), half period after the suction side vortex shedding (t/T = 0.25), the periodic flow is in opposition of phase, the counterclockwise pressure side vortex, just before the shedding instant, possesses the maximum of negative vorticity. Positive vorticity is now entering in the base region from the suction side and the clockwise vortex begins to form.
3.4. Normal Reynolds stresses The possibility of separating the random fluctuations from the periodic part of the signal allows the evaluation of Reynolds stress normal components due to random fluctuations only. Fig. 5 shows that streamwise and cross-stream normal Reynolds stresses are rather high in the wake region, with maxima of equivalent turbulence intensity larger than 22 and 26 per cent, respectively. Different distributions, with a double peak structure for the streamwise normal component and a single maximum centred in the wake for the transverse normal component, and different intensities cause large turbulence local anisotropy. A characteristic feature of the
749
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'
y/D s ~"s"
~./J~~'~o
f 0Fj Co
-1 '
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~
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"
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3
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-1
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. . . .
'
1
. . . .
'
2
.
.
.
.
.
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Fig. 6 Time averaged normal stresses due to the periodic velocity fluctuations. Reynolds stress pattern is the low turbulence intensity of both components in a narrow zone in proximity of the trailing edge surface. The periodic nature of the wake flow characterises also the turbulence quantities, as shown by the periodic change of curvature of all the contours. Considering the momentum equations for the phase averaged flow [2], only the true Reynolds stress terms, which result from the turbulence models in numerical predictions, appear. On the contrary, if global (time-averaged) equations are considered, additional terms due to the correlation of the large-scale periodic fluctuations have to be evaluated and introduced into the momentum balance [2]. Figure 6 shows the distributions of the streamwise and cross-stream time-averaged normal stresses 6 2 and 0 2 due to the velocity periodic fluctuations only. The periodic normal stress patterns reflect the distribution of the corresponding velocity fluctuations with a couple of peaks of different magnitude on the two sides of the wake for ~2 and a centred core of larger values for L72. . These correlations are lower compared with those due only to random fluctuations and their rate of decay in streamwise direction is sharper, as can be expected by the different mechanisms of dissipation. Nevertheless from the present results it is clear that the periodic contribution to the diffusive term should not be neglected, when considering the momentum balance of the mean flow in the near wake of the blade. 4. C O N C L U S I O N S Time-varying properties of the flow in the near wake of a large scale turbine blade in cascade have been experimentally investigated in detail by means of a two-component laser Doppler velocimeter and analysed by using a phase-locked ensemble averaging technique allowing the decomposition of the flow into mean, periodic and random contributions. Due to the turbulent state of the boundary layers at the separation points, the time-mean recirculating flow zone at the trailing edge is appreciably shorter compared with the ones of preceding investigations on circular cylinders in the "shear layer transition region". The rather high value for the Strouhal number (S = 0.26) of the present experiment is consistent with the short separation bubble and with the experimental values for cylinders in the range of Re D larger than 106. The structure of the periodic flow is qualitatively in agreement with that described by Cantwell and Coles in the near wake of a cylinder at ReD= 140000, but the amplitude of the periodic fluctuations in the present case are weaker.
750 Reconstruction of the time-evolution of the periodic vector field and the associated periodic vorticity reveals the existence of a regular vortex street with peaks of opposite vorticity located very close to the wake centreline. These results support entirely the model of vortex formation proposed by Gerrard. Normal Reynolds stresses associated to the random fluctuations are larger and decay more slowly in streamwise direction compared with the normal stresses associated to the large scale periodic motions. However in the base region of the blade wake the two contributions are comparable and therefore stresses due to periodic velocity fluctuations cannot be neglect in the time-averaged momentum balance. The present results provide an organised data base suitable for unsteady code validation.
ACKNOWLEDGEMENTS This work was performed under the BRITE-EURAM Contract AER2-92-0048, endorsed by MTU Munich and SNECMA Paris. These supports are gratefully acknowledged. NOMENCLATURE c chord D trailing edge diameter f frequency t time T period of one vortex shedding cycle U velocity x streamwise direction at the cascade outlet y cross-stream direction at the cascade outlet
03 vorticity Subscripts s in the streamwise direction n in the cross-stream direction 2 at the cascade outlet Superscripts and overbars ' fluctuating component time averaged -ensemble averaged
REFERENCES 1. G. Cicatelli and C. H. Sieverding,. AGARD PEP 85th Symposium on Loss Mechanisms and Unsteady Flows in Turbomachines, Derby, 1995. 2. B. Cantwell and D. Coles, J. Fluid Mech. 136 (1983) 321. 3. D.A. Lyn, S. Einav, W. Rodi, J.-H. Park, J. Fluid Mech, 304 (1995) 285. 4. G. Cicatelli and C. H. Sieverding, ASME Paper No. 96-GT-359 (1996). 5. M. Ubaldi, P. Zunino, U. Campora and A. Ghiglione, ASME Paper No. 96-GT-42 (1996). 6. W.C. Reynolds and A.K.M.F Hussain, J. Fluid Mech. 54 (1972) 263. 7. D. Modarress, H. Tan and A. Nakayama., Fourth International Symposium on Application of Laser Anemometry to Fluid Dynamics, Lisbon, 1988. 8. A. Boutier, Lecture Series 1991-05, VKI, Brussels (1991). 9. C.H. Sieverding and H. Heinemann, ASME Paper No. 89-GT-296 (1989). 10. P. Zunino, M. Ubaldi, U. Campora and A. Ghiglione, Proc. 2nd European Conference on Turbomachinery- Fluid dynamics and Thermodynamics, Antwerpen (1997) 247. 11. A. Mc. Killop and F. Durst, Third International Symposium on Application of Laser Anemometry to Fluid Dynamics, Lisbon, 1986. 12. C. Williamson, in Fluid Vortices (S. I. Green ed.), Kluwer, 1995. 13. R. D. Blevins, in Fluid Vortices (S. I. Green ed.), Kluwer, 1995. 14. J. H. Gerrard, J. Fluid Mech., 25 (1966) 401.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
751
E x p e r i m e n t a l guidelines for retaining energy-efficient axial f l o w rotor cascade operation u n d e r off-design c i r c u m s t a n c e s J. V ad and F. Bencze Department of Fluid Mechanics, Technical University of Budapest Bertalan Lajos u. 4 - 6., H-1111 Budapest, Hungary. Email" [email protected] 1. INTRODUCTION Axial flow turbomachines are optimised for a given flow rate - total head rise parameter couple, i.e. the geometry of rotor cascade and guide vanes are designed such a way that the machine is expected to achieve a maximum hydraulic efficiency at its design point [e.g. 1, 2]. However, the operating state of a turbomachine usually differs more or less in a practical application from the design point, resulting in reduced hydraulic efficiency. This is partly due to the fact that for off-design conditions the flow through the rotor cascade sections declines from the flow according to the design conditions for which the geometry of cascade sections have been optimised. It would be desirable to design rotor cascades of which geometry is quasi-optimal also under off-design circumstances, i.e. the turbomachine would be characterised by high hydraulic efficiency at the design point and also over a wide surrounding flow rate range. This demand must be emphasised from the viewpoint of rational use of input energy in the case of high pressure (high performance) turbomachines. There are a number of publications dealing with detailed investigation on the flow field developing in axial flow turbomachines of design as well as off-design points [e.g. 3-6]. These studies usually focus on the fluid mechanical behaviour of the particular machinery (which is only one test rotor in most cases) and do not outline generally applicable guidelines for design of rotors retaining high hydraulic efficiency also under off-design circumstances. The present paper aims to satisfy this need, on the basis of the following idea: since the geometry of rotor blades is fixed (in general, not even the stagger angle is adjusted), quasi-optimum operation of the "elemental rotor cascades" building up the rotor can be retained only if the threedimensional (3D) rotor throughflow changes with operating state such a way that the modified stream surfaces relative to the blading pass the rotor fitting to sloping blade sections of which geometry corresponds better to the off-design flow conditions from efficiencypoint of view. It is clear that the radial flow developing within the rotor must have a key role in this "selfadjusting" process. The role of radial velocity components in cascade operation has been discussed in [7]. This paper reports detailed and systematic experimental investigation of three high pressure axial fan rotors, at design and off-design operating points, with special regard to operating state dependent radial fluid re-arrangement inside the rotors. On the basis of the experiments, guidelines are presented as a contribution for design of rotors retaining high hydraulic efficiency also at off-design operating points.
752 2. EXPERIMENTAL FACILITY, INSTRUMENTATION, AND TEST FANS The experimental program was carried out in the axial flow fan facility at the Department of Fluid Mechanics, Technical University of Budapest. The facility, instrumentation and measurement method is described in detail in [8-10]. The three rotors on investigation, designated with BUP-26, BUP-29 and BUP-103, are characterised with the following common parameters: 0.630 m casing dia, hub-to-casing ratio v=0.676, N=12 straight (unswept) circular arc plate blades, design flow coefficient c/~=0.5, average tip clearance r=3 mm. They were run at a speed of 1100 rpm. The non-free vortex design of the fans has been described in [2, 7]. The details of the airfoil geometry and fi~her design parameters are given in Table 1. Figure 1 shows the performance of the rotors measured with stator. The performance curves will be discussed in Section 4. A single-component laser Doppler anemometer (LDA) system is connected to the fan facility. A measurement method has been elaborated for 3D non-simultaneous LDA velocity measurements outside the rotor cascades, as discussed in [10]. The LDA measurements reported in the paper were performed with isolated rotors, i.e. no prerotator or straightener blades were installed. The 3D flow field downstream of the rotors was probed in planes perpendicular to the duct axis (Fig. 2). Measured at hub radius, these planes were located at an axial distance of 8 mm upstream of leading edges Of BUP-29 and downstream of trailing edges of rotors BUP-29 and 103, and at 5 mm downstream of trailing edges of BUP-26. The flow field downstream of one selected blade passage was measured from hub to tip for each rotor. The central angle of the annular sector measurement region was set to 35 ~ (1 1/7 pitch) to measure one blade wake region contiguously and to check the periodicity in measurement results. Considering the length of LDA probe volume (3 mm) and uncertainty of angular data readout (0.15~ the measurement range was resolved to cells of 5.0 percent blade height (5.0 mm) in radial direction and 3.3 percent pitch (1.0 ~ in circumferential direction. The three velocity components were determined from the ensemble averages of 3 X 100 LDA data for each cell. Error estimation for the LDA measurements has been briefly discussed in [7]. The estimated LDA experimental uncertainty is summarised in Table 2.
1 0.9
....
i
..... B
I :29 .............................................. I
CASING WALL ~ CASCADE MIDPLANE ]~ ~ i LDA INLET ~; BUP-26, "1" ~" 29 "q " ~i PLANES: FLOW ~ /~B%Ue103,1.1:/,i/~ /BUP-26 ~1"-1 i g//BUP-29 NOSE Iii ] ~ i ~ //BUP-103
"I'~
0.8 0.7
I~.ON-U.
0.6
I; I I
~
/~|
:Ix
. . . . .
0.5
\
A PLANE: BUP-29
0.4 0.3 0.2 0.1 0.2
0.3
0.4
0.5
0.6
0.7
Figure 1. Characteristic and efficiency curves
\ D U C T AXIS
Figure 2. LDA measurement planes
753 Table 1 Rotor airfoil ~eometlT and desil~n parameters BUP-26 BUP-29 tip root mid tip root mid R 0.676 0.833 0.990 0.676 0.833 0.990 ~3D 0.430 0.495 0.561 0.419 0.494 0.573 ~3D 0.446 0.578 0.714 0.492 0.665 0.852 (g/t) 1.219 0.984 0.833 1.534 1.237 1.047
rcamber/mm y/deg
318.0 46.8
396.2 40.5
480.3 36.2
360.7 47.9
425.5 42.2
Table 2 General uncertain~ of the presented LDA measurements c, Bias limit (relative errorX100) 3.5 Precision limit (relative errorX100) 2.0 Uncertaint), (relative errorX100) 4.1 *absolute error, based on (uc/100)
493.1 38.3
c~ 0.6 1.3 1.5
root 0.676 0.410 0.529 1.830
BUP-103 mid 0.833 0.493 0.747 1.476
tip 0.990 0.581 0.991 1.249
397.4 48.8
441.4 44.0
486.3 40.5
c~ 0.3* 0.1" 0.32*
3. LDA MEASUREMENTS AND DISCUSSION During the LDA measurements, the rotors were operated at design and off-design flow rates as indicated in Fig. 1 with triangles and designated as operating points D (design flow rate), L (flow rate lower than design), and H (flow rate higher than design). The Reynolds numbers (based on the airfoil chord length and relative inlet flow velocity at midspan) for the three rotors under circumstances L, D and H were within the range 3.05-105 + 5.03.105. 3.1. Pitchwise-resolved data
In order to give a lifelike view on the nature of flow processes occurring in the rotor, secondary flow has been derived from the detailed 3D flow data measured downstream of the bladings. Using the defmition and calculation method of Inoue et al. [4], the secondary flow was obtained as a velocity component perpendicular to the relative theoretical flow direction. The theoretical flow was considered as an ideal flow pattern corresponding to the design concept based on two-dimensional stationary cascade data [7, 9]. Vector plots of the secondary flow are shown in Fig. 3. Considering the relatively high hub-to-tip ratio and low blade pitch values, authors felt admissible to transform the annular sector measurement region into a rectangle during representation. The plots are extended to a 40 deg angular range by means of copying the 6...11 deg section to the end of the measured range in order to get a better view of the flow structure. In Fig. 3, the blading moves from "the left to the right" (pressure side on the left, suction side on the right within the 30 deg blade passage outlet region, bounded by well-recognizable blade wake zones). The special flow phenomena developing near the annulus walls (vortices etc.) are not discussed here, their description can be partly found in [9]. Instead, this paper focuses on the radial fluid rearrangement within the region farther from blade passage endwalls - termed the
754 main flow. For operating points D, a vortical secondary flow pattem filling the largest part of the measurement range can be observed for each rotor in Fig. 3. It consists of suction side outward and pressure side inward radial flow branches, linked with an overturning zone near the hub and an undertuming zone near the casing wall. As already discussed in [7], this secondary flow (abbr. NFVO flow for "Flow due to Non-Free Vortex Operation") develops according to the spanwise change in blade circulation, which is a feature of non-free vortex operating rotors. The NFVO flow is weaker for operating points H, according to the reduced spanwise change in blade circulation. For points L, a drastic outward flow exists near the casing wall and the NFVO flow loses its circular pattern due to a strong blockage effect of the fluid stagnating near the casing wall. Fig. 3 suggests that the 3D flow pattern and thus, the radial fluid re-arrangement developing within the main flow is highly dependent upon the operating state of the rotor.
Rotor BUP-26: Operating point D"
Operating point L:
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,.....I,I.~.,.
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20
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0 10
20
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Operating point H: 9 " - ~ _ ~ - ~ , , ~ ' ~ ~ - - _ ~ ~ - c ~
v. tzztt~t ,it iliiit;: " 9 0 ~ ~ ! i ......... ...........-'<-/~}~i,~t
40~ R
0 10
20
30
40~
Figure 3. Secondary flow vector maps. Scaling: linear section of 10 ~ on the horizontal axis corresponds to 0.4 uc secondary velocity In order to get additional information for a better understanding of flow processes developing in rotating axial flow cascades of non-free vortex operation, fluctuations in the outlet velocity field have also been studied in detail. The method applied for 3D LDA
755 measurements was a non-simultaneous one by which only the axial velocity was determined independently from the other velocity components. Thus, authors felt reasonable to study the fluctuating behaviour of the axial velocity only. This investigation has been proved to be sufficient for refinements in the qualitative interblade flow model. For characterisation of velocity fluctuation, the following definition has been used: Zu x _- 4
c7__
(1)
-~2
r
,for which the ensemble average and mean square values have been calculated from the LDA measurement data obtained in each measurement cell. Considering the reasonably high number of LDA data and the small-size, monodisperse seeding particles applied, Tuxhas been accepted by authors as a quantity which is suitable for representing the turbulent nature of internal flow. Figure 4 shows the contour plots of Tu,, downstream of the three test rotors, at the design point t/~D=0.5. (The representation format of Fig. 3 is also valid for Fig. 4.) As the plots show, the flow near the annulus wall and in the blade wakes is characterised by higher fluctuation values, behaving as expected and also pointed out by earlier measurements [4]. The more the blade load and thus, the performed total head rise is (i.e. rotors BUP-26, BUP29 and BUP-103, in this sequence), the wider the zones of relatively high Tu x values representing the blade wakes are. The flow in the blade boundary layers appears to influence the main flow and this effect manifests itself in higher Tug values in the main flow close to the blade wakes. Main flow zones of lowest turbulence level are located approximately in the central part of the blade passage region, embedded in regions of higher turbulence level originated from the blade passage endwalls. The average turbulence level in the main flow increases with blade load. Since the inlet flow conditions are identical for the three rotors, it must be concluded that the increased turbulence level in the main flow is originated predominantly from a main flow- blade boundary layer flow interaction.
1.00I~_.7~~
Rotor BUP-26: ~
0.90
Rotor BUP-29: 1 . 0 0 ~ ~ ~ ~ _ ~ 0 . 9 0 ~ - " 0.(05 ~
t
R
~
0 0t t9 10
20
30
40~
0 10
20
Rotor BUP-103" 0.90~~~'~~1"00~~~-~
"/l~~0~.2
0 0t/
30
40~
( ~ ~ ~ / f
0 10
20
30
40~
Figure 4. Comour plots of outlet axial velocity fluctuation Tu x for operating points D
3.2. Pitchwise-averaged data Since the rotor cascade design methods are based in most cases on two-dimensional (2D) stationary cascade measurement data corresponding to pitchwise-averaged velocity components assumed upstream and downstream of the rotor [1, 2, 7], discussion on the pitchwise-averaged flow measurement results is of great importance from design viewpoint.
756 Earlier experiments using a miniaturised Prandtl probe upstream of the rotors showed that the inlet axial velocity profile remains nearly identical for the three rotors at the same flow rate and can be approximated very well with a linear function for the nose cone configuration applied (i.e. very thin annulus wall boundary layers have been experienced). Therefore, LDA measurements on the inlet axial velocity profile has been carried out only for rotor BUP-29 in order to determine the axial velocity gradients. The LDA measurement points and the fitted linear profiles are shown in Fig. 5. The inlet flow rates calculated from these linear profiles agree very well with the measured flow rates, confirming again the appropriateness of linear approximation. ,
,
|
ROTOR BLADING EXAMPLE: Solidity, camber curvature and stagger angle decreases with radius. Presence of energy-efficient cascade sections:
i
CASING ....................... -IP" H
. . . .ww-I .
r0 I
0.7
0.75
I
I
!
0.8
0.85
0.9
I
:._?.-;-22222~:ST
.............. -q--..................... ]'r3 D II~ L
HUB
0.95R
Figure 5. Axial velocity data at inlet
Figure 8. Scheme of energy-efficient conical cascades
Figure 6 presents the pitch-averaged ideal total head rise coefficient and axial flow coefficient profiles along the radius, which were calculated from the measured outlet LDA data. During averaging, the ideal total head rise coefficient values were weighed with the axial flow coefficients. With modelling assumption of conical stream tubes through the rotor, a lifelike indicator of radial fluid arrangement is the conicity of stream tubes, which is characterised with the cone half-angle of stream tubes (see Fig. 8 above, next to Fig. 5 for explanation):
e=tan"[(r3-ro)/h ]
(2)
where r0 is the radius at which a particular stream tube enters and r3 at which it exits the rotor. If measured inlet and outlet axial velocity profiles are considered, h is the axial distance between the inlet and outlet measurement planes, which is approximated now as the blade width. Related r3 and r0 values can be calculated with numerical integration of the continuity equation, with knowledge of the inlet (Fig. 5) and outlet (Fig. 6) axial velocity profiles: I bo Ro dR0 = I b3 R3 dR3 V
V
The calculated c distributions are shown in Fig. 7 versus the mean radius.
(3)
757 1.1 1 BUP-26 0.9 ..... g//3(R) 0.8 0.7 0.6 0.5 0.4
BUP-29
/"
s
0.9 0.8 0.7
.
.
.
.
aCRJ
.
1.11.
0.9 ..... 0.8
/
"'~' "
,*"
06
0.6
0.5 0.4
0.4
/./ /--" ,D
0.3
0.7 0.75 0.8 0.85 0.9 0.95 R
0,7 0.75 0.8 0.85 0.9 0.95 R
0.7 0.75 0.8 0.85 0.9 0.95 R
~l/3(R)
0'f
0.3
0.3
L //'"
BUP-103
/r E
Figure 6. Pitch-averaged outlet data e' 2
///'~-'"
1.5
e'
g
2
2
1.5
1.5
1 0.5
0.5
0.5
/
.
-0.5 0.65
,,
~; 7 5- ~-8 8L ~ !/ _~._.''''~'/' ~b - ~ 8 ~203
.
.
.
.
.
.
g
BUP-103
" _...~.
-0.5
0 H ...... .... 0.65 0.7 0.75 0.8 0.85 019 0.95 /~o3
0.65 0.7 0.75 0.8 0.85 0.9 0.95 R03
Figure 7. Spanwise distributions of stream tube cone half-angles
4. RADIAL FLUID RE-ARRANGEMENT AND THE CONCEPT OF ENERGYEFFICIENT CONICAL CASCADES On the basis of measurement results presented in the previous section, the following qualitative model can be established for the radial re-arrangement of fluid within axial flow rotors of non-free vortex operation: the spanwise gradient in ideal total head rise (and thus, in blade circulation) according to non-free vortex operation results in characteristic, complex flow phenomena in the blade boundary layers (suction side radial outward and pressure side inward flow), driving a 3D secondary flow (NFVO flow, Figs. 3, 4) filling the dominant flow region within the blade passage and manifesting itself in torsion of interblade stream surfaces. According to a thickened boundary layer on the blade suction side and also due to an effect of centrifugal forces acting on the blade boundary layers, the radial outward flow will be dominant within the blade passage, manifesting itself in expanding stream tubes at almost the entire span (within the main flow) if a conical stream tube model is applied (positive c values, see Fig. 7). (It must be noted here that the torsion of interblade stream surface segments can be neglected, i.e. rotationally symmetric stream surfaces can be assumed through the rotor from design point of view, under the conditions discussed in [7].) A blockage effect of the hub boundary layer continuously thickening through the rotor also contributes to the expansion of stream tubes below midspan. The radial outward flow is
758 counterbalanced only near the casing wall, due to the displacement effect of the casing wall boundary layer thickening through the rotor and resulting even in the contraction of stream tubes (negative c values, see Fig. 7). As Fig. 7 suggests, the radial re-arrangement of fluid is generally dependent on the operating state of the rotor, according to the operating pointdependent NFVO flow and displacement effects (which are also affected by the NFVO flow since it influences the formation of annulus wall boundary layers). The modelling assumption of conical stream tubes through the rotor allows the establishment of a concept for retaining quasi-optimum operation of rotor cascade sections even under off-design circumstances, from this point onwards referred to as "concept of energy-efficient conical cascades". This concept is also applicable for rotors of free vortex design and is outlined as follows. Let us consider an elemental cascade of the rotor, which fits under design circumstances to the conical stream tube of cone half-angle eD, entering the rotor at r0 and exiting it at r3D within the main flow (Fig. 8, next to Fig. 5). The geometry of the elemental conical cascade can be optimised with knowledge of design (D) condition of b03D----(b0Dd- b3D)/2' and ~r (in this example, no pre-swirl is assumed). The optimisation can be carried out on the basis of 2D stationary cascade data applied to conical throughflow stream tubes [7], which method is an improved version of the classic design methods assuming cylindrical stream tubes through the blading [1, 2], and results in optimum (g/t)D, YD and (ReamberD)-1 values. If unswept blades
are designed, the blade can be optimised for the above sketched throughflow process approximately such a way that the conical cascade of optimum geometry must be projected onto a cylindrical surface of radius r03D = (r0 + r3D)/2 t73. Throttling the turbomachinery below its design flow rate results in changed conditions for the flow entering the rotor at r0 (condition L, cp. Fig. 6): ~03L
-
-
b03D--Ab03 and ~r = ~3D +A~q
(4)
It is demonstrable with use of the basic design relationships [e.g. 1] that an optimum elemental cascade geometry determined for condition L compares with that for condition D as follows (cp. Table 1): (s
> (s
and usually )'L > 7'0 and (RcamberL) -1 > (RcamberD) -1 9
(4a)
Such modified elemental cascade geometry can come "virtually" into being only if the flow entering the rotor at r0 fits to a conical stream tube of e L r e D. For example, if (g/t)D, YD and (RcamberD)-I decreases with r a d i u s - a n d this is the case for rotors BUP-26, BUP-29 and
BUP-103, see Table 1 - , s L < c D (i.e. r03L < r03D) must be performed in order to ensure that the stream tube passes the rotor along sloping, "quasi-optimum" blade sections of increased mean solidity, stagger angle and camber curvature. If the machine is operated at a flow rate higher than design (condition H), the train of thought is just the opposite (cp. Fig. 6): ~03H -- ~003D-at-Ab03 and ~r
-- ~r
-- A ~r and the corresponding optimum geometry
(5)
(g/t)H < (t/t)D , and usually ?'n < 7'o and (R~amberH)-'< (Rc~mberD)-' (5a) ,giving that eg. for the rotors discussed in this paper, e H > c D (i.e. r03H > to3D) must be ensured.
759 Of course, an extensive research work and thorough quantitative analysis is still needed on the possibilities for design utilization of the above effects, to be discussed in another paper. Now an experimental, rather qualitative justification is attempted at present state of the concept, through a simultaneous evaluation of hydraulic efficiency curves in Fig. 1 and stream tube cone half-angle distributions in Fig. 7. Rotor BUP-26 has an acceptably high hydraulic efficiency at the design flow rate, which suggests that it represents a quasi-optimum geometry for the throughflow at tPD. Nevertheless, the hydraulic efficiency is rapidly reduced for off-design flow rates (the efficiency curve has a peak). This experience can be explained with analysis of Fig. 7: though condition c L < e D is fulfilled within the dominant part of the main flow, e D - e L has relatively small values, and has even a zero transition, representing effects which act contrary to the concept of energy-efficient conical cascades, e H > e D is fulfilled for a limited range only, and an undesirable zero transition occurs in e H - e D as well. Rotor BUP-29 has the highest hydraulic efficiency at ~ = 0.5 and also over a wide surrounding flow rate range (the efficiency curve has a "high plateau"). Fig. 7 shows that e L < e D is fulfilled for the entire main flow, with e D - e L values higher than those valid for BUP-26. Though condition e H > e D is failed to be fulfilled for rotors BUP-29 and BUP-103 (Figs. 7b, c), the reader is notified now that s values for these rotors are less than optimum [7] (s was kept constant with span for simplification in manufacturing) and thus, failure in fulfilment of condition e H > e D may be advantageous in these particular cases (preventing the virtual s value from further reduction). Small but positive e D - e L values are present in the case of BUP-103 (Fig. 7). This experience, together with consideration in the previous paragraph suggests why the efficiency curve of also BUP-103 has a plateau, though lower than that for BUP-29, which lag is probably due to the considerable reduction of cascade solidity relative to the optimum values determined in design. 5. SUMMARY The aerodynamic behaviour of high pressure axial flow fans has been investigated through global and LDA measurements. Secondary flow vector plots, turbulence properties (velocity fluctuations) and profiles of pitchwise-averaged velocity components have been studied in detail. The generally applicable concept of energy-efficient conical cascades has been sketched out and illustrated through evaluation of the experimental results. After elaboration of appropriate quantitative models regarding the influence of geometrical and aerodynamic parameters on radial fluid re-arrangement within the rotor, this concept can be built in the design methods of axial flow rotating cascades in order to retain energy-efficient rotor operation even under off-design circumstances. ACKNOWLEDGEMENT The experimental work has been carried out and the concept of energy-efficient conical cascades has been outlined and was discussed in the paper by J. Vad, whose work was supported by a J~.nos Bolyai Hungarian National Research Grant, Ref. No. BO/00150/98. Authors acknowledge the support of OTKA (Hungarian National Foundation for Science and Research) program T 025361, under the co-ordination ofF. Bencze.
760 NOMENCLATURE c absolute velocity h rotor width g blade chord N blade number r radius R = r/rc dimensionless radius Reamber camber radius of curvature t = 2 r n / N blade pitch Tug axial velocity fluctuation ur reference velocity (rr co) s conical stream tube half-angle y blade stagger angle (measured from the circumferential direction) flow coefficient (area-averaged axial velocity in the annulus divided by ur cp = c x/uc local axial flow coefficient (Pr -- C'r/Uc
local radial flow coefficient
~3 = 2 R C3u/Uc local ideal total head rise
coefficient (assuming zero inlet swirl) ~t average total head rise coefficient (total head rise in the annulus normalised by
m /2)
p fluid density v hub-to-casing ratio r average tip clearance corotor angular speed r/h hydraulic efficiency SUBSCRIPTS AND SUPERSCRIPTS c casing wall L,H,D lower/higher/design r, u, x radial, tangential, axial 0 rotor inlet plane 3 rotor exit plane ^ pitch-averaged
REFERENCES 1. 2.
Wallis, R. A.: Axial Flow Fans. Newnes, London (1961) Somly6dy, L., Berechnung der Beschaufelung und der Str6mung axial durchstr6mter Wirbelmaschinen mit Anwendung von Gittermessung. Periodica Polytechnica, No. 3. Budapest (1971) 3. Dring, R. P., Joslyn, H. D., and Hardin, L. W., An Investigation of Axial Compressor Rotor Aerodynamics, ASME J. Eng. for Power, Vol. 104, pp. 84 - 96. (1982) 4. Inoue, M., and Kuroumaru, M., Three-Dimensional Structure and Decay of Vortices Behind an Axial Flow Rotating Blade Row, ASME J. Eng. Gas Turbines Power, Vol. 106, pp. 561 - 569. (1984) 5. Meixner, H. U., Vergleichende LDA-Messungen an ungesichelten und gesichelten Axialventilatoren, Ph.D. Thesis, University of Karlsruhe, Germany (1994) 6. Lakshminarayana, B., Zaccaria, M., Marathe, B., The Structure of Tip Clearance Flow in Axial Flow Compressors, ASME J. Turbomachinery, Vol. 117 (1995) 7. Vad, J., and Bencze, F., Three-Dimensional Flow in Axial Flow Fans of Non-Free Vortex Design, Int. J. Heat Fluid Flow (accepted in 1997, Ref. No. HFF 6038) 8. Vad, J., Bencze, F., Fiiredi, G., and Szombati, R., Fluid Mechanical Investigation on Axial Flow Fans, 10th Conference on Fluid Machinery, Budapest, pp. 500 - 509. (1995) 9. Vad, J., and Bencze, F., Secondary Flow in Axial Flow Fans of Non-Free Vortex Operation, 8th Int. Symp. Applications Laser Techniques to Fluid Mechanics, Lisbon, Portugal. Vol. 1, pp. 14.6.1. - 14.6.8. (1996) 10. Vad, J., and Bencze, F., Laser Doppler Anemometer Measurements Upstream and Downstream of an Axial Flow Rotor Cascade of Adjustable Stagger, 9th Int. Conf. Flow Measurement (FLOMEKO), Ltmd, Sweden, pp. 579 - 584. (1998)
11. Heat Transfer
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
On p r e d i c t i o n of t u r b u l e n t c o n v e c t i v e rib-roughened rectangular cooling ducts
763
heat transfer
in
Arash Saidi and Bengt Sund~n Division of Heat Transfer, Lund Institute of Technology, 221 00 Lund, Sweden
1. ABSTRACT A numerical investigation has been carried out to predict local and mean thermal-hydraulic characteristics in rib-roughened ducts. The Navier-Stokes and energy equations, and a low-Re number k-~ turbulence model are solved. Two methods for determination of the Reynolds stresses, eddy viscosity model (EVM) and explicit algebraic stress model (EASM), are used. The numerical solution procedure uses a collocated grid, and the pressure-velocity coupling is handled by the SIMPLEC algorithm. The computations are performed with the assumption of fully developed periodic conditions. The ribbed duct configuration is identical to the experimental setup of Rau et al. (1998) with only one wall having ribs, rib to hydraulic diameter ratio equal to 0.1 (e/Dh=0.1) and pitch to rib height ratio of 9 (P/e=9), see Figure 1. The Reynolds number is 30,000. The calculated mean and local thermal-hydraulic values are compared with these experimental data and prediction capabilities of the two turbulence models (EVM and EASM) are discussed.
2. INTRODUCTION To increase the performance of gas turbine cycles it is desirable to raise the temperature of the gas entering the turbine. However, this increase is limited by the mechanical and thermal properties of the materials used in gas turbine blades. In the 60s, cooling methods for blades were introduced. More recently internal cooling ducts have been applied. There are various types of such ducts, but in this report rectangular ducts with embedded ribs are considered. In the past both experimental and numerical investigations have been carried out but many features of the flow and heat transfer at various conditions still remain to be found. Experimental investigations by Han (1988) gave some information concerning influence of rib pitch and height on the overall Nusselt number and friction factor. Effects of rib shape were studied by Liou and Hwang (1993). The importance of model orientation was considered by Johnson et al. (1994) and Parsons et al. (1995). Detailed investigations of local flow and thermal fields have only recently become available and the studies by Hwang and Liou (1997), and Rau et al. (1998) have provided
764 interesting and useful results. Some experimental investigations including rotation effects are also available. Numerical investigations appear frequently nowadays but more research is needed to improve the reliability and accuracy in the predictions. Recent works are those of Acharya et al. (1993), Prakash and Zerkle (1995), Iacovides (1996), Rigby (1998), and Saidi and Sund6n (1998). In the present investigation, prediction of local and mean thermal-hydraulic characteristics is considered for a particular ribbed duct. The ribbed duct geometry is identical to that of Rau et al. (1998) and thus experimental data are available for comparison. Two different turbulence models are considered and one purpose of this work is to reveal the performance and evaluate these models. 3. P R O B L E M S T A T E M E N T
The problem to be solved is the fluid flow and heat transfer in a ribbed duct but only one wall has ribs (Figure 1). In this paper the pitch to rib height ratio is 9, (P/e=9).
Z
I'.
tY
Flow direction
Dh
X
<
P
>
Figure 1. One module of the one-sided ribbed duct, aspect ratio is one. 4. M A T H E M A T I C A L MODEL
The governing equations are the steady state continuity, the time averaged Navier-Stokes and the energy equations for turbulent flow, i. e., 0 Ox-~ (oUj) - 0 (1) 0
-0Xj - (oUjUi)-
0 [ OU i OUj ] 0 OP - 0x i + ~jxj L~t( 0xj + -V----)| Oxi J + Oxj ( - P u i u J )
(2)
765
axj (pUjT)=
(3)
~jxj P-r Oxxj- p u j t
In this case incompressible flow is considered and the density is thus constant. In many cases, the assumption of fully developed periodic conditions is applicable in the ribbed ducts and mean velocity and thermal fields field are periodically repeated, and in the considered experimental case this point is explicitly mentioned by the investigators (Rau et al., 1998). The procedure for handling fully developed periodic flow and heat transfer was introduced already by Patankar et al. (1977) and has then been applied extensively. This procedure is followed in this study. One introduces, p* P =-~x + (4) where ]3 resembles the non-periodic pressure gradient and P* is the periodic part of pressure in the main flow direction. In the energy equation a constant heat flux boundary condition is applied because this prevailed in the experimental setup.
4.1. T u r b u l e n c e m o d e l Several models have been suggested for calculation of turbulent flow and heat transfer, see e.g., Wilcox (1993). For engineering calculations, two equation models accompanied with the eddy viscosity concept have become popular because they provide reasonable overall results in many cases although not providing correct local distributions of turbulence generated secondary flows in ducts. Abe et al. (1994) proposed a low-Re number version of the k-~ model. Their model has the following advantages: 1) it uses a velocity scale near the wall other than the friction velocity and this velocity scale solves the singular point problem close to the reattachment in the separated flow cases, 2), numerical experimentation showed that this model does not have the initial value problem as other models in periodic boundary conditions, see Rokni and Sund~n (1997). This model can be summarized as,
ax---j(pUjk)- ~jxj (la + ~
---pe -OUiUj Oxj
' 'E .t ';1 -C l Oxj (pUjl3)= ~jxj (
~2 OUi C~2f~P k puiuj ~-xj
(5)
(6)
k2 ~tt = C~f~p
(7)
766
f~ = 1 - e x p ( - ~-~)
5 Re 2 1 + Ret3/4 exp - ( 2 - ~ )
(8)
fE= 1 - e x p ( - 3 - ~ )
Ret )2 1-0.3exp-(-6-ff
(9)
where f~ and f~ are d a m p i n g functions. Two different approaches for d e t e r m i n a t i o n of the Reynolds stresses are adopted in this study. The first one is the eddy viscosity model (EVM) and the second one is the explicit algebraic stress model (EASM) of Speziale and Xu (1995). In EVM the "Reynolds stresses" are calculated as: 2 ~tt u i u j = ~ kSij - 2 ~ Sij (10) P . In EASM these stresses are calculated as : 2 , k2 , k3 uiuj = ~ kSij -(Zl ~( z 2~jij g - ~ (Sik 0)kj + Sjk 0)ki) (11) , k3
1
+0~3 -~-2-(Sik Skj - ~ Skl SklSij ) la3k ~1- 2 (z2 g
(~jijS~ij)1/2
_ 1 a2 k (co--~co--~)m/2 2a 1 (~i = O~i(11,~)
(12) (13) (14)
where Sij and coij are the m e a n rate of strain and m e a n vorticity tensors, and al.2.,3 are model variables. For detailed description of the EASM, see Speziale and Xu (1995). 4.2. B o u n d a r y
conditions
Periodic b o u n d a r y conditions are applied at the inlet and outlet of every periodic module (which is equal to one pitch in the ribbed wall duct). This condition is defined as, (U,V,W are velocity components, P, k and e are pressure, t u r b u l e n t kinetic energy and t u r b u l e n t energy dissipation, respectively) 9 q)(x,y,z)=r +L,y,z) (15) r The b o u n d a r y conditions at walls are imposed as:
(16)
767 U=V=W=k=O
(17)
and qw=COnstant ~w=2 ~ p (~) k~/~n) 2
(18) (19)
where n represents the normal wall distance. 5. N U M E R I C A L S O L U T I O N P R O C E D U R E
A computer code based on the finite volume technique is modified and applied to solve the governing equations. The code uses a collocated grid arrangement and employs the Rhie and Chow (1981) method to interpolate values of velocity at the control volume faces. The SIMPLEC algorithm handles the coupling between pressure and velocity. Convective fluxes are determined by the hybrid scheme in all equations. An interpolation technique similar to that used by Obi et al. (1991) is employed to avoid oscillatory solutions that might otherwise appear from inappropriate coupling of the mean velocity and Reynolds stress fields in an EASM case. A non-uniform grid has been chosen, and grid refinement has been applied in the near wall regions. As a low Reynolds number formulation is applied, it is important that the n § -value of the grid points closest to the walls is of the order of unity. In this work the n+-values were in the range of 0.3-0.8 for all considered cases. The calculations were terminated when the absolute residuals of all the variables became less than 10 .6. The under-relaxation factors for all calculations were set to values in the range of 0.5-0.6. The non-periodic pressure gradient is specified to create a flow field. Through the rate of mass flow, the Reynolds number is coupled to ]3. Thus various values of [3 correspond to various Reynolds numbers. Grid refinement studies have been carried out, in both normal and axial directions. The presented results are obtained by a 60X 28X 58 points grid and the results (mean Nu and f values) changed less than one percent as further refinement to 70X 28X 58 points was applied (grid refinement was already applied in the normal to wall directions, namely Y and Z directions). The CPU times for calculations with EASM were 70% higher than those of EVM. 6. R E S U L T S 6.1. M e a n v a l u e s : The case considered is a one-sided ribbed duct of pitch to rib height ratio of 9 and rib height to duct hydraulic diameter of 0.1 for a square-sectioned duct at Re=30,000, see Figure 1. This case is identical to t h a t of Rau et al. (1998). The obtained mean values of the friction factor and the Nusselt numbers are
768 compared with the experimental data for the two considered models as shown in the Table 1. ( f and Nu values are normalized with respect to the experimental correlation for the same duct without ribs). As is evident in the table, the friction factor prediction becomes more accurate by using the EASM method. The heat transfer predictions show that both models have the same prediction quality. They have predicted the heat transfer increase in the floor between ribs within an acceptable error but over-predicted the Nusselt number on the smooth side wall.
Table 1. C o m p a r i s o n of m e a n t h e r m a l - h y d r a u l i c p r e d i c t i o n ( c o m p a r e d to R a u et al. (1998)) Turb. Model
fifo, error
EVM EASM
3.79,-27% 5.34, +2%
Nu/Nuo. ,error (Floor between ribs) 2.10,-4% 2.14,-2%
NtffNuo,error (Side wall) 1.90, +22% 1.97, +27%
6.2. Local v a l u e s and flow fields: In order to provide a detailed comparison between the two models, the flow fields and local values are examined. Figure 2 shows the vector representation of the velocity components (floor section) in an XY plane located at Z/e=0.25. In both methods the entrained mean stream flow turns towards the side wall in the front of the next rib. As can be seen in the experiment the m a x i m u m of the U-velocity component downstream the recirculation zone and upstream the next rib does not occur at the symmetry line but somewhere closer to the side wall. This feature is not reflected at all in the EVM but it exists in the EASM prediction of the flow field. x
m
q
Experiment EASM EVM Figure. 2. Velocity vectors in the floor section between ribs, XY plane, Z/e=0.25. This m a x i m u m can be explained by consideration of the secondary flow pattern downstream the recirculation zone behind the rib. The secondary flow vectors in a cross-sectional plane are shown in Figure 3 (a YZ plane). The secondary flow has a sort of impingement at the corner, and then turns towards the symmetry line but before reaching it, disappears, or in other
769 words amplifies the primary flow or U-velocity component. This explains the shift in the maximum point of the U-velocity profile shown in Figure 2.
.
.
.
.
.
.
,.
.
.
. .
. .
.
.
.
.
. '.
.
~
9
1
,
i
,
'.
~
~
'. ",. . . . . .
~
~
~
,.,
....
,,,i,
.
.
.
.
,,
,..,..,..,.,,,.
,,
.
.
.
.
,,
"~
~
x
\
~ % ~. % % 'l,%lllWitl
.
.
.
.
.
.
.
.
~
.
l
l
.
.
1
---
'
..........
~. \
% 't.'%l,l,',lt~.
,~ t ~ ,i ~, I llltillfWI
.
.
x
~Q9
,,I
........
,.
. . . . i,
~
x
'
.
.
1
.
l>
t
.
I
I
I,
r
I 1 i llllr1~111
,
q
I
~
rltltM~
I i I 1 llllllt~
l llj t lJtt , tl
/_
Figure 3. Secondary flow vectors (EASM). Figure 4. Vorticity iso-lines (EASM) Experimental investigations proved the existence of a large vortex cell in each half cross sectional plane of a one-sided ribbed duct (Rau et al., 1998). In Figure 4 the vorticity isolines are depicted in a half cross section of the duct (a YZ plane) and this vorticity plot certainly shows a large vortex cell in the EASM calculation (all calculations were done for a half duct considering existing symmetry line, the left side end of the Figure 3 and Figure 4). On the other hand, the EVM method could not predict this large vortex cell. The vertical velocity component (W) is measured at the symmetry line (see Figure 5). This velocity component (W), which is negative just downstream the rib, can be related to the flow entrainment from the main-stream to the recirculation zone. This flow brings the cold air from the center in to contact with the wall and increases the heat transfer. As can be seen in the figure, both methods have almost the same ability to predict these values but the location of minimum of this velocity component is not correctly predicted.
770 W/U r o2
........... EASM 9
0.1
.'""Ira
EVM EXPERIMENT (Rau et al., 1998)
0.0 .....9
........ ""
-0"ii,I .... , .... I , , ,I, I I , I,, .... , .... , .... , .... -02
I
2
4
3
5
6
7
X/e
Figure 5. Normalized W-velocity component at symmetry line at Z/e=l plane. The Nusselt number enhancement factor along the symmetry line on the ribbed floor is shown in Figure 6. The EASM method presents a relatively better prediction capability, although both methods failed to predict the correct location of the maximum enhancement. N~4Uo
3.0 2.5 n
n
n
•
2.0 1.5 1.0 0.5
1
2
3
4
5
6
7
•
Figure 6. Nusselt number enhancement at the symmetry line between ribs. CONCLUSIONS A numerical investigation is carried out for prediction of the thermalhydraulic characteristics in a ribbed duct. Two different methods for
771 determination of the Reynolds stresses are used. The considered case was chosen similar to the experimental run ofRau et al. (1998). The investigation has shown that EASM can considerably improve the mean friction factor prediction. The superiority of EASM over EVM in the prediction of the friction factor could be partly explained by considering the calculated flow field and comparing it with the experimental one. Then, it is evident that there are some flow field phenomena existing in this problem, which were successfully predicted by EASM, but not by EVM. A displacement of the maximum streamwise velocity toward the smooth side wall is predicted by EASM, which is in a qualitative agreement with the experiment, but the EVM did not capture this shift. Another interesting difference between the two prediction methods is the existence of a large vortex cell in the half cross section of the duct in the EASM prediction. This large vortex cell was experimentally observed. The mean values of the heat transfer enhancement were successfully predicted by both methods on the wall between ribs (rough wall) but they both failed, i.e., over-predicted these values over the smooth side wall. REFERENCES
1. Abe, K., Kondoh, T. and Nagano, Y., 1994, "A New Turbulence Model for Predicting Fluid Flow and Heat Transfer in Separating and Reattaching FlowsI. Flow Field Calculations", Int. J. Heat Mass Transfer, Vol. 37, pp. 139-151. 2. Acharya, S., Dutta, S., Myrum, T.A. and Baker, R.S., 1993, "Periodically Developed Flow and Heat Transfer in a Ribbed Duct", Int. J. Heat Mass Transfer, Vol. 36, pp. 2069-2082. 3. Han, J.C., 1988, "Heat Transfer and Friction Characteristics in Rectangular Channels with Rib Turbulators", ASME J. Heat Transfer, Vol. 110, pp. 321-328. 4. Hwang, J. and Liou, T., 1997, "Heat Transfer Augmentation in a Rectangular Channel With Slit Rib-Turbulators on Two Opposite Walls", ASME J. ofTurbomachinery, Vol. 119, pp. 617-623. 5. Iacovides, H., 1996, "Computation of Flow and Heat Transfer Through Rotating Ribbed Passages", Biennial Colloqium on Computational Fluid Dynamics, UMIST, pp. 3-19:3-24. 6. Johnson, B. V., Wagner, J. H., Steuber, G. D. and Yeh, F. C., 1994, "Heat Transfer in Rotating Serpentine Passages with Selected Model Orientations for Smooth or Skewed Trip Walls", ASME J. Turbomachinery, Vol. 116, pp. 738744. 7. Liou, T. and Hwang, J., 1993, "Effect of Ridge Shapes on Turbulent Heat Transfer and Friction in a Rectangular Channel", Int. J. Heat Mass Transfer, Vol. 36, pp. 931-940.
772 8. Obi, S., Peric, M. and Scheuerer, G., 1991,"Second-Moment Calculation Procedure for Turbulent Flows with Collocated Variable Arrangement", A/AA J., Vol. 29, pp. 585-590. 9. Parsons, J. A., Han, J. C. and Zhang, Y., 1995, "Effect of Model Orientation and Wall Heating Condition on Local Heat Transfer in a Rotating Two-Pass Square Channel with Rib Turbulators", Int. J. Heat Mass Transfer, Vol. 38, pp. 1151-1159. 10. Patankar, S. V., Liu, C. H. and Sparrow, E. M., 1977, "Fully Developed Flow and Heat Transfer in Ducts Having Streamwise-periodic Variations of Cross-sectional Area", ASME J. Heat Transfer, Vol. 99, pp. 180-186. 11. Prakash, C. and Zerkle, R., 1995, "Prediction of Turbulent Flow and Heat Transfer in a Ribbed Rectangular Duct with and without Rotation", ASME J. Turbomachinery, Vol. 117, pp. 255-264. 12. Rau, M., Cakan, M., Moeller, D. and Arts, T., 1998, "The Effect of Periodic Ribs on the Local Aerodynamics and Heat Transfer Performance of a Straight Cooling Channel", ASME J. of Turbomachinery, Vol. 120, pp. 368-375. 13. Rhie, C. M. and Chow, W. L., 1983, "Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation", A/AA J., Vol. 21, pp. 15251532. 14. Rigby, D. L., 1998, "Prediction of Heat and Mass Transfer in a Rotating Ribbed Coolant Passage with a 180 Degree Turn", ASME paper 98-GT-329. 15. Rokni, M. and Sund~n, B., 1997, "Improved Modeling of Turbulent Forced Convective Heat Transfer in Straight Ducts", ASME HTD-Vol. 346, pp. 141-149. 16. Wilcox, D. C., 1993, "Turbulence Modeling for CFD", DCW Industries Inc., USA. 17. Saidi, A. and Sund~n, B., 1998, "Calculation of Convective Heat Transfer in Square-Sectioned Gas Turbine Blade Cooling Channels", ASME paper 98GT-204. 18. Speziale, C.G. and Xu, X-H., 1995, "Towards the Development of SecondOrder Closure Models for Non-Equilibrium Turbulent Flows", 10 ~h Syrup. on Turbulent Shear Flows, Penn. State University, pp. 23-7:23-12.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 1999 Elsevier Science Ltd.
773
The m e a s u r e m e n t o f local wall heat transfer in stationary U-ducts o f strong curvature, with smooth and rib roughened walls. H. Iacovides, D.C. Jackson, G. Kelemenis, and B.E. Launder Department of Mechanical Engineering, UMIST, Manchester, UK
Abstract
The paper presents some of our recent experimental investigations of convective heat transfer in flow through stationary passages relevant to gas-turbine blade-cooling applications. Local Nusselt number measurements in flows through round-ended U-bends of square cross-section, with and without artificial wall roughness are presented. Our earlier LDA measurements of flows through these passages are first briefly reviewed and then the liquid-crystal technique for the measurement of local wall heat transfer inside passages of complex geometries is then presented. Tightly curved U-bends generate strong secondary motion and cause flow separation at the bend exit, which substantially raise turbulence levels. Wall heat transfer is significantly increased, especially immediately downstream of the U-bend, where it is over two times higher than in a straight duct. The local heat-transfer coefficient around the perimeter of the passage is also found to vary considerably because of the curvature-induced secondary motiol~ The introduction of surface ribs, results in a further increase in turbulence levels, a reduction in the size of the curvature induced separation bubble and a complex flow development after the bend exit with additional separation regions along the outer wall. Heat-transfer levels in the straight sections are more than doubled by the introduction ofn'bs. The effects of the bend on the overall levels of Nusselt number are not as strong as in the smooth U-bend, but are still significant. The effects of the bend on the perimetral variation of local heat-transfer coefficients within the ribbed downstream section are also substantial. 1.
INTRODUCTION
The demand for in~rovements in the efficiency and power output of gas-turbines, has led engine designers continuously to seek to raise operating temperatures. In order to preserve the structural integrity of rotating blades, elaborate cooling systems have been evolved in which relatively cool air is extracted from the compressor and, through the central shaft, is fed to cooling passages inside the turbine blades and nozzle guide vanes. The flow inside these cooling passages is complex and highly three-dimensional, influenced by the presence of sharp U-bends, artificial rib-roughness and also by the rotation of the blades. In order to optimise the cooling process, the engine designer needs to have accurate information on the detailed flow development and its effects on local wall heat transfer. Because of the geometrical complexities of the cooling passages and the presence of rotation, most of the experimental heat-transfer data available until recently, though of considerable value, were nevertheless confined to averaged values. Due to the
774 lack of detailed data on local flow and heat transfer, it has neither been possible to gain a clear understanding of the hydrodynamic and thermal behaviour in such passages, nor to develop and validate numerical flow solvers that can be reliably used for the simulation of blade cooling flows and the consequent thermal stresses that arise. It is the second of these two issues, the provision of local flow and thermal data for validation of numerical flow solvers, that provides the motivation for the work reported here.
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This investigation forms part of a more extensive experimental effort in our group [ 1], aimed at producing such CFD-validation data for flows through idealised blade-cooling passages under stationary and rotating conditions. As shown in Figure 1, we here focus on the thermal development through stationary, round-ended U-bends of strong curvature of square crosssection, both with smooth walls and with the inner and outer walls of the straight sections roughened with staggered ribs of square cross-section. Though the geometries of these passages are somewhat idealised, the presence of strong curvature and rib roughness leads to the 180"
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LDA measurements for case I, from reference [2].
Figure 3. LDA measurements for cases II and III, [4] and [3].
775 development of most of the important flow features found in real blade-cooling passages. The local flow development through these passages has been the subject of earlier investigations in our group, [2-4]. As can be seen in Figure 2, for the passage with smooth walls the LDA measurements [2], revealed that the flow development is dominated by the strong streamwise pressure gradients at the bend entry and exit. A separation bubble is formed along the inner wall, half-way around the bend, which extends to about two diameters in the straight downstream sectiorL Because of the flow separation, turbulence levels are greatly increased at the bend exit. As can also be seen in Figure 2, due to the curvature-induced secondary motion, the flow on the mid-plane and on a parallel plane close to the flat wall are very different. The introduction ofrl'bs, [3] and [4], naturally increases turbulence in the straight sections. This results in the delay of flow separation along the inner wall of the bend. As seen in Figure 3, Case III, downstream of the bend, as the faster fluid along the outer wall encounters the first rib, a large separation bubble is formed along the outer wall, behind the first rib. Consequently the length of the separation bubble along the inner wall is reduced. Removal of the first downstream rib from the outer wall, Case II in Figure 1, displaces the outer-wall separation bubble further downstream and reduces its size. This is accompanied by an increase in the length of the separation bubble along the inner wall, however. Here heat-transfer measurements that we have recently obtained using the liquid-crystal technique are reported and, through comparisons with the earlier LDA measurements, the mechanisms by which the flow development influences wall heat transfer are discussed. 2.
APPARATUS
The thermal measurements have been obtained using air as the working fluid. As shown in Figure 4, on open-loop system is employed, with a fan drawing air into the experimental model through a contracting inlet section and a combination of fine wire meshes and a honeycomb section, placed there to ensure uniform and symmetric entry conditions. The working section is connected to the fan through a long orifice plate section, used to measure |- BLEED CONTROL PUMP \ 9 VALVE FLOW the mass flow rate. , ..... STRA~.'rENE,~ The experimental model is made ~"~ - ' -of 10mm thick perspex. The passage ........................... -I ~,\ has a square cross-section with a .~. / ~' " ORIFICE PLATE //" ; diameter of 50mm. The ratio between j /- / the bend mean radius and the duct diameter, Rc/D, is 0.65 and the upstream and downstream straight / - / / y CONTRACTION / .--~-~[ ./"I ~ ~ A I R FILTER sections are 10 diameters long. The ~.7"-// .. fibs, for cases II and III, are of a square : ---/" ~'~qJI FLOW IN cross section. The ratio between the rib HONEYCOMB TmLET height and the duct diameter, e/D, is 0.1 and the rib spacing, P/e, is 10. In Figure 4. Apparatus both the upstream and the downstream sections the ribs closest to the bend are at a distance of 0.45D from the bend entry and exit respectively.
776 3.
HEAT-TRANSFER MEASUREMENTS.
As is well documented [5] and [6], the molecular structure of thermochromic liquid crystals, over a certain temperature range, depends on temperature. Changes in molecular structure in turn affect the wavelength of visible light absorbed by the liquid crystals and hence, over a certain temperature range, the colour of the liquid crystals can be used to determine their temperature. In heat-transfer experiments where air is the working fluid, most groups have adopted the transient liquid-crystal technique, in which the surface under investigation is covered with a layer of liquid crystals and is exposed to a hot air stream. As the surface temperature gradually rises from the initial ambient conditions, the movement of the colour contours of the liquid crystals along the surface is monitored. Then, at each location along the surface, from the time needed for the wall temperature to rise from its initial value to that of the crystal-colourchange temperature, the solution of the one-dimensional transient heat-conduction equation produces the value of the heat-transfer coefficient, h. In this study, however, because of our intention to employ this technique subsequently in experiments in rotating passages using water as the working fluid, the steady state technique was employed. This is because in water, due to the higher coefficients of heat transfer involved, the thermal time constant (a;= PwCpJqflla2) is too short to measure accurately. The internal surfaces of the inner and outer walls of the experimental model are covered with a 0.013mm thick, electrically heated, stainless steel foil which provides a constant-heat-flux thermal boundary condition. The ribs were made of perspex and the heating foil underneath the ribs was short circuited. The rib surfaces could thus be considered as thermally insulated. A thin layer of liquid crystals is then applied over the surface of the heating foil. Once steady thermal conditions are reached, the resulting contours of the yellow colour are also contours of known wall temperature, determined through a prior calibration, under conditions of uniform wall heat flux. Electrical measurements provide the wall heat flux, qw and, from the overall energy balance, the fluid bulk temperature, TB, can also be determined : m Cp (dTB/dZ) = P qw
(1)
where m is the mass flow rate, z the streamwise direction and P the heated perimeter. The local Nusselt number along each colour contour can then be calculated, since : Nu - (qw Dh)/[ k (Tw-TB)]
(2)
By repeating the above procedure for a number of heating rates, detailed mapping of the local Nusselt number over each heated wall can be constructed. The liquid crystals employed, were micro-encapsulated crystals manufactured by Merck, with a nominal colour-change band between 29.5~ and 31 ~ for the entire visible spectnma, and a nominal change of 0.1~ across the yellow colour, which was used to determine the wall terr~mttme. At each heating rate, the image of the resulting colour contours was recorded using a CCD camera at a 45 ~ angle as shown in Figure 5, digitized and stored on a PC. In order to minimise heat losses, the experimental model was covered with thermal insulation. The thermal
777 insulation of the top wall was removed for only the few seconds needed for the camera to record the image. The digitized image was then converted into a hue-saturationand-intensity format and the pixels with a hue value corresponding to that of the yellow colour were identified. Software developed for this study then corrected the co-ordinates of the selected pixels, accounting for distortions caused by the ~ ' ~ tion camera lens and angle. The information from each image was thus reduced to provide the co-ordinates of the constantwall-temperature contour. The value of h along the contour line was then calculated, using equations (1) and (2). Finally, the Perspex Wall Black Paint contour lines produced by different heating Figure 5. Heating and Viewing arrangements. rates were brought together and, through interpolation, the continuous variation of the Nusselt number over the heated safface was produced. Side-averaged values of the Nusselt number were also subsequently computed. Error analysis indicates that the uncertainty in the local Nusselt number is +7.4% and in the side-averaged values +5.5%. While in real blade-cooling passages the cooling fluid would remove thermal energy from all four walls, in these experiments only the inner and outer walls have been heated. This does not however diminish the usefulness of the data in CFD validation, or invalidate any conclusions reached on how the flow development influences wall heat transfer. Ideally the heat transfer pattern should be symmetric about the centre-plane of the duct, y=0. While contour lines, discussed in the next section, are not perfectly symmetric, the main anomalies occur in regions where Nu is not changing rapidly, so discrepancies associated with the asymmetric contours are not important. 4.
RESULTS AND DISCUSSION
Three sets of data are presented, one for each of the three geometries shown in Figure 1, all at a Reynolds number of 95,000. 4.1
Case I
The local Nusselt number measuremems in the U-bend with smooth walls, Figure 6, provide an opportunity to assess the effects of strong curvature on heat transfer. In the upstream section, Nusselt-number levels along both walls are fairly uniform and close to the value returned by the Dittus-Boelter correlation: Nu=0.023 Re ~ Pr ~
(3)
778 Within the U-bend heat transfer levels rise along the outer wall, with the maximum level at each axial location occurring at the centre line of the outer wall. The general rise in Nusselt number is consistent with the flow acceleration along the outer wall, as shown in Figure 2. Within the bend the higher heat transfer levels along the centre line of the outer wall, are probably caused by the secondary motion, which causes relatively cool fluid fxom near the duct centre to impinge along the cemre-line of the outer wall. This fluid heats up as it moves out to the comer regions, leading to a reduction in the heat transfer coefficient in the two comers. Downstream of the bend, along the outer wall, heat-transfer coefficients continue to rise, reaching a maximum value more than twice that prevailing upstream, at about three diameters after the bend exit. As can be seen in [2], the location of the maximum Nusselt number coincides with the region where the highest turbulence levels along the outer wall have been measured. Beyond this point, as the flow along
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Figure 6. Nusselt number contours for the smooth U-bend, Case I. Re=95,000 and Pr=0.71 the outer wall starts to slow down and the turbulence levels begin to reduce, Nusselt numbers also start to diminish. Even after six diameters from the bend exit, however, the Nusselt number is still more than 50% higher than that upstream. Along the inner wall, Nusselt number levels rise very rapidly after the bend exit, reaching their maximum values, similar to those along the outer wall, about two diameters after the exit, which is very close to the reattachment point, Figure 2. In contrast to the measured distribution along the outer wall, along the inner wall, the Nusselt numbers in the corner regions are higher than those on the centre line. This is also consistent with the curvature-induced secondary motion which, along the thermally insulated top and bottom walls, transports relatively cool fluid to the comer regions of the inner wall. As this fluid moves towards the duct centre, it is heated and also slowed down by the inner wall, causing a strong reduction in wall heat transfer. At about three diameters downstream of the bend, the variation in Nusselt number between the centre line and the comer regions, at the inner wall, is about 60%. As the flow begins to recover after r e - a t t a c ~ Nttsselt number levels along the inner wall begin to decrease, though, as for the outer wall, even alter six diameters from the bend exit, they are
779 still significantly higher than those of the upstream region. The strong streamwise changes in the mean flow and the rise in turbulence levels caused by the tight U-bend thus lead to substantial increases in the heat-transfer coefficient. Moreover, curvature-induced secondary motion, as well as contributing to the increase in wall heat transfer, also influences the distribution of the local Nusselt number on the inner and outer walls. The influence of the bend on heat transfer is still significant after six diameters from the bend exit. 4.2 Case II
The effects of rib-roughness on heat transfer can be seen in the Nusselt number contours of Figure 7. In the upstream section, there appear to be enough rib intervals for thermal conditions to repeat themselves over each interval. Nusselt numbers become highest in the middle of each fib interval, by which point, according to the LDA data of Figure 3, the flow has re-attached, after separating behind the upstream rib. Nusselt number levels in the comer regions are higher than in the centre of each wall, probably because the top and bottom walls are thermally insulated. The actt~ levels of the heat-transfer coefficient are typically twice as high as those in the smooth Ubend (Case I). This substantial increase must be related to the high levels of turbulence caused by the presence of fibs. Within the bend, heat-transfer coefficients display a monotonic rise along A
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C, Figure 7. Nusselt number contours for the fibbed U-bend, Case II. Re=95,000 and Pr=0.71 the outer wall but, in contrast to the smooth U-bend, Nusselt number levels along the centre-line are higher than those in the comers. The levels of Nu along the outer wall are higher than the corresponding levels for the smooth U-bend, but not as high as the highest levels encountered in the ribbed upstream sections. Downstream of the bend, along the outer wall, Nu levels fall after the first diameter, where, as shown in Figure 3, the flow along the outer wall starts to decelerate.
780 Over the first downstream rib interval along the outer wall, where a large separation bubble is generated, the Nusselt number rises to levels higher than those encountered upstream of the bend. Heat transfer coefficents gradually fall, over the subsequent rib intervals, tending towards the levels measured in the upstream section. Along the inner wall, over the first downstream rib interval, which is partially within the region of bend-induced separation, heat-transfer levels are especially high in the comer regions. Over the two subsequent rib intervals, the Nusselt number rises even further, to levels higher than those along the outer downstream wall. Beyond the third downstream rib, heat-transfer coefficients begin to fall towards those of the upstream sections. The overall effect of the U-bend and of the ribs on wall heat transfer, can be more easily assessed through the plots of the side-averaged Nusselt numbers, shown in Figure 8. As expected, the introduction of ribs raises the levels of the heat-transfer coefficient in the upstream section by more than a factor of 2. The round-ended U-bend still influences heat transfer levels downstream of the bend, especially along the inner wall, but its effects are not as strong as in the smooth U-bend. 600 500 400
2110 100
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500 400 z 300 200 Fully-
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Figure 8.
BEND
Outer W all
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Axial variation of side-averaged Nusselt number of a smooth and a rib-roughened Ubend, Cases I and II. : Smooth Duct, 9Ribbed Duct.
4.3 Case III Finally the local Nusselt measurements of Figure 9 show how the inclusion of an additional rib along the outer downstream wall, at a distance of 0.95 diameters from the bend exit, affects wall heat transfer. Some of the images for the higher heating rates were, unfortunately, lost and consequently some regions have not been fully mapped. In these regions, shown as shaded in Figure 9, the Nusselt number reaches values that are higher than those indicated by the levels of the contours present. Along the outer wall, differences in the thermal behaviour between Cases II and III begin to appear over the second half of the bend, where the additional downstream rib causes an earlier rise in the Nussett number. After the bend, because the first downstream rib is now closer to the bend, the reduction in Nusselt number after the first diameter si eliminated. Over the first rib interval, now one diameter closer to the bend, the Nusselt nmnber is higher, than for Case II, and the distribution is different, with the highest levels occurring at the downstream end of the
781 interval This must be caused by the larger separation bubble over the first rib interval in Case III. Beyond the first rib interval along the outer wall, the thermal behaviour becomes similar to that of Case II.
Figure 9. Nusselt number contours for the ribbed U-bend, Case III. Re--95,000 and Pr=0.71 Along the inner wail, while the Nusselt number dism"aution is similar to that for Case II, over the first two rib intervals the levels are about 10% higher. The changes observed are consistent with the changes that the additional rib causes to the flow development, Figure 3.
5.
CONCLUDING REMARKS
The results presented provide data for local wall heat-transfer in idealised blade cooling passages that should be useful for CFD code validation. The data have also revealed how, in strongly curved and ribbed passages, the flow development influences wall heat transfer. In flows through smooth U-bends the main flow features are the flow separation from the inner wall at the bend exit and the curvature induced secondary motion. Flow separation causes a substantial increase in turbulence levels, especially over the first three downstream diameters. The secondary motion and the higher turbulence levels increase the Nusselt number through the bend, with the peak values, more than twice as high those upstream, along both the inner and outer walls, reached between two and three diameters after the bend exit. Moreover, the secondary motion induces strong perimemfl variations in Nusselt number, around 60%, along both the inner and outer walls. Bend effects on heat transfer remain appreciable even beyond seven diameters l~om the exit. The introduction of fibs substantiaU~ raises turbulence levels which, for the case of the round-ended U-bend, reduces the size of the separation bubble along the inner wall. The flow development downstream of the bend becomes very complex, as the flow, highly distorted by the bend, encounters ribs along the inner and outer walls. It is also found to be sensitive to the distance of the first downstream ribs to the bend exit. Wall heat transfer is substantially raised within the bend and also within the upstream and downstream sections. While the effects of the U-bend on heat transfer levels are not as strong as in the case with smooth walls, they are
782 nevertheless still significant. Furthermore, the effects of the bend on the perimetral variation in Nusselt number after the bend exit are also highly evident. Moreover, the rather steep streamwise variation of Nusselt number in the space between ribs suggests that, even though the blade as a whole is more effectively cooled by surface ribbing, important thermal stresses may nevertheless be encountered. The non-linear nature of the laws that govern the fluid motion lead to the development of complex flow patterns in blade cooling passages, which can neither be anticipated nor deduced by simply combining the separate effects of rotation, strong curvature and rib roughness. Detailed experimental data, or use of reliable flow solvers is thus necessary to determine local Nusselt numbers of the quality needed for modem design. ACKNOWLEDGEMENTS
Funding for the work presented has been provided by the EPSRC, ABB(Switzerland), EDF(France), EGT, and Rolls-Royce pie. The authors gratefully acknowledge both the financial support and expert technical advice received. The authors also acknowledge technical support provided by Mr D. Cooper, Mr J Hosker and Mr M. Jackson. Authors' names are listed alphabetically. REFERENCES
1.
2.
3.
4. 5. 6.
H. Iacovides, D.C. Jackson, G. Kelemenis, B.E. Launder and Y-M Yuan,1998, Recent progress in the experimental investigation of flow and local wall heat transfer in internal cooling passages of gas-turbine blades, Proc 2nd EF Conference on Turbulent Heat Transfer, pp 7.14-7.28, Manchester, UK Cheah S C, Iacovides H, Jackson D C, Ji H H and Launder B E, 1996, LDA Investigation ofthe Flow Development Through Rotating U-ducts, ASME Journal of Turbomachinery, 118, 590-596. Iacovides H, Jackson D C, Kelemenis G, Launder B E and Y-M Yuan, 1996, LDA Study of the flow development through an orthogonally rotating U-bend of strong curvature and n"a-roughen~ walls, Engineering Turbulence Modelling and Experiments 3, 561-570,Ed W Rodi and G Bergeles, Elsevier, H. Iacovides, D. Jackson, B.E. Launder and Y-M. Yuan, 1998, An experimental study of a rib roughened rotating U-bend flow, In preparation Baughn J W and Yan X, 1992, Local Heat Transfer Measurements in Square Ducts with Transverse Ribs, ASME HTD-Enhanced Heat Transfer, 202, 1-7 Jones T V, Wang Z and Ireland P T, 1992, Liquid Crystal Techniques, ICHMT, Int Symp. on Heat Transfer in Turbomachinery, Athens, Greece.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
783
Studies o f Turbulent Jets impinging on M o v i n g Surfaces K Knowles a and T W Davies b aHead, Aeromechanical Systems Group, Cranfield University, RMCS, Shrivenham, Swindon, Wilts., SN6 8LA, UK. Tel +44 1793 785 354 Fax +44 1793 785 260 bprofessor of Thermofluids Engineering, School of Engineering and Computer Science, University of Exeter, Exeter, Devon, EX4 4QF, UK.
Measurements have been made of the flow produced by a single, round, turbulent jet impinging on a flat moving surface. The wall jet profile formed on the retreating side of the impingement region (where the belt surface was travelling in the same direction as the wall jet) had a very thin inner boundary layer region. On the approach side (where the surface was moving towards the impingement point, against the wall jet) the initial development seemed very similar to the case with the surface stationary. At a radial position of about r/Dn = 4, an increase in the wall jet half-thickness (the height above the surface to half peak velocity) was noticeable over the stationary case and at r/Dn = 7 there was a sudden rapid increase. This appeared to be due to separation of the wall jet inner boundary layer due to the increased shear stress. Measurements of wall jet momemum flux were consistent with this. Numerical modelling, using a commercial CFD package (PHOENICS v 2.1 and also v 3.1) and the low Reynolds number Lam-Bremhorst turbulence model with Yap correction, has shown similar trends. Heat transfer characteristics derived from these CFD studies have agreed with previous experimental data, including the skewing of the radial distribution of local Nusselt number, with the peak values shifted to the approach side of the impingement region. NOTATION
Dn I-I. Mf r
V Vg Vn Vm Y Y1/2
Nozzle exit diameter Height of nozzle above ground Wall jet momentum flux (in V-direction) Radial co-ordinate measured from nozzle centre-line Wall jet velocity Moving ground velocity Nozzle exit velocity Local maximum wall jet velocity Co-ordinate distance above ground Height to half peak velocity (Y at V = Vm/2)
784 1. INTRODUCTION Impinging jets are widely used in many engineering applications involving heating, cooling or drying. Some of these applications involve jets impinging on moving surfaces, such as belts or rotating drums. There is relatively little data in the literature on such flows, other than some global studies of heat or mass transfer characteristics e.g. Martin, (1977), Polat, (1993). There seem to have been few previous studies of the detailed flowfields involved with jets impinging on moving surfaces. The present study has been motivated by two requirements: to be able to design an optimum array of impinging jets for heat transfer control; and to be able to understand the mechanisms involved under vertical take-off aircratt moving over the ground (which produces a jet flowfield equivalent to an impinging jet on a moving surface - Knowles et al., 1992). 2. EXPERIMENTATION For the experimental phase of this work a dedicated jet impingement rig was set up (Myszko, 1997). This consisted of a centrifugal fan driven by a 3.73kW electric motor which provided air at a pressure of 55mbar(g) to a small settling chamber (340mm wide, 140mm deep and 150ram high, externally) containing baffles and filter materials. On the lower face of this settling chamber was mounted a simple conical nozzle of exit diameter (Dn) 12.7ram. The centre-line of the nozzle was vertical and the entire nozzle/settling chamber assembly was mounted on a steel framework (which was 2m high by 3m square) in such a way that it could be varied in height. Below the nozzle was a rolling road (as used for racing car model ground simulation) which consisted of a 1.219m-wide rubber belt driven over a 1.524m-long platen or table via two rollers. The drive roller was powered by a 20kW electric motor and a feedback control system maintained the belt speed to 0.25% under the conditions used here. Flapping of the belt was prevented by the application of suction through holes in the platen under the belt; this could be varied in four sections (front, le~ side, right side and middle and back) using butterfly valves. This same arrangement of compressor and settling chamber had previously been used for studies of a jet impinging on a fixed surface (a varnished wooden board of 3m x 2m - see Myszko & Knowles, 1996 or Knowles & Myszko, 1998). These earlier studies had confirmed the symmetry and top hat profile of the nozzle exit flow and had shown the nozzle exit turbulence intensity to be about 5% (based on nozzle exit velocity). It should be pointed out that this level of turbulence is typical of the sort of industrial applications in which we are interested. Further details of the free jet characteristics, based on crossed-wire hot-wire anemometry measurements, are contained in Myszko (1997). For the present study, only the wall jet flow was investigated using, initially, a pitotstatic probe to measure mean velocities. The probe was traversed using a three-axis linear traverse frame which was found to give an accuracy of +0.050mm in each direction. The total traverse area available was lm x lm x 0.Sin. To allow for possible slight mis-alignments the traverse axis system was calibrated with respect to the nozzle and rolling road centre-lines. The nozzle, rolling road and probe traverse system are shown in Fig 1. Probe measurements were taken in the flowfield plane of symmetry (i.e. along the centre-line plane of the nozzle and rolling-road) on either side of the jet impingement region, both with the road stationary and moving at 10ms-1. Where the road and wall jet are moving in the same direction is referred to as the retreating side (Vg=-10ms-1);
785 where they are in opposite directions is the approach side (Vg = +10msl). Wall jet mean velocity measurements were taken for nozzle heights of 4, 8 and l0 D, at the radial positions shown in Table 1. Table 1 - Radial locations of probe traverses for fixed and moving ground (H/D, = 4, 8, 10) r/Dn V~ 10 0 -10
1 1.5 2 2.5
3 3.5
x x x x x x
x x x
x x x
x x x
x x x
10.5 12 14 16 18 20
4
4.5
5
6
7
8
9
9.5
10
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
Throughout these tests the jet was run at a nozzle pressure ratio of 1.05 (NPR is the ratio of nozzle stagnation pressure, p0, to atmospheric pressure, pa; NPR = 1.05 gives a nozzle exit velocity of about 90msl). The nozzle had an exit diameter (Dn) of 12.7mm, giving a jet Reynolds number (based on nozzle exit conditions) of 9 x 104. Detailed heat transfer measurements for single and multiple cold air jets impinging on a hot rotating cylinder have been reported by Journeaux, (1990), and are used here to test the plausibility of the CFD predictions of this system using PHOENICS. Much of the data relates to an air jet emerging from a one inch ( 2.54 cm) diameter round nozzle at 20 C with a speed of around 60 ms 1 giving a jet Reynolds number based on nozzle diameter of 105. These conditions are typically encountered in North American paper manufacturing plants. A range of cylinder surface speeds and nozzle-to-surface distances were used during the study but here we use only data for the case when the dimensionless distance between the nozzle and the surface was H/D, =2 and when the ratio of surface speed to jet exit speed (Vg/Vn) was either 0.3 or 0.6.
3. N U M E R I C A L M O D E L L I N G We report here only two examples of our modelling work. Firstly a single round isothermal air jet impinging on a fiat stationary hot plate was modelled in Cartesian coordinates. The computational domain and objects of the Cartesian model are shown in Figure 2, where full use is made of symmetry to reduce computational effort. The quadrant used for the nozzle was approximated in the Cartesian system as shown in Figure 3. The impact plate was divided into sections to allow some spatial resolution of the local heat transfer rate. In the y direction, the sections W 1 to W4 have a width of 89nozzle diameter and the sections W5 to W9 have a width of 2 nozzle diameters. Plate section W l 0 covers the remainder of the domain where very little heat transfer occurs and is necessary to reduce computational end effects. (More than 90% of heat transfer occurs within 10 nozzle diameters of the stagnation point). The computational grid consisted of (40, 130, 40) cells in the (r,z,y) directions, also nonuniformly distributed according to the complexity of the flow and concentrated around the jet boundaries and plate surface. The turbulence models used were the Lam-Bremhorst with and without the Yap correction. Secondly the same air jet system was modelled but this time the impact surface was in motion normal to the centreline of the jet flow. This required a different model arrangement which is shown in Figure 4, once again making full use of symmetry.
786 Here the plate is divided into 10 strips each 2D wide distributed symmetrically about the stagnation point, and two end strips each 6D wide to reduce end effects. The computational grid had (40,196,40) cells with heavy cell concentration around critical areas. Two impact surface speeds were considered; 30% and 60% of the jet exit speed. Computation was carried out on a Pentium II personal computer and run times for satisfactory convergence were around 40 hours. 4. RESULTS AND DISCUSSION Wall jet velocity measurements revealed, for the fixed ground case, the familiar nondimensional shape, with an inner boundary layer, a peak velocity at about Y/Y1/2 - 0.2 and then a free shear layer with a decaying velocity. At low values of r/D, these profiles were affected by proximity to the impingement region but by r/Dn = 4 (for Hn/Dn = 10, earlier at the lower nozzle heights) the mean velocity profiles had become self-similar. This was consistent with the earlier work of Knowles and Myszko (1998, 1996). With the rolling road running at 10ms-1, the wall jet profiles on the retreating side of impingement revealed a much thinner boundary layer than with the fixed ground (Fig 5a). For most of these data the peak velocity measured was the lowest traverse point: at low radii this meant that the true peak velocity was probably not being resolved due to the very thin boundary layer region, whereas at larger radii (typically r/D,>8) this was consistent with the wall jet peak velocity having decayed to the ground speed. On the approach side of the impingement zone the high wall shear stresses caused the wall jet to be much thicker than with the fixed ground. Typical wall jet profiles for this region are shown in Fig 6. At large radii it appears that the wall jet has separated; this is seen more clearly in the detailed profiles of Fig 7. From this figure it can be seen that a point of inflection develops in the wall jet velocity profile between r / O n - - 8 and 9.5. The thickness of the wall jet can be quantified by the distance from the ground to where the outer shear layer velocity has decayed to half the peak wall jet velocity. This "half-thickness" is plotted against radial position in Fig 5(a). This clearly shows an apparent increase in wall jet thickness at r/D, = 8 on the approach side. Fig 5(b) plots the radial decay of peak wall jet velocity for the three cases of moving ground approach side, moving ground retreating side and fixed ground. The approach side shows a more rapid decay of peak velocity, with an increasing decay rate aiter r/D, = 8. On the retreating side it can be seen that the peak velocity is heading asymptotically to the ground speed of 10 ms"1 as radial position increases. The lower recorded values of Vm at low r/Dn reflects the point made earlier about not being able to resolve the very thin boundary layer on the retreating side close to the impingement region. For this reason the recorded values of Yv2 are too high in this area. The separation of the wall jet on the approach side of impingement in the case of the moving ground plane is further emphasised by the momentum flux plots of Fig 8. This Figure also shows that separation occurs at about the same radial position for all the nozzle heights tested and that the lower the nozzle height above the ground the lower the wall jet momentum flux. For CFD model validation purposes values of local Nusselt number were predicted for the case of impingement on a flat stationary surface. The spatial resolution of the local Nusselt number was fixed by the choice of surface strip width as shown in Figure 2. Surface averaged values of Nusselt number plotted in Figure 9 were obtained from the local values by integration and it can be seen that the Lam-Bremhorst with Yap correction model gives slightly better agreement with the experimental data of
787 Journeaux (1990) than the Lam-Bremhorst model. With the rather coarse surface divisions chosen it was not possible to resolve the finer detail of the Nusselt number distribution in the vicinity of the stagnation point, but for use in design the predicted average value of the Nusselt number over a region of 10Dn (eg for this jet Reynolds number, Nuav = 100) is quite acceptable. The second example of heat transfer rate prediction is that where the surface is in motion under an impingement flow ( Figure 4). The radially averaged predicted Nusselt number distributions for impingement on a stationary plate, and on a plate moving at 30% and 60% of the nozzle exit velocity shows an increasing skewness of the distributions with surface speed show the same trend as the experimental data of Journeaux (1990). At low plate speeds the overall effect of surface motion on total heat transfer is generally small; the heat transfer peak is shifted towards the direction of the approaching plate. At plate speeds greater than half the jet exit speed measureable increases in total heat transfer appear and the flow on the approach side of the stagnation zone is predicted to separate in the same position as that detected experimentally in the aerodynamic study of the wall jet region, as shown in Figure 11. CONCLUSIONS Aerodynamic measuremems on a single, isothermal jet impinging on a moving surface (with a surface to jet velocity ratio of about 0.11) have confirmed a strong asymmetry between the advancing and retreating sides of the impingement zone. On the centreline of the retreating (where the wall jet and the impingement surface are moving in the same direction) the wall jet was extremely thin and beyond r/Dn = 8 to 10 the peak velocity decayed to the surface speed. On the approach side the wall jet was thicker than with a fixed ground, and between r/Dn = 8.0 and 9.5 the wall jet separated from the surface this separation position did not vary significantly with nozzle height (in the range Hn/D, = 4 to 10). For many practical applications of impingement heating, cooling or drying, the estimation of the average heating or drying rate over the surface of interest is often carried out on the basis of the correlation between surface average Nusselt number and jet exit velocity Reynolds number for a stationary surface, i.e. Nuav = 0.133.Re 0.71 (r/D,) -0.63 Davies et al (1997). However these practical applications often involve multiple interacting jet arrays impinging on moving surfaces and these systems create heat transfer rates which are higher than those associated with single jets. Here we have shown that it is possible to predict the heat transfer performance of the basic single jet impingement system with reasonable accuracy and in work to be published we will extend these predictions to the more practical systems involving jet arrays. We have also shown that the model predicts the measured effects of surface motion and that for most practical values of surface/jet speed ratio the effect of surface speed on average heat transfer rate for round jets is negligble. Note that this is not the case for slot jet impingement systems, Pekdemir et al, (1998) because of the major flow disruption which is created by the incidence of an air curtain on the flow induced by plate motion. REFERENCES Davies, T.W. (1997), "Convective heat transfer from a hot rotating cylinder with jet impingement", Proc. 5 th UK Heat Transfer Conference, Journeaux, I.J. (1990), "Impinging Jet Heat Transfer and Thermal Deformation for Calender Rolls", PhD Thesis, McGill University.
788 Knowles, K., Bray, D., Bailey, P.J. and Curtis, P. (1992), "Impinging Jets in Crossflow", Aeronautical J., 96 (952) 47-56, February. Knowles, K. and Myszko, M. (1998), "Turbulence Measurements in Radial Wall Jets", Experimental Thermal and Fluid Science, 17, 71-78. Martin, H. (1977), "Heat and mass transfer between impinging gas jets and solid surfaces", Advances in Heat Transfer, 13, 1-60. Myszko, M. (1997), "Experimental and Computational Studies of Factors Affecting Impinging Jet Flowfields", PhD Thesis, Cranfield University, RMCS. Myszko, M. and Knowles, K. (1996), "Radial Wall Jets - Turbulence Measurements", in: Engineering Turbulence Modelling and Experiments 3, eds. W. Rodi and G. Bergeles, 453-462. Pub. Elsevier Science B.V., Amsterdam. Pekdemir, T. and Davies, T.W. (1998), "Mass transfer from stationary circular cylinders in a submerged slot jet of air", Int J Heat Mass Transfer, 41 (15), 23612370. Polat, S. (1993), "Heat and mass transfer in impingement drying", Drying Technology, 11(6), 1147-1176.
Figure 1. Jet Impingement Rig, showing rolling road, probe traverse flame, nozzle, settling chamber and air supply pipe.
789
~._
20Dn
Noz.z.l~
Opening
/!
Y
H/Dn Openings
k21 _L ....., -1_2 71 ...... r-~.
9~ / w 2 / w 4 wt w3
Plate
Sections
Figure 2. Segment of computational domain
Figure 3. Nozzle model
2.00
28.00 r Y(mm) ~
+r/D,
i
1.20
o.,o
/
= 6.0
r/D n -
7.0
o- r/D a =
9.5
---a- r / D , = 10.0 +
r/D a -
10.5
20.00
0.00
,
J
0.0
. . . .
i
4.0
80.0r
,
8.0
,
,
L
,
--A- V = -10.0 , , El . , ,
12.0
r/I
16.0
Co) -'~
9m
n'B 60.0
i 20.0
-It
YI =
--o-
12.00
-10.0
+v,:
16.00
00
10.0
vz =
-
8.00 40.0
..'--
4.00 ~o.o :30.0.I
?~
--2_~_ ~- --e _~
10.0
0.0 I 0.0
~
,
~
I 4.0
.
~
i 8.0
,
. ,...,
r/t
,
I 13.0
~
,
I 16.0
i
~
J
I 20.0
Figure 5. Wall jet development varying surface speeds at HdD.=IO (a) growth of half thickness; (b) decay of peak velocity
0.00 _~J~~~A 0.00 4.00 8.00 12.00 16.00 20.00 V (ms -])
Figure 6. Detailed wall jet velocity profiles on approach side of impingement zone for Hn/Dn= 10 showing separation
790
Opening/[
Opening
"~/I
Y
~
Nozzle ~
~
l
T,H/Dn /_~Openingsl /1
~
6Dn Z
32Dn
Figure 4. Segment of computational domain for impingement on moving plate
2.50 r/D u r/D= r/D n r/D u r/D.. r/D u r/D u r/D u r/D u r/D, r/D, - ~ - r/D u - e - r/D= r/D u --X-- r / D u
Y/Yv2 2.00
1.50
= = = = = = = = = =
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 6.0 7.0
-- B,O
9.5 = 10.0 - 10.5
1.00
0.50
0.00
I
0.00
,
..
,
0.20
,
- J l m m = , , ~
0.40
I-
0.60
,
it~'~,vw'~5~-
0.80
,
1.00
I
1.20
V/V,,
Figure 7. Non-dimensional wall jet profiles on the approach side of the impingement region for Hn/Dn= 10
791 1.0
(kg ms "2)
Mf
Vg = 10ms -1
0.8 x=
0.4
J~P---- -
~::I---B_
_
~ -~<----
----_-",....~.
"
0.2
•
Hn/D n =
o
Hn/D n =
O.O
'~ 9
Hn/D n =
4.0
O. 0
_1__._____~
0.0
3.0
10.0
i ......
4.0
t__
J
,
6.0
I
8.0
,
I
,
i
10.0
12.0
r/Dn
Figure 8. Radial variation of wall jet momentum flux for varying nozzle height
Radially averaged Nusselt Number (HIDn=2, Re=100000) 300 z O}
<~
....................................................................................................................................................................................................
250 200 150
loo
~ :.2 2--22"L'L-_- ..:.: ._:._.._. . . . . . . .
5O
"'l ...............
i
0
5
10
r'
15
i
"
20
rlDn -------- Journeaux
Lam-Bremhorst . . . . . .
Lam-Bremhorst + Yap]
Figure 9. Radial distributions of average Nusselt number for a single round jet impinging on a stationary flat plate
792
Surface Motion Average Nusselt Number Profiles (i-YDn=2, Rle=100000, Surface Motion -->) 250 200 150
,=
100
~.~,
50
!
r,/ \x
L-'L'C -
\ !
-15 -50-
-10
-5
0
5
10
15
-.... rlDn
VgNn = 0 ....
Vg/Vn = 0.3 . . . . . . .
V g N n = 0.6 . . . . .
exp. ]
2o
The experimental data are referred to Vg/V. = 0.64 and Re = 21400
Figure 10. Radial distributions of average Nusselt number for a single round jet impinging on a stationary and a moving plate.
Figure 11. Predicted velocity vectors on the approach side of a moving plate with jet impingement, showing flow separation.
12. Combustion Systems
This Page Intentionally Left Blank
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
795
Turbulence modelling in joint PDF calculations of piloted-jet flames J. Xu and S.B. Pope ~ ~Sibley School of Mechanical and Aerospace Engineering, Cornell Univesity, Ithaca, N.Y., 14853, U.S.A. The objective of this work is to investigate the pressure transport modelling in PDF methods. Calculations are conducted on a piloted-jet non-premixed flame. The pressure transport model is analyzed and explored from the aspects of the model constant and its influence on the scalar flux. Also, the importance of modelling the pressure transport in PDF methods is discussed in comparison with the modelling of turbulence frequency. 1. I n t r o d u c t i o n Computational methods based on theoretical grounds for turbulent combustion have become common tools for engineering designs of gas turbine combustors and furnaces [1,29]. Among these methods, the notable advantage of PDF methods that they can treat convection and reaction without any closure models makes them very attractive [19]. The foundation of PDF methods is the probabilistic description of turbulent flows. Consider a low Mach number reacting flow without radiation and gravity, and with a constant reference pressure P0. The one-point joint PDF f of velocity and compositions at location x and time t is defined by f ( V , r x, t) = (5(V - U ) 5 ( r - r
(1)
where r and V are the sample spaces for compositions r - {r a = 1, 2 , . . . , or} and velocity U, respectively. Here, angled brackets '<( )" denote means. Then, the mean of any function of the velocities and compositions can be obtained from f. For example, the Reynolds stress (uiuj) and scalar flux (uir are respectively calculated by
dVdr (uir
-
i T (V~ - ( U i ) ) (r
(2)
-(Ca))/(V,r162
(3)
where u ~ - U~ - ( U i ) , and r - r - ( r With the use of standard techniques, the joint PDF transport equation can be derived
Of Of p (r --~ + p (r V~Ox-~
O (P) Of 0 Oxi OV~ § ~ [p (r S~(r
v +)s] +
~ [f(ou~ ovoovj P- xj v, r
O~Ox,If
(4)
796 where p(~) is fluid density which is uniquely determined by the compositions; the fluid pressure has been decomposed into two parts: the mean (P) and the fluctuational pressure p; T~j is the viscous stress; J~ represents the molecular diffusion of the c~th species; and finally S~ denotes the source of compositions due to chemical reaction. The terms on the right-hand side of Eq.(4) represent the transport by viscous stresses and dissipation, the transport in composition space by the molecular diffusion, the pressure rate-of-strain correlation (PRS), and the pressure transport. They contain conditional expectations that cannot be expressed by the one-point PDF. Hence, models are to be built for these terms. In contrast, the left-hand side consists of closed terms: convection, transport due to the mean pressure gradient and chemical reaction. It is remarkable that the convection and reaction terms can be treated essentially without approximation. The modelled equation can then be solved by Monte Carlo methods for the PDF's [19] from which a tremendous amount of statistical information can be obtained. These features of PDF methods, in conjunction with other techniques for evaluating chemical reaction rate, say ISAT [21], provide the opportunity of accounting for the interaction between turbulent fluctuations and chemical reactions more accurately, and thus of giving better predictions of turbulent combustion than the Reynolds averaged Navier-Stokes (RANS) equations based turbulence models [9,11]. In PDF methods, the modelling of molecular diffusion process leads to mixing models such as IEM [4] and EMST model [28]. The modelling of the PRS term is similar to those in the second-moment closure (SMC) models. The models for this term include the simplified Langevin model (SLM), the generalized Langevin model (GLM)[7,2O,3O] and the wave-vector model (WVM) [25]. To provide a time scale, models are developed for turbulence frequency such as the stochastic model [25]. These models have been applied and proven to perform very well in turbulent flames [14,15,24]. However, historically, very little attention has been paid to the pressure transport modelling in both RANS models and PDF methods. In RANS models, this term is either neglected or presumably taken into account by the turbulent transport model. Recently, several models have been suggested in the frame of RANS models [3,6]. In PDF methods, the modelling of the pressure transport term is more important since the turbulent convection is represented exactly. Van Slooten et al. recently derived a pressure transport model for velocity, and found that it can greatly improve the calculation of temporal mixing layer (TML)[25]. In this work, a piloted-jet methane flame, i.e., the L flame in [13] is used as a test case to investigate the pressure transport model further. The PRS term is modelled by a GLM model, i.e., the LIPM model [20]. A fast chemistry assumption is found to be suitable for this flame [14], and therefore the IEM mixing model is accordingly adopted. In the next section, PDF methods are briefly described. Section 3 focuses on the pressur~ transport model. Results of testing the pressure transport model are presented in Section 4. Conclusions are drawn in the final section.
2. B a c k g r o u n d PDF methods achieve closure for conditional expectations in Eq.(4) through modelling the behavior of fluid particles by stochastic differential equations (SDE) in accordance
797 with Monte Carlo methods [19]. The flow is represented by an ensemble of stochastic particles. Each particle has its own position X*, velocity U* and composition r For the convenience of discussion, all models presented here are expressed in terms of conventional means, e.g., (Ui). However, it is understood that in the calculations made later, these means are replaced by the corresponding density-weighted means, say (U~) by U~ - (pU~)/(p), without changing the expression of the model. This is the same as the common approach adopted in the modelling of variable density flows [8]. Basically, velocity models are built based on the SDE of the Langevin type. The model considered here is the GLM [7] with an additional term T* denoting pressure transport
dU;=
1 O (P} dt + Giy (U] - (Uj)) dt + T:dt + ffCok~dWi,
(5)
where Co - 2.1, turbulence energy is k - ~1 (UiUi) ~-~ is t h e c o n d i t i o n a l t u r b u l e n c e fiequency (to be discussed later) and W is an isotropic Wiener process. In Eq.(5), the tensor G models the PRS term. Its general form reads [20]
aij -- ~-~ (0~1(~ij --]- ct2bij +
ct3bij ) + Hijkl o
(6) OXl
where
Hgjkt =/31~ijSm + t~25gkSy~+/~3~il~jk + 715~jbm + V2~ikbjl + '73(~ilbjk +v4bij(~kl + "?'5bik(~jl + "?'6bil(~jk.
(7)
The 12 model constants are generally determined as functions of the scalar invariants of
b~y, S~j and t2~j defined respectively by bij
(ltiltj) 2k
1 ~ij 3
Sij
'
1 ~
0 (Ui) -}Oxj Oxi
~ij
'
.
~
Oxj
(8)
Oxi
Pope has shown that by specifying the tensor G, a GLM corresponds to a realizable Reynolds Stress Model in terms of modelling the PRS [20]. One of such models is the Lagrangian isotropization of production model (LIPM) which is a Lagrangian version of the IPM model of Launder et al. [10]. The model constants of LIPM are given in [20]. To provide a time scale, a stochastic model for turbulent frequency has been devised for the particle turbulence frequency w* [25]: do.)* -- --C3@d* -- @ d ) ) a d t
- S ~ w * d t + [2C3C4 (o.))~'~o.)*]1/2 dW,
(9)
where W is an independent Wiener process, and the source term S~ is of the form P S~ - C~2 - C~1 k---~'
(10)
where P - -(uiu5) ~Ox5 is the production of k. Model constants C~1 ~ C~2 C3 and C4 are specified in Table 1. An appropriate physical interpretation of turbulent frequency is that the product of k (w) equals the turbulence energy dissipation c. To account for the external intermittency effects, a conditional-mean turbulence frequency is defined f~ -- Ca (w*[w* > (w)),
(11)
798 Table 1: Model Constants
Cwl Cw2 C3 0.44
0.9
1.0
C4
Cr
Cf/
0.25
2.0
0.6893
where the model constant is chosen such that f~ equals (co} for fully developed homogeneous turbulence (Table 1). Hence, more precisely, the mean dissipation becomes e = kf~. Since the task is to investigate the turbulence modelling associated with PDF methods, and the chemistry of flame considered is near equilibrium (section 4.1), the particle thermochemical state is characterized only by the mixture fraction defined by Zi-
Zi2
(12)
= Z i l - Zi2'
where the subscript 1 and 2 refer to fuel and oxidant respectively, Zi is the mass fraction of element i. Density, temperature and mass fraction are retrieved as functions of the mixture fraction. Accompanying with this model, the IEM mixing model d~* - - ~1Ccf~(~* -
(~))dt,
(13)
can give reasonably accurate results. Hence, 4~ reduces to a single scalar ~ with Sr = 0. 3. P r e s s u r e T r a n s p o r t o~[i(~lv,r The term T* in Eq.(5) models the transport due to the fluctuating pressure ov~ox~ O(puj) i, Eq.(4) which corr po.ds to - -[ol,o,I qu tio.. Ozj + Ox~ ] i. the a ynola - tr Traditionally, the modelling of the pressure transport term is considered as a minor topic both in RANS models and PDF methods. It is either neglected or modelled together with turbulent transport process in RANS models [9]. Recently, the pressure transport has been discussed by Fu [6], and Demuren et al. [3] for RANS models. They proposed models based on the model first suggested by Lumley [12]
1 (UiUkUk)
(pu~) - -~
,
or
1 [O(uiukuk) + O(UjUkUk)] , Tij- -~ Ozj
Oz~
(14)
which is derived from homogeneous turbulence. Early in the development of PDF methods, Pope put forward the model [17] 1
(15)
This model is consistent with Lumley's model Eq.(14). However, it has not been applied in any PDF calculations, because a tractable numerical implementation of the modelled term has not been devised [25]. Recent calculations of TML using PDF methods by Van Slooten et al. [25] show that the pressure transport becomes dominant at the edge of free shear layer. Ignoring this term leads to very poor predictions of even the mean velocity profiles. Since PDF methods
799 express the turbulent transport in exact form, modelling of the pressure transport becomes crucial at the edge of free shear layer where the turbulent transport is less important. Therefore, Van Slooten et al. suggested the following model [25]
T: - Cpt ( u~u~ 2k
1) Ok Oxi '
(16)
where the model constant Cpt = 0.2 is recommended corresponding to the LIPM model. Application of this model in TML does improve the comparison of PDF calculations with DNS results. But, its performance, such as the influence on scalar fields, needs to be examined further in reacting flows. Given Eq.(16), the pressure transport term in Eq.(4) becomes
02 OViOxi If (pl V, r
0 ~-~ ((T~I V, ~ ) f ) -
1 Ok 0 -Cpt2kOx~OV~ If ( ( 2 k - ukuk)] V, r
.(17)
o This model is similar to Pope's model Eq.(15) except for that ~-~7~ ( ) is taken place by 1 ok Ox, ( )" The reason is, as shown by Van Slooten et al. [25], that Eq.(15) is not able to be implemented in the velocity model of the Langevin equation type for high Reynolds number turbulence where the viscous effect is neglected. The counterpart of the pressure transport model for Reynolds stress is derived from Eq.(17) with the use of Eq.(2)
Tij = Cpt
Ok
2k
Ok +
-
0 (UiUkUk} + -
3
~
Oxj
k3/-'7-- + ~
Ox,
k3/2
"
It might be concerned that Tij does not process a form of true transport process. However, the decomposition in the second line of Eq.(18) implies that T~j actually stands for the outcome of two processes: the first term is a true transport process; the second term is a transport-like process of the normalized triple correlation with coefficient OF'3k3~2 If the second term is approximately homogeneous (similar to the algebraic stress model (ASM) assumption [23]), T~j is identical to the Lumley's model (with Cpt = 0.6). In PDF methods, the velocity model combined with a particular mixing model yields a particular model for scalar flux [20]. This connection also provides a way to justify the velocity model: the velocity model should lead to a consistent model for the scalar flux [5,18,22]. The influence of the pressure transport model Eq.(16) on scalar fields can be understood in conjunction with the transport equation for the scalar flux 0 (u~C') + (Uj)
Ot
Oxj
= Pfi +
~
-ef
"
(19)
The four terms on the right-hand side represent production, turbulent transport, pressure scrambling, and dissipation, respectively. For the convenience of discussion, the pressure scrambling term is expressed explicitly. The modelling of the pressure scrambling is subject to much more uncertainty than is the case for Reynolds stresses. It is appreciated in SMC models that there is very little
800 advantage to decompose it into two parts as it is done for the Reynolds stress equations. Also, it is common to model this term as a linear function of (ui~'}. From the velocity model and Eq.(13), Pope has shown that the LIPM model leads to the following model for pressure scrambling (in absence of T*)
-I~ 'Op }~
- ~ (\ C r1
3C0) ft (ui~'}-~-~
[~Sij-~-~-~ij-- ~f~jlbli](Uj~'} .
(20)
On the other hand, T~* yields an additional term in the scalar flux equation
Z~ -- Cpt(~k~tk~'}2kOXiO__k Cpt 03 (UkUk~'}OXi Cptk3/2Oxi ((Uk~k~'})k3/2'
(21)
which apparently can not be included in the Eq.(20). To seek for an alternative interpretation, we decompose the pressure scrambling into two parts
_{~,op / o( }, b-~z~}- nf + T~, nf- \P~-~z~
T[--( ~oxi )}"
(22)
Then, it is appropriate to state that 1-If is modelled by the right-hand side of Eq.(20), and the transport-like term T/~ is modelled according to Eq.(21). Two observations are worth emphasizing for T[: 1. T[ is in a linear form of ~', and thus satisfies the consistent conditions for scalar field modelling [18]. This then justifies the validity of velocity pressure transport model T~* in terms of consistency.
2. T[ tends
to transport the scalar fluctuations against the energy gradient as well. It affects the scalar variance @'~'} through the influence on the scalar flux (ui~'}.
4. R e s u l t s and D i s c u s s i o n s 4.1. Test Case
Piloted-jet methane flames have been the target of several numerical modelling studies using PDF methods [14,15,24]. The flame considered here is the L flame of which the jet, pilot and coflow velocities are, respectively, Uj = 41re~s, Up = 24m/s and Uc = 15m/s. Detailed description of the experiment is given by Masri et al. [13]. As experiment indicates, this flame is blue up to 60Rj (Rj is the jet radius) with very little extinction. This favors the usage of IEM mixing model and flamelet model. Following the discussion of Xu and Pope [31] about the numerical accuracy in the joint PDF calculations, we use total 1600 cells and about 3.2 z 105 particles in the computational domain (80Rj z 15Rj). Boundary conditions are also specified according to [31]. 4.2. M o d e l T e s t i n g s The calculation results using the standard model constant (Table 1) without pressure transport are compared to the available experimental data (neither (uv) nor scalar fluxes are measured) in Fig.1. For abbreviation, only profiles at x/Rj = 40 are presented. Overall, the comparison seems satisfactory except for (~'~'} whose disagreement is mainly due to the simple mixing model used here.
801
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6
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Figure 1. Comparison of mean profiles at x / R j - 40. Lines: PDF calculation, solid, Cpt - 0.0, C~1 - 0.44; dashed, Cpt - 0.2, C~1 - 0.44; dash-dotted, Cpt - 0.0, C~I - 0.56; Symbols: experimental data of Masri et al. Attention, however, is drawn to the noticeable difference between the calculation and the experiment at the edge of shear layer: the calculation overpredicts the mean velocity, the mean mixture fraction, and in particular the turbulence energy k. Similar problems have also been detected in previous calculations of the L flame using the more advanced EMST mixing model and the ISAT algorithm [24]. This deficiency may cause significant error in the prediction of composition mass fractions since the stoichiometric value of mixture fraction in the L flame is about 0.035 which lies in the edge region of the turbulent shear layer. It is then conjectured that the inclusion of pressure transport is able to remedy this problem [24]. This argument is based on the observation that in the TML calculation, by doing so, the mean velocity profile at the edge of shear layer is dramatically improved. The mechanism behind this is argued to be that the pressure transport Eq.(16) pushes the high energy particles up the energy gradient such that the turbulence excursions into the non-turbulent region is prevented [25]. A calculation is then conducted first by including the pressure transport with the recommended model constant Cpt = 0.2 suitable for LIPM [25]. Results are plotted in Fig.1. There is almost no improvement over the calculation of Cpt = 0.0. The scalar flux is also insensitive to pressure transport model for Cpt = 0.2. In fact, even in the TML calculation, it is found that modelled pressure transport is relatively much smaller than the estimate from DNS data [25]. But, the similar performance does not appear here. Therefore, we investigate the performance of the model with larger values of the model constant. Calculations with Cpt = 0.2, 0.5 and 1.0 are then compared in Fig.2. It is apparent that there is no significant difference between Cpt = 0.2 and Cpt = 0.5, while Cpt = 1.0 gives rise to unacceptable results. To inspect the behavior of the pressure transport model more carefully, we turn to examine the effect of pressure transport on the energy transport equation which reads
Dk = 7~ - ~ + D + 7-, Dt
(23)
where besides neglecting viscous effect, the pressure transport term 7- appears in addition
802 to the standard model of k [11]
l Ok _CptO
(24)
,-i-R
Here, again we separate 7- into two parts: "YTand "YRto demonstrates explicitly that T is equivalent to a true pressure transport process (Lumley's model) if TR is negligible. Fig.3 plots T along with "IT and "YR.It is obvious that TR is of about the same magnitude as TT and T, and thus can not be neglected. In particular, TT and TR are cancelled out in the region of turbulence energy peak. Therefore, T does not act like a transport process, which deteriorates the performance of the model. However, T indeed becomes a sink of energy at the edge so as to prevent the particles of high energy from spreading out. The reason that the pressure transport model performs here not so well as it does in TML calculations may be due to the fact that the piloted-jet flame is physically similar to turbulent round-jet flows whose modelling has been shown to be different from the modelling of plane flows [16]. The current results are rather consistent with the findings in the P D F calculations of non-reacting swirling jet [26] that the pressure transport indeed plays a secondary role compared to the turbulence frequency. A conventional way to overcome this so-called round-jet anomaly of turbulence modelling is to modify the dissipation equation, such as the modification suggested by Pope [16]. To examine this idea, we use a simple approach in that the model constant C~1 is changed from 0.44 to 0.56. This is very similar to what Dally et al. [2] adopted in their axi-symmetric jet and bluff-body flame calculations. The result of this modification without pressure transport shows a little improvement of the mean velocity profile, but the turbulence energy peak is underestimated (Fig.l). 5. C o n c l u s i o n s A pressure transport model introduced by Van Slooten et al. for the joint P D F methods [25] has been discussed, and investigated using a piloted-jet methane non-premixed flame as the test case. The influence of this transport model on scalar fields is discussed in conjunction with the IEM mixing model, and the model is verified to be valid in terms of the consistency requirement of modelling scalar fields. This model is shown to be equivalent to the Lumley's pressure transport model if the normalized triple correlation is approximately homogeneous. However, the results demonstrate that this is not true, and the model does not perform so well as it did in the previous calculation of temporal mixing layer. A model for pressure transport in the true transport form and a more accurate model for the turbulence frequency are desired. The source term S~ defined here is derived from the conventional model of dissipation, and thus should be considered as a provisional choice. A model that depends on both the mean rate of strain and the mean rotation may be more appropriate [27]. This is to be explored in future.
Acknowledgments This work is supported by Air Force Office of Scientific Research Grant No. F4962097-1-0126.
803 REFERENCES
.
3. .
5. 6. 7. 8.
10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.
S. M. Correa. Twenty-Seventh Syrup. (International) on Combust., 1998. The Combustion Institute. B. B. Dally, D. F. Fletcher, and A. R. Masri. Combust. Theo. Modelling, 2:193, 1998. A. O. Demuren, S. K. Lele, and P. A. Durbin. Center for Turbulence Research: Proceedings of the Summer Program, 1994. C. Dopazo. Phys. Fluids A, 18(2):389, 1975. P. A. Durbin and Y. Shabany. Fluid Dyn. Res., 20:115, 1997. S. Fu. Comput. Fluids, 22(23):199, 1993. D. C. Haworth and S. B. Pope. Phys. Fluids, 29(2):387, 1986. W. P. Jones. In P. A. Libby and F. A. Williams, editor, Turbulence Reacting Flows, Academic Press, 1994. B. E. Launder. In J. L. Lumley, editor, Whither Turbulence? Turbulence at the Crossroads, page 439. Springer-Verlag, New York, 1989. B. E. Launder, G. J. Reece, and W. Rodi. J. Fluid Mech., 68(3):537, 1975. B. E. Launder and D. B. Spalding. Mathematical Models of Turbulence, Academic Press, London, 1972. J. L. Lumley. In Advances in Applied Mechanics, Vol. 18, page 123. Academic Press, 1978. A. R. Masri and R. W. Bilger. Twenty-First Syrup. (International) on Combust., page 1511, 1988. The Combustion Institute. A. R. Masri and S. B. Pope. Combust. Flame, 81:13, 1990. A. T. Norris and S. B. Pope. Combust. Flame, 100:211, 1995. S. B. Pope. AIAA J., 16:279, 1978. S. B. Pope. Phys. Fluid, 24:588, 1981. S. B. Pope. Phys. Fluid, 26:404, 1983. S. B. Pope. Prog. Energy Combust. Sci., 11:119, 1985. S. B. Pope. Phys. Fluids, 6(2):973, 1994. S. B. Pope. Combust. Theo. Modelling, 1:41, 1997. S. B. Pope. J. Fluid Mech., 359:259, 1998. W. Rodi. Turbulence Models and Their Application in Hydraulics - a State of the Art Review, International Asso. for Hydr. Res., 1980. V. Saxena and S. B. Pope. In Twenty-Seventh Syrup. (International) on Combust., 1998. The Combustion Institute. P. R. Van Slooten, Jayesh, and S. B. Pope. Phys. Fluid, 10(1):246, 1998. P. R. Van Slooten and S. B. Pope. Flow, Turb. and Combust., 1998. (submitted). C. A. Speziale and T. B. Gatski. J. Fluid Mech., 344:155, 1997. S. Subramaniam and S. B. Pope. Combust. Flame, 115:487, 1998. R. M. Stubbs and N.-S. Liu. AIAA Paper 97-3114, July 1997. H. A. Wouters, T. W. J. Peeters, and D. Roekaerts. Phys. Fluids, 8(7):1702, 1996. J. Xu and S. B. Pope. J. Com. Phys., 1998. submitted.
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Figure 2. Effects of model constant Cpt ( z / R j - 40) with C~1 - 0.44. Solid lines: Cpt - 0.0; Dashed lines: @t - 0.2; Dash-dotted lines" Cpt - 0.5; Long dashed lines: Cpt - 1 . 0 .
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Figure 3. Profiles of pressure transport in energy equation ( x / R j - 40). (a) Cpt - 0.2; (b) ~pt- 0.5; Solid line: T; Dashed-dotted line: TT; Dash line" TR.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
805
Towards a General Correlation of Turbulent Premixed Flame Wrinkling K. Atashkari, M. Lawes, C.G.W. Sheppard, and R. Woolley. School of Mechanical Engineering, The University of Leeds, Leeds, LS2 9JT, United Kingdom. 1. INTRODUCTION The turbulent burning velocity (ut), defined as the rate of entrainment of unburned mixture into the flame brush, is an important parameter that has been used to characterise the combustion rate of turbulent premixed flames. Bradley et al. (1992a) correlated all available data on turbulent burning velocities in terms of dimensionless groups involving thermochemical and turbulence parameters. For this, a universal dimensionless psd has been proposed to characterise turbulence over a wide range of turbulence Reynolds numbers (Abdel-Gayed et al. 1987). This has been used to quantify the temporal development of a turbulent flame following spark ignition. Although the turbulent burning velocity is well predicted by empirical correlations, it does not, by itself, quantify the burning rate since it is a measure of the consumption of unburned gas, rather than the production of burned gas, nor does it yield details of turbulent flame structure or wrinkling. The present work is a first attempt to produce a general correlation of flame wrinkling characteristics in terms of dimensionless groups. An understanding of how turbulence wrinkles a flame is essential if one is to understand fully the structure and burning rate of turbulent flames. Flames within the wrinkled laminar regime (Borghi, 1985) are considered and the complexities of flame quenching due to high rates of stretch are avoided. 2. EXPERIMENTAL APPARATUS AND TECHNIQUES To explore generality, three very different experimental systems were employed. A turbulent V flame burner provided a steady state flame at modest levels of turbulence, a fan stirred combustion vessel provided a wider range of operating conditions under non-steady conditions, and an optically accessed engine provided data in a complex engineering application. In all cases, turbulence was characterised using laser Doppler velocimetry (LDV). In the V burner and engine, the integral length scales were obtained using an autocorrelation technique in conjunction with Taylor's hypothesis. However, in the fan stirred vessel, in which the mean velocity was near zero (rendering Taylor's hypothesis inapplicable) a two point correlation method was adopted. Turbulence characteristics and flame operating conditions, for the three rigs, are presented in Table 1. 2.1 V burner
This apparatus has been described and characterised elsewhere (Bradley et al. (1992b)
806 and Atashkari (1997)). An unconfined premixed methane-air flame, at an equivalence ratio, ~b, of 0.84 was stabilised on a 1 mm diameter rod, mounted on the exit plane of a 50 mm diameter cylindrical burner. Turbulence was generated by a wire mesh grid with 1 mm diameter wire spaced at 3 mm, and mounted 50 mm upstream of the rod. The turbulence decayed with distance from the burner as indicated in Table 1. In addition to laser Doppler velocimetry, mean and turbulence velocities have been measured in reacting flows by particle imaging velocimetry (PIV). This technique had the advantage of revealing information on the interaction between turbulence and flame wrinkling. This apparatus has the experimental convenience of facilitating rapid collection of large data sets at well controlled operating conditions. However, its usefulness is restricted by its limits of operation due to the need for flame stabilisation. 2.2 Fan stirred combustion vessel Premixed methane-air mixtures were ignited in a high pressure/high temperature spherical fan stirred combustion vessel (bomb) (Bradley et al. 1998) at an initial pressure and temperature of 1 bar and 300 K. The 380 mm diameter vessel had extensive optical access via 3 pairs of orthogonal windows of 150 mm diameter. Turbulence was generated by four fans, each driven by an independently controlled electric motor up to a maximum speed of 10 000 rpm. For all experiments, the combustible pre-mixture was ignited by a centrally located spark electrode. An average ignition energy of 23 mJ was supplied to the spark gap. The turbulence was near uniform and isotropic within the field of view of the windows. The integral length scale was not influenced by the fan speed except at the lowest speeds, where it increased slightly. The mean velocity was nearly zero and the rms. turbulence velocity increased linearly with fan speed. 2.3 Engine Engine data were generated using an extensively modified single cylinder JLO L3273 ported engine with a bore diameter of 80 mm, stroke of 74 mm and a compression ratio of 7.6" 1 (Hicks et al. 1994). The original cylinder head was replaced by a modified one in which a fused quartz window permitted full bore optical access. Windows were also placed in the side walls of the head to allow laser sheet access to the combustion chamber. The engine was operated at 1500 rpm with stoichiometric iso-octane-air mixtures and, to minimise the concentration of residual burnt gases from previous firing cycles, was skip fired every fifth cycle. The inlet ports were configured to produce a low turbulence flow field with minimal large scale bulk flow motions.
Table 1 Operating characteristics of the three rigs. Rig Fuel ~ u L (ms "l) (mm) V Burner methane 0.84 0.5-0.7 3 Bomb methane 1.0 0.56 23 Bomb methane 1.0 1.12 20 Engine Iso-octane 1.0 2.25 1.6 a
Maximum attained values, used in Figure 5 (b).
a
ul
(ms -1) 0.27 0.36 0.36 0.75
u '/ul
Zak a
a 'k
1.9-2.7 1.6 3.1 3
(mm) 5.0 21.7 27.7 7.4
(mm) 2.0 7.9 11.3 2.3
807
2.4 Laser sheet imaging The same laser sheet imaging technique was used in all three experimental rigs. The 510.6 nm pulsed beam from a copper vapour laser was formed into a thin laser sheet with a combination of spherical and cylindrical optics. The laser sheet was used to illuminate seed particles that were introduced into the flow field and the Mie scattered light was recorded by a camera for subsequent processing. For the V burner and engine, titanium dioxide seed of approximately 0.2 ~tm diameter was utilised. The number density and optical properties were such that a good contrast between Mie scattering from within unburned and burned gas was attained. Hence, the flame edge contour was readily defined. However, this seeding method proved unsatisfactory in the bomb due to flow dynamics which caused rapid dispersion of seed onto the chamber walls. Within the bomb sub-micron tobacco smoke was found to be more effective although the image contrast remained poorer than attained in the other rigs. Two cameras were used to record the Mie scattered light. A Cordin rotating drum camera using TMAX 3200 ASA film was used for the V burner and engine. The camera speed was set to capture images at a laser frequency of 10 kHz. A Kodak EktaPro HS Motion Analyser (Model 4540), operated at a framing rate of 4.5 kHz, recorded bomb flame images, which were stored onto video tape. In each case, a 510.6 nm bandpass (10 nm FWHM) interference filter was placed directly in front of the camera lens to block extraneous light. Digitised flame contours, were extracted from the film or video images by computer image processing of 'good quality' images, or hand tracing of those of lower quality. The resolution of the flame contours was a function of the optical system for each experiment and of the processing technique. Flame contours could be resolved, for the V burner, bomb and engine respectively to better than 0.4, 0.8 and 0.2 mm. The digitised flame contours were expressed as a series of amplitudes, aj, given by the distance of the local flame front from, and at a normal to, the mean flame front (Hicks et al. 1994). For the bomb and engine, the mean flame front was assumed to be a circle whose centre was at the centroid of the flame. This usually corresponded to the point of ignition. The mean flame front on the V burner was defined using a first order least squares fit through all of the points, aj-. In all cases, the points, aj, were taken equidistant along the mean flame contour. The rms amplitude of flame wrinkling, a 'k, was determined directly from the series of aj . Here, the subscript k indicates values at a given instant during flame development following ignition. Flames on the V burner were assumed to be fully developed at all times (a 'k - a '). A characteristic size of flame wrinkles was obtained from the autocorrelation of the series, aj to yield an integral scale of flame wrinkling, L~k, in a similar way to that used to obtain L. Finally the spectral content of the flame wrinkles was obtained from the Fourier transform of the series, aj, to yield the Power Spectral Density (psd) (e.g. Newland, 1993) in which S(k) is the spectral coefficient at a wave number k, (= 1/g ), where g is the wrinkle wavelength. The maximum resolvable wave number depended on the resolution of the imaging technique (in mm/pixel). To avoid aliasing, the maximum wave number, within the sample should be less than half the imaging resolution. In the present work, this criterion was unlikely to be satisfied, especially at conditions of high u' and low L. However, any aliasing at high frequency was unlikely to have a serious effect on the conclusions drawn from the analysis.
808
3. RESULTS
AND
DISCUSSION
Typical examples of flame edge contours derived from flames in each rig are shown in Fig. 1. Shown in Fig. l(a) are two consecutive flame contours from both sides of the symmetrical V flame. Also shown are velocity vectors after subtraction of the mean velocities in both the horizontal and vertical directions to reveal details of the turbulence structure. A qualitative view of the interaction between turbulent 'eddies' and flame wrinkling can be obtained from Fig. l(a). However, care must be taken in this because the spatial resolution of the PIV data is---2 mm, almost the size of the integral length scale, and the small scale Flame 1
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-60 -60'-zl0'-20'
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Figure 1. Examples of flame contours. (a) Two V bumer flame edges with the associated velocity field (b) Sequential contours from the bomb at u ' / u z - 1.6 at 1.1, 3.8, 7.8, 11.1 and 14.4 ms from ignition. (c) Sequential contours from the bomb at u '/uz = 3.1 at 0.9, 2.2, 4.4, 6.2 and 8.4 ms from ignition. (d) Flame edge contour from the engine at 1o after top dead centre.
809 turbulent structure cannot be discerned from this data. Shown in Figs. 1(b) and 1(c) are flame tracings from deflagrations in the bomb at different times after ignition. The ratio of u '/Ul is 1.6 and 3.1 in Figs. l(b) and l(c) respectively. It is clear from both these figures that the amount of wrinkling increases during flame growth. The flame shown in Fig. l(b) is nearly spherical, especially in the early stages. However, that in Fig. l(c), obtained at higher turbulence than in the former, has an elongated structure that probably is due to significant convection of the embryonic flame kernel during and immediately after, ignition. In Fig. 1(d) a single flame edge contour from the engine, taken 1o after top dead centre, is shown. At this point the cylinder pressure was 17 bar. There are clear differences between the wrinkling structure of the flames on each rig and, in particular, smaller wrinkles were obtained in the engine flame of Fig. 1(d) than in any of the other flames. Probable reasons for this are higher turbulence, smaller length scale and smaller laminar flame thickness (due to the higher pressure) in the engine than in the other rigs. Shown in Fig. 2 are the temporal developments of the wrinkling parameters a 'k and Lak for the developing flames similar to those presented in Figs. 1(b) and l(c). Data from two explosions at each condition are presented. Both a 'k and Lak increase with time from ignition. This phenomenon is very similar to that first reported by Lancaster (1976) in which the turbulent burning velocity increases as the flame grows. Abdel-Gayed et al. (1987) show that the burning velocity can be related to the effective rms turbulence velocity which is given by the portion of the turbulence spectrum with frequencies greater than the reciprocal of the elapsed time from ignition of the developing flame kernel. A similar explanation might be used here to interpret Fig. 2. Only eddies smaller than the flame, or with a fast enough turn over time, can significantly wrinkle it. As the flame grows, progressively larger eddies will affect the flame structure and, hence, both the amplitude and mean wavelength of the wrinkles increase. Since both the amplitude and wavelength of wrinkles are likely to be related to the length scales of turbulence, it appears reasonable to normalise the lengths in Fig. 2 by the integral length scale of turbulence, L.
35
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Figure 2. Variation of a 'k and Lak with time from ignition for Mie scattered images taken in the bomb.
810 0.7 1.5
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9 9 9
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t/r
Figure 3. Variation of normalised a 'k and L~k with normalised time for Mie scattered images in the bomb. Similarly, since the rate of flame development is related to the integral timescale of turbulence, r, (Abdel-Gayed et al. (1987)) an appropriate normal 9 parameter for time is r. The result of this normal 9149 is shown in Fig. 3. Both a '~ and Lak for each value of u '/ul collapse onto the same curves, suggesting, for the conditions studied, a generality of flame wrinkling development. The maximum attained values of a'k and Lak from each experimental condition are shown in Table 1. In all cases L~k is significantly larger than a 'k, a consequence of the flame tending to 'smooth out' the turbulent wrinkles. It is interesting to note that, with the exception of the engine flame, values of L~k are approximately equal to L. Support for the above 'spectral' explanation of the development of flame wrinkling is presented in Fig. 4. Shown in Figs. 4(a) and 4(b) are power spectral density functions (psd) obtained from three flame contours at different times during the development of the flames in Figs. 1(b) and 1(c). All functions exhibit the same general trend. It is interesting to note that as the magnitude of flame wrinkling increases, as shown by a 'ilL in Fig 3(a), the magnitude of S(k), at a given wavenumber, remains unchanged. Instead, progressively smaller wavenumbers (larger length scales) contribute to the flame wrinkling. In all cases, the slope of the spectrum is approximately-2.7. At high wavenumbers there is a slight decrease in this slope which might be due to aliasing of higher wavenumbers, or to noise generated in the flame edge tracing process. Power spectral densities are compared for all three rigs in Fig. 5(a). Those for the bomb are the most developed ones from Fig. 4. At the lower wavenumbers, spectra from the engine are considerably lower than those from the bomb (Spectra at low wavenumbers were not available in the V burner due to the relatively small size of the flame). The probable reason for this is that the integral length in the engine (1.6 mm) is much smaller than that in the bomb (-20 mm). The amplitudes of flame wrinkling in the engine and in the V burner are similar within the mid wavenumber range. However, even though the integral length scales of turbulence
811
10000
"
A
a a ~ 9
1000
a ,,
100
+
1.1 m s
9
7.8ms
a
14.4 ms
A~ 9 ~a~
10000
= 1.6
u'/ul
""
A,,
AA
1000
u '/u I = 3.1
A
+
""~.
100
9
9 "
em
lO
1.3 ms 5 . 8 ms 9 . 3 ms
9
~
1o 9 %AAA 9 A +
0.1
~
o~X +
1
r~
9162
0.1
-~
0.01
0.01 9
§
1E-3
. . . . .
I
.
.
.
.
.
.
.
.
I
0.01
'
'
'
'
.~++
1E-3
' ' " I
0.1
.....
!
.
0.01
1
.
.
.
.
.
"1
.
.
.
.
.
+ .
++
.
0.1 k (mm 1)
k (ram -1) Figure 4. Power spectral densities for two bomb explosions at time from ignition.
.
u '/ul
= 1.6 and 3.2 at increasing
are comparable in these rigs, at high wavenumbers the amplitude is higher in the engine than in the V burner. This might be due to the higher rms velocity in the engine than in the V burner. This would impart more kinetic energy into the high wavenumbers and, hence, increase wrinkling However, an alternative, or co-existing explanation is that propagating turbulent flame fronts are thought to 'smooth out' any turbulent wrinkling of a scale that is smaller than the laminar flame thickness. Since, the flame thickness of the engine flame is thinner than those in the V burner and bomb, this might explain the observation of small scale wrinkling in the engine.
10000
2
10 x
1000
o x
• ~"~--o
1 X
100
x
X 9
D ~a9::,
n
X
0.1
•
~
o
XIiL~ 0
10 ~
~ 0.1 0.01
~- 1E-3
o
V Bumer
n
Bomb, u'/u t = 1.6 ~ Bomb, u Y u I = 3.2 ~ox Engine c~x~
1E-3
x
1E-4
.....
0.01
o
1E-4
[]
1E-5 x
i
0.01
. . . . . . . .
i
. . . . . . . .
0.1 k (mm -I )
!
1
V Burner Bomb, u'/u i = 1.6 Bomb, u'/u I - 3.2 Engine
~215 -G 9 ~x ~9~o 2 • Nx
~
. . . . . . . .
10
1E-6 ..... 0.1
. . . . . . . .
|
. . . . . . . .
1
I
,
,
,
10 kLak
Figure 5. Comparison of psd's from the 3 experimental systems. (a) raw psd's. (b) Nondimensionalised with the wrinkling parameters a 'k and Lak.
812 The lower amplitude of flame wrinkling at the middle and high wavenumbers within the bomb than within the other rigs, probably is because of the much wider range of turbulence scales present in the bomb. Hence, flame wrinkling also is distributed over a wider range. In the case of the engine and the bomb, wrinkle scales larger than either the mean radius of the flames (or in the bore of the engine) were observed. These large scales may be a result of large 'bulk' flows present in the vessels, but may also be due to an inherited nonspherical flame shape produced early during flame development. Previous workers have compared the gradient of the psd of flame wrinkling with that for isotropic turbulence within the inertial sub-range which is -5/3 (Hicks et al. 1994, and Wirth et al, 1993). Wirth et al. suggested that the magnitude of the gradient depends on u '/IA l with the gradient approaching that of the psd of turbulence as u '/1Al is increased. Gradients of the psd's in the present work (neglecting the upper and lower extremes of wavenumber) are plotted in Fig. 6 as a function of u' and of u'/ul. Although there is some scatter, Fig. 6 supports the proposal by Wirth et al.
1
u'/u t
2 '
I
3
'
4
I
10
'
9
X
1
-2.6
0.1
x~ X
X(~'(.O 0 0
9
x r
X
~
9
,~~-2.8
0.01
xxlFe'~m Eo
1E-3
o~..~
-3.0
~
1E-4
~
1E-5 1E-6,
-3.2
oo
X
.
0.'5
1E-7
9
.
1.'0
.
.
1.'5 u' (m/s)
210
2.5
Figure 6. Variation of spectral gradient with u' and u '/ul.
1E-8 0.01
~e... V Bumer %~%o Bomb, u ' / u I - 1.6 Bomb, u ' / u t = 3.2 Engine . . . . . . . .
|
0.1
. . . . . . . .
|
1 kL(u'/u)
. . . . . . . .
|
% . . . . . .
10
Figure 7. Non-dimensionalised wrinkling psd in terms of turbulence parameters.
A universal non-dimensional psd of turbulence in terms of the turbulent integral length scale and u' was proposed by Abdel-Gayed et al. (1987). A similar approach was taken here using the wrinkling parameters a'k and Lak and this is demonstrated in Fig. 5(b). The four resultant psd's overlay each other at all except the higher wavenumbers (kLak > 3) and so, demonstrates some generality of the wrinkling psd's for flames in each rig. However, a difficulty with this method of normalisation is the uncertainty of the appropriate values of a '~ and Lak to be used in the normalisation. In the analogous normalised spectrum of turbulence, fully developed values of u' and L are used. However fully developed values of a '~ and Lak are not available for the bomb and engine data. It would be unreasonable to assume that a general non-dimensional spectrum would result unless the full spectrum were used. This sets a limit on the appropriateness of the present normalisation since the full spectrum can be
813 obtained only from large flames. Moreover, the spectrum is not fully predictive since one requires, in advance, values of a 'k and Lak. Flame wrinkling is primarily the result of the action of turbulent eddies acting upon the flame surface, which burns locally at the laminar burning velocity. Ultimately, the wrinkling seen across the flame front should scale with that of the turbulent flow field and ut. Hence a more appropriate normalisation to that in Fig. 7 should be in terms of these. It is proposed that the appropriate normalisation factor is L(u '/ul) which addresses both the size of turbulent eddies and the relative velocities of the local flame surface and turbulence. The spectra of Fig. 5 are expressed in this form in Fig. 7. For dimensional consistency, the normalisation of S(k) is by (L(u '/ul)) 3. Figure 7 provides a reasonable correlation of flame wrinkling in terms of readily measurable turbulence and flame parameters. Clearly, it is not totally general since it neglects the observed small variation in the gradient of the psd (Fig. 6). This is reflected in the amount of scatter in the results for different conditions. Nevertheless, it provides valuable insight into the mechanisms of turbulent premixed flame wrinkling. It allows a comparison between the wrinkling data obtained under very different operating conditions. The integral length scale in the bomb is about an order of magnitude greater than that in the other rigs and this provides a useful contrast, the effects of which can be seen in Fig. 6. Although the actual wrinkling wavelengths, as seen in Figs. 1 and 5, are smaller in the engine than in the bomb, in comparison with the turbulence structure the bomb provides data at relatively smaller scales. Whilst the study of small scale wrinkling (relative to L) in the engine and V burner is limited by the resolution of the imaging technique, the bomb allows study at smaller scales. Conversely, the engine flame, with its large value or flame radius/L provides a useful vessel for the study of large scale wrinkling (again, relative to L). The present work presents a first attempt at a general correlation of flame wrinkling. Limitations of the analysis are the neglect of flame stretch, including strain and curvature, and the effect of flame thickness. One would expect flame stretch to have a significant affect on flame wrinkling at higher turbulence levels than were studied here. This is still to be investigated. However, it is remarkable that flames studied in the engine, which because of its higher pressure would have a significantly different flame thickness from those in the other vessels, yielded similar wrinkling characteristics. 4. CONCLUSIONS 9 The wrinkling characteristics of premixed flames have been studied in three very different experimental systems. These cover fully developed, steady state flames on a burner, and transient flames following spark ignition in an engine and an explosion vessel. The integral length scale of turbulence was varied by more than an order of magnitude. 9 Both the rms amplitude of flame wrinkling, a'k and its integral length scale, Lag, were found to increase with time following spark ignition. The rate of development of wrinkles was found to be well predicted by the dimensionless groups: a 'k/L, Lak/L and t/~. 9 Power spectral density functions of flame wrinkling were obtained and the same general features were observed in those from each of the three vessels. 9 The increasing size of flame wrinkles, during flame development, was shown to be due to contributions from progressively larger length scales, rather than from an increase in the magnitude of smaller scales. 9 Power spectra from the three rigs were expressed in dimensionless form involving the groups" S(k)/(L(u '/bll)) 3 and k(L(u '/ul). In terms of these groups, spectra from each rig had
814 the same general form. It is clear from the spectra that, in relation to the turbulence scales, the combustion bomb, with its large integral length scale of turbulence, is useful in providing insight into small scale wrinkling (relative to L). However, the engine, with a small value of L, is more useful for the study of larger scale wrinkling. Although the effects of flame stretch and flame thickness were neglected, remarkable consistency was obtained between data in the engine at a pressure of 17 bar and that in the other rigs at atmospheric pressure. One would expect the flame thickness to be significantly different in these cases. ACKNOWLEDGEMENTS The authors thank Zahurul Haq and Lynne Gillespie who obtained the experimental measurements from the bomb and engine respectively. Financial support was given by the EPSRC, the Kodak HS4540 was supplied by the EPSRC loan pool. Thanks are due to Oxford lasers who provided valuable assistance and equipment loans. REFERENCES Abdel-Gayed R.G., Bradley D. and Lawes M., (1987) Turbulent burning velocities: a general correlation in terms of straining rates, Proceedings of the Royal Society of London, A 414, pp. 389-413. Abdel-Gayed R.G., Bradley D., Lawes M. and Lung F.K-K. (1987) Premixed turbulent burning during the early stages of an explosion, Twentyfirst Symposium(International) on Combustion, The Combustion Institute, Pittsburgh, pp. 497-504. Borghi, R., (1985) in Recent Advances in the Aerospace Sciences (C. Bruno and C. Casci, Eds.), Plenum Press, New York. Bradley D., Lawes M., Scott M.J., Sheppard C.G.W., Greenhalgh D.A. & Porter F.M, (1992) Measurement of temperature pdfs in turbulent flames by the CARS technique, Twentyfourth Symposium (International) on combustion, The Combustion Institute, Pittsburgh, pp. 527-535. Bradley D., Lau K-C. and Lawes M. (1992a) Flame stretch rate as a determinant of turbulent burning velocity, Philosophical Transactions of the Royal Society of London A, 338, pp. 559-587. Bradley, D., Hicks, R. A., Lawes, M., Sheppard, C. G. W. and Woolley, R., (1998) The measurement of laminar burning velocities and Markstein numbers for iso-octane-air and isooctane-n-heptane-air mixtures at elevated temperatures and pressures in an explosion bomb. Combustion and Flame, 115, pp. 126-144. Hicks RA, Lawes M, Sheppard C. G. W. and Whitaker B.J. (1994) Multiple laser sheet imaging investigation of turbulent flame structure in a spark ignition engine, SAE paper 941992, presented at the Fuels and Lubricants Meeting, Baltimore, October. Lancaster, D.R. (1976) SAEpaper 760159. Newland, D.E. (1993) An introduction to random vibrations, spectral and wavelet analysis, 3rd Ed., Longman Scientific and Technical. Wirth, M., Keller, P. and Peters, N. (1993) A flamelet model for premixed turbulent combustion in SI-engines, SAEpaper 932646.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
815
M i x i n g in isotropic turbulence with scalar injection M. Elmo and J.-P. Bertoglio ~ and V.A. Sabel'nikov b* ~Laboratoire de M~canique des Fluides et d'Acoustique UMR 5509, Ecole Centrale de Lyon, 36 Av. Guy de Collongue, 69130 Ecully, France bTSAGI, Zhukovsky, Moscow Region 140160, Russia The mixing of a passive scalar is investigated in a DNS of homogeneous isotropic turbulence. A new technique to force the scalar field is developed. This technique permits to obtain statistically steady states associated with different levels of mixing. The probability density function of the scalar and the conditional scalar dissipation are studied. Both quantities are found to strongly depend on the level of mixing. It is also shown that the ratio between the integral scalar length scale and the integral velocity length scale affects the scalar statistics. 1. I N T R O D U C T I O N The mixing of a passive scalar by turbulence is of interest for many practical applications. For turbulent reactive flows, a description of the scalar field by low order statistical moments is insufficient to account for the non linear nature of the chemical reaction term. An estimation of the probability density function (p.d.f) of the scalar is more appropriate since the p.d.f approach allows to take into account a chemical reaction in a closed form. W h e n writing the equation for the p.d.f P(F, t) of a scalar c (without chemical reaction): 0 P(F t ) -
o--/
'
02
or~ [< ~ / ~ - F, t > P(r, t)]
(1)
the conditional scalar dissipation < ec/c = r > appears. This term offers the essential difficulty in constructing a closure for the p.d.f equation. It needs to be modelled to make a practical use of equation (1). It is therefore important to obtain information on P(F, t) and < ec/c = F > and on the different parameters that can affect their statistical distributions. This will help asserting the validity of existing models (see for example [1] and [2]) or establishing the basis of models which can portray in details the mixing mechanisms and be applied to a large range of practical situations. The mixing of a passive scalar was studied in Direct Numerical Simulation of turbulence by several authors. The case of a decaying scalar field in isotropic turbulence has been extensively investigated [3]. DNS has also been used to analyse the effect of a uniform mean scalar gradient [4] [5]. For several practical problems, such as non premixed *also: Laboratoire de Combustion et de D~tonique UPR 9028, Centre d'l~tudes A~rodynamiques et Thermiques 43, Route de l'A~rodrome, 86036 Poitiers Cedex, France
816 combustion, it is important to understand situations in which the mixing is not complete. Another important issue is that the statistical properties of the scalar seem to be strongly dependent on the way the fluctuations are "injected" in the fluid [6] or on the way they are produced by the fluid motion in the case of the presence of a mean scalar gradient [4] [7]. One can also suspect that the ratio between the scalar and velocity integral length scales may play a major role in the form adopted by the p.d.f. These considerations had led us to perform a DNS of an isotropic turbulence in which an unmixed scalar is injected at a given length scale and regularly in time. A new technique to force the scalar field was therefore developed. This technique has the advantage of providing a way to control the integral length scale ratio and to generate statistically steady states the scalar p.d.f of which can continuously be varied from nearly bi-modal distributions associated with a high level of unmixing to roughly Gaussian distributions corresponding to fully mixed situations. The results obtained with this forcing technique applied to incompressible isotropic turbulence at a Reynolds number (based on the integral length scale) equal to 90 and a Schmidt number equal to 1 are presented in this paper.
2. N U M E R I C A L
METHOD AND INJECTION TECHNIQUE
The Navier Stokes equation and the convection diffusion equation for the scalar c are integrated using a pseudo-spectral method. These equations are solved in a three dimensional cubic domain of size L with periodic boundary conditions on the velocity and scalar fields. The time stepping scheme is a second-order Runge Kutta method and the simulations are performed at resolution of 1283 grid points. A random forcing is applied in the low-wave number range of the spectrum in order to maintain a statistically stationary velocity field. This stochastic forcing can be alternatively white-noise or time correlated (generated using a Langevin equation). The scalar forcing technique consists in refreshing the field by operating an injection of fluctuations in physical space. This operation is repeated periodically in time, with a period Ti. The basic outline of the forcing can be described as follows. In the computational domain (of size L), n subboxes of size 1 are randomly selected. In one half of these subboxes (n/2), c(x) = +1 is imposed, whereas c(x) = - 1 is imposed on the other half. This procedure essentially leads to a forcing function the p.d.f of which is bi-modal. The choice of 1/L allows to keep control upon the integral scalar length scale Lc and correlatively on the ratio Rl - L_~ where L~ is the velocity integral length scale. L~ and L~ are L,~ respectively defined by:
= Y fo
=
2 fo E (k)ak
where E~,(k) and Ec(k) are respectively the velocity and scalar spectra. A characteristic time of injection is T ~ - Ti * (v_.t_) where Vt is the volume of the computational domain nv I and vs is the volume of a forced subbox. The choice of the time scale ratio Rt = T,.~/Tt,,,.b (T,,,.b being the eddy turnover time of the turbulent field) governs the level of mixing.
817 3. R E S U L T S
Fig.1 shows the time evolution of the scalar r.m.s value cr = v/< c2 > for two time scale ratios R t = 0.35 and R t = 2.2 (Rl being fixed to 0.71). A statistically stationary state is reached after a transient behaviour observed on a time of order R t * Ttu,.b. A larger r.m.s value is obtained for R t = 0.35 which is directly connected to the fact that more unmixed scalar fluctuations are injected in this case. Fig.2 shows the time evolution of the scalar r.m.s for a given time scale ratio ( R t = 2.2) and two length scale ratios Rt = 0.71 and Rl = 0.47. It can be observed that the stationary level of a is lower for Rl = 0.47, indicating that the mixing is faster when the scalar is injected at smaller scales. This result is consistent with the study of WARHAFT ~: LUMLEY 1978 [8] who have shown that the length scale ratio affects the exponent of the power law of a decaying scalar.
1.0
.,=o35 _
r
.s"
Rt=2.2
0.8 _ -
_
~
,.o
R1=0.71 R!--0.47
......
0.4
_ s
0.6
. . . . . . .
0.5
A
u~
E
E 0.3
~-" 0.4 0
/
I
t
v I0
/
0.2
/
0.2
0%.0
0.1
,:0
,:0
~:0
,:0
' ~0
t/Tturb
Figure 1. Time evolution of the scalar r.m.s value for two time scale ratios, R t - 0.35 and R t - 2.2 (Rt = 0.71).
0.0
00
.
.
,:0
.
2:o
.
.
310
t/Tturb
.
,:0
5o
Figure 2. Time evolution of the scalar r.m.s value for two length scale ratios, Rl = 0.71 and R1 - 0.47 ( R t = 2.2).
The statistical results presented below are for stationary states. They are averaged in time to improve the statistical sampling. In Fig.3 the kinetic energy spectrum E u ( k ) is plotted, as well as the spectrum of the scalar fluctuation E r for two length scale ratios (Rl = 0.47 and Rl = 0.71; at R t = 2.2). Although the Reynolds number is too low for an inertial range to be present on E ~ , ( k ) , it can be observed that a K - ~ behaviour is detected on the scalar spectrum (CoRRSIN 1951 [9]) in the case corresponding to the larger injection scale (Rt = O. 71). Fig.4 shows the stationary p.d.f of the scalar for three time scale ratios, R t = 0.35, R t - 1.1 and R t - 2.2. For R t - 0.35 two peaks are clearly observed close to +r__ _ 1.4, r which correspond to the extremal fluctuations F = +1. Such a behaviour is associated with a high level of unmixing. The peaks are much smaller when R t = 2.2 and the p.d.f of the small amplitude fluctuations is higher indicating a better mixed situation. The p.d.f for R t = 1.1 is also shown, giving an example of an intermediate situation of mixing in
818
which the peaks at F = 4-1 are still pronounced but the p.d.f is nearly uniform. In Fig.5, the p.d.f for Rt = 2.2 is replotted in semi-logarithmic axis and compared to a Gaussian law. The agreement with a Gaussian distribution is good over a large range of scalar fluctuations ( - a < F < +a).
.
.
.
.
.
.
.
.
|
10 3
.
.
.
.
.
.
.
......
.
|
.
.
.
.
.
.
.
1.0
.
Eu E c (RI=0.71) E c (R1=0.47)
|
0.8 /
101
I\
0.4
10 -3
.
\
:,1 .k;..
5/3
10 -1
"\
/ I
I\
0.6
10 ~
Rt=o.35 Rt=l.1 i Rt=2.2
|
',/I
JI
.-2'
I 1-;;'X ...../W1
,:',L.I.," \
I ;i i I,,q
9d
!
!:
0.2
.
10 0
10~
K (wave number)
.
.
.
.
.
.
10 2
Figure 3. Kinetic energy spectrum. Scalar spectrum, Rl - 0.71 and Rl = 0.47 ( R t = 2.2).
0.0 -3.0
.(!1
, -1.0
,
, 1.0
Lit k
%,I
3.0
Figure 4. Probability density function of the scalar for three time scale ratios, Rt 0.35, R t - 1.1 and R , - 2.2 (Rl = 0.71).
a~ o ~~/ c F > Fig.6 shows the conditional scalar dissipation < r F > = < D-g-2i~,ox Oc Oc normalized by the mean dissipation rate < ec > = < D g~-;~g~-~ > where D is the molecular diffusion coefficient. It can be observed that the conditional scalar dissipation strongly departs from a uniform distribution for unmixed situations ( R t = 0.35 and R t = 1.1) whereas in the case of the lowest forcing ( R t = 2.2) it tends to become roughly uniform over a large plateau corresponding to small and moderate scalar fluctuations. It is interesting to point out that high values of the p.d.f of the scalar, corresponding to the effect of injection at c = +1, are associated with low values of the conditional dissipation. This suggests the existence of blobs of nearly uniform scalar concentration in the field. These blobs are reminiscent of the injection subboxes. It is also interesting to notice the correspondence between the flat shape of the conditional scalar dissipation (for Rt = 2.2) and the Gaussian distribution of the scalar p.d.f already observed in Fig.5. This is in agreement with the theoretical result of SINAI ~1; YAKHOT 1989 [10] which shows that if the p.d.f is Gaussian then the conditional dissipation is uniform. Fig.7 shows the p.d.f of the scalar for two length scale ratios, Rl = 0.71 and Rz = 0.47 (Rt - 2.2). The two peaks close to +r_~ = 2 are lower when Rz = 0.47 indicating that the p.d.f is associated with a better mixed situation than for Rl = 0.71. This result is in agreement with the fact that the mixing is improved when the fluctuations are injected at smaller scales, as previously observed for the r.ms value (Fig.2). In Fig.8, it can be observed that the conditional dissipation is also affected by the length scale ratio. Fig.9 shows ~, the conditional scalar dissipation weighted by the scalar p.d.f and nor-
819 malized by the mean dissipation rate, for three time scale ratios ( R t = 0.35, R t = 1.1 and R t = 2.2). This quantity represents the contribution to the scalar dissipation associated with a given value of the scalar fluctuation. Obviously the shape of this quantity is similar for the three time ratios considered although they are associated with very different degrees of mixing. In Fig.10, it can be observed that the shape of the weighted conditional dissipation remains unaltered when the length scale ratio is changed.
100
4.0
'
'
'
9
v
--'----Rt'=2.2 N~--~- Gaussian
9
i
A
Rt=0.35 9 Rt=l.1
.....
3.0 V
t
d.
I
9
I
~"
oI I
I ~
"~'~ 2.0
9I t
-'P
V
s S"
1.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
0.0 -3.0
3.0
Figure 5. Comparison of the probability density function with a Gaussian distribution, R t = 2.2 and R t - 0.71. '
"" ....
\\"
F/o
1.0
3.0
i
R'!=0.71
......
R1=0.47
0.8
"ID
-1.0
2.0
A
~
i/1 ; !I
Figure 6. Normalized conditional scalar dissipation for three time scale ratios R t = 0.35, R t - 1.1 and Rt - 2.2 ( R I - 0.71).
1.0
9
Rt=2.2
RI=0.71
R1=0.47 1.5
0.6
V
0.4
0.2
0.0 -3.0
,%J, , -1.0
. F/o
. ;, 1.0
to
V
3.0
Figure 7. Probability density function of the scalar for two length scale ratios, R t - 0.71 and R t = 0.47 ( R t = 2.2).
0.5
0.0 -3.0
-1.0
1.0
3.0
Figure 8. Normalized conditional scalar dissipation for two length scale ratios, Rt = 0.71 and Rl - 0.47 ( R t = 2.2).
820 1.5
|
|
...... RI=0.71
Rt=0.35 ,-
1.0
" ' -t
Oll
0.8
.....
Rt=l.1
......
Rt=2.2
R1=0.47
~,~,
,7/
1.0
%
;7/
~.,,I
ha,'1
0.6
,Y
0.4
0.2
0.0 -3.0
",:,'
0.5
-71
/11
-1.0
\'i:~', ,,
F/a '
110
3.0
Figure 9. Weighted conditional scalar dissipation for three time scale ratios Rt - 0 . 3 5 , Rt - 1.1 and Rt - 2.2 (Rl = 0.71).
0.0 -3.0
-1.0
F/a
1.0
3.0
Figure 10. Weighted conditional scalar dissipation for two length scale ratios Rl 0.71 and R l - 0.47 ( R t - 2.2).
4. C O N C L U S I O N The scalar injection technique proposed in the paper has been used to study turbulent mixing in situations where the mixing is not complete. Both the scalar p.d.f and the conditional dissipation were found to strongly depend on the level of mixing. The length scale at which the scalar fluctuations are injected in the turbulent field also appears to affect the statistical results. An interesting point is that the weighted conditional dissipation seems to be less sensitive to the degree of mixing. This suggests a relationship between the p.d.f and the conditional dissipation reflecting the fact that low values of dissipation are associated with high probability density of the scalar. More results are necessary before attempting to close the p.d.f equation by expressing the conditional dissipation. It is also important to notice that the influence of the Reynolds and Schmidt numbers should be investigated. REFERENCES
.
3. 4. 5. 6. 7. 8. 9. 10.
V.R. Kuznetsov and V.A. Sabel'nikov, Turbulence and Combustion, P.A. Libby Hemisphere Publishing Corporation (1990) H. Chen, S. Chen and R.H. Kraichnan, Physical Review Letter 63 (1989) 2657. V. Eswaran and S.B. Pope, Phys. Fluids, 31 (1988) 506. M.R. Overholt and S.B. Pope, Phys. Fluids, 8 (1996) 3128. A. Pumir, Phys. Fluids, 6 (1994) 2118. F.A. Jaberi, R.S. Miller, C.K. Madnia and P. Givi, J. Fluid Mech. 313 (1996) 241. Jayesh and Z. Warhaft, Phys. Fluids, A 4 (1992) 2292. Z. Warhaft and J.L. Lumley, J. Fluid Mech. 88 (1978) 659. S. Corrsin, Journal of Applied Physics 22 (1951) 460. Ya.G. Sinai and V. Yakhot, Phys. Rev. Letters, 63 (1989) 1962.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
821
Modelling Turbulent Diffusion Flames with Full Second-Moment closures using Cubic, Realizable Models W.T. Chan and Y. Zhang Thermodynamics and Fluid Mechanic Division, Mechanical Engineering Department, UMIST, PO BOX 88, Manchester, M60 1QD, England
Predictions of turbulent jet diffusion flames with full second-moment closure are presented. Special attention is given to the modelling of the pressure-scalar correlations. Calculations are compared with experimental data and along with the predictions of other researchers. It is demonstrated that the anisotropy of turbulence plays an important role in determining the structure of a turbulent flame. And it is found that the performance of jet flame calculations could be greatly improved by using a cubic, fully realizable model.
1. I N T R O D U C T I O N Historically, a two-equation approach was the most widely adopted variant for flame predictions [1,2] as it offers relative simplicity and often leads to reasonable results in simple flow cases. However, this type of a gradient-transport concept approach does not resolve the anisotropy of turbulence and does not respond well to flow situations such as flows containing effects of strong streamline curvature, buoyancy, swirl and severe adverse pressure gradient. Due to these weaknesses, a more universal approach, such as second-moment closure (SMC), has gradually replaced this type of approach. Recently non-linear modelling approaches, which could guarantee a realizability constraint and a two-component limit, are available and provide additional favour to flame predictions [3-6]. So far, applications of non-linear calculations are mainly for the pressure-strain correlations. Although attempts have been made in the past to extend the use of non-linear approach on to the pressure-scalar correlations, the understanding of this kind of approach is still not clear [7]. This paper focuses on the modelling of the pressurescalar correlations, by the use of cubic, fully realizable model.
2. M A T H E M A T I C A L DESCRIPTIONS AND M O D E L L I N G A P P R O A C H The numerical computations are obtained with a finite volume elliptic solver, namely a version of the UMIST code TEAM [8]. Governing equations are written in Favre average form due to the large density variations occurred in turbulent flames. The mixture fraction Z is used as a conserved property of the flow. In this section, modelling approaches for turbulent flame predictions will be discussed.
822 2.1. Turbulence Model The turbulence model used in this study is based on a well-developed second-moment closure (SMC) [9] of constant density flows. For variable density flows, extra terms have to be added. Hence the stress equation can be written as:
%## ##
(
_ du~uj _
p ~ -
-- "Puiu k
dt
-}
3U- : _ - - ~ . OU i 1-. I. ---I-~jfi +uTNj) Ox~ pu ju k o5c~
-- "J~i -t-"i~jJ--t
r o
J (~i
i olTX~-"
OqXk~k
,.
-Jr-L----tOTXk Pblkbliblj-sr- p blj(~ik-I- pXLli'(~jk-- -pv 07Xk Alternatively, it may be written in a more universal way as
Co = Pij + Fo + Oij + G i j - e i j +do Similarly, for the scalar fluxes
--
P
dui z -dt
_'- . . o~ _'-9--'~o7lj, "~ . oTp ~ c) P -"~- zttf i -- Z -- -- Z Ox~ 07Xj 07Xi OqXi
-- Pblibl k --7t-PUkZ
(
-- (,~, -I'-V )
or
J
. axkaZ Jr" _ ~k
Ciz =: eiz --I- Fiz ]-Oiz
'
~'P UiUk Z ~]
+Giz --eiz +diz
From left to right in Equation 1 and 2, the forth terms represent redistribution, ~ij and pressure correlation. Each term is split into two parts ~)ij -- Oijl + 0(]2
;
(~iz by
Oiz -" Oizl -Ji- Oiz2
where c~ijJ, Oizl and 002, Oiz2 can be described as 'slow part' for return to isotropy in the absence of strain and as 'rapid part' for the rapid distortion of initially isotropy turbulence respectively. The modelling approach is adopting a cubic model which is first proposed by Fu [10] and later modified by Craft [11]. These terms are already simplified in the current calculations and are written as
cIa+claiak
823 1 ) P~k Oij2----0.6 Pq--~,jPkk + 0 . 3 e a q ~l ~ . . . .
~, <
Tt,, ~
u,u,~
where aij is the anisotropic tensor
uiuj a~ = k
2(3.. 3 :~
and
A2
- a~176
For ~izl and ~iz2, two modelling approaches are being investigated. Model 1, a linear process [ 12,13], which reads
e .'->-; ~ . . c?U ~ Oiz ----Clz ;UiZ +C2zUkZ Oq,X k Model 2, a non-linear process, which is on the basis of a proposal by Craft and Launder [9]
Oizl ----Clzl ;[UiZ ~l +0.6A2)+Cljtaikukz +Clzmaikakjujz ]-cljvRka o oaxj
alternatively, we split the above equation into four parts for simplicity in later analysis.
Oizl -- OizlA+ OizlB + OizlC + OizlD For the rapid part, "-~ 1 P~k 2.'S'-;. (OU k a O l l Oiz2-0"8u"kz"O-Ui-o.2u~z "3U1' +-OiXk &i 6 k UiZ --TUk z ailt---~-i + OqXk
1~ 3U m Og, 1~,( Pmk+2amkPim +-~UI~ Z ail~aml OqXl + &m -- ~ UiZ t aim k +
3 a,,,,(ao, eo,~ a mkUi Z -~+
20
o3q
-- amiUkw'Z-'~n'J~
-2---ago, 7am,tU,Z --d;, +u,z ~)-u, Z tam,~+am, ~ r n tt 3U k
[
.
_
~
3 0 i ~ ~'-"nttr
30 i
30 i
1
824
where the time scale ratio, R, is related to both the mechanical time scale and the scalar time scale. R_2ke7
gZ The scalar dissipation rate ez can be approximated using either a constant value for R or its own transport equation [see Ref. 11 for more details]. The fifth terms, in Equation 1 and 2, identify mean pressure gradient contribution, Gij and Giz, which have zero values in constant density flow. In our case, ideal gas law is assumed and follows Jones (4) that yields a linear relationship between density and mixture fraction. Hence H ,~
/Y
---.3P uiz z OP Hi & j -- Z'~ & j
The scalar variance "~,,2 z" is described by its own transport equation, d z "2 ~
~ H R i)Z ~
2ez + C z Ox k ~,
Ox~
And for Giz, we assume that p=p(Z). It can then be simply described by the pdf.
10 tT~Xi 0
The sixth terms represent dissipation by viscous process, eij and Eiz, Eiz is zero in an assumption of local isotropy behaviour. For eij, e is calculated via its own transport equation. Additional term is included in order to take into account of large density variations. m
dt -
-k-
c ~l e - c ~2 + d , - U i "k
11
OqXj
Pt is the generation rate of turbulence energy =P~/2. The diffusion, dij and diz, (the last terms in Equation 1 and 2) are modelled via the generalised gradient diffusion hypothesis (GGDH) [14] as
--
C~ mUkl.I l e
Ox~
diz - -~k
cz --UkUt
The full set of model constants and coefficients used in the current study is listed in Table 1.
12
825 Table 1: Model Constants and Coefficients Cl Cl / Clz 3.1
1.2
C2z
3
0.5
Clzl
Clzll
1.7(1+1.2
-0.8
(A2A) 1/2) R 1/2
(A2A) 1/2 C l ziil
ClzrV
R
CcI
Ct:2
Cs, Cz
1.1
0.2A 1/2
Either2 or fromthe
1.45
1.8
--0.18-0.25
transporteq. of c,
2.2. Combustion Model Flamelet approach is applied for combustion modelling. Global one-step mechanism of methane is adopted. The temperature and species are computed in the mixture fraction space [15]. And the density of the mixture p(Z) can be calculated using the ideal gas relationship
p(z):
where
P
Ror(Z)E r,( z )
Ro is the universal gas constant and P
is the pressure.
To determine the mean density of the hot flow, A presumed probability density function (PDF) P(Z,x,t) is adopted and a standard ]3-function has been chosen
~ Zt~-l(1- Z )fl-ldz
13
Then the mean density can then be computed by
15 The mean temperature is written as 16 And to take into account the radiation effect, the empirical approach of Fairweather et al [16] is adopted to correct the computed temperature profile.
826 3. RESULTS AND DISCUSSION Two axisymmetrical jet flame test cases have been chosen to investigate the behaviour and the predictability of the current SMC model. Test Case I is a pure methane flame of Reynolds number around 20000 issued into a single coflowing air stream and stabilised by a large ring of pilot flames. The case was investigated experimentally, computationally, and with a combination of both by Masri et al [17,18] and Chen et al [5]. Test Case II is a natural gas flame issued vertically upwards from a burner nozzle into double coflowing annulus air streams, and it is stabilised by a ring of tiny pilot flames around the nozzle exit. This flow, has a Reynolds number around 10000, was studied experimentally by Stroomer [19] and its prediction was attempted by Peeters [20]. 3.1 Test Case I - Pure M e t h a n e Jet F l a m e
Figure 1 and 2 show the radial profiles of mean axial velocity U and the root mean square of axial velocity Urms at various axial locations. Model 1 and 2 (see Equation 5 and 6, Section 2.1), and the predictions of Ref. 16 are compared with the experimental data. At the upstream region, X/D=20, all models could produce similarly good agreement with the measured data except that the model of Ref. 16 shows under-predictions near the centreline region. Further downstream, X/D=50, slight under-predictions are found from all models. Since both Model 1 and 2 are employing cubic methodology for the stresses, they all show reasonable agreement with the measured turbulence profiles plotted in Figure 2. The turbulence predictions of Ref. 16 are good in the upstream region but not so in the downstream region. The overall hydrodynamic results indicate that different scalar flux models will not have great influences on the hydrodynamic field. However the scalar flux models will affect the flow density. This can be illustrated by plotting the density profile across the axis, X/D=20, shown in Figure 3. The figure shows that all the models, including those of Chen et a l ' s PDF modelling approach [3], give a wider mixing zone but predict the near centreline region reasonably well. Excellent agreement is found at the centreline region if cubic model for the pressure-scalar correlations is employed. For the scalar field, Figure 4 shows predicted and measured radial profiles of mean mixture fraction Z at various axial locations. The figure clearly illustrates the advantages of using cubic model (Model 2) for the pressure-scalar correlations. Although, at the upstream region, the cubic model has a slight disagreement with the experimental data, it still produces much better results than the linear model (Model 1). The discrepancy in the near nozzle region is probably due to intensive mixing of fuel and pilot flames, which present combustion model could not address adequately. Since the trend of the mixture fraction is correctly predicted, the discrepancy found in the density profiles shown earlier indicate that the chemical kinetic in the flame is strongly non-equilibrium. The predicted and measured radial profiles of mean temperature T is shown in Figure 5. Because the better results are obtained for the mixture fraction, the cubic approach could have a more accurate description of the PDF and hence provide more realistic prediction than that of the linear approach. The general behaviour of the predicted profiles is very consistent with the density profiles where a wider mixing zone is predicted. Notice that the peak values of temperature are predicted similarly by both models. However, distinctive values are found near the centreline region and after the mixing region. Compared with Model 1, the pressure-scalar correlation is not only related to the velocity gradient but also related to the gradient of mixture fraction in Model 2. To investigate the influence of the scalar gradient, ~)izl in Equation 6 is separated into four parts and plotted across
827 the flame at X/D=20 in Figure 6. Clearly the figure indicates OizlA is the main contributor. Although the magnitudes of q)i~/eand O;~IDare smaller, they still could not be ignored (Notice that C~i~IDare the minor contributions in both plane and round, non-reacting, jet cases studied by Craft [11] ). More information on the overall influence of the cubic model is obtained by plotting the budget profiles of u"z" for both linear and cubic models shown in Figure 7 and 8. Cubic model produces a lower ~i~.~ m~aagnitude than that in the linear model. However q~iz2is in the same order for both models. In v"z" budget profiles (shown in Figure 9 and 10), cubic model generates a much lower values of ~1 and has a positive magnitude of ~z2 while linear model has generated very small values (around zero). On the whole, cubic model gives larger sink of Oiz a n d ~ after theemixing zone, leads to smaller scalar flux predictions and sharper gradient of both u"z" and V "Z ".
3.2 Test Case II - Natural Gas Jet Flame
The centreline profiles of mean axial velocity U and the radial profiles of normal stress component Urms are shown in Figure 11 and 12 respectively. The results demonstrate the superiority of second-moment closures over the traditional eddy-viscosity approach. However the spreading rate of the flame is still high. Figure 13 shows the radial profiles of mean temperature T. Similar to Test Case I, the agreement is good across the entire flame width. Presumably, both the scalar and its variance predictions should be more accurate by using cubic model but unfortunately there is no experimental data for this case to support this argument.
4. CONCLUSION Two models for modelling pressure-scalar correlations are analysed and tested for turbulent jet diffusion flames. It is found that the cubic, fully realizable model, which includes the invariants of the anisotropy tensor and the gradient of the scalar, will enhance the performance of the predictions o_~fthe jet fl_~ames. Based on the current results, the linear model produces l ~ h a r p rg~ient of u"z" and v"z" than that of the cubic models. And from the budget profiles of u"z" and v"z", the net effect of the cubic model is seemed to remove the turbulence from the flames. The over-prediction of temperature that occurs in far downstream region may be attributed to the effects of non-equilibrium chemistry and radiant heat losses. The current mathematical model fails to take these effects into account. Nevertheless the use of cubic, fully realizable model, which could resolve anisotropy of turbulence for both turbulent stresses and scalar fluxes, will be an advantage. Also it may overcome many turbulence modelling difficulties occurred in most of the practical combustion system.
The research was partly sponsored by EPSRC under the grant number GR/L60722 and a separate EPSRC scholarship to the first author. The very helpful advice given by Dr. T.J. Craft is gratefully acknowledged by the authors. REFERENCES
1. Lockwood, F.C., and Naguib, A.S., Combustion and Flame 24:109-124 (1975) 2. Jones, W.P., and Whitelaw, J.H., Combustion and Flame 48:1-26 (1982) 3. Chen, J.Y., Kollmann, W., and Dibble, R.W., Sandia Report, SAND89-8403 (1989)
828
4. Jones, W.P., Turbulent Reacting Flows, Academic Press:309-374 (1994) 5. Chert, J.Y., Kollmann, W., Turbulent Reacting Flows, Academic Press:211-308 (1994) 6. Pfuderer, D.G., Neuber, A.A., Fruchtel, G., Hassel, E.P., and Janicka, J., Combustion and Flame 106:301-317 (1996) 7. Chart, W.T., and Zhang, Y., 8th Biennial Colloquium on CFD (1998) 8. Huang, P.G., and Leschziner, M.A., Dept. Mech. Eng., UMIST, 1983. Report TFD/83/9(R) 9. Craft, T.J., and Launder, B.E., Int. J. Heat and Fluid Flow 17:245-254 (1996) 10. Fu, S., Ph.D. Thesis, Faculty of Technology, Univ. of Manchester (1988) 11. Craft, T.J., Ph.D. Thesis, Faculty of Technology, Univ. of Manchester (1991) 12. Monin, A.S., Izv. Atm. Oceanic Phys. 1:45 (1965) 13. Owen, R.G., Ph.D. Thesis, The Pennsylvania State University (1973) 14. Daly, B.J., and Harlow, F.H., Phys. Fluids, vol 13:2634-2649 (1970) 15. Rogg, B., RUN- 1DL, Cambridge University (1993) 16. Fairweather, M., Jones, W.P., Ledin, H.S., and Lindstedt, R.P., Proc. 24 th Symp. (Int.) on Combustion (1992) 17. Masri, A.R., and Bilger, R.W., in 21 st (Int.) on Combustion, (1988) 18. Masri, A.R., and Pope, S.B., Combustion and Flame 81:13-29 (1990) 19. Stroomer, P., Ph.D. Thesis, Technische Universiteit Delft (1995) 20. Peeters, T., Ph.D. Thesis, Technische Universiteit Delft (1995)
45 '" ,, "~ k
U30
15
Exp. DataX/D=20 o X/D=50 + Model 1 ...... Model 2 Masri Predi. - ....
+ + ~ + "~,,,, + ~++
'"'""',,
~
A
"
"
"
urms
5
iii
,.g,,
60 0.02
r(m)
20
--"
[]
Chen:Five-scalar, four step
[] [] [] _ []
trained
equilibrium model
~
...... Model Model
0.6
~
[]
~
,o'"
:z ..... ~:.. ............
O o
2
[] []
~
0.04-"- 20
Data
~ ~ hn:aFno~rm-s c a I. . . . . . . >" 0.8
0.03
Figure 2: Radial Profiles of Normal Stress Component Urms (m/s)
Figure 1" Radial Profiles of Mean Axial Velocity U (m/s)
Exp.
60
0.01
X/D 0.03
[]
i;,~ ~i\
~
00
r(m)
Exp.DataX/D=20 <> X/D=50 § Model 1 ...... Model2 - MasriPredi. - ....
.," ........... ".,. !....... + "',. ~ , i ' "
~
~
Q
0.4 0.2 0
O
O.OO5
O.O1
O.O1 5
0.02
0.025
r (m)
Figure 3" Radial Profiles of Mean Density (Kg/m3) at X/D=20
0.03
829
1
I "'~o Z I ,\ 0 7 5 1 -"\o I ",\ 0,5t
%'.+ \ "" \+ "". \
.,,t
Exp. Data X/D=IO o X/D=20 + X/D=30 u Model1 .....
u ~ o -.. x'~o
',,.,~
........ ~ .
1500
Exp, DataXJD=IO o
I
X/D=30
............ ~ . . . o <>
"" o o
o .....
......
Model ..... Model2
~ ....
~176
o
"'
Model2 - -
0.25
00
30
/
o.oo5 ~ . . ~
0.01 ~ o.o15
r (m)
~ 0.02 0,025
0.01 ~ 0.015 ~
X/D
""
10
3O
/
o.oo5 ~ _
...,-""Y~o
/
0o
~
r(m)
0.025
""
X/D
10
Figure 5: Radial Profiles of Mean Temperature T (K)
Figure 4: Radial Profiles of Mean Mixture Fraction Z
300
200
~)izl
i
oo
Parts
o
-100 -200
......
(~izlB
""~izlC
-300 ~"
-400
.
r
Total
=
~iz2
l~izl -500
,
o
,
0.005
,
O.Ol
O.Ol 5
0.02
r(m)
Figure 6: Budget Profiles of ~)izl at X/D=20
400 ............................................................................................................................................................................................................ 403 .............................................................................................................................................................................................................. ..'".... 300 .,."" , 330/ .,
,";..:::/ ,././'f':'x """", "'"'",.,.,., ..... ""..., ..............
.,,., o) rn -~
ii
2OO
/ :.
100
..':":':
".,,
230 "",.... ".,.., ,.,. ,.., -.,
.--_ ,_._r
o
~
'"~-: . . . . . . .
-
-
m
..
x o-
.;
,.
~
~
=
""~_ . . . . . =_
..
=
,'7"
"~-1O0 O9 -2130
Cuz --~ Duz
Puz
"~ -100-
uz ~-Duz
9
9
Puz
-2100-
x
-3130 -,40C 0
,,
Q035
Q01
Guz --~ ~uzl --,,--@uz2
Q015
Q02
Q025
. Guz --~ ~uzl--,,--@uz2i
-~0-
0
QC05
Q01
Q015
O~
r(rrl
Figure 7: Budget Profiles of Scalar Flux u"z" at X/D=20 (Using Cubic Approach)
Figure 8: Budget Profiles of Scalar Flux u"z" at X/D=20 (Using Linear Approach)
0025
830
400 ...............................................................................................................................................................................................................
.....,'"",,.,., ,.,," ",.
300 200-
:'
"~ 100m
200-
".,,.,
, ,.-.-., ,...,. . . . . . . . ',.,,,,..,.
,... m
~
o.
It.
",,,,
-100or) -200-
'"_:
:
~
Cvz
:
,
:
-
_~ ii
. o
~
Dvz ......Pvz
.
.
.
.
x ~ ~ ~ i ~xx .
. .
C
:. .
.
.
.
v
.
.... : ....., ............ .
.
.
.
:
z
:
:
"
~ Dvz ...... Pvz
03 -200
x
-320
x Gvz --*--Cvzl --"-r
-300-
Gvz--~vzl--,,-~vz2
-40C
-40C
0
QO05
Q01
Q015
Q(]25
Q(]2
0
QO05
Q01
Q015
Exp. Data Model 1 Model 2 Predi.
Peeters
10
0
Q13~5
Figure 10: Budget Profiles of Scalar Flux v"z" at X/D=20 (Using Linear Approach)
Figure 9: Budget Profiles of Scalar Flux v"z" at X/D=20 (Using Cubic Approach)
0
QcO
r(rrl
r(nl
20
7JD
30
o - ....
40
50
Figure 11" Centreline Profiles of Mean Axial Velocity U (m/s)
Exp. Data X/D=25 X/D=42 Model 1 Model 2 Peeters Predi.
UlmS
o ~
~
§
2~f
o + ...... -- ....
20OO
1500
........... .....~
;./
........;"
,.,o ""-,,
~+
~
Exp. Data X/D=25 o X/D--.42 +
:::::, ~.
.
M ~ l ~ ......
+ ",'~;,..
',,,
Model 2 -Peeters Predi. - ....
': \ "., '.. +',, ,, +',,
1000
~'~," ~ ~ ' , "...
"....
§
III
5O :)05
"'""~"':l:
500
,. . . . .
5O
oO
40 ....
0.015 ~
r(m)
....
...,./,'~30 0.025 g u
Figure 12: Radial Profiles of Normal Stress Component Urms (m/s)
X/D
' r(m)
0.02 0 025 "
30 0.03 0.035
20
Figure 13: Radial Profiles of Mean Temperature T (K)
X/D
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
831
Advanced Modeling of Turbulent Non-equilibrium Swirling Natural Gas Flames A. Hinz ~, T. Landenfeld, E. P. Hassel and J. Janicka ~Fachgebiet Energie- und Kraftwerkstechnik, Technische Universit~it Darmstadt Petersenstr. 30, D-64287 Darmstadt, Germany The current work presents numerical simulations of a confined strongly swirling nonpremixed natural gas-air flame, the TECFLAM swirl burner. The focus is directed towards the modeling of combustion processes with exhibiting non-equilibrium effects and their interaction with turbulence Therefore, two reaction models are examined to study the turbulent combustion processes. Overcoming the limitations of a laminar flamelet formulation the chemistry mechanism described in terms of intrinsic low dimensional manifolds (ILDM) is applied via multidimensional presumed probability density function (PDF) closure and Monte Carlo PDF method. Comparisons for the velocity field are made with experimental data obtained from LDV measurements. The predictions of temperature and species concentrations are compared with data obtained from probe sampling and hot wire thermometry technique. 1. I N T R O D U C T I O N Modern and elaborated techniques of turbulence and combustion modeling are essential to describe the turbulent reactive flow in technical combustion systems with high accuracy. Statistical turbulence models and especially the second order moment closures (SMC) are up to now the only feasible approach to model reactive flows with high anisotropies of turbulence and strong streamline curvature [1-3]. Combustion models beyond the assumption of infinitely fast chemistry have become available on different levels of complexity [4]. The focus of this work is directed towards the simulation of complex flames with SMC in coupling with the well known flamelet approach [5] and also with a chemistry described on intrinsic low dimensional manifolds (ILDM) developed by g a a s and Pope [6]. A standard way of implementation via presumed PDFs is used and in addition to that, a Monte Carlo PDF method, promoted by Pope [7], is employed here with ILDM chemistry. This investigation is the first attempt to apply SMC and the Monte Carlo PDF method to confined strongly swirling natural gas flames with a central recirculation zone. Attention in this study is drawn to a confined strongly swirling natural gas flame from the TECFLAM project [8] (Figure 1), for which valuable modeling experience has been gathered over the last two years, e.g. [9-11]. The description of the chamber can also be found in these references. The confined flame with a Reynolds number Re = 42000 and a thermal power of 150 kW represents an object close to engineering applications.
832 The swirl is generated by a movable-block and can be characterized by a swirl number of Sth = 0.95 based on a definition of Leuckel [12]. The turbulent flow field has been extensively measured with the LDV technique by Kremer et al. [13,14]. The temperature and species measurements have been performed by Schmittel et al. [15] using suction probes and thermocouples.
Figure 1. TECFLAM swirl burner exit geometry and swirl generator
2. A P P L I E D M O D E L S A complete statistical model for the description of turbulent flames consists of a turbulence closure and a chemistry mechanism. These two are combined by an appropriate closure assumption for the treatment of the turbulence-chemistry interaction. 2.1. T u r b u l e n c e closure
The Favre-averaged Navier-Stokes equations are closed by solving for the unknown Reynolds stress tensor whose transport equation for the statistically stationary case reads
a(-~:'~'j~,~) N
Ozk
_(
= p -ui "k ~
_ O a i u~~ , j + P'OU i + P' 2u - u~uk Ozk J --~ Ozk Ozk ~ Oxi P~j eij --2/395ij lIij ~ r
r
o~ ~ p,~," o~ ~ o [~ o(-~.u'j) _ ( p u- i ~Uj "ttk Jr- P' Ui" ~jk -~ Oz~ -# Oz~ ~ Ozk
+ 12 r 5ik
-"'"
(i) )1 9
Cijk
Throughout the paper, the variables xj, uy, k, ~, p, p and v represent the spatial coordinate, velocity, turbulent kinetic energy, dissipation rate, density, pressure, and kinematic
833 viscosity, respectively. The non-linear second moment closure for Hij of Speziale et aI. [16] is employed together with the turbulent transport model for Cijk of Daly and Harlow [17] and the fluctuating density-velocity correlation, p'-a i", is modeled with a closure of Jones [18]
rlij - -2cz-fig
u i .aj= 2k
15ij 3
9
) J
bij
(2) 9 Pkk b , bkzbkl 3 (~iJ)-CI-P--2 - i j
+ Cnl~ (bikbkj ~
r~
II
II
k ~ aui uj C~jk = -C~ ~puku~ Oz~
'
I !1
P ui
_
_
_C b
-
(3)
Oxk
The original unaltered model coefficients used in this study are given by C1 = 1.7,
C2 = -0.4167,
6'3 = 0.4125 ,
C; - 1.8,
Cnz = 4.2,
Cs - 0.22,
C4 = -0.0167Cb = 0.2326.
0.65v/bk~b~k,
2.2. C h e m i c a l m o d e l
In this study, two chemical models are applied. First, a laminar flamelet model is used with an extended libary capturing strain rates up to the blow-off limit. The coupling with the turbulence field is performed via a presumed joint PDF of mixture fraction and strain rate consisting of the product of a/3-function and a quasi-Gaussian respectively, assuming statistical independence [19]. Additionally, a chemical mechanism based on the ILDM method [6] is used. In this work, the ILDM tables are parametrized by mixture fraction f, assuming equal molecular diffusivities, and two reaction progress variables, namely the mass fractions Yco: and YH~O. These three variables define the current state on the low-dimensional manifold and their rate of change the motion along this hyper-surface. The description via two reaction progress variables is equivalent to a global two-step mechanism except for the advantage of using locally always the two slowest, hence most appropriate, reactions. Two different ways of implementing the ILDM model within a PDF approach are considered. At first, a three-dimensional PDF is defined via a presumed joint PDF and secondly, a Monte Carlo method solving for the joint PDF itself is employed. 2.3. P r e s u m e d
PDF
modeling
The mean of some arbitrary scalar T(f, Yco~, YH~O) can be obtained from an integration over the composition space =
fff
yoo,
roo, r. o) f dYco2 dYH2O.
(4)
Statistical dependence of the variables f, Yco2, YH:O is overcome by introducing nor-
834 malized quantities Y~o2~---
Yco= eq Yco=
and
rI~.o
--
rH,O eq , YH~O
(S)
where the superscript eq indicates the values at chemical equilibrium. Now, a threedimensional P D F is defined via three presumed ~-PDFs, assuming statistical independence of the normalized formulation -
1"1"1"/01/~(/,01/01
* , Y H*= O ) P ( f ) P ( Y c o = ) Yco~
-
*
Z,* P(YH2o)dfd co~ dYH2 0 *
9
(6)
Depending
on the normalized means and variances of mixture fraction and the progress . y. --7--variables, f'~, f..2 . Yc02, c02 .2 , YH20, YH2 0.2 , a six dimensional look-up table is obtained in a pre-processing step. Thus, the mean source terms of the progress variables and other relevant thermochemical properties are evaluated more efficiently than by an online integration. Hence, during the CFD simulation transport equations of the means and variances of the reaction progress variables, modeled with gradient-diffusion approximation for "~i -.xz. x~ and
ill" Yc,,,2
(7) Oxj
- Oxi
~
~ Oxi Oxi
_ e ~-'.2 2p~-Vo +
2Y~"w",
(8)
are solved. Transport of mixture fraction is described by the same equations with the source term being identical zero. In all equations the eddy-viscosity assumption, pt Cpk2/~, Cp - 0.09, is applied. The values of the turbulent Prandtl numbers are GI - 0.83, and G2 - 0.7. In order to access the normalized look-up table, the means and variances of f and Ya have to be transformed in a similar manner to (5). Appropriate assumptions lead to
Y2 = Y~
~~
(9)
, -
~yeq) ~ y-,_Z2(y_@) .
(10)
The factors (1/Yeq'--"~ and (1/Yeq) 2 are obtained through P D F integration (6) just as a11 other statistical quantities. The source terms ~b~ and Y~ w~ are retrieved from H
--
-
The components
so
" "'
(Y~
of the look-up table So and
v'~+~'-Y'~
1/,+~,
" !
S 17"c,)
Y~S~
have been determined
(11) through
( so )(Eq.6)( ~ )
835
2.4. P D F transport equations In contrast to these common statistical moment closure approaches, it is in general possible to solve for the joint composition PDF embodying an infinite number of statistical moments. In this work, the transport equation of the joint PDF of the three scalars r = (f, Yco2, YH~O) is solved to describe chemical reactions without modeling. Since finitedifference methods for the discretization are not feasible for more than two scalars, a Monte Carlo method is used to solve for the joint PDF. This embodies a discrete representation of the joint PDF by an ensemble of notional particles with specified properties, namely f, Yco2, and YH20. The transport equation for the mass weighted joint PDF/5r is given by x, t)
0
-
x, t)] -
o[
o
(-
tr -
x, t)
]
(13)
where the expression (alb) denotes the mean of a conditioned on the event b. The terms on the left hand side need no modeling where the emphasis is on the closed description of the chemical source term. The first term on the right hand side is closed via a gradient diffusion approximation and reads
O"1 O X j
"
In this work, the molecular mixing in composition space r represented by the second term on the right hand side, is closed by the Modified Curl's model [20]. For the approach of joint composition PDF, the mean velocity field and the turbulent quantities have to be provided externally. Following this route, the PDF transport equation is solved using fractional time steps. The contributions of convection, molecular mixing and chemical reactions are determined consecutively in three steps. Since in the Eulerian description, applied here, the ensembles of particles remain fixed at each grid node, the convection is simulated by exchanging particle properties across the cell surfaces. The other two steps are carried out for each individual particle. By undergoing these processes, every particle represents a state in scalar space moving to some new position. 3. N U M E R I C A L
SETUP
3.1. Discretization and solution m e t h o d Equations are discretized with the Finite-Volume-Method and a resolution of 80 • 59 grid points in axial and radial direction respectively, condensed near the burner. The SIMPLE algorithm [21] ensures conservation of mass and momentum, for the convective terms the QUICK scheme [22] gives second order accuracy. For the Monte Carlo PDF calculations 100 particles are used in each cell. For computational efficiency, a local time step method is implemented for this model [23]. This yields a memory need of about 40 Mb where the ILDM look-up table is already included. The
836
presumed PDF approach also requires 40 Mb of RAM memory and a CPU time of about 15 hours on a Alpha LX533 machine. In order to investigate the impact of the chemistry mechanism on the species concentrations and temperature, the Monte Carlo simulation is performed in a post-processing step. According to the common experience being a computationally expensive method, the CPU time required for these post-processing computations ranges at about 60 hours. The velocity and turbulence fields stem from the computation using ILDM chemistry and the presumed joint PDF and serve as input for the Monte Carlo method. The density evaluated from the Monte Carlo computation is not fed back to the CFD calculation. Hence, this approach provides a configuration where only one submodel is changed. The more general hybrid method [24], a coupling of the Monte Carlo and the CFD code in a iterative solution procedure, leads to severe difficulties of obtaining a stably burning flame. 3.2. I n i t i a l a n d b o u n d a r y c o n d i t i o n s The computational domain resembles the size of the combustion chamber which has the length L = 1.2 m and diameter D = 0.5 m. The burner geometry is depicted in Figure 1. Inlet profiles are taken from the LDV measurements at an axial plane 6 mm downstream of the burner throttle and adjusted such that correct volumetric flow rates for fuel and air are assured. Dissipation rate at the inlet is modeled with a mixing length l proportional to the annulus width to give ~- C3/4k3/2/I. Standard wall functions are employed assuming the logarithmic law in the boundary layer. As a first step, the simulations are carried out under adiabatic conditions, thus neglecting convective and radiative heat loss across the wall The PDF at the wall is set to a Dirac function assuming zero-gradient condition for the mean. This seems to be a first approximation holding for the fact of vanishing turbulence close to the wall and hence, a very narrow PDF. In order to enhance the evolution of a stably burning flame, the entire field is initially assumed to be in chemical equilibrium at stoichiometric conditions.
4. RESULTS
AND
DISCUSSION
Results show comparisons of experimental data and numerical simulations using the flamelet model (denoted by Flamelet), the ILDM chemistry in combination with multidimensional presumed PDFs (denoted by 13-PDF) and in combination with a Monte Carlo solution procedure for the PDF transport equation (denoted by MC-PDF). It may be noted that currently the experimental data are restricted to mole fractions for the species. For the computations, mass fractions are used which are then converted for comparison. 4.1. Flow field Attention is first drawn to the flow field, which according to the LDV measurements is assumed to be axisymmetric. The streamwise distribution of mean velocity (Figure 2, right) indicates a central recirculation zone (CRZ) extending from the bluff body to at least 300 mm downstream. Both models are able to predict the CRZ although distinct differences are apparent close to the nozzle. The radial distribution of the mean velocity components and the turbulent shear stress
837 u"v" at the axial plane x - 60 mm are depicted in Figures 3 and 4. The fl-PDF is superior
in predicting the mean flow. The shear stress is under-predicted in the inner shear layer and over-predicted in the outer. 2500
10 /,
2000
............................................................... '7
~_~1500
I000
0
:,
I
:1
9
-10
"""
500
!,
0
,
i
I00 200 Axial distance [ram]
:
-15 I -20 [ 0
!
300
%
.......
.
-" ~
~6
" ~ r
"" ! ! 100 200 Axial distance [mini
!
300
Figure 2. Axial distribution of mean temperature and axial velocity. Symbols: Flamelet , ;3-PDF , MC-PDF - . . , Exp. 9 30 i
'
25 ~ 20~
-,o ;~,~ . / ~
25
Axial distance x=6Omm
o
,
-20 ~
0
i:
Axial distance x=60mm
15
;'/
_
/~N/~.\
20
z~
IO
q
5
/
0 ! I I I 50 I00 150 200 Radial distance [mm]
',,,,. ~ ~ 0
9
50 100 150 200 Radial distance [mini
Figure 3. Radial distribution of the mean axial and radial velocities, ~ and ~. Symbols: Flamelet ~ , fl-PDF---, Exp. 9
4.2.
Temperature
and scalar fields
Figure 2 (left) shows the streamwise distribution of mean temperature and Figures 5 and 6 the radial profiles of CH4, CO2, and CO at the axial plane x = 70 mm. The Monte Carlo PDF results show a superior agreement over the other two models for temperature and CO2. Considering the adiabatic boundary conditions, the deviations in the outer regions are expectable. The flamelet model shows a large overprediction for CO. It is notable that the two presumed PDF approaches show similar results for the temperature and the major species. Hence, more elaborated chemistry models will only improve the predictions of the minor species. In the current work, the ILDM method yields better results for CO with both, the presumed and also the Monte Carlo PDF methods.
838
1~8 IP-
,,"~' , ,~
Axial distance- I x = 60 mm l
,, 41, ,';'7 "._%/
0~ 0
!
!
!
,,",N-x
i, .+ ,o'~ 1o
:,
I
50 L 40 !-
!
-I0 l -20 F
I
50 100 150 200 Radial distance [mm]
Axial distance
,,'.Z " , \
x
60
=
mm
"'. ..
"/
0
!
!
I
50 100 150 200 Radial distance [mm]
Figure 4. Radial distribution of mean tangential velocity, @, and shear stress component u'v'. Symbols" Flamelet ~, ~-PDF---, Exp. 9 2500
0.1
, .~- ) ~
2000
-~_1+oo~".... ~ ' ~
,~
1000
I F
I
500 b 0
~
-: 9 0"~'; "---'6 ~ 3,+'+#'-
,
,
50
100
9
+
00+
.L,-... .t:..;,'.
~
9, 9
9 . 9_ .--
,
.................... -++~176I' "~~~.'~-'+-.+/ ....... 9
9
~+
Axial distance
150
Axial distance x170 m I
o 0.02
+=+Om~
ol
,
,
0
50
100
200
I
,.,**...../.. .",_._," " O__.
~~
~0.04 ~
,
~
Radial distance [mm]
150
200
Radial distance [mm]
Figure 5. Radial distribution of mean temperature and mole fraction of CO2. Symbols: Flamelet ~, ~-PDF---, MC-PDF ..-, Exp. 9 0.04
0.1
~= co 0.08
A
/'/\\ r I
Axial distance x = 70 mm
X x
o$
0.03 .o
.~ 0.06
0.02 0.04
r
~9
o 0.02 ~,-'-';~,.'
0
!
".1,.. _"-.=-.~
1
:
Axial distance x = 70 m m
I
50 100 150 200 Radial distance [mm]
Figure 6. Radial distribution of CH4 and CO ~-PDF---, MC-PDF..., Exp. 9
_ ,,--,
~.... ,,,
0.01 ~BaT 0
\
........ . . . .
0
9
/
\/',
..."'"..OXl .-'X"-.
"
"iii;2
I
"9
",.>.v..-.
I,
-
." . . . . . . . t_._. . . . . . . . :._. . . . . I . . . . . . .
50 100 150 200 Radial distance [mm]
mole fractions. Symbols:
Flamelet
839
4.3.
Local
extinction
An inherent feature of the Monte Carlo P D F method is the ability to describe local extinction effects because the full statistical information is contained. Figure 7 shows scatter plots of t e m p e r a t u r e at the axial locations x - 10 m m and x - 70 m m above the burner. Additionally, the conditional mean is plotted. At x - 10 m m a significant a m o u n t of local extinction can be observed which represents a severe test for the turbulence-chemistry interaction model. T h e t e m p e r a t u r e is plotted over the mixture fraction f , thus separating of the effects of the chemistry model from those of the turbulence model. The local extinction combined with the complex, impinging-jet like flow field near the bluff body emphasizes the need for more sophisticated models. Exhibiting less local extinction at x = 70 mm, the deviations further downstream seem to be more dependent on the turbulence rather t h a n on the chemistry model.
2500 2000
2500
Axial distance x-10mm
=':
2000
. .
~1500
Axial distance x = 70 m m
1500
'~
r
1000
-'-.';~--:-.,
~ ~--:
I000
':"'"'''-~
'
' o o .
5OO
500 0
O.2
0.4 0.6 0.8 Mixture Fraction
1
I
0
0.2
I
I
0.4 0.6 0.8 Mixture Fraction
1
Figure 7. Scatter plots of t e m p e r a t u r e and its mean conditioned on the mixture fraction resulting from the Monte Carlo simulation with ILDM chemistry.
5. C O N C L U S I O N S
The ILDM reduced chemistry has been applied to a strongly swirling combustor flow. Computations were carried out with both, a presumed PDF integration and a Monte Carlo PDF method. The results prove the ILDM model to overcome the shortcomings of simple chemical models like the laminar flamelet, presented in this work as reference calculation. Moreover, the Monte Carlo PDF method provides a tool to distinguish effects of the chemistry and the turbulence model. Further experimental data from Rayleigh/Raman measurements are needed to investigate the local extinction effects and, hence, the stabilization mechanism. The presumed PDF approaches show similar results for the major species. Significant deviations between ILDM and the flamelet model have only been observed for minor species, e.g. CO.
840 6. A C K N O W L E D G E M E N T S The authors are grateful to P. Schmittel at University of Karlsruhe for providing the experimental data, and to the Deutsche Forschungsgemeinschaft (GRK 91/2) and BMBF (TECFLAM) for the financial support. REFERENCES
Sloan D., Smith P. and Smoot L., Prog. Energy Combust. Sci. 12:163-250 (1986). Breussin F., Pigari F. and Weber R., in 26 ta Symposium (Int.) on Combustion, pp. 2. 211-217 (1996). Jones W.P., in Libby P.A. and Williams F.A., eds., Turbulent Reacting Flows, pp. 309-374, Academic Press, London, San Diego, New York (1994). 3rd International Workshop on Measurement and Computation of Turbulent Nonpremixed Flames, Boulder, CO (1998). Peters N., Progress in Energy and Combustion Science 10:319-339 (1984). Maas U. and Pope S., in 24 th Symposium (Int.) on Combustion, pp. 103-112 (1992). Pope S., Combust. Sci. Technol. 25:159-174 (1981). "TECFLAM Swirl Burner Data Archive," TU Darmstadt, FG Energie- und Kraftwerkstechnik, http://www.tu-darmstadt.de/fb/mb/ekt/tecflam (1998). Landenfeld T., Kremer A., Hassel E.P. and Janicka J., in Eleventh Symposium on Turbulent Shear Flows, pp. 18-1- 18-6 (1997). i0. Landenfeld T., Kremer A., Hassel E., Janicka J., Sch~ifer T., Kazenwadel J., Schulz C. and Wolfrum J., in 27 'a Symposium (Int.) on Combustion (1998). II. Habisreuther P., Schmittel P., Idda P., Eickhoff H. and Lenze B., in VDI Berichte 1313, pp. 127-132 (1997). 12. Leuckel W., "Swirl Intensities, Swirl Types and Energy Losses of Different Swirl Generating Devices," Tech. Rep. G02/a/16, IFRF (1967). 13. Kremer A., Hassel E.P. and Janicka J., VDI Engineering Research 8 (1997). 14. Kremer A., Landenfeld T., Hassel E.P. and Janicka J., in Verbrennung und Feuerungen, 18. Deutsch-Niederl~indischer Flammentag, VDI Berichte 1313, pp. 139-144. VDI Verlag (1997). 15. Schmittel P. and Lenze B., in Verbrennung und Feuerungen, 18. Deutsch-Niederl~indischer Flammentag, VDI Berichte 1313, pp. 121-126. VDI Verlag (1997). 16. Speziale C., Sarkar S. and Gatski T., J. Fluid Mech. 227:245-272 (1991). 17. Daley B. and Harlow F., Phys. Fluids 13(11):2634-2649 (1970). 18. Jones W., in Prediction Methods for Turbulent Flows, vol. 1979-02, Von Karman Institute for Fluid Dynamics (1979). 19. Sanders J. and Lamers A., Combust. Flame 96(1/2):22-33 (1994). 20. Janicka J., Kolbe W. and Kollmann W., J. Non-equilib. Thermodyn. 4:47-66 (1979). 21. Patankar S.V. and Spalding D.B., Int. J. Heat Mass Trans. 15:1787-1806 (1972). 22. Leonhard B.P., Comp. Meths. Appl. Mech. Eng. 19:59-98 (1979). 23. Laxander A., Ph.D. thesis, Universit~it Stuttgart (1996). 24. Nooren P., Wouters H., Peeters T., Roekaerts D., Maas U. and Schmidt D., Combustion Theory and Modelling (1997). Io
.
.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
841
Effects of turbulence length scale on flame speed: a modelling study A. Lipatnikov and J. Chomiak ~* ~Department of Thermo- and Fluid Dynamics, Chalmers University of Technology, 412 96, Gothenburg, Sweden The influence of the integral length scale L of turbulence on the flame speed St has been numerically investigated by simulating the planar and statistically spherical premixed flames which propagate in homogeneous and iso-tropic turbulence. The simulations have been performed using the Turbulent Flame Speed Closure Model discussed briefly in the paper. The results of the simulations show that, for statistically spherical flames, the dependence of St on L reverses as the flame kernel grows. This prediction is supported by a survey of known experimental data. 1. I N T R O D U C T I O N Although turbulent flame speeds St were the main objective of numerous experimental and theoretical studies into premixed turbulent combustion in 5 recent decades, a reliable approach to an evaluation of St has not been elaborated, yet. Even the concept of a turbulent flame speed is sometimes questioned by highlighting the substantial sensitivity of St to flame geometry and the method of measurements. This sensitivity yields the strong scatter in the published data. Moreover, even certain qualitative trends in the behavior of St have not been clarified yet, for example, the dependence of St on the integral length scale L of turbulence. On the one hand, most extensive published experimental data bases imply that combustion speed increases with L. Indeed, G/ilder [1] has analyzed an extensive experimental data base obtained by various teams at moderate turbulence. The best fit approximation implies that St ",, L 1/4 [1]. Karpov and Severin [2] and gido et al. [3] have amassed two extensive data bases by processing pressure records measured in fan-stirred bombs (two slightly different methods employed by these authors have yielded some combustion velocities Ut rather than flame speed). These data bases are reasonably well approximated by expressions resulting in Ut '~ L~ [4] and Ut "., L ~ respectively. Finally, the well-known Leeds data base is approximated by an expression resulting in St ,~ L ~ [5]. It is worth noting that Karpov and Severin [2], Kido et al. [3], and Bradley et al. [5] have varied the r.m.s, turbulent velocity u', the unperturbed laminar burning velocity SL,O associated with a planar steadily propagating flame, and the molecular transport properties of mixtures in wide ranges, but they have not varied L. So, the above relations *This work was supported by the Chalmers Combustion Engine Research Center (CERC).
842 between St, Ut, and L result from processing the experimental data in terms of Utlu' and St/u' as functions of u'/SL,O and L/hL where 5L = a,,/SL,o is the laminar flame thickness and a~ is the molecular heat diffusivity of the unburned mixture. On the other hand, direct experimental investigations of the influence of L on flame speed have shown contradictory trends. Whol and Shor [6] and Talantov et hi. (see Ref. [7] and references therein) have measured the speeds of premixed flames stabilized in channels of different sizes. Both investigations have shown that St increases with channel size. Ballal and Lefebvre [8] and Ballal [9] have investigated premixed combustion in small-scale grid-generated turbulence in a channel. They have reported that St increases with L when u'/SL,O < 2 but St decreases with L when u'/SL,o > 3. The latter result has been critically discussed by Kuznetsov and Sabel'nikov [7]. Recent experiments performed by Leisenheimer and Leuckel [10] in two fan-stirred bombs have shown that Ut, evaluated from the pressure diagrams, increases markedly with bomb size. Ting et al. [11] have varied L between 2 and 8 mm by using different grids to generate turbulence and have observed the decrease of St with L. The above brief review of experimental data shows that the influence of turbulence length scale on flame speed is an intricate issue. The goal of this paper is to contribute to a clarification. For this purpose, simulations of planar and statistically spherical premixed turbulent flames were performed at various L by using the Turbulent Flame Speed Closure Model (TFSCM) put forward by Zimont [12] and developed recently [13-16]. The model is briefly summarized in the next section but the detailed discussion of it is given elsewhere [13-15]. Then, the results of the numerical simulations are reported and analyzed in order to clarify the issue under consideration. 2. T U R B U L E N T F L A M E S P E E D C L O S U R E M O D E L 2.1. Model equations The model employs the reduction of combustion chemistry to a single reaction and characterizes the combustion process by a single progress variable (c = 0 in the unburned gas and c = 1 in the products) following the well-known Bray-Moss method [17]. The model yields the following closed balance equation for the mean progress variable 0~
at
0
0 [
) 0~]
+ -~xj (fi~j~) = Oxj fi(a + Dt
#(1-~) 0 + t~(X + Dt/ab) e x p ( - ~ )
Ap,,u' [uLr~] ~/4 { 1 + Tr' [exp ( - ~ )
- 1] }~/~{ (j=lk
+
(11
~Xj Oq~2)2} 1/2
where the well-known approximation [18]
Dt = D r , 0 [ 1 - e x p ( - ~ ) ]
(2)
of time-dependent turbulent diffusivity Dt is used. Here, the t is time counted beginning with ignition; xj and uj are the coordinates and flow velocity components, respectively; p is the gas density; r~ = tr o and r' = Dt,o/U n are the chemical and turbulent time
843 scales, respectively; subscripts u and b label the unburned and burnt gas, respectively; the Reynolds averages are denoted by overbars and the Favre averages, such as ~ = ~-~, are used. The r.m.s, turbulent velocity u' = ~/2k/3, integral turbulent length scale L = CDU'3/~, and the steady turbulent diffusivity Dt,o = Cuk2/(~ac) are evaluated using, for example, the standard k - e turbulence model [19], where Co, Cu, and ac are constants, k and e are the turbulent kinetic energy and its dissipation rate, respectively. The Favre averaged temperature is linked with the progress variable as follows [17] T = T~(1 - ~
+7~),
(3)
where 7 = P,,/P~ is the heat release parameter. In addition to the two turbulence characteristics (k and e or u ~ and L), the TFSCM includes a single constant A and a set of physico-chemical characteristics, such as SL,O, ,r and the activation temperature O of a single global combustion reaction. The time scale t~ of this reaction is calculated so that it yields the known value of SL,O for the planar steadily propagating flame at u ~ = 0. 2.2. M o d e l f e a t u r e s
To show the basic features of the model in a clear manner, let us consider the limit behavior of Eq. 1 in the simplest case of a planar, one-dimensional flame. For the limit of weak turbulence (u' -+ 0), Eq. 1 is reduced to the standard balance equation of thermal laminar flame theory [20] 0 0 0-7 (ze) + ~ ( p u c )
0 [~ 0E] ~(1-E) exp(_O ) = ~ ~ + t~
(4)
For the opposite limit case of strong turbulence (u ~ >> UL and Dt >> x), the first source term on the right-hand side (RHS) of Eq. 1 is reduced by the ratio of Dt/xb and the last source term dominates. Then, Eq. 1 is reduced to
o--i (fiE) + ~ (pilE) = ~
Dt-~x
+ p,,St I V~[
(5)
in the laboratory coordinate system, or to
0 i) i) 0-7 (ze) + ~ (z~e) = ~
D t . ~x
(6)
in the coordinate system moving from x = +c~ to x = - c ~ with a speed of
{ "fex ( )lJ}
s, = S,,o 1 + T
(7)
Here, ~ = fi - St and St,o is associated with the fully developed turbulent flame speed ,
=
.
(8)
Two points are worth emphasizing. First, Eq. 6 yields a permanent growth of the turbulent thickness St, controlled by the turbulent diffusion law. This feature is the core
844 of the TFSCM: in fact Eq. 5, was proposed by Zimont [12] in order to model a regime of turbulent combustion, characterized by the growing $t. Experimental data reviewed elsewhere [16,21] shows that such a regime occurs in many combustion devices. Second, the turbulent flame speed St is incorporated into the model through Eqs. 7 and 8, so the model uses a certain submodel for St in order to close the balance equation [12] (a similar idea was employed recently by other authors [22-24] in a different manner). This feature of the model has given its name TFSCM although, as is shown below, the flame speed predicted by the model for non-planar flames can differ from Eqs. 7 and 8 even qualitatively. Equation 7 accounts for the development of St due to the fact that as a kernel grows after ignition, it experiences a wider range of the turbulence spectrum. This submodel is discussed in Ref. [15]. Equation 8 accounts for the effects of both turbulence and mixture characteristics on fully developed flame speed. The same expression has been suggested by various authors [21,25-29] using substantially different approaches. A similar expression has been found to be the best fit of an extensive experimental data base associated with moderate turbulence [1]. A close expression (St ", u ' g a -~ where g a ,,, (u'/SL,o) 2 R e t ~/2 and Ret = u ' L / u are the Karlovitz and Reynolds numbers, respectively, u is the mixture viscosity), well approximates the data bases of Bradley et al. [5] and Karpov et al. (see Ref. [4]) in the case of Le = tolD ~_ 1, where D is the molecular diffusivity of the deficient reactant and Le is the Lewis number. Equation 8 predicts that the turbulent flame speed is controlled by the only physicochemical characteristic, that is the chemical time scale re. This feature is supported by the experiments performed by Kido et hi. [3] and offers the opportunity to account for the following important effect. Both old experiments discussed elsewhere [27] and recent investigations [30] show that St increases with pressure P despite the substantial decrease in SL,O. Numerous models employing SL,O as the only physico-chemical characteristic of the mixture cannot predict such pressure effects. On the contrary, the TFSCM is able to do so. For example, Kobayashi et hi. [30] have shown that St is roughly constant in the range of P = 1 - 30 bar, despite the strong decrease in SL,O. For small u'/SL,o, this effect is associated with laminar flame instabilities [30] but it has not been explained for high u'/SL,o. According to Eqs. 7-8 applicable to high u'/SL,O, pressure may affect St only through rc = a,,/S~, o. Since SL,O ": p-~/2 [30] and to, ,-, p - l , Eq. 8 predicts the constant St,o, in agreement with the measurements. 2.3. Validation The TFSCM has been reliably validated [15,31] for statistically spherical kernels expanding in a premixed turbulent gas. Tests performed on the basis of the data of Karpov and Severin [2], Bradley et al. [32], Kido et hi., Mouqallid et al. [33], Groff [34], and Hainsworth (see Ref. [35] and references therein) have shown that the model predicts the growth of flame radius and the development of flame speed at various initial pressures, temperatures, mixture compositions, and u' with the same value A = 0.4. Each aforementioned experimental study used to test the model was performed at a constant L, whereas the objective of this paper is the influence of L on St. For this reason, we have singled out an experimental data base amassed by the different teams but at different values of L. The data base, collected in Fig. 1, encompasses the experimental
845 results of Ting et al. [11] (CH4/air mixture with the equivalence ratio F = 0.70), Bradley et al. [32] (CH~/~ir, F = 0.83), Mouqallid et al. [33] (C3Hs/air, F = 0.75), Groff [34] (C3Hs/air, F = 1.0), and Hainsworth (CH4/air, F = o.80- see Ref. [351). All the measurements were performed for expanding, statistically-spherical flames during a period characterized by a slight pressure rise. Flame kernels were ignited by a spark in homogeneous, isotropic, stationary [32,34] or decaying [11,33,35] turbulence. The flame radius was evaluated using Schlieren movies. All the experiments were modeled by unsteady, spherically symmetrical balance equations for the mass fractions of the fuel and oxidant, closed by the TFSCM and supplemented by the mass conservation equation, the enthalpy balance equation, the k - e turbulence model, and the ideal gas state equation. The full set of governing equations, boundary and initial conditions, input parameters, and the ignition submodel are reported, in detail, elsewhere [15]. It is worth emphasizing that we simulated all the experiments using the same governing equations and the same value A = 0.4 for the only constant of the TFSCM. Since the Schlieren images are associated with the leading edge of flames, the computed flame radius F! was equal to the radius of the surface where ~ = 0.1. The flame speed was calculated to be st = d F ! / d t .
,I"
0.04
~ 0.03
/
/
24
i~
~16
E o~
.
t /
'
E
.4:1
/ /
- -
2
,-r
0.01
'
A L---8 mm L--20 mm 9 L---2 m m - - - L--4 mm
o
'
_
9 "
..I
........
-""
1
t..~.....~ .st," . - - -
~-"....":.-'"
I
L--8mm
0.02
0.00 0.001
o L--2 m m o L---4 mm
""
J
i
8
04
0.006
0.011 Time,
0.016
s
Figure 1. Turbulent flame radius growth. Symbols show experimental data. Curves have been computed. 1 - u ' = 1.73 m/s, L = 20 mm [32]; 2 - u ' = 0.8 m/s, L = 5 [33]; 3 - u' = 2.0 m/s, L = 25 mm [34]; 4 - u' = 1.93 m/s, L = 3 mm [35]; 5 - u' = 1.0 m/s, L = 8 mm [11].
0
I
2
3
4
R . M . S . turbulent velocity, u', rrVs
5
Figure 2. Flame speed vs r.m.s turbulent velocity at different length scales L. Open and filled symbols have been computed for the spherical flames and correspond to F! = 20 mm and F/ --4 cr respectively. Solid lines show second-order fits to the open symbols. Broken lines have been computed for steady-limit planar flames.
Figure 1 shows that the model well predicts the results of all the measurements performed at substantially different values of L, ranging from L = 3 mm [35] to L = 25 mm [34]. These tests, supplemented with an extensive set of tests discussed elsewhere [15,31], support the use of the TFSCM for the purposes of this study.
846 3. R E S U L T S A N D D I S C U S S I O N To simplify the problem and to focus on the discussion on the influence of L on St solely, most of the following simulations have been performed under a constant pressure and for frozen turbulence. In other words, k - e balance equations are not solved and u' and L are assumed to be stationary and uniform. Numerical tests have shown that this simplification does not alter the qualitative trends discussed below. The following results have been obtained by varying u' and L, other things being equal. The initial conditions correspond to the stoichiometric iso-octane/air mixture at temperature To = 358 K and pressure P = 1 bar. The input physico-chemical characteristics of the mixture are as follows: O = 15000 K, tr = 0.525 #s, Sz,o = 0.43 m/s, to= = 0.257 cm2/s. For each set of the initial conditions, the simulations were performed as long as the flame moved the distance equal to 0.5 m, approximately. The simulations were performed both for spherical and planar flames. In the latter case, the mixture was ignited at the left boundary and the symmetry conditions were set at this boundary. For the planar flames, the speed reached the steady-limit value close to 7(SL,0 + St,o) where the term 3' = p,,/pb resulted from the hot product expansion and St,o was determined by Eq. 8. The computed steady-limit planar flame speeds are presented vs. u' for various L by the broken lines in Fig. 2. In line with Eq. 8, a higher St is associated with a larger L, other things being equal. On the contrary, the flame speeds computed for moderately small (~I = 20 mm), statistically spherical flames under the same initial conditions show the opposite trend, that is a higher St is associated with a smaller L (see open symbols or solid curves in Fig. 2). A comparison of the open symbols and broken curves shows that both the value of St and the behavior of St when varying the turbulence length scale, depend substantially on flame geometry and size. This dependence explains, in part, the scatter of the measured data on St and, on the face of it, questions the usefulness of the turbulent flame speed concept. However, some arguments supporting the concept are discussed below. The opposite effects of L on St, predicted for the moderately small, statistically spherical flames and for steady-limit planar flames, appear to be in line with the aforementioned experimental data. When processing the extensive experimental data bases of Karpov and Severin [21, Kido et al. [3], and Bradley et al. [5] in terms of St/u' (or Ut/u') as a function of u'/SL,o and 8L/L, the resulting dependence of St/u' on L follows the dependence of St/u' on 8L by virtue of dimensional reasoning (it is worth keeping in mind that L was not varied in these experiments). Since there are no reasons to suggest that the effect of the laminar flame thickness on the turbulent flame speed depends on the geometry and transient behavior of the turbulent flame; the resulting dependence of St/u' (or Ut/u') on 8L, and, hence, on L should be associated with fully-developed planar flames. For such flames, the computations predict the increase of St with L, in line with the aforementioned empirical approximations. On the contrary, Ting et al. [11] have investigated moderately small, statistically spherical turbulent flames and have reported a decrease of St with L. For such flames, the above computations predict the same behavior. Finally, Leisenheimer and Leuckel [10] have investigated large statistically spherical flames and have reported an increase of St with vessel size and, hence, with L. As we shall discuss in the following, for statistically
847 1,5
,
9
,
1.2
0.05
,
"" -'-~-"~":"~'~
~-~:-'2.-~; ~'~--
E
...... 0.04 . . . . . .
d
._~ 0.03
'o., I/.;,.;/;~
......
/ ~.'~/.~
o.~~
/ ,:~,~"
0.0
o.ooo
,--4 mm
"o 0.02
~_=io;~;~
"~ 0.01 I--
L--8 m m
r
L=IO mm
o.~o2 o.;o,
o.~o,
Time, t, s
o.~
o.olo
Figure 3. Turbulent flame speed development calculated from Eqs. 7 and 8 at u' = 2.36 m/s. Other initial conditions correspond to the results shown in Fig. 2.
L-2 mm L=4 mm L--8 m m L=IO mm L=20 mm
.~4fi
0.00 0.000
0.002
..--"~ . ..-'~.~
.....
0.004
0.006
Time, t, s
0.008
0.010
Figure 4. Turbulent diffusivity Dt v s . time t. The results have been calculated from Eq. 2 at u' = 2.36 m/s. Other initial conditions correspond to the results shown in Fig. 2.
spherical turbulent flames, as the kernel grows, the flame speeds tend towards the values presented by broken lines in Fig. 2. As a result, a higher St is associated with a larger L if the kernel is large enough, in line with the results of Leisenheimer and Leuckel [10]. Thus, for statistically spherical, turbulent flames, the dependence of St on L reverses as ~1 increases. What physical mechanisms can control this effect? Variations in L can affect the predictions of the model: (1) through St calculated from Eqs. 7 and 8 and used in the last source term on the RHS of Eq. 1, and through (2) the turbulent diffusivity calculated from Eq. 2. The dependencies of St(t) and Dr(t) calculated from Eqs. 7-8 and Eq. 2 for various L are presented in Figs. 3 and 4, respectively. The turbulence length scale affects St in two opposite directions. On the one hand, St,o is increased by L according to Eq. 8. On the other hand, the turbulent time scale r' = Dt,o/u '2 ,-, L/u' is increased by L. Hence, the ratios of t/r' and St/St,o are reduced by L. The latter effect dominates at small t but relaxes with time. For the conditions of Fig. 3, the range of 20 mm < ~1 < 40 mm, typical for the laboratory experiments, is associated with 4 ms < t < 8 ms. In this range, the latter effect dominates but the former effect tends to compensate it (see dotted and short-dashed curves in Fig. 3). For L = 2 and L = 4 mm, the results shown in Fig. 3 cannot explain the increase of St with L shown in Fig. 2 (compare open diamonds and circles), especially as the increase of St with L has been computed at F! = 40 mm, too. Similarly, the turbulence length scale affects Dr(t) in the same opposite directions. However, the dependence of fully-developed turbulent diffusivity on L (Dt,o " L) is much stronger than the dependence of fully-developed turbulent flame speed (St,o " L1/4). As a result, for diffusivity, this effect prevails over the decrease in t/r' and Dt/Dt,o by L; and a higher Dr(t) is associated with a larger L. The effects become more pronounced as the flame develops. The increase in Dr(t) by L can affect flame speed by means of the mean flame brush curvature and finite flame thickness mechanisms well-known for laminar flames. The speed Sb = drs/dt of spherically expanding laminar flames is known to differ from
848 0.04
flatne mdius,'c~=0.9
'
/
'/
flame radius, cn=O.5 // / / E flame radius, c~=0.1 ./ / / : . . . . flame brush thickness // 0.03 / / / / , / // ./ 0
"~ 0.02
,
//
#
0.01
/
.~m 15 E
///
/..,
/ "~
/"
lO
,/
o/~
= u'=0.2 m/s
.............
u. s
~~;---
0.00 0.000
0.002
0.004
A u'=0.6 m/s * u'=1.2 m/s
. .-- o - " "
0.006
Time, s
0.008
0.010
Figure 5. Turbulent flame brush thickness
6t and the radii of various iso-scalar surfaces (?:1 =const) vs. time t elapsed after ignition. u' = 2.36 m/s, L = 20 mm, other initial conditions correspond to the results shown in Fig. 2, k - ~ turbulence model has been employed in these computations.
0
0.0
o u'---3 m l s o u'--4 m l s 0.1
0.2
Thickness/Radius
0.3
0.4
Figure 6. Turbulent flame speed St vs the ratio of the mean flame brush thickness 6t to the mean flame radius ~I. Symbols have been computed,curves linearly approximate the computed results. L = 8 mm, other initial conditions correspond to the results shown in Fig. 2.
7SL,O and this difference is well approximated by the following linear relation [36] Sb = 7SL,0 (1 -- 71n77 -12~L)r!
(9)
if Lewis number is equal to unity. The difference between Sb and 7SL,o is caused by the effects of (1) the flame curvature, and of (2) a higher (as compared with Pb) gas density averaged over the enflamed volume. Both of these effects reduce flame speed at the finite 6L/~1 but they relax as 6L/~1 tends to zero. Typically, the difference is small enough because the thickness 6L of the laminar flames is much less than the flame radius r I. The same physical mechanisms, the mean flame brush curvature and higher averaged density, affect turbulent flames too but the corresponding variations in the turbulent flame speed should be much stronger as compared with the laminar case because the mean flame brush thickness 6t, controlled by turbulent diffusivity, is substantial even for quite large flames (see Fig. 5). These mechanisms will reduce flame speed at the finite ~t/r I but this effect will relax a s ~t/r I tends to zero. By analogy with laminar flames, one may assume that turbulent flame speed increases linearly with ~t/r I due to the influence of the emphasized mechanisms. In fact, such an assumption corresponds to the first order approximation when expanding turbulent flame speed with respect to a perturbation parameter, such as the dimensionless curvature, ~t/rl, of the mean flame brush. The results of the simulations, processed in terms of St vs the ratio of ~t/rl, support this hypothesis (see Fig. 6) and show that the effect can be strong enough even for sizeable flames. For example, when ~1 = 20 mm (the right edges of the curves in Fig. 6) the flame speed is approximately less by 2 times as compared with the largest flames. The simulations have shown that the contribution of the time-dependence of St determined
849 from Eq. 7 to the increase of flame speed is of minor importance when f / > 20 mm. Based on the substantial increase of flame speed with 6t/rl, shown in Fig. 6, three points are worth emphasizing. First, these results clarify the effect of L on the speed of statistically spherical, turbulent flames. For moderately small flames, the effect of $t/r! on St is of substantial importance. An increase in L results in increasing Dt (see Fig. 4), enhancing the effect, and decreasing St. As the flame grows, the effect relaxes and the dependence of St on L reverses and is controlled by Eq. 8. Second, similar to the well-elaborated methods for spherical laminar flames, the approximating straight lines shown in Fig. 6 can be used to evaluate fully developed turbulent flame speed that is assumed to be equal to the intersection between these lines and the ordinate axis. The values evaluated using this method are indicated by filled symbols in Fig. 2 and agree very well with the results computed for steady-limit planar flames. This agreement supports the proposed method. Thus, although the speeds measured for expanding turbulent flames substantially depend on flame geometry and size, such experimental results can be employed to evaluate fully developed turbulent flame speed, provided that flame thickness is also measured. Third, the effect under discussion can contribute to the leveling-off of St(u') (see open squares in Fig. 2), observed in numerous measurements [2,3,5]; in addition to the contribution made by local stretching and quenching phenomena, discussed elsewhere [5]. Indeed, if L is large enough, then, Dt and 6t are also large, the decrease in the flame speed by 6t/fl will be well-pronounced and augmented by u', whereas St(t) given by Eq. 7 will always be increased by u'. The counter-action between these two effects results in a leveling-off of the computed data presented for rl = 20 mm by open squares in Fig. 2. 4. C O N C L U S I O N S The results of simulations of expanding, premixed, turbulent flames, performed using the Turbulent Flame Speed Closure Model, highlight the importance of transient phenomena and, especially, of the effects associated with the substantial thickness of the flame brush (flame brush curvature and density averaged over the enflamed area). These effects substantially reduce the speed of moderately large, statistically spherical flames and can result in decreasing the flame speed with L and leveling-off the dependence of the flame speed on u'. As the flame kernel grows, the effects relax and the dependence of flame speed on the turbulence length scale reverses. A method of evaluating fully-developed turbulent flame speed in experiments with expanding kernels is proposed. REFERENCES
1. O.L.G/ilder, 23rd Symp. (Int.) on Combust., The Combustion Institute, Pittsburgh (1000) 743. 2. V.P.Karpov and E.S.Severin, Combust. Explos. Shock Waves 16 (1980) 41. 3. H.Kido, T.Kitagawa, K.Nakashima, and K.Kato, Memoirs of the Faculty of Engineering, Kyushu University, 49 (1989) 229. 4. A.N.Lipatnikov, in Advanced Computation and Analysis of Combustion, G.D.Roy, S.M.Frolov, and P.Givi (eds.), ENAS Publisher, Moscow, (1997), 335.
850 5. D.Bradley, A.K.C.Lau, and M.Lawes, Phil. Trans. R. Soc. London, A338 (1992) 359. 6. K.Whol and I.Shor, Ind. Eng. Chem., 47 (1955) 828. 7. V.R.Kuznetsov and V.A.Sabel'nikov, Turbulence and Combustion, Hemisphere Publ. Corp., New York, 1990. 8. D.R.Ballal and A.H.Lefebvre, Proc. R. Soc. London, A344 (1975) 217. 9. D.R.Ballal, Proc. R. Soc. London, A367 (1979) 353. 10. B.Leisenheimer and W.Leuckel, Combust. Sci. Technol., 118 (1996) 147. 11. D.S.-K.Ting, M.D.Checkel, R.Haley, and P.R.Smy, SAE Paper 940687 (1994) 1. 12. V.L.Zimont, in Chemical Physics of Combustion and Explosion Processes. Combustion of Multi-Phase and Gas Systems, OIKhF, Chernogolovka, (1977) 77 (in Russian). 13. V.L.Zimont, A.N.Lipatnikov, Chem. Phys. Reports 14, (1995) 993. 14. V.P.Karpov, A.N.Lipatnikov, V.L.Zimont, 26th Symp. (Int.) on Combust., The Combustion Institute, Pittsburgh, (1996) 249. 15. A.N.Lipatnikov and J.Chomiak, SAE Paper 972993 (1997) 1. 16. A.N.Lipatnikov and J.Chomiak, Modeling of Turbulent Flame Propagation, Annual Report 98/2, Chalmers University of Technology, 1997. 17. K.N.C.Bray and J.B.Moss, Acta Astronautica 4 (1977) 291. 18. J.O.Hinze, Turbulence, 2nd Edition, McGraw-Hill, New York, 1975. 19. B.E.Launder and D.B.Spalding, Mathematical Models of Turbulence, Academic Press, London, 1972. 20. Ya.B.Zel'dovich, G.Barenblatt, V.Librovich, and G.Makhviladze, "The Mathematical Theory of Combustion and Explosions," Plenum Publ. Corp., New York, 1985. 21. A.G.Prudnikov, M.S.Volynskii, and V.N.Sagalovich, Mixing Processes and Combustion in Jet Engines, Mashinostroenie, Moscow, 1971 (in Russian). 22. M.Wirth and N.Peters, 24th Symp. (Int.) on Combust., The Combustion Institute, Pittsburgh (1992) 493. 23. H.G.Weller, S.Uslu, A.D.Gosman, R.R.Maly, R.Herweg, and B.Heel, Symp. COMODIA 94, JSME, Yokohama (1994) 163. 24. H.P.Schmidt, P.Habisreuther, and W.Leuckel, Combust. Flame, 113 (1998) 79. 25. V.R.Kuznetsov, Izv. Akad. Nauk SSSR, Mekh. Zh. Gaza 5 (1976) 3 (in Russian). 26. V.A.Frost, in Combustion and Explosion, Nauka, Moscow, (1977) 361 (in Russian). 27. V.L.Zimont, Combust. Explos. Shock Waves 15 (1979) 395. 28. O.L.G/ilder, 23rd Symp. (Int.) on Combust., The Combustion Institute, Pittsburgh (1990) 835. 29. W.T.Ashurst, M.D.Checkel, and D.S.-K.Ting, Combust. Sci. Techn., 99 (1994) 51. 30. H.Kobayashi, T.Tamura, K.Maruta, and T.Niioka, 26th Symp. (Int.) on Combust., The Combustion Institute, Pittsburgh (1996) 389. 31. A.Lipatnikov, J.Wallesten, and J.Nisbet, Symp. COMODIA98, JSME, Tokyo, (1998) 239. 32. D.Bradley, M.Lawes, M.J.Scott, and E.M.J.Mushi, Combust. Flame, 99 (1994) 581. 33. M.Mouqallid, B.Lecordier, M.Trinite, SAE Paper 941990 (1994) 1. 34. E.G.Groff, Combust. Flame, 67 (1987) 153. 35. S.B.Pope and W.K.Cheng, 21st Symp. (Int.) on Combust., The Combustion Institute, Pittsburgh (1986) 1473. 36. P.Clavin, Progr. Energy Combust. Sci., 11 (1985) 1.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
851
Application of a Lagrangian PDF Method to Turbulent Gas/Particle Combustion M. Rose, P. Roth ~ and S.M. Frolov, M.G. Neuhaus b* ~Institut f/ir Verbrennung und Gasdynamik, Gerhard-Mercator-Universitfit, 47048 Duisburg, Germany bN.N. Semenov Institut for Chemical Physics, Russian Academy of Science, Moscow, Russia A mathematical model for two-phase turbulent reactive flows is presented and applied to calculate the combustion of a dust jet under conditions of homogeneous, isotropic turbulence. Contrary to commonly used Euler/Euler or Euler/Lagrange methods this model is based on considering both phases in Lagrangian manner. The mechanical and thermodynamical properties of the two-phase mixture are calculated along the trajectories of two kinds of "particles" representing the gas and the dispersed phase. Similar to Monte-Carlo methods for solving a high dimensional joint velocity-composition probability density function, the turbulent gas phase is described by means of stochastic calculus. The deterministic equations for individual solid particles can be treated directly. In this approach, the interaction between both phases is restricted to the vicinity of solid particles by the definition of an "action-sphere" which is attached to every solid particle. Applications of the method indicate that it is capable of providing information on the local structure of combustion zones with species formation and transport. The results show that the method is applicable independent of the combustion modes in the gas phase. 1. I N T R O D U C T I O N Many flow phenomena in nature and engineering applications involve the interaction between a fluid carrier phase and a dispersed solid or liquid phase. In general, this interphase processes include a multifacetted exchange of mass, momentum, and energy, the latter process being of fundamental importance in chemically reacting flows, involving highly energetic materials. Due to the geometrical size of the flow system and the condensed phase, and the velocity range most of these flows are turbulent. A large variety of different mathematical models for two-phase turbulent reactive flows are known, based either on considering two penetrating and interacting continua in Eulerian manner, or on considering the carrier phase in Eulerian and the dispersed phase in Lagrangian manner. In the models, chemical energy release is usually provided by both homogeneous reactions in the gas phase and heterogeneous reactions at the surface of *This work was supported by the German Science Foundation and by the German-Russian Scientific Exchange Program on Physical Chemistry of Combustion.
852 the combustible condensed phase (CP-) particles. Turbulence is represented by enhanced transport coefficients in the gas phase as compared to lalninar flow. A standard k-e model for homogeneous flow is often used for these purposes. Attempts were made to involve statistical approaches into the models. Berlemont et al. [1] apply a presumed PDF approach to model gas-phase velocity fluctuations in the vicinity of CP-particles. Among other approaches, discrete vortex methods [2] and LES [3] are worth to be mentioned. Fundamental aspects of (liP-particles dispersion dynamics in a turbulent flow have been studied by DNS [4-6]. This paper deals with test implementations of a new method for modelling two-phase turbulent reactive flows. The development of this method is described in detail in [7]. Contrary to existing approaches, the method is based oil considering both interacting continua in Lagrangian manner. It combines the benefits of particle methods for cap culating the joint velocity-composition PDF of a turbulent reactive gas phase with the possibility to follow the processes of heat and mass transfer in the vicinity of the trajectories of either individual CP-particles or groups of CP-particles. The method therefore incorporates the effect of CP-particles on the local flow structure, as there are physical aspects of CP-particle evaporation (volatilization), heterogeneous chemistry, and radiation absorption.
2. G E N E R A L E Q U A T I O N S FOR T W O I N T E R A C T I N G P H A S E S 2.1. Modelling A s s u m p t i o n s In this formulation, the two phases of the gas/particle mixture have their own density, velocity, and temperature and interact due to the exchange of mass, momentum, and energy. For the sake of computational capability, some assumptions are made, concerning the thermochemical properties of the gas phase, the particle phase and the interphase fluxes.
For the gas phase it is assumed that the ideal gas law holds and that heat flux is governed by molecular diffusion. Volume forces are neglected in the gas phase and the mean pressure is assumed to be uniform all over the volume, but fluctuations in pressure do exist. For the condensed phase it is supposed that the volume fraction of particles is neglibibly small and that the number of CP-particles is constant. They are assumed not to interact with each other and CP-particles can volatilize resulting in release of CO. Also volume forces are neglected for the condensed phase. In the interphase fluxes, heat transfer by radiation is neglected, and it is assumed that drag force is the only interphase force.
2.2. P s e u d o C P - P a r t i c l e s and G P - P a r t i c l e s As a matter of fact, the properties of CP-particles are not smeared over the volume, as it is implied when the condensed phase is considered as a pseudo-continuum. Instead, every CP-particle has its own individuality. Due to the large number of condensed phase particles in realistic two-phase systems, it is impossible to simulate each CP-particle on its own. To make the problem computationally capable, we apply the concept of pseudo CP-particles Pk (k - 1 , . . . , Np), as introduced in [7]. Pseudo~CP-particles represent groups of npk individual CP-particles. They move in physical space and interact with its gaseous surrounding like real particles, but the local effect of a pseudo CP-particle on the
853 gas is amplified by the number npk of individual real particles in the pseudo particle. The dynamics of the reactive gas phase is also calculated by a particle method, which is used efficiently for solving a high dimensional transport equation for a joint velocitycomposition probability density function (PDF). This method involves a large number of gas phase particles (GP-particles), Gi (i = 1 , . . . , Na), which represent local realizations of the turbulent flow field. The evolution of the gas phase properties, which are represented by averages over GP-particle properties, is given by the modelled conservation equations in Lagrangian form given below. 2.3. A c t i o n S p h e r e The intrinsic feature of interphase exchange processes is the finite dynamic depth of interphase fluxes. For example, temperature and vapor concentration profiles around a quiescent, evaporating droplet have definite widths, which dependent on time [5,8]. Under flow conditions, the dynamic effect of a particle on the surrounding gas is localized in the vicinity of "perturbed" streamlines. Clearly, mass, momentum, and energy fluxes to/from the gas phase vanish at some finite distances from the CP-particle surface. This implies that the interphase fluxes should be localized in the vicinity of CP-particles, rather than smeared out over the volume. The characteristic depths of the fluxes are different for mass, momentum and energy exchange processes, their proper estimation is a special task. In the present case we assume the characteristic depths of these processes to be the same and independent of location and time. To describe the vicinity of a pseudo CP-particle/Sk, an action sphere f~a (/Sk) of action radius ra is defined, which is attached to the particle, as shown in Fig. la. Interphase
Figure 1. (a) Action sphere attached to a pseudo condensed phase (CP) particle (b) Interaction of three pseudo CP-particles with GP-particles located inside their action spheres.
fluxes are localized in this action sphere. A pseudo CP-particle/Sk is influenced by a GPparticle Gi only if Gi is inside the action sphere of/sk. The interaction of three pseudo
854 CP-particles with GP-particles is illustrated in Fig. lb.
2.4. Conservation Equations in Lagrangian Form In the Lagrangian terminology, the gas phase is represented by Na GP-particles Gi of volume Va~, with mass pa~ Va,. The partial mass of species l in GP-particle Gi is P~a~ Va~. Furthermore, each GP-particle has its own velocity vc;,, density pa,, and enthalpy ha,. The condensed phase is represented by Np pseudo CP-particles, each having its own mass mp~, velocity vpk, and internal energy h p . The number of individual real particles, represented by a pseudo CP-particle/5~ is n p . The trajectories of pseudo CP-particles and the evolution of mass, momentum, and energy along their trajectories become in the Lagrangian frame [7]: D pk xpk Dt = v pk , mPk
__fdrag
DP~vPk D--------~=
Pk
D pk mpk vol _ Dt = -aJPk '
mPk
h~t vk '
(1)
D pk hp~ .co~v ~ o l (~vol_ hp~) -a~ h-~t ~h~t ,(2) Dt =-qPk + pk Pk
where wvol h-et are the rate of mass release due to volatilization and heterogeneous P~ and w P~ reactions respectively. The term fdrag pk represents drag forces acting on the surface of the condensed phase particle, @COYtV is the heat flux, and the terms tzr o t -hp~, and tzh~t account for heat of volatilization an~ heterogeneous reactions, respectively. The governing equations describing the GP-particle trajectories and the conservation equations of mass, species, momentum, and energy in Lagrangian form are: D a' xa, Dt
=
__~7
Dt
pa~
Da'va' Dt
pa~
D a~ hG, Dt
D a' (pa, Va, ) Dt
v~,,
"l + 3~, "l + 9J~,
~
E
,~
~
['t,~, volt~
(cdvol
~
~
+ ~ het ,~),
(3)
.hett~), + ~,
(4)
Pk aien(Pk)
=
V(pE-T)+
E
( fdr a 9 vol ~p~,_~,~ +v~,~( aj ~,~ + ~ het ~)),
(5)
Pk cien(Pk)
-
(0con~ § fd,'%g
L hom - V ' q a , di f f +'~a~ +
Pk ai~a(Pk)
+
~_. npkwa~lpk (~vot
- ' o at,. vol -
va,
. v ~
lv2p~ + -~
lv2a , -~
)
,
(6)
Pk Giea(Pk)
where V.j~, for 1 - 1,... , Ns, is the diffusion flux of gas species l, and 2a,'l and coa,hetzpkis the mass production rate due to homogeneous and heterogeneous reactions, respectively. The .het Pk - ~2~=1 Ns wa~ .hettPk and ojvol Ns a~:~k account for the total mass production terms wa, a, Pk - ~21=1 rate due to all heterogeneous reactions and volatilization, respectively. The fluctuating pressure is p. E and ~- are the unit tensor and the shear stress tensor, respectively. The
855 fdrag
t e r m - c , Pk denotes the drag force between gas particle Gi and pseudo CP-particle /Sk. hom are the diffusioanl heat flux in the gas phase and the energy source due V - q adi~f f and h G, :cony to homogeneous reactions in a gas particle Gi, respectivly. The term qG, Pk accounts for convective heat transfer between gas particle Gi and pseudo C,P-particle Pk. Summations in Eqns. (3) and (6) are taken over all pseudo CP-particles s which are influencing GPparticle Gi.
3. I N T E R P H A S E FLUXES Equations (1), (2) and (4)-(6) for pseudo CP-particles and GP-particles are coupled through the interphase fluxes of mass, momentum, and energy. Simple explicit relationships for modelling these fluxes are adopted as outlined in [7] . For simplicity, the dispersed solid particles are assumed to have thermochemical properties similar to those of coal.
3.1. Mass Flux, Drag Force, and Energy Flux Devolatilization of particles and heterogeneous reactions are both described by simple one-step mechanisms of the form" Coal -+ CO
(devolatilization),
Coal + 2 02 + C02 + 2 CO
(het. reaction).
The rates of pseudo CP-particle volatilization/heterogeneous reaction is given by the commonly used Arrhenius-type expressions. In this study, momentum fluxes are represented by drag forces. The drag force between a pseudo CP-particle/Sk and the surrounding gas in Eqn. (2) is taken in the form: fdrag
7r d2- 1
I%-vb l,
(7)
where fia and 9a are the density and velocity of the gas, averaged over all gas particles inside the action sphere of pseudo CP-particle/Sk. The mean drag coefficient ~D is given by the Stokes law. Interphase energy exchange is represented by convective heat flux between a pseudo CP-particle Pk and GP-particles Gi inside the action sphere of/Sk. It is taken in the form" v
d
-
,
(s)
where Nu is the Nusselt number, taken Nu - 2.0 for simplicity, and To and ~a are the average gas temperature and heat conductivity of gas in the vicinity of pseudo CP-particle Pk, respecitevly. ".conv The drag force fdra9 -c, Pk and heat exchange rate qa~ Pk between pseudo CP-particle /Sk and GP-particle Gi in Eqns. (5) and (6) are also calculated by formulas (7) and (8), respectively, but using the averaged values of density, velocity, and temperature in the vicinity of GP-particle Gi.
856 4. M O L E C U L A R
FLUXES IN GAS PHASE
The basic difficulty in solving Eqs. (1) to (6) under turbulent flow conditions lies in the treatment of the molecular fluxes and the fluctuating pressure gradient. The effects of convection in physical space and chemical reaction require no modelling and can be treated straightforward. For modelling the molecular fluxes V . jlGi and V " "~'::f the model of Dopazo [9 10] is ~tGi used, which is based on the simple representation of relaxation to the local mean values of species concentration and energy. The terms in Eq. (5), representing viscous stress V(T) and fluctuating pressure gradient V ( p E ) , are modelled by a simple stochastic Langevin equation similar to the approach of Pope [10]. The gas-phase reaction is taken in the form [11]: 2 CO + 02 -+ 2 CO2 , with rates of species formation/depletion in Arrhenius type form. The term h h~ a~ in Eq. (6) is calculated with regard to the heat of reaction. 5. R E S U L T S OF T E S T I M P L E M E N T A T I O N 5.1. Initial and B o u n d a r y C o n d i t i o n s The goal of the current study is to simulate two-phase combustion in a uniform pressure reactor under simple conditions of constant, isotropic, homogeneous turbulence. The computational set up used is given in Fig. 2.
Figure 2. Sketch of boundary conditions: open volume of air enriched with CO-devolatilizing reactive particles.
Preheated air of initial temperature T O - 800 K and mean pressure p0 _ 1 bar flows with constant mean velocity of 2.5 ms -1 through an open volume of size 5 cmx2.5 cmx0.1 cm. The airflow contains in its lower part (z <_ l cm) CP-particles of 50 #m in diameter. The material density of particles and loading ratio is 1000 k g / m 3 and 1509/rn 3, respectively. Turbulence is described by constant values of turbulent kinetic energy k and its dissipation rate e with the values (k - 0.4 J k g - 1 , c - 35 J k g - l s -1) for low turbulence intensity and (k - 1.6 Jkg -1, c - 140 J k g - l s -1) for more intensive turbulent mixing. 5.2. R e s u l t s of C a l c u l a t i o n s The effect of turbulence intensity on temperature and (',O mass fraction in a turbulent reacting gas/particle mixture was investigated. The upper part of Fig. 3 shows the distribution of temperature in the x/y-plane 20 ms after particles start to devolatilize CO into the preheated ambient air.
857
Figure 3. Distribution of temperature (upper part, 300 K _< T _< 1950 K) and CO mass fraction (lower part, 0 _< [CO] <_ 0.4) in the x/z-plane for low (left) and high (right) turbulence intensity 20 ms after particles start to devolatilize.
The left and right parts of the figure correspond to low and high turbulence intensity, respectively. At this time, the mean characteristics of the flow field have become stationary, i.e. spreading angle of the "flame" and mean gas/particle-mixture compositions at different cross-sections normal to the x-axis. Independent of turbulence intensity, high chemical activity indicated by light colours in the T-picture is restricted to the vicinity of CP-particles in the inflow section, x _< 1 crn. Further downstream, turbulent transport influences significantly the evolution of the temperature field. For lower turbulent mixing, the spreading angle of the flame front in the flow direction is much lower than in the case of higher turbulence. In the latter case, also the distance between isothermals in the z-direction is greater than for lower turbulence intensity. This represents a much higher flame brush thickness. In addition, the more erratic form of the isothermals for higher turbulence indicate a much more wrinkled and corrugated shape of the flame. Characteristic distributions of the CO mass fractions are shown in the lower part of Fig. 3. In both parts of the figure, the amount of unburnt CO in the inflow region is high, which is represented by light colours. For higher turbulence, the enhanced transport in the z-direction leads to a much better mixing of CO with fresh air further downstream, resulting in a much lower amount of CO at the right border of the volume. In the case of lower turbulence intensity, devolatilized CO is not burned completely and leaves the volume mostly concentrated near z - 0 cm. A quantitative description of the results illustrated above is given in Figs. 4 to 6. Figs. 4 shows the mean temperature and CO mass fraction along the x-axis. The values
858
=ooo[
=
low turbulence intensity
I
1800
nor,
: -"
[~
o~
low turbulence intensity high turbulence intensity
30
91600 ~1400
~ 20
,~ 12oo
~176
10
1000 8000
1
2
3 X / cm
4
5
O0
1
2
3
4
5
X / cm
Figure 4. Temperature distributions (left) and CO mass fraction (right) along x-axis averaged for 0 c m <_ z <_ 0.5 c m .
were obtained by averaging in the z-direction between z=0 cm and z - 0 . 5 cm, i.e. the originally particle loaded flow part. The inflow region z _< 1 cm is characterised by peaks in temperature and CO concentration up to 1400 K and 40 %, respectively. The very high heat release results from homogeneous burning of devolatilized CO and heterogeneous particle burning in the fresh surrounding air. A local minimum in temperature and CO concentration follows the zone of individual particle burning at about z - 1 cm and z - 0.5cm, respectively. From about z - 1.5cm the graphs of temperature and CO mass fraction become more continuous, indicating that the chemical activity is no longer concentrated to the vicinity of individual particles. The combustion mode changes from diffusion/devolatilization controlled reaction to partially premixed reaction in this region. In the case of high turbulence intensity, the temperature remains nearly constant at a value of about 1400 K for z >__ 1.5 c m , whereas the temperature further increases up to 1800 K for less intensive turbulent mixing. At the same time, the mass fraction of CO decreases down to less than 5~ at the exit, whereas for low turbulence more than 12% of the total mass is unburned CO. The effect of different turbulence intensities on the flow field becomes even more significant when looking at the distribution of temperature and CO mass fraction in the upper section of the flow field at z=1.75 cm, see Fig. 5. In the case of high turbulence intensity, the left part of the figure shows for z _> 1.5 cm a significant continuous increase in temperature, resulting in an average outlet temperature which is about 200 K higher than for lower turbulent mixing. The CO mass fraction shown in the right part of Fig. 5, is always less than 1%, and has a maximum at z - 3 c m in the case of high turbulence. This is exactly the position where the turbulent flame brush influences the flow field, whereas in the case of low turbulence intensity only very little CO reaches this upper flow section by diffusion and turbulent transport. The profiles of temperature and CO mass fraction in the vertical outlet region are given in Fig. 6. The left part shows that lower turbulence intensity causes more heat release in
859 1050 -
--"
low turbulence intensity
low turbulence intensity high turbulence intensity
o~ 0.8
1000
c
.o 950
0.6
E c
900
0 o
0.4
c
E
E 0.2
850
800 :
1
2
3
4
5
0 ;
X/cm
,
d.
' -" 1
i
__d -
I
2-
-3
--
I
4
5
X/cm
Figure 5. Temperature distributions (left) and CO mass fraction (right) along x-axis averaged for 1.5 c m <_ z <_ 2 c m .
the originally particle loaded flow part z _< I c m and less one for z _> i c m compared to the case with high turbulence intensity. The corresponding temperature graph shows a high gradient around z = l cm, connecting the lower temperature region of about 800 K with a high temperature region of about 1900 K. The temperature graph of the corresponding high turbulence case is more smooth with less extreme values ranging between 1000 K and 1500 K. It becomes obvious from the right part of Fig. 6, that the amount of unburned CO leaving the flow section for z _< 0.5cm is very high in the case of low turbulence intensity. Furthermore, the temperature is also high in this flow part, but the expected high reactivity is restricted by a lack of oxidizer.
2.5
2.5 -"--
--
low turbulence intensity high turbulence intensity
low turbulence intensity high turbulence intensity
1.5
1.5
1
0.5
0.5
0
I
800
,
I
,
I
~
I
11 O0 1400 1700 mean temperature / K
i"'I
2000
0
I
0
3
6
9
12
mean CO mass fraction / %
Figure 6. Profiles of temperature (left) and CO mass fraction (right) at the right end of the flow volume averaged for 4.5 c m <_ z <_ 5 c m .
860 6. C O N C L U S I O N A new method for modelling two-phase turbulent reactive flows has been described and applied to isotropic, homogeneous turbulence. Contrary to existing approaches, the method is based on considering both interacting continua in Lagrangian manner. In this formulation, gas and condensed phase properties are calculated along the trajectories of "particles" representing the system. The present example of combustion modelling of a dust jet clearly indicates the influence of different degrees of turbulence intensity on the flow field. At a low rate of turbulent mixing, the resulting reaction volume is much more narrow than for higher turbulence, the latter leading to a more corrugated and thicker flame brush. For the initial conditions of this study, higher turbulence leads to a much better consumption of fuel due to enhanced mixing of devolatilized fuel and oxidizer from the surrounding fresh air. The calculations indicate that the method is capable to provide information on the local structure of combustion zones with species formation and transport. Furthermore it is applicable independent of the combustion mode in the gas phase, i.e. no assumption regarding the degree of premixedness is required. REFERENCES
1. Berlemont, A., Grancher, M.-S., Gousebet G.: On the Lagrangian Simulation of Turbulence Influence on Droplet Evaporation; Int. J. Heat Mass Transfer: 34, 11, pp. 2805 (1991). 2. Hansell, D., Kennedy, I.M., Kollmann, W.: A Simulation of Particle Dispersion in a Turbulent Jet; Int. J. Multiphase Flow: 18, 4, pp. 559 (1992). 3. Yeh, F., Lei, U.: On the Motion of Small Particles in a Homogeneous Isotropic Turbulent Flow; Phys. Fluids: 3, 11, pp. 2571 (1991). 4. Riley, J.J., Patterson, G.S.: Diffusion Experiments with Numerically Integrated Isotropic Turbulence; Phys. Fluids: 17, pp. 292 (1974). 5. Squires, K.D., Eaton, J.K.: Preferential Concentration of Particles by Turbulence; Phys. Fluids: 3, 5, pp. 1169 (1991). 6. Nigmatulin, P.I.: Dynamics of Multiphasr Media; Moscow, Nauka, 1, 1987 (in Russian) 7. Rose, M. and Roth, P. and Frolov, S.M. and Neuhaus, M.G. Lagrangian Approach for Modelling Two-Phase Turbulent Reactive Flow; Advanced Computation & Analysis of Combustion, editors: G. D. Roy, S. M. Frolov, P. Givi, pp. 175-194 (1997). 8. Xu, Ch.R., Fu, W.B. Study on the Burning Rate of a Carbon Particle under Forced Convection Conditions; Comb. Sci. and Tech.: 124, pp. 167-182 (1997). 9. Dopazo, C.: Recent Developments in PDF Methods, in Turbulent Reactive Flows, pp. 375-474, editors: Libby, P.S. and Williams, Academic Press (1994). 10. Pope, S.B.: PDF Methods for Turbulent Reactive Flows; Prog. Energy Combust. Sci.: 11, pp. 119 (1985). ll. Frolov, S.M., Basevich, V.Ya., Neuhaus, M.G., Tatschl, R.: A Joint Velocity Scalar PDF Method for Modeling Premixed and Nonpremized Combustion, In: Advanced Computation and Analysis of Combustion; G.D. Roy, S.M. Frolov, P. Givi, Moscow (1997).
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 1999 Elsevier Science Ltd.
861
Large Eddy Simulation of a Nonpremixed Turbulent Swirling Flame N. Branley and W.E Jones Department of Chemical Engineering and Chemical Technology, Imperial College of Science,Technology and Medicine London, SW7 2BY, England Large Eddy Simulation (LES) of an unconfined nonpremixed turbulent swirling methane flame has been performed. An approximate dynamic localization model has been used to close the Favre-filtered Navier-Stokes equations. The instantaneous thermochemical state of the flow is characterized in terms of a strictly conserved scalar. Laminar flamelet data provides the thermochemical model. The influence of subgrid scale (SGS) fluctuations in the scalar field on the resolved scale density, temperature and species concentrations is accounted for through the use of an assumed shape subgrid pdf; in this instance a fl function has been employed. The subgrid scale scalar variance must be known to evaluate the fl-sgpdf; this is provided by means of a simple algebraic model. The predictions show fair agreement with measurements in the near burner region, though the simulation has failed to reproduce the correct shape of the recirculation zone. 1. I N T R O D U C T I O N An understanding of turbulent reacting flow is essential in the design of many engineering devices such as furnaces, gas turbines and internal combustion engines. In many of these applications, swirl is added to the flow to stabilize the flame and augment mixing. The resultant flow patterns are characterized by regions of strong streamline curvature and recirculation zones. Increasingly, Large Eddy Simulation (LES) is being applied to turbulent flows of engineering interest and it seems natural to ask the question, can LES make a contribution in the field of turbulent reacting flow modelling? As pointed out by Pope [ 1], LES does not overcome the closure problem associated with non-linear reaction rates. As with Reynolds averaged approaches, the closure problem can be treated by solving a transport equation for the probability density function describing subgrid scale fluctuations of the scalar fields. Such an approach has been considered by a number of authors who variously refer to this pdf as the "large eddy" pdf [2], the "subgrid" pdf (sgpdf) [3], and the "filtered density function" [4]. The cost of its calculation in an LES is considerable since the sgpdf varies with the filter width and with time. Colucci et al. [4] have shown that some account must be made for the subgrid fluctuations of the scalars when evaluating the reaction rate term if serious errors are to be avoided. Some authors have investigated the possibility of using a similarity model for the subgrid component of the reaction rate [5],[6]; however, as noted by Pope [ 1], reaction and diffusion dominate the small scale structure of the composition field which invalidates the assumption of similarity between the subgrid scales and the smallest resolved scales upon which these models rest. Treatment of the reaction rate term can be avoided in nonpremixed combustion by invoking
862 the conserved scalar formalism [7], whereby mixing of fuel and oxidant is characterised by a strictly conserved scalar, the mixture fraction, and reaction is assumed to be fast. This methodology has been employed previously in the simulation of a turbulent jet of hydrogen burning in a coflowing stream of air [8]. In order to assess the performance of LES when applied to reacting flows of more practical importance, a swirling methane flame, measured by Kremer et al. [9], is the focus of the present study. In the next section the governing equations and models will be presented, followed in Section 3 by a short description of the calculation method and the computational parameters of the simulation. The results of the simulations are presented in Section 4, followed by the conclusions of the investigation. 2. G O V E R N I N G E Q U A T I O N S A N D M O D E L L I N G
The LES equations can be obtained by applying a spatial filter to the flow equations. The spatial filter of a function f - f(x, t) is defined as its convolution with a filter function, G, according to f ( x , t) -
G(x - x'; A(x))f(x', t)dx'
(1)
where the integration is carried out over the entire flow domain, f2. The filter function has a width A which may vary with position. To ensure that the sgpdf has the properties of a pdf, G must be positive definite [10]. In turbulent reacting flows large density variations occur which must be properly accounted for. In the context of time or ensemble averaging the most straightforward method of accounting for these variations is through the use of density weighted, or Favre averages; in LES this approach is the only means of rendering the problem tractable. Here, a "Favre filtered" quantity is denoted by a tilde (-) and is defined by f(x, t) - e_f. p Applying the Favre filter to the Navier-Stokes equations gives
O-p~z~ O-p~z~(zj=
c9~
0
1
~
(
Orij ,
where Sij is the Favre filtered strain rate, Sij _ 7~ o_% Oxi + Oxj ,], and Tij is the unknown or subgrid scale stress, defined as -
(uT
j -
(2)
residual (3)
In (2) the molecular viscosity, #, is treated as constant. The Boussinesq-type model used by Smagorinsky has been employed here to parameterize the deviatoric part of the residual stress according to
Tij - 89
- -2-pC-A21SI (Sij- 35ijSkk)
(4)
IZsgs
where #~g~ is the subgrid scale eddy viscosity and loci - V/2~ijSij. -A is proportional to the filter width A, and is equated here to the cube root of the grid cell volume. The approximate
863 localised dynamic model of Piomelli and Liu [ 11 ] has been used to calibrate (4), whereby C is obtained from C(x, t) - (C*oq~ - L~'~)S~
(5)
where the caret (:) denotes the application of a testfilter of width ~. Lij is the subtest scale Leonard stress, and aij - 2-ff~21SISij. The superscript a denotes the anisotropic part of the tensor. C* is the C-field from the previous timestep. The testfilter has been defined here as a volume average over adjacent cells so that A ~ 3A. When spatially filtered, the transport equation for the mixture fraction, ~, takes the form 0~
Ot
=
Oxk
,
Oxk
~
(6)
Oxk
where Sc is the Schmidt number. The filtering introduces an unknown subgrid scalar flux of ~, Jk, defined as
Jk - -fi(ua~- ~za~)
(7)
which must be modelled. The usual form of model used for the subgrid flux of a scalar is (see for example Schmidt and Schumann [12]):
#sgs O~
SCsgsOXk
(8)
where ~-~sgsis the subgrid scale Schmidt number. The use of a subgrid scale Schmidt number implies that the subgrid scale flux of momentum and of the scalar have similar length and velocity scales. Consequently SCsgs must be O (1) for this to be accurate. A constant value of 1.0 has been used here. A reaction model must be supplied to calculate the species concentrations and temperature from the element concentrations obtained by solving equation (6). In this work laminar flamelet data at a strain rate of 200s -1 has been used to express the thermochemical variables in terms of the mixture fraction in polynomial form, p - ~(~),
T-
7~((),
Y,~ - I~'~(().
(9)
Due to the nonlinearity of the relations (9), fluctuations in ~ at the subgrid scale require the use of the sgpdf to determine the resolved scale species mass fractions and temperature. Denoting the density weighted sgpdf of a single conserved scalar b y / 5 ( r x, t), where r is the sample space of ~c, the resolved scale density, for example, can be determined from -1
(/o1
)
(10)
Though there is little, if any, information available on the form of the sgpdf, several LES studies (for example, that of Jim6nez et al.[13]), have established that the/3-function can be used successfully to represent it. The r-function has, without ad hoc modifications, all the properties of
864 a pdf and is defined as ~ a - 1 ( 1 _ ffj)b-1 /1~(~; X, t) -- fo 1 v a - l ( 1 _
V)b-ldv
(e)
where a - ~ ~
- 1 , and_ b -
(11) L~a. Once ~ has been obtained from equation ((6))the
subgrid scale mixture fraction variance ~s~s must be evaluated in order to fix the shape of the ~/-sgpdf. By assuming local equilibrium in the subgrid scales, an estimate for ~s~ may be obtained: (12) This approach is comparable to that employed by Smagorinsky to approximate the subgrid scale turbulent kinetic energy. In a previous study [8], a value for B~ of 0.09 was found to work well and has been employed here. 3. COMPUTATIONAL DETAILS 3.1. Numerical Scheme The filtered flow equations, written in boundary fitted coordinates, were discretised using a finite volume approach. Second order central differences were used for all spatial derivatives, except the convection terms in the filtered mixture fraction transport equation, for which a TVD scheme was used. This was necessary to ensure that the mixture fraction remained bounded in the range [0, 1]. Whilst it is well known that such schemes introduce dissipation near extrema in order to preserve monotonicity, their use in the present context seems unavoidable. For the momentum equations, the second order central difference approximation of the convective terms is energy conserving. A colocated structured grid arrangement has been used with Rhie and Chow[ 14] pressure smoothing. A fully implicit scheme has been employed with a two step second order time accurate approximate factorisation method to ensure mass conservation. The method is described in detail by Branley [15].
3.2. Grid Design and Boundary Conditions A schematic of the computational domain is shown in figure (1), which also shows the geometry of the burner. The domain had dimensions Lz = Lxv -- 10De and was discretised using 793 gridnodes. A constant gridline expansion expansion ratio of %v[z=0 = 1.13 was used in the inlet plane to concentrate gridnodes around the bumer. The geometry of the air and fuel slots was approximated by castellation; 68 gridnodes were used to resolve the fuel slot and 523 the air slot. The value of %v was reduced with distance downstream, having a value of 1.02 at the outflow. In the z direction, an expansion ratio of 7z -- 1.03 was used. Mean inflow velocity profiles were constructed by extrapolating measurements of the mean velocity at z - 3.7ram to the inlet plane in a manner which reproduced the swirl number (0.95), Reynolds numbers (8000 in the fuel stream and 42900 in the air stream) and mass flow rates of the experiment. Instantaneous inflow conditions were generated by adding to this a Gaussian random number field scaled on the turbulence intensity. A convective outflow condition was used for all variables. An inward normal velocity of 0.5ms -1 was specified on the side walls of the domain to approximate entrainment through these boundaries.
865
Figure 1. (a) Schematic of the computational domain. de=26mm, Di=30mm, De=60mm.
(b) Burner geometry; di=20mm,
4. RESULTS The flow was started impulsively at t = 0 and the equations integrated for 12.3Atftow,where Atytowis the flow through time based on the bulk velocity at the outflow. The flow was ensemble averaged over the last 4.1Atytow,by sampling the flow field once per time step for a total of 104,000 steps. The profiles plotted in figures 5 to 7 were obtained by further averaging in the azimuthal direction. All distances are given in metres (m). The centreline variation of mean axial velocity is plotted in figure 2 against the measured values. The predicted zero mean axial velocity contour has been plotted in figure 4; also shown are the locations where the measured mean axial velocity takes a value of zero. From figures 2 and 4 it is clear that a more complex system of recirculation zones has become established in the simulation than that observed experimentally. Downstream of the burner a recirculation zone forms between the two air jets which, in the experiment, is attached to the burner wall. In the simulation however, the fuel jets have been bent away from the centreline only slightly with the result that an additional recirculation zone has formed between them, positioned at the wall of the burner. The reversed flow downstream of this zone is deflected radially outwards by the fuel jets and extends into the region between the fuel and air streams. The measurements suggest that the recirculation zone may extend into the burner, which could enhance the radial spreading of the fuel jets. To simulate this would necessitate extending the computational domain into the burner, which cannot be achieved using the current numerical scheme. Though this may not be the only cause of discrepancy between the present simulation and the measurements, it is clear that the airstream inflow conditions do not encourage the fuel jets to spread and this has a major influence on the shape of the recirculation zones formed. Figure 3 compares the centreline variation of mean turbulent kinetic energy, < k >, with the measured values. Whilst the lack of smoothness in the predictions indicates that a longer integration time is required, the overall shape of the curve is not expected to change with further averaging. Although the variation of < k > on the centreline will be affected by the shape of the recirculation zone, the simulation does reproduce overall the correct level of turbulent kinetic
866 energy. In figure 5 profiles of predicted mean axial, radial and azimuthal velocity components are compared with measured values at various downstream locations. Despite the discrepancies mentioned above, in the near burner region, away from the centreline, the agreement between the predicted mean axial velocity, < Uz >, and the measurements is quite good. The predicted spreading of the air streams is in accord with the measurements upto around 30mm downstream, though the fuel jets have penetrated further into the domain than in the experiment. The swirl component of the predicted mean velocity, < u o > is also in good agreement with the measurements to the same downstream distance. The predicted mean radial velocity, < ur >, however, is in poor agreement with the measurements close to the centreline in the near burner region, as would be expected from the pattern of recirculation zones implied in figures 2 and 4. Downstream of around 30mm all three components of the velocity are in serious error. The axial, azimuthal and radial intensities are shown in figure 6 for various downstream locations. The errors in the predicted mean velocity field will clearly affect the distributions of the turbulent intensites; however, overall levels are reproduced in the simulation at almost all of the downstream measuring stations. The underprediction in < u tz2 > ~1 at 3mm may be due to the rather crude method used in the simulation to generate instantaneous inflow velocity profiles. 1 The underpredictions in < u tr2 > 5, however, are more likely to be due to the incorrect pattern of recirculation zones found in the simulation. ! ! The predictions and measured values of two components of the Reynolds stress, < u z u ,. > and < U z! U to > are presented in figure 7 at several downstream locations. The predictions of t ! < U z U r > show poor agreement throughout; these errors in the radial turbulent transport are ! ! likely to be due to the errors in the mean velocity field. The predicted < U z U o > component of the Reynolds stress, on the other hand, shows fairly good agreement with the measured data upto around 18mm downstream. 5. CONCLUSIONS LES of a swirling nonpremixed methane flame has been performed. The pattern of recirculation zones found in the simulation differs considerably from that found in the experiment. This seems to be due, at least in part, to the failure of the fuel jets to spread radially in the near burner regions. Arguably, this arises because of the inability, in the present work, to simulate negative axial velocity in the airstream at the burner exit. The level of agreement between measurements and predictions is reasonable for mean velocities and intensities in the near burner region away from the centreline, however, and this gives rise to the hope that, if the present difficulty in treating the inflow boundary could be overcome, the quality of the predictions would improve.
Acknowledgements The authors acknowledge gratefully the support of British Gas plc and the EPSRC under grant number GR/L98257.
867 REFERENCES 1. S.B. Pope. Computations of turbulent combustion: Progress and challenges. In TwentyThird Symposiumon Combustion, pages 591-612. The Combustion Institute, 1990. E Gao and E.E. O'Brien. A large-eddy simulation scheme for turbulent reacting flows. Physics of Fluids A, 5(6):1282-1284, 1993. 3. J. Rev6illon and L. Vervisch. Subgrid-scale turbulent micromixing" Dynamic approach. AIAAJournal, 36(3):336-341, March 1998. EJ. Colucci, EA. Jaberi, and E Givi. Filtered density function for Large Eddy Simulation of turbulent reacting flows. Physics of Fluids, 10(2):499-515, 1998. 5. EA. Jaberi and S. James. A dynamic similarity model for large eddy simulation of turbulent combustion. Physics of Fluids, 10(7):1775-1777, July 1998. EE. DesJardin and S.H. Frankel. Large eddy simulation of a nonpremixed reacting jet: Application and assessment of subgrid-scale combustion models. Physics of Fluids, 10(9):2298-2314, September 1998. 7. R.W. Bilger. Turbulent flows with non-premixed reactants. In EA. Libby and EA. Williams, editors, Turbulent Reacting Flows, chapter 3, pages 65-113. Springer-Verlag, 1980. Topics in Applied Physics, Vol. 40. 8. N. Branley and W.P Jones. Large eddy simulation of turbulent non-premixed flame. In Eleventh Symposiumon Turbulent Shear Flows, pages 4.1-4.6, Grenoble, France, September 1997. 9. A. Kremer, E.E Hassel, and J. Janicka. Velocity measurements in a strongly swirling natural gas flame. Engineering Research, 113(1-2):198-211, 1997. 10. A.W. Cook and J.J. Riley. A subgrid model for equilibrium chemistry in turbulent flows. Physics of Fluids, 8(6):2868-2870, 1994. 11. U. Piomelli and J. Liu. Large Eddy Simulation of rotating channel flows using a localized dynamic model. Physics of Fluids, 7(4):839-848, 1995. 12. H. Schmidt and U. Schumann. Coherent structure of the convective boundary layer derived from large eddy simulation. Journal of Fluid Mechanics, 200:511-562, 1989. 13. J. Jim6nez, A. Limin, M.M. Rogers, and EJ. Higuera. A-priori testing of sub-grid models for chemically reacting nonpremixed turbulent shear flows. In E Moin and W.C. Reynolds, editors, Proceedings of the Summer Program 1996, pages 89-110. Center for Turbulence Research, Nasa Ames Research Center, Stanford University, 1996. 14. C.M. Rhie and W.L. Chow. Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAAJournal, 21(11):1525-1532, 1983. 15. N. Branley. Large Eddy Simulation of Nonpremixed Turbulent Flames. PhD thesis, Department of Chemical Engineering, Imperial College, London SW7, 1999. In preparation. .
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Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
871
L a r g e E d d y S i m u l a t i o n of a B l u f f B o d y S t a b i l i s e d F l a m e S. Sello and G. MariottP aENEL Ricerca- Area Generazione, Via Andrea Pisano 120, 56122 Pisa- Italy The paper proposes a LES formulation for the simulation of flows with combustion, based on Favre averaging of LES filtered quantities. The classical Smagorinsky model is employed as sub-grid scale model. A transport equation for a conserved scalar is solved to simulate the mixing of fuel and oxydiser and a fast chemistry assumption is adopted for the representation of the chemical reaction. The model is applied to the analysis of a bluff body stabilised flame, which is actually a quite complex situation even if at relatively small scale. The work shows that the LES approach is already able to provide much better results than conventional turbulence models for the simulation of the mixing process, in particular when it is dominated by the dynamics of the largest turbulence structures. In the reactive case some more investigation is necessary, in particular concerning the level of detail necessary to provide a reasonably realistic model of the chemistry. 1. I N T R O D U C T I O N Numerical simulation has recently received considerable attention as a useful research and designing tool for a better understanding of the physical processes of combustion. The traditional approach to the turbulence and combustion modelling is however not completely satisfactory when non stationary phenomena become important. On the other hand the use of the Large-Eddy Simulation (LES) appears now as a powerful and reliable approach for the prediction of turbulent and transitional reactive flow fields. Indeed, in LES calculations only the large turbulence scale are resolved, where the energy and momentum exchanges are dominant; the small more homogeneous sub-grid scales are modeled, with a lower computational and storage effort than in the direct numerical simulations. In this work preliminary results of LES simulations of reactive compressible flows are presented. The mathematical formulation and its numerical implementation are summarized in the next section. The geometrical and physical configuration investigated as example of application of the code, are presented in the section 3 with a sample of results. Some final comments will be made in Section 4. 2. M A T H E M A T I C A L
MODEL AND NUMERICAL
IMPLEMENTATION
The mathematical formulation implemented in the finite-volume code "C3-LES" is based on a LES approach for compressible flows with the modelization of the anisotropic part of the sub-grid scale Reynolds stress tensor by an eddy viscosity assumption using
872 the classical Smagorinsky model [1]:
(I)
~ ( ~ , ~j) = ( ~ j > ~ - <~)~<~j)~ = -2~7~<s~j)~ where the SGS eddy viscosity is given by:
.7 ~-
(2)
and the magnitude of the large-scale strain rate tensor:
l(O(Ui)F
O{Uj)F)
We note that this choice is physically satisfactory due to the galilean invariance property of the 2-rid order generalized moment ~-F(Ui, uj) [2]. In the classical model the Smagorinsky coefficient Cs assumes a constant value supplied as an input parameter a priori. In (2) 11 is the filter scale lenght. For compressible flows we must pay particular attention to the LES level spatial filtering (with suffix f) and the mass-weighted operator, analog to Favre averaging (with suffix F) defined as [3], [4]: <(I))F =
(3)
In this approach, which allows a great simplification for several closure terms in the equations, we assume as usual that each physical quantity may be decomposed in its filtered part and its subgrid component: O = (q>)y + ~". The sub-grid modeling for the stress tensor allows both the energy dissipation from the large to the small scales and the accurate computation of the large coherent structures of the turbulent flow. For the reactive case of interest in combustion problems, we need also a model for the representation of the chemical reactions. Here we adopted a fast chemistry assumption (equiliobrium chemistry) where a conservative scalar, the mixture fraction ~ , is derived from a filtered transport equation [5]: 0 ((P)I(~)F)0t
+ 0 ((P)I(~)F(Uj)F)oxj
+ 0 (
= 8X~0 ,(PD O(~)F)ozj.
(4)
Here the term pD is assumed invariant to filter operation. As usual, filtering introduces new terms in the equations that have to be modeled. Here the unresolved term is the following (Galilean invariant) generalized central 2-nd order moment [6]:
The SGS model adopted for the scalar { in the present LES formulation is the so called eddy diffusivity model [5], [7]:
~(~, ~) _
_ . } . 0<~/~
Oxi
For compressible flows it is more convenient to define: #sgs (fl)fl:f ~g~--#t~g~/Sct,where tsgs is given by (2) and Sct is the turbulent Schmidt number. Thus the filtered transport equation for the scalar ~ (eq.(4)), assumes the following final form: _
o(<~>:
o(
Oxj
0[(. = Oxj
~ + ~''"
) 0<~>~ Oxj
(5)
873 As noted by Yoshizawa [8], the simultaneous presence of both the Favre average F and the LES filter f is necessary in order to formulate the above filtered equation in a similar way to the incompressible flows case. In the present combined LES - assumed PDF (Probability Density Function) approach for combustion problems, we need a second filtered equation for the transport of the "variance" or, more precisely, for the following generalized central 2-nd order moment: TF(r () -- ( ( 2 } F - (r After long and tedious mathematical steps we derived the following transport equation for T [9]:
0 ((p)r Ot
()) + v ((p):(~)~(r r
v [ ( ~ + . ~ ) w~(r r -
_1/2 _0.489(p))/2 %gs
( ( ( ) + 2psg~ (V(()F)2
(6)
where e~o~ is the subgrid scale dissipation derived by the Smagorinsky assumption e~o~ = -7f(ui, uj)(Sij}F. The solution of eqs. 5-6 allows the derivation of a given termochemical quantity of interest, like temperature, species concentration, etc. when coupled with a proper assumed-PDF: 1
PDFF(() - ~
+(x)
fo
pPDF(p,()dp
(7)
where P D F F ( ( ) is the so-called PDF Favre, here assumed to have a beta-function structure [10]" ca-1 (1 _ r
PDFF ((; (r
a -- (()F
TF((, ()) -
~-F(r r
(8)
B(a,b) .
.
.
.
a
'9
B(a, b) -
r(a + b)
where F is the gamma function. Following Cook and Riley [11] this subgrid-scale PDF contains as first two moments the Favre filtered values: {~)F and TF((, (). It can be noted that this assumed "large Eddy" PDF is necessary also for the closure of the model, since the LES filtered density (p)f is needed at each time step. The above approach has been demonstrated to be both practical and accurate for Large Eddy Simulations of non premixed turbulent reacting flows with equilibrium chemistry [11] The above LES formulation was implemented in the nonstationary finite volume code C3- LES which uses a fractional step scheme as numerical method to the integration of the differential equations. The accuracy of the numerical method is 2-nd order in space for the momentum and 1-st order for the scalar. 3. LES C O M P U T A T I O N
FOR INERT AND REACTIVE
FLOWS
3.1. D e s c r i p t i o n of t h e e x p e r i m e n t After a first application of the code to the study of a simple jet flame, which provided satisfactory results, and which for shortness is not discussed here, it was decided to apply it to more complex geometries, typical of industrial problems. We considered
874 a laboratory-scale axisymmetric flame of methane-air in a non confined bluff-body configuration. Measurement of velocities, turbulent correlations and mixture fraction rate have been performed by Gaz de France and Sandia National Labs both in isothermal and reacting conditions. More precisely the burner consists of a 5.4 mm diameter methane jet located in the center of a 50 mm diameter cylinder (Fig. 1).
MEAN AXIAL MIXTURE FRACTION PROFILE (C3-LES)
IK.~Im 0.7mml 0.6 nmm
0.9
i!tJ /I i)
0.8
0.5
1
E;
0.3 0.2
0.1
_, 52 mm
F 0
, \
,
mmm 0.0~ Streamwmse variable
0.1
,
I methane r 5.4 mm
Figure 2. Axial profile of mixture fraction in the isothermal case. Prediction of k-e and Figure 1. Sketch of the experimental config- LES models against experimental values. uration.
Air is supplied through a 100 mm outer diameter coaxial jet around the 50 mm diameter bluff-body. The Reynolds number of the central jet is 7000 (methane velocity = 21 m/s) whereas the Reynolds number of the coaxial jet is 80000 (air velocity = 25 m/s). The flame power is about 20 kW. The test has already been proposed in the framework of international benchmarks [12], [13], and it has been revealed as a very challenging situation for the traditional turbulence models, like k-c, not only in the reactive case, but also in the isothermal one. In Fig. 2 the results of a k-e computation are compared to the experimental data (in the same figure some results from LES computation are anticipated). The significant discrepancies here shown are very similar to those ones found in previous RANS simulations presented at the ERCOFTAC Workshop and decribed in [12]. Actually the k-c computation predicts a sharp fall of the mean mixture fraction of CH4 in correspondence of the first stagnation point along the axis. The experiments clearly point out the presence of non stationary turbulent structures moving along the axis, which contributes to increase the average methane mixture fraction downstream the stagnation point. This is a typical effect not captured by Reynolds averaged turbulence models.
875 3.2. Setup of the LES Computation In order to allow the free development of the large turbulence structures, the simulation needs to be performed in all the three spatial dimensions. The grid, composed by 235x108 nodes in each azimuthal plane, is uniform towards the axial direction but properly stretched toward the radial direction. Due to constraints in memory and CPU time, the computational region is restricted to a 90 ~ cylindric sector, assuming cyclic conditions at the azimuthal boundaries, and introducing only 7 nodes along the third direction. These last assumptions certainly force the features of the spatial spectrum resolved, in particular in the axis region, but an accurate evaluation of the related error was not performed. To try to overcome the well-known problem of overestimation of the turbulent kinetic energy in the vicinity of the bluff-body for the classical Smagorinsky model, the mesh in this region is refined using a logaritmic law. The filter lenght (1/in (2)) was chosen as I / = ~AxAyAz. At the inlet of the domain, a homogeneous and isotropically turbulent velocity profile based on a given experimental spectral energy behaviour is imposed [9]. In particular, we impose here a cut-off in the range of frequencies at 40 kHz due to consistency with the assumend time step. The magnitude of the inlet axial velocity fluctuations is assumed about 5.5% for the air jet and about 6.3% for the fuel jet. Moreover, for radial and azimuthal directions the magnitude of turbulent fluctuations is confined to about 5% of the above axial component. A totally nonreflecting condition is applied at the outlet:
Ou +
Ou
019
"Tn - o;
0-7 +
Op - o,
where c is the sound speed and un the normal component of the velocity at the outlet. An important topic to address in LES computations is the relevance of the SGS turbulence model with respect to the features of the numerical scheme employed. More precisely, it is very important to establish what is the practical role and the impact of the SGS model in the whole numerical simulation. We verified that if the Smagorinsky constant Cs is set sufficiently small, the trend of the residuals diverges; the presence of the SGS term is therefore essential in our LES simulation to avoid a sharp divergent numerical behaviour. In the simulation presented in this work we use, as Smagorinsky coefficient, the value Cs = 0.05, following a previous extensive parametric computation. 3.3. T h e I n e r t Case The isothermal case, due to the above discussed reasons, is not a simple preliminary item, but is on the contrary a particularly interesting case. The performed LES computation includes about 53000 time steps. The first 15000 are discarded since they are necessary to reach a stationary regime; this was monitored looking at the statistical properties of both the total kinetic energy and the Turbulent Kinetic Energy (TKE) present in the system (the TKE level in stationary regime is about the 10% of the total kinetic energy). All the results shown in this section are therefore obtained by the averaging of the instantaneous values recorded during 38000 time steps. This corresponds to a real time of 0.57 s, which is about two orders of magnitude longer that the time of the dominant period. The velocity axial profile (Fig. 3, left) shows the presence of two stagnation points. The comparison with the experimental data shows good agreement both for the location
876 of the first point and the location of the minimum negative velocity, even if its value is underestimated. The second stagnation point is predicted to be too near to the burner. In Fig. 4 (left) the mean velocity vector field is shown. The flow pattern is correctly predicted, with the presence of two counter-rotating eddies, the first one generated by the methane jet and the second one, larger, generated by the external air. As already anticipated in Fig. 2, the LES simulation predicts a deeper stretching of the CH4 mixture fraction pattern along the axis, with a related spatial increase of the methane concentration, in a better agreement than the k-c model with the experimental measurements. This result can only be explained by the capability of the LES method to capture the vortex shedding occurring in the central region. The almost periodic dynamics of the eddies is well evidenced in a multiscale analysis through a wavelet time series computation of the TKE of the system. Fig. 7 shows a further comparison between LES calculation and experimental data: the mean TKE field in the middle azimuthal plane. The qualitative pattern prediction is quite satisfactory even if in the simulation we detect a significative reduction for the intensities along the radial direction and thus a more intense turbulence activity concentrated along the axis, where the peak is well located. An extensive 2D wavelet analysis ranging from 566 Kc and 8Kc, where K~ is the estimated Kolmogorov scale (here about 0.3 mm), suggests that this lack of turbulence activity in radial direction may be due to a reduced contribution coming from the largest scales of motion. 3.4. T h e R e a c t i v e Case As for the isothermal case, the results here shown are derived by averaging each quantity (except for temperature) through 26000 time steps of computation in statistically stationary regime. In this case this corresponds to 0.4 s of real time. The time simulated is a bit shorter than in the inert case because the nonlinear iterations required inside each time step in order to stabilize the more complex solution lead to a higher need of computational time. It is interesting to note that in this case the level of TKE in stationary regime is about the 7% of the total kinetic energy; in comparison to the isothermal case the turbulence does not increase as much as the total energy. The mean velocity axial profile (Fig. 3, right) shows the disappearence of the two stagnation points detected in the inert case. This is coherent with the experimental measurements, even if the minimum of the velocity is again strongly underestimated. Furthermore, probably due to the flattening of the minimum, there is a lack of the phase shift of the velocity profile, a negative feature of most of the RANS simulations [12]. Fig. 4 (right) shows the mean velocity vector field. The flow pattern is always characterized by the presence of two counter-rotating eddies, but the inner one is now much smaller and compressed at the fuel jet inlet, near the bluff-body wall. Fig. 5 shows the radial profiles of the mean mixture fraction at two axial points, compared with the experimental data. Since the measurements provide the concentrations of CH4, CO, CO2, H2O and 02 chemical species, the concentration of the simplest conserved scalar (nitrogen) was assumed to be given by the difference between the unity and the sum of all measured species. The agreement is quite satisfactory, even if the central jet is more persistent than in the experimental case. In Fig. 6 the axial profiles of the mixture fraction and of its variance and of the
877 temperature are shown. It must be noted that differently from all other quantities, the values for temperature are computed in post-processing, and are therefore not derived from a run-time statistics. They are simply evaluated from the averaged fields of the mixture fraction and of its resolved variance, neglecting the sub-grid values. These plots confirm that the fuel jet penetrates too deeply into the field, leading to a delayed mixing and hence to a delayed combustion. The high value of the temperature peak is due to the assumption of chemical equilibrium, which implies a very fast heat release. Also the comparison between the computed and measured mean TKE field, shown in the Fig. 8, is consistent with the conclusions of the above discussion. Here there is a persistent high level of turbulence activity along the axis. The situation is completely different from that one of the RANS computations, where the peak along the axis is predicted too early and too weak [12]. A 2D wavelet analysis ranging from 566Kc and 8Kc (in this case the estimated Kolmogorov scale is about 0.26 mm) confirms the extended contribution coming from the largest scales of motion for the reactive case. Reasons for the above discrepancies with the experimental data may be various. It seems that fluid-dynamic aspects are dominant as far as the general flow pattern is concerned (penetration of the jet, size of the recirculations zones), while chemical aspects become more important for quantitative aspects (e.g. temperature peak). Also the use of approximated values for some medium properties, which are certainly depending on temperature, may affect the results. 4. C O N C L U S I O N S A comparison of the numerical results with experimental measurements for the above test case, clearly suggests good performances of the C3-LES code for reliable and accurate LES simulations of turbulent isothermal mixing processes. The great advantage of this method, when compared to the results of traditional RANS codes, is that it allows to look at phenomena from a point of view which is qualitatively different. Since the mixing process is one of the critical aspects of the design of modern combustion chambers for lean premixed combustion in gas turbines, it derives that a LES code may be even today already useful as a designing tool. As far as the reactive case is concerned, the agreement between computed and experimental data is not as good as hoped. The discrepancies already present in the inert case are emphasized when moving to the reactive one. This implies that a higher level of accuracy is required to face this kind of problems, involving many aspects of the simulation: SGS model, numerical values of the constants, boundary conditions, computational grid, temperature dependency of many physical properties. Possible advantages of more complete and flexible LES models, like the dynamic eddy viscosity model, should be explored in order to better capture the energy transfer between the various scales. In particular, the SGS approach based on the classical Smagorinsky model, embedded in the framework of the LES-Favre filtering-averaging formalism for compressible reactive flows, needs more attention being a critical aspect for the whole formulation. For that reason a significant effort is still required both from the theoretical and from the numerical point of view to correctly use and interpret the results of Large Eddy Simulations of turbulent reacting flows. A big effort should also be done in order to improve the modelling of some physical
878 aspects of the combustion; the hypothesis of chemical equilibrium represents very often, and even in this case, a very strong approximation of reality. REFERENCES
1. J. Smagorinsky, "General Circulation Experiments with the Primitive Equations. I The Basic Experiment", Monthly Weather Review Vol. 91, (1963). 2. C.G. Speziale, "Galilean Invariance of Subgrid-Scale Stress Models in the Large Eddy Simulation of Turbulence", J. Fluid Mech., Vol. 156, (1985). 3. A. Favre, "Statistical Equations of Turbulent Cases in Problems of Hydrodynamics and Continuum Mechanics", SIAM, Philadelphia, (1969). 4. G. Erlebacher and M. Y. Hussaini, in Large Eddy Simulation of Complex Engineering and Geophysical Problems, B. Galperin and S. A. Orzag, Eds, Cambridge University Press, Cambridge, (1993). 5. P. Moin, K. Squires, W. Cabot, and S. Lee, "A Dynamic Subgrid Scale Model for Compressible Turbulence and Scalar Transport", Phys. Fluids A 3 (11),(1991). 6. M. Germano, "Averaging Invariance of Turbulent Equations and Similar Subgrid Modeling", CTR Manuscript 116 Stanford University, August (1990) 7. G. Erlebacher, M. Y. Hussaini, C. G. Speziale and T. A. Zang, T.A., ICASE Report 90-76, Langley Research Center, NASA, (1990). 8. A. Yoshizawa, "Simplified Statistical Approach to Complex Turbulent Flows and Ensemble- Mean Compressible Turbulence Modeling", Phys. Fluids 7 (12) (1995). 9. S. Sello, "Introduzione di una Formulazione LES nel Codice a Volumi Finiti C3 (C3LES)", Rapporto Topico CISE-SMA-96-18 (1996). 10. R. W. Bilger, "Turbulent Flows with Nonpremixed Reactants", in Turbulent Reacting Flows Topics in Applied Physics, Vol.44, (1980). 11. A. W. Cook and J. J. Riley, "A Subgrid Model for Equilibrium Chemistry in Turbulent Flows, Phys. Fluids, 6 (8), August (1994). 12. EDF-Direction des Etudes et Researches, First A.S.C.F. Workshop Final Results, Chatou, France October 17-18 (1994). 13. D. Garreton and O. Simonin, "Aerodynamics and Steady State Combustion Chambers and Furnaces", Ercoftac Bulletin, June 1995, pp. 29-35. FIGURES 2s [----..~,
25 20
10 5
~i
9
9
0
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.
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.
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Figure 3. Axial profiles of axial velocity in the isothermal (left) and reactive (right) cases.
879
MEAN
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,
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Figure 4. Velocity vector field in isothermal (left) and reactive (right) cases.
O. O.
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Figure 5. Radial profiles of mixture fraction in the reactive case on traverses 2 cm (up) and 4 cm (down) downstream the inlet.
,
400 h = ~ = = p =, o
,
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, - o
/ \
~,~
0.02
"
A
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Axial Coordinate (m)
,ooo
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',
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Figure 6. Axial profile of mixture fraction (solid line) and variance (dashed line) and of temperature in the reactive case.
880
Figure 7. T K E field in the isothermal case: computed values on left, measured on right.
Figure 8. T K E field in the reactive case: computed values on left, measured on right.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
881
I n v e s t i g a t i o n o f the e f f e c t o f t u r b u l e n t f l o w b e h a v i o u r a n d m i x i n g c o n d i t i o n s o n the c o m b u s t i o n p r o c e s s in the h o m o g e n e o u s b u r n o u t z o n e o f a s m a l l s c a l e w o o d heater by numerical simulations and measurements S. Unterberger, H. Knaus, H. Maier, U. Schnell, Klaus R.G. Hein Institute for Process Engineering and Power Plant Technology (IVD), University of Stuttgart, Pfaffenwaldring 23, 70569 Stuttgart, Germany
The main objective of the present work is to investigate the mixing and homogeneous combustion process within the reaction zones of small scale wood heaters with a thermal capacity of up to 15 kW for the purpose of optimising the combustion process when using natural wood logs. For a basic understanding of the formation and decomposition of different gas components and the dependence on mixing intensity, and therefore on the turbulent behaviour of the reacting flow, numerical studies were carried out using the simulation program AIOLOS. A comparison of results obtained during the numerical modelling studies with detailed experimental data shows a good correspondence both for isothermal conditions as well as for combustion. Using the validated numerical model as an engineering tool, parameters such as mixing conditions, combustion air distribution and furnace geometry, and therefore the combustion process in the wood heater, can be optimised for a complete bum-out and for reduced emission of unburned components, such as CO, hydrocarbons and soot. 1 INTRODUCTION One way of reducing the global CO2 emission problem is to substitute wood for fossil fuels in heat and power generation to a certain extent, an option also appropriate for small scale combustion systems for domestic heating purposes. However, the combustion of natural wood logs in small scale stoves with a thermal capacity in the range of up to 15 kW for domestic heating purposes, often involves higher emissions of CO, unburned hydrocarbons and particles. In order to improve the emission behaviour of small scale wood heating appliances a basic understanding of the combustion process is essential. Therefore the project "Newly Designed Wood Burning Systems with Low Emissions and High Efficiency"* [1 ] started in January 1996. The characterisation of the combustion process requires detailed information about gas concentrations and temperatures. In addition, a special interest is taken in the mixing conditions of combustible gases and burnout air and the turbulent behaviour of the reacting flow within the reaction zones of the investigated firing systems. Facilitating the insight into the complex interlinked phenomena of chemical reactions and turbulent flow field behaviours as well as for the investigation of parametric effects, the simulation program AIOLOS was used as an effective tool for detailed analysis of the combustion process. The numerical modelling studies were based on boundary conditions determined by profile measurements of gas concentrations, temperatures, velocities and turbulence intensities within the burnout zone of selected test stoves. This work is partly funded by the European Commission DG XII within the non-nuclear JOULE-III programme under the contract JOR3-CT95-0056
882 2 DESIGN AND COMBUSTION BEHAVIOUR OF THE TEST STOVE A tile stove heating insert with a thermal capacity of 10 kW representing the state of the art of wood log combustion systems was selected as a test stove. The design of the stove and the area investigated by the mentioned profile measurements is shown in Figure 1.
Figure 1.
Investigated test stove, thermal capacity 10 kW
Figure 2.
Course of main gas components in the flue gas
The commercially available stove consists of two spatially separated reaction zones. In the first reaction zone, both the wood fuel is gasified and the gases released by the wood logs start to combust. In the secondary reaction zone, preheated air is added to the gases for almost complete burnout. The injection of the secondary air is realised in two ways: by means of three nozzles before the throat and by a slot in the middle of the throat. One typical characteristic of batch-wise wood log combustion in small scale firing systems with a natural draft chimney is the change of firing conditions during a burn cycle. Figure 2 shows a typical burn cycle of the investigated test stove, characterised by analysis of the main flue gas components. The burn cycle can be divided into three different combustion phases: start-up, main burning and char-burning phase. The start-up phase lasts from reloading of a new batch of wood logs to the maximum CO2 concentration in the flue gas. Due to low temperatures in the reaction zones, low fuel burning rate and a high amount of excess air, the start-up phase involves high emissions of unburned components, such as CO and hydrocarbons. The following main burning phase is characterised by a high CO2 content in the flue gas and low emissions of unburned components. The almost complete combustion can be put down to the fact of increased temperatures in the stove and a lower excess air ratio compared with the conditions during the start-up phase. For a duration of approximately 20 to 30 minutes, the combustion conditions and the flow field within the reaction zones remain nearly constant. The crossover from the main burning phase to the char burning phase is indicated by a rapid increase of the CO emissions. Due to the complete gasification of the wood logs, the combustion changes from homogeneous gas phase reactions to a heterogeneous combustion process involved with a lower fuel burning rate. As a result of a higher amount of excess air at the end of the burn cycle and the associated lower temperatures the concentration of CO increases to very high values.
883
3 EXPERIMENTAL INVESTIGATIONS
3.1 Experimental set-up The experimental test field consisted of measurement devices for gas concentrations, temperatures and velocities. The gas analysis system was designed to measure at the same time the gas concentrations of the main components CO2, CO and 02 in the flue gas as well as in the burnout zone. Additionally H20 and hydrocarbons could be measured either in the flue gas or in the burnout zone. The suction probes for the profile measurements were arranged at different positions in the secondary reaction zone. The arrangement of the measuring points is shown in Figure 3. Profile measurements were carried out by measuring successively at the positions 1 - 11 for a duration of approximately 1 minute per single probe. Ifll
1
111 I11 ill 11/
Ill
1t]1 d illl [!!!1
If
kl III/ gl III/ IT Ill[ I llll I I1 I!i lill ~1 II i l I/
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(nozzles)
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...... ! [ [ I [ !
,/z=....!80. mm
-- ! ! ! [ [// I..... I ....ill II 11[
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t
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t~ / . : . . . : : , :
- ~ : ~ : : : L...............z=-20 .. mm
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.........
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~
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....................z='20 ........ mm 90 m m
~1 I I IT, I f I k LI 3 [I
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Figure 3. Measuring points of: gas concentrations and temperatures (left), velocities at isothermal conditions (centre) and velocities for combustion (right) Gas temperature measurements were done by using Ni-CrNi thermocouples which were installed inside the gas sampling suction probes. The measured gas temperatures were corrected according to a formula obtained by using a suction pyrometer. Wall temperatures were measured by thermocouples at two measuring points in the burnout combustion chamber as well as the temperature of the preheated secondary air before entering the nozzles and the slot. The LaserDoppler Velocimeter (LDV) system used for the flow field characterisation consisted of a 2W argon-ion laser used with a DANTEC/invent 2D FiberFlow system facilitating the simultaneous measurement of two perpendicular velocity components in x- and z-direction, respectively.
3.2 Determination of boundary conditions and profile measurements Boundary conditions of velocities at isothermal conditions as well as boundary conditions of gas composition, temperatures and velocities at combustion conditions were determined at the different inlets of the burnout zone. In order to get reliable gas concentration, temperature and velocity data during combustion, results of several burn cycles were used to calculate mean values. For the non-isothermal LDV measurements a reduced number of measuring points was used in order to scan a single velocity profile within a short period of time due to the instationary combustion process. An overview of the boundary conditions of the different inlets used as basis for the numerical modelling studies is given in Table 1. Concerning the velocity boundary conditions of the burnout zone the influence of the velocity component v on the three dimensional computation of the investigated area was supposed to be negligible due to almost two dimensional, directed flow fields at all inlets.
884
Table 1. Overview of boundary conditions for modelling studies at different conditions Conditions Inlet N~ z[mm] u v w Tu CO CO2 02 CH4 [m/s] [m/s] [m/s] [%] [pbw] [pbw] [pbw] [pbw] Isothermal Burnout 21 -85:-100 -0.25 l) 0 -0.04 45.4 . . . . . . zone 22 -100 :-130 -0.50 l) 0 -0.11 3 9 . 1 . . . . . . 23 -130:-150 -0.351) 0 -0.18 28.4 . . . . . . 24 -150:-180 -0.20 ~) 0 -0.15 27.2 . . . . . . - _
- -
. .
- .
Main combustion phase
t
[oc] 20 20 20 20
Nozzles
3
-90
2.38
0
0.34
7.0
. . . . . .
Slot
41 42 43 44 45
-35 : -27 -27:-25 -25:-24 -24:-23 -23:-22
0.26 2.07 1.89 1.89 1.42
0 0 0 0 0
0.1 0.8 1.3 1.3 1.3
7.0 7.0 7.0 7.0 7.0
. . . . .
Burnout zone
21 22 23 24
-85:-100 -100 : -130 -130:-150 -150:-180
-1.252) -2.252) -1.002) -0.252)
0 0 0 0
-0.04 -0.11 -0.18 -0.15
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0.72
1020
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0.0
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200
. . . . .
. . . . .
. . . . .
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. . . . .
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- -
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Slot
41 -35 :-27 0.16 0 0.11 7.0 42 -27:-25 0.87 0 0.58 7.0 43 -25:-24 1.60 0 1.18 7.0 0.0 0.0 23.3 0.0 200 44 -24:-23 1.97 0 1.65 7.0 45 -23:-22 1.08 0 0.94 7.0 ,~,2~ averaged values, used boundary conditions are defined by different functions for isothermal and combustion conditions taking into account the measured velocity profile in the inlet area of the burnout zone For the modelling studies at combustion conditions the temperature of the surrounding walls was set to an uniform value of twail = 600 ~ Facilitating a comparison of measured and calculated results velocity profile measurements were carried out at isothermal conditions and for combustion tests, respectively. Additionally gas concentration and temperature profile measurements at different positions in the burnout zone according to Figure 3 were carried out during several burn cycles.
4 NUMERICAL MODELLING STUDIES The present calculations were performed with the finite-volume code A I O L O S [2]. This simulation program has been developed for the numerical calculation of stationary, turbulent reacting flows. The SIMPLEC-method [3] is applied to compute the pressure. The pressureweighted interpolation method (PWIM) [4] is used to prevent the decoupling of velocities and pressure on the non-staggered grid. The convection fluxes are approximated by the upwind or higher-order scheme.
4.1 Transport equations and turbulence modelling The conservation equations of mass, momentum, scalar quantities and turbulence quantities and their discretisation are solved, presupposing high Reynolds-numbers and steady-state flow conditions. It is assumed that the flow field is weakly compressible which means that the density depends only on temperature and fluid composition but not on pressure.
885
4.1.1 Finite volume discretisation A body-fitted grid is used to represent the secondary reaction zone of the investigated test stove. Therefore the conservation equations have to be formulated in general curvilinear coordinates. In this paper, the formula proposed by Peric [5] is used which provides a strong conservative form of the conservation equations. 4.1.2 k,e-turbulence model The k,e-model is based on a first order turbulence model closure according to Boussinesq. In analogy to laminar flows, the Reynolds stresses are assumed to be proportional to the gradients of the mean velocities. Transport equations for the turbulent kinetic energy and the turbulent dissipation are developed from the Navier-Stokes equations assuming an isotropic turbulence. The implementation of this model and the parameters used can be found in [6]. 4.1.3 Differential Reynolds-stress model (RSM) Flows with non-isotropic turbulence cannot adequately be described by a two-equation model like the k-~ model. Under such flow conditions, turbulence should be described by a differential equation for each Reynolds-stress component to get a realistic representation of the stress anisotropy. The second moment closure adopted here is that of Schneider [7] which was extended to a general co-ordinate system by Knaus [8]. 4.2 Kinetic model The combustible volatiles are converted by the following reactions {C 1} - {C3 }:
{C1}" CmH n + [ 2 + c t -n] -02 4 1 {C21" CO + ~ O 2 --~ CO 2
-~ m C O + [ 2 +or 4 ] H 2 + ~ -n- H 2 0 2
1 {C3}" H 2 + ~ O 2 --), H 2 0 z
The distribution of water and hydrogen as products of the hydrocarbon combustion described by reaction {C1 } is calculated from the water-gas-shift equilibrium. All reactions are treated as irreversible reactions and the kinetic rates of the reactions {C 1} - {C3 } are taken from [9, 10]. The Eddy Dissipation Concept (EDC) is used for treating the interaction between turbulence and chemistry in flames [ 11 ]. The method is based on a detailed description of the dissipation of turbulent eddies. In the EDC the total space is subdivided into a reaction space, called the 'fine structures' and the surrounding fluid. In the presented reaction scheme the reactions {C 1}, {C2}, and {C3} are treated as taking place only in these fine structures, i.e. only on the smallest turbulent length scales. 4.3 Heat transfer The discrete ordinates method in a S4-approximation is used to solve the radiation transport equation. Since the intensity of radiation depends on absorption, emission and scattering characteristics of the medium passed through, a detailed representation of the radiative properties of a gas mixture would be very complex and currently beyond the scope of a 3D-code for the simulation of industrial combustion systems. Thus, contributing to the numerical efficiency, some simplifications are introduced, even at the loss of some accuracy. The absorption coefficient of the gas phase is assumed to have a constant value of 0.2/m. The wall emissivity was set to 0.65 for the ceramic walls and to a value of 0.15 for the glass pane inserted in one side wall for optical access.
886
5 COMPARISON OF EXPERIMENTAL RESULTS AND SIMULATION STUDIES A comparison of experimental data obtained by profile measurements in the burnout zone with simulation results carried out on a body fitted grid consisting of 59.900 nodes was done. 5.1 Isothermal conditions The flow field in the burnout zone of the test stove was investigated at isothermal conditions by LDV measurements and numerical simulations using the k,s-model and the Reynolds-stress model. Figure 4 shows the comparison of the calculated flow fields with measurement data. The shown profiles are situated in the centre of the burnout zone.
Figure 4.
Calculated and measured flow field at isothermal conditions
The characteristics of the secondary air injection by the slot and the three nozzles as well as the location of the recirculation zone in the upper part of the burnout zone are well predicted. However, the size of the recirculation zone is predicted too small and the main flow is not as much deflected to the wall on the right hand side as shown in the measurements. Besides the more global representation of the results above, detailed comparisons of simulation results with velocity measurements at different positions are shown in the following Figures 5 and 6. For the location of the different measuring positions see Figure 3. 2 "~ .=_ 1.5
2-
k-epsilon ............... RSM zx me,as.
E
"~ 1.5 t,'-
zx
8.
'-
"
0
8 o.5 Z~ ~>
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A -0.5
-60
,
i
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Figure 5.
i
-20
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.,,. •
o
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,
i
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,
,
,
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,
,
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'
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,
,
,
,
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-0.5
0
1'0 2'0
3'0 4'0
x in mm
5'0
6'0
Calculated and measured velocities at different positions in the burnout zone (left: z = 100 mm, right: z =-20 mm)
The comparison of the velocity values in z-direction shows a good correspondence between calculated and measured results. The flow field characteristics in areas with high velocity gradients, for example the secondary air injection at the slot (z = -20 mm), are qualitatively well
887
predicted. However, deviations between calculated and measured absolute velocity values are found. The differences for the calculated velocity fields using the k,e- and the Reynolds-stress turbulence model are negligible fo~' the presented profiles at isothermal conditions. 0.5 l"
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I
60
Figure 6. Calculated and measured RMS values at different positions in the bumout zone (left: z = 100 mm, right" z = -20 mm) The comparison of RMS values calculated by using different turbulence models and measured data (Figure 6) shows more fundamental differences as mentioned for the velocity profiles. The RMS values for the k,e-model are computed from the turbulent kinetic energy k assuming isotropic turbulence conditions and therefore, they are the same for the x- and z-directions. The measured RMS values in the two directions differ up to a factor of two leading to a relative poor prediction of the RMS values by the k,e-model. For the RSM the values are qualitatively in good agreement with the measured ones. The absolute RMS values are too small in areas where the maximum mean velocity values occur. This corresponds to the underpredicted mean velocity values in these areas.
5.2 Combustion conditions Further experimental and numerical investigations during combustion were carried out in order to get a basic understanding of the flow field behaviour and of the formation and decomposition processes of different gas components. The experimental velocity data shown in the following figures are obtained during two reference burn cycles BC1 and BC2. The profile measurements were carried out during the main burning phase in a period between 20 and 30 minutes after reloading a new batch of wood logs. 6
k-epsilon ........... R S M /k meas.,BC1 V meas. BC2
"~ 5 !:: ._c ~) c-
4
8. 3 E
-
k-epsilon ........... " RSM /k meas.,BC1 V meas.,BC2
o0
~5
._c
~4 E
F
v
' o............ ; 2'0 ,;o :2Z
F igure 7.
x in mm
~z
R E 3 8 ~'2 8
V
V
U
& ~ ~o ~o
0~0 -10 ,
0,
1,0~,0
x in mm
, 30
~,0
, 50
, 60
Calculated and measured velocities at different profiles in the burnout zone during combustion (left: z = 180 mm, right: z = -90 mm)
888 In Figure 7 a comparison of calculated and measured velocities is presented in the same way as for isothermal investigations. The characteristics of the measured velocity profiles are qualitatively well predicted. The differences between calculated and measured data are similar to the investigations at isothermal conditions. The maximum values of the mean velocities are underestimated by both the k,e-model and the RSM. In a further step the modelling studies for combustion conditions were used for obtaining information about the mixing quality of the burnout air distribution by the slot and the three nozzles. Figure 8 shows results of the mixing ratio between combustible gases and burnout air calculated in the centre of the burnout zone. The penetration depth of the burnout air injected by the nozzles is approximately half of the combustion chamber width combined with locally high
mixing intensity. Figure 8. Calculated mixing ratio between combustible gases and burnout air (left: k,e-model, right: RSM) The lower output momentum of the burnout air by the slot which is due to the relatively large outlet area results in a lower penetration depth and a lower mixing intensity. The injected air is deflected without complete mixing to the left hand side of the burnout zone. Therefore, a streak is formed on the right hand side of the burnout zone, retaining approximately the initial concentration. The characterisation of the combustion process within the burnout zone of the test stove is done by studying the measured and calculated gas concentration and temperature fields. In Figure 9a and 10a contour plots are shown for the computed CO-concentrations and temperatures using the Reynolds-stress turbulence model. Additionally, the positions for gas and temperature measurements are depicted. For these positions, measured values are opposed to the computed values in Figures 9b and 10b. The comparison of the numerical simulations with measurement data shows that the characteristics of the computed gas concentration and temperature fields are well in accordance with the measurement data. The CO-concentration field, computed in the middle of the burnout zone shows a rapid reduction of the initially high values due to the injection of secondary air by the slot and the nozzles and the high temperatures within this part of the burnout zone.
889
Figure 9a.
Calculated concentration field of CO (RSM)
Figure 9b. Comparison of measured and calculated CO-concentrations
Related to the asymmetric arrangement of the air distribution and therefore by the insufficient mixing quality the high CO-content of the streak at the right hand side of the burnout chamber can not be reduced while passing the throat of the burnout zone. In the upper part of the burnout zone lower temperatures and a reduced mixing intensity prevent a further reduction of remaining CO.
Figure 10a. Calculated temperature field (RSM)
Figure 10b. Comparison of measured and calculated temperatures
The fluctuations of measured temperatures at the different measuring positions are due to changes in the flow field and combustion conditions in the burnout zone depending on the burn course and on different boundary conditions. Differences between measured and calculated data, especially for the temperature measurements, can be put down to the fact that the predicted size of the recirculation zone formed in the upper left part of the burnout zone is too small, as already mentioned during the discussion of the isothermal investigations. The affected measuring points are the positions 1, 2, and 3, which are also the positions where the biggest differences between measured and computed temperatures are found. Furthermore, taking into account the strong influence of the wall temperature on the heat transfer calculation and therefore on the temperature field and burnout of the gasification gases the assumption to use a uniform wall is involved with uncertainties. Wall temperature measurements at numerous additional positions facilitating the definition of a wall temperature distribution for the surrounding walls could improve the numerical results. However, the experimental determination of gas and wall temperatures is often involved with uncertainties and measuring errors.
890 6 SUMMARY The turbulent flow and homogeneous combustion process within the burnout zone of a commercially available tile stove heating insert have been investigated. Numerical modelling studies on the basis of detailed experimental boundary conditions were carried out in order to characterise the flow field at isothermal conditions as well as the gas concentration, temperature and flow field in the reaction zone for combustion of natural wood logs. A good accordance between the computed results and measurement data was found. However, differences between theoretical and experimental data related to the absolute values might be reduced by using a more detailed grid with an increased number of nodes facilitating the reproduction of the existing gradients of the gas concentrations, temperature and velocities. The use of higher order discretisation schemes, for example QUICK, MLU, could also contribute to a further improvement. The developed numerical model can be used as an engineering tool for optimising the air distribution arrangement and therefore the combustion process within the investigated test stove.
REFERENCES
[ 1]
Maier, H., Unterberger, S., Struschka, M., Baumbach, G., Hein, Klaus R.G.: Joint European Project: Development of newly designed wood burning systems with low emissions and high efficiency, 1st European Conference on Small Burner Technology and Heating Equipment, Ztirich, 25-26 September 1996 [2] Schnell, U.; Schneider, R.; Magel, H.-C.; Risio, B.; Lepper, J.; Hein, K.R.G.: Numerical Simulation of Advanced Coal-fired Combustion Systems with in-furnace NOx Control Technologies. 3rd Internat. Conference on Combustion Technologies for a Clean Environment, Lisbon, 3-6 July 1995 [3] Van Doormal, J.P., Raithby, G.D.: Enhancements of the SIMPLE Method for Predicting Incompressible Fluid Flows. Numerical Heat Transfer Vol. 7 (1984), S. 147-163 [4] Rhie, C.M., Chow, W. L.: Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Sparation, AIAA Journal, Vol 21, Nr. 11 (1983), S. 1525-1532 [5] Peric, M.A.: A Finite Volume Method for the Prediction of Three-Dimensional Fluid Flow in Complex Ducts. Ph.D. Thesis, Mech. Eng. Dept., Imperial College of Science and Technology, London, 1985 [6] Launder, B.E., Spalding, D.B.: The Numerical Computation of Turbulent Flows.,Comp. Meth. Appl. Mech. Engrg., Vol. 3 (1974), S. 269-289 [7] Schneider, R., Risio, B., G6rres, J., Schnell, U., Hein, K.R.G: Application of a Differen-tial Reynolds-Stress Turbulence Model to the Numerical Simulation of Coal-Fired Utility Boilers. 3rd International Symposium on Coal Combustion, 1996, Beijing, China [8] Knaus, H., F6rtsch, D., Schnell, U., Hein, K.R.G.: Vergleich von Simulations-ergebnissen mit Messungen aus einem 252 MW Kraftwerk zur Validierung mathematischer Feurraummodelle, VDI Bericht 1390, 1998, S. 29 - 41 [9] Zimont, V.L., Trushin, Yu.M.: Total Combustion Kinetics of Hydrocarbon Fuels. Combustion, Explosion, and Shock Waves, Vol. 5, Nr. 4 (1969), S. 391-394 [10] Howard, J.B., Williams, G.C., Fine, D.M.: Kinetics of Carbon Monoxide Oxidation in Postflame Gases. 14th Symposium (Int.), The Combustion Institute, 1972, S. 975-986 [11] Magnussen, B.F., "The Eddy Dissipation Concept". XI Task Leaders Meeting -Energy Conversion in Combustion, IEA, 1989
13. Two-Phase Flows
This Page Intentionally Left Blank
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
893
P r o p o s a l of a R e y n o l d s Stress M o d e l for G a s - P a r t i c l e Turbulent Flows and its A p p l i c a t i o n to C y l o n e Separators Osamu KITAMURA and Makoto YAMAMOTO Mechanical Engineering Dep&rtment, Science University of Tokyo 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, JAPAN
Abstract
A Reynolds stress model for predicting gas-particle two phase turbulent flows has been developed in order to take account of interactions between dispersed particles and fluid turbulence. The effects of dispersed particles on fluid turbulence are added to the Reynolds stress model for single-phase turbulent flows. The model coefficients are determined based on the experimental data of the gas-particle round jet and the fully-developed vertical pipe flow. The proposed model can satisfactorily reproduce the mean flow and turbulent properties of the gas-particle round jet, the fully-developed pipe flow and the swirling flow. Finally, the model is applied to the gas-particle highly swirling flow in the cyclone separator in order to predict the particle collection performance. It is shown that the present model can reproduce the flow field, whereas it slightly underestimates the p&rticle collection efficiency for fine particles.
1. I n t r o d u c t i o n
The cyclone separator is a device for separating solid particles from contaminated gas streams, and ha,s long been used in industrial applications such as power generations, gas turbines, chemical processes and so forth. The highly swirling flow field inside a cyclone is very complex, including the interactions between the particulate and fluid phases. Although a lot of empirical theories for estimating the performance of cyclones have been developed so far [1], these models strongly depend on the experiments that were used to develop the model equations and are not successful for a, wide va,riety of cyclones. Some numerical studies for predicting the cyclone performance have been also carried out recently [1,2]. However, the interactions between the particulate and fluid phases were not considered in these approaches, and that leads to poor predictions for cases with moder&te and high particle concentrations. On the other hand, it has been experimentally reported that fluid turbulence is strongly attenuated by the presence of particles [3,4]. A lot of gas-particle turbulence models based on both Eulerian and Lagrangian approaches have been proposed to predict the phenomenon of turbulence modulation for the last 15 years [5-9]. However, almost all the investigations have adopted the k-e model with the isotropic eddy viscosity assumption for the fluid turbulence, which should be inappropriate for highly swirling turbulent flows in cyclone separators. The purpose of the present study is to simulate the gas-particle highly swirling flow in the cyclone separator and numerically predict the particle collection performance. First, a Reynolds stress model is developed for predicting the gas-particle turbulent flows in the
894 Eulerian formulation. The effects of particles on fluid turbulence are taken into account in the Reynolds stress model. The model constants are optimized, using the experimental data, of the gas-particle round jet and the fully-developed vertical pipe flow. Then the model performance is verified for the gas-particle swirling flow. Finally, the proposed model is applied to the highly swirling turbulent flow in the cyclone separator in order to predict the flow field and the collection performance. It is shown that the flow field can be a,ccurately reproduced, whereas the collection efficiency for fine particles is underestimated by the present model.
2. Governing Equations In the present study, the following assumptions are made for predicting gas-particle turbulent flows. (1) The particle phase is dilute enough (the particle phase volume fraction Cp << 1) to be treated as a continuum and comprised of the mono-dispersed particle size distribution. Thus, particle-particle collisions are negligible, but two-way interactions between the fluid phase and the particle phase should be considered. (2) The fluid flow is incompressible and fully turbulent. The effects of Brownian motion and molecular diffusion on the particle phase are neglected in comparison with turbulent diffusion. (3) The particle density is much larger than the fluid one. Therefore, the drag force due to momentum exchanges between both phases is influential. The virtual mass force, the Basset force and forces due to the pressure gradient in the surrounding fluid are neglected. (4) The triple correlations containing particle concentration fluctuations are negligible. The governing equations for mean motions are described in the Eulerian formulation as follows. The fluid phase" OUi -0 (1) Oxi lop 02Ui . . . ,00xi . . ~- U--Ox2
0 -jOx (UjUi)
0 Oxj tti~tj
1(
~pT.; pc(Ui - Vi) -nt- ptc('tti - vi)
)
(2)
The particle phase" Ox,
0
0
Ox---~(pcViVj)--~--~xj(tOc~)--~
~zj
+
-
(3)
o
0
1(
(ViYcVj)-nt-~xj(VjycVi) - -~p flc(Ui-Vi)--~fltc(tti - vi)
) (4)
as follows. where rp' is given by the characteristic time of particles rp(- ppdp/18pu) 2 !
24
,
_=
d,lu
7"p -- rp RepCD
-
(5)
u
The Reynolds number for particles R % is defined by the relative velocity between the fluid and the particle. The drag coefficient Cz) is given by 24 (1 + 0.1 5 R (tO~.678 ) CD---~%
(6)
which is based on Stoke's drag law. In the above equations, Ui and Vi denote the Reynolds-averaged velocity components
895 of the fluid and the particle phases, respectively. P is the mean pressure, u and p are the kinematic viscosity and density of the fluid phase, dp, pp and pc are the particle diameter, density and m e a n concentration of the particle phase, ui, vi and P'c denote the fluctuating components of each variable. The Reynolds stresses uiuj, vivj and the correlations Ycui, fllcViare unknown variables which should be modeled. Modeling of these terms is presented in the following section.
3. Turbulence M o d e l i n g 3.1 F l u i d - P h a s e Turbulence In the present study, the Reynolds stress model proposed by Gibson and Launder [10] is adopted for the fluid turbulence. This is standard in a single-phase turbulent flow. The transport equations for the Reynolds stresses uiuj and the dissipation rate r are expressed as follows.
o~
~
c~~--Ef-x,]~ ouj
. . . Pu . -'uiu;~k
+ P~J + ~ J -
5 ~j~ +
'~;,~
oui
u~j- - - 7Ox~ --
(8)
(~ij = (I)ij(1) Jr- (I)ij(2) na (~ij(lw) -~- (~ij(2w)
r
2
(I)ij(1) -- _C1-~('lti'tt j - ~r
(9)
]c )
(~o)
2 r
--
3
3
k 3/2
3
C2(Okm(2)nkrtmSij -- -~Oki(2)nknj -- -~Okj(2)nkni)
0 (Ukr
t~X k
(11)
3
,
Oij(2w)
~
0 (
kOc)
Ce lt l lt m -e l~x l
(7)
c
k 3/2
Clxle
c2
(14)
-JI- Ce l --~P -- Ce 2 --~ -Jl--Se
C 1 -1.8, C 2 - 0 . 6 , C ~ - 0.5, C ; - 0.3, C h - 0 . 2 2 , C e - 0 . 1 8 , C e l - 1.44, C ~ 2 - 1.92, C t - 2.5
(15)
where Pij is the production of each Reynolds stress, (~ij is the redistribution of Reynolds stresses, and xl means the normal distance from a wall. @,j and ,b~ are the source terms arising from interactions between the fluid phase and the particle phase. These terms are exactly expressed by Spi:l __
(16)
tic {,tti(,ttj _ Vj) Jr-'Uj(tti - Yi)}
p~-p
1 {Oui O p~p Oxk Oxk which need to be modeled further.
c%i 0
o~ o~ (p,~(u~- ~))
}
(~7)
896 3.2 F l u i d - a n d P a r t i c l e - P h a s e T u r b u l e n c e I n t e r a c t i o n
Modeling of the source effects of particles on fluid turbulence is based on the nonisotropic expression of the source model of Chen and Wood [6]. 1 7-pc 5'pi ~ =
p~.p
The source term in the c equation is modeled by c
1
where C~p is a model constant. 3.3 P a r t i c l e - P h a s e T u r b u l e n c e
The Reynolds stresses for the particle phase Vi~)j a r e modeled by extending Boussinesq's eddy viscosity approximation because it is numerically stable to calculate using that expression. -
+
-
+
(20)
Here, the model proposed by the Chen and Wood [6] is adopted for expressing the turbulent diffusivity for the particle phase. 1 r'p -- Ut l -t- 7 p c / C t k
'
r,t -
C,- c
-2 ui
, C, -
0.09
(2 1)
where ut denotes the eddy viscosity for the fluid phase, and Ct is a model constant. The correlations of particle concentration fluctuations are given by introducing the assumption of gradient diffusion as follows. Pc i -
lit Opc
(22)
(7t Ox i up Opt
O'p Ox i
where Schumidt numbers at and cr~ take the following values, respectively. o't = 0.7, o'~, = 0.7
(24)
In the above model equations, two constants C~p and Ct remain unknown. These empirical constants were determined through numerical optimization to obtain good agreement with the experimental data of the gas-particle round jet of Modarress et al.[ll] and the fully-developed vertical pipe flow of Tsuji et al.[12] as described below. C~p - 1.0, Ct - 0.13
(25)
4. N u m e r i c a l P r o c e d u r e
The gas-particle round jet, fully-developed pipe flow and swirling flows including cyclone flows were calculated to evaluate the turbulence model performance. The governing
897 equations for the round jet were simplified with the boundary-layer approximation, and then solved parabolically. The other flows were solved elliptically by the explicit time marching scheme, i.e. MAC method. The third-order upwind differencing scheme [13] was used for convection terms, considering numerical stability and accuracy. The other terms were discretized by the second-order central differencing scheme. The wall functions used for single-phase turbulent flow calculations were adopted for fluid-phase turbulent quantities at the wall boundaries. For the particle-phase, slip conditons were imposed on the velocity components at the walls. 5. Results and Discussion 5.1 R o u n d Jet First, the gas-particle round jet flow was calculated to determine the model coefficients C~p and Ct. Our predictions are compared with the experimental da.ta of Modarress et al.[ll]. Table 1 lists the experimental conditions in which the mass loading # is defined as the mass flow rate of particles normalized by that of gas. 50 grid nodes in the lateral direction were used to obtain grid-independent solutions. Fig.l(a) compares the predicted and measured mean velocity decays along the centerline. The present model can satisfactorily reproduce the mean velocity decays of both phases for Case 1. The predictions also indicate that the reduction of mean velocity decays becomes more remarkable with the increase of the mass loading. In Fig.l(b), mean velocity profiles of the downstream position x/D=20 for Case 2 are plotted. It is clear that the particle velocity is higher than the gas velocity due to the inertia effect of particles. Figs.l(c) and (d) show the turbulent shear stresses and the streamwise velocity fluctuations, respectively. The Reynolds stresses are slightly overpredicted for the single-phase case using the Reynolds stress model. However, the decrease in the turbulent quantities, which is caused by the presence of particles, is well represented by the present model. 5.2 P i p e Flow In this section, the numerical results for the fully-developed vertical pipe flow measured by Tsuji et al.[12] are presented. The experimental conditions are shown in Table 2. 60 grid nodes were assigned in the radial direction, after it was confirmed that the 60-node grid results are virtually grid-independent. Fig.2(a) shows the calculated and measured streamwise mean velocities of the fluid phase for Case 1. The gas velocity for the two-phase case is smaller than the single-phase velocity since the gas phase is dragged by the particle phase due to the gravitational effect on particles. The agreement between the predictions and the experimental data is reasonable. The effect of the mass loading on the gas turbulence intensities is shown in Fig.2(b). This figure indicates that the addition of particles decreases turbulence intensities in the whole region, and this effect is enhanced with increasing the mass loading. It could be possible through a parametric study to get better agreement with the experimental data of the turbulence intensities by lowering the value of C~p. However, the modification of the model constant leads to much worse results for the round jet cases mentioned in the previous section. Therefore, it was concluded that the combination of the model constants C~p=l.0 and Ct=0.13 is the best possible choice in the present model. 5.3 Swirling Flow The gas-particle swirling flow for which detailed experimental data are available by Sommerfeld et al.[14] was calculated to evaluate the present model performance. Fig.3 shows the flow configuration where the central primary jet is loaded with particles and
898 Table 1 Experimental conditions for the round jet (Modarress et al. [11]) dp Case
1
dp
Pp
(~.~) (]~g/m 3)
Re
1.68 x 104
#
2990
5O
2 '
1
'
=l,~.
Table 2 Experimental conditions for the vertical pipe flow (Tsuji et al. [12])
I
'
1
'
Case
0.32
1
3.3
x 104
0.85
2
2.3
x 104
I
0.8
_
I
.
.
.
.
IIl~i..-~ 1 ~
0
~"-..,.~!
O &
single phase gas phase
9 ....... ,
particle phase
I
o
~b
'
'
i
'
cal.
0 - --
single phase
o---
gasphase
9 .......
particle phase ( . = 0.85)
O
x/D = 20
'
2.1
0.4
= 0 ~,
" ~ ~ ~
#
1020
I
exp.
I
\
)
0.5/1.3
~::k~
0.5
(kg/rr/3
200
-,
:::2
Pp
(#m)
Re
2b
"""
o
a'o
o:1
,/~
o12
z/V
(b) Streamwise mean velocity
(a) Centerline mean velocity decays ~2
0.03
.
.
.
.
!
z/D = 20
.
.
.
.
i
]
i
x /` ~u
0'4"]
exp. cal. 0
v, = 0.0
A
~ = 0.32
0.02
= z u l n ' '
.
exp.
.
.
!
~:
0--:~
.
cal.
u=o.o .=0.32
A---
i .-]
0.2
!
0.01
[
0
,
i
|
,
0'.1 ' ,,/=. . . .
012
o 6
9
,
,
i
o~ I
i
i
i
,/,
,
o12
(d) Streamwise velocity fluctuation (c) Turbulent shear stress Fig.1 Computational results of the round jet (Symbols" Modarress et al. [11]) |
.
.
.
.
I
.
.
.
.
I
0.15
t
.
.
.
i
.
.
.
.
.
I
exp. cal.
@. 0.1
0
. --- o.0
A
.=0.5
I-1 . . . . .
.=1.3
0.5 exp. cal. f-I ~
I
I
9 ....... gas phase
o [
,
L
i
,
o'5
0.05
I
single phase
I L
i
,/~
~
,
4
~.-~--" []
i
0-
I
0
~,
[]
.
.
[]
.
.
[]
D
''
dp = 200.m
|
0.5
.
.
.
.
,-/R
t
1
(b) Streamwise turbulence intensity (a) Streamwise mean velocity Fig.2 Computational results for the vertical pipe flow (Symbols 9Tsuji et al. [12])
899 the annular jet provides the swirling flow. The flow conditions and particle properties are given in Table 3. The Reynolds number is based on the nozzle diameter and the bulk velocity at the nozzle exit. The computational region extends to 5D downstream from the nozzle exit. The grid-independent solutions were obtained on the grid of 61 x61 nodes because it was confirmed that increasing the nodes from 51 x51 to 61 • makes very little differences in predicted axial and tangential velocities. Fig.4 shows the gas-phase mean velocity vectors. The central recirculation bubble as well as the corner one can be reproduced, which quantitatively corresponds to the experimental results of Sommerfeld et al.[14]. The development of the decaying mean velocities for the gas and the particle phases is compared with the experiments in Fig.5, indicating that the particles lag behind the gas phase due to the inertia effect on particles. The numerical results in comparison with the measurements at six locations in the axial direction are presented in Figs.6(a)'~(d). The agreement of the predicted axial and tangential velocities for both phases with the experimental data is fairly good as shown in Figs.6(a) and (b), although it is seen in Fig.6(c) that the gas-phase turbulence intensities are slightly underpredicted. Fig.6(d) depicts the development of the particle mass flux, which show reasonable agreement with the experiments.
5.4 Cyclone Separator Finally, the proposed model was applied to the gas-particle highly swirling flows in several cyclone separators. Fig.7 illustrates the swirling flow field in the cyclone chamber. As the gas enters through the inlet, it is subjected to a swirling motion. After the flow reaches the bottom part of the chamber, it reverses its direction and swirls upward to exit from the upper part. Dust particles included in the swirling gas flow are separated under the influence of centrifugal forces. The cyclone geometries and operating conditions are summarized in Table 4. Case 1 refers to the cyclone for which mean and fluctuating velocities of both axial and tangential components are measured by Boysan et a1.[15], while Case 2 refers to the cyclone for which collection efficiency data for dust particles are available by Stairmand [16]. The Reynolds numbers based on the inlet bulk velocity and the cyclone diameter are 4.00x10 s and 2.06x10 s for Casel and Case 2, respectively. The particle material used in Case 2 is silica, sand with its density being 2000kg/m a. The swirling flow for Case 1 was previously calculated in order to validate the present Table 3 Experimental conditions for the swirling flow (Sommerfeld et al. [14]) Gas flow Mass flow rate of primary jet Mass flow rate of secondary jet Inlet Reynolds number Swirl number
Particle phase Particle mass flow rate Particle mass loading Particle properties Particle diameter Particle material density
9.9 g/.s 38.3 g/s 5.24 • 104 0.47
0.34 g/.s 0.034 45 #m 2500 kg/m 3
Table 4 Cyclone geometries and operating conditions
Case
D
2
200.0 203.2
D~
Dimensions (ram) Do H h
100.0 7 2 . 0 800.0 101.6 7 6 . 2 812.8
300.0 304.8
Qin S
a
b
100.0 100.0 40.0 101.6 101.6 40.6
(.~/~)
Data
0.120 0.062
Boysan et al. [15]
Stairmand [16]
900
Fig.3 Schematic of the swirling flow
Fig.4 Gas-phase mean velocity vectors ~t
Fig.5 Centerline mean velocity decays
~to
%-.
%-.
0~t
oO 9
-
,0!
tc
~t1~
(a) Gas-phase mean velocity
~~
,0t
'
"t~
'
to'~~176176 ~' r (m)
., (.'1
~
(b) Particle-phase mean velocity
0.r ~
r (,n)
l~ I ~..._g.~.~_D_g_ ooo
1
- ~ ; (,,,)
!!!
05
't
!
'i Io:~~"I
0~~
~ ~ \ i
~;~., 2_- o.;hm
i
(c) Gas-phase turbulence intensity (d) Particle mass flux Fig.6 Comparison of computational results with the experimental data
(Symbols" Sommerfeld et al. [14])
oI~
901 turbulence model. Fig.8 compares the computational results obtained with 100x110 grid with the experimental data. for Case 1. The agreement is satisfactory. In particular, the non-linear change in tangential velocity components can be accurately reproduced using the Reynolds stress model. Fig.9 shows radial profiles of particle mean concentrations for Case 2. It is evident that fine particles (dp=l, 2pro) are uniformly distributed in the whole region of the cyclone chamber. On the other hand, coarser particles are strongly affected by centrifugal forces and tend to be easily trapped at the wall. It should be noted that ahnost all particles with 10#m diameter are collected above the height of x/H=0.50 without reaching the bottom part of the chamber. Fig.10 depicts the particle collection performance curve based on the flow fields predicted with the present model. The collection efl:iciency is defined as the ratio of the mass flux of particles through the inlet and the mass flux of particles through the exit. There is a disagreement in the collection efficiency for finer particles. This underprediction may be caused by the excessive particle dispersion due to the assumption of Boussinesq's isotropic eddy viscosity for the particle-phase turbulence. Therefore, further investigatios should be needed to improve the model performance. 6. S u m m a r y A Reynolds stress model was proposed for predicting the gas-particle turbulent flows. The effects of dispersed particles on fluid turbulence were considered in the Reynolds stress model. The proposed model can satisfactorily reproduce the experimental data of the gas-particle round jet and the fully-developed vertical pipe flow. The present model was also applied to the gas-particle swirling flow in order to investigate the model performance. The predictions show good agreement with the ineasurements in the case. Finally, the model was applied to the gas-particle highly swirling turbulent flow in the cyclone separator. The predictions were compared with the experimentM data. of the particle collection efficiency. It was shown that the present model can accurately reproduce the flow field, whereas it underestimates the collection efficiency for fine particles. This result is caused by the overestimation of particle dispersion using the eddy viscosity assumption for the particle-phase turbulence. Therefore, further investigations are needed to improve the model performance. References 1. W.D. Griffiths and F.J. Boysan, Aerosol Sci., 27 (1996), 281. 2. L.X. Zhou and S.L. Soo, Powder Tech., 63 (1990), 45. 3. R.A. Gore and C.T. Crowe, Int. J. Multiphase Flow, 15 (1989), 279. 4. J.D. Kulick et al., J. Fluid Mech., 277 (1994), 109. 5. S.E. Elghobashi and T.W. Abou-Arab, Phys. Fluids, 26 (1983), 931. 6. C.P. Chen and P.E. Wood, Can. J. Chem. Eng., 63 (1985), 349. 7. M.A. Rizk and S.E. Elghobashi, Int. J. Multiphase Flow, 15 (1989), 119. 8. O. Simonin, Proc. 8th Symp. on Turbulent Shear Flows (1991), 7.4.1. 9. Y. Sato et al., Trans. ASME, J. Fluids Eng., 118 (1996), 307. 10. M.M. Gibson and B.E. Launder, J. Fluid Mech., 86 (1978), 491. 11. D. Modarress et al., AIAA J., 22 (1984), 624. 12. Y. Tsuji et al., J. Fluid. Mech., 139 (1984), 417. 13. T. Kawamura and K. Kuwahara, AIAA Paper, (1984), 84-0340. 14. M. Sommerfeld and H.H. Qiu, Int. J. Heat and Fluid Flow, 12 (1991), 20. 15. F. Boysan et al., I. Chem. E. Syrup., 69 (1983), 305. 16. C.J. Stairmand, Trans I. Chem. Eng., 29 (1951), 356.
902
De
4~~
o
40
20
40,
.~--~,;...o,., i
~~ ' ~ ~ 3
Fig.7 Schematic of the cyclone separator g'-
.......
20
i :x:o60.33. X
0.1
,oL:~... i .... ,..o,.,
....
,-(m)
W
0'.1
~
I
~" . . . . . . . . .
~
= =0.15 m
0:,
,. (,,.,.)
Fig.8 Computational results of the swirling flow in the cyclone separator (Symbols" Boysan et al. [15])
~
g--
.-.
~o
....It0.
....( o ~ . o ~ . . ' " ol
....foo.i /
~ ....[
,-(m) dp -
o.1
. . . . . . . 0:, oi
,,(m)
. . . . . . .
l#m
.... ....~o-t to.~ oo,f~
! ~" ~
o11
,-(m)
or
dp - 2 # m
,/
,-(m)
ol
oI
;;.:0.
dp - 5 # m
dp -
o.1
10#m
Fig.9 Particle mean concentration profiles
oooS&o&&8oo . . . . . .
100
-
O O"~~
-
0
~9
i
50
"5 L)
4
~
,
,
0
Measurements (Stairmand, 1951)
:
Prediction
I
10
,
,
,
~
I
20
Pmicle diameter dp (/~m)
Fig.10 Particle collection efficiency (Symbols" Stairmand [16])
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 1999 Elsevier Science Ltd.
903
A C R W M o d e l for Free Shear F l o w s T.L. Bocksell and E. Loth Department of Aeronautical and Astronautical Engineering, University of Illinois at UrbanaChampaign, 306 Talbot Laboratory, 104 South Wright St., Urbana, IL 61801, USA 1. I N T R O D U C T I O N Computational models used to simulate particle diffusion in turbulence employing a stochastic Lagrangian particle methodology can fall into several categories. These can be arranged from the least computationally intensive to the most computationally intensive as: discontinuous random walk (DRW) models, continuous random walk (CRW) models, and stochastic differential equation (SDE) methods. The first two models are characterized by having a Reynolds-Averaged Navier-Stokes (RANS) solution for the continuous-phase and computing the trajectory of a large number of particles in the flow to obtain mean particle diffusion information. Both the DRW and CRW models include an eddy-particle interaction model that computes the length and time scales of an eddy from the gas-phase RANS solution. These scales, along with a random number generator, are used to simulate the chaotic effect of the turbulence on the particles. The CRW model correlates the turbulence statistics in time, and this one in particular utilizes a Markov chain to correlate the velocity fluctuations with the ones from the previous time step. Other more detailed CRW models make use of a larger number of earlier time steps such as Berlemont et al [1]. Maclnnes & Bracco [2] have pointed out serious deficiencies with conventional CRW models with respect to spurious drift, and the present work seeks to appropriately design and validate these models so that they can be used effectively in engineering calculations. Since current RANS solutions for engineering problems typically predict locally isotropic turbulence, the present stochastic model was designed to also employ such an isotropic flow solution. Other issues associated with particle diffusion are the crossing trajectories effect, continuity effect, and inertial-limit effect. The crossing trajectories effect is associated with particle inertia and the inability of particles to follow the fluid exactly. As a result larger particles will cross an eddy rather than be captured by it. Defming the eddy-interaction time scale, "l:int, to be a function of an estimate for the eddy lifetime, "re, and an estimate for the eddy traversal time, zt, can capture this effect. The continuity effect arises under the condition where the relative velocity of the particle is much larger than the root-mean-square of the fluctuation velocities. It stems from the two-fold increase in eddy length scale in the longitudinal direction as compared to the lateral direction as noted by Graham [3]. The resulting particle diffusion is thus anisotropic. The inertial-limit effect arises under the conditions of negligible gravity for high Stokes numbers. For these conditions, the fluid Lagrangian time scale no longer controls the fluid time scale seen by the particles, but is instead controlled by the moving Eulerian time scale. Also, Stock [4] asserts that the ratio of the Lagrangian to moving Eulerian time scale is
904
~fmE =2.8 a:fL
(1)
This value is based on the numerical simulation of Wang & Stock [5] using a specific number of Fourier modes for isotropic homogeneous turbulence. However, the numerical simulation of Elghobashi & Truesdell [6] of the grid generated turbulence experiment of Snyder & Lumley [7] showed that the time scales for the Lagrangian correlation and the moving Eulerian correlation in zero gravity were nearly identical. An experiment performed by Loth & Stedl [8] in a turbulent mixing layer measured the Lagrangian and moving Eulerian time scales and also found that these two scales (when based on the velocity fluctuations) were nearly equal. Loth & Stedl [8] explained their results in terms of the structural evolution of an eddy. The transport of a fluid point away from the moving Eulerian path is composed of two motions" an eddy translation and an eddy rotation. The Lagrangian correlation due to translation alone may be expected to increase the resulting time scale since the eddy is most coherent along its own path. However, the effect of the rotational component is to make positive velocity fluctuations become negative as they rotate around the eddy so that the velocity fluctuations can de-correlate. The effect of the two mechanisms can cancel each other out such that the overall correlation of the pure Lagrangian motion is about the same as the moving Eulerian correlation. Wells & Stock [9] noted experimentally that for zero gravity conditions, the Stokes number only increased diffusion slightly, thus suggesting XfL = "l;fmE . Based on these results, it may be more reasonable to approximate the two time scales as equal. A separate issue is the independent influence of non-dimensional parameters on particle diffusion in free shear flows. There has been substantial theoretical work in the diffusion of particles in turbulent flows as reviewed by Stock [4]. Recently Loth [10] developed a theoretical diffusion model for particles in a shear layer that discussed three critical nondimensional parameters: local Stokes number (ratio of particle time scale to local eddy time scale), eddy Froude number (ratio of hydrodynamic forces to gravitational forces), and drift parameter (ratio of terminal velocity to turbulent intensity of the surrounding fluid velocity). Of these three parameters, two are independent. The goals of this paper are to validate and evaluate the CRW methodology for homogeneous and inhomogeneous turbulence through comparison to detailed experimental data. Next, the CRW model is used to study long time particle diffusion in isotropic turbulence and understand the influences of particle Stokes number and eddy Froude number on particle diffusion. 2. M E T H O D O L O G Y The equations of motion goveming a particle in a turbulent flow can be written in each Cartesian coordinate (i) as ^
dxpi dt = upi where
(2)
Xpi is the particle position and Upi is the particle velocity. The equation for Upi used
herein is
905
dupi_ 3 0fl fi +u i-Upil( fi dt - 4dp 9p
-Upi)fD+g i
(3)
and 10pis the particle density, [of is the surrounding fluid density, dp is the particle diameter, is gravity, and ufi is the fluid velocity for which the overbar indicates a Reynolds average and the prime indicates the fluctuation component of the velocity. Co is the coefficient of drag which can be given by the composite function used by DeAngelis et aL [11] as C D = 12R~e4p(1 + 1Re2/3)
0.424
Rep < 1000
(4)
R ep > 1000
Or can be simply given as the linear, Stokesian drag: C D = 24/Rep for Rep < 1. Gosman & Ioannides [12] arrived at (3) by assuming that forces associated with freestream pressure gradients, virtual mass, and Basset history are negligible due to a heavy particle assumption, such that 9p )) 9f. The CRW model integrates (2) and (3) by approximating the turbulent fluctuation velocities through a stochastic sampling of a Gaussian distribution with a standard deviation of ~/2k/3, where k is the turbulent kinetic energy from a RANS gas-phase solution ask = ufi' u 'fi/ 2. These fluctuation velocities are correlated in time according to (where repeated indices are not summed) ufi(t+At) =(tiufi(t)+ 1-t~
Yi
k +t~Ufi
(5)
with (6)
.nt.i-
where Yi is a Gaussian random number with variance of one and 6~fi is the drift correction velocity necessary to account for a mean pressure gradient due to inhomogeneous turbulence. Interestingly, we have found that the empirical expression for "l:int,i used in (6) provides better prediction with experiment than the more smoothly varying expression, 'lTint -2 = ,i;e-2 + 'iTt,i -2. In (6), Ze and 17t are computed from the RANS gas-phase solution as k I;e = C e - and E (7) Ae i 'l:t,i = Vrel ' with A e i = Cc iCA C3/4 k3/2 ' ' E
Vrel,i and C c i = l + ~ ' [Vrel,il
906 Where C e and C A are empirical coefficients and will be discussed w and Cc, i represents the continuity effect in a similar method as proposed by Graham [3]. Note, the eddy lifetime is not dependent on Stokes number as suggested by Stock [4], since we have assumed l:fL = "~fmE" As discussed in w1, the CRW model can have a spurious mean drift of particles if a mean pressure gradient exists due to inhomogeneous turbulence. To correctly account for this mean pressure gradient, a mean drift velocity is added to the fluctuation velocities computed from (5). As discussed by Maclnnes & Bracco [2], this mean drift correction velocity can be written in terms of the gradient in the kinetic energy as
8ufi = 1 At 2 a___k_k 2 3 ~x i
(8)
o
The effects of this term will be examined and validated for inhomogeneous turbulence in the next section. 3. T E S T F L O W S Validation of the model was performed using two basic free shear flows (a turbulent wake and a turbulent axisymmetric jet) for which detailed experimental data is available. In addition, a 1-D numerical test flow was designed to evaluate the drift correction of (8) in the context of a highly simplified inhomogeneous turbulent flow. The number of particles used in each simulation was varied over a significant range to ensure statistical convergence. The time steps were kept uniform (independent of particle position) for the CRW simulations and were sufficiently small to provided accurate temporal resolution. For the first test, Snyder & Lumley [7] conducted a turbulent, grid-generated wake experiment, and the mean-square diffusion of four types of particles was measured. This grid-generated flow is nearly homogeneous in the two directions perpendicular to the flow direction, meaning the drift correction term can be neglected for lateral diffusion. 5.0 4.5 4.0 3.5 r 3.0 2.5 2.0 ~ 1.5 1.0 0.5 0.0
f
o HollowGlass ACornPollen S: c l i d G l ass
//cY/ ~
g
w
ij,,~
//
~
Solid Glass
.I,,U
_ at.~_..----m
I
I
I
I
0.1
0.2
0.3
0.4
0.5
Time [s]
Figure 1. Plot comparing the results from the CRW simulation of diffusion in grid-generated turbulence to the expefimentof Snyder & Lumley [7].
907 Schuen et al. [13] notes that due to somewhat arbitrary definitions for eddy length and time scales, CRW models typically require calibration. Coefficients for the eddy time and length scales appearing in the CRW model as Ce and CA were calibrated to agree with the Snyder & Lumley [7] experiment. The resulting values for C e and C A are 0.27 and 1.6 respectively and Figure 1 shows the comparison of the CRW simulation of particle diffusion in grid-generated turbulence to the data of Snyder & Lumley [7], where the agreement is quite reasonable for short and long time diffusion. Specifying a simple 1-D flow with isotropic inhomogeneous turbulence accomplished the validation of the drift term for the CRW model. The numerical test flow was defined as uf = 1,
vf = 1,
k = 0.3 exp(-115(0.3 - y)2)
(9)
for 0 _< y < 0.6 and 0 < x g 16 with periodic boundary conditions in the y direction. This represents a Gaussian distribution in the lateral direction with peak in the center of the domain. Since the mean flow is divergence free, a uniform initial distribution of fluid particles should remain uniform for this flow. To test this, fluid particles (of negligible mass and volume) were injected uniformly at the inlet (x=0) and resulting density profiles were computed at various planes downstream. It was determined that 50,000 particles was sufficient for the numerical test flow study. Figure 2 shows the normalized density profiles at a plane far enough downstream so that the normalized density profiles are independent of x. The tendency of the CRW model without the drift correction term is a non-physical accumulation of particles in regions of low turbulence intensity (near the edges of the domain) and a depletion of particles in the center. In general, the drift correction worked better than those used for DRW models such as Bocksell & Loth [ 14]. 0.6
\
0.5
Without
Drift
Correction
0.4
y 0.3 0.2 0.1 0.0 0
0.5
I
I
I
I
I
I
1
1.5
2
2.5
3
3.5
L
P/Po Figure 2. Particle concentration profiles of the numeric test flow showing the effect of the drift correctionterm. Next, particle diffusion in an axisymmetric jet was simulated to compare with concentration measurements of Yuu et al. [15]. Bocksell & Loth [14] describe details of the axisymmetric jet mean velocities as and turbulent intensities as functions of the radial and
908 axial coordinates. For the axisymmetric jet flows, 100,000 particles were used in computing the centerline particle concentration profiles; 20,000 particles (in conjunction with 15 collection bins evenly distributed in the radial direction) were used to compute the radial particle concentration profiles. The particle initial conditions for the axisymmetric jet were determined by Schuen et al. [13] as described by Bocksell & Loth [14]. Figure 3 shows the ratio of particle concentration to centerline particle concentration versus radial position for the CRW model along with the data of Yuu et al. [15]. Figure 3 corresponds to a jet discharge velocity of 50 m/s, a jet diameter of 8 mm, and an initial particle Stokes number of 15 at an axial location of z/djet = 50.
,,o 0.8 t~"
~-
x'~",
- ...... ~ "
o,
o
C R W with driftcorrectionC R W without drift correction Experimental data
.-~ 0.6 Iii
r~ca
0.4
o
0.2
0
0.05
o
"'""
0.1
.....
. .....
0.15
0.2
r/z Figure3. Comparison of the CRW results for the radial particle concentration of an axisymmetricjet compared to the experimental data of Yuu et al. [15]. The overall match of the CRW model with drift corrections to the data of Yuu et al. [15] is good. The effect of the spurious mean drift correction suggested by MacInnes and Bracco [2]. Neglecting the drift correction term results in particles incorrectly accumulating in the outer region of the jet (away from the centerline) where the smallest turbulence intensity is located. A second test condition at 100 m/s described by Bocksell & Loth [14] yielded similar results. Another improvement to make the model more computationally efficient is to filter out the high frequency fluctuations that do not significantly contribute to particle diffusion. The resulting filtered solution can significantly reduce CPU. The filtering is accomplished by defining a localtime step to be a function of the eddy interaction time scale as ~
At
=
m
I~int,il
(1 0)
Ncutoff In this manner the fluctuation velocities computed by (6)will only be due to the largest energy containing eddies. The minimum acceptable value of Neutoff was determined with several simulations by matching the lateral diffusion to within 1% of that found using the statistically converged global time step. The results showed rapid convergence to the global timestepping diffusion curve with Ncutoff of four or more. This is a much smaller Ncutoff than
909 expected based on DRW tests, and possibly because the CRW introduces randomness at every time step compared to every "[int for the DRW model. An interesting feature of the acceleration scheme for the CRW model is the resulting spectra of the turbulence seen by the particles. Figure 4 shows a comparison of the spectra of the fluctuation seen by particles for two disparate values of Ncutoff: 4 and 1024. 1000
~, 100
~1
10
[,,.
1
9" , ,
0.1
,.
. . . . . . .
Ncutoff = 1024 Ncutoff = 4
I
0.01 q;Lp
0.1 'l~Lp
l'lTLp
10 'lTtp
100 "lTLp
1000 q~Lp
Frequency Figure 4. Comparison of the spectra of the velocity fluctuations for the CRW model for two values of Ncutoff. While Ncutoff = 1024 has captured more of the complete spectra of the turbulence, Ncutoff = 4 has only captured the fluctuations associated with the largest structures in the turbulent field. This is remarkable since the resulting mean-square diffusion curves for these two spectra are nearly identical. 4. P A R A M E T R I C
STUDIES
Finally, the CRW model was used to study the effect of mean local particle Stokes number (S) and mean local eddy Froude (Fr A) number on particle diffusion in isotropic homogeneous turbulence. In order to further simplify the physical interpretation, a Stokesian (linear) drag law was used rather than the composite function shown in (4). The mean local particle Stokes and mean local eddy Froude numbers are defined as follows: m
~=
dppp , 18gf~e
and
FrA =
k 61gilAe '
(1 1)
where < > indicates an ensemble average over all the particles at a particular time from injection. The equation for the diffusion of a particle (ep) and the lateral diffusion ratio (D] 1) are
1 ((xp - Xpo )2 )
ep,lat = "~ ~'~"
and
Dll =
ep,lat es
(12)
910 where es is the scalar diffusion (computed by using a tracer particle), x is the lateral coordinate direction, and Xpois the lateral injection location. The mean diffusion for each set of particle conditions was then obtained by fitting a line (in the least squares sense) to the mean-square diffusion curve over the region where the mean-square diffusion had become linear with time. The simulations ensured that long-time linear diffusion would be reached by simulating the particles for at least eight particle relaxation times, "Cp, where the particle relaxation time is defined as d2pp
(13)
18vfpf The particle diffusion results for the isotropic turbulent flow are shown in Figure 5 where the mean diffusion ratio, D 11, is plotted versus S for a range of FrA . m
10
!
e
|
~
9
o
~
&
e
[]
9
0.1
e
Ik
o 0
[]
0
OFr=~ &
o Fr = 1.0
El
!:3 Fr = 0.1 Fr = 0.01
9
m,
9 Fr = 0.001 0.01 0.001
9
A
I
I
I
I
0.01
0.1
1
10
100
Figure 5. Plot showing the effect of particle Stokes number and eddy Froude number on long time particle diffusion in isotropic turbulence. m
The resulting diffusion curves indicate that as S increases beyond a value of approximately unity, the diffusion is generally reduced monotonically. This expected result simply stems from the increased particle response time relative to the eddy time scale such that the particle's large inertia prevents it from being deflected by the eddy. However, the results show a strong dependence on FrA . The reduced diffusion as eddy Froude number decreases, for a fixed Stokes number, is consistent with Loth [10], which noted that lower Froude numbers were associated with a greater likelihood of eddy traversal time to dominate 1;int . This in turn causes the particles (with a fixed ratio of eddy lifetime to particle response time) to undergo more frequent but less pronounced trajectory fluctuations thereby yielding less diffusion. For zero gravity (Fr ~ oo), the inertial-limit effect does not exhibit an increase in Dll as Stokes number increases as modeled by Stock [4]. This is simply due to our assumption that 1;fL = 1;fmE.
911 When the data from Figure 5 is plotted as D 11 versus S/F-~A, which is the ratio of particle terminal velocity to fluid velocity fluctuations, the data collapse to a single curve as shown in Figure 6. 10
a
0.1
~
vFr=
I!
8
/,
!
1.0
aFr=0.1 A Fr = 0.01 m F r = 0.001 0.01 0.01
I 0.1
i 1
I 10
i 100
10t
Figure6. Plot showing the effect of particle Stokes number and eddy Froude number on long time particle diffusion in isotropic turbulence. This indicates that the velocity ratio, not the time scale ratio, is the dominant description of long-time turbulent diffusion. Note that in Figure 5, the inertial-limit case corresponds to the curve for FrA --~ oo and D 11 remains near unity for all particle Stokes numbers. This is again consistent with the inertial-limitbeing dominated by the ratio ZfrnE/ZfL' which is assumed to be unity. It should be recognizedthat previous simulations which noted reduced diffusion as Stokes number increased for zero-gravity might not have been taken over sufficient time averaging to ensure long-time dispersion (since the averaging time increases with Stokes number) or were for cases in which the flow was not truly homogeneous and isotropic. 5. C O N C L U S I O N S A CRW model was designed to simulate particle diffusion in free shear flows. These models included enhancements to correct for the spurious mean drift of particles based on the work of Maclnnes & Bracco [2]. The resulting models provided reasonable particle diffusion predictions for the Snyder & Lumley [7] wake experiment and the Yuu et al. [15] turbulent axisymmetricjet experiment. In addition, a parametric study examiningthe effects of S and FrA on particle diffusion ir isotropic turbulence was conducted, where the results showed significant influence of S anc FrA . In general, increases in Stokes number result in reduced diffusion for a fixed edd Froude number, as did decreases in Froude for a fixed local Stokes number. When plottin Dll versus S/~A (corresponding to the ratio of particle terminal velocity to fluid fluctuatic velocity), the data collapse to a single curve, where Dll decreases monotonically S / ~ A =1.
912
6. A C K N O W L E D G E M E N T S This work was supported by NASA Lewis Research Center (LeRC) under contract NAS3-97011 with Tom Irvine as technical monitor and by the Office of Naval Research (ONR) under grant N00014-96-1-03412 with Dr. Edwin Rood as technical monitor. Also, NCSA computer facilities were used in this research.
REFERENCES 1 Berlemont, A., Desjonqueres, P., Gouesbet, G., "Particle Lagrangian simulation in turbulent flows," Int. J. Multiphase Flow 16, 1990. Maclnnes, J.M. & Bracco, F.V. "Stochastic Particle Dispersion Modeling and the TracerParticle limit," Physics Fluids A, 12, 1992, pp. 2809-2824. Graham, D.I. "Improved Eddy Interaction Models With Random Length and Time Scales," Int. J. Multiphase Flow, 24 (2), 1998, pp. 335-345. Stock, D.E. "Particle Dispersion in flowing Gases - 1994 Freeman Scholar Lecture," Journal of Fluids Engr., 114, 1992, pp. 100-106. 5 Wang, L.P., & Stock, D.E. "Numerical simulation of heavy particle dispersion: time step and nonlinear drag considerations," Journal of Fluids Engr., 114, 1992, pp. 100-106. Elghobashi, S., & Truesdell, G.C. "Direct simulation of particle dispersion in a decaying isotropic turbulence," J. Fluid Mech., 242, 1992, pp. 655-700. Snyder, W.H. & Lumley, J.L. "Some Measurements of Particle Velocity Autocorrelation Functions in a Turbulent Flow, "Journal of Fluid Mechanics 48, 1971, pp. 41-71. 8 Loth, E. & Stedl, J.L. "Taylor and Lagrangian Correlations in a Turbulent Free Shear Layer," Experiments in Fluids, 25, 1998. 9 Wells, M.R. & Stock, D.E. 'q'he effects of crossing trajectories on the dispersion of particles in a turbulent flow," J. Fluid Mech., 136, 1983, pp. 31-62. 10 Loth, E. "An Eulerian Model for Turbulent Diffusion of Particles in Free Shear Layers," AIAAJournal 36, No. 1, Jan. 1998, pp. 12-17. 11 DeAngelis, B.C., Loth, E., Lankford, D., Bartlett, C.S., March-April "Computations of Turbulent Droplet Dispersion for Wind Tunnel Tests," AIAA Journal of Aircraft 34, No. 2, 1997, pp. 213-219. 12 Gosman, A.D. & Ioannides, E. "Aspects of Computer Simulation of Liquid-Fueled Combustors," AIAA-81-0323, 1981. 13 Schuen, J-S., Chen, L-D., & Faeth, G.M. "Evaluation of a Stochastic Model of Particle Dispersion in a Turbulent Round Jet, "AIChE Journal 29, 1983, pp. 396-404. 14 Bocksell, T.L., & Loth, E. "An Enhanced Discontinuous Random Walk Model for Particle Diffusion in Wakes and Jets," ICMF, June 1998, Lyon, France. 15 Yuu, S., Yasukouchi, N., Hirosawa, Y., & Jotaki, T. "Particle Turbulent Diffusion in a Dust Laden Round Jet, "AIChE Journal 24, 1978, pp. 509-519.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All rights reserved.
T e s t of a n E u l e r i a n - L a g r a n g i a n g a s - s o l i d p i p e flow
913
s i m u l a t i o n of w a l l h e a t t r a n s f e r in a
R. Andreux, P. Boulet and B. Oesterl~ Laboratoire Universitaire de M6canique et Energ6tique de Nancy (L.U.M.E.N.) E.S.S.T.I.N.- Universit~ Henri Poincar~, Nancy 1 2 rue Jean Lamour, F-54500 VANDOEUVRE, France. Tel: (33) 383 50 33 39 - Fax: (33) 383 54 21 73 - email: [email protected]
1.
ABSTRACT An Eulerian-Lagrangian model is proposed to numerically predict the heat transfer between a vertical pipe wall and a turbulent gas-solid suspension. The whole problem lies in the combined solution of mass, momentum, and energy equations written for each phase. Closure equations devoted to turbulence and heat transfer simulation are also needed for the continuous phase. This problem is treated using a k-~ model and a turbulent Prandtl number model for the gaseous phase. The coupling terms standing for the fluid-particle interactions are taken into account. The simulation of the dispersed phase dynamics is addressed using a Lagrangian model, taking the particle-wall and particle-particle interactions into account. Additionally, the temperature of each particle is tracked using a model for the convective heat exchange between the two phases. The accuracy of the thermal and dynamic solutions has been tested for particles of diameter 200 pm and 500 pm, and for a wide range of loading ratios (0 to 20). Corresponding velocity and turbulent kinetic energy profiles are presented for the dynamic modelling validation, while Nusselt numbers are calculated for the study of the thermal problem.
2.
INTRODUCTION Addition of particles in a gas turbulently flowing in a pipe is well known to have an influence on the dynamic flow characteristics and on the heat transfer between the wall and the suspension. Among the factors affecting these phenomena, the loading ratio, the particle characteristics, the inter-particle collisions, the particle-turbulence interactions as well as the flow regime have been identified. At low loading ratios for example, it has been experimentally noticed that the turbulent intensity tends to decrease for small particles, while it tends to increase for large particles. Simultaneously, the heat transfer between the wall and the suspension seems to decrease at low loading ratios and to increase at higher loading ratios. This last phenomenon has been especially observed through experimental data by Depew and Farbar [1] and Jepson et al.
914 [2], among others, who reported measurements of the suspension Nusselt number as a function of the loading ratio for several kinds of particles. The aim of this paper is to present a numerical simulation of a turbulent gas-solid flow in a vertical pipe, heated by a constant wall heat flux. The injected particles are assumed to be spherical with a diameter of 200 pm or 500 ~m. The predictions of the velocity and temperature fields are based on an EulerianLagrangian approach. In fact, the work is in keeping with various studies which aim to simulate these kinds of flows more and more accurately by means of Eulerian or Lagrangian modelling. For example, in the two-fluid approach suggested by Han et al. [3], the turbulent Prandtl number is expressed as a function of the solid-to-fluid heat capacity ratio and the ratio of the turbulent time scale to the particle thermal relaxation time. Treating a similar problem, Avila and Cervantes [4] have used an Eulerian-Lagrangian approach based on a standard k-~ model and on an enthalpy balance for the continuous phase, while the LSD (Lagrangian-Stochastic Deterministic) model was used to predict the particulate phase motion, inter-particle collisions and wall-particle interactions being neglected, however. Boulet et al. [5] have recently combined the two approaches, using a two-fluid model in order to simulate the dynamic and thermal characteristics, closure being achieved on the basis of Lagrangian simulation results. In the present Eulerian-Lagrangian formulation, one of the objectives was to bring special care to the interactions between the turbulence and the particles. The formulation thus takes into account the appropriate coupling terms in the various equations. The fluid flow is predicted by means of a standard k-~ model, whereas the thermal problem is solved using a turbulent Prandtl number expression by Kays [7] for the closure. This formulation allows the large increase of the turbulent Prandtl number in the near-wall region to be taken into account. The dynamic features of the particle flow are predicted through an improved version of the Lagrangian simulation presented by Oesterl6 and Petitjean [6], where particular attention is paid to the effects of collisions undergone by particles. For the present purpose, it has been supplemented by temperature tracking based on the energy balance of each particle along its trajectory. In the following sections, the needed assumptions and the mathematical formulation of the problem are first presented. Then, the numerical procedure is described, including the axial and radial grid definition. Finally, after comparing predictions concerning the fluid and the particle dynamics to experimental data by Tsuji et al. [8], the numerical heat transfer results are validated against the experimental data by Depew and Farbar [1] and Jepson et al. [2]. t
NOMENCLATURE np surface area of a particle Co drag coefficient specific heat particle diameter g gravitational acceleration transfer coefficient hp heat around particles
k m np
turbulent kinetic energy solid loading ratio number of particles (/m 3) N u Nusselt number N u L asymptotic Nusselt number Nup particle Nusselt number
915
Nuo pure air Nusselt number Pr Prandtl number Prt turbulent Prandtl number Qw wall heat flux Re pipe flow Reynolds number Rep particle Reynolds number Ret turbulent Reynolds number
z
Greek symbols a solid volume fraction dissipation rate of turbulent kinetic energy ct thermal eddy diffusivity Von Karman constant K 2f fluid thermal conductivity
r R
radial co-ordinate pipe radius Spk kinetic energy source term Sp~ dissipation rate source term Spu momentum source term SpT heat source term due to fluid particle exchange T temperature Tm bulk average temperature U+ fluid velocity in wall unit velocity at the pipe centre Uc fluid velocity Uz friction velocity Ur particle velocity Vz Vp volume of a particle y+ wall distance in wall unit 4.
FORMULATION
4.1
Dynamic problem
axial co-ordinate
v
p
laminar cinematic viscosity eddy viscosity density particle relaxation time
Subscripts and superscripts f fluid property p particle property t turbulent quantity w property at the wall averaged quantity fluctuating quantity
The basic assumptions used in the mathematical formulation of the dynamic models are the following: 1. the gas is incompressible 2. the gas-solid suspension is flowing upward in a vertical pipe 3. the flow is quasi-developed in the axial direction (convection terms being neglected and velocity profiles being only altered due to fluid-particle interactions) 4. particles are solid, spherical, with a constant diameter. Under above conditions, the k-cmodel may be written as follows: Momentum equation: - 0 -
dP
-
rr
;,0 -
r ~ r L ;o,(1-~)v,
Turbulentenergyequation:ld[r(l__lvtd# 1 Dissipation rate equation:
1 d r(l_~)v,
r dr
d-~
;rJ
~[du, 1
o'~ ~ r +C~(1-a')vt ~L
dr
j+S,u=0
[~]2
-c
dr J -C2~-a)-~ +Sp;=O
__
916 --2
Closure equation:
k vt = C , _
Commonly adopted values are taken for the model constants C,, C~, C 2, crh and or. The coupling terms Spu, Sp~ and Sp~ are given by: "SPu : aPP (~-<)z-p
4 ppdp r ~ : - ~ p t C v [ ~_..uzz[
where
is the particle dynamic
relaxation time. 9 Sph ~ S'p,,.u' = S~,,.Uz - S~,,,.Uz
9Sp~ = C - ~ ~ k
as proposed by Berlemont et al. [9] with C - 1.9
The simulation of the dispersed phase dynamics follows the Lagrangian code presented by Oesterl6 et Petitjean [6], which is based on a probabilistic technique to treat the inter-particle collisions. The effect of turbulent fluctuations is simulated by an eddy interaction model. 4.2
Thermal formulation The basic assumptions used in the mathematical formulation of the thermal problem, allowing the study of the heat transfer between the heated pipe and the suspension, are the following ones: 1 each particle has a uniform temperature and constant characteristics 2 the heat transfer by conduction due to particle-particle interactions and wall-particles interactions is negligible 3 the heat flux at the pipe wall is uniform 4 axial heat transfer is negligible compared to radial one 5 temperatures considered are sufficiently low to neglect the radiative transfer
Under above conditions, the thermal problem for the fluid flow is modelled as follows: Energy balance"
(1--a)ptCptuz
1 d rdr
Oz
r(1 - ~ ) ( l t + p t C , t e t ) - - ~ r
+ S'T
Closure relation: The expression of the turbulent Prandtl number Pr t is given by Kays [7] as: prt =
1
0.5882 + 0.228 The coupling term
- O.0441
1 - exp
S~rwhich represents
-vt
v
the heat exchange between the
particulate phase and the continuous phase, is given by:
917
SpT
= 5~h------x-P(Tpp - T-~t) d~
where hp, the heat transfer coefficient characterising the convection around particles, is deduced from a corresponding particle Nusselt number: Nup = hp dp 2r defined as follows: Nu~ = 1 + (1 + Rep Pr) ~ 9 O
~ 0.4, ~ 1 Nup = aep 1 + Rep Pr
9 100 <_Re p <_ 2000"
Nu~ = 0.752Rep ~
Pr~
1 + Rep Pr
_It_ 1
Pr~
+1
The t e m p e r a t u r e of each particle is calculated along its trajectory, by solving the following energy balance" p , V , Cp a T , dt = hpA~ (Tf - Tp )
Finally, the associated boundary conditions are" at the pipe inlet:
and at the wall"
T f - Tp - To ;
Qw =
- 2r - ~ r
at the axis:
Or r=O
=0
r=R
5.
NUMERICAL P R O C E D U R E Equations are numerically solved using a finite difference-scheme. The geometry of the here-studied pipe may be divided into two parts, the adiabatic inlet section and the heated section. At the end of the first part the flow is assumed to be dynamically fully developed. We are primary interested in the second section, where the wall is heated, thus leading to a thermally developing flow. This a r r a n g e m e n t is similar to the classical set-up used in the abovementioned experimental works. 5.1
N u m e r i c a l grid
In order to accurately handle the heat transfer problem, it is necessary t h a t the mesh width be small enough near the wall. A logarithmic scheme as proposed by Wassel and Edwards [10] is used for the radial co-ordinate definition:
918 with i = [1;...; n,. ], where nr is the number of nodes in the radial direction, A=is the Von Karman constant and R e t denotes a turbulent Reynolds number defined as R e t = ~u r, R
u~ being the friction velocity. For the axial co-ordinate, the following v grid definition, suggested by Azad and Modest [11], is adopted in the heated section: exp z(i) =
r
~-J-ln
Lnz
1
+1 -1
with i = [1;...; n z ], where nz is the number of nodes in the axial direction, c is a constant and L designates the heated pipe length. Numerical calculations have been carried out using the above-described co-ordinates, with nr-50, nz=lO, c=lO. Such a low number of grid nodes in the axial direction was chosen in order to reduce the storage needs for the Lagrangian simulation, after comparison through Eulerian computation [5] has shown no significant difference between nz=lO and n z = l O 0 . As the fluid phase dynamics is only solved in the core region (between the pipe centre and a node located at y+ = 30) using the above-presented k - e model, some further dynamic data are needed in the near wall region, in order that the heat transfer problem may be treated very close to the wall. This difficulty is surmounted by using an interpolation scheme between the wall and the calculated values at y+ = 30. However, instead of applying a simple interpolation to calculate the velocity, the standard three-layer formulation has been preferred in the near-wall region. Similarly, the eddy viscosity close to the wall is calculated using the Van Driest formula as modified by Spalding [12]:
1
vt --~v exp(Icu+)-l-lcu + (K'u+)2,2l ....(~:u+)33.tfor y+ < 40 5.2 Numerical technique Equations are numerically handled using a finite difference technique. An iterative numerical method is applied in order to allow all the coupling terms between the two phases to be taken into account. Numerical tests have been carried out in order to give the optimum scheme to obtain the convergence of the two-way approach. The dynamic problem is finally treated according to the following steps: 1. seeking of a solution for the continuous phase starting with simple particle phase characteristics (constant velocity and concentration) 2. tracking of particles injected in the corresponding air flow
919 3. solution for the continuous phase loaded with particles with properties calculated above 4. successive t r e a t m e n t of previous steps until satisfactory convergence criteria are obtained. In the hereafter presented examples, three successive loops have been carried out between the Eulerian and the Lagrangian simulations, 5000 to 15000 particle trajectories being calculated in each Lagrangian simulation. Any additional coupling iteration did not introduce significant alteration of the dynamic results in case of moderate loading ratios (less than 3.6). After computing the two-phase flow dynamics, the t h e r m a l problem in the heated part of the pipe is dealt with, repeating the two following steps successively: 1. seeking of a solution to the energy balance on the continuous phase 2. calculation of the particle temperature along their trajectories. In the examples presented in the following section, two loops between Eulerian and Lagrangian t r e a t m e n t s were found to be sufficient to yield stable numerical results for the thermal problem. However, more coupling-iterations would be needed for larger loading ratios. 6.
NUMERICAL VALIDATION
6.1
Dynamic characteristics
First validation tests have been carried out for the dynamic characteristics, numerical results being compared to the experimental data of Tsuji et al. [8]. Velocity and streamwise turbulent intensity profiles are presented in figures 1 to 4. The studied cases correspond to an air flow loaded with particles with a density of 1020 kg/m 3 and a diameter of 500 ~m. The pipe diameter is 3.05 cm. Velocity profiles are plotted on figure 1 for a loading ratio of 2 and a mean fluid velocity of 8 m/s. Results are normalised, dividing by the fluid velocity at the pipe centre. The same presentation is kept in figure 2, for a larger loading ratio (m = 3.6) and a similar mean fluid velocity of 7.89 m/s. Particle velocity profiles are especially well reproduced thanks to the use of a realistic virtual wall model and inelastic bouncing conditions. However, the accuracy of the fluid flow prediction can be seen to decrease with increasing loading ratio, particularly in the near-wall region (figure 2). Two major causes may be invoked to explain this problem. First of all, the use of the present standard k-e model does not allow to take into account the anisotropy of the turbulence, which is obvious near the wall. Secondly, the influence of the particles on the fluid phase velocity is only qualitatively obtained, since the fluid velocity is not altered as expected, considering the experimental data on figure 2. Since the particle velocity seems to be in agreement with experimental data, the observed inaccuracies might be attributed to particle concentration profiles. More tests should therefore be carried out in order to investigate the effect of the concentration distribution upon turbulence and fluid velocity alteration. In particular, the simulation of the inter-particle collisions is expected to significantly affect the particle dispersion, thus influencing the fluid dynamics through coupling terms. Figures 3 and 4 are devoted to axial fluctuating velocity results, calculated in similar conditions" particles are injected in an air flow with a mean velocity of
920 10.8 m/s. The loading ratio is 1.1 for figure 3 and 3.4 for figure 4. In a g r e e m e n t with the previous analysis concerning the velocities, the results are satisfactory at low loading ratio, whereas inaccuracies a p p e a r w h e n more particles are injected. .2
.............
.2
-}
1.0
......... ~........-.......................................................
1.0 "
0.8
-x-x
.x.x
xx
0.8a'.r
,h
0.6
20.6 -
0.4
0.4 -
0.2
_
0.2-
0.0
1
0.0
0.5
,
0.0
I
1.0
0
0.5
r/R
r/R
Fig. 1: Velocity profiles (the thin line stands for the fluid and the heavy line for the particles). Comparison with Tsuji's experiments (symbols). ( R e .~ 16, 000; m = 2. O; dp = 5 0 0 / a n )
Fig. 2: Velocity profiles (the thin line stands for the fluid and the heavy line for the particles). Comparison with Tsuji's experiments (symbols). ( R e ~ 16, 000; m = 3. 6; dp = 5 0 0 / a n )
.1 ............................................................=......
0.1 i.................................................................................................... •
0.08 -
0.08 r
0.06 -
0.06
0.04:
"~ 0.04
0.02 -
0.02
0
0 0
0.5
1
r/R
................ 0
', . . . . . . . . 0.5
1
r/R
Fig. 3: Streamwise fluid fluctuating velocity profiles. Comparison with Tsuji's experiments (symbols)
Fig. 4: S t r e a m w i s e fluid fluctuating velocity profiles. Comparison with Tsuji's experiments (symbols).
( R e = 22, 000; m = 1.1; dp = 500/am)
( R e ~ 2 2 , 0 0 0 ; m = 3.4; d p = 5 0 0 ~tm)
6.2
Thermal
results
The t h e r m a l problem solution has been tested by m e a n s of comparisons with d a t a by Depew and F a r b a r [1] and Jepson et al. [2]. Suspension Nusselt n u m b e r s have been calculated using the following definition:
921 Nu =
2RQw
r(Tw - T , . )
Twand T,,being the wall temperature
and the bulk average temperature,
respectively. _
I[ [] o
o/ o
4O Z~
Z~
1.5-
m=O
m=0,52 . . . . . .
m=0,52
m=2 36
m=2 36
"0
3O
20
m=O
0 ,00
. . . . . . . . . . . .
0
. . . . . . . . . . . .
O.5
1
I
I
1
m
1.5
....
I...............
_
I
2
2.5
Fig. 5: Nusselt number as a function of the loading ratio. Comparison with Depew and Farbar's experiments (symbols). (Re ~ 13,500; dp = 200 pm)
0
10
20/D30
40
50
Fig. 6: Nusselt number as a function of axial position. Comparison with Depew and Farbar's experiments (symbols). (Re ~ 13,500; r - 200 pm).
Figures 5 and 6 illustrate the comparison with the experiments by Depew and Farbar, who used 200 pm spherical glass beads. In figure 5, the Nusselt number at the end of the pipe, where the flow is fully developed, is presented as a function of the loading ratio. The agreement is good but it must be noticed that the corresponding loading ratios are relatively small. Some decrease in the Nusselt number is first observed, then the numerical results indicate that NUL increases with increasing loading ratio. In figure 6, local Nusselt numbers are plotted as a function of the axial location along the pipe, for several loading ratios. Qualitative agreement is obtained, showing the ability of the simulation to reproduce the thermal entry length increase due to the presence of particles. Results are even very good for m = 0.52. Unfortunately, predictions become inaccurate for the largest loading ratio, again indicating the limitations of the present numerical code, as stated above in the section devoted to the dynamic solution validation. Figure 7 presents similar tests performed in a case experimentally studied by Jepson et al. [2]. Sand particles with the same diameter (200 ~m) were injected in an air flow. Comparisons are performed for the asymptotic Nusselt numbers, which are referred to the corresponding pure air flow Nusselt number. In agreement with previous observations, the general variations are qualitatively obtained, but results become inaccurate for loading ratios exceeding about 5. Considering the first tests performed using the present simulation, a qualitative description of the dynamic and thermal behaviour of the suspension has been obtained. However, the agreement between numerical results and
922 e x p e r i m e n t a l ones has not been found satisfactory at high loading ratio. This corresponds to flows where the turbulence is significantly modified by particles. We therefore consider t h a t this p h e n o m e n o n is still m i s r e p r e s e n t e d by our simulation at this moment, as previously noticed. Some further investigations m u s t be carried out in this field. Moreover, the closure equations used in the present model are not satisfactory, since the turbulence anisotropy and the particle influence on the fluid phase t u r b u l e n t P r a n d t l n u m b e r are not yet t a k e n into account. This will also be two major investigation subjects to be developed soon. _
1~
-
1.61.4-
-
Fig. 7: Nusselt n u m b e r as a function of loading ratio. Comparison with Jepson et a l . experiments (symbols). (Re ~ 23,000; d p = 200 ~m)
•
1.2-
I
0.8 0
i
10
20
30
m
7.
CONCLUSION
An E u l e r i a n - L a g r a n g i a n model has been presented in order to predict the heat tr an s f er between a t ur bul ent l y flowing gas-solid suspension and a vertical pipe with a uniformly heated wall. A k-~model combined with a complete particle tracking has been used, including coupling t e r m s to simulate the fluid-particle interactions. Although the presented preliminary tests indicate encouraging results, the model becomes unsatisfactory as loading ratio rises. F u r t h e r study is now necessary in order to extend the validity range of the code to higher loading ratios. S u b s e q u e n t investigations will be carried out, particularly in order to improve the formulation of the closure equations and the coupling t e r m s used for the continuous phase. REFERENCES 1. C.A. DEPEW and L. FARBAR, Trans. ASME, J. Heat Transfer 85C, 164-171 (1963). 2. G. JEPSON and A. POLL & W. SMITH, Trans. Instn Chem. Engrs, 41,207-211 (1963).
3. K.S. HAN and H.J. SUNG & M.K. CHUNG, Int. J. Heat Mass Transfer, 34, n~ 69-78 (1991). 4. R. AVILA and J. CERVANTES, Int. J. Heat Mass Transfer, 38, n'll, 1923-1932 (1995). 5. P. BOULET, B. OESTERLE and A. TANIERE, ICMF'98, P613, Lyon, France, (June 1998). 6. B. OESTERLE and A. PETITJEAN, Int. J. Multiphase flow, 19, n~ 199-211 (1993). 7. W.M. KAYS, Trans. ASME, 116, 284-295, (May 1994). 8. Y. TSUJI, Y. MORIKAWA and H. SHIOMI,. J. Fluid Mech., 139, 417-434 (1984) 9. A. BERLEMONT, P. DESJONQUERES and M.S. CABOT, ASME FEDSM' 97, 97-3577 (1997). 10. A.T. WASSEL and D .K. EDWARDS, J. Heat transfer, Paper n ~ 76-HT-Q (1976). 11. F.H. AZAD and M.F. MODEST,. Int. J. Heat Mass Transfer, 24, n~ 1681-1698 (1981). 12. D. SPALDING, Conf. Int. Dev. in Heat Transfer, Boulder-Colorado, PartII, 439-446 (1961).
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
923
A n a l y s i s a n d D i s c u s s i o n o n t h e E u l e r i a n D i s p e r s e d P a r t i c l e E q u a t i o n s in Nonuniform Turbulent Gas-Solid Two-Phase Flows S. Lain ~ and R. Aliod b ~MVT-UST, Fachbereich Verfahrenstechnik, Martin-Luther-Universit/it Halle-Wittenberg, D-06099 Halle (Saale), Germany bFluid Mechanics Department, E.U. Polit~cnica de Huesca. University of Zaragoza, Carretera Zaragoza km. 67, 22071 Huesca, Spain The work presented here analyses the underlying mechanisms in the continuum equations for non-colliding particles in dilute nonuniform two-phase flows. The study is carried out for axisymmetric particle-laden gas jets, which constitute a good example of nonuniform flow. Equations of radial and axial momentum and turbulent kinetic energy for the particulate phase are divided into their basic terms and analysed separately. As a result, for high inertia particles, the modelling of the "interaction terms" (terms due to gas-particle interaction) reveals itself as the crucial point, as long as they drive the existing equilibrium in the Eulerian particle equations. The hypotheses used to model the dispersed phase Reynolds stresses are in accordance with the previous theoretical work of Reeks [1]: the normal stresses in the streamwise direction are enhanced over the corresponding ones in the other directions and the former prevail over the shear stresses in the limit of great inertia particles, too. This particle Reynolds stress modelling is found to be directly related with the correct prediction of the dispersion of the corresponding volume fraction profile, c~d, in the considered experiments. Moreover, the information obtained from the balance of present contributions in the radial momentum equation confirms a previously used closure of radial relative velocity as proportional to the gradient of particle void fraction [2-4]. This effect is the ultimate responsible for the spreading of c~d along the jet. Comparisons of numerical calculations with the experiments of Mostafa et al. [5] are provided showing reasonable agreement in all available variables. 1. I N T R O D U C T I O N Nowadays, two approaches are mainly used to describe the dispersed phase in a twophase flow (solid, droplet or bubble suspensions). In the so called Lagrangian method the discrete elements are tracked through a random fluid field by solving their equations of motion. In the second methodology, both phases are handled as two interpenetrating continuums and are governed by a set of differential equations representing conservation laws; this approach is named as Eulerian. In this last context, for establishing the dispersed elements equations, two possibilities come out. First, the second phase is considered as a fluid for all effects. This corresponds to the well-known two-fluid model. Second, the
924 non-continuous phase is thought of as a cloud of material elements, whose behaviour is driven by a probability density function (PDF), depending on each element variables, that responds to a kinetic transport equation similar to the Maxwell-Boltzmann one. The continuum equations for the second phase are obtained as the statistical moments of such PDF-evolution equation. In spite of the lack of a complete agreement about the final form of the equations and the constitutive relations used for the dispersed phase the Eulerian strategies continue to be attractive from an engineering point of view because of their simplicity and computational efficiency. However, the traditional closures, even giving approximated values for the mean fields, fail in the predictions of particle turbulent quantities specially in nonuniform flows. To overcome this fact considerable effort has been devoted during the last years to develop turbulence closures at the level of second moments of the particulate phase [1,6,7], but those are still in the research stage. Following this investigation direction, the relevant underlying mechanisms in the discrete element continuum equations in a dilute nonuniform gas-solids jet flow is identified as one objective pursued in this paper. To achieve this end, the momentum and fluctuating energy equations are divided into their basic terms and analysed separately. It will be shown the importance of modelling the normal stresses and it is found that they are directly implied in the spreading of the particles along the jet. A collateral result is the confirmation of a closure for the radial relative drift velocity introduced previously by some authors [2-4]. The employed modelling of particulate Reynolds stresses is consistent with previous experimental [5,8] and theoretical [1] works, which establish in the case of great inertia elements, that the normal stresses in the streamwise direction dominate over the stresses in the other directions and over the shear stresses, too. In addition, a Boussinesq closure for these last stresses can be adopted in the former limit by means of using the concept of fluctuating diffusivity coefficient that is formulated here as proportional to the particle mean square velocity in the required direction and its response time. Finally, the predictions are compared with the measurements of Mostafa et al. [5] giving a reasonable agreement in all availables mean quantities. Finally, the predicted particle normal stresses are showed versus the experiments of Hishida et al. [8], too. 2. G O V E R N I N G
EQUATIONS
Using the Dispersed Elements PDF-Indicator Function ensemble conditioned average [9,10], the following Eulerian model equations for both phases, gas and solids, in the context of isothermal dilute flows [11], are considered (the superscript d refers to the dispersed phase): GAS PHASE
Mass conservation equation: +
-
0
(1)
925 Momentum conservation equation:
2c~
)] 6ij]
(2)
+ [o~ (# + #T)[Ui,y + Uj,i]],, - If) + f y
Fluctuating kinetic energy equation k: (3)
+ o~ [P - pe] - I W ,i
O'k
,i
Dissipation rate of turbulent kinetic energy equation r
I,~ + I,~176
(.+
]
+ OL~ [ G 1 p
--
Cs
"3t-I ~
(4)
DISPERSED PHASE
Mass conservation equation:
Momentum conservation equation: (6) Fluctuating kinetic energy equation k d
__
l~.--52~.td. k7~a __ 5vi?_)i
+ c~d7~a + I W
(7)
,j
The conditioned ensemble averaged main variables are the velocities of continuous and dispersed phases, U and V, the respective volume fractions, c~ and c~d, and the pressure, P. The respective velocity fluctuations are defined with respect to U, V and are denoted as u', v'. The densities of fluid phase, p, and solids, pd, are supposed to be constant and p denotes the fluid viscosity. In the momentum equations, (2) and (6), I D is the interaction term due to the aerodynamic drag, which is modelled as proportional to the ensemble averaged relative velocity:
i~ - vo~9
~ (vj - vj)
(8)
Spherical particles with constant diameter, dp, are considered. Therefore, a standard drag law is used with CD -- 18#f(Rep)/(pdd~), where f ( R e p ) - 1 + 0.15Re ~ is a correction function of the particle Reynolds number, Rep = p l U - V ] dp/p. The volumetric forces take into account the weight and the flotability- f v _ pgj and fdV _ c~d(pd _ p)gj, where 9 is the gravity.
926 The turbulence of the continuous phase is modelled following the k - ~ strategies, whilst the Eulerian dispersed phase equations are obtained as the statistical moments of a postulated Boltzmann-like evolution equation for the dispersed elements PDF. The two evolution equation for k and c for the gas phase are solved in order to obtain the gas eddy viscosity #T p C , k/c. ak and a~ are the respective Schmidt numbers. P is the standard production term also found in single phase flow.The interaction-modulation terms I W and I ~ provide information about the presence of the second phase. I W express the fluctuating work interchanged between the two phases and is written as: -
I W - C D a d p d ( k O - kd]; \
O--
/
k
7Z
7-L+ CD1;
TL -- 0.4--~
(9)
The closure for I ~ can be found in [9]. 3. D I S P E R S E D
PHASE TURBULENCE
Although particle fluctuating transport r d (here r denotes any generic variable of the discrete elements) can not exclusively be related, in general, to the mean velocity shear (Boussinesq-Prandtl hypothesis), for the limit of large inertial particles in simple shear flows, Reeks theoretical work [1] shows that the Boussinesq-Prandtl hypothesis is feasible for momentum transport. In this work, Reeks split up the particle Reynolds stresses in two components: A homogeneous component, whose structure is the same as if the local carrier flow were homogeneous, and a deviatoric component involving terms proportional to the mean shear of both the dispersed and carrier flows. However, for long particle response times the deviatoric component of the shear stresses dominate over the homogeneous component reaching a finite value of - - i1~ S d, where ~ is the long-time particle diffusion coefficient in the transverse direction and S d the shear gradient of the dispersed phase. In addition, in this limit the diffusivity momentum coefficient, #d, is said to be proportional to ~ . Also, in this case of great inertia particles, Reeks shows that the normal stresses in the streamwise direction are enhanced over the corresponding ones in the other directions and that the former prevail over the shear stresses. According to the previous considerations, the following closure of the Reynolds stresses is proposed [11]: -p
-
2
+
S,
+
+
vj, ]
(10)
Recalling that the long term diffusion coefficient ~ is usually written as the product of the particle mean square velocity and its response time, the dispersed phase turbulent viscosity is defined as: pd -- CppdkdCD 1
(11)
Cp is a coefficient retaining the fact that, on the one hand, the general shear flows of interest are far from satisfying the constant shear and/or homogeneity and, on the other hand, that the solids have finite inertia. For the case of great inertia particles (but not infinity) a value of the order 10 .2 for Cp can be devised and has been adopted after comparison with a set of experiments.
927 The anisotropy parameters C (i) i = 1, 2, 3 are present because in a general two-phase flow the dispersed elements velocity fluctuations can be far from isotropy. In fact, in the configuration of axisymmetric particle-laden jets, considering the data of [5], an estimation for C (i) can be performed: C (x) = 2.6; C (r) = C (~) = 0.2 ((x, r, ~) are, respectively, the axial, radial and azimuthal coordinates in the jet). Consequently, the turbulent transport of k d in the right-hand side of (7) is expressed as: d
- p d k Id vjt d - ~# k j
d
(12)
ok
Here, a kdis the turbulent Schmidt number for k d, which was assigned to 0.3 in the present work. Finally, the contribution ~i~d - - p duiuj ~ d t z vi,j is a production term analogous to that appearing in the continuous phase equation (3). 4. A N A L Y S I S A N D D I S C U S S I O N
4.1. Analysis of the equations for the dispersed phase The nonuniform configuration considered in this study has been the unconfined axisymmetric turbulent particle-laden jet of Mostafa et al. [5], which is briefly described below. The transport equations given in section 2 have been solved by an elliptic finite-volume method. This method as well as the computational domain and boundary conditions employed are described in detail in [11]. The particle volume-fraction at the inlet was approximately 1.0 • 10 -4, thus justifying the absence of inter-particle collisions in the model, and the mass loading LR-0.2, defined as the ratio of particle-to-gas mass flow rate at the inlet plane. In a first stage, we are interested in the underlying mechanisms that rule out in the continuum equations describing the particulate phase. To analyse them, the equations (2)-(4) and (6)-(7) are divided in four global contributions: Convection = Diffusion
+ Source + Interaction
(13)
Here, the sources have been split in two categories. On the one hand the terms denoted by I x in the system (1)-(7), which constitute the so called I n t e r a c t i o n contribution, and, on the other hand, the rest of source terms, grouped in the denomination S o u r c e . The interaction terms in the continuous phase add only a modulation to the present equilibrium in single phase flow. For this reason they will not be showed here. The situation for the solid phase needs to be considered apart. Snapshots for the terms defined in (13) for a representative section of the jet ( X / D - 12.45, where X is the axial coordinate downstream the nozzle and D is its diameter) are shown in Fig. 1. The main character shown by the equation of axial momentum is the convectioninteraction equilibrium, since the diffusion and source contributions are pretty small compared with the former ones. Apparently, the convection terms, representing the acceleration, are equal to those forces responsible for the interchange of momentum between the phases.
p~o pd
"(T T) :~u!sn Xq punoj .gl!m3a s! ~A4 :[oj uoTssa:[dxa ue (17T) mosd "e(~I.:)OlaAaat.~,13[a~IIe!p~3s aq:l s! ~ - ~//-- ~A4 'o:[OH
p ~q(,)O-~ ~ "A4aDp ~~ :paqs!lqmsa aq uea uln!aq!l!nba aa~3ui!xoadd~3 :~u!~olloJ aq~ 'maaa qm3a jo aan~,an:[as aqa o~ sasola :~uptoo, I "uI:[a~ uo!:paauoa aqa dq paaelnpouI 'sui:[aa uo~.aaeaa~u! pue aa:mos aqa aa~ suo!~nq!a~uoa ~m3aalas asoui aqa 'STX~3 .ga~amui.gs aq~, jo avau auoz aq~, puodafl "~,old uop, -~nba mn~uaulOUi p3!p~3s aq~ jo uo!laadsu! aq~ tuoaj pa~,m3a~xa aq m3a uo!~3m:[qu! a:[oIAI
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929 (15) is directly related to other closures appeared in the literature ([2,3] and more recently [1,9,4]). In the next subsection we show how this behaviour implies the spreading of the c~d profile along the jet. Finally, we want to point out that the peaks of the diffusion and source contributions near the symmetry axis are only due to the inclusion into the sources of the extra diffusion term --2#dVr/r that appears in cylindrical coordinates. In the k d equation the behaviour of the different contributions is similar to the axial momentum ones. The convection term is roughly compensated with the interaction term, while the diffusion and source terms modulate this balance. Therefore, the change in the turbulent kinetic energy is mainly due to the interchange of fluctuating work between the phases. It must be mentioned that the source term corresponds to the specific production term for the dispersed phase, p d which is small with respect to the dominant terms, as it can be seen in the associated plot in Fig. 1. 4.2. R e l a t i o n s h i p b e t w e e n the radial relative velocity a n d t h e evolution of O/d profile along t h e jet To illustrate this fact, it is necessary to perform the following change of variables in the equations system (1)-(7)" tim _ O~fl + OLd p d
V~m =
a p U i + o~dfldyi
(16)
pm
Here, p~n is the average If Wi - Ui - Vi is the and V~ - V~TM - ~ p W i / p equation for c~d, in the
density of the gas-solid flow, and V(n is its mass-weighted velocity. relative velocity, it is possible to write: Ui - Vim + adpdWi/pm m. Therefore, introducing these expressions in (5) the following stationary case and for constant p and pd, is obtained:
p m~vi~,~p a d,i - [PaPdadWi] ,i + pm'i P~adVi
(17)
(pmV(n) , i - 0, which results from the addition of the stationary equations (1) and (5) and the continuity constraints. Substituting the expression for radial relative velocity (15) in (17) a diffusive term proportional to (papd(~d,r),r is found. This contribution, in axisymmetric jet flows, is much more important that the corresponding one implying derivatives in the axial direction and it is the direct responsible for the spreading of dispersed phase volume fraction profile along the jet. It can be summarized, from the analysis of the contributions in the equations for the dispersed phase, that the modelling of the interaction between both phases is essential as long as they drive the behaviour as a continuum of the discrete elements. Moreover, the expression of the particle normal stresses, making use of the anisotropy parameters, is directly related to the spreading of the c~d-volume fraction distribution along the jet by means of introducing an explicit diffusion term in the continuity equation for the dispersed phase, which confirms several closures used previously for the radial relative velocity. As an additional point, the modelling (10) provides the correct prediction of the evolution of the c~d profile in the experiments of Mostafa et al. [5], which will be shown next.
930 Axial Velocities (m/s) 1-D..
4.00
[] \x
3.00
.~9 o ',\ "~g \\ "~,, ~ + ~
2.00
\
9
',\
v"b,,--~ "lk~,,
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/'~, ~ ", - ..... \ ",,,
~"~,.,,
1.00
Turbulent Kinetic Energy (m2/s2)
Exp. fluid X/D=6.2 Exp. part. X/D=6.2 Exp. fluid X/D=12.45 Exp. part. X/D=12.45 Fluid X/D=6.2 Part. X/D=6.2 Fluid X/D=12.45 Part. X/D=12.45
0.60 _
0.40 0.30
7o
-
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1 []
.DD_D_[]~ ~
0.02 0.04 0.06 Radial Distance (m)
0.08
0.00 0.00
0.08
~
0.00
0.00
0
,
,
9 9
6e-05
0.08
9Exp. part. X/D=6.2 [] Exp. part. X/D=12.45 - Part. X/D=6.2 - Part. X/D=12.45
5e-05 4e-05 3e-05
g
,
Volume Fraction o~d
9Exp. fluid X/D=6.2 [] Exp. fluid X/D=12.45 - Fluid X/D=6.2 Part. X/D=6.2 - Fluid X/D=12.45 . . . . . . Part. X/D=12.45
~
Xx\
0.04
,,,
0.02 0.04 0.06 Radial Distance (m)
Shear Stresses (m2/s2)
0.12
\
0.10
0.00 0.00
0.16
[] o + -
\,\O
0.20
,,,
o
Exp. fluid X/D=6.2 Exp. part. X/D=6.2 Exp. fluid X/D=12.45 Exp. part. X/D=12.4~ Ftuid x/D=6 2 Part. X/D=6.2 - Fluid X/D=12.45 9. . . . . . Part. X/D=12.45 9
,IF~, 9 ~' \ " 9/ \ 9 / \ /,.-"'_"'~,'-z~ 9
0.50
" "'D,,.
[]
"',,\
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1e-05
0.02 0.04 0.06 Radial Distance (m)
0.08
0 0.00
.
.
.
.
.
.
0.01 0.02 0.03 Radial Distance (m)
.
.
.
.
--,--
0.04
Figure 2. Radial profiles for the different variables in the experiment of Mostafa et al. [5]. The calculated profiles of particle shear stresses have been scaled by a factor of three in order to get a higher resolution in the same plot than the gas shear stresses.
5. C O M P A R I S O N
WITH
EXPERIMENTS
In this section the obtained results employing the equations (1)-(7), including the corresponding closures are compared with the experiments of Mostafa et al. [5]. The experimental configuration was characterized by an air jet laden with glass particles flowing downwards, without inlet swirl, issuing from a 25.3 mm diameter into a 457 mmsquare cage assembly. Data were obtained at several axial positions: 15, 25, 35, 50, 75, 150 and 300 mm from the exit plane. Besides, the data at the exit plane served as initial conditions for the numerical calculation. Because the authors are interested in the application to gas turbines, the majority of the test sections are located close to nozzle, where the turbulent jet is not completely developed. For this reason, only the last two stations will be used to compare with the
931 results of the model (1)-(7). The calculations together with the experimental data are presented in Fig. 2 for the cross sections 150 and 300 mm downstream the nozzle, corresponding to the ratios X / D = 6.2 and 12.45. All the quantities are plotted in absolute values. Only the calculated particle shear stresses, for that any experimental measurement is provided, are scaled by a factor three in order to be shown in the same plot than the gas shear stresses. In Fig. 2 a reasonable good agreement between calculations and measurements in all available mean quantities can be observed. The centerline values and the shape of the axial velocity profiles are well captured, especially in the cross section far downstream. The calculated maximum values for the turbulent kinetic energy and shear stress of the gas phase are close to the experimental values, but a little bit displaced to the symmetry axis. The good results obtained for the dispersed phase fluctuating kinetic energy are due to the existence of an own transport equation for this variable, (7), with a specific production contribution. Special attention is directed to the satisfactory agreement in the solid volume fraction profiles, where the anisotropy parameters appearing in the expression of the normal stresses in the closure (10) have played a crucial role.
Normal Particle Stresses (m2/s2) 0.30
9
2
9 ~ 1 7 6 1 7 6 197 6 ~ - " ' . <> i--"" v ,~ ~,,,\
0.20
(> k
\ k _
9Exp. u' X / D = 6 . 2 [] Exp. v '2 X / D = 6 . 2 o Exp. u '2 X / D = 1 2 . 4 5 + Exp. v '2 X / D = 1 2 . 4 5
"", \'\ \
Normal Particle Stresses (m2/s2)
'\,
- u '2 X / D = 6 . 2 - v '2 X / D = 6 . 2 - u '2 X / D = 1 2 . 4 5
3.00
9
9
2.00
9 9
9o,@"
"'"
o
'
o ~
9
',
""'~'"'O:~
;5 ~ ' ' ' ' ~> o
0.10
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i\
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0.02 0.04 0.06 Radial Distance (m)
0.08
0.00
.
0.q )o
,
.
--
,
,
T
U'2 X / D = 2 0 v '2 X / D = 2 0
. . . . .
9
,p
1.00
0.00 0.00
9Exp. u '2 X / D = 1 0 [] Exp. v '2 X / D = 1 0 * Exp. u '2 X / D = 2 0 + Exp. v '2 X / D = 2 0 . . . . . . u '2 X / D = 1 0 .... v '2 X / D = 1 0
9
00
9149
9
% 0 '%
i \,
0 0
1 " : ~ - ~ +
O.Ol
0
<> o 9 +
+
+
+
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Radial Distance (m)
Figure 3. Normal particle Reynolds stresses for the experiments of Mostafa et al. [5] (left) and Hishida et al. [8] (right) based in the formula (10). u '2 corresponds to the axial streamwise direction and v '2 to the transversal direction.
In Fig. 3 the normal components of the Reynolds stresses are compared with the refered values provided for Mostafa et al. [5] (left plot). The same is showed in the right picture for another experiment performed by Hishida & Maeda is a gas-solids jet with coflow. The details are given in [8]. In both diagrams the same values for the anisotropy parameters presented in section 3 are used. A reasonable agreement is found in both cases, especially in the sections far downstream, where the jet is completely developed. This concerns the
932 axial and radial particle velocity fluctuations. At this stage, it could be suggested that the values for the parameters C (i), i - x, r, ~ are roughly the same for all jet flows laden with high inertia particles. This would be similar to what is generally accepted for the gas rms velocities, which keep the approximate relation u x,2 "u r,2 "u~, 2 _ k . k / 2 . k/2. Nevertheless, the validity of this assertion should be confirmed by additional experimental measurements. 6. C O N C L U S I O N S A study on the continuum equations that describe the particulate phase in a nonuniform dilute turbulent gas-solid jet has been performed. It is concluded, that in the case of great inertia particles the modelling of discrete elements normal stresses drives the spreading of the solid volume fraction along the jet. If a consistent expression is given for the former, based on the anisotropy parameters, a confirmation of several closures previously used for the radial relative velocity is found. Moreover, this work reveals the importance of the formulation of the interaction terms (i.e., forces and fluctuating work exchanged between both phases) for achieving the correct evolution of the mean quantities involved.
Acknowledgement We want to acknowledge to Dr. G. Kohnen his useful comments and suggestions.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
M.W. Reeks, Phys. Fluids A 5 (1993) 750. M. Ishii, Thermo-fluid dynamic theory of two-phase flow, Eyrolles, 1975. S.L. Lee and M.A. Wiesler, Int. J. Multiphase Flow 12 (1986) 99. O. Simonin, Proc. 5th Workshop on Two-Phase Flow Predictions, Erlangen (Germany), 1990. A.A. Mostafa, H.C. Mongia, V.G. McDonell and G.S. Samuelsen, AIAA J. 27 (1989) 167. K. Hyland, O. Simonin and M.W. Reeks, Proc. 3rd Int. Conf. Multiphase Flow, Lyon (France), 1998. P. F~vrier and O. Simonin, Proc. 3rd Int. Conf. Multiphase Flow, Lyon (France), 1998. K. Hishida and M. Maeda, Japanese J. Multiphase Flow 1 (1987) 59. R. Aliod and C. Dopazo, Part. Part. Syst. Charact. 7 (1990) 481. A. Prosperetti and D.Z. Zhang, J. Fluid Mech. 267 (1994) 185. S. Lain, Ph.D. University of Zaragoza, Spain (1997).
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
933
E x p e r i m e n t a l r e s e a r c h o f m o d i f i c a t i o n o f g r i d - t u r b u l e n c e b y r o u g h particles. M. Hussainov a, A. Kartushinsky a, G. Kohnen b and M. Sommerfeld b aEstonian Energy Research Institute, Paldiski mnt. 1, Tallinn, EE0001, Estonia bInstitut fur Mechanische Verfahrenstechnik und Umweltshutztechnik, Martin - Luther Universit~it Halle - Wittenberg, D-06099 Halle (Saale), Germany
Abstract
Experimental results of the modification of the grid-generated turbulence by rough solid particles in a vertical downstream channel flow are presented here. Glass particles with an average diameter of 700 ~tm are used in the experiments, while the mass loading was set to 0.05 kg dust&g air and 0.1 kg dust/kg air. The two grids with a mesh size of 4.8 mm and 10 mm made from circle rods with a sizes of 2 mm and 5 mm, respectively, were generating isotropic turbulence in the initial period. The mesh Reynolds numbers for these grids, based on a mean air velocity of U=9.5 m/s, were ReM=3045 and ReM=6340, respectively. The results of the mean velocity distribution of the carrier flow, the turbulence intensity expressed in the root-mean-square along the flow and in the different cross-sections downstream, which were obtained by using LDA, are presented. The turbulence decaying along the flow and the spectra in different centerline points downstream are shown as well. 1. I N T R O D U C T I O N For gas - solid two-phase flows as well as for gas - liquid systems the influence of the dispersed phase on the continuous phase is very important with respect to the objective of the whole process under consideration. The mechanism of turbulence modification induced by the motion of solid, liquid or gaseous particles within a flow are not very well understood in most situations. Due to the variety of parameters involved with this issue it is very difficult to extract a reasonable amount of information from a single experiment. Moreover, the quality (or the accuracy) of the results obtained by experimental investigations has a tremendous impact on the expected success of any model used to predict turbulence modification in such two-phase flow systems. Basically, there are two classes of reviews available in the literature, where an attempt was made to classify turbulent two-phase flows. One of them is due to Gore & Crowe [2], where the most relevant experimental investigations are analysed according to the change in turbulence intensity depending on a length scale ratio Dp/LE, where Dp represents the particle diameter and LE the integral length scale of the flow, respectively. This analysis resulted in a criterion for Dp/LE in order to decide, if the particles attenuate the single phase turbulence (values below Dp/LE-0.1) or augment it (values above Dp/LE -- 0.1). In a later publication of
934 Gore & Crowe [3] different parameters were suggested, which may have an influence on the change of turbulence intensities, such like the density ratio 9p/9, the flow and particle Reynolds numbers, Re and Rep, Dp/LE, the relative turbulence intensity, and the volume concentration. There are no quantitative results available until now concerning the parameters listed above with respect to turbulence attenuation and augmentation. Hetsroni [5] in his review argued, that such a criterion for determining turbulence attenuation/augmentation could be found solely in the particle Reynolds number Rep in such a way, that for Rep>400 wake effects become predominant leading to an enhanced turbulence production and hence to turbulence augmentation. Besides, different results are obtained for wall-bounded turbulent shear flows and free shear flows. Schreck & Kleis [10] experimentally studied the mechanisms of turbulence decay behind the grid in water flow loaded by two types of particles, namely plastic and glass beads with a particle size about 650 lam. In this work the fluid velocity was around 1 m/s with the mesh Reynolds number ReM=l.56"104 based on a mesh size of M=17.8 mm. Their experiments have shown, that both kind of particles did attenuate the turbulence of the water flow for various volume fractions of the dispersed phase within the range of 0.4-1.5%. The spectral energy density behaved different depending on the velocity component under consideration. Comparing the results for the laden two-phase flow with the corresponding one for the singlephase flow they found, that the growth of the energy took place at large wave numbers for the axial velocity component. For the transversal component the opposite could be observed, namely a reduction of the spectral density of the energy at this range of wave numbers for both types of particles (plastic and glass spheres). The distribution of the total turbulent energy, namely E=Eu+2Ev (Eu and Ev are the energy of the axial and transversal component of the velocity fluctuations, respectively), has shown a decrease of the spectral density of the energy E in this wave number range. Kulick et al. [8] studied a two-phase flow in a vertically downward channel flow configuration. They used comparatively small particles, namely copper particles with an averaged diameter of 70 lam and two charges of glass particles each having 50 and 90 lam, respectively. For mass loadings up to 40 % a reduction in turbulence intensity was observed. The work of Lance & Bataille [9] was devoted to the investigation of the influence of particles (air bubbles) on the turbulence intensity in a water flow behind the grid. They observed enhancement of turbulence caused by the bubbles. The distribution of the spectral energy density reveals a reduction in the small wave number range due to the bubbles, whereas the large wave numbers were fed with energy and took on higher values compared with the single phase flow. 2. EXPERIMENTAL SET-UP The experiments were carried out in a disconnected vertical two-phase wind-channel with closed test section (Figure 1). The wind-channel consists of an intake (1) with a diameter of l m, followed by a contraction (2) in order to lower the streamwise turbulence level. Its contour was profiled by the Vitoshinsky formula according to Gorlin [4] characterized by an area ratio of 6.25, a transition duct (3), that smoothly changed from a round inlet cross-section to a square outlet cross-section, the test section (5), and the diffuser (6). The test section was made of plexiglass and had a length of 2 m. At the initial upper cross-section its dimensions
935
9\ 8
7
E
?C/AT
! [L. . . . 2 i m W ~
Optics
+
r -i
Figure 1. Flow facility: 1-intake; 2-contraction; 3-transition duct; 4-turbulence grids; 5-test section; 6-diffuser; 7-seeding generator; 8-particle feeder; 9-cyclone separator. were 200 x 200 mm, downstream it increased gradually up to 205 x205 mm at the exit. Such a geometry of the test section allowed to compensate partially the growth of the boundary layer along the walls and, thus, to keep the flow core to be uniform along the test section. There was a possibility to install the optical glass windows along the full length of the test section in order to have the access of the LDA-technique. Air was flowing in the downward direction with a mean velocity of U=9.5 m/s. This downflowing air was created with the help of a suction fan. The particles of the dispersed phase were brought into the flow by a special particle feeder (8) installed in the intake (1). They were separated from the gas flow in the diffuser (6) and then conveyed by a vertical pneumatic transport system into the cyclone separator (9) within the circuit for recovering the solid phase. The reservoir of feeding the solid phase, (8), was loaded by particles taken from the cyclone separator directly before the beginning of the experiments. The capacity of this reservoir was 40 liters. In this way 90 kg of glass beads were available in order to carry out the experiment in a continuous way for 20 minutes, while the mass loading of the solid phase was about 0.1 kg/kg. Particles made of titanium dioxide TiO2 having a particle size in the range of 0.5-2 gm were used as particle
936 tracers for visualizing the air flow. The particle-tracers were brought into the flow through the intake (1) by using the seeding generator (7). The operation principle of this generator was based on the principle of fluidized bed. A turbulence generating grid (4) was placed at the inlet of the test section (5). Two types of grids with square meshes were used. Their characteristics are presented in Table 1. Table 1 Characteristics of the turbulence producin~ ~rids used in the experiments Grid Mesh size M [mm] M/d ReM Solidity Sh/S 1 4.8 2.53 3045 0.487 2 10 2.5 6340 0.49
3. MEASUREMENT TECHNIQUE The forward-scattering Laser Doppler Anemometer (LDA) was used to measure the instantaneous velocities of the gas and particles in the two-phase flow. The LDA consists of two channels, each channel for the respective phase of the dispersed flow. The transmitting unit of the LDA channels with a 26 mW and 10 mW laser formed two different measurement volumes in order to distinguish between the velocities of the two groups of particles, the TiO2 seeding particles and the glass particles with an averaged diameter about 700 lam. The channels were organized by the module principle. Hence, it was possible to easily get access to the system depending on the peculiarities of the problem to be solved. The tuning of each LDA channel for registration of its own particles was based on an amplitude discrimination of the Doppler signals by means of changing the parameters characterizing the measurement volume and the geometrical conditions of the receiving optics, and by varying the sensitivity of the photomultiplier as well. The parameters of the LDA system for studying the gaseous flow laden with solid particles are summarized in Table 2. During the experiments the gaseous phase channel and the channel of the dispersed phase were tuned by the maximum data rate of Doppler signals from tracer and glass particles. The experiments carried out in a single-phase wind-channel have shown a continuous registration of Doppler signals from particle-tracers by the channel of the carrier phase. In the case of the two-phase flow, when 700 gm glass particles were brought into the flow, micron tracer particles sedimented on the surfaces of the glass particles coating their surfaces and resulting in the modifying their optical properties. This means, that we dealt with a situation, where the scattering properties of the glass particles have been modified during the experiments leading to a degradation of the Doppler signals in the direction of the signal detection. As a result, a continuous registration of signals in the channel of the dispersed phase took place only in the case, when the glass particles were still clean. This situation has slightly improved after decreasing the fringe spacing of the interference field to 24.83 gm and, correspondingly, increasing the total number of fringes in the measurement volume to 66. Consequently, only clean glass particles with an averaged diameter of 700 gm were used. The instantaneous velocities of the gaseous phase were measured by the LDA system using the parameters presented in Table 2. 15 series or measurements were carried out in order to obtain the data rate for calculations of the power spectra of fluctuation velocities of the
937 gaseous phase at a certain distance from the generating turbulence grid along the axis of the flow. Each series of the measurements included 2000 samples of the axial component of the instantaneous velocity of the gas. Those series were registered only, if the registration times were less or equal 0.4 seconds. The initial assembly of samples of the instantaneous velocities included 30000 samples with an average interval between each single sample of 150 laS, which results in the average frequency of 6.6 kHz for the registration. Each series of the instantaneous velocities was filtered by amplitude and time discrimination. Table 2 Parameters of the optical system of LDA for studying the gas flow laden with solid particles The channel of the Gaseous phase Dispersed phase
Transmitting optics Wavelength of the laser [~m] Focal length of the front lens [mm] Beam separation [mm] Diameter of the measuring volume [lam] Fringe spacing [~tm] Fringe number
0.6328 300 35.0 87.9 5.43
0.6328 300 3.85 1651 49.30 (24.83)
16
33 (66)
3.0 o 58
5.8 ~ 58
250
422
Receiving optics Off-axis angle Focal length of collimating lens [mm] Distance from the measuring volume to the receiving optics [~m]
4. RESULTS AND DISCUSSIONS The main investigations were performed in different cross-sections of the channel at a distance from the grid X=63, 95, 126, 251, 383, 503, 692, 1264 mm. These distances normalized with the respective grid mesh size correspond to the following values: 9for grid 1 (M=4.8 mm): X/M= 13.1, 19.8, 26.3, 52.3, 79.8, 104.8, 144.2, 263.3; 9for grid 2 (M=10 mm): X/M=6.3, 9.5, 12.6, 25.1, 38.3, 50.3, 69.2, 126.4. The energy spectra for the streamwise velocity fluctuations of the gas have been determined at the axis of the channel for the laden and unladen flow in the above mentioned cross-sections.
4.1 Mean velocity distributions The basic series of experiments were carried out, when the mean velocity of the gaseous flow (air flow) was about 9.5 m/s. Small deviations from this mean flow velocity were taken into account during processing of the experimental data in the way, that all averaged values of fluctuations U~ms and power spectra of velocity fluctuations were corrected to the mean velocity 9.5 m/s. Figure 2 depicts the profiles of the averaged streamwise velocities and their fluctuation values above the grid for the unladen and laden flow. It is evident, that the non-uniformity of the flow for the averaged velocities was less than or equal to 0.5% and for the fluctuation
938
values less than or equal to 6% for the two-phase flow. Figure 3 presents the profiles of the above-mentioned parameters and the particle mass distribution in the cross-section X=293 m m of grid 1 (M=4.8 mm).
1.1~ ,
T i
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i
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.
.
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o71
-
8 ............~ - : . - . - ' ~,2
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i ~
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02. Q1
o
-if) JE) -43 -33 -33 -10 0 10 33 33 43 93 83 y, mm
Figure 2. Profiles of the averaged velocities Figure 3. Profiles of the averaged velocities, and their fluctuations in the initial cross-section, their fluctuations and the particle mass loading at X=293 ram, grid 1. 4.2 Intensity of turbulence distributions U 2
Figure 4 and 5 present the behaviour of U
m
*2
U 2 2 U rms
along the channel behind the two grids
for the laden and unladen flow.
600
~e = 0.051 9 500
a~ = 0
U2
4o0
.
~e = 0.087 9 0
a~ = 0.099 ~e = 0.047
36O
~e
~/
a~=O
u2____ ~
9
UP2 300 200
~
20
40
60
5O 0 80
0 . 0 ~ = 0 ~
l
100
120
X/M
140
160
180 200 220
Figure 4. Modification of the turbulence intensity behind grid 1. Unladen flow conditions (thin line) versus laden flow conditions (bold line).
~
9
::::
~e= 0.087 = 0.064 Ee= 0.089
100
~e = 0.085 0
~e =
= 0.066
150
~e = 0.087
~ ~e = 0.063
lOO
=0.051
~
0
10
20
30
40
50
X/M
60
70
80
90
100 110
Figure 5. Modification of the turbulence intensity behind grid 2.
The decay curves show, that the turbulence intensity of the flow behind the grid decreases up to the fixed cross-section (X/M--90 for grid 1 (M=4.8 mm) and X/M--40 for grid 2 (M=10
939 mm) and it remains almost constant more downstream. This constant level of turbulence intensity approximately corresponds to the turbulence of the flow in a channel without a gridgenerating turbulence and is conditioned by the initial turbulence. From these figures one also can conclude, that the loading ratio of the flow about 0.05 kg/kg leads to a decrease of the turbulence intensity of the gas phase. This decrease is more pronounced for grid 1 (M=4.8 mm) in comparison with grid 2 (M=10 mm). According to these experimental results the turbulence intensity of the single-phase was about 3.5% at the final stage of the turbulence decay. This is essentially larger than it was found by other experimentalists (e.g. Hinze [6]) and may be caused by the high initial turbulence level of the flow, which was in front of the grid due to the presence of the special particle feeder in the intake (8, Fig.l). The energy decay curves are adequately represented by: 2 -C
-A
.
(1)
U rms
The corresponding constants for the different flow conditions can be found in Table 3. Table 3: Constants in the deca~, curves for the single-and two-phase flow for different l~rids Single-phase flow Two-phase flow A C Grid A C 1 (M=4.8mm) 0.22 8.19 1.24 8.92 2 (M=10 mm) 3.09 11.42 3.7 12.55 Unfortunately, the mass loading was different for each data point on Figures 4 and 5. There were some reasons for that. Each series of experiments were conducted separately in laboratory. In fact, the circuit for recovering the solid phase was disconnected. Besides, a capacity of the receiving bin of a cyclone separator (9, Fig. 1) was limited and therefore each test series had a time limit which were characterized by its own loading ratio of the flow. Some time was required for the flow adaptation in order to reach the same magnitude of mass loading of flow, but, in this case we would lost a definite mass of the glass particles and have no time to measure the fluctuating velocity properly. During the measurements we skipped the flow adaptation process and, therefore, there was some difference in mass loading between various test series. 4.3. Energy spectra The initial energy spectra of the streamwise velocity fluctuations of the gaseous phase were smoothed with the averaging procedure over ensemble of neighbouring values. For the given energy values they are calculated as follows: 1 i+4
E'(f i ) -
~s
E(f i ), i - 5, N - 4,
(2)
2i-1 s t ( f i ), i -
(3)
j=i-4 1
E'(fi ) - 2 i - 1 j=,
1,2,3,4,
N
1
E'(fi ) - 2(N - i ) +
~_ E(fi),i _ N _ 3,N,
1 j=2i_ N
(4)
940 where E(f i ) and E'(f i ) are the initial and smoothed energy density at the frequency fi, respectively. Such smoothing allows to imagine the influence of particles on the energy spectra as a whole, although at that time the information about the minor details of the spectrum vanishes. Figures 6 and 7 show the smoothed energy spectra for the single and twophase flows for grid 1 at X=383 mm and X=1264 mm. For comparison the respective smoothed energy spectra are shown in Figures 8 and 9 for grid 2 at the same positions. One can recognize, that the influence of mass loading in the case of the grid 2 is not as strong as for the grid 1. In the high-frequency range of the spectrum (frequencies higher than 1000 Hz) a decrease of the turbulence energy was observed in the case of grid 1 almost in all considered cross-sections. 1.E-(~ 84
1.ECs
'~": ''"~'::'"+?I":": :"":h"i:iii i!!'-:-:-i: :-'"::-'--:":-:""::--:":F":' I-!!!!t :F-I":--H: '-":"-' ~'i-+'::"!:i"!' 1 '::-":i":"::-i:"4:' '-"i?i: :[ 1.E-03 84
1.E(B
1.EO4
1.EO4
::::::::::::::::::::::::::::::: ::::i::i::::::i:: ~`~`~.~.~i~i~.~i~i~`~!~`~:~:~i~
E',m2/s 1.E-C6
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E',m2/s 1.E~
i
1.E-~
1.E06 1.E07
1.E-07 10
100
f,Hz
1000
1(3330
1.E-03
1.E-04
E', m2/ s
i!"ii iiiliili
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1.~
!
Ni]i
f,Hz
1030
1Q330
..............i i~ ........... ~........................ ::::~.::..::.+.:.~~:::::: I:.:::::i::::~::~:::~:-:: .:I+.:~.++.~:.::::5 ~.:.~::::.-- a~(/57 t[it:?:~ ....================================== ::~
I ::ii:iT:iliI:~
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iiiiiiiiiiiiiiiiiiiiiiiiiiii ' iitli
..............~:,:-:::-:,:.::-:~,:,:,:,:,:~-::,:,:i ................. ............. i:::;,:.:i ................ -~,...-,~.:-~,~.:~,~,:~:~h~.~!+~,!~+:!-~,~i ~-
1,e-
....
...............r........i i.......iiii -
1.Eo3 iiiiiiiiiiiiill~ ~ ~
I.E-05 ............... i......... iT!iii ,........................... iii ....... i ~: ............ ii ]iii![
ii
.......... ..... 100
Figure 7. Smoothed energy spectra for the single and two-phase flows (X=1264 mm) grid 1 (M=4.8 mm).
Figure 6. Smoothed energy spectra for the single and two-phase flows (X=383 mm), grid 1 (M=4.8 mm) 1.E-02
.......... 10
~?~:~i~}~i~}~i~?~:~:~:~:~..:~?~i~i~!~)?
...............i.........,Iiiiigl ..........il......iiiil~I~i~lW
10
100 f,Hz
1000
10000
Figure 8. Smoothed energy spectra for the single and two-phase flows (X=383 mm), grid 2 (M= 10 mm)
10
103 f, Hz
1030
10330
Figure 9. Smoothed energy spectra for the single and two-phase flows (X= 1264 mm) grid 2 (M= 10 mm).
For a detailed analysis of the spectrum shape in the frequency range of the energy containing eddies, 400-2000 Hz, the spectra were approximated by the power function y - Ax -n . The dependencies of the exponent n on the distance from the grid are presented in
941 Figure 10 for grid 1 and in Figure 11 for grid 2. For comparison a value for n=5/3 (dotted curve) related to Kolmogorov's law (the inertial subrange) is included there as well. As follows from Figure 10, the pronounced decrease of the turbulent kinetic energy of the carrier fluid takes place also in the range of the energy containing eddies for grid 1 (M=4.8 mm). Figure 11 shows, that in the case of grid 2 (M=10 mm) there is no significant effect of mass loading on the modulation of the turbulent kinetic energy in the given range.
2.5
[] ,,
,,
~e=O ,
2.5
--I--ae>O
I
2
_.
1.5 n
n
1.5 -~ I
1 0.5
0.5
-I I
0
200
400
600
800
1000
1200
X, m m
Figure 10. Dependence of exponent n in the power function on the distance from grid 1.
1400
ol 0
200
400
600
800 X, m m
1000
1200
1400
Figure 11. Dependence of exponent n in the power function on the distance from grid 2.
Next, the most relevant parameters related to the dispersed phase are presented in Table 4. The Stokes number was based on the turn-over time. To determine the volume concentration let us use the formula [3- pa~U/(ppUs), where ~e is the integral mass loading of the flow. Further, one can find the ratio between the inter-space particle distance between the particles and the particle size ~ / 8 - (n/6[3) 1/3 - 1 by the results of Kenning & Crowe [7]. Table 4 Relevant parameters characterizin~ the dispersed phase Grid 1 Grid 2 ae [kg/kg] 0.09 0.09 U-Us [m/s] 1.5 2 Rep 70 93 Iu [S] 4.5 4.5 StE 2.47.102 1.34.102 ~,/8
118
118
8/qK
6.31
5.22
According to Gore & Crowe [2], the character of the influence of the particles on turbulence modulation of the carrier fluid in depends roughly on the parameter 8/LE, which represents the ratio of the particle size to the integral length scale of the turbulence. If 8/LE>0.1 turbulence enhancement takes place, otherwise, 8/LE<0.1, turbulence is attenuated. In our case 8/LE=0.06 for grid 1 and 5/LE=0.03 for grid 2. Thus, the particles must attenuate
942 the turbulence of the carrier flow that was observed in the experiments. On the other hand, the diagramme of Elghobashi [1] shows the influence of the inter-particle spacing normalized with the particle diameter on turbulence modulation as well. In our notation this ratio is described by X/8 and takes on a value of ~/8=118. This is close to the limit of X/8=100 or values below, where the two-way coupling regime is of considerable importance. Hence, we are in a transition region, where we can expect to have some influence of the particles on the turbulence of the carrier phase. Based on the calculated values of StE in combination with the parameter X/8 we shift along the vertical axis in the diagramme of Elghobashi [ 1] and tend to drive to the domain, where the influence of particles on turbulence modulation would take place resulting to turbulence enhancement, which was not observed in our experiments. One result of the present investigation is the detection of an increase of the decay rate of the turbulence intensity in the two-phase flow, which is accompanied by the reduction of spectral energy in the high-frequency range. This result agrees with the data by Schreck & Kleis [10], who observed an increase in the energy dissipation rate and a reduction of the spectral energy in the high-wave number end. CONCLUSION The conducted experimental investigations may be considered to be the first stage of studies, since it revealed the influence of particles on turbulence modulation in case for diluted two-phase flow with low particle mass loading. Apparently, one can expect the generation of turbulence by particles with an increase of the particle mass loading. In this connection it would be expediently to continue experimental investigations of the influence of the particles on the turbulence decay of the carrier fluid for comparatively higher mass loadings (o~--0.3). Moreover, the influence of the Stokes number, mainly influenced by a variation of the particle size and the material density, on turbulence modulation of the carrier phase is of major interest as well. ACKNOWLEDGEMENT The authors would like to extend our appreciation to the Volkswagen - Stiftung for supporting the present work.
REFERENCES 1. Elghobashi, S., Appl. Sci. Res., 48, (1991) 301. 2. Gore, R.A. and Crowe, C.T., Int. J. Mult. Flow, 15, No. 2, (1989) 279. 3. Gore, R.A. and Crowe, C.T., J. Fluids Engng., 113, (1989) 304. 4. Gorlin, S.M., (in Russian), (1970). 5. Hetsroni, G., Int. J. Multiphase Flow, 15, No. 5, (1989) 735. 6. Hinze, J.O., McGraw Hill, New York, (1975). 7. Kenning, V. M. and Crowe. C. T., Int. J. Multiphase Flow, 23, No. 2, 403, (1997). 8. Kulick. J.D., Fessler, J.R. and Eaton, J.K., J. Fluid Mech., 277, (1994) 109. 9. Lance. M. and Bataille, J., J. Fluid Mech., Vol. 222, pp. 95 - 118, 1991. 10. Schreck, S. & Kleis, S.J., J. Fluid Mech., 249, (1993) 665.
Engineering Turbulence Modelling and Experiments - 4 W. Rodi and D. Laurence (Editors) 9 1999 Elsevier Science Ltd. All fights reserved.
943
Interaction b e t w e e n leading and trailing elongated bubbles in a vertical pipe flow C.Aladjem Talvy, L. Shemer and D. Barnea Department of Fluid Mechanics and Heat Transfer Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel
The motion of elongated air bubbles in a vertical pipe filled with water is studied quantitatively using video imaging of the flow and subsequent digital image processing of the recorded sequence of images. Experiments are carried out to determine the influence of the separation distance between two consecutive bubbles (liquid slug length) upon the behavior of the trailing bubble in vertical slug flow. The details of the trailing bubble acceleration and merging process are observed and the instantaneous parameters of the trailing bubble, such as its shape, velocity, acceleration, etc., are measured as a function of the separation distance. The leading bubble is found to be unaffected by the trailing elongated bubble.
1. INTRODUCTION Slug flow pattern is encountered in many industrial two-phase gas-liquid flow applications being one of the basic gas-liquid flow patterns, which take place naturally inside pipes over a wide range of flow parameters. It is characterized by periodic alteration of liquid slugs and long bullet-shaped bubbles. In vertical pipes, the flow pattern is axially symmetric on the average. The elongated (Taylor) bubble occupies the central region of the pipe, while a thin liquid falling film flows around it adjacent to the pipe wall. Each slug sheds liquid in its back to the subsequent film, which is then injected into the bubble wake as a circular wall jet, producing a complicated vortical flow in the bubble wake. The details of the movement of those elongated (Taylor) bubbles are extremely important for developing reliable theoretical models of liquid-gas slug flow in pipes (Taitel and Barnea 1990). The rise velocity of a single elongated bubble in stagnant and moving liquid in a vertical pipe has been studied by numerous researchers, both theoretically and experimentally (Dumitrescu 1943; Davies and Taylor 1949, Nicklin et al. 1962, Collins et al. 1978; Mao and Dulder 1991). However, this information is sufficient only when fully developed slug flow is considered. In such a flow, the liquid slug ahead of any Taylor bubble is long enough so that the trailing bubble is uninfluenced by the wake of the preceding one. However, for the case of undeveloped slug flow, the bubbles are influenced by the wake of their predecessor. This leads to their coalescence and to the variation of the flow structure along the pipe (Moissis and Griffith 1962; Shemer and Bamea 1987, Taitel and Barnea 1990). Pinto and Campos (1996) applied differential pressure transducers to investigate the coalescence process of pairs of Taylor bubbles rising in a vertical pipe.
944 Recently, Polonsky et al. (1999) studied in detail the two-dimensional velocity distribution in the wake of a rising Taylor bubble. The velocity field in the near wake region is found to be highly turbulent due to the strong shear as a result of the penetration of the falling film into the liquid slug. Away from the near wake region, the velocity profile gradually becomes less disturbed, until it attains a fully developed profile. The objective of the present study is to perform controlled experiments on the interaction between two consecutive elongated bubbles rising in a water column as a function of the lengths of the two Taylor bubbles and the separation between them. In these experiments, digital image processing and Particle Image Velocimetry (PIV) technique are employed. The results of the present experiments contribute to an accurate estimation of the stable liquid slug length.
2. EXPERIMENTAL FACILITY AND PROCEDURE Experiments are performed in a 4m long transparent Perspex pipe with an internal diameter of 2.5cm. The pipe is filled with tap water, and a desired sequence of bubbles of prescribed lengths and intervals can be injected using central compressed air line and computer-controlled valves. The pipe is equipped veith three, 1m long, rectangular transparent boxes filled with water. The boxes serve to reduce image distortion and to provide cooling against illumination. The flow field is illuminated either by a thin Ar laser-generated light sheet or by a number of halogen lamps. The flow visualization is performed using two black-and-white interlaced NTSC video cameras. The cameras are mounted at two locations along the pipe with slightly overlapping fields of view. The computer synchronizes the operation of the cameras and of the bubble-injecting valves. The process of bubbles propagation and coalescence viewed by the two video cameras is recorded by the computer using a frame grabber and the appropriate custom-written sol, ware. The images are digitized at the rate of 30 frames/s and stored on a hard disk. The interlaced video images are then split into odd and even fields. The spacing between the lines is filled using linear interpolation between the gray levels in two adjacent lines. Hence, the effective rate of data acquisition is 60 images/s. In order to study the velocity field between the bubbles, the Particle Image Velocimetry (PIV) technique is used. Small neutrally buoyant polystirol spheres 20-40 lam in diameter are added to water. The particles contain fluorescent dye and emit light in the yellow-orange domain of the spectrum when illuminated by the laser. The double exposure necessary for PIV is obtained using either image interlacing or a polarizing shutter. More details about the experimental procedure and the data processing are given in Polonsky (1998).
3. EXPERIMENTAL RESULTS In the experiments, couples of bubbles of different lengths and with a large range of distances between them (from 60 pipe diameters to zero) are tested. These data are used to reconstruct and understand the process of the interaction between the two Taylor bubbles. An example of a recorded series of the two interacting bubbles is given in Fig. 1.In order to obtain better spatial resolution in the vertical direction, the cameras are rotated by 90 ~, so that
945 in the video images the bubbles appear to move horizontally from lett to fight. Fig. 1 represents a sequence of flames obtained with a time interval of 1/60s between the consecutive images. In order to emphasize the bubble shape, edge enhancement is performed in this Figure. It can be clearly seen here that the trailing bubble accelerates and eventually coalesces with the leading one. Note also that the shape of the nose of the trailing bubble, which propagates in the wake of the leading one, is highly disturbed and unsteady.
Q
F_~_-_'-'-""-r"
O
lip u - _ - -
" - i
-_-2--------?_ _" --
8
I["..'-
-
-
-
-r-
-
.
.....
I
I
I
Q
i Figure 1. Example of a recorded sequence of coalescence of two consecutive Taylor bubbles, illumination by the halogen lamps. The leading bubble is 4.2D long. 3.1 Leading
bubble
movement
The effect of the trailing bubble on the movement of the leading one is analyzed first. The rise of the leading bubble is proved to be essentially undisturbed by the approaching trailing
946 bubble, even when the coalescence is imminent. This can be observed qualitatively in Figure 1, where the leading bubble appears to propagate with a constant velocity. The image processing technique allows performing accurate measurements of the instantaneous velocity of each bubble. 300 E 250 E
z~
"5 _o
200
_~
150
e~
=k
"~ 100 e~
~
Trailing bubble at _ the distance of 6D 50
................................
ICoalescence I
_J
0
I
I
I
I
I
I
0.5
1
1.5
2
2.5
3
3.5
Time, s
Figure 2. The leading bubble rise velocity obtained at different stages of the two bubbles approach.
6
--
Leading
r/I
4-0
n
II~
bubb,e 1]i/ ,
lil ',1
2 --
0
2
4
~l bubble
6 8 Frequency, Hz
10
12
14
Figure 3. Spectra o f the bubble bottom oscillations for a single Taylor bubble and the leading
bubble just before coalescence. The temporal variation of the instantaneous velocity of the leading bubble determined from the bubble nose shitt between two consecutive images is plotted in Fig. 2. The relative location of the trailing bubble is also indicated in this figure. It can be clearly seen that the velocity of the leading bubble remains essentially constant. This finding supports the conjecture that the translational velocity of a Taylor bubble is determined solely by the flow field ahead of it. In stagnant liquid, the propagation velocity of a single bubble is equal to the
947 drift velocity Uo given by Uo=0.35(gD) ~/z, where D is the pipe diameter and g the acceleration due to gravity, Dumitrescu (1943). In the present case, for D--25mm, Uo=17.3cm/s, in agreement with the mean value obtained in Fig. 2. A single Taylor bubble propagating in a vertical pipe is characterized by vigorous resonant oscillations of the bubble bottom (Polonsky et al. 1999). The bubble bottom movement of the leading elongated bubble followed by the trailing bubble, up to the moment of coalescence, was therefore studied here as well, and compared with the behavior of a single bubble. Power spectra of the resulting oscillations of the bubble bottom (in the frame of references moving with the bubble drift velocity) are presented in Fig. 3. Comparison of the two curves in this figure indicates that both spectra have a pronounced resonant character, with a nearly identical peak frequency. The trailing Taylor bubble does affect neither the mean nor the time-dependent parameters of the leading Taylor bubble. The verification of this conjecture is essential in the present study, since it allows to assume constant known rising velocity of the leading bubble and thus to calculate the instantaneous distance between the two bubbles, even when the leading bubble is out of the frame. 3.2 Flow field ahead of the trailing Taylor bubble
a)
r'~':"'
t,
i~l*-- i I;~"
b)
,
, [ ""
e)
UP:../',
'
,.i
: m
,d. dr162 " ' ~ "
.,',,..~.',._
f
.,,~It'l"
.-
fll
, 4
-
,11
'~
I "
/,
7 " "
9 '
,.?"
"
't '".
;
VELOCITY SCALE
t
/[
Figure 4.2D flow field in the liquid adjacent to the bubble bottom
250cm/s
~
948 For a Taylor bubble rising in a moving liquid, Nicklin et al. (1962) suggested that the translational velocity of the bubble is a superposition of its drift velocity Uo and the contribution due to the mean liquid velocity Ur.: (1)
U, - CUr. + U o
The value of the factor C in (1) depends on the velocity profile in the liquid ahead of the bubble, and can be seen as the ratio of the maximum to the mean velocity in the profile. Hence, for turbulent flows, C 1.2, while for the laminar pipe flow, C 2 (Nicklin et al. 1962, Collins 1978, Grace and Cliff 1979). A trailing Taylor bubble entering the wake of the leading bubble is affected by the flow field in the wake ahead of it (Shemer and Barnea 1987). The presence of a strong annular jet in the leading bubble's wakes region results in a negative (downstream) velocity at the pipe cross-section periphery, and thus a high upward velocity in the central region of the pipe. It is generally assumed that the bubble nose follows the local maximum velocity in the pipe cross-section (Collins 1978). This fact explains the acceleration process of the trailing bubble relative to the leading one, which propagates in the stagnant liquid.
9
^
.
4
"
v
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~
t
.
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,'
~
i
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"""
.
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.
-.~
.
.
.
.
.
'
9
.
.
r
t" 9
.
.
,.
.
.
' ~
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.
~
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,
9
,
,
v
i
i
A
v
~
i
.,
A VELOCITY SCALE
T /
2.5cm/s
Figure 5.2 D flow field in the far-wake region (22D and 30D from the bubble bottom) The trailing bubble motion therefore cannot be analyzed without the understanding of the flow upstream of it. In figures 4a-4c the recorded consecutive PIV images of the flow behavior behind a rising Taylor bubble are presented. More detailed study of the flow in the near wake region is currently under research, the pictures in these figures represent only preliminary qualitative results. Fig. 5a, b depicts the flow field in the far wake of the bubble. Note the velocity scales in figs. 4 and 5 differ by 2 orders of magnitude. The flow upstream of the trailing bubble results from the leading bubble wake. In the region directly behind the bubble bottom, the wake is characterized by the presence of torroidal vortices created by the impact of the downward liquid film entering the upward moving liquid slug. These vortices cause a strongly disturbed unsteady flow in the liquid slug. Due to viscous dissipation, they gradually lose intensity with the distance from the leading
949 bubble bottom. Polonsky et al (1999) demonstrated that the vorticity remaines noticeable even at distances up to 60D from the Taylor bubble. 3.3. Oscillations of the trailing bubble nose If the bubble nose follows indeed the local maximum velocity in the pipe cross-section, its shape should be affected by the vortices encountered during its approach towards the leading bubble. Figure 6 represents a sequence of instantaneous nose outlines for two cases: rising of a single Taylor bubble and of a trailing bubble in the wake of a leading one. a)
b)
,13))'l:iiS~))'ii))t~'i~i))).,!,/;),.))) ) ) ) ))l ))'~)'i 11ll ) t l l ) )lli)lt))\~ ~,i~.\\~q,;:~~ Figure 6. Consecutive outlines of the bubble nose: a) single propagating Taylor bubble, b) trailing Taylor bubble up to coalescence Fig 6 demonstrates that while in the case of a single rising Taylor bubble, the outlines are equidistantly distributed and remain round shaped; the trailing bubble nose interfaces do not retain a constant shape due to the influence of a leading bubble wake. The effect of vortices in the wake region of the leading bubble on the nose of the trailing bubble is further studied in fig. 7. When the trailing bubble approaches the leading one, the nose of the trailing bubble starts "swinging" from one side of the pipe to the other. Figs. 7 a 7c demonstrate that the oscillations of the nose become stronger as the distances between the bubbles decreases. Certain increase in the trailing bubble velocity as it approaches the leading bubble can be seen in fig. 7c.
a)
b)
. . . .
C)
......
'I
i!
I
Figure 7. Trailing bubble nose outlines, a) 7 to 5.5D behind the leading bubble; b) 6 to 4.5D behind the leading bubble; e) 4 to 2D behind the leading bubble.
950 The image processing technique applied here allows determining temporal variation of the instantaneous radial position of the tip of the trailing bubble nose. The results plotted in fig. 8 provide an additional way to demonstrate the increase the amplitude of the nose oscillations with decreasing distance between the bubbles. The observations, however, indicate that the final impact between the bubbles occurs always around the pipe axis region (of. fig. 1). At the final stage of the pursuit, the trailing bubble becomes sharp-pointed due to the intense toroidal vortex close to the leading bubble bottom.
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I
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-2
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3.4 Trailing bubble propagation velocity In undeveloped slug flow, the translational velocity of a Taylor bubble, which is in the wake region of the leading bubble, can not be determined using (1) and varies in time and space. For modeling undeveloped slug flow and determining slug length distribution, the information on the variation of the trailing bubble axial velocity with distance from the leading bubble thus becomes crucial. Vortices upstream of the trailing bubble in the liquid slug can either accelerate its propagation or slow it down, depending on the instantaneous radial location of the bubble tip and the rotating sense of the vortex (cf. figs. 4 and 5). Since the strength of these vortices varies significantly with the distance from the leading bubble bottom, the distance to the leading bubble apparently strongly affects the axial trailing bubble velocity. The oscillations of the trailing bubble velocity therefore become more violent as the trailing Taylor bubble approaches the leading one. The measured instantaneous approach velocity of the trailing bubble just prior to the coalescence is presented in fig. 9. The rate of approach may locally become negative, but on the average, the acceleration towards the leading bubble is extremely high: just before the coalescence, the trailing bubble can attain velocity exceeding l m/s, more than 7 times that of the leading bubble.
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Figure 9. The variation of the instantaneous velocity of the trailing Taylor bubble prior to coalescence. Note that when the contact between the bubbles is established, the bubbles do not coalesce immediately and propagate for some time with the velocity of the leading bubble (cf. Fig. 1).
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Figure 10. Variation of the trailing bubble velocity with the separation distance between the bubbles The velocity of propagation of the trailing Taylor bubble, normalized by that of the leading one, is presented in fig. 10 as a function f the separation distances between the bubbles. The
952 model prediction of Moissis and Griffith (1962), as well as the correlation of the experimental data by Pinto and Campos (1996) are shown in this figure. At the distance of few pipe diameters, a considerable acceleration of the trailing bubble is obtained. In this region the present results are close to those obtained earlier. The current measurements, however, extend to substantially larger distances than those investigated before. It appears that interaction between the bubbles exists at distances exceeding the generally assumed stable liquid slug length (16D for a vertical flow). The present experimental results indicate an average rate of approach of-~10mm/s for distance between bubbles in the range of 60 to 30D. In other words, two Taylor bubbles separated initially by a liquid slug of length below 60D will end up colliding if the pipe is sufficiently long.
4. SUMMARY Interaction between two consecutive Taylor bubbles rising in stagnant liquid is studied quantitatively using the image processing technique. Certain conclusion regarding the process of the bubbles approach and merging are drawn on the basis of the flow visualization. The 2-D flow field in the liquid between the two bubbles is then measured using the PIV technique, and the variation of the vortex structure of the flow is studied up to a separation distance of about 60D. The variation of the propagation velocities of the two bubbles is measured. It is found that the trailing bubble does not affect the leading one. The trailing bubble, on the other hand, is sensitive to the velocity distortion in the wake of the leading bubble even at distances exceeding 50 pipe diameters. The information regarding the propagation velocity of the trailing bubble as a function of the separation distance from the leading one is essential for modeling slug length distribution in pipes (Barnea and Taitel 1993). REFERENCES
Barnea, D. and Taitel, Y. (1993) Int. J. Multiphase Flow, 19, 829-838 Collins, R, de Moraes, F.F., Davidson, J.F., and Harrison, D. (1978)J. Fluid Mech., 89, 497-514. Grace and Clif~ (1979) Chem. Eng. Sci.34, 1348-1350. Davies, R.M. and Taylor, G.I. (1949) Proc. R. Soc. London, Ser. A 200, 375-390. Dumitresku, D.T. (1943) Z Angew. Math. Mech., 23, 139-149. Mao, Z.-S. and Dukler, A. (1990) J. Comp. Phys., 91, 132-160. Moissis, D. and Griffith, P. (1962)J. Heat Transfer, Trans. ASME, Series C, 2, 29-39. Nieklin, D.J., Wilkes, J.O., and Davidson, J.F. (1962) Trans. Inst. Chem. Eng., 40, 61-68. Polonsky, S. (1998)Ph.D. Thesis, Tel-Aviv University. Polonsky, S., Barnea, D., and Shemer, L. (1999) To appear in Int. J. Multiphase Flow. Pinto, A.M.F.R. and Campos, J.B.L.M. (1996) Int. J. Multiphase Flow, 51, 45-54. Shemer, L. and Barnea, D. (1987) Physicochem. Hydrodyn., 8, 243-253. Taitel, Y. and Barnea, D. (1990) Advances in Heat Transfer, 20, 83-132.
953
Author Index Aanen, L. 371 Abu-Ghannam, B. J. 533 Akker, H. van der. 257 Alekseenko, S. V. 637 Aliod, R. 923 Amielh, M. 453 Andersson, H. I. 349 Andreux, R. 913 Anselmet, F. 453 Antonia, R. A. 423 Arakawa, C. 155 Atashkari, K. 805 Atmane, M. A. 521 Aubrun, S. 491 Barnea, D. 943 Ballmann, J. 649 Batten, P. 19 Bauer, W. 113 Behzadi, S. A. 279 Bencze, F. 751 Bertoglio, J. P. 289, 815 Besnard, D. 37 Bilsky, A. V. 637 Bocksell, T. L. 903 Boisson, H. 491 Bonnet, J. P. 207, 501 Booij, R. 339, 415 Boulet, P. 913 Bousquet, N. 125 Branley, N. 861 Brodie, G. H. 627 Buffat, M. 237 Burr, R. 269 Cazalbou, J.-B. 125 Chan, W. T. 821 Chartrain, D. 501 Chassaing, P. 125 Chauve, P. 197 Chemobrovkin, A. 555 Chomiak, J. 841 Chumakov, Yu. 607 Coratekin, T. 649 Coustols, E. 689
Craft, T .J. 73 Davidson, L. 227 Davies, T. W. 783 Delville, J. 207, 443 Derksen, H. 257 Doisy, A. M. 501 Druault, Ph. 207 Duursma, R. 135 Elmo, M. 815 Essen, T van. 135 Ewing, D. 461 Ferr6, J. A. 393 Franke, M. 659 Friedrich, R. 83, 247 Frolov, S. M. 851 Fujiwara, H. 155 Fu, S. 659 Fulachier, L. 453 Geissler, W. 679 Gessner, F. B. 709 George, J. 521 Giralt, F. 393 Gltick, M. 269 Grotjans, H. 269 Gr6tzbach, G. 165 Guilbaud, M. 501 Ha Minh, H. 491 Haag, O. 113 Had~.i6, I. 587 Han, Y. O. 471 Hangan, H. 381 Hanjali6, K. 135, 587 Hassel, E. P. 831 Hein, K. R. G. 881 Hennecke, D. K. 113 Hinz, A. 831 Houra, T. 481 Hussainov, M. 933 Hutter, K. 93 Htittl, T. J. 247
954 Iacovides, H. 773 Ichimiya, M. 597 Ishigaki, T. 511 Itoh, M. 699 Jackson, D. C. 773 Janicka, J. 93, 831 Johansson, A. V. 175 Jones, W. P. 861 Kao, P. L. 491 Kartushinsky, A. 933 Kaufmann, F. 543 Kavanagh, P. 533 Keffer, J. F. 393 Kelemenis, G. 773 Kenjere~, S. 135 Kidger, J. W. 73 Kim, Y. S. 471 Kitamura, O. 893 Knaus, H. 881 Knoell, J. 103 Knowles, K. 783 Kobayashi, M. 699 Kohnen, G. 933 K6ngeter, J. 361 Kopp, G. A. 381,393 Lain, S. 923 Lakshminarayana, B. 555 Landenfeld, T. 831 Launder, B. E. 73,773 Lawes, M. 805 Le Penven, L. 237 Leschziner, M.A. 19 Lipatnikov, A. 841 Liu, W. P. 627 Liu, X. 721 Loth, E. 903 Loyau, H. 19 Ltibcke, H. 659 Lukowicz, J. V. 361 Lucas, J. F. 453 Maier, H. 881 Majumdar, S. 309 Manceau, R. 319 Mariotti, G. 871 Markovich, D. M. 637
Martinuzzi, R. 381 Maruyama, T. 217 Matsuo, Y. 155 May, N. E. 329 Menter, F. 269 Meri, A. 197 Merzkrich, W. 63 Michard, M. 289 Muto, Y. 511 Nagano, Y. 481 N'Diaye, M. 237 Neuhaus; M. G. 841 Nigim, H. H. 533 Nikolskaja, S. 607 Nitsche, W. 619 Norberg, C. 227 Oesterl6, B. 913 Olsen, J. F. 423 Pailhas, G. 689 Pantano, C. 187 Parneix, S. 319 Parpais, S. 289 Perot, B. 145 Peters, N. 49 Pettersson-Reif, B. A. 349 Picard, C. 443 Pope, S. B. 795 Raddaoui, M. 197 Rajagopalan, S. 423 Rajani, B. N. 309 Rettich, T. 63 Riess, W. 731 Romano, G. P. 403 Rose, M. 851 Roth, P. 851 Ruiz-Calavera, L. P. 679 Rung, T. 659 Sabatino, A. 403 Sabel'nikov, V. A. 815 Sadiki, A. 93 Saidi, A. 763 Sagaut, P. 207 Sarkar, S. 187 Sauvage, Ph, 689
955 Schiele, R. 543 Schiestel, R. 197 Schneider, F. 63 Schnell, U. 881 Schubert, A. 649 Schulz, A. 543 Sello, S. 871 Semenov, V. I. 637 Sentker, A. 731 Shao, L. 187 Shemer, L. 943 Sheppard, C. G.W. 805 Shur, M. 669 Sltick, M. 269 Sohankar, A. 227 Sommerfield, M. 933 Spalart, P. R. 3,669 Steelant, J. 577 Strelets, M. 669 Sund6n, B. 763 Talvy, C. A. 943 Taulbee, D.B. 103 Thiele, F. 659 Tinapp, A. F. 619 Travin, A. 669 Touil, H. 289 Tsuji, T. 481 Tucker, P. 299
Ubaldi, M. 741 Uijttewaal, W. S. J. 339, 415 Unterberger, S. 881 Vad, J. 751 Vemet, A. 381 Wagner, C. 83 Wang, H. 145 Watkins, A. P. 279 Weatherly, D. C. 567 Wengle, H. 197 Westerweel, J. 371 Wikstr6m, P.M. 175 Williams, K.E. 709 Wittig, S. 543 Woolley, R. 805 W6mer, M. 165 Xiong, W. 63 Xu, J. 795 Xue, L. 659 Yamamoto, M. 893 Ye, Q.-Y. 165 Zhang, X. 433 Zhang, Y. 821 Zunino, P. 741
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