Frontiers of Computational Fluid Dynamics

10.4 Vorticity Confinement Two formulations of VC have been developed which have similar properties: The first, “VCI”, involves first derivatives of velocity [31, 341, while the second, “VC2” involves second derivatives [27]. The two versions will be described below. The basic principle in VC, as in the one-dimensional convecting scalar example described above, is that there is a solution with a stable structure that can be propagated indefinitely. Two terms control this structure: One acts to contract it, and one to expand it, so that the structure remains close to an equilibrium solution as it propagates. This solution is stable to perturbations caused by discretization errors in other terms such as convection. Although the VC equations can be written as a discretization of a pde (described below), the resulting solution at the small scales (within the structure), is not meant to be an accurate or even approximate solution of the original pde. This is because VC is meant to capture, or model the small scale features over only a couple of grid cells, so that the discretization “error” is 0(1) there. As explained in Section 10.2, the features essential for the problem, however, are still preserved. As such, the captured feature is actually a non-linear solitary wave that “lives” on the grid lattice. There is currently a large amount of work being done on intrinsically discrete-r difference,as opposed to j h i t e difference,equations (see, for example Ref [41]). In smooth regions (or large scales), on the other hand, VC can be made to automatically revert to conventional CFD where the pde’s are then accurately and efficiently approximated.

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For thin vortical regions, we use essentially the same approach as in the onedimensional example above: “Confinement” terms are added to the conventional, discretized momentum equation. Although we use a primitive variable, and not a vorticity, formulation, we will see that if we look at the resulting vorticity transport equation (for the VC2 version, to be described below), it has the identical “Confinement” terms as a multidimensional extension of the one-dimensional scalar transport equation described in Section 10.3. In this way, vorticity is transported in a compact way, with no numerical spreading. For general unsteady incompressible flows, the governing equations with Vorticity Confinement are discretizations of the continuity and momentum equations, with added terms: (10.18)

v.i=o -vp 1 + [pv’q -a

G

(10.19)

P

where is the velocity vector, p is the pressure, p is the density, and p is a diffusion coefficient that includes numerical effects due, for example, to discretization of the first right hand side term (convection term). (We assume that the Reynolds number is large and that physical diffusion is much smaller than the added terms). For the last term, & , E is a numerical coefficient that, together with p , controls the size and time scales of the convecting vortical regions or vortical boundary layers and s’ is defined below. For this reason, we refer to the two terms in the brackets as the “confinement terms”. The vector 3 is different for the two VC formulations, and is defined below. Equation 10.19 involves constant ,Ll and E , which is sufficient for many problems. If these are not constant, such as, for example, when the grid spacing is not constant or there are multiple vortical scales, then these quantities can be taken inside the differential operators in the corresponding terms, to maintain momentum conservation (in the VC2 formulation). As in the one-dimensional example the pair of confinement terms, which represent spreading, or positive diffusion and “contraction”, or negative diffusion, together create the confined structures. Stable solutions result when the two terms are approximately balanced. In this way, corrections are made each time step to compensate for any perturbations to the vortical structure caused by convection in a non-constant external velocity, discretization error in the convection operator, or the pressure correction. The parameters ,U and & determine the thickness of the resulting vortical structure and the relaxation rate to that state. In general, for boundary layers and convecting vortex filaments, computed flow fields external to the vortical regions are not sensitive to the internal structures, and hence to the parameters E and p , over a wide range of values. For example, a general thin, concentrated vortex will physically tend to evolve to an

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axisymmetric configuration [22]. Further, even a rapidly rotating non-symmetric configuration will be approximately axisymmetric when averaged over a short time [23]. Then, it is well known that the flow outside an axisymmetric twodimensional vortex core is independent of the vortical distribution, and hence will not depend on & and ,U as long as the core is thin (and the filament curvature is large, so that the flow is approximately two-dimensional in a plane normal to the filament). Therefore, the issues involved in setting these parameters will be similar to those involved in setting numerical parameters in other standard computational fluid dynamics schemes, such as artificial dissipation in many conventional shock-capturing schemes, which, as explained, are closely analogous. Further, for turbulent wake flows, preliminary studies suggest that E can be used to parameterize finite Reynolds number effects, since it controls the intensity of the smallest resolved vortical scales (this is the subject of current research [l 11). An important feature of the Vorticity Confinement method is that, for incompressible flow, the Confinement terms are non-zero only in the vortical regions, since both the diffusion term and the “contraction” term vanish outside those regions. Thus, even if there is a second order isotropic numerical diffusion associated with the convection operator, and the diffusion operators are only second order, outside the vortical regions the resulting accuracy of these terms can be third or fourth order, since this diffusion is just the negative curl of the vorticity. A final point concerns the total change induced by the VC correction in mass, vorticity and momentum, integrated over a cross section of a convecting vortex. It can be shown [20] that mass is conserved because of the pressure projection step in the solver, and vorticity is explicitly conserved because of the vanishing of the correction outside the vortical regions, and that, (in the VC2 formulation) momentum is also exactly conserved [27]. Momentum conservation, of course, results from the added terms having a spatial derivative operator in front. In the one-dimensional example, this allowed us to write what we termed an “Ehrenfest” relation for the motion of the pulse centroid. We have not proven this for confined convecting vortices, but we also believe it to be true in that case. Then, the vortex centroids will move with a weighted average of the velocity of the “background” flow, with no effect due to self-induced flow (at least for two-dimensional cases with constant background velocity). This has been demonstrated numerically [ 12, 281. (Errors due to the lack of momentum conservation in the VC1 formulation have been shown to be small in most cases, discussed below.) Many basic numerical methods could be used for space and time discretization. We use a simple first order Euler integration in time and second order in space. In conventional CFD schemes higher order methods often must be used, usually to reduce numerical diffusion and hence spreading for thin vortical regions. Vorticity Confinement eliminates this problem for many cases and avoids the boundw condition complexity and computational cost of the higher order methods. (It should be mentioned, however, that the second confinement, or contraction term involves a larger difference stencil than the other terms).

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Another numerical issue involves the regularity of the grid. It is important to realize that, since a convecting vortex or separated boundary layer is captured directly on the grid, over a few grid cells, large grid aspect ratios or rapidly varying cell sizes should not be used. If these are avoided, VC will result in a dynamics that is close to rotationally and translationally invariant. These issues also occur, of course, in shock capturing. Some corrections can be made, however, to accommodate non-uniform grids if the aspect ratio is not too large.

10.4.1 Basic Formulation The two different formulations, VC1 and VC2, have somewhat different dynamics, since they differ in the order of the derivative in the contraction term. The one developed initially (VC1) has been described in a number of publications and only a few details will be presented here;

10.4.1.1 VC1 Formulation This formulation involves an expression for the “contraction term”, s’ that does not explicitly conserve momentum: ;=liX6 (10.20) (10.21) (10.22) This term essentially convects vorticity within a thin vortical region along its own gradient 2 , from the edge, or regions of lower magnitude, toward the center, or region of larger magnitude. As the structure contracts and the gradient increases, the “expansion” term, which is a linear diffusion, increases until a balance is reached. (This is a well-known property of convection-diffusion phenomena.) Due to the rapid rotation of convecting concentrated vortices, any nonconservative momentum errors are almost completely canceled and the method has proved to be sufficiently accurate for many problems. A technicality in applying this method is often overlooked by people using it: this was described in earlier papers [31]. Since vorticity is convected along h , upwind (in h ) values of w should be used in the contraction term to avoid creating “downwind” values of vorticity with an opposite sign. This is easily accomplished with weighting factors at each node that depend on ri and unit vectors to neighboring grid nodes. Most of the VC results presented in the literature use the VC1 formulation. However, they do not involve very slow background flow and do not involve the momentum conservation issue discussed below. An important point, however, is

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that exact momentum conservation, in some cases, may not be as important as other features (such as, in our case, ensuring that a convecting vortex remain thin) and should not be regarded as an absolute requirement (see, for example, the basic CFD textbook - Tannehill, Anderson and Pletcher [39], pg. 60).

10.4.1.2 VC2 Formulation Only for very accurate long-term trajectory determination of vortices convecting in a very slow background velocity field has the momentum-conserving VC2 formulation been found to be necessary (for incompressible flow). This ensures that the contribution of the self-induced velocity to the vortex motion is completely canceled. The incompressible fluid dynamic equations for the VC2 formulation are discretizations of

V.+O

(10.23)

and

$+' = $ " ' A t ? . ( < ~ ) + V z p $ + $ x ( E G " ) or a similar form:

;"+' = 4 " -&.(G<)-Vx(p&"

-&")

(10.24)

where

6"=axsin

(10.25)

and

5; =I&;

I+&

(10.27)

Equation 10.24 has some numerical advantages over the form just above it, since the same difference operator acts on and G . Also, the second confinement term (10.26) is the sum over the stencil which consists of the central node (where $ is computed) and its neighboring ( - 1) nodes, and 6 is a small positive constant ( 10' ) to prevent problems due to finite precision. When two oppositely directed vortices are close to each other, there can be grid cells in between in which 6is not well-defined, which may cause oscillations. To prevent this, if the scalar product of any of the other vorticity vectors in the stencil with the central node is negative, $ is set to zero.

-

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To see the effect of VC2, we take the curl of Eq. 10.24. We then get a transport equation for vorticity. For example, in two dimensions, a,m = -v. (m;) + v2[pm-m(W)] (10.28) This equation, including the confinement term, is exactly a multi-dimensional, rotationally invariant generalization of the one-dimensional scalar advection equation shown to be effective in Section 10.3. Of course, the solution will still reflect the four-fold symmetry of the grid. This effect, however, vanishes rapidly away from a vortical region. Further, the rotating flow around a vortex core actually allows a simpler discretization of Eq. 10.28, compared to an axisymmetric convecting passive scalar distribution. This is explained in Ref. [27]. Equation 10.26 is a harmonic mean. It is chosen to weigh the small values in the stencil more heavily. As is well known, this term vanishes when any of the values of its argument vanishes, preventing creation of values of opposite sign (for a range of parameters). Using VC, the total vorticity in a region surrounding a vortex is conserved, since it is a local term. This means that the vorticity cannot diverge due to this term, since the maximum absolute value cannot be greater than the absolute value of sum when all values have the same sign. (This is also a property of the one-dimensional scalar advection example.) There are a large number of alternative forms that would work as well as the harmonic mean. We believe that the term should have a smooth algebraic form, however, to give smooth results. This should be more appropriate for multidimensional applications than forms involving logic functions, such as “minmod”, which give good results in one-dimensional applications. As discussed above, terms such as 10.26 have previously been used as limiters, but in onedimensional compressible flow, and, to the author’s knowledge, not for multidimensional vorticity. (The VC method was developed independently, as a multidimensional, rotationally invariant “Confinement” method specifically for thin vortical regions).

10.4.1.3 Boundary Conditions First, we describe general features involving the use of VC for enforcing boundary conditions and creating model boundary layers (BL’s). We then describe two approaches: the use for immersed BL’s, and the use with surface - conforming grids. Both involve no-flow-through conditions. They also involve no-slip conditions. This latter is important for problems with separation because they ensure that the resulting separating BL has a well-defined vorticity. Also, both involve coarse, inviscid-size grids. As in the use of VC for thin, convecting vortical regions, they result in a simple, very economical BL model. This model does not involve determining a detailed time-averaged velocity profile, which would require a very fine, body-fitted grid: Instead, it models the profile over only a few coarse grid cells. As such, it is meant to be useful for blunt body flows with

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massive separation, where this detailed profile, as well as skin friction, are of secondary importance compared to the location and strength of the separating BL. With a conventional CFD solution without VC, this vorticity would quickly convect and diffuse away from the surface regions due to the large numerical errors at the boundary resulting from the coarse and possibly non-conforming grid, destroying the accuracy of the outer solution. However, the use of VC confines vorticity to 1-2 grid cells along the surface, when it is attached. Just outside this layer the velocity is smooth and close to tangent to the adjacent surface. This simple boundary layer can still separate, however, especially at edges and in regions of strong adverse pressure gradient. Even though we are concentrating on the VC2 version, the use of the VC1 version should be mentioned here since it has a very simple interpretation for attached boundary layers. In this case, the vector 2 in Eq. 10.21 is defined to be locally normal to the surface in the boundary layer region. Then, VC1 is simply a combination of positive diffusion (which spreads the vorticity away from the surface) and convection of vorticity towards the surface. This has proven to be a very robust and efficient way of modeling the boundary layer, combining a tangential smoothing for the external velocity and a normal ”compression” of the vorticity. A number of results have been presented which demonstrate the effectiveness of this approach [9,11,12,27]. a. Immersed boundary layer model To enforce no-slip boundary conditions on immersed surfaces, first, the surface is represented implicitly by a smooth “level set” function, “F’,defined at each grid point. This is just the (signed) distance from each grid point to the nearest point on the surface of an object - positive outside, negative inside. Then, at each time step during the solution, velocities in the interior are set to zero. In a computation using VC, this results in a concentrated vortical region along the surface. The important point is that no special logic is required in the “cut” cells, unlike many conventional schemes: only the same VC equations are applied, as in the rest of the grid. Also, unlike conventional immersed surface schemes, which are inviscid because of cell size constraints, there is effectively a no-slip boundary condition which results in a boundary layer with well-defined total vorticity and which, because of VC, remains thin. This method is effective for complex configurations with separation from sharp comers. Also, even with no special boundary layer model terms, it can approximately treat separation from smooth surfaces, as shown in Section 10.5. b. Conforming grid, turbulent boundary layer model We are beginning to develop detailed models for turbulent boundary layers, within the VC framework. This involves modeling the evolution of the Confinement parameters so that, for example, separation is accurately predicted, even on smooth

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surfaces. Results of some new, exploratory studies will be described in the next section. There is still research to be done on these models, but the capabilities of the basic approach appear to be very promising. A simple way to begin this work is to use surface-conforming grids. An important point, however, is that this VC-based method is fundamentally different from conventional RANS schemes, which typically use an eddy viscosity (EV) type of term and discretize a (modified) Navier-Stokes type of equation on a very fine grid in order to model the time-averaged velocity. A very important feature of VC here is that it greatly expands our modeling capability, compared to EV - type schemes: Typically, these latter schemes can only accommodate positive values of EV. If the EV is negative over significant regions of space and time, they tend to diverge due to numerical instability [ 141. This means that the modeled BL can only directly be made to expand, or diffuse, and not to contract. (Of course, slower expansion rates can be obtained and smaller BL thickness, but a finer grid is then required and a smaller value of the EV.) VC, on the other hand, can directly model contraction; unlike a conventional scheme with a negative eddy viscosity, VC will not diverge. This is very useful, for example, in separation from a smooth surface at low Re: physically, the separating layer then tends to transition and quickly reattach. A contraction term such as VC easily models this effect [7]. Another point is that, for conforming grids a scaling factor must be applied to ,u and E which depends on the (varying) cell size [33]. This is not a problem, since inviscid-size grids are used that do not have large aspect ratios. A final point involves the use of VC is in retarding separation in adverse pressure gradient regions. For example, as is well known, a turbulent BL tends to separate later (in an adverse pressure gradient) than a laminar one. VC can easily be used to simulate this, again without very fine grids. [ 121

10.4.2 Comparison of the VC2 Formulation to Conventional Discontinuity Steepening Schemes In this section, we first reformulate the one-dimensional scalar pulse equation of Section 10.3 as a discontinuity steepening method for velocity contact discontinuities. The result is similar to forms that have been developed over a number of years to keep gradients, such as contact discontinuities, steep and overcome the smoothing resulting from the convection terms. As explained above, these schemes have typically involved one-dimensional compressible flows and utilize a number of different “limiters.” If we consider the integral of the one-dimensional pulse of Section 10.3 (and change the sign), we have a propagating step function that remains steep (see Figure 2). We then have: (10.29) @Jn = SjVJ?= VJ? -vJ:, Eq. (10.6) then becomes

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a; = 0;({JjVY})

(10.30)

Partially summing Eq. (10.7) overj, we then have (for constant c) V

(v;+l -v;-, ) + pdj’Vi”- €6,a; (10.31) 2 In this form Confinement is similar to conventional one-dimensional steepening schemes. However, real flows never have a single point at which they have a steep gradient, and exactly constant properties everywhere else. In general, there are O(1) (smooth) gradients away from the discontinuity. These smooth gradients are also acted on by the steepener, causing errors, unless special logic is used to cutoff the steepener. Using such one-dimensional schemes along each coordinate axis to keep a convecting vortex core compact would cause the same problem, since the velocities vary inversely with the radius away from the core (see Figure 3). However, we do not do this! The important point is that we should not use the exact one-dimensional Confinement terms, but should only keep their basic structure - that they are functions of first derivatives of velocities. In developing a formulation for multiple dimensions, we should then use only rotationally invariant quantities. The only quantities, for example, in 2-D incompressible flow, which are first derivatives of a velocity are: + W=VXijl, (10.32) v,n+’= v;

--

and -3

D = V.q’ (10.33) where k denotes the out-of-plane direction. But D = 0 for incompressible flow, so we have only one choice: a=a((W). (10.34) This eliminates any problems with gradients away from the core since w +0 , even though both a,v and a,u will be 0(1) there. In addition, it results in a much simpler formulation. Finally, for a vortex filament in three-dimensions, we consider it sufficient to confine in a two-dimensional plane normal to the vortex, as depicted in Figure 4. In this way we arrive at the momentum conserving VC formulation.

10.4.3 Computational Details for the VC2 Formulation The details of the discretization are probably not important, and different forms have been used in the past for different applications. Just one particular form is described below. For each time-step (n),the following computations are executed:

Step a: Convection

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As in conventional incompressible “split velocity” methods 1201,a convection-like computation is used to treat the momentum equations. This is a discretized conservative version of

;= ’ ;”

+

- A t V . ( q+ n +qn )

(10.35) involving a second order central difference. For the x-component, for example,

(10.36)

An intermediate vorticity,

6‘,is then computed by taking the curl of G‘ .

Step b: Confinement Vorticity Confinement is used to compute a velocity increment such that the solution relaxes to thin vortical structures (after a small number of initial time steps): +” q =&At?x(pc$’-Ei?’). (10.37) In Eq. 10.26, a harmonic mean based on the central node and 26 nearest neighbors is used for i6’ :

Simpler formulations with only 8 nearest nodes give very close results for many cases; Further research is needed to determine the minimum number of nodes necessary. Equation 10.37 is then discretized with a second order difference scheme.

Step c: Pressure Computation The pressure is computed such that the velocity at time step IZ free: v.;n+l

=o.

+ 1 is divergence(10.39)

This involves solving a Poisson equation

v”=-v.;”

(10.40)

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as in a conventional “split velocity” procedure. A conventional “box” scheme with a staggered grid is used to compute the right hand side. The pressure p is then given by

(10.41) For cases where the pressure is to be computed on a near by surface immersed in the grid, a tri-linear extrapolation method is used to compute the pressure on specified points on the actual surface.

Step d: Velocity Update The velocity at the next time step is then computed using, again, a box scheme, or staggered grid: ;“+I

=

T+Qj

This agrees with the momentum equation (to fist-order in

(10.42)

At ).

Step e: Enforcement of Boundary Conditions Boundary conditions are handled as described in Section 10.4.1.3. For a nonconforming grid, i.e. immersed surface, each time step the velocity is set to zero at each grid cell inside the body. With conforming grids, a simple no-slip condition is applied at the surface, with no-flow -through conditions. (For the VC1 formulation, the vector i? is defined locally to be normal to the surface.)

10.5 Results 10.5.1 Wing Tip Vortices Measurements had been made in a towing tank at the Institute fuer Luft and Raumfahrt in Aachen for a wing with flaps, as described in Fig 5, at a Reynolds number of 55,000. The trajectories and strengths of the shed vortices were determined over a distance of 60 full spans using laser velocimetry (this is described in [ 161). A preliminary three-dimensional computation of this flow was performed using VC. To save computer memory only one half of the (symmetric) flow was computed and the problem was semi-parabolized by breaking it up into streamwise blocks of 1.35 spans. Upstream boundary conditions for each block were taken from downstream ones of the previous block. This introduced only a negligible error compared to a full three-dimensional computation since the upstream influence of curvature of the vortices over this distance was very small. The flow computation was initiated using measured velocities just behind the wing, and flow from only a half span was treated with symmetry conditions.

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Vorticity magnitude is presented in Figure 6 comparing experiment with computation for four downstream locations. The symmetry plane is on the right, and the (weaker) flap vortex can be seen to spiral around the main tip vortex. Sensitively tests were done for the Confinement parameters. Two widely separated vortices are not sensitive to these values. This is true, for example, for the descent rate of the main tip vortex under the influence of the other tip (image) vortex. Since the flap and tip vortex are close, compared to their diameters, some (small) influence on the internal structure of each is expected. This is minimized since VC enables the vortices to be captured over only a few grid cells. Also, the close proximity of the boundary will induce an effect. This is reflected in only a small influence of the Confinement parameters on the angle of rotation of the flap vortex around the tip vortex. Details are described in Ref. [16].

10.5.2 Cylinder Wake This study involves two distinct properties of VC. In one, VC is used to treat immersed surfaces in a non-conforming, regular Cartesian grid. In the other, VC is used as a new type of LES model. The immersed surface use is documented and results have been validated for a number of cases. Both uses are areas of current research. Results are shown for a three-dimensional circular cylinder, based on the VC2 formulation. Results for the VC1 formulation were shown in Ref. [12]. (A square cylinder was also treated in that study.) Both VC formulations compared very well with each other and with experiment. The computations that had previously been done using the VC1 formulation, as well as for the one presented here, were for Re = 3900. A coarse, uniform Cartesian grid 18 1x 121x 61 was used with an immersed boundary for the cylinder that was only 15 cells in diameter. Computed isosurfaces of vorticity using the VC1 formulation were shown in Ref. [12], together with the mean and rms averages of the streamwise velocity in the wake. Comparisons with experimental data were also shown. In that work, as here, only one parameter was adjusted; the confinement strength, which was used to simulate Reynolds number effects. This was constant throughout the field. Results for the VC2 formulation are presented below: Vorticity magnitude isosurface are shown in Fig. 7.a, where the isosurface magnitude has a value of VI of the maximum. Plots corresponding to computed average streamwise velocity along lines behind the cylinder are shown in Fig. 7.b and rms streamwise velocity fluctuations are presented in Fig. 7.c. The lines where the measurements were taken are shown in Fig. 7.d. The agreement with experiment is seen to be good. The pressure distribution on the cylinder surface also compares very well with experiment data, as can be seen in Fig. 7.e. The important point here is that only by adjusting one parameter E , which was constant throughout the field, the computed results agreed closely with experiment

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for all six curves plotted in Figs. 7.b and 7.c. Additional comparisons with experiment at different Reynolds number will be required to calibrate the Reynolds number dependence of this parameter. It must be emphasized that the spanwise instabilities and chaotic behavior that result when E is increased are only from three-dimensional effects, as in physical turbulence, and are not due to numerical instabilities: Extensive studies have been done over a much wider range of E values than that studied here for flows where no instabilities are expected, which only show stable flow. These involved vortex pairing over a two-dimensional cylinder and isolated, shed wing-tip vortices in three-dimensions.

10.5.3 Dynamic Stall The main goal of this project was to create a computational method for dynamic stall that was very efficient in computer resources so that it could be used as a routine engineering tool. Besides efficiency, one of the constraints was to formulate VC as a modification to be implemented in an existing NASA Accordingly, unlike in the description in Section compressible code-"TURNS". 10.4, a compressible formulation was used. This can be found in Ref. [9, 101. The flow was subsonic and the results do not depend significantly on this difference. For simplicity of presentation, we only have described the incompressible formulation in this paper. Flow was simulated over a pitching VR12 airfoil at Re = 1.9x106 undergoing dynamic stall, where the boundary layer suddenly separates. This causes a rapid loss of lift. Details of this study are described Refs [9] and [lo]. The main point here is that this case is similar to many blunt body flows in that the separation location and time are the important properties of the boundary layer (BL). Other quantities, such as skin friction, are of secondary importance for these flows. Our approach is to define a VC-based dynamic model that directly results in the correct behavior for these quantities and where the BL is efficiently defined over only a small number of inviscid-size grid cells. As explained in Section 10.4, no attempt is made to model a detailed time-averaged version of the physical velocity profile, as in conventional turbulent BL Reynolds Averaged NavierStokes (RANS) equation schemes. This effort would require very fine grid cells near the surface and much more computing power. Further, a thin convecting vortex sheet typically results from the separation. This is also captured efficiently with the VC approach. Preliminary results from Ref. [9, 101 showing velocity vectors at several times during the pitch cycle are shown in Figure 8. Also shown are computed C, values and comparisons with wind tunnel measurements. The model used in this study required VC parameters that were prescribed separately for each of several phases during the cycle, but were constant throughout the grid (except for a grid scaling

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factor). A more advanced model that dynamically computes these parameters is being developed. An important point is that the two-dimensional dynamic stall computations required only about 10 minutes per cycle on a standard (1.7 GHz) PC. Conventional RANS schemes apparently require far more computational resources [6]. These quick runs are a big help in the development of VC-based models.

10.6 Other Studies The three cases presented above demonstrate uses of VC for simulating selected phenomena. Out of the large number of results obtained in the last several years, three additional results will be mentioned because they are recent and demonstrate additional uses of VC-simulation of boundary layer separation at a sharp comer (missile base flow), vortex propagation and interaction with an airfoil (blade vortex interaction) and simulation and visualization of turbulent flow (for special effects). As mentioned in the introduction, a comprehensive review of general applications of VC is under preparation. Some recent results can be seen at http://www.flowanalysis.com.

10.6.1 Missile Base Drag Computation Boundary layer separation at the base of a missile has been computed using a compressible version of VC [6, 26, 371. (The title of Ref. 1261 uses the word inviscid to state only that an inviscid-size grid was required with corresponding short computing times, even though the computation was viscous.) In these papers it is demonstrated that VC can capture the essential features of the separating boundary layer, keeping it thin and maintaining its circulation, while allowing economical coarse grids to be used with no-slip conditions, as explained in Section 10.4.1. It can be seen in the papers that the results agree with experiment. The only other accurate alternative is the much more expensive RANS approach, since conventional, economical inviscid approaches lead to inaccurate boundary layers and expansion fans.

10.6.2 Blade Vortex Interaction (BVI) The ability of VC to economically simulate propagation of concentrated vortices for BVI has made possible a recent parametric study of two-dimensional BVI cases [24, 251. This study also utilized a compressible version of Vorticity Confinement. In these papers, it is also demonstrated that there is excellent agreement between the computations and experiment.

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10.6.3 Turbulent Flow Simulations for Special Effects For special effects, the important aspect of a turbulent simulation is, of course, that it look turbulent, which means that it include visible small scale effects. Of course this, by itself, is not sufficient for engineering purposes, but could be thought of as a prerequisite, especially if small scale phenomena are important in the problem. VC1 has been found to simulate small scale phenomena more effectively and economically than most other schemes. Ron Fedkiw has performed excellent computations and visualizations with this in mind [13].

10.7 Conclusions The Vorticity Confinement (VC) method has been presented in more comprehensive detail than has been previously available. Although the basic ideas are somewhat different than conventional CFD, there is some commonality with a number of well-known computational methods, such as shock-capturing. Extensive use of analogies with these methods is made to explain the basic motivation. The main goal of VC is to efficiently compute complex high Reynolds number incompressible flows, including blunt bodies with extensive separation and shed vortex filaments that convect over long distances. Almost all of the vortical regions in these flows are turbulent. This means that, for any feasible computation, they must be modeled. The remainder of the flows are irrotational and are defined once the vortical distributions are. Further, these vortical regions are often very thin. For these reasons, the basic approach of VC is to efficiently model these regions. The most efficient way to do this appears to be to develop model equations directly on the computational grid, rather than to first develop model partial differential equations (pde’s) and then attempt to accurately discretize them in these very thin regions. Some of the goals are readily achieved, especially where the essential features of the main flow are not sensitive to the internal structure of thin vortical regions. Then, VC can easily be used to capture these regions over only a couple of grid cells and propagate them, essentially as nonlinear solitary waves that “live” on the computational lattice. Flows with these features, that are treatable with the present state of VC, include blunt bodies with separation from edges and other welldefined locations. These configurations include complex geometries that can be easily “immersed” in uniform Cartesian grids using VC. These flows also include vortex filaments which can convect, with no numerical spreading, even over arbitrarily long times, and which can merge automatically with no requirement for special logic.

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More difficult goals involve separation from smooth surfaces, which depends on the turbulent state of the boundary layer. These clearly require more detailed modeling, including parameter calibration. A large amount of effort has been expended over a number of years by a large number of workers to develop turbulent pde-based models for conventional CFD schemes, such as RANS and LES. We are currently starting to develop corresponding VC-based models. Preliminary results of these studies, some of which are presented, suggest that very large computer savings can be achieved.

10.8 Acknowledgements Funding from a number of sources is gratefully acknowledged: Primarily, the Army Research Office and the Army Aeroflightdynamics Directorate, which have supported the development of Vorticity Confinement from its beginning. Additional help has been provided by the Institute fuer Luft und Raumfahrt at the Technical University of Aachen, and the University of Tennessee Space Institute. Also, numerous discussions with Frank Caradonna at Moffett Field and with Stanley Osher and others at UCLA are acknowledged.

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10.9 Bibliography Bohr, T., Jensen, M., Paladin, G., and Vulpiani, A., Dynamical Systems Approach to Turbulence, Cambridge, 1998. Boris, J., and Book, D., “Flux-Corrected Transport: I SHASTA, A Fluid Transport Algorithm That Works,” Journal of Computational Physics, Vol. 11, 1973. Braun, C., “Application of Vorticty Confinement to the Flow over a 6:1 Ellipsoid at High Angles of Attack” Master thesis, Institut fur Luft- und Raumfahrt, RWTH Aachen, Germany, 2000. Costes, M., and Kowani, G., “An Automatic Anti-diffusion Method for Vortical Flows Based on Vorticity Confinement”,Aerospace Science and Technology, Vol. 7,2003. Dadone, A., Hu, G., Grossman, B., “Towards a Better Understanding of Vorticity Confinement Methods in Compressible Flow,” AIAA-2001-2639. AIAA Anaheim meeting, June 2001. Dietz, W., “Application of Vorticity Confinement to Compressible Flow,” AIAA-2004-0718. AIAA Reno meeting, January 2004. Dietz, W., “Analysis, Design, and Test of Low Reynolds Number Rotors and Propellers,” SBIR Final Report, September 2004. Dietz, W., Fan, M., Steinhoff, J., Wenren, Y., “Application of Vorticity Confinement to the Prediction of the Flow Over Complex Bodies,” AIAA2001-2642. AIAA Anaheim meeting, June 2001. Dietz, W., Wang, L., Wernen, Y., Caradonna, F., and Steinhoff, J., “The Development of a CFD-Based Model of Dynamic Stall,” AHS 60th Annual Forum, Baltimore, MD, June, 2004. Dietz, W., Wenren, Y., Wang, L., Chen, X., and Steinhoff, J., “Scalable Aerodynamics and Coupled Comprehensive Module for the Prediction of Rotorcraft Maneuver Loads,” SBIR Final Report, May 2004. Fan, M., and Steinhoff, J., “Computation of Blunt Body Wake Flow by Vorticity Confinement,” AIAA-2004-0592. AIAA Reno meeting, January 2004.

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Fan, M., Wenren, Y., Dietz, W., Xiao, M., Steinhoff, J., “Computing Blunt Body Flows on Coarse Grids Using Vorticity Confinement,” Journal of Fluids Engineering, Vol. 124, No. 4, pp. 876-885, December 2002. Fedkiw, R., Stam, J., and Jensen, H. W., “Visualizations of Smoke”, Proceedings of SIGGRAPH 2001, Los Angeles, 2001. Ferziger, J., “Large Eddy Simulation,” Simulation and Modeling of Turbulent Flows, Ed. By T. B. Gatski, M.Y. Hussaini and J.L. Lumley, Oxford. 1996. Grinstein, F., and Fureby, C., “Implicit Large Eddy Simulation of High-Re Flows with Flux-Limiting Schemes,” AIAA-2003-4100. AIAA Orlando meeting, June 2003.

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Haas, S., “Computation of Trailing Vortex Flow Over Long Distances Using Vorticity Confinement,” Master thesis, Institut fur Luft- und Raumfahrt, RWTH Aachen, Germany, 2003. Harten, A., “The Artifical Compression Method for Computation of Shocks and Contact Discontinuities 111, Self-Adjusting Hybrid Schemes,” Mathematics of Computation, Vol. 32, No. 142, April 1978. Hu, G., Grossman, B., and Steinhoff, J., “A Numerical Method for Vortex Confinement in Compressible Flow,” AZAA-Journal, Vol. 40, October 2002. Katz, J., and Plotkin, A., Low Speed Aerodynamics, Cambridge University Press, New York, 2001. Kim, J., and Moin, P., “Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations,” Journal of Computational Physics, Vol. 59, No. 2, 1985. Lax, P. D., “Hyperbolic Systems of Conservation Laws 11,” Comm. Pure Appl. Math, Vol. 10, 1957. Melander, M. V., McWilliams, J. C., and Zabusky, N.J., “Axisymmetrization and Vorticity-gradient Intensification of an Isolated Two-dimensional Vortex through Filamentation,” Journal of Fluid Mechanics, Vol. 178, 1987. Misra, A., Pullin, D., “A Vortex-based Stress Model for Large-eddy Simulation,” Physics of Fluids, Vol. 9, No. 8, August 1997.

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Morvant, R., Badcock, K., Barakos, G., and Richards, B., “Aerofoil-Vortex Interaction Simulation Using the Compressible Vorticity Confinement Method,” 29th European Rotorcraft Forum, Friedrichshafen, Germany, September 2003. Morvant, R., “The Investigation of Blade-Vortex Interaction Noise Using Computational Fluid Dynamics,” PhD Dissertation, University of Glasgow, United Kingdom, February 2004. Robinson, M., “Application of Vorticity Confinement to Inviscid Missile Force and Moment Prediction,” AJAA-2004-0717. AIAA Reno meeting, January 2004. Steinhoff, J., Dietz, W., Haas, S., Xiao, M., Lynn, N., and Fan, M., “Simulating Small Scale Features In Fluid Dynamics And Acoustics As Nonlinear Solitary Waves,” AIAA-2003-0078. AIAA Reno meeting, January 2003. Steinhoff, J., Fan, M., and Wang, L., “Vorticity Confinement - Recent Results: Turbulent Wake Simulations and a New, Conservative Formulation,” Numerical Simulations of Incompressible Flows,Ed. By M. M. Hafez, World Scientific, 2003. Steinhoff, J., Fan, M., Wang, L., and Dietz, W., “Convection of Concentrated Vortices and Passive Scalars as Solitary Waves,” SIAM Journal of Scient@c Computing, Vol. 19, December 2003. Steinhoff, J., Mersch, T., Underhill, D., Wenren, Y., and Wang, C., “Computational Vorticity Confinement: A Non-Diffusive Eulerian Method for Vortex Dominated Flows,” UTSI preprint, 1992. Steinhoff, J., Mersch, T., Wenren, Y., “Computational Vorticity Confinement: Two Dimensional Incompressible Flow,” Developments in Theoretical and Applied Mechaltics, Proceedings of the Sixteenth Southeastern Conference on Theoretical and Applied Mechanics, 1992. Steinhoff, J.; Puskas, E.; Babu, S.; Wenren, Y.; Underhill, D., “Computation of thin features over long distances using solitary waves,” AIAA Proceedings, 13th Computational Fluid Dynamics Conference, pp. 743-759,1997.

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Vorticity Con$nement Steinhoff, J., and Raviprakash, G., “Navier-Stokes Computation of BladeVortex Interaction Using Vorticity Confinement,” AIAA-95-016 1. AIAA Reno meeting, January 1995. Steinhoff, J., Senge, H., and Wenren, Y., “Computational Vortex Capturing,” UTSI preprint, 1990. Steinhoff, J., and Underhill, D., “Modification of the Euler Equations for “Vorticity Confinement” - Application to the Computation of Interacting Vortex Rings”, Physics of Fluids, Vol. 6, August 1994. Steinhoff, J., Wenren, Y., and Wang, L., “Efficient Computation of Separating High Reynolds Number Incompressible Flows Using Vorticity Confinement,” AIAA-99-33 16-CP 1999. Suttles, T., Landrum, D., Greiner, B., and Robinson, M., “Calibration of Vorticity Confinement Techniques for Missile Aerodynamics: Part I - Surface Confinement,” AIAA-2004-0719. AIAA Reno meeting, January 2004. Szydlowski, J., and Costes, M., “Simulation of Flow Around a NACA0015 Airfoil for Static and Dynamic Stall Configurations Using RANS and DES,” 4th Decennial Specialist’s Conference on Aeromechanics, January 2004. Tannehill, J., Anderson, D., and Pletcher, R., Computational Fluid Mechanics and Heat Transfer, Taylor & Francis, 1997. Van Leer, B., “Towards the Ultimate Conservative Difference Scheme. 11. Monotonicity and Conservation Combined in a Second-Order Scheme,” J. Comp. Phys., Vol. 14, 1974. Von Neumann, J., and Richtmyer, R.D., “A Method for the Numerical Calculation of Hydrodynamic Shocks,” J. Appl. Phys., Vol. 21, No. 3, 1950. Wang, C., Steinhoff, J., and Wenren, Y., “Numerical Vorticity Confinement for Vortex-solid Body Interaction Problems”, AIAA Journal, Vol. 33, August 1995. Wenren, Y., Fan, M., Wang, L., and Steinhoff, J., “Application of Vorticity Confinement to the Prediction of Flow over Complex Bodies”, AIAA Journal, Vol. 41, May 2003.

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[44] Wenren, Y., Steinhoff, J., and Robins, R., “Computation of Aircraft Trailing Vortices,” SBIR Final Report, June 1995. [45]

Wenren, Y., Fan, M., Dietz, W., Hu, G., Braun, C., Steinhoff, J., Grossman, B ., “Effeicient Eulerian Computation of Realistic Rotorcraft Flows Using Vorticity Confinement: A Survey of Recent Results,” AIAA-200 1-0996. AIAA Reno meeting, January 2001.

Vorticity Confinement

228 With Confinement c. 1 Pass (900 Timesteps)

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Figure 5. Schematic of water tow tank experiment. Two vortices are shed from each side of the wing, one from the wingtip and an inner one from the flap.

231

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Figure 7.a Vorticity Isosurface for Flow over Cylinder with Vorticity Confinement ( ,U = 0 . 1 5 , ~ = 0.325)

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Part 111

Flow Stability and Control

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Chapter 11

Flow Control Applications with Synthetic and Pulsed Jets R. Agarwal,' J. Vadillo,' Y. Tan,' J. Cui,' D. Guo,' H. Jain' A. W. Cary,2 & W. W. Bower2

11.1

Abstract

In recent years, a promising approach to the control of wall bounded as well as free shear flows, using synthetic and pulsed jet actuators, has received a great deal of attention. A variety of impressive flow control results have been achieved experimentally by many researchers, including the vectoring of conventional propulsive jets, modification of aerodynamic characteristics of bluff bodies, control of lift and drag of airfoils, reduction of skin-friction of a flatplate boundary layer, enhanced mixing in circular jets, and control of external as well as internal flow separation and of cavity oscillations. More recently, attempts have been made to simulate numerically some of these flow fields. Several of the above mentioned flow fields have been simulated numerically using the Reynolds-Averaged Navier-Stokes (RANS) equations with a turbulence model, and a limited few using Direct Numerical Simulation (DNS). In the simulations, both the simplified boundary conditions at the jet exit, as well as the details of the cavity and lip, have been included. In this article, we describe the results of simulations for five different flow fields dealing with virtual aeroshaping, thrust-

'

Department of Mechanical and Aerospace Engineering, Washington University, St. Louis, Missouri 63130. 'The Boeing Company, St. Louis, Missouri 63166.

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Flow Control with Synthetic and Pulsed Jets

vectoring, interaction of a synthetic jet with a turbulent boundary layer and control of separation and cavity oscillations. These simulations have been performed using the RANS equations in conjunction with either one- or a twoequation turbulence model.

11.2 Introduction In recent years, there has been great emphasis on the development of advanced aerodynamic technologies, based on fluidic modification of aerodynamic flow fieldslforces, that can cover multiple flight conditions without the need of conventional control surfaces such as flaps, spoilers and variable wing sweep. The fluidic modification (or flow control) can be accomplished by employing microsurface effectors and fluidic deviceslactuators dynamically operated by an intelligent control system. These new active flow control (AFT)technologies have the potential of resulting in radical improvement in aircraft performance and weight reduction. A variety of impressive flow control results have been achieved experimentally by many researchers using pulsed or synthetic (oscillatory) jet actuators including the vectoring of conventional propulsive jets, modification of aerodynamic characteristics of bluff bodies, control of lift and drag of airfoils, reduction of skin-friction of a flat plate boundary layer, enhanced mixing in circular jets and control of external as well as internal flow separation and cavity oscillations. A synthetic jet is formed by employing an oscillatory surface within a cavity. It is created entirely from the fluid that is being controlled. It is generated with a piezoelectric diaphragm in a periodic manner. Flow enters and exits the cavity through an orifice in a periodic manner. The unique feature of the synthetic jet is that no fluid ducting is required. Synthetic jets have been shown to exert significant control authority in many applications and have the additional benefit of being compact with zero net mass flux. An excellent review of synthetic jets and some of their applications has recently been given by Glezer and Amitay [ 5 ] . A pulsed jet, on the other hand, has both a steady and an oscillatory velocity component, with net mass flow and momentum injected into the flow. In this paper, some results of numerical simulations on flow control using synthetic or pulsed jets for five different flow fields dealing with virtual aeroshaping, thrust-vectoring of a propulsive jet, the interaction of a synthetic jet with a turbulent boundary layer, control of recirculation region behind a backward facing step and control of cavity oscillations are presented. The

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numerical simulations are compared with the experimental data wherever possible.

11.3 CFD Flow-solvers Employed Two different flow solvers have been employed in the five flow field simulations reported in this paper in Section 3. Both are well known RANS solvers, called INS2D [ 111 and WIND [2]. “INS2D’ is an incompressible Reynolds-Averaged Navier-Stokes (RANS) solver developed at NASA Ames Research Center. It is employed in the simulations reported in parts (b), (c) and (d) of Section 3, which correspond to the vectoring of a propulsive jet using a synthetic jet, the control of separation behind a backward facing step using a synthetic jet, and the interaction of a synthetic jet with a turbulent boundary layer respectively. INS2D solves the continuity equation and the RANS equations for incompressible flow in two-dimensions in generalized coordinates for both the steady-state and time varying flow using the method of pseudocompressibility. The convective terms are computed using an upwind third-order accurate differencing scheme based on flux-difference splitting. The difference equations are solved using a Generalized Minimum Residual Method (GMRES) which is stable and is capable of running at very large pseudo-time steps leading to fast convergence for each physical time step. The code has two turbulence models for computing the eddy viscosity - the Spalart-Allmaras (SA) one-equation model [ 141 and Menter’s Shear Stress Transport (SST) two-equation model [9]. SA model is employed in all the computations performed with r”NS2D, reported in Section 3-parts (b), (c) and (4. The computations reported in Section 3, part (a) virtual aerodynamic shape modification of an airfoil using a synthetic jet and part (e) control of subsonic cavity shear layer using pulsed blowing, a multi-zone structured-grid compressible RANS solver WIND is employed. WIND [2] has been developed and is now supported by the NPARC Alliance, a partnership between NASA Glenn Research Center (GRC), Boeing, Pratt and Whitney, and the USAF Arnold Engineering Development Center (AEDC) which was formed to provide a national applications-oriented flow simulation code. WIND also has several turbulence models - SA model, SST model and the Large Eddy Simulation (LES) model. Recently the code was modified by Bush and Mani [3] to include the Detached Eddy Simulation (DES) formulation of Spalart [15] for the SA model and its extension to k-E model by Strelet [16]. This extension allows for combining the RANS and LES models in a consistent manner in the flow field. Although WIND is a 3-D code, the simulations reported in parts (a) and (e) of Section 3 are in 2-D.

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In all the simulations except in Sections 3(b) and 3(d), the boundary conditions for the actuator are applied at the exit of the jet; the influence of the cavity of the actuator is not included. In the simulation of the interaction of a synthetic jet with a turbulent boundary layer described in Section 3(d), the influence of the cavity of the actuator is included. The following boundary conditions are employed at the bottom of the cavity: v(x,y = const, t ) = A sin0 t (11.1) u ( x , y = const,t) = 0 (11.2) (11.3) In Eq.(ll.l), A=UJ b/W, where UJ is the velocity amplitude of the synthetic jet, and W and b are the width of the cavity and the jet-orifice respectively.

11.4 Results & Discussion In this part, we present typical selected results showing the effect of synthetic and pulsed jet on five different flow fields. For each case, detailed results are available in the cited references.

(a) Virtual Aerodynamic Shape Modification of an Airfoil Using a Synthetic Jet Actuator f X J Recently, it has been shown experimentally that the pressure drag of an airfoil at low angles of attack can be significantly reduced with a minimum change in lift by modification of the apparent aerodynamic shape of the airfoil. This virtual aerodynamic shape modification can be achieved by creating a small domain adjacent to the upper airfoil surface (downstream of the leading edge) which displaces the local streamlines sufficiently to modify the local pressure distribution. In a recent experiment [I] Glezer and his co-workers deliberately created such an interaction domain adjacent to the upper surface of a 24% thick Clark-Y airfoil by employing a synthetic jet actuator placed immediately downstream of a surface-mounted passive obstruction of small dimensions. We have performed numerical simulations of this experimentally observed fluidic modification of airfoil pressure distribution leading to reduced pressure drag [ 181. For performing the computations, CFD solver WIND was employed. Computations were performed for subsonic flow past a 24% thick Clark-Y airfoil with a triangular bump on the upper surface with and without a synthetic jet. Reasonably good agreement was obtained between the computations and the experimental data. Figure 1 shows the two-zone grid around the airfoil with a bump and synthetic jet.

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Figure 2 shows the comparison of computed pressure distribution with experimental data for the baseline Clark-Y airfoil (without bump) at Mach 0.1, chord Reynolds number Re, = 381,000 and angle of attack a=3". Figure 3 shows a typical comparison of computed pressure distribution with experimental data for the Clark-Y airfoil with a bump and synthetic jet of frequency f=850Hz and momentum coefficient Cp =1.2e-3; reasonably good agreement is obtained. Reference [ 181 provides extensive details of a range of computations performed for achieving the maximum drag reduction with minimum change in lift Figures 4 and 5 show the variation of lift and drag coefficient respectively with the frequency of the oscillatory jet. These figures show that a significant reduction in drag can be achieved, with minimal change in lift, at higher frequencies of the jet.

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Figure 1: Two-Zone Grid Around a 24% Thick Clark-Y Airfoil with a Bump

Figure 2: Computed and Experimental Pressure Distributions on Clark-Y Airfoil; M=0.085, Re=381,000, a=3.0".

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Flow Control with Synthetic and Pulsed Jets

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R.Agarwal, et al.

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(b) Vectoring Control of a Primary Jet with Synthetic Jets Recently it has been shown experimentally [ 131 that thrust-vectoring control of a subsonic primary jet exiting from a nozzle can be accomplished by a single or multiple synthetic jet actuators. We have performed extensive numerical simulations [6] to quantify the influence, on the vectoring angle of the primary jet (of given width and exit velocity), of various parameters of the synthetic jets, namely the width, the velocity amplitude, the fiequency, the location with respect to the primary jet, the angle, and the numbers. These simulations have provided information leading to optimum values of these parameters for achieving the maximum vectoring benefit. These simulations have also helped in clarifying the physical mechanism responsible for vectoring. The results of numerical analyses were employed in determining the requirements for the synthetic jet devices for controlling the thrust vectoring of an F-18 aircraft (these results are unpublished and cannot be included here). Here we present the results of a typical simulation; additional simulations and other details are given in Reference [6]. Figure 6 shows the schematic of a typical configuration (both primary jet and synthetic jet) employed in the numerical study. It also shows the Schlieren photographs of the unforced primary jet (without synthetic jet) and the forced primary jet (with synthetic jet) obtained from the experiments of Smith and Glezer [ 131. For a given set of flow parameters of the primary jet and synthetic jet, a 30" vectoring of the primary jet was observed in the experiment as shown in Figure 6. Our

Figure 6: Vectoring of a Primary Jet(PJ) with a Synthetic Jet (SJ); Uave=5.16 d s , ReuO=383, e l 1 2 0 Hz, dpull/h=3.556

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Flow Control with Synthetic and Pulsed Jc

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forced PJ (without cavity)

Figure 7: Computation of Vectoring of a Primary Jet(PJ) with a Synthetic Jet (SJ); Uave=5.16 m/s, ReuO=383, e l 1 2 0 Hz, dpull/h=3.556

dpuII= distance between PJ and SJ h = width of SJ H = width of PJ U,, = velocity amplitude of SJ U,, = velocity of the PJ Re% = Reynolds number of SJ f = frequency of SJ

computations reproduce the experimental results of Smith and Glezer in great detail. Figure 7 shows a typical result of numerical simulation; it should be noted that the inclusion of cavity of the synthetic jet is very important in obtaining the “correct” results from the numerical simulation. Figures 8-10 respectively show the influence of three parameters of the synthetic jet, namely dpull = distance between the synthetic jet and the primary jet, f = frequency, and Uamp = velocity amplitude, on the vectoring angle of the primary jet. Keeping two of the parameters fixed, there is an optimum value of the remaining parameter that gives the maximum vectoring benefit. These computations were performed by modifying the incompressible Reynolds-Averaged Navier-Stokes solver INS2D.

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Figure 8: Variation of Vectoring Angle of Primary Jet with dpun;Reu,,=383, U,=7.0 d s , e l 1 2 0 Hz

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Figure 9: Variation of Vectoring Angle of Primary Jet with f; Reuo=383, Uav,=7.0 d~ dp,l+h=3.556 ,

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Flow Control with Synthetic and Pulsed Jets

Figure 10: Variation of Vectoring Angle of Primary Jet with U,,; e l 1 2 0 Hz, d,,ll/h=3.556

U,,=7.0

m/s,

(c) Control of Recirculatinr!Flow Region Behind a Backward Facing; Step Using Synthetic Jets Recently it has been shown experimentally by Rediniotis et a1 [lo] that the recirculating flow region behind a backward facing step in a channel can be significantly reduced by employing a synthetic jet on the step. We have performed a numerical study [S] to investigate the influence of a single as well as multiple synthetic jets on the separated flow region behind the step. Numerical simulations have allowed us to perform an extensive parametric study, the results of which are documented in Reference [S]. Figure 11 shows the results of a typical calculation modeling the experimental configuration and flow parameters employed by Rediniotis et a1 [ lo] in their experiment. The computed velocity profiles compare reasonably well with experimental profiles. More importantly, the computed and experimental reattachment lengths are in excellent agreement as shown in Table 1. These computations were performed employing a two-zone grid with the Reynolds-Averaged Navier-Stokes solver INS2D.

(d) Interaction of a Synthetic Jet with a Flat Plate Turbulent Boundary Laver r41 Recently Honohan [7] has experimentally studied the interaction of a synthetic jet with a turbulent boundary layer. He considered two different sets of

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a l Blowing Stroke of SJ

x (mm) 9 Velocity Vectors for P e l Suction Stroke of $9

Figure 11: Vorticity Contours and Velocity Profiles for Flow over a Backward Facing Step with a Synthetic Jet(SJ) Employed on the Vertical Face of the Step; Average Inflow Velocity=lO c d s , Velocity Amplitude of SJ=40 c d s , Frequency of SJ=20 Hz, SJ Width=2 111111, Synthetic Jet is between 6 to 8 mm from the Bottom Wall

252

Flow Control with Synthetic and Pulsed Jets

Table 1: Reattachment Length for Various Flow Conditions of Figure 11 Flow condition

Reattachment length

Steady flow without syntheticjet Peak blowing stroke of syntheticjet Peak suction stroke of synthetic jet

8h

Experimental Reattachment Length 7.95h

1.6h

1.6h

0.6h

0.6h

parameters for the turbulent boundary layer and the synthetic jet (shown in Table 2) such that in the first case the discrete vortices of the synthetic jet scale with the entire turbulent boundary layer and in the second case they scale with the inner viscous sublayer of the approaching turbulent boundary layer. In this paper, we present some of the results of numerical simulation with experimental data for Case 1. More detailed results for this case and the results for Case 2 are given in Reference [4].

Table 2: The parameters used in the experiments of Honohan16

#‘

Case

U, (ds)

1 2

8.0 10.0

Case

Sr

1 2

,057 ,046

=(pJbuO2)’(pcomco2)

6 (mm) 12.7 31.0

Re

C,

1214 .23 406 .026

0 (mm) 1.4 3.2 (mm) 1.91 0.51

UJ (mh) 31.4 39.3

f

LO

(Hz) (mm) 300 33 1130 11

h/b

W/b

H/b

4.9 13

21 6.3

38 94

Sr =2 A f bl/2 / UJ = b/ Lo

First, the baseline case (without jet) is computed. Figure 12 shows the streamwise velocity profiles at the synthetic jet location obtained from the experiment, present computation and one-seventh power law. This velocity profile is imposed as a boundary condition on the left boundary in the simulation. Computed velocity profile agrees with the experiment quite well near the wall. However, there are minor differences in the region 0.2 < yl6 < 0.5. The cause for

R. Agarwal, et al.

253

this difference was investigated thoroughly. It was not found to be related to the fineness of the grid, order of the numerical scheme or turbulence model. Boundary layer thickness 6 is the same in both the computation and the experiment at the synthetic jet location and downstream of the synthetic jet. On the whole, the computed and experimental velocity vectors for the flat plate turbulent boundary layer are in good agreement as shown in Figure 13. Figure 15 shows the simulations and their comparison with the experimental results for velocity vectors and vorticity fields at every 6O0(=xI3 radians) during one time period (=2n) of the synthetic jet. On the whole, the simulations and experimental results are in good agreement, however significant differences are observed, especially in the velocity vectors in the downstream region close to the synthetic jet. These differences are more pronounced at @=O, n/3 and 5x13; that is during the early part of the “blowing” cycle and later part of the “suction” cycle. However, the generation of discrete vortices of the synthetic jet, their evolution and convection downstream during the “blowing” is reasonably well captured in the simulations when compared to the experimental data except at both @=O and x/3. The strength of the vortices downstream of the synthetic jet is much stronger in the experiment compared to the computations. During the suction cycle, the discrete pair of vortices is formed inside the cavity. Again, at both @=5n/3 and 2n (same as +O), the strength of the vortices observed in the experiment downstream of the synthetic jet is much stronger than in the computations. Based on these results, it can be concluded that simulations capture the physics of interaction of the synthetic jet with turbulent boundary layer reasonably well for the most part, however significant quantitative differences with experiment remain especially in the vicinity downstream of the jet during the early part of the “blowing” cycle and later part of the “suction” cycle. The resolution of these differences will require closer examination and evaluation of both experimental data and the simulations. Similar results were obtained for Case 2 as reported in Reference [4]. Nevertheless in spite of these differences between the simulations and experimental results, the nature of the influence of the synthetic jet on crossflow turbulent boundary layer is well captured in both the simulations and the experiments. The synthetic jet alters the pressure gradient near the wall, results in substantial reduction in vorticity thickness downstream of the jet (compared to unforced flow) and reduces the thickness of the turbulent boundary layer due to the presence of a favorable pressure gradient resulting from the displacement effect of the interaction domain.

254

Flow Control with Synthetic and Pulsed Jets

( e ) Control of Subsonic Cavity Shear Layer Using Pulsed B l o w i n g u We have performed time-accurate numerical simulations of subsonic laminar as well as turbulent flow over a 2-D cavity employing the multizone flow solver WIND [2]. TOvalidate the code for these type of calculations, a test case reported in the paper of Rowley et. al. [12] was employed. Figure 16 shows the vorticity contours at four different time-stages of one-cycle of cavity oscillations. These calculations for vorticity as well as time-averaged streamwise profiles (not shown here) are in excellent agreement with the calculations of Rowley et al. To break the structure of the shear layer over the cavity, we have employed pulsed blowing at the leading edge of the cavity in a direction normal to the streamwise flow. Numerical results with pulsed blowing show significant success in breaking the structure of the shear layer, as shown in Figure 17. The pulsed jet Mach number in the computations is Mj= 0.2 + O.lsin(0.902t) where 0 2 =780Hz is the frequency of the second Rossiter mode. Figure 18 shows the buffeting pressure on downstream edge of the cavity with and without pulsed blowing. It is clear that the pulsed blowing significantly reduces the pressure oscillations. Detailed flow field solutions for this case and for a (2x1) and a (15x1) cavity are given in Reference

1~

0.8 0.6 -

3.

0.4

0.2 -

Figure 12: Streamwise Velocity Profile at the Synthetic Jet Location in Case 1

R. Agarwal, et al.

255

Simulated Velocity Vectors

Experimental Vorticity Contours and Velocities

Simulated Vorticity Contours Figure 13: Vorticity Contours and Velocity Vectors without SJ in Case 1

Figure 14: Simulated Vorticity Contours in the Entire Cavity at @=O for Case 1

256

Flow Control with Synthetic and Pulsed Jets

(a. 1) Simulated Velocity Vectors

(a.2) Experimental Vorticity Contours and Velocities

-1

0

1

a

3

4

(a.3) Simulated Vorticity Contours (a>o=o

Figure 15: The Phase-Locked Vorticity Contours and Velocity Vectors in Case 1

25 7

R.Aganual, et al.

(b.1) Simulated Velocity Vectors

(b.2) Experimental Vorticity Contours and Velocities

-1

0

1

2

3

4

(b.3) Simulated Vorticity Contours (b) @=d3

(c. 1) Simulated Velocity Vectors

(c.2) Experimental Vorticity Contours and Velocities

258

Flow Control with Synthetic and Pulsed Jets

-1

0

1

2

4

3

(c.3) Simulated Vorticity Contours (c)@=2r1/3

(d. 1) Simulated Velocity Vectors

(d.2) Experimental Vorticity Contours and Velocities 1

0

-1

0

1

2

3

(d.3) Simulated Vorticity Contours ( 4 @=n

4

R. Agarwal, et al.

259

(e. 1) Simulated Velocitv Vectors

(e.2) Experimental Vorticity Contours and Velocities

-1

0

1

2

3

4

(e.3) Simulated Vorticity Contours (e) $=4z/3

(f. 1) Simulated Velocity Vectors

(f.2) Experimental Vorticity Contours and Velocities

260

Flow Control with Synthetic and Pulsed Jets

-1

0

1

2

3

4

(f.3) Simulated Vorticity Contours ( f ) +=5n/3 Figure 1S(contd.): The Phase-Locked Vorticity Contours and Velocity Vectors in Case 1 ( x and y are normalized by 6, the vorticity scale is given in Figure 14)

11.5 Conclusions In this paper, some results on flow control using synthetic and pulsed jets for five different flow fields dealing with virtual aeroshaping of an airfoil, thrust-vectoring of a propulsive jet, interaction of a synthetic jet with a flat plate turbulent boundary layer, control of recirculation region behind a backward facing step and control of cavity oscillations, have been presented. These applications demonstrate that the synthetic and pulsed jets can be employed as effective flow control devices. Simulations and experiments have reasonable agreement in capturing the overall features of the flow fields. All the five cases presented in this paper have been for low speed flows. Additional experimental data as well as numerical simulations are needed to cover realistic flow configurations for practical applications.

11.6 Acknowledgments The financial support provided by Boeing-St. Louis to J. Vadillo, Y. Tan and D. Guo is gratefully acknowledged. The authors would also like to acknowledge Prof. Glezer, Dr. Amitay and Dr. Honohan of Georgia Tech, for providing experimental information on Case 3(a), 3(b) and 3(d) and for many helpful discussions.

261

R. Aganval, et al.

Without pulsed blowing

With pulsed blowing ---, prrG:"--

-v*.Z.-*

---.

*.-

2 %

C ) 'ikmiesR. contoun a, 1 . 4 I ...........................................

.

'

-

"

I

x

*

. ) I

Figure 16: Vorticity Contours for Unsteady Flow over a (4x1) Cavity; M,=0.6, Ree=58 3.

Figure 17: Vorticity Contours for Unsteady Flow over a (4x1) Cavity with pulsed jet; M,=0.6, Re,5 8.8.

262

Flow Control with Synthetic and Pulsed Jets

Figure 18: Buffeting Pressure on Downstream Face of (4x1) Cavity without and with Pulsed Blowing

11.7 Bibliography 1. Amitay, M., Horvath, M., Michaux, M., and Glezer, A,, “Virtual Aerodynamic Shape Modification at Low Angles of Attack Using Synthetic Jet Actuators,” AIAA Paper 01-2975,2001. 2. Bush, R.H., “The Production Flow Solver of the NPARC Alliance,” AIAA Paper 88-0935, 1988. 3. Bush, R. H. and Mani, M., “A Two-Equation Large Eddy Stress Model for High Sub-Grid Shear,” AIAA Paper 2001-2561,2001. 4. Cui, J., Agarwal, R., Cary, A. W., “Numerical Simulation of the Interaction of a Synthetic Jet with a Turbulent Boundary Layer,” AIAA Paper 2003-3458, 33rd AIAA Fluid Dynamics Conference and Exhibit, Orlando, Florida, 23 - 26 Jun 2003. 5 . Glezer, A. and Amitay, M., “Synthetic Jets”, Annu. Rev. Fluid Mech., Vol. 34, 2002, pp.503-529. 6. Guo, D., Cary, A.W., and Agarwal, R.K., “Numerical Simulation of Vectoring Control of a Primary Jet with a Synthetic Jet,” AIAA Paper 2002-3284, lst AIAA Control Conference, St. Louis, MO, 24-27 June 2002. 7. Honohan, A.M., “The Interaction of Synthetic Jets with Cross Flow and the Modification of Aerodynamic Surfaces,“ Ph.D. Thesis, Georgia Institute of Technology, May 2003. 8. Jain, H., Agarwal, R.K., and Cary, A. W., “Numerical Simulation of the Influence of a Synthetic Jet on Recirculating Flow Over a Backward Facing

R. Agarwal, et al.

9. 10.

11.

12.

13. 14. 15.

16. 17.

18.

263

Step,” AIAA Paper 2003-1125, AIAA 4lStAerospace Sciences Meeting and Exhibit, Reno, NV, 6-9 January 2003. Menter, F.R., “Zonal Two-Equation k-o Turbulence Models for Aerodynamic Flows”, AIAA Paper 93-2906, 1993. Rediniotis, O.K., KO, J. and Yue, X., “Synthetic Jets, Their Reduced Order Modeling and Applications to Flow Control,” AIAA Paper 99-1000, 37” Aerospace Sciences Meeting and Exhibit, Reno, NV, Jan. 1999. Rogers, S.E. and Kwak, D., “An Upwind Differencing Scheme for the TimeAccurate Incompressible Navier-Stokes Equation,” AIAA Journal, Vol. 28, NO. 2, 1990, pp. 253-262. Rowley, C.W., Colonius, T., and Basu, A.J., “On Self-sustained Oscillations in Flows over Rectangular Cavities,” J. Fluid Mech., Vol. 455, 2002, pp. 315346. Smith, B.L. and Glezer, A., “Vectoring and Small-Scale Motions Effected in Free Shear Flows Using Synthetic Jet Actuators,” AIAA Paper 97-0213, 35” AIAA Aerospace Sciences Meeting, Reno, NV, January 1997. Spalart, P. R. and Allmaras, S. R., “A One-equation Turbulence Model for Aerodynamic Flows,” La Recherche Aerospatiale, Vol. 1, 1994, pp. 5-21. Spalart, P. R., Jou, W. H., Strelets, M., Allmaras, S. R., “Comments on the Feasibility of LES for Wings, and on a Hybrid RANSILES Approach,” lst AFOSR Int. Conf. on DNSLES, Ruston, LA, 4-8 August 1997. Strelets, M., “Detached Eddy Simulation of Massively Separated Flows,’’ AIAA Paper 2001-0879, 3 9 AIAA ~ Aerospace Sciences Meeting and Exhibit, Reno, NV, 8- 11 Jan. 200 1. Tan, Y., Agarwal, R. K., Bower, W. W., and Cary, A. W., “Flow Control of Shear Layer Over Cavities Using a Pulsed Jet and Aero-optical Analysis,” AIAA Paper 2004-0928, AIAA 42”dAerospace Sciences Meeting, Reno, NV, 5-8 Jan 2004. Vadillo, J.L. and Agarwal, R.K., “Numerical Study of Virtual Aerodynamic Shape Modification of an Airfoil Using a Synthetic Jet Actuator,” AIAA Paper 2003-4158, 331d AIAA Fluid Dynamics Conference, Orlando, FL, 23-26 June 2003.

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Chapter 12

Control of Flow Separation over a Circular Cylinder Using Electro-Magnetic Fields: Numerical Simulation Brian H. Dennis' and George S. Dulikravich2

12.1

Nomenclature magnetic flux density, kg A-' s-' specific heat at constant pressure, K-l m2 s - ~ average rate of deformation tensor, s-l material derivative, s-l electric displacement vector, A s rn-' electric field intensity, k g m s - ~A-' electromotive intensity, kg rn sP3A-l magnetic field intensity, A rn-l unit tensor electric current density, A rn-2

'1 E k= H

&+gxB

I

-

J =J,

+gel ~~

'Mechanical and Aerospace Engineering, University of Texas at Arlington, Arlington, Texas, 76019 'Mechanical and Materials Engineering, Florida International University, 10555 W. Flagler St., Miami, FL, U.S.A. 33174

265

266

Flow Separation Control with Electro-Magnetic Fields electric conduction current, A m-' total magnetization per unit volume, A m-l magnetomotive intensity, A m-l pressure, kg m-1 sp2 total polarization per unit volume, A s m P 2 total electric charge per unit volume, A s mP3 conduction heat flux, kg s - ~ heat source per unit volume, kg m-l s - ~ time, s Cauchy stress tensor, kg m-l sp2 absolute temperature, K internal energy per unit mass, m2 s-' fluid velocity,m s-' electric permittivity, kg-' mP3s4 A' electric permittivity of vacuum, kg-' m-3 s4 A2 polarization electric permittivity, kg-' mP3s4 A2 relative electric permittivity fluid viscosity, kgm-ls-l electric potential, V m modified hydrostatic pressure, kgm-'sP2 fluid density, kg mP3 deviator part of stress tensor, kg m-l sP2 magnetic permeability, kg m A-' s - ~ magnetic permeability of vacuum, kg mA-2s-2 relative magnetic permeability magnetization magnetic permeability,kg mA-2s-2 magnetic susceptibility based on B electric susceptibility vorticity, s-l

12.2

Introduction

In recent years there has been a growing interest in the simulation of coupled physics or multidisciplinary phenomena. Advances in computer processor technology has recently allowed researchers to consider large systems of differential equations representing complex coupled problems. An example of a multidisciplinary analysis is the simulation of fluid flow under the influence of externally applied electro-magnetic fields. The study of fluid flows containing electric charges under the influence of an externally applied electric field and negligible magnetic field is known as electrohydrodynamics or EHD. The study of fluid flows without electric charges and influenced only by an externally applied

B. H. Dennis and G. S. Dulikruvich

267

magnetic field is known as magnetohydrodynamics or MHD 1181. Numerous publications are available dealing with the EHD and the MHD models [19, 131, their numerical simulations, and applications [6, 4, 3, 5 , 7, 17, 9, 8, 11. Although fairly complex, the existing mathematical models for EHD and MHD often represent unacceptably simplified and inconsistent models of the actual physics [ll]. The study of fluid flows under the combined influence of the externally applied and internally generated electric and magnetic fields is often called electro-magneto-fluid dynamics (EMFD) [12, lo]. However, the mathematical model for such combined electromagnetic field interaction with fluid flows is extremely complex and requires a large number of new physical properties of the fluid that cannot be found in open literature. Thus, a somewhat simplified mathematical model should be used in actual numerical simulations of fluid flows under the combined influence of the externally applied electric and magnetic fields. In the case of incompressible fluids, such a non-linear model termed second order lectromagnetohydrodynamics (EMHD) was derived by KO and Dulikravich [15]. This is a second order theory that is fully consistent with all of the basic assumptions of the complete EMFD model [12, lo]. The basic assumptions are that the electric and magnetic fields, rate of strain, and temperature gradient are relatively small. Furthermore, terms of second order and higher in the average rate of deformation tensor are neglected as in the case of conventional Newtonian fluids. Only the terms up to second order in d,E,B, V T are retained. Because of the unavailability of the complete EMHD model until recently and because of the considerable complexity of even simpler versions of the EMHD model, it is still hard to find publications dealing with the combined influence of electric and magnetic fields and fluid flow. The objective of this paper is to present numerical results for the flow over a circular cylinder that is under the influence of combined electric and magnetic fields. The results presented here indicate that electro-magnetic fields can be used to eliminate the flow separation in steady flow. In addition, results also show that electremagnetic fields can be used to eliminate periodic vortex shedding in the case of unsteady flow. These simulations were performed using a simplified EMHD model for the case of two-dimensional planar flows for electrically conducting incompressible fluids. The equations were discretized with the Least-Squares Finite Element Method (LSFEM) and solved on a single processor workstation. The numerical results will be presented for both steady and unsteady laminar flows of homocompositional Newtonian fluids. The accuracy of the numerical method was also verified against a simple analytical solution for magnetohydrodynamics.

It should be pointed out that similar effects on the flow-field around a circular cylinder were numerically predicted and experimentally verified by a research team from Germany [20, 211. However, their arrangement of magnets and electrodes was entirely different from the arrangement presented in this paper indicating that there are multiple configurations of magnets and electrodes

268

Flow Separation Control with Electro-Magnetic Fields

capable of producing the same flow-field alterations.

12.3

Second Order Analytical Model of EMHD

A full system of partial differential equations governing incompressible flows under the combined effects of electromagnetic forces [15] is summarized in this section by using the constitutive equations which have been derived through the second order theory. Specifically, polarization and magnetization vectors are defined as (12.1) which indicates a medium with purely instantaneous response [16]. The deviator part of the stress tensor is defined as

Electric current conduction vector is defined as

while thermal conduction flux is defined as

Then, Maxwell's equations become

(12.5) (12.6) (12.7)

269

B. H. Dennis and G. S. Dulikravich while the Navier-Stokes equations become

(12.9)

+

&.---.D(E&

B DB

Dt

pm

Dt'

Notice that in this EMHD model the physical properties of the incompressible fluid, Xei X B , Pv,01, 0 2 , 0 4 , 0 5 , 0 7 , 61, 6 2 , 6 4 , 6 5 , 6 7 , K 8 , Q, can be either constants or functions of temperature only.

12.4

Least-Squares Finite Element Method

The system of partial differential equations described in this section 12.3 is discretized using the Least-Squares Finite Element Method (LSFEM). We first look at the LSFEM for a general linear first-order system [14, 21

( 12.12)

Lu=f

where

d

d

L = A1 - + A2- =ax =ay

+= A3

(12.13)

for two-dimensional problems. The residual of the system is represented by R(a)= & f - --

H.

(12.14)

We now define the following least squares functional I over the domain R

1(a)=

H(aIT . R(u)dx dy

( 12.15)

270

Flow Separation Control with Electro-Magnetic Fields

The weak statement is then obtained by taking the variation of I with respect to and setting the result equal to zero.

(12.16) Using equal order shape functions,

for all unknowns, the vector

is written

as

u = ~ & { U l , U 2 , U 3 : ...,

T

(12.17)

i=l

where {u1,u2,ug,...,~,}~ are the nodal values at the ith node of the finite element. Introducing the above approximation for g into the weak statement leads to a linear system of algebraic equations

m = F -

where K - is the stiffness matrix, vector.

12.4.1

uis the vector of unknowns, and

(12.18) is the force

Nondimensional First Order Form for Simplified EMHD

The full system of partial differential equations describing EMHD flows contain many parameters that refer to physical properties that are not known at this time. Rather than complete numerical simulations with guessed values of these parameters, we chose to work with only those terms for which the material properties are known. In this case, we simplify the equations by retaining only source terms that contain ril and 01 since these values are available for various fluids. Use of LSFEM for systems of equations that contain higher order derivatives is usually difficult due to the higher continuity restrictions imposed on the approximation functions. For this reason it is more convenient to transform the system into an equivalent first order form before applying LSFEM. For the case of electro-magneto-hydrodynamics, the second order derivatives are transformed by introducing vorticity, g,as an additional unknown. In addition, we assume the flow is unsteady but isothermal and without charged particles. In this case the energy and charge transport equations are not required and source terms associated with charges are dropped. We also consider only electrostatic and magnetostatic fields.

2 71

B. H. Dennis and G. S. Dulikravich

v

=0

(12.19)

B*

=0

(12.20)

v x g*

=0

(12.21) (12.22)

'

g*

at + g * .vg*+ &v x g*+ v p * - g g * x B* x B* a?'

-M&* g*

-

x

V.B* Vx

B* = Rm< x B* -+ B&* V .E* v x E*

vq5* where g* = gv01, g* = g)Ov,',

=0

(12.23) =0

(12.24)

=0

(12.25) =E* (12.26)

B* = BB,', E* = EL0 A&',4*

=

q5Aq5ClI

IC* = 5 L , y* = y Li'. Here, LOis the reference length, 210 is p* = the reference speed, Aq50 is the reference electric potential difference]and Bo is the reference magnetic flux density. For convenience the * superscript will be dropped for the remainder of the paper. The nondimensional numbers are given by:

It should be noted that the electric potential is introduced as an additional variable due to the convenience of applying physically meaningful boundary conditions for electrodes. For the electric field equations] the first order form of Maxwell's equations does not include electric potential. Since the most common boundary conditions for static electric fields are given in terms of potential, it is necessary to add the equation (12.26) for electric potential] 4. We now write the above system in the general form of a first-order system (12.12). Although the entire system written in (12.19)-(12.26) can be treated by LSFEM, it was found to be more economical to solve the fluid and electromagnetic field equations separately, in an iterative manner. Here, a general form first order system is written for the fluid system (12.19)- (12.21) and denoted by the superscript f l u i d . Here the time derivative term in the fluid equations is approximated using the backward-Euler scheme. (12.28)

A first-order system is also written in general form for the electro-magnetic field equations (12.22)-(12.26)and is denoted by the superscript em. In addition, the nonlinear convective terms in the fluid equations are linearized with Newton's

272

Flow Separation Control with Electro-Magnetic Fields

method leading to a system suitable for treatment with the LSFEM. For the two-dimensional case we specify the z component of the magnetic field and assume the x and y components are zero. For many engineering applications, the magnitude of Rm and B2 is typically small so we expect the current-induced magnetic field in x - y to be negligible compared to the magnitude of the externally applied magnetic field. The x and y components for velocity, 2, and electric field, E, are left as unknowns while their z components are assumed to be zero. For simplicity, we only consider flows that do not contain free charged particles.

fe

rn={

ii.y.-iL}

(12.30)

A solution satisfying all of the above systems of equations can be found by using a simple iterative process. First, the system given in (12.30) is solved for the electric field. The system in (12.29) is solved with the electric field and velocities from the previous time step. Here, quantities taken from the previous iteration are designated with the subscript 0. These equations may be iterated at each time step if the problem is very nonlinear. In that case the iteration at each time step is repeated until a specified convergence tolerance is reached. The reduction of the residual norm of both systems by 3.5 orders of magnitude is usually achieved in less than 5 iterations.

273

B. H. Dennis and G. S. Dulikravich

12.4.2 Verification of Accuracy It is difficult to verify the accuracy of an EMHD code. This is due to the absence of analytical solutions for such equations. However, analytical solutions do exist for MHD flows. Here we will use such an analytic solution to validate the MHD portion of the code. The accuracy of the LSFEM for MHD was tested against analytic solutions for Poiseuille-Hartmann flow [13]. The Poiseuille-Hartmann flow is a 1-D flow of a conducting and viscous fluid between two stationary plates with a uniform external magnetic field applied orthogonal to the plates. Assuming the walls are at y = f L and that fluid velocity on the walls is zero and that the fluid moves in the x-direction under the influence of a constant pressure gradient, then the velocity profile is given by

u(y) = --

aBy"dx

cosh(Ht) - c o s h ( y ) sinh(H t )

(12.31)

The movement of the fluid induces a magnetic field in the x-direction and is given by s i n h ( y ) - f sinh(Ht) cosh(Ht) - 1

(12.32)

A test case was run using the parameters given in Table 12.1 and with a mesh composed of 2718 parabolic triangular elements. Figure 12.1 shows the computed and analytical results for the velocity profile. Figure 12.2 shows the computed and analytical results for the induced magnetic field. For both cases, one can see that the agreement between the analytical solution and the LSFEM solution is excellent.

Ht Rm Lo (m) 210 (rn s-1)

(5

,-1

S -1)

Bo (TI p (Hm-l) dpldx ( P a m-l) c (0-lm-l)

10.0 6 x 10W7 1.0 0.6 0.01 1.0 1 x 10-6 0.6 1.0

Table 12.1: Parameters used for Poisuille-Hartmann Flow Test Problem

274

Flow Separation Control with Electro-Magnetic Fields

7.00E-01 6.00E-01 5.00E-01

-

4.00E-01

- 3.00E-01 2.00E-01 1.00E-01 O.OOE+OO -1.00E-01

0.5

1

1.5

t

Figure 12.1: Computed and exact values for velocity

12.5 Numerical Results The LSFEM formulation for EMHD will now be demonstrated for the electromagnetic control of flow over a circular cylinder. The configuration of the electrodes and magnets is illustrated in Figure 12.3. In this configuration, the cylinder is divided into two electrodes, one on top and bottom, with magnets placed slightly downstream in the wake region. This configuration is in reality 3-D, but can be approximated in 2-D by specifying the z-component of the magnetic and computing the x - y components of the electric and flow fields. The known magnetic field is assumed to be uniform and is applied into the x - y plane. The fluid is considered to be electrically conducting and flows over a circle with unit diameter. We computed both steady and unsteady flow cases to observe the effect of the electro-magnetic fields on the flow patterns. The relevant nondimensional parameters are shown in Table 12.2 for the two test cases. The hybrid triangular and quadrilateral mesh shown in Figure 12.4 was used for all computations. The mesh is composed of 818 parabolic elements with 2458 nodes. No slip boundary conditions were applied to the cylinder surface while free stream conditions were applied at the inlet and top and bottom of the outer domain. Conditions on the outlet boundary were left free. Electric potentials are specified on the surfaces of the electrodes, thus creating a potential difference

275

B. H. Dennis and G. S. Dulikravich

0

0.5

1

Y+L (m)

1.5

2

Figure 12.2: Computed and exact values for induced magnetic field that forces current to flow through the electrically conducting fluid. Figure 12.5 shows the computed distribution of the electric potential as well as the electric field lines for the electrostatic field. The source terms in the EMHD system directly involve the electric field intensity, E , so we expect that the shape of the field lines will have a strong influence on the flow pattern. In the present case, constant potentials are used on each electrode so the field lines are distributed smoothly across the domain. In the first case, the steady flow at Re = 37 is computed with no electric or magnetic field. Figure 12.6 shows the streamlines and pressure distribution for this classical flow. When the electric and magnetic fields are applied to the flow, the velocity and pressure distribution has been changed dramatically as shown in Figure 12.7. One interesting note about this result is that although the separation behind the cylinder has been removed, the pressure on the back of

1.0 x 10-

1.0 x 10-

Table 12.2: Non-dimensional numbers used for test cases

276

Flow Separation Control with Electro-Magnetic Fields

-

Uniform Flow

Figure 12.3: Test configuration for magnets and electrodes

5

>

0

5

0

10

X

20

Figure 12.4: Hybrid unstructured mesh used

the cylinder is significantly decreased. This low pressure is due to the increased flow velocity that occurs just behind the cylinder when the body force due to the electro-magnetic field is present. In addition, if the polarity of the electrodes is reversed, the opposite effect is observed. Figure 12.8 shows that the separation strength is actually enhanced by the reverse polarity electro-magnetic field. However, although the separation is stronger, the pressure on the back wall of the cylinder increases. Once again, this increase in pressure is due to the energy inserted into the flow through the external electro-magnetic field. In the second case, the unsteady flow at Re = 100 is first computed with no electric or magnetic field. The flow is computed from an initially uniform flow field until a periodic state is reached. For this case a time step At = 0.1 was used and the periodic flow was reached after 900 time steps. Figure 12.9 shows the particle traces at one instant around 1400 time steps. At this point the characteristic vortex shedding pattern can be clearly seen. The case was run

B. H. Dennis and G. S. Dulikravich

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Figure 12.5: Computed electric field lines and electric potential contours again, but after 1200 time steps the static electro-magnetic field was activated. By 1400 time steps the particle traces in Figure 12.10 clearly show the effect of the electro-magnetic field on the wake structure. By the 2000 time step mark the flow reaches a steady, time-independent state. The flow becomes more stable due t o the elimination of the periodic vortex shedding as shown in Figure 12.11. The combined electric and magnetic field in this configuration have a strong damping effect to the point of completely suppressing the vortex shedding typically seen at Re = 100. The resulting flow field is steady and is shown in Figure 12.12.

12.6

Conclusion

A unified theoretical model of simultaneously applied and interacting electric and magnetic fields and incompressible homocompositional viscous fluid flows has been expressed as a coupled sequence of first order partial differential equation systems. These systems were discretized in 2-D using a least-squares finite element method and integrated on an unstructured computational grid. Numerical results are in excellent agreement for the test case of a steady laminar flow between infinite parallel plates with simultaneously applied uniform vertical electric field and a uniform horizontal magnetic field. The method was used to simulate the flow over a circular cylinder with and without an externally applied electric and magnetic field. Results show that a certain arrangement of

2 78

Flow Separation Control with Electro-Magnetic Fields

Figure 12.6: Pressure field and streamlines for steady flow with no electric field and no magnetic field electrodes and magnets can be used to eliminate flow separations in steady flow and suppress vortex shedding in unsteady flows.

12.7

Acknowledgements

This work was performed while the primary author held the position of Visiting Associate Professor at the University of Tokyo. The primary author gratefully acknowledges support from the Graduate School of Frontier Sciences at the University of Tokyo. The second author is grateful for the partial support provided by the NSF grant DMS-0073698 administrated through the Computational Mathematics Program.

12.8 Bibliography [l] Colaco, M. J., Dulikravich, G. S., & Martin, T. J. Optimization of wall electrodes for electro-hydrodynamic control of natural convection effects during solidification. Materials and Manufacturing Processes, 19(4), 2004.

[2] Dennis, B. H. Simulation and Optimization of Electromagnetohydrodynamic Flows. PhD thesis, Pennsylvania State University, University Park, PA, Dec. 2000.

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Figure 12.7: Pressure field and streamlines for steady flow with electric field and magnetic field

[3] Dennis, B. H. & Dulikravich, G. S. Magnetic field suppression of melt flow in crystal growth. International Journal of Heat & Fluid Flow, 23(3):pp. 269-277, 2002. [4] Dennis, B. H., Egorov, I. N., Han, Z.-X., Dulikravich, G. S. & Poloni, C. Multi-objective optimization of turbomachinery cascades for minimum loss, maximum loading, and maximum gap-to-chord ratio. International Journal of Turbo & Jet-Engines, 18(3):201-210, 2001. [5] Dulikravich, G. S. Electro-magneto-hydrodynamics and solidification. In D. A. Siginer, D. De Kee, and R. P. Chhabra, editors, Advances in Flow and Rheology of Non-Newtonian Fluids, Part B, volume 8 of Rheology Series, chapter 9, pages 677-716. Elsevier Publishers, 1999.

[6] Dulilcravich, G. S., Ahuja, V., & Lee, S. Modeling three-dimensional solidification with magnetic fields and reduced gravity. International Journal of Heat and Mass Tbansfer, 37(5):pp. 837-853, 1994. [7] Dulikravich, G. S., Choi, K.-Y., & Lee, S. Magnetic field control of vorticity in steady incompressible laminar flows. In D. A. Siginer, J. H. Kim, S. A. Sheriff, and H. W. Colleman, editors, Symposium on Developments in

Electrorhwlogical Flows and Measurement Uncertainty, A S M E WAM’94, 1994. Chicago, IL, Nov. 6-11, 1994, ASME FED-Vol. 205/AMD-Vol. 190, (1994), pp. 125-142.

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Flow Separation Control with Electro-Magnetic Fields

Figure 12.8: Pressure field and streamlines for steady flow with reversed electric field and magnetic field [8] Dulikravich, G. S., Colaco, M. J., Dennis, B. H., T. J. Martin, T. J., Egorov-Yegorov, I. N., & Lee, S . 4 . Optimization of intensities, and orientations of magnets controlling melt flow during solidification. Materials and Manufacturing Processes, 19(4):695-718, 2004. [9] Dulikravich, G. S., Colaco, M. J., Martin, T. J., & Lee, S. Magnetized fiber orientation and concentration control in solidifying composites. J. of Composite Materials, 47(15):pp. 1351-1366, 2003.

[lo] Dulikravich, G. S. & Lynn, S. R. Unified electro-magneto-fluid dynamics (emfd): Introductory concepts. International Journal of Non-Linear Mechanics, 32(5):913-922, 1997. [ll]Dulikravich, G. S. & Lynn, S. R. Unified electro-magneto-fluid dynamics (emfd): Survey of mathematical models. International Journal of NonLinear Mechanics, 32(5):923-932, 1997. [12] Eringen, A. C. & Maugin, G. A. Electrodynamics of Continua and Complex Media. Springer-Verlag, New York, 1990.

IX; Fluids

[13] Hughes, W. F. & Young, F.-J. The Electromagnetodynamics of Fluids. John Wiley and Sons, New York, 1966.

[14] Jiang, B.-N. The Least-Squares Finite Element Method. Spring-Verlag, Berlin, 1998.

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6

:t

Time=l40.3 sec

3

2

1

*

o 1

-2

-3 -4 -5

-6

I

I

0

10

X

0

Figure 12.9: Particle traces for unsteady flow with no electric field and no magnetic field [15] KO, H.-J & Dulikravich, G. S. A fully non-linear model of electromagneto-hydrodynamics. International Journal of of Non-Linear Mechanics, 35(4):pp. 709-719, 2000. [16] Lakhtakia, A. & Weiglhofer, W. S. On the causal constitutive relations of magnetoelectric media. In 1995 IEEE International Symposium on Electromagnetic Compatibility. Atlanta, GA, August 14-18, (1995), pp. 611613. [17] Meir, A. J. & Schmidt, P. G. Analysis and finite-element simulation of mhd flows, with an application to seawater drag reduction. In Proceedings of the International Symposium on Seawater Drag Reduction, Newport, July 22 - 23, pages 401-406, 1998. [18] Stuetzer, 0. M. Magnetohydrodynamics and electrohydrodynamics. Physics Of Fluids, 5(5):534-544, 1962. [19] Sutton, G. W. & Sherman, A. Engineering Magnetohydrodynamics. McGraw Hill, New York, 1965. [20] Weier, T., Gerbeth, G, Mutschke, G, Lielausis, O., & Platacis, E. Exper-

iments on cylinder wake stabilization in an electrolyte solution by means of electromagnetic forces localized on the cylinder surface. Experimental Thermal and Fluid Science, 10:84-91, 1998. [21] Weier, T., Gerbeth, G., Posdziech, O., Lielausis, O., & Platacis, E. Some results on electromagnetic control of flow around bodies. In Proceedings of

Flow Separation Control with Electro-Magnetic Fields

282

EM Flew Applled

.

0.3

:

0.2

:

.

-0.1 u1

g0

> -0.1

:

-0.2 7

-0.3

:

-0.4

-0.5

I

75

, , , ,

I 100

, , , ,

I 125

, , , ,

8 , 150

, ,

,

I

,

175

time (see)

Figure 12.10: Time variation of v-component of velocity in the wake

the International Symposium on Seawater Drag Reduction, Newport, July 22 - 23, pages 395-400, 1998.

B. H. Dennis and G. S. Dulikravich

283

6

5

Time=140.3 sec

4

3 _ * .

.

1

..

..

1

*

D 1 -1 .3

4 -5

-6

I

0

I

x

10

10

Figure 12.11: Particle traces for unsteady flow with electric field and magnetic field turned on at 120 s

Figure 12.12: Pressure field and streamlines for Re=100 steady flow induced by electric field and magnetic field

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Chapter 13

Bifurcation of Transonic Flow Over a Flattened Airfoil Alexander G. Kuz’min’

13.1 Introduction At high subsonic freestream velocities, there exist local supersonic zones near the upper and/or lower surfaces of the airfoil. If the curvature of the airfoil is small in the midchord region, then the flow may exhibit several supersonic zones near each surface. The zones typically merge when the freestream Mach number M , increases and they split into smaller ones when M , decreases. Recently, Kuz’min and Ivanova [8, 5, 121 studied inviscid transonic flow over a bump in a channel and demonstrated an instability associated with the splitting/merging of the supersonic zones. The instability implied an abrupt change of the structure of steady flow under small perturbation of boundary conditions. The concept of the structural instability has contributed to understanding the non-uniqueness of transonic flow revealed in previous years (Jameson [7], Bafez and Guo [3, 41, Caughey [l]). For symmetric airfoils, a link between the non-uniqueness and instability was analyzed in [lo, 6 , 141. A few asymmetric airfoils were discussed in [ll,91. In this paper, we pursue further analysis of inviscid transonic flow over a symmetric airfoil. The dependence of the lift coefficient C, on freestream con‘Institute for Mathematics and Mechanics, St. Petersburg State University, 28 University Ave., Petrodvorets, St. Petersburg 198504, Russia.

285

Bifurcation of Transonic Flow

286

ditions, the stability of symmetric solutions, and the hysteresis with respect to variation of the angle of attack are discussed.

13.2

Problem statement and a numerical method

We consider two-dimensional inviscid compressible flow over the airfoil y(z) = *0.06.\/1

-

(2x - 1)4(1- d21’,

O < X l l ,

(13.1)

whose curvature attains a minimum of 0.0126 at x = 0.482. A C-type far-field boundary of the computational domain is placed at 15 to 18 chord lengths from the airfoil. The Mach number Mm and the angle of attack a are given on the far-field boundary, while the classical slip condition is prescribed on the airfoil. To obtain numerical solutions of the problem, we employed the NSCKE finite-volume solver in which the Euler equations are discretized in space on unstructured meshes using the Roe scheme [13]. The second-order accuracy is obtained with a MUSCL reconstruction using Van Albada type limiters. The numerical modeling of inflow and outflow boundary conditions is based on the Steger-Warming flux vector splitting technique [15]. The time integration was performed with an explicit four-stage Runge-Kutta scheme. Steady-state solutions were calculated using the local time stepping strategy. The initial data were either an uniform state defined by the freestream conditions or a steady flow field obtained previously for other values of Mm and a. Computations have been performed primarily on a triangular mesh of 733 x 215 grid points which clustered near the airfoil and in the vicinity of shock waves. Test computations on a coarser mesh of 489 x 143 showed that the error in the location of shock waves increased considerably. On the other hand, a refinement of the primary mesh yielded just insignificant corrections in the calculated flow field, while leading to a disproportionate increase of the CPU time necessary to obtain steady solutions. The method was verified by computation of transonic flow in a channel and comparison with solutions of the Euler equations obtained with a scheme E N 0 2 [8, 51. The results obtained on similar meshes were in excellent agreement. Also, the accuracy of the method was confirmed by computation of a benchmark problem for the NACA 0012 airfoil at Q = 1.25 deg [2].

13.3

Analysis of the lift coefficient as a function of Mm

A plot of the lift coefficient versus Mm calculated at Q = 0 with the method outlined above is shown in Figure 1. The left branch of the plot was obtained

287

A . Kuz’rnin

cL

c

I

I

I

I

I

I

I

,

I

I

I

‘M,

0.820 0.824 0.828 0.832 Figure 13.1: Lift coefficient Ch as a function of M , for airfoil (13.1) at

(I:

= 0.

by computing the flow at M, = 0.818 and thenincreasing the Mach number step-by-step to 0.8275. At each step, the previous steady state was used as initial data. The calculated flow field is symmetric with respect to the z-axis and involves two couples of supersonic zones (Figure 2a). The right branch of the plot shown in Fig. 1 was obtained by computing the flow at Mm = 0.836 and then reducing the Mach number step-by-step to 0.829. The corresponding flow is symmetric and exhibits a single supersonic zone on each surface of the airfoil. The singular Mach number M, M 0.8283 is determined as the limit to which the right and left branches can be extended without restructuring the flow. For initial data given by the symmetric flow depicted in Fig. 2a, the perturbation a = 0.07 deg resulted in an abrupt amalgamation of the supersonic zones on the upper surface and relaxation to an essentially asymmetric steady state. The obtained structure 1+2 (with a single supersonic zone on the upper surface of the airfoil and two supersonic zones on the lower one) is preserved if a returns from 0.07 to zero (Figure 2b). This structure persists when the freestream Much number varies in the interval Mmin

< Mm < Mmax

(13.2)

with Mmin M 0.8228 and M,, x 0.8315 (see the upper branch in Fig. 1). Similarly, a negative perturbation of (I: entails transition to the lower branch of the plot shown in Fig. 1. Dependence of interval (13.2) on the mesh refinement is demonstrated in can Figure 3. It shows that the absolute error in the values of M,in and M, be estimated by 0.0003. Therefore, the last digits in the obtained values are

288

Bifurcation of Transonic Flow

0

0.4

0.8

X

0

0.4

0.8

X

Figure 13.2: Mach number contours in transonic flow over airfoil (13.1) at M , = 0.824, a = 0: a) symmetric steady state corresponds to point A2 in Fig. 1, b) asymmetric steady state corresponds to point A1 in Fig. 1.

rather questionable. However, we do not round off the values to avoid extra errors.

A comparison of Fig. 1 with the results obtained for a similar airfoil with (1 - do)instead of (1 - d2)in (13.1) [6, 141 shows that both the length of interval (13.2) and the maximum of C, in the case at hand are greater than those in (6, 141. This is attributed to the smaller curvature of the airfoil in the midchord region. At the same time, both the length of interval (13.2) and the maximum of CL are smaller than those for an airfoil with the flat middle part considered in [3]. Figure 4 displays the lift coefficient as a function of M , at a = 0.1 deg. In this case the interval (13.2) expands since Mminshifts to 0.8206 while M,, does to 0.8333.

289

A . Kuz’min

0.829 0.827 0.825 Mmin

0’823: ,

,

NO. of.grid

Figure 13.3: Shifts of the endpoints of interval (13.2) with mesh refinement.

13.4

Analysis of stability with respect to variation of ctl

If one takes the asymmetric flow shown in Fig. 2b for initial data and reduces Q from 0 to -0.04 deg keeping M , = 0.824 fixed, then computations demonstrate the splitting of the supersonic zone on the upper surface and transition to the structure 2+2 (with two supersonic zones adjacent to each surface of the airfoil). This corresponds to a jump from the point A1 indicated in Figure 5b to the left endpoint of the middle branch of the plot as shown by the arrow. After that, the return to Q = 0 yields a shift along the middle branch to the point A2, while further decrease of Q to -0.06 deg results in an abrupt amalgamation of the supersonic regions adjacent to the lower surface and transition to the structure 2+1. The latter implies a jump to the lower branch of the plot in Figure 5b. Then restoration of a = 0 yields a shift along the lower branch to the point A3. As M , increases from 0.824 to 0.8255, the middle branch of the plot CL(Q) shrinks and rotates to a nearly vertical position (Figure 5c). Hence, for the symmetric flow, the stability range with respect to variations of Q becomes very narrow. At M , = M, the middle branch disappears, while the hysteresis associated with transitions from the upper branch to the lower one and vice versa attains its maximum (Figure 5d). In fact, the symmetric flow is unstable not only at M , = M, but in a certain interval of freestream Mach numbers enclosing M,. For example, in the interval 0.8248 f Moo 5 0.8305 (indicated by the dashed segments in Fig. 1) the replacement of Q = 0 by Q = 50.04 deg is sufficient to trigger the abrupt

290

Bifurcation of Transonic Flow

cLl 0.1

'.-P

" l - O -0.2 -

0.820 0.824 0.828 0.832 0.836 Moo

Figure 13.4: Lift coefficient CL versus M , for airfoil (13.1) at a = 0.1 deg. transition to the asymmetric state 1+2 or 2+1. Therefore, in practice the range of Mach numbers, in which steady symmetric flow over the airfoil cannot exist, appears to depend on the freestream turbulence level. At M , = 0.831 > Ms the plot of CL versus a is similar to that shown in Figure 5c. For Mach numbers 0.818 < M , < Mminand Mmax< M , < 0.836, the hysteresis is observed in two separate intervals of the a-axis (Figure Sa,e,f).

13.5

Summary of the results

The lift coefficient CL as a function of two variables, a and M,, may be illustrated by a surface in the space ( a ,M,, CL). A fragment of the surface residing in the upper half-space CL > 0 is presented in Figure 6 . The surface exhibits a slight seam that goes from point El to Ez on the upper part of the surface, and from point E3 to E4 on the lower right part. The seam corresponds to the flow pattern in which the oblique shock meets the foot of the shock terminating the local supersonic zone. Figure 7 demonstrates both positive and negative parts of the surface CL(CY, M,) in the vicinity of the origin. Figure 8 shows bifurcation curves obtained by prcjecting the edges of the above surface onto the plane ( a ,M , ) .

13.6

Conclusion

For the airfoil (13.1) flattened in the midchord region, the numerical analysis revealed multiple solutions of the Euler equations in certain ranges of the

A. Kuz’min

291

freestream parameters M , and a. The symmetric flow was proved to be unstable with respect to small perturbations of a for freestream Mach numbers which are close to the singular value M,.

This work was supported by the Russian Foundation for Basic Research, under grant 03-01-00799. The author is grateful to B. Mohammadi for providing the NSC2KE solver for transonic flow simulation.

13.7 Bibliography [l] Caughey, D.A., Studies in Unsteady Transonic Flow and Aeroelasticity,

Proc. IUTAM Symp. Transsonicum IV, H. Sobieczky (ed.), Kluwer, 2003, pp. 41-46. [2] Delanae, M., Geuzaine, Ph. & Essers, J. A., Development and Application of Quadratic Reconstruction Schemes for Compressible Flows on Unstructured Adaptive Grids, AIAA Paper 97-2120, 1997, pp. 250-260. [3] Hafez, M. M. & Guo, W. H., Nonuniqueness of Transonic Flows, Acta Mechanica 138, 1999, pp. 177-184.

[4] Hafez, M. M. & Guo, W. H., Some Anomalies of Numerical Simulation of Shock Waves, pt I: Inviscid Flows, Computers and Fluids 28, no. 415, 1999, pp. 701-719. [5] Ivanova, A. V., The Structural Instability of Inviscid Transonic Flow in a Channel, J. Engineering Physics and Thermophysics 76, no. 6 , 2003, pp. 58-60. [6] Ivanova, A. V. & Kuz’min, A. G . , Non-Uniqueness of Transonic Flow over Airfoils, Izvestiya Akademii nauk, Mekhanika zhidkosti i gaza (transl.: Proc. Academy of Sciences of Russia, Mechanics of gases and liquids), no. 4, 2004, pp. 152 - 159.

[7] Jameson, A., Airfoils Admitting Non-Unique Solutions of the Euler Equations, AIAA Paper 91-1625, 1991. [8] Kuz’min, A. G., Interaction of a Shock Wave with the Sonic Line, Proc. IUTAM Symposium Transsonicum IV, H. Sobieczky (ed.), Kluwer, 2003, pp. 13-18.

[9] Kuz’min, A. G., Instability of Transonic Flow over Airfoils at Singular Freestream Mach Numbers, Proc. IV European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004, P. Neittaanmaki, T. Rossi, K. Majava, and 0. Pironneau (eds.), 2004. ~

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[lo] Kuz’min, A.

G. & Ivanova, A. V., The Structural Instability of Inviscid Transonic Flow, Preprint 01-04, Research Inst. for Maths and Mechanics, St. Petersburg State University, 2004.

[ll] Kuz’min, A. G. & Ivanova, A. V., The Structural Instability of Transonic Flow over an Airfoil, J. Engineering Physics and Thermophysics 77, no. 5, 2004, pp. 144 - 148. [12] Kuz’min, A. G. & Ivanova, A. V., The Structural Instability of Transonic Flow Associated with Amalgamation/Splitting of Supersonic Regions, Theoretical and Comput. Fluid Dynamics 17, Springer, 2004 (in press). [13] Mohammadi, B., Fluid Dynamics Computation with NSC2KE: an UserGuide, Release 1.0, INRIA Tech. Report RT-0164, 1994. [14] Semyonov, D. S., Regimes of the Instability of Transonic Flow over an Airfoil, Mathematical Modeling 16, Moscow, 2004 (in press). [15] Steger, J. & Warming, R. F., Flux Vector Splitting for the Inviscid Gas Dynamics Equations with Application to Finite-Difference Methods, J. Comp. Phys. 40, no. 2, 1983, pp. 263-293.

A . Kuz'min

293

a)

0.2 -

4:

0- h

-0.4 -0.2

0

0.2

0.4

a,deg

Figure 13.5: Lift coefficient C, versus the angle of attack a for airfoil (13.1): a) M , = 0.820, b) M , = 0.824, c) M , = 0.8255, d) Mm = 0.8283, e ) M , = 0.834, f ) M , = 0.836.

Bafurcataon of Transonic Flow

2

Figure 13.6: A fragment of the surface C L ( M ~ ,residing ~) in the half-space CL > 0: a), b) - views from two different view angles.

295

A . Kuz’min

Figure 13.7: A fragment of the surface C L ( M ~ a ), in the vicinity of the origin.

0.1

0

-0.1

- 0.2

0.820

0.824 0.828

0.832 0.836

Figure 13.8: Bifurcation curves showing the freestream conditions at which the restructuring of the flow occurs. The dashed segments indicate the range of symmetric flow instability with respect to the perturbation a = f 0 . 0 4 deg .

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Chapter 14

Study of Stability of Vortex Pairs over a Slender Conical Body by Euler Computations Jinsheng Cai', Her-Mann Tsai', Shijun Luo2, and Feng Liu2

14.1

Abstract

The formation and stability of stationary symmetric and asymmetric vortex pairs over a slender conical combination of a circular cone and a flat-plate delta wing in an inviscid incompressible flow at high angles of attack without sideslip are studied by Euler computations. The Euler solver is based on a parallel, multi-block, multigrid, finite-volume method for the three-dimensional, compressible, steady and unsteady Euler equations on overset grids. Stationary vortex configurations are first captured by running the Euler code in its steadystate or time-accurate mode. After a stationary vortex configuration is obtained, a transient asymmetric perturbation consisting of small suction and blowing of short duration on the left- and right-hand sides of the wing is introduced to the flow and the Euler code is then run in the time-accurate mode to determine if the flow will return to its original undisturbed conditions or evolve into a 'Ternasek Laboratories, National University of Singapore, Kent Ridge Crescent, Singapore,

119260. 'Department of Mechanical and Aerospace Engineering, University of California, Irvine,

CA 92697-3975

297

298

Stability of Vortex Pairs

different steady or unsteady solution. Details of the vortex core is examined to assess the usefulness of Euler computations in resolving the vortex structure. The computational results agree well with the stability theory developed by Cai, Liu, and Luo (J. of Fluid Mech., vol. 480, 2003, pp. 65-94) and available experimental data. The agreement corroborates the conclusion that an absolute type of hydrodynamic instability can be a mechanism for the vortex instability, and demonstrates the usefulness of the Euler method for the stability study of the vortex-dominated high-angle-of-attack flows over sharp-edged bodies and for the simulation of the primary vortex cores.

14.2 Introduction An initially symmetric vortex pair over a slender pointed body becomes asymmetric as the angle of attack is increased beyond a certain value, causing large asymmetric side forces and moments even at zero sideslip. The mechanism of the breaking of symmetry of such vortex flows is yet not clear. A great deal of disagreement exists regarding the understanding, prediction, and control of the vortex asymmetry despite much experimental, theoretical, and computational effort spent on the topic. The subject has been reviewed by Hunt [22], Ericsson and Reding [18],and Champigny [lo]. It is found by numerous experimental observations [15, 23,471 and numerical studies [12, 13, 26, 201 that a micro-asymmetric perturbation close to the nose tip produces a strong flow asymmetry at high angles of attack. There seems little doubt that the vortex asymmetry is triggered, formed, and developed in the apex region, and the after portion of forebody and the after cylindrical body (if any) have little effect on the asymmetry over the apex region. The evolution of perturbations at the apex plays an important role in determining the flow pattern over the entire body. Since the apex portion of any slender pointed body is nearly a conical body, high angle-of-attack flow about conical bodies has been studied analytically. Using a separation vortex flow model of Bryson[2], Dyer, Fiddes and Smith [16] found that in addition to stationary symmetric vortex flow solutions there exist stationary asymmetric vortex flow solutions over circular cones when the angle of attack is larger than about twice of the semi-vertex angle even though the separation lines are postulated at symmetric positions. The stability of these stationary vortices were later investigated analytically by Pidd and Smith.[35] The disturbances which they treated in their stability analysis were spatial rather than temporal. A small change in the positions of the originally stationary symmetric/asymmetric conical vortices is introduced into the flow near the body tip. Under the assumptions of the slender-body theory, the initial rate of change of this disturbance in the downstream direction is calculated. If all disturbances decay, the solution is stable, while if any disturbance grows, the solution is unstable. This kind of instability mechanism is commonly called the

J. Cai, H-M. Tsai, S. Luo tY F. Liu

299

convective instability. Dagani[l2, 131 and Levy, Hesselink, and Degani[26] studied the separation vortices over a 3.5 caliber tangent ogive-cylinder body of revolution at low speeds by numerical computations using a time-accurate Navier-Stokes method. They found that it is necessary to maintain a fixed small geometric disturbance near the body apex in order to obtain and keep an asymmetric vortex pattern in their numerical computations. Once the small artificially introduced "imperfection" at the nose-tip was removed, the vortices would return to symmetry. This finding in numerical computation coupled with an experimental observation of Degani and Tobak [14] led them to believe that a convective instability mechanism similar in concept to that studied by Pidd et al.[35] was responsible for the onset of asymmetry of the otherwise would-be symmetric vortices over a slender body of revolution with a pointed nose. The experiments of Degani and Tobak showed that the vortex pattern over an ogive-cylinder body depends continuously and reversibly on a controlled tip disturbance at all angles of attack 30 - 60". In a separate numerical study, Hartwich, Hall, and Hemsch [20] reported an asymmetric vortex flowfield solution of the incompressible three-dimensional turbulent Navier-Stokes equations for a 3.5-caliber tangent-ogive cylinder at an angle of attack of 40" without the imposition of a fixed geometric asymmetry in the computations. It was claimed that the asymmetric solution is triggered by the computer round-off error in the computations. Thus asymmetries can be induced by a transient disturbance. This route to asymmetry is referred to as an absolute instability. There are experimental observations supporting the notion of absolute instability, for example, the existence of bi-stable configurations when the roll angle is varied for ogive cylinders at incidence angles in the range of 50 - 60" [15, 471 and hysteresis effects[34, 11. The absolute instability mechanism is also studied theoretically. Using the simplified separation-vortex flow model of Legendre[24], Huang and Chow [21] succeeded in showing analytically that the vortex pair over a slender flat-plate delta wing at zero sideslip can be stationary and is stable under small conical perturbations. Using the same flow model, Cai, Liu and Luo [5] developed a stability theory for stationary conical vortex pairs over general slender conical bodies under the assumption of conical flow and classical slender-body theory. The disturbances which they treated in the stability analysis were temporal or transient rather than spatial. Small displacements are introduced to the stationary vortex positions and then removed. The displaced vortices are assumed to remain conical. The disturbances are of a global nature rather than a localized nature. Cai, Luo, and Liu[6, 8, 71 extended the method described in Ref. [5] to study the stability of stationary asymmetric vortex pairs over slender conical bodies and wing-body combinations with and without sideslip. In their studies, perturbations are introduced at an initial time. If the two vortices of the vortex pair return to their original stationary positions after the initial action of the

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Stability of Vortex Pairs

perturbations, the vortex pair is stable; if any one vortex of the pair runs away from its original position, the vortex pair is unstable; and if the vortices move periodically around the stationary point or stay at the disturbed position, the vortex pair is neutrally stable. This pertains to the absolute type of instability discussed in the above. A different view was offered by Ericsson[l7], who claims that there is no direct evidence for a hydrodynamic instability. Because experiments are always dominated by their geometric micro-asymmetries, a perfectly symmetric flow state is never achievable, which would be a prerequisite for showing the effects of perturbations. He argues that the mechanisms for flow asymmetry are asymmetric flow separation and/or asymmetric flow reattachment. However, there are some observations that are difficult to reconcile with this view. For example, symmetric vortex flows exist when the angle of attack is low. Flow asymmetry appears only when the angle of attack is increased over a certain value. Moreover, the asymmetric vortex flow over a delta wing with sharp leading-edges observed by Shanks[40], which Ericsson suggested to be induced by asymmetric reattachment, was more likely caused by the hydrodynamic instability in the presence of a short vertical fin mounted in the lee-side incidence plane of the model as shown by Cai et a1.[5]. When viewed from a static (fixed) angle of attack perspective, the convective instability concept and the absolute instability concept are quite different. They may lead to different stability conclusions for the same vortex flow field. For the case of circular cone which Pidd et a1.[35] studied, they showed that the stationary symmetric conical vortex pairs are convectively stable in a narrow band of the incidence parameter, while the stationary asymmetric conical vortex pairs are convectively stable with insignificant exceptions. However, Cai et al. [7, 81 proved that both symmetric and asymmetric stationary conical vortex pairs are absolutely unstable regardless if symmetric or asymmetric separation lines are postulated. Thus, stable conical symmetric and asymmetric vortex pairs may exist over circular cones under certain conditions in terms of convective stability, while stable vortex pairs over circular cones are either nonexistent or must be of non-conical configurations in terms of absolute stability. Satisfaction of both the convective and absolute type of stability conditions is logically necessary for any configuration of a conical symmetric or asymmetric vortex pattern to persist in a flow. Recently, Cummings, Forsythe, Morton, and Squires[ll] gave an alternative explanation for high angle of attack asymmetry. The convective instability hypothesis states that any level of asymmetry including the symmetric case is possible as the angle of attack is increased into a high range. The absolute instability hypothesis states that as the angle of attack is increased to a certain level, a bifurcation will take place which will produce one of two mirror-image asymmetric solutions, and neither intermediate asymmetric solutions nor symmetry are possible. jFrom the perspective of the behavior of the dynamic system

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as the angle of attack is increased, the convective instability could be called as an unstable bifurcation, and the absolute instability is a bifurcation. The interaction between the body motion and the forebody flow field is not fully understood at this moment. The fundamental mechanism responsible for the vortex asymmetry is still a subject of debate. It is noted that to predict the behavior of the separation vortex flow is also extremely difficult with computations since computations are not perfect simulations either. While some researchers are content with allowing their numerical algorithms to supply the perturbations physically needed to trigger vortex-flow instabilityl20, 30, 411, it is much more desirable to use an algorithm that does not add any unknown level of perturbation. It would be superior to have the perturbation added explicitly as a geometric or flow field disturbance. Levy, Hesselink and Degani[25] showed that inherent biases contained in a certain algorithm generate anomalous asymmetries in the flow. To perform numerical studies of vortex-flow stability, the numerical algorithm is required to be symmetric in order to obtain the basic stationary symmetric or asymmetric vortexflow solutions when appropriate initial and boundary conditions are assigned. Flows at high angles of attack are especially sensitive to artificial viscosity or numerical dissipation. Hartwich[lS] demonstrated that excessive numerical dissipation due to a first-order accurate difference scheme in the exit boundary condition suppresses symmetry breaking. In his Navier-Stokes computations of a supersonic viscous flow over a 5" half-angle cone at an angle of attack of 20°, Thomas[43] found that an inadequate grid resolution near the body tip gives rise to a spurious asymmetry. As high gradients exist in the regions of shear layer separation and the vortex cores in addition to areas in the vicinity of the body surface, appropriate grid topology with sufficient grid density in these regions are vital to the computational studies. For sharp-edged geometry, the separation point is fixed at the leading edge, independent of the Reynolds number. The dissipation introduced by numerical methods of the Euler equations should mimic the physical viscosity and cause separation. Just as the separation point is insensitive to the real viscosity, it should also be insensitive to the artificial viscosity. Once the separation and the generated vorticity are established, the dynamics of the vortex motion, i.e. its interaction with neighboring body surfaces, are essentially inviscid and thus adequately described by the Euler equations. Although secondary vortices brought about by viscous effects on the leeside of the wing are not modeled in the Euler computations, their effects on the primary vortices are small. Therefore, use of Euler codes for the stability study of vortex flow over slender sharp-edged bodies is an attractive alternative to using the Navier-Stokes codes that require greater computational resources and suffer from the empiricism of turbulence modeling. The computation model considered in this article is conical. Neither trailing edge effects nor vortex breakdown will be studied. In the following sections, the numerical method and the flow model are

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described. The low-speed conical-flow assumption is validated and based on the validation a conical grid is implemented for computing the stationary symmetric and asymmetric vortex flow and their stability. The computed primary vortex core as an important player in the flow is studied. The computational results are compared with known theoretical analysis and available experimental data. Lastly conclusions are drawn.

14.3

The Euler Solver and the Flow Model

It is known that the Euler solver can capture automatically the shear layer separated from the swept sharp leading edge and its spiral roll-up into a vortex core in the flow field. Although secondary features of the vortex are absent in the Euler solutions, the gross dominant characteristics of the flow field, i.e. the primary vortex configurations and their interaction with the body surface are reproduced. The present Euler solver is based on a parallel, multi-block, multigrid, finitevolume code for the three-dimensional, compressible, Euler/Navier-Stokes equations. Only the Euler option of the code is used. The basic numerical method uses a central difference scheme with a blend of second- and fourth-order artificial dissipation and explicit Runge-Kutta-type time marching. The resulting code preserves symmetry. Unsteady time-accurate computations are achieved by using a second-order time-accurate implicit scheme with dual-time stepping. The solver has been validated for a number of steady and unsteady cases [27, 28, 29, 391. A newly developed overset-grid techniques[3] is implemented to facilitate the grid refinement in the domain of high vorticity. Figure 14.1 illustrates the slender conical wing-body model consisting of a circular cone body and a flat-plate delta wing. Two separation vortices, the body rectilinear coordinate system (2,y, z ) and the vortex cylindrical coordinate system (a,r,O) are also shown. The shear layer connecting the vortex and the leading edge of the wing (not shown in the Figure) was neglected in the theory of Ref. [5] but is considered in the computations in this paper. The free stream Mach number M , in the computations is set at 0.1 to approximate an incompressible flow. Reference [4] reported a number of computations for both flat-plate delta wings and also wing-body combinations. We will restrict our discussion in this paper to a configuration that has a body-radius to wingsemi-span ratio y = b/s = 0.7 and a semi-apex angle 6 = 8" for the delta-wing under flow conditions of zero side-slip ( p = 0) and an angle of attack a! = 28". High angle-of-attack flows over slender bodies are characterized by the Sychev similarity parameter[42] K = t a n a / t a n E rather than by a or E individually. For the present model K = 3.7833.

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Figure 14.1: Slender conical wing-body combination and separation vortices.

14.4

Computational Grid and Boundary Conditions

It is known that a subsonic flow over a conical body cannot be strictly conical. However, if the conical body is slender, the flow is nearly conical. This was observed in water tunnel for a triangular thin wing of E = 15" at a = 20" by Wed6 in 1961 as shown in Reference [45]. It is also proved by Navier-Stokes computations of, e.g. Thomas, Kirst and Anderson [44] and our previous studies in Ref. [4]. In principle, a conical flow can be solved in a two-dimensional plane with the appropriate modified equations. However, the present studies maintain the use of a three-dimensional code on a three-dimensional grid to allow calculation of not perfectly conical flows. For the nearly conical flows studied in the present paper, however, a conical grid may be used. An overset conical grid is designed to match the local flow gradients and to facilitate the parallel processing of the computations. In Fig. 14.2, part (a) gives the grid on the incidence plane, and part ( b ) shows the grid on the right-hand half exit plane although a full crossplane grid is used in the following computations. The far-field lateral boundary is a conical surface which shares the same apex with the conical body and is at 25s distance away from the body axis where s is the local wing semi-span. Grids are bunched into one point at the body apex. No numerical difficulties are encountered at the vertex point since a cell-centered finite-volume method is used. Only a few grid lines are needed in the longitudinal direction for conical flow calculations. However, very fine grids in the radial and circumferential

Stability of Vortex Pairs

Figure 14.2: Conical grid for a wing-body combination of a flat-plate delta wing and a circular-cone body, E = 8", y = 0.7, ( a ) on the incidence plane, ( b ) on the right-half exit plane. Only every 4th line is shown in the radial and circumferential directions for clarity.

directions in the cross planes must to be used to resolve the vortical flowfield for the purpose of stability studies. A close-up view of the full grid on the exit plane is shown in Fig. 14.3. The grid consists of three layers: the iqner layer has 5 x 177 x 581 grid points in 8 blocks; the intermediate layer has 5 x 49 x 385 grid points in 2 blocks; and the outer layer has 5 x 49 x 257 grid points also in 2 blocks. The inner layer has two sub-layers, each of which has 4 blocks. The upper half of the inner sub-layer has 5 x 81 x 387 grid points, and the upper half of the outer sub-layer has 5 x 97 x 387 grid points. The total number of the grid points is 671,475. The computing time for one iteration in double (64 bit) precision is about one second on an 8-processor parallel cluster computer consisting of AMD Athlon XP1600+ CPUs. Zero normal velocity boundary condition is applied on the wing and body surface. Kutta condition at the sharp leading edges of the wing is satisfied automatically in an Euler code. Characteristics-based conditions are used on the upstream boundary of the grid. On the downstream boundary, all flow variables are extrapolated. Computations are performed starting from a uniform freestream flow until the maximum residual of the continuity equation is reduced by more than 11 orders of magnitude. Such a stringent convergence criterion is needed especially for stability studies of high angle-of-attack flows as is pointed out by Siclari and Marconi[41].

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Figure 14.3: Close-up view of the conical grid for a wing-body combination of a flat-plate delta wing and a circular-cone body in the exit plane, only every 4th line is plotted in the figure for clarity, E = 8", y = 0.7. Table 14.1: Three grids of different densities.

Grid Fine Baseline Coarse

Inner layer 5 x 209 x 641 5 x 177 x 581 5 x 89 x 387

Intermediate layer 5 x 65 x 385 5 x 49 x 385 5 x 49 x 321

Outer layer

5 x 49 x 257 5 x 49 x 257 5 x 49 x 225

The grid densities used for the calculations represent those determined based on a grid refinement studies and a balance of available computing resources. The grid given above is considered as a baseline grid. Computations are also performed on a fine grid within the limit of of our available computing resources and also a coarse grid as listed in Table 14.1. Stationary symmetric solutions for the flow model are obtained by the Euler solver in the steady-state mode with the free-stream flow as initial solution using the three different grids. The computed position of the vortex-core center, x / s and y f s, and the computed total velocity U/Um and pressure coefficient C, at the vortex-core center are compared in Table 14.2. The solution on the coarse grid yields significant position shifts of the stationary vortex center and rather large changes of the flow parameters, while the solution on the fine grid gives negligible position shifts and insignificant flow parameter changes compared

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x/s 0.5734 0.5741 0.6066

y/s 0.9178 0.9174 0.9156

U/Um 2.7742 2.7288 2.4357

C, -9.4063 -9.0903 -7.1355

to the solutions on the baseline grid. Hence, the baseline grid is used in the following calculations.

14.5

Stationary Symmetric and Asymmetric Solutions and Their Stability

Stationary vortex configurations are first captured by running the Euler code in its steady-state mode. After a stationary vortex configuration is obtained, a small transient asymmetric perturbation consisting of suction and blowing of short duration on the left- and right-hand side of the wing is introduced to the flow and the Euler code is then run in the time-accurate mode to determine if the flow will return to its original undisturbed conditions or evolve into a different steady or unsteady solution. The former case indicates that the original stationary vortex configuration is stable while the latter case proves it unstable or neutrally stable. The stationary symmetric and asymmetric vortex configurations and their stability for this flow model have been analyzed analytically in Ref. [7] by the conical slender body theory presented in Ref. [5].

14.5.1

Temporal Asymmetric Perturbations

Asymmetric perturbations consisting of suction and blowing through two narrow conical slots on the upper surface of the wing are applied because they are found to be the most unstable modes of motion in the theory [5]. The suction and blowing slots are symmetrically located approximately beneath the vortex cores. The perturbations are activated in the initial time period 0 < t < 1 of the time-accurate Euler computation, where t is a non-dimensional time. On the right-hand side of the wing, looking toward the downstream direction, the air-blowing velocity Vj is defined as follows.

The blowing velocity on the left-hand-side slot is defined in the same way except with the direction reversed. Two configurations of the same type of asymmetric

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Figure 14.4: Pressure contours on a cross flow plane of a symmetric solution over a wing-body combination of a flat-plate delta wing and a circular-cone body, E = 8", y = 0.7, (Y = 28", /3 = 0". perturbations are used. Perturbation ( A ) has y1 = 0.73s, yz = 0.78s, and Vo = 2.0Um, and Perturbation ( B ) has y1 = 0.90s, y2 = 0.95s, and Vo = -1.33Um. The instantaneous maximum blowing momentum flux occurs at t = 112. The instantaneous maximum blowing momentum flux coefficient based on the wing area is c/, = 0.1, and c/, = 0.04, respectively, for the two configurations. The corresponding air-blowing force is about one order of magnitude less than the normal force acting on the body at high angles of attack. The suction and blowing slots of Perturbation ( A ) are located closer to the wing root compared to those for Perturbation ( B ) . In addition, the blowing velocities have opposite directions, i.e., there is an exchange between suction and blowing. The locations and directions of the suction and blowing velocities of Perturbation ( A ) are marked by the arrows in Fig. 14.6.

14.5.2

Stationary Symmetric Vortex Flow

In order to investigate stationary positions of symmetric and asymmetric vortex pairs and their stability, the full flow-space including both sides of the incidence plane has to be considered. A stationary vortex flow is first searched for by running the Euler solver in its steady mode over the full space of the flow. With a uniform free-stream flow as the initial solution and on the three-layer baseline grid, the computations are run in double precision until the maximum residual is reduced by 11 orders of magnitude. A steady-state solution is found to be indeed symmetric with respect to the incidence plane. Figure 14.4 gives

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Figure 14.5: Contours of the longitudinal velocity component in four cross-flow planes of a symmetric solution over a wing-body combination of a flat-plat delta wing and a circular-cone body, c = 8", y = 0.7, a = 2 8 O , p = 0. the computed pressure contours on a cross-flow plane. The centers of the two vortex cores are clearly seen to be symmetrically located at z / s = 0.5741, y l s = +0.9174. Figure 14.5 shows the computed contours of the longitudinal velocity component in four cross-flow planes along the wing-body combination. In this figure the shear layer connecting the the wing leading edge and the vortex core emerges. Clearly, this solution represents a stationary symmetric vortex flow, which is subjected to a stability examination in the following discussions.

14.5.3

Stability of the Stationary Symmetric Vortex Flow

To investigate the stability of the stationary vortex flow, the symmetric solution obtained above is then used as a new initial condition at t = 0, Perturbation ( A ) is activated at 0 < t < 1 as shown by the two arrows in Fig. 14.6. The time-accurate Euler code is used to simulate the evolution of the disturbed flow. In the time-accurate Euler computation, 50 real-time steps are taken in every unit increment of t. For each real-time step the pseudo-steady-flow computation is performed until a reduction of four or higher orders of magnitude in the maximum residual is reached. Figure 14.6 gives the pressure contours in a cross-flow plane of the computed asymmetric solution. It is seen that the disturbed flow does not return to its starting stationary configuration even after the initial disturbance has long disappeared. Instead, it wanders farther and farther away until it reaches a new steady-state solution, where the two lines with circles mark the trajectories of the two vortex centers and the contours are

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Figure 14.6: Pressure contours in a cross-flow plane of the asymmetric solution after Perturbation ( A ) is applied to the symmetric stationary vortex solution over a wing-body combination of a flat-plate delta wing and a circular-cone body, E = 8", y = 0.7, a = 28", p = 0; and trajectories of the perturbed vortex core.

' 0

8

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24

32

40

48

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Time(t)

Figure 14.7: Vortex core position vs. time after Perturbation ( A ) is applied to the symmetric stationary vortex solution over a wing-body combination of a flat-plate delta wing and a circular-cone body, E = 8", y = 0.7, a = 28", /3 = 0.

31 0

Stability of Vortex Pairs

Figure 14.8: Contours of the longitudinal velocity component in four cross-flow planes of the asymmetric solution over a wing-body combination of a flat-plate delta wing and a circular-cone body, E = 8", y = 0.7, a = 28", ,B = 0. constant pressure lines of the newly obtained steady-state solution at the end of the time-accurate computation. Figure 14.7 shows the vortex core position X / S and y / s vs. the dimensionless time t. The new steady-state solution is highly asymmetric. The left vortex moved a small distance downward toward the wing surface while the right vortex wandered significantly upward away from the wing surface and inboard compared to the original symmetric solution. The left, lower vortex core is at x / s = 0.3448 and y / s = -0.9127. The right, upper vortex core is at x / s = 1.6320 and y / s = 0.5992. Figure 14.8 shows the contours of the longitudinal velocity component in four cross-flow planes along the wingbody combination. Notice that the disturbances are only imposed for a short duration 0 < t < 1 while the computation is continued without any externally imposed disturbance or asymmetry from t = 1 until t = 64 (see the abscissa of Fig. 14.7.) This can only be explained by the fact that the initial symmetric vortex solution (obtained under a very stringent convergence criterion) is not a stable configuration and in addition the new asymmetric solution is another stationary vortex configuration.

14.5.4 Stability of the Stationary Asymmetric Vortex Flow The above computations demonstrate the existence of a stationary asymmetric vortex configuration in addition to the symmetric one for this high angle of attack condition. It remains to be seen whether this asymmetric vortex pair is stable under small perturbations.

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'...

Figure 14.9: Pressure contours in a cross-flow plane of the asymmetric solution after application of Perturbation ( A ) with reverse suction/blowing to the asymmetric solution over a wing-body combination of a flat-plate delta wing and a circular-cone body, E = 8", y = 0.7, a = 28", p = 0; and trajectories of the perturbed vortex core.

To do this, time is reset to zero for the time-accurate Euler computations to continue with the new stationary asymmetric solution as the initial condition. Perturbation ( A ) is again imposed for the initial unit time period (0 < t < 1). Just for variety, the suction and blowing directions are exchanged this time. The new directions are shown by the arrows in Fig. 14.9. The computed vortex core positions vs. time t are shown in Fig. 14.10. It is seen that the solution goes back to the initial asymmetric solution. The trajectories of the vortices during the flow evolution are shown by the solid lines around the vortex centers in Fig. 14.9. Although the excursions of the vortices last a rather long time period, especially for the upper vortex, both vortices return to their original locations, and thus the stationary asymmetric vortex pair is stable under small perturbations.

14.5.5

A Mirror-Image of the Asymmetric Vortex Flow

Another question is whether the stationary asymmetric solution obtained above depends on the specific initial perturbations. To answer this question, a different disturbance, Perturbation ( B ) ,is applied to the converged symmetric solution of Fig. 14.5. Perturbation ( B ) has the same functional form as Perturbation ( A ) except it has the suction and blowing locations interchanged and moved closer to the wing tips and its maximum velocity V , is smaller. The

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312

!

!

!

!

!

I

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I

25

30

35

40

45

Y

g > 0 '

5

10

15

20

50

Time ( t )

Figure 14.10: Vortex core position vs. time after application of Perturbation ( A ) with reverse suction/blowing to the asymmetric solution over a wing-body combination of a flat-plate delta wing and a circular-cone body, E = 8", y = 0.7, a = 2 8 O , ,8 = 0.

.-

.

,. i., \.

'i. /

Figure 14.11: Pressure contours in a cross-flow plane of the asymmetric solution after application of Perturbation ( B )to the symmetric solution over a wing-body combination of a flat-plate delta wing and a circular-cone body, E = 8", y = 0.7, Q = 28", ,8 = 0; and trajectories of the perturbed vortex core.

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-.

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16

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64

UI

2,

f c H

8

n

s

0.8

0 X

0.4

> 0 '

24

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Time(t)

Figure 14.12: Vortex core position vs. time after application of Perturbation ( B )to the symmetric solution over a wing-body combination of a flat-plate delta wing and a circular-cone body, E = 8", y = 0.7, Q = 28", p = 0. same time-accurate computations are performed as before. Again, the initial converged solution breaks away from symmetry under the initial (short time) asymmetric disturbance and finally converges to an asymmetric vortex configuration as shown by the pressure contours in Fig. 14.11. The trajectories of the vortex cores from the initial symmetric positions to the stationary asymmetric positions are marked by the circled lines in Fig. 14.11. The time history of the dimensionless coordinates of the two vortices are shown in Fig. 14.12. This time, because of the reversed directions of the suction and blowing, the left vortex moves up and to the right while the right vortex moves down, eventually forming a perfect mirror image of the solution shown in Figs. 14.6 and 14.7 that are obtained by using perturbation ( A ) . Since perturbation ( B ) is different from perturbation ( A ) not only in the directions of suction and blowing but also in their strength and locations, the computation demonstrates that the flow asymmetry is independent of the initial disturbance except for the possibility of an exact mirror image. Evidently, the mirror-imaged stationary asymmetric vortex pair must also be stable. Thus, the flow is bi-stable.

14.5.6

Symmetry Nature of the Present Euler Solver

The above computations also demonstrate the symmetry nature of the algorithm and the computer code of the three-dimensional time-accurate Euler solver used for the present studies. Such a symmetry nature is highly desirable for the numerical study of flow instability. Otherwise, the stationary symmetric vortex flow obtained in Fig. 14.5 would be elusive because asymmetry in the numerical

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Stability of Voytex Pairs

computations could supply the needed perturbations to trigger the physical instability and thus drive the flow toward the asymmetric solution.

14.5.7 Comparison with Theoretical Predictions on Stability A number of wing-body combinations were studied in Ref. [7] by the analytic method of Cai et a1.[5]. For the case of the computation model, i.e. the wingbody combination of a flat-plate delta wing and a circular-cone body, € = 8", y = 0.7, Q = 28", ,B = 0, and K = 3.7833, the analytical results are summarized here. 1. Both symmetric and asymmetric stationary vortex pairs exist. Relative to the symmetric vortices, the upper vortex of the asymmetric pair locates a little outboard and the lower one locates a little inboard. 2. The stationary symmetric vortex pair is stable under symmetric perturbations, while unstable under anti-symmetric perturbations. 3. The stationary asymmetric vortex pair is stable under both symmetric and anti-symmetric perturbations. And the upper vortex is less stable than the lower one under the anti-symmetric perturbations.

The present Euler computations of the vortical flows over the wing-body combination agrees completely with the analytical predictions on the formation of stationary symmetric and asymmetric vortex pairs and their stability under small perturbations, even including a longer time and a greater region for the upper vortex to travel in coming back to its undisturbed position than the lower vortex as shown in Fig. 14.10, which indicates that the upper vortex is less stable than the lower as predicted by the theory.

14.5.8 Comparison with Experimental Data on Stability The present computational result is compared with the low-speed wind-tunnel experimental data of a forebody strake/fuselage configuration made by Murri and Rao[32]. The tested forebody is a conical combination of a circular-cone and a flat-plate delta wing of y = 0.77, and E = 12'. The afterbody is a circular cylinder. The Reynolds number based on the fuselage diameter is Re = 1.9 x 10'. The tests showed that at K = 3.7833 or Q = 38.54", there exists steady nonzero yawing moment. It indicates that a stable stationary asymmetric vortex pair prevails over the flowfield. It is noted that the y value of the test model is 10% greater than that of the computation model. However, it is close enough to be used to confirm the present computational results of the existence of the stationary stable asymmetric vortex pair.

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The two mirror-imaged bi-stable asymmetric vortex configurations over a sharp-nosed slender body of revolution at high angles of attack, like those over the present computation model shown in Figs. 14.9 and 14.11 were observed in a number of wind tunnel experiments. For example, Zilliac et al. [47] measured the overall side force of a 3.5 calibre ogive-cylinder body of revolution using a six-component strain gauge balance at a Reynolds number based on the cylinder diameter, Re = 3 x lo4 in a low-speed and low-turbulence-levelwind tunnel. The semi-apex angle of the ogive nose is 16.26". The overall side force coefficient against roll angles is a square-wave curve for 50" < a < 60" or equivalently 4 < K < 6. In this case the asymmetry has only two stable states, or bi-stable states. The side force coefficient switches abruptly from the constant positive value to the constant negative value of the same magnitude or vice versa over the whole range of roll angle, and no intermediate side force coefficient is found. It was established that microvariations in the tip geometry of the test model have large influence on the downstream development of the flowfield, and the existence of a bi-stable lee flowfield is a result of instability. Ng and Malcolm [34] conducted an experiment in a flow visualization water tunnel. The test model is a 6%-scale F/A-18 forebody. The model nose is nearly axisymmetric and has a semi-apex angle of about 30". Above an angle of attack of about 60" or K = 3, flow visualization revealed that the flow becomes bistable. The yawing moment can be switched between two essentially steady values by a transient and small-amount mass-flow jet blowing from nozzles located on the leeward side and near the nose tip. Keeping the blowing on after switching only increases the yawing moment by a relatively small amount. Similar results are observed at a = 65" and 67.5" or K 4. It is noted that the present computational model is a slender wing-body combination with y = 0.7, while the above two test models are slender body alone, i.e. y = 1. However, the observations of bi-stable flowfield for the test models corroborate qualitatively the computational and theoretical results. N

14.6

Structure of the Vortex Core

The above computed flow fields show closed and nearly concentric circular contours over the leeward side of the body in the cross-flow planes. These represent the cores of the primary vortices formed from the spirally rolling-up of the shear layer separated from the leading edge of the wing. The structure of the computed vortex cores are examined in this section and compared with experimental observations.

14.6.1

Computational Result

The right vortex of the stationary symmetric solution presented in the previous section is chosen as a typical example. The center of the core is located at

31 6

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07

06

2 05

0.4

08

09

1

11

Figure 14.13: Local and detail view of the conical grid in the vortex-core region on a cross-flow plane for a wing-body combination of a flat-plate delta wing and a circular-cone body, E = 8", y = 0.7.

x / s = 0.5741,y / s = 0.9174. The local but detailed grid configuration used to resolve the vortex core region is shown in Figure 14.13. The contours of the the longitudinal vorticity component longitudinal velocity component u,/U,, wzs/U, and the total pressure loss coefficient C,t are given in Figures 14.14, pt is the 14.15, and 14.16, respectively, where CPt = (pt - p t , ) / ( p , U & / 2 ) , local total pressure and p t , is the free-stream total pressure. All contours of the three flow parameters are nearly concentric circles about the vortex center. The values of all of the three flow parameters increase toward the center. To further examine the structure of the vortex core, two orthogonal grid lines are taken across the vortex core. The intersection of the two grid lines is the grid point closest to the vortex center. The distributions of various flow parameters along the radial and circular grid lines are plotted versus the distance, T , from the intersection point of the two grid lines. Figure 14.17 gives the distributions of uz/Uw and w,s/U, versus r / s along the radial (from the center body) and the circular (around the body) grid lines. Figure 14.18 shows the distributions of Cpt and the static pressure coefficient C, versus r / s along the two grid lines. Figure 14.19 gives the distributions of the velocity components ua/U, and ue/U, versus r / s along the two grid lines. Here the velocity is decomposed into components along the directions of the cylindrical coordinates ( a ,T , O ) , where the axis a coincides with the vortex-core axis. It is seen that the distribution for each flow parameter along the two perpendicular grid lines nearly coincide, and they are almost symmetric with respect

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Figure 14.14: Contours of the longitudinal velocity component in the vortexcore region on a cross-flow plane of a symmetric solution over a wing-body combination of a flat-plate delta wing and a circular-cone body, E = 8", y = 0.7, = 28", ,B = 0, A(uZ/Um) = 0.2.

07

-

06

-

/

s : 05

-

Figure 14.15: Contours of the longitudinal vorticity component in the vortexcore region on a cross-flow plane of a symmetric solution over a wing-body combination of a flat-plate delta wing and a circular-cone body, E = 8", y = 0.7, a = 28", ,I3 = 0, A ( w Z s / U m )= 5.

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Figure 14.16: Contours of the total pressure loss coefficient in the vortex-core region on a cross-flow plane of a symmetric solution over a wing-body combination of a flat-plate delta wing and a circular-cone body, E = a", y = 0.7, QI = 28", ,8 = 0 , AC,, = 0.2.

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I

4

Figure 14.17: Distributions of the longitudinal component of velocity and vorticity versus distance from the vortex-core center along the radial and circular grid lines passing through the vortex-core center on a cross-flow plane of a symmetric solution over a wing-body combination .of a flat-plate delta wing and a circular-cone body, E = a", y = 0.7, a = 28", ,8 = 0.

J. Cai,H-M. Tsai, S. Luo tY F. Liu

319

-

-'0'00.4

-0.3

-0.2

-0.1

0

0.1

r- -

0.2

-I - - -

0.3

0.4

Figure 14.18: Distributions of the total pressure loss coefficient and the static pressure coefficient versus distance from the vortex-core center along the radial and circular grid lines passing through the vortex-core center on a cross-flow plane of a symmetric solution over a wing-body combination of a flat-plate delta wing and a circular-cone body, E = 8", y = 0.7, CY = 28', p = 0.

to the origin r / s = 0, indicating that the flow in the vortex core is nearly axisymmetric and conical. The vorticity increases sharply toward the core center and reaches a maximum value at the center. Fig. 14.17 shows that the edge of the highly rotational region is located at r, = 0.2s, which is the radius of the rotational or vortex core. Inside the rotational region viscous diffusion has smoothed out completely the gradients of the velocity distribution, and a shear layer can no longer be detected. Inside the vortex or rotational core, the static and total pressure decrease toward the vortex center and reach minimum values at the vortex center. The axial velocity component, ua increases toward the vortex axis, and reaches a maximum value at the vortex axis, while the circumferential velocity component, UQ, first increases toward the vortex axis, and after reaching a maximum value near the vortex axis, decreases sharply to zero at the vortex axis. The location of the maximum U Q defines the edge of a subcore, and inside this subcore large gradients of velocity and pressure prevail and numerical viscous forces dominate. This subcore is a viscous subcore in which an artificial total pressure loss results in. From Figures 14.19 and 14.18 the radius of the subcore, r,, = 0.08s. The computed radial velocity component is one order of magnitude less than the other two velocity components, and thus is not shown. In the vortex core, the radial velocity component is pointed to the core axis. It first increases

320

Stability of Vortex Pairs

-2.0

-0.4

-

I - - -1- -I - - - -

- -

L-

-

-1-

I - - -1- ~--

- -

I , , , , l , , , , l , , , , l , , , , I , , , I I I ,I , I , , , ,

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Figure 14.19: Distributions of the axial and circumferential components of velocity versus distance from the vortex-core center along the radial and circular grid lines passing through the vortex-core center on a cross-flow plane of a symmetric solution over a wing-body combination of a flat-plate delta wing and a circular-cone body, 6 = 8", y = 0.7, Q = 28", p = 0. toward the vortex axis, and then decreases sharply to zero at the core axis.

14.6.2

Comparison with Experimental Data

It is conceivable that the vortex core plays an important role in the flow-body interactions. A comparison of the computed vortex core with known experimental data is carried out in this subsection. So far there is no experimental data for wing-body combination known to the present authors. A pair of available experimental data on sharp-leading-edge delta wing alone is used. The two test models had practically the same geometry and were set at about the same angle of attack, but at significantly different Reynolds numbers. Test 1: A vortex core over a sharp leading-edge, flat-plate delta wing was measured by Carcaillet, Manie, Pagan, and Solignac[S]using a three-dimensional laser velocimeter and a five-hole pressure probe in an 1 m research low-speed wind tunnel. The tested wing has E = 15" at a: = 20", and the Reynolds number based on the root chord, CO, is 0.7 x lo6. The data of the measurement cross-flow ~ used here. The measured distributions of the static pressure, plane at 6 0 % is total pressure loss, streamwise velocity component, and circumferential velocity component along a traverse passing through the primary vortex core center were given in Figure 11 of Ref. [9]. They are similar to the corresponding computed distributions reported in Figures 14.18 and 14.19.

321

J. Cai, H-M. Tsai,S. Luo & F. Lau

Table 14.3: Comparison of computed vortex-core parameters with test data. Flow Variables ( u z )max/Um

/ urn slum

(ug)maz (Wz)max

(Cp)min (Cpt)min

rc/s rscls

Computed 2.7 1.3 53 -9.2 -2.9 0.2

Test 1 3.0 1.4 152 -11 -2.8 0.2

0.08

0.08

Test 2 3.0 1.3 224 -13 -4.8 0.2 0.03

Test 2: Verhaagen and Kruisbrink [46] measured the flow properties of the conical part of a leading-edge vortex using a five-hole pressure probe in a lowspeed and turbulence level of about 0.05% wind tunnel. The model is a sharp leading-edge flat-plate delta wing of E = 14' at a = 20.4'. The Reynolds number is 3.8 x lo6, based on the model root chord length, CO. The measurement cross. measured distributions of the the axial vorticity, flow plane was at 5 0 % ~The axial velocity component, circumferential velocity component, static pressure, and total pressure loss along traverses passing through the primary vortex core center were shown in Figures 1, 10 and 11 of Ref. [46]. They are also similar to the corresponding computed distributions reported in Figures 14.17, 14.19, and 14.18. It is seen that the trends of the computed distributions agree well with the test results. The magnitudes of various characteristic flow parameters are tabulated and compared in Table 14.3. Most of the important flow parameters are predicted very well by the Euler computations. In particular, the predicted level of total pressure loss is quite realistic. This agreement between the calculations and experiments may be fortuitous, considering it is a purely spurious numerical artifact. However, similar observations were found in other free-vortex flow simulations over sharp edge delta wings using Euler methods, e.g. Murman and Rizzi[31], Rizzi [38], and Powell, Murman, Perez, and Baron [36]. They performed systematic studies in which various computational parameters were changed. In particular grid spacing and artificial damping coefficients were changed by an order of magnitude. They found that the magnitude of the total pressure loss was insensitive to all the computational parameters although the vortical region was more diffused on coarser grid and/or with high damping constants. Rizzi[38] claimed that the invariance of total pressure loss with the grid size appears to result from a singularity in the solution. Moreover, Rizzetta and Shang [37] reported that total pressure contours from the Euler solution were virtually identical to those of the Navier-Stokes calculations, except for the zone of secondary flow not re-

322

Stability of Vortex Pairs

produced in the inviscid result. Just as the separation at a sharp leading edge is insensitive to viscosity, the total pressure loss in the subcore is insensitive to viscosity. Both the sharp edge of the wing and the center of the vortex core are singular points of the Euler differential equations. Numerical dissipation smoothes out the singularities. The generation of vorticity about a sharp edge and the total pressure loss at the center of vortex core are both insensitive to the actual magnitude of the numerical dissipation, as long as there is some. It is noted that the computed maximum vorticity is lower than the experimental data. In fact, the experimental results of the maximum axial vorticity from different investigators vary quite substantially as pointed out by Nelson and Visser [33]. From an examination of the grid resolution used in the experimental measurement of each investigation, they found that the highest derived vorticity values correspond to the finest grid resolution and vice versa. The lower value of the computed maximum axial vorticity may be due to the insufficient computational-grid resolution. According to Fig. 14.13, there are approximately 100 x 40 grid points along radial and circumferential directions, respectively, lying in the cross section of the vortex core, and about 50 x 20 grid points in the subcore. It seems that the grid is still not fine enough to resolve the flow in the subcore.

14.7

Summary and Conclusions

We have presented a brief review of the different theories on the stability of symmetric and asymmetric separation vortices over slender bodies at high angles of attack. Both convective and absolute stability mechanisms have been proposed in the literature. Previously, the authors developed an absolute stability criterion for slender conical bodies. We present in this paper a time-accurate three-dimensional Euler code using overset grid to study the problem of vortex stability. Complete three-dimensional flows at typical flow conditions that vary from stable to unstable regimes as determined by the theoretical analysis are computed by the Euler flow solver. Stationary vortex configurations are first captured by running the Euler code in its steady-state or time-accurate mode. After a stationary vortex flow configuration is obtained, a transient asymmetric perturbation consisting of suction and blowing of short duration on the leftand right-hand side of the wing is introduced to the flow and the Euler code is run in time-accurate mode to determine if the flow will return to its original undisturbed condition or evolve into a different steady or unsteady solution. Features of the vortex core structure resolved by the Euler computation are also examined. The following conclusions are drawn from the computations. 1. The Euler method can be used to effectively study the stability of separation vortices over wing-body combinations. The computed Euler solutions automatically satisfy the Kutta condition at the sharp leading

J. Cai, H-M. Tsai, S. Luo €9 F. Liu

323

edge of slender delta wing, capture the free shear layer shed from the leading edge, and develop it into a compact and coherent rotational core in the leeside of the wing. Fine grids, high accuracy with 64bit computations, and stringent convergence criteria must be used to resolve the tight vortices. The numerical algorithm should also be symmetric in order to preserve symmetry of the computed stationary symmetric vortex configurations. 2. Limited but judiciously chosen computational cases agree well with previous theoretical analysis by the authors and some experimental observations, and thus support the conclusion that an absolute type of hydrodynamic stability is responsible for the breaking of symmetry of many typical vortex flows over slender conical bodies at high angles of attack. 3. The Euler computations reproduces the essential features of the vortex core. A clear core of significant rotational flow and a subcore based on the variation of the tangential velocity in the vortex can be identified. The basic features of the vortex core are well modeled by the Euler methods. In particular, the predicted total pressure loss in the subcore is realistic despite the inviscid nature of the flow model. It appears that the numerical viscosity in the solution of the Euler equations adapts itself near a singularity such as the sharp leading-edge of the wing or the center of an inviscid vortex in a way to yield solutions that reasonably approximate the real viscous flow.

14.8 Bibliography [l] Bernhardt, J. E. & Williams D. R. Proportional control of asymmetric forebody vortices. A I A A Journal, 36( 11):2087-2093, November 1998.

[2] Bryson, A. E. Symmetrical vortex separation on circular cylinders and cones. J. Appl. Mech. (ASME), 26:643-648, 1957. [3] Cai, J., Tsai, H-M., & Liu, F. An overset grid solver for viscous computations with multigrid and parallel computing. AIAA Paper 2003-4232, June 2003. [4] Cai, J., Tsai, H-M., Luo, S. & Liu, F. Stability of vortex pairs over slender conical bodies-theory and numerical computations. AIAA Paper 20041072, January 2004. [5] Cai, J., Liu, F., & Luo, S. Stability of symmetric vortices in two-dimensions and over three-dimensional slender conical bodies. J. Fluid Mech., 480:6594, April 2003.

Stability of Vortex Pairs

324

[6] Cai, J., Luo, S., & Liu, F. Stability of symmetric and asymmetric vortex pairs over slender conical wings and bodies. AIAA Paper 2003-1101, January 2003. [7] Cai, J., Luo, S., & Liu, F. Stability of symmetric and asymmetric vortex pairs over slender conical wings-body combinations. AIAA Paper 20033598, June 2003. [8] Cai, J . , Luo, S., & Liu, F. Stability of symmetric and asymmetric vortex pairs over slender conical wings and bodies. Physics of Fluids, 16(2):424432, February 2004. [9] Carcaillet, R., Manie, F., Pagan, D., & Solignac, J. L. Leading edge vortex flow over a 75 degree-swept delta wing - experimental and computational results. ICRS 86-1.5.1, September 1986. [lo] Champigny, P. Side forces at high angles of atack. why, when, how? La Recherche Aerospatiale, (4):269-282, 1994. [ll]Cummings, R. M., Forsythe, J. R., Morton, S. A., & Squires, K. D. Computational challenges in high angle of attack flow prediction. Progress in

Aerospace Sciences, 39(5):369-384, May 2003. [12] Degani, D. Effect of geometrical disturbance on vortex asymmetry. A I A A Journal, 29(4):560-566, April 1991.

[13] Degani, D. Instabilities of flows over bodies at large incidence. A I A A Journal, 30( 1):94-100, January 1992. [14] Degani, D. & Tobak, M. Experimental study of controlled tip disturbance effect on flow asymmetry. Physics of Fluids A , 4(12):2825-2832, 1992. [15] Dexter, P. C. & and Hunt, B. L. The effects of roll angle on the flow over a slender body of revolution at high angles of attack. AIAA Paper 1981-0358, 1981. [IS] Dyer, D. E., Fiddes, S. P. & Smith, J. H. B. Asymmetric vortex formation from cones at incidence a simple inviscid model. Aeronautical Quarterly, 33, Part 4:293-312, November 1982. ~

[17] Ericsson, L. E. Sources of high alpha vortex asymmetry at zero sideslip. Journal of Aircraft, 29(6):1086-1090, Nov-Dec 1992. [18] Ericsson, L. E. & Reding, J. P. Asymmetric flow separation and vortex shedding on bodies of revolution. In M.J. Hemsh, editor, Tactical Missile Aerodynamics: General Topics, Progress in Astronautics and Aeronautics, volume 141, pages 391-452, New York, 1992. American Institute of Aeronautics and Astronautics.

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[19] Hartwich, P. M. Symmetry breaking in vortical flows over cones - theory and numerical experiments. A I A A Journal, 32(5):1013-1020, May 1994. [20] Hartwich, P. M., Hall, R. M., & Hemsch, M. J. Navier-stokes computations of vortex asymmetries controlled by small surface imperfections. Journal of Spacecraft and Rocket, 28(2):258-264, Mar.-Apr. 1991. [21] Huang, M. K. & Chow, C. Y. Stability of leading-edge vortex pair on a slender delta wing. A I A A Journal, 34(6):1182-1187, June 1996.

[22] Hunt, B. L. Asymmetric vortex forces and wakes on slender bodies. AIAA Paper 1982-1336, 1982. [23] Lamont, P. J. Pressures around an inclined ogive cylinder with laminar,transitional,or turbulent separation. A I A A Journal, 20( 11):1492-1499, November 1982. [24] Legendre, R. Ecoulement au voisinage de la pointe aavant d’une aile a forte flBche aux incidences moyennes. La Recherche Aeronautique, Bulletin Bimestriel, De L’Ofice National D’Etudes E t De Recherches Aeronautiques, Jan-Feb 1953. [25] Levy, Y., Hesselink, L., & Degani, D. Anomalous asymmetries in flow generated by algorithms that fail to conserve symmetry. A I A A Journal, 33(6):999-1007, June 1995. [26] Levy, Y., Hesselink, L., & Degani, D. Systematic study of the correlation between geometrical disturbances and flow asymmetries. A I A A Journal, 34(4):772-777, April 1996. [27] Liu, F. & Jameson, A. Multigrid navier-stokes calculations for threedimensional cascades. A I A A Journal, 31(10):1785-1791, October 1993. [28] Liu, F. & Zheng, X. A strongly-coupled time-marching method for solving the navier-stokes and k-w turbulence model equations with multigrid. J. of Computational Physics, 128:289-300, 1996. [29] Liu, F. & Ji, S. Unsteady flow calculations with a multigrid navier-stokes method. A I A A Journal, 34( 10):2047-2053, October 1996. [30] Marconi, F. Asymmetric separated flows about sharp cones in a supersonic stream. In Proc. of the 11th Intern. Conference on Numerical Methods in Fluid Dynamics, pages 395-402, July 1988. [31] Murman, E. M. & Ftizzi, A. Applications of euler equations to sharp edge delta wings with leading edge vortices. In Proceedings of A G A R D Conference on Applications of Computational Fluid Dynamics in Aeronautics, pages 15-1-15-13, Parais des Congress, Aix-en-Provence, France, 7-10, April 1986.

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Stability of Vortex Pairs

[32] Murri, D. G. & Rao, D. M. Exploratory studies of actuated forebody strakes for yaw control at high angles of attack. AIAA Paper 87-2557, September 1987. [33] Nelson, R. C. & Visser, K. D. Breaking down the delta wing vortex-the role of vorticity in the breakdown process. AGARD CP-494, pp. 21-1-2115, 1990. Vortex Flow Aerodynamics. [34] Ng, T. T. & Malcolm, G. N. Aerodynamic control using forebody blowing and suction. AIAA Paper 91-0619, January 1991. [35] Pidd, M. & Smith, J. H. B. Asymmetric vortex flow over circular cones. In Vortex Flow Aerodynamics, A G A R D CP-494, pages 18-1-11, July 1991. [36] Powell, K. G., Murman, E. M., Perez, E. S., & Baron, J. R. Total pressure loss in vortical solutions of the conical euler equations. A I A A Journal, 25(3):360-368, March 1987. [37] Rizzetta, D. P. & Shang, J. S. Numerical simulation of leading-edge vortex flows. A I A A Journal, 24(2):237-245, February 1986. [38] Rizzi, A. Three-dimensional solutions to the euler equations with one million grid points. A I A A Journal, 23(12):1986-1987, December 1985. [39] Sadeghi, M., Yang, S., & Liu, F. Parallel computation of wing flutter with a coupled navier-stokes/csd method. AIAA Paper 2003-1347, January 2003. [40] Shanks, R. E. Low-subsonic measurements of static and dynamic stability derivatives of six flat-plate wing having leading-edge sweep angles of 70" to 84". NASA TN D-1822, 1963. [41] Siclari, M. J. & Marconi, F. Computation of navier-stokes solutions exhibiting asymmetric vortices. A I A A Journal, 29( 1):32-42, January 1991. [42] Sychev, V. V. Three-dimensional hypersonic gas flow past slender bodies at high angle of attack. Journal of Maths and Mech. (USSR), 24:296-306, 1960. (431 Thomas, J. L. Reynolds number effects on supersonic asymmetrical flows over a cone. Journal of aircrajl, 30(4):488-495, Ju1.-Aug. 1993. [44] Thomas, J. L., Kirst, S. T., & Anderson, W. K. Navier-stokes computations of vortical flows over low-aspect-ratio wings. AIAA Journal, 28(2):205-212, February 1990. [45] Van Dyke, M. A n Album of Fluid Motion. Parabolic Press, 1982. photo 90, page 54.

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327

[46] Verhaagen, N. G. & Kruisbrink, A. C. H. Entrainment effect of a leadingedge vortex. AIAA Journal, 25(8):1025-1032, August 1987. [47] Zilliac, G . G., Degani, D., & Tobak, M. Asymmetric vortices on a slender body of revolution. AIAA Journal, 29(5):667-675, May 1991.

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Chapter 15

Effect of Upstream Conditions on Velocity Deficit Profiles of the Turbulent Boundary Layer at Global Separation Oleg S. Ryzhov'

15.1

Introduction

Separation of an incompressible flow is one of L e most intricate p--3nomena of fluid mechanics. For many decades since the boundary-layer concept has been put forth by Prandtl [13], it was not amenable to rational analytical treatment. An extension of Prandtl's ideas proposed by Stewartson [19], Neiland [ll]and Messiter [lo] in the form of the triple-deck flow pattern for laminar motion offered the clues to understanding the origin of recirculating incompressible fluid bubbles around small local obstacles. Sychev [20] and Smith [17] were the first to successively resolve the problem on massive separation of the laminar boundary layer from a smooth body such as a circular cylinder. The turbulent boundary layer presents a severe challenge owing to the necessity of introducing a closure hypothesis in order to make the mathematical 'Department of Mechanical and Aeronautical Engineering, University of California at Davis, Davis, California 95616

329

330

Turbulent Boundary Layer Separation

description complete. Irrespective of any kind of this hypothesis, Yajnik [22] and Mellor [8] developed an asymptotic expansion for mean turbulent motion. A novel feature of the theory is that it deals with an undetermined system of equations. Nevertheless, their results cover the velocity deficit law and the logarithmic law by von K k m h Independently, Sychev & Sychev [21] mounted an effort to attack the problem on massive turbulent separation from a smooth surface without making recourse to a particular closure hypothesis. This work was later criticized by Melnik [9]in his analysis of marginal turbulent separation leading to a singularity of the Goldstein's type. Recently Barenblatt, Chorin & Prostokishin [3] discussed asymptotic properties of non-zero-pressure-gradient boundary layer in the context of incomplete similarity set forth in Barenblatt

PI.

The experimental data for turbulent, non-zero-pressure-gradient boundary layers are available in Schubauer & Klebanoff [16], Coles [6], Samuel & Joubert [15], MaruSiC & Perry [7] and Castillo & Walker [4], to mention only the basic papers on the subject. A model suggested in Perry et al. [la] on the basis of wind-tunnel tests is of particular importance for our purposes. However, direct measurements of the turbulent boundary-layer velocity and skin friction profiles at the threshold of massive separation are not reported in the literature referred to above.

15.2

Singular inviscid pressure gradient

The breakaway of a free streamline from the body surface is typical of massive separation. This breakaway creates a singularity in the pressure gradient which governs the velocity field. According to the results available in Sobey [18], the pressure gradient (15.1) along the length 2 < 0 upstream of the point of separation, be it laminar or turbulent. However, a fundamental distinction takes place between the laminar and turbulent flows. The Brillouin-Villat condition imposes a severe constraint

b+ = O

(15.2)

on laminar inviscid pressure distribution. As a result, theoretically separation would occur from the forebody at an angular coordinate 8, M 55" (Sobey [18]). This estimate does not correlate with experimental data (Achenbach [l]).Sychev [20]supposed the flow outside the boundary layer at laminar separation to obey (15.1) but with b i depending on the Reynolds number R through

b 2l = b l5 ( R )= by'R-A 2

+0

as R + m.

(15.3)

331

0. S. Ryzhov The full triple-deck solution obtained by Smith [17] shows that

hio) M 0.22. 2

Thus, h i -+ 0 as R + DC) in keeping with inviscid analysis. The Brillouin-Villat condition (15.2) does not hold for the turbulent flow. Experimental data provide strong evidence that in fully turbulent environments

h i = O(1)

(15.4)

and the breakaway point is shifted far downstream at an angular coordinate 8 M 120" (Achenbach [l]). This roughly corresponds to the inviscid limit solution where the free streamline becomes at infinity parallel to the direction of the oncoming stream.

15.3

Governing equations

Let x and y denote the local coordinates along the body surface and normal to it, respectively, u and v stand for the corresponding mean velocity components, and u',v' imply pulsations. The two-dimensional equations of motion in the turbulent boundary layer may be written as

au av -ax+ - a y

au

au

ax

ay

u-+v-

= 0,

(15.5a)

=

(15.5b)

where v designates, as usual, kinematic viscosity, and the pressure along the body comes from (15.1). The viscous term on the right-hand side of the momentum equation is small away from the near-wall region and will be neglected below. This system of equations is not closed owing to an additional term T~~ = - (u'v') entering (15.5b). Our aim is to work out an asymptotic approach independent of any specific assumptions on the turbulence model relating the Reynolds stresses to the quantities of the mean motion. A similar idea has been applied first by Yajnik [22] and Mellor [8] to attached boundary layers and channel flows. If turbulent pulsations are considered to be small and discarded, (15.5b) reduces to an equation for a thin inviscid boundary layer introduced by Cole & Aroesty [5] in connection with the blowhard problem. Thus, the mathematical foundation of transition and turbulent separation is common to both phenomena. The continuity equation (15.5a) suggests the introduction of the stream function such that

Turbulent Boundary Layer Separation

332

(15.6a) 'u

=

a+

--

(15.6b)

axc'

As a result, the x-momentum equation (15.5b) reduces to

ap +-aTzy ay

a+az$ ayaxay ax a y 2

a$ a2+ - -- - --

ax

(15.7)

with v@ omitted. As is often the case in the multistructured boundary-layer theory, the velocity field is divided in two domains, each consisting of sublayers with different properties according to the role played by turbulent pulsations. In the first domain of preseparated motion, the pressure distribution comes from (15.1), whereas in the second domain, where separation occurs, the pressure derives from a specific inviscid/inviscid interaction and is not known in advance. The nature of the inviscid/inviscid interaction is akin to that controlling the soliton formation in the earlier nonlinear stage of transition (Ryzhov & BogdanovaRyzhova [14]). The domain of preseparated flow analyzed in what follows can be affected by both the upstream conditions and singular pressure gradient at separation.

15.4 Inviscid sublayer 1 The distinction between (15.3) and (15.4) is crucial. Laminar preseparated flow can be treated by using an approach based on linearization with b+ taken as a small parameter. Then, the scaling of a domain centered about the breakaway point comes naturally and provides reference lengths and gauge functions typical of the triple-deck theory (Sychev [20];Smith [17]). No linearization is admissible when treating turbulent separation controlled by a strong singularity (15.4) in the pressure gradient. The velocity field in the inviscid sublayer of potential preseparated motion derives from the Bernoulli integral

U; (x) - 4b; (-x); - 2bl

(-X)

+ ... = 1

(15.8)

whence, to the third-order accuracy in x, we have an expansion

+

U1 (x) = 1 2b; (-x)i

+ (bl - 2 b i ) (-x) + ...

showing the singular behavior of the derivative affect the second-order velocity field.

(15.9)

%. Thus, the Reynolds stresses

333

0. S. Ryzhov

15.5

Outer turbulent sublayer 2

Here the form of asymptotic expansions

$ =

$20

=

720

7

(Y) (y)

+(-44

+ ...

$21

(y) + (-1

$22

(Y)

+ ...

(15.104 (15.10b)

is dictated by the singularity entering the velocity distribution in (15.9). Substitution of (15.10a,15.10b)into (15.9) shows that the zero-order function $20 (y) stemming from the upstream history of the boundary layer remains arbitrary, whereas $21 comes from d+20 d$2l

dY dY Hence the first-order approximation

d2*20 -~7@21

= 2b+.

for the velocity field does not depend on the turbulent Reynolds stresses at all. However, the next approximation reads

Thus, the Reynolds stresses affect the second-order velocities. This is the main distinction of the outer turbulent sublayer 2 from the inviscid potential sublayer 1 where turbulent pulsations play no role in the second-order velocity field. The matching of the asymptotic expansions (15.9) and (15.10a, 15.10b) which hold in the inviscid sublayer 1 and the outer turbulent sublayer 2, respectively, is easily achieved since

15.6

Outer turbulent sublayer 3

A solution in this sublayer located beneath is determined by the behavior of the velocity field in the limit y + 0. Let $20

--t

aoyQfl

+ ...,

as y

Then it follows from (15.10a) and (15.11) that

---f

0.

(15.12)

334

Turbulent Boundary Layer Separation

I

2bi

$ -+ a o y a + l +

+ ... + ( a + 1) aoAy" + ...

...+ [ ( a+ 1) (1: 2a)ao (-z)'

(15.13) to the first-order accuracy. Thus, a value of a determines the flow pattern in the turbulent sublayer 3. Dominant impact of the singular pressure gradient. In this case, the first term in the square brackets becomes comparable in magnitude with the 1 leading-order term on the right-hand side of (15.13) provided that y (-z) zz . A self-similar variable for the sublayer 3 is introduced by

-

1

(=--- Y (-X)"

1

1

a > -2

(15.14)

Dominant impact of upstream conditions. This regime comes into operation if the second term in the square brackets is comparable in magnitude with the leading-order term on the right-hand side of (15.13). Then y (-z)i and a self-similar variable

-

(=-

Y 1 1

(-X)%

1 a<2

(15.15)

does not depend on a at all. The pressure gradient is in balance with upstream conditions. Both definitions given by (15.14) and (15.15) coincide when a takes the value that makes the asymptotics in (15.13) invalid. Instead we have

( 15.16) The limit of $20 (y) as y -+ 0 turns out to be the deciding factor in establishing the disturbance field located below the turbulent sublayer 2.

15.7

Pressure-dominated flow pattern

We seek a zero-order solution in the form U+l

$ = (-z)

4a $30

(<)

(15.17)

where ( is defined by (15.14). Substitution of (15.17) into (15.7) with the Reynolds stress term on the right-hand side omitted leaves us with

(15.18)

0. S. Ryzhov

335

This equation can be integrated by passing to $30 as a new independent variable and taking (15.19)

as a desired function. As a result, (15.18) reduces to

whence (15.20) on the strength of (15.19), C being an arbitrary constant to be determined from matching with the first-order solution for the outer turbulent sublayer 2. In the limit as 6 -+ 00,we derive from (15.13) the two-term asymptotics 4(1-a)b$ 1 - 2a that leads to a unique value

c = ( a+ 1 ) Z uo* entering (15.20). It is worth comparing the above solution with that derived by Sychev & Sychev [21] in von Mises variables. In the notation adopted here, this solution reads

with the exponent a

--+ 00

in (15.12). Accordingly, the stream function

exponentially grows with y + 00 and cannot be matched with (15.10a) through the limit condition (15.13).

Turbulent Boundary Layer Separation

336

Xin -

20.5 21.0 21.5 22.0 23.0

(xS

92.4 86.4 80.4 74.4 68.4 62.4 56.4 50.4 44.4 38.4 32.4

216 222 228 234 240 246 252 258 264 270 276 -

U -

UCC

(&)* x 102

0.495 0.483 0.450 0.442 0.404 0.409 0.396 0.368 0.357 0.314 0.276

6.00 5.44 4.10 3.82 2.66 2.80 2.46 1.83 1.62 0.97 0.58

- x)in

Table 15.1: From G.B. Schubauer & P.S. Klebanoff, NACA Rept. 2133, 1950.

15.8

Comparison with experiment

On the basis of physical arguments and experimental data Perry et al. [12] proposed a model with three regions in the wall layer. In testing the proposed model, they discovered in the mean velocity distributions extensive half-power regions, some extending as far as the free stream. This correlates with (15.16) provided that a = Theoretically, a = is not a unique value of the exponent in (15.12), as we saw the other values, both a > and a < are also possible. However, the regime with a = $ is notable because the strong pressure gradient strikes a balance with upstream conditions in determining the disturbance pattern in the turbulent boundary layer. One more comparison to validate the above theory can be made with experiments by Schubauer & Klebanoff [16], whose data related to the preseparated flow are summarized in Table 15.1 and shown in Figure 15.1. In their work, X , is apparently a position x ~ f= , 25.7feet = 308.4in of fully developed separation, and U , designates the oncoming stream velocity. In the above notation, the'best-fit line

i.

i

(k)*

x 10' = -2.417

i

i,

+ 0.086 ( x , - x )

agrees with this theoretical prediction exhibiting some scatter.

15.9

Conclusion

The asymptotic approach applied to preseparated turbulent flow is based to leading order on a system of equations for a thin inviscid boundary layer. The

337

0. S. Ryzhou

(XS

- X),"

Figure 15.1: Dependence based on Table 1 strong singular pressure gradient determined by the free-streamline potential theory creates the driving mechanism. A similar approach with the pressure gradient induced by the inviscid/inviscid interaction leads to a soliton-bearing model (Ryzhov & Bogdanova-Ryzhova [14]). Thus, there exist profound parallels between the nonlinear stages of transition and fully turbulent flows. However, the disturbance pattern upstream of separation can essentially depend also on the history of the oncoming stream. Preliminary comparisons with experimental data substantiate theoretical predictions regardless of a turbulence model.

15.10 Bibliography [l] Achenbach, E. 1968 Distribution of local pressure and skin friction around a circular cylinder in cross-flow up to R = 5 x lo6. J. Fluid Mech. 34,

625-639. [2] Barenblatt, G.I. 2003 Scaling. Cambridge University Press. [3] Barenblatt, G.I., Chorin, A.J. & Prostokishin, V.M. 2002 A model of turbulent boundary layer with a non-zero pressure gradient. Proc. US Nat. Acad. Sci. 99, 5772-5776.

[4] Castillo, L. & Walker, D.J. 2002 Effect of upstream conditions on the outer flow of turbulent boundary layers. AIAA J. 40, 1292-1299.

338

Turbulent Boundary Layer Separation

[5] Cole, J.D. & Aroesty, J. 1968 The blowhard problem - inviscid flows with surface injection. Int. J. Heat Mass Transfer 11,1167-1183. [6] Coles, D. 1956 The law of the wake in the turbulent boundary layer. J. Fluid Mech. 1, 191-226. [7] MaruSiC, I. & Perry, A.E. 1995 A wall-wake model for the turbulence structure of boundary layers. Part 2. Further experimental support. J. Fluid Mech. 298,389-407. (http://www.mame.mu.oz.au/ivan). [8] Mellor, G.L. 1972 The large Reynolds number, asymptotic theory of turbulent boundary layers. Int. J. Engng. Sci. 10, 851-873.

[9] Melnik, R.E. 1989 An asymptotic theory of turbulent separation. Computers & Fluids, 17,165-184. [lo] Messiter, A.F. 1970 Boundary-layer flow near the trailing edge of a flat plate. SIAM J. Appl. Math. 18,241-257. [ll] Neiland, V. Ya. 1969 Contribution to the theory of separation of a laminar boundary layer in a supersonic stream. Izv. Akad. Nauk SSSR, Mekh.

Zhidk. i Gaza (4), 53-57 (in Russian; English translation: Fluid Dyn. (4), 33-35, 1972). [12] Perry, A.E., Bell, J.B. & Joubert, P.N. 1966 Velocity and temperature profiles in adverse pressure gradient turbulent boundary layers. J. Fluid Mech. 25,299-320. [13] Prandtl, L. 1905 Uber Flussigkeitsbewegung bei sehr kleinen Reibung. In: Verhandl. I11 Intern. Math. Kongr. Heidelberg, pp. 484-491, Leipzig, Teubner. [14] Ryzhov, O.S. & Bogdanova-Ryzhova, E.V. 1997 Forced generation of solitary-like waves related to unstable boundary layers. Adv. Appl. Mech. 34,317-417. 1151 Samuel, A.E. & Joubert, P.N. 1974 A boundary layer developing in an increasingly adverse pressure gradient. J. Fluid Mech. 66, 481-505. [16] Schubauer, G.B. & Klebanoff, P.S. 1950 Investigation of separation of the turbulent boundary layer. NACA TN No. 2133. [17] Smith, F.T. 1977 The laminar separation of an incompressible fluid streaming past a smooth surface. Proc. R. SOC.Lond. A356,443-463. [18] Sobey, I.J. 2000 Introduction to interactive boundary layer theory. Oxford University Press.

0. S. Ryzhov

339

[19] Stewartson, K. 1969 On the flow near the trailing edge of a flat plate 11. Mathematika 16, 106-121. [20] Sychev, V.V. 1972 On laminar separation. lzv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza (3), 47-59 (in Russian; English translation: Fluid Dyn. (3), 407-417, 1974). [21] Sychev, V.V. & Sychev, Vik.V. 1980 On turbulent separation. Zh. vychisl. Mat. i mathem. Fiz. 20, 1500-1512 (in Russian; English translation: USSR Comput. Math. Math. Phys. 20, 133-143, 1980). [22] Yajnik, K.S. 1970 Asymptotic theory of turbulent shear flows. J. Fluid Mech. 42, 411-427.

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Chapter 16

Hypersonic Magnet o-Fluid-Dynamic Interactions J. S. Shang'

16.1

Abstract

The most effective plasma-fluid-dynamic interaction for flow control is derived from an electromagnetic perturbation to the growth rate of a shear layer and amplified by the ensuing strong viscous-inviscid interaction. Computational efforts adopting a drift-diffusion and a simple phenomenological plasma model have shown the effectiveness of using electro-fluid-dynamic interaction as a hypersonic flow control mechanism. The numerical results are fully substantiated by experimental observations. The magneto-fluid-dynamic interaction introduces an added mechanism in the Lorentz force as a flow control mechanism. However, this approach also incurs additional challenges due to the Hall effect for computational simulations. The electromagnetic force for separated flow suppression has shown to be able to energize the retarded shear layer in the viscous interacting region on a very limited scope. Numerical simulations based on a simple phenomenological plasma model have shown the feasibility for the intended purpose. However, the effectiveness of this mode of magneto-fluiddynamic interaction requires further validation. 'Research Professor, Mechanical and Materials Engineering Department, Wright State University, Dayton, OH

341

342

16.2 B

D E J n

S U

2,Y ,z

U Pm E ff

16.3

Hypersonic Magneto-Fluid-Dynamics

Nomenclature Magnetic flux density Electric flux density Electric field strength Electric current density Charged particle number density Stuart number, aB2L/pu velocity vector Cartesian coordinates Conservative variables Magnetic permeability Electric permittivity Electrical conductivity

Introduction

Recent magneto-fluid-dynamics (MFD) research has made impressive progress in basic understanding for interdisciplinary science [ll, 27, 26, 1, 22, 2, 171. This added physical dimension is derived from the behavior of an electrically conducting flow medium. The drastically altered transport property of a fluid medium is originated from the excitation of higher internal degrees of freedom of air molecules via ionization. Using electromagnetic force to enhance the aerodynamic performance was advocated by Sears et a1 [25] in the late fifties and just a few years ago by the Ajax Program in Russia [7, 121. Innovative and attractive ideas have been put forward but are limited by our ability to analyze these extremely complex physics. The assessment of these revolutionary concepts at best, is inconclusive. However, the rejuvenate interest in MFD offers some realizable opportunities for flow control using electromagnetic forces. In these endeavors, the flow medium must be electrically conducting and the weakly ionized gas is often encountered in hypersonic flow. However, ionization fraction is still relatively low and the electrical conductivity is limited to a value around 100 mho/m for reentry conditions [25]. Therefore, the concentration of charged particles must be enriched. The energy required to generate a volumetric plasma that has a sufficient charged particle number density for strong MFD interaction is enormous [34,21, 15, 24, 191. Even if the simpler electronic collision mechanism is employed for plasma generation, the ionization potential always underestimates the energy requirement because the nonequilibrium energy cascades to vibration excitation, recombination and attachment processes [15, 241. In the past, the strong MFD interaction is further enhanced by an external applied magnetic field. The relative magnitude of electromagnetic force and the fluid inertia of the interact-

J . S. Shang

34 3

ing phenomenon is described by an interaction parameter, the Stuart number, S = u B ’ L / ~ u [34, 211. Although the presence of a magnetic field introduces additional complications by the Hall effect and the ion slip of the partially ionized gas, the MFD interaction is dominant when the magnetic field intensity is high and the fluid inertia is low. Since the magnetic field intensity is maximal at the magnetic poles and the inertia is minimal in the inner region of a shear layer, the most intensive MFD interaction always occurs near a solid surface. The description of weakly ionized gas is the most challenging issue for both experimental and computational research efforts [20, 31, 161. In most laboratory experiments, the plasma is generated by electron collision with embedded electrodes on control surfaces [26, 1, 221. This mechanism is frequently adopted at a specific location for enriching the charged particle number density in applications. The generated plasma consists of electrons in a highly excited state, but the heavy ions retain the thermodynamic condition of thier surroundings [15, 241. Most important of all, the surface plasma generated by electronic collision is far from a stable thermodynamic equilibrium state. For flow control using electromagnetic forces the degree of ionization, electrical conductivity, current density, and electrical field strength of the plasma are critical. Unfortunately, all the well-known methods in high-temperature thermodynamics with chemical kinetics that have developed for hypersonic flow are inapplicable for describing the weakly ionized gas that is generated by the electron collision process. A drift-diffusion plasma model by Surzhikov and Shang has successfully simulated a glow discharge in a magnetic field [32, 331. The numerical results substantiate the classic observations that an order of magnitude higher charged particles number density is adjacent to the electrodes, thus a much greater electrical conductivity s above the plasma sheath region [15, 241. This behavior is also observed in direct current discharge measurements in a Mach 5 plasma channel [20, 31, 161. This condition of a combined high electrical conductivity and strong applied magnetic field greatly ensures a strong electromagnetic-fluiddynamic interaction. Indeed, this interaction using the surface glow discharge has demonstrated the applicability for hypersonic flow control. However, those simulations are limited by the extremely high altitude environment where the pressure is only in a few mm of Hg. To apply plasma flow control to a wide range of applications, the limitation of plasma generation to low-pressure or low-density conditions must be alleviated. There are numerous surface plasma generation processes that can reliably function in a high-pressure environment t o augment the low electrical conductivity in a weakly ionized gas [3, 61. Viscous-inviscid interactions are one of the unique features of hypersonic flows, in that the presence of the boundary layer near a sharp leading edge is no longer ignorable as in the other flow regimes. The displacement thickness of a shear layer produces an outward flow deflection that leads to the formation of compression waves and eventually coalescence into a shock wave. The induced shock wave in turn modifies the boundary-layer structure to close the

Hypersonic Magneto-Fluid-Dynamics

344

interacting loop. Classic hypersonic flow theory describes the inviscid-viscous interaction over a sharp leading edge by Hayes and Probstein as the pressure interaction [13]. The magnitude of the induced pressure is characterized by a single interaction parameter X = M 3 ( C / R e y ) l I 2 . The dependence of the pressure interaction parameter on the cube of the free stream Mach number strongly amplifies its importance in hypersonic flows. Several experimental and computational investigations have been focused exploiting this possibility by introducing the electromagnetic effect for flow control [20, 31, 161. This control mechanism can be instantaneously actuated in microseconds to deflect the flow over a fixed control surface, emulating a deflected flap. Preliminary results from both experimental and computational efforts have shown promise that the chain of events, including the magneto-fluid-dynamic and inviscid-viscous interactions, constitutes a very effective hypersonic flow control using a simple surface glow discharge. To pursue this promising innovation, the present effort attempts to describe a basic concept that may renew a frontier for fluid dynamics research. In this process, some accomplishments for developing an effective plasma actuator for hypersonic flow control and possible research opportunities are delineated.

16.4

Governing equations

The governing equations of classical Magnetohydrodynamics (MHD) consist of the time-dependent compressible Navier-Stokes equations and the Maxwell equations in the time domain. The plasma is a gas medium in which the long range Coulomb force dominates the global behavior of the charged particles. A critical length scale of plasma is the Debye shielding length. Within this distance, the surrounding charged particles effectively shield any two particles from interacting with each other. As a consequence, the plasma is globally neutral and the conductive electrical current aD/& is negligible up to the microwave frequency [34]. Invoking these traditional MHD approximations, the remaining governing equations degenerate into Faraday induction law, the reduced Ampere circuit law, and two Gauss's divergence laws for electric and magnetic flux density, as well as the full complement of the compressible Navier-Stokes equations. After some rearrangement, the MHD governing equations become:

aP

-

at dB

+v

-

at

'

(16.1)

(pu) = 0

+ V . (uB

-

Bu) = -V x [(V x B/pe)/O]

-

T ) ] = (V x B / p e ) ' / a

(16.2)

(16.4)

34 5

J. S. Shang

The above nonlinear partial differential equations consist of eight dependent variables: five fluid dynamics and three electromagnetic variables ( B , E , and J ) . In the inviscid limit, the system of equations further reduces to the so-called ideal MHD equations, which constitute a non-strictly hyperbolic differential equation system; in addition these equations are nonconvex [4]. A total of four different waves propagate in a MHD field, the Alfven, slow and fast plasma waves are transverse waves, as well as a longitudinal acoustic wave to make the wave structure more complex [4, 231. The formulation of the wave dominant equation of motion also becomes very complicated, because the Lorentz force is perpendicular to both the magnetic flux density B and electric current density J . As a consequence, the eigenvalue associated with the normal magnetic flux density has a null value. The remaining seven eigenvalues of the MHD equations can also locally degenerate to coincide with each other, depending on the relative magnitude and polarity of the magnetic field [4, 23, 281. For most aerospace applications, the magnetic Reynolds (Bern = p,ouL) is much less than unity, which means the induced magnetic flux density is negligible in comparison with the externally applied field [34, 211. Under this circumstance, Faraday’s induction law of the Maxwell equations can be decoupled from the rest. This approximation now shifts the emphasis from the study of electromagnetic wave motion to magneto-aerodynamic interaction. In this formulation, the Lorentz force and Joule heating appear as source terms in the modified Navier-Stokes equations. The resulting governing equations and the initial values and boundary conditions are substantially simplified. A very large group of numerical simulations have been obtained using this system of governing equations and have demonstrated their ability to predict accurately a wide range of magneto-aerodynamic interactions [31, 32, 33, 8, 14, 18, 9, 10, 291.

aP

-

+ v . (pu) = 0

* at

+ V . (puu+?If-?)

at -ape +V. at

(16.5) -

J xB

[peu+Q+u.($-?)]

=0

-E.J=O

(16.6) (16.7)

Since the basic partial differential system consists mainly of the compressible Navier-Stokes equations with the electromagnetic variables in source terms that will not modify the eigenvectors, all traditional flux splitting formulations and solving procedures for the Navier-Stokes equations are directly usable. Accordingly, the system of equations can be easily cast into the flux vector form and solved by numerical procedures developed by the CFD community [9,10,29,35].

(16.8) where U = U ( p ,pu, pv, pw,pe)

34 6

Hypersonic Magneto-Fluid-Dynamics

The boundary conditions for the aerodynamic variables are straightforward; the no-slip condition is imposed for all velocity components, a constant wall temperature or adiabatic wall condition describes the condition on solid surfaces, and the vanishing pressure gradient condition provides the value of density locally. In the far field, the flow is required to return to its unperturbed state beyond the shock envelope. In hypersonic flow the traditional no-change conditional applies at the downstream far field. For the electromagnetic variables, in general, the tangential component of the electrical field intensity and the normal component of the magnetic flux density are continuous across media interfaces.

16.5

Plasma models

The long-range Coulomb force dominates the collective behavior of charged particles; a basic property of a partially ionized gas is its tendency towards electrical neutrality. This intrinsic characteristic, and the great differences in mass between electrons and ions, directly affect the kinetics of plasma. Two fundamental mechanisms of charged particle movement are the drift velocity and ordinary diffusion [24]. This behavior is independent from how the gas discharge is generated. Partially ionized gas produced by thermal collision has been extensively studied in a high temperature environments, and the modeling of the plasma has been built on the Lighthill’s model for a dissociating gas and the Saha equation for ionization [34, 211. The equilibrium degrees of dissociation or ionization of a high-temperature gas can be computed by their respective characteristic temperature and the associated partition functions. Ionization by electron collision is widely used for flow control, the physical model however is not as well known. Part of the reason is that the physical phenomenon is very complex and involves interaction at the atomic structure level of gas and solid. Nevertheless, an electric field of sufficient intensity generates electron-ion pairs by electron impact ionization of the neutral gas. As a consequence of the electrical conductivity, an electric current flows through the external circuit that supplies the electrodes [15, 241. In the discharge region between electrodes, the current consists of conduction and displacement electrical current components. In a DC field, only the conductive current flows and consists of electron and ion components. The electron component is the result of the avalanche growth of electron number density produced by secondary emission from the electrodes. In an AC field, the displacement current increases with frequency, while the importance of the secondary emission diminishes. The most widely adopted dielectric barrier discharges (DBD) generally operate at atmospheric pressure, the discharge in the electrode gap behaves like a streamer and the random transition filament quenches by the current limitation due to the localized charge build-up on the dielectric layers and are restored by the AC field [3, 61. From a phenomenological viewpoint, the driving forces of the motion of

J. S. Shang

34 7

charged particles are the drift velocity and diffusion, in which the different diffusion velocities between electrons and ions restrain electron diffusion. This component of diffusion is referred to as the ambipolar diffusion. Again in selfsustaining plasmas, the rate of change for charged number density in a control volume is mainly balanced by generation through ionization and depletion by recombination. The continuity equations for species concentration of the twocomponent plasma are given by the drift-diffusion theory [32, 331.

ane ~

at

+ V . re

= a ( E ,P)lr,l - pn,ni

(16.9) (16.10)

The electrical field intensity in the low Magnetic Reynolds number limit must satisfy the charge conservation equation and the global neutral condition to be compatible with the drift-diffusion formulation or the invoked generalized Ohm's law.

+

+

V . [ ( p i pL$)nE (D,*- De)Vn]= 0

+

(16.11)

+

where re = -DeVn, n,peE and l?i = -DiVni nipiE are the electron and ion fluxdensities, respectively. In this formulation, a ( E , p ) and ,B are the first Townsend ionization coefficient and recombination coefficient, p, and pi are the electron and ion mobilities, and D, and Di axe the electron and ion diffusion coefficients, respectively. In most electrodynamic formulations, the electrical field intensity is replaced by an electrical potential function:

E

= -V4

(16.12)

The compatibility conditions, Eqs. (16.11) and (16.12), lead to the well-known Poisson equation for the electrical potential 4 [32, 33, 8, 14, 18, 9, 10, 291. The electrical current density, by definition, is given by:

J

= e(ri -

re)

(16.13)

In the presence of a magnetic field, the Hall effect can be included by explicitly modifying the mobility of ions and electrons, and the diffusion and ionization coefficients of the drift-diffusion theory [28, 81. pe

~2

pe/(l+H,2) = b e +pi)/ [(I + H,")pi +pel =

De/(l+Hz) D,* = (0,peD+)/(Fe a ( E ,P ) = a ( E ,P ) / ( l + H,") De

(16.16)

=

+

(16.14) (16.15)

+ pi)

(16.17)

(16.18)

34 8

Hypersonic Magneto-Fluid-Dynamics

where He = wem,C and Hi = w i m i C are the Hall parameters for the electron and ion, and we and wi are the Cyclotron or Larmor frequencies of electron and ion respectively [21, 15, 241. Here, C is the speed of light. For the drift-diffusion model, the appropriate boundary condition on the cathode requires enforcement of zero electrical potential and the number density of electrons is proportional to the coefficient of secondary electron emission [32, 331. It is assumed that the anode reflects all ions; the ion number density vanishes, and the electrical potential is prescribed by the difference over the electrodes. On the dielectric surface, the charged number density is negligible and the outward normal gradient of the electrical potential is zero. For all electronic collision ionization, the basic discharge structure is sustained by an electric field. For numerical simulation purposes, the plasma domain can be completely described by the electrical field strength and electrical conductivity. These rudimentary data are routinely collected by most experimental observations [20, 161. Gaitonde adopted a simple phenomenological model for surface discharge to investigate the stability of an entropy layer, a shock-boundary-layer interaction, and the performance of a three-dimensional scramjet [9, 10, 29, 351. For the maximum flexibility, a modified Gaussian distribution was adopted to describe the electrical conductivity between electrodes. In the simple phenomenological plasma model, the Hall effect and ion slip can be included by the generalized Ohms equation [21].

E

= g . [E = u x

B

-

ahe(J x B )

+ ai,(J

x B x B)]

(16.19)

where ffhe and ais are coefficients associated with the Hall effect and ion slip respectively. In view of the great disparity of the characteristic speeds between the sonic speed of fluid dynamic motion and speed of light of the electrodynamics, the governing equations are solved loosely coupled. In this approach, the much-shortertime-scale electrodynamic equations are solved iteratively with the magnetoaerodynamic equations. The electrodynamic equations consist of the continuity equations for the charged particle species and the conservation of charge number density equation, in most circumstances, the Poisson equations for the electrical potential. The SOR (Successive Over Relaxation) scheme has been adopted to solve the governing equations in a pentadiagonal matrix system [31, 32, 331. All numerical results of the magneto-aerodynamic and electrodynamic equations are obtained with a nominal second-order spatial and temporal resolution. The computed charged particle number density over the embedded electrodes in a wedge model in a Mach number 5.15 stream is presented in Fig. 16.1. An electrical field of 1.2 kV sustains the glow discharge at an ambient pressure of 78.4 Pa (0.59 Torr). In the studied configuration, the cathode is placed upstream of the anode with a distance of 2.22 cm from the leading edge and the distance separating the electrodes is 3.81 cm. The overall dimensions of the identical electrodes are 0.64 cm in width 3.18 cm in length. The computed ion

J. S. Shang

34 9

Figure 16.1: Ion number density distributions over electrodes number density distribution above the anode is in excellent agreement with experimental observation [20, 161. However, there is a large discrepancy between computational and experimental results for the ion number density profile directly above the cathode. Nevertheless, the difference between these two results is confined within one order of magnitude that is similar to the measurement disparity between data obtained by microwave absorption techniques and Langmuir probing. The cause of the discrepancy is still being studied.

16.6

Elect ro-Fluid-Dynamic Interact ion

The behavior of inviscid-viscous interaction near a sharp leading edge wedge is clearly displayed in the Schlieren picture of Fig. 16.2. This image is recorded at a free stream Mach number of 5.15, a density of 5 x 1O3kg/m3, and a static temperature of 43 K. The model has an overall dimension of (3.81 x 6.67) cm. Under these conditions, the Reynolds number based on the length of the wedge model is 1.08 x lo5, and the boundary layer over the flat upper surface of the wedge model is expected t o be laminar [20, 161. The growth of the displacement thickness of the boundary layer clearly deflects the stream outward and induces a pressure over both the upper and lower model surfaces. The inviscidviscous interaction even generates an oblique shock over the flat plate surface of the wedge model. At the trailing edge of the model, the pressure interaction parameter with the Chapman-Rubesin constant of unity is 0.65. In this photograph, the boundary layers leave the wedge model surface and continue downstream as a free shear layer.

350

Hypersonic Magneto-Fluid-D ynamics

Figure 16.2: Schlieren of Wedge in hupersonic flow

The viscous-inviscid interaction near the sharp leading edge of a slender body reveals that a perturbation to the growth rate of the displacement thickness of a shear layer can be greatly amplified. The ensuing pressure interaction further magnifies the perturbation to produce a significant pressure plateau for flow control. This non-intrusive electromagnetic perturbation appears to be very attractive, because embedded electrodes with an applied electric field can ignite a glow discharge and an applied transverse magnetic field also adds the Lorentz force to the MFD interaction. In fact, the glow discharge produces two distinct perturbations to the structure of a shear layer. First, the electrode heating raises the surface temperature and second, the Joule heating increases the gas temperature above the surface in the glow discharge domain. Both mechanisms produce a thermal perturbation to the shear-layer structure. A direct current discharge over a sharp leading edge wedge in a Mach number 5.15 hypersonic stream is displayed in Fig. 16.3. The plasma is generated between two electrodes embedded in the flat plate surface. A total electrical current of 50 mA is maintained by an applied electric field of 1.2 kV in the external circuit. The maximum electron number density of the plasma is 3 x lOI2/cm3, and the electrode temperature is estimated to be 600 K [24, 201. At an ambient pressure of 0.59 Torr and a static temperature of 43 K, the air density is and the electrical con1.33 x lOI7/cm3; the degree of ionization is 2.25 x ductivity is on the order of 1 mho/m. In this sense, the glow discharge provides a truly weakly ionized gas over the electrodes. The electromagnetic field modifies the growth rate of the displacement thickness in two aspects; changing the kinematic field structure and initiating a heat

J. S. Shang

351

Figure 16.3: DC discharge over wedge in hypersonic flow

exchange in the wall region. The Lorentz force and Joule heating can be manipulated to alter the profile of the boundary layer. When electrodes for plasma generation are embedded in the model surface, the substantial local plasma heating is derived from two distinct sources. One of them is the volumetric Joule heating, the other source of heat release is the conduction by the heated electrodes. For glow discharge, the electrodes often attain a surface temperature approaching 600 K [24, 201. Since the heating mechanisms are vastly different, the characteristic time scale for conduction heating is in milliseconds and the Joule heating occurs on a much shorter time scale. The plasma heating will affect the thermal and velocity profiles of the boundary layer. The computed temperature contours of the electro-aerodynamic interaction are depicted in a composite presentation. Figure 16.4 consists of computed results for a simple hypersonic boundary layer, the same boundary layer with electrode heating, and finally the glow discharge over a wedge model. It is clearly displayed that the conduction and Joule heating introduce a significant perturbation to the structure of a hypersonic shear layer. The effect of Joule heating is much more pronounced than electrode heating. Both computations using plasma models, either by the drift-diffusion theory or the simpler phenomenological approach, produce reasonable agreement with measurements obtained by a shield stagnation temperature probe [20, 161. Figure 16.5 compares the dimensionless temperature (normalized by the freestream temperature of 43 K) over the electrodes. The most outstanding feature of the computational results is that the Joule heating is observed to dominate, resulting in gas temperatures near the wall that are hotter than the surface

352

Hypersonic Magneto-Fluid-Dynamics

/ Flat Plate

I GlowDischarge 1

Figure 16.4: Temperature Contours of electrode and Joule heating

temperatures of both cathode and the anode. In Fig. 16.6, the effect of Joule heating transforms the perturbation to the boundary-layer structure into an intensified viscous-inviscid interaction is depicted by the pressure profiles above the wedge surface. The observation is made easier by a direct contrast with the computed results of a classical hypersonic pressure interaction, in which the induced pressure is concentrated mostly near the leading edge of the wedge. The compression waves coalesce rapidly to form an oblique shock near the leading edge. On the other hand, the Joule heating triggers additional compression waves above the cathode and anode. These waves coalesce and eventually merge with the oblique shock wave originating from the sharp leading edge. The resulting shock wave produces a higher-pressure rise over the wedge surface. This induced pressure plateau is clustered near the leading edge of the wedge to become effective means of producing a pitching moment for hypersonic vehicle control. The hypersonic flow field in the leading edge region of a plate is dominated by the favorable pressure condition due the pressure interacting phenomenon [13]. The electro-aerodynamic interaction alters the expanding flow with multiple compression and expansion domains adjacent to the electrodes. The boundarylayer structure must respond to the streamwise pressure gradient. The surface shear force is calculated using the far downstream value as the reference, and the familiar inverse square-root decay with increasing distance from the leading edge is clearly exhibited. The surface shear stress in terms of skin-friction coefficient C f is depicted in Fig. 16.7 to reveal the boundary-layer structural response to the plasma flow control mechanism. The surface shear decreases upstream and

353

J. S. Shang

aza 0.18 a.16

m

0.14 0.12

1%

I____e____

Comp [Wthndo) Comp (Anode) olts(hthodo)

me

D

WtalAnoda)

0.10

am an6 an4 an2

ana SJJ

ua

711 ED

sa

Ian

iia

i2.a 136 i 4 n 160 %.a 178 i8.a

TrdTlnf

Figure 16.5: Stagnation temperature profiles comparison

Figure 16.6: Pressure profiles in EFD interaction

Hypersonic Magneto-Fluid-Dynamics

354 1211

1i5 10.0

911

811 711

rB u

€4 611

411

311 211

111 0.0

00

0.1

02

oa 0.4

06

06

0.7

O.B

09

i n 1.i

iz

UL

Figure 16.7: Surface shear stress in EFD interactions increases downstream of the electrodes, according to the local surface pressure gradient. The relative importance of the surface conduction and volumetric Joule heating is confirmed by experimental and computational results using plasma models. In Fig. 16.8, solutions to the magneto-aerodynamic equations using the drift-diffusion model are compared with measurements. The counterpart of the similar comparison is also given in Fig. 16.9, based on the phenomenological plasma model. Both numerical results exhibit reasonable agreement with data. More important, the glow discharge induces a bona fide electro-aerodynamic interaction that is not possible by electrode heating alone. At a very low level power supply of 60 watts, the electro-aerodynamic interaction generates an equivalent surface deflection of one degree. From this result, a scaling of power required for the plasma actuator per electrode length is 18.90 watts per cm per degree. In essence, the combined computational and experimental investigations have demonstrated that the electro-aerodynamic interaction induced by a glow discharge over a wedge surface in a hypersonic stream can be adopted as an effective flow control mechanism. However, the plasma generation process restricts the effective application range only to a low ambient pressure environment.

16.7

Magneto-Fluid-Dynamic Interaction

The full impact of magneto-aerodynamic interaction requires plasma in the presence of an externally applied magnetic field [25, 71. A transverse magnetic field

J. S. Shang

355

4.6 411

3s

-

-

E

f

25-

211

15

-

Figure 16.8: Effect of EFD on surface pressure (Drift-Diffusion Model)

----3--

{ E

B

ElectrDdQHearlng Jou!eHsaling

Dam

" 2.6

211

1.6

111

OR

0.1

02

03

0.4

05

06

0.7

OS

09

I11

1.1

I2

Figure 16.9: Effect of EFD on surface pressure (Phenomenological Model)

356

Hypersonic Magneto-Fluid-Dynamics

exerts a profound effect on the plasma, and especially alters characteristics of the plasma structure including that of electrode sheaths [4, 231. The basic phenomenon is governed by reduced electron mobility in a magnetic field. In a parallel plate glow discharge, the discharge column continuously drifts in the direction according to the polarity of a transverse magnetic field [32, 331. For glow discharge in hypersonic flow control, very little is known about the electrodynamic structure, but can only be analyzed from the fundamental collision process and try to gain a better understanding from experimental observations. The present numerical simulation duplicates an experimental arrangement, electrodes are embedded in the model wedge surface parallel its leading edge to generate a conductive current vector aligned to the airflow [20,16]. A transverse magnetic field is then applied across the plasma channel perpendicular to the electric current. By this arrangement, the transverse magnetic field generates a Lorentz force, J x B that either expels ( J x B > 0) or restrains ( J x B < 0) the charged particles to the electrodes. It is anticipated that for the case where the charged particles are expelled from the plate surface, the momentum exchange between ions and neutral particles by inelastic collisions enhances the subsequent viscous-inviscid interaction. Conversely, the restrained discharge particle motion suppresses the intensity of the pressure interaction. Preliminary experimental observations have confirmed this finding, but uncertainty is evident over a wide range of magnetic field strengths [20, 161. In essence, this net result of magneto-aerodynamic interaction reveals the compensating effects between the constricted surface discharge and acceleration by the Lorentz force. The surface plasma in experiment already operates in the abnormal discharge regime; the externally applied magnetic field creates additional discharge instability that hinders experimental observation [15, 241. At testing condition, the electron Larmor frequency at B = 1.0 Tesla is 1.76 x 10l1 radian/s, and the electron-heavy particle collision frequency is estimated to be in the range from 3 . 9 10g/s ~ to 2.3 x 1O1'/s. Under these conditions, the behavior of the plasma in the hypersonic MHD channel is not collisionally dominant and the maximum and effective Hall parameter ranges approximately from 7.6 to 45.1. In order to alleviate the uncertainty, the simulations are limited to a lower intensity of magnetic field, 0.2 T < B < 0.2 T , then the Hall parameter is reduced to the order of unity. From experimental observation, the predominant effect of an externally applied magnetic field to the glow discharge is the suppression the visible discharge over the cathode layer [20,16]. In the absence of an externally applied magnetic field, the glow discharge is most visible over the cathode layer and is concentrated above the dielectric surface between the electrodes. The plasma has also been convected downstream by the hypersonic stream beyond the wedge model. The discharge becomes more uniformly extended over the electrodes by the expelling Lorentz force. The visible plasma domain above the cathode layer is significantly suppressed by the applied magnetic field through the reduced degree of ionization. When the

357

J . S. Shang 4s

4a

3s

-

-

oa

. .

......... .......... ....,...

FIR1 Plat0

-

JXBcO. b n 2 T JXBSD. BtP.2T

" ::I

mta, B 4 . P l

I

1.0

mg842T

211

311

40

%a

6.0

711

81

9.0

10.0

X (om)

Figure 16.10: Effects of MFD on surface pressure

magnetic polarity is reversed, the glow discharge is confined to a narrow layer over the model surface. In numerical simulations, this effect of the Lorentz force is easily detected from the computed charged particle density contours over the electrodes as shown in Fig. 16.10. In the absence of an external magnetic field, the charged particle density is mostly concentrated over the cathode and anode layers. The highest degree of ionization of the plasma model reaches a value of 8.11 x 1O1'/cc near the electrodes and this value is compatible with those measured. However, the computed result under predicts the data above the cathode layer. In the presence of an externally applied magnetic field, the discharge pattern is substantially altered. The expelling Lorentz force ( J x B > 0) indeed pushes the charged particles away from the electrodes and enlarges the discharge domain. The degree of ionization is also reduced at the electrodes by 24.1% in comparison with the glow discharge without the external magnetic field. The trend of increasing discharge domain is reversed with the opposite polarity of the magnetic field. The charged particles are strongly constricted to the electrodes with a maximum value of 1.06 x 101'/cc in the cathode and anode layers and the discharge domain is significant reduced. In a glow discharge field, the electric field force always dominates over the J x B acceleration, and the plasma is nearly collisionless, and yet the effect of the Lorentz force is clearly demonstrated. Nevertheless, the effect of the Lorentz force is easily detected from the computed surface pressure over the electrodes in Fig. 16.10. The computed result generally overpredicts the pressure measurements and the data also exhibits an unusually large data scattering band

358

Hypersonic Magneto-Fluid-D ynamics

due the unstable glow discharge pattern in the presence of an applied and uniform magnetic field. The expelling Lorentz force ( J x B > 0) indeed pushes the charged particles away from the electrodes and enlarges the discharge domain. As anticipated, the magnetic field generates a much stronger electromagnetic perturbation which can be amplified by the viscous-inviscid interaction. As a consequence, the induced surface pressure is greater than the electroaerodynamic interaction under identical electric field strength. The trend of increasing discharge domain is reversed with the opposite polarity of the magnetic field ( J x B < 0). The induced surface pressure is also diminished accordingly by the suppression of the outward deflection of the streamline. However, there is also an increased effort required to maintain computational stability. The numerical results using the drift-diffusion model capture the difference between the applied magnetic fields of opposite polarity, and the phenomenological model is less accurate in predicting the induced pressure [as]. MFD interaction has also been applied for separation flow control [2, 17, 351. In these applications, the electromagnetic force is introduced into a bifurcating flow field not as a small perturbation, but to energize the retarded shear flow to overcome an adverse pressure gradient. The MFD separated flow control derives from an active Lorentz force, thus the control mechanism requires the presence of a strong external applied transverse magnetic field. The most illustrative and classic viscous-inviscid interaction is encountered in flow over a compression ramp [30, 51. In this flow field, the oncoming boundary layer separates upstream of the compression ramp. The thickened shear layer induces a family of compression waves that eventually coalesce into an oblique shock wave over the ramp. The separated flow region can be extensive and leads to a high-pressure plateau and a hot spot at the reattachment. In this respect, the MFD separated flow control suppresses the viscous-inviscid interaction and alleviates the degraded aerodynamic performance. However, there will not be any leverage obtainable by the viscous-inviscid interaction to amplify the magnetcaerodynamic perturbation [24, 19, 201. Using plasma for separated flow control is based on a different mechanism than that of the plasma actuator, relying on the electromagnetic perturbation and amplified by viscous-inviscid interaction. Flow separation is one of the most drastic fluid dynamic bifurcations; the transition of dynamic states is the result of an adverse pressure gradient in the streamwise direction. The suppression of the separated flow is achieved by energizing the shear layer to overcome the adverse normal stress or the pressure gradient. Thus the energy require for flow control is explicit and the advantage via viscous-inviscid interaction no longer exist. All known separated flow controls are associated with shock-boundary-layer interaction [17, 351, The most recent numerical simulation utilizes a simple phenomenological model for separated flow suppression over a 24-degree compression ramp at a Mach number of 14.1 and a Reynolds number of 103,680

J. S. Shang

359

based on the distance from the leading edge to the corner [35]. In the numerical simulation, a surface plasma domain was imposed over the entire corner region (0.8 < x / L < 1.2). A uniform and transverse magnetic field is imposed orthogonally to the two-dimensional flow and the Stuart number is assigned a constant value of unity. The surface temperature of the plasma domain is increased from a constant value of 297.2 to 555.6 K to simulate the electrode heating during the plasma generation. For the phenomenological plasma model, the electrical conductivity at the outer edge of the plasma domain reduces to a value of 0.001 mho/m. The parameter controlling the electrical field intensity is described as k = -E,/u,B = 1.5. Since this model does not necessarily represent the detailed physics but to represent a feasibility study, both the model and the numerical simulation are therefore not optimized. The velocity profiles of the Navier-Stokes (based line) and the magnetoaerodynamics equations at the compression corner (x/L=l.O) are presented side-by-side in Fig. 16.11. The computed viscous inviscid interaction has led to a separated flow region upstream to the compression corner, the separated flow region in fact has a physical dimension from x / L = 0.36 to x / L = 1.42 to agree with experimental observation and previous computations [35]. For the magneto-aerodynamic computation, this plasma domain is described by a strip of 0.8 < x / L < 1.2 and vertically extending to y / L = 0.1. The electromagnetic field, mainly the Lorentz force, accelerates the streamwise velocity component to completely eliminate the reversed flow region. Meanwhile, the strong acceleration and high value of Joule heating near the surface also produced a spike in the skin-friction and Stanton number distributions in the compression corner. In any event, the separated flow region is completely suppressed in the numerical simulation [35]. In Fig. 16.12, four pressure distributions are given. They depict solutions to the inviscid, shock-boundary-layer interaction, and magneto-aerodynamics equations. The two solutions of the magneto-aerodynamics equations are obtained at different vertical plasma domains. The vertically enlarged plasma domain previously mentioned is able to eliminate the separated flow for the shock-boundary-layer interaction. The numerical simulation indicates that the Lorentz force accelerates the inner region of the shear layer to override the highpressure plateau downstream. However, the computed result has not received any substantiation from experimental observation as yet.

16.8

Concluding Remarks

The plasma actuator using electromagnetic perturbation to the shear layer structure and amplified by the viscous-inviscid interaction in a hypersonic stream has been fully substantiated by experimental observations and numerical simulations with two plasma models. The basic mechanism is the combined effects of the conductive electrode and volumetric Joule heating. The time scale of

360

Hypersonic Magneto-Fluid-Dynamics

0.15

0.14 0.12

WHD No Sep.

0.1 1

I I

I I I I \

0.05 0.04

\

,

\

0.03 0.01

5 /

' i

- _,_ -

7

-

_ _ - - --

u 0.5

,, -

7

1 1

Figure 16.11: Velocity profiles withlwithout MFD control

X

Figure 16.12: Surface pressure distributions withlwithout MFD control

J. S. Shung

361

the Joule heating is estimated in the microsecond range that is orders of magnitude shorter than the conducting electrode heating and ensued convection. The electro-fluid-dynamic interaction by a plasma actuator embedded in a fixed and non-movable plate surface behaves as if the plate is executing a pitching movement. Both experimental and computational results have shown the effectiveness of flow control induced by a pressure plateau that can be scaled as 18 watts per degree per cm of electrode dimension. The magneto-fluid-dynamic interaction exhibits even greater amplification with an applied external magnetic field than the electro-fluid-dynamic interaction. The drift-diffusion plasma model has been successfully applied to numerical simulations and the Hall effect is explicitly included into the formulation. From both experiments and computations, the Hall effect significantly constricts the mobility of charged particles and alters the surface discharge pattern. In addition to that the resultant discharge instability obscures the measurement and computational accuracy. Two-dimensional computations for magneto-aerodynamic interactions using a simple phenomenological plasma model demonstrate the possibility to completely suppress the separated flow over a compression ramp at a Mach number of 14.1. Experimental verification of this phenomenon is not as yet accomplished.

16.9 Acknowledgment The sponsorship by Dr. J . Schmisseur and Dr. F. Fahroo of the Air Force Office of Scientific Research is deeply appreciated. The invaluable contributions by Prof. S.T. Surzhikov of the Russian Academy of Science, Drs. D.V. Gaitonde and R. Kimmel of Air Force Research Laboratory, as well as Prof. J . Menart of Wright State University are sincerely acknowledged.

16.10 Bibliography [l]Artana, G., D’Adamo, J., Lger, L., Moreau, E., & Touchard, G., ”Flow

Control with Electrohydrodynamic Actuators,” AIAA Journal, Vol. 40, 2002, pp. 1773-1779. [2] Bityurin, V., Klimov, A., & Leonov, S., Assessment of a Concept of Ad-

vanced Flow/Flight Control for Hyersonic Flights in Atmosphere, AIAA 99-4820, Norfolk, VA, Nov. 1999. [3] Boeuf, J.P., Plasma Display panels: Physics, Recent Developments and Key Issues, J. Physics D; Applied Physics, Vol. 36, 3006, pp. R53-R79.

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[4] Brio, M. & Wu, C.C., An Upwind Differencing Scheme for the Equations of Ideal Magnetohydrodynamics, J. Comp. Physics, Vol. 75, 1988, pp 400422. [5] Dolling, D.S., Fifty years of Shock-Wave/Boundary-Layer Interaction Research: What Next? AIAA J. Vol. 39, Aug. 2001, pp.1517-1531. [6] Elisson, B. & Kogelschatz, U., Nonequilibrium Volume Plasma Chemical Processing, IEEE Trans. Plasma Science, Vol. 19, 1991, pp 1063-1077. [7] F’raishtadt, V.L., Kuranov, A.L., & Sheikin, E.G., Use of MHD Systems in Hypersonic Aircraft, Technical Physics, Vol. 11, 1998, p. 1309. [8] Gaitonde, D.V., Development of a Solver for 3-D Non-ideal Magnetogasdynamics, AIAA 99-3610, June 1999. [9] Gaitonde, D., Higher-Order Solution Procedure for Three-dimensional Non-ideal Magnetogasdynamics, AIAA J., Vol. 39, No. 1, 2001, pp. 21112120. [lo] Gaitonde, D.V., Three-Dimensional Flow-Through Scramjet Simulation with MGD Energy-Bypass, AIAA 2003-0172, January 2003.

[11] Ganiev, Y., Gordeev, V., Krasilnikov, A., Lagutin, V., Otmennikov, V., & Panasenko, Aerodynamic Drag Reduction by Plasma and Hot-Gas Injection, J. Thermophysics and Heat Transfer, Vol. 14, No. 1, 2000, pp.10-17. [12] Gurijanov, E.P. & Harsha, P.T., Ajax: New Direction in hypersonic technology, AIAA Preprint 96-4609, Nov. 1996. [13] Hayes, W.D. & Probstein, R.F., Hypersonic Flow theory, Academic Press, 1959. [14] Hoffmann, K.A., H-M Damevin, & J-F Dietiker, Numerical Simulation of Hypersonic Magnetohydrodynamic Flows, AIAA 2000-2259, June 2000. [15] Howatson, A.M., An Introduction to Gas Discharges, 2nd Edition, Pergamon Press, Oxford, 1975. [16] Kimmel, R, Hayes, J., Menart, J., & Shang, J.S., Effect of Surface Plasma Discharges on Boundary Layer at Mach 5, AIAA 2004-0509, Reno NV, January 5-8, 2004. [17] Leonov, S., Bityurin, V., Savelkin, K., & Yarantsev, D., Effect of Electrical Discharge on Separation Processes and Shock Position in Supersonic Airflow, AIAA 2002-0355. Reno NV, January 2002. [18] MacCormack, R.W., A Conservative Form Method for Magneto-Fluid Dynamics, AIAA 2001-0195, Reno NV, January 2001.

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[19] Macheret, S.O., Shneider, M.N., & Miles, R.B., Magnetohydrodynamics and Electrohydrodynamic Control of Hypersonic Flows of Weakly ionized Plasmas, AIAA J., Vol. 42, July 2004, pp. 1378-1387. [20] Menart, J., Shang, J.S., Kimmel, R., & Hayes, J., Effects of Magnetic Fields on Plasma Generated in Mach 5 Wind Tunnel, AIAA 2003-4165, Orlando FL, June 2003. [21] Mitchner, M. & Kruger, C.H., Partially Ionized New York, 1973.

Gas,John Wiley & Sons,

I221 Post, M.L. & Corke, T.C., Separation Control Using Plasma Actuators: Stationary and Oscillating Airfoils, AIAA 2004-0841, January 2004. [23] Powell, K.G., Roe, P.L., Myong, R.S., Gombosi, T., & Zeeuw, D.D., AIAA 95-1704-CP, 1995, pp. 661-671. [24] Raizer, Yu. P., Gas Discharge Physics, Springer-Verlag, Berlin, 1991. [25] Resler, E.L., Sears, W.R., The Prospect for Magneto-aerodynamics, J. Aero. Science 1958, Vol. 25, 1958, pp. 235-245 and 258. [26] Roth J. R., Sherman, D.M., & Wilkinson, S.P., Electrohydrodynamic flow Control with a Glow-Discharge Surface Plasma, AIAA J. Vol. 37, 2000, pp. 1166-1172. [27] Shang, J. S., Plasma Injection for Hypersonic Blunt Body Drag Reduction, AIAA J. Vol. 40, No. 6, 2002, pp. 1178-1186. [28] Shang, J.S., Shared Knowledge in Computational Fluid Dynamics, Electromagnetics, and Magneto- Aerodynamics, Progress in Aerospace Sciences, Vol. 38, 2002, pp. 449-467. [29] Shang, J.S., Gaitonde, D.V., & Updike, G.A., Simulating MagnetoAerodynamic Actuator for Hypersonic Flow Control, AIAA 2004-2657, Portland, OR, June 2004. [30] Shang, J.S. & Hankey, W.L., Numerical Solution for Supersonic Turbulent Flow over a Compression Ramp, AIAA J. Vol. 13, Oct. 1975, pp. 13681374. [31] Shang, J.S & Surzhikov, S.T., Magneto-Aerodynamic Interaction for Hypersonic Flow Control, AIAA 2004-0508, Reno NV, January 5-8, 2004. [32] Surzhikov, S.T. & Shang, J.S., Glow Discharge in Magnetic Field, AIAA 2003-1054, 41st Aerospace Science Meeting, Reno NV, 6-9 January 2003.

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[33] Surzhikov, S.T. & Shang, J.S., Two-Component Plasma Model for Twodimensional Glow Discharge in Magnetic Field, J. Comp. Physics, Vol. 199, 2004, pp. 437-464. [34] Sutton, G.W. & Sherman, A., Engineering Magnetohydrodynamics, McGraw-Hill, New York, 1965. [35] Updike, G.A., Shang, J.S., & D. V. Gaitonde, Hypersonic Separated flow Control Using Magneto-Aerodynamic Interaction, AIAA 2005-0164, Reno NV, January 2005.

Part IV

Multiphase and Reacting Flows

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Chapter 17

Computing Multiphase Flows Using AUSM+-up Scheme Meng-Sing Liou' and Chih-Hao Chang'

17.1

Abstract

An extension of the AUSM+-up scheme to calculations of multiphase flow at all speeds and all states of compressibility is presented in this chapter. Two approaches, namely mixture and two-fluid models, for describing multiphase flows will be described in detail. The first approach considers the multiphase fluid as a mixture described by a real fluid equation of state and applies thermodynamic equilibrium for treating liquid-vapor phase transitions. The second approach is nonequilibrium model which solves each phase separately via transport equations, with pressure equilibrium between phases. The stratified flow model is proposed to include the inter-phasic effects in the cell-interface fluxes. The new AUSM+-up scheme is employed in both approaches. Numerical results have shown that the new scheme is effective in simulating some rather challenging numerical problems involving liquid and vapor. The calculations are robust and stable and the results are accurate in comparison with analytical, experimental and other computational results, in which phase interfaces and their evolution are captured sharply, without resort to special treatment for the interfaces. Propulsion Systems Division, NASA Glenn Research Center a t Lewis Field, Cleveland, OH 44135,USA

367

A USM+-up scheme for multiphase flow

368

17.2

Introduction

Two phase flows can be found in broad situations in nature, biological systems, and industry devices and they often involve diverse and complex mechanisms. While physical models may be specific to certain situations, the mathematical formulation and numerical treatment for solving the governing equations can be general. Based on the continuum mechanics, we shall treat the fluid as a mixture consisting of two interacting phases (or materials) occupying the same region in space at any given moment. Hence, we will require information concerning each individual phase as needed in a single phase, but also the interactions between them. These interaction terms, however, pose additional numerical challenges. For example, the mathematical equations are not guaranteed to be hyperbolic in time for all conditions. Moreover, due to disparate differences in time and velocity scales, fluid compressibility and nonlinearity become acute, further complicating the numerical procedures. A family of AUSM schemes [16, 11, 25, 41 have been developed in the past ten years. These schemes have been proven to be accurate, simple, robust, and easy to extend to other types of conservation laws, thus providing an attractive alternative to other existing schemes; a summary is given recently in [la]. This paper is to provide extensions suitable for solving the general multiphase flow problem for arbitrary flow speeds and arbitrary levels of compressibility. Prior efforts have focused on solving the “mixture”-model equations [14, 151. This paper will specifically address the solution of the so-called two-fluid model equations. Discrete equations will be derived, revealing the numerical fluxes that were not included previously. The scheme will be analyzed on a term-byterm basis to illustrate various interphase interactions, within the cell (control volume) and between cells. Recently, the AUSM-family has been extended to the multiphase flow calculations, e.g., in Refs. [14, 15, 5, 171. Paillkre et a1.[17] solved a system of two-fluid models with interfacial source terms included. Several features that are different from the usual equations for aerodynamic flows add complexity significantly. Contrary to the single phase flow, a major difficulty in dealing with multiphase flow is the lack of a common set of governing equations in which various levels of approximation and modeling of phase exchange terms have been proposed. The models proposed are based on experimental data accumulated over the years and carried out for specific configurations and flow regimes. Hence, they are highly phenomenological from the physics point of view. Moreover, the resulting system has severe mathematical consequences, for example: nonconservative form, stiffness due to disparity in length and time scales, complex representation of thermodynamic state variables, non-hyperbolicity, etc. As a result. these have direct implications on every aspect of numerical solution, in accuracy, efficiency and stability. Most of all, it imposes severe constrains on

M.-S. Liou and C.-H. Chang

369

robustness of a multiphase flow code. The extension from the single phase flow code is everything but straightforward. Hence, we have seen in the past decade active research attempting to devise a robust, accurate and general numerical method for solving multiphase flow equations. To overcome the possible difficulty of non-hyperbolic nature of the equations, a number of additional model terms have been proposed. They are generally adhoc (albeit with some physical considerations) in nature and devised primarily for numerical purposes. Mutliphase flow is characterized by the presence of phase-separating interfaces, be it in micro or macro scales, across which different (i.e., discontinuous) fluid and dynamic properties exist. For example, in the gas-liquid fluid systems, three distinct flow regimes have been classified according to the topological structure of the interfaces [9, 261: (1) separated flow, such as stratified and annular flows, (2) mixed flow, such as slug and bubbly annular flows and (3) dispersed flow, such as bubbly and droplet flows. These different flow regimes generally are described with different constitutive laws. Since a two-phase flow can evolve continuously from one regime to another, mathematical description of a two-phase flow in a general situation is thus difficult, if not impossible. A general practice is to make use of certain averaging processes, be it time, space, or statistical, by which a macroscopic description is deduced, resulting in loss of describing fine structures.

17.3

Governing Equations (Models) for Multiphase Flows

Two fundamentally different formulations for two-phase flows are often employed; they are mixture model and two-fluid model. The former, including thermodynamic equilibrium model and drift-flux model, considers the flow systems as a mixture and the governing equations are similar to the single phase equations with additional information delineating phase distributions. The latter considers each phase separately by employing governing equations for each individual phase with coupling (interaction) terms representing transfer of mass, momentum and energy between phases. In what follows we will give a brief account of these approaches. For more indepth discussion of models and mathematical properties, the interested reader can find it in some excellent references, for example [23, 241.

17.3.1

Thermodynamic Equilibrium Model [14]

This model belongs to the mixture model and specifically assumes that all phases are in equilibrium, thereby having identical velocity components, pressure and temperature among phases. This model is often called in the literature, includ-

A U S M f -up scheme for multiphase flow

370

ing in our previous paper [14], homogeneous equilibrium model. It is deemed more appropriate to call it a thermodynamic equilibrium model because the conditions for describing the co-existence of two phases2 is solely based on the principle of thermodynamic equilibrium. It allows transition from one phase to another through thermodynamic equilibrium. Hence, co-existence of phases can be dealt with solely from thermodynamic principles, such as Maxwell principle or assumption of being at saturation. The set of equations for thermodynamic equilibrium model appears exactly identical to that of single phase fluid flow. For clarity, we will present only equations in one space dimension and neglect viscous effects. The inclusion of corresponding terms in other dimensions are similar and straightforward. However, terms peculiar to multiphase flow, independent of spatial dimensions, will be maintained whenever needed, especially later in two-fluid model.

aQ dF -+-=o. at ax

(17.1)

The conservative variables are given in Q = [p, pu, pEIT. The inviscid fluxes are F = [pu, pu2 p , puHIT. The notation and definitions of variables are rather standard, hence omitted herein. It however must be emphasized that these variables are that of the mixture. Additional condition is imposed when the fluid lies in the two-phase region, otherwise it is described by equation of state of either phase, say liquid or vapor. A description of this model can be found in [14]. The thermodynamic state equation describing multi-phases can be included in the single-phase real fluid model,

+

where the compressibility factor Z(P,2’) is a complex algebraic function derived from empirical data or the Van der Waals-type formulas. Typical isotherms for the Peng-Robinson equation [18] are plotted in Fig. 17.1 on a pressure-density diagram. Clearly indicated is the vapor regime, where pressure varies nearly linearly with density, and the liquid regime, where large pressure changes are required t o induce a density change. For a given pressure and temperature, the solution of Eq. (17.2) returns one or three values of the compressibility factor Z or density, the former corresponding to the singlephase region (either liquid or vapor) and the latter corresponding to the twophase region, where vapor and liquid co-exist. The corresponding densities for a pressure within the two-phase region are shown as points A-C. A and C represent saturated vapor and liquid states, while B is physically meaningless. For a particular temperature, the “allowable” two-phase region is bounded by the pressure values at D and E, which are local extrema. The loci of these 21n this connection, a system that consists of two phases is said t o be heterogeneous.

371

M.-S. Liou and C . - H . Chang 2.OE+07

t.BE+07

T c T, 1.5E+07

1.2E+07

-

single-phase liquid

p (Nlm"1 1 OE+07

-

single-phase vapor

7.5E+06

-

metastable vapor

5.OE+06

-

2.5E+06

-

1 /4( /

n

two-pKe

upper (vapor) spinodal vapor pressure metastable liquid lower (liquid) spinodal

region

1 i 1 , , I I I II I L I ~ I II I ~II 20 25 10

5

p (Cmo~elm~)

I

I

Z

IJ 30

Figure 17.1: Pressure vs. molar density (isotherm below critical temperature).

pressure values for temperatures between the triple and critical points define liquid and vapor spinodal curves, dividing the two-phase region into metastable vapor, unstable, and metastable liquid regions. Information in the unstable parts of the two-phase region bears no physical meaning and is not useful. For densities between the spinodal values (D-B-E), it can be shown that the acoustic eigenvalues are complex, meaning that the Euler system is not hyperbolic in time and that conventional time-marching procedures for integrating the equations are ill posed. It is also of note that the liquid spinodal pressure may be negative for high molecular-weight liquids at lower temperatures, implying that the simulated expansion of a liquid might produce reasonable densities, but unphysical pressures, in the metastable region. In the metastable branches, A-D and C-E, they are physically feasible and have unique and possibly attractive properties. The states on the metastable vapor branch are also referred to as undercooled because the vapor exists at a temperature lower than the saturated vapor temperature corresponding to a pressure on A-D. Similarly, the states on C-E are called supersaturated they can reach, without evaporating, a pressure lower than the saturated liquid pressure corresponding to a temperature on C-E. Moreover, at a particular pressure between the liquid and vapor spinodal points, the system is in equilibrium. This pressure is known as the vapor pressure pvap(T) and is directly related to the temperature through the ClausiusClapeyron equation. Or this is equivalent to setting that the vapor and liquid

372

A USM+ -up scheme for multaphase flow

250000 -

300000

200000

-

150000

-

~

- - - -

----

50000

i

I I

I I closeup view of liquid I, I two phase juncture I I I

a2(m/s) I 100000

Peng Robinson acousticeigenvalue theoretical result

-

-

-

vapor

two phase

-50000

Figure 17.2: Square of sound speed a2 vs. molar density (octane at 350 K), taken from [15]. fugacities attain equal values. f ( G , T , p v a p ) = f(zi,T,Pvap).

(17.3)

By iteration, the vapor pressure m a p is calculated as a function of temperature. Hence, the number of independent variables for the two phase mixture is reduced from two to one, which is consistent with the Gibbs' phase rule.[lO]This sets the line A-B-C in Fig. 17.1. Given updated values for the density and temperature at a grid point as determined from a time-integration method, a specific point can be located along this line through the introduction of a scaled parameter, e.g., (vapor) void fraction, (17.4) where superscripts "I" and "II" refer to saturated liquid and vapor states, i.e., points C and A, respectively. Then, the associated thermodynamic quantities for the mixture, e.g., internal energy or enthalpy, can be determined accordingly.[14] One of the unique features in multiphase flow is that the mixture has a very small speed of sound, on the order of meters per second, as seen in Fig. 17.2.

M.-S.Liou and C.-H. Chang

3r3

Hence, a shift to a locally “supersonic” flow condition during a phase transition is a distinct possibility. An example of this approach showing its capability for describing multiphase flow involves a water flow over a hemispherical cylinder. The flow can undergo cavitation if the pressure difference (cavitation number, K = 2(pm - pvap)/pmU&) is low enough. Figures 17.3 and 17.4 correspond to cavitating water flow over a hemisphere/cylinder geometry. Two flow models are used. The first is the thermodynamic equilibrium model based on the Sanchez-Lacombe state equation [21]. The second employs additional equation for describing the transport of a phase property, such as volume or mass fraction, thereby allowing non-equilibrium effects. In [15], the equation employed is

spy, +--dpY,u at

dX

- 0,

(17.5)

where Y, is the mass fraction of vapor. Figure 17.3 compares density contours of both models. As shown, both result in a sharp capturing of the cavitation bubble interface but differ in their predictions of the collapse of the cavity in the ”wake” region. Figure 17.4 shows that the finite rate model, if calibrated carefully, can yield predictions superior to the equilibrium model.

17.3.2 Two-fluid Model The basic idea of this model is that each phase is described by its own set of mass, momentum, and energy equations and the interactions between phases are handled through source terms. It can be further divided into one-pressureand two-pressure models, depending on whether pressure is postulated to be identical for both phases. The one-pressure assumption also implies that certain physical phenomena, such as gravity, surface tension, viscosity, etc. are neglected. Considering the fluids to be compressible, immiscible and inter-penetrating to each other, the multifluid, single-pressure model for multiphase flow consists of six equations for a 1D system: (17.6) where the vector quantities are

0

pauH (17.7) where subscripts “1” and “g” denote the liquid and gas (vapor) phases, respectively.

374

AUSM+-up scheme for multaphase pow

X

Figure 17.3: Density contours: liquid water flow over a hemisphere/cylinder geometry, taken from [15]. Note that pressure is common for both phases in the single-pressure model. More physics (flow structures) can be included through the source terms, but it is beyond the scope of this paper to consider this issue. Nevertheless, they must satisfy the internal consistency constraint, if the source terms are solely based on the inter-phasic transfers of momenta and energy, i.e.,

sz + s, = 0

(17.8)

It is instructive t o point out that the six equations in Eqs. (17.6) and (17.7) contain 10 unknowns, five for each phase ( a ,p , u,e , p ) , and there are four additional auxiliary equations including two equations of state, one constraint a1 ag = I , and the one pressure assumption pi = pg = p. Hence, the algebraic system for the unknowns is closed. This system of equations, Eqs. (17.6) and (17.7), presents two numerical difficulties: (1)the source terms are in non-conservative form and (2) the system is non-hyperbolic in general. The source terms if not treated carefully can give rise to serious numerical difficulties, due to the strong stiffness resulting from these terms, manifested by disparate time and length scales. The numerical consequences may be that discontinuities propagate at wrong speeds or afflicting with oscillations. As a result of non-hyperbolicity, an initial-valued problem

+

375

M.-S. Liou and C.-H. Chang

1

0.8

0 A

0.6

--- - -

_--

=0.4 0 0.2

K=0.4 (equilibrium) K=0.3 (equlibrium) Kz0.2 (equilibrium) K=0.4, Rouse and McNown data K=0.3, Rouse and McNown data Kz0.2, Rouse and McNown data - K=0.4 (finite rate) - K=0.3 (finite rate) - Kz0.2 (finite rate)

-

c

0-

-

-0.2

-0.4-

I

I

I

I

I

I

1

I

I

l

i

l

2

l

l

I

I

1

3

/

1

1

I

4

I

I

I

I

I

I

I

5

sld Figure 17.4: Surface pressure distributions: liquid water flow over a hemisphere/cylinder geometry, taken from [15]. becomes ill-posed in the sense of Hadamard [6], which means that the solutions do not depend continuously on the initial data. Several attempts to remove the non-hyperbolicity difficulty have been proposed in the past by adding regularization terms to the model, for example, the interfacial pressure correction term [l],the two-pressure model [20] or the virtual mass term [22]. Even so, an implicit operator or additional numerical dissipation was still necessary to make the calculation stable. As a result, excessive smearing was usually found in the solution. A weak formulation for the multifluid model has been developed [24] based on the Roe-type approximation. But the weak solution is not unique in the sense that it is dependent on the choice of path for constructing the dissipation matrix. In addition, the eigensystem of the multifluid model is very complicated and it is difficult to find the analytical form of the associated eigenmatrix. As a result, it is cumbersome or complicated, especially when the equation of state (EOS) is complicated, to use the Roe-type or Godunov schemes to calculate the numerical fluxes under the two-fluid system.

376

A U S M " - u p scheme for multiphase flow

Gas

Liquid

Gas

_A _ _ - - Liquid

Figure 17.5: Illustration of the inter-phasic terms between different phases. For dealing with the non-conservative terms in the multifluid model, all previous studies assumed that transfers of momentum and energy between phases only took place within the same cell. That is, only the inter-phasic terms marked by A in Fig. 17.5 are considered and they will cancel with each other. This is correct when the void fraction function is continuous. However, when a contact discontinuity exists in the flow field, we find that the inter-phasic term between neighboring cells (marked by B in Fig. 17.5) must be taken into consideration. Hence, in what follows we propose to employ the stratified fluid model to clarify issues requiring in multiphase flow calculations additional attentions absent in single phase flow.

17.3.3 Multiphase Stratified Fluid Model The stratified flow model was introduced in [2] to describe each discretized cell in which each fluid is confined within a separated control volume. The interfaces between the like and unlike phases are defined on the control surface. Consequently, the numerical flux on the interface between different phases, either within the same cell or between neighboring cells can be taken into account. The stratified flow model has been applied to several one dimensional cases in our early work [2], including the air-water shock tube problem, Ransom's faucet problem and the phase separation problem, etc. In this paper, we will extend the stratified flow model to solve two dimensional flows. The stratified model, by defining different fluids in separate regions, is very amenable to the practice of finite-volume method since the conservative law can be applied to each region of fluid respectively. Hence, we shall use it to construct discrete equations governing flow motions of each respective fluid. In addition, although the derivation of Eq. (17.6) is based on the assumption that the void fraction function aiis continuous, we shall make it more general and allow the void fraction function to be piecewise continuous, thus recognizing an inter-phasic interface at the cell boundary. We begin our approach by defining the control volume of each phase according to the stratified flow model. Figure 17.6 illustrates the continuum concept

377

M . 3 . Liou and C.-H. Chang

Fluid 2

_ - - - -- - - - _ _ _ Fluid 1

Figure 17.6: Illustration of one dimensional stratified flow, the continuum concept. of one dimensional stratified flow reduced from the two dimensional flow. Two different fluids, separated by the void fraction function ai, are considered to flow between two parallel plates. The flow is assumed to depend only on the longitudinal variable x and time t. We also assume ai to be a piecewise continuous function for the case of contact discontinuity. From a general discrete version of the stratified flow model, we can define the control volume of gas a c g h within mesh cell, as illustrated in Fig. 17.7. The dashed line in the figure represents the reconstructed function of ai. Three different types of the interfaces can be found in Fig. 17.7, including liquid-liquid (cd and and gas-liquid interfaces (bc, the gas-gas (ab and 3), cg and fs). Based on this illustration, we can define the effective length of each type of interface at the cell boundaries, j f 1/2 as the following.

g),

&g-g

= min ( ( @ g ) L , ( a g ) R )

6,-i

= max

(0, -Aag)

61-,

= max

(0, -Aa,) = rnax (0, Aa,)

(17.9)

where

(17.10)

A(.) = ( ' ) R - (')L and

hg-,

+

6l-l

+ 6,-1 + 61-,

=1

(17.11)

with the mutual-exclusive property 6g-l

.61-,

=0

(17.12)

We have 6,-1 represents the effective length of the gas-liquid interface with gas fluid on its left side and liquid on the right side. The definition for 61-, is similar, but with liquid on its left side and gas on its right side.

378

A USM+-up scheme for multiphase flow

i

j-112

j+1/2

Figure 17.7: Illustration of one dimensional stratified flow, the discrete concept. The governing equation for each fluid within the control volume can be written as

J aipi dV + $(aipi?)

.?ids = 0

& J aipiv'dV + $(aipi??)

. ?ids

+$

rip 6 d S = 0

(17.13)

& J ~ ~ p i E ~ d V + ~ & J a i d V + $ ( a i p. i i~d S i ~=) 0 where the integrations are taken over and are the entire control volume V and its associated control surfaces S containing both fluids. Then, Eq. (17.13) can be discretized and written in one dimension as

where 6t is the time difference operator, and &(.) = (.)n+l - (.)". The source term, Oil is the inter-phasic transfer term, such as that involving p*, and is added to the above equation for numerical purposes. The fluxes at cell boundaries, represent the summation of convection and pressure fluxes, C F : and F[, respectively. They are written as

CFf

=

(FC). z 3+1/2 - (Ff)j-1/2

(17.15) (17.16)

where

M . 3 . Liou and C.-H. Chang

379

It is noted that (F:)j+l12 corresponds to the pressure force exchanged between neighboring cell through the cell boundary, and (F:)j represents the pressure force exchanged between different fluids within the same cell. We will discuss these terms separately in the following sections.

17.3.4 Convection fluxes, Ffjhtl,2 As shown in Fig. 17.7, the gas fluid can only flow into cell j through the gas-gas and gas-liquid interface on the cell boundaries, and no gas or water will flow through the inter-phasic interface g . The convection fluxes across the gas-gas interfaces (2and 8)can be given by the AUSM+-up scheme [13, 21 as 6g-g

[ ( a g ) 1 / 2 ( M ~ ) 1 / 2 ( P ~ ) L-k / R( D ~ ) g ] ( $ g ) L / R

(17.19)

and the “L/R” state is chosen by the following rule:

where $ = (1,u,

(17.20) and ‘gL/R =

{

&L’

QgR,

if [ ... ] in Eq. (17.19) > 0, otherwise.

(17.21)

The cell interface Mach number M I / , and the pressure diffusion term Dp along with other functions specific to the AUSM-type schemes will be give in Appendix for completeness. However, the interested reader can find other details concerning these schemes in [11, 131. The convection fluxes of gas on the inter-phasic interfaces (bc and are determined by the position and velocity of the gas to the interface. When gas is on the left side of the inter-phasic interface with 6,-1 # 0, the gas will flows across the interface if ( u g )>~0. Therefore the convection flux on the interface is 69-1 max(0, ( U g ) L ) ( & ) L (17.22)

fs)

Similarly, when gas is on the right side of the inter-phasic interface, the convection flux is (17.23) 61-9 min (0, ( u g ) R ) ( $ g ) R Note that we have assumed the gas and liquid fluid to be inter-penetrating to each other. Therefore, we can neglect the effect of liquid fluid when we consider the convection flux of gas fluid. The summation of Eqs. (17.19), (17.22) and (17.23) gives the convection flux of gas on the cell boundary. The convection flux for the liquid can be derived similarly. Then we can get the general form of the convection flux F f as:

Ft

=

62-2

+ 62-2‘ + 621-i

[(.2)1/2(M2)1/2(Pi)k

ma@, ( 4 L ) (1Cli)L d n ( 0 , ( u i ) ~( $)2 ) ~

+ (OP,i]

($i)lC

(17.24)

380

AUSM+-up scheme for multiphase flow

where a' = 1 if i = g, and 'i = g if i = 1.

17.3.5 Pressure fluxes, F:j*l,2 and F f j On the cell boundary, the pressure force acts on the control volume of gas through the gas-gas and gas-liquid interfaces. Consider the cell boundary between cells j and ( j l),the pressure flux on the gas-gas interface ($) can be given by the AUSM+-up scheme [13, 21 as:

+

(F:-g)j+i/2

= (hg-g)j+i/2[P&,)

((Mg)L)

PL

+ PG)( ( M g ) R ) PR + (DUlg]3+1/2 .

(17.25) Notice that there is always a gas-gas interface on the control surface of gas. But only when an inter-phasic interface is in contact with the control volume (dg-l > 0), the pressure on the inter-phasic interface can be imposed on it. Therefore for cell j , we define the pressure flux on the inter-phasic interface as: (F:-l)j+l/2

= (bg-l)j+l/2

[P&)(Mg-l)pL

+ PG)(Mg-l)PR]3+1/2 ,

(17.26)

(17.27) Again, the functions P* and D, are given in the Appendix. As a first attempt, we just took a simple approach by applying the same parameter M g - l , evaluated by a simple averaging, to both the splitting functions P+ and P-. Then the pressure flux ( F , P ) j + l / 2 for cell j will be the summation of Eqs. (17.25) and (17.26). Notice that the pressure flux of Eq. (17.26) explicitly exchanges the pressure force between the gas in cell j and the liquid in cell ( j 1). The pressure flux ( F f ) j - 1 / 2 for cell j can be derived similarly, in fact it is exactly the reverse of that at j 1/2. Then for cell j , we can define the pressure fluxes in a general form as:

+

+

and

M.-S. Liou and C.-H. Chang

381

It can be shown that for the case of a moving contact discontinuity with a constant pressure and velocity. The pressure force applied on each fluid will be automatically balanced, yielding that C Ff = 0. Therefore, only the convection fluxes will be left in Eq. (17.14), ensuring to capture the contact discontinuity exactly by our method.

17.3.6 The interfacial pressure correction term The original multifluid model of Eq. (17.6) (p* = 0 ) is ill-posed. Additional interfacial pressure correction term p* is required to make it well-posed. In the present work, the interfacial pressure correction term proposed by Bestion [l] will be adopt. We have (17.31) where Qg and Qz are the states of fluids, and 0 is a positive constant. Equation (17.6) will become well-posed if the pressure correction term p" is included with 0 2 1. Referring to Eq. (17.6), the interfacial pressure correction term is imposed on the inter-phasic interfaces. So we follow this idea and apply p* to all the inter-phasic interfaces on the control surface. We can write Oi in a general form as: oi = (O,OZ, U20i)T (17.32) where

17.3.7 Time integration We use the four-step Runge-Kutta method for the time integration. The time discretization of Equation (17.14) can be written as:

=

(Wi,M i , &i)T (17.34)

where the superscripts m and s are the index for the present and next time step, and w is the parameter constant for each sub-step of the Runge-Kutta method. To calculate the primitive variables of the next time step, the equations of state

382

A USMf-upscheme for multiphase flow

(EOS) of fluids need to be employed to the above equation in order to decode the variables. For the problems studied in this paper, the following EOS are used to represent th gas and liquid phases. It turns out that the particular liquid EOS, called stiffened gas model, contains ideal gas EOS as a special case. The stiffened gas model [8]is expressed as

(17.35) el

=

( y j+ P, PI

and the parameters for the water [17] are yl = 2.8,

J (Cp)zM 4186 -

p , = 8.5 x lo8 Pa

kg . K '

(17.36)

Note that pa is zero for an ideal gas and is a very large constant for the liquid fluid. It makes the coupling between the pressure and density field very weak for the liquid and the variation in density is extremely insignificant even a very large pressure gradient is imposed on the flow. On the other hand, a small oscillation in density field will result in a violent jump in the pressure. As a result of this EOS, a quadratic equation is obtained for the pressure, (p')'

+ ( ( - A + B + ("19 - l ) p m ) p"

-

AC

+

- l ) B p m= 0

(17.37)

where

(17.38)

c

=

rzP,+(yz

-

l)pm

The pressure p" is the positive root of the above equation. Then the other primitive variables can be derived easily,

& - 1Ms2

a;

=

9

2

w, , a ; = 1 - a ;

It is noted that the coefficients in Eq. (17.37) are extremely large, as p , for liquid is included. The ratio between the largest and smallest coefficient

383

M . 3 . Liou and C.-H. Chang

Initial condition

Transition state

Steady state

Figure 17.8: Illustration of the water faucet problem. may be as large as 1016 in some of our test cases. Large numerical error may appear in the results even we use double precision in the calculation. Therefore, an additional Newton iteration method is used t o improve the accuracy of p s . The Newton iteration method is standard and will not be discussed here. In general, two or three iterations are enough to drive the numerical error of p s under lop5. However we can still find that the numerical error of other variables are amplified when ai + 0. Therefore, a lower limit of a( about lop6 to lo-' is usually applied in our simulation.

17.4

Calculated Examples and Discussion

17.4.1 Ransom's faucet problem Referring to Fig. 17.8, the faucet problem introduced by Ransom [19] consists of air and a water jet within a channel. In the beginning, the water column surrounded by the air is moving with a constant speed 10 m/sec and is exerted by the gravitational force, accelerating the water downward (gravity effect on air is negligible). The width of the water column is getting narrower as the water is accelerated. As shown in Fig. 17.8, the void fraction wave is moving toward the outlet, and the flow will become steady when its wavefront moves out of the computational domain. The inflow boundary conditions are specified as follows. The void fraction of air ag = 0.2, the velocity of both air and water ug = u1 = 10 m/s, the

A USM+-up scheme for multiphase flow

384 0.5

,

,

,

,

1

,

,

,

Analytical, t

0.45 1 -.-iB-.t = 0.1s

I -

,

1

,

,

,

,

1

,

,

,

,

0.5s

-

--la-- t=0.2s -..B.*t 0.3s

I I I

-

I \

0.150'

I

'

I

2

'

'

' '

I

4

' ' ' '

I

6

'

I

I

I

8

X Ransom's water faucet problem, time = 0.5 sec, N=500

~=2.0,Dp=0.0,Du=0.0,CFL=0.5,RK-4

Figure 17.9: Time evolution of the void fraction profile in the water faucet problem. temperature Tg = TZ = 300" K, and the pressure is extrapolated from the interior point. In the outflow boundary, we set the pressure p = lo5 Pa, and all of the other primitive variables are extrapolated from the flow inside. Figure 17.9 shows the time evolution of the void fraction profile for the faucet ~ K., = 0, as there is no sharp pressure water problem. We set F = 2.0, and I C = gradient in the flow. The Courant number CFL of 0.5 and a mesh of 500 cells are used for the computation. The analytical solution is derived by assuming that the water is incompressible and the effect of air and pressure variation may be neglected. [17] The analytical solution of void fraction is given below. The computation result is very close to the analytical solution. There is no obvious oscillation in the profile.

Figure 17.10 demonstrates how the interfacial pressure correction term p* affects the multifluid model. Different values of 0 are applied on the Faucet

M . 3 . Liou and C.-H. Chang '

1

1

,

~

385 ,

'

0.55 - -Analytical - --m-- o=o oz2.0 0.5 - -.*--..- .-..5.0 -

1

,

,

,

,

,

1

-+--o=lO.O (T=

'

,

1

'I II 'I II

,

,

'

-

Irn II

Figure 17.10: The void fraction profiles based on different a in the water faucet problem. problem. When a = 0, the multifluid model is ill-posed mathematically. A large overshoot in the void fraction is seen at the wave front in the figure. The implementation of the interfacial pressure correction term (a > 1) can effectively eliminate the overshoot. However, the void fraction profile is smeared as a increases, indicating that additional numerical dissipation is introduced by the interfacial pressure correction term.

17.4.2 Air-water shock tube problem The second case involves a more critical initial condition in which essentially pure air and water are located separately on either side of the diaphragm and a very large pressure difference is imposed on the flow. We have ( p ,ag, u i , T i )= ~ (10' Pa, 1 - E , 0 m/s, 308.15 KO) (p, as,ui,T ~ )= R (lo5 Pa, E , 0 m/s, 308.15 KO)

(17.41)

where E = 1.0 x lop7. The result is presented in Fig. 17.11, with CFL = 0.5, a = 2 and K~ = K, = 1.

386

AUSM+-up scheme for multaphase flow

As we have mentioned in the previous section, the numerical error will be amplified when ai -+ 0. The Auid states of the disappearing phase can be neglected since the phase only occupies a vanishing portion of the fluid. Under this situation, the profile of velocity and temperature of fluid can become erroneous when ai -+ E . However, it would not influence the averaged (mixture) quantities of the flow, because its effect is so insignificant. This is seen in Figure 17.11 where we present the averaged profiles weighted by the void fraction. We can see both the results on coarse and fine meshes match very well. Notice the pressure/void waves near the transition region disappear in this case since its corresponding eigenvalue approaches to ui when ai -+ 0. As a result, these waves coincide with the contact discontinuity. The contact discontinuity and the shock wave are captured sharply by the present method, with no visible oscillations.

17.4.3

Shock-bubble interaction problem

The interaction of a moving underwater shock with an air bubble is studied. The initial condition is basically same as the case used by Hankin [7]. An air bubble (diameter 6.0 mm) is immersed in the water with its center at the origin. The incoming shock is initially located at z = -4.0mm. The fluid states before the shock are p = 1.013250x 105Pa, u = O.Om/s and T = 292.98K1and the fluid states behind the shock are p = 1.6 x lOgPa,u = 661.81m/sl and T = 595.14K. The Mach number of the shock wave is M = 1.509. The simulation is computed on a mesh of about 149,000 cells. The time evolution of the simulation result is presented in Fig. 17.12. We observe that, after the water shock wave hits the bubble, a strong reflected rarefaction wave is developed and a relatively weak shock is transmitted into sec). The strength of the shock in the air bubble the air bubble (t = 0.6 3 . 0 , ~ is relatively weak (no more than 0.1% the strength of the incoming shock), making it difficult to identify the pressure contours within the bubble. However, the shock wave can be clearly seen in the Mach number contours which is not shown here. A water jet generated by the rarefaction wave continuously pushes the bubble into a crescent shape. The bubble finally breaks up and the water ~ The collision jet collides with the still water ahead of the bubble (t = 3 . 6 sec). generates several shock waves propagating in all directions radially. At the same time, the separated air bubbles are compressed into a very small volume ~ due to the extremely high pressure (O(3 x 108Pa)) imposed on it (t = 4 . 8 sec). Afterwards, the shock wave begins to disperse outward and its strength reduces, resulting in an expanding of the air baubles due to the relief of pressure from the surrounding water. This case clear demonstrates the capability of our method in capturing a complex interface and shock wave system, without serious smearing found in the result.

-

M . 3 . Liou and C.-H. Chang

387

l.lE+09

-N = 5 W

1E+09 9E+W 8E+08 7E+08 6E+O8

n'

5E+08 4E+08 3E+O8 PE+08 1E+08

0

-1c+oe~

'

'

'

'

'

X

'

'

'

'

10 4

X 400

+-

350

8-

+ F

300

a" 250

3

'

3w

0

"

'

I

"

'

'

'

"

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;'

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;'

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;'

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(a,p,u,T),:(a,p,u,T),=(1-~,2.0e7,0,308.15):(~,1.0e7,0,308.15),~=1.OxlO~~ cs= 2.0,D, = 1.O,D, = 1.O,CFL = 0.5,lter = 400, N = 500

Figure 17.11: Profiles of states for the air-water shock tube problem. The (1.0 x lo9, 1- E , 0,308.15), ( p , a g ,ui,T ~ ) = R initial condition is ( p , agrui,q )=~ (1.0 x 105,~,0,308.15), E = 1.0 x lo-' and N = 5000.

388

A USMf-up scheme for multaphase flow

Figure 17.12: Pressure and void fraction contours for the shock-bubble interaction problem.

M.-S. Liou and C.-H. Chang

17.4.4

389

Shock-water column interaction problem

In this case, we have a water column (diameter 6.4 mm) at the origin and an incoming air shock wave at the position z = -4.0mm initially. The initial condition is based on the case in the paper by R. Nourgaliev, et al. [3]. The fluid states before the shock are p = 1.0 x 105Pa, u = O.Om/sec, T = 347.0K, and after the shock are p = 2.3544 x 105Pa, u = 246.24m/sec, T = 451.2K. A mesh of about 186,000 cells is used in the simulation. Figure 17.13 shows the time evolution of the simulation result. We find that when the incident shock hits the water column, part of the shock wave is transmitted into the water and part of it is reflected. Since the incompressibility of the water is much larger than the air, the shock wave within the water column moves much faster than the shock wave in the air(t = 4.0psec). Also, the pressure wave in the air can be transmitted into the water easily, but it’s very difficult for the pressure wave to be transmitted from water to air, as in the previous case. As a result, the pressure wave transmitted into the water column is basically confined within the water and very little of it is “leaked” out. We find the waves bounce back and forth within the water quickly, and the pressure field of air is essentially not influenced by what has happened inside the water column, before the two branched incoming shock waves merge again (t 2 18.5psec).

17.5

Concluding Remarks

Extension of the AUSM-family schemes for calculations of multiphase flow at all speeds and at all states of compressibility has been described in this paper. We presented two approaches for describing multiphase flows. The first approach considers the multiphase fluid as a mixture described by a real fluid equation of state and applies thermodynamic equilibrium principles for treating liquidvapor phase transitions. This approach is simple, yet powerful, for naturally describing existence of phase change, co-existence of several phases. Additional transport equation can be included to account for nonequilibrium effects. The second approach is based on nonequilibrium model which solves each phase separately via transport equations, with pressure equilibrium between phases. The stratified flow model was proposed for the construction of numerical fluxes in which the inter-phasic effects were included at the cell-interface fluxes. The new AUSM+-up scheme was detailed and employed in both approaches. Numerical results showed that the modifications based on the new scheme were effective in simulating some rather numerically challenging problems involving liquid and vapor. The problems considered were numerically stiff because of a large disparity in density, velocity, and speed of sound, thus spanning small to large Mach numbers. The calculations were found to be robust and stable and the results were accurate in comparison with analytical, experimental and

390

AUSM+-up scheme for multaphase flow

Figure 17.13: Pressure and void fraction contours for the shock-water column interaction problem.

M.-S. Liou and C.-H. Chang

391

other published computational results. Phase interfaces and their evolutions were captured sharply, without resort to special treatment for the interfaces.

17.6

Acknowledgments

The first author would like to thank: (1) Jack R. Edwards of North Carolina State University for several enlightening collaborations, (2) Nam Dinh of University of California at Santa Barbara for his encouragement and inspiration on research of multiphase flow, and (3) Richard A. Blech of NASA Glenn Research Center for his management support on the subject presented herein.

17.7

Bibliography

[l] Berger, M. J. & Oliger, J. The physical closure laws in the cathare code.

Nuclear Engineering and Design, 124:229-245, 1990. [2] Chang, C.-H. & Liou, M . 3 . A new approach to the simulation of compressible multifluid flows with ausmf scheme. AIAA Paper 2003-4107, 2003.

[3] Dinh, N., Nourgaliev, R. & Theofanous, T. Direct numerical simulation of compressible multiphase flows: Interaction of shock waves with dispersed multimaterial media. Technical report, 5th International Conference on Multiphase Flow, 2004. [4] Edwards, J. R. A low-diffusion flux-splitting scheme for Navier-Stokes calculations. Computers & Fluids, 26:635-659, 1997. [5] Edwards, J. R., Franklin, R. K., & Liou, M . 3 . Low-diffusion flux-splitting methods for real fluid flows at all speeds. AIAA Journal, 38:1624-1633, 2000.

[6] Hadamard, J. Lectures on Cauchy's Problem in Linear Partial Diflerential Equations. Dover Publications, New York, 1952. [7] Hankin, R. K. S. The Euler equations for multiphase compressible flow in conservation form: Simulation of shock-bubble interactions. Journal of Computational Physics, 172:808-826, 2001. [8] Harlow, F. H. & Amsden, A. A. A numerical fluid dynamics calculation method for all flow speeds. Journal of Computational Physics, 8:197-213, 1971. [9] Ishii, M. Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles, Paris, 1975.

AUSM+-up scheme for multiphase flow

392

[lo] Kestin, J. A Course in Thermodynamics. Waltham, MA, 1966.

Blaisdell Publishing Co.,

[11] Liou, M.-S.

A sequel to AUSM: AUSM+. Journal of Computational Physics, 129:364-382, 1996. Also NASA TM 106524, March 1994.

[12] Liou, M.-S. Ten years in the making-ausm-family. AIAA Paper 20012521-CP, 15th AIAA CFD Conference, June 11-14 2001. 1131 Liou, M.-S. A further development of the ausm+ scheme towards robust and accurate solutions for all speeds. AIAA Paper 2003-4116-CP, 16th AIAA CFD Conference. June 23-26 2003.

[14] Liou, M.-S. & Edwards, J. R. Ausm schemes and extensions for low Mach and multiphase flows. VKI Lecture Series 1999-03, VKI, Belgium, 1999. [15] Liou, M.-S. & Edwards, J. R. Ausm-family schemes for multiphase flows at all speeds. In M. M. Hafez, editor, Numerical Simulations of Incompressible Flows, pages 517-543. World Scientific, 2003. [16] Liou, M.-S. & Steffen, C. J., Jr. A new flux splitting scheme. Journal of Computational Physics, 107:23-39, 1993. Also NASA TM 104404, May 1991. [17] Pailkre, H., Core, C., & Garcia, J. On the extension of the AUSM+ scheme to compressible two-fluid models. Computers and Fluids, 32:891-916, 2003. [18] Peng, D.-Y. & Robinson, D. C. A new two-constant equation of state. Ind. Eng. Chem. Fundam., 15(1):59-64, 1976. [19] Ransom, V. H. Numerical benchmark tests. In G. F. Hewitt, J. M. Delhay, and N. Zuber, editors, Multiphase Science and Technology, Vol. 3. Hemisphere Publishing, Washington, DC, 1987. [20] Ransom, V. H. & Hicks, D. L. Hyperbolic two-pressure models for twophase flow. Journal of Computational Physics, 751498-504, 1988.

[all

Sanchez, I. C. & Lacombe, R. H. An elementary molecular theory of classical fluids: Pure fluids. Journal of Physical Chemistry, 80(21):23522362, 1976.

[22] Stadtke, H., Franchello, G., & Worth, B. Numerical simulation of multidimensional two-phase flow based on flux vector splitting. Nuclear Engineering and Design, 17:199-213, 1997. [23] Stewart, H. B. & Wendroff, B. Two-phase flow: Models and methods. Journal of Computational Physics, 56:363-409, 1984.

M.-S. Liou and C.-H. Chang

393

[24] Toumi, I., Kumbaro, A., and Paillere, H. Approximate Riemann solvers and flux vector splitting schemes for two-phase flow. VKI Lecture Series 1999-03, 30th Computational Fluid Dynamics, von Karman Institute, Belgium, 1999. [25] Wada, Y. & and Liou, M.-S. An accurate and robust flux splitting scheme for shock and contact discontinuities. SIAM Journal on Scientific and Statistical Computing, 18:633-657, 1997.

[26] Wallis, G. B. One-Dimensional Two-Phase Flow. McGraw-Hill, New York, 1969.

Appendix 17-A Numerical Flux Formulas In this paper, we use the following definitions in the numerical flux formulas. Let

*

M f i ) W = $ ( M IMI), M $ ) ( M ) = &+(Mf 1 ) 2

(17.42)

Then, the interface Mach number for each phase is defined as

and the split pressure functions are defined as

Finally, the pressure and velocity diffusion terms used in the mass flux and pressure flux are given by

D,

= K,

AM max( 1 - M 2 ,0) (PL - P R ) . , a

K,

= 1.0

(17.46)

D,

= K,

?;)(a) pG)(a)

K,

= 1.0

(17.47)

p a ( ' U ~- 'UR);

where

A M = MG)( M L )- M;) ( M L )- M(4)( M R )+ M G ) ( M R )

(17.48)

The parameters u, p and M are the simple averages of the values at the L and R states.

394

Chapter 18

A Finite-Volume Front-Tracking Method for Computations of Multiphase Flows in Complex Geometries Metin Muradoglul

18.1

Introduction

Simulation of multiphase flows is notoriouL-Jr difficult mainly due to the presence of deforming phase boundaries. A variety of numerical methods have been developed and successfully applied to a wide range of multifluid and multiphase flow problems[l9, 24, 25, 29, 341. In spite of this success, significant progress is still needed especially for accurate computations of multiphase flows involving strong interactions with complex solid boundaries. The most popular approaches to compute the multiphase flows are classified into four categories: The first category is the front capturing method such as the Volume-Of-Fluid (VOF)[10, 241 and the level-set[lg, 25, 281 methods. In these methods, the front is captured indirectly through the volume-fraction distribution (VOF) or the zero-level-set of the distance function. The constrained 'Department of Mechanical Engineering, Koc University, Rumelifeneri Yolu, Sariyer 34450 Istanbul, Turkey.

395

396

A Finite- Volume/Front- Packing method

interpolation profile (CIP) method of Yabe[34] also belongs to this category. The second category is the boundary-fitted grid methods in which a separate, boundary-fitted grid is used for each phase[33]. The third class is the Lagrangian methods with moving grids[ll]. The fourth approach used in the present study is the front-tracking method[29, 321. The front-tracking method developed by Unverdi and Tryggvason[32]is based on the one-field formulation of the NavierStokes equations and treating different phases as a single fluid with variable material properties. In this approach, a stationary Eulerian grid is used for the fluid flow and the interface is tracked explicitly by a separate Lagrangian grid. The immersed boundary method developed by Peskin[2l] is used to smoothly discretize the jumps in material properties and to treat the effects of surface tension. The front-tracking method combined with a finite-difference flow solver has been successfully applied to a wide range of multiphase flow problems but almost all in relatively simple geometries[29] except for Udaykumar et a1.[31]. The method has been reviewed recently by Tryggvason et al. [29]. It is of great importance to be able to accurately model strong interactions between bubbles/drops and the curved solid boundary in many engineering and scientific applications such as microfluidic systems[27], pore-scale multi-phase flow processes[l7, 181 and biological systems[7, 221. The front-tracking method has many advantages such as its simplicity and lack of numerical diffusion. However, its main disadvantage is probably the difficulty to maintain the communication between the Lagrangian marker points and Eulerian body-fitted curvilinear or unstructured grids. In the present study, a finite-volume/front-tracking method is developed to compute dispersed multiphase flows in complex geometries using body-fitted curvilinear grids. An efficient and robust tracking algorithm is developed for tracking the front marker points in body-fitted curvilinear grids. The tracking algorithm utilizes an auxiliary regular Cartesian grid and it can be easily extended to unstructured grids. The front-tracking methodology is extended to body-fitted curvilinear grids and is combined with a newly developed finite-volume method to facilitate accurate and efficient modeling of strong interactions between the phases and complex solid boundary. The finite-volume method is based on the concept of dual (or pseudo) time-stepping method and is developed for time-accurate computations of incompressible laminar flows. The method is implemented for computations of two dimensional (plane or axisymmetric) dispersed multiphase flows in complex geometries[l4, 151. The method is first tested for an oscillating drop and the results are compared with the analytical solutions. The method is also validated for the motion of the drops falling due to gravity in a straight channel studied earlier by Han and Tryggvason[8]. It is found that the present results are in very good agreement with the results obtained by Han and Tryggvason[8]. The method is then applied to compute the buoyancy driven motion of drops in constricted channels studied experimentally by Hemmat and Borhan[9]. Finally the planar two-dimensional

M. Muradoglu

397

version of the method is used to compute the chaotic mixing in a drop moving through a winding channel[l6]. In the next section, the governing equations are briefly reviewed and are transformed into an arbitrary curvilinear coordinate system. Then the finitevolume/front-tracking algorithm is described in Section 3. The results are presented and discussed in Section 4 and some conclusions are drawn in Section 5. An analysis for selection of artificial compressibility parameter is presented in the Appendix.

18.2

Mat hematical Formulation

The governing equations are briefly described in this section only for an axisymmetric flow but noting that the equations for a two dimensional planar flow can be expressed essentially in the same form. The incompressible flow equations for an axisymmetric flow can be written in the cylindrical coordinates in the vector form as

af d g af, ag, ag + + - = - f - + h, + f b , dt dr dz dr dz -

(18.1)

where

and

In Eqs. (18.1)-(18.3), r and z are the radial and axial coordinates and t is the physical time; p, p and p are the fluid density, the dynamic viscosity and pressure; w, and w, are the velocity components in r and z coordinate directions, respectively. The viscous stresses appearing in the viscous flux vectors are given by

The last term in Eq. (18.1) represents the body forces resulting from the buoyancy and surface tension and is given by fb =

-.(Po

- p)G -

s,

ralcnb(x - x f ) d s ,

(18.5)

398

A Finite- Volume/Front- Tracking method

where the first term represents the body force due to buoyancy with po and G being the density of ambient fluid and the gravitational acceleration, respectively. The second term in Eq. (18.5) represents the body force due to the surface tension, and S, xf,n, K, n, S and ds denote the Dirac delta function, the location of the front, the surface tension coefficient, the twice of the mean curvature, the outward unit normal vector on the interface, the surface area of the interface and the surface area element of the interface, respectively. In Eq. (18.1), the fluids are assumed to have constant properties so that the density and viscosity of a fluid particle remain constant, i.e., (18.6)

&+

= u.V. As can be seen in where the substantial derivative is defined as Eq. (18.1), the continuity equation is decoupled from the momentum equations since it does not have any time derivative term. In order to overcome this difficulty and to be able to use a time-marching solution algorithm, artificial time derivative terms are added to the flow equations in the form

where r is the pseudo time. The solution vector w, the incomplete identity matrix I1 and the preconditioning matrix rW1 are given by

W =

{ 7r~",

},

I1=

r'uz

[

0 0 0 0 i 0 1 , 0 0 1

r-1=

[ f :], 0

(18.8)

P

where p is the preconditioning parameter to be determined and it has dimensions of velocity. Note that the relation q = pI1w has been used in Eq. (18.7). Based on the analysis presented in the Appendix, the preconditioning parameter ,O is given by (18.9) where 'GO is a constant of order of unity, Uref and !are the velocity and length scales, respectively, and the Reynolds number Ree is defined as Reg = U,,f!/po. Equation (18.7) can be transformed into a general, curvilinear coordinate system

E

= E(r, z ) ,

77 = d r , z ) ,

(18.10)

and the resulting equations take the form

dphw a h F dhG r-1-dhw + 11+-+-=ar at at aV

dhF,

dhG, ++ h(h, + fb), at a7

(18.11)

399

M. Muradoglu Auxilary Uniform r CartesianGnd

r"urvilinear Grid "

Figure 18.1: Three types of grids used in the computations. The governing equations are solved on a fixed Eulerian curvilinear grid and the interface between different phases is represented by a Lagrangian grid consisting of connected marker points. An auxiliary uniform Cartesian grid is used to maintain communication between the curvilinear and Lagrangian grids. where h = r p ? - rqzC represents the Jacobian of the transformation. The vectors

hF = z,f-

r7g,

hG = -zcf

+ rtg,

(18.12)

and

represent the transformed inviscid and viscous flux vectors, respectively.

18.3

Numerical Method

Once the material properties and surface tension forces are determined, any standard time-marching algorithm can, in principle, be used to solve Eqs. (18.11) since these are in the same form as the usual continuum flow equations. Three types of grids used in the present method are sketched in Fig 18.1. A fixed curvilinear grid is used to solve the conservation equations (Eqs. (18.11)) while a Lagrangian grid of lower dimension is used to track the interface separating different phases. An auxiliary uniform Cartesian grid is used to maintain computationally efficient communication between the curvilinear and Lagrangian grids.

400

18.3.1

A Finite- Volume/Front-Tracking method

Integration of the Flow Equations

Following Caughey[5], a two-parameter family of numerical scheme to solve Eq. (18.11) can be written as

(18.14) where (. . .)P and (. . .)" denote the pseudo and physical time levels, respectively. The parameter cp governs the approximation to the physical time-derivative and 6 determines the level of implicitness of the method in pseudo time. Note that when a steady state is reached in pseudo time, we have wp + wn+l. Three combinations of cp and 0 are of particular interest and correspond to different approximations in the physical time as summarized in Table 18.1. In the present study, the three-point backward implicit method is used throughout. Note that the terms h, and f b are treated explicitly in the pseudo time in the present formulation although it is possible to include h, into the implicit operator. The increment AT represents the time step for sub-iteration while At represents the physical time step which is usually different. The correction AwP = wP+' - wp is computed according to

The sub-iteration to solve Eq. (18.15) is linearized as

(hF)p+l = (hF)p+ APAwP + O(Ar2), ( ~ L G ) ~ "= (hG)p BpAwp + O ( A r 2 ) , (hF,)p+l = (hF,)p + AEAw? + O(Ar2),

+

(hG,)p+l = (hG,)p + B[Aw:

+ O(A?)),

(18.16)

4 01

M. Muradoglu

cp

6'

0 0 1

1 1/2 1

Order of accuracy First order Second order Second order

Scheme Implicit Backward Euler Method Implicit Trapezoidal Method Three-Point Backward Implicit Method

Table 18.1: Physical time-integration schemes. where the inviscid and viscous Jacobian matrices are defined as

Ap={%-r' Bp={x)y; dhF

dhG

with WE = aE and wV = then be written as

dhF,

dhG, B:={bw,}

(18.17)

F.A linearized approximation to Eq. (18.15) can Aw'

-Rp, (18.18)

where

s

=

r-l

-+ p ( 2 + + 1 ,

AT

(18.19)

2At

and

dh(F - F,)

dh(G - G,) - hh,] drl dh(F - F,) dh(G - G , ) +

+

P

+ 8hh;l"

- h(h,

arl

+ fb)]

n.

(18.20)

=

In Eq. (18.19), ,&' is approximated as @'+' pn+l in the solution process. Equation (18.18) represents a linear system of equations that can be solved for the correction Awp by a variety of methods but, following Briley and McDonald[2] and Beam and Warming[l], it is factorized for computational efficiency a s

. .

(18.21) which can be solved in two steps using a block tridiagonal solver when the spatial derivatives are discretized by three-point approximations.

402

A Finite- Volume/Front-Tracking method

The spatial derivatives are approximated using a cell-centered finite-volume method that is equivalent to second order central differences on a uniform Cartesian grid and fourth order numerical dissipation terms similar to that of Caughey[4] are added to the right hand side of Eq. (18.21) to prevent the odd-even decoupling. Note that the numerical dissipation terms are treated explicitly in pseudo time. Since the accuracy in pseudo time is not of interest, in addition to the preconditioning method, a multigrid method similar to that of Caughey[5] and a local time-stepping method are used to further accelerate the convergence rate in pseudo time. A diagonalized version of the AD1 method similar to that of Pulliam and Chaussee[23] has also been implemented in which only the convective terms are treated implicitly and all other terms are treated explicitly in pseudo time. A plane two dimensional version of the diagonalized scheme is described by Muradoglu and Gokaltun[l4]. Since all the cases studied in the present paper are essentially in the Stokes’ limit, it is found that block diagonal version is more efficient than the diagonalized version. Therefore the block diagonal version of the AD1 method is used in the present study.

18.3.2

Front-Tracking Met hod

In the front-tracking method developed by Unverdi and Tryggvason[32], the interface between different phases are represented by a Lagrangian grid with connected marker points as shown in Fig. 18.1. The marker points can be considered as fluid particles moving with local flow velocity. In order to maintain communication between the Lagrangian and fixed curvilinear grids, it is necessary to determine the locations of the marker points in the curvilinear grid at every physical time step. Although it is a simple task to determine locations of the marker points in a uniform Cartesian mesh, it is substantially more difficult to track them in a general curvilinear or in an unstructured grid. To overcome this difficulty and to keep tracking computationally feasible, a new tracking algorithm has been recently developed and found to be very robust and computationally efficient[15]. The tracking algorithm utilizes an auxiliary uniform Cartesian grid as sketched in Fig. 18.1 and reduces the particle tracking on a curvilinear grid to a particle tracking on a regular Cartesian grid with a look-up table. It is emphasized here that the tracking algorithm is not restricted to the structured grids and can be easily adapted for unstructured grids. Since the particle tracking on a regular Cartesian grid is a trivial task, the new tracking algorithm is computationally efficient and makes the front-tracking method a feasible tool for computations of dispersed multiphase flows in complex geometries using curvilinear or unstructured grids. In addition, since the tracking algorithm is general, it can be used to track Lagrangian points in other methods such as the particle-based Mote Carlo method widely used in solving the probability density (PDF) model equations of turbulent reacting flows[l3]. The

M. Muradoglu

403

details of the tracking method can be found in Muradoglu and Kayaalp[l5]. Since the flow equations are solved on the curvilinear grid but the surface tension is computed on the front, it is necessary to convert the surface tension into a body force by an appropriate distribution function. This involves an approximation to the delta function on the curvilinear grid in a conservative manner. Let q5f be an interface quantity per unit surface area, it should be converted into the grid value q5g given per unit volume. To ensure that total value is conserved in the smoothing, we must have

(18.22) where Av is the volume of the grid cell. Following Tryggvason et a1.[29], this consistency condition is satisfied by writing (18.23) where 41 is a discrete approximation to the front value 4fl 4ij is an approximation to the grid value q5gl As is the area of front element 1 and wfj is the weight of grid point ij with respect to element 1. For consistency, the weights must satisfy = 1, (18.24)

CWZ ij

but can be selected in different ways[29]. In the present study, the weight for the grid point ij for smoothing from x p = ( r p ,zp) is defined as a tensor-product kernel in the form Wij(Xp) = K(TT)K(TZ), (18.25) where T, = Irp - rijl/rmax and T, = IzP - ZijI/zmax.Note that rmax and z,, are the maximum distance of the grid nodes on which the front quantity q5ij is to be distributed in T and z directions, respectively. The functional form of K used here is

$ - 8(1 - ? ) f 2 K(F) =

;(l - 3

1

0

3

if i 5 0.5 if 0.5 < i 5 1.0 otherwise,

(18.26)

which is symmetric about i = 0, and piecewise cubic with continuous first and second derivatives. The weights wij are normalized to satisfy the consistency condition given by Eq. (18.24). The weights wij are also used to interpolate grid values such as velocity field from the curvilinear grid onto the front points. The material properties such as density and viscosity are computed according to

(18.27)

4 04

A Finite- Volume/Front- Tracking method

where the subscripts o and d refer to the ambient and the drop fluids, respectively. The indicator function 4 is defined such that it is unity inside and zero outside of the drops and, following Tryggvason et al.[29],it is obtained by solving the Poisson equation

0’4

= vh.vh4,

(18.28)

where Vh is the discrete version of the gradient operator. The jump Vh4 is distributed on the neighboring grid cells using the Peskin distribution[29] and Eq. (18.28) is then solved on the uniform grid in the vicinity of each drop. After computing the indicator function on the uniform grid, it is interpolated onto the curvilinear grid using bilinear interpolations. In fact, it is possible to efficiently solve the Poisson equation on the curvilinear grid but the above procedure seems to be robust and produces sufficiently smooth solutions for the problems studied in the present work. The surface tension on each front element is computed following the procedure described by Tryggvason et al.[29]. The surface tension on a small front element can be computed as

SF,

= ]A,

rulcnds.

(18.29)

Using the definition of curvature of a two-dimensional line, i.e., Kn = ds/ds and accounting for the axisymmetry of the problem, Eq. (18.29) can be integrated to yield

bF,

= r 4 s z - S L ) - Asm,,

(18.30)

where e, denotes the unit vector in the radial direction. The tangent vector to the curve s is computed directly by a Lagrange polynomial fit through the end-points of each element and the end-points of adjacent element in the same way as described by Tryggvason et a1.[29].

18.3.3 The Overall Solution Procedure The finite-volume and front-tracking methods described above are combined as follows. In advancing solutions from physical time level n (t, = n . At) to level n 1, the locations of the marker points at the new time level n 1 are first predicted using an explicit Euler method, i.e.,

+

+

%+;’

= Xp”

+ Atvp”,

(18.31)

where X, and V, denote the position of front marker points and the velocity interpolated from the neighboring curvilinear grid points onto the front point

4 05

M. Muradoglu

X,, respectively. Then the material properties and surface tension are evaluated using the predicted front position as -n+l

,,n+l

= p(X,

);

-n+l

pn+l = p ( X ,

1;

-n+l

$+l

=f&

1.

(18.32)

+

The velocity and pressure fields at new physical time level n 1 are then computed by solving the flow equations (Eq. (18.11)) by the FV method for a single physical time step and finally the positions of the front points are corrected as At x;+l = x; + -(v; + v;+1). 2

(18.33)

After this step the material properties and the body forces are re-evaluated using the corrected front position. The method is second order accurate both in time and space. All terms except f b in Eq.(18.11) are treated implicitly in physical time so that the physical time is determined solely by the accuracy considerations and stability constraint mainly due to surface tension. Perfect reflection boundary conditions are used at the solid boundary for the front marker points, i.e., the front marker points crossing the solid boundary due to numerical error are reflected with respect to the inward normal vector back into the computational domain. If the front marker point is close to the boundary, the front properties are distributed onto curvilinear cells in a conservative manner, i.e., the weights are defined only for the cells within the computational domain and are normalized to satisfy the consistency conditions given by Eq. (18.24). The grid properties are interpolated onto the front point in a similar manner. The Lagrangian grid is initially uniform and is kept nearly uniform throughout the computations by deleting small elements and splitting the large elements in the same way as described by Tryggvason et al.[29]. The initial front element size is typically set to 0.75A1, where A1 is the minimum size of the curvilinear grid cell. During the simulation, in each physical time step, the elements that are smaller than 0.5Al are deleted and the elements that are larger than A2 are split in order to keep the Lagrangian grid nearly uniform and to prevent the formation of wiggles much smaller than the grid size.

18.4

Results and Discussion

The method is first validated for the classical problem of the oscillation of a drop immersed in another fluid. The oscillation is induced by an initial perturbation to the surface configuration of the drop and is damped due to viscous dissipation. The computational results are compared with the analytical solution in terms of the oscillation frequency and damping rate. It is also validated for the motion of a freely falling drop in a straight channel studied earlier by Han and Tryggvason[8] using a finite-difference/front-tracking (FD/FT) method. The method is then applied to compute the motion of freely rising drops due to

A Finite- Volume/Front- Tracking method

406

buoyancy in a continuously constricted channel studied experimentally by Hemmat and Borhan[9]. Finally the planar two-dimensional version of the method is used to compute the chaotic mixing in a drop moving through a winding channel[161.

18.4.1

Oscillating Drop

The first test case concerns with the classical problem of an oscillating drop. The oscillation is induced by initial perturbations to the drop surface configuration and is damped due to viscous damping. Consider a drop of liquid with perturbed radius in the form T = a

( :

i+-(i+3cos2e)

1

(18.34)

where a is the equivalent drop radius and is magnitude of the perturbation. Note that this form of deformation conserves the volume of the drop of radius a[6]. When the drop is perturbed] surface tension tries to draw the interface back into a spherical configuration causing the surface to oscillate. Lamb[l2] showed that the oscillation frequency is given by (18.35) with the period (18.36) where n is the mode of oscillation with n = 2 being the primary mode and 0 is the surface tension coefficient. Neglecting the effects of the ambient fluid viscosity, the viscous damping time constant is given by (18.37) where v is the kinematic viscosity of the drop Auid. Figure 18.2 shows the evolution of the deformation parameter y defined as the ratio of the drop radius in the axial direction to the drop radius in the radial direction. The initial perturbation is set to C = 0.05. The computational domain is 6a in the radial direction and extends to 12a in the axial direction] and is resolved by a 192x 384 Cartesian grid stretched in the axial and radial directions to better resolve the flow in the vicinity of the drop. Periodic boundary conditions are used in the axial direction and no-penetration, perfect slip boundary conditions are used in the radial direction. The density and kinematic viscosity of the drop fluid are 50 and 400 times larger than the density and kinematic viscosity of the ambient fluid, respectively. The time is nondimensionalized by the theoretical period for the lowest (n = 2) mode. The theoretical damping rate is also plotted in

4 07

M. Muradoglu

Figure 18.2: The deformation y versus nondimensional time for an oscillating drop immersed in the host-fluid with the perturbation coefficient = 0.05, pd/po = 50 and pd/po = 400. The theoretical rate of decay is shown by dashed lines below and above the oscillating curve.

<

Fig. 18.2. It can be seen in this figure that both the oscillation frequency and damping rate are predicted well with the present method. For example, the error between the computed and theoretical oscillation frequencies is found to be less than 1%at time t / r t h = 2.0 showing the accuracy of the present method for this standard test case.

18.4.2 Buoyancy-Driven Falling Drop in a Straight Channel The second test case concerns with buoyancy-driven falling drops in a straight channel studied earlier by Han and Tryggvason[8]. The physical problem and computational domain are sketched in Fig. 18.3. As can be seen in this figure, the ambient fluid completely fills the rigid cylinder and the drop that is denser than the ambient fluid accelerates downward due to gravitational body force. The problem is governed by four nondimensional parameters[8], namely the Eotvos number Eo (interchangeably called the Bond number, B o ) , the Ohne-

A Finite- Volume/Front- Tracking method

Ambient Fluid Po

dl

*bo

Z

Figure 18.3: Schematic illustration of the physical problem and computational domain for a buoyancy-driven falling drop in a straight channel. sorge number Ohd, the density and viscosity ratios defined as

I**

=

Pd

-

PO

(18.38)

where Ap = Pd - p, is the density difference between the drop and the ambient fluids, gz is gravitational acceleration and d is the initial drop diameter. The Ohnesorge number based on the ambient fluid is defined similarly as Oh, = The subscripts d and 0 denote the properties of the drop and ambient fluids, respectively. The nondimensional time is defined as

e.

t t* = -

IhG.

(18.39)

In all the computations presented in this section, the computational domain is 5d in radial direction and is 15d in the axial direction. No-slip boundary conditions

M. Muradoglu

409

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I I

I I

I

I

I

I

I I

I I

I I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I I I

I I

I I

I I

I

I

I

I

I

I

I I

I

EO= 12 (Present)

I

I

EO= 12 (FD/FT)

I I

I

I

EO = 24 (Present)

Figure 18.4: Evolution of drop for Eo = 12 (left plots) and Eo = 24 (right plots). The present results are plotted on the left in each group. The gap between two successive drops in each column represents the distance the drop travels at a fixed time interval and the last interface is plotted at t* = 39.6 and t* = 44.01 for Eo = 12 and Eo = 24 cases, respectively.

are applied on the cylinder walls and axisymmetry conditions are applied on the centerline. The drop centroid is initially located at (r,, z,) = (0,124. The computational domain is resolved by a 128 x 768 regular Cartesian grid. The grid is stretched in the radial direction to have more grid points close to the centerline. The Ohnesorge number, the density and viscosity ratios are kept constant at Ohd = 0.0466 (Oh, = 0.05), pd/po = 1.15 and pd/p, = 1 in all the results presented here. In Fig. 18.4, the evolution of the drop for Eo = 12 and E o = 24 is presented together with the results obtained with the finitedifference/front-tracking (FD/FT) method[8]. As can be seen in this figure, the present results are in a very good qualitative agreement with the results obtained with FD/FT method. To quantify the accuracy of the present method, the velocity is nondimensionalized by &Z and the nondimensional velocity of the drop centroid V, is plotted in Fig. 18.5a together with the FD/FT results. It is clearly seen in this figure that the present results are also in a very good quantitative agreement with those of the FD/FT method demonstrating the

410

A Finite- Volume/Front- Backing method

Figure 18.5: (a) Centroid velocity (b) percentage change in drop volume versus

t* for Eo = 12 and Eo = 24 cases. The dashed lines are the present results and solid lines are the FD/FT results. accuracy of the present method. Finally the percentage change in drop volume is plotted in Fig. 18.5b as function of nondimensional time both for Eo = 12 and Eo = 24 cases and are again compared with the results of the FD/FT method. This figure shows that the volume changes in the present and FD/FT methods are of the same order and the maximum volume change at the end of the simulation is smaller than 2.5% in the present method. The relatively large volume change in the present method compared to the FD/FT method may be attributed to the larger numerical error in the present results partly due to larger physical time steps and partly due to the interpolation and distribution algorithms.

18.4.3 Buoyancy-Driven Rising Drops in a Continuously Constricted Channel The previous test cases have confirmed the accuracy of the present method. This test case concerns with the buoyancy-driven motion of viscous drops through a vertical capillary with periodic corrugations studied experimentally by Hemmat and Borhan[S]. The computational setup is sketched in Fig. 18.6a. The capillary tube consists of a 26 cm long, periodically constricted cylindrical tube with 6 corrugations. The average internal radius of the tube is R = 0.5 cm, and the wavelength and amplitude of the corrugations are h = 4 cm and A = 0.07 cm, respectively. The suspending fluid is a diethylene glycol-glycerol mixture denoted by DEGG12. A variety of UCON oils are used as drop fluids. The properties of the drop and suspending fluids are summarized in Table 18.2 where the same label is used for each system as that used by Hemmat and Borhan[9]. A com-

M. Muradoglu

411

plete description of the experimental set up can be found in [9]. A portion of a coarse grid containing 8 x 416 grid cells is plotted in Fig. 18.6b to show the overall structure of the body-fitted grid used in the simulations. The average rise velocity of buoyant drops as well as the drop shapes are computed and the results are compared with the experimental data[9] for a range of the governing parameters, viz. the dimensionless drop size, K , defined as the ratio of the equivalent spherical drop radius to the average capillary radius, the ratio of the drop to the suspending fluid viscosities, A, the corresponding ratio of fluid densities y,and the Bond number B o = ApgZR2/a,representing the ratio of buoyancy to interfacial tension forces; A p and u denote the density difference and interfacial tension between the drop and suspending fluid, respectively, and gz is the gravitational acceleration. In all the results presented in this section, the drops are initially located at z = 1.5h in the ambient fluid that fully fills the cylindrical tube and is initially in the hydrostatic conditions. Symmetry boundary conditions are applied along the centerline and no-slip boundary conditions are used at top, bottom and lateral surfaces of the cylindrical tube. Drops are initially stationary and start rising due to buoyancy. The drops are initially spherical. The results are expressed in terms of non-dimensional quantities denoted by superscript "*". The dimensionless coordinates are defined as z* = z / h and r* = r / R . Time and velocity are made dimensionless with Tref= and

Kef = "

p y 2

, respectively.

First a qualitative analysis of the shapes of the drops are shown in Fig. 1 8 . 6 ~ . In this figure, a sequence of images for the evolution of the shapes of viscous drops through constricted channel are plotted for the DEGG12 system with the nondimensional drop sizes K = 0.54, 0.78 and 0.92. The computations are performed on a 32 x 1664 grid, the physical time step is At* = 1.641 and the residuals are reduced by three orders of magnitude in each sub-iteration. As can be seen in these figures, when a large drop ( K > 0.7) reaches a constriction, its leading edge follows the capillary wall contour and squeezes through the throat. Once the leading meniscus clears the throat, its rise velocity increases as it enters the diverging cross-section while the trailing edge of the drop remains trapped behind the throat similar to the experimental observations[9]. The drop shapes, the velocity field and pressure contours in the vicinity of the drop are plotted in Fig. 18.7 for a DEGG12 drop with K = 0.92 while it passes through the throat

System DEGGl2

po (mPa.s)

pd

Po

Pd

CT

(mPa.s)

(kg/m3)

(kg/m3)

(N/m)

87

115

1160

966

0.0042

Table 18.2: Two-phase system used in the computations.

A Finite- Volume/Front- D-acking method

Ambient Fluid P,.IJ.O

h

Figure 18.6: (a) Schematic illustration of the computational setup for a buoyancy-driven rising drop in a constricted channel. (b) A portion of a coarse computational grid containing 8 x 416 cells. (c) Snapshots of buoyant drops of DEGG12 system for drops sizes K = 0.54, 0.78 and 0.92 from left to right, respectively. The gap between two successive drops in each column represents the distance the drop travels at a fixed time interval and the last interface is plotted from left to right at t* = 2831.3, 3693.0 and 5416.4, respectively.

and just after the throat to better show the overall quality of the solution. Finally the vertical drop tip location scaled by the corrugation wavelength and the drop tip rise velocity scaled by the reference velocity Vref are plotted against the non-dimensional time in Fig. 18.8 for the DEGG12 system for various drop sizes. The retardation effect of the constrictions is clearly seen in these figures for large drops, i.e., K. > 0.7. It is also seen that the drops quickly accelerate and reach a periodic motion in all the cases.

M. Muradoglu

413

Figure 18.7: Velocity vectors (right portion) and pressure contours (left portion) in the vicinity of the DEGG12 drop with 6 = 0.92 while it passes through (a) the throat and (b) the expansion regions. Grid: 32 x 1664, At* = 1.641.

18.4.4 Chaotic Mixing in a Drop Moving through a Winding Channel The final test case concerns with computational modeling of the chaotic mixing in a drop moving through a winding channel in a planar two-dimensional setting[l6]. The mixing inside droplets by chaotic advection has been used to perform kinetic measurements with high temporal resolution and low consumption of samples[3, 261. The purpose of this test problem is to show the ability of the method to compute dispersed multiphase flows in complex geometries involving strong interactions between the deforming drop phase and the solid walls. The channel geometry is sketched in Fig. 18.9 and a complete description of the problem can be found in Muradoglu and Stone[l6]. The computational domain is resolved by a body-fitted grid containing 1024 x 64 grid cells. The volume flow rate is specified at the inlet based on a fully developed channel flow and the pressure is fixed at the exit. The flow is initialized as a single phase steady flow using the ambient fluid properties and a cylindrical drop is then places in the ambient flow as shown in the sketch. The molecular mixing is ignored and only the mixing by chaotic advection is considered. The mixing patterns are visualized by passive tracer particles which are initially distributed on random uniformly inside the drop. The particles that initially occupy the lower half of the drop are identified as “red” while the rest are “blue”. The

A Finite-Volume/Front-Tracking method

414

0 t

(4

500

1000 1500 2000 2500 3000

3500

4000

t

(b)

Figure 18.8: The non-dimensional vertical positions (left plot) and the nondimensional rise velocities (right plot) of the drop tip plotted against the nondimensional time t* for the drops of DEGG12 system with K. = 0.54,0.78 and 0.92. Grid: 32 x 1664, At* = 1.641. tracer particles are advected in the same way as the front marker points. The governing non-dimensional parameters are identified as the Reynolds number number Re = poUidc/po,the capillary number Ca = p 0 U t / u , the viscosity ratio X = pd/po, the density ratio y = pd/po, the ratio of the drop size to the channel inlet height C = d d / d c where po and Pd are the ambient and drop phase fluid densities, po and ,ud are the ambient and fluid viscosities, respectively. Based on the inlet velocity and the corrugation wave length, the nondimensional physical time is defined as t* = tUi/L. The snapshots of the mixing patterns taken at the nondimensional time 18.0 x lop3 and frames t" = 0,3.3 x lop3, 7.3 x lop3, 11.3 x 10F3,14.7 x 22.0 x are plotted in Fig. 18.10 to demonstrate the evolution of mixing patterns in the drop while it moves through the model winding channel. The nondimensional numbers are set to Ca = 0.025, Re = 6.6, C = 0.7576, and X = 1.0 for this case. The mixing patterns are enlarged in the top plots of Fig. 18.10 to better show the details of the mixing process. As can be clearly seen in this figure, the chaotic advection occurs in 2D drop as it moves through a winding channel and the mixing patterns qualitatively resemble the actual 3D mixing patterns[20].

18.5

Conclusions

A finite-volume/front-tracking (FV/FT) method has been developed for computations of dispersed multiphase flows in complex geometries. The method is based on the one-field formulation of the flow equations and treating the different phases as a single fluid with variable material properties. The flow equations

M. Muradoglu

415

Figure 18.9: The sketch of the channel used in the computations. The volume flow rate is specified at the inlet based on a fully developed channel flow and the pressure is fixed at the exit. The flow is initialized as a single phase steady flow using the ambient fluid properties and a cylindrical drop is then placed instantly in the ambient flow. are solved by a FV method on a body-fitted curvilinear grid and a separate Lagrangian grid is used to represent the interfaces between different phases. The FV method is based on the concept of dual-time stepping and time integration is done by a block diagonal alternating direction implicit (ADI) scheme. A novel tracking algorithm that utilizes an auxiliary uniform Cartesian grid is developed to track the interfaces on the curvilinear grid and is found to be robust and computationally efficient. The method is implemented to solve two-dimensional (plane or axisymmetric) dispersed multiphase flows and has been successfully applied to several test cases including the classical problem of a vibrating drop, buoyancy-driven rising drops in a continuously constricted channel and the chaotic mixing in a planar two-dimensional drop moving through a winding channel. It is found that the present method is a viable tool for accurate modeling of dispersed multiphase flows in complex geometries.

18.6

Bibliography

[l]Beam, R. M. & Warming, R. F. An implicit factored scheme for the com-

pressible Navier-Stokes equations, AIAA J. 16, 393 (1978). [2] Briley, W. R. & McDonald, H. Solution of the tree-dimensional compressible Navier-Stokes equations by an implicit technique, Lecture notes an physics 35, 105, New YorkVerlag (1974). [3] Bringer, M. R., Gerdts, C. J., Song, H., Tice, J. D., & Ismagilov, R. F. “Microfluidic systems for chemical kinetics that rely on chaotic mixing in droplets”, Phil. Trans. R. SOC.Lond. A , 362, 1087, (2004).

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A Finite- Volume/fiont- Tracking method

Figure 18.10: Snapshots of mixing patterns taken at the nondimensional time frames t* = 0,3.3 x lop3,7.3 x lop3,3.3 x 18 x and 22 x from left to right, respectively. The top plots are enlarged versions of the corresponding scatter plots shown in the channel (bottom plots). (Ca = 0.025, Re = 6.6, X = 1.0, C = 0.76, Re = 6.6, K = 0.33, Grid:1024 x 64.) [4] Caughey, D. A. Diagonal implicit multigrid algorithm for the Euler equations. AIAA J., 26, 841 (1988). [5] Caughey, D. A. Implicit multigrid computation of unsteady flows past cylinders of square cross-section, ComputersfYFluids30,940 (2001). [6] Che, J. H. Numerical simulation of multiphase flows: Electrohydrodynamics and solidification of droplets, Ph.D. Thesis, The University of Michigan, Ann Arbor, (1999). [7] Fauci, L. & Gueron, S. Eds. Computational modeling in biological fluid dynamics, Springer- Verlag, New York (2001). [8] Han, J. & Tryggvason, G. Secondary breakup of axisymmetric liquid drops: I. Acceleration by a constant body force, Phys. Fluids 11(12), 3650 (1999). [9] Hemmat, M. & Borhan, A. Buoyancy-driven motion of drops and bubbles in a periodically constricted capillary, Chem. Eng. Commun. 150, 363 (1996).

[lo] Hirt, C. W. & Nichols, B. D. Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys. 39, 201 (1981). [ll]Hirt, C. W., Amsden, A. A., and Cook, J. L. An arbitrary LagrangianEulerian computing method for all flow speeds, J . Comput. Phys. 135,203 (1997). [12] Lamb, H. Hydrodynamics, Dower Publishers, New York, (1932).

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[13] Muradoglu, M., Jenny, P., Pope, S. B., & Caughey, D. A. A consistent hybrid finite-volume/particle method for the PDF equations of turbulent reactive flows, J . Comput. Phys. 154(2), 342, (1999). [14] Muradoglu, M. & Gokaltun, S. Implicit multigrid computations of buoyant light drops through sinusoidal constrictions, ASME J. Applied Mech. 71, 1, (2004). [15] Muradoglu, M. & Kayaalp, A. D. “An Efficient Tracing Algorithm for Finite-Volume/Front-TrackingComputations of Dispersed Multiphase Flows in Complex Geometries,” J . Comput. Phys., to be submitted, (2004). [16] Muradoglu, M. & Stone, H. A. “Chaotic Mixing in a Plug Moving through a Winding Channel: A Computational Study” Phys. Fluids, to be submitted, (2004). [17] Olbricht, W. L. & Leal, L. G. The creeping motion of immiscible drops through a converging/diverging tube, J. Fluid Mech. 134,329 (1983). [18] Olbricht, W. L. Pore-scale prototypes of multiphase flow in porous media, Ann. Rev. Fluid Mech. 28, 187 (1996). [19] Osher, S. & Fedkiw, R. P. Level set methods: An overview, J. Comput. Phys. 169 ( 2 ) , 463 (2001). [20] Ottino, J. M. “The kinematics of mixing,” Cambridge, UK: Cambridge

Univ. Press, (1989).

[21] Peskin, C. Numerical analysis of blood flow in the heart, J. Comput. Phys. 25, 220 (1977). [22] Pozrikidis, C., Ed. Modeling and simulation of capsules and biological cells, Chapman & Hall/CRC (2003). [23] Pulliam, T. H. & Chaussee, D. S. A diagonal form of an implicit approximate factorization algorithm. J. Comput. Phys. 39,347 (1981). [24] Scardovelli, R. & Zaleski, S. Direct numerical simulation of free-surface and interfacial flow, Ann. Rev. Fluid Mech. 31,567 (1999). [25] Sethian, J. A. & Smereka, P. Level set methods for fluid interfaces, Ann.

Rev. Fluid Mech. 35, 341 (2003). [26] Song, H., Tice, J. D., & Ismagilov, R. F. “A microfluidic system for controlling reaction networks in time,” Angew. Chem. Int. Ed., 42,768, (2003).

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A Finite- Volume/Front- Tracking method

[27] Stone, H. A., Stroock, A. D., & Ajdary, A. Engineering flows in small devices: Microfluidics toward lab-on-a-chip, Ann. Rev. Fluid Mech. 36, 381 (2004). [28] Sussman, M., Smereka, P. & Osher, S. A level set approach for computing solutions to incompressible two-phase flows, J. Comput. Phys. 144, 146 (1999). [29] Tryggvason, G., Bunner, B., Esmaeeli, A,, Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S., & Jan, Y.-J. A Front-Tracking Method for the Computations of Multiphase Flow, J. Comput. Phys. 169 (2), 708 (2001).

[30] Turkel, E. Preconditioning methods for solving incompressible and low speed compressible flows, J. Comput. Phys. 72, 277 (1987). [31] Udaykumar, H. S., Kan, H.-C., Shyy, W. & Tran-Son-Tay, R. Multiphase dynamics in arbitrary geometries on fixed Cartesian grids, J. Gomput. Phys. 137, 366 (1997).

[32] Unverdi, S. O., & Tryggvason, G. A front-tracking method for viscous incompressible multiphase flows, J. Gomput. Phys. 100, 25 (1992).

[33] Ryskin, G. & Leal, L. G. Numerical simulation of free-boundary problems in fluid mechanics, Part 2. Buoyancy-driven motion of a gas bubble through a quiescent liquid, J. Fluid Mech. 148, 19 (1984). [34] Yabe, T., Xiao, F., & Utsumi, T. The constrained interpolation profile (CIP) method for multiphase analysis, J. Comput. Phys. 169(2), 556 (2001).

Appendix 18-A Optimal Artificial Compressibility in the Stokes Limit The artificial compressibility parameter /3 should be specified in such a way that it gives the best asymptotic convergence to a steady state in pseudo time. It is well known that ,O must be proportional to the velocity scale in convection dominated flows[30],i.e., if Re >> 1. An analysis is presented here to determine the optimal value of /3 in the Stokes’ limit. In the Stokes’ limit, the flow equations become

v.v vp-pv2v

0, = 0.

=

(18.40)

After adding the pseudo time derivative terms, Eqs. (18.40) become

P

dV x

+

vp-pv2v=o.

(18.41)

Then taking a divergence of the momentum equation yields d dT

p-p.

V)

+ v2p

-

p V ( 8 . V) = 0.

(18.42)

Substituting the continuity equation (18.43)

into Eq. (18.42), we obtain (18.44)

where v = p / p is the kinematic viscosity. With an assumption of periodic boundary conditions, the spatial Fourier transform of Eq. (18.44) is given by d2p + Y X 2 ;dfit ; + dr2

= 0,

(18.45)

where fi is the Fourier transform of p, x is the wave number vector and x2 = x.x. Equation (18.45) is in the same form as a mass-spring-damper system. Then looking for a solution in the form

fi = &ear, 419

(18.46)

A Finite- Volume/fiont- Tracking method

420

where ljo is a function of space only, and substituting Eq. (18.46) into Eq. (18.45) yields the following characteristic equation for a Q2

+ ux2a+ (px)2= 0,

(18.47)

which can be solved to give (18.48)

Since (xp)’ is always positive, the real parts of the both roots are negative, i.e., it is always forced to a steady state. However, the optimal damping is obtained if the terms in the square root are smaller or equal to zero, which is satisfied for

P2 2

p1 x ) 2 .

(18.49)

Now let the length scale l be specified as

l = x-1 max

(18.50)

where xmaxis the maximum wavenumber, then the artificial compressibility parameter can be specified as

a

2 1 p2 = 1 (./e) 2 = uref 4Rez ’

(18.51)

where Uref is a reference velocity and R e f = U,,p?/p is the Reynolds number based on Uref. Combining this expression with the optimal value of p in high Reynolds number case, the artificial compressibility parameter can be specified as

(18.52) where rcp is a constant of order of unity. Equation (18.52) gives a nearly optimal value for P in the entire range of Reynolds numbers although it may require to tune up the constant rcp for some cases for the best convergence.

Chapter 19

Computational Modeling of Turbulent Flames Stephen B. Pope'

19.1

Introduction

Turbulent combustion presents a formidable challenge to computational modeling. Depending on the fuel, of order 10, 50, or 1,000 chemical species may be involved; and the fuel reacts to form combustion products (and trace amounts of pollutant species) through a complex set of highly non-linear chemical reactions. This occurs in a turbulent flow containing a large range of length scales and time scales, which renders direct numerical simulations intractable for many decades to come. However, good progress is being made in statistical modeling approaches and in the associated computational algorithms. In this paper we review recent progress in PDF methods for turbulent reactive flows, focusing on the work at Cornell on non-premixed turbulent combustion. Following an overview of PDF methods, recent calculations of two flames are described in Section 19.2, and then the important issue of modeling turbulent mixing is discussed in Section 19.3. Modeling approaches to turbulent reactive flows [22][11]can be broadly categorized according to two attributes: first, how the flow and turbulence are represented; and, second, how the turbulence-chemistry interactions are modeled. The principal approaches to the flow and turbulence are [27]: Reynolds-averaged Navier-Stokes (RANS) turbulence modeling; large-eddy simulation (LES); and direct numerical simulation (DNS). At present, RANS is the dominant approach 'Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853-7501

421

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Turbulent Flames

used in applications, whereas LES is the focus of much research [30][23]. While DNS is a powerful research tool [37], its range of applicability is severely limited (by computer power), much more so for reactive flows than for inert flows. It is important to appreciate that the turbulence-chemistry interactions require modeling both in RANS and in LES [25]. The large-scale turbulent mOtions play the dominant role in the transport of momentum, heat and species, and consequently these are well-represented in LES by the resolved fields. But in reactive flows, especially combustion, the essential processes of molecular mixing and reaction occur on the smallest (sub-grid) scales, and therefore require statistical modeling in LES, as in RANS. This paper is concerned with PDF methods [24][10][11],i.e., approaches for modeling turbulence-chemistry interactions through the solution of a transport equation for the joint probability density function (PDF) of the fluid composition (and other variables). The primary advantages of PDF methods are: that they are generally applicable (as opposed to being confined to homogeneouslypremixed or two-stream non-premixed problems) ; the turbulent fluctuations of the fluid variables considered are completely represented through their joint PDF; and that arbitrarily complex and non-linear chemical reactions can be treated without approximation. The two most widely used PDF methods in the RANS setting are the composition PDF method, and the velocity-frequency-compositionmethod. In the former, a RANS turbulence model (e.g., k-E or Reynolds stress) is used, and the turbulent transport term in the PDF equation is modeled as gradient diffusion. In contrast, a complete closure is provided by the modeled transport equation for the joint PDF of velocity, composition and turbulence frequency [36]: separate mean momentum and turbulence-model equations are not required; and turbulent convective transport is in closed form, so that the gradient diffusion approximation is avoided. In the LES setting, the filtered density function (FDF) [25] represents the distribution of compositions (on all scales), and conceptually it represents the PDF conditional on the resolved flow field [ll]. The combined LES/FDF approach has been developed in recent years based on the composition FDF [13], and also on the velocity-composition FDF [31]. In the next section, recent PDF calculations of two non-premixed turbulent flames are reviewed in order to illustrate the ability of the method to represent accurately finite-rate turbulence-chemistry interactions. Then, in Section 19.3 we discuss the status of the modeling of molecular mixing, which is the principal modeling issue in PDF methods.

19.2

PDF Calculations of Turbulent Flames

In non-premixed turbulent flames, whether or not finite-rate chemical effects are significant depends on the Damkohler number, Da, i.e., the ratio of char-

423

S.B. Pope

acteristic mixing and reaction timescales. At high Da, simple models based on equilibrium chemistry or steady laminar flamelets can be successful (e.g., [3]). But as Da decreases, departures from equilibrium and flamelet behavior becomes pronounced, local extinction and eventually global extinction [2] occur. Several recent experiments have investigated such phenomena over a range of Da, achieved either by varying the jet velocities [2][8] or by varying the temperature and hence the reaction timescale. It is a good test and challenge for models to reproduce the observed behavior over the range of Da.

19.2.1 Piloted Jet Flames The Barlow & Frank [2] flames have, deservedly, received much attention. The value of these experiments lies in: the design of the burner; the quality of the diagnostics; and, the range of conditions covered. By varying the jet and pilot flame velocities a series of six flames (designated A , B , . . . , F ) of decreasing Da are produced. Flame D is turbulent with little local extinction; flame F exhibits significant local extinction and is quite close to global extinction; whereas flame E is in between. There have been many calculations of flame D using several different approaches, but relatively few calculations of the more challenging flames E and F[1]. Velocity-composition-frequencyjoint PDF calculations of the Barlow & Frank flames are described by Lindstedt et a1.[19], Xu & Pope [39] and by Tang, Xu & Pope [35]. In the latter works, the methane and N O chemistry is described by a 19-species augmented reduced mechanism [34], and the mixing by the EMST mixing model [32]. Detailed comparisons between the calculations and experimental data (including conditional means and PDFs) are given in [39][35]. Here we present results just of the “burning index” BI, which is an overall measure of finite-rate effects. The burning index can be based on different species and is locally defined: a value of 1 corresponds to complete combustion, and a value of 0 to complete extinction. Figure 19.1 shows the burning index based on COZ and CO as a function of axial distance for the three flames D , E and F . As may be seen, in general, the calculations represent accurately the minimum BI observed at 30 jet radii (due to local extinction) followed by the downstream recovery (due to reignition). Furthermore, the increase in local extinction between flames D, E and F is calculated accurately.

19.2.2

Lifted Jet Flame in a Vitiated Co-Flow

Cabra et al. [4] have studied experimentally the new configuration of a lifted flame formed by a Hz/N2 jet issuing into a vitiated co-flow (at around 1045K). It is hypothesized that the stabilization mechanism for this flame is substantially different from that of lifted flames in cold co-flows. Specifically, mixing between the fuel and vitiated co-flow near the jet exit leads to a hot, lean mixtures which subsequently autoignite, thus anchoring the flame.

Turbulent Flames

424

1.5

1.5

7

6 - 1

8 cd

0.5

M

0

0

50 x/R , I

(4

100

n ”

0

50

“/Ri

100

(b)

Figure 19.1: Burning index based on (a) COz and (b) CO against axial distance in flames D, E and F. Comparison of PDF calculations (lines) and experimental data (symbols).

In further experiments using this burner at the University of Sydney the temperature of the vitiated co-flow has been varied over a narrow range (1010K1045K) which results in lift-off heights between 45 jet diameters to 5 jet diameters (as the temperature is increased). There have been several PDF calculations of these flames [4][20] [la][5]. From the latter work, we show in Fig. 19.2 (a) the calculated lift-off height using two chemical mechanisms for hydrogen combustion: the Mueller mechanism [21]; and the Li mechanism [18] in which a few rates and enthalpies of formation are updated. It appears that the calculations with the Li mechanism are in excellent agreement with the experimental data. It is important to appreciate, however, that the experimental uncertainties in the temperature of the vitiated co-flow combined with the marked sensitivity of the lift-off height to this quantity results in experimental error bars that are larger than the differences between the calculations. What can be concluded is that both PDF calculations reproduce the experimental lift-off heights (within the error bars) and that they exhibit a sensitivity to the details of the chemical mechanism. Figure 19.2(b) shows the calculated lift-off heights using the Mueller mechanism and the three most widely used mixing models, namely: the interaction by exchange with the mean (IEM) model [38][9];the modified Curl (MC) model [7][14];and the Euclidean minimum spanning tree (EMST) model [32].As may be seen, for this flow, there is no great sensitivity to the choice of mixing model. This issue is discussed further in Section 19.3.

425

S.B. Pope

Commrison of dinerent mechanisms (EMST C W . 0 )

comparisonof dmerenl mixlng mDdels

Figure 19.2: Lift-off height versus co-flow temperature for the hydrogen jet flame in a vitiated co-flow (a) comparison of the Li and Mueller chemical mechanisms (b) comparison of MC, IEM and EMST mixing models.

19.3

Modelling of Turbulent Mixing

In turbulence research on inert flows, there have been numerous studies of scalar mixing in which the primary focus is on the mean, variance and derivative statistics of a single inert scalar (e.g., temperature excess). The issues of mixing in turbulent combustion are significantly more involved. Typically, there are of order 20 compositions; the effect of molecular mixing on the shape of the PDF is important (not just the decay rate of the variances and covariances); fluctuations in mixing rates are significant in effecting local extinction; and (especially in premixed combustion) reaction and mixing can be strongly coupled. The three most commonly used mixing models, IEM, MC and EMST, have known theoretical deficiencies, but at the same time, in some circumstances, they can yield quantitatively accurate results. A current objective of research in this area is to delineate the range of applicability of these different models. The calculations of the lifted flame described above illustrate a case in which all three mixing models yield similar results. In contrast Ren & Pope [29] studied a partially stirred reactor (PaSR) in which radically different behavior is observed. As an example, Fig. 19.3 shows scatter plots of temperature versus mixture fraction given by the three models for a case of hydrogen combustion. In addition to the chemistry, the reactor is characterized by the residence time T~,,, and the turbulent mixing time T ~ T ,For ~ ~small . values of T ~ ~ (e.g., ~ T ~ < 1/20) ~ the ~ PaSR T approximates ~ ~ a~PSR and the three mixing models yield essentially the same results. But for larger values of T,~~/T,,, the scalar variances become significant and, as observed in Fig. 19.3, the shapes of the

/

T

Turbulent Flames

426

m

I b)

.

.

.

I

Figure 19.3: Scatter plots of temperature versus mixture fraction in a PaSR using different mixing models (a) IEM (b) MC (c) EMST.

Figure 19.4: Mean temperature conditional on stoichiometric against residence time for a PaSR using different mixing models. joint PDFs predicted by the models can be quite different. For the same PaSR test case, Fig. 19.4 shows the conditional mean temperature (at stoichiometric) as a function of the residence time for fixed rmix/rre, = 0.35. As in a PSR, blow-out occurs at a critical value of rres,as indicated by the asterisks in Fig. 19.4. As may be seen, the three models have significantly different critical residence times with EMST being most resilient, and IEM being least. Similar conclusions were drawn by Subramaniam & Pope [33] in a significantly different test case. An interesting development in the context of turbulent mixing is the development of the multiple mapping conditioning (MMC) approach [17]. This can be considered to be a marriage between the conditional moment closure [16] and the amplitude mapping closure [6][26]. For a turbulent reactive flow involving n, species, it is hypothesized that (in the composition space) all compositions lie on a manifold of dimension n, < n,. (This hypothesis is also explored by

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Pope [28].) The manifold is parameterized by nr reference variables to which the mapping closure is applied. As a basic test of the mapping closure aspects of MMC, an analytic solution is obtained for the joint PDF of two scalars evolving in isotropic turbulence from a symmetric triple-delta-function initial condition. The shapes adopted during the evolution of the joint PDF are in excellent agreement with those obtained from DNS by Juneja & Pope [15]. Mixing models should remain an active area of research for some time, since they are a crucial element in PDF methods in both RANS and LES approaches, and current models have several well-appreciated shortcomings. There are also questions to be answered about the performance of the existing models. In the theory of non-premixed turbulent combustion, extinction and ignition events are associated with large and small values, respectively, of the scalar dissipation. How is it that these phenomena can be accurately calculated using existing mixture models which do not explicitly represent the distribution of scalar dissipation?

19.4

Acknowledgment

It is a great pleasure to dedicate this paper to David A. Caughey on the occasion of his sixtieth birthday. This work is supported by the Air Force Office of Scientific Research under Grant No. F-49620-00-1-0171,and by Department of Energy, grant number DE-FG02-90ER14128.

19.5

Bibliography

[ 11 Anon.,

International Computation of

Workshop Turbulent

on Measurement and Nonpremixed Flames. http://wuw.ca.sandia.gov/TNF/abstract.html,2004.

[2] Barlow, R. S. & Frank, J. H., Proc. Combust. Inst, 27:1087-1095, 1998. [3] Barlow, R. S., Smith, N. S. A., Chen, J.-Y., & Bilger, R. W., Combust.

Flame, 117:4-31, 1999. [4] Cabra, R., Myhrvold, T., Chen, J.-Y., Dibble, R. W., Karpetis, A. N., & Barlow, R. S., Proc. Combust. Inst, 29:1881-1888, 2002. [5] Cao, R., Pope, S. B., & Masri, A. R., (in preparation), 2004. [6] Chen, H., Chen, S., & Kraichnan, R. H., Phys. Rev. Lett., 63:2657-2660, 1989. [7] Curl, R. L., AIChE J., 9:175-181, 1963.

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[8] Dally, B. B., Flectcher, D. F., & Masri, A. R., Combust. Theory Modelling, 21193-219. 1998. [9] Dopazo, C. & O’Brien, E. E., Acta Astronaut., 1:1239-1266, 1974.

[lo] Dopazo, C., In P. A. Libby and F. A. Williams, editors, Turbulent Reacting Flows, chapter 7, pages 375-474. Academic Press, London, 1994. [ll]Fox, R. O., Computational models for turbulent reactive flows.Cambridge University Press, New York, 2003.

[la] Gordon, R., Masri, A. R., Pope, S. B., & Goldin, G. M.,

Proc. Combust.

Inst, 30:(submitted), 2004. [13] Jaberi, F. A., Colucci, P. J., James, S., Givi, P., & Pope, S. B., J . Fluid Mech., 401:85-121, 1999. [14] Janicka, J., Kolbe, W., & Kollmann, W., J. Non-Equilib. Thermodyn, 4~47-66, 1977. [15] Juneja, A. & Pope, S. B., Phys. Fluids, 8:2161-2184, 1996. [16] Klimenko, A. Y. & Bilger, R. W., Prog. Energy Combust. Sci., 25:595-687, 1999. [17] Klimenko, A. Y. & Pope, S. B., Phys. Fluids, 15:1907-1925, 2003. [18] Li, J., Zhao, Z., Kazakov, A., & Dryer, F. L., Technical report, Fall Technical Meeting of the Eastern States Section of the Combustion Institute, Penn State University, University Park, PA, 2003. [19] Lindstedt, R. P., Louloudi, S. A , , & VBos, E. M., Proc. Combust. Inst, 28:149-156, 2000. [20] Masri, A. R., Cao, R., Pope, S. B., & Goldin, G. M., Combust. Theory Modelling, 8:l-22, 2004. [21] Mueller, M. A., Kim, T. J., Yetter, R. A., & Dryer, F. L., Int. J. Chem. Kznet., 31:113-125, 1999. [22] Peters, N., Turbulent Combustion. Cambridge University Press, 2000. [23] Pitsch, H. & Steiner, H., Phys. Fluids, 12:2541-2554, 2000. [24] Pope, S. B., Prog. Energy Combust. Sci.,11:119-192, 1985. [25] Pope, S. B., Proc. Combust. Inst, 23:591-612, 1990. [26] Pope, S. B., Theor. Comput. Fluid Dyn., 2:255-270, 1991.

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[27] Pope, S. B., Turbulent Flows. Cambridge University Press, Cambridge, 2000. [28] Pope, S. B., Flow, Turbulence and Combustion, (in press), 2004. [29] Ren, Z. & Pope, S. B., Combust. Flame, 136:208-216, 2004. [30] Sankaran, V. & Menon, S., Proc. Combust. Inst, 28:203-209, 2000 [31] Sheikhi, M. R. H., Drozda, T. G., Givi, P., & Pope, S. B., Phys. Fluids, 15:2321-2337, 2003. [32] Subramaniam, S. & Pope, S. B., Combust. Flame, 115:487-514, 1998. [33] Subramaniam, S. & Pope, S. B., Combust. Flame, 117:732-754, 1999. [34] Sung, C. J., Law, C. K., & Chen, J.-Y., Proc. Combust. Inst, 27:295-304, 1998. [35] Tang, Q., Xu, J., & Pope, S. B., Proc. Combust. Inst, 28:133-139, 2000. [36] Van Slooten, P. R., Jayesh, & Pope, S. B., Phys. Fluids, 10:246-265, 1998. [37] Vervisch, L. & Poinsot, T., Ann. Rev. Fluid. Mech., 30:655-691, 1998. [38] Villermaux, J. & Devillon, J. C., In Proceedings of the 2nd International Symposium on Chemical Reaction Engineering, pages 1-13, New York, 1972. Elsevier. [39] Xu, J. & Pope, S. B., Combust. Flame, 123:281-307, 2000.

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Part V

Education

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Chapter 20

Educating the f i t ure: Impact of Pedagogical Reform in Aerodynamics David L. Darmofal'

20.1

Introduction

In recent years, engineering curriculum reform has received serious attention from industry, government, and academic groups as the need for change in engineering education has become a well-recognized problem. Within aerodynamics, the need for re-engineering the traditional curriculum is critical. Aerodynamics has been revolutionized by the development and maturation of computational methods. At the same time, educational research in the sciences has demonstrated that learning can be significantly improved using pedagogical methods that differ from the standard lecture approach. These factors cast significant doubt that the traditional aerodynamics curriculum and pedagogy remain the most effective education for the next generation of aerospace engineers. This paper describes a five-year effort to reform the undergraduate aerodynamics education at the Massachusetts Institute of Tecnology. The decision to pursue educational reform in aerodynamics was stimulated not only by the external forces mentioned above but also by personal experiences teaching the subject. In particular, we had found that our students had limited abilities to deal with aerodynamic problems that were different than the specific situations 'Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachussetts 02139.

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covered in the course. For example, the semester prior to modifying our curriculum, a final exam was developed to assess the students ability (1) to apply concepts to different situations than encountered during the semester, and (2) to integrate concepts and apply them in a more complex, open-ended problem, i.e. the type of problems they would face as practicing engineers. The student performance on the exam was very poor and neither ability was demonstrated. Although we thought our students were achieving a deep level of conceptual understanding through our teaching, they were not. As a result, in the final exam, we assessed skills that the students did not have a good opportunity to develop through the subject’s pedagogy. Since we felt strongly that conceptual understanding and an ability to integrate concepts to solve complex problems was a primary goal in our subject, we resolved to change our teaching. The reform of the curriculum largely focused on three issues, specifically: 1. The application of active learning to enhance conceptual understanding. 2. The integration of theoretical, experimental, and computational techniques into a modern aerodynamics curriculum. 3. The use of a semester-long aerodynamic design project to provide educational motivation and authentic learning experiences. The initial two years of this work was described by Darmofal et aZ.[3]. This paper describes the current pedagogy which has been refined since that initial report and, more importantly, includes a variety of data demonstrating the improvements resulting from the new pedagogy.

20.2

Course Overview

Aerodynamics (M.I.T. subject number 16.100) is one of a set of upperclass subjects that undergraduates have the option of using to complete their degree requirements. The course is offered once a year in the fall semester and during the past five years the enrollment has been approximately 40 students. Prior to this course, students have some exposure to basic fluid dynamics including conservation principles, potential flows, and some incompressible aerodynamics including thin airfoil theory and lifting line. The main objective of this subject is that students acquire the ability to formulate and apply appropriate aerodynamic models to estimate the forces on realistic three-dimensional configurations. The major topics covered are: 2D/3-D potential flows (incompressible to supersonic) including panel and vortex lattice methods; boundary layer theory including the effects of transition and turbulence; shock waves and expansions fans; 2-D/3-D Euler and Navier-Stokes computations including some basic turbulence modeling; and, wind tunnel testing.

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20.3

435

Conceptual Understanding and Active Learning

Within cognitive science, the constructivist model of learning has become popular as it explains a large body of experimental evidence on learning and problemsolving[l3, 111. The constructivist model of learning argues that individuals actively construct their knowledge through testing concepts on prior experience, applying these concepts to new situations, and integrating the concepts into prior knowledge. A difficult situation arises when new (presumably correct) concepts conflict with existing (presumably incorrect) concepts. Unless the learner has been given strong reasons to reject these misconceptions, these new concepts will be difficult to accommodate and learning is generally superficial and short-term. In the area of physics education, Halloun and Hestenes investigated the impact of initial knowledge on student performance in a first course in physics[8]. To do this, they developed a diagnostic test to assess a student’s knowledge of Newtonian concepts. The diagnostic was carefully designed to include misconceptions that students frequently possess from personal, everyday experience with motion. The results showed that initial knowledge (as assessed by a pretest using the diagnostic) was the dominant indicator of performance in the physics courses while factors such as the specific instructor, academic major, high school mathematics, gender, and age had no impact. Furthermore, the post-test performance on the diagnostic showed that the overall performance on the exam, while better than in the pre-test, was quite poor. Thus, while students could perform well on the usual course exams that determined their grades, the conventional instruction they received did little to alter their misconceptions about mechanics. The importance of pre-existing knowledge and the constructivist model of learning casts considerable doubt on traditional instruction. Traditional teaching uses a transmittal approach in which students are assumed to gain knowledge while passively listening to lectures giving rise to the analogy between students and blank slates. This style of teaching is in direct conflict with a constructivist view of learning as it does not account for the need for students to actively confront their misconceptions such that they may be replaced by a more advanced understanding. Thus, simply improving the quality of the presentation of concepts within a lecture will not result in a greater understanding, rather, the constructivist model of learning suggests that a more substantial change in pedagogy is required to address misconceptions. One strategy for strengthening conceptual learning is a set of pedagogical methods known as active learning. Bonwell & Eison define active learning as instructional strategies that involve “students in doing things and thinking about the things they are doing”[l]. By this definition, a traditional lecture in which students passively listen to presented material is not an active learning

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Educating the Future

strategy. Rather, active learning requires some type of student engagement of the material during class. Utilizing the physics exams developed by Halloun and Hestenes, Hake showed that active learning methods had a statisticallysignificant improvement on learning gains compared to traditional lecturing in a study of six-thousand students from a wide variety of college (and some high school) campuses[6]. Furthermore, educational research shows that this active learning can also increase confidence, enjoyment of a subject, and inter-personal skills[l2]. During the Fall 1999 semester, we began using peer instruction, an active learning approach developed in physics by Mazur[9, 21. In this approach, conceptual questions (referred to as ConcepTests by Mazur) are given to students in class with time for individual reflection. After a check to see how well students have understood a question, small group discussions may be held. In addition] the instructor will usually clarify misconceptions and lead students in further exploration of the concept often giving a mini-lecture. In a typical class, two-to-three concept questions are usually discussed. Several options exist for measuring the class understanding. In 16.100, we have found the use of a handheld personal remote to be very effective. The personal remotes have several advantages over hand-raising or flash cards including anonymity of student responses and the efficient generation of assessment data to analyze aggregate performance. The use of peer instruction in a set of sophomore aerospace engineering courses is also discussed by Hall et al.[7] To illustrate a typical concept question, consider the generation of lift. The generation of lift on an airfoil is filled with many misconceptions due to the (usually inaccurate) folklore regarding how airplanes fly and further complicated by the knowledge gained in previous courses. In discussing lift generation, a series of concept questions are used concentrating on understanding lift generation through momentum changes, streamline curvature, and reaction forces. The first question involves the impingement of a water jet on a cylinder as shown in Figure 20.1. Although many students believe the jet will cause the cylinder to be propelled away from the stream, in actuality] the object will rotate into the stream. A simple momentum balance leads directly to the connection between force (lift) generation and momentum change. When we use this question, we include an in-class demonstration that clearly demonstrates the cylinder being drawn into the stream. This question is then followed by a series of questions connecting the concept of flow turning to force generation] and extending the ideas to understand the loss of lift at stall when the airfoil no longer turns the flow as effectively. Our experience with concept questions has shown that the students must have some experience with the material prior to class. Otherwise] discussing concepts and misconceptions is nearly impossible since students are not likely to have encountered much of the material prior to the course. To address this, reading assignments and graded homework are given that are due prior to dis-

D. L. D a m o f a l

Water jet

A jet of water impinges a cylinder attached to a pendulum. Which way will the pendulum swing? 1. Into the water (clockwise) 2. Away from the water (counter-clockwise)

3. Not enough information

Figure 20.1: Concept question example

cussing the material in class. The use of pre-class homework is a significant shift from traditional engineering pedagogy in which homework is assigned and due only after discussing the material in class. Not only is the pre-class homework critical to the success of active learning in the classroom but it also encourages student self-learning. Furthermore, by scanning the homework assignments, student misconceptions and common difficulties can be detected immediately rather than only week(s) after discussing material. With the improved student preparation, the classroom becomes a significantly more active environment with increased faculty-to-student and student-to-student discussions on the subject’s concepts. In addition to changing our in-class pedagogy, we have also modified our exams from a written to an oral format. While written exams can only analyze the information that appears on paper, i.e. the final outputs of a student’s thought process, an oral exam is an active assessment which can provide greater insight into how students understand and relate concepts. Also, oral exams are adaptive to each student. If a student is stuck or has misunderstood a question, the faculty can help the individual. As opposed to a wasted assessment opportunity, the dynamic adaptivity of an oral exam raises the likelihood of an effective assessment. Finally, practicing engineers are faced daily with the realtime need to apply rational arguments based on fundamental principles. By using oral exams, this ability can be directly assessed.

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20.4

Integration of Theory, Computation, and Experiment

Our past aerodynamics curriculum had a significant focus on theoretical aerodynamics with some limited exposure to experimental and computational aerodynamics. However, as suggested by Murman & Rizzi[lO], “today’s aerodynamics engineer needs to be fluent in modern CFD methods and tools, and must know how to utilize them in conjunction with theory and experiment for aerodynamic analysis and design.” The difficulty is how best to integrate computational methods into the mainstream aerospace curriculum. We envision that the large majority of aerospace engineers will only have experience with the results of a CFD calculation, some engineers will be endusers of CFD, and a very small fraction will be involved in some aspect of CFD development. Thus, our general philosophy for integration of computational aerodynamics into our undergraduate course is that the underlying aerodynamic approximations embodied by a computational tool must be well understood by a modern aerodynamicist, however, the details of the numerical methods are less important. For example, we expect students to understand that a threedimensional, compressible Euler calculation can model shock waves but, being inviscid, is not appropriate when viscous effects might be critical. By contrast, we do not expect students to understand what a second-difference artificial dissipation operator is, or how flux-difference splitting differs from flux-vector splitting . Computational and experimental methods are integrated in the course through the use of a design project (described in more detail in Section 20.5). The project requires that the student teams develop validated aerodynamic models for the required operating conditions. To do this, students perform both computational simulations and wind tunnel tests. Furthermore, since the student teams are required to reduce and correct the raw wind tunnel data, they begin to appreciate how wind tunnel testing is as much of a model as purely theoretical or computational techniques. In the process, students quickly learn that neither computational methods nor experiments are capable of providing reliable predictions for all applications, and understanding both the agreement or lack thereof between simulations and experiment is a crucial role for an aerodynamicist.

20.5

Project-based Learning

Typically, aerodynamics and other advanced engineering topics are taught with a significant focus on theory but little opportunity to apply theory especially to problems that approach the complexity faced in the design of modern aircraft. As a result, students perceive they are learning material ’just-in-case’ they may need it later in their careers. In the project-based approach used in 16.100,

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the knowledge is immediately being applied. Another advantage of including a project-based approach is that it increases the richness of the pedagogical techniques in the course. This variety of learning experiences has been recognized as a key principle of effective teaching[4, 51. Over the past five years, two design projects have been developed: one based on a military fighter aircraft and another on a blended-wing body commercial transport aircraft. Both of these projects have an initial modeling phase in which student teams develop and validate aerodynamic models for a baseline configuration, followed by a design phase in which the models are used to improve the aerodynamic performance. A key feature of our design project implementation is weekly project work sessions. The goal of these project sessions is to provide a scheduled block of time in which the course staff (typically one faculty member and a teaching assistant) can interact with the teams as they begin to tackle the project. These two-hour sessions are held in a large electronic classroom with approximately 25 computers or roughly one computer for every two students. We have found that this ratio of computers-to-students is effective in promoting collaboration. At the beginning of the semester, these project sessions are often used to provide information to the students about the project, clarify requirements, and introduce the various computational tools and experimental facilities. However, later in the semester, the role of the staff tends more towards coaching and trouble-shooting. The student teams consist of approximately four students. Each team submits an interim and a final written report that is the basis for their grades. For the interim report, which is due roughly 2/3’s through the semester, the teams are required to fully describe all of the aerodynamic models they have developed including their validation studies. The final report focuses on using these validated models for design (in addition to correcting any errors found in the interim report). A best-practice that we have found for the design phase of the project is to require the teams to make a hypothesis on what design changes are likely to improve their ability to meet the design requirements based on their conceptual understanding of aerodynamic performance, prior to performing any re-designs. Then, the final design phase becomes a study of whether the proposed design modifications have the desired effects; if not, the students are required to explain why their initial hypothesis was incorrect.

20.5.1

Military Aircraft Design Project

During the summer of 1999, we contacted several industry and government representatives requesting a design project that could serve as the semester-long theme of our aerodynamics course later that fall. Lockheed Martin Aeronautics Company (LMAC) proposed a project based on a typical re-design scenario encountered in the military aircraft industry. Specifically, the student teams

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Educating the Future

were to develop aerodynamic models of an F-16-like wing-body geometry at several critical operating conditions and then use these aerodynamic models in a wing design trade study. The project was used for three semesters, Fall 1999, 2000, and 2001. This project includes flight regimes from low subsonic to supersonic speeds including some at high angles of attack. The performance metrics of interest were: 1. The take-off distance at sea level conditions assuming the angle of attack is limited to a maximum of 25 degrees to avoid the tail striking the ground. 2. Radius of action (i.e. range) for Mach 0.9 cruise at 10K ft 3. Dash time estimate from Mach 0.9 to Mach 1.2 at 10K ft

For the subsonic (i.e. take-off) regime, a 1/9th scale wind tunnel model was built and tested in the low-speed tunnel at M.I.T. At high speeds, experimental data was available from previous LMAC tests. The design phase of project focused on improving the take-off, cruise, and dash performance through introduction of leading and trailing-edge flaps, and variations in wing sweep and span.

20.5.2

Blended-Wing Body Design Project

For the Fall 2002 semester, a new design project was developed in collaboration with The Boeing Company based on the Boeing Blended-Wing Body aircraft design. The goal of this project was to redesign the baseline configuration to improve the static stability while minimizing drag and maintaining balance. Specifically, two flights conditions were considered: transonic cruise and lowspeed approach. In approach, leading and trailing edge devices were permitted to be active, while in cruise, the aircraft was required to be clean. As in the fighter aircraft project, low speed wind tunnel tests were performed to provide validation data for the aerodynamic models.

20.6

Results

Quantifying the impact of pedagogical change on learning is difficult. Our approach is to take data from a variety of sources and draw our conclusions from the aggregate. While any single source is suspect, taken together, the results are more conclusive.

20.6.1

Effectiveness of Pedagogy

During the past three years (Fall 2001-2003), the pedagogy as described above has remained nearly the same with only minor adjustments. The student ratings

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60 50 40

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50 30 a 20 10

Not Effective

Effective

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Lecture

Project

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a z

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Effective

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Figure 20.2: Comparison of student evaluations from 2000 and post-2000 (20012003) semesters for reading/homework, lecture, and project effectiveness

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of the effectiveness of the pedagogy (specifically, reading/homework, lecture, and project) are shown in Figure 20.2. For the post-2000 semesters, the majority of students rated all aspects of the pedagogy as very effective, though the project effectiveness is rated somewhat less highly than either the reading/homework and lecture (this observation on project effectiveness is also consistent with the student comments discussed in Section 20.6.3).

20.6.2

Impact of Pre-Class Homework

The use of challenging pre-class homeworks was found to significantly increase the effectiveness of the lectures. In the Fall 2000 semester, while the pedagogy was as described above, the pre-class homeworks were designed to encourage reading but did not require significant engagement of the material. As a result, the students were not sufficiently prepared for in-class active learning. In fact, the student feedback from the Fall 2000 semester led directly to the decision to increase the homework difficulty. The result of the increased homework difficulty is that the students found not only the reading/homework but also the lecture to be more effective. For example, as shown in Figure 20.2, the percent of students rating the reading/homework and lecture as very effective shows a statistically-significant increase from the 2000 to the post-2000 semesters. The use of challenging pre-class homework assignments also had a favorable effect on the student exam performance. During the Fall 2000 and 2003 semesters, a written final exam was given. Both final exams consisted of five questions of which three were identical. The remaining two questions were different but of similar difficulty. The three identical questions assessed different skills, specifically conceptual, synthesis, and quantitative abilities. The student performance on these three questions is shown in Figure 20.3. 0

0

0

Conceptual Question: This question focused on the prediction of lift and drag using different types of flow models (e.g. 2-D potential flow, 3-D potential flow, 3-D Euler, and 3-D Navier-Stokes). Students were given different drag polars and lift curves and asked to identify the model that was used to generate each. As can be seen in Figure 20.3, the performance on this question was nearly the same in both years. Synthesis Question: This question focused on modeling the aerodynamic forces on a refueling boom of a tanker, and required not only recognition of the important physical effects but also some ability to quantitatively model these effects. While the percentage of top scores is similar, the percentage of lowest score (i.e. 0-60%) improved from around 40% to less than 20% from 2000 to 2003. Quantitative Question: This question focused on the use of an integral boundary layer method to estimate boundary layer growth in a duct flow. The difference in these results shows a substantial improvement from 2000

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.

.

80-100

Synthesis question

70 60

70

Quantitativequestion

1I 2000

Figure 20.3: Comparison of exam performance from 2000 and 2003 for questions assessing conceptual, synthesis, and quantitative skills.

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to 2003. This improvement is not surprising, however, since in 2000 the students had a less opportunity to perform this type of more detailed, quantitative analysis. Thus, by effectively combining homework (or similar application activities) with concept-based active learning, students can achieve high-levels of performance in skills ranging from quantitative application to synthesis of multiple concepts.

20.6.3

Student Comments

In addition to the effectiveness ratings of the pedagogy, students were given open response questions that asked ‘What were the best parts of the course?’ as well as ‘How could the course be improved?’ In this section, we present some of these comments and summarize the main conclusions. The open-response questions show that students are often initially hesitant about pre-class homework, but by the end of the semester they recognize the benefits of this technique. Some of the comments include: Doing homework before the lectures is good ... makes actual learning in lectures possible. 0

Prof. Darmofal forces you to learn the subject material by assigning homework that he has not covered in lecture, therefore I have to force myself to read the text and go to ofice hours. W h e n he does go over in lecture after the Pset is due, I did absorb the material much better. The teaching methods are outstanding ... making us read before the p-set is good form.

0

I was initially opposed to the idea that I had to do reading & homework before we ever covered the subjects. Once I transitioned I realized that it made learning so much easier!!

I was skeptical at first of new techniques like [concept question], homework o n material that hasn’t been learned in lecture. I n the end, it worked out very well. This has been a course where I really felt like I got m y money’s worth.

These comments also reinforce the impact of pre-class homework on the effectiveness of the lectures. Another common theme in the open-response questions is the student satisfaction with the oral exams. Many students find the oral exam to be a more accurate representation of their understanding than more traditional written exams. In fact, several students have said that the oral exams were the best parts of the course. Of the 21 comments made about oral exams in the open response evaluations, 19 were favorable, only one was negative, and one suggested a modified implementation. Some typical comments are: The oral exam was a different learning assessment approach that I liked a lot. 0

T h e oral exams are a n excellent measure of understanding.

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44 5

Oral exams [are the best part of the subject], I think these gave a good opportunity to show what you understand. I really like the idea of the oral final. Even though it is scary, it really shows how much you know about the subject, better than any exam would.

One of the most challenging aspects of the new pedagogy has been the implementation of the team project. In the first place, the project has multiple facets (in particular the wind tunnel experiments and the computational simulations) that must be successfully managed. Furthermore, keeping ten or more teams of four students functioning effectively can be highly time-consuming for both the faculty and the students. The open-response questions for the past three years clearly show both the benefits as well as the difficulties of the team project. During this time, 31 positive comments were made about the project with only 2 negative comments; however, 29 students suggested the need to improve the implementation. Typical comments include: I n a project, you have to take what you learn und directly apply to something. This is more effective than a problem set because it is on a larger scale - while on a problem set you may only perform a calculation once, a project makes you do that many more times. You begin to understand why and when what you are doing is applicable on a much deeper, intimate level.

I think the team projects are really good. There are some kinks which need to be worked out and possibly explained sooner, but they really bring us to an understanding of what elements are necessary to incorporate theory into design. The projects were verg interesting. Learning how to use computational tools and seeing how all the theory and testing is used in conjunction to gain accurate results was very useful and enjoyable. M y group floundered for a while with the project. I n the end we got everythhing to come together, but it was tough to get through. I’m not sure that I would have wanted it any other way, now that I look back on it. I learn best when I struggle with material for a while, provided I have enough time to finally understand it. I had just enough time for the project.

The students have perceived the educational benefits of applying the material they are learning in class on a complex problem; furthermore, several students (including other comments not shown here) note that the project allowed them to better appreciate how theory and computation compliment experiments in aerodynamic design. However, the effectiveuse of projects remains a challenging issue t o address.

20.7

Outlook

In response to external and internal forces of reform, we have re-engineered our undergraduate aerodynamics curriculum. The key ingredients of the reformed pedagogy include:

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active learning combined with pre-class homework to increase conceptual understanding, oral exams to assess onceptual understanding, 0

a semester-long team project stressing the roles of experiment, theory, and computation in modern aerodynamic design.

The results as measured by student evaluation and performance demonstrate that improvements in pedagogical effectiveness and learning have been achieved. Furthermore, the critical role of adequate student preparation on the effectiveness of active learning was demonstrated. While students often expressed an initial hesistancy with respect to the less-traditional aspects of the pedagogy, they eventually found the methods to be highly effective.

20.8

Acknowledgements

This work over the past five years has involved significant interactions both internal and external to MIT. The initial re-design of the curriculum began as a collaboration with Prof. Earl1 Murman. The military aircraft project was developed with Lockheed Martin by Mike Love and Dennis Finley. The Blended Wing Body was developed with Bob Liebeck (a Boeing employee and a Professor of the Practice at MIT). During the Fall 2000 semester, 16.100 was co-taught with Professor Steve Ruffin who was on sabbatical from Georgia Tech. This work was partially supported by the National Science Foundation, the MIT Alumni Fund for Educational Innovation, and the MIT MacVicar’s Faculty Fellow program.

20.9

Bibliography

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[a] Crouch, C.H. & Mazur, E. Peer Instruction:

Ten Years of Experience and Results. American Journal of Physics, 69, 2001.

[3] Darmofal, D.L., Murman, E.M., & Love, M. Re-engineering Aerodynamics Education, AIAA Paper 2001-0870, January 2001. [4] Felder, R.M. & Silverman, L.K. Learning and Teaching Styles in Engineering Education. Engeering Education, 78(7), 1988. [5] Felder, R.M., Matters of Style. ASEE Prism, 6(4), 1996.

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[lo] Murman,

E.M. & Rizzi, A. Integration of CFD into Aerodynamics Education. Frontiers of Computational Fluid Dynamics - 2000. Editors: D.A. Caughey and M.M. Hafez. 2000.

[ll]National Research Council. How People Learn: Brain, Mind, Experience,

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