(v and w are the fluctuating radial and azimuthal velocities and < . > is the statistical average), could not be measured, and the energy dissipation could not be directly calculated. Therefore P&L neglected the pressure diffusion in the energy budget and obtained the dissipation profile as the closing balance, whereas HC&G estimated energy dissipation from measurements assuming isotropy or axisymmetry of small scales, and obtained the pressure diffusion as the remaining term. Considering these experimental weaknesses, numerical simulations may appear as one appropriate way to describe exhaustively the turbulence developing in jets, because they give access to all flow quantities. The limitations in this case are due to the computational resources, and might originate from the numerical methods and the turbulence models involved in the simulations. Direct Numerical Simulation (DNS) can be used for flows at low Reynolds numbers, as shown by Mansour et al. [5] for a turbulent channel flow. However, for flows at higher Reynolds numbers, typically those investigated experimentally, one must make use of Large Eddy Simulation (LES), where only the turbulent scales larger than the grid size are calculated. Dejoan & Leschziner [6] for example recently computed in this way the energy budget in a plane turbulent wall jet at a moderate Reynolds number. In the present work, a circular jet at Mach number M = uj /c0 = 0.9 (c0 is the sound speed in the ambient medium) and at Reynolds number ReD = 1.1 × 104 is simulated by LES. This Reynolds number corresponds exactly to the Reynolds number of the experimental jet of P&L. The LES is performed using numerical schemes with low dissipation and low dispersion [7], and is based on the use of selective filtering as subgrid modelling [8, 9, 10]. The computational domain includes 44 million nodes, and is sufficiently large to contain a part of the jet self-similarity region. The aim is to investigate the turbulence properties in this region by calculating, directly from the LES data, turbulence quantities such as the second- and third-order moments of velocity fluctuations, the pressure-velocity correlations, and all the terms in the kinetic energy budget. The present results are systematically compared to the P&L data, when those are available. This will allow us to assess the LES accuracy, and to discuss the assumptions made by P&L, as well as by HC&G, for evaluating the energy terms across the jets. Reliable informations on jet turbulence physics are also expected to be obtained.
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2 Simulation parameters An isothermal round jet at Mach and Reynolds numbers M = uj /c0 = 0.9 and ReD = 1.1 × 104 is simulated. The LES is performed on a Cartesian grid using numerical schemes with spectral-like properties [7], optimized in the wavenumber space to be low-dissipative and low-dispersive for waves discretized by more than four grid points. The spatial discretization is taken into account by explicit finite-differences and selective filtering, using both eleven-point stencils. The time advancement is carried out using a six-stage low-storage Runge-Kutta algorithm. The selective filtering is applied explicitly every second iteration to remove the grid-to-grid oscillations without affecting the resolved scales, but also to provide the effects of the subgrid energydissipating scales. The LES approach based on explicit filtering has been successfully used by different authors for various flows [8, 11, 12, 13]. It has been shown in particular that the filtering does not artificially decrease the effective flow Reynolds number unlike subgrid models based on eddy-viscosity [9]. The effects of the filtering have been also investigated from the energy budget [10]. The computational domain is discretized by a Cartesian grid of 651×261× 261 nodes, and extends up to 182 radii in the axial direction and to 33 radii in the transverse directions. As in our previous simulations [8], the boundary conditions are non-reflective. A sponge zone is used at the outflow, restricting the physical part of the domain to 150 radii downstream. This is illustrated by the vorticity snapshot of Figure 1, where the development of the flow in axial direction is also clearly visible. As for the jet initial conditions, mean velocity profiles are imposed at the jet inflow, while random velocity disturbances are added to seed the turbulence. The axial mesh spacing is Δx = r0 /4, and the transverse mesh spacing is Δy = Δy0 = r0 /8 near the centreline, but Δy = r0 /4 for y ≥ 7r0 . The time step is Δt = 0.85Δy0 /c0 . In the present study, all the terms in the energy budget are calculated explicitly from the LES data as in [14]. For their statistical convergence, the simulation time is necessarily important. The results presented in this paper are obtained from a simulation of 350000 time steps, leading to a physical time of T uj /D = 16700. The statistics are computed during the final 270000 time steps, or T uj /D = 12900. Note finally that the present LES required 22.4 Gb of memory and 2100 CPU hours using a Nec SX5.
3 Results The development of the jet flow towards self-similarity is investigated in figure 2 where centreline profiles of mean and turbulence quantities are represented. In the self-preserving jet, the centreline mean velocity uc and the jet half-width δ0.5 are indeed respectively given by uc /uj = B × D/(x − x0 ) and δ0.5 = A × (x − x0 ), where B and A are two constants, and x0 denotes a
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virtual origin. The self-similarity of the turbulent flow is also reached when the turbulence intensities display constant values on the jet axis. The variation along the jet axis of the mean axial velocity is shown in figure 2(a). For x ≥ 50r0 , the centreline velocity appears to decrease following the x−1 law, with a decay constant of B = 6.4. A fair agreement with the results of P&L is also observed, which, however, reflect a slightly lower decay constant of B = 6.06. For brevity, the evolution of the jet half-width is not presented here, but note that it is found to increase linearly for x ≥ 50r0 , at the spreading rate A = 0.086. For comparison, P&L obtained A = 0.096. The axial variations of turbulence intensities are shown in figure 2(b). Both the axial and the radial components appear to increase up to about /uc = 0.2 x = 100r0 , where self-similar values of urms /uc = 0.24 and vrms seem to be reached. These evolutions along the jet axis compare well with the data of P&L, which are also plotted in the figure and exhibit self-similar /uc = 0.19. values of urms /uc = 0.24 and vrms The present profiles support that, for the simulated jet, the self-similarity of the mean flow is observed for x ≥ 50r0 , but that the similarity of the turbulent flow is found farther downstream, for x ≥ 100r0 . This trend agrees with experimental results of Wygnanski & Fiedler [3] or of P&L. The profiles across the jet of the mean axial and radial velocities normalized with uc , obtained over the range 70r0 ≤ x ≤ 130r0 , are shown in figures 3(a) and (b). The LES mean axial velocity agrees very well with the self-similarity profile measured by P&L. A good similarity is also seen for the mean radial velocity profile predicted by the LES and the profile obtained by P&L from the mean axial velocity using the continuity equation. The negative values of the mean radial velocity for large distances from the axis indicate the entrainment of the surrounding fluid into the jet flow. The radial profiles of turbulence intensities, computed over the range 100r0 ≤ x ≤ 140r0 , are presented in figure 4. Both their shapes and their magnitudes compare well with the results of P&L in the self-preserving jet. The agreement is in particular very good for the axial component urms and for the Reynolds stress <−u v >.
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For the evaluation of the different terms in the energy budget, LES provides all turbulent quantities that are necessary, including those that are often not available from the experiments. Two examples are given in figure 5 with the pressure-velocity correlations and third-order moments of velocity. The radial profiles of pressure-velocity correlations and
, computed from the LES data, are shown in figure 5(a). These results are of interest since these correlations cannot be measured experimentally, and are therefore usually evaluated thanks to turbulence models. They are for instance estimated in HC&G from the energy dissipation curves assuming isotropic or axisymmetric turbulence. The third-order moments of velocity fluctuations < v 3 > and < v w2 > computed by LES are presented in figure 5(b). The profile of across
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/(ρc u3c ). (ρc and uc are the mean centerline density and velocity.) (b) Third-order moments of velocity fluctuations, /u3c : —— LES, o P&L, /u3c : – – – LES. The LES profiles are averaged over 100r0 ≤ x ≤ 140r0 .
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the jet is found to be very close to the P&L measurements. The profile of shows a similar shape, but has a magnitude which is about half of the magnitude of the previous moment. With this result in mind it may be useful mentioning that the assumption = is frequently made in calculations of the turbulent diffusion term in the energy budget using experimental data. The budget of the turbulent kinetic energy in the jet self-similarity region is now presented. All the terms of the budget, including those derived from the filtered compressible Navier-Stokes equations [10] and those due to the explicit selective filtering [14], are computed directly from the LES fields. The profiles across the jet of the main energy terms are shown in figure 6. These terms correspond to mean flow convection, production, dissipation, turbulent diffusion and pressure diffusion, the dissipation term being here the sum of the viscous dissipation and of the filtering dissipation. The LES results are compared to the experimental results of P&L for a jet at the same Reynolds number. There is an excellent agreement for the four curves associated with convection, production, dissipation and turbulent diffusion. The pressure-diffusion term calculated by LES, albeit not negligible, is found to be small with respect to the other energy terms. The P&L hypothesis that this term can be neglected in the evaluation of the energy budget then appears relevant. The present results finally cast doubt on the energy budget obtained experimentally by HC&G for a jet at a higher Reynolds number where, using various turbulence modellings, dissipation is found to be about twice as large as in P&L, and pressure diffusion is of the order of mean flow convection. 0.02 0.015 0.01 0.005 0 −0.005 −0.01 −0.015 −0.02
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4 Concluding remarks The first LES results of a round jet at Reynolds number ReD = 11000, that is being performed to investigate the self-similarity region of the flow, are presented in this paper. These results, including mean flow and turbulence properties, are shown to compare very well with the measurements of P&L for a jet at the same Reynolds number. The agreement is particularly striking for the turbulent kinetic energy budget, for which P&L used different assumptions. By providing all flow quantities, LES gives us an opportunity to evaluate the turbulent features, such as pressure-velocity correlations, that might not be available experimentally. The computational cost is however quite high, and the present simulation is to be continued to have fully converged triple correlations of velocity and energy terms. Further results will also deal with the component energy budgets.
Acknowledgments The authors gratefully acknowledge the Institut du D´eveloppement et des Ressources en Informatique Scientifique (IDRIS - CNRS) for providing computing time and for its technical assistance.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
Panchapakesan NR, Lumley JL (1993) J. Fluid Mech. 246:197–223 Sami S (1967) J. Fluid Mech. 29(1):81–92 Wygnanski I, Fiedler H (1969) J. Fluid Mech. 38(3):577–612 Hussein HJ, Capp SP, George WK (1994) J. Fluid Mech. 258:31–75 Mansour NN, Kim J, Moin P (1988) J. Fluid Mech. 194:15–44 Dejoan A, Leschziner MA (2005) Phys. Fluids A, 17(2):025102 Bogey C, Bailly C (2004) J. Comput. Phys. 194(1):194–214 Bogey C, Bailly C. (2005) Computers and Fluids (available online) Bogey C, Bailly C (2005) AIAA Journal 43(2):437–439 Bogey C, Bailly C (2005) In proceedings of Turbulence and Shear Flow Phenomena-4 (2):817–822 To appear in Int. J. Heat Fluid Flow (2006) Stolz S, Adams NA, Kleiser L (2001) Phys. Fluids 2001 13(4):997–1015 Mathew J, Lechner R, Foysi H, Sesterhenn J, Friedrich R (2003) Phys. Fluids 15(8):2279–2289 Rizzetta DP, Visbal MR, Blaisdell GA (2003) Int. Journal for Numerical Methods in Fluids 42(6):665–693 Bogey C, Bailly C (2003) In proceedings of Direct and Large-Eddy Simulation V 23–30
The dependence on the energy ratio of the shear-free interaction between two isotropic turbulences D. Tordella and M. Iovieno Dipartimento di Ingegneria Aeronautica e Spaziale, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy – [email protected], [email protected]
Summary In the absence of kinetic energy production, the influence of the initial conditions on turbulent transport can be characterized by the presence of an energy gradient or by the concurrency of an energy and a macroscale gradient. In this work, we present a similarity analysis that interpret a new result on the subject recently obtained by means of numerical experiments on shearless mixing (Tordella & Iovieno, 2005, a–b). In short, the absence of a macroscale gradient is not a sufficient condition for the setting of the asymptotic Gaussian state hypothesized by Veeravalli and Warhaft (1989), where, regardless of the existence of velocity variance distributions, turbulent transport is mainly diffusive and the intermittency is nearly zero up to moments of order four. In fact, we observed that the intermittency increases with the energy gradient, with a scaling exponent of about 0.29. The similarity analysis, which is in fair agreement with the previous experiments, is based on the use of the kinetic energy equation, which contains information concerning the third order moments of the velocity fluctuations. The analysis is based on two simplifying hypotheses: first, that the decays of the turbulences being mixed are almost nearly equal (as suggested by the experiments), second, that the pressure-velocity correlation is almost proportional to the convective transport associated to the fluctuations (Yoshizawa, 1982, 2002).
1 Numerical results on the energy mixing. Second and higher order velocity moments A few aspects of the interaction of different decaying homogeneous and isotropic turbulences in absence of mean shear are described in the laboratory experiments by Gilbert (1980) and Veeravalli and Warhaft (1989). In
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this particular flow configuration the two turbulences external to the mixing have the same mean velocity but different turbulent kinetic energy and have been experimentally investigated by means of turbulence generated by grids with different size, but with the same solidity. This produces two homogeneous turbulences with the same mean velocity but with different energy and scale. In this paper, we consider the influence on turbulent transport of initial conditions characterized by the presence of an energy gradient and the absence of a macroscale gradient, see fig.1(a)-(b) and fig.2. The simulations were carried out by means of either direct numerical or large eddy simulations. The direct numerical simulations here presented were carried out by means of a new technique for the parallel dealiased pseudospectral integration of the Navier-Stokes equations (Iovieno et al., 2001). The boundary conditions are periodic in all directions. Two computational domains, a (2π)3 cube with 1283 points, and a 4π(2π)2 parallelepiped with 256 × 1282 points, were used to obtain an estimate of the numerical accuracy. In the initial condition, the two turbulence fields are matched by using the Briggs et al.(1996) technique. The same numerical method was used to implement the large eddy simulations, which were carried out by using the Intrinsic Angular Momentum (IAM) subgrid scale model (Iovieno & Tordella, 2002). This model is based on the proportionality of the turbulent diffusivity to the intrinsic moment of momentum of the finite element of a fluid. The IAM correctly scales the eddy diffusivity νδ , with respect to both the filtering length and the dissipation rate, and introduces a differential equation – the intrinsic angular momentum
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equation – to follow the evolution of νδ . This is particularly advantageous in the case of nonequilibrium turbulence fields, since it adds a degree of freedom to the subgrid modelling. It is recalled that the basic definition of a longitudinal integral scale that permits a direct measure and does not depend on the flow global Reynolds number is ∞ 1 0 Rii (r, t)dr , (1) (t) = 3 i Rii (0, t) where Rii is the longitudinal velocity correlation, see Batchelor (1953). The integral length approximation deduced from the hypothesis of statistical equilibrium, i.e. = E 3/2 /, should be applied with caution whenever the Re does not allow the great divergence of scales to be obtained that the universal equilibrium theory requires. In fact, at the relatively low Reynolds numbers, typical of the current literature, /E 3/2 = f (Re).
(2)
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Function f is of order 1, but is not yet completely known. Simulations of homogeneous and isotropic turbulence in the periodic box and laboratory experience, see Batchelor and Towsend (1948) and the collection of experimental data in Sreenivasan (1998), show that, in the low Re number range, its value almost halfs when the Re quadruple. In this paper, we use definition (1), principally because it is not affected by the actual value of Re and because it evidences that the integral scale does not depend on the level of kinetic energy but on the spectral distribution of energy over the wave numbers. Furthermore, definition (1) implies that turbulences which have similar spectra, but a different overall kinetic energy, see fig.1 b, have the same spatial macroscale.
Fig. 2. Kinetic energy contours. Plots at 3 < t/τ < 4.
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The dependence of the turbulence mixings with a macroscale ratio L = 2 = 1 on the initial conditions has been considered and documented through 1 single-point statistics (Tordella & Iovieno, 2005, Iovieno & Tordella, 2002 – here and in the following subscript 1 and 2 refer to the high/low energy regions, respectively). It is seen that the statistical distributions of orders higher than the second highly depend on the initial values of the ratio of energy, E = E1 /E2 . If the energy ratio is far from unity, no Gaussian behaviour is observed up to order four. The asymptotic state for the shearless turbulence, where the velocity variance follows the form of an error function and the velocity fluctuations are Gaussian, which was attributed by Veeravalli and Warhaft(1989) to the L ≈ 1 type of mixing and, in particular, to Gilbert’s experiment (where, because of the very low energy gradient exploited, it was very difficult to show the weak, eventual removing of the velocity statistics from the Gaussian behaviour) was not observed. On the contrary, the mixing is very intermittent. If the lateral penetration is considered in terms of the position of the maximum of skewness and kurtosis distributions, it is observed, that, when L = 1, the intermittency increases with the energy ratio with a scaling exponent that is almost equal to 0.29, see fig.s 3(a,b). Independently of the values of the control parameters a set of common properties exists for all the studied mixings. First, the statistical distributions become selfsimilar after nearly a decay of three time units. Second, in the self-similarity region of the decay, the lateral spreading rate is on average close to 0.15. Third, the kinetic energy distribution has a common shape (see, (13) below). Fourth, all the mixings are very intermittent, as the skewness S and kurtosis K distributions show, see fig.s 3(a) and 3(b). 1.0 0.8 0.6
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2 Similarity analysis We analyze here the consequences of the observation that in all the numerical mixing experiments a self-similar state appears to exist. In the time interval where the near–similarity is reached (t/τ > 3), to carry out the similarity analysis, we considered the second moment equations for the velocity fluctuations (u, in the inhomogeneous direction x, v1 , v2 in the plane normal to x), ∂t u2 + ∂x u3 = −2ρ−1 ∂x pu + 2ρ−1 p∂x u − 2εu + ν∂x2 u2
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The two mixed turbulences decay in a similar way, as shown by the numerical simulations (Tordella & Iovieno, 2005). In the decay laws E1 (t) = A1 (t + t0 )−n1 , E2 (t) = A2 (t + t0 )−n2
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the exponents n1 , n2 are close each other, which assures the constancy of E with respect to the time variable. In this analysis, we suppose n1 = n2 = n = 1, a value which corresponds to Rλ 1 (Batchelor & Townsend, 1948). In the absence of energy production, the pressure-velocity correlation has been shown to be approximately proportional to the convective fluctuation transport (Yoshizawa, 1982, 2002) u3 + 2v12 u , a ≈ 0.10, (7) 2 moreover all experiments show no appreciable difference in the second order moments in the mixing, i.e. u2 vi2 , so that u3 − v12 u 2ρ−1 p∂x u and consequently 3a ≈ 0.25. (8) −ρ−1 pu = αu3 , α = 1 + 2a In the initial value problem here considered, the moment distributions are determined by the coordinates x, t, and by the energy E and the macroscale of the two mixing turbulences. Thus, through dimensional analysis −pu = aρ
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L = 1 (t)/ 2 (t). It should be noticed that, if n = 1, E, L, ϑ1 = n/f (Rλ1 ) and R1 ∝ t1−n are constant (Batchelor 1953). By inserting relation (9) in (3), it is possible to deduce that Δ(t) ∝ 1 (t). By putting Δ(t) = (t), one obtains: 1 ∂ϕuuu ν 1 ∂ϕuu ∂ 2 ϕuu + (1 − 2α) − = − η 2 ∂η f (Rλ1 ) ∂η Af (Rλ1 )2 ∂η 2 2 ϕε = ϕuu − f (Rλ1 ) u
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Given the lateral boundaries of the mixing, which correspond to homogeneous conditions for the turbulence, one can observe that the rhs of (10) must be an odd function of η. It is zero in homogeneous (equilibrium) turbulence, while the previously mentioned experiments (Tordella & Iovieno, 2005) suggest that this rhs could be modelled by means of a diffusive term, so that ∂ 2 ϕuu 2 ϕεu − ϕuu = β f (Rλ1 ) ∂η 2
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where β is a constant of proportionality; β = 0 corresponds to the hypothesis of local equilibrium. In the following, by simply writing f instead of f (Rλ1 ), 3/2 the skewness, S = ϕuuu /ϕuu , reads η −3 f ν ∂ϕuu ϕuu2 ∂ϕuu dη + − βf S= η (1 − 2α) 2 −∞ ∂η A1 f ∂η
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By representing the second moments with the fitting curve given by the experimental distributions (Veeravalli-Warhaft, 1989 and Tordella & Iovieno, 2005) 1 − E −1 3 1 + E −1 ϕuu = − erf(η) (13) 2 2 2 one obtains 12 2 3 4ν 1 − E −1 f 1− S= √ + 4β e−η × A1 f 2 π 4(1 − 2α) 2 − 32 1 − E −1 1 + E −1 − erf(η) (14) 2 2 Figure 4 shows the good agreement of the modelled variance and skewness distributions (relations 13 and 14) with the experimental data. The intermittency parameter associated to the lateral penetration of the mixing is compared in fig.5 with the values given by the present similarity law. It can be observed that the scaling exponent deduced from the experiment (Tordella & Iovieno, 2005), which is approximately equal to 0.29, is correctly represented. It should be noticed that such scaling is independent from the energy-dissipation model (11), because the model coefficient β does not influence the shape of the skewness distribution (14) and does not modify the position of the skewness maximum,
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which appears to be a function of the energy ratio E only. However, β determines the value of the maximum of the skewness distribution, and β ≈ 0.08 gives the best fit with experimental data by Tordella & Iovieno (2005). The other parameters that appear in figures 4 and 5 are α = 0.25 (see equation 8) and f (Rλ1 ) = 0.65. This value has been obtained for Reλ1 = 45 from Sreenivasan (1998). 1.0
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(xs-xc )/Δ
1.00
LES, IAM model, Reλ = 45 LES, IAM model, Reλ = 450 DNS, Reλ=45 Data Fitting Similarity solution
0.75 0.50 0.25
Δ
0.00 0.00
0.25
0.50
*(E/E 1 )
Fig. 6. Position xs of the maximum of the skewness distribution as a function of the normalized initial gradient of energy ∇∗ = ∂/∂(x/Δ), ∇∗ (E/E1 ) (1 − E −1 )/2.
3 Conclusions The present similarity analysis confirms our numerical experiment result where the turbulent transport is highly intermittent for shear-free decaying homogeneous isotropic interacting flows with kinetic-energy ratios far from unity in contrast to a Gaussian asymptotic state.
References 1. Batchelor GK, Townsend AA (1948) Proc. Roy. Soc. 193, 539–558. 2. Batchelor G. K. 1953 Cambridge University Press. 3. Briggs DA, Ferziger JH, Koseff JR, Monismith SG (1996) J. Fluid Mech. 310, 215–241. 4. Gilbert B. 1992 J. Fluid Mech. 100, 349–365. 5. Iovieno M, Cavazzoni C, Tordella D (2001) Comp. Phys. Comm., 141, 365–374. 6. Iovieno M, Tordella D (2002) Phys. Fluids, 14(8), 2673–2682. 7. Sreenivasan KR (1998) Phys. Fluids 10(2), 528–529. 8. Tordella D, Iovieno M (2005a) J. Fluid Mech., to appear. 9. Tordella D, Iovieno M (2005b) 22th IFIP TC 7 Conference on System Modeling and Optimization, Torino, July 18-22, 2005. 10. Yoshizawa A (1982) J. Phys. Soc. Japan 51, 2326. 11. Yoshizawa A (2002) Phys. Fluids 14(5), 1736–1744. 12. Veeravalli S, Warhaft Z (1989) J. Fluid Mech. 207,191–229.
Part VI
Multiphase Flows
Orientation of elongated particles within turbulent flow J. J. J. Gillissen, B. J. Boersma, and F. T. M. Nieuwstadt Laboratory for Aero and Hydrodynamics, J.M. Burgers Centre, Leeghwaterstraat 21, 2628 CA Delft, The Netherlands [email protected]
1 Summary Direct Numerical Simulation (D.N.S.) is used to obtain a turbulent channel flow field. Massless, non-interacting rigid rod like particles of sub-Kolmogorov length scale are present in the fluid. The particle position and orientation vectors are transported and rotated by the flow and they are subjected to a diffusion process caused by Brownian motion. The orientation of the particles is solved numerically by integrating the Fokker Planck equation, being the continuity equation for the particle orientation distribution function. The results are compared with a second method, in which moments of the distribution function are solved. Computing moments is cheaper than computing the complete function, but closure relations are needed to obtain a closed set of equations. The performance of the closure relation, developed by Cintra & Tucker (1995), is investigated for the turbulent channel flow case. The second and fourth moments obtained from both methods are compared in terms of mean and rms values. Mean values agree very well and rms values differ considerably.
2 Introduction Under certain turbulent conditions, the drag of long-chained polymer solutions in pipe flow is smaller than that of the pure Newtonian solvent (Virk, Mickley & Smith 1970). This phenomenon is refered to as drag reduction. Besides flexible polymers, drag reduction has also been observed using stiff polymers (Sasaki 1991) and macroscopic rigid-rod like particles (McComb & Chan 1985). Compared to Newtonian flow, drag reduced flow exhibits different turbulent structures. For instance, the Reynolds stress is more anisotropic with a larger value for the normal streamwise component and smaller values for all others. Also, the near wall coherent structures have larger length and time scales.
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The physical processes distinguishing a turbulent polymer flow from a Newtonian flow are not yet fully understood and numerical simulation is a useful tool to gain better insight. Many researchers studied the interaction between elastic polymers and turbulent flow using D.N.S. in combination with constitutive models for the polymer stress, (see for instance Ptasinski et. al. 2003). Similar research regarding rigid polymers was carried out by Paschkewitz et. al. (2004). Their studies show that polymer elasticity (storage and release of mechanical energy) is not a necessity for drag reduction. The visous anisotropy introduced by the orientation of the polymer seems to be the key property for polymer induced drag reduction. Within our project, we use numerical tools to study the interaction between rigid rod like particles and turbulent fluid flow. Our modeling technique is based on a statistical approach, involving a distribution function, governing the density of particles over the particle orientation angles. From this function, the forces acting between particles and fluid can be extracted. A direct computation of this function within turbulent flow has proven to be very expensive. Computational costs can be reduced considerably by adopting an approximation involving the moments of this function. The present work deals with the validity of using this approximation in predicting particle orientation and corresponding particle stress. To this end a comparison is made between a direct computation of the distribution function and a moment calculation within turbulent channel flow.
3 Orientation density equation 3.1 Fokker Planck equation In this work we compute flow induced particle orientation within turbulent channel flow. The distribution of particle orientation angles is governed by a function f . The particles translational and rotational velocities transport f according to an advection equation over the spatial and orientational coordinates, referred to as Fokker Planck equation: ∂f ˙ ) + ∇ · (xf ˙ ) = 0. + ∇p · (pf ∂t
(1)
Here x is the particle position vector and p is the particle orientation unit vector, which is defined using the spherical coordinate sytem. We use ∇p and ∇ to discriminate between gradients over the spherical coordinates θ and φ and the spatial coordinates x, y and z and ‘·’ signifies the material time derivative. The non interacting particles are very small so that particle inertia is neglegible and the fluid velocity on the particle length scale may be assumed a linear function of space. Under these conditions it follows that x translates with the undisturbed fluid velocity u at the particle centre.
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Jeffery (1922) solved the Stokes equations of motion for the flow surrounding an ellipsoidal particle, where the undisturbed fluid velocity is stationary and has a linear dependence on space. We use Jefferies results for very large aspect ratio particles, which states that the particle rotational velocity equals the projection of the undisturbed fluid velocity on the plane normal to the particle axis. The particles are assumed sufficiently small to experience Brownian motion caused by thermal fluctuations of the solvent molecules. In the mean Brownian motion gives rise to a diffusion velocity in the direction opposite to the gradient of f (Bird et. al. 1977). Including Brownian effects the particle translational and rotational velocities can be written as: x˙ = u −
L2 ∇ ln f, λ
p˙ = p · ∇u · (δ − pp) −
1 ∇p ln f, λ
(2)
where λ is the diffusion time scale, L the particle length, δ the second order unit tensor and ∇u the velocity gradient of the undisturbed fluid velocity. 3.2 Moment approximation The particle orientation distribution function f depends on both spatial and orientational coordinates and numerical computation is expensive. With diminishing Brownian diffusion, the computation becomes increasingly expensive and other methods must be adopted to compute the orientational configuration of the particles. One alternative is to solve the second moment of f instead of f itself. The second moment is defined as the orientational average of the second order dyad of the particle orientation vector: π 2π dφ dθ sin θf pp. (3) pp = 0
0
This is a symetric second order tensor with unit trace. Its equation of change is found by multiplying the Fokker Planck equation (1) with pp and subsequently integrating over θ and φ (see Bird et. al. 1977): ∂ pp + u · ∇ pp − (∇u)T · pp − pp · ∇u = ∂t L2 2 6 1 δ − pp + ∇ pp . −2∇u : pppp + λ 3 λ
(4)
The moment approach reduces the angular dependence of f (θ, φ) to five numbers. However, it also causes a problem. The equation of change for pp (4) contains the fourth moment pppp . In fact the equation for each nth moment contains the (n + 2)th moment. I.e., there are always more unknowns than equations, which means that ad hoc relation must be used to close the set of equations.
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Many closure relations for the fourth moment have been posed in literature. In this work we use the orthotropic closure approximation (Cintra & Tucker 1995). It is based on the fact that the principal axis of pppp match those of pp . In the principal frame pp and pppp contain two and three independent components when symmetry and normalization are taken into account. The closure is constructed by posing functions for the three independent components of pppp in terms of the two independent components of pp . If the principle values of pp are denoted a11 , a22 and a33 and the coordinates are chosen such that a11 ≥ a22 ≥ a33 , then all possible configurations fall within the bold triangle plotted in Fig. 1. The corners of the triangle
Fig. 1. Possible orientational configurations on the plane spanned by the two largest eigenvalues of the second moment of f fall within the bold triangle.
correspond to special states for which the distribution function is known: A is complete allignment in the 1-direction, B an evenly distribution on the 12plane and C is a uniform distribution over all directions. Since f is known at these points, the independent components a1111 , a2222 and a3333 of pppp are known too. The simplest closure is constructed by linear interpolating between these values, giving: ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ −0.15 1.15 −0.1 a1111 ⎝ a2222 ⎠ = ⎝ −0.15 ⎠ + ⎝ −0.15 0.15 ⎠ a11 . (5) a22 a3333 0.6 −0.6 −0.6 Our main goal is to examine the performance of this closure relation under turbulent flow conditions. To this end we compare numerical solutions of the moment equation with the Fokker Planck equation in a turbulent channel flow.
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4 Closure performance in turbulent channel flow 4.1 Numerical method Turbulent channel flow is the pressure driven flow between two flat, parallel, no slip surfaces. Our D.N.S. code for solving the space and time depencies of this flow type utilizes fast Fourier transforms to obtain derivatives in the homogeneous directions and second order finite differences for the wall normal direction. Time integration is achieved with an explicit scheme. A detailed explanation of the solver is given by Ptasinski et. al. (2003). To validate the moment approximation we have carried out two simulations (i) direct integration of the Fokker Planck equation (1) and (ii) integration of the moment equation (4) closed with the orthotropic closure relation (5). Spatial derivatives in (1) and (4) are compted with fast Fourier transforms in streamwise and spanwise directions and second order central differences in the wall normal direction. Time integration is achieved with the second order Adams Bashfort scheme. The orientational dependence of (1) is treated with a Galerkin method (Stewart & Sorensen 1972), expanding f in spherical harmonics: f (x, p, t) =
l N
m Am l (x, t)fl (p).
(6)
l=0 m=−l m Here Am the spherical harmonics of l are the expansion coefficients and fl degree l and order m. The equation of change for the coefficients is found by inserting the expansion into (1) and setting the inner products of (1) with each fnq to zero. This method provides high accuracy with a relative small number of expansion coefficients.
4.2 Simulation parameters The channel dimensions in the streamwise (x), spanwise (y) and wall normall (z) direction are 5H × 2.5H × H, with H the channel height. The number of grid points in x, y and z are 96 × 96 × 96. The grid is non-uniform in z with a 1.04 increase in cellvolume per cell away from the wall. The grid size in wall units at the wall and the channel center are: 13.0×6.5×1.2 and 13.0×6.5×4.7 respectively. The order of approximation of f is N = 12, which corresponds to 91 orthogonal modes. The Reynolds number based on channel height and bulk velocity equals 3800. The Weissenberg number, defined as the dimensionless product of the Brownian time scale λ with the mean shear rate γ at the wall, equals: W e = γλ = 50. particle length is chosen such that L2 /(λHUb ) = 0.007, with Ub the bulk velocity. We found that for these parameters, a time step of (ΔtUb )/H = 1 × 10−3 ensures a numerical stable solution. Statistics are obtained from 100 samples collected during the integration time interval of (ΔT Ub )/H = 200.
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Fig. 2. Mean and rms values of components of second and fourth moments of f versus wall distance within turbulent channel flow. Comparison between Fokker Planck (—) and moments equation (- -).
4.3 Results Fig. 2 shows the mean and rms values of components of the second and fourth order moments of f versus wall distance. Wall distance is presented in wall unit, where 250 wall units correspond to the full channel height. Near the wall, the flow is approximately a steady shear and particles are strongly orientated in the streamwise direction. Here the second and fourth order moments are highly anisotropic and deviate relatively little from their mean values. Further away from the wall turbulent fluctuations enhance the rms values and move the mean values towards isotropy. The moment approximation (dashed lines) is compared with the direct integration of f (solid lines). It is found that mean values are predicted very well, while the rms values are systematically too large. This paper deals with the validation of the moment approximation for predicting particle orientation in turbulent flow. The moment approximation will be used in later work to study the effect of particles on the characteristics of turbulent flow. The forces between rigid rod like particles and fluid is governed by the following stress tensor (see Bird et. al. 1977):
Orientation of elongated particles within turbulent flow
1 6 pp − δ . τ = αμ 2∇u : pppp + λ 3
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(7)
In this equation, μ is the solvent kinematic viscosity and α is a dimensionless parameter depending on particle concentration. The performance of the moment approximation in predicting correct particle stress behavior is examined in Fig. 3, where the stress is plotted versus wall distance. The stress is computed using α = 1 and it is scaled with wall shear stress. Again the moment approximation performance is tested by comparing its results with those obtained from the direct computation of f . Mean values agree reasonably well and rms values differ considerably, with the largest discrepancies occurring near the wall for the normal streamwise component.
Fig. 3. Mean and rms values of components of stress tensor versus wall distance within turbulent channel flow. Comparison between Fokker Planck (—) and moments equation (- -).
5 Conclusion Direct numerical simulation of a turbulent channel flow was carried out and the distribution function of the particle orientation angles was computed by a direct integration of the corresponding continuity equation. Due to limitations in computational resources, the simulation was carried out for a very low Weissenberg number, being the dimensionless product of the orientational diffusion time scale and the fluid rate of strain at the wall. A cheap alternative for computing particle orientation was investigated. The method, based on computing moments of the distribution function, requires a closure relation to obtain a closed set of equations. The performance of the closure approximation for the fourth order moment of the distribution function, developed by Cintra & Tucker (1995), is examined for the turbulent channel flow case. Consistency is determined by comparing second and fourth
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order moments and corresponding particle stress in terms of mean and rms values as functions of wall distance. Mean values agree very well, while rms values differ considerably, with the largest difference occuring near the wall for the streamwise normal stress component. Shapes of the rms profiles show correct behavior. We conclude that the moment approximation enhances stress fluctuations but does not change qualitative trends.
6 Acknowledgments This work is supported by The Research Council of Norway.
References 1. Bird, R. B., Hassager, O., Armstrong, R. C. & Curtiss, C. F. 1977 Dynamics of polymeric liquids. Volume 2. John Wiley & Sons. 2. Cintra, J. S. & Tucker, C. L. 1995 Orthotropic closure approximations for flow-induced fibre orientation. Soc. Rheol. 39 (6), 1095-1122. 3. Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. Roy. Soc. A 102, 161-179. 4. McComb, W. D. & Chan, K. T. J. 1985 Laser-Doppler anemometer measurements of turbulent structure in drag-reducing fibre suspensions. J. Fluid Mech. 152, 455-478. 5. Paschkewitz, J. S., Dubief, Y., Dimitropoulus, C. D., Shaqfeh, E. S. G. & Moin, P. 2004 Numerical simulation of turbulent drag reduction using rigid fibres. J. Fluid Mech. 518, 281-317. 6. Ptasinski, P. K., Boersma B. J., Nieuwstadt, F. T. M., Hulsen, M. A., van den Brule, B. H. A., & Hunt, J. C. R. 2003 Turbulent channel flow near maximum drag reduction: simulations, experiments and mechanisms. J. Fluid Mech. 490, 251-291. 7. Sasaki, S 1991 Drag reduction effect of rod-like polymer solutions. Influences of polymer concentration and rigidity of skeletal back bone. J. Phys. Soc. 60 (3), 868-878. 8. Stewart, W. E. & Sorensen, J. P. 1972 Hydrodynamic interaction effects in rigid dumbbell suspensions. II. Computations for steady shear flow. Trans. Soc. Rheol. 16 (1), 1-13. 9. Sureshkumar, R., Beris, A. & Handler, R. 1997 Direct numerical simulation of turbulent channel flow of a polymer solution. Phys. Fluids 9 (3), 743-755. 10. Virk, P., Mickley, H. & Smith, K. 1970 The ultimate asymptote and mean flow structures in tom’s phenomenon. Trans. ASME E:J. Appl. Mech. 37, 488493.
On the closure of particle motion equations in large-eddy simulation Maria Vittoria Salvetti1 , Cristian Marchioli2 , and Alfredo Soldati2 1
2
Dipartimento Ingegneria Aerospaziale, Via G. Caruso, 56122, Pisa (Italy) [email protected] Centro Interdipartimentale Fluidodinamica e Idraulica and Dip. Energetica e Macchine, Via delle Scienze 208, 33100, Udine (Italy) [email protected], [email protected]
A closure model for lagrangian tracking of particles, starting from LES flow data, is presented. The basic idea is to reconstruct the velocity field from the knowledge of its filtered values on a coarse grid, by means of fractal interpolation. Validation is carried out by means of a priori tests for turbulent channel flow. DNS data are filtered through a cut-off filter; different filter widths are considered. Three sets of particles having different relaxation times are tracked. Particle statistics are computed, with and without fractal interpolation, and compared to those obtained starting from the DNS flow fields.
1 Introduction The mechanisms of particle transfer in turbulent boundary layers are of great importance in many engineering and environmental applications. Direct Numerical Simulation (DNS) together with Lagrangian particle tracking has been widely used to investigate and quantify these mechanisms, Clearly, DNS is limited to low Reynolds numbers, while the simulation of turbulent flows at higher Reynolds numbers can be tackled using Large-Eddy Simulation (LES). As far as the fluid dynamic part is concerned, the closure problem of LES equations has been deeply investigated and several Sub-Grid Scale (SGS) models have been proposed and tested. Let us assume, for the moment, a simplified simulation setting in which one-way coupling between the two phases exists (i.e. the fluid dynamics governing equations are unchanged) and only the effect of the drag force on particles is considered. The particle motion is driven by the fluid velocity field at particle position, u(xi , t). Since only the filtered field, u(xi , t), is available from LES, a closure model should in principle be needed to retrieve the SGS velocity fluctuations. However, this point has received little attention in the literature, especially if compared with the huge amount of work devoted to
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the closure problem for the fluid dynamic part, and it is still an open question whether SGS terms significantly affect particle motion. Previous studies indicate that the effects of SGS velocity fluctuations on particle motion are negligible; see, for instance, the work by Armenio et al. [2] on dispersion statistics of both tracer and finite-inertia particles in LES of turbulent channel flow. However, the a priori and a posteriori tests recently carried out for the same channel flow by Keurten and Vreman [3] showed that LES is not able to predict turbophoresis if SGS effects are ignored in the equation of particle motion. Also, Fevrier et al. [4] have shown that LES filtering has an effect on particle motion which depends on the ratio of particle size to the filtered spatial scales. Thus, the importance of SGS terms for different particle inertia and different grid resolutions still requires systematic investigation. The present paper is a contribution to the study of the closure problem in Lagrangian particle tracking one-way coupled with LES. To this purpose, we use a well validated and tested numerical set-up, namely a pseudo-spectral code for the Eulerian simulation of the plane channel flow coupled with a Lagrangian tracking algorithm (see [1] for an example of application). The aim of the present study is twofold. First, a priori tests for particleladen turbulent channel flow are used to investigate the importance of the SGS velocity fluctuations in predicting the statistical properties of the dispersion process for different inertia particles and for different grid resolutions. Second, a closure model for the equations of particle motion is proposed and its performance analysed repeating the a priori tests. The idea is to reconstruct the SGS fluid velocity fluctuations by means of the fractal interpolation technique, previously used by Scotti and Meneveau [5] to construct SGS models for the Navier-Stokes equations.
2 Methodology Channel Flow Simulation Particles are introduced in a pressure driven incompressible and Newtonian turbulent flow of air. The flow is bounded by two infinite flat parallel walls with origin of the coordinate system located at the channel centre and the x, y and z axes pointing in the streamwise, spanwise and wall-normal directions respectively. Periodic boundary conditions are imposed on the fluid velocity field both in streamwise and spanwise (homogeneous) directions and no-slip boundary conditions are enforced at the walls. The flow field is calculated by integrating the mass and momentum balance equations in dimensionless ) form (obtained using the channel half-width, h, and the shear velocity, uτ = τw /ρ, τw being the wall shear stress and ρ the fluid density). The flow is driven by a mean pressure gradient. The equations are discretized through a pseudo-spectral method, using Fourier representations
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for the homogeneous directions and a Chebyshev representation for the wallnormal (non-homogeneous) direction. A two-level explicit Adams-Bashforth scheme for the nonlinear terms and an implicit Crank-Nicolson method for the viscous terms were employed for time advancement. Further details of the method have been previously published [6]. In the present study, we consider air with ρ = 1.3 kg/m3 and ν = 2.0 · 10−5 m2 /s. The pressure gradient is such that the shear Reynolds number, Reτ = huτ /ν, is equal to 150. The Reynolds number based on mean velocity and half duct width is 2100. Calculations have been performed in dimensionless units (indicated by the superscript +), the Reynolds number being the one parameter to scale the flow. The computational domain is 1885 × 942 × 300 wall units in x, y and z respectively. The present DNS uses 128 × 128 Fourier modes in the homogeneous directions and 129 Chebyshev polynomia in the wall-normal direction. Thus, the computational grid is made of 128 × 128 × 129 points, corresponding to a spatial resolution of Δx+ = 15 and Δy + = 7.5. In the wall-normal direction the grid resolution is not homogeneous; the first collocation point is at z + = 0.05 from the wall, while in the center of the channel Δz + = 3.7. Statistics of the flow field (not shown here) match closely with those of the work by Lyons et al. [7], calculated for the same Reynolds number over a slightly different domain. The time step used was chosen equal to Δt+ = 0.03 in wall time units. Particle Equation of Motion We assume that particle number density and particle size are both small to neglect particle feedback onto the gas flow. Particle-particle interaction due to their inertial force is neglected and particles are assumed pointwise, rigid, spherical and obeying the following vectorial Lagrangian equation of motion: dv Cd (u − v = dt τP
in which v is particle velocity, u is fluid velocity at particle location calculated )/ReP is the with 6th-order Lagrange interpolation, Cd = 24(1 + 0.15Re0.687 P Stokes drag coefficient, τP = d2P ρP /18ρ is the particle relaxation time (dP and ρP being particle diameter and density respectively). The LHS of Eq. 1 represents particle inertia and the term on the RHS of Eq. 1 represents the effects of the Stokes drag. In the present simulations, 105 flyash particles, characterized by a particle-to-fluid density ratio equal to 769.23, have been released at randomly chosen locations within the computational box. Particle trajectories were tracked individually for an interval of 1190 wall time units (corresponding to 1.72 s). The initial velocities of the particles were set equal to the interpolated fluid velocities at each particle location. Particles are elastically reflected away from the wall when their centre is less than a distance dP /2 from the boundary. The fluid forces acting on each single particle are calculated with a 6th-order Lagrange interpolation. In
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τp+ 1 5 25
dp (μm) 20 46 100
d+ p 0.153 0.342 0.765
Table 1. Parameters of tracked particles: superscript + indicates dimensionless variables. Note that the dimensionless particle relaxation time τp+ corresponds to the particle Stokes number St.
Table 1, parameters of the tracked particles are reported. The time step for particle tracking is equal to 0.45 wall time units, less than half the relaxation time of the smaller particles.
3 Fractal interpolation The aim of fractal interpolation is to reconstruct the velocity field u(xi , t) from the filtered field u. This is done by iteratively applying an affine mapping procedure to u. Let us consider for simplicity a 1D case, by introducing the non dimensional coordinate ξ such that ξ = (x−xi−1 )/2Δ in the interval [xi−1 , xi+1 ]. Considering the space C[0, 1] = {u C[0, 1] | u(0) = ui−1 , u(1) = ui+1 }, the map Wi is defined as follows: Wi [u](ξ) = di,1 (2 · ξ) + qi,1 (2 · ξ)
(2)
Wi [u](ξ) = di,2 (2 · ξ − 1) + qi,2 (2 · ξ − 1)
(3)
Eq. 2 holds for ξ [0, 12 ], while Eq. 3 for ξ [ 12 , 1] and: qi,1 (ξ) = [ui − ui−1 − di,1 · (ui+1 − ui−1 )] · ξ + ui−1 · (1 − di,1 )
(4)
qi,2 (ξ) = [ui+1 − ui − di,2 · (ui+1 − ui−1 )] · ξ + ui − di,2 · ui−1
(5)
By iteratively repeating this procedure, starting from a coarse LES-like grid on which u is defined, a signal can be reconstructed on a finer DNS-like grid. The characteristics of the reconstructed signal depend on the two stretching parameters di,1 and di,2 . It can be shown [5] that these parameters are related to the fractal dimension of the signal; the values recommended in [5] are used in this work, i.e. di,1 = 2−1/3 and di,2 = −2−1/3 . Generalization of the fractal interpolation procedure to 2D or 3D is straightforward (see [5] for details).
4 Results In this Section, we present results of a priori simulations with and without fractal interpolation. In these simulations, we solved for the particle equations of motion using different filtered fluid velocity fields. These fields were
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obtained through explicit filtering of the DNS velocity applying a cut-off filter in the wavenumber space; different filter widths are considered and the corresponding coarsening factors (CF), with respect to the DNS resolution, are 2, 4 and 8 in the homogeneous directions. In the wall-normal direction data are not filtered, since often in LES the wall-normal resolution is DNS-like. Particle statistics are compared with those obtained from the DNS velocity fields to investigate the importance of the SGS velocity fluctuations in predicting the statistical properties of the dispersion process for different particle inertia and for different grid resolutions. 4.1 A priori tests without fractal interpolation Fig. 1 shows mean particle statistics for the three particle sets coinsidered as obtained from a priori tests without fractal interpolation. Specifically, streamwise velocity profiles are shown in Figs. 1(a-c) whereas streamwise, spanwise and wall-normal rms velocity fluctuation profiles are shown in Figs. 1(d-f), Figs. 1(g-i) and Figs. 1(l-n) respectively. All profiles were obtained averaging in time and space (over the homogeneous directions). Color code in Fig. 1 is the following: black lines refer to particles tracked in the unfiltered DNS fluid velocity field; purple, light blue and green lines refer to particles tracked in filtered DNS fluid velocity fields restricted to LES grids with coarsening factors of 2, 4 and 8 respectively. It is apparent that filtering the fluid velocity has a small, if not negligible, effect on the mean particle velocity but a large impact on the turbulent velocity fluctuations. Particle velocity fluctuations are severely reduced, in particular for large filter widths corresponding to coarser LES grids (light blue and green lines). In particular, the reduction of the wall-normal velocity fluctuations near the wall for the a priori LESs (see Figs. 1(l-n)) is worth noting because it corresponds to a reduction of particle turbophoretic drift (namely, particle migration to the wall in turbulent boundary layers) and, in turn, to a reduction of particle accumulation in the near-wall region [3]. The above results seem to confirm that (i) LES can not provide correct quantitative estimate of turbophoresis if SGS effects are not properly included into the equation of particle motion [3] and that (ii) filtering has an effect on particle motion which depends on the ratio of particle size to the filtered spatial scales [4]. 4.2 A priori tests with fractal interpolation Test simulations have been performed to verify if a priori LES results could be improved retrieving part of the filtered SGS contributions through reconstruction of the SGS fluid velocity field by fractal interpolation. Preliminary results are reported in Fig. 2, which shows the same particle statistics as in Fig. 1 with the same line color code. Differences between the two figures are due solely to the use of fractal interpolation, which appears
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Fig. 1. Particle statistics for a priori simulations without fractal interpolation: a-c) mean streamwise velocity; d-f) streamwise, g-i) spanwise and l-n) wall-normal rms velocity fluctuations. CF indicates the coarsening factor of the LES grid.
to have little effect on the mean particle velocity (see Figs. 2a-c) but a large impact on the turbulent velocity fluctuations. For the smaller particles (St = 1, Figs. 2d, 2g and 2l), the statistical properties of the reconstructed velocity signal are not satisfactory as they become much larger than those obtained from DNS. This may be due to the fact that the motion of this size particles is more affected by the filtered velocity fluctuations of the underlying flow field: we believe results can be
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Fig. 2. Particle statistics for a priori simulations with fractal interpolation: a-c) mean streamwise velocity; d-f) streamwise, g-i) spanwise and l-n) wall-normal rms velocity fluctuations. CF indicates the coarsening factor of the LES grid.
improved by fine tuning of the statistical properties of the reconstructed signal at the specific shear Reynolds number of our flow (Reτ = 150) based on exact calculation of the stretching parameters, which is currently under way. For the larger particles (St = 5, Figs. 2e, 2h and 2m and St = 25, Figs. 2f, 2i and 2n), the unfiltered velocity statistics are approximately recovered even on the coarser LES grid (green lines). It is not clear yet if this improvement can be
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ascribed to the use of fractal interpolation or it is just a result of the inertial filtering of SGS velocity fluctuations by the larger particles.
5 Concluding remarks This paper presents the first results obtained by our research group in the frame of a project dealing with the closure problem in Lagrangian particle tracking one-way coupled with LES. Particle statistics from a priori simulations indicate that the importance of SGS velocity fluctuations in predicting the properties of particle dispersion depends both on particle inertia and on Eulerian grid resolution. Specifically, it is confirmed that well-known phenomena such as turbophoresis and nearwall particle accumulation can not be predicted accurately by LES if SGS effects are ignored in the equation of particle motion. A closure model for particle motion equation is proposed, in which the SGS fluid velocity fluctuations are reconstructed by fractal interpolation [5]. Preliminary results seem promising but further effort is required to improve the capability of the model to retrieve the SGS contribution to the fluid velocity field. Longer simulations have to be run to evaluate in more detail the performance of the fractal interpolation technique when the steady state for depositing particles is approached. Fine tuning of the stretching parameters used to reconstruct the filtered velocity signal is also required to match the statistical properties of the unfiltered velocity. Finally, the effect of the specific filter (cut-off versus top-hat) used to obtain the filtered velocity field needs investigation.
References 1. Marchioli C, Soldati A (2002) Mechanisms for particle transfer and segregation in a turbulent boundary layer. J. Fluid Mech. 468: 283–315. 2. Armenio V, Piomelli U, Fiorotto V (1999) Effect of the subgrid scales on particle motion. Phys. Fluids 11: 3030–3042. 3. Kuerten JGM, Vreman AW (2005) Can turbophoresis be predicted by largeeddy simulation? Phys. Fluids 17 (art. no. 011701). 4. Fevrier P, Simonin O, Squires KD (2005) Partitioning of particle velocities in gas-solid turbulent flows into a continuous field and a spatially-uncorrelated random distribution: theoretical formalism and numerical study. J. Fluid Mech. 528: 1–46. 5. Scotti A, Meneveau C (1999) A fractal model for large eddy simulation of turbulent flow. Physica D 127: 198–232. 6. Soldati A, Banerjee S (1998) Turbulence modification by large scale organized electrohydrodynamic flows. Phys. Fluids 10: 1742–1756. 7. Lyons SL, Hanratty TJ, McLaughlin JB (1991). Large-scale computer simulation of fully developed turbulent channel flow with heat transfer. Int. J. Numer. Methods Fluids 13:999-1028.
Dispersion of Circular, Non-Circular, and Swirling Spray Jets in Crossflow Mirko Salewski and Laszlo Fuchs Lund University, Division of Fluid Mechanics, Ole R¨ omersv. 1, P.O. Box 118, 22100 Lund, Sweden [email protected]
Multiphase jets in crossflow are investigated using Large Eddy Simulation (LES). The multiphase flow is handled in an Euler/Lagrange framework with two-way coupling. Atomization, droplet breakup, and droplet collision are modeled. The simulation is validated against experimental results found in the literature. The trajectory of the counter-rotating vortex pair (CVP) is shown to lie below the trajectory of the droplets. This leads to lateral dispersion of the droplets depending on the strength of the CVP. If the momentum flux ratio is large, the penetration is stronger and the CVP is also strong, leading to the split of the main droplet trajectory into two branches. The droplet dispersion is compared for various nozzle geometries and injection of a swirling liquid jet. Swirl is shown to have profound effects on the lateral and vertical dispersion of the spray.
1 Introduction Spray jets in crossflow are relevant to fuel/air mixing in many industrial applications, e.g. gas turbines. Liquid fuel jets are injected in a crossflow of air where the liquid jet atomizes to small droplets. The dispersion of the droplets is a central issue as it ultimately determines the mixture fraction profiles after evaporation and thereby has impacts on combustion efficiency and pollutant formation. The flow field is inherently unsteady and three-dimensional. The ratio of the momentum of the injected jet to the momentum of the crossflow is a central parameter of the problem. The momentum flux ratio J is defined in Eq.1 where ρ is the density, U the velocity, and the indices ”j” and ”cf” stand for jet and crossflow, respectively. J=
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Fig. 1. Diagonal view snapshot of the spray volume fraction (dark) and the CVP (light), visualized with λ2 vortex visualization [7]
Fig. 2. Top view snapshot of the spray volume fraction (dark) and the CVP (light), visualized with λ2 vortex visualization [7]
The injected jet forms a shear-layer with the crossflow which is inherently unstable. The injected jet is bent by the crossflow, and with it also the orientation of the shear-layer. Additionally, the crossflow induces a flow within the jet. As the jet bends one notes the formation of a counter-rotating vortex pair (CVP). Ahead of the injected jet, one finds a stagnation point, leading to the division of the incoming flow. This forms a vortex that has the shape of a horseshoe. The tail of this vortex is lifted up and interacts with the CVP and the boundary layer that is present near the jet-injection plane. A qualitative picture of the main features of the jet in crossflow (JICF) can be obtained by Large Eddy Simulation (LES), showing the formation of the unsteady CVP. Fig.1 shows a snapshot of the CVP (light), visualized with the λ2 vortex visualization technique [7], and the spray volume fraction (dark). This instantaneous picture shows that the droplets follow higher trajectories than the CVP. Fig.2 shows the corresponding top view. The two vortices of the CVP, which are convected downstream, are evident. For this two-phase case one notes that the droplets ride on top of the CVP and they disperse laterally away from the center-plane due to the presence of the CVP. The two branches of droplet trajectories are visible in Fig.2. Experimental investigations of the spray JICF flow field are numerous. Most researchers correlate trajectories of the main spray body (e.g.[8]). The lateral dispersion of the spray [5] or the spray volume flux distribution (e.g. [10], [13], [9]) receive attention in recent years. The CVP is widely documented for single-phase flows. For two-phase flows, however, the CVP is seldom mentioned. Most researchers investigate this flow field with experimental techniques, e.g shadowgraphy [12], PDA [13], or Mie-scattering [10]. It is difficult to measure the CVP in optically dense sprays. This measuring difficulty makes numerical simulation an even more essential tool for understanding the problem and for understanding the limited experimental data. This work focuses on the mutual interaction of the spray and the CVP, namely: The injected
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jet (spray) induces the CVP and the formed droplets interact closely with the CVP as they are transported downstream. The droplet dispersion is compared for various nozzle geometries, momentum flux ratios, and swirl levels. Round nozzles are compared to square ones. For round nozzles, hollow cone injection and injection with various levels of swirl are investigated. Especially swirl is shown to have profound effects on the lateral dispersion of the spray.
2 Case Description The baseline case has been investigated using PDA by Becker and Hassa [2]. The geometry of the computed case is shown in Fig.3, where D is the diameter of the injected (fuel) jet. The injected fuel is selected to be octane. The ratio of channel height and nozzle diameter is 90. The Reynolds number for the gas phase is 260000. The liquid Weber number at the nozzle exit is on the order of 1000, the aerodynamic Weber number is more than 10. At such We numbers the spray rapidly atomizes. The momentum flux ratio of the baseline case is J = 6. The channel has dimensions of (x,y,z)=(90,90,540) nozzle diameters. The transverse jet is injected at (45,0,405) nozzle diameters from the inlet plane. At the upper and lower walls no-slip boundary conditions are applied, in lateral direction periodic boundary conditions. The channel is divided into two parts: For the first half of a length of 270 nozzle diameters periodic boundary conditions are imposed so as to simulate an infinitely long channel. This first part of the channel is used to set the inflow boundary conditions for the second half where the liquid jet is injected. The outflow of the second half of the channel has a flux conserving zero-gradient boundary condition. At the lateral walls of the channel, periodic conditions are assumed. The initial parcel positions and velocities are set at the liquid jet nozzle. It is assumed that the droplets bounce elastically at solid walls and are destroyed at the outlet.
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3 Governing Equations and Numerical Methods The numerical simulation is done in an Euler/Lagrange framework. Twoway coupling between the continuous and dispersed phases is applied for the momentum equations. The non-dimensional continuity (Eq.2) and momentum (Eq.3) equations for incompressible flow of Newtonian fluids with constant diffusivities are described in Eulerian framework. ∂uj =0 ∂xj ∂p 1 ∂ 2 ui ∂ui ∂ui ˙ + uj =− + + Fs,i ∂t ∂xj ∂xi Re ∂x2j
(2) (3)
˙ are the source terms for the momentum equations. The source terms Fs,i provide for the coupling from the liquid phase to the gas phase. The grid is stretched to improve the near-wall resolution. The spatial discretization is done with a fifth-order weighted essentially non-oscillatory (WENO) scheme [6] for the convective terms to handle the strong flow field curvature near the jet injection. The diffusive terms are discretized with a fourth-order central difference scheme. The flow solver uses a third-order Runge-Kutta scheme. No explicit model for subgrid scale turbulence is applied. The dispersion of poly-dispersed spray droplets is computed using Lagrangian particle tracking with the stochastic parcel method [1]. Atomization and droplet breakup are modeled by a combination of the breakup model by Reitz and the Taylor analogy breakup (TAB) model [3]. Droplet collision is also modeled. More details about the modeling of the dispersed phase are found in [11].
4 Results Fig.4 and Fig.5 give an overview over the spray volume flux distribution in a distal cross section of the channel (z/D = 90 nozzle diameters downstream) for two different momentum flux ratios. The outer boarder of the spray may be defined by a low value of the iso-contour of the liquid volume fraction (0.1%, say). Measured by this iso-contour, the spray penetration increases strongly with the momentum flux ratio. The lateral dispersion is, however, only weakly dependent on it. This is consistent with previous findings [5],[2] which are obtained in the near-field. The distribution of the flux of the liquid phase for a momentum flux ratio of J = 6 in the cross-sectional plane (Fig.4) has a shape close to circular. For a higher momentum flux ratio (J = 41) the distribution shows two peaks which correspond to two well separated tails of liquid droplets (Fig.5). This
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Fig. 4. Volume flux distribution in a Fig. 5. Volume flux distribution in a channel cross-section (z/D=90) for jet channel cross-section (z/D=90) for jet to cross-flow momentum ratio J=6 to cross-flow momentum ratio J=41
indicates that for high momentum flux ratios the main volume flux trajectory is split into two branches. The reason for this lateral dispersion is the CVP as is elaborated later. For low momentum flux ratios, the CVP is weak, while for high momentum flux ratios the CVP is stronger. Thus, for low momentum flux ratios (J = 6, Fig.4), dispersion due to turbulence dominates dispersion due to the CVP, while for higher momentum flux ratios (J = 41, Fig.5) dispersion due to the CVP dominates. In snapshots two branches can be observed for all momentum flux ratios. For low momentum flux ratio these fluctuate more than the mean distance between them. Hence they are smeared out on average. For the high momentum flux ratio, the two branches are also found on average. This finding is significant with respect to fuel placement in gas turbines: The main droplet trajectories determine the location of fuel rich zones with accelerated N Ox production. The downstream behavior of the spray can be characterized by different parameters. Fig.6 shows the location of the center of the CVP projected onto an vertical plane as computed using different number of grid points. The trajectory of the CVP is defined to be locus of the vertical velocity maxima in the center-plane. This is motivated by the fact that vertical velocities are largest between the two vortices of the CVP. A comparison with the trajectory of λ2 minima shows congruence. One may also define a trajectory as a mean streamline from the nozzle. This trajectory is not suitable for a multiphase flow case as the induced jet in the continuous phase is weak near the boundary. Here the Stokes number of the droplets is still large, and the momentum transfer between the phases is small. The trajectories based on the mean streamline are therefore flat. The trajectory of the induced jet is distinct from the trajectory of the CVP as shown below. The trajectory of the spray can be defined as
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the locus of liquid volume fraction maxima. It is important to note that this trajectory is a three-dimensional curve. This three-dimensional curve can be projected on a vertical plane. If, on the other hand, the curve is projected on a horizontal plane, the lateral dispersion of the droplets can be studied. Fig.7 shows the local maxima of liquid volume fraction in a horizontal plane, revealing two branches related to the splitting of the spray body.
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The numerical accuracy can be assessed in Fig.6 and Fig.7. The three stretched grids have 1.9, 3.4, and 3.8 million cells. The high Reynolds number (260000) means that the boundary layer even on stretched grids cannot be resolved: On the finest grid y + ∼ 15. The number of parcels is about 50000. The computed results are shown not to be sensitive to the Reynolds number and the number of parcels in this range (figures not included). The trajectories become higher with grid refinement due to less numerical diffusion. Secondly, the trajectories of the maximum vertical velocities have peaks near the nozzle (z/D ∼ 2). The droplets accelerate the gas in this region and induce a jet. The peaks in the curve occur due to this induced jet. This shows that the CVP is initiated at the bifurcation point of the induced jet. The bifurcation point is observed for single-phase JICF using PIV [4] and LES [14]. Fig.8 shows a comparison of a computed volume fraction profile to a corresponding PDA measurement found in the literature [10]. The curves are vertical profiles in the center-plane 178 nozzle diameters downstream. The simulations show a lower level of dispersion as compared to the experimental one. One may speculate about this discrepancy; an important source of dispersion is the level of turbulence and its spectral content upstream of the injected jet. In the LES we have assumed to have an infinitely long channel. This is of course not so in the experiments. It should be noted that the two curves match within the experimental and numerical uncertainty.
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An important parameter that determines the spreading of the droplets is the momentum flux ratio. Fig.9 depicts a top view of the spreading of the droplets (i.e. the center of the peak locations) in the main flow in the transverse plane. Increasing the momentum of the injected jet leads to a larger (mean) spreading of the droplets at all downstream locations. This spreading is the
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result of the large coherent structures that are formed by the shear-layer between the crossflow and the injected jet. Further spreading is introduced by turbulence which is stronger the stronger the shear-layer is. Fig.10 depicts the trajectories of the spray defined by liquid volume fraction maxima for various momentum flux ratios. As expected, higher momentum flux ratios yield higher trajectories. Fig.11 shows the corresponding trajectories of the CVPs for the same momentum flux ratios. A comparison between Fig.10 and Fig.11 shows that the CVP trajectories are lower than the droplet trajectories, i.e. the droplets ride on top of the CVP. The rotation of the flow about each vortex of the CVP induces flow upward between the CVP, flow inward towards the center-plane below the CVP, and flow outward away from the center-plane on top of the CVP. This flow transports the droplets away from the centerplane and results in the lateral dispersion in two main branches. This lateral dispersion grows with the momentum flux ratio since the CVP becomes more pronounced (Fig.9).
Fig. 12. Volume flux distribution in Fig. 13. Volume flux distribution in a channel cross section (z/D=90) for a a channel cross-section (z/D=90) for a swirl number of S = 0.5 swirl number of S = 1.0
The addition of swirl to the liquid jet is investigated in Fig.12 and Fig.13 which display the volume flux distribution for the same momentum flux ratio (J = 41) and location (z/D = 90) as in Fig.5 which depicts the corresponding U swirl-free case. The swirl number is defined as S = tangential Uaxial . Clearly, swirl induces asymmetry into the volume flux distribution (which is not necessarily noticeable for other momentum flux ratios). The right branch penetrates higher than the left branch. Additionally, in the right branch there is more lateral dispersion. Addition of swirl offers the opportunity to control the lateral dispersion independent of the momentum flux ratio. Thus one can tailor various spray volume flux distributions by choosing pairs of the momentum flux ratio and the swirl number.
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Fig.14 and Fig.15 show the effect of changing the nozzle shape and swirl number on the vertical and lateral droplet trajectories. The momentum flux ratio is J = 41 for all cases. All nozzles have identical exit plane areas. Trajectories for round and square nozzles differ less than the numerical inaccuracy and can therefore not be distinguished in the trajectory parameter. The nozzle shape is, however, important for the spatial development of the flow itself. Hollow cone injection gives a slightly lower trajectory and enhances lateral dispersion. Fig.14 shows also the different heights of the two branches of the trajectories for swirling flow. The right and left branches for the swirling cases are projected on the center-plane separately, and the asymmetry in penetration of the right and left branches is clearly revealed. Also the lateral dispersion (Fig.15) is asymmetric in the swirling injection case, as shown by the different x/D location of the two branches.
5 Conclusions The momentum flux ratio of the jet and the crossflow determines the penetration of the spray, CVP trajectories, and the lateral dispersion of the spray. The main droplet trajectories are above the CVP trajectory. If the momentum flux ratio is large, the spray penetrates far into the main flow, the CVP is strong, and the mean droplet trajectory splits into two branches. For weaker momentum flux ratios, the trajectories are flatter and the splitting into two branches cannot be observed as turbulent dispersion is stronger than the dispersion due to the CVP. Hollow cone injection yields slightly lower trajectories and more lateral dispersion. Addition of swirl to the jet lowers the spray penetration and enhances the lateral dispersion. The volume flux distribution becomes asymmetric due to the addition of swirl.
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6 Acknowledgments The project was supported by the Swedish Energy Agency (STEM) within the TPE program. Computational resources were provided by LUNARC computing center at Lund University.
References 1. A.A. Amsden, P.J. O’Rourke, and T.D. Butler. Kiva-ii: A computer program for chemically reactive flows with sprays. Technical Report LA-11560-MS, Los Alamos National Laboratory, 1989. 2. J. Becker and C. Hassa. Breakup and atomization of a kerosene jet in crossflow at elevated pressure. Atomization and Sprays, 11:49–67, 2002. 3. D. Caraeni, C. Bergstr¨ om, and L. Fuchs. Modeling of liquid fuel injection, evaporation and mixing in a gas turbine burner using large eddy simulation. Flow, Turbulence and Combustion, 65:223–244, 2000. 4. I.M Ibrahim, S. Murugappan, and E.J. Gutmark. Penetration, mixing and turbulent structures of circular and non-circular jets in cross flow. AIAA paper, 2005-0300, 2005. 5. T. Inamura and N. Nagai. Spray characteristics of liquid jet traversing subsonic airstreams. J. Propulsion and Power, 13(2):250–256, 1997. 6. G.-S. Jiang and C.-W. Shu. Efficient implementation of weighted eno schemes. J. Comput. Phys., 126:202–228, 1996. 7. J. Joeng and F. Hussain. On the identification of a vortex. J. Fluid Mech., 285:69–94, 1995. 8. M.Y. Leong, V.G. McDonell, and G.S. Samuelsen. Effect of ambient pressure on an airblast spray injected into a crossflow. J. Propulsion and Power, 17(5):1076– 1984, 2001. 9. R.K. Madabhushi. A model for numerical simulation of breakup of a liquid jet in crossflow. Atomization and Sprays, 13:413–424, 2003. 10. M. Rachner, J. Becker, C. Hassa, and T. Doerr. Modelling of the atomization of a plain liquid fuel jet in crossflow at gas turbine conditions. Aerospace Science and Technology, 6:495–506, 2002. 11. M. Salewski, D. Stankovic, and L. Fuchs. A comparison of a single and multiphase jets in a crossflow using les. In Proc. of ASME Turbo Expo 2005, volume GT2005-68150, 2005. 12. P.-K. Wu, K.A. Kirkendall, R.P. Fuller, and A.S. Nejad. Breakup processes of liquid jets in subsonic crossflows. J. Propulsion and Power, 13(1):64–73, 1997. 13. P.-K. Wu, K.A. Kirkendall, R.P. Fuller, and A.S. Nejad. Spray structures of liquid jets atomized in subsonic crossflows. J. Propulsion and Power, 14(2):173– 182, 1998. 14. L.L. Yuan, R.L. Street, and J.H. Ferziger. Large-eddy simulation of a round jet in crossflow. J. Fluid Mech., 379:71–104, 1999.
Direct Numerical Simulation of Droplet Impact on a Liquid-Liquid Interface Using a Level-Set/Volume-Of-Fluid Method with Multiple Interface Marker Functions E.R.A. Coyajee∗ , R. Delfos, H.J. Slot, and B.J. Boersma J.M. Burgerscentre, Delft University of Technology, Leeghwaterstraat 21, 2628 CA Delft, The Netherlands. Summary. In this paper, a new approach is proposed for the simulation of noncoalescing interfaces using an existing interface-capturing method (Mass Conserving Level-Set method [17]). Separate marker functions are used for the interfaces of different bodies of the same fluid. The method is applied to the problem of the gravity driven impact of a droplet on a liquid-liquid interface. Results show that our approach accurately captures multiple interfaces in a single cell. Overlap of the different interfaces is found to be extremely small and to converge upon grid refinement. Key words: Multiple interfaces, Volume-Of-Fluid, Level-Set, droplet, liquid-liquid two-phase flow
1 Introduction For the numerical simulation of droplets in a viscous fluid, some of the most commonly used methods either belong to the class of front-tracking or frontcapturing methods. Using any of these methods to simulate droplet interaction, one is confronted with the wide range of length scales involved. When two droplets collide, a thin film of the surrounding liquid persists in the gap between the droplets, requiring a certain drainage time before coalescence may take place. Depending on the size of the droplets and the material properties of the fluids, the thickness of the thin film can be much smaller than the radius of the droplet. Effort can be done to resolve the fluid motion in the thin film by extensive local grid refinement, usually at the expense of large computational cost. However, if the ratio of the film thickness and the droplet radius is very small, it can also be argued that solving the fluid motion in the thin film is of no importance to the macroscopic behavior of the ∗
Corresponding author, e-mail: [email protected]
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droplets. In the past this approach has been used by [7]. Now, if the thin film is not resolved, simulating the collision of droplets with a front-tracking or front-capturing method may give qualitatively very different results, because of different interface representations. In the class of front-capturing methods, the interface is defined implicitly by a marker function on the fixed grid. As a result, standard front-capturing methods, such as Level-Set (LS) [14], Volume-Of-Fluid (VOF) [10, 11, 12] or combined LS/VOF methods [13, 17] are not capable of representing multiple interfaces in a single computational cell. When the interfaces of different droplets collide, they merge naturally, resulting in automatic coalescence of the droplets. In case of front-tracking, the interface is defined explicitly by means of a set of logically connected particles [15]. Consequently, multiple interfaces can easily be represented in a single cell and droplet collision without coalescence is naturally simulated. To merge interfaces, special effort needs to be done. In the region of merger, interface particles are deleted while the remaining ones need to be reconnected. In conclusion, although front-tracking enables simulation of interfaces both with and without merger, standard front-capturing methods are limited to automatic merger of interfaces upon collision. Therefore, front-capturing methods have been combined with particle tracking in recent years. Combined particle/VOF [1, 9] and particle/LS [6] methods have been proposed to capture interfaces at sub-cell resolution, however at the cost of introducing additional complexity compared to the original VOF or LS methods. In this work, we propose a different strategy to simulate non-coalescing interfaces within an existing LS/VOF method (The Mass Conserving LevelSet (MCLS) method [17]). The concept is to use separate marker functions for the interfaces of different bodies of the same fluid. It will be shown that only small adjustments to the original MCLS method are required, adding little complexity to the code. To evaluate our approach, it is applied to the problem of the gravity driven impact of a droplet into a liquid-liquid interface. Essentially, this problem is equivalent to the problem of the collision of two droplets, one of finite and one of infinite radius. Physical experiments of droplet impact on a liquid-liquid interface are also performed for validation of computational results.
2 Governing Equations In our method, two-phase flow is described by a set of equations for fluid and interface motion. The motion of incompressible Newtonian fluids is described by the Navier-Stokes equations: ∇ · u = 0, ρ (ut + u · ∇u) = −∇p + ∇ · μ ∇u + ∇uT + f ,
(1) (2)
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where ρ, μ, p, u = (u, v, w)T and f respectively represent density, viscosity, pressure, the velocity vector and a volume force. The interface Γ between fluids is described by the zero level of a marker function, commonly denoted φ, i.e. Γ (t) = {x | φ(x, t) = 0}. Away from the interface φ, the so-called Level-Set function, is required to be a distance function such that φ < 0 in phase ‘1’ and φ > 0 in phase ‘2’. Consequently, the density and viscosity can be prescribed from: ρ(φ) = ρ1 (1 − H(φ)) + ρ2 H(φ),
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In (2), the volume force f may represent gravity as well as surface tension forces. The gravity force is equal to ρg, with the gravity vector g = (0, 0, −g). A continuum description of the discontinuous surface tension force acting at an interface between two fluids: σκ(φ)∇H(φ),
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3 Computational Method 3.1 MCLS Method The governing equations are spatially discretized on a uniform Cartesian mesh. A standard staggered arrangement of variables is used. Velocity components are located on the cell faces, whereas pressure, density, viscosity and LS function are defined at cell centers. Boundary conditions on the 3D rectangular computational domain are Dirichlet conditions for the velocity (in this work u = 0, i.e. no-slip) and Neumann for the LS function (∇φ · n = 0).
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Fluid motion For the spatial discretization of the NS equations (1,2), central second order finite differences are used. Following the MCLS method [17], the discontinuous viscosity and surface tension force are regularized near the interface by replacing the discontinuous Heaviside function (5) in the equations for viscosity (4) and surface tension force (7) with a continuous representation: ⎧ if φ < −α ⎨0 φ (9) Hα (φ) = 12 (1 + α + π1 sin( πφ )) if |φ| ≤ α , α ⎩ 1 if φ > α where α = 3/2 h, h being the uniform mesh width. Also following the MCLS method, the density is kept sharp using the approach described in [8]. Time integration of the NS equations is performed using a pressure correction method. The convection terms in (2) are treated with the second order Adams-Bashforth method, the diffusion terms are treated with implicit Euler. Both systems of equations from the Poisson and Helmholtz problem (resulting from respectively the pressure correction method and the implicit integration of the diffusion terms) are solved iteratively by a Conjugate Gradient solver. An adaptive time step criterion is used, based on the CFL condition for convection and a restriction for capillary stability due to the surface tension force [14]. As our time-integration procedure treats viscous terms in (2) implicitly, these do not impose a restriction on the time step. Interface Advection Because numerical methods to solve (6) are diffusive, the LS function cannot be advected while conserving the volume enclosed by the zero LS. Therefore, the MCLS method combines LS advection with volume conserving VOF advection, to advance the interface from time step n to n+1. For details of the following description we refer to [16]: I. Given φn , compute volume fractions ψ n in each computational cell Ω using the relation: H(φL )dx , (10) ψ= Ω dx Ω where φL is the linearized LS function. Using the linearized LS function, analytic evaluation of the integrals in (10) corresponds to a piecewise linear interface construction in each cell. As a consequence, (10) computes volume fractions without the need to explicitly reconstruct the interface as in traditional VOF methods. However, the LS and VOF interface representations should remain coupled, which is enforced at the end of the advection step (IV.). II. Advect ψ n toward ψ n+1 , solving: ψt + ∇ · (uψ) = 0,
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by means of an operator-split approach [13]. Because (10) is able to compute volume fraction in any rectangular domain, it can be used to compute the volume fluxes (for the second term in (11)), without the need for explicit interface reconstruction. III.Advect φn toward φ∗ . First order methods are used for spatial and temporal discretization of (6), to keep gradients of the LS function smooth. IV. Correct φ∗ to obtain φn+1 such that H(φn+1 )dx n+1 Ω L − (12) ψ <ε dx Ω in each computational cell Ω at the end of each timestep. In this work, tolerance ε = 1 · 10−5 . In [5], increased levels of spurious currents, introduced by the LS/VOF correction procedure, were reported. As a remedy, we use the curvature filtering procedure suggested in [16]. 3.2 Implementation of Multiple Interface Representations To extend the MCLS method to deal with multiple domains of the same fluid without automatic merger of interfaces, multiple marker functions are defined. I.e. we still simulate two-phase flow, however using more than one LS function to represent separate volumes of the same fluid. Simulating K droplets, we introduce the same number of LS functions, φ1 , . . . , φK , such that each droplet k has a boundary defined from Γ k (t) = {x | φk (x, t) = 0} and an interior domain Ω k (t) = {x | φk (x, t) > 0} (figure 1). To calculate material properties, a common LS function to all interfaces, denoted φc , is determined: φc (x, t) = max φ1 (x, t), . . . , φK (x, t) , (13) giving ρ = ρ(φc ) and μ = μ(φc ) in (2). For representation of the surface tension force in (2), contributions from each of the interfaces need to be taken into account. Following the Continuous Surface Force methodology [2, 3], the surface tension force is obtained by summation of the separate contributions of all interfaces: 1 κ(φ1 )∇H(φ1 ) + . . . + κ(φK )∇H(φK ) . We
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4 Results To test the implementation of multiple interface functions, the method is used to simulate the impact of a buoyant droplet onto a liquid-liquid interface. In this problem, a droplet is released in a container of a heavier liquid. The droplet will accelerate upward due to buoyancy until it impacts on an interface between the heavier liquid and a layer of the same liquid as the droplet. Finally, after all fluid motion has come to rest, the droplet rests in a steady state configuration against the liquid-liquid interface. In our computation, material properties (table 1) correspond to refractive index matched mixtures of glucose/water for the heavy liquid and mineral oils for the droplet and top-layer as used in our experiment. For nondimensionalization (section 2), we use the properties of the droplet liquid, ρ2 and μ2 , the droplet diameter, D = 0.01 m, and droplet Stokes terminal velocity, Ust = 0.51 m/s (determined from the Rybczynski-Hadamard equation [4]). As a result we obtain a Reynolds number Re = 87.9, Weber number W e = 89.9 and Froude number F r = 2.66. Note: In the simulation, the droplet never reaches the free terminal Stokes velocity. Therefore, the actual magnitude of Re, W e and F r during our simulation is considerably lower. The computational domain and initial configuration of the phases is given in figure 2, where u(x, 0) = 0. To investigate convergence of our numerical code, simulations are performed on four different grids: 48, 72, 96 and 144 cells in each coordinate direction. In figure 3, snap-shots of the interfaces between the fluids over a midsection of the domain are presented at different times in the computation.
Table 1. Material Properties.
Table 2. Overflow on different meshes.
ρ1 1376 kg/m3 μ1 0.08 Pa s 865 kg/m3 ρ2 μ2 0.0036 Pa s σ 0.04 N/m
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Fig. 3. Time evolution of droplet impact. Figures display contours of the interface over a mid-section of the domain at t = 0, 3.7, 7, 11, 15.4, 40 (963 cells).
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rapidly. However, different convergence behavior is found for periods of high or low aspect ratio: When the aspect ratio is low, the maximum curvature over the interface perimeter (at the sides of the droplet) is much larger than the maximum curvature when the aspect ratio is close to unity and therefore more demanding to resolve. In figure 6, steady state configurations of the computation and the experiment are presented (1443 cells). In the experiment, the steady state aspect ratio measures 0.68 ± 0.02, whereas the aspect ratio we determine from our computations at 60 dimensionless time units is considerably lower for
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Fig. 6. Steady state configuration for computation (1443 cells) (left) and experiment (right).
all meshes. In figure 5, the aspect ratio at the end of computation appears to converge irregularly. Therefore, the results are not extrapolated toward a converged steady state aspect ratio for an infinitely dense mesh. To assess the performance of the multiple interface function approach, we monitor the combined volume fractions of droplet and top-layer in a cell. If the combined volume fraction, defined ψ c = ψ 1 + ψ 2 , is greater than unity, then the overflow, defined ψ c − 1, can be used as a measure of overlap of interfaces. In table 2, the maximum overflow is given, where the maximum is taken over all computational cells and all time steps of each computation. Table 2 shows that the magnitude of overflow is very small. Also, the amount of overflow is found to decrease rapidly with increasing resolution. Finally, we note that the maximum magnitude of overflow occurs only during a very small portion of the total time of simulation.
5 Conclusion We have introduced multiple marker functions in an existing LS/VOF method for the representation of multiple interfaces in a single computational cell. The approach is applied successfully to the problem of droplet impact on a liquidliquid interface. Visualizations of computational and experimental results for the droplet steady state configuration show good agreement. In the future, comparison between computational and experimental results will be extended to include time evolution of interface geometry and fluid motion. The concept of multiple marker functions could be applied to other frontcapturing methods as well. Because it is currently used in a combined LS/VOF method, an additional study could be conducted to investigate its application to plain LS or VOF methods.
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Acknowledgments This work was financially supported by the Computational Science Research Program of the Dutch Organization for Scientific Research (NWO). For the use of computing facilities, we acknowledge the National Computing Facilities Foundation (NCF). E.C. is obliged to Dr. Brady for hosting him at Caltech, where this work was completed.
References 1. E. Aulisa, S. Manservisi, and R. Scardovelli. A mixed markers and volume-offluid method for the reconstruction and advection of interfaces in two-phase and free-boundary flows. J. Comp. Phys., 188:611–639, 2003. 2. J.U. Brackbill, D.B. Kothe, and C. Zemach. A continuum method for modeling surface tension. J. Comp. Phys., 100:335–354, 1992. 3. Y.C. Chang, T.Y. Hou, B. Merriman, and S. Osher. A level set formulation of eulerian interface capturing methods for incompressible fluid flows. J. Comp. Phys., 124:449–464, 1996. 4. R. Clift, J.R. Grace, and M.E. Weber. Bubbles, Drops, and Particles. Academic Press, Inc., London, United Kingdom, 1978. 5. E.R.A. Coyajee, M. Herrmann, and B.J. Boersma. Simulation of dispersed twophase flow using a coupled Volume-of-Fluid/Level-Set method. In: Center for Turbulence Research - Proceedings of the Summer Program 2004, 315-327, 2004. 6. D. Enright, R. Fedkiw, J. Ferziger, and I. Mitchell. A hybrid particle level-set method for improved interface capturing. J. Comp. Phys., 183:83–116, 2002. 7. A. Esmaeeli and G. Tryggvason. Direct numerical simulations of bubbly flows. part 1. low reynolds number arrays. J. Fluid Mech., 377:313–345, 1998. 8. X.-D. Liu, R. Fedkiw, and M. Kang. A boundary condition capturing method for poisson’s equation on irregular domains. J. Comp. Phys., 160:151–178, 2000. 9. J. Lopez, J. Hernandez, P. Gomez, and F. Faura. An improved plic-vof method for tracking thin fluid structures in incompressible two-phase flows. J. Comp. Phys., 208:51–74, 2005. 10. D. Lorstadt and L. Fuchs. High order surface tension vof-model for 3d bubble flows with high density ratio. J. Comp. Phys., 200:153–172, 2004. 11. Y. Renardy and M. Renardy. Prost: A parabolic reconstruction of surface tension for the volume-of-fluid method. J. Comp. Phys., 183:400–421, 2002. 12. M. Rudman. A volume-tracking method for incompressible multifluid flows with large density variations. Int. J. Numer. Meth. Fluids, 28:357–378, 1998. 13. M. Sussman and E.G. Puckett. A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows. J. Comp. Phys., 162:301–337, 2000. 14. M. Sussman, P. Smereka, and S. Osher. A level set approach for computing solutions to incompressible two-phase flow. J. Comp. Phys., 114:146–159, 1994. 15. S.H. Unverdi and G. Tryggvason. A front-tracking method for viscous, incompressible multi-fluid flows. J. Comp. Phys, 100:25–37, 1992. 16. S.P. van der Pijl. Computation of bubbly flows with a Mass-Conserving Level-Set method. PhD thesis, Delft Universtity of Technology, 2005. 17. S.P. Van der Pijl, A. Segal, C. Vuik, and P. Wesseling. A mass-conserving levelset method for modelling of multi-phase flows. Int. J. Numer. Meth. Fluids, 47:339–361, 2005.
Direct numerical simulation of particle statistics and turbulence modification in vertical turbulent channel flow A. Kubik and L. Kleiser Institute of Fluid Dynamics, ETH Z¨ urich, Switzerland [email protected] Summary. Results are presented for the behavior of particle-laden gases in a moderate Reynolds number vertical channel down flow. The effects of particle feedback on the gas-phase turbulence and particle concentration are studied. A direct numerical simulation including models of wall-particle interaction was conducted. It is confirmed that particle feedback causes the turbulence intensities to become more non-isotropic as the particle loading is increased. The particles tend to increase the characteristic length scales of the fluctuations in the streamwise velocity, which reduces the transfer of energy between the streamwise and the transverse velocity components. The particle concentration exhibits a maximum close to the wall and a slight increase in the middle of the channel.
1 Background Particle-laden channel flows are found in a variety of natural settings and in many bio-fluidmechanical phenomena, and they are of great practical importance due to their frequent occurrence in engineering applications. Numerous experimental and computational studies related to particle transport in turbulent flows were reported in the literature. There have been a few applications of DNS to particle-laden channel flows, such as in [11] and [14]. Those studies usually focus on rather low Reynolds numbers (Reτ ≈ 120, based on friction velocity and channel half-width). Kulick et al. [10] reported results of experiments on particle-laden flows in a vertical channel down flow at Reτ ≈ 644. LES results for a channel flow at approximately the same Reynolds number are also available, e.g. [7], [15]. In the present study, a vertical turbulent channel flow (with gravity pointing in the mean flow direction) at moderate Reynolds number Reτ ≈ 210 is investigated by means of DNS.
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2 Methodology, numerical methods, and parameters The Eulerian-Lagrangian approach was adopted for the calculations in which the fluid phase is described by 3D time-dependent field equations whereas the particles are tracked individually. The two phases are coupled, as the fluid phase exerts forces on the particles and experiences a feedback force from the dispersed phase. The feedback force is added as an effective body force to the Navier-Stokes equations for the fluid phase. The equations are solved together with the incompressibility constraint using a spectral–spectral-element Fourier–Legendre code [17] with periodic boundary conditions in the wallparallel directions and no-slip condition at the channel walls. The coordinates in the channel are labeled as x for the streamwise, y for the wall-normal and z for the spanwise direction. The trajectories of the particles are calculated simultaneously in time with the fluid phase by integrating the equation of motion for each particle. Maxey and Riley [12] provided a modified Basset-Boussinesq-Oseen (BBO) equation for describing the particle motion. The lift force [13] has been used as an ad hoc added component of the total hydrodynamic force. According to [1] and [8] the BBO equation for particles in channel flow can be simplified to include only drag and gravity; lift force, however, becomes significant close to the walls. Empirical and analytical corrections for the drag and lift were necessary to accommodate moderate Reynolds numbers [5], [16] and the proximity of walls [6], [2], [16]. Particle-wall collisions are modeled taking into account the elasticity of the impact and particle deposition for low-velocity particles in regions of low shear [9]. Particle-particle collisions are omitted in this study and the parameter range for the calculations is restricted such as to keep this assumption valid.
3 Results In fig. 1(a), the mean fluid velocity u+ f is plotted in wall units against the distance from the closest channel wall y + for particle mass loadings (ratio of total particle mass to fluid mass in the computational domain) φ = 0, 0.1, and 0.2. The particles in this simulation have a Stokes number of St = 3.0 (based on the particle relaxation time and the turbulence timescale k/ at the channel centerline) and a particle-to-fluid density ratio of ρp /ρf = 7458, they are equivalent to the 70 μm copper particles in [10]. The mean velocity profile is unchanged by the presence of particles, as constant fluid mass flow was enforced in the simulation. (When forcing via constant pressure gradient is employed, the particles increase the mean fluid velocity, an effect which is enhanced with increasing particle loading.) The mean particle velocities are shown in fig. 2(a). It may be seen that the particles lag the fluid in the core of the channel, but lead in the near-wall region, resulting in profiles that are flatter than those of the fluid phase. This
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Fig. 2. (a) Mean particle velocity profiles in wall units for different particle mass loadings φ and mean fluid velocity profile for unladen flow. (b) Fluctuation intensities of the particles for different particle mass loadings and Reynolds stresses of the fluid (·)f , φ = 0; (·)p , φ = 0.1; (·)p , φ = 0.2; St = 3.0, for unladen flow. ρp /ρf = 7458.
phenomenon has also been observed in [10]. The slip velocity can be quite large, this leads to relatively large particle Reynolds numbers Rep (based on particle diameter). For both the copper particles in fig. 2(a) and the glass particles to be discussed later, the average Rep in the viscous sublayer was about 5. For the smaller copper particles, Rep was about 2. In the near-wall region the difference between the fluctuation intensities of the particles and the fluid Reynolds stresses is substantial, too (fig. 2(b)). (up up )1/2 /uτ are increased in respect to (uf uf )1/2 /uτ , this trend is most eminent at the maximum, which retains its location but almost doubles its value. The wall-normal fluctuating velocity of the particles near the wall is relatively large compared to the fluid. This provides partial explanation for the divergence of the particle velocity from the decreasing fluid velocity in this region and is consistent with the particle trajectory statistics in the y-z-plane, where high-speed particles move towards the wall and rebound, still carrying
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much of their streamwise momentum. (wp wp )1/2 /uτ show a decreased, leveledoff behavior. Fig. 1(b) shows the fluid turbulence intensities versus y + . The effect of the particles on the fluctuation intensities is non-isotropic: the disparity between the normal and spanwise fluctuations to the streamwise ones increases with φ. The maxima of the three intensities become broader as the particle mass loading increases. This behavior is also consistent with the experimental observations of [10].
Fig. 3. (a) Effect of particle inertia on the fluid turbulent intensities, φ = 0; φ = 0.2, St = 3.0, ρp /ρf = 7458; φ = 0.2, St = 1.0, ρp /ρf = 7458. φ = 0; φ= (b) Effect of density ratio on the fluid turbulent intensities, φ = 0.2, St = 3.0, ρp /ρf = 2119. 0.2, St = 3.0, ρp /ρf = 7458;
Fig. 3(a) displays the fluid turbulence intensities as a function of y + for particles with the same density (ρp /ρf = 7458) but two different values of St (St = 3.0 and 1.0). The particles with higher inertia, or, equivalently a larger particle diameter, have a smaller effect on the turbulent fluctuations than the smaller particles. A possible explanation for this phenomenon is that the particles with St = 1.0 respond more strongly to the coherent eddies and, hence, have a stronger feedback effect on the fluid-phase turbulence. In reality particle-particle collisions may also play a role in producing the phenomenon. Li et al. [11] showed that the collision rate rises an order of magnitude when the particle response time (equivalent to the Stokes number) is reduced to a third, while retaining the same mass loading. Particle-particle collisions can be expected to randomize the particle velocities and to increase the slip velocity between the particles and surrounding fluid. Thus, it is plausible that they would tend to increase the interaction between the particles and the fluid. Fig. 3(b) shows the fluid turbulence intensities as a function of y + for particles of different density, ρp /ρf = 7458 and ρp /ρf = 2119, corresponding to copper and glass, respectively; for both cases St = 3.0. The heavier particles have a slightly bigger influence on the velocity fluctuations than the lighter ones. Studies of the forces acting on the particles [1] and [8] indicate that there
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is only scant influence of the density ratio on the individual forces, provided that the particle-to-fluid density ratio is not as low as O(1). Thus, the feedback force exerted by the particles on the fluid must be independent of the particle density, as long as the latter is not low. The particles have noticeable effect on the streamwise correlation of the streamwise velocity. In fig. 4(a) the quantity Ru u (Δx+ ) = < uf (x+ + Δx+ , y + , z + , t+ ) × uf (x+ , y + , z + , t+ ) > / < uf >, 2
is plotted as a function of the streamwise separation Δx+ , the angle brackets denote averaging over time and a wall-parallel plane at a given y + . The discrepancy between the computed correlations for φ = 0 and 0.1 are comparable to the statistical uncertainty in the correlations. However, for φ = 0.2, the correlation is significantly higher than for φ = 0 for all Δx+ . The presence of the particles has a smaller influence on the spanwise correlation of the streamwise velocity, as can be seen in fig. 4(b). For all mass loadings, the correlation has a minimum for a spanwise spacing at approx. 50 wall units and a maximum at 100 wall units, consistent with the streak spacing in the viscous sublayer. Fig. 7(a) and (b) show instantaneous contours of the streamwise velocity fluctuations in the wall-parallel plane at y + = 38. Apparently, the particles have a smoothing effect on the fluctuations. 1
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Particles in the size range examined in this study have a tendency to accumulate near the walls even though they were homogeneously dispersed in the channel at the beginning of each simulation. This is apparent in fig. 6(a) which displays the particle concentration as a function of y + , averaged over wall-parallel planes at the final time of the simulation, and normalized by the initial particle concentration, which was uniform. In each case, the particle concentration near the wall (indicated by the value displayed in the plot) is significantly larger than in the core. Due to their inertia particles tend to move
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closer to the walls than the fluid elements that bring them into or near the viscous sublayer. Some particles strike the wall and rebound after the impact leaving the near-wall region, but other particles lack sufficient momentum to reach the closest wall and are confined to the viscous wall region for long periods of time. Therefore, the particles tend to have a higher residence time in regions close to the wall than in the channel core. Several other numerical studies, e.g. [3], [11], report this accumulation of particles near the walls of a vertical channel for a broad range of particle characteristics. The slight increase of particle concentration in the middle of the channel (y + ≈ 210) can be partly explained by the turbophoresis phenomenon [4]. Turbophoresis originates from small random steps taken by a particle in response to the fluid turbulence. Given a gradient in the the intensity of the turbulence, particles are inclined to migrate to regions of lower turbulence intensity, since they have a longer residence time in those regions. But the particles in this study have large values of St which limits their response to local turbulence and causes them to move along roughly straight lines over comparatively large distances. The concentration near the wall is larger for φ = 0.2 than for φ = 0.1. The particle velocity fluctuations close to the wall (fig. 2(b)) abate when the particle load rises. Thus the particle residence time in the wall-near regions increases with increasing φ even when St remains the same. The accumulation of particles in near-wall regions is diminished with increasing load when the particle-particle collisions are allowed for [11]. In fig. 6(b), the concentration profiles of for calculations with and without a wall-influence model are compared. The ’wall-influence model’ consists here of the corrections for forces in near-wall regions, elasticity of the impact and the possibility of deposition of low-velocity particles in regions of low shear. It may be seen that the particles exhibit a much greater tendency to accumulate near the walls when wall-influence models are eliminated. This is partly counterintuitive, since the increase in the drag coefficient slows the particles down and should increase their residence time in the near-wall regions. The corrected
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Fig. 6. (a) Normalized particle concentration, φ = 0.1 and φ = 0.2, St = 3.0, ρp /ρf = 7458. (b) Normalized particle concentration, φ = 0.1, St = 3.0, wall-influence model without the model. ρp /ρf = 7458,
Fig. 7. Normalized particle concentration, St = 3.0, ρp /ρf = 7458. (a) φ = 0.1. elastic, e = 1 and inelastic wall-particle collisions, e = 0.9. (b) φ = 0.2.
lift force reverses partly this tendency, since it is directed outward. Inelastic wall-particle collisions can also cause particles to accumulate near the walls. (Which would also increase the particle concentration for simulations with the discussed model.) However, for the cases considered in the present study, almost the same results were obtained when the coefficient of restitution e was chosen to be 0.9 instead of 1 as may be seen in fig. 7(a) and 7(b). Deposition is responsible for most of the effect, it eliminates the slowest particles reducing therefore the residence time of the particles and their total number.
4 Conclusions This paper has presented DNS results for the behavior of particle-laden turbulent flows in a vertical channel. For small mass loadings φ, there is a decrease in the fluctuation intensities of the normal and spanwise components of velocity while the streamwise component increases slightly. Although there is no significant increase in the spanwise streak spacing, the streamwise two-point
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correlation of the streamwise velocity indicate a suppression of small-scale fluctuations. The particle concentration exhibits a maximum close to the wall and a slight increase in the middle of the channel that appear to be caused, in part, by turbophoresis.
References 1. V. Armenio and V. Fiorotto. The importance of the forces acting on particles in turbulent flows. Phys. Fluids, 13(8):2437–2440, 2001. 2. H. Brenner. The slow motion of a sphere through a viscous fluid towards a plane surface. Chemical Engineering Science, 16:242–251, 1961. 3. J. W. Brooke, K. Kontomaris, T. J. Hanratty, and J. B. McLaughlin. Turbulent deposition and trapping of aerosols at a wall. Phys. Fluids A, 4(4):825–834, 1992. 4. M. Caporaloni, F. Tampieri, F. Trombetti, and O. Vittori. Transfer of particles in nonisotropic air turbulence. 32, 1975. 5. R. Clift, J. R. Grace, and M. E. Weber. Bubbles, Drops, and Particles. Academic Press, New York, 1978. 6. H. Fax´en. Die Bewegung einer starren Kugel l¨ angs der Achse eines mit z¨ aher Fl¨ ussigkeit gef¨ ullten Rohres. Arkiv Mat. Astron. Fys., 17(27):1–28, 1923. 7. K. Fukagata, S. Zahrai, S. Kondo, and F. H. Bark. Anomalous velocity fluctuations in particulate turbulent channel flow. Int. J. Multiphase Flow, 27:701–719, 2001. 8. A. Kubik and L. Kleiser. Forces acting on particles in separated wall-bounded shear flow. Proc. Appl. Math. Mech., pages 512–514, 2004. 9. A. Kubik and L. Kleiser. Particle-laden turbulent channel flow and particle-wall interactions. Proc. Appl. Math. Mech., 2005. Submitted. 10. J. D. Kulick, J. R. Fessler, and J. K. Eaton. Particle response and modification in fully turbulent channel flow. J. Fluid Mech., 277:109–134, 1994. 11. Y. Li, J. B. McLaughlin, K. Kontomaris, and L. Portela. Numerical simulation of particle-laden turbulent channel flow. Phys. Fluids, 13(10):2957–2967, 2001. 12. M. R. Maxey and J. J. Riley. Equation of motion for a small rigid sphere in nonuniform flow. Phys. Fluids, 26(4):883–889, 1983. 13. P. G. Saffmann. The lift on a small sphere in a slow shear flow. J. Fluid Mech., 22:385–400, 1965. 14. M. Soltani and M. Ahmadi. Direct numerical simulation of particle entrainment in turbulent channel flow. Phys. Fluids, 7(3):647–657, 1995. 15. Q. Wang and K. D. Squires. Large eddy simulation of particle-laden turbulent channel flow. Phys. Fluids, 8(5):1207–1223, 1996. 16. Q. Wang, K. D. Squires, Chen M., and J. B. McLaughlin. On the role of the lift force in turbulence simulations of particle deposition. Int. J. Multiphase Flow, 23:749–763, 1997. 17. D. Wilhelm, C. H¨ artel, and L. Kleiser. Computational analysis of the twodimensional–three-dimensional transition in forward-facing step flow. J. Fluid Mech., 489:1–27, 2003.
Large-eddy simulation of particle-laden channel flow J.G.M. Kuerten Department of Mechanical Engineering, Technische Universiteit Eindhoven P.O.Box 513, NL–5600 MB Eindhoven, The Netherlands [email protected] Summary. In this paper LES of particle-laden turbulent channel flow is studied for the dynamic eddy-viscosity model (DEM) and the approximate deconvolution model (ADM), each with a subgrid model based on defiltering of the fluid velocity in the particle equation of motion. The results show that this defiltering substantially improves the results for the particle behavior. Moreover, the results of ADM correspond better with results of direct numerical simulation than the DEM results.
1 Introduction In many turbulent flows particles are transported. The motion of the particles is affected by the fluid flow, for example by a drag force. If the particles are small compared to the smallest length scales of the fluid flow, a point-particle description can be employed [1]. The fluid is then modelled as a continuous phase, while for each particle an equation of motion is imposed. Direct numerical simulations (DNS) of particle-laden flows in simple geometries have been carried out in this way [2, 3, 4]. For single-phase flows large-eddy simulation (LES) has gradually become a more and more powerful tool. The development of more accurate subgrid modelling strategies, such as dynamic modelling [5] and approximate deconvolution models [6], has demonstrated the large potential of LES. In the last decade, LES has also been applied to particle-laden flows. The particle equation of motion always contains the fluid velocity and in an LES only the resolved part of the fluid velocity is known. In many examples the equation of motion for the particles is solved with the filtered fluid velocity [4, 7, 8] without incorporating a model for the subgrid scales. When the particle relaxation time is large compared to the Kolmogorov time scale this approach is justified [4]. However, in [9, 10] it has been shown that a phenomenon as turbophoresis, where particles move towards the walls of a channel by the effect of the turbulence, cannot accurately be predicted if the subgrid scales in the fluid velocity
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are disregarded in the particle equation of motion. The results depend on the subgrid model applied, but even for an ’optimal’ subgrid model, a substantial difference between the DNS and LES results remains, especially for particle relaxation times of the same order as the Kolmogorov time. It has also been shown [9, 10] that results improve in case a defiltered fluid velocity [11] is used in the particle equation of motion and an adequate subgrid model, such as the dynamic eddy-viscosity model (DEM) [5], is applied. However, the defiltering procedure proposed in [9] remains somewhat arbitrary, since the filter adopted has no relation with the filter applied in the LES. In contrast, in the approximate deconvolution model (ADM) [6] an explicit inverse filter is used in the LES itself. Application of the deconvolved velocity to both the fluid and particle equations would yield a more uniform method. Therefore, in this paper it will be studied whether the deconvolved fluid velocity of the ADM can also be used in the particle equation of motion. Results will be compared with results from DNS and from LES with the dynamic eddy-viscosity model. The paper is organized as follows. In the next section, the equations of motion and numerical methods for particles and fluid are formulated. In Sect. 3 results are shown for DNS and for the two different LES models, including results obtained with the defiltering procedure. Finally, in Sect. 4 some conclusions are stated.
2 Governing equations and numerical method In this section first the equations of motion and numerical methods for fluid and particles are described. Moreover, the defiltering procedure is elucidated. 2.1 Fluid The flow considered in this paper is incompressible turbulent channel flow. The Navier-Stokes equation is solved in the rotation form [12] ∂u + ω × u + ∇P = νΔu + F , ∂t
(1)
where ω = ∇ × u is the vorticity, P = ρpf + 12 u2 , ν is the kinematic viscosity, p the fluctuating part of the pressure and ρf is the fluid density. Finally, F is the mean pressure gradient, chosen constant in time and space in such a way that τ = 150, the Reynolds number based on the friction velocity, uτ , Reτ = Hu ν where H is half the channel height. In the DNS all relevant length- and time scales are resolved. In the streamwise and spanwise directions periodic boundary conditions are applied. Therefore, the use of a pseudo-spectral method is very convenient. In the two periodic directions a Fourier-Galerkin approach is applied, whereas a Chebyshevcollocation method is adopted in the wall-normal direction.
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The time integration is performed with a combination of the CrankNicolson method for the viscous and pressure terms and a third-order RungeKutta method for the other terms. The nonlinear term is calculated in a pseudo-spectral way and the 3/2-rule is applied. The computational domain has a size 2H in the wall normal direction, 4πH in streamwise direction and 2πH in spanwise direction. The number of Chebyshev collocation points equals 129, whereas 128 Fourier modes are used in both periodic directions. In the LES calculations an equation for a spatially filtered fluid velocity u is solved. Filtering of the Navier-Stokes equation for the fluid velocity leads to the turbulent stress tensor τij = ui uj − ui uj , which depends on the unfiltered fluid velocity and hence is unknown in an LES. In a large-eddy simulation the turbulent stress tensor is replaced by a subgrid model. In this work two subgrid models are considered: DEM [5] and ADM [6]. In the dynamic eddy-viscosity model τij = −Cd Δ2 |S(u)|Sij (u), where Sij is the rate of strain tensor, |S| = 12 Sij Sij and Δ is the filter width. The parameter Cd is dynamically adjusted to the local structure of the flow by the introduction of a test filter with filter width 2Δ and application of Germano’s identity [5]. The dynamic coefficient is averaged over the homogeneous directions. As a test filter the top-hat filter is applied. The basis of ADM is replacement of the unfiltered velocity by an approximation found from defiltering the filtered velocity, according to: τij = u∗i u∗j − ui uj , where u∗i
= QN u i =
N
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(3)
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and G is the filter kernel. Details of this filter and its implementation in the LES can be found in [6]. In order to represent the effects of the subgrid scales an extra regularization term is added to the Navier-Stokes equation: ∂u + ω ∗ × u∗ + ∇P = νΔu + F − χ(I − QN G)u , ∂t
(4)
where χ is dynamically adjusted in such a way that the kinetic energy contained in the smallest resolved scales remains constant in time [6]. The numerical method used for the LES is the same as for the DNS. The turbulent stress tensor is treated in the same way as the other nonlinear terms in the Navier-Stokes equation. The number of Chebyshev collocation points in the wall-normal direction equals 33, the number of Fourier modes in the streamwise direction equals 32 and in the spanwise direction 64. To conclude this section, the rms of the wall-normal velocity component, which has a large influence on the particle behavior, is shown in Fig. 1. The DNS results cannot directly be compared with the LES results. Therefore, the
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filtered DNS result is also included. It can be seen that filtering reduces the velocity fluctuations. In the near-wall region the filtered DNS result agrees quite well with the LES result for DEM. Near the center of the channel the agreement is less. On the other hand, the agreement of ADM with the filtered DNS is quite good over the whole channel width. Finally, it can be observed that the results without subgrid model (coarse grid DNS, which is stable at the low Reynolds number considered here) correspond better with the unfiltered than with the filtered DNS.
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2.2 Particles Particles are described by an equation of motion for each particle. In the present work only the drag force will be considered, which has been justified in [13]. Particle-particle interaction and the effect of particles on the fluid will be disregarded. Hence, the equation of motion for a particle i with instantaneous position xi , velocity vi and mass mi reads: u(xi , t) − vi dvi = (1 + 0.15Re0.687 ), p dt τp
(5)
where u(xi , t) is the fluid velocity at the position of the particle. The particle relaxation time τp quantifies the drag by the fluid on the particle and is given by: τp = ρp d2p /(18ρf ν), where dp is the particle diameter. In the simulations shown here, ρp /ρf = 769.23 and three different diameters are investigated: dp,1 /H = 1.02 × 10−3 , dp,2 /H = 2.28 × 10−3 and dp,3 /H = 5, 10 × 10−3 . This corresponds to Stokes numbers, defined as St = τp+ , of 1, 5 and 25.
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In the particle-laden simulations (5) is solved with the second-order accurate Heun method. In order to find the fluid velocity at the particle position, an interpolation has to be made. In this work, in the two periodic directions fourth-order Lagrange interpolation, and in the wall-normal direction, fourth-order Hermite interpolation is applied. The particle-laden simulations start from a statistically stationary state of fully-developed turbulence with 100,000 particles of each Stokes number randomly distributed over the channel. Particles collide elastically with the walls. Solution of (5) gives accurate results in DNS, but in the particle-laden LES simulations the fluid velocity present in (5) is unknown and simulations with the fluid velocity replaced by the filtered fluid velocity result in substantial deviations in statistical particle quantities compared to DNS. Subgrid effects on the particle motion can be incorporated by introducing a subgrid model in (5). In [9, 10] a subgrid model was proposed in which a defiltered fluid velocity instead of the filtered fluid velocity of the LES is used in (5). In DEM an approximate inverse of the top-hat filter is applied in order to defilter the velocity field. The inversion is performed in Fourier space for the two periodic directions, whereas in the wall-normal direction the inverse is approximated with a Taylor series up to second order in the filter width. In ADM the deconvolved velocity field u∗ defined in (3) is used with N = 5, just as in the filtered Navier-Stokes equation.
3 Results In this section results of the particle simulations are presented. DNS results will be compared with LES results for the two subgrid models DEM and ADM, both with the filtered and with the defiltered fluid velocity adopted in (5). Also results without subgrid model are included. A good subgrid model should yield results better than without subgrid model. The discussion will be restricted to two quantities: the rms of particle velocity fluctuations and particle concentrations. The former quantity is of interest for particle dispersion, whereas the latter is directly related to the phenomenon of turbophoresis. Although the mean wall-normal fluid velocity component equals zero, the mean wall-normal particle velocity is initially unequal to zero. This particle transport mechanism is caused by the inhomogeneity of the turbulent velocity fluctuations [14]. It can be shown from a perturbation expansion that for small values of the Stokes number vy = upy − τp
∂ 2 u , ∂y y
(6)
where up is the fluid velocity at the particle position. Initially, upy = 0 and vy will be directed towards the walls. After a long time a stationary particle concentration will be reached, in which vy = 0. Equation (6) shows that the
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particles are then preferentially located at positions where the fluid velocity is directed away from the wall. The statistically stationary particle concentration is reached at t+ ≈ 16, 000. As a first result in Fig. 2 the particle concentration near the wall is shown as a function of time for St = 1. To this end the computational domain is divided in 40 equidistant strips parallel to the walls and the number of particles in the strips closest to both walls is counted. The particle con-
DNS no model DEM DEM inverse ADM ADM inverse
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Fig. 2. Concentration of particles close to the wall as a function of time for St = 1.
centration is normalized in such a way that for a uniform distribution c = 1. Although these instantaneous results fluctuate in time, the results of ADM are generally better than the DEM results. Moreover, the defiltering improves the results substantially. In Fig. 3 the particle concentration is shown in the statistically stationary state for St = 5 and St = 25. The results is averaged over time from t+ = 16, 000 till 18, 000. For this quantity the result without subgrid model agrees quite well with the DNS result. This is not so surprising, since turbophoresis is caused by the rms of the velocity fluctuations in wall-normal direction, and this quantity is quite well predicted by the no-model results, as shown in Fig. 1. From Fig. 3 it can also be concluded that ADM performs better than DEM and that defiltering improves the results, especially for ADM at St = 5 and for DEM at St = 25. As a final result the root mean square of the streamwise particle velocity is plotted for St = 1 in Fig. 4. ADM with defiltering gives the best agreement with the DNS results. The DEM gives too high and no subgrid model too low velocity fluctuations. The agreement between inverse ADM and DNS is
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even better for the spanwise and wall-normal velocity fluctuations (not shown here), especially at the higher two Stokes numbers. For all quantities studied defiltering yields a substantial improvement and ADM is better than DEM.
4 Conclusions In this paper LES of particle-laden turbulent channel flow is studied for several subgrid models. It is shown that for the two subgrid models considered defiltering the fluid velocity in the particle equation of motion leads to quite good agreement with DNS results of particle concentration and velocity fluctuations. Moreover, the approximate deconvolution model yields better results than the dynamic eddy-viscosity model, for which some quantities are not better predicted than without subgrid model.
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3.5
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3
v x,rms
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Fig. 4. Rms of streamwise velocity for St = 1.
Acknowledgement This work was sponsored by the Stichting Nationale Computerfaciliteiten for the use of supercomputer facilities, with financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek. The author is grateful to Dr. A.W. Vreman for many discussions on this topic and to Dr. S. Stolz for providing information on ADM.
References 1. Maxey MR, Riley JJ (1983) Phys Fluids 26: 883–889 2. Marchioli C, Soldati A (2002) J Fluid Mech 468: 283–315 3. Marchioli C, Giusti A, Salvetti MV, Soldati A (2003) Int J Multiphase Flow 29: 1017–1038 4. Uijttewaal WSJ, Oliemans RVA (1996) Phys Fluids 8: 2590–2604 5. Germano M, Piomelli U, Moin P, Cabot WH (1991) Phys Fluids A3: 1760–1765 6. Stolz S, Adams NA, Kleiser L (2001) Phys Fluids 13: 997–1015 7. Yeh F, Lei U (1991) Phys Fluids A3: 2571–2586 8. Wang Q, Squires KD (1996) Int J Multiphase Flow 22: 667–683 9. Kuerten JGM, Vreman AW (2005) Phys Fluids 17: 011701 10. Kuerten JGM (2005) A subgrid model for large-eddy simulation of particleladen channel flow. In: Sommerfeld M (ed) Proceedings of the 11th workshop on two-phase flow predictions, ISBN 3-96010-767-4 11. Kuerten JGM, Geurts BJ, Vreman AW, Germano M (1999) Phys Fluids 11: 3778–3785 12. Canuto C, Hussaini MY, Quarteroni A, Zang TA (1988) Spectral methods in fluid dynamics. Springer, Berlin 13. Armenio V, Fiorotto V (2001) Phys Fluids 13: 2437–1440 14. Reeks MW (1983) J Aerosol Sci 14: 729–739
High Resolution Simulations of Particle-Driven Gravity Currents Eckart Meiburg1 , F. Blanchette1 , M. Strauss2 , B. Kneller3 , M.E. Glinsky4 , F. Necker5 , C. H¨artel5 ,, and L. Kleiser5 . 1
2 3 4 5
Dept. of Mechanical and Environmental Engineering University of california Santa Barbara Santa Barbara, CA 93106, USA [email protected] Nuclear Research Center, Negev, Beer Sheva, Israel University of Aberdeen, Scotland BHP Billiton Petroleum, Houston, USA Institute of Fluid Dynamics, ETH Zurich, Switzerland
Summary. High-resolution simulations of particle-driven gravity currents are presented. The governing equations are integrated numerically with a high-order mixed spectral/spectral-element technique. In the analysis of the results, special emphasis is placed on the sedimentation and resuspension of the particles, and on their feedback on the flow dynamics. Resuspension is modeled as a diffusive flux of particles through the bottom boundary. The conditions under which turbidity currents may become self-sustaining through particle entrainment are investigated as a function of slope angle, current and particle size, and particle concentration.
1 Introduction Particle-driven gravity currents are driven by a horizontal density difference caused by differential loading with suspended particles. From a practical point of view, it is of great interest to predict such features as the speed, run-out length, and sedimentation and resuspension behavior of particle-driven gravity currents For such predictions, in the past often simplified integral models or theoretical approaches based on shallow-water theory were employed, e.g. ([1]- ). The empirical input required for the validation of these models to a large extent stems from laboratory experiments in which global flow features such as the speed or height of propagating fronts are studied in prototype configurations. However, while the overall shape of the current and its front velocity are relatively easily monitored, velocity and concentration fields within the front are difficult to measure accurately in the laboratory. Here, highly resolved numerical simulations can help provide the missing information needed to close some of the gaps in knowledge. For density-driven gravity currents, we
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presented the first such high-resolution simulations [4, 5]. Particle-driven gravity currents were tackled by high-resolution simulations only recently by our group [6, 7]. In the present paper, we focus on the extension of this work to resuspending flows.
2 Governing Equations We concentrate on dilute, incompressible fluid flows laden with small, monodisperse particles whose density, ρ˜p , is significantly higher than the constant density of the fluid, ρ˜, (a tilde denotes a dimensional quantity here). The dispersed phase is assumed to be sufficiently dilute to have a negligible volume fraction (˜ c 1) so that the presence of particles can be neglected in the continuity equation and particle-particle interactions may also be neglected. In addition, the particles are assumed to have a much smaller aerodynamic response time than typical fluid flow time scales. Hence the particle velocity is given by the sum of the fluid velocity and the constant particle settling speed. To describe the particulate phase, we consider an Eulerian transport equation for the local particle volume fraction, c. The motion of the fluid phase, obeys the incompressible Navier-Stokes equations for a constant density fluid, augmented by a forcing term that accounts for the force that particles exert on the carrier fluid. We render the equations dimensionless, using the initial ˜ c0 )1/2 , ˜ g h/2˜ half-height of the current, h/2, and the buoyancy velocity u ˜b = (˜ ρp − ρ˜)/˜ ρ is the reduced gravitational acceleration and c˜0 is where g˜ = g(˜ the initial (uniform) particle concentration in the suspension. For strictly two-dimensional flows, we eliminate pressure terms by considering a stream function-vorticity description of the fluid motion[4]. Three dimensionless parameters govern the system. The dimensionless set˜s /˜ ub quantifies the relative importance of particle seditling velocity us = u mentation, where the particle settling speed, u ˜s , can be determined from the ˜ ν , and P´eclet particle size and density [8]. The Reynolds number, Re = u ˜b h/˜ ˜ κ, quantify the ratio of inertial terms to viscous and diffunumber, P e = u ˜b h/˜ sive terms, respectively, with ν˜ the fluid viscosity and κ ˜ the particle diffusion constant. Typical turbidites parameters yield a current Reynolds number of order 106 , which is well beyond the current reach of direct numerical simulations. However, as was shown in our earlier work [9] and experimentally [10], provided Re > o(1, 000), variations in Re only have a small effect on the overall features of the flow. Therefore, the simulations discussed below were carried out with a reduced Reynolds number Re = 2, 200. We also set the value of the reduced P´eclet number P e equal to that of the reduced Reynolds number. We use a lock-release model, where heavy fluid is initially confined to a small region, 0 ≤ x1 ≤ xf and 0 ≤ x2 ≤ 2. In order to model complex geometries, we allow for a spatially varying gravity vector [11]. A curvilinear coordinate system is thus simulated but second order curvature terms are
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Fig. 1. Structure of a particle-driven gravity current visualized by isosurfaces of concentration, c = 0.25. Results obtained from a 3D simulation for a Reynolds number of re = 2, 200 and a dimensionless settling velocity of us = 0.02. Here a mesh of 1440×200×221 grid points was used.
neglected. The resulting approximation was shown to be valid if the ratio of the height of the flow to the radius of curvature of the bottom surface is everywhere small [11]. The particle concentration flux at the boundaries, F is set to zero at the top, left and right walls. However, particles are allowed to deposit and reenter suspension at the bottom boundary: F = (−cus cos θ + Rs us ), where Rs is a measure of the resuspension flux empirically relating resuspension flux and particle Reynolds number and bottom shear velocity [12]. The influx of particles due to resuspension is therefore modeled as a turbulent diffusive flux, as small scale turbulent motions bring deposited particles into suspension. The numerical integration of the governing equations in two or three dimensions in the flow domain is accomplished by a numerical scheme based on spectral and spectral-element discretizations in space, along with finite differences in time, see [4, 6, 9] and [13] for details. We illustrate the spatial structure and temporal evolution of a threedimensional particle-driven front progressing over a horizontal surface in Fig. 1. Isosurfaces of constant particle concentration at different times are shown. Here the height and lateral size of the flow domain were set to H = l3 = 2 and we set Rs = 0 for simplicity. In order to enhance the breakdown of the flow into a 3D state after the release, weak turbulent disturbances in the velocity field are superimposed at time t=0 in the neighborhood of the interface.
3 Validation We have developed 2D simulations of particle-driven gravity currents and compared them to 3D simulations. While 2D simulations are appealing because of the lesser number of grid points required, they will also miss important
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flow features, such as the break-up of the interfacial vortices into turbulence. Comparing two- and three-dimensional simulations, we found that the early flow evolution produced essentially the same mean flow, despite the intense turbulence seen 3D simulations [4]. In contrast to the early flow stages, more substantial differences between the 2D and 3D results appear in the later flow evolution. A striking feature of the 2D flow is the emergence (and persistence) of large and energetic vortices which are absent in the 3D case. Although the 2D flow at late times differs more significantly from the 3D flow, we found that the final runout length of the front and the final deposition pattern of sediment are very similar in both cases. We also compared the deposit measured from an experimental particledriven current by DeRooij & Dalziel [14] to that obtained via our simulations. Our simulations make use of a reduced Reynolds number (2,200 compared to 10,000 in the experiment). The results were seen to be in fairly good agreement as to the extent and elevation of the deposit both for 2D and 3D simulations [6]. Differences were largest near the left hand wall and were probably attributable to variations in the initial conditions: continuous stirring in the experiment versus quiescent suspension in our simulations. Note that here resuspension was included but turned out to be negligible. Unfortunately, to the best of our knowledge, lock-release experiments in which resuspension plays a significant role have not yet been published in the literature. However, it may be seen that the size and density (d˜ ∼ 100μm, ρ˜p ∼ 2.5g/cm3 ) of particles subject to reentrainment at slope angles of order 5◦ for a current of typical velocity (∼ 1m/s) in our simulations are commensurate with available experimental data [12], [15]. In conclusion, the present results show that the early stage of the flow which is largely determined by the dynamics of the initial collapse is well described by a two-dimensional model. Here, three-dimensional effects, although important locally, have only a minor influence on overall features of the flow. On the other hand, caution should be exerted when 2D simulations are employed to study aspects of the long-time flow evolution, which may develop somewhat differently in 2D and 3D settings.
4 Results Influence of Resuspension Figure 2 shows an example of a strongly resuspending current. Here the slope angle is sufficiently large, θ = 5◦ , and the particle settling speed sufficiently small, us = 0.02, that the amount of resuspended particles exceeds that of deposited particles. Unresolved turbulent motions, modeled as a diffusive effect, are responsible for the high concentration observed below x2 ≈ 0.1, while the increase in particle concentration at higher levels is mostly attributable to the
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Fig. 2. Evolution of the particle concentration at time 5, 10 and 15, of a strongly resuspending gravity current. The color code is: 0.1 < c ≤ 0.5 yellow, 0.5 < c ≤ 0.8 green, 0.8 < c ≤ 1 red, 1 < c ≤ 3 cyan and 3 < c black. Other parameters are θ = 5◦ , c˜0 =0.5%, us = 0.02, Re, P e = 2, 200, xf = 2 and H = 2. Here a mesh of 1025×385 grid points was used, which corresponds to approximately 20 points in the lower boundary layer.
advection of the concentration through resolved fluid motions. As the current propagates downslope, its mass and velocity increase. In the early stages of motion (Fig. 2a), the current resembles a non-eroding gravity current and only a small boundary layer at the bottom exhibits a larger particle concentration than c˜0 . In both cases, mixing with clear fluid dilutes the upper part of the current and the vortices shed behind the front are very similar. In the presence of resuspension, the particle concentration increases near the front and the formation of a massive head is observed (Fig. 2b). The volume of the head does not change significantly as the current progresses, but it becomes progressively heavier as resuspended particles accumulate near the front and it thus propagates faster, generating further erosion. Self-Sustainment Criteria We now wish to characterize the conditions under which a gravity current is self-sustaining, i.e. increases in size as it travels downslope. We focus our attention on the effects of the initial (dimensional) height of heavy fluid, particle concentration and particle radius. For given flow parameters, we find that there exists a critical slope angle, θc , above which currents are erosional and increase in size and below which currents are depositional and eventually come to rest. We present in Fig. 3 the dependence of the critical slope angle θc on the heavy fluid height and initial particle concentration in (a) and particle radius in (b). The dimensional settling speed, buoyancy velocity, particle Reynolds number and resuspension factor are computed using the empirical ˜ d, ˜ and c˜0 . As expected, large critformulas [8, 12] with varying values of h, ˜ As either c˜0 ical slope angle are associated with small values of c˜0 and h. ˜ ˜ or h increases, the typical velocity of the current u ˜b = (˜ g h˜ c0 )1/2 increases,
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Fig. 3. a) Critical self-sustaining angle θc as a function of initial heavy fluid height, ˜ (solid line, top scale) and the initial particle concentration c˜0 , (dashed line, bottom h, scale). b) Dependence of the critical angle on particle radius (solid line) and 50/Rs on particle radius (dashed line). Parameters used in these simulations are Re = ˜ = 1.6m. P e = 2, 200, xf = 2, H = 4, d˜ = 100μm, c˜0 = 0.5% and h
and currents may thus generate larger bottom shear stresses. The influence ˜ particle reentrainment will affect low particle of c˜0 is weaker than that of h; concentration currents more readily as the relative particle concentration will then become larger, thus partially counteracting the fact that low values of c˜0 reduce the current velocity. Figure 3b shows that increasing the particle radius renders resuspension more difficult since large particle radii cause Rs to decrease. For comparison, we show the dependence on particle size of a typical value of the inverse of the resuspension factor (50/Rs , scaled for plotting purposes). Both curves are nearly parallel, indicating that Rs is the determinant factor in the selfsustaining quality of a current. In the parameter regime investigated here, we therefore find, by fitting the curves shown in Figs. 3a-b, that currents are self-sustaining if ˜ c˜0 5/3 h ˜ c0 )5/3 ( 1.6m sin θc sin θc (h˜ 0.5% ) ≈ 1
(1)
where K is a constant determined by the critical angle associated with our ˜ = 1.6m and c˜0 = 0.005. default parameter values d˜ = 10−4 m, h Broken Slope Currents We conclude by presenting an application of our model to resuspending turbidity currents traveling down a slope of varying angle. To simulate the base of the continental slope, we selected a geometry where the initial slope is 5◦ and the surface away from the source is horizontal.
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tip position
Fig. 4. a) Front velocity (solid line) and suspended mass (dashed line) as a vs the position of the nose of a current propagating over a broken slope. b) Dependence of the deposit height on the distance from the left wall at various times for the same current. Left of the first vertical dotted line the slope angle is θ = 5◦ , right of the second line it is θ = 0◦ . Other flow parameters are as in fig. 2.
In the early stages of motion, the current is erosional and its concentration increases near the lower boundary. However, upon reaching the horizontal bed, the current becomes depositional and eventually comes to rest. Figure 4a illustrates the dependence of the mass of suspended particles and front velocity on the position of the current tip. As the current travels downslope, its mass increases through erosion of the bed. The suspended mass continues to increase even after the nose has reached the flat surface, as most of the heavy fluid is still traveling downhill. At later times, all the heavy fluid overlies a horizontal surface and the current becomes depositional, causing the mass to decrease. The front velocity, after the initial slumping phase, increases while overlying a surface of sufficiently large slope angle. When the nose reaches the corner, the front velocity starts to decrease, showing that the local slope angle readily influences the front velocity. As the current spreads, the velocity keeps decreasing as particles are deposited. The corresponding deposition pattern is presented in Fig. 4b for different times. Particles are deposited near the left wall before the eroding character of the current develops as it moves downstream. The depth of the eroded region then remains nearly constant over the region of large slope angle. Near the corner, the current enters a depositional regime and leaves a deposit of maximum height at the beginning of the flat region. The deposit then decreases with distance from the corner. If a current transports sufficient particles, the geometry of the bottom surface may therefore be significantly altered. The cumulative effect of successive turbidity currents could then displace or create large topographic features and have important geological consequences.
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5 Conclusion We have discussed results from high-resolution computational studies of particle-driven gravity currents. The primary objective of the work is to analyze those flow features that cannot easily be studied theoretically or experimentally. We put special emphasis on the energy budget of the flow, on a comparison of two- and three-dimensional results, and on the dynamics of the resuspension process. Of particular interest is the ability of the flow to become self-sustaining, as a function of the bottom topography and the governing flow parameters. The computations demonstrate the existence of a critical slope angle that depends on both particle size and concentration. An important aspect to incorporate in future research is the polydispersity of the suspended particles, as well as the presence of particle-particle interactions, which may become important at the higher concentration levels near the particle bed. Efforts in these directions are currently underway.
Acknowledgments The authors wish to thank Vineet Birman and gratefully acknowledge the financial support of BHP Billiton Petroleum and the NSF.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Bonnecaze R, Huppert H, Lister J, (1993) J Fluid Mech, 250:339–369 Dade W, Huppert H (1995) Sedimentology 42:453–471 Hogg A, Ungarish M, Huppert H (2000) Eur J Mech B-Fluids 19:139–165. H¨ artel C, Meiburg E, Necker F (2000) J Fluid Mech 418:189–212 H¨ artel C, Carlsson F, Thunblom M (2000) J Fluid Mech, 418:213–229 Necker F, H¨ artel C, Kleiser L, Meiburg E (2002) Int J of Multiphase Flow 28:279–300 Necker F, H¨ artel C, Kleiser L, Meiburg E, accepted for publication. in J Fluid Mech Dietrich W, (1982) Water Resources Res 18 (6):1615–1626 Blanchette F, Strauss M, Meiburg E, Kneller B, Glinsky M (2005) accepted for publication in J Geophys Res Parsons J, Garc´ıa M (1998) Phys of Fluids 10:3209–3213 Blanchette F, Piche V, Meiburg E, Strauss M, accepted for publication in Computers & Fluids Garc´ıa M, Parker G, (1993) J Geo Res, 98:4793–4807 H¨ artel C, Kleiser L, Michaud M, Stein C (1997) J Eng Math 32:103–120. de Rooij F, Dalziel S (2001) Spec Publs in Ass Sediment 31:207–215 Garc´ıa M (1994) J Hydr Engr, 120:1240–1263
Implicitly-coupled finite difference schemes for fictitious domain simulation of solid-liquid flow; marker, volumetric, and hybrid forcing Hung V. Truong1 , John C. Wells1 , Gretar Tryggvason2 1
2
Departement of Civil & Environmental Engineering, Ritsumeikan University Noji Higashi, Kusatsu 525-8577 Japan [email protected] Department of Mechanical Engineering, Worcester Polytechnic Institute, USA
Summary. We propose fictitious domain algorithms for accurately and economically capturing the momentum exchange between dispersed solids and a liquid carrier phase. Instead of solving separately the momentum equations for each phase, we model the solid-fluid system as a variable density fluid mixture with artificial body forces applied inside particles to recover rigid motion. These rigidity constraint forces can be evaluated volumetrically within grid cells [1] and/or at Lagrangian markers forming a rigid template [2]. We present three variants: volumetric, marker and hybrid forcing schemes, and test them in the context of a freely moving disk in 2D Couette flow and a sphere dropped in a 3D bath. The new hybrid scheme appears to combine the strengths of the basic methods well. Key words: Particle-fluid simulation, immersed boundary, direct forcing
1 Introduction and governing equations Our group aims to simulate “bedload transport” of sediment in water, whereby a turbulent boundary flow induces particles to roll, slide, and saltate over deposited sediment. Available continuum models remain unverified; to be credible, we believe it necessary to track the motion of O(1000) 3D particles in a well-resolved turbulent flow field. To cover the important transition between hydraulically smooth and rough boundaries, particle Reynolds numbers in a range of roughly 20-1000 must be handled accurately. We report here on development of our basic solver. Consider a single free rigid particle submerged in a domain filled with an incompressible Newtonian fluid. One might briefly integrate the motion of the particle by modeling it as an extremely viscous droplet of fluid whose density matches the modeled particle. To handle longer times, it would be necessary to add an elastic restoring force f within the particle, and the governing equations would then be D(ρu) = ∇.(pI + μ(∇u + (∇u)T )) + ρg + f Dt
,
∇·u=0
(1)
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Such viscous and elastic forces, which serve to resist particle deformation, would be referred to as “feedback forcing” in the parlance of fictitious domain methods [3]. Alternatively, one might apply f in such measure as to recover a “target velocity” up that satisfies perfectly rigid motion within the particle up = Up + Ω p × r . Then f is referred to as “direct forcing” [4]; this is the approach taken herein. Formally f=
D(ρ up ) − [∇.(pI + μ[∇u + (∇u)T ) + ρg ] Dt
,
(2)
and the immediate issue is how to calculate the evolution of Up and Ω p . In the present context of a variable density solver, this issue is resolved easily thanks to the constraint that f only redistributes (angular) momentum within each particle ;
f dV = 0
,
r × f dV = 0
(3)
There has been considerable development of fictitious domain algorithms based on constant-density N-S solvers, perhaps because of the availability of fast pressure solvers. When the Stokes number is less than O(10) , the relative importance of particle inertia is sufficiently small that one can first solve the constant density Navier-Stokes equations for the entire particle-fluid domain based on known particle data, and then solve the particle equations based on the current fluid velocity field [1, 2]. Our recent comparisons of direct forcing methods in 2D flow [5] show however that as Stokes number increases, such “constant-density explicit coupling” methods become less accurate then variable-density solvers in capturing the initial acceleration of particles. Also, the latter yield more accurate particle trajectories in Couette flow and Poiseuille flow, which are important test cases for the highly sheared flow we are targeting. Note that multigrid methods for solving the pressure in the variable density case [6] have made computational time a minor issue. Accordingly, the present paper is restricted to methods based on variable-density flow solvers. We compare the performance of algorithms for calculating f based on volumetric and “marker” forcing; the latter is generally more accurate, but more costly. To combine their strengths, we combine these two approaches into a “hybrid” scheme, which is shown to be quite successful for the present test cases.
2 Numerical algorithms We employ a standard variable density Navier-Stokes solver based on a staggered grid to solve the equations [6]: central differences in space, forward Euler stepping in time and a SMAC treatment of pressure. Equation (1) is discretized from time level n to time level n+1 as: ρn+1 un+1 − ρn un + An = Fn − ∇h φn+1 + fn+1 Δt
(4)
where Δt is the time step, φn+1 is the pseudo-pressure correction, the subscript h denotes a discretized differential operator, An = ∇h · ρn un un is momentum advection andFn = ∇h · μn (∇h un + (∇h un )T ) − ∇h pn + (ρn − ρf )g is the intensive fluid force. Substeps for the fluid (F#) and particle (P#) phases are as follows: F1) Compute the momentum advection An and the intensive fluid force Fn .
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P1) Update particle cordinates, then calculate the new density(viscosity) field. ˜ by subtracting the moF2) Obtain the “fractional-step” momentum densityρn+1 u ˜ = ρn un + Δt (Fn − An ) mentum advection term An : ρn+1 u ˜. and thence determine the fractional step velocity u Δt ˜ ˜ = u ˜ − ρn+1 ∇φ, where φ F3) Project u ˜ to obtain a solenoidal velocity field: u satisfies the elliptic equation :∇h · n+1
n
∇h φ ρn+1 1
∇h · u ˜ Δt
=
. The pressure field is
n+1
then updated by: p =p +φ P2) Within each particle, calculate and apply the rigidity constraint forces. Details of substeps treating the particle phase are presented in the following subsections, for volumetric, marker and hybrid forcing respectively.
2.1 Variable density based Implicit Volumetric forcing (VIV) Equation (3) state only that the net (angular) forcing must be zero, but the modelled form of f has not been secified. In the “volumetric” direct forcing approach [1] , the rigidity constraint force density at each grid cell is: f = αρ
˜ ˜) (up − u Δt
(5)
where α is the solid volume fraction, depicted in Figure 1, and up is the target velocity at the corresponding grid point in a grid cell. Applying (3):
αρ
˜ ˜) (Up + Ω p × r − u =0 Δt
,
r × αρ
˜ ˜) (Up + Ω p × r − u =0 Δt
(6)
where h is the grid spacing and d is dimension of the space, then
˜ ˜ αρ u Up = αρ
˜ r × αρ u ˜ Ωp = αρ r · r
,
(7)
In this volumetric forcing approach, the forcing f is proportional to the volume fraction α, thus the new velocity in a momentum cell is the mass-averaged velocity of the partial step mixture velocity and the rigid velocity at the same location (VIV-e). As addressed in [2], such a method is unstable when particles cross streamlines. To smooth temporal and spatial changes in material properties, one can alternatively determine α from a smoothed distribution. In the present computations, the halfwidth of the smoothing fringe is set to 2 grid spacings (VIV-s)[5]. Particle steps P1) and P2) are detailed below for disks (spheres): P1) We use forward Euler steps to update the particle center, then update density (viscosity) fields;
n = Xn Xn+1 p p + Δt Up
,
ρn+1 μn+1
= (1 − αn+1 )
ρf μf
+ αn+1
ρp ρ μf ρfp
P2) The particle target velocity Un+1 and angular velocity Ω n+1 are calculated by p p eqs. (7), and the momentum density field is rigidified according to: ρn+1 un+1 = ˜ + Δ t fn+1 where fn+1 is calculated by eqs. (5). ρn+1 u p p
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Solid Partice
Grid cell
(1-α)r f V αrpV
Fig. 1. Solid volume fraction α in cell and a possible distribution of markers in the VIM-n scheme.
2.2 Variable density based Implicit Marker forcing (VIM) Suppose the rigid particle is represented by a set of markers Xm which are distributed nearly uniformly through the particle domain (VIM-n, Figure 1) or only at the particle surface (VIM-1). The term “marker forcing” is used when a value at a marker is forced to a desired value by applying a singular force at its position. Because the markers no longer coincide with the grid points, interpolation and spreading is necessary. In the present implementations, Peskin’s discretized delta function δ [7] was employed. The resulting forcing density in a cell is specified by: f(x) =
δ (Xm − x)F(Xm ) hd
F(Xm ) = mp
˜ m )) (up (Xm ) − U(X Δt
(8)
where F(Xm ) is the Lagrangian force at a marker, mp is the mass of the control area/volume of a marker, which is typically set proportional to the solid cell’s mass, ˜ m ) is the marker i.e. mp = ρp hd if each grid cell contains one marker, and U(X velocity which is obtained by : ˜ m) = U(X
uδ(Xm − x)hd
(9)
Analogously to the VIV scheme, the particle target velocity is given by: Up =
˜ U(Xm ) Nm
Ωp =
˜ m) (Xm − Xp ) × U(X (Xm − Xp ) · (Xm − Xp )
(10)
and then the marker target velocity is calculated by : up (Xm ) = Up + Ω p × (Xm − Xp )
(11)
P1) Step the particle center, calculate volume fraction and material properties like = Xn+1 + Δt Un step VIV/P1) above. Update markers positions: Xn+1 m m p. n+1 and Ω by eqs. (10), then calP2) Calculate the particle target velocity Un+1 p p (Xn+1 culate the marker target velocity un+1 p m ) at every marker by eqs. (11). The n+1 is calculated by eqs. (8) and added to the rigidity constraint force density fp ˜ + Δ t fn+1 fluid; ρn+1 un+1 = ρn+1 u . p The number of markers must be large enough to enforce rigid motion, i.e. at least one per grid cell. Interpolation and spreading are performed frequently at each marker, so the method is time consuming when there are many large particles.
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Table 1. Numerical and computational parameters in the simulation. Boundary conditions are wall-bounded for Case D# and periodic in x for Case C. Case C D3 D31 D130
Domain [0,8]×[0,4] [0,2]x[0,1]x[0,1] [0,4]x[0,1]x[0,1] [0,6]x[0,1]x[0,1]
Xp0 4.0 0.5 “ “
Yp0 2.0 0.5 “ “
Zp0 0.5 “ “
2R 2.0 0.4 “ “
ν 0.01 0.2 0.05 0.02
|g| 0.0 981.0 “ “
ρp /ρf 1.0 1.14 “ “
Uw 0.1 0.0 “ “
2.3 Hybrid scheme; VIV-VIM By combining VIM-1 with VIV-e, we aim to get a new scheme, which is as accurate as the VIM-n scheme but which incurs less computational cost. In this “VIV-VIM” scheme, volume forcing is applied to the core region, comprised of cells for which α = 1, while marker forcing is applied to markers distributed on the particle’s nominal surface. The rigidity constraint force density is specified as: f = αρ
˜ m )) ˜) (up (Xm ) − U(X (up − u δ (Xm − x) m p + Δt Δt
(12)
Determination of the rigid target velocity, and the numerical scheme, is similar to the previous two schemes.
3 Numerical results and discussions We have implemented the above numerical schemes for a single freely moving disk in 2D Couette channel flow (Case C) and a single sphere dropped in a quiescent 3D fluid flow(Case D#); conditions are summarized in Table 1 below.
3.1 A single freely moving disk in 2D Couette channel flow In this test problem we simulate a single neutrally buoyant circular disk placed initially in the centerline of a 2D Couette flow that is taken periodic in x. The channel dimensions, material properties and calculation parameters are summarized in Case C, Table 1 at Re=40 where Re is the bulk Reynolds number based on the shear gap and wall velocity Re = HUw /ν. Grid spacing was varied from 1/2 to 1/8 and the corresponding CFL ranged from 0.2 down to 0.15. The disk’s initial velocity and the initial velocity field are set to zero. Physically, the disk will migrate to an equilibrium height at the channel centre [9] with the terminal velocity equal to the average velocity of the two walls, and such a behavior was indeed observed in our computations when the walls moved in opposite directions at equal speeds. However, with the lower wall fixed and the upper one moving at a speed of Uw to the right, the disk migrated to an incorrect equilibrium position nearer the fixed wall , as shown in Figure 2a for different numerical schemes presented above.
20
VIV-e VIV-s V I M-1 V I M-n VIV-VIM
0.505 0.5 0.495
(a)
15 10
E xact value
(b)
Error(% ) , equilibrium height
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5
0.49 0.485 0.48
VIV-e VIV-s V I M-1 V I M-n VIV-VIM 2nd order
0.475
10
-1
20
30
40
0.6
0.5
0.4
(c)
15 10
5
VIV-e VIV-s V I M-1 V I M-n VIV-VIM 2nd order
0.5
0.4
0.3
0.2
0.2
0.1
(d)
VIV-e VIV-s V I M-1 V I M-n VIV-VIM 2nd order
Grid spacing 0.6
0.3
30 25 20
Error(% ) , terminal Velocity
10 0
10
Error(% ) , mean Inlet Velocity
10 1
Grid spacing
X*
0.47
Grid spacing 0.
0.6
0.5
0.4
0.3
0.2
0.
Fig. 2. (a) Trajectories of a neutrally buoyant disk in 2D Couette flow (a) at h=1/4 (corresponded to 8 grid points per particle diameter); (b) disk’s equilibrium height, (c) mean average velocity at the inlet of the channel (c) and disk’s terminal velocity (d) at h=1/2, 1/4, 1/6 and 1/8. The arrow in plot (b) indicates the grid resolution used for (a), which corresponds roughly to our target simulation of thousands of 3D particles.
Figure 2b, c, d presents the grid-refinement test. As shown in the figure, values predicted by the VIM-1 scheme deviate most from exact solutions at the finest resolution. This implies that using only surface markers to perform rigidification is not sufficient for most circumstances. The non-rigid velocity field within the particle domain. In the latter, the non-rigid velocity field in the particle core region may cause stronger deformation of the particle after each fluid step, thus increasing the local forcing at each marker. Also, errors may result when calculating the particle target velocity. When the markers are distributed regularly over the entire particle domain as in VIM-n scheme, the predicted values are much better, probably because such errors are reduced substantially. The VIV-e scheme’s predicted equilibrium height at the finest resolution, which deviates 0.60% from the centre of the channel, is larger than the 0.44% of the VIM-n scheme. Suggesting that the rigidity and boundary conditions at the particle surface is satisfied better in the VIM-n scheme than in the VIV-e scheme. Another known drawback of volumetric forcing schemes using exact volume fraction is that the calculated hydrodynamic force acting on the particle is oscillatory [2], which causes non-physical oscillation in the particle trajectory. The VIV-s scheme, although it
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Table 2. Comparison between our calculated and terminal velocities with those of ref. [10] for the dropped sphere. Re is our calculated terminal Reynolds number defined as : Re = 2RUT /ν. Case D3 D31 D130
Terminal velocity(UT ) Calculation ref. [10] Exp. 1.4 1.746 1.673 3.94 4.448 4.573 6.48 5.89 6.946
Relative error w. exp.(%) Calculation ref. [10] 16.3 4.36 13.84 2.73 6.71 15.17
Re 2.8 31.52 129.6
gives non-oscillatory solutions for some circumstances, e.g. when neglecting the rotational motion, is found here to be inadequate as an overall treatment for volumetric forcing approaches. Moreover, averaging and forcing over the smeared particle domain may slow down the convergence of the solution. Among the VIM-1, VIM-n, VIV-e and VIV-s schemes, VIM-n is the most accurate and the fastest to converge. However, VIM-n is also the most expensive due to the interpolation and spreading tasks at each marker, especially for large particles, i.e. O(20) grid spacings per particle diameter. Fortunatelly, we observe from Figure 2 that the VIV-VIM scheme is even more accurate than VIV-e and VIM-n schemes. The superior performance of VIM-VIV could be explained by the combinations of the strengths and eliminations of the weakness of VIV-e and VIM-1 schemes. In VIM-n, interpolations are required for getting values at any marker inside the particle domain, and the accuracy of these interpolated values depends on the interpolation method. In the VIV-e scheme, no interpolations are required for getting values at the grid points inside the particle domain, i.e. grid points which belonging to the cells having unity volume fraction, thus the order of accuracy for these values is the same as that of the fluid solver.
3.2 A sphere dropped in a 3D bath The hybrid VIV-VIM scheme has been used to simulate the motion of a rigid sphere dropped from rest in an incompressible viscous fluid. Numerical and computational parameters are given in Table 1, Cases D#. The grid spacing is h = 1/32, equivalent to 12.8 grid points per particle diameter. Comparison between our calculated terminal velocities with the data collected in [10] is reported in Table 2. Even at this coarse resolution, the difference of less than 7% (Case D130) from the experiment data, at a terminal Reynolds number of 129.6, supports the scheme’s accuracy at moderate particle Reynolds numbers.
4 Conclusion In the present paper, several numerical schemes building on the same variable density Navier-Stokes solver and direct forcing approach with different forcing strategies have been tested in 2D Couette flow. Our tests show that VIV schemes are easy-to
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implement and computationally efficient, but less accurate in imposing the boundary condition at the interface, VIM schemes account better for boundary condition but require significant CPU time for particle calculation steps when there are a large number of particles. The VIV-VIM scheme, which has been shown to be the best one with the combined strong points from VIV and VIM schemes, has been further tested by dropping a sphere in a 3D quiescent bath. Even at very coarse grid resolution, the VIV-VIM scheme can predict quite well the terminal velocity at a terminal Reynolds number greater than 100.
Acknowledgments This work has been supported by a grant-in-aid for scientific research from the Japanese MEXT Ministry, and by the Environmental Flows Research Group at Ritsumeikan University. We thank Dr. T. Kajishima for useful discussions.
References 1. T. Kajishima, S. Takiguchi, H. Hamasaki, Y. Miyake, Turbulence structure of particle-laden flow in a vertical plane channel due to vortex shedding,JSME Int. Journal, 2001, Series B, Volume 44,No. 4, 526-530 2. M. Uhlmann, An immersed boundary method with direct forcing for the simulation of particulate flows, J. Comp. Phys. 209, 2005, 448-476. 3. D. Goldstein, R. Handler, and L. Sirovich, Modeling a no-slip boundary with an external force field, J. Comp. Phys. 105, 1993, 354-366. 4. J. Mohd-Yusof, Combined immersed boundary/B-spline methods for simulations of flows in complex geometries, Annual Research Briefs,Center for Turbulence Research, Stanford, CA, 1997,317-327. 5. H.V. Truong, J.C. Wells, G. Tryggvason, Explicit vs. implicit particle-liquid coupling in fixed-grid computations at moderate particle Reynolds number, FEDSM2005-77206, ASME FED Summer Meeting, 2005, Houston, TX, USA. 6. G. Tryggvason, B.Brunner, A.Esmaeeli, D.Juric, N.Alrawashi, W.Tauber, J.Han, S.Nas and Y.J.Jan, A front tracking method for the computations of multiphase flow, J. Comp. Phys. 169, 2001, 708-759. 7. C.S. Peskin, The immersed boundary method, Acta Numerica 11,2002, 1-39. 8. N. Sharma,N. A. Patankar, A fast computation technique for the direct numerical simulation of rigid particulate flows, J. Comp. Phys. 205, 2005, 439-457. 9. J. Feng, H.H. Hu, D.D. Joseph, Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 2. Couette and Poiseuille flows, J. Fluid Mech. 277, 1994, 271-301. 10. R.Glowinski,T.W. Pan, T.I. Hesla, D.D Joseph, J. Periaux, A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies : application to particulate flow, J. Comp. Phys. 169, 2001, 363-426.
Part VII
Wall Models for LES
Development of Wall Models for LES of Separated Flows Michael Breuer1 , Boris Kniazev1 , and Markus Abel2 1
2
Institute of Fluid Mechanics, University of Erlangen–N¨ urnberg, D–91058 Erlangen, Germany breuer/[email protected] Institute of Physics, University of Potsdam, D–14415 Potsdam, Germany [email protected]
Summary. A key technology for the application of LES to high Reynolds number flows of engineering interest is an appropriate wall modeling strategy which bridges the near–wall region. Standard wall models deliver reliable results for attached but not for separated flows. In the present paper statistical evaluations of wall–resolved LES data were used to propose new wall models being able to predict separated flows reliably. As test cases the flow over a periodic arrangement of hills at Re = 10, 595 and the plane channel flow at Re = 10, 935 were considered. First a–posteriori results show that the new models provide accurate predictions for attached and separated flows. Finally, possible generalizations of the new models are discussed. Key words: LES, wall models, separated flows, statistical analysis
1 Introduction LES is a highly promising technique for the prediction of complicated turbulent flows including large–scale flow phenomena such as massive separation [1]. However, despite several years of development, most flows cannot be computed by LES because of the still extremely large resources required, especially for technical applications. One possibility to reduce the costs are appropriate wall models, which bridge the near–wall region and allow to place the first grid point far away from the wall avoiding the expensive DNS–like resolution. Wall models such as the classical one of Schumann [2] work well for flows within simple geometries, e.g., the plane channel flow where the averaged quantities required can be calculated. Schumann’s model reads: τ w = [ |τ w | / |vtan | ] vtan . Hence, a phase coincidence of the instantaneous wall shear stress τ w and the tangential velocity vtan at the wall–nearest grid point is assumed. More complex three–dimensional flows with large pressure gradients or local separation are commonly not well reproduced. For such flows the mean values ( |τ w | , |vtan | ) required are not accessible and the validity of the law of the wall used is questionable. The model of Werner and Wengle [3]
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can be defined in the form τ w = f (|vtan |) vtan . Thus it uses the function f (|vtan |) which is expressed by a law of the wall applied to the instantaneous quantities instead of the time–averaged ones used by Schumann. Therefore, this wall model is also not able to predict separated flows correctly. The main reason is the dependence of the wall shear stress τ w on a number of additional quantities such as the pressure gradient ∇p.
2 Wall Modeling Using Statistical Data Analysis The objective of the present work is to develop wall models which are more generally applicable and can be used for predictions of separated flows. For this purpose, possible generalizations of the classical wall models were tested which take only instantaneous quantities into account, leading to the approach τ w = f (v, ∇p) vtan . To find promising dependences f (v, ∇p) an a–priori data analysis was carried out, inferring conditional statistics [4, 5] from highly resolved LES data [6] obtained by the finite–volume code LESOCC [1, 7]. Hence the procedure is as follows: 1. data generation on a fine grid, 2. data analysis, 3. model derivation, 4. a–posteriori tests on a coarse grid. As the first reasonably complex test case (besides channel flow) the flow over periodic hills was chosen [8, 9]. In contrast to the original experimental setup by Almeida et al. [10], a modified configuration was used, which is better suited for numerical simulations. Data from a wall–resolved reference solution [6] with about 12.4 × 106 control volumes (CVs) were analyzed. In order to obtain reliable statistics, eight data blocks separated in time were considered. Every data block consisted of 125 time steps. The time interval Δt between two time steps included in the block was about 0.01 flow–through time, which corresponds to 50 time steps done within the fine LES. Only points belonging to one grid layer were taken into account. The appropriate grid layer was chosen in such a way, that it provided the best approximation of the first layer close to the wall of the coarse grid with about 1.3 × 106 CVs used for the a–posteriori tests of the wall models. Only the points located in the first half of the computational domain were taken into account, i.e., the area containing the separation region (see Fig. 2). Before using complicated functions f (v, ∇p) simple dependences should be tested. The first choice are the projections of the vectors v and ∇p on lines as well as the absolute value of the projections of these vectors on planes, i.e., f (vnor ), f (|vtan |), f (∂p/∂n), f (|∇ p|) and f (∂p/∂t). Here n denotes the unit vector in wall–normal direction. The unit vector t = vtan /|vtan | points to the direction of the tangential velocity vtan and |∇ p| denotes the absolute value of the projection of the pressure gradient ∇p on the tangential plane. Consequently, five different formulations are possible. The model based on the function f (|vtan |) is known as Werner and Wengle’s model [3] and hence not investigated here. To demonstrate that it is not reasonable to use the function f (|∇ p|) or f (∂p/∂t), the special case of the
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viscous sublayer is considered in which the direction of the tangential velocity vtan and the pressure gradient ∇ p should coincide. In this case it follows that ∇ p = (∂p/∂t)t. If the wall distance Δn is small, the wall shear stress τ w can be closely approximated using the tangential velocity vtan and the pressure gradient ∂p/∂t: ∂p Δn vtan − t = f (∂p/∂t) vtan . (1) τw = μ Δn ∂t 2 This relation derived by integrating the momentum equation in the direction of the tangential velocity vtan shows that appropriate functions f (∂p/∂t) or f (|∇ p|) do not exist. Thus, the following new models based on the variables vnor and ∂p/∂n remain and are considered at first: Model VN: τ w = f (vnor ) vtan ,
Model PN: τ w = f (∂p/∂n) vtan . (2)
It should be noticed that although the pressure gradient ∇ p is exceedingly important for separated flows, it is not used explicitly by the new models. However, this pressure gradient is accounted for implicitly since the variables vnor and ∂p/∂n strongly depend on it. The scatter plots |τ w |/|vtan | = f (vnor ) and |τ w |/|vtan | = f (∂p/∂n) provided by the reference solution [6] were smoothed using the polynomial regression. Polynomials of order 9 turned out to be sufficient for all applications. The curve–fits for these relations exhibit local minima. The reason for this observation may be due to the fact that Schumann’s assumption concerning the alignment of τ w and vtan is not always reasonable for flows with separation. This misalignment was approved by an additional stochastic analysis. The curve–fits were approximated by quadratic functions f (vnor ) = (vnor − a)2 + b and f (∂p/∂n) = c(∂p/∂n + d)2 + b used in the wall models with a = 0.03, b = 0.009, c = 0.081, d = 0.041 (see Fig. 1). Thereby an exceedingly accurate approximation in the vicinity of the minima is very important. Otherwise the calculations are not able to predict the separation and reattachment lines precisely.
3 Results of A–posteriori Tests for the Hill Flow The predictions based on the improved models VN and PN produced larger separation regions and generally fitted better to the reference solution than the results of classical models. Fig. 2 displays the streamlines of the time– averaged flow and Fig. 3 shows profiles of the mean dimensionless velocities as well as turbulent statistics at x/h = 2 which is located in the main recirculation region. The difference between the results provided by the classical and new models is essential. The calculations with the new models predict the separation line very closely, reproduce well the form of the separation region and provide overall much better statistics for the mean velocities as well
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Fig. 2. Mean streamlines for Schumann’s model and both new wall models VN and PN in comparison to the reference solution. Werner and Wengle’s model delivers similar results as Schumann’s model, i.e., xsep /h = 0.434, xreatt /h = 3.655.
as the Reynolds stresses. Especially, the new model PN delivers results in close agreement to the reference solution. Note that the grid used for these a– posteriori tests overall is about 10 times coarser than that of the wall–resolved LES and especially near the lower wall the cell height is 15 times larger. Hence the first grid point is far outside the viscous sublayer.
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4 Generalization of the New Wall Models for Hill Flow The next step is to generalize the wall models devised. In the following only the model PN is considered since the derivation for the model VN is analogous. As expected, the results of the statistical analysis of the data for the lower and upper walls of the hill flow case are completely different (see Fig. 4). Contrarily,
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Fig. 4. Comparison of the functions f (∂p/∂n) for both walls of the hill flow case and the plane wall of the channel flow at two different wall distances. 300
Wall Distance dn/h = 0.013, Re= 5,600 Wall Distance dn/h = 0.027, Re= 5,600 Wall Distance dn/h = 0.066, Re= 5,600 Wall Distance dn/h = 0.013, Re=10,59 5 Wall Distance dn/h = 0.027, Re=10,59 5 Wall Distance dn/h = 0.066, Re=10,59 5
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the functions f (∂p/∂n) obtained for the upper wall of the periodic hill flow and the wall of the plane channel are essentially the same. To investigate the dependences of the function f (∂p/∂n) on the Reynolds number Re and the wall distance n, series of statistical evaluations were carried out. Among others the functions Re · f (∂p/∂n) were considered. It was found that the functions Re · f (∂p/∂n) corresponding to Re = 10, 595 and Re = 5600 can be transformed one into another using a simple shift by a constant, which does not depend on the wall distance n (see Fig. 5). For the functions Re · f (vnor ) analogous results were obtained. Therefore, it seems to be possible to find the dependences of the functions Re·f (vnor ) and Re·f (∂p/∂n) on the Reynolds number Re and the wall distance n, if more reference cases are taken into account. Thereby a consideration of the hill flow at higher Re can provide an important evidence about the dependences desired.
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The generalizations of the wall models devised should contain scaled variables. It is questionable, whether the classical scaling variable y + can be used for flows in which uτ varies according to the position and vanishes at the separation line. But this problem disappears, if the plane channel flow is considered. Thus, it is reasonable to investigate the plane channel flow first.
5 Generalized New Wall Models for Plane Channel Flow The functions f (vnor ) and f (∂p/∂n) were found by analyzing the dependences of the variable |τ w |/|vtan | on vnor and ∂p/∂n, respectively. A scaling constant s = |vtan |/|τ w | can be chosen so that the mean value of the variable s |τ w |/|vtan | is equal to 1. It should be noticed that the scaling constant s depends on y + . For Re = 10, 935 and 22,776 the dependences of the scaled quantity s |τ w |/|vtan | on the variable ∂p/∂n were statistically evaluated. The curves obtained for varying y + values intersect in one point but are not exactly the same for different Reynolds numbers (not shown here). Therefore, the weak dependence on Re (see Fig. 6) should be the object of further investigations. It should be kept in mind that the evaluation error grows with the Reynolds number because the magnitude of the fluctuations increases. Thus, the dependences found by the statistical evaluation are always weaker than the physical dependences and the deviation grows with the Reynolds number. Moreover, it should be studied if an appropriate scaling of the variables vnor and ∂p/∂n can be found in order to reduce the difference between the functions corresponding to the same wall distance y + but different Re. The results of a–posteriori tests using the new wall models at Re = 10, 935 did not show remarkable advantages in comparison to the classical wall models due to the fact that Schumann’s wall model already provides satisfactory predictions for the plane channel flow, which can not be significantly improved.
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6 Conclusions Using statistical evaluations of highly resolved reference data, new wall models denoted VN and PN were proposed which read τ w = f (vnor ) vtan and τ w = f (∂p/∂n) vtan . These new models take only instantaneous quantities into account. Based on a–posteriori tests, it was shown that in contrast to classical wall models the new models are able to predict the separated flow over periodical hills at Re = 10, 595 reliably. Furthermore, they reveal the same order of accuracy as the classical models for the plane channel flow at Re = 10, 935. For the lower wall of the hill geometry it was shown that the difference between the function Re · f (∂p/∂n) for two different Re is constant and does not depend on the wall distance y. For the channel flow the difference between the functions f (∂p/∂n) for different Re obtained at the same y + is small and can be explained by the error made in course of the statistical evaluations. Analogous results were obtained for the function f (vnor ). Generalizations of these models are the objective of further investigations.
References 1. Breuer, M. (2002) Direkte Numerische Simulation und Large–Eddy Simulation turbulenter Str¨ omungen auf Hochleistungsrechnern, Habilitationsschrift, Univ. Erlangen–N¨ urnberg, Berichte aus der Str¨ omungstechnik, ISBN: 3–8265–9958–6. 2. Schumann, U. (1975) Subgrid–Scale Model for Finite–Difference Simulations of Turbulent Flows in Plane Channels and Annuli, J. of Comput. Physics, 18:376– 404. 3. Werner, H., Wengle, H. (1993) Large Eddy Simulation of Turbulent Flow Over and Around a Cube in a Plate Channel, 8th Symp. on Turb. Shear Flows, Durst et al. (eds.), pp. 155–168, Springer Verlag, Berlin. 4. Adrian, R. J. (1979) Conditional Eddies in Isotropic Turbulence, Phys. of Fluids, 22(11):2065–2070. 5. Langford, J., Moser, R.D. (1999) Optimal LES Formulations for Isotropic Turbulence, J. of Fluid Mech., 398:321–346. 6. Breuer, M. (2005) New Reference Data for the Hill Flow Test Case, personal communication, http://www.hy.bv.tum.de/DFG-CNRS/. 7. Breuer, M. (1998) Large Eddy Simulation of the Sub–Critical Flow Past a Circular Cylinder: Numerical and Modeling Aspects, Int. J. for Num. Methods in Fluids, 28: 1281–1302. 8. Jakirli´c, S., Jester–Z¨ urker, R., Tropea, C. (eds.) (2001) 9th ERCOFTAC/IAHR/COST Workshop on Refined Flow Modeling, Darmstadt University of Technology, Germany, Oct. 4–5, 2001. 9. Manceau, R., Bonnet, J.-P. (eds.) (2002) 10th Joint ERCOFTAC (SIG– 15)/IAHR/QNET–CFD Workshop on Refined Flow Modeling, Universit´e de Poitiers, France, Oct. 10–11, 2002. 10. ERCOFTAC Database, Classic Collection, Test case C18, 2D Model Hill Wake Flows, http://cfd.me.umist.ac.uk.
Anisothermal Wall Functions for RANS and LES of Turbulent Flows With Strong Heat Transfer Antoine Devesa1 and Franck Nicoud2 1
2
CERFACS, 42 Avenue Coriolis, 31057 Toulouse cedex 1, France [email protected] Universit´e Montpellier II, CNRS UMR 5149, I3M – CC 51, Place Eug`ene Bataillon, 34095 Montpellier cedex 5, France [email protected]
1 Introduction Various types of flows (e.g. in aeroengines, nuclear reactors) present very large temperature differences, implying density variations. In those flows, the correct prediction of thermal fluxes at the walls is a crucial problem, because materials can be submitted to contraction and dilatation phenomena that can damage them. As it is well known in the field of LES and DNS, the computational cost of wall bounded flows strongly depends on the modeling used for the walls, thus on the mesh refinement in the near wall regions. Consequently, DNS or wall-resolved LES can only be used for moderate friction Reynolds numbers, while using low order wall-modeling allows to resolve high Reynolds number flows with a reasonable cost. Reliable wall-modeling in highly anisothermal configurations at low Mach numbers (i.e. under low compressible fluid approximation) has not been reached yet, the main reason being that the wall fluxes depend on all the details of the turbulent flow in the vicinity of the solid boundaries and that measuring such details for high temperature flows is very challenging from an experimental point of view. Still, relevant reference data are required to support/test the physical assumptions made during the development of new wall functions. In this context, DNS appears as a good candidate to generate such data. DNS results from Nicoud [1, 2] in combination with the use of the Van Driest tranformation [3, 4] are used in Section 2 to derive an anisothermal wall function for RANS. Section 3 describes how the anisothermal properties of this wall function can be integrated in a more complex wall model, namely the TBLE model [5], in the framework of LES. Both models have been
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implemented in the Trio U CEA (French Nuclear Energy Agency) code and first results have been produced.
2 Derivation of an anisothermal wall function for RANS simulations 2.1 About the Van Driest transformation The Van Driest transformation [3] is based on the calculation of a modified dimensionless wall coordinate UV+D by: UV+D =
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Note that the mass-weighted quantities are considered here, based on the Favre decomposition [6]. The dimensionless wall coordinate U + stands for /Uτ , where Uτ is the friction velocity. U This transformation has been verified in various turbulent channel flow, particularly in the DNS results of Nicoud [1, 2], even in the case of very high temperature gradients, to match the standard logarithmic law [7], so that one can admit that: UV+D =
1 ln y + + C κ
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The DNS results of a periodic turbulent channel flow with variable density, at a friction Reynolds number Reτ = 180, from Nicoud [1, 2] show that the ratio between the local temperature T and the wall temperature Tw is a simple linear function of the product P rt Bq U + . P rt denotes the turbulent Prandtl number, and Bq = Tτ /Tw (Tτ is the friction temperature): 1 Tw = + C − Pr 1 t Bq U T
(3)
where C1 is a constant. Even if C1 = 1 has been proven to be a reasonable choice [8], this constant will be expressed by identification later on in this section. At this point, for algebraic reasons, we choose C1 as (K being a constant): (4) C1 = 1 − Prt Bq × K Inserting Eq. (3) in the Van Driest tranformation, and using the notation ΔT = (T − Tw )/Tw , we can integrate Eq. (1): 1 2 (U + + K) √ KΔT + = ln y + + C (5) 1 + ΔT − 1 + + UV D = ΔT U +K κ
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2.2 Determination of the constant K When ΔT → 0, it is trivial to demonstrate that the anisothermal wall function recovers the standard law for the velocity, for all values of K. In terms of temperature, the standard law valid for the quasi-isothermal cases is Kader’s law [9] (obtained from a flat plate configuration). In the logarithmic layer, the Kader formula leads to:
with:
T + = 2.12 ln y + + β(Pr) 2 β(Pr) = 3.85Pr1/3 − 1.3 + 2.12 ln(Pr)
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The identification of the two functions given by the relation T + = Prt (U + + K) from Eq. (3) and Eq. (6) has been made at y + = 100, in an arbitrary way. We find: β(Pr) 2.12 1 ln(100) (7) + − K = K(P r) = −Cu,vd + Prt Prt κ The constant K has then be expressed, and the anisothermal wall function that has been derived here can be described in its final form by the set of three equations: Eqs. (3)–(5)–(7). 2.3 Expression of the turbulent kinetic energy k and dissipation In a RANS simulation, using the k − model, the use of a wall function implies that the values of k and are adapted from the wall characteristics, namely Uτ and the wall-normal velocity gradient. In the anisothermal regime, the expression of the turbulent kinetic energy is: ρw Uτ2 ) (8) k= ρ Cμ This expression of k is the standard expression for the turbulent kinetic energy in the first off-wall point, multiplied by the density ratio, which is simply 1 + ΔT . Consequently, in quasi-isothermal flows, the density ratio tending to unity, the expression of k recovers the standard one. As the logarithmic law is not valid in anisothermal flows, this velocity + gradient is recalculated from Eq. (5). Noting B = 1 + (K(P r)Uτ ΔT )/(U K(P r)Uτ ), the dissipation can be expressed as: √ −1 ρw 2 2B 1/2 K(P r)B −1/2 2 1 + ΔT U κy = − + (9) + K(P r)Uτ ρ τ ΔT Uτ ΔT Uτ U In the vicinity of ΔT → 0, the density ratio has yet been identified to tend to unity. The parenthesis term of Eq. (9) in a quasi-isothermal regime becomes Uτ−1 . It is consequently found that the dissipation recovers the standard expression.
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2.4 Academic test case and results The main academic test case investigated to validate the anisothermal wall function is a turbulent bi-periodic channel flow. Firstly, the considered flow is quasi-isothermal, i.e. submitted to a small temperature gradient near the wall. Then, three anisothermal cases, with increasing temperature gradients, are presented. The change of the temperature gradient is obtained by changing the wall temperature ratio T2 /T1 , T1 being the bottom wall temperature (cold) and T2 the top wall temperature (hot). The friction Reynolds number Reτ = yUτ /νw is equal to 20, 000, which is a high enough value to generate a turbulent flow representative of engineering applications. Unfortunately, few reliable experimental or numerical results are available for such a value. The first off-wall point, where the wall function applies, is located around y + = 100. A total of 33 nodes composes the wall normal refinement. The simulations were performed with the CEA homemade CFD Trio U code [10], validated elsewhere, following a Low Mach number method, that is in a Quasi-Compressible approach (QC), accounting for dilatable fluids without solving the acoustics. • Quasi-isothermal case: Here, we verify that the anisothermal wall function really recovers a standard behaviour. The temperature ratio is close to 1 (T2 /T1 = 1.01). The dimensionless U + and T + profiles across the channel flow are displayed in Fig. 1. No difference between the different calculations is detected. Moreover, the velocity profiles fit well the standard logarithmic law. The same result is obtained for the temperature, comparing the results to Kader’s law. 30
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• Anisothermal cases: Three anisothermal flows were investigated, with T2 /T1 = 1.33, 1.66 and 4. The values of the friction velocity and friction temperature are listed in Table 1. The standard wall function over-estimates the wall heat flux, which is an established fact. The coupling between the dynamics and thermal properties
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is observed by the significant changes of Uτ in the different cases. Fig. 2 shows the dimensionless velocity and temperature profiles for the case T2 /T1 = 4. Note that the logarithmic and Kader’s laws are not valid in this case, but are shown as a reference. For large temperature gradients, the change of wall-modeling leads to significant differences in the T + profiles, thus in the assessment of qw .
T2 /T1 = 1.01 T2 /T1 = 1.33 T2 /T1 = 1.66 T2 /T1 = 4
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Anisothermal WF 0.4039 -0.0483719 0.3925 -1.40449 0.3807 -2.50365 0.3105 -6.94711
Standard WF 0.4024 -0.0484043 0.3982 -1.43095 0.3934 -2.58128 0.3657 -7.29286
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3 Adaptation of the TBLE model to anisothermal flows 3.1 The TBLE model and its anisothermal version The TBLE consists in embedding a one-dimensional fine grid between the wall and the first off-wall coarse mesh point. It was introduced in 1996 by Balaras [5] and has been increasingly investigated by other researchers [11, 12, 13]. Thanks to the 1-D mesh refinement, the wall shear stress is evaluated directly by the velocity gradient at the wall, after reaching the converged solution of the boundary layer equations in the fine mesh.
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The TBLE method is based on the resolution of four equations: two for the streamwise and spanwise momentum components, one for the temperature and the state equation. If y is the wall-normal direction, the set of equations reads: P0 = ρRT ∂ ∂Uk ∂ρUk = Fk + (μ + μt ) ∂t ∂y ∂y ∂ ∂T ∂T = Fth + (λ + λt ) ρcp ∂t ∂y ∂y
(10) (11) (12)
P0 is the thermodynamic pressure of the Low Mach number approach, which is constant over the whole domain and appears in the state equation (10). The subscript k denotes the velocity component (k = 1 or 3). Fk and Fth denote the source terms for the momentum and the temperature. In order to adapt the TBLE model to high temperature gradient configuration, the mixing length turbulence model basically used [5] needs to be adjusted so that the momentum equation recovers the Van Driest transformation. Without temporal and source terms, Eq. (11) reads (in the logarithmic region, where μ μt ): d dU μt =0 (13) dy dy Eq. (13) is known to correspond to the logarithmic law. As this standard law is not valid in the anisothermal regime, we need to use an adapted turbulence model (noted μta ), so that Eq. (2) is true, that is equivalent to: d dUV D μta =0 (14) dy dy The direct differentiation of Eq. (1) gives a simple relation between U and UV D and, hence, Eq. (14) can be rewritten as: dU d 1 μt √ =0 (15) dy 1 + ΔT dy Eq. (15) finally shows that a suitable expression for μta is a mixing length model formulation, multiplied by an anisothermal factor, function of the temperature difference between the local TBLE mesh point considered and the wall. 3.2 Preliminary results and expectations A first step in the use of this dilatable version of the TBLE model has been to test it off-code. Note that its implementation in the Trio U code has been done, even if no results can be discussed at the moment.
Anisothermal wall functions for RANS and LES
387
From a linear profile that is imposed between the first off-wall LES point and the wall onto the fine mesh, the TBLE model converges towards a turbulent profile as given by the solution of the simplified Navier-Stokes and energy equations. In a quasi-isothermal regime, the dimensionless velocity and temperature profiles were analysed to check the ability to recover respectively the standard logarithmic law [7] and the Kader formula [9]. Fig. 3 shows that the TBLE model fits very well these two reference laws. 20
15 +
+
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+
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Fig. 3. Dimensionless velocity (left) and temperature (right) profiles given by the TBLE model after convergence.
Our expectations concerning the dilatable version of the TBLE model is that it gives at least the same results as the anisothermal wall function in high temperature gradient cases. Also, we expect that the additional cost will remain low.
4 Conclusions and future work In this paper, we present an original method for the derivation of anisothermal wall models. A first one dedicated to RANS simulations is an anisothermal wall function in the classical law-of-the wall concept, while the second is a zonal approach particularly suited for LES. As the Van Driest transformation has been proven to be valid even in very high temperature gradient cases, particularly by the use of DNS results, we except that these two anisothermal models will give valuable results in strong heat transfer configurations, compared to the classical wall models. The TBLE approach, in the framework of LES, is meant to provide information of the unsteady near-wall behaviour of the velocity and temperature fields. But obviously, the unstationarity that can be captured by the wall function is limited to a certain frequency that is unknown for the moment. We propose to investigate in a further work this frequency limit, via direct simulation of a channel flow, by scanning a range of frequencies of passive scalar
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temporal variations and analyzing the impact on the near-wall behaviour. Such a work is under progress.
5 Acknowledgements The authors want to acknowledge the financial support and scientific expertise from CEA (French Nuclear Energy Agency), by funding the first author’s ongoing PhD thesis and providing access to the CEA in-house CFD code: Trio U. The authors are also grateful to CINES for the access to computing resources.
References 1. Nicoud F. and Poinsot T. DNS of a channel flow with variable properties. In Banerjee S. and Eaton J., editors, 1st International Symposium on Turbulence and Shear Flow Phenomena, Santa Barbara, USA, 1999. 2. Nicoud F. Conservative high-order finite difference schemes for low-mach number flows. Journal of Computational Physics, 158, 2000. 3. Van Driest E.R. Turbulent boundary layers in compressible fluids. Journal of Aeronautical Sciences, 18(3), 1951. 4. Huang P.G. and Coleman G.N. Van Driest transformation and compressible wall-bounded flows. AIAA Journal, 32(10), 2004. 5. Balaras E., Benocci C., and Piomelli U. Two-layer approximate boundary conditions for large-eddy simulations. AIAA Journal, 34(6), 1996. 6. Favre A. Statistical equations of turbulent gases. Problems of hydrodynamics and continuum mechanics, 1969. 7. Von K´ arm´ an T. Turbulence and skin friction. Journal of Aeronautical Sciences, 1, 1934. 8. Bradshaw P. Compressible turbulent shear layers. Annual Review of Fluid Mechanics, 9, 1977. 9. Kader B.A. Temperature and concentration profiles in fully turbulent boundary layers. International Journal of Heat and Mass Transfer, 24(9), 1981. 10. Calvin C., Cueto O., and Emonot P. An object-oriented approach to the design of fluid mechanics software. ESAIM: M2AN (Mathematical Modelling and Numerical Analysis), 36(5), 2002. 11. Cabot W. and Moin P. Approximate wall boundary conditions in the large-eddy simulation of high Reynolds number flow. Flow, Turbulence and Combustion, 63, 2000. 12. Diurno G.V., Balaras E., and Piomelli U. Wall-layer models for LES of separated flows. In Modern simulation strategies for turbulent flows. Ed. B. Geurts, (Philadelphia, Edwards), 2001. 13. Wang W. and Moin P. Dynamic wall modeling for large-eddy simulation of complex turbulent flows. Physics of Fluids, 14(7), 2002.
Wall Layer Investigations of Channel Flow with Periodic Hill Constrictions Nikolaus Peller1 , Christophe Brun2 and Michael Manhart1 1
2
Fachgebiet Hydromechanik, Technische Universit¨ at M¨ unchen, Arcisstrasse 21, 80333 M¨ unchen Germany [email protected], [email protected] Laboratoire de M´ecanique et d’Energ´etique, Polytech’Orl´eans, 8 rue L´eonard de Vinci, 45072 Orl´eans France [email protected]
1 Abstract The paper presents near-wall investigations for turbulent attached and separated flow in a channel flow with periodic hill constrictions at Re = 5600. An extended scaling for the streamwise velocity profile is introduced which takes into account wall shear stress and streamwise pressure gradient at the same time. A parameter α is defined representing the relative contributions of wall shear stress or pressure gradient, respectively. By this parameter it is possible to classify certain positions in the flow. Since wall shear stress and pressure gradient are extremely unlikely to be become zero at the same time in the average flow field this scaling can also be applied in regions of separation and reattachment.
2 Introduction The near wall behaviour of turbulent flows is of special interest when wall models for Reynolds Averaged Navier-Stokes Simulations (RANS) or Large Eddy Simulations (LES) have to be developed. The design of explicit wall models, i.e. when no special transport equations based on simplified boundary layer equations are solved, is dependent on some universal behaviour of velocity profiles close to the wall. If such a universal behaviour was known, then the layer described by this universal behaviour could be bridged by the simulation leading to considerable savings of CPU time and memory requirements. The search for universal near wall behaviour is therefore one of the key elements in designing efficient solvers for high Reynolds number flows in complex geometries.
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When the turbulent flow is attached, as e.g. in zero pressure gradient boundary layers or in channel flow, the region in the vicinity of the wall is mainly governed by the wall shear stress. Dimensional considerations lead to the classical law of the wall based on inner coordinates which are defined by using the wall shear stress. It has been shown by many experiments and Direct Numerical Simulations (DNS) that this classical law of the wall is valid for the profiles of the mean velocity component in streamwise direction. In more complex flow situations as e.g. when a strong streamwise pressure gradient causes flow separation or attachment, this well known law of the wall is no more a valid description of the flow profile in the vicinity of the wall [4, 9, 1, 2]. If e.g. a turbulent boundary layer along a flat plate is approaching a line of separation, deviations from this law of the wall can be observed well before the mean separation line [11]. More severe, when the wall shear stress is going to zero, i.e. at the points of separation or reattachment, the inner coordinates based on wall shear stress are stretched to infinity. This problem has been addressed by several studies taking the streamwise pressure gradient into account. As already stated by Stratford [14, 13, 16] and Simpson [11, 10] the velocity profile close to the wall has to be described by both, the wall shear stress and the pressure gradient. Skote and Henningson [12] proposed a wall scaling based on either wall stress or pressure gradient, depending on which one is the larger one. In the context of wall models for LES, Manhart [5] found out that a combination of both quantities is necessary to predict the instantaneous velocity behaviour close to the wall. In the present paper, we propose a universal polynomial law of the wall that is based on the combination of wall shear stress and pressure gradient. In comparison to a scaling proposed by Simpson [10] our scaling circumvents problems when the wall shear stress approaches zero. We evaluate the behaviour of this formulation based on DNS of turbulent flow through a channel with periodic smoothly contoured constrictions at Reynolds numbers Re = 2808 and 5600. The simulations were performed in the framework of the FrenchGerman research group “LES of complex flows” and serve at providing a data base for studying basic turbulent flow phenomena such as near wall behaviour or fine scale processes which are relevant for LES.
3 Numerical method and test case The simulation is carried out with a Finite Volume code [6] for incompressible Navier-Stokes equations on a non-equidistant staggered Cartesian grid. The advancement in time is described by a Leapfrog time step. The Stone Implicit Procedure (SIP) is used for the solution of the Poisson equation. An Immersed Boundary technique [15] is used to represent the periodic hill surface in the Cartesian mesh. A third order least squares interpolation at the body surface [7] allows a smooth representation of the body. The boundary conditions are periodicity in streamwise and spanwise directions and no-slip at the hill
Wall Layer Investigations of Channel Flow w. Periodic Hill Constrictions
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Fig. 1. Instantaneous streamwise velocity in a plane z = const.
Channel DNS Coarse DNS
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Fig. 2. Kolmogorov length scale of plain channel flow and periodic hill flow at x/h = 0.0 for Re = 2808.
Fig. 3. Comparison of velocity scales uτ , up and uτ p ; (Re=5600).
surface as well as the channel top. The sizes of the computational grid are 9.0h in streamwise direction, 4.5h in spanwise direction and 3.035h in vertical direction (Figure 1). The flow is driven by a pressure gradient adjusted to keep the bulk velocity ub at the desired value. The grids for both simulations at Re = 2808 and Re = 5600 are refined close to the wall where small turbulent scales must be resolved. The stretching factors are kept below 3%. In order to find the proper grid spacings, we use estimations based on wall shear stress for the wall normal resolution and estimations based on the Kolmogorov scale in the core of the flow [8]. A first evaluation of these criteria has been obtained by preliminary simulations with a relatively coarse grid. The Kolmogorov scale for the case Re = 2808 with fine and coarse grid can be seen in Figure 2. The finally computed values confirm the resolution estimations from the coarse grid and show that the resolution requirements are more stringent than for plane channel flow at the same bulk Reynolds number. Based on the wall shear stress the grid spacing in y-direction remains below Δy + = 1.4 at the lower wall. The development of the friction velocity uτ in streamwise direction for Re = 5600 can be seen in Figure 3. The resulting number of grid cells equals 47 · 106 for the Re = 2808 case and 233·106 for the Re = 5600 case. The computations were performed on the
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super computer Hitachi SR8000 at the Leibniz Computing centre of Munich with a fully MPI parallelised code [6].
4 Near wall scaling We consider the boundary layer hypothesis, including pressure gradient effects. If equilibrium assumed the Navier Stokes equations for the streamwise velocity component reduce then to [14, 16] U (y) =
1 ∂p 2 τw y+ y . μ 2μ ∂x
(1)
Following Simpson [10], we define a pressure gradient based velocity up in addition to the standard friction velocity uτ . These two reference velocities write 1/2 τw μ ∂p 1/3 . uτ = up = 2 (2) ρ ρ ∂x With these two we construct a combined velocity scale uτ p = u2τ + u2p
(3)
which never goes to zero (Figure 3) and prevents singularity behaviours in ∂p is small. The ratio α ∈ [0, 1] defined by regions where either τw or ∂x α=
u2τ u2τ + u2p
(4)
describes which of the two effects is preponderant (Figure 4, including the sign of the friction and the sign of the pressure gradient). With these definitions, a combined universal scaling law can be derived from equation (1) through the relevant normalisation which has the form ∂p 1 (1 − α)3/2 y ∗ 2 (5) U ∗ (y) = sign(τw )αy ∗ + sign ∂x 2 with U∗ =
u uτ p
y∗ =
yuτ p . ν
(6)
This relation constitutes a generalisation of the law proposed by Simpson [11, 10]. For flows with zero pressure gradient (α = 1), we obtain the classical law of the wall with u∗ = u+ and y ∗ = y + . For flows with zero wall friction (α = 0), we obtain a parabolic law accounting for the pressure gradient effect only 1 (7) U ∗ (y) = y ∗ 2 . 2 For realistic flow configurations, both contributions are blended in equation (5). For αb ≈ 0.3 and y ∗ ≈ 1 both contributions are equally weighted.
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5 Physical flow features and near wall behaviour We study the possible application and range of validity of the universal law of the wall (equation 1) in the case of channel flow with periodic hill constrictions. In this test case, flow separation occurs from the smoothly contoured surface of the hills. The separating boundary layer evolves to a free shear layer that is highly active and leads to strong oscillations of the separation and reattachment points (Figure 1). Fr¨ ohlich et al. [3] point out that the reattachment position is strongly dependent on the point of flow detachment. Because of the length of the domain, the reattachment region is located on the channel bottom before the next hill. In what follows, only the Re = 5600 case is considered. The recirculation region of the time averaged flow field can be identified from Figure 4 where α is multiplied with the sign of τw (equation 4). At positions where uτ = 0 the parameter α becomes zero. In the case of up = 0, α yields one. Table 1 gives an overview of the characteristic locations where α = 0 (uτ = 0) or α = 1 (up = 0). In the traditional scaling U + and y + problems arise at points of separation and reattachment where the wall shear stress and uτ = 0 become zero (Figure 3). This situation is reflected in α = 0 (Figure 4). In such case, the standard scaling is not applicable, because the physical length for e.g. y + = 1 goes to infinity(Figure 5). On the other hand the proposed law yields reasonable values for y ∗ = 1 at all positions in the flow (Figure 5) since uτ p is never zero.
1
0.5(1- α)
(3/2)
α +/- αb
0.8
0.1
scale1: yp=1 scale2: ys=1
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*
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4. Values · sign(τw ) and Fig. for α ∂p 3/2 1 (1 − α) ·sign( ∂x ) plotted along the bottom wall. 2
5
6
7
8
9
x/h
Fig. 5. Physical wall distance for y + = 1 and y ∗ = 1; (Re=5600).
Table 1. Positions for α = 0 or α = 1 x01 x02 x03 x04 x11 x12 x13 α = 0: 0.17 4.99 7.00 7.33 α = 1: 1.91 7.66 8.74
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Nikolaus Peller, Christophe Brun and Michael Manhart
In what follows two characteristic positions in the flow will be analysed which refer to (i) flow separation (x01 = 0.17), and (ii) a position within the tiny recirculation region at x/h = 7.25. At x01 = 0.17 the average flow separates from the hill and the friction velocity becomes zero. It can be expected that instantaneous wall shear stresses in this region are also small (α → 0). Figure 6 shows some instantaneous velocity profiles normalised with the friction velocity for this location. As expected great scattering of the profiles occurs due to the small values of uτ . As mentioned before, for such regions the traditional scaling cannot be applied. Normalising the velocity profiles at the same position as before with the proposed scaling based on uτ p reduces the scattering (Figure 7). There are two branches where profiles cluster, one for positive and one for negative u+ and u∗ , resp.. These branches refer to forward and backward flow which should have equal probabilities to occur at the point of separation. Note, that uτ and uτ p are always positive by definition. When looking at the averaged velocity profile at that location the effect is even more striking (Figure 8). When scaled by the classical wall units, the profile strongly deviates from the classical law of the wall well before y + = 1. With the proposed scaling, close agreement with equation (5) can be achieved up to y ∗ = 3. Also, the average velocity profile for the proposed scaling is in close agreement with the analytical profile given by equation (5) (Figure 8). 40 40
35
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25 20 *
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20 10
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Fig. 6. Instantaneous wall layer profiles at x01 = 0.17 in traditional scaling (scale 1); Re=5600; αmean ≈ 0.
scale2
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Fig. 7. Instantaneous wall layer profiles at x01 = 0.17 in proposed scaling (scale 2); Re = 5600; αmean ≈ 0.
In the tiny recirculation zone at x/h ≈ 7.25, the value α assumes α ≈ 0.17 (Figure 4) rendering that the flow is strongly influenced by the pressure gradient. Figure 9 shows the averaged near-wall velocity profiles for both scalings. Since τw is negative in this region the classical law of the wall predicts an analytical profile with backflow velocity. It cannot capture the change from backflow to forward flow occurring below y + = 1. In contrary to that the positive pressure gradient in the proposed scaling, is able to capture change in flow direction what makes it a promising candidate for wall shear stress
Wall Layer Investigations of Channel Flow w. Periodic Hill Constrictions 50
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scale1: analytical scale1: numerical scale2: analytical scale2: numerical
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Fig. 8. Averaged vel. profile at x01 = 0.17; scale 1(traditional) and scale 2(proposed); Re = 5600; α ≈ 0.
0.1
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Fig. 9. Average vel. profile at x/h = 7.25; scale 1(traditional) and scale 2(proposed); Re=5600; α ≈ 0.17.
predictions in very thin recirculation events that can not be represented well by the numerical grid.
6 Conclusions A modified law of the wall for the viscous sublayer has been introduced which accounts for the effects of both, the wall shear stress and the pressure gradient. This law consists of a scaling based on these two effects and a universal family of velocity profiles derived purely on physical arguments. The proposed scaling circumvents the problems of the classical law of the wall when the shear stress is zero by including the effects of the pressure gradient in the definition of velocity scale which makes it extremely unlikely to become zero. A linearparabolic law constitutes a family of universal velocity profiles u∗ depending on the new dimensionless wall distance y ∗ and the ratio between wall shear stress and pressure gradient effects, α. We analysed the performance of this new formulation in separating turbulent flow established in a channel with periodic hill constrictions. The data were computed by DNS and show the following. (i) In the average flow field the combined friction-pressure velocity scale uτ p never assumes zero. Scaling based on this quantity therefore does not render inapplicable at e.g. the separation point. But also the instantaneous values of uτ p in the region of separation generally do not assume zero and consequently velocity profiles based on this scaling parameter show significantly less scattering than in the classical formulation. (ii) Since the pressure gradient can have the opposite sign of the wall shear stress its contribution can lead to a change from e.g. forward to backward flow. This has been observed in the tiny statistical recirculation region where the classical scaling deviates much faster. A parametric study for α ∈ [0, 1] should be undertaken to check the universality of the proposed law of the wall and determine its domain of validity in term of y ∗ .
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References 1. E.W. Adams, J.P. Jahnston, and J.K. Eaton. Experiments on the structure of turbulent reattaching flow. Tech. Rep. MD-43, Thermosciences Division, Department of Mechanical Engineering, Stanford University, Stanford, CS, 1984. 2. W.J. Devenport and E.P. Sutton. Near-wall behaviour of separated ad reattaching flows. AIAA J., 29(25), 1991. 3. J. Fr¨ oehlich, C. P. Mellen, W. Rodi, L. Temmerman, and M. Leschziner. Highly resolved large-eddy simulation of separated flow in a channel with streamwise periodic constrictions. J. Fluid Mech., 526:19–66, 2005. 4. Hans-Jakob Kaltenbach. A priori testing of wall models for separated flows. Phys. Fluids., 15(10):3048–3068, 2003. 5. M. Manhart. Analysing near-wall behaviour in a separating turbulent boundary layer by DNS. In B. Geurts, R. Friedrich, and O. M´etais, editors, Direct and Large-Eddy Simulation IV. Kluwer Academic Publishers, Dordrecht, 2001. 6. M. Manhart, F. Tremblay, and R. Friedrich. MGLET: a parallel code for efficient DNS and LES of complex geometries. In Jensen et al., editor, Parallel Computational Fluid Dynamics 2000, pages 449–456, Amsterdam, 2001. Elsevier Science B.V. 7. N. Peller, A. Le Duc, F. Tremblay, and M. Manhart. High-order stable interpolations for immersed boundary methods. International Journal for Numerical Methods in Fluids, submitted. 8. N. Peller and M. Manhart. Turbulent Channel Flow with Periodic Hill Constrictions. Notes on Numerical Fluid Mechanics and Multidisciplinary design. Springer, 2005. 9. R. Ruderich and H.H. Fernholz. An experimental investigation of a turbulent shear flow with separation, reverse flow, and reattachment. J. Fluid Mech., 163:283–322, 1986. 10. R.-L. Simpson. A model for the backflow mean velocity profile. AIAA J., 21:142, 1983. 11. R.L. Simpson, Y.T. Chew, and B.G. Shivaprasad. The structure of a separating turbulent boundary layer. Part 1. Mean flow and Reynolds stresses. J. Fluid Mech., 113:23–51, 1981. 12. M. Skote, D.S. Henningson, and R.A.W.M. Henkes. Direct numerical simulation of self-similar turbulent boundary layers in adverse pressure gradients. Flow, Turbulence and Combustion, 60:47–85, 1998. 13. B.S. Stratford. An experimental flow with zero skin friction throughout its region of pressure rise. J. Fluid Mech., 5:17–35, 1959. 14. B.S. Stratford. The prediction of separation of the turbulent boundary layer. J. Fluid Mech., 5:1–16, 1959. 15. F. Tremblay, M. Manhart, and R. Friedrich. LES of flow around a circular cylinder at a high subcritical reynolds number. In B. Geurts, R. Friedrich, and O. M´etais, editors, Direct and Large-Eddy Simulation IV. Kluwer Academic Publishers, Dordrecht, 2001. 16. F.M. White. Viscous fluid flow. Mc Graw-Hill, 1974.
A Multi-Scale, Multi-Domain Approach for LES of High Reynolds Number Wall-Bounded Turbulent Flows M. U. Haliloglu and R. Akhavan∗ University of Michigan, Ann Arbor, MI 48109-2125, USA ∗ Corresponding author. E-mail: [email protected] Summary. A new approach for LES of high Reynolds number wall-bounded turbulent flows is presented. The proposed method utilizes the quasi-periodicity of the near-wall turbulent structures to compute the near-wall region in a minimal flow unit at high resolution. This minimal flow unit is then repeated periodically or quasi-periodically and matched to a full-domain but coarse resolution LES in the outer layer. The proposed multi-scale, multi-domain (MSMD) approach has been implemented in LES of turbulent channel flow using a patching collocation spectral domain-decomposition method. The Nonlinear Interactions Approximation (NIA) model [1] is employed as the subgrid-scale (SGS) model. Simulations are performed at Re τ ≈ 1000, 2000, 5000 and 10,000 with a resolution of 32 × 32 × 17 in the near-wall region and 32 × 64 × 33 in the outer layer, independent of the Reynolds number. The predictions are found to be in good agreement with the law of the wall and the available experimental data. These results suggest that the MSMD approach provides a simple and computationally efficient method for accurate LES of high Reynolds number wall-bounded turbulent flows.
Key words: Large eddy simulation, wall modelling, spectral domain decomposition methods, turbulent channel flow, wall-bounded turbulent flows.
1 Introduction Large-eddy simulation (LES) is growing in recognition as a useful technique for the computation of turbulent flows in complex engineering environments. Nevertheless, there are a number of remaining issues which need to be addressed before LES can be adopted as a routine engineering tool. One major stumbling block in LES of wall-bounded flows is the treatment of the near-wall region. This is the site of the bulk of turbulence production in wall-bounded flows. Hence, its accurate computation is critical to the success of LES. The difficulty arises from the fact that the large-scale eddies in this region have a small lateral scale, of size on the order of ∼ 30 − 50 wall units. Resolving
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these eddies requires a fine spatial resolution. If this fine resolution is extended throughout the computational domain, as is done in traditional LES, 3 the computations require O(Re 2.5 τ −Re τ ) grid points, depending on the numerical discretization scheme. This makes computations of high Reynolds number wall-bounded turbulent flows prohibitively expensive. A number of approaches have been proposed in the literature to alleviate these prohibitive resolution requirements. Broadly speaking, these can be classified into one of two categories: zonal methods and wall-modelling approaches. In zonal methods, the flow field is everywhere computed based on the solution of the LES equations. The computational savings derive from using either a finer grid (grid-embedding methods) [2, 3, 4, 5] or a smaller domain size (multi-block approaches) [6] in the near-wall region. These approaches, while accurate, are still prohibitively expensive for application to high Reynolds number flows. In wall-modelling approaches, the flow in the near-wall region is computed based on either analytical results or the solution of a simplified set of equations. Examples include hybrid RANS/LES methods [7], Detached Eddy Simulation (DES) [8], and wall-modelling approaches based on One Dimensional Turbulence (ODT) [9]. While these methods provide superior predictions compared to pure RANS, they each have their own limitations in complex turbulent flows [10]. The shortcomings of existing approaches suggests a need for alternative approaches to wall-modelling. Much has been learned about the structure of wall turbulence in the recent years. It is known that the near-wall dynamics consists of a sequence of quasi-periodic events, which are repeated randomly in time and space [11]. The basic near-wall structure is an asymmetric horseshoe vortex with head extending to a distance of ∼ 200 wall units from the wall. The vortex has a streamwise extent of ∼ 200 − 500 wall units, and a spanwise extent of ∼ 200 wall units as measured at the head of the horseshoe [12, 13]. Recent work [14, 15] has shown that these structures coalesce to form larger vortex packets of size ∼ 5000 wall units in the streamwise and ∼ 1000 wall units in the spanwise directions, respectively (corresponding to ∼ 2δ and ∼ 0.4δ in a boundary layer at Re τ ≈ 2200 [15]). These findings suggest a new approach to wall-modelling. In this approach the near-wall region is solved in a minimal flow unit large enough to accommodate only one packet of vortical structures. This unit is then repeated periodically or quasi-periodically and matched to a full-domain but coarseresolution LES in the outer layer. In this study, we present an implementation of this multi-scale, multi-domain (MSMD) approach in a turbulent channel flow using a spectral patching collocation domain decomposition method. The organization of the paper is as follows. In section 2, a description of the MSMD approach is provided. In section 3, the proposed MSMD approach is tested in LES of turbulent channel flow at 1000 ≤ Re τ ≤ 10, 000. A brief summary and conclusions are provided in section 4.
A multi-scale, multi-domain method for LES
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2 Description of the MSMD Approach In the MSMD approach, the computational domain is partitioned into nonoverlapping sub-domains in the wall-normal direction. The near-wall region is solved at fine resolution in a computational domain large enough to accommodate only one packet of vortical structures. This solution is then repeated periodically (in the near-wall region), or quasi-periodically (as it is passed to the outer layer), and matched to a full-domain but coarse-resolution LES in the outer layer. Based on the available experimental data on the size of vortex packets [15], the near-wall domain needs to be of minimum size ∼ 5000 wall units in the streamwise, ∼ 1500 − 2000 wall units in the spanwise, and ∼ 200 − 250 wall units in the wall-normal directions, respectively. x3 (z)
x2 (y) Ω1
x1 (x)
Ω2 2h
Ω3
L1
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Fig. 1. A schematic of the channel and implementation of the MSMD approach.
In this study, we discuss the implementation of the MSMD approach in LES of a turbulent channel flow. The channel is bound by solid walls in the wall-normal (z) direction and is assumed to be periodic in the streamwise (x) and spanwise (y) directions. For simplicity, the multi-domain approach is applied only in the spanwise direction. In the streamwise direction, the flow is solved in the full domain in both the near-wall and outer layers (see Figure 1). This approach remains viable for channel flows up to Reτ ∼ 10, 000. For higher Reynolds numbers, the MSMD approach needs to be implemented in both the streamwise and spanwise directions. 2.1 Governing Equations The governing equations for LES of incompressible flow are given by ∂ 1 ∂p ∂ 2 ui ∂τij ∂ui + (ui uj ) = − +ν − , ∂t ∂xj ρ ∂xi ∂xj ∂xj ∂xj ∂ui =0 ∂xi
(1) (2)
where the overbar denotes filtering with a spatial filter of characteristic width Δi in the xi -direction, ui is the large-scale velocity field, p is the resolved pressure, and ν is the kinematic viscosity. The Nonlinear Interactions Approximation (NIA) model [1] is used as the SGS model. In NIA, the subgrid-scale stress (including its non-deviatoric part) is parameterized as
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τij ≡ (ui uj − ui uj ) = (ˇ ui u ˇj − u ˇi u ˇj ) − 2CΔ |S|S ij ,
(3)
where C(x, t) is the Smagorinsky model coefficient, Δ is the characteristic width of the LES filter typically defined as Δ = (Δ1 Δ2 Δ3 )1/3 , S ij = 1/2(∂ui /∂xj + ∂uj /∂xi ) is the large-scale rate of strain tensor, and |S| = (2S ij S ij )1/2 . Filtering is performed explicitly using a discrete top-hat filter and is applied only in the homogeneous directions. In equation (3), u ˇi represents the de-convolved velocity, computed by exact de-convolution up to the maximum wavenumber resolved in LES. The model constant C(x, t) is computed using a dynamic procedure. Further details are given in [1]. Equations (1) and (2) are solved using a spectral patching collocation method [16]. To proceed, the velocity field in each sub-domain, Ωs , is represented in terms of Fourier series in the streamwise and spanwise directions and mapped Chebyshev polynomials in the wall-normal direction as, ui,s (x, y, z, t) = |m|≤Ms /2 |n|≤Ns /2 vi,s (m, n, z, t) exp[iαmx + iβny]. Time advancement is performed using a fractional-step splitting method [17]. With these discretizations, the solution of equations (1) and (2) at each time step reduces to an advection step plus a sequence of solutions of elliptic equations for each Fourier mode in the pressure and viscous steps. The general elliptic problem to be solved is of the form d d qs vi,s + γvi,s = fi,s , (4) Lvi,s ≡ dz dz where qs (z) and fi,s (z) are known variables, and γ is a (negative) constant. Depending on the nature of qs (z), the elliptic equations (4) are solved using either a direct [18] or an iterative [19, 20] method. 2.2 Interface Boundary Conditions The most critical issue in the implementation of the MSMD approach is establishing the connectivity between the near-wall and outer layers. In a traditional patching collocation method with conforming grids, the internal interface boundary conditions for a second-order problem typically enforce the continuity of the solution and its first derivative at the interface. In a spectral patching collocation method utilizing Fourier series in the directions parallel to the wall, these can be specified mode by mode in the Fourier space as vi,s (z, t) = vi,s+1 (z, t) dvi,s+1 (z, t) dvi,s (z, t) = dz dz
for i = 1, 2, 3,
(5)
for i = 1, 2, 3,
(6)
where the subscripts s and s + 1 denote neighboring sub-domains. In the MSMD approach, however, the formulation of the problem does not provide sufficient degrees of freedom to enforce both conditions (5) and (6) simultaneously, because there are large-scale modes in the outer layer which have no counterpart in the near-wall layer and small-scale modes in the near-wall region which have no counterpart in the outer layer. The simplest
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boundary conditions one can impose on these non-common modes is to set them equal to zero at the internal interface. Thus one would retain boundary conditions (5)-(6) for the modes which are common between neighboring subdomains, while for the non-common modes one would set vi,s (z, t) = 0 vi,s (z, t) = 0
for i = 1, 2, 3; in the near-wall layer (s = 1, 3), for i = 1, 2, 3; in the outer layer (s = 2).
(7) (8)
In practice, boundary conditions (7) work well for the non-common small-scale modes arising from the near-wall layer. However, boundary conditions (8) are inappropriate for the non-common large-scale modes arising from the outer layer. The basic problem is that such boundary conditions inhibit the natural development of the large-scale modes in the outer layer. As a result, the computations cannot take full advantage of the larger domain size employed in the outer layer. The simplest internal interface boundary conditions one can impose on these non-common large-scale modes, which would preserve their natural development, are the Neumann boundary conditions dvi,s =0 dz
i = 1, 2, 3; in the outer layer (s = 2).
(9)
These boundary conditions prove satisfactory when applied to the streamwise and spanwise components of the velocity, but lead to numerical instability when applied to the wall-normal component of the velocity. To overcome this limitation, an approximation of (9) is adopted as the internal interface boundary conditions for the non-common large-scale modes. These approximate boundary conditions can be expressed as vi,s (z, t) = Ai vi,s (z ± δz, t − Δt)
for i = 1, 2, 3 in the outer layer,
(10)
where vi,s (z ±δz, t−Δt) denotes the Fourier coefficient representing the largescale mode of interest at the previous time step from grid points a distance of δz from the internal interface within the outer layer, and 0 < Ai ≤ 1 are scaling factors. With the internal interface placed at z + ≈ 200−250 wall units from the walls, best results are obtained with Ai = 1 for i = 1, 2; Ai = 0.5 for i = 3, and δz/h ≈ 0.05. Numerical tests were performed to ensure that the solution is insensitive to the exact choice of δz as long as δz/h ≈ 0.05 − 0.1.
3 Results & Discussion The proposed MSMD approach has been implemented in LES of turbulent channel flow at 1000 ≤ Reτ ≡ uτ h/ν ≤ 10, 000, corresponding to 20, 000 ≤ Rem ≡ Um h/ν ≤ 266, 667. Here, uτ and Um denote the wall-friction and bulk velocities, respectively, and h is the channel half-width. Computations were performed in channels with periodicity lengths of Lx /h = Ly /h = π in the streamwise and spanwise directions, and employed 2, 4, 8 and 16 periodic near-wall units at Reτ ≈ 1000, 2000, 5000 and 10,000, respectively. This gives a spanwise extent of 1500 − 2000 wall units for the near-wall layer in all the
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A multi-scale, multi-domain method for LES Reτ 1000 2000 5000 10,000 10,000H
Rem 20,000 42,667 133,333 266,667 266,667
Present Study 5.08 × 10−3 4.20 × 10−3 3.10 × 10−3 2.50 × 10−3 2.73 × 10−3
Dean’s Correlation 5.16 × 10−3 4.27 × 10−3 3.21 × 10−3 2.70 × 10−3 2.70 × 10−3
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% Error 1.6% 1.6% 3.4% 7.4% 1.0%
Table 1. Average skin-friction coefficient predicted by the MSMD approach compared to values from Dean’s correlation.
simulations. In the streamwise direction, computations were performed in the full domain in both the inner and outer layers. The interface was placed at z + ≈ 200 − 250 in all the simulations. Simulations were performed with a resolution of 32 × 32 × 17 in the near-wall layer and 32 × 64 × 33 in the outer layer at all Reynolds numbers. Initial conditions for all the simulations were obtained from a single-domain DNS of a stationary turbulent channel flow at Re τ ≈ 570. Results were validated against the experiments of [21] and [22]. The skin friction coefficients predicted by LES are compared to values from Dean’s correlation [23] in Table 1. The average skin friction coefficient is predicted to within 1.6% of values from Dean’s correlation at Re τ ≈ 1000 and Re τ ≈ 2000, to within 3.4% at Re τ ≈ 5000, and to within 7.4% at Re τ ≈ 10, 000. The larger errors at Reτ ≈ 10, 000 arise from the very coarse streamwise grid spacings (Δx+ ≈ 1000) at this Reynolds number. If the resolution in the x-direction is doubled, the skin-friction coefficient is predicted to within 1% of the value from Dean’s correlation, as demonstrated by the case 10,000H in Table 1. This higher resolution would not be needed had the MSMD approach also been implemented in the streamwise direction. The predicted mean velocity profiles are shown in Figure 2 (a). The profiles show good agreement with the available experimental data and the logarithmic law of the wall. The slope of the logarithmic layer is accurately predicted in both the inner and outer layers, and no jump discontinuity is observed across the interface. plus subgrid) second order statistics, / full . (resolved / . The computed using ui uj ≈ ui uj + τij [1], are shown in Figures 2 (b)-(e). Both the r.m.s. turbulent fluctuations and the Reynolds stress are in good agreement with the available experimental data in the outer layer. However, the second order statistics are under-predicted in the near-wall region. This is to be expected and reflects the absence of large-scale modes in the nearwall solution. Similar features can be observed in a priori tests in which the large-scale near-wall modes are set to zero. In general the MSMD approach is designed to give an accurate solution in the outer layer, but only approximate results in the near-wall region.
4 Summary and Conclusions A multi-scale, multi-domain (MSMD) approach for LES of wall turbulence is presented. The method solves the near-wall region in a minimal computational
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domain at fine resolution. This solution is then repeated periodically or quasiperiodically and matched to a full domain but coarse-resolution LES in the outer layer. The proposed method has been implemented in LES of turbulent channel flow at 1000 ≤ Re τ ≤ 10, 000. Simulations are performed with a resolution of 32 × 32 × 17 in the near-wall layer and 32 × 64 × 33 in the outer layer at all Reynolds numbers. This suggests a fixed cost for the MSMD approach, independent of Reτ . With increasing Reτ , however, the size of one grid cell in the outer layer could exceed the full width of the near-wall layer in the spanwise direction. Beyond this Reτ , a near-wall domain larger than the minimal unit is required to provide some overlap between the inner and outer layers, resulting in a cost which grows as ∼ Re τ . While the present implementation of the MSMD approach is in a periodic channel, the method is general and can also be applied to non-periodic geometries. In this case, the near-wall solution would be periodic, but that in the outer layer non-periodic.
References 1. Haliloglu, M.U. & Akhavan, R. (2004) in Direct and Large-Eddy Simulation V, edited by R. Friedrich et al. (Kluwer Academic Publishers), 39-48. 2. Kallinderis, Y. (1992) J. Comput. Phys. 98, 129–44. 3. Shariff, K. & Moser, R. (1998) J. Comput. Phys. 145, 471–88. 4. Kravchenko, A.G., Moin, P. & Shariff, K. (1999) J. Comput. Phys. 151, 757–89. 5. Kang, S. (1996) PhD dissertation, The University of Michigan, Ann Arbor. 6. Pascarelli, A., Piomelli, U. & Candler, G.V. (2000) J. Comput. Phys. 157, 256–79. 7. Balaras, E., Benocci, C. & Piomelli, U. (1996) AIAA J. 34, 1111–19. 8. Nikitin, N.V., Nicoud, F., Wasistho, B., Squires, K.D. & Spalart, P. R. (2000) Phys. Fluids 12, 1629–1632. 9. Schmidt, R.C., Kerstein, A.R., Wunsch, S. & Nilsen, V. (2003) J. Comput. Phys. 186, 317–55. 10. Piomelli, U. & Balaras, E. (2002) Annu. Rev. Fluid. Mech. 34, 349–374. 11. Panton, R.L. (1997) Adv. in Fluid Mech. 15, Comput. Mech. Publications. 12. Robinson, S.K. (1991) Annu. Rev. Fluid. Mech. 23, 601–39. 13. Robinson, S.K. (1991) NASA TM-103859. 14. Adrian, R.J., Meinhart, C.D. & Tomkins, C.D. (2000) J. Fluid Mech. 422, 1–54. 15. Tomkins, C.D. & Adrian, R.J. (2003) J. Fluid Mech. 490, 37–74. 16. Canuto, C., Hussaini, M.Y., Quarteroni, A. & Zang, T.A. (1988) (SpringerVerlag). 17. Yakhot, A., Orszag, S.A., Yakhot, V. & Israeli, M. (1989) J. Sci. Comput. 4, 139–58. 18. Israeli, M., Vozovoi, L. & Averbuch, A. (1993) J. Sci. Comput. 8, 135–49. 19. Zanolli, P. (1987) Calcolo 24, 201–40. 20. Funaro, D., Quarteroni, A. & Zanolli, P. (1988) SIAM J. Numer. Anal. 25, 1213–36. 21. Comte-Bellot, G. (1963) Ph.D. Thesis, University of Grenoble. 22. Niederschulte, M.A., Adrian, R.J. & Hanratty, T.J. (1990) Expt. in Fluids 9, 222–30. 23. Dean, R.B. (1978) J. Fluids Eng. 100, 215–23.
Part VIII
Complex Geometries and Boundary Conditions
Modeling turbulence in complex domains using explicit multi-scale forcing A. K. Kuczaj1 and B. J. Geurts1,2 1
2
Department of Applied Mathematics and J.M. Burgers Center for Fluid Dynamics - University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands, [email protected] Department of Applied Physics - Eindhoven University of Technology, P.O. Box 513, 5300 MB Eindhoven, The Netherlands
Abstract: A new computational framework for the numerical simulation of turbulent flows through complex domains and along irregular boundaries is presented. The geometrical complexity is included by introducing explicit fractal forcing. This involves the agitation of a spectrum of length-scales and forms an integral part of the flow modeling. The potential application of such a modeling approach is illustrated by the evaluation of the turbulent mixing of a passive scalar field, driven by this turbulent flow. The surface-area and wrinkling of level-sets of the scalar field are monitored showing the influence of the forcing localization on the mixing efficiency. Keywords: turbulence, complex domains, multi-scale forcing, passive scalar mixing
1 Introduction Various important multi-scale phenomena in turbulent flows are caused by the interactions between the flow and geometrically complex objects placed inside the flow-domain. The origin of the flow perturbations that arises on many different scales of motion comes from the geometrically complex boundaries as occur, e.g., in case of a flow through a porous region (Fig. 1a). In literature, two approaches are used to capture the influence of these perturbations. These incorporate either the explicit boundary modeling, precisely describing its intricate shape, or a rather general macroscopic approximation in terms of effective boundary conditions. However, these methods either suffer from a lack of incorporated scales or are computationally not feasible. We propose a different modeling approach to flows which undergo simultaneous perturbation over a broad range of scales by the interaction with a geometrically complicated object. The emergence of self-similar spectra in turbulence which do not follow the well known Kolmogorov −5/3 slope [1] was observed experimentally in flows over tree canopies [2]. This motivated us
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to adopt explicit multi-scale fractal forcing [3] which captures the dominant geometrical complexities of the flow domain. In fractal forcing, turbulence is forced over a whole range of length-scales in a way that mimics a powerlaw forcing in spectral space (Fig. 1b). This offers the possibility of modeling the dynamic consequences of complex domain boundaries without the need to explicitly account for their intricate geometrical shape. The simultaneous disturbance of the flow over a spectrum of length-scales is approximated by a broad-band distribution of forcing intensities. This method can also incorporate cases in which only part of the domain is occupied by a complex obstruction, as sketched in Fig. 1c. In fact, by introducing an ‘indicator’ function to locate the complex object within the flow domain, the forcing can accommodate such spatial localization. This way the method could be used for global modeling of complicated geometry flows. However, a proper development of this methodology requires extensive examination of the influence of forcing on the energy dynamics, the spatial structures and the flow characteristics.
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The aim of this paper is to present the general framework of multi-scale forced turbulence simulations. Forcing in computational models of turbulence has been solely directed toward maintaining a quasi-stationary state. It allows a study of inertial range dynamics corresponding to the classical Kolmogorov theory. We extend the application of such explicit forcing to simultaneously perturb the flow over a wide spectrum of length- and time-scales. Specifically, we consider the incompressible Navier-Stokes equations with the fractal forcing working as a stirrer controlled by its strength and size of disturbed scales. This computational modeling is illustrated with passive scalar mixing in the forced turbulent flow. Special attention is devoted to the mixing efficiency of a tracer by monitoring the surface area, curvature and wrinkling of level-sets of the scalar fields. The changes in mixing efficiency caused by the broad-band forcing are directly quantified using the level-set method developed in [4]. The application of broad-band forcing leads to distortion of the classical Kolmogorov energy spectrum picture reminiscent of a spectral
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short-cut feature observed experimentally [2]. The production of additional large- and small-scale flow-features by the forcing enhances wrinkling and surface-area growth which plays an important role in turbulent mixing. The organization of this paper is as follows. The Navier-Stokes equations and the passive scalar transport are briefly presented in §2 within our new framework. The results illustrating the possible application of broad-band forcing are collected in §3 with concluding remarks in §4.
2 Forced turbulence and passive scalar transport Turbulence is described by the non-dimensional incompressible Navier-Stokes equations which govern the evolution of velocity u(x, t) and reduced pressure p(x, t) subject to an external forcing F(x, t). The evolution obeys: ∂t u(x, t) + (u(x, t) · ∇)u(x, t) = −∇p(x, t) + Re−1 ∇2 u(x, t) + F(x, t), ∇ · u(x, t) = 0, (1) where Re is the Reynolds number. The equation for the passive scalar T (x, t) has the form: ∂t T (x, t) + (u(x, t) · ∇)T (x, t) = (ScRe)−1 ∇2 T (x, t).
(2)
The non-dimensional molecular diffusivity is denoted by 1/(ScRe) in terms of the product of the Schmidt (Sc) and Reynolds numbers. Applying the Fourier transform F and using the incompressibility constraint we obtain the following system of equations for the Fourier-coefficients of the velocity (u) and scalar fields (T ) [5]: & ∂t + Re−1 k 2 u(k, t) + W(k, t) = F(k, t), (3) −1 2 ∂t + (ScRe) k T (k, t) + Z(k, t) = 0, where k = |k| denotes the length of the wavevector k. The nonlinear & term W(k, t) = F [(u(x, t) · ∇)u(x, t) + ∇p(x, t)] and the convective term Z(k, t) = F [(u(x, t) · ∇)T (x, t)] are computed pseudo-spectrally which involves transforming them back to the physical space to perform the products [6]. Beside the computational advantage of the Fourier expansion in periodic domains, this representation of the solution directly identifies the different length-scale contributions. This decomposition is helpful in the forcing definition presented next. We used a slightly modified fractal forcing compared to the one proposed in [3]. The interactions of a fluid with a truncated fractal-like object are approximated through the induced drag force caused by the contact of that object with the flow. A power-law dependence on wavenumber was derived in which it is assumed that the drag is proportional to the surface area which
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causes the blockage effect of the fluid. This surface area scales with a coefficient β, connected to the fractal dimension Df by β = Df − 2 [3]: F(k, t) = k β f e(k, t).
(4)
The unit vector e is constructed on the basis of velocity and vorticity Fourier coefficients: k × u(k, t) u(k, t) +ı , (5) e(k, t) = γ |u(k, t)| |k||u(k, t)| 2
where γ is the normalization parameter to satisfy |e| = 1. The strength of the forcing term is controlled by specifying the desired energy input rate εw distributed over the set of forced modes KF expressed by: f = εw / k β |u(k, t)|. (6) KF
To solve (3) we used the four-stage compact-storage Runge-Kutta method [7]. The spectral discretization was fully dealiased by spectral truncation and a phase shift scheme. We applied exact integration of the viscous and diffusive terms [6]. To illustrate and quantify the influence of the fractal forcing on the turbulent dispersion of a passive scalar field we adopted the level-set integration method proposed in [4]. This method allows to determine, e.g., the surface-area, curvature and wrinkling of scalar level-sets. To quantify the evolving scalar level-set we may monitor its surface-area or surfacewrinkling. The latter is obtained by integrating |∇ · n| over the scalar levelset, with n the unit normal vector on that set. To present the application of the explicit broad-band forcing method along with the adopted procedure for the passive scalar evaluation we performed some relatively simple idealized numerical experiments which will be presented in the next section.
3 Forcing and mixing efficiency In this section we first show the results of applying broad-band forcing in turbulence. Then, we turn to turbulent passive scalar mixing discussing briefly the influence of forcing on mixing time and mixing quality. We performed numerical simulations at Reynolds number Re = 1067 where the length and time scales are defined by the size of the computational box L = 1 and the energy input rate εw . The resolution requirements were satisfactorily fulfilled as kmax η ranges from 2.3 to 3.5 (η - Kolmogorov scale) using a resolution of 1283 and 1923 grid-cells, which yields after dealiasing 60 and 90 ‘active’ wavenumbers, respectively. For the passive scalar with Sc = 0.7 used in the simulations this gives kmax ηOC = 3.0 . . . 4.5 (ηOC - Obukhov-Corrsin scale). The fractal dimension used in the simulations was Df = 2.6 yielding a surface area scaling β = 0.6. The wavenumbers were rescaled by L/(2π).
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As a reference point we used large-scale forcing with εw = 0.15 which keeps the system energetically in a quasi-stationary state. We refer to this as case A15 in which the forced modes are KF : k ≤ 1.5. The application of additional perturbation to the flow in higher wavenumber bands produces more spatial scales which change the characteristics of the transport. To study this influence we specified two regions where we applied supplementary fractal forcing with εw = 0.45. The first region is situated near the largest scales of the flow (B451 case - KF : 4.5 < k ≤ 8.5) and the second one at some distance from the largest scales (B452 case - KF : 12.5 < k ≤ 16.5) to enhance even smaller scales. Finally, we also considered mixing in case εw = 0.6 in the large scales only (A60 case - KF : k ≤ 1.5). 10−0
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The energy spectra averaged over 10 independent realizations of the initial condition and over a time-interval of 10 eddy-turnover times are presented in Fig. 2. Time-averaging was started after allowing the flow to develop for 5 eddy-turnover times. The additional energy input in the higher wavenumber bands causes a nonlocal modulation in the energy spectrum. This leads to a different dynamics of the flow (Fig. 2a). There is a clear cross-over in the spectra of the broad-band forced cases. At low wavenumber the spectra coincide with the A15 case while at high wavenumbers the tails overlap the A60 case. Rescaling the energy and length-scales yields Fig. 2b. This gives information about the energy distribution over the different scales. Broad-band forcing is seen not to lead to a significant change in the total energy present in the flow. Mainly the high-k forcing changes the distribution of energy over the scales of motion. Specifically, broad-band forcing produces smaller scales and in this way the dissipation of energy increases. The energy is present in different scales of motion and by changing the strength and spatial location
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of the forcing, we have the possibility to control its distribution and hence its transport properties. We next consider the consequences of the different forcing strategies for the efficiency with which the passive scalar is stirred. The nature of stirring comes from the convective turbulent flow properties which drive the scalar while on the other hand mixing is strongly influenced by molecular diffusion. These two physical mechanisms work simultaneously and are responsible for the final result which is commonly called ‘mixing’. To study turbulent mixing properties we simulated the spreading of a passive tracer at the earlier mentioned Schmidt number of 0.7. Initially, we start with a spherical distribution of tracer of radius 3/16 with the radial distribution as a step function softened by a Gaussian profile at the edge and internal concentration equal one.
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Fig. 3. Snapshot of velocity field iso-surfaces (above) and passive scalar concentration (below) at t = 0.5 for: a) A15, b) B451 , c) B452 .
The changes in the flow properties in the different cases have consequences for the quality of mixing as we can observe in Fig. 3. Consistent with the length-scale ranges that are forced we observe more small-scale features in the velocity fields and correspondingly more localized ‘wrinkling’ of the level-sets of the passive scalar. To quantify this first impression given by these snapshots we define the growth parameter of the surface-area A(t) at time t of the selected level-set as ϑA (t) = A(t)/A(0). Similarly, we may monitor the growth ϑW of the surface-wrinkling W (t).
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We obtain an enhancement of the mixing for the broad-band cases (B451 , B452 ) compared to the large scale forcing case at εw = 0.15 (A15). The mixing is more efficient not only in terms of the quality as indicated by the maxima of the individual curves in Fig. 4a, but also in terms of the time needed to reach a similar level of mixing. Comparing the different cases, similar levels of surface-area are obtained in about half the time for the broad-band forced cases. So, ‘investing’ extra energy by agitating high-k bands does produce additional mixing. However, if we compare these results to the energetically equally costly A60 case with energy inserted only in the low-k band then the situation is quite different. The growth parameter for the area reaches its maximum value both sooner and at a higher value. This can be explained because the ratio of the initial tracer volume and the whole computational domain is 1 : 36 and the Schmidt number is comparatively small. Convective spreading of the tracer dominates over the decay caused by molecular diffusion and, hence, larger scales play a crucial role. 4
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In many cases, e.g., in combustion processes not only the surface-area but other quantities like the small-scale wrinkling are important to keep the chemical reactions at an optimum level. The wrinkling growth parameter in Fig. 4b is in general a measure of the average local surface complexity. It exhibits a maximal value for the broad-band forcing situated near the largest scales. The higher band forcing needs to compete more directly with the viscous effects and it was found less effective in producing surface-area, which is more related to ‘sweeping’ motions over ‘reasonable’ distances. In contrast we observe here that more localized distortions of the scalar level-sets are less affected by this competition with viscosity. The level-set integration method is
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effective in quantifying these general impressions. This may help to identify optimal stirring procedures to which future research will be devoted.
4 Conclusions We presented a methodology for the numerical investigation of turbulent flow which undergoes simultaneous forcing over a broad range of scales as a result of interaction with complex domain boundaries. We have shown that with a relatively simple model we can mimic some basic properties of very complex flows. The application of fractal forcing causes an enhancement of energy dissipation in the system producing the so-called spectral short-cut feature observed experimentally [2]. The passive scalar field driven by the forced flow was examined using a level-set evaluation approach to quantify basic general characteristics of mixing quality and efficiency. We illustrated it by performing numerical simulations of a tracer decay in turbulence forced at different length-scales. The results show successful application of the evaluation method in searching an optimal state between surface area and its wrinkling needed in many technological processes. Specifically, it was found that broad-band forcing causes additional production of smaller scales in the flow and for a small initial size of a tracer and Schmidt number this effect is directly responsible for the wrinkling area enhancement, while the surface area of a tracer is mainly governed by the large-scales in this case. The natural extension of the presented method will be the use of passive scalar forcing and the spatial localization of the broad-band forcing region to which future attention will be given.
Acknowledgments This work is part of the research program ”Turbulence and its role in energy conversion processes” of the Foundation for Fundamental Research of Matter (FOM). The authors wish to thank the SARA Computing and Networking Services in Amsterdam for providing the computational resources.
References 1. 2. 3. 4. 5. 6. 7.
Kolmogorov A.N. (1941) C.R. Acad. Sci. URSS, 30:301–305 Finnigan J. (2000) Ann. Rev. Fluid Mech., 32:519–571 Mazzi B., Vassilicos J.C. (2004) J. Fluid Mech., 502:65–87 Geurts B.J. (2001) JoT, 2(17):2–24 McComb W.D. (1991) Physics of Fluid Turbulence, Oxford University Press Canuto C., et al. (1988) Spectral Methods in Fluid Dynamics, Springer Verlag Geurts B.J. (2003) Elements of direct and large-eddy simulation, R.T. Edwards
Direct and Large-Eddy Simulations of a Turbulent Flow with Effusion S. Mendez1 , F. Nicoud1 and P. Miron2 1
2
CERFACS, 42, avenue Gaspard Coriolis. 31057 Toulouse cedex 1. France. [email protected] TURBOMECA, 64511 Bordes Cedex, France [email protected]
Abstract LES results are reported for the flow created by an infinite effusion plate, with staggered holes inclined at an angle of 30 deg to the main flow. Two original methods are proposed to generate the flow in a periodic configuration. Results for mean velocity and velocity fluctuations are compared with measurements made on a large-scale isothermal test rig.
1 Introduction In almost all the systems where combustion occurs, solid boundaries need to be cooled. One possibility often chosen in gas turbines is to use multiperforated walls to produce the necessary cooling. In this approach, fresh air coming from the casing goes through the perforations and enters the combustion chamber [4]. The associated micro-jets coalesce to give a film that protects the internal wall face from the hot gases. The number of submillimetric holes is far too large to allow a complete description of the generation/coalescence of the jets when computing the 3D turbulent reacting simulation within the burner. Effusion is however known to have drastic effects on the whole flow structure. As a consequence, new wall functions for turbulent flows with effusion are required to perform predictive full scale computations. One major difficulty in developing wall functions is that the boundary fluxes depend on the details of the turbulent flow structure between the solid boundary and the fully turbulent zone. Unfortunately, the configuration of full-coverage film cooling is difficult to treat: • •
the perforation imposes small scales structures that are out of reach of current experimental devices and the thermal conditions in actual gas turbines make any measurement very challenging. the number of holes make accurate simulations difficult to perform.
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This explains the lack of data concerning full-coverage film cooling through discrete holes. Most available aerodynamic measurements deal with largescale isothermal flows [4, 6, 13]. On the contrary, experimental studies about the thermal behaviour (evaluation of cooling effectiveness and heat transfer coefficient) do not provide flow measurements [2], or too coarse ones [12]. RANS (Reynolds Average Navier-Stokes) simulations of full-coverage film cooling with several rows (10 rows for Harrington et al. [5]) have been performed. However, as noted by Acharya et al. [1], the issues of modeling errors and the non-universality of the turbulence models do not allow to consider RANS codes as predictive tools. Large-Eddy Simulations (LES) or Direct Numerical Simulations (DNS) only treated the configuration of a single jet in crossflow. Moreover, numerical simulations often study the case of large hole length-to-diameter ratios, while effusion cooling for combustion chamber walls is done through short holes, because of the small thickness of the plates in aeronautical applications. This makes necessary to compute both the aspiration and the injection sides of the perforated plate, in order to avoid unrelevant assumptions on the flow in the hole. In order to gain precise information about the behaviour of the flow near a perforated plate, Large-Eddy Simulations are performed: DNS or Wallresolved LES can be used to generate precise and detailed data of generic turbulent flows under realistic operating conditions, with no limitations due to the size of the configuration or to difficulties to realise measurements in a hot flow. In this paper, two computational methodologies are proposed to perform calculations of an effusion flow and results are compared with the TURBOMECA experimental database, generated on the LARA test rig [6]. Present results only concern the injection side of the flow.
2 Computational Methodology For effusion cooling studies, experimental test rigs are generally divided into two channels: one represents the combustion chamber, with a primary flow of hot gases and the other represents the casing, with a secondary flow of cooling air. The plate between the channels is perforated. Because the pressure is higher in the casing side, cooling air is injected through the perforated plate. Results in the open literature show that the effusion flow highly depends on the configuration of interest: the flow generated by a ten-row plate would be different from the one generated by a twenty-row plate. This situation is hardly tractable from a modeling point of view and we decided to simplify the problem by considering the asymptotic case where the number of rows is infinitely large. The present simulation is then designed to reproduce this asymptotic case of a turbulent flow with effusion around an infinite plate. This choice presents several advantages: the infinite plate can be reduced to a domain containing only one perforation, with periodic boundary conditions to reproduce the whole geometry of an infinite plate, as it is suggested in
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Fig. 1 and the difficult question of the inlet and outlet boundary conditions in turbulent simulations (see [8]) is avoided. Modeling an infinite plate by a periodic domain implies that an additional length-scale is introduced. Tests on the size of the domain, for example by considering four holes, have to be performed to make sure that the computations to not depend on this size. However, experiments show that the jets do not seem to interact together: for usual operating points in effusion cooling applications, jets are shown to be rapidly mixed with the main flow [12]
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With such a periodic calculation domain, the objective is to have information about the structure of the flow far from the first rows, when the film is established. However, this periodic option raises a problem: natural mechanisms that drive the flow, such as pressure gradients, are absent. The flow has to be generated artificially. The purpose of next sections is to describe two treatments used to drive the flow. 2.1 Main flows For a classical channel flow, a volumetric source term is added to the momentum conservation equation in order to mimic the effect of the mean streamwise pressure gradient that would exist in a non-periodic configuration. This is a very classical method for channel or pipe flow simulations. The source term is usually constant over space. For example, it can have the following form: S(ρ U ) =
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treatment is done for one channel. This approach can be generalized in the case of an effusion configuration, having a source term of the previous form for each channel: therefore, the source term on momentum is constant by part, with two distinct values for the cold and the hot sides. No source term is applied in the hole. 2.2 Injection In experiments, channels are bounded by impermeable walls at the top and at the bottom. If used in conjunction with periodic boundary conditions in the tangential directions, this outer condition prevents the flow from reaching a statistically steady state with effusion, because the net mass flux through the perforation tends to eliminate the pressure drop between the cold and the hot domains. In order to sustain the secondary effusion flow in periodic LES, two different strategies are investigated: First option CST method: Constant Source Terms In this method, boundary conditions at the top and at the bottom are walls. Exactly as it is done for momentum, constant (over each half of the domain) source terms on pressure and density are used in order to drive these quantities towards prescribed target values consistent with the operating point that is being studied. As temperature, pressure and density are linked together via the state law, controlling the couple (P,ρ) allows to control the couple (P,T). They have the same form as for the momentum source term. Second option BC method: Boundary Conditions This method is based on the use of boundary conditions. In this method, the physical boundaries that bound the experiment are replaced by characteristicbased [11] freestream boundary conditions. These boundary conditions allow to define pressure, density and velocity at the top and the bottom boundaries in order to impose the appropriate mean vertical flow rate.
3 Details of Numerical Simulations The computational domain is designed as the smallest domain that can reproduce the geometry of an infinite plate with staggered perforations, as it is shown in Fig. 1. All simulations are carried out with the AVBP code [9]. It is a fully explicit cell-vertex type code that solves the compressible Navier-Stokes equations on unstructured meshes for the conservative variables (mass density, momentum and total energy). AVBP is dedicated to LES and DNS, and it has been widely used and validated in the past years in all kinds of configurations. The present simulations are based on the WALE sub-grid scale model [10].
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A coarse grid (grid 1) is first used to evaluate the ability of the two methods (CST and BC) to generate an effusion flow. It contains 150,000 tetrahedral cells. Finer computations made on a grid containing 1,500,000 tetrahedral cells (grid 2) are then performed. In this second grid, fifteen points describe the diameter of the hole and on average the first off-wall point is situated at y+ ≈ 5. The numerical scheme is the Lax-Wendroff scheme (second order in time and space) for the calculations on grid 1 and the TTGC scheme [3] (third order in time and space) for grid 2: this scheme was specifically developed to handle unsteady turbulent flows with unstructured meshes.
4 Results and Discussion 4.1 Operating point Both methods are tested and results are compared with experimental measurements obtained by Laser Doppler Anemometry. The configuration corresponds to the geometry studied by Miron [6]. The study focuses on the case of a large-scale isothermal plate, with a hole diameter of d=5 mm (0.5 mm is a common value for gas turbines applications). Holes are spaced 6.74 diameters apart in the spanwise direction and 5.84 diameters apart in the streamwise direction. The thickness of the plate is 10 mm and holes are angled at 30 deg with the plate: they are short holes, with a length-to-diameter ratio of 1.73. This is an isothermal experiment: both the “primary flow”, which represents the burned gases flow inside the combustion chamber, and the “secondary flow”, which represents the cold air coming from the compressor, are at the same temperature. The main aerodynamical parameters, given for the region upstream the perforations, are summarized here: • • •
The Reynolds number for the primary flow (based on the duct centerline velocity and the half height of the rectangular duct) is Re=17750. The Reynolds number for the secondary flow (based on the duct centerline velocity and the half height of the rectangular duct) is Re=8900. The pressure drop across the plate is 42 Pa.
In simulations, these parameters are fixed. The behavior of the flow in the hole results from the calculations. Numerical results are compared with measurements performed at the ninth row of the experiment. Further details about this experiment can be found in [6] and [7]. 4.2 Comparisons between experimental and numerical results Measurements on the LARA test rig show that effusion jets interact to form a film that protects the plate from the primary flow (hot gases in real cases). Only the structure of the film far from the first rows is interesting for comparisons with the simulations. After a few rows, downstream the hole, mean
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streamwise velocity profiles are characterized by two peaks: the first one, next to the wall, represents the jet core. It is the result of the interaction between the jet coming out of the hole and the film. The second peak results from the interaction of all former jets with the main flow: it represents the film core.
Fig. 2. Mean streamwise velocity profiles (U) and mean vertical velocity profiles (V) 3d downstream the hole, on the centerline plane.
Figure 2 shows the mean velocity profiles at 3 diameters downstream the hole center, on the centerline plane. Symbols correspond to measurements, continuous lines to LES with the BC method and dotted lines to LES with the CST method, using grid 1. Both methods show their ability to reproduce the general topology of the flow, even on a coarse grid . Even if they are very different, the two methods give close results. Furthermore, measurements and results of the computations are very similar. They show the same global behaviour and even the same orders for the mean velocity: the strength and the penetration of the jets are well reproduced, even if results with the BC method show a better agreement with experimental measurements. Finer computations were performed with the BC method. Improvements on the physics of the flow are observed (not shown): the velocity field in the jet shows a realistic form: separation of the jet due to the sharp-edged, inclined inlet along the downstream portion of the hole is reproduced. At the hole outlet, the jet lift-off and the entrainment process [13] are observed. Figure 3 shows the mean and the root mean square velocity profiles in the two directions available in the LARA database (streamwise and vertical). Profiles are shown at 3 diameters downstream the jet, on the centerline plane (symbols show the measurements, continuous lines the LES with the BC method, using grid 2). Figure 3.a shows two different trends for the mean streamwise velocity U: the behavior in the near-wall region is quite well reproduced and the velocity peak due to the jet is situated as in the experiment. This ability to describe the near-wall region is crucial . Larger differences can be found above the jet. It is believed that it is mainly due to the difference between the cases of interest. Simulations characterize the flow around an infinite perforated plate, while measurements are made at the ninth row of the test rig. This effect of accumulation for mean streamwise velocity is coherent with what is observed
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experimentally [6]. The number of rows has an effect on the mean velocity profile above the peak due to the jet: the velocity of the film core tends to increase with the number of rows. For quantities that are not directly affected by the effect of accumulation due to the infinite configuration, comparisons between experiments and simulations show good agreement, as shown for the mean vertical velocity or for the root mean square velocities.
5 Conclusion To compute realistic cases of effusion cooling in the generic configuration of an infinite plate, two original methods are proposed. Although the methods are very different, their results are similar. Computational velocity fields are compared with measurements coming from the TURBOMECA experimental database generated by LDA on the LARA experiment. This configuration is an isothermal effusion flow through large-scale holes. Coarse computations reproduce the general topology of the flow. A second finer grid is used to perform simulations with the BC method. The use of this grid leads to improvements on the physical structure of the flow and the mean velocity levels are modified. Velocity gradients at the wall are correctly predicted. Mean streamwise velocity shows an effect of the infinite plate, with differences on the film core prediction, compared to the experimental data. RMS velocity levels show a good general behaviour, with good prediction of the trends observed in the experiment.
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Acknowledgements The authors are grateful to the European Community for funding this work under the project INTELLECT-DM (EU Project AST3-CT-2003-502961), and to the Centre Informatique National pour l’Enseignement Sup´erieur (CINES) for the access to supercomputer facilities.
References 1. S. Acharya, M. Tyagi, and A. Hoda. Flow and heat transfer predictions for film cooling. HEAT TRANSFER IN GAS TURBINE SYSTEMS. Annals of the New York Academy of Sciences, 934:110–125, 2001. 2. F. Bazdidi-Tehrani and G.E. Andrews. Full-coverage discrete hole film cooling: investigation of the effect of variable density ratio. Journal of Engineering for Gas Turbines and Power, 116:587–596, 1994. 3. O. Colin and M. Rudgyard. Development of high-order Taylor-Galerkin schemes for unsteady calculations. J. Comp. Physics, 162(2):338–371, 2000. 4. R.J. Goldstein. Advances in Heat Transfer. Academic Press, New-York and London, 1971. 5. M. K Harrington, M. A. McWaters, D. G. Bogard, Lemmon C. A., and K. A. Thole. Full-coverage film cooling with short normal injection holes. ASME TURBOEXPO 2001. 2001-GT-0130, 2001. ´ 6. P. Miron. Etude exp´erimentale des lois de parois et du film de refroidissement produit par une zone multiperfor´ ee sur une paroi plane. PhD thesis, Universit´e de Pau et des Pays de l’Adour, 2005. 7. P. Miron, C. Berat, and V. Sabelnikov. Effect of blowing rate on the film cooling coverage on a multi-holed plate: application on combustor walls. In Eighth International Conference on Heat Transfer. Lisbon, Portugal, 2004. 8. P. Moin and K. Mahesh. Direct numerical simulation: A tool in turbulence research. Annu. Rev. Fluid Mech., 30(539-578), 1998. 9. V. Moureau, G. Lartigue, Y. Sommerer, C. Angelberger, O. Colin, and T. Poinsot. Numerical methods for unsteady compressible multi-component reacting flows on fixed and moving grids. J. Comp. Physics, 202(2):710–736, 2005. 10. F. Nicoud and F. Ducros. Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow, Turbulence and Combustion, 62(3):183–200, 1999. 11. T. Poinsot and S. Lele. Boundary conditions for direct simulations of compressible viscous flows. J. Comp. Physics, vol.101(1):104–129, 1992. ´ 12. S. Rouvreau. Etude exp´erimentale de la structure moyenne et instantan´ ee d’un film produit par une zone multiperfor´ ee sur une paroi plane. Application au refroidissement des chambres de combustion des moteurs aeronautiques. PhD thesis, E.N.S.M.A. et Facult´e des Sciences Fondamentales et Appliqu´ees, 2001. 13. S. Yavuzkurt, R.J. Moffat, and W.M. Kays. Full coverage film cooling. Part 1. Three-dimensional measurements of turbulence structure. J. Fluid Mech, 101:129–158, 1980.
Nasal Airflow in a Realistic Anatomic Geometry R. van der Leeden1 , E. Avital1 and G. Kenyon2 1
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Faculty of Engineering, Queen Mary, University of London, Mile End Road, London E1 4NS, UK Whipps Cross University Hospital, Leytonstone, London E11 1NR, UK
Email corresponding author: [email protected] Summary. A realistic anatomic geometry of the human nasal cavity is used for simulating respiratory airflow. To mimic the function of the lungs, an oscillating pressure difference is used as the driving force of the airflow. Flow patterns during the inhalation and expiration phases are found to have similar main pathways, but it is also showed that the olfactory region is better ventilated during the inhalation phase. Temperature and humidity fields during inhalation are analyzed, showing a slightly deeper penetration for the ambient temperature.
1 Introduction The main physiologic functions of the nose that are of importance for respiration are filtering and conditioning air during inhalation, preserving heat and reducing water loss during expiration and its nasal resistance. Especially the conditioning process, that consists of warming and moistening the inhaled air, is so well performed that the inhaled air (hot or cold, dry or wet) is brought to proper body temperature, and is moistened to near 100 percent saturation before it leaves the nasal cavity and reaches the lungs. This ensures optimal conditions for respiratory gas exchange in the lungs. The main passageway for air to travel into or out of the lungs is the nasal cavity, a region of complex anatomic structure part of the upper respiratory system. It extends from the nares (nostrils) anteriorly to the nasopharynx (the passage connecting the back of the nasal cavity to the top of the throat), and is divided medially by the nasal septum into the left and right nasal chamber. The complex geometry of each nasal chamber is mainly due to the presence of three turbinates; the superior, middle and inferior turbinate, see Fig. 1(a) and Fig. 1(b). Turbinates are small, scroll-like bony structures, protruding into the nasal chamber and lined by mucous membrane. The mucous membrane – a type of tissue that is well-vascularized and highly mucosal – greatly aides the process of filtration, heating and moistening of the
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inhaled air. The purpose of the turbinates is to greatly increase the surface area, and therefore the contact area of the mucous membrane exposed to the inhaled/exhaled air. The turbinates may also create turbulence, and both factors increase the ability to allow more efficient conditioning of the inhaled/exhaled air. On the other hand the onset of turbulence can increase the nasal resistance, and therefore the uncomfortable feeling of the patient. Studies of nasal aerodynamics have been hampered due to the difficulty of an in-vivo study. Experimental studies [3, 6, 9, 10] using enlarged scale models or reconstructed cavities from cadavers have shed some light on nasal aerodynamics, but have often ended with different conclusions as to the nature of flow and on the distribution of the nasal air stream. Nevertheless a threshold of about 220 ml/s has emerged as required for the flow rate in order for turbulence to develop. Computational studies [1, 4, 7, 8, 11] have been performed since early ’90s, but studies based on a realistic nasal geometry are ‘just’ emerging. Often these studies are associated with some assumptions on the flow field, e.g. steady or fully turbulent, and mostly prescribe uniform or parabolic velocity distributions at the inlet and a zero gradient at the outlet are used, which can be far from the real physical dynamics.
(a) Sagittal view
(b) Coronal view
Fig. 1. Anatomical features within the human nasal cavity.
In this study we aim to investigate a regular breathing cycle by imposing a periodic pressure difference over the left nasal chamber. This is done in order to gain a better understanding of the flow regime in the turbinate region and its dependency on the geometry of the nasal cavity. Furthermore, signs of flow transition as energy transfer to higher modes will be investigated as well as the penetration of the ambient temperature and humidity into the nasal cavity.
2 Geometry Description As part of an ongoing study, computed tomography (CT) scans of the nasal cavity of different persons were examined for anatomic abnormalities. The most common abnormalities in the nasal cavity are septal deviation and turbinate hypertrophy (swelling of the turbinates). For the purpose of this study, a nasal cavity without anatomic abnormalities was selected. From the CT scan data an anatomically detailed, three-dimensional reconstruction of the left nasal chamber was generated (see
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Fig. 2) by using AMIRATM , a commercial software package for processing 3D image data. An uniform rectangular grid, containing up to 10 million cells, was used to capture the geometry. The transverse plane (the xz-plane) contained 400x128 grid points, while in the normal y-direction 256 grid points were used. The disadvantage of using a uniform rectangular grid in the current situation was that more then 50 percent of the grid cells were outside of the flow domain. However, future work will include the use of grid mapping, which should reduce that ratio. For the discussion of the results in Section 4, medical terminology is used to name certain areas within the nasal cavity. To avoid indistinctness concerning the terminology, a brief description of these medical terms will be given. An illustration is given in Fig. 3, which depicts a typical cross-section in the turbinates’ region. Because the superior turbinate is much smaller in comparison to the middle and inferior turbinates, the superior turbinate has been omitted during the generation of the geometry.
Olfactory Slit Middle Meatus Middle Turbinate Middle Airway Inferior Meatus Inferior Turbinate Inferior Airway
Fig. 2. Surface geometry of the enclosure of the nasal cavity.
Fig. 3. Airways within the nasal cavity in the turbinates’ region, shown in a typical coronal cross-section.
The airway enclosed by the middle turbinate and the sidewall of the nasal cavity is called the middle meatus, while the inferior meatus is the airway enclosed by the inferior turbinate and the sidewall. The airways located underneath the middle and inferior turbinates are referred to as middle and inferior airway respectively. Furthermore, a very important area within the nasal cavity is the olfactory slit, the area near the nasal roof containing the nerve cells responsible for olfaction (sense of smelling).
3 Numerical Method The incompressible Navier-Stokes equations are solved on a staggered rectangular grid using a second-order central scheme for the diffusion terms, and a second-order WENO scheme [5] for the convection terms (formulated in advection form). Due to the type of discretization the simulations should be considered as of the MILES approach [2]. A fully implicit time marching algorithm is used in order to overcome
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the severe time step constraint of an explicit time marching, which was found to be too restrictive due to the CFL-condition. To ease the computational burden of solving the elliptic equations resulting from the implicit time marching algorithm, a directional splitting technique is used. The time accuracy is of first order, however the time step is kept well below 0.1% of the time period imposed by the function of the lungs. Reducing the time step by half showed a negligible effect on the results discussed in Section 4. The projection method is used in order to satisfy continuity where the pressure equation is solved using a bi-conjugate gradient solver. Also, three scalar transport equations are modeled. One scalar transport equation is used to illustrate the flow evolution and therefore has the same diffusion coefficient as the momentum equation. The two other scalar equations are used to describe the transport of temperature and humidity. The shape of the geometry is captured using the ghost-point method. The values of the velocity on these points are determined using no-slip wall boundary conditions, and the projection method boundary condition for the pressure. For the passive scalar, temperature and humidity fields, Dirichlet wall boundary conditions are used. The flow is driven by an oscillating pressure difference between the naris and the nasopharynx to mimic the function of the lungs. Zero value and zero-gradient conditions are imposed at the inflow and outflow boundaries respectively for the transverse velocities U and W (x-, and z-direction respectively). The state of the condition is decided according to the direction of the normal velocity V (y-direction) at the naris or nasopharynx. Similarly, zero-gradient conditions are used for the temperature and humidity at the outflow boundary, whereas for the inflow boundary, the body wall conditions for the nasopharynx, and normalized atmospheric conditions for the naris are used. A similar approach is used for the passive scalar which has an inflow value of zero at the nasopharynx and an inflow value of one at the naris.
4 Results Simulations were performed by imposing a periodically oscillating pressure difference between the naris and the nasopharynx with a time period of 5 seconds and an amplitude of 35 mm H2 O. This level of pressure difference is typical for the lower range of quiet breathing [3]. The simulation started from a no flow condition and the pressure difference was raised in a sinusoidal motion. The flow was allowed to run one cycle before statistics were accumulated for several breathing cycles. Four sections within the nasal cavity have been chosen to discuss the results. Section 1 is the nasal valve region, an area in front of the turbinates, section 2, 3 and 4 are the anterior, middle and posterior part of the turbinates’ region respectively. Typical instantaneous streamwise velocity contours in the coronal cross-section are shown in Fig. 4 and Fig. 5 for the inhalation and expiration phase respectively. Both phases show quite similar flow patterns. It is evident that a significant amount of flow is going through the inferior airway and the airway along the nasal septum, which is in in agreement with the previous studies of [3, 8, 11]. Furthermore, the flow patterns in Fig. 4 and Fig. 5 show a very good qualitative agreement with the experimental findings of [3]. Quantitative comparisons between the two phases show that in most regions of the nasal cavity higher streamwise velocities are reached during expiration. The dominance of the air flow during expiration can also be
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Fig. 4. Instantaneous streamwise velocity contours during inhalation, shown in coronal cross-sections of the turbinates’ region.
Fig. 5. Instantaneous streamwise velocity contours during expiration and in the same cross-sections as in Fig. 4.
derived from the flow rates, which is up to 165 ml/s during inhalation and 210 ml/s during expiration. These values are lower or in the borderline of the threshold quoted earlier as required for turbulence to develop. During inhalation very low streamwise velocities are observed in the inferior meatus, while during expiration these velocities are much higher. This large difference is likely to be caused by the fact that the passageway from the naris to the inferior meatus is much smaller in comparison to the passageway from the nasopharynx to the inferior meatus. The middle meatus doesn’t show a considerable difference between the two phases. On the other hand, the olfactory slit shows a large difference in the streamwise velocities, as can clearly be seen from Fig. 4 and Fig. 5. During inhalation the streamwise velocity levels are more raised towards the olfactory slit, and also much higher streamwise velocities are observed. This can be an indication that the anatomy of the nasal cavity is build
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to support the function of olfaction. This idea is also strongly supported by the fact that the olfactory slit shows a positive vertical velocity V during inhalation, meaning that air is moving upward towards the olfactory nerve cells. The surrounding region shows negative values of V (downward flow), which can be seen in Fig. 6. Power spectra of the U velocity component and the weaker V and W components near the nasal valve region were analyzed to show the dominance of the forcing frequency and its harmonics, see Fig. 8. However in the inferior meatus and towards the end of the region, in the middle airway, the W component was dominated by the first harmonic while the V component actually lacked that mode, see Fig. 9. This indicates that the energy transfer to the harmonics involved interactions between the velocity components. On the other hand, signs of secondary energy transfer as noticeable sub-harmonics have not been identified. Typical instantaneous humidity contours during the inhalation phase are shown in Fig 7, where the low levels in Section 4 point to the strong suppression function of the turbinates. The good similarity with the flow patterns in Sections 1 and 2 are due to the dominance of the convection mechanism. The raised peak in Section 1 as compared to the streamwise velocity contours in Fig 4 is due to the action of the stream-normal velocity causing the more evenly distributed flow in the turbinates region seen in Fig 4. The temperature field shows similar contour levels, however slightly higher levels than of the humidity are observed in Section 4. This is caused by the diffusion coefficient being about 10% lower in the temperature equation than in the humidity equation.
Fig. 6. Instantaneous normal velocity contour during inhalation.
Fig. 7. Humidity contours shown in cross-sections as in Fig. 4.
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5 Conclusions Simulations for a realistic anatomic geometry, reconstructed from CT scans, were performed for a single nasal chamber. The numerical method used an implicit time marching algorithm and had a global second order accuracy in space. The flow was driven by imposing a moderate sinusoidal oscillating pressure difference between the nostril and the nasopharynx. Inspiratory and expiratory flows were analyzed to identify dominant flow paths through the inferior airway and the airway along the nasal septum. Also support was found for the fact that the olfactory slit is more ventilated during the inhalation phase. Signs of flow transition of energy transfer to higher frequencies and some dependency on interaction between the velocity components were identified. However, noticeable secondary energy transfer was not identified, which may be due to the flow rates that did not exceed the quoted threshold obtained by experiments. Temperature and humidity fields were seen to be strongly affected by the turbinates and the outside temperature showed a slightly deeper penetration than the humidity. Future work will focus on simulating different geometries of the nasal cavity, which may explain the difference in the identification of the main flow pathways between the current study and for example that of [8]. A more accurate presentation of the forcing pressure time variation than a simple sinusoidal motion, and increasing the pressure amplitude may enhance flow transition to turbulence. Further investigation of these effects and the response of the temperature and humidity fields is also planned.
Acknowledgments The authors gratefully acknowledge the JLO research scholarship for their support. Also, the authors like to thank R. Jones and P. Richards of St. Bartholomew’s Hospital London for providing the CT data sets.
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References 1. Elad D., Liebenthal R., Wenig B.L. Einav S. (1993) Analysis of air flow patterns in the human nose. Medical & Biological Engineering & Computing 31:585–592 2. Fureby C., Grinstein F.F. (1999) Monotonically integrated large eddy simulation of free shear flows. AIAA 37:544–556 3. Hahn I., Scherer W., Mozell M.M. (1993) Velocity profiles measured for airflow through a large-scale model of the human nasal cavity. Journal of Applied Physiology 75:2273–2287 4. H¨ orschler I., Meinke M., Schr¨ oder W. (2003) Numerical simulation of the flow in a model of the nasal cavity. Computers & Fluids 32:39–45 5. Jiang G.S., Shu C.W. (1996) Efficient Implementation of Weighted ENO schemes. J. Comp. Phys. 126:202–228 6. Kelly J.T., Prasad A.K., Wexler A.S. (2000) Detailed flow patterns in the nasal cavity. Journal of Applied Physiology 89:323–337 7. Naftali S., Schroter R.C., Shiner R.J., Elad D. (1998) Transport Phenomena in the human nasal cavity. A computational model. Annals of Biomedical Engineering 26:831–839 8. Sarangapani R., Wexler A.S. (2000) Modeling particle decomposition in extrathoracic airways. Aerosol Science and Technology 32:72–89 9. Schreck S., Sullivan K.J., Mo C.M., Chang H.K. (1993) Correlations between flow resistance and geometry in a model of the human nose. Journal of Applied Physiology 75:1767–1775 10. Simmen D., Scherrer J.L., Moe K., Heinz B. (1999) A dynamical and direct visualization model for the study of nasal airflow. Archives of Otolaryngology Head & Neck Surgery 125:1015–1021 11. Zhao K., Scherer P.W., Hajiloo S.A. Dalton P. (2004) Effect of anatomy on human nasal air flow and odorant transport patters. Chemical Senses 29: 365–379
Large-eddy simulation of a purely oscillating turbulent boundary layer S. Salon1 , V. Armenio2 and A. Crise1 1
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Istituto Nazionale di Oceanografia e di Geofisica Sperimentale - OGS, B.go Grotta Gigante 42/c, 34010 Sgonico (TS), Italy [email protected] Universit` a degli Studi di Trieste - Dip. di Ingegneria Civile, sez. Idraulica, P.le Europa 1, 34127 Trieste, Italy [email protected]
1 Summary A numerical study of the Stokes boundary layer in the turbulent regime is performed and herein discussed in view of the relevant literature. The study is carried out by Large eddy simulations in conjunction with a plane-averaged dynamic mixed subgrid-scale model. As in canonical boundary layers, the grid resolution in the spanwise direction results critical to correctly reproduce the main characteristics of the flow. Specifically, resolved LES (according to the definition given for canonical wall-bounded flows) is proved to be able to reproduce correctly the main features of the flow field. Consistently with literature experimental studies, the presence of a log-layer was detected in the central phases of the oscillatory cycle, where turbulence is fully developed. At the value of Reynolds number herein investigated turbulence was observed to decay during the late deceleration and to switch on again in the mid acceleration phases. The analysis of the turbulence structure showed that the flow field resembles a canonical boundary layer in the phases of fully developed turbulence. In the remaining of the cycle, the vertical turbulent kinetic energy was found to decay and to re-energise much faster than the other two components, giving rise to an highly anisotropic behaviour.
2 Introduction The study of the purely oscillating (Stokes) boundary layer (SBL) is of considerable relevance in different research fields, such as coastal and offshore engineering, bio and geophysical fluid dynamics. In SBL, the flow is driven by a zero-mean, harmonic velocity field, and the Reynolds number, usually of the outer defined as Reδ = U0 δS /ν, depends on the maximum amplitude ) velocity U0 and the thickness of the laminar layer δS = 2ν/ω, where ν is the kinematic viscosity of the fluid and ω the angular frequency of the oscillation.
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Theoretical, experimental and numerical studies (see among the others [1, 2, 3, 4, 5, 6, 7, 8] showed that the SBL experiences four different flow regimes, depending on the value of Reδ . Fully developed turbulence is present throughout the cycle at Reδ ≈ 3500 ([2]), although [4] showed that turbulence is present in most of the phases already at Reδ ≈ 1800. Due to the burdensome computational requirements needed to simulate the turbulent regime, so far the SBL has been investigated numerically in the intermittent regime for Reδ = 800 − 990 (direct simulations by [5, 6, 7, 8]), mainly with the aim to understand the triggering of turbulence and the evolution of the near-wall coherent structures. Over the last decade, large-eddy simulation (LES) has demonstrated to be a very robust tool to accurately simulate equilibrium as well as nonequilibrium turbulent flows. As regards turbulent flows subject to periodic forcing, LES was successfully employed by [9] to study the pulsating flow in a channel in the current dominated regime (ac = 0.7 with ac the ratio between the maximum velocity of the oscillating part and the maximum velocity of the steady component). So far, no numerical simulations of the turbulent SBL have been carried out by means of DNS or LES. The goal of this work is to fill such gap, providing insights on the characteristics of the turbulent regime, and in particular comparing our results with the experimental investigations of [2]. The flow field is studied using LES in conjunction with a plane-averaged dynamic-mixed SGS model.
3 The problem formulation In the present work we reproduce test 8 of [2], corresponding to Reδ = 1790. According to [4], at such Reynolds number turbulence is present in most of the cycle. The oscillating flow is driven by an harmonic pressure gradient aligned with the x-direction: dP (t) = −U0 ωcos(ωt) (1) dx where t is the dimensional time. The filtered, non-dimensional governing equations of an oscillating boundary layer driven by an harmonic pressure gradient are reported in [10]. The equations are solved by means of a second-order accurate finite difference scheme that adopts the semi-implicit, fractionalstep method of [11]. The subgrid-scale stresses are parameterised using the dynamic-mixed model extensively discussed and validated in [12]. Since turbulence is homogeneous in the streamwise and spanwise directions, periodic boundary conditions are there implemented, while no-slip and stress-free conditions are enforced respectively at the bottom wall and at the top boundary. The initial condition is represented by a well developed turbulent flow previously simulated (forced by a steady pressure gradient): from such field we removed the mean velocity field holding the three-dimensional
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fluctuating components, and started the oscillatory motion. The statistics were accumulated ensemble-averaging over the xy planes of homogeneity and in-phase over half cycle. The horizontal dimensions of the computational box, Lx ≈ 50δS and Ly ≈ 25δS , were chosen large enough to allow the two-point correlation function of the velocity and pressure fields decaying completely within half length of the domain [8]. The height of the domain was chosen equal to Lz = 40δS , about twice the vertical extension of the SBL at the value of Reδ herein investigated, with the cells clustered in the wall region. Several grids were employed with increased resolution in the horizontal directions (see Table 1). As regards the grid spacing in the vertical (wall-normal) direction, in all cases about 30 cells were located within δS and the first velocity point closer to the bottom wall was placed at z + = 1. Finally, to estimate z ∗ , we used the maximum value of τw along the oscillatory cycle as measured in test 8 of [2]. Table 1. Computational parameters of the simulations at Reδ = 1790. The maximum friction coefficient is defined as cf = 2τw,max /ρU02 ; cf0 = 0.044 was obtained by [2]. case Lx /δS , Ly /δS , Lz /δS nx , ny , nz C1 C2 C3 C4
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4 Results All the grids are able to reproduce the succession of accelerating and decelerating phases, in agreement with the experimental observations. However, C1 fails to correctly replicate the experimental wall shear stress (Fig. 1). The marginal resolution of C1 is not able to suitably resolve the streaks in the spanwise direction: this causes the value of the friction coefficient cf = 2τw,max /ρU02 to be underestimated by about 26% with respect to the value given by [2] (see Tab. 1). The cf of grid C2 appears under-predicted by only 5%, and very accurate results are obtained with grids C3 and C4, which respectively give cf /cf 0 = 1.03 and cf /cf 0 = 1.01. The improvement obtained with grids C3 and C4 with respect to the results of case C2 is particularly evident during the sharp transition to turbulence observable around 40◦ in Fig. 1. Grid convergence is obtained with grid C3, and a further increase of the resolution in the streamwise direction (C4) does not significantly improve the results. The evolution along the acceleration phases of the wall shear stress exhibits a two-slope behaviour: the first part (up to about 40◦
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for the value of Reδ herein investigated) is characterised by a somewhat sinusoidal behaviour, that is relative to the part of the cycle where the turbulent kinetic energy (not shown) is around its own minimum and turbulence is in the decaying phases of the cycle. The second part is characterised by a rapid increase of the wall stress that corresponds to the transition to turbulence and the increase of the turbulent kinetic energy. This behaviour, that is consistent with the experimental data of [2], is very well reproduced by the computations C3 and C4. After that the maximum value has been reached at about 80◦ the wall stress decays along the phase of deceleration. Again the fine-grid simulations (C3-C4) predict very well the behaviour above discussed. The mean vertical profiles of the streamwise velocity with grids C2, C3 and C4 (Fig. 2) closely agree with those obtained in the experiments. Moreover, a log-layer is clearly observed between 60◦ and 150◦ . The presence of the log-layer u+ (z + ) = (1/κ) log(z + ) + A was also detected in the experiments of [2]: ...the higher Re, the earlier the logarithm layer comes into existence. At Reδ = 3464 the authors observed a log-layer with κ = 0.4 and A = 5, values nearly equal to those of a canonical steady boundary layer, to span the half cycle from about 15◦ to about 150◦ . This range of phases is narrower when the Reynolds number decreases. Our results are consistent with those of the experimental investigations relative to the test 8 at Reδ = 1790, with a log-layer detectable from about 45◦ to 150◦ . A cross-comparison between our results and the exp. data of [2] confirms that although the phase of transition to turbulence changes with Reδ , the last phase at which a log-layer is nearly independent on the Reynolds number. The analysis of the evolution throughout the cycle of the Reynolds shear stresses (Fig. 3) and of the turbulent intensities (not reported) shows that the beginning of the turbulence activity occurs between 30◦ and 45◦ , and the maximum level of turbulent kinetic
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energy is reached between 90◦ and 105◦ . As previously stated, grid C1 is not able to resolve the near-wall streaks responsible for the development of u w that, as a consequence, is strongly underestimated especially at 45◦ , when the near-wall turbulent shear stress strongly increases up to four times than that observable at 30◦ . The general under-estimation of the Reynolds shear stress in the wall region obtained with simulation C1 explains the low values of the wall stress shown in Fig. 1. As regards the results of simulations C2-C4, the agreement between numerical and experimental data is very good, except for the last two phases characterised by a strong decrease of turbulence where the magnitude of the Reynolds stress appears very small. Similar results where obtained for the turbulent intensities. Analogous disagreements were found by other authors when comparisons between DNS and experimental data were carried out (see for example [7]). It is difficult to speculate on the disagreement between numerical and experimental second-order statistics. A possible source of disagreement can be related to the experimental setup of [2], that is different from the domain configuration used in the numerical simulations: specifically, the physical experiments were carried out in a low-aspect-ratio duct, that is known to produce secondary recirculations and a spanwise variation of the Reynolds stresses; second, a steady streaming was observed in the experimental device ([13]) that makes the flow field to behave as a pulsating one, more than a purely oscillating field. The analysis of the turbulent spectra and of the instantaneous velocity and vorticity field has shown that during the late-deceleration/early-acceleration part of the half cycle, turbulence appears to be less energetic and the shape of the coherent structures to differ substantially from those developing along the rest of the half cycle. One question of interest is to understand whether the shape of turbulence (intended as the shape of the energetic structures that are prevalent in the flow field) changes along the half cycle. An effective tool for understanding such features is the Lumley’s invariant map ([14]). Figure 4 shows the vertical distribution of the turbulent states along half cycle. Two different regions in the flow field can be detected, namely a nearwall region (z < 5δS ) and an outer one (z > 5δS ). In the near-wall region, the decay of turbulence experienced from 150◦ to 30◦ does not lead to an isotropization, rather, since the vertical kinetic energy decays much faster than the horizontal one, we observe the generation of pancake-like turbulence (see for example phase 165◦ in Fig. 4). The outer region behaves similar to the near-wall one as regards the switch-on and the decay of turbulence along the acceleration and deceleration phases of the cycle. However, a main difference occurs in the shape of anisotropy, indeed, from 45◦ to 135◦ the outer region behaves similarly to that of a canonical boundary layer, since turbulence tends toward isotropy going far from the wall. On the other hand, from 135◦ to 30◦ the opposite is true, in that two components of turbulence, namely the vertical and the spanwise ones, tend to decay much faster than the streamwise one, thus going toward the onecomponent turbulence.
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5 Conclusions In the present work a numerical study of a Stokes-boundary layer in the turbulent regime at Reδ = 1790 was carried out using LES with a plane averaged dynamic mixed model. The results of the numerical simulations were compared with data of test 8 of the experiments of [2]. Different computational grids were used: the fine grid simulations C3-C4 supply a good estimation of the wall shear stress as well as of the turbulent intensities. The vertical profiles of the mean streamwise velocity and of the Reynolds shear stresses appear to be well reproduced during the central phases of the half cycle where developed turbulence is present. On the other hand, disagreements between numerical results and experimental data appeared in the phases of the cycle characterised by decaying turbulence, possibly caused by the presence of a steady streaming in the experimental device and by the fact that the authors used a low aspect-ratio duct that substantially differs from an infinitively large domain. Our fine-grid computations correctly simulated the evolution of turbulence along the succession of acceleration and deceleration phases in the cycle
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of oscillation. Specifically, a rapid increase of turbulent kinetic energy was observed around 30◦ - 45◦ ; turbulence appeared fully developed between 60◦ and 150◦ , with mean velocity profiles characterised by the presence of a log-layer, similar to that of a canonical steady boundary layer. Decaying turbulence was observed at the end of the deceleration phases, from 150◦ to 180◦ and the beginning of acceleration. The analysis of the energy spectra (not shown here), of the coherent structures and of the map of anisotropy of the Reynolds stresses has shown the presence of two different regions in the flow field, namely a near-wall region (z < 5δS ) and an outer one (z > 5δS ), where the shape of turbulence changes, depending on the phases of the cycle. In the phases when turbulence is fully developed, the turbulent structures resemble those of a canonical channel flow. Conversely, when turbulence decays, pancake-like structures are observed in the near-wall region, due to the fact that vertical kinetic energy tends to decay and to re-energise much faster than the horizontal components, whereas cigar-like turbulence characterises the outer region, where the streamwise kinetic energy dominate over the other two components. As already proved by [9] the plane-averaged dynamic SGS model is able to simulate periodically driven turbulent flows even when part of the cycle of oscillation is in a relaminarized state. Therefore, this study demonstrates that the resolved LES (according to what established in steady wall turbulence) here adopted, in conjunction with a mixed-dynamic subgrid scale model, is able to grasp the relevant features of the purely oscillating flow.
References 1. 2. 3. 4. 5.
6. 7. 8. 9. 10. 11. 12. 13. 14.
Blondeaux P., Seminara G. (1979) Rendiconti Accad Naz Lincei 67:407–417 Jensen B.L., Sumer B.M., Fredsøe J. (1989) J. Fluid Mech. 206:265–297 Akhavan R., Kamm R.D., Shapiro A.H. (1991a) J. Fluid Mech. 225:395–422 Sarpkaya T. (1993) J. Fluid Mech. 253:105–140 Spalart P.R., Baldwin B.S. (1987) Direct simulation of a turbulent oscillating boundary layer. In: Andr´e JC et al. (eds) Turbulent Shear Flows 6. Springer, Berlin Heidelberg New York Akhavan R., Kamm R.D., Shapiro A.H. (1991b) J. Fluid Mech. 225:423–444 Vittori, G., Verzicco, R. (1998) J. Fluid Mech. 371:207–232 Costamagna P., Vittori G., Blondeaux P. (2003) J. Fluid Mech. 474:1–33 Scotti A., Piomelli U. (2001) Phys. Fluids 13:1367–1384 Salon S. (2004) Turbulent mixing in the Gulf of Trieste under critical conditions. PhD Thesis, Univ. of Trieste, Trieste (Italy) Zang Y., Street R.L., Koseff J.R. (1994) J. Comp. Phys. 114:18–33 Armenio V., Piomelli U. (2000) Flow Turb and Comb 65:51–81 Fredsøe J., Sumer B.M., Laursen T.S., Pedersen C. (1993) J. Fluid Mech. 252:117–146 Lumley J.L. (1978) Computational modelling of turbulent flows. Adv. Appl. Mech. 18:123–176
Large Eddy Simulation of a Turbulent Channel Flow With Roughness S. Leonardi1 , F. Tessicini2 , P. Orlandi1 and R.A. Antonia3 1
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Dipartimento di Meccanica e Aeronautica, Universit` a La Sapienza, Via Eudossiana 16, I-00184, Roma [email protected] Dep. of Aer. Imperial College of London, London UK Discipline of Mechanical Engineering, University of Newcastle, NSW 2308 Australia
Summary. Large and Direct Numerical Simulations (LES, DNS) of a turbulent channel flow with square bars on one wall have been carried out at Re = 10400. Two sub-grid models have been used: Smagorinsky with Van Driest damping and Dynamic. There is satisfactory agreement between the two types of simulations for the pressure and skin friction on the wall and the rms streamwise velocity. Comparison for the rms normal and spanwise velocities is poor but the sub-grid models are a significant improvement relative to the no model (which corresponds to an unresolved DNS). A further DNS at Re = 18000 has been performed with the aim of comparing the results with the experiment by Hanjalic & Launder [1]. The Reynolds number dependence (Re ranging from 4200 to 18000) has been discussed. The pressure on the wall and hence the form drag does not depend on Re and the velocity profile changes slightly for Re > 10400.
1 Introduction Flows over rough surfaces are of interest in many practical applications, ranging from shipbuilding and aviation, the flows over blades in different types of turbomachines and the flows over vegetated surfaces in the atmospheric surface layer. In all these cases, the Reynolds number is high and the roughness is very small relative to the characteristic length of the outer flow. Jim´enez [2] claimed that numerical or laboratory experiments should have at least δ/k > 50 and k+ = kuτ /ν in the fully rough regime (δ represents either the radius of the pipe, the thickness of the boundary layer or the half–width of a duct, and k+ is the height of the roughness elements in wall units). Therefore, numerical simulations require a large number of points. For this reason, to date, numerical simulations have been carried out only at low Reynolds numbers (Re = Uc h/ν < 10000, h is the channel half-width, Uc is the centerline velocity and ν is the kinematic viscosity) e.g. DNS, [3], [4], [5] and LES, [6]. Although these simulations have provided useful results, it is important to increase the Reynolds number. In the present paper, LES and DNS results of a turbulent channel flow with square bars on the bottom wall and a smooth upper wall are
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discussed. One of the aims is to compare the results with those of the experiment by Hanjal´ıc & Launder [1] for a turbulent channel flow with square bars on the bottom wall with λ/k = 10, where λ is the distance between successive elements. DNSs have been carried out at Re = 10400 and Re = 18000. The computational box is 8h × 2.125h × 6.25h in the streamwise (x) wall–normal (y) and spanwise (z) direction respectively. The additional 0.125h increase in the channel height is due to the cavity height where the square elements (k = 0.125h) are placed. In this context, the development of reliable LES sub–grid models remains an important objective. The grid used for the LES, (240×160×49 in x, y, z respectively), is much coarser than that used for the DNS (513 × 177 × 193 and 769 × 161 × 193 for Re = 10400 and Re = 18000 respectively). The models used are the standard Smagorinsky model with Van Driest damping (Cs = 0.1, hereafter SM10) and the dynamic model (DYN). To underline the effect of the sub–grid model, a simulation without model has been carried out (NOM). The latter would correspond to an unresolved DNS.
2 Numerical Procedure The non-dimensional Navier-Stokes and continuity equations for incompressible flows are: ∂Ui ∂P 1 ∂ 2 Ui ∂Ui Uj =− + +Π , + ∂t ∂xj ∂xi Re ∂x2j
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(1)
where Π is the pressure gradient required to maintain a constant flow rate, Ui is the component of the velocity vector in the i direction and P is the pressure. The Navier-Stokes equations have been discretized in an orthogonal coordinate system using the staggered central second-order finite-difference approximation. Here, only the main features are recalled since details of the numerical method can be found in Orlandi [7]. The discretized system is advanced in time using a fractional-step method with viscous terms treated implicitly and convective terms explicitly. The large sparse matrix resulting from the implicit terms is inverted by an approximate factorisation technique. At each time step, the momentum equations are advanced with the pressure at the previous step, yielding an intermediate non-solenoidal velocity field. A scalar quantity Φ projects the non-solenoidal field onto a solenoidal one. A hybrid low-storage third-order Runge-Kutta scheme is used to advance the equations in time. The roughness is treated by the efficient immersed boundary technique described in detail by Fadlun et al. (2000). This approach allows the solution of flows over complex geometries without the need of computationally intensive body-fitted grids. It consists of imposing Ui = 0 on the body surface which does not necessarily coincide with the grid. Another condition is required to avoid that the geometry is described in a stepwise way. Fadlun et al. (2000) showed that second-order accuracy is achieved by evaluating the velocities at the closest point to the boundary using a linear interpolation. This is consistent with the presence of a linear mean velocity profile very near the boundary even for turbulent flows, albeit at the expense of clustering more points near the body.
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3 Results and discussion The square element on the bottom wall induces a separation at the trailing edge of the elements (x/k 1 Fig.1). In agreement with the results of Leonardi et al. [4] obtained for larger elements (DNS k = 0.2h) the flow reattaches on the bottom wall at about x/k 5 (where x = 0 is taken at the leading edge of the element). On the other hand, LES simulations predict a larger recirculating region, with a smaller intensity. As the next element is approached, a separation occurs at about x/k 9, one roughness height upstream of the element. The LESs, in this case, yield a good approximation for the Cf with respect to the DNS. This behaviour is due to the non–uniform grid used for the LES with a larger number of points very near the element, and a very coarse resolution within the cavity. The element leads to a large increase of velocity and a presence of the friction peak at the leading edge of the element. Above the crest, as shown in Leonardi et al. [4] for λ/k > 8 a separation occurs. LES and NOM are not able to reproduce this separation which was also observed in the experiment of Liu, Kline and Johnston [9]. Pressure distributions along the horizontal and vertical walls are shown in Fig. 2 over one wavelength. Very near the element (0 < x/k < 0.25 and 1.25 < x/k < 1.5) LES and DNS results are in good agreement. At the center of the cavity, larger differences are found. This is again due to the non-uniform grid used. Since the difference between pressure distributions on the vertical walls, corresponds to the form drag for this roughness element, approximately, the LES and DNS yield values of the form drag that agree. On the other hand, NOM yields a different pressure distribution over most of the wavelength and a different (smaller) form drag. The
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pressure distributions for Re = 10400 and Re = 18000 are close to each other. The form drag, is Pd = 6.4610−3 and 6.610−3 for Re = 10400 and Re = 18000 λ respectively (Pd = λ−1 0 P n · xds, angular brackets denote averaging in time and z, P is normalised by ρUc2 ). This is an extension of the results of Leonardi et al. [4]. In a previous paper, they defined Cd = Pd /k and showed that for several values of λ/k, Cd does not depend on Re (which was varied between Re = 4200 to Re = 10400) and on k (in the range 0.1h to 0.2h). For large values of λ/k (e.g. λ/k > 3), the total drag is almost entirely due to the form drag. Therefore, the value of the friction velocity, Uτ ≡ (Pd + Cf )1/2 , does not change with the Reynolds λ number, (Cf = λ−1 0 Cf ds). As a consequence, we believe that, for this type of roughness, the flow physics near the wall can be investigated through numerical simulations at moderate Reynolds numbers. The mean velocity distribution shown in figure 3 for different Reynolds numbers, are compared with the measurements of Hanjalic & Launder [1] at Re = 18000. The agreement between experiment and DNS is satisfactory. The DNS results show that by increasing Re the maximum velocity is shifted upwards (towards the smooth wall). However, whereas the changes to the velocity profile are large between Re = 4200 and Re = 10400, only slight differences are observed between Re = 10400 and Re = 18000. Therefore, the dependence on the Reynolds number, for intermediate values of Re is weak even in the outer layer, so that DNS is a useful tool for providing insight into rough flows.
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where C and κ are constants and ”+” denotes normalization by either Uτ or ν/Uτ . The origin for y is at 0.15k above the bottom wall. With respect to the smooth wall the velocity profile is shifted downward by a factor ΔU + , known as the roughness function. In Figure 4, the mean velocity profiles in wall units for DNS, LES and NOM are compared to the smooth wall distribution by Moser, Kim & Mansour [10]. As expected, the mean velocity profile is shifted downward, and the agreement between LES and DNS is reasonable. The roughness function is indeed due essentially to the increase of Uτ . For this value of λ/k, Uτ is mostly due to the pressure distribution which was shown to be similar for DNS and LES (Fig.2). On the other hand, the pressure drag for NOM was different from that relative to the DNS, then larger differences to the velocity profile are expected. Even if Uτ does not change, the roughness function for Re = 18000 is larger than that for Re = 10400. As the origin in y is the same, and k+ increases, the velocity distribution is shifted downward. In fact, Perry, Schofield & Joubert [11] showed that, for large λ/k (k-type roughness), ΔU + = κ−1 ln k+ + B.
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The value of k+ is 80, 103 and 180 for Re = 4200, 10400 and Re = 18000 respectively. The corresponding values of ΔU + are 12.9, 13.5 and 14.8 respectively, in agreement with equation 3 and B = 2.2. For these values of k+ we are in the fully rough regime (Bandyopadhyay [12]).
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Turbulent intensities are shown in Figure 5 normalised by Uc2 . For uu , both the Large Eddy Simulations performed with Smagorinsky and dynamic models agree reasonably well with the DNS. However, for the other two stresses, the agreement is poor, especially for ww . Near the rough wall (y/h = −1), there is reasonable agreement for vv but significant differences can be discerned in the inner part of the channel. Perhaps surprisingly, the agreement between DNS and LES is not satisfactory near the upper smooth wall. Since sub-grid models work well for a smooth wall, this result should mean that the grid is too coarse to simulate the interaction between the two walls. In fact, roughness increases the communication between the wall and the outer layer. The improvement brought by the sub-grid models that have been tried is encouraging. Indeed, with respect to NOM, Large Eddy Simulations compare much better with the DNS results. While uu on the rough–wall is about the same as that on the smooth wall, vv and ww increase by about 2.5 times. This means that isotropy is better approximated over rough wall, as noted by Smalley et al. [13].
4 Conclusions Direct and Large Eddy Simulations have been performed for a turbulent channel flow with square bars on the bottom wall with a pitch to height value of λ/k = 10 at Re = 10400. For the wall pressure, skin friction and rms streamwise velocity the agreement
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between DNS and LES is satisfactory. On the other hand, for the rms spanwise and normal velocity, the agreement is poor. The improvement brought by the sub grid model is encouraging. The DNS at Re = 18000 showed a reasonable agreement with experimental results by Hanjalic & Launder [1]. The Reynolds number dependence for intermediate Reynolds is very weak, then we speculate that DNS can be very useful in the study of rough flows.
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References 1. Hanjalic & Launder. Fully developed asymmetric flow in plane channel. J. Fluid Mech. 51, 1972, 301–335. 2. Jim´enez, J. Turbulent flows over rough walls. Ann. Rev. Fluid Mech. 36, 2004, 173–196. 3. Miyake Y., Tsujimoto K. & Nagai N. Numerical simulation of channel flow with a rib–roughened wall. J. Turb. 3, 2002, 35. 4. Leonardi, S., Orlandi, P., Smalley, R.J., Djenidi, L. & Antonia, R.A. Direct numerical simulations of turbulent channel flow with transverse square bars on one wall. J. Fluid Mech. 491, 2003, 229-238. 5. Ashrafian A. & Anderson H.I. (2003). DNS of Turbulent Flow in a Rod– Roughened Channel. Proceedings of the Turbulent and Shear Flow Phenomena 3, Sendai, Japan. N. Kasagi, J. K. Eaton, R. Friedrich, J. A. C. Humphrey, M. A. Leschziner, T. Miyauchi. Vol I, 2003, 117–123. 6. Cui J., Virendra C. Patel & Ching-Long Lin. Large–eddy simulation of turbulent flow in a channel with rib roughness. Int. J. of Heat and Fluid Flow 24, 2003, 372–388. 7. Orlandi, P. Fluid flow phenomena, a numerical toolkit. Kluwer Academic Publishers. 2000. 8. Fadlun E.A.,Verzicco, R., Orlandi P. & Mohd-Yusof, J. Combined immersed boundary finite-difference methods for three-dimensional complex flow simulations. J. Comput. Phys. 161, 2000, pp.35-60. 9. Liu, C.K., Kline, S.J. and Johnston, J.P. An experimental study of turbulent boundary layers on rough walls. Report MD–15, 1966, Department of Mechanical Engineering, Stanford University. 10. Moser R.D., Kim J. & Mansour N.N. Direct numerical simulation of turbulent channel flow up to Reτ = 590. Phys. Fluids 11, 1999, 943-945. 11. Perry, A. E., Schofield, W. H. & Joubert, P. N. Rough wall turbulent boundary layers. J. Fluid Mech. 37, 1969, 383–413. 12. Bandyopadhyay, P.R. Rough–wall turbulent boundary layers in the transition regime. J. Fluid Mech. 180, 1987, pp.231–266. 13. Smalley, R.J., Leonardi, S., Antonia, R., Djenidi, L. & Orlandi, P. Reynolds stress anisotropy of turbulent rough walls layers. Expts in Fluids 33, 2002, 31-37.
Part IX
Flow Control
Bimodal control of three-dimensional wakes Philippe Poncet Laboratoire MIP, Dept GMM, INSA Toulouse, 135 avenue de Rangueil, 31077 Toulouse Cedex 4 [email protected]
Summary. This paper investigates control strategies for drag reduction of threedimensional wake generated by a circular cylinder at Reynolds number Re = 300, such flows presenting mode B instabilities whose main feature is streamwise fingershaped eddies. The control is performed thanks to a field of tangential velocities on the cylinder. One first focuses on two-dimensional velocity fiels (spanwise invariant), using both a clustering genetic algorithm (see [7]) and Newton algorithm in Fourier space with five Fourier modes. Besically the same field comes out, whatever the control technique used. A square-root regression of the drag reduction versus amplitude of the control leads to the formulation of an efficiency criterion. One then considers a class of spanwise harmonic perturbation of this quasi-optimal profile, leading to a two paramater optimization problem, involving amplitude and wavelength of the perturbation. A cartography of the efficiency with respect to these two parameters is finally obtained, showing regions of interest.
1 Methodology One considers the full three-dimensional Navier-Stokes equations in their velocity-vorticity (u, ω) formulation and in the context of external flows: ∂ω + u · ∇ω − ω · ∇u − νΔω = 0 (1) ∂t with ∇ · u = 0 and ∇ × u = ω in the domain, and u = 0 on boundaries, ν being the kinematic viscosity and the velocity u satisfying far field condition lim u(x) = U∞ ex , ex being the streamwise basis vector. |x|→∞
The numerical scheme used is an hybrid vortex in cell method, fully described in [5, 3], performing direct numerical simulation of equation (1). As a summary, a time splitting algorithm is used in order to split apart convective and diffusive effects. The diffusive part is solved using a large finite-difference stencil based on particle strength exchange methods, and the Chorin algorithm in the context of cylindrical geometry is used for kinematic boundary conditions.
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The remaining convective part is exactly the Euler equations satisfying incompressibility and with only no-flow-through boundary condition u·n = 0, where n is the inward normal field to body. It is solved using a Lagrangian method involving particles of vorticity-location-volume (ω i , xi , vi ), changing the Euler equation into a classical dynamical system: dω i = (ω · ∇u)x=xi , dt
dxi = u(xi ) , dt
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the velocity field being built by differentiation of stream, that is to say by solving Poisson equations on a grid, with back and forth interpolations between particles and grid (cf. [5]). In order to avoid holes or accumulation of particles, which is sometimes reported as a drawback of Lagrangian methods, frequent high order remeshing onto a uniform lattice is performed. The Lagrangian formulation makes transport terms u · ∇ω in the Euler equations vanish from the dynamical system (2). The transport stability condition, which is extremely restrictive in practice, disappears as well and allows to use time steps hundred times larger than conventional methods, without significant lack of accuracy. This is especially interesting to reach large time scales and perform control beyond transient regimes. This numerical scheme has been validated in many contexts, for examples: up to Re = 9500 for 2D wakes in complex geometries [4], and up to Re = 1400 in 3D with the dynamic of an oblique ring interacting with a flat body, from [3] and represented on figure 1. Three-dimensional wakes have been especially investigated in [8, 9] up to Re = 1000. The understanding of the physics of wakes behind circular cylinders has considerably evolved during the last fifteen years, and is mainly governed by the Reynolds number Re = U∞ D/ν, D being the cylinder diameter. The transition from steady to periodic flow occuring around Re = 47 is a very well-known feature of wakes. The transition from 2D to 3D, breaking the spanwise invariance feature by means of two modes of instabilities, has been intensively tracked during the mid-nineties, numerically (see [12]), by Floquet analysis (see [1]) and experimentally (see [13]). Around Re = 190, the wake turns spontaneously three-dimensional in large wavelength (close to four diameters) called mode A instability. Above Re = 260 the large 3D structures do not appear anymore and are replaced by finger-shaped instabilities of shorter wavelength (less than a diameter) called mode B. This kind of instabilities is the dominant feature up to Re = 2000, superposing with turbulence as inertial range grows, and is responsible of a drop of 25% of the drag coefficient. In the present work, one considers cylinder wakes presenting mode B instabilities at Re = 300, whose vorticity is plotted on figure 2. The drag coefficient is computed over a large interval of time, plotted on figure 3, showing the high stability of the numerical scheme presented above.
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Fig. 1. Oblique annular vortex impinging into a wall at Re = 1400 (Courtesy of G.H. Cottet).
Control of such wakes is performed by means of tangential field of velocity on body in the azimuth direction, that is to say vector eθ in standard cylindrical coordinates. This field of velocity is usually called profile. Section 2 describes how an optimal profile for two-dimensional flows has been built and how it affects three-dimensional flows. Since drag decreases as energy of control increases, the drag reduction cannot be used as cost function for minimization, and an efficiency criterion is defined also in section 2. Spanwise modulation of this profile is then introduced in section 3. Preliminary computations from [6] of short duration, thus having possibly transient effects, and involving only low mode numbers, have shown that drag reduction tends to be larger as mode number increases. Section 4 investigates this phenomenon with long runs and accurate statistics, showing that efficiency increases up to mode 3 and the decreases, independently of energy level.
2 Two-dimensional control of 3D wakes One considers a control by means of a field of velocity tangential to body, here a circular cylinder. The piecewise constant formulation of such fields
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Fig. 2. Vorticity (to the top) and streamwise velocity (to the bottom) isovalues of uncontrolled flow at Reynolds number Re = 300.
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Fig. 3. Drag coefficient of a 3D wake at Re = 300. Early stage is periodic, until instabilities become dominant, leading to more chaotic values.
is usually called “belt actuators”. Applying velocity u to the body surface involves a non-dimensional kinetic energy defined by 1 ∗ u(x)2 ds (3) Ec (u) = 2 σ(∂Ω) 2 U∞ ∂Ω where σ(∂Ω) is a measure of the body: σ(∂Ω) = LπD is the body surface in 3D, L being the cylinder width, and σ(∂Ω) = πD in 2D. Concerning two-dimensional flows, an optimal profile of piecewise constant tangential velocities has been carried out in [7], using an energy-improved genetic algorithm operating over 16 panels, called clustering genetic algorithm (CGA). Such profiles being not continuous, whether considering the best or the most probable element of the population, one has symmetrized and smoothed this profile, using a composite sinus-rational regression (see [10] for example), obtaining the function Vc (θ) plotted on figure 4.
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Fig. 4. Function Vc (θ) used for 2D control, obtained by smoothing profile from Clustering Genetic Algorithm.
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Fig. 5. Optimal modes (left picture) and 2D profile (right picture) for control of 3D flow using a Newton algorithm with five Fourier modes (Courtesy of Roland Hildebrand).
As often for genetic algorithms, global optimality is questionable. In order to validate the profile obtained by smoothing the CGA profile, and show that it is not dependent on the numerical method or its parameters, one has provided an other optimal control: the profile, function of angle, is searched as a combination of five Fourier modes and the minimization is performed by a Newton algorithm, involving 31 parallel runs for each step of minimization (1 central run, 5 for the gradient and 25 for the Hessian matrix). At the same energy level as the CGA, one finds a very strong similarity between profiles, for both maximal value and shape, plotted on figure 5. In order to investigate control of three-dimensional wakes, one first considers a 3D flow controlled by this 2D profile with various amplitudes, that is to say C Vc (θ) corresponding to an non-dimensional energy C 2 Ec∗ (Vc ). Figure 6 shows plots of drag coefficients with respect to time for different values of amplitude C from 0.1 up to 2.0, and as qualitatively expected the drag reduction increases as the energy involved in the control increases. It has been shown in [6] that if energy is sufficiently high, the flow comes back to its nominal two-dimensional state. Figure 6 also shows the mean post-transient drag reduction with respect to the amplitude C, and exhibits a square-root regression, discussed in [11]. This means that an efficiency criterion can be introduced: ) 0 − CD )/ Ec∗ (4) Ef f = (CD 0 = 1.374 is the uncontrolled two-dimensional drag coefficient at where CD Re = 300 for the present computations. As a consequence, the efficiency depends only on profile shape and not on the energy level for this 2D profile, thus provides a suitable cost function for three-dimensional control.
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3 Bimodal control of 3D wakes In order to introduce three-dimensionality in the control, one adds a spanwise harmonic perturbation to the optimal 2D profile, with a rescaling in order to conserve energy. Vector orientation is kept azimuthal, that is to say parallel to basis vector eθ in usual cylindrical coordinates. In order to reduce control space dimension, as a first approach, one also considers a profile with stagnation points and without local reverse velocity. This leads to the function ) (5) u(θ, z) = C 2/3 (1 + sin(kz)) Vc (θ) which depends on integer mode k and real amplitude C. Such a profile involves constant mode and mode of number k (being consequently a bimodal control).
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Fig. 8. Isolines of efficiency with respect to mode k and amplitude C.
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This three-dimensional profile is at the same level of energy as 2D profile, that is to say C 2 Ec∗ (Vc ). Typical flow obtained with such a control is represented on figure 7, here for k = 4 and C = 1.0. After having checked that the final drag does not depend on control activation time, one has computed efficiencies for k = 1..8 and from C = 0.1 up to C = 1.1. In order to exhibit the couple mode/amplitude of highest efficiency, a diagram of isoefficiency is plotted on figure 8. According to this C −k cartography, the maximum efficiency is found to be for modes 2 and 3, with amplitude around 0.65. The first noticeable fact is that nothing significant occurs at mode k = 4 which is the main growth mode of instability (mode B) at this Reynolds number. Indeed, a stronger interaction
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Fig. 10. Vorticity field in the optimal region (k = 3, C = 0.6).
between natural instabilities and a control at the same wavelength could have been expected. Moreover, signification of local extrema at low energy is questionable due to the lack of periodicity of the drag coefficient signal for the uncontrolled flow, thus for low amplitude controlled flow (at larger energies signal is periodic). The only significant low-energy minimum occurs at k = 1 and C = 0.22: this reduces speed difference between body and far field, and consequently reduces strain, which locally comes to consider a flow at Re = 220. At such a Reynolds number, instability of mode A is the main mode of instability, creating large structures (see figure 9) and influencing dramatically the drag coefficient [2]. Note that this control does not interact with mode A, but allows it to exist by generating the right window of strain value. To the opposite, high mode control fields (k above 6) lead to non-optimal stable states for which the flows are strongly separated (see figure 10). Furthermore, for amplitudes above C = 0.76, zones of larger velocities than far field appear on body, and consequently physics of wakes is replaced by physics of jets as energy increases. Such a modification of the physics is probably responsible for the lack of efficiency at high energy.
4 Conclusion A highly stable numerical method has been used for three-dimensional direct numerical simulation of the wake behind a circular cylinder. The problem
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of control of drag coefficient has then been investigated, considering velocity profiles tangential to body. An efficiency criterion been defined, and used for optimization since the drag reduction has been shown to be not suitable as a cost function. Optimal two-dimensional profile has been put forward for 2D and 3D flows. An harmonic perturbation of this profile has then been used to perform control of 3D wakes, and the impact of this perturbation wavelength is discussed. It appears that efficiency increases as wavelength decreases down to wave number 3, that is to say about one diameter, with an optimal amplitude C = 0.65. Smaller wavelengths make efficiency decrease. The question of optimal profiles with different combination between constant mode and harmonic perturbation, or multi-modal optimization is left open, work being under progress. The author would like to thank Georges-Henri Cottet, Petros Koumoutsakos, Roland Hildebrand and Michele Milano for their invaluable help in the elaboration of this work. The computational resources have been provided by CalMiP (CICT, Toulouse, France), LMC (Grenoble, France) and INSA (Toulouse, France).
References 1. Barkley D., Henderson R. D. (1996) J. Fluid Mech. 332:215–241 2. Blackburn H. M., Henderson R. D. (1999) J. Fluid Mech. 385:255-286 3. Cottet G.-H., Koumoutsakos P. (2000) Vortex Methods, Theory and Practice. Cambridge University Press 4. Cottet G.-H., Poncet P. (2002) J. Turbulence 3(038):1–9 5. Cottet G.-H., Poncet P. (2003) J. Comp. Phys. 193:136–158 6. Cottet G.-H., Poncet P. (2004) Comput. Fluids 33:687–713 7. Milano M., Koumoutakos P. (2002) J. Comp. Phys. 175:79-107 8. Poncet P. (2002) Phys. Fluids 14(6):2021–2024 9. Poncet P. (2004) J. Fluid Mech. 517:27–53 10. Poncet P., Koumoutsakos P. (2005) Intl. J. Offshore Polar Eng. 15(1):1–7 11. Poncet P., Cottet G.-H., Koumoutsakos P. (2005) CR Mecanique 333:65–77 12. Thompson M., Hourigan K., Sheridan J. (1996) Exp. Therm. Fluid Sci. 12:190– 196 13. C.H.K. Williamson (1996) J. Fluid Mech. 328:345–407
DNS/LES of Active Separation Control Julien Dandois1 , Eric Garnier1 and Pierre Sagaut2 1
2
ONERA, Applied Aerodynamic Department, BP 72, 29 av. de la division Leclerc, 92322 Ch tillon Cedex, France ’ [email protected] ` ` ` Pierre et Marie Laboratoire de ModElisation pour la MEcanique, UniversitE Curie, Boite 162, 4 Place Jussieu 75252, Paris cedex 5, France and ONERA, CFD and Aeroacoustics Department
Summary. Numerical simulations of active separation control by means of a synthetic jet are carried out to demonstrate that, if the oscillating frequency is well adapted, separation can be suppressed for a velocity ratio between the jet and the flow lower than one. The chosen test case is a rounded ramp at a Reynolds number based on the step height of 28275. The incoming flow is fully turbulent with Rθ = 1410. In a first step, the results of a Large-Eddy Simulation (LES) are validated in comparison with a Direct Numerical Simulation (DNS). In a second step, the effect of a synthetic jet at a Strouhal number based on the step height and the free-stream velocity of 0.11 is investigated using LES. The results show that the separation is reduced by 48% for a velocity ratio between the jet maximum velocity and the crossflow of 0.5.
Key words: DNS, LES, separated flow, synthetic jet, active separation control
1 Introduction Active flow control is one of the most active fields in applied aerodynamics. The interests are practical as well as fundamental. Among all available actuators for flow control, synthetic jets appear to be promising since they have been proven to effectively control separation [15], enhance mixing [3] and vector thrust [17]. The advantage of synthetic jets face to steady blowing or suction is that they need less momentum by one or two orders of magnitude to produce equivalent effects [15]. They also do not require a complex plumbing system because the momentum expulsion is only due to the periodic motion of a diaphragm or a piston on the lower wall of a cavity. Moreover, a better efficiency can be achieved by a coupling with natural flow instabilities. The present paper deals with the control of flow separation. Active separation control over ramps or backward-facing steps has been studied
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experimentally by Bhattacharjee et al. [1], Chun & Sung [4, 5], Yoshioka et al. [21, 22], Seifert & Pack [16] and Narayanan & Banaszuk [11] and numerically by Wengle et al. [20], Neumann & Wengle [12], Dejoan & Leschziner [6] and Krishnan et al. [9]. Bhattacharjee et al. [1], Chung & Sung [4], Yoshioka et al. [21, 22], Wengle et al. [20] and Dejoan & Leschziner [6] have observed on their backward-facing step a reduction of the recirculation length of about 30% for an optimal Strouhal number based on the step height Sth = f.h/U∞ 0.2 with U∞ the inflow velocity. This frequency correspond to the mixing layer instability at Stθ = 0.012 with θ the momentum thickness of the incoming boundary layer (see fig. 6 of Chun & Sung [4]). Chun & Sung [5] have found a higher optimal Strouhal number Stθ = 0.025 on their step and attribute this discrepancy to their lower Reynolds number. Neumann & Wengle [12] have performed a LES of separation control over a rounded step. They have also found an optimal Strouhal number Sth 0.2 but this frequency does not correspond to the optimum frequency Stθ = 0.012 found for sharp-edged backward-facing steps. Seifert & Pack [16] have found an optimal reduced frequency F + = f.L/U∞ = 1.6 which gives, scaled by the hump model thickness Sth = 0.43. As can be seen from this brief review, the optimal Strouhal number is different on sharp-edged and rounded steps and can not be found a priori. This justifies a parametric study on each particular configuration. The motivation of the paper is to demonstrate numerically that separation can be drastically reduced on a smooth ramp by a synthetic jet with an output velocity lower than the free-stream one. This point is of particular importance for transonic applications since present synthetic jets actuators have a limited output velocity. Moreover, accurate prediction of the mean separation point is still a challenging problem over smooth ramps because separation is caused by an adverse pressure gradient contrary to backward-facing steps. This is the reason why advanced numerical methods like direct numerical simulations (DNS) and large-eddy simulations (LES) have been performed. In a first part, a LES of the uncontrolled case is validated in comparison with a DNS. Then, the effect of a synthetic jet on the separation is studied by means of LES in order to limit the computing cost.
2 Flow Configuration Figure 1 presents the flow configuration. The step height is h = 20mm and the maximum slope is 35◦ . This academic configuration has been designed for its representativeness of separated flows encountered in industrial applications like intakes. Nevertheless, to avoid difficulties of internal flow simulations, the top wall is suppressed. The Mach number is 0.3. The stagnation pressure is 21300 Pa and the stagnation temperature is 288 K. The Reynolds number based on the momentum thickness is Rθ = 1410 as in the direct numerical simulation (DNS) of a turbulent boundary layer by Spalart [18]. The boundary layer thickness is 0.5h. The Reynolds number based on the step height and
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the free-stream velocity is Reh = 28275. The synthetic jet orifice consists in a two-dimensionnal slot of width d = 1/3δ. The coordinate system is the following: x is oriented in the streamwise direction, y is vertical and z is in the spanwise direction. The origin is located at the beginning of the ramp. 6
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3 Numerical method The FLU3M code is a finite-volume solver for the compressible Navier-Stokes equations. For LES, the equations are obtained following [19]. The subgrid scale model is the selective mixed scale model detailed by Sagaut [14]. The time integration is carried out by means of the backward scheme of Gear which is second-order-accurate. The time step is 0.0025 h/U∞ . The spatial scheme is the scheme proposed by Mary & Sagaut[10]. It is based on the AUSM+(P) scheme (see [7]). Two grids have been used for the computations. Both grids have the same spatial extent. The streamwise length of the computational domain is 16h (5h upstream of the ramp and 8h downstream), the spanwise length is 4h and the vertical length is 6h upstream. The fine grid with Nx = 1008, Ny = 68 and Nz = 400 cells is employed for the DNS. Grid spacings are: Δx+ = 16, Δy + = 1 and Δz + = 12. The friction velocity used for the scaling in wall unit is taken at the entry of the computational domain. The complete grid comprises 28 106 cells. The LES grid has 345 × 68 × 268 points. Grid spacings are: Δx+ = 50, Δy + = 1 and Δz + = 18. A view of the LES grid in the x-y plane is provided in figure 1. Computations have been performed on four NEC/SX6 processors with the parallel version of FLU3M. The CPU time for one through-flow with the freestream velocity is 144 hours for the DNS and 36 hours for the LES. For the
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L / L0
controlled case, a transient time equivalent to five periods was necessary to reach the periodic regime. After that, statistics were collected on ten periods. To simulate the diaphragm movement, a blowing/suction condition with a top-hat distribution which varies sinusoidally in time is implemented on the whole cavity’s bottom surface: u(x, t) = U0 ∗ cos(2πf t) with U0 = 16.8 m.s−1 and f = 580 Hz or F + = f.L/U∞ = 0.5 with L the separation length without control computed from DNS. This reduced frequency has been shown to be the most effective in other papers [2, 11] and by an URANS+SA 2D preliminary study (see fig. 2). As it can be seen, the flow is very sensitive to the actuation frequency. For example, a forcing at F + = 0.5 reduces the separation length by 76% whereas there is no effect at F + = 0.6. It will be shown in the following in comparison with the LES of controlled flow that the URANS 2D computation with the Spalart-Allmaras turbulence model overestimates the effect of the synthetic jet because of the larger vortices sizes. 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
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The actuator velocity amplitude U0 has been calculated in order to have a momentum coefficient Cμ equal to 1% at the orifice: Cμ =
ρj dUj2 2 ρ∞ LU∞
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with d the actuator slot width, Uj the synthetic jet velocity at the orifice, L the uncontrolled separation length and U∞ the inflow velocity. The synthetic jet velocity amplitude Uj is equal to 0.5 U∞ . For these computations, a realistic inflow boundary condition has been used. Numerically, this is achieved by the adjunction of a new block from which turbulent fluctuations are reinjected at its entry and added to a RANS boundary layer profile. This domain is initialized with a previous flat plate LES computation.
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4 Results 4.1 LES validation The DNS computation is used here to validate the LES results. Figure 3 displays the mean streamwise velocity u in the entire computational domain. The DNS and LES computations lead to very similar results for the mean velocity profile. The main differences, which occur in the recirculation zone, are lower than 6%. The time-averaged streamwise positions of separation xs and reattachment xr are xs /h = 0.53 (0.51) and xr /h = 3.93 (3.96) for the DNS and the LES respectively. Compared with the DNS, the LES separation length (xr −xs )/h is overestimated by only 1.6%. For comparison, a Reynolds-averaged simulation with the Spalart-Allmaras turbulence model and the rotation correction by Dacles-Mariani et al. [8] gives xs /h = 0.55 and xr /h = 6.3. The separation point is well predicted but the reattachment is too far downstream which leads to an overestimation of the separation length by 70%. Concerning the streamwise velocity fluctuation urms (see fig. 4), it can be seen that the level of fluctuations of the LES is in very good agreement with the DNS in particular in the separated region.
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Figure 5 displays a comparison of the instantaneous turbulent structures for the two cases. Because of the turbulent incoming boundary layer, twodimensionnal spanwise structures are not visible in the Q-criterion isovalue but they could be seen either on a pressure isovalue or by using the first POD modes (see [13]). The turbulent boundary layer structures interact with the shear-layer spanwise structures which explains that these vortices are not as clearly visible as in the case of a laminar or transitional incoming boundary layer. Figure 5 also shows finer turbulent structures in the DNS than in the LES. 4.2 Controlled flow Figures 6 and 7 display the effect of forcing on the mean streamwise velocity u and its fluctuations urms . The separation length is reduced by 48% for the controlled case at a frequency of Sth = 0.11 (F + = 0.5 and Cμ = 1%. At x/h = 1, it can be seen that the separation process is delayed in the controlled case. At x/h = 2, the backflow velocity amplitude is lower in the controlled case than in the baseline one. Downstream of x/h = 4, the boundary layer in the forced case redevelops earlier and the velocity is larger near the wall.
2 Fig. 5. Q-criterion isovalue Q = 4U∞ /h2 colored with streamwise vorticity (left: DNS, right: LES).
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Figure 7 shows that the level of fluctuations of the forced case is higher at x/h = 0 and x/h = 1 due to the synthetic jet blowing. After the reattachment, urms is higher in the forced case than in the baseline case near the wall because of the transport of the fluctuations created by the actuator. These results differ with Neumann & Wengle’s [12] observation of a lower level of urms in the controlled case downstream of reattachment. This may be due to the higher level of excitation in our case (Uj /U∞ = 0.5 instead of 0.2 in [12]). 2
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5 Conclusion A large-eddy simulation of a controlled flow over a smoothed ramp has been performed to prove the effectiveness of synthetic jets in reducing separation length. A LES of the separated flow has been validated in comparison with a 28 million cells DNS. The separation length has been reduced by one half for a velocity ratio of 0.5 and a reduced frequency F + = 0.5. Future work will focus on the understanding of the great sensitivity of the flow to the actuation frequency.
References 1. S. Bhattacharjee, B. Scheelke, and T. R. Troutt. Modification of vortex interactions in a reattaching separated flow. AIAA J., 24(4):623–629, 1986. 2. A. Brunn and W. Nitsche. Separation control by periodic excitation in a turbulent axisymmetric diffuser flow. J. Turb., 4(9), 2003. 3. Y. Chen, S. Liang, K. Aung, A. Glezer, and J. Lagoda. Enhanced mixing in a simulated combustor using synthetic jet actuators. AIAA Paper 99-0449, 1999. 4. K. B. Chun and H. J. Sung. Control of turbulent separated flow over a backwardfacing step by local forcing. Exps. Fluids, 21:417–426, 1996. 5. K. B. Chun and H. J. Sung. Visualization of a locally-forced separated flow over a backward-facing step. Exps. Fluids, 25:133–142, 1998. 6. A. Dejoan and M. A. Leschziner. Large eddy simulation of periodically perturbed separated flow over a backward-facing step. Int. J. Heat and Fluid Flow, 25:581–592, 2004.
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7. J. R. Edwards and M.S. Liou. Low-diffusion flux-splitting methods for flows at all speeds. AIAA J., 36(9):1610–1617, 1998. 8. J. Dacles-Mariani and G. G. Zilliac and J. S. Chow and P. Bradshaw. Numerical/Experimental study of a wingtip vortex in the near field. AIAA J., 33(9):1561–1568, 1995. 9. V. Krishnan, K. D. Squires, and J. R. Forsythe. Prediction of separated flow characteristics over a hump using rans and des. AIAA Paper 2004-2224, 2004. 10. I. Mary and P. Sagaut. LES of a flow around an airfoil near stall. AIAA J., 40(6):1139–1145, 2002. 11. S. Narayanan and A. Banaszuk. Experimental study of a novel active separation control approach. AIAA Paper 2003-60, 2003. 12. J. Neumann and H. Wengle. LES of controlled turbulent flow over a rounded step, In R. Friedrich et al. (eds), Direct and Large Eddy Simulation V. pages 557–564, 2004 Kluwer Academic Publishers. 13. J. Neumann and H. Wengle. Coherent structures in controlled separated flow over sharp-edged and rounded steps. J. Turb., 5(22), June 2004. 14. P. Sagaut. Large-Eddy Simulation for Incompressible Flows, An Introduction, 2nd ed. Springer, Berlin, 2002. 15. A. Seifert, T. Bachart, D. Koss, M. Shepshelovich, and I. Wygnanski. Oscillatory blowing: a tool to delay boundary layer separation. AIAA J., 31(11):2052–2060, 1993. 16. A. Seifert and L. G. Pack. Active flow separation control on wall-mounted hump at high reynolds numbers. AIAA J., 40(7):1363–1372, 2002. 17. B. L. Smith and A. Glezer. Jet vectoring using synthetic jets. J. Fluid Mech., 458:1–34, 2002. 18. P. R. Spalart. Direct simulation of a turbulent boundary layer. J. Fluid Mech., 187:61–98, 1988. 19. A. W. Vreman. Direct and large eddy simulation of the compressible turbulent mixing layer. PhD thesis, University of Twente, Twente, 1995. 20. H. Wengle, A. Huppertz, G. B¨ arwolff, and G. Janke. The manipulated transitional backward-facing step flow: an experimental and direct numerical simulation investigation. Eur. J. Mech. B - Fluids, 20:25–46, 2001. 21. S. Yoshioka, S. Obi, and S. Masuda. Organized vortex motion in periodically perturbed turbulent separated flow oer a backward-facing step. Int. J. Heat and Fluid Flow, 22:301–307, 2001. 22. S. Yoshioka, S. Obi, and S. Masuda. Turbulence statistics of periodically perturbed flow over backward facing step. Int. J. Heat and Fluid Flow, 22:393–401, 2001.
Direct Numerical Simulation of a Spatially Evolving Flow from an Asymmetric Wake to a Mixing Layer Sylvain Laizet and Eric Lamballais Laboratoire d’Etudes A´erodynamiques UMR 6609, Universit´e de Poitiers, CNRS T´el´eport 2 - Bd. Marie et Pierre Curie B.P. 30179 86962 Futuroscope Chasseneuil Cedex, France [email protected] Summary. In this paper, the flow obtained behind a trailing edge separating two streams of different velocities is studied by means of direct numerical simulation. The influence of the shape of the trailing edge, that can be either blunt or bevelled, is considered through the analysis of the destabilizing mechanisms and their resulting effects on the spatial development of the flow. It is shown that the use of a bevelled trailing edge leads to a conventional mixing-layer dynamics with moderate effects of the wake component. In contrast, a self-excited flow is obtained behind a blunt trailing edge where usual features of the wake dynamics are recovered. In terms of receptivity, it is found that the flow behind a bevelled trailing edge is strongly sensitive to upstream perturbations. The inverse behaviour is observed for the flow behind a blunt trailing edge. These differences are interpreted in terms of convective/ absolute stability. The vortical organization obtained in each case is discussed using vorticity visualization.
1 Introduction It is well known that the spatial development of turbulent wakes or mixing layers is strongly influenced by the dynamics of large scale vortices. For a wake flow, the alternating large structures (called Karman vortices) are initially created through a vortex shedding mechanism occurring immediately behind the body considered. For a mixing layer flow, the formation of primary structures leads to co-rotating KelvinHelmholtz vortices. In terms of stability, the mechanisms responsible of these two vortex families are of different nature. In typical wake flows, the vortex shedding is a self-excited phenomenon that can be interpreted in terms of global instability [7]. In contrast, Kelvin-Helmholtz formation in spatial mixing layers admits a strong sensitivity to upstream conditions as a consequence of the convectively unstable character of this type of flow. In many applications, the flow over a given geometry leads to an asymmetric wake that can be viewed as a flow with a double component wake/mixing-layer.
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Compared to a conventional wake, a strong breaking in the symmetry can change deeply the spatial development of the flow. The same can be said by comparison with a pure mixing-layer, the wake component being able to introduce very different mechanisms in the vortex generation behind the geometry. In order to investigate the dynamics of this type of hybrid flow, we consider here the general flow configuration where two independent streams of velocity U1 and U2 are flowing on opposite sides of a semi-infinite plate of thickness h (see figure 1 for a sketch of the flow geometry). The asymmetry of the flow can be quantified through the dimensionless parameter 1 −U2 defined as usually for a conventional spatial mixing layer. Just behind the λ= U U1 +U2 trailing edge, a wake takes place while further downstream, the decreasing velocity deficit can lead the flow to transition from a wake regime to a mixing layer regime. The persistence of the wake is naturally dependent on the asymmetry parameter λ but also on the more of less marked of the wake component that can be linked to the shape of the trailing edge. An experimental investigation of the case λ = 0.2 was carried out by [3] who have compared the flow obtained with a bevelled trailing edge with the one generated downstream a thick splitter plate (blunt trailing edge). In the first case, a Conventional Mixing Layer was observed (CML case) while in the second one, a composite wake/mixing-layer flow was noticed and designated as a Thick Mixing Layer (TML case). The goal of the present numerical study is to investigate by DNS a flow configuration similar to the experimental set-up of [3] in order to better understand the reasons of the deep changes introduced by the shape of the trailing edge, in the vicinity of the trailing edge as well as further downstream. An improvement of the understanding of these fundamental mechanisms should be helpful in the context of flow control via the passive or active modification of trailing edge geometry.
2 Numerical methods and computational flow configuration A numerical code fully based on sixth-order compact finite difference schemes and a Cartesian grid is used to solve the incompressible Navier-Stokes equations. The incompressibility condition is ensured via a fractional step method introducing a Poisson equation for the pressure. An original characteristic of the present code, called “Incompact3d”, is that this equation is directly solved in the framework of the modified spectral formalism. More precisely, our Poisson solver is only based on 3D Fast Fourier Transforms (FFT3D) despite the use of inflow/outflow boundary conditions. This very direct solving technique is obviously possible for periodic and free-slip boundary conditions, but also when Dirichlet conditions for the velocity are combined with homogeneous Neumann conditions for the pressure (see for instance [14], [12] for the basic principles of the spectral solving of a Poisson equation based on cosine expansion). Note that homogeneous Neumann conditions provide only second order accuracy in the corresponding direction, even if high-order finite difference schemes are used. This formal drawback is considered here as being of secondary importance, the excellent behaviour of our sixth-order schemes being preserved outside from the near inflow/outflow regions. In addition, the full spectral treatment of Poisson equation offers three major advantages: (i) it allows the very easy staggering of the grid pressure; (ii) it allows the introduction of a grid stretching in one
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direction by preserving the direct nature of the solver ([4], [1]); (iii) its cost can be less than 10% of the overall computational expense (for regular and collocated grids) while never overtaking 20% (for stretched and staggered grids). Concerning the former point, we follow here our recent conclusions of a previous work [8] by using staggered pressure grid in order to avoid the excitation of pressure oscillations due to the immersed boundary treatment. More details about the present computational methodology can be found in this study [8]. The trailing edge is modelled using an immersed boundary technique where the specific direct forcing method of [11] is employed. Basically, the principle of this method is to calibrate the forcing in order to verify the no-slip condition at the wall of the body while trying to ensure the regularity of the velocity field across the immersed surface by creating an artificial flow (with a mass source/sink) inside the body. This particular procedure was found to improve significantly the results when high-order schemes are used for the spatial differentiation (see [11] for more details). The governing equations are directly solved in a computational domain Lx ×Ly × Lz = 63h × 96h × 9h discretized on a Cartesian grid of nx × ny × nz = 561 × 257 × 80 points except for 2D simulation where nz = 1. The stretching of the grid in ydirection leads to a minimal mesh size of Δy min ≈ 0.03h. Inflow/outflow, free-slip and periodic boundary conditions are used in x, y and z directions respectively. The geometry parameters correspond to the experimental conditions of [3] concerning the shape of the trailing edges (see figure 1), the boundary layers thicknesses δ1 = 0.57h and δ2 = 0.50h and the velocity ratio λ = 0.2. Present DNS show two significant differences with respect to experiments. First, the Reynolds number 2 )h is reduced from 14400 to 400, and secondly, both inflow boundRe = (U1 −U ν ary layers are laminar (mean Blasius-like profiles) whereas they are turbulent in experiments. Consequently, only a qualitative agreement with experiments can be expected.
3 Preliminary 2D simulations Despite their artificial nature, 2D DNS performed without any inflow perturbation can be instructive in terms global stability characteristics for each case (CML or TML). The resulting flows are illustrated in figure (2ac). Both flow configurations Lx U1 l=10h h Blunt trailing edge (TML case)
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Fig. 1. Flow configuration and computational domain.
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lead to a quasi-steady behaviour where vortices are formed periodically and convected downstream. These established states are obtained after a transient stage where the computed flow has to adapt to initial conditions and to evacuate them. By performing several DNS based on various initial conditions (strongly, weakly or not at all perturbed), we verified that both flows can forget their initial conditions and finally reach the steady-periodic behaviour illustrated in figure (2abcd). In the TML case, a vortex shedding can be clearly identified through the presence of coand counter rotating vortices in the near-edge region while for the CML case, only vortices of negative spanwise vorticity (ωz < 0) are created significantly further downstream. In first analysis, the behaviour of the TML case seems to be related to a globally unstable situation leading to a self-excited dynamics. More precisely, both blunt and bevelled trailing edges introduce a velocity deficit and a near-wake region that is absolutely unstable in a local analysis. However, it can be expected that the resulting pocket of absolute instability is clearly more reduced in the CML than for the TML case, in such a way that no global modes can exist in the former case. The same 2D DNS were performed again using inflow perturbations correlated in time and space. By the term correlated, we mean that the noise used to define inflow perturbations is not purely white (non-correlated data) while being of large band-width in the spectral space. This type of inflow treatment was previously shown to improve the realistic character of a spatial flow in the inlet region. Using these inflow conditions, more realistic results are obtained for the CML case whereas minor modifications are obtained for the TML case (see figure 2cd). The
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main improvement concerns the location of the primary Kelvin-Helmholtz that can appear more upstream in agreement with experimental observations. The main frequency f associated to the primary (Kelvin-Helmholtz or Karman) vortex passing lead to St = 0.17 for the TML case and St = 0.21 for the CML case, where St is the Strouhal number defined by St = f h/Uc . The Strouhal number obtained for the TML case is consistent with [6] who found St ≈ 0.11 at Re = 128 while highest values 0.2 ≤ St ≤ 0.24 are found for significantly higher Reynolds numbers [2, 3]. Concerning the global overall of the TML flow, in agreement with previous observations of [6, 3], a shifting of the mixing layer centre towards the high velocity region is well recovered in present results (see figure 2ab), whereas the inverse tendency occurs very slightly for the CML case. Finally, it seems possible to conclude from these DNS 2D that the wake component is strongly dominated by the mixing-layer component everywhere in the CML flow, where primary and secondary (pairing) instability phenomena can be clearly observed as it can be expected for a conventional mixing layer. In contrast, both components play a various role in the TML case depending on the x-location. In the near-edge region, a typical wake behaviour can be observed while further downstream, vortices of positive vorticity are progressively damped, the Karman street becoming gradually asymmetric and leading finally (for x > 40h) to a spatial mixinglayer flow. This downstream evolution for an asymmetric wake to a mixing-layer is consistent with the previous DNS of [6]. These behaviours will be discussed more in detail in the next section by considering a more realistic flow configuration.
4 Results of 3D simulations In this section, 3D DNS of TML and CML cases are presented and compared to the experimental results of [3]. Both calculations use exactly the same perturbed inflow data. For these two simulations, a fully 3D dynamics is obtained with the presence of large scale coherent structures. For the CML case, in a similar way than in the previous section, no wake component can be identified qualitatively in the vortex dynamics observed via animations. An illustration of the flow is presented in figure 3cd through vorticity visualizations. The formation of Kelvin-Helmholtz vortices submitted to parings further downstream can be easily identified. A similar main frequency is obtained for the primary instability with St = 0.22. Moreover, it can observed that 3D motions are in qualitative agreement with reference studies on the spatial mixing layer (see for instance [5]) where the typical flow topology involving longitudinal structures stretched between large spanwise rolls is well recovered here. Despite the fully 3D nature of the development of the flow, no self similar turbulent state can be expected, the present computational domain being to short in the x-direction. This view is clearly confirmed by the analysis of turbulent statistics (non presented results) that clearly shows the signature of a transitional mixing layer where the vertical/spanwise velocity fluctuations are over/under-predicted respectively by comparison with self-similar turbulent mixinglayer results. The preliminary observation of vorticity visualisations (see figure 3ab) obtained in TML case suggests clearly that the mechanisms associated to the self-excitation of the flow allows the reaching of a highly 3D turbulent state immediately behind
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the blunt trailing edge. In first analysis, this behaviour is consistent with the experimental observations of [3] who also have detected the occurrence of vortex shedding in the near trailing-edge region (see figure 4 for a visual comparison between experimental and DNS visualisations covering an identical sub-domain). The tendencies already observed in DNS 2D results are recovered here, especially the deviation of the layer towards the high velocity region and the strongly marked wake dynamics that evolves progressively towards a mixing-layer dynamics. This change of regime associated to the spatial development of the flow can be identified qualitatively by observing the gradual disappearance of large scale spanwise structure in the slow part of the flow. Downstream from x ≈ 40h, only big rolls of negative vorticity (i.e. of the same sign than the vorticity associated to the mixing layer velocity profile) can be visually detected in animations. Despite the lack of any wake signature in the vortical organization of the flow for x > 40h, the spatial development of the flow does not correspond to the one expected for a mixing layer. The more drastic change concerns the pairing that can never be observed for the present TML case, this tendency being already present in 2D DNS results. It is well known that pairings contribute strongly to the expansion of a mixing layer. Here, in the absence of any paring, we find a spreading rate dδ/dx ≈ 0.024 significantly weaker than for a conventional mixing layer. This behaviour is in complete contradiction with the experimental results of [3] who find an important increase of the TML expansion compared with the CML one, the growth rate being almost doubled when a blunt trailing edge is used. The consequence of this difference can be directly observed by comparison between mean velocity profiles obtained experimentally and computationally at the same location x = 42h (see figure 5), the mixing layer thickness being underestimated by a factor 2 in present DNS. We cannot propose any simple explanation of this disagreement between experiments and present calculations. The main difference between present DNS and
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Fig. 3. Vorticity isosurfaces for DNS 3D results. TML (top) and CML (bottom) cases in side or upper views.
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experiments concerns the Reynolds number and the inflow conditions that are not realistic enough here to allow the establishment of self-similar conditions as quickly as in experiments. Therefore, more investigations seem to be required by considering the influence of the Reynolds number using LES [9], the impact of the use of realistic inflow conditions (that can be based on DNS/LES data of turbulent boundary layers using the technique of [10]) or the effects of the size of the computational domain in z-direction1 . Naturally, the poor agreement between present DNS and experiments concerning the growth rate of the TML is recovered in other turbulent statistics like Reynolds stresses (non presented results), the main difference being that DNS leads to a more 2D flow than the one reported in experiments, the more spectacular consequence of this tendency being the overestimation of transverse velocity fluctuations that are more than twice stronger in DNS compared with measurements.
Fig. 4. Experimental (left, [3]) and vorticity (right, present DNS) visualizations of the TML. To end this section, let us consider the velocity deficit associated to the wake component in the TML case. The spatial stability characteristics of a generic composite mixing-layer/wake profile U (y)/Um = 1 − f sech2 (y/δ) + λ tanh(y/δ) has been considered by [13]. For the present velocity ratio λ = 0.2, [13] have shown that if f > 0.95, the corresponding flow become absolutely unstable. For present DNS results, it is worth noting that the mean velocity profile can be very well fitted (via the relevant adjusting of δ) by the generic profile of [13] in the near trailing edge region (see figure 5). Moreover, the wake deficit parameter can be evaluated from the mean velocity field in each x-location through the relation f (x) = (Um − miny [ u (x, y)])/Um . The longitudinal evolution of this parameter is presented in figure 5. An acceptable agreement is found between DNS and experiments. For present results, using the criterion f > 0.95, the region 0 < x − l < 1.6h is found to be absolutely unstable. Note that if the same procedure is repeated for the CML (non-presented curve), the absolutely unstable zone of the flow is found to be significantly smaller, the condition being 0 < x − L < 0.2h in this case. This change of the size of the pocket of absolute instability, depending of the shape of the trailing edge, suggests once again that the TML flow is globally unstable (leading to self-excitation) contrary to the CML flow. 1
We have considered the influence of the spanwise length Lz by increasing its value by 50% (while keeping constant the mesh size) but we did not see any significant effect on the results discussed here for both TML and CML cases.
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Fig. 5. Left: Longitudinal evolution of the wake deficit parameter f (x) = (Um − miny [ u (x, y)])/Um . Centre: Comparison of the mean velocity profile at x − l = 1.6h (solid line) with the generic profile used by [13] (dotted line). Right: Comparison of the mean velocity profile at x − l = 42h (solid line) with its experimental counterpart [3]. Acknowledgments: Calculations were carried out at the IDRIS. We are grateful to Jalel Chergui, Jean-Marie Teuler and Laurent Perret for their precious help.
References 1. E.J. Avital, N.D. Sandham, and K.H. Luo. Int. J. Numer. Methods Fluids, 33:897–918, 2000. 2. D.R. Boldman, P.F. Brinich, and M.E. Goldstein. J. Fluid Mech., 75:721–735, 1976. 3. C. Braud, D. Heitz, G. Arroyo, L. Perret, J. Delville, and J.-P. Bonnet. Int. J. Heat and Fluid Flow, 25(3):351–363, 2004. 4. A.B. Cain, J.H. Ferziger, and W.C. Reynolds. J. Comp. Phys., 56:272–286, 1984. 5. P. Comte, J.H. Silvestrini, and P. B´egou. Eur. J. Mech. B/Fluids, 17(4):615– 637, 1998. 6. D.A. Hammond and L.G. Redekopp. J. Fluid Mech., 331:231–260, 1997. 7. P. Huerre and P.A. Monkewitz. Ann. Rev. Fluid Mech., 22:473–537, 1990. 8. S. Laizet and E. Lamballais. In Proc. IV Escola de Primavera de Transi¸caoe Turbulˆencia, Porto Alegre, RS, Brazil, 2004. 9. M. Lesieur, O. M´etais, and Pierre Comte. Large-eddy simulation of turbulence. Cambridge University Press, 2005. 10. T.S. Lund, X. Wu, and K.D. Squires. J. Comp. Phys., 140:233, 1998. 11. P. Parnaudeau, E. Lamballais, D. Heitz, and J.H. Silvestrini. In Proc. DLES-5, Munich, 2003. 12. P.N. Swarztrauber. SIAM Review, 19:490–501, 1977. 13. D. Wallace and L.G. Redekopp. Phys. Fluids, 4(1):189–191, 1992. 14. R.B. Wilhelmson and J.H. Ericksen. J. Comp. Phys., 25:319–331, 1977.
Part X
Heat Transfer
LES of Flow and Heat Transfer in a Round Impinging Jet M. Hadˇziabdi´c and K. Hanjali´c Department of Multi-scale Physics, Delft University of Technology Prins Bernhardlaan 6, 2628 BW Delft, The Netherlands [email protected] Summary. Large-eddy simulations of a round impinging jet have been performed aimed at gaining a better insight into flow and turbulence structure and their imprints on the heated target wall. The Reynolds number Re = 20000 and the orificeto-plate distance H/D = 2, where D is the jet-orifice diameter, were chosen to match the experiments of Baughn and Shimizu [1] (also [4],[5]). The simulations, performed with the in-house unstructured finite-volume code T-FlowS, confirmed the experimentally detected double peaks in the Nusselt number and the negative production of turbulence energy in the stagnation region. The LES revealed also some other phenomena such as a strong jet flapping around the stagnation point and the unsteady flow separation at the onset of the wall-jet formation. These events seem to be the main cause of the Nu-number nonuniformity, and were linked to the break-up of the shear-layer-generated ring vortices prior or after their impingement.
1 Introduction Impinging flows have been researched extensively because of their industrial importance, but more recently also because of the interesting physics with a multitude of phenomena occurring in a relatively simple flow configuration. Despite continuous research efforts, the physics of the vortex and turbulence structure, and the heat transfer mechanism, are not fully understood. Three distinctive regions can be identified in an impinging jet: the free-jet zone, the stagnation- and the wall-jet region. Initially, the jet issuing from an orifice into a stagnant ambient fluid bears the features typical of a free jet, with a strong shear layer developing at the jet edge. As the jet approaches the wall, the impermeability condition and the consequent build-up of the static pressure begins to affect the jet. A stagnation region is formed in the centre of the impingement zone. The wall deflects the jet in the radial direction and a wall-jet is formed further downstream. In the deflection region the flow is subjected to a strong acceleration that causes a local thinning of the newly formed boundary layer. Subsequently, unlike in a plane (slot) jet, in a round impinging jet the fluid is subjected to radial spreading and a consequent linear deceleration of the radial wall jet.
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The distribution of the Nusselt number depends on several factors. In contrast to the laminar case, in turbulent impinging jets the Nusselt number exhibits a peak in the stagnation point or close to it. The stagnation Nu number depends on the inflow conditions and the orifice-to-plate distance, i.e. whether the incoming jet has a potential core (fluid issuing from a plenum chamber through a contoured nozzle) and whether this core has reached the target plate, or the jet is fully turbulent originating from a developed turbulent pipe flow, as in the present work. Depending on the Reynolds number and nozzle-to-plate distance, a second peak in the Nusselt number may appear at r/D > 1. Plausible physical explanations of the Nusselt peaks is still lacking, despite a number of hypotheses published in the literature. For example, [3] attributed the second peak in Nusselt number to transition from a laminar to turbulent boundary layer in the wall-jet region, which appears reasonable if the incoming jet has a potential or laminar core. [10] found a relatively high level of turbulent kinetic energy even in the stagnation region and argued that the second maximum is a result of high turbulent kinetic energy convected from the jet-edge region. Other interesting phenomena have also been reported in the experimental studies by [1], [14], [9], [11], [7] and others. However, experimental techniques are limited to point and plane measurements. While providing invaluable data on mean-flow properties and turbulence statistics, experiments cannot still generate sufficiently detailed three-dimensional time-dependent information to complete the physical picture of flow and turbulence, especially very close to the wall where they are most needed for proper understanding of the heat transfer mechanism. Most numerical simulations (LES and DNS) of the impinging jets reported in the literature focus on computationally less demanding low Reynolds numbers and plane (often confined) geometries (slot jets), e.g. [15], [2], [13], [12] and others. However, the LES of low-Re-number jets, especially with the potential inlet core, face the problem of capturing and maintaining the instabilities and turbulence development, prompting some researchers to impose an artificial forcing or a periodic excitement at the jet inlet, e.g. [13]. On the other hand, no reliable computer simulations (DNS and LES) of unexcited round impinging jets at Reynolds numbers of 20000 and higher can be found in literature. We report on LES of a natural (unexcited) round turbulent jet issuing from a fully-developed pipe flow at Re = 20000 and impinging normally on a flat plate at a distance H/D = 2 (where D is the jet-orifice diameter) using the in-house unstructured finite-volume code T-FlowS.
2 Numerical procedure The configuration considered and the solution domain are shown in Fig. 1. A fluid jet issues from a fully developed turbulent pipe flow, which was generated by a separate LES. The velocity components in the axial (z), radial (r) and
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azimuthal (θ) directions are denoted by W, U and V. At the top boundary three types of conditions have been explored: convective and pressure boundary condition and a constant inflow velocity of 1% of the bulk velocity. The convective boundary condition, based on the hyperbolic convection equation, was used for the outlet boundary. At the wall, no-slip condition was used for the velocity, and a constant heat flux was specified for the energy equation. The sub-grid scale stresses were modelled by the dynamic model of Germano, [6]. The central-difference scheme was used to discretize diffusive and convective terms in the momentum equations. However, because of well-known numerical instability associated with the central-difference scheme on collocated grids, the second-order accuracy upwind-biased scheme QUICK ([8]) was applied locally in the small zone for r/D < 0.5 and z/D < 0.15 where earlier the oscillatory numerical wiggles in the velocity field were observed. Since the affected region is characterized by a high pressure, we expect that the numerical diffusion resulting from the QUICK scheme does not significantly affect the overall accuracy. The discretisation of the energy equation was similar to that for the momentum equations except that the QUICK scheme was used throughout the whole domain to avoid overshoot that can lead to negative temperatures. The grid used was of the hybrid type with triangular prisms in the region r/D < 0.5 and hexahedral cells in the rest of the domain. The wall-nearest value of z + at the impingement plate was smaller than 1.0. The grid contained 154 cells in the axial direction with a strong hyperbolic clustering at both ends. The fine mesh resolution in the axial direction in the jet-entrance region is needed to properly resolve the initial jet-edge shear layer. The number of cells in the azimuthal direction was 504 for r/D > 0.5. The total number of cells was 9.9 million. Simulations were also performed on a twice finer mesh, but only for one quadrant with symmetry conditions on the cut-sides, using about 5 million cells. Although the simulation over one-quadrant cannot capture the full 3D vortex dynamics, the grid refinement lead to some improvement in the wall Nusselt number.
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Fig. 2. Contours of the instantaneous velocity, (a) - side view, (b) - top view
3 Impinging jet structures The issuing jet interacts with the ambient fluid triggering the growth of the shear layer, which eventually leads to the jet core being consumed, see Fig. 2(a). Instabilities of the Kelvin-Helmholtz type develop in the initial shear layer, generating a street of vortex rings with frequency parameterized by the Strouhal number St = fWDb = 0.64. The initially axisymmetric ring vortices stretch and get distorted with visible nests of highly concentrated vorticity, Fig. 3(a-c). Eventually, the ring vortices break up and the vortical nests evolve into large-scale turbulent eddy structures, Fig. 3(c-d). For z/D = 1.25, and closer to the wall, the radial-velocity field is highly nonuniform and asymmetric. The large-scale eddies that were originally part of the ring-vortex structure, strike on the impingement wall. As a result, the radial velocity sharply increases. Fig. 2(b) shows the instantaneous velocitymagnitude field near the wall (z/D = 0.05). Locally, the peak velocity can be twice as high as in the nearby regions. The flow structures in the stagnation region were difficult to identify. No organized eddy structures were observed. High static pressure and the wall-blocking effect cause a transfer of energy from the turbulence field back into the mean flow, as indicated by the negative turbulence energy production, Fig. 5(c). The centre of the stagnation region is influenced by the dynamics of large scale eddies which come from the jet mixing layer. Fig. 2(b) reveals that the iso-lines of the instantaneous velocity in the wall-parallel plane at z/D = 0.05 from the impingement wall are stretched in the direction perpendicular to a regions of high velocity. These distortions are associated with the flow acceleration in the wall-jet region, enhanced by the impact of large eddies advected from above. A consecutive series of snapshots of the instantaneous-velocity field and corresponding streamlines in the horizontal plane at z/D = 0.005 above the impingement wall, reveals an oscillatory movement of the impingement point, Fig. 3(b). It is interesting to notice that the impingement ”point” temporarily evolves into a line and even splits into two separate impingement regions, Fig. 3(d). These oscillations, which cannot be identified as turbulence, nevertheless contribute to the
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Fig. 3. (a): Vortex distortion illustrated by the vorticity-magnitude in horizontal plains, (b): A time sequence of velocity field and instantaneous streamlines at z/D = 0.005
rms of the velocity fluctuations, and are most probably the explanation of the sharp peak in the radial component u ˜ very close to the wall, Fig. 5(b). As the flow deflects from the impingement plate, it evolves into a radial wall jet, subjected initially to a strong acceleration and a subsequent thinning of the boundary layer. Due to radial spreading, the round wall-jet is then subjected to a strong deceleration, azimuthal stretching and a roll-up of the eddy structures with the rotation axis aligned with the azimuthal direction. The iso-surface of the instantaneous pressure field reveals large deformed and locally broken ring-like vortex structure at position r/D ≈ 1.7, Fig. 4(a) and Fig. 4(b). The strong azimuthal stretching of the ring vortices affects the flow in the near-wall region. As a result, the counter-rotating vortices are formed very close to the wall, creating local wall-attached recirculating zones - bubbles. Fig. 9(a) shows the velocity-vector field in the wall-jet region. It can be seen that the wall-attached eddies are rolled up between the plate surface and the large-scale toroidal vortex at the edge of the jet. These wallattached vortices are subjected to further stretching before they are destroyed and entrained into the turbulent wall-jet further downstream. The presence of the wall-attached vortices increases the turbulence level in this region. The Q-criterion was used as a vortex identifier in the impingement and wall-jet regions. The wall-jet region is populated by elongated near-wall streaks, see Fig. 4(c). The streaks are also subjected to stretching in the azimuthal direction due to the radial-flow spreading. As a consequence, a single streak occasionally splits into two parts and forms two separate streaks. However, the opposite trend was also observed at several locations, where two streaks merged and formed a single larger streak. While the streak splitting can be explained by the effect of radial flow spreading, a mechanism behind the streak merging is not clear.
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Fig. 4. Iso-surfaces of the instantaneous pressure: (a)- top view, (b)- side view, (c): Iso-surfaces of Q = −0.1
4 Velocity and stress fields Fig. 5(a) shows the axial and radial mean-velocity components at the station r/D = 0.0. On the centreline, the mean-axial velocity (W ) remains nearly constant for z/D > 1.0. Below z/D = 1.0 the flow decelerates due to the presence of the impingement wall. An indication that the jet core is longer than the distance between the nozzle and the impingement wall are the constant values of the rms of the velocity fluctuations on the centreline down to z/D ≈ 0.1, Fig. 5(b). Hereon, the wall-normal component w ˜ begins to decrease to zero wall value, whereas the radial component u ˜ increases sharply reaching a peak at z/D ≈ 0.01 and then falls to zero at the wall (not visible in Fig. 5(b)). Both, the experiments and LES, show negative production of turbulence kinetic energy in the stagnation region, Fig. 5(c). The budget of the turbulence kinetic energy (not shown) indicates that the major positive term in the stagnation region is the pressure diffusion, but it is unlikely that this can generate such a high peak in the rms of the wall-parallel fluctuations. Instead, we argue that the main contribution comes from non-turbulent jet flapping and oscillations in the stagnation point, illustrated in Fig. 3(b). Apart from the sharp peak in u ˜ very close to the wall, the turbulence level in the stagnation region is relatively low, Fig. 6(a), but it recovers in the wall-jet region by shear production in the near-wall layer, enhanced by flow acceleration and subsequent deceleration, as well as by the contribution of the large-scale vortical structures originating from the break-up of the ring vortices. The increase in k is especially noticeable at the downstream edge of the wall bubble, Fig. 6(b).
5 Heat transfer The distribution of the instantaneous surface temperature is presented in Fig. 7(a) and of the corresponding instantaneous Nusselt number in Fig. 7(b), both corresponding to the instantaneous velocity field shown in Fig. 2(b). The lowest temperature area is located in the stagnation region due to the efficient
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Fig. 7. (a): Instantaneous surface temperature, (b): Instantaneous Nusselt number.
cooling effect of jet impingement. The region of high heat-transfer coefficient is not circular, but also stretched in a direction which nearly corresponds to the stretching direction of the instantaneous velocity iso-lines shown in Fig. 2(b). Between the positions r/D = 1.3 and r/D = 1.8, the radially-stretched islands of low Nusselt number occur. These are probably the regions where the instantaneous separation occurs, resulting in a decrease of the heat-transfer coefficient.
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The time histories of the wall temperatures at two positions, r/D = 0 and r/D = 1.0, are shown in Fig. 8(a) and Fig. 8(c), respectively. Both signals show oscillating behavior. The temperature recording in the stagnation region is clearly affected by turbulence, but more regular oscillations are discernible than in the signal at the position r/D = 1. Although the temperature signal at r/D = 0 does not show a precise periodicity, it clearly indicates the jet flapping or precessing. The time interval between the two local maxima, considered as a period, corresponds to the observed frequency of the generating ring vortices in the jet shear layer. This confirms the assumption that the temperature fluctuations in the stagnation region are determined by the impingement of the large-scale vortices which originate from the jet shear layer. As the radial distance from the stagnation point increases, the mean Nusselt number starts to decrease and reaches its local minimum at r/D = 1.3, Fig 10(b). This position coincidences with a location where distinct separation bubbles were found. The separation causes the thickening of the thermal boundary layer, as seen in Fig. 9(b) and to a local minimum in the mean Nusselt-number distribution. Subsequent flow reattachment and an increase in turbulence production leads in turn to the enhancement of the local heat transfer and most likely to the occurrence of the second peak in the Nusselt number. Fig. 10(a) shows a set of the instantaneous Nusselt number over the wall. The oscillations with highest amplitudes occur in the stagnation region and the region around the second maximum. The latter can be attributed to the enhanced turbulence at this locality.
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The present LES predicted the mean Nusselt number with moderate success, see Fig. 10(b). While the agreement between the experimental data of [1] and the LES results is very good in the stagnation region and some distance downstream, the mean Nusselt number in the wall-jet region is underpredicted. Insufficient mesh resolution is the most probable reason for a relatively poor prediction of the heat-transfer coefficient in the region beyond r/D = 1.7.
6 Conclusion Detailed LES studies of a round jet impinging normally on a flat surface at different temperatures have been conducted, using an in-house unstructured finite-volume code. This flow poses a major challenge to LES because it is a wall-attached, fast-evolving flow without strong forcing, but with a sequence of multiple events: impingement, jet deflection with strong acceleration, radial spreading and deceleration. In order to capture the major turbulence features and the structural interaction, a large mesh (close to 10 million cells) had to be employed, making the simulation very expensive. The capture of the wall friction and heat transfer, especially around the impingement of the jet shear layer, is a critical issue. The simulations were also found to be sensitive to the grid resolution in the jet shear layer, especially in the initial zone, where a strong interaction of the issuing jet with the ambient fluid generates instabilities and an abrupt increase in the turbulent kinetic energy. A correct prediction of the turbulence level in the jet shear layer is a necessary prerequisite for a faithful representation of the flow field and heat transfer in the impingement and wall-jet regions. Despite some shortcomings (e.g. still insufficient grid resolution for r/D > 2), the present LES provided new information and revealed several interesting phenomena such as a strong flapping/precessing of the jet centre and intermittent flow separation on the impingement wall. Because the turbulence intensity in the impingement centre is relatively low, we argue that the jet flapping is most probably responsible for the high Nusselt number in the stagnation region. The flow separation at the wall, caused by a roll-up of the vortical structure that originated from the break-up of the ring vortices, seems to be the main cause of the non-monotonic distribution of
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the Nusselt number. A local dip in the Nusselt number at r/D ≈ 1.4 coincides with the instantaneous local separation and the consequent thickening of the thermal boundary layer. The subsequent wall-jet reattachment coincides with the second Nusselt-number maximum. This contrasts with the conventional notion that the second maximum in the mean Nusselt-number distribution is caused by the impingement of the jet shear layer.
References 1. Baughn, J.W. and Shimizu, S. (1989) Heat transfer measurement from a surface with uniform heat flux and an impinging jet, Journal of Heat Transfer, 111, 1096, 1989 2. Beaubert, F. and Viazzo, S (2003) Large eddy simulations of plane turbulent impinging jets at moderate Reynolds numbers, International Journal of Heat and Fluid Flow, 24. 512-519 3. Colucci, D. and Viskanta, R (1996) Effect of Nozzle Geometry on Local Convective Heat Transfer to a Confined Impinging Air Jet, Exp. Therm. Fluid Sci., 13, 71-80 4. Cooper, D., Jackson, D.C., Launder, B.E. and Liao, G.X. (1993) Impinging jet studies for turbulence model assessment-i. flow field experiments, Int. J. Heat Fluid Flow, 36, 2675-2684 5. Geers, L., Hanjalic, K., Tummers, M. (2004) Experimental investigation of impinging jet arrays, Experiments in Fluids, 2004, 36, 946-958 6. Germano, M., Piomelli, U., Moin, P. and Cabot, W. H. (1991) A dynamic subgrid-scale eddy viscosity model, Phys. Fluids A, 3, 1760-1765 7. Kataoka, K. (1990) Impingement heat transfer augmentation due to large scale eddies, Heat Transfer 1990, Proc. 9th Int. Heat Transfer Conference, 1, 255-273 8. Leonard, B. P. (1979) A stable and accurate convective modelling procedure based on quadratic upstream interpolation, Comp. Meth. Appl. Mech. Enginnering, 19, 59-98 9. Livingood, J. N. B. and Hrycak, P. (1973) Impingement Heat Transfer from Turbulent Air Jets to Flat Plates - a Literature Survey, NASA TM, X-2778 10. Lytle, D. and Webb, B. W. (1991) Secondary heat transfer maxima for air jet impingement at low nozzle-to-plate spacing, Experimental Heat Transfer, Fluid Mechanics and Thermodyanmics, Elsvier, New York 11. Nishino, K. and Samada, M. and Kasuya, K. and Torii, K. (1996) Turbulence statistics in the stagnation region of an axisymmetric impinging jet flow, Int. J. Heat and Fluid Flow, 17, 93-201 12. Olsson, M. and Fuchs, L. (1998) Large eddy simulation of a forced semiconfined circular impinging jet, Physics of fluids, 10, 476-486 13. Tsubokura, M., Kobayashi, T., Taniguchi, N. and Jones, W. P (2003) A numerical study on the eddy structures of impinging jets excited at the inlet, International Journal of Heat and Fluid Flow, 24, 500-511 14. Viskanta, R. (1993) Heat Transfer to Impinging Isothermal Gas and Flame Jets, ¨ ı134 Exp. Therm. Fluid Sci., 6, 111 A` ’ 15. Voke, P.R. and Gao, S. (1998) Numerical study of heat transfer from an impinging jet, International Journal of Heat and Mass Transfer, 41, 671-680
Direct Simulations of a Transitional Unsteady Impinging Hot Jet X. Jiang1 , H. Zhao2 , and K. H. Luo3 1
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Mechanical Engineering, School of Engineering and Design, Brunel University, Uxbridge UB8 3PH, UK [email protected] School of Engineering and Design, Brunel University, Uxbridge UB8 3PH, UK School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK
1 Introduction Impinging jets are encountered in a broad range of applications such as cooling and drying. The study of impinging jets is of great interest in the context of near-wall fluid flow and heat transfer. The impinging configuration covers a broad range of flow phenomena, such as large-scale structures, wall boundary layers with stagnation, large curvature involving strong shear and normal stresses, wall heat transfer and small-scale turbulent mixing. In many applications, the near-wall flow and heat transfer processes are highly unsteady. Although there have been a substantial number of studies in the literature, impinging jets are still not fully understood due mainly to the highly unsteady nature and the great difficulty of performing detailed numerical and experimental investigations. Moreover, most previous studies were concerned with isothermal impinging jets while research on the more important nonisothermal impinging jets was limited. Impinging jets deserve more research efforts because of their great importance in both fundamental study and practical applications. Due to the complex flow phenomena involved but the geometric simplicity, flow control techniques can be tested and developed from the impinging flow configuration. For instance, external flow perturbations including acoustic excitations play significant roles in the dynamics of impinging jets [1-4]. Impinging jets can be effectively used to examine the effects of external perturbations on dynamics of near-wall fluid flow and heat transfer, and consequently to develop flow control techniques. A full understanding of impinging jets needs both temporally and spatially resolved diagnostics. Direct numerical simulation (DNS) provides an effective means to investigate unsteady impinging jets. In this study, the unsteady flow and temperature fields of an impinging round jet have been investigated by performing three-dimensional (3D) DNS. The DNS solves the compressible
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time-dependent 3D nondimensional Navier-Stokes equations using highly accurate numerical methods and high-fidelity boundary conditions. The DNS aimed at better understanding of the jet unsteady dynamics. A comparative study of the flow characteristics of undisturbed and disturbed impinging jets has been carried out to examine the effects of external perturbations.
2 Mathematical formulation The physical problem considered is a heated round jet issuing into an open boundary domain that impinges on a flat surface. Simulations have been performed using a recently developed code for spatial DNS of non-reacting and reacting jets [5,6]. The compressible time-dependent Navier-Stokes equations in the Cartesian coordinate system x, y, z have been employed, where the z−axis is along the streamwise direction for the head-on impingement and the x − 0 − y plane is the domain inlet where the jet nozzle exit locates. Major reference quantities used in the normalization of the governing equations are the centerline streamwise mean velocity at the jet nozzle exit (domain inlet), jet nozzle diameter, and the ambient temperature, density and viscosity. The flow field governing equations are supplemented by the ideal-gas law. The equations are solved using a sixth-order compact finite-difference scheme for evaluation of the spatial derivatives [7]. The time-dependent governing equations are integrated forward in time using a third-order RungeKutta scheme. Boundary conditions for the 3D spatial DNS of impinging jets represent a challenging problem. In this study, the Navier-Stokes characteristic boundary condition (NSCBC) [8] has been utilized for the inflow and wall boundary conditions, while nonreflecting characteristic boundary condition has been used to specify the open boundaries with the ambient field. The wall is assumed to be at constant ambient temperature, impermeable and satisfying the no-slip condition. The amplitudes of characteristic waves at the impinging wall are estimated by the local one-dimensional inviscid relations for walls [8], where the near-wall density is allowed to change with the characteristic wave variations at this boundary. At the domain inlet (jet nozzle exit), the streamwise mean velocity has been specified with a hyperbolic tangent profile. The jet originates from the center of the inlet domain. For the impinging jet with external perturbation, an unsteady disturbance in a sinusoidal form is artificially added to the mean velocity profile at the domain inlet. The time-dependent velocity components at the domain inlet z = 0 are given by u = Asin(2πf0 t), v = Asin(2πf0 t), w = w[1 + Asin(2πf0 t)], where the perturbation amplitude A is varied between 0 to 4%. The nondimensional frequency (Strouhal number) of the unsteady disturbance is f0 = 0.30, which has been chosen to be the unstable mode leading to the jet preferred mode of instability [9]. The flow field is initialized using velocity and temperature fields that vary linearly between their values at the domain inlet and those at the wall.
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3 Numerical results and discussion Direct numerical simulations have been performed with different upstream velocity conditions. Results to be discussed have been taken from four cases: Case A - a baseline case without external perturbation; Case B - a perturbed case with the perturbation amplitude A = 1%; Case C - a perturbed case with A = 2% and Case D - a perturbed case with A = 4%, respectively. To allow sufficient resolution, the impinging jet examined has a relatively low Reynolds number of Re = 1000, at which the impinging flow may display transitional behaviour [10]. In the simulations, the considered jet Mach number is M = 0.3, Prandtl number is P r = 1, and the ratio of specific heats is γ = 1.4. For the hot jet, there is a temperature ratio of T0 /Ta = 2 at the inlet, where subscripts 0 and a represent the center of the domain inlet (jet nozzle exit) and the ambient environment respectively. The wall temperature is Tw = 1. The dynamic viscosity is chosen to be temperature-dependent according to μ = μa (T /Ta )0.76 . The dimensions of the computational box used are Lx = Ly = 12 and Lz = 6. The grid system used is of 180 × 180 × 120 nodes with a uniform distribution in each direction. In this study, a grid independence test was performed and further refinement of the grid to 240 × 240 × 120 did not lead to appreciable changes in the flow field solution. The Courant-FriedrichsLewy number used to limit the time step is 2.0, which has been tested to give time-step independent results.
Fig. 1. Instantaneous isosurfaces of vorticity magnitude at t = 80 of cases A & D
3.1 Instantaneous flow and temperature fields Figure 1 shows the instantaneous isosurfaces of vorticity magnitude. From the 3D plots, it is evident that the impinging jet deflects from the wall and then convects along the wall surface. The vorticity maxima occur at locations near the stagnation region. The maximum vorticity corresponds to locations
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with the strongest velocity deflection. In Fig. 1, an important feature is that vortical structures are observed for the perturbed case D. Vortical structures are important features of impinging jets [3,11,12]. The primary vortices in the primary jet stream caused by the Kelvin-Helmholtz type shear instability are evident for case D. Fig. 2 shows the instantaneous temperature contours of the four cases. Vortical structures in the jet shear layers are observable for the three perturbed cases. Vortical structures are typical characteristics of transitional flow. The results indicate that the impinging jet displays transitional behaviour when an external perturbation is present at the jet upstream locations. From Fig. 2, it is obvious that the vortical level of the impinging jet is strongly affected by the external perturbation amplitude, and a larger perturbation leads to a more vortical flow field and a stronger tendency of flow transition to turbulence.
Fig. 2. Instantaneous temperature contours on x = 6 plane at t = 80
Fig. 3. Instantaneous streamwise velocity and temperature profiles at t = 80 along the jet centerline (x = 6, y = 6)
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Flow vortical structures greatly affect the instantaneous flow and temperature fields. Fig. 3 shows the instantaneous profiles of the jet centerline z−velocity component w and temperature. It can be observed that external perturbations lead to spatial oscillations in both the velocity and temperature profiles. Larger external perturbations lead to larger amplitudes of the spatial oscillations. For case D with 4% external perturbation, the spatial oscillations at the jet centerline can be as high as 20% for the velocity and 5% for the temperature. From Fig. 3, it can be seen that the temperature gradients of the impinging jet near the wall are very large, while the velocity profile does not drop as sharply as the temperature profile in the near wall region. This is because the thermal boundary layer near the wall starts its formation in the stagnation region, while the jet velocity deflects nearby and spreads in the cross-streamwise direction near the wall surface. 3.2 Oscillatory behaviour of the impinging jet Spatial oscillations have been observed for the perturbed impinging jet, as shown in Fig. 3. The oscillatory behaviour is associated with the vortical structures in the primary jet stream. Fig. 4 shows time traces of the velocities at a location along the jet centerline, from which the temporal oscillation of the impinging jet is evident. For case D, the velocity variation at that location can be as high as 50% of the mean velocity. The perturbation applied at the jet nozzle exit propagates along the primary jet stream and reflects at the wall. The continuous supply of perturbation leads to multiple reflections that interact with each other, which subsequently lead to large oscillations in the primary jet stream. In Fig. 4, Fourier spectra corresponding to the velocity traces are also shown. It is observed that larger external perturbation leads to larger amplitude of the velocity oscillation. The amplitude of the velocity oscillation is more than doubled when the external perturbation amplitude is doubled. It is clear that the oscillation of the perturbed impinging jet is dominated by the frequency of the external perturbation that leads to the Kelvin-Helmholtz type shear layer instability. For the relatively low Reynolds number considered, the impinging jet does not contain appreciably higher frequencies due to the lack of vortex breakdown and small scales. The oscillatory behaviour also occurs for the temperature field due to the coupling between momentum and energy transfer. Fig. 5 shows time traces of the temperatures at two locations along the jet centerline of case C. It can be seen that temperatures at both locations vary appreciably with time, indicating an appreciable temporal oscillation. It is noticed that the temperature at z = 4 is generally lower than that at z = 2, due to the mixing of the hot jet with the ambient fluid. It is also noticed that there is a phase difference between the temperature variations at the two locations. This is mainly because of the convection of vortical structures in the primary jet stream associated with the spatial and temporal evolution of the impinging jet.
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Fig. 4. Velocity history and its Fourier spectra at (x = 6, y = 6, z = 4)
Fig. 5. Time traces of temperatures at (x = 6, y = 6, z = 2 & z = 4) of case C
3.3 Time-averaged flow and heat transfer characteristics Time averaging of the results has been performed to examine the mean flow properties. The time interval used for the averaging is between t1 = 60 and t2 = 80. Fig. 6 shows the time-averaged jet centerline streamwise velocity and temperature profiles. The time-averaged profiles do not differ significantly among the four cases. This is mainly because the external perturbations applied at the jet nozzle exit are periodic and the flow is not fully turbulent. However, it is still noticeable that vortical structures enhance the mixing between the hot jet and its surroundings that lead to faster decay of the temperature. It is observed that the temperature gradients of the impinging jet near the wall are very large, while the velocity gradients are smaller. Figure 7 shows the time-averaged wall shear and normal stresses in the x = 6 plane of the four cases. The differences between the four cases are insignificant because the wall jet structures are similar for the four cases due to the re-laminarization effects of the wall (the flow velocity decays after the impingement). It is observed that the shear stresses are zero at the stagnation point, where the largest normal stresses (negative value) occur. The largest wall shear stress occurs at the velocity deflection region within one diameter to the impinging point. The maximum shear stresses are approximately 30 times larger than the maximum normal stresses.
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Fig. 6. Time-averaged streamwise velocity and temperature profiles along the jet centerline (x = 6, y = 6)
Fig. 7. Time-averaged wall shear and normal stresses (τ yz and τ zz ) in plane x = 6
Fig. 8. Time-averaged Nusselt number at the wall in plane x = 6
To examine the wall heat transfer characteristics, Fig. 8 shows the timeaveraged Nusselt number at the wall in plane x = 6. It is well known that the Nusselt number for wall heat transfer has a bell shape distribution, where the stagnation Nusselt number is predominately determined by the flow Reynolds number and the nozzle-plate distance plays a secondary role. In Fig. 8, the bell shape distribution is evident. The stagnation Nusselt number for case A is approximately 25. For the three perturbed cases, the stagnation Nusselt number is slightly lower and it also decreases slightly with increasing
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perturbation amplitude. This is because the perturbation induced vortical structures enhance the mixing of the hot jet with the ambient fluid, which leads to lower temperature gradients near the wall. An experimental stagnation Nusselt number around 31 was reported for the same Reynolds number but a distance-to-diameter ratio 2 [12]. This value is higher than the figure obtained in this study with larger nozzle-to-plate distance. However, it is known that the Nusselt number increases with a decrease in the nozzle-to-plate distance [10]. Therefore the Nusselt number obtained here is expected to be reasonable.
4 Summary Effects of external perturbation on the flow and temperature fields of the impinging jets have been examined by a comparative DNS. The simulations focused on a relatively low Reynolds number Re = 1000 and a moderate nozzle-to-plate distance of 6 jet diameters. The results have revealed that the unsteadiness associated with the external perturbation leads to vortical structures in the jet shear layers of the primary jet stream emerging from the nozzle. Due to the interaction with the wall, these vortical structures lead to significant spatial and temporal oscillations. The amplitude of these oscillations increases with the increasing amplitude of the external perturbation. The vortical structures that are induced by the external perturbations also enhance the mixing between the hot jet and the ambient fluid. The computational cases involving external perturbations show transitional behaviour in the primary jet stream, while the unperturbed case does not have a transitional tendency. It has been observed that the wall shear stresses are much larger than the normal stresses. The Nusselt number distribution is in agreement with the available experimental data for similar conditions.
References 1. Akiyama T., Yamamoto K., Squires K.D., Hishida K. (2005) Int J Heat Fluid Flow 26:244-255 2. Hwang S.D., Cho H.H. (2003) Int J Heat Fluid Flow 24:199-209 3. Tsubokura M., Kobayashi T., Taniguchi N., Jones W.P. (2003) Int J Heat Fluid Flow 24:500-511 4. Liu T., Sullivan J.P. (1996) Int J Heat Mass Trans 39:3695-3706 5. Jiang X., Luo K.H. (2003) Combust Flame 133:29-45 6. Jiang X., Luo K.H. (2001) Theor Comp Fluid Dyn 15:183-198 7. Lele S.K. (1992) J Comput Phys 103:16-42 8. Poinsot T.J., Lele S.K. (1992) J Comput Phys 101:104-29 9. Hussain A.K.M.F., Zaman K.B.M.Q. (1981) J Fluid Mech 110:39-71 10. Viskanta R. (1993) Exp Therm Fluid Sci 6:111-134 11. Chung Y.M., Luo K.H., Sandham N.D. (2002) Int J Heat Fluid Flow 23:592-600 12. Angioletti M., Di Tommaso R.M., Nino E., Ruocco G. (2003) Int J Heat Mass Trans 46:1703-1713
Part XI
Aeroacoustics
Basic Sound Radiation from Low Speed Coaxial Jets Mikel Alonso1 and Eldad J. Avital2 Department of Engineering, Queen Mary, University of London, Mile End Road, E1 4NS London, UK. e-mail: [email protected] , [email protected]
The basic sound radiation generated by large-scale structures is calculated for circular and elliptical coaxial jets. A hybrid method is used in which the hydrodynamics is computed using an incompressible Large-Eddy Simulation (LES) procedure and the generated sound field is calculated using Lighthill’s analogy. The sub-grid scale (SGS) stress is modelled explicitly using the MixedTime-Scale (MTS) model. For the coaxial jet the inner to outer velocity ratio (U1 /U0 ) is 0.4 although 0.6 and 0.2 are also considered. The Reynolds number (Re) based on the outer diameter of the jet is 6000. When compared with the single circular and elliptic jet the coaxial jet shows an extended potential core as expected. The sound radiation is calculated and analysed using the acoustic power spectrum and Lighthill’s quadrupoles Root-Mean-Square (RMS). The addition of a secondary jet increases the acoustic power output as U1 /U0 approaches unity.
1 Introduction Coaxial jets are of interest in several engineering applications such as jet exhaust noise control or mixing in combustion systems. The research on subsonic coaxial jets in the past is not as extensive as for subsonic single jets. Important work has been done on the mean flow characteristics of subsonic circular coaxial jets by, for example, Chigier and Be´er [1] and Williams et al. [2] and on full-scale jet engine nozzles by Bushell [3]. Ko and Kwan [4] presented a detailed experimental study of the initial region of coaxial jets where this region was divided into the initial merging, the intermediate and fully-merged zones. They studied the effect of the velocity ratio U1 /U0 (inner to outer, i.e. primary to secondary) on these regions and a comprehensive study was presented in Ko and Au [5] for a wide range of U1 /U0 < 1. A detailed experimental study of the vortex structure and dynamics of the near field in a coaxial jet is presented by Dahm et al. [6]. In this study there is disagreement with previously presented work that described coaxial jets simply as a combination of several
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single jets. A more recent experimental study of coaxial jets by Bitting et al. [7] presents results for circular and square coaxial nozzles. An interesting conclusion was that the internal unmixed region diminished when U1 /U0 decreases. Non-circular jets are generally known to show enhanced mixing and axis-switching [8]. As expected, the square jet configuration showed enhanced mixing. However, no axis-switching was observed, which was attributed to the initial highly turbulent shear layer and to the lack of coherence in the inner mixing region. In [2] the main objective was to study the effect of U1 /U0 on the jet noise. As the secondary jet velocity was increased the noise was in excess of that of the primary jet alone. They suggested that when U1 /U0 was less than about 0.4 the noise generated was assumed to be proportional to the difference of the square of the jet velocities raised to the fourth power. Olsen and Friedman [9] studied experimentally the jet noise emitted by circular coaxial jets for a variety of geometric and flow parameters. Two years later, Balsa and Gliebe [10] developed a prediction method for estimating aerodynamic and noise characteristics of coaxial jets. Their experimental results agree qualitatively with theoretical predictions at all but high frequencies, where refraction is important. One of their major conclusions was that the noise reduction obtained by coaxial jets comes primarily from a reduction in turbulence intensity. In 1998 Fisher et al. [11] produced Part I of their experimental study on subsonic coaxial jet noise that corresponded to unheated primary flow. In this study the jet was divided in three regions and the noise for each region was calculated. They compared their results to existing data and concluded they achieved a good agreement for a wide range of U1 /U0 and angles of observations. Recently, Jiang et al. [12] studied the sound generation in subsonic axisymmetric coaxial jets. This is the only computational simulation study found for subsonic coaxial jets. Direct Numerical Simulation (DNS) was used to calculate the detailed sound source structure and the near sound field. An acoustic analogy based on Lilley’s third-order wave equation was also used to investigate the sound generation. The Mach number was 0.8 and Re number was 2500. Three cases were compared: a single cold jet, coaxial cold jets and a single hot jet. The velocity ratio was 0.4. They concluded that the secondary jet tended to weaken the vorticity field downstream and as a result, the size of the sound source was reduced. This lead to the conclusion that the sound radiation of coaxial jets was lower than that of the single jet for this Mach number. In the present work coaxial circular and elliptic jets with U1 /U0 of 0.4 and Re number of 6000 are studied. The results are compared to the corresponding single jets. Additional runs for U1 /U0 of 0.2 and 0.6 are also carried out. The next two sections present the hydrodynamic and aeroacoustic formulations. They are followed by a discussion in Section 4, and a summary in Section 5.
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2 Hydrodynamic Formulation The non-dimensional incompressible Navier-Stokes (NS) equations in a conservative formulation are marched in time to resolve the jet flow using the LES method. This is implemented in our in-house code Lithium. A finite difference method is used to discretise the governing equations using a staggered Cartesian grid. The convection terms are calculated using a one point biased upwind fourth-order scheme based on a five point stencil and a fourth order interpolation where required. The diffusion terms are calculated using a fourth order staggered scheme. The projection method is used for the time marching along with a third order Runge-Kutta method. The pressure solution is found by solving the Poisson equation using an iterative procedure based on the Fast Fourier Transform scheme as described in Avital [13]. In the filtered momentum equation the SGS stress tensor (τij ) is modelled using the MTS model [14].This SGS model is based on the explicit SGS kinetic energy (κSGS ) and the concept of scale similarity. The essential the time √ point is that scale TS is defined as the harmonic average of Δ/ κSGS and CT /|S| where CT is 10. The first term represents the characteristic time of the small-scale motion corresponding to the cut-off scale, and the second term represents the characteristic time of the large-scales. A passive scalar was also simulated for flow illustration where the turbulent Prandtl number (P rt) was taken as one. The size of the computational domain is 50R x 30R x 30R (R is the radius of the jet) in the x, y and z directions respectively. Algebraic grid mapping in the streamnormal and spanwise directions is used to cluster points near the jet centre with a stretching ratio of 1.5. The corresponding number of grid points is 297 x 129 x 129. The stretching technique is further described in Avital et al. [15]. As an inflow boundary condition the jet is subjected to a streamwise velocity profile perturbation whose phase is randomised in space, i.e. εcori sin (ωi t + 2πgrand ) (1) u = Uin 1 + i
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∗ U1 r r R0 R1 ]) + (1 − tanh[b2 − − ∗ ]) R0 r 2 R1 r
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A random phase factor (0 < grand < 1) is used to enhance the occurrence of small scale structures and to prevent phase locking between the random frequencies. The variables εcori and ωi are, respectively, the amplitude and the frequency of the disturbance components. In (1) the mean inflow velocity is adapted from [16], and represents a typical profile at the beginning of the mixing region of a circular jet with primary and secondary radii R1 and R0 respectively. b2 is taken as 0.25R0 /θ where θ is the momentum thickness of the jet. Note that it has been adapted for the coaxial jets. In (2) r∗ = |r −2R0 |
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and R1 = (U1 /U0 )R0 . The inflow disturbance for the passive scalar is taken in the same way but without the disturbance. Zero inflow condition is taken for the streamnormal and spanwise velocities. On the streamnormal and spanwise sides of the computational domain a constant pressure condition is used along with a free slip condition for the velocities. Convective boundary conditions were used at the outflow side of the domain along with a buffer zone located in the last 10R. The calculations of the convective terms were gradually replaced by a first order scheme and the normal diffusion terms were gradually made zero. The source term in the pressure equation was also made gradually zero following [17] in order to remove the ellipticity coming from the pressure field. Therefore only in the first 40R the flow can be viewed as physical.
3 Acoustic Formulation The aeroacoustic properties are calculated using the acoustic-analogy approach with the Green’s function solution method [18]. Due to the compact approximation in this method the computational resources can be dedicated to enlarging the computational domain in order to include a sufficient portion of the sound source for more accurate predictions. Lighthill’s acoustic analogy is used, which is based on a free-space Green function solution of the linear wave equation in a stationary medium, leading to [18]: ρ0 xi xj ∂ 2 ui uj dy (3) ρ(x, t) − ρ0 = 3 4 4 π c0 |x| V ∂t2 The integral term in the equation is that of a quadrupole source. The quadrupole can be either longitudinal when i = j or lateral when i = j. The direction i = 1 points in the axial streamwise direction, i.e. θ = 00 where θ is the spherical angle [19]. Boundary corrections to account for vortical structures leaving the computational domain were used following [18]. Second order schemes were used for the spatial integration and time differentiation.
4 Results The results are presented for Reynolds number of ReD = 6000, where D is the jet diameter (outer diameter in the case of the coaxial jet). The elliptic jet has an equivalent inflow area to the circular jet and it has an aspect ratio of 2 having the major axis along the z-axis. The inflow disturbance is composed of four discrete frequencies in the Strouhal range 0.175 < StD < 0.425 with a peak amplitude of 2%. The instantaneous vorticity contours of the coaxial jet shown in Fig. 1, reveal a pattern similar to that described in [7], where significant interactions
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Fig. 1. Instantaneous vorticity contours at time 340. Elliptic coaxial jet with U1 /U0 =0.4. Axial locations from right to left 12R, 17R and 22R (12 contours from maximum to minimum); vorticity lines are also shown in the xy plane.
between the inner and outer shear layer can be seen in the non-circular case. The axis-switching phenomenon can be observed for 12R < x < 17R (Fig. 1), which was not found in previous studies [7]. As described in [7] it is difficult to observe a clear axis-switch due to the weakly coherent nature of the mixing structures. The statistics were accumulated for the time period of 240 to 340 once the initial transient structures left the computational domain. The downstream evolution of the mean centreline velocity (Ucl ) and the streamwise evolution of the centreline RMS are shown, respectively in Figs. 2(A) and 2(B). In the circular case the single jet shows the shortest potential core of 14R to 15R with RMS values of 0.17. The RMS of Ucl shows three stages: an initial stage of linear growth for the first 10R to 12R of the jet, an intermediate stage of strong mixing at 12R to 22R which coincides with the end of the potential core and a final stage that shows a non-monotonic decline of the centreline RMS and mean Ucl . These results agree qualitatively with those previously reported in existing literature, where for a low Re number circular jet, the potential core end is approximately at 14R depending on inflow conditions, and the fluctuations are of the order of 15-17%, see [20] and [21]. In the case of the coaxial jets the potential core length increases as U1 /U0 increases, as was also observed in [4]. The end of the potential cores are located at 18R, 22R and 24R for the velocity ratios 0.2, 0.4 and 0.6 respectively. As U1 /U0 is increased the end of the potential core becomes smoother and the peak of the centreline RMS decreases to about 16.5%, 14.5% and 12%, for the velocity ratios 0.2, 0.4 and 0.6 respectively. The elliptic jet shows a shorter potential core (Fig. 2(A)) than the circular jet as expected [19]. Interestingly, the elliptic jet’s Ucl with velocity ratio U1 /U0 =0.2, is similar to that of the single jet. However, the RMS has higher values.
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The acoustic power spectrum is shown in Fig. 3 for the circular and elliptic configurations. The coaxial jet with U1 /U0 =0.4 shows an overall stronger acoustic output in the circular case. In the elliptic case for U1 /U0 =0.4 the jet has a stronger output for some frequencies but otherwise is similar to the single jet. The U1 /U0 =0.6 jet behaves similarly. The effect of a secondary jet seems to be stronger for the circular jet. The single jet has a stronger acoustic output than the U1 /U0 =0.2 jet for many frequencies, especially around StD = 0.3 which coincides with the single jet’s most unstable frequency [20]. The coaxial circular jet shows stronger acoustic output than the single circular jet as U1 /U0 is increased. This was also observed in [2] where as U1 /U0 was increased the noise was in excess of the central jet alone in cold subsonic conditions. In [2] it was concluded that the addition of unheated secondary air to an unheated primary jet could result in an increase in noise output. In Fig. 4 the statistical sound source distribution is presented. This figure shows the sound source fluctuation, by means of the RMS of Lighthill’s longitudinal xx quadrupole along the spanwise direction of the jet. It is presented
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for several streamwise locations. Upstream of the jet, at 5R, the fluctuations are weak in all cases. These results can be related to the centreline RMS seen in Fig. 2. The double peak structure in the elliptic coaxial case (Fig. 4(B)) can also be related to the two peaks shown in Fig.1 where the vorticity is strong at both sides of the centreline at 20R.
5 Summary In the present study incompressible LES was coupled with Lighthill’s acoustic analogy to investigate the flow pattern and basic sound radiation of circular and elliptical coaxial jets. The potential cores in the single jets were found to be shorter than in their coaxial counterparts in agreement with previous studies. Axis-switching occurred in the elliptic coaxial simulations. The mixing in the coaxial jet was found to be stronger as expected. The addition of a secondary jet was found capable of increasing the emitted acoustic power as the velocity ratio approaches one, which agrees qualitatively with previous results on cold low speed coaxial jets. However, a reduction in the low frequency noise was found for the low velocity ratios in the circular jet. On the other hand the effect of the secondary stream in the elliptic jet was not as strong as in the circular jet.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
Chigier N.A., Be´er J.M. (1964) Trans ASME, J Bas Eng 86:788–804 Williams T.J., Ali M.R., Anderson J.S. (1969) J Mech Eng Sci 11:133–142 Bushell K.W. (1971) J Sound Vibr 17(2):271–282 Ko N.W.M., Kwan A.S.H. (1976) J Fluid Mech 73(2):305–332 Ko N.W.M., Au H. (1985) J Sound Vibr 100(2):211–232 Dahm W.J.A., Frieler C.E., Tryggvason G. (1992) J Fluid Mech 242:371–402 Bitting J.W., Nikitopoulos D.E., Gogineni S.P., Gutmark E.J. (2001) Exp Fluids 31:1–12 Grinstein F.F., Gutmark E., Parr T. (1995) Phys Fluids 7(6):1483–1497 Olsen W., Friedman R. (1974) AIAA Paper 74-43:1–17 Balsa T.F., Gliebe P.R. (1977) AIAA J 15(11):1550–1558 Fisher M.J., Preston G.A., Bryce W.D. (1998) J Sound Vibr 209(3):385–403 Jiang X., Avital E.J., Luo K.H. (2004) AIAA J 42(2):241–248 Inagaki M., Kondoh T., Nagano Y. (2002) Trans Japan Soc Mech Eng b 68(673):2572–2579 Avital E.J. (2005) Int J Num Meth Fluids 48(9):909–927 Avital E.J., Sandham N.D., Luo K.H. (2000) Int J Num Meth Fluids 33:897–918 Michalke A. (1984) Prog Aerosp Sci 21:159–199 Mittal R., Balachandar S. (1996) J Comp Phys 124(2):351–367 Avital E.J., Sandham N.D., Luo K.H., Musafir R.E. (1999) AIAA J 37(2):161–168 Alonso M., Avital E.J. (2004) AIAA Paper 2004-3027 Crow S.C., Champagne F.H. (1971) J Fluid Mech 48(3):547–591 Moore C.J. (1977) J Fluid Mech 80:321–367
Use of surface integral methods in the computation of the acoustic far field of a turbulent jet. P. Moore1 , B.J. Boersma2 Laboratory for Aero and Hydrodyamics Delft University of Technology Mekelweg 2, 2628CD, the Netherlands. [email protected] Summary. A high order compressible DNS code is developed and the acoustic results of a Mach 0.8, Reynolds number 4,000 turbulent jet are used for validation of our implementation of the porous Ffowcs Williams-Hawkings (FW-H) formulation. Validation proceeds by matching the near field acoustic component of the DNS output with predictions from the porous FW-H formulation using the DNS results as sources for the acoustic surface integrals. This represents the determination of the far sound field of a turbulent jet from first order principles only. An advantage of the porous FW-H formulation is that it should allow the placement of contour surfaces in non-linear regions. For jet flows, this would then allow the use of a closed contour surface, providing the advantage of versatility over formulations that require open surfaces. However surface placement can not be made arbitrarily, and regions of strong non-linearity must still be avoided. The calculation was performed with a contour surface that passed through the jet after the region of initial breakup (a strongly non-linear region) necessitating the removal of a contour section at the inflow region. The contour section in the outflow region (more weakly non-linear) provided no problems. A completely closed surface is possible when the region of jet breakup is fully within the contour.
1 Introduction Lighthill in 1952 [9] provided a theoretical basis to the understanding of how turbulent flow generates acoustics with his acoustic analogy theory. His theory was formulated by rearranging the Navier Stokes equations into the form of an inhomogeneous wave equation, then identifying the inhomogeneous terms as acoustic sources. This theory was successful in predicting several empirical relationships for turbulence produced aeroacoustics. Recent advances in computing power have made direct numerical simulation of turbulence possible. When coupled with Lighthill’s acoustic analogy (or another acoustic analogy such as the Ffowcs, Williams-Hawkings formulation (FW-H) [16, 6] ), direct far-field noise prediction from first principles
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is possible. This type of calculation has only been available in the last four or five years for jets, where the large spatial extent, and large range of scales, especially at high Reynolds numbers, make direct numerical simulation time consuming. While direct noise prediction is possible with Lighthill’s acoustic analogy, formulations that rely on surface integrals are far more efficient in far-field acoustic calculations, and recent attention has focused on these methods. To date, the Kirchoff surface method, and the porous FW-H method appear to be the most promising in this regard as applied to jets. Only a handful of these calculations have appeared in literature, including [1, 5, 8, 13, 14, 15]. The main problem faced by these calculations is that even with modern super computers, it is impossible to directly model all the relevant scales of the flow at higher Reynolds numbers, and impossible to capture all acoustic sources and propagation effects within the extend of a DNS. Commonly the scales problem is approached by performing LES - the large scale structures are solved directly and the effect of the smaller scales is modeled. The jet simulation is made large enough to capture the most significant noise sources, while propagation effects outside the scope of the DNS/LES are modeled (using a geometric extension for example [11]) or ignored. Because the jet is not fully computed, the linear wave equation in general will not be valid in the outflow region of the jet computation. This poses a problem for the Kirchoff surface method (which requires linearity on a surrounding contour surface), normally handle by opening the contour section at the downstream end. The porous FW-H formulation does not require linearity on the contour surface and should be able to handle surfaces placed in such regions, although strongly non-linear regions still need to be avoided. In this paper we perform a low Reynolds number (4,000) jet simulation. At this Reynolds number we are reasonably confident of capturing the relevant turbulent scales without resorting to LES. Further we anticipate to have the means of low Reynolds number experimental validation in the near future from an in-house experiment. For now we seek to demonstrate self consistency between the acoustic predictions of the porous FW-H formulation and those of the compressible DNS, in an overlap region where both results are available. A partially open contour surface is used (at the inflow side), although this is necessary only because the chosen contour did not fully enclose the region of jet breakup. Closure of the contour at the outflow section provided no problem. Finally we demonstrate the ability to directly make far-field acoustic prediction, with a view to future experimental validation.
2 Simulation of compressible flow A 10th order numerical method utilising a staggered compact finite difference formulation is used to solve the compressible Navier Stokes equations. A detailed description of this numerical method is given in [4]. The advantage of
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this method is that it is highly accurate and stable, even when not all scales are fully resolved. Our calculation proceeded at a Reynolds number of 4,000 (based on jet inflow diameter and velocity), a Mach number of 0.8 (based also on jet inflow velocity) and with 160×128×128 grid points. No LES was done to resolve the effect of the small scales, but we anticipate this will not pose a significant problem at low Reynolds numbers. The DNS part of the calculation was run on a supercomputer, utilising 8 processors for about 2 weeks. Simulation consisted of three time sections: running long enough to be sure that all transient effects associated with jet startup have convected out of the flow domain, a second period to obtain averaged flow quantities, and a third period to output source data for later acoustic post processing. In comparison, post aeroacoustic calculation of time series and wave patterns took only a few days on a standard single processor PC.
3 The porous Ffowcs Williams-Hawkings method The porous FW-H formulation relates the far-field acoustic pressure to flow quantities on a surface enclosing the noise producing turbulence. It is in reality a linear extension of flow quantities to the far-field, utilising the simple wave nature of aeroacoustics away from source regions. The derivation and theory behind this formulation can be found in [6, 16]. The FW-H formulation with a stationary surface (adequate for a stationary, developed jet) can be written as 1 p= 4π
∂ ∂t
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1 ∂ ρ0 Un Lr Lr dS + dS + dS 2 r c0 ∂t s r ret S r ret ret
where Un =
ρun ρ0
Li = (p − p0 )δij nj + ρui un , nj is the unit normal in the vector direction j, ui are perturbation velocities, ρ is density, p is pressure, r is distance from integration point to observer, subscript 0 represents mean value of given quantity and ”ret” denotes evaluation at retarded time t = t − cr0 . We do not make corrections for sources outside the control surface. We use a rectangular control surface S as illustrated in figure (1), and obtain flow data every time step once the flow is fully developed. The integrand quantities, Un dS, Lx dS, Ly dS, Lz dS are output at each discrete patch on the integral surface at every time step (for later retarded integral summation).
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4 Self consistency validation of the compressible DNS code and porous FW-H routines and discussion of control surfaces The intention of our group is to develop our own aeroacoustic code, validated by in-house low Reynolds number experimental results. These experimental results are not yet ready for publication, so in this paper, we proceed by demonstrating self-consistency between the compressible part of our acoustic code, and its extension to the far-field by the porous FW-H formulation, in an intermediate area, where data from both calculations is available. This intermediate area lies within the original area simulated by the DNS code, but far enough from the central acoustic producing region to allow an integral surface to capture the most important noise sources inside. 4.1 Necessity for open contour in inlet region Initial calculations using a complete contour were completely inaccurate. However, calculations performed on a contour opened at the inflow region were quite reasonable. In figure (1) it can be clearly seen that the left side of the contour is crossing the inflow section of the jet through the region of jet break up. It turns out that the porous Ffowcs Williams-Hawkings formulation can not handle the strong non-linearity of this region (a more recent calculation indicates that expanding the contour expanded to fully include the jet breakup region provides accurate results when closed). To determine the radius of the contour section we wished to discard, we evaluated the overall sound pressure levels (OSPL) at a specific location for a range of diameters in figure (2). We determined that the unphysical contributions began when the diameter of the hole is around 1.6Dj , evidenced by
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the beginning of non-linear change in the resulting pressure, as a function of D2 of the hole. Next a visual comparison was made. Shown in figure (3) is a contour plot of p − p0 obtained by DNS, extended to the far field using the porous FW-H formulation. The main wave pattern appears to develop continuously from the DNS region to the porous FW-H region. Some discrepancy occurs, which appears to be caused by the reflection of waves on the DNS part of the calculation from the DNS wall boundary. Work will need to be done in the future to eliminate such reflections, by implementing non-reflective boundary conditions and extending the calculation domain somewhat further in the normal direction. There are also acoustic waves generated at the inflow region from our boundary conditions there. Further work also needs to be done to eliminate these waves. A quantitative comparison is made between noise spectra at specific locations, obtained through DNS, and through the porous-FWH method. We obtained time series at two specific locations, using both techniques and converted to spectra. The resulting overall pressures were all within 0.5 decibels, when the inflow contour section was removed. Comparison at point 1: (21.12, 10.48, 0.0)Dj , left, and at point 2: (26.86, 10.48, 0.0)Dj , right, is shown in figure (4). The corresponding overall pressures for point 1 were 122.89dB and 123.37dB from the porous-FWH and DNS obtained results respectively, while for point 2 these figures were 122.88dB and 123.23 dB respectively. The peaks of the spectra corresponded reasonably in both position and height. Calculations over the three contours match at low frequencies, with divergence at large frequencies due to increased grid spacing. The difference to
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Fig. 3. Pressure perturbation contours obtained solely by DNS (lower region) extended to the far field using porous FW-H formulation).
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the DNS is caused largely by an insufficient time series length for an accurate Fourier transform, but wave reflections inside the DNS and the removal of a contour section also played a role. While we plan to address these issues in a future publication, the results shown here are at least encouraging.
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5 Far field acoustic sound prediction Finally, we discuss far-field acoustics for the noise computation, and demonstrate that they conform qualitatively with expected results. Figure (3) visualises the near pressure field (as obtained by DNS) extended to the far field by use of the porous FW-H methodology. The wave pattern in the far-field is centered on the region of the jet breakup, with a wave amplitude maximum at around 300 above and below the jet axis. Directional plots of the overall sound pressure level (OASPL) in the far field confirm the preferred radiation direction. The lower intensity at angles close to zero degrees is known as the cone of silence, an expected feature of the radiation field. OSPL at 60Dj, 120Dj and 180Dj 115 Contour A Contour B Contour C
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6 Concluding remarks and future work A subsonic (Ma 0.8) Reynolds number 4000 jet was simulated using a partially-resolved DNS, and the far-field acoustic sound field determined by post processing of the DNS results using the porous FW-H formulation. A limited validation consisted of demonstrating reasonable self consistency between the compressible part of the DNS calculation with the predicted near field obtained by the porous FW-H method. Contour placement proved to be important for the porous FW-H formulation calculations, and results obtained with contour sections in regions of strong non-linearity were inaccurate. Accurate results were still obtained by removing these contour sections, although better surface placement would provide the ideal solution. Surface placement in the outflow region, where non-linearity exists, but is weaker, provided no problems.
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Finally we demonstrated the ability of our code to determine far-field aeroacoustics, obtaining directivity plots consistent with expected behavior. Our intention in developing this code is to eventually compare the results with in-house aeroacoustic measurements at lower Reynolds numbers, to verify the porous Ffowcs Williams-Hawkings and other aeroacoustic formulations.
References 1. Andersson, N., Eriksson, L.E. and Davidson, L.. 2004, A Study of Mach 0.75 Jets and Their Radiated Sound Using Large Eddy Simulation, AIAA Paper, 2004-3024 2. Bodony, D.J., Lele, S.K., 2004, Jet Noise Prediction of Cold and Hot Subsonic Jets Using Large-Eddy Simulation, AIAA Paper, 2004-3022 3. Boersma, B. J. and Lele, S. K., 1999, Large Eddy Simulation of a Mach 0.9 Turbulent Jet, AIAA Paper, 99-1874 4. Boersma, B. J., 2005, A staggered compact finite difference formulation for the compressible Navier-Stokes equations, Journal of Computational Physics, 208, 675-690. 5. Constantinescu, G.S. and Lele, S.K., 2001, Large Eddy Simulation Of A Near Sonic Turbulent Jet and Its Radiated Noise, AIAA Paper, 2001-0376 6. Di Francescantonio, P., 1997, A New Boundary Integral Formulation for the Prediction of Sound Radiation, J. Sound and Vibration, 202, No. 4, 491-509. 7. Freund, J. B., 2001, Noise Sources in a Low-Reynolds-number Turbulent Jet at Mach 0.9, Journal of Fluid Mechanics, 438, 277-305. 8. Gr¨ oschel, E. R., Meinke, M. and Schr¨ oder, W., 2005, Noise Prediction for a Turbulent Jet Using and LES/CAA Method, AIAA Paper, 2005-3039 9. Lighthill, M. J., 1952, On Sound Generated Aerodynamically: I. General Theory, Proc. R. Soc. London A, 211 No. 12, 564-587. 10. Pilon, A. R., Lyrintzis, A. S., 1996, An improved Kirchoff method for jet aeroacoustics, AIAA Paper, 96-1709 11. Pilon, A. R., Lyrintzis, A. S., 1997, Refraction Corrections for the Modified Kirchoff Method, AIAA Paper, 97-1654 12. Prieur, J., Rahier, G., 2001, Aeroacoustic integral methods, formulation and efficient numerical implementation, Aerosp. Sci. Technol., 5. 457-468 13. Shur, M.L., Spalart, P.R., Strelets, M.Kh., Travin A.K. 2003, Towards the Prediction of Noise from Jet Engines, Int. J. Heat and Fluid Flow, 24, 551-561. 14. Uzun, A., Blaisdell, G.A., Lyrintzis, 2003, 3-D Large Eddy Simulation for Jet Aeroacoustics, AIAA Paper, 2003-3322. 15. Uzun, A., Blaisdell, G.A., Lyrintzis, 2004, Coupling of Integral Acoustics Methods with LES for Jet Noise Prediction, AIAA Paper, 2004-0517. 16. Williams, J. E. F. and Hawkings, D. L., 1969, Sound Generated by Turbulence and Surfaces in Arbitrary Motion, Philosophical Transactions of the Royal Society, A264, 321-342
Helmholtz decomposition of velocity field of a mixing layer: Application to the analysis of acoustic sources M. Cabana, V. Fortun´e, P. Jordan, F. Golanski, E. Lamballais, and Y. Gervais Laboratoire d’Etudes A´erodynamiques - University of Poitiers 40 avenue du Recteur Pineau - 86022 POITIERS Cedex - FRANCE [email protected] Summary. Direct numerical simulation of a two-dimensional mixing layer is performed and a Helmholtz decomposition of the velocity field is carried out. The associated acoustic field is evaluated and analyzed using the Lighthill equation and a decomposition of acoustic sources, with the aim to better understanding the mechanisms of sound production. A first classification of the source terms is proposed and the acoustic contributing terms are highlighted.
1 Introduction The reduction of noise radiated from turbulent flows is a crucial problem in many engineering applications. The development of numerical prediction models in aeroacoustics requires a good knowledge of physical mechanisms of sound production by turbulent flows such as mixing layers or jets. Over the past decade, direct numerical simulations (DNS) and large eddy simulations (LES) have been used (see [1, 2, 3] for instance) to investigate this topic. These techniques are advantageous for aeroacoustics studies because they allow the determination of both the hydrodynamic field and the corresponding acoustic field simultaneously. Thanks to the large amount of spatio-temporal data obtained from a DNS of the flow, it has become feasible to associate theoretical approaches and simulations aiming to investigate the mechanisms of sound production. Examples of this are the studies of Freund [1] and Sandham et al. [4], where Lighthill’s analogy [5] is used to analyse jet noise. Despite recent progress in this field, the sound generation mechanisms present in subsonic turbulent shear flows remain at present relatively poorly understood. In this work, we propose to analyse results from a mixing layer simulation, with the aid of the Lighthill’s acoustic analogy [5] and a decomposition of acoustic sources which yields ten terms comprising in particular vorticity and dilatation. In addition, a Helmholtz decomposition [6, 7] of the velocity field is carried out. This decomposition into solenoidal and compressible components will allow to separate their respective contribution in acoustic sources. The objective is to examine how compressible DNS results, associated with this analysis methodology, can improve the
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understanding of physical mechanisms of sound production in free shear flows. First results show which source terms have a dominant contribution in the radiated sound field.
2 Lighthill’s analogy and decomposition of acoustic sources Following Lighthill, the mass and momentum conservation equations ∂ρ + ∇ · (ρu) = 0 ∂t
(1)
∂ρu + ∇ · (ρu ⊗ u) = − ∇p ∂t
(2)
can be combined to obtain the classical equation describing the propagation of acoustic waves from an inviscid flow developing in a medium at rest
∂2ρ − c20 Δρ = ∇ · (∇ · (ρu ⊗ u)) + ∇ · ∇ p − c2o ρ ∂t2 ! "# $ ! "# $ I
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where u is the velocity field, ρ and p are the density and the pressure respectively and c0 is the ambient speed of sound. The acoustic source term (I) is associated with the compressible turbulent velocity field and involves complex physical mechanisms of sound generation and interactions between acoustic and dynamic fields while the source term (II) is related to temperature inhomogeneities in the flow. For isothermal flows, the contribution in the acoustic field of the source term (II) is negligible and our attention is focused here on the contribution of source term (I).
A first decomposition of acoustic source terms In order to better understand the description of the various mechanisms included in term (I), the equations (1,2) can be reformulated (see also [9, 10]). The momentum equation (2) is first developed as ∇u2 ∂ρu + ρu∇ · u + ρ + ρ (∇ ∧ u) ∧ u + (u ⊗ ∇ρ) u = − ∇p, ∂t 2
(4)
then the divergence of equation (4) is taken, and after some additional developments, the following equation is derived ∂ρu + ρu∇ (∇ · u) + ρ (∇ · u) (∇ · u) + u (∇ · u) ∇ρ ∂t ∇u2 Δu2 + ∇ρ + uρ ∇ ∧ (∇ ∧ u) + u∇ρ ∧ (∇ ∧ u) +ρ 2 2 − ρ (∇ ∧ u) (∇ ∧ u) + u∇ρ (∇ · u) + u (∇∇ρ) u + (u ⊗ ∇ρ) : ∇u = − Δp.
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Subtracting the first time derivative of equation (1) from equation (5), the following form of the wave equation is obtained
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∂2ρ − c20 Δρ = ρu∇Θ + ρΘ2 + 2Θu∇ρ + ρΔu2 /2 + ∇ρ∇u2 /2 ! "# $ !"#$ ! "# $ ! "# $ ! "# $ ∂t2 1
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(6) where ω = ∇ ∧ u is the vorticity and Θ = ∇ · u is the dilatation. The decomposition (6) of acoustic sources in terms 1 to 10 is of interest as it identifies interactions between velocity, vorticity, dilatation and density fields. A first a priori classification of the source terms can be proposed from their analytical form [10]. The vorticity field, representative of vortical motion, is included in the terms 6 and 8, and the kinetic energy in the term 4, so they are thought as ‘hydrodynamic’ production terms. The dilatation field, representative of acoustic phenomena, appears in the terms 1, 2 and 3 and thus they can be viewed as ‘acoustic’ response terms. All the terms 5, 7, 9 and 10 involve the density gradient, which can be influenced by acoustic phenomena as well as compressibility effects, so it is more difficult to classify them in ‘hydrodynamic’ or ‘acoustic’ terms. In a first approach, the terms 5, 7, 9 and 10 are just called hydrodynamic-acoustic ‘mixed’ terms. Such a decomposition was applied to acoustic sources generated by the rollup process in a mixing layer [10] and the acoustic field radiated from each source was evaluated with the aid of the solution of equation (6). We observed that all the ‘hydrodynamic’ terms 4, 6, and 8 have a significant contribution, but also that the ‘mixed’ terms 5 and 7 played a part in the radiated sound. On the contrary, the ‘acoustic’ terms 1, 2, 3 and the ‘mixed’ terms 9 and 10 were shown to have a negligible contribution to the farfield. The decomposition can be continued about the ‘mixed’ terms, in order to better identify underlying mechanisms by means of a Helmholtz decomposition of the velocity field.
Helmholtz decomposition In the so-called Helmholtz decomposition approach, the velocity field u is decomposed into solenoidal us and compressible uc components, as u = us + u c with
(7)
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∇ ∧ uc = 0.
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In such a decomposition, the divergence-free component is rotational, while the compressible one is irrotational. It must be noted that the Helmholtz decomposition does not allow to access directly the acoustic component of turbulent field because non-acoustic, compressibility effects are included in the compressible part uc . However, by introducing the Helmholtz decomposition we can hope to improve identification of various effects included in the acoustic source terms of (6), especially the so-called ‘mixed’ terms.
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3 Numerical simulations 3.1 Numerical methods and flow configuration In this work, the computation is performed by a code which solves the compressible Navier-Stokes equations, by Direct Numerical Simulation (DNS). Spatial derivatives are evaluated using sixth order compact finite differences, time advancement being performed by a fourth order Runge-Kutta scheme. The code simulates in a rectangular domain the temporal development of a two-dimensional mixing layer, where a periodic boundary condition is imposed in the streamwise x direction. As shown previously by [3], the temporal mixing layer is an interesting virtual laboratory that allows an easy investigation of the sound emission mechanisms. In the transverse y direction, we can choose to apply a free-slip or a non-reflecting boundary condition. The free-slip boundary condition does not allow a good representation of the open physical domain, because the acoustic waves will be reflected at the boundary of the computational domain. These parasitic reflections will then contaminate the entire domain and so the computed acoustic field. Consequently a large transverse extent is necessary in this case to examine results before the first wave reaches the boundary. On the contrary, the use of a non-reflecting boundary condition allows evacuation of the acoustic waves without spurious reflections and so to avoid a large transverse extent. Finally, the initial mean velocity profile takes the form of a hyperbolic tangent and an incompressible disturbance field is added to initiate the transition process (see [3] for more details).
3.2 Computation of solenoidal and compressible parts of velocity field If the system is transformable into Fourier space (flow with periodic or free-slip directions), the solenoidal us and the compressible uc parts of the velocity field are obtained by simply projecting each Fourier coefficient onto its wavenumber vector. When the flow has inhomogeneous directions, as real flows, the use of Fourier transforms is no longer possible, in the sense of physics. In addition, if the mesh is stretched over one direction, it becomes also difficult to use classical Fourier transforms. So it is useful to develop a more general methodology, available for more complex physical configurations. According to the definition (9), uc = ∇B
(10)
where B is a scalar field. So, another method allows us to access directly the compressible component, with the aid of the resolution of the following Poisson equation ΔB = ∇ · u
(11)
using ∇B ≈ u, as Neumann boundary condition in the far acoustic field. The Poisson equation (11) can be solved to obtain uc . Golanski [8] has developed a Poisson solver which combines modified spectral methods in two directions with a compact finite differences scheme in the third direction. The use of this solver is advantageous because it is based on fully compatible numerical methods with our DNS code. In a first step, we impose free-slip boundary conditions in the transverse direction and so Fourier transforms are used to obtain a reference solution, with the solenoidal
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and the compressible parts of the velocity field. In a second step, the method based on the resolution of (11) is applied to extract both parts of the velocity field. Results from both approaches are presented and compared in the next section.
4 Results In this work, we simulate the temporal evolution of an isothermal mixing layer, with a Mach number M = 0.8 and a Reynolds number Re = 400. Two calculations are carried out: simulation 1 uses free-slip boundary conditions and simulation 2 uses non-reflecting boundary conditions. The computational domain in the simulation 1 and 2 is (15.32, 150) and (15.32, 60) respectively, and the uniform grid resolution uses 150 × 1501 points and 150 × 601 respectively. The time step is Δt = 0.025.
4.1 Hydrodynamic fields and acoustic source terms ∂v Time evolution of vorticity (ωz = ∂x − ∂u ) is presented in Fig. 1(a). We can see first ∂y the roll-up phase leading to the formation of two Kelvin Helmholtz vortices, then vortex pairing occurs and a single vortical structure remains in the computational domain. Time evolution of dilatation field (Fig. 1(b)) highlights the generation of quadrupole structures associated with vortices. The acoustic source terms of (6) are shown in Fig. 1(c-l). We notice that the range of spatial fluctuations is very different for the various terms. As expected, the shape of the ‘production’ terms resembles that of the vorticity field and the shape of the ‘acoustic’ terms is very similar to that of the dilatation field, with quadrupole patterns. The ’mixed’ terms exhibit common trends to both ‘production’ and ‘acoustic’ terms. We state also about all these results that they are exactly the same in the simulations 1 and 2. Now we are interested in the temporal evolution of compressible and solenoidal parts of velocity field. Both parts are obtained from the two methods presented in section 3, for the purpose of comparison and mutual validation. Iso-contours of the longitudinal compressible part ucx are presented in Figs. 2-a and 2-b, from simulation 1 and Fourier transforms, and from simulation 2 and resolution of Poisson equation, respectively. We mention for both calculations that the constitutive relations (8) and (9) are checked, with an accuracy close to the machine accuracy, except near the boundaries in the simulation 1 associated with the resolution of Poisson equation. This can be due to the use of the Neumann boundary condition. It can be seen that ucx presents positive and negative peaks and the shape of isocontours is similar to a dilatation field, with quadrupole patterns. We note that an excellent agreement is found between solutions from both calculations. Some little differences are noted in iso-contours at t = 33, probably due to the Neumann condition, which slightly perturbes the solution. To complete comparisons, iso-contours of the transverse solenoidal part usy are presented in Figs. 2-c and 2-d, from simulation 1 and 2 respectively. We note also a good agreement between solutions obtained from the two calculations. As in the previous case, we observe some departures for t = 33, due to Neumann’s boundary condition. Comparisons of iso-contours of longitudinal solenoidal usx and transverse compressible ucy parts are not shown here but they exhibit similar trends.
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Fig. 1. Time evolution of vorticity (a), dilatation (b) and acoustic source terms (c-l) in simulation 2.
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Fig. 2. Time evolutions of longitudinal compressible part of velocity ucx (a,b), and transverse solenoidal part of velocity usy (c,d), from simulation 1 and Fourier transforms (a,c) and from simulation 2 and resolution of Poisson equation (b,d). Intervals of contour lines are of ±0.00125 in a) and b) and of ±0.0125 in c) and d).
4.2 Acoustic farfield In the following, we focus our attention on the temporal evolution of the mean acoustic density ρ , measured at the bottom boundary of the computational domain. ρ is averaged over the periodic direction x. Figure 3-a presents time evolutions of acoustic density from direct acoustic computation and from Lighthill’s equation (3), obtained with the aid of Green’s functions. The temporal variations of both curves are clearly identical. In addition the contributions of terms I and II are provided and it can be seen that the contribution of term I is largely dominant, as expected. Figure 3-b shows results obtained from solution of (6). We observe that the sum of the terms 4, 5, 6, 7 and 8 provides the most of the acoustic radiation. The sum of terms 1, 2, 3, 9, and 10 has only a little contribution to the acoustic farfield. Notice that ‘mixed’ terms are present in both contributing and non-contributing terms. The introduction of Helmholtz decomposition in ‘mixed’ terms could provide a more meaningful decomposition and so it could be feasible to better understand the physics of the sub-terms.
5 Conclusions Numerical estimations of acoustic source terms associated to the Lighthill equation have been obtained from a compressible DNS of a temporal mixing layer, with the view to better understand the mechanisms of sound production in free shear flows. It was shown that ‘mixed’ terms play a significant part to the acoustic far field, but the reasons for that remain unclear. In addition a new tool was developed to extract both solenoidal and compressible parts of the velocity field, in a configuration
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Fig. 3. Time evolution of acoustic density ρ at y = −Ly /2, from simulation 2. a) direct computation (- -), term I (—), term II (- · -), terms I+II (....). b) term I (—), terms 4+5+6+7+8 ( + ), terms 1+2+3+9+10 (− − −). The (....) and (- -) curves in view (a) are superimposed. with free boundary conditions. The continuation of this work will include results from Helmholtz decomposition in the source terms and we hope so to enhance the description of underlying mechanisms of each term. Finally, it will be relevant to perform the same investigations for three-dimensional simulations of the flow, in order to better understand the sound emission mechanisms under consideration.
References 1. Freund J.B. (2001) J. Fluid Mech 438:277–305 2. Bogey C., Bailly C., Juv´e D. (2002) Theoret Comput Fluid Dynamics 16(4):273–297 3. Fortun´e V., Lamballais E., Gervais Y. (2004) Theoret Comput Fluid Dynamics 18(1):61–81 4. Sandham N.D., Morfey C.L., Hu Z.W. (2004) Turbulence simulation for aeroacoustics applications In: Proceedings of 10th European Turbulence Conference, Barcelona, Spain 5. Lighthill M.J. (1952) Proc Roy Soc A 211(1107):564–587 6. Kraichnan R.H. (1953) J Acoust Soc Amer 25(6) 7. Erlebacher G., Hussaini M.Y., Kreiss H.O., Sarkar S. (1990) Theoret Comput Fluid Dynamics 2:73–95 8. Golanski F. (2004) M´ethode hybride pour le calcul du rayonnement acoustique d’´ecoulements anisothermes ` a faibles nombres de Mach, PhD thesis, Poitiers University, France 9. Jordan P., Fortun´e V., Gervais Y., Lamballais E. (2004) Analysis of acoustichydrodynamic interactions using DNS. In: Proceedings of International Congress on Acoustics, Kyoto, Japan 10. Cabana M., Jordan P., Fortun´e V. (2005) A decomposition of the Lighthill source term. In : Proceedings of Euromech 467 : Turbulent flow and noise generation, Marseille, France
Large-Eddy Simulation of acoustic propagation in a turbulent channel flow Pierre Comte1 , Marie Haberkorn2 , Gilles Bouchet2 , Vincent Pagneux3 , and Yves Aur´egan3 1
2 3
LEA, 43 rue de l’A´erodrome, F86000 Poitiers [email protected] IMFS, 2 rue Boussingault, F67000 Strasbourg [email protected] LAUM, 2 avenue Olivier Messiaen, 72085 Le Mans Cedex 9 [email protected]
Summary. Large-Eddy Simulation of a turbulent pulsating channel flow is performed in order to investigate the origin of a critical Strouhal-number-range, in which the modulus of the wall-shear impedance (that can be interpreted in terms of sound attenuation) is lower than in the laminar regime. Comparisons with linearized calculations, in which a passive scalar oscillating at the walls is advected by a non-pulsating flow, confirm the non-linear origin of this critical range.
1 Introduction As shown experimentally in [7], sound propagation in turbulent ducted flows can be investigated either by introducing a long-wave pulsation of the mean flow, or by making the wall oscillate without pulsating the mean flow. Results obtained with both techniques collapse quite well, relatively independently of the forcing amplitude. The most relevant quantity to sound propagation τ) is the (non-dimensional) attenuation factor ατ = Re(Z co ρL , in which the wall shear impedance Zτ is the ratio of the complex amplitudes of the wall shear and the forced velocity. In general, attenuation is larger in a turbulent flow than in a laminar flow (ατ /ατStokes > 1) except in a rather small range of pulsations (Strouhal number ω + = ων/u2τ ∈ [5 10−3 , 3 10−2 ]) the origin of which has remained unclear. Two types of explanations have been proposed, the second of which is essentially linear: a) energy feedback from turbulence to the deterministic pulsating part, due to the synchronization of the bursting events in the boundary layers [9] b) diffraction of the shear wave due to the wall-normal gradient of eddyviscosity (in a RANS or phase-averaged sense)[7]. The suitability of Large-Eddy Simulation to tackle this problem of identification has been demonstrated in Ref. [8], in which incompressible pulsating
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channel-flow simulations were performed for a wide range of frequencies of the driving pressure gradient, with thorough comparison between LES and DNS. Among the results, the Stokes regime was retrieved beyond ω + = 0.1 for the AC component of the flow, extracted by phase averaging. At lower forcing frequencies, the non-deterministic part (denoted above with a double prime superscript) was found to develop unsteadiness, with quasi-total cyclic relaminarisation. The critical zone was retrieved, and an eddy-viscosity model was proposed for the ”turbulent Stokes length”, namely, the penetration thickness of the shear waves into the boundary layers. This model yields lt+ lS+ for + + + 2 high frequencies ) (κlS /2 1), and lt κ(lS ) at lower frequencies. Here, arm´ an constant and laminar Stokes length, κ and ls = 2ν/ω denote the K´ respectively, and we have ω + = ων/u2τ = 2/(lS+ )2 . Our motivation here is to evaluate the possibilities of improving upon existing diffusion models in use in the acoustic propagation community, by means of lower cost LES and within a fully-compressible framework capable of taking into account arbitrarily large density and temperature gradients. Because of the complexity of the physics involved, we will nevertheless restrict ourselves to low Mach numbers and nearly isothermal situations. We proceed as follows: 1. we validate our results against [8] and the associated experimental background. 2. we reduce the forcing amplitude from A0 = 70% to A0 = 20% to get closer to acoustically-forced configurations, and check whether cyclic relaminarisation is needed to observe the critical zone. This is not so easy, because the effet to capture is rather subtle. We also check whether the presence of the critical zone is accompanied by overall drag reduction. 3. we check whether the critical zone can be retrieved in a linearized framework, either by solving for the advection, by a non-pulsating flow, of a passive scalar oscillating at the walls, or by solving a diffusion model with prescribed profiles of eddy-viscosity.
2 Methodology The filtered compressible Navier-Stokes equations are solved in the macrotemperature [3] closed form, by means of an explicit McCormack-type method which solves the convective terms with 4th-order accuracy, and the Filtered Structure-Function subgrid-scale model is used, as in Ref. [1]. With respect to [8], the time-averaged Reynolds number has been reduced from Reτ = 395 to Reτ = 180, to minimize the influence of the SGS model. The domain size is (Lx, Ly , Lz ) = (2πh, 2h, πh) with (Nx , Ny , Nz ) = (64, 109, 64). The mesh spacing is uniform in the streamwise and spanwise directions (hereafter denoted x and z, respectively), with Δx+ = 19 and Δz + = 10. Hyperbolictangent stretching is used in the wall-normal y direction, with the first mesh line away from the wall located at y + = 0.2.
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The forcing term Q(t) = Q0 [1 − A0 sin(ωt)] prescribes the flow rate, which enables us to investigate the influence of the pulsation on the time-averaged drag, in contrast with the prescribed-shear forcing used in [8]. Two different forcing amplitudes have been considerered, as in the experiments [9], namely, 70% and 20% (relative to the centerline velocity of the laminar Poiseuille profile of same flowrate as the mean flow). Up to the difference in Reynolds numbers, the first case has been treated in [8]. To the best of our knowledge, the second case at lower amplitude, and therefore closer to acoustic propagation, has not been simulated yet. As in Ref. [8], flow decomposition is performed, after the spin-up of each simulation, in terms of phase average at the forcing pulsation: for each variable f , this reads f (x, y, z, t) = f Φ (y, t ) + f ”(x, y, z, t) with Nx Nz N 1 f (x, y, z, t + nT ) , t ∈ [0, 2π/ω] (1) f Φ (y, t ) = N Nx Nz n=1 i=1 k=1
This indeed corresponds to a triple decomposition, into mean, cylic and nondeterministic parts, as introduced in [2]: f (x, y, z, t) = f (y) + f (y, t) +f ”(x, y, z, t) "# $ ! f Φ (y,t )
The wall friction is thus decomposed, yielding τw = μw
.
(2)
∂u ∂y
w
= τw +
Aτ cos(ωt + Φτ ) + τw ”, in which Aτ and Φτ minimize τw ”. We then have Ac (τw ) τ i(Φτ −Φu ) = A , in which Φu denotes the phase shift of the cenZτ = A Ao e c (uw ) terline velocity wrt Q(t). ) Amplitude Ao is eliminated using the laminar Stokes solution AτStokes = ων μw Ao , which yields ατ ∝ Aτ Aτ cos (Φτ − Φu ). Stokes
3 Pulsed simulations The parametres of the LES undertaken are summarized in Table 1 below, together with, in italics, those of cases considered in [8]. N stands for the number of forcing periods during which the statistics have been considered, using phase and spanwise averaging. The corresponding CPU time, on a NEC SX-5 vector supercomputer, is also mentioned for information. A steady simulation has also been performed, not only for validation purposes, but also for the simulation of the advection of several passive scalars, forced harmonically at the walls with different frequencies, as detailed in section 4. As in Ref. [8], it is found that, for A0 = 70%, the non-deterministic part u exhibits unsteadiness when the forcing frequency is sufficiently low: Fig. 4 shows the time evolution of the vortex structure during a cycle, starting, from top to bottom, at the beginning of the acceleration phase. In contrast, unsteadiness is not visible at ω + ∼ 0.01. and (Φτ − Φu ) for the whole set of numerical and Figure 1 shows Aτ Aτ Stokes experimental results we know of. The solid line corresponds to the experiments
` P. Comte, M. Haberkorn, G. Bouchet, V. Pagneux, Y. AurEgan
524
Table 1. Parametres. Ao 0.7 0.7 Ref. [8] 0.7 0.7 Ref. [8] 0.7 0.7 Ref. [8] 0.2 0.2
ω ω+ 0.054 0.00604 0.0016 0.108 0.0107 0.01 1.08 0.093 0.1 0.108 0.0097 1.08 0.0941
ls+ 18.2 35 13.68 14 4.63 4.4 14.37 4.61
lt+ 138.25 504.7 79.14 63.16 10.78 9.89 87.03 10.69
N CPU hours 6 700 15
880
20
120
7 48
410 280
50
2.5
40
Φτ Φu
1.5
τ
A /A
τ
Stokes
2
1
30
20
10
0.5
0 0
0
10 0.02
0.04
0.06
+
ω
0.08
0.1
0.12
0
0.1
+
ω
0.2
Fig. 1. Amplitude (left) and phase shift (right) of deterministic wall shear: —— [7] A < 5% ; • [8] A = 70% ; + [9] A = 70% ; + [9] A = 20% ; • present LES, A = 70% ; ◦ present LES, A = 20%.
of [7], which were performed in a pipe flow (as [5] and [6]), whereas the other ones were in plane channel flows. This might have been thought to be a reason for the dispersion observed, but the channel-flow numerical results of [8] fall right on the those of [7], which is not so clear for ours. However, the trends are retrieved for both forcing amplitudes, namely, the asymptotic Stokes-like at low ω + , due to behaviour at high ω + , the ω −1/2 behaviour of Aτ Aτ Stokes
the fact that AτStokes ∼ ω 1/2 when Aτ goes to a quasi-steady finite limit at < 1. vanishing ω + , and the critical zone in which Aτ Aτ Stokes Figure 2, analogous to Fig. 15 in [8], confirms that cyclic relaminarisation occurs at forcing amplitude 70%, but not really at 20%, at least for ω + ≥ 10−2 . This suggests quite convincingly that cyclic relaminarisation is not required for the presence of the critical region. On the other hand, the relative indifference of this critical region to the forcing amplitude is reminiscent of a linear process, in which the shear wave would be diffused and maybe refracted by the turbulent flow, without the latter being necessarily affected. Note also that significant drag reduction is observed, for sufficiently low forcing frequency and sufficiently high amplitude, as shown in Figure 3.
LES of acoustic propagation in a turbulent channel flow +
ω ∼0.006
ω ∼0.01
160
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140
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120
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80
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60
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20
2
Φ
2 τ
/
+
+
ω ∼0.1
525
0 10
0
0
10
0
0
10
0
+
y
Fig. 2. Phase-averaged evolution of the trace of the Reynolds stress tensor. Forcing amplitude 70% (black), 20% (medium grey) and 0% (pale grey). Profiles are at T /8 apart and offset by 20 units in the wall-normal direction. The dash-dotted and dashed lines are at y + = ls+ and y + = 2lt+ , resp. 22 20
16
τ
/
18
14 12 10 8 6 4 2 0
0
10
10
+
y
1
10
2
ω+ 0.006 0.01 ” 0.1 ”
Ao 0.7 0.7 0.2 0.7 0.2
Rew 27200 13600 1100 1400 100
h/ls Cf ref. 9.9 -11.7% · · · 13.16 -6.2% −− ” -1.5% −− 40.9 0% — ” ” —
Fig. 3. Mean velocity profiles. That at zero forcing amplitude is also plotted. It is collapsed with − − −.
Although not unanimously observed, this is consistent with the experimental findings of Ref. [4]
4 Linearized calculations and eddy-viscosity arguments To check on the linear origin of the critical zone, simulations of the advection of a passive scalar c, forced harmonically at the walls and advected by a
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Fig. 4. Evolution of the coherent structures during a cycle, for ω + = 0.006. Left: contours of u at y + = 10. Right: isosurface Q = 0.6(ub /h)2 .
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50
1.4
45
Φτ Φu
Aτ /Aτ
Stokes
40
1.1
35 30 25 20 15
0.8 0
0.05
0.1
10 0
0.05
0.1
+
ω
+
ω
Fig. 5. Amplitude (left) and phase shift (right) of deterministic wall shear: —— [7] A < 5% ; 2 passive scalar LES ; ∗ linear model κt (y). 0.16
0.1
0.14
νt/(h uτ)
νt/(h uτ)
0.12 0.1 0.08
0.05
0.06 0.04 0.02 0 0
20
40
60
80
y
100
120
140
160
180
0 0
20
40
+
60
80
100
y
120
140
160
180
200
+
Fig. 6. Profiles of eddy-viscosity in time and phase averages, for ω + 0.006 (left) −u v and ω + 0.1 (right). Black crosses: νt (y) = ∂u /∂y in steady LES. Grey solid line: same in unsteady calculations Other symbols: νt Φ (y, t) =
spaced phase angles t . Portion y
+
−u”v” ∂u φ /∂y
for 8 equally-
≥ 140 irrelevant due to 0/0-type numerical errors.
non-pulsed turbulent flow, have been performed. The results have been compared with those of a semi-analytical diffusion model, with a prescribed profile v Φ of effective eddy-diffusivity obtained from κt (y) = −c ∂c Φ /∂y . and (Φτ − Φu ) are shown in Figure 5. The same trends Plots of Aτ Aτ Stokes are observed for both calculations, namely, the recovery of the asymptotic regimes at low and high ω + , but the critical region is lost. Refs. [7, 8] strongly support the idea that the effect of turbulence on the shear waves can be modelled by means of an eddy-viscosity, in a RANS sense. Figure 6 shows strong dependance of νt on the phase angle at low frequency.
5 Conclusions LES of a pulsating plane channel flow have been performed to shed light on the critical frequency range where sound attenuation is lower than in the
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laminar regime. This has been retrieved in a reasonable agreement with the LES of Ref. [8] and the experimental data of Refs. [7, 9]. Up to 11% drag reduction has been found, as in Ref. [4]. The results at a forcing amplitude of 20%, which are original to the best of our knowledge, showed that cyclic relaminarization was not needed to observe the critical zone. The asymptotic behaviours of the shear-wave impedance at low and high forcing frequencies have been retrieved in LES of the advection by a steady flow of a passive scalar oscillating at the walls, but not the critical zone. Finally, eddy-viscosity profiles have been computed, showing strong variation with the phase angle at low frequencies, and phase-dependent analytical models are in progress, in order to check whether the critical zone can be retrieved with more realistic eddy-viscosity profiles than the rigid-wall model proposed in Ref [7], which is in use in current sound-propagation codes in industrial configurations. Acknowledgement: most of the CPU time used was allocated free by IDRIS, the French CNRS supercomputing center.
References 1. F. Ducros, P. Comte, and M. Lesieur. Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a flat plate. J. Fluid Mech., 326:1–36, 1996. 2. A.K.M.H. Hussain and W.C. Reynolds. The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech., 41(2):241–258, 1970. 3. M. Lesieur and P. Comte. Favre filtering and macro-temperature in large-eddy simulations of compressible turbulence. C. R. Acad. Sci., 329(IIb):363–368, 2001. 4. C.R. Lodhal, B.M. Sumer, and J. Fredsøe. Turbulent combined oscillatory flow and current in a pipe. J. Fluid Mech., 373:313–348, 1998. 5. Z-X. Mao and T.J. Hanratty. Studies of the wall shear stress in a turbulent pulsating pipe flow. J. Fluid Mech., 170:545–564, 1986. 6. M. Peters, A. Hirschberg, A. Reijnen, and A. Wijnands. Damping and reflection coefficient measurements for an open pipe at low mach and low helmoltz numbers. J. Fluid Mech., 256:499–534, 1993. 7. D. Ronneberger and C.D. Arhens. Wall shear stress caused by small amplitude perturbations of turbulent boundary-layer fow : an experimental investigation. J. Fluid Mech., 83(3):433–464, 1977. 8. A. Scotti and U. Piomelli. Numerical simulation of pulsating turbulent channel flow. Physics of Fluids, 13(5):1367–1384, 2001. 9. S. Tardu, G. Binder, and R.F. Blackwelder. Turbulent channel flow with largeamplitude velocity oscillations. J. Fluid Mech., 267:109–151, 1994.
Numerical Methodology for the Computation of the Sound Generated by a Non-Isothermal Mixing Layer at Low Mach Number F. Golanski1 , C. Moser, L. Nadal, C. Prax, and E. Lamballais Laboratoire d’Etudes A´erodynamiques UMR CNRS 6609 - University of Poitiers - France [email protected] Summary. The sound generated by an isothermal and a non-isothermal mixing layer is computed by two methods : an aeroacoustic direct computation and an aeroacoustic hybrid method composed of a quasi-incompressible simulation coupled with the linearized Euler’s equations. A flow configuration that allows quantitative comparisons in the space-time domain is established. The good agreement found between the evolution of the computed flows allows comparisons of the radiated fields in terms of acoustic amplitude and directivity patterns, opening the analysis of incompressible acoustic source terms.
1 Introduction The capability of accurately predicting the sound generated by turbulent flows will be a key factor in many industrial fields, the more obvious being the aeronautic industry. In order to fulfil this goal, an improvement of the understanding of aeroacoustic phenomena is required. The principle of direct numerical simulation (DNS) of sound, which consists in solving the compressible Navier-Stokes equations without any turbulence or acoustic modelling, seems to be the way to reach this goal. However, this solution is too expensive to be applied, even a few decades later, to industrial applications. Nevertheless it can provide, in some academic cases, useful reference results for the development of tools for engineers. The computational cost of DNS in computational aeroacoustics (CAA) is dramatically increased by acoustic features when dealing with low Mach number flows where the speed of sound is much higher than the characteristic speed of the flow while acoustic pressure fluctuations are very much lower than aerodynamic ones. It appears to be particularly detrimental when noting that for most free shear flows the influence of acoustic fluctuations on the dynamics of the flow is negligible. Among the flows that are supposed to be of interest, the ones involving large temperature fluctuations are not well known yet in an aeroacoustic context. Starting from this knowledge deficit, the present study aims to improve the understanding and the capability to accurately compute the sound generated by non-isothermal
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free shear flows at low Mach numbers. To do that, we use two different approaches to calculate the same problem. First, a DNS provides reference results. Then, the same problem is modeled with an aeroacoustic hybrid approach composed of two distinct stages based on the computation of the flow dynamics (CFD step using “quasi-incompressible´ı´ı DNS) followed by the calculation of the acoustic propagation with the aid of a specific code solving the linearized Euler equations (LEE). Two important features are expected from this approach. First, the CFD and propagation parts are both reduced problems (compared to the full aeroacoustic question) and the numerical efficiency for each of them leads to a global computational cost lower than the original one. Second, this splitting can provide a favorable medium to perform a physical analysis of the phenomena involved.
2 Flow configuration and numerical schemes 2.1 Flow configuration: the mixing layer This study, which relies on a two-dimensional non-isothermal mixing layer, takes place in a progression in the field of numerical investigations about the noise produced by turbulent flows. This methodology which consists in relying on DNS to investigate physics has been successfully applied to a temporal mixing layer [6, 8]. This paper can be seen as an extension to a more general and realistic configuration than the one of [8]. Indeed, the properties of acoustic radiations induced by the periodicity in the streamwise direction lead to a lack of generality which is not matched to the most part of actual flows.
O
Fig. 1. 2D Non-isothermal mixing layer configuration. The flow configuration, presented in figure 1, is characterized by a ratio U1 /U2 = 2 and T1 /T2 = 1 and 2. The Reynolds and Prandtl numbers are respectively Re = ρ2 ΔU δω 0 /μ2 = 400 and P r = μ2 cp /k = 0.75, where ΔU = U1 − U2 , δω 0 is the inflow vorticity thickness, μ and k are respectively the dynamic viscosity and the thermal conductivity. The superscript above indicates dimensional quantities. In the following, all the expressions are dimensionless and ΔU , δω 0 , δω 0 /ΔU , T2 and ρ2 are chosen as speed, length, time, temperature and density references.
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531
Our objective in this study is to obtain accurate quantitative comparisons of deterministic nature in a given flow configuration. To do that, we have chosen to simulate a mixing layer forced on two distinct harmonic frequencies. Then the development of the mixing layer is controlled and the vortex pairing is the main acoustic source mechanism [4]. In previous studies [2, 1] based on Large Eddy Simulation, the authors introduced a harmonic perturbation based on the most unstable mode of the hyperbolic tangent profile at the inflow given by the linear instability theory of [10]. Our study, based on DNS, corresponds to a much lower Reynolds number than those cited above and the growing of the mixing layer is increased by the viscous diffusion. Moreover, to avoid an excessive intrusion of the forcing in the inflow boundary condition, we chose to add the perturbation in the physical domain just downstream of the inflow boundary. In this context, it must be emphasized that a too high forcing amplitude in the DNS would inevitably produce acoustic waves in the whole domain. Thus, to avoid these problems, the DNS is forced with two harmonic perturbations at the frequencies f = 0.132ΔU/δω (x1 ) (with the amplitude α = 2.5 10−4 ) and f /2 (with the amplitude α/2), x1 being the position where δω (x1 ) 2δω0 . The effects of this perturbation are then automatically matched to destabilize the flow at x1 80. The same process is used in the hybrid method. It has been shown in [3] that pressure-feedback loops produced by numerical outflow boundary conditions can generate self-sustained oscillations in incompressible flows. However, the forcing amplitude chosen in the compressible DNS is of the same order of this self-sustained perturbation. So, to ensure that our forcing effectively controls the vortex development, we chose to destabilize the incompressible simulations with a much higher forcing amplitude. In such a way, the feedback effect is produced at a negligible level compared to the forcing effects.
2.2 Numerical schemes This study involves three distinct numerical codes developed in the Laboratoire d’Etudes A´erodynamiques. All are based on the same numerical schemes. For spatial discretizations, a sixth order compact finite difference scheme of [9] is implemented on a Cartesian grid. Since the observation of acoustic waves implies the use of very large physical domains, DNS and LEE grids are stretched in both directions. The temporal integrations are performed using a fourth order Runge-Kutta scheme in the compressible DNS and LEE solving. For accuracy and stability reasons, a specific treatment of the density term is combined with a third order RungeKutta scheme for the quasi-incompressible simulations (see [7] for more details). The size of the computational domain is Lx = 500 and Ly = 1000. The mesh consists of 1861 × 981 points in the DNS and 1101 × 801 in the hybrid method.
3 The direct numerical simulation The DNS code uses the characteristic-based formulation of the full compressible Navier-Stokes equations [11] where the flow equations are decomposed into several wave modes of propagation
532
F. Golanski, C. Moser, L. Nadal, C. Prax, and E. Lamballais ∂p ∂t ∂u ∂t ∂v ∂t ∂s ∂t
p ρc + X + X− + Y + + Y − + 2 Cv 1 + 1 ∂τ1 j =− (X − X − ) + Y u + 2 ρ ∂xj 1 1 ∂τ2 j = − X v + (Y + − Y − ) + 2 ρ ∂xj R ∂qi − +Φ = −(X s + Y s ) + p ∂xi =−
−
∂qi +Φ ∂xi
(1) (2) (3) (4)
where X ± and Y ± are acoustic waves, X s and Y s are entropy waves and X u and Y v are transport terms defined in Cartesian coordinates
X ± = (u ± c)
1 ∂p ∂u ± ρc ∂x ∂x
, Y ± = (v ± c)
1 ∂p ∂v ± ρc ∂y ∂y
(5)
∂s ∂s ∂v ∂u , Ys =v and X v = u , Yu =v (6) ∂x ∂y ∂x ∂y qi is the heat flux (modeled by Fourier’s law), τij is the viscous stress tensor and Φ is the viscous dissipation function. Furthermore, the system is closed by the thermodynamic relations for an ideal gas Xs = u
ρ = p 1/γ e−s/Cp ,
T =
p ρR
and
c2 =
γp ρ
(7)
where γ = cp /cv is the ratio of specific heats at constant pressure and volume.
3.1 Boundary and initial conditions We use the conceptual model based on characteristic analysis [13] to specify the boundary conditions. At the inflow boundary there are acoustic, entropy and vorticity waves entering the domain. To estimate these waves, we consider an inflow with fixed total entropy and enthalpy [11]. At lateral boundaries non-reflecting boundary conditions are simply imposed by setting to zero the incoming acoustic waves. Similarly to [4], a buffer zone is applied at the outflow boundary : the vorticity and entropy waves are dissipated by adding damping terms to the governing equations. Compressible DNS is particularly sensitive to initial conditions, especially when focusing on acoustics. The non-isothermal case led us to be extremely attentive to this point. The computation is initiated with a hyperbolic-tangent profile for the mean flow velocity. The density comes from the perfect gas law ρo = po /To R, where po = 1/γ is the pressure and To is the temperature, defined by To = 1/(γ − 1) in isothermal flows and by the Crocco-Busemannrelation in non-isothermal ones. The 1/γ
initial state of entropy is defined as so = cp ln po /ρo . In non-isothermal mixing layers, the characteristic formulation is used to reduce the initial transverse thermal instability in the mixing region. Supposing that the initial pressure corresponds to a steady state, i.e ∂po /∂t = 0 and subsequently replacing the acoustic wave pair Y ± (Eq. 5) in the Eq.(1) results in
∂qy o ∂po ∂vo vo + γpo = (γ − 1) − + Φo ∂y ∂y ∂y
(8)
Sound Generation by a non-isothermal mixing layer
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Assuming that Φo can be neglected at the initial instant, the integration of (8) in the y-direction gives the initial condition vo = (1 − γ)qy o + v∞ where qy o is the transverse heat flux and v∞ an integrating constant.
4 The hybrid method As already mentioned in the introduction, a hybrid method has been developed for comparison with the DNS presented above. The initial and inflow conditions are deduced from the DNS to facilitate the comparisons. This hybrid method is composed of a “quasi-incompressible” DNS for the computation of low Mach number flows and of the LEE for the propagation of acoustic waves. The LMNA (for Low Mach Number Approximation) has been chosen here to provide the equivalent of an incompressible model for variable density flows. This model, given by [5], considers the variations of density related to thermal fluctuations whereas compressible effects are removed. The LEE are used here to take account of mean flow effects on acoustics such as convection and refraction. The LMNA, fully detailed in [5], is obtained by expanding the flow quantities (ρ, ui , p, T ) of the compressible Navier-Stokes equations (mass, momentum, total energy conservation equations and the perfect gas law) in power series of the small parameter ε = γM 2 : (0)
ρ = ρ(0) + ερ(1) ; ui = ui
(1)
+ εui
; T = T (0) + εT (1) and p =
p(0) + p(1) . ε
(9)
The equation for the pressure is obtained in accordance with the developments of ρ, T and the perfect gas law. The lowest order gives (0)
∂ρ(0) ui ∂ρ(0) + ∂t ∂xi
(0) (0)
(0)
∂ρ(0) ui ∂t
=0
+
∂ρ(0) ui uj ∂xj
(10) (0)
=−
∂p(1) 1 ∂τij + ∂xi Re ∂xj
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(0)
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∂ui ∂ 2 T (0) 1 = ∂xi ReP rT (0) ∂x2j p(0) = ρ(0) T (0) .
(12) (13)
LEE for a small perturbation over a steady mean flow are solved [8] by forcing the momentum conservation equation with the acoustic source terms
Si = −
(0)
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∂ρ(0) ui uj + ∂xj
.
This definition is valid for both isothermal and non-isothermal flows. The Tam & Dong [12] non-reflecting boundary conditions are implemented in the LEE with a buffer zone at the outflow.
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5 Numerical results The results of the methods presented in this paper applied to the flow mentioned in section 2 are presented here. The Mach number M = ΔU ∗ /c∗2 is 0.25 for both isothermal and non-isothermal configurations.
Comp. Isothermal LMNA
Comp. Non-isothermal LMNA
Fig. 2. Vorticity fields. From top to bottom : isothermal compressible DNS ; isothermal LMNA ; non-isothermal compressible DNS ; non-isothermal LMNA. 5 contours in [−0.7 : 0]. 3 contours in[0 : 0.3] (Non-isothermal cases).
Figure 2 shows vorticity visualizations obtained from DNS and hybrid codes in both isothermal and non-isothermal configurations. Comparisons between the plots obtained in isothermal cases (top of the figure) show an excellent agreement in terms of vortex dynamics. Due to the attention given to the calibration of the inflow excitation (already discussed in section 2) the vortex pairing is observed almost at the same position in both configurations. Consequently, the noise emission from these two results, despite their different nature, can be compared with relevance, the main acoustic sources being located in the same region of the flow. The following two visualizations in figure 2 present non-isothermal cases. Once again, despite the use of different numerical codes based on different governing equations, a good agreement is found. Particularly the counter-rotating vorticity (dashed lines) is well represented even with the LMNA simulation which does not take compressible effects into account. These results clearly show that non-isothermal mixing layer characteristics are well modeled by a LMNA simulation. Figure 3 shows the divergence of the velocity fields obtained from the CFD simulation in both isothermal and non-isothermal configurations and from the hybrid method in the isothermal case. A rather close match is obtained between the two isothermal images. The sources seem to be located at the same place either for the DNS simulation or for the hybrid method. This is due to the vortex pairing which occurs at the same position Ω(230, 0) in both cases. Moreover, not only wave fronts are found to be very close to each other but also magnitudes and wave lengths.
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Consequently these two images evidence that the hybrid method used in this paper models properly non-uniform convection and refraction effects in the inhomogeneous acoustic domain. However closer scrutiny reveals the existence of silent zones in the hybrid method result. This is probably an interference phenomenon between the main wave propagation and the propagation of spurious waves created by the outflow damping zone. A technique to suppress this unphysical phenomenon is currently under progress. In figure 4 the value of the root mean square of the divergence field (based on time average) along a circle with origin Ω and radius R = 200 is plotted on a directivity diagram. From this point of view the similarities are less obvious because of the interferences already mentioned. However, disregarding this phenomenon, the predictions of directivities from DNS or from hybrid method are found to be in acceptable agreement. In figures 3 and 4 are also plotted the non-isothermal DNS results. The global shape of the wave fronts seems to be correct. Notably, the wave magnitude ratio as well as the wavelength ratio between the fast/hot and the slow/cold streams are consistent since the wavelength is smaller and the magnitude
Fig. 3. Dilatation fields. Isothermal mixing layer : Left, compressible DNS ; Middle, Hybrid method. Non-isothermal mixing layer : Right, compressible DNS. The entire computational domain (including buffer zones) is plotted.
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higher on the cold side of the mixing layer, as it can be expected. Nevertheless, if we compare this work with [6] and [7] in which the authors computed temporal mixing layer in both isothermal and non-isothermal cases, one problem comes to light. Indeed, the non-isothermal mixing layer has been found by [6, 7] to be noisier than the isothermal one, contrary to the present results in spatial configuration. We are currently investigating this problem by suspecting that non-isothermal DNS could require highest spatial resolutions in the mixing region of the flow in order to accurately predict realistic noise generation. Additional calculations based on finer grids are required to conclude about this point. Acknowledgments. Compressible DNS calculations were performed on the NECSX5 supercomputer at the IDRIS (Institut du D´eveloppement et des Ressources en Informatique Scientifique), which we thank for providing computational time.
References 1. M. Billson, L.-E. Erikson, and L. Davidson. Acoustic souce terms for the linear Euler equations on conservative form. AIAA paper, 02-2582, 2002. 2. C. Bogey, C. Bailly, and D. Juv´e . Computation of flow noise using source terms in linearized Euler’s equations. AIAA Journal, 40(2):235–243, 2002. 3. J.C. Buell and P. Huerre. Inflow / outflow boundary conditions and global dynamics of spatial mixing layers. Technical report, Center for Turbulence Research, Stanford, USA, 1988. 4. T. Colonius, S. K..Lele, and P. Moin. Sound generation in a mixing layer. J. Fluid Mech., 330:375–409, 1997. 5. A.W. Cook and J.J. Riley. Direct numerical simulation of a turbulent reactive plume on a parallel computer. J. Comp. Phys., 129:263–283, 1996. 6. V. Fortun´e, E. Lamballais, and Y. Gervais. Noise radiated by a non-isothermal, temporal mixing layer. Part I : Direct computation and prediction using compressible DNS. Theoret. Comput. Fluid Dynamics, 18(1):61–81, 2004. 7. F. Golanski, E. Lamballais, and V. Fortun´e. Noise radiated by a non-isothermal, temporal mixing layer. Part II : Prediction using DNS in the framework of low Mach number approximation. Theoret. Comput. Fluid Dynamics, to be published. 8. F. Golanski, C. Prax, E. Lamballais, V. Fortun´e, and J.-C Vali`ere. An aeroacoustic hybrid approach for non-isothermal flows at low Mach number. Int. J. Numer. Methods Fluids, 45:441–461, 2004. 9. S. K. Lele. Compact finite difference scheme with spectral-like resolution. J. Comp. Phys., 103:16–42, 1992. 10. A. Michalke. On the inviscid instability of the hyperbolic-tangent velocity profile. J. Fluid Mech., 19:543–556, 1964. 11. J. A. Sesterhenn. A characteristic-type formulation of the Navier-Stokes equations for high order upwind schemes. Comp. and Fluids, 30:37–67, 2001. 12. C.K.W. Tam and Z. Dong. Radiation and outflow boundary conditions for direct computation of acoustic and flow disturbances in a nonuniform mean flow. AIAA paper, 95-007, 1995. 13. K. W. Thompson. Time dependent boundary conditions for hyperbolic systems, II. J. Comp. Phys., 89:439–461, 1990.
A Hybrid LES-Acoustic Analogy Method for Computational Aeroacoustics K. H. Luo and H. Lai School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, UK [email protected]
1 Introduction Accurate numerical prediction of aerodynamically generated sound is a great challenge for three main reasons: (1) The presence of an extremely wide range of scales makes any finite computational domain insufficient; (2) The energy of emitted sound is a tiny fraction of the flow energy, and any error in flow simulation leads to a larger error in sound prediction; (3) The treatment of numerical boundaries assumes particular importance. In response, Computational Aeroacoustics (CAA) has experienced rapid development over the past few decades. With current high-performance parallel computers, it is now possible to compute the sound field directly by solving the time-dependent compressible Navier-Stokes equations over a large computational domain [1, 2]. Such direct numerical simulation (DNS), however, is restricted to simple prototype aeroacoustics problems, often in two-dimensions (2D). For realistic three-dimensional (3D) aeroacoustic problems, there have been few reliable and economical numerical tools. A promising strategy is to develop hybrid methods which combine a flow solver for computing the sound source field with an acoustic solver for the acoustic far-field. The acoustic solver can be any of the extended Kirchoff method, acoustic analogies and the linearized Euler equations (LEE). Such hybrid CAA methods provide the flexibility of selecting the most appropriate methods to compute the sound source and the acoustic fields, respectively, to suit different aeroacoustic problems. Regarding the flow solver, DNS is highly desirable but very expense. The Reynolds averaged Navier-Stokes (RANS) flow solver is economical and capable of calculating any complex flow, but the review of Sinha et al. [3] concluded that RANS suppresses the acoustic field and underpredicts dynamic loads. A suitable compromise in accuracy and cost is large eddy simulation (LES), which simulates the sound-emitting, unsteady large-scale motions directly but models the small-scale motions that are less efficient in sound generation. In the present study, a hybrid LES-acoustic analogy method for CAA has been developed and assessed in a cavity flow. The Ffowcs Williams-Hawkings
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(FW-H) acoustic analogy is used for its generality. Cavity flow embodies many elements of complex flows including free shear layers, wall layers, flow recirculation, flow separation, wakes, trailing votices, flow-structure interaction and acoustic feedback, all in one simple configuration, which makes it an ideal geometry for testing models. Recently, a hybrid DNS-acoustic analogy (FWH) method in 2D was applied to cavity acoustics [4]. LES of cavity flow was reported in [5]. A combination of 3D LES and FW-H acoustic analogy has not been reported before, which therefore is the focus of the present paper.
2 A Hybrid LES-Acoustic Analogy Method 2.1 Acoustic analogies To compute the acoustic far-field, there are several approaches, including the extended Kirchoff method, acoustic analogies and LEE. Among these, LEE is the most expensive method computationally, as it has to solve a group of linearized equations on a continuous computational mesh that covers the whole acoustic field of interest. The Kirchhoff integration and acoustic analogies, on the other hand, solve only a scalar equation at discrete locations or directions corresponding to the observer positions, and are consequently very efficient computationally. The Kirchhoff method, however, is very restrictive in the sense that the integration surface must be placed in the linear acoustic zone. In comparison, acoustic analogies are the most versatile and economical. The idea of acoustic analogy was pioneered by Lighthill [6]. Curle [7] extended the Lighthill analogy to include the effects of solid boundaries. Ffowcs Williams and Hawkings [8] obtained a most general form of the Lighthill analogy by incorporating arbitrary motion of aerodynamic surfaces. In the FW-H formulation, the integration surface can be permeable, allowing mass, momentum and energy to pass through it. This feature offers a high degree of flexibility in positioning the integration surface, which is of great practical importance. For the current cavity flow, we start with the differential form of the FW-H equation for sound propagation in a uniform background flow [4]:
∂2 ∂2 ∂2 ∂2 − c2∞ + Ui Uj + 2Ui 2 ∂t ∂xi ∂xj ∂xi ∂t ∂xi ∂xi =
where
[H(f )ρ ]
∂ ∂ ∂2 [Qδ(f )] [Tij H(f )] + [Fi δ(f )] + ∂xi ∂xj ∂xi ∂t
' ( Tij = ρ(ui − Ui )(uj − Uj ) + p − c2∞ (ρ − ρ∞ ) δij − σij ∂f Fi = − [ρ(ui − 2Ui )uj + pδij + ρ∞ Ui Uj − σij ] ∂xj ∂f Q = (ρui − ρ∞ Ui ) ∂xj
(1)
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correspond to the quadrupole, dipole and monopole sources. f = 0 defines the integration surface Σ, which separates the sound source region (f < 0) from the acoustic field (f > 0). H(f ) and δ(f ) are the Heaviside and Dirac delta functions, respectively. Ui is the uniform background velocity. c∞ is the free-stream sound speed and σij the viscous stress tensor. t and xi are the temporal and spatial coordinates in the observer domain. All other symbols have their usual meanings. Applying the Fourier transformation to Eq. (1) and integrating the resulting equation convoluted with the Green function, the integral form of the FW-H analogy in the frequency domain is obtained: → → ∂G(− x |− y , ω) → → − y , ω) dΣ F i (− p ( x , ω)H(f ) = ∂yi f =0 → → → + iω Q(− y , ω)G(− x |− y , ω)dΣ f =0
+
→ → ∂ 2 G(− x |− y , ω) → y , ω) dV Tij (− ∂yi ∂yj
(2)
f >0
→ → where − x and − y are space vectors in the observer and the sound source domains, respectively. The above equation in the frequency domain avoids the evaluation of the retarded time, which is a great advantage from the computational point of view. Moreover, if the quadrupole source region is enclosed in the integral surface, the volume integration in Eq. (2) can be omitted. In principle, Eq. (2) is valid for 3D as wells as 2D acoustic calculations. For the latter case, the 2D Green function is used, and the surface and volume integrals are reduced to line and plane integrals, respectively. 2.2 Large Eddy Simulation The non-dimensional Favre-filtered Navier-Stokes and energy equations for unsteady compressible flow are solved, supplemented by a subgrid-scale (SGS) model. In order to focus on the integration of an LES solver and an acoustic solver, the standard Smagorinsky eddy-viscosity model is employed in the momentum equations. The filtered energy equation contains at least six unknown SGS terms, the modelling of which is not yet established. Here we adopt the simplified treatment of Larcheveque et al. [5]. As the Smagorinsky model does not have a proper behaviour as solid walls are approached, the length scale in the model is corrected by the Van Driest damping function: l = Cs Δ[1 − exp(−y + /25)3 ]0.5 , where Cs is a constant set to 0.17, Δ the filter width and y + the normalised distance to the wall. The LES code used was previously designed for simulating shock/boundarylayer interaction (SBLI) [9]. An entropy-splitting approach is employed, which improves the non-linear stability and minimizes numerical dissipation. The finite difference scheme has fourth-order accuracy in spatial discretization and
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a third-order temporal resolution. The code uses multi-block meshes and can handle complex geometries. It is parallelised using the MPI algorithm and optimized for massively parallel computers. The 3D cavity configuration studied includes cavity side walls, with a length-to-depth ratio L/D = 5, width-to-depth ratio W/D = 1. The computational mesh consists of the cavity block (BL1) and the block outside cavity (BL2). The upstream and downstream boundaries are located at 4D from the leading and trailing edges of the cavity, respectively. The upper horizontal boundary is set at 7D so that the computational domain includes a portion of the acoustic field. The spanwise length of BL2 is set as W2 = 2W1 . The grid points in x−y−z directions are 151×61×61 in BL1 and 301×121×121 in BL2. Non-reflecting boundary conditions are applied at the inflow, top and outflow boundaries. On solid walls, an isothermal wall condition is prescribed and a no-slip condition is imposed on velocity components. At the inflow, the mean velocity is specified using a power law. Small disturbances with magnitude up to 4% of the mean streamwise velocity is added at the inflow boundary.
3 Results and Discussions The present cavity configuration was used in a series of experimental measurements in the UK’s DSTL. The case M219 under concern had an inflow Mach number 0.85 and a Reynolds number ReD = 1.36 × 106 [10]. In the present study, LES at ReD = 5 × 103 and 1.36 × 106 were conducted. The low ReD case was intended for comparison with a DNS. Figure 1 shows the 3D vortical structures defined by the iso-surface of the Q criterion [11] at three instants during a period corresponding to the second Rossiter mode. Large spanwise vortices are generated periodically near the cavity leading edge, due to the Kelvin-Helmholtz instabilities. These vortices are typically 2D spanwise rollers, which become 3D structures close to the cavity trailing edge. After impinging on the cavity trailing edge and rear walls, strong wave reflections are produced, which propagate upstream to excite the shear layers at the leading edge, thus forming an acoustic feedback loop. In the region downstream of the cavity trailing edge, vortices become aligned mostly in the streamwise direction, similar to the streaks found in boundary layers. However, flow separation and reversal are also observed near the trailing edge. The structures observed in two LES cases are quite similar to those of our DNS [12], which used a refined mesh of 21 million grid points. Figure 2 shows the sound pressure level (SPL) at a series of monitoring points on the cavity floor along the line z/D = 0.625. The SPL is defined as: SPL (dB) = 20log10 (prms /2 × 105 Pa), where prms is the root-mean-square sound pressure. Both the DNS and the LES data at ReD = 5000 agree with the experimental data reasonably well, despite the ReD difference. However, our earlier result of a DNS at M = 0.85 and ReD = 1000 showed poor agreement with the same experimental data. It seems that the ReD effects are large
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Fig. 1. 3D vortex structures defined by the Q criterion at three instants (t0 , t0 + 1/3Δt2 and t0 + 2/3Δt2 ), where t0 = 100 is an abitrary starting time and Δt2 is the period of the second Rossiter mode. 180 6
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in the very low ReD range, but become less important when ReD reaches a certain value. Interestingly, the higher ReD LES case does not show improved comparison with experimental data, possibly due to the larger numerical errors involved as the same grid resolution was used in both LES cases. The FW-H analogy was implemented in both 2D and 3D formulations, and extensively validated against prototype 2D and 3D acoustic problems. It was then applied to calculating the acoustic field using the LES data at ReD = 5000 to compute the sound source. The schematic of the hybrid LESacoustic analogy method is shown in Fig. 3. In the 3D calculation, the integration surface is located at a distance s above the cavity, beyond which the quadrupole term in Eq. (2) is assumed to be negligible. Therefore, only the dipole and monopole source terms are used in the acoustic calculation. In the 2D calculation, the integration surface is reduced to a line in the central plane in the spanwise direction. In both 2D and 3D calculations, LES data at 128 time instants are sampled and used during a period of the second Rossiter mode. In order to investigate the effect of placing the integration surface, three distances of s = 0.5D, 0.8D and 1D are tested in the 2D acoustic calculation. In the 3D calculation, only one distance of s = 0.8D is used. Figure 3 also shows the 2D and 3D results of the hybrid method in comparison with the LES data in the overlapping zone at three monitoring points denoted by P1, P2 and P3, as there is no experimental data for comparison. At all points, the 3D FW-H prediction of the instantaneous pressure follows the same trend as the LES data. However, there are deviations between them. One possible cause may be that the sampling period for performing the acoustic analogy is too short. In retrospect, the sampling period should be at least as long as the period of the first Rossiter mode. Another cause for the discrepancy might be that the monitoring points are too near to the acoustic source so that the pressure fluctuations are strictly not the far-field sound that the FW-H analogy is intended to predict. Finally, the integration surface at s = 0.8D may be too close to the sound source that the quadrupole term should not be neglected. The last point is not supported by the 2D results: there are considerable differences between results from s = 0.5D and s = 0.8D, but the results of s = 0.8D and s = 1D completely overlap. The directivity of sound, expressed in terms of the SPL distribution, predicted by the 2D and 3D formulations of the hybrid method is shown in Fig. 4. Qualitatively, the 2D and 3D predictions of directivity are surprisingly similar, giving a SPL peak value at around θ = 75◦ . The similarity in the SPL distribution does not change much, as the radius is increased. However, the magnitude of SPL in 2D prediction is significantly higher than in 3D prediction, and the difference increases with the radius. This quantitative discrepancy is due to the different propagating behaviours of sound waves in 2D and 3D. In 2D, the wave front is cylindrical and the sound pressure decays according to p ∼ 1/r0.5 . In 3D, however, the wave front is spherical and the sound pressure decays according to p ∼ 1/r. These trends are correctly predicted in our calculations using the hybrid method as shown in Fig. 5.
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4 Conclusions A hybrid CAA method has been developed, in which LES is used to compute acoustic sources and the FW-H acoustic analogy to calculate the acoustic field. The hybrid method has been tested in a cavity flow of a Mach number 0.85 in a realistic 3D configuration. The integral form of the FW-H equation is solved in the frequency domain starting from a permeable surface above the cavity to give the acoustic field, using sound sources provided by LES. The hybrid CAA method has been assessed in both 2D and 3D implementation. The directivity of the cavity sound field is predicted correctly by both 2D and 3D formulations. Regarding the SPL, the 2D and 3D formulations of the hybrid CAA method give the right rates of decay with respect to the distance from the sound source for 2D and 3D acoustic problems, respectively. However, while in many cases, 3D flow problems may be approximated to a good accuracy by their 2D counterparts, such simplification will lead to serious over-prediction of SPL in acoustic calculations. Thus, unless the sound source is strictly 2D in nature, 3D acoustic calculations must be performed to correctly predict the sound field.
5 Acknowledgement The work was funded by the UK EPSRC and MOD/DSTL under Grant No. GR/R85303/01. The authors are grateful to Dr. Trevor Birch and Dr. Graham Foster at DSTL for their support. The computing resources were from the UK Turbulence Consortium under the EPSRC Grant No. GR/R64964/01.
References 1. Freund JB, Lele SK, Moin P (2000) AIAA J 38 (11):2023-2031. 2. Jiang X, Avital E, Luo KH (2004) J Sound & Vibration 270 (3): 525-538. 3. Sinha N, Dash SM, Chidambaran N and Findlay D (1998) AIAA paper No. 98-0286. 4. Gloerfelt X, Bailly C, Juve D (2003) J Sound & Vibration 266:119-146. 5. Larcheveque L, Sagaut P, Mary I, Labbe O (2003) Phys. Fluids 15 (1):193-210. 6. Lighthill MJ (1952) Pro. R. Soc. Lond. A 211:564-587. 7. Curle N. (1955) Proc Roy Soc London A231 (1955):505˜ n514. 8. Ffowcs Williams JE, Hawkings DL (1969) Phil Trans Roy Soc A 264 (1151):321˜ n342. 9. Sandham ND, Yao YF, Lawal AA (2003) Intl J Heat & Fluid Flow 24:584-595. 10. De MJ, Henshaw C (2000) Tech. Rep. RTO-TR-26 AC/323(AVT)TP/19, pp. 453-472. 11. Hunt JCR, Wray AA, Moin P (1988) Proc Summer Program, CTR, Stanford, CA, pp. 193˜ n208. 12. H. Lai, K. H. Luo (2005) 4th Intl Symp on Turb & Shear Flow Phenomena (TSFP4), June 27-29, Williamsburg, Virginia, USA.
Numerical simulation of wind turbine noise generation and propagation Drago¸s Moroianu and Laszlo Fuchs Lund University, Division of Fluid Mechanics, Ole R¨ omersv. 1, P.O. Box 118, 22100 Lund, Sweden [email protected]
Summary. The acoustical field behind a complete three dimensional wind turbine is considered numerically. Noise generated by the spatial velocity variation, force exerted by the blade on the fluid, and blade acceleration is taken into account. The sources are extracted from a detailed, instantaneous flow field which is computed using Large Eddy Simulations (LES). The propagation of the sound is calculated using an acoustic analogy developed by Ffowcs Williams and Hawkings. It is found that the near field is dominated by the blade passage frequency. The results, also include ground effects (sound wave reflections).
1 Introduction A major issue related to the use of wind-turbines, besides the visual impact, is the generation of noise by the turbine. Usually the sound is called noise when it is disturbing and unwanted. However, the differences in human perception lead to difficulties in quantifying noise. Several studies have investigated the annoyance of noise produced by wind turbines [20, 19] ending with recommendations about the level of noise and the distance from the wind turbine that can be acceptable for a human. Different methods to model the flow field around wind blades and compute the acoustics simultaneously exist. One approach is to compute the compressible Navier-Stokes equations for the whole domain of interest. In this way the computations can be assured to also include the influence of the sound waves on the flow field. However, due to the requirement of a large computational domain to solve the acoustic problem, such an approach demands large resources such as storage of data, computational speed and post-processing of data. Another approach, which has become more popular in recent years, is a hybrid one and comprises two steps: in the first step the sources of noise from an assumed or computed flow field are calculated, and in the second step the
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acoustic waves generated by the noise sources are transported through the whole computational domain. Different models are used to compute the first step in the above mentioned method for the flow past a wind turbine [4, 12]. Presently, industry codes mainly use the blade element momentum theory (BEMT)[9, 11] for calculating the wind turbine power and rotor loads. There are some limitations inherently included in this model: the assumption of an inviscid two-dimensional flow, the wake is represented as a set of radially independent stream-tubes, no interaction of the flow in adjacent stream-tubes, and no interaction between the wake and the free stream. Several experimental and numerical investigations have been conducted [6, 7, 8], in order to improve RANS numerical prediction of a flow past a shrouded wind turbine. Increasing the complexity of the geometry, Noerstrud [18] used a standard RANS with k- turbulence model for a large diameter wind turbine. The approach includes a Runge-Kutta time stepping and a finite volume space discretization. As mentioned above, the flow field and the noise sources are first computed and then the acoustic field is calculated. Different methods have been developed for acoustic computations. A major step in this field was made in 1952 when Sir James Lighthill published his “acoustic analogy”[13, 14, 15]. The theory describes aerodynamic noise generation provided by a flow. Mathematically, the approach resides in an inhomogeneous wave equation derived from the compressible Navier Stokes equations. The wave equation contains the noise source in the form of an inhomogeneous term. An extension of this method was proposed by Ffowcs Williams and Hawkings [5] in order to include the effect of solid surfaces on sound generation. Starting with the Ffowcs Williams-Hawkings equation, Brentner and Farassat [1] investigated the aerodynamically generated sound by helicopter rotors. Mihuaescu has developed numerical methods for solving these equations for arbitrary domains [16, 17]. When only some specific acoustical modes are of interest, highly efficient numerical methods may be used [2].
2 The Acoustical Problem This paper presents results on noise prediction, obtained with a method based on an acoustic analogy. The flow and the acoustic field are computed separately. This approach has the advantage that for low and intermediated Mach number flows, the sound pressure waves travel much faster than the flow, leading to a decoupling of the acoustics from the flow. This decoupling enables to solve the flow and acoustics on different grids and with different methods. The governing equations for the flow field computations, used in the present approach as the first part of the decoupled problem, are the conservation of mass and momentum with the incompressibility assumption:
Wind turbine noise
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i
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=0 2 ∂p ui ∂ui + uj ∂x = − ρ1 ∂x + ν ∂x∂j ∂x j i j
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(1)
For the second part of the decoupled problem, acoustic wave propagation, Ffowcs-Williams Hawkings equation (2) is integrated using the source terms extracted from the flow field eq. 1 and transferred to the acoustical grid by spatial averaging. 2 ∂ 2 [Tij H(f )] ∂ 2 ρ 2 ∂ ρ − c = 0 ∂t2 ∂xi ∂xi ∂xi ∂xj ! "# $ quadrupole
∂ ∂f ∂ ∂f ρ0 ui δ(f ) ΔPij δ(f ) − + ∂xi ∂xj ∂t ∂xi "# $ ! "# $ ! dipole
(2)
monopole
∂f where H(f ) is the Heavyside function, ni = ∂x are the normal component i to the surface f = 0 of the moving body, Pij = pδij − σij is the stress tensor, Tij = ρui uj is Lighthill’s tensor, and ρ0 is the reference density. In this form, equation (2) identifies three distinct sound sources. The first term is called monopole and accounts for the sound produced by an accelerating body. The second term is called dipole and accounts for the sound produced by a constant velocity moving body. The last term is called quadrupole and in an isothermal flow, accounts for the sound produced only by turbulence. Traditionally, time-averaged flow quantities have been used in order to compute the different source terms. Using the mean fluctuation, usually leads to a significant underestimation of the volumetric acoustic source term. For this reason, only Large Eddy Simulation (LES) or Direct Numerical Simulation (DNS) can offer details of the flow dynamics that are required for proper acoustical source terms computation. As an alternative to DNS high spatial resolution, LES may provide ways to perform computations on coarser grids. The drawback is that it has to model the small scales effect on the larger resolved ones through a so called Sub-Grid-Scale (SGS) model.
3 Numerical Methods The LES solution is computed using a commercial code (Fluent1 ). The discretization of the space filtered equations corresponding to equations (1), is performed using an upwind second order finite-volume scheme. The effect of the small scales on the large resolved ones is taken into account with the help of implicit SGS model. The flow computational domain is discretized using an unstructured tetrahedral body fitted mesh. The time integration is performed using an implicit second order discretization. 1
http://www.fluent.com
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The propagation of the acoustical waves is computed by numerically integrating equation (2) on a Cartesian grid, using an in-house code. The solver includes both spatial and temporal discretization of second order. The source terms are computed from the solution of equations (1) in each time instant. The solution procedure presented in this paper is composed of two steps. First, in each time instant the flow field is computed by integrating equations (1). Second, from the flow variables, the three acoustical source terms are computed, transferred to the acoustical grid by spatial (Gaussian based) averaging of the source terms, and used to integrate equation (2). This approach has the advantage that the sound propagation is computed on a larger domain with a smaller computational effort than the calculation of the flow field.
4 Problem Definition A three blade complete wind turbine, rotating with the angular speed of 60[rpm] in a wind of 10[m/s] is considered. The geometry is described in Fig. 1a),b). The boundary conditions used for this geometry are the following (see Fig. 1): inflow boundary condition on face abcd - normal velocity with a magnitude of 10 [m/s] and a turbulence intensity of I = 4% is imposed, outflow boundary condition on face ef gh - constant pressure and a turbulence intensity of I = 4 % is imposed, free stream boundary conditions on faces bcgf , dcgh and adhe,
Fig. 1. a) Isometric view of the wind turbine; b) Front view of the wind turbine.
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the rest of the surfaces (abf e and the surface defining the wind turbine) are wall boundary conditions. The flow computational domain is included in a box with the following dimensions: L1 = L2 = 1.54 · D and L3 = 0.38 · D; where D is the rotor diameter. The position of the rotor above the ground H (see Fig. 1b)) is H D = 0.8. The cross section profile is from a NACA 4415 airfoil. The blade is twisted to keep a constant angle of attack of 7[deg] which will provide a lift coefficient of CL = 1.2 and a drag coefficient of CD = 0.01[3]. The spatial discretization of the domain was based on an estimation of the Taylor length scale outside the rotor (integral length scale has the same order of magnitude as the rotor diameter) and near the blade (the integral lenght scale has the same odrer of magnitude as the blade chord). The chosen value for the time step was Δt ∼ 2.77 · 10−3 [s]. This will provide frequencies up to 1kHz, a rotation of one degree per time step for the rotor, and a local CFL number below 1. Data sampling will influence the statistical accuracy of the average (A). For a given error s , the number of the required data samples or numerical time steps, is given through: 2s =
1 σ 2 N A
(3)
where N is the number of samples, σ is the standard deviation and A is the mean. The estimated statistical error is below 5%. The computational domain of the acoustical problem is much larger than the computational problem of the flow field, and the far-field of the acoustical field is depicted schematically in Fig. 2. The boundary conditions that are set for the acoustical problem are as follows: on all boundaries, non-reflecting conditions are set, whereas on the ground, fully reflective conditions are imposed.
Fig. 2. Far field of acoustical domain.
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5 Results 5.1 Flow Computations As noted the flow contributes mainly at regions of larger gradient in the flow which are close to the leading- and trailing-edges of the blade. These regions contribute to the quadrupole source. The “thickness” source (monopole) and the “loading” source (dipole) are located close to the blade surface. Besides leading and trailing edge, also the tip of the blade is an important generator of shear. These vortices are formed due to the pressure difference between pressure and suction side of the blade and they are extended backward, transported by the flow in a spiral mode, as seen in Fig. 3a). They interact with the mast and with the vortices that are developing behind it.
Fig. 3. a) Tip vortices; b) Hub vortices.
In Fig. 3a) can be noticed a hub vortex. In fact, isolating the zone (Fig. 3b)), it can be seen that there is not only one but three vortices, twisted one over each other and spinning in opposite direction to the blades as it can be seen also experimentally [10]. Two monitoring points (x2 /D = 0; y2 /D = 0; z2 /D = 0.5; x9 /D = 0; y9 /D = −0.14; z9 /D = 0.49) are positioned in the tip vortex. Although there is a wide range of frequencies in the spectra (Fig. 4) they are dominated by the blade passage frequency or a harmonic of it. 5.2 Acoustic Computations For the acoustic wave propagation in the far field acoustical domain (Fig. 2), there have been defined twelve monitoring points, along two lines. First line (x/D = 2; z/D = 1.3) starts from the tip of the rotor (y/D = 2.5) and extends downstream in a plane containing the mast. The second line (y/D = 2.5; z/D = 0.1) extends across the stream starting at the mast. The spectra of acoustic density fluctuation show that in the near field, the frequency range is broader and shrinks towards smaller values with the
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Fig. 4. Turbulent kinetic energy spectrum: a) Point a; b) Point b
Acoustic denstiy fluction spectra non-dim amplitumde
1 Y1 Y6
0.01 0.0001 1e-06 1
10 frequency/fBPF
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non-dim amplitumde
1 Z1 Z6 0.01 0.0001 1e-06 1
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Fig. 5. Acoustic density fluctuation spectra and the decay of frequency.
distance from the rotor. This is an expected behavior and is caused by the production of the turbulence originating in the shear near the rotor blades which gets dissipated with the distance from its source. Except the blade passage frequency f = 1 and its harmonics, the spectra have a peak in the high frequency part (Fig. 5). This peak is shifted towards smaller values when the distance from the wind turbine is increased and after a certain value (∼ 3D) remains constant. Although the instantaneous acoustic density fluctuation field has a similar distribution from blade to blade, some differences can be noticed. These differences (Fig. 6a) are caused mainly by the influence of the mast over the flow. As seen in the same figure, the ground reflected waves interact with the propagating waves from the wind turbine.
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Fig. 6. a) Cross sectional wave reflections by the ground at y = 2.5D; b) Stream wise sectional wave reflections by the ground x = 4D (side view).
Fig. 7. a) Cross section sound pressure level distribution at y = 2.5D (front view); b) Stream wise section sound pressure level distribution x = 4D (side view).
In Fig. 6b), the strong interference of the propagating waves with the ground reflected waves is observed, though there is a negligible influence of the mast. The corresponding sound pressure level (SPL) field is depicted in Fig. 7. As expected, in the far field the SPL intensity decreases as if the wind turbine is a point source. The isocontours become almost spherical at a distance of about 6D. The interferences from the ground in the far field are also weak and cannot be observed in the sound pressure levels. 5.3 Conclusions The focus of the present work has been on the computation of the flow around a wind turbine as well as on the spreading of the sound that it generates. The vortex dynamics have been found to be in agreement with experiments. In order to compute the noise generated by the turbine, an acoustic analogy has been used. The acoustical field close to the wind turbine is dominated by the rotation frequency of the blades (BPF). In the far field the spectrum is influenced by the ground so different modes are stronger. There is a decay in the amplitude of the sound with the distance and far from the wind turbine, the acoustical waves spread spherically as if the are coming from a point source. The influence of the mast in the process of generation and propagation of sound is not so strong. To asses the influence of terrain irregularities, air dissipation or interaction with neighboring wind turbines, further work is required.
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6 Acknowledgments Computational resources were provided by LUNARC computing center at Lund University, this contribution is highly appreciated.
References 1. Brentner, K. S., Farassat, F., ”Modeling Aerodynamically Generated Sound of Helicopter Rotors”, Progress in Aerospace Sciences 39, pp. 83-120, 2003. 2. M. Caraeni, D. Caraeni and L. Fuchs ”Fast algorithm to compute resonance frequencies of a combustion chamber”, AIAA-Paper 2003-3225, 2003. 3. Mark Drela, Harold Youngren ”XFOIL 6.94 User Guide”, 2001 4. John A. Ekaterinaris, Max F. Platzer, ”Computational prediction of airfoil dynamic stall”, Prog. Aerospace Sci. Vol. 33, pp. 759-846, 1997, Elsevier Science Ltd, PII: S0376-0421997)00012-2. 5. Ffowcs Williams, J. E., Hawkings, D.L. ”Sound Generation by Turbulence and Surfaces in Arbitrary Motion”, Philosophical Transactions of the Royal Society of London, Vol. 264, No. A 1151, pp. 321-342, 1969. 6. F. Bet, H. Grassmann, ”Upgrading conventional wind turbines”, Renewable Energy 28 (2003) 71-78, 29 November 2001. 7. H. Grassmann, F. Bet, G. Cabras, M. Ceschia, D. Cobai, C. DelPapa, ”A partially static turbine - first experimental results”, Renewable Energy 28 (2003) 1779-1785, 6 February 2003. 8. H. Grassmann, F. Bet, M. Ceschia, M.L. Ganis, ”On the physics of partially static turbines”, Reneweable Energy 29 (2003) 491-499, 27 July 2003. 9. H. Glauert, Airplane propellers, ”Aerodynamic Theroy”, vol. IV, division L, Springer, Berlin, 1935. 10. C.G. Helmis, K.H. Papadopoulos, D.N. Asimakopoulos, P.G. Papageorgas, ”An experimental study of the near-wake structure of a wind turbine operating over complex terrain”, Solar Energy, vol. 54, no. 6, pp. 413-428, 1995; 0038092X(95)00009-7. 11. J.M. Jonkman, ”Modeling of the UAE Wind Turbine for Refinement of FAST AD”, Technical Report, NREL National REneweable Energy Laboratory, NREL/TP-500-34755, December 2003. 12. J. Gordon Leishman, ”Challenges in Modeling the Unsteady Aerodynamics of Wind Turbines”, AIAA 2002-0037 13. Lighthill, M. J., ”On Sound Generated Aerodynamically: I. General Theory”, Proceedings of the Royal Society of London, Series A, Vol. 211, pp. 564-587, 1952. 14. Lighthill, M. J., ”On Sound Generated Aerodynamically: II. Turbulence as a Source of Sound”, Proceedings of the Royal Society of London, Series A, Vol. 222, pp. 1-32, 1954. 15. Lighthill, M. J., ”Waves in Fluids”, Cambridge University Press, ISBN 0-52129233-6, 1978. 16. Mihuaescu M., Fuchs L., ”Evaluation of the sound generated by an unsteady flow field, using a hybrid method”, Proceedings of The 12th International Conference on Fluid Flow Technologies, T. Lajos and J. Vad (Eds.), Vol. 1, pp. 465-471, 2003.
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17. Mihuaescu M., Sz´ asz R. Z., Fuchs L., ”Modeling of the Acoustical Field due to a Jet Engine with Ground Effect”, Proceedings of The Tenth International Congress on Sound and Vibration, A. Nilsson and H. Boden (Eds.), Vol. 2, pp. 677-684, 2003. 18. Helge Noerstrud, Torstein Thorsen, Ivar Oeye, ”Computational Modelling of a large diameter wind turbine”, Conference on Modelling Fluid Flow (CMFF’03), The 12th International Conference on Fluid Flow Technologies Budapest, Hungary, September 3-6, 2003. ¨ 19. K. Persson Waye, E. Ohrstr¨ om, ”Psycho-acoustic characters of relevance for annoyance of wind turbine noise”, Journal of Sound and Vibration (2002) 250(1), 67-73. 20. ”External industrial noise-guidelines 1978:5”. The Swedish Environmental Protection Agency ISBN 91-38-04488-9.
Large Eddy Simulation for Computation of Aeroacoustic Sources in 2D-Expansion Chambers Gustavo Rubio, Wim De Roeck, Wim Desmet and Martine Baelmans Katholieke Universiteit Leuven, Department of Mechanical Engineering, Celestijnenlaan 300B, B-3001 Leuven, Belgium. [email protected]
Abstract Expansion chambers are often installed in flow duct systems to attenuate the noise generated by combustion engines, fans,... However, in certain circumstances, they can become flow-excited noise generators rather than silencers. In the present paper, a Large Eddy Simulation approach is used to predict the flow-induced noise in a 2Dexpansion chamber in order to gain more insight in the physics of the flow-acoustic interaction responsible for the tonal noise components such as tailpipe and chamber resonances. An analysis of the mean energy flux of the fluctuating variables makes it possible to identify the main regions of sound generation due to a flow-acoustic feedback coupling.
1 Introduction In exhaust ducts, expansion chambers are commonly installed to attenuate the noise emitted by e.g. IC engines and compressors. The energy of this engine noise is concentrated around the harmonics of the engine firing frequency. Their contribution is dominant for low to medium engine speeds due to the relative inefficiency of aerodynamically generated noise sources at low Mach number. When the engine speed increases, flow noise effects become more important and can even become the dominant source of exhaust noise [1]. In this framework, expansion chambers can become flow-excited sound generators rather than silencers. Expansion chambers are traditionally designed to damp out engine noise and flow noise sources, generated at flow discontinuities or singularities inside the exhaust system upstream to the expansion chamber. In contrast to internally generated flow noise, these sound attenuation principles are quite well understood [2], [3]. Therefore, the purpose of the work presented here, is to numerically predict and gain more insight in the internally generated tonal flow
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noise sources in expansion chambers. These tonal noise components can either be generated by a flow-acoustic feedback-loop inside the expansion chamber or by an excitation of chamber and tailpipe acoustic modes. In this paper these flow-acoustic interactions are investigated for a simple 2D-expansion chamber. Although the flow field inside should ideally be calculated with a 3D simulation in order not to neglect the inherent 3D nature of turbulence, a 2D Large Eddy Simulation (LES) is used. The major goal of the present research is to investigate whether tonal flow-noise phenomena in expansion chambers can be predicted using 2D LES techniques rather than to give an accurate 3D prediction of the turbulent flow characteristics inside an expansion chamber. As this tonal noise is expected to be generated by the large 2D vortical structures on one hand and by acoustic waves on the other. The paper has the following outline: a first section discusses the sound generating mechanisms in expansion chambers. Subsequently, the numerical method used to solve the LES-equations is described. Next two sections are devoted to the problem definition and to a discussion of the results. The major conclusions are summarized in a final section.
2 Noise Sources in Expansion Chambers A distinction between three different kinds of internal flow noise sources can be made. First, the turbulence inside the confined jet in the tailpipe generates a broadband noise. Due to the fact that in this study a 2D LES is used, these sources are not predicted in a precise way. Since this type of flow noise source has a quadrupole nature with a radiated acoustic power that scales with M 8 , it is, for moderate Mach numbers, a less effective noise source than the other two sources, which are monopole or dipole like (scaling with M 4 resp. M 6 ). The latter sources are generated by a flow-acoustic feedback-coupling inside the expansion chamber or by the excitation of the acoustic resonances of the chamber and the tailpipe. The basic mechanism of this flow-acoustic coupling is well-known [4]: when a flow leaves a sharp backwards facing edge it separates and forms a thin shear layer or vortex sheet. Such sheets are very unstable and quickly roll up to form a train of vortices. In the case of the chamber resonances the vortex sheet is perturbed by the acoustic waves reflected at the downstream edge of the expansion chamber. Based on experimental results on cavity flows, Rossiter [5] derived the following semi-empirical formula for the Strouhal number St of this phenomenon: n−γ fL (1) = St = U∞ M + 1/k with f the frequency, L the length of the cavity, U∞ the free-stream velocity, n the mode number, M = U∞ /c0 the undisturbed Mach number, c0 the speed of sound, k = Uconv /U∞ the ratio of the convection velocity of the vortices (Uconv ) to the free-stream velocity and γ a factor to account for the lag time
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between the passage of a vortex and the emission of a sound pulse at the trailing edge of the cavity. Another possible source of tonal noise is the excitation of the purely acoustic resonances by the flow. The acoustic resonance frequencies for the tailpipe and the expansion chamber can be obtained using the following formula [1]: (1 − M 2 ) fL = nL (2) St = U∞ 2M Lc where Lc the characteristic acoustic length of the expansion chamber or tailpipe. For the expansion chamber Lc = L, for the tailpipe Lc = (Lt + l) where Lt is the length of the tailpipe and l ≈ 0.525D an end-correction [3]. This formula is only valid for 1D propagation of acoustic waves. Above the first transversal resonance frequency of both tailpipe and expansion chamber other resonances occur due to the 2D acoustic propagation. The remainder of the paper will concentrate on purely internally generated flow noise. As such, the present work will not handle the noise sources generated upstream of the expansion chamber. Also broadband jet noise generated at the exit of the tailpipe is not investigated since this kind of noise is, at moderate Mach numbers, quite inefficient due to its quadrupole nature [6]. As such, both the acoustic field and its underlying flow phenomena can be considered to be 2-dimensional for these tonal phenomena. Therefore, the the flow domain should be calculated with a 3D simulation. Nevertheless the flow calculation in the present paper is carried out with a 2D unsteady flow calculation with a LES subgrid scale model. As a result, further turbulent phenomena in the flow field and its related dissipation is not appropriately described. In this way, the largest 2D eddies can be assumed to be modelled fairly accurate, and so are the major low frequency tonal components, which can be reasonably predicted. Broadband components, on the other hand, are generated by the 3D turbulence and will not be predicted in a precise way with a 2D unsteady flow calculation.
3 Flow Domain Modeling The compressible LES equations for a viscous flow are obtained from a decomposition of the variables of the Navier-Stokes equations (ρ, ui ) into a Favrefiltered part (ρ, u i ) and an unresolved part (ρ , ui ) that has to be modeled with a subgrid scale model: ui ∂ρ ∂ρ + =0 ∂t ∂xi ∂ρ ∂p ∂( τij + τij,SGS ) ui ui u j ∂ρ + =− + ∂t ∂xj ∂xi ∂xj ∂ ui p ∂ ui ( qi + qi,SGS ) ∂ρ e ∂ρ eu i τij + τij,SGS ) ∂( + =− + − ∂t ∂xi ∂xi ∂xj ∂xi
(3) (4) (5)
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where ρ, u i and p are the resolved density, velocity components and pressure. For a perfect gas, the total energy per mass unit e is defined as p/(γ − 1)ρ + 22 )/2 and γ the ratio of specific heats. ( u21 + u The viscous stress tensor τ ij is modeled as a Newtonian fluid, and the heat flux q i is modeled with Fourier’s law. Dynamic molecular viscosity and molecular conductivity are kept constant. The subgrid scale stress tensor τij,SGS , and the subgrid scale heat flux qi,SGS , reproduce the dissipative effects of the unresolved scales by using a turbulent viscosity μt , and a turbulent Prandtl number P rt . These are then modeled as τij,SGS = 2μt S ij and qi,SGS = −(μt cp / P rt )∂ T /∂xi , with T the temperature and S ij = (1/2)(∂ ui /∂xj + ∂ uj /∂xi ) To determine the turbulent viscosity, a Smagorinsky model is used, where 2 μt = ρ(Cs Δ) 2S ij S ij . The Smagorinsky constant Cs and the turbulent Prandtl number are set to 0.1 and 0.5 respectively. The filter size Δ is locally set to the cube root of the cell volume. To take into account the scale reduction that occurs near walls, the filter size Δ is weighted with the normal wall coordinate y + in the way proposed by Van Driest: Δ = Δ(1 − exp(−y + /A)). The filtered compressible Navier–Stokes equations are implemented in a finite volume code. They are integrated in time using a fourth-order explicit Runge-Kutta scheme. Convective and viscous terms are discretized using central second-order schemes.
4 Problem Description The expansion chamber, studied in the present paper, is shown in figure 1. The inlet pipe has a length of 15D with D the height of the inlet pipe. In this way, the boundary layer is fully developed at the end of the inlet pipe. The length and height of the expansion chamber equal 3D. The tailpipe has the same height as the inlet pipe and has a length of 12D. The Reynolds number of the computation based on D is 2500 and the Mach number at the inlet is 0.5.
Reflecting Outlet B.C
Soft Inlet B.C 10D Y
6D 3D
D
D 12D
15D 3D X
Sponge Zone
Fig. 1. Schematical overview of the computational domain and detail of the mesh.
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The mesh size has to be taken sufficiently small due to the fact that second-order schemes are used to capture acoustic phenomena. Taking the pipe height, D, as the characteristic length of the bigger scales, the Kol3/4 mogorov length scale (lη ) can be estimated as D/lη ∼ ReD [7]. The mesh that is used in the present study is approximately of size δ ∼ 4lη . For DNS, the mesh size has to be δ ≤ 2lη in the free-shear region and y + ≤ 1 near the walls. The mesh used for the present calculations is close to that limit, but still belongs to the LES range so that the Smagorinsky subgrid scale model is used. The computations are started from a laminar Blasius boundary layer profile in the whole expansion chamber, and are run until a statistically steady-state regime is reached. A structured multi-block mesh, containing in total 62.000 cells, is used for the 2D LES-computation. The mesh is refined near the walls and in the region of the mixing layers (fig. 1). Towards the outlet and inlet section, the mesh is progressively stretched to dissipate vortical structures and high-frequency acoustic disturbances before they reach these boundaries. On all solid walls, no-slip and no-penetration isothermal conditions are imposed with ∂p/∂n = 0, n being the direction normal to the wall. At the upstream boundary, characteristic soft inflow conditions, proposed by Kim and Lee [8], are applied together with a sponge zone with a length of 10D to obtain an anechoic inlet condition [9]. At the outlet, the subsonic, reflecting outflow condition of Poinsot and Lele [10] is implemented. This boundary condition is applied instead of the non-reflecting one since, at low frequencies (below the transversal cut-off frequency of the tailpipe), the area discontinuity has an acoustic impedance close to 0 and can thus be regarded as reflecting [3] (p = 0). Near the boundary a sponge zone is needed to damp out vortical structures. This sponge zone with a length of 6D may cause the outlet boundary to become partly non-reflecting and may result in an underestimation of the tailpipe resonances. A more exact boundary condition can be obtained by enlarging the domain, by incorporating a small exit region or by setting the ratio of the amplitudes of incoming and reflecting waves in the characteristic boundary condition formulation equal to the exact reflection coefficient.
5 Discussion of the Results The time-averaged vorticity contours and streamlines are shown in figure 2. The mean flow streamlines are nearly horizontal between inlet and outlet pipe and two large vortical structures can be identified, filling the second half of the expansion chamber. This flow pattern is similar to the one of a cavity oscillating in shear-layer mode [11]. As such, the presence of Rossiter modes are expected to appear. Figure 3 shows the pressure spectrum for two points (marked with crosses on fig. 1), one inside the expansion chamber at (2.4D, 2.2D) and one inside the
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3D 3D
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0 0 -1D
0
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Fig. 2. Mean vorticity contours (a) and streamlines based on mean velocities (b).
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Fig. 3. Pressure spectrum for a point inside the expansion chamber (a) and inside the tailpipe (b). Peaks in polygon: 1st Rossiter mode, peaks in square: chamber resonances, peaks in circle: tailpipe resonances.
tailpipe at (7.5D, 1.6D) where the origin (0, 0) is taken at the lower upstream corner of the expansion chamber. Clearly all different tonal phenomena, as described in section 2, can be observed. Tailpipe resonances occur, according to eq. (2), at a Strouhal number of 0.06 and its higher harmonics. The tailpipe resonances are the tonal components with the lowest frequency in the noise spectrum and are marked with circles in figure 3. These resonances are clearly more visible for the point inside the tailpipe. It appears that tailpipe resonances occur at St = 0.12n, twice the predicted frequency. This is probably caused by the fact that the standing wave pattern is caused by the impedance jump at the start of the sponge zone and rather than at the outlet of the pipe. The characteristic acoustic length of the tailpipe is thus the total length of 12D minus the length of the sponge zone 6D. This results, according to eq. (2), to resonance frequencies around St = 0.12n. The first acoustic resonance at St = 0.12 is shown in figure 4a. In this figure also the starting point of the sponge zone is indicated. It is clear
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that the resonances are caused by reflections at the beginning of the sponge zone. Previous experimental research [4] pointed out that tailpipe resonances are the dominant source of noise in expansion chamber applications. The fact that these findings are not apparent in figure 3 can be caused by the introduction of a sponge zone, applied at the end of the tailpipe to damp out vortical structures near the outlet. As already pointed out, acoustic reflections occur at the beginning of this buffer treatment. The reflection coefficient is smaller than unity, which causes the standing waves to be less dominant as compared to experiments where the reflection coefficient is near unity. Expansion chamber resonances occur according to eq. (2) at a Strouhal number around 0.75 and its higher harmonics. The observed resonance is slightly higher (around 0.80). This is caused by the fact that the Mach number in eq. (2) should be the local mean flow Mach number inside the chamber which is slightly lower than the 0.5, which yields the resonance at 0.75. Other dominant peaks at higher frequencies in the pressure spectra (Fig. 3) are caused by the higher harmonics of this frequency or by the two-dimensional behavior of the acoustic waves inside the cavity above St = 1.0. According to eq. (1) (with k = 0.57 and γ = 0.25) the first Rossiter mode occurs at St = 0.33 and the second mode at St = 0.78. The first mode can be observed at St = 0.38. The slight overestimation of eq. (1) was confirmed in previous research [11]. The second Rossiter mode is coincident with the purely acoustic resonance of the expansion chamber. This coincidence is responsible for an amplification of this phenomenon and causes the dominant peak in the pressure spectrum at St = 0.8 and its higher harmonics. The interaction between the flow-acoustic feedback coupling and acoustic resonance is shown in figure 4b, where the pressure at St = 0.8 is shown inside the expansion chamber. The second Rossiter mode is observed by the fact that the two pressure maxima along the length of the chamber, caused by the vortical movement typical for this mode, are present. More insight in the flow-acoustic interaction is gained by considering the energy balance equations. Nelson et al. [12] showed that the mean energy flux associated with the fluctuating variables can be expressed with the vector H (ρu) , with H = E + p/ρ0 + 1/2u2 the stagnation enthalpy, u the velocity vector and E the internal energy, the values indicated with a prime are the fluctuating variables and those with a bar are time-averaged. The mean energy flux associated with the fluctuating variables contains sources and sinks, which represent the transport of energy from the fluctuating variables to the mean flow field. These sources and sinks do not completely counteract each other. This results in the generation of acoustic waves, which are radiated into the far field and in vorticity waves which are convected through the tailpipe. The local energy production term ∇ · H (ρu) is shown in figure 4(c). This figure clearly illustrates the flow-acoustic feedback-interaction. Energy concentrated into the vortices interacts with the acoustic field at the outlet
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(a)
(b
c)
Fig. 4. Pressure contour of the first acoustic resonance in the tailpipe at St = 0.12 (a) and of the interaction between acoustic resonance and second Rossiter mode inside the cavity at St = 0.80 (b).Contours of the local energy production term ∇ · H (ρu) inside the expansion chamber.(c)
zone of the expansion chamber and at the place where the vortex at the downstream edge interacts with the mean flow field in the middle of the cavity. At these places, the maxima in the energy contours are an indication for the formation of acoustic waves by energy extracted from the mean flow. At pressure minima the opposite energy transfer takes place. In the middle of the cavity, the energy maximum more or less completely counteracts with the energy minimum. At the back of the chamber, on the other hand, this counteraction is not complete and results into the formation of acoustic waves which are radiated into the far-field and the separation of vortices, which are convected downstream in the tailpipe. Thus, the main region of sound generation inside the expansion chamber is near the downstream corners where the tailpipe begins.
6 Conclusions In this paper 2D Large Eddy simulations are used to determine the aeroacoustic tonal sources inside a 2D expansion chamber. Since only second-order numerical schemes are used to solve the computational domain, the mesh size has to be sufficiently small in order to avoid dispersion errors which may seriously contaminate the flow-acoustic interaction mechanisms that causes tonal noise phenomena. Although, a 3D LES is necessary to predict accurately the broadband noise mechanisms and to fully validate the results with previous experimental research, 2D vortical structures and acoustic feedback is governing the tonal noise. Indeed, the obtained results show a good agreement
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with experimental and analytical formulas for the resonant behavior of expansion chamber. Both purely acoustic as coupled flow-acoustic resonances can be predicted with the present method. For the expansion chamber geometry, studied in this paper, the major sound generating mechanism is an interaction between the second Rossiter mode and the first acoustic resonance of the expansion chamber. Through analysis of the mean energy flux of the fluctuating variables, it is clear that the main region of sound generation inside the expansion chamber is near the downstream corners where the tailpipe begins. The sound levels of the tailpipe resonances are, as compared to previous experimental research, probably underestimated due to a buffer treatment at the outlet boundary. This also causes a shift of the resonance frequencies inside the tailpipe. It appears that the characteristic acoustic length is decreased with the length of the sponge zone and that tailpipe resonances are merely caused by reflections at the start of the sponge zone rather than by the impedance jump at the exit of the tailpipe. The use of a more accurate representation of the outlet boundary through an impedance formulation or an extension of the computational domain will be investigated in the future. Further research will also focus on the prediction of the broadband noise generation in such devices. For this purpose, 3D simulations are needed, since broadband noise is mainly generated by smaller turbulent structures, which are inherently 3 dimensional. Acknowledgements The research work of Wim De Roeck is financed by a scholarship of the Institute for the Promotion of Innovation by Science and Technology in Flanders (IWT).
References 1. Desantes J. M., Torregrosa A. J., Broatch A. (2001) Experiments on Flow Noise Generation in Simple Exhaust Geometries, Acta Acoustica 87:46–55 2. Munjal M. L. (1987) Acoustics of Ducts and Mufflers. Wiley-Interscience, New York 3. Davies P. A. O. L. (1988) Practical Flow Duct Acoustics. Journal of Sound and Vibration 124:91–115 4. Davies P. A. O. L. (1981) Flow-Acoustic Coupling in Ducts. Journal of Sound and Vibration 77:191–209 5. Rossiter J. E. (1964), Wind Tunnel Experiments on the Flow over Rectangular Cavities at Subsonic and Transonic Speeds. Royal Aircraft Establishment, Techical Report 64037 6. Torregrose A. J., Broatch A., Climent H., Anders I. (2005) A Note on the Strouhal Number Dependance of the Relative Importance on Internal and External Flow Noise Sources in IC Engine Exhaust Systems. Journal of Sound and Vibration 282:1255–1263 7. Pope S. B. (2000) Turbulent Flows. Cambridge University Press
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8. Kim J. W., Lee D. J. (2000) Generalized Characteristic Boundary Conditions for Computational Aeroacoustics. AIAA Journal 38:2040–2049 9. Colonius T., Lele S. K., Moin P. (1993) Boundary Conditions for Direct Computation of Aerodynamic Sound. AIAA Journal 31:1574–1582 10. Poinsot T. J., Lele S. K. (1992) Boundary Conditions for Direct Simulations of Compressible Viscous Flow. Journal of Computational Physics 101:104–129 11. De Roeck W., Rubio G., Reymen Y., Meyers J., Baelmans T., Desmet W. (2005) Towards Accurate Flow and Acoustic Prediction Techniques for Cavity Flow Noise Applications. AIAA-paper 2005-2978 12. Nelson P. A., Halliwell N. A., Doak P. E. (1983) Fluid Dynamics of a Flow Excited Resonance, Part II: Flow Acoustic Interaction. Journal of Sound and Vibration 91:375–402
Part XII
Variable Density Flows
High resolution simulation of particle-driven lock-exchange flow for non-Boussinesq conditions James E. Martin1 , Eckart Meiburg2 and Vineet K. Birman2 1 2
Dept. of Mathematics,Christopher Newport Univ., Newport News, VA 23606 Department of Mechanical and Environmental Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106
Summary. The structure and dynamics of particle-driven lock exchange flows are examined by means of simulations of the variable density, incompressible NavierStokes equations. A large starting contrast is taken in the bulk density or instead in the interstitial fluid density. We record the streamwise liftoff location for the dense current as a function of the chosen density contrast for settling velocities beyond the range of validity of present day models.
1 Introduction Gravity currents are flows driven predominantly by horizontal density differences. For one, the density difference may be caused by thermal effects. Additionally, differential loading of suspended particles will further contribute to the density contrast. Spectacular examples of such particle-driven gravity currents are the pyroclastic flows associated with the ash fall out of volcanic eruptions. The contrasts in density and temperature in pyroclastic flows are immense, with the bulk density of the erupting mixtures as large as 4 to 18 kg m−3 (compared to a standard sea-level atmospheric density of 1.25 kg m−3 ) and the eruption temperature as much as 1000◦ C[1]. [2] perform direct Navier-Stokes simulations of particle-driven gravity currents. However, their use of the Boussinesq assumption limits their investigation to very small density contrasts and precludes the formation of interesting flow phenomena such as reverse buoyancy currents. The relatively few investigations of particle-driven gravity currents that have accounted for potentially non-Boussinesq conditions do so either on the basis of the box model approach [3] or the shallow water equations [4]. Studies along these lines can be quite powerful in terms of providing information on the global properties. However, they do not allow for a detailed investigation of the structure of these currents. Consequently, the goal of the present investigation is to deepen our
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understanding of particle-driven gravity currents while paying particular attention to the ramifications of large density contrasts.
2 Basic Equations For this investigation we use the classic lock exchange configuration as our flow geometry. To begin each simulation, we uniformly seed only the left half of the simulation domain (x < 0) with particles of uniform density and diameter, but negligible inertia. Our setup is such that there is a bulk density decrease from left to right across the starting separating membrane. This initial configuration results in a discontinuity in pressure across x = 0 and a predominantly horizontal flow once the membrane is removed. If we let the dimensional densities of the fluid and the particles be denoted by ρ˜ and ρ˜p , respectively, and the dimensional volume fraction of the particles by s˜ (throughout this discussion, tildes will be used to denote dimensional quantities), then the bulk density, ρ˜s , is given by ρ˜s (x, z) = [ρ˜p − ρ˜(x, z)] s˜(x, z) + ρ˜(x, z) .
(1)
We will focus on dilute, fluidised suspensions with volume fractions well below one percent. In this case particle-particle interactions can be neglected. ρ itself is expressed in terms of the fluid concentration, c˜, as ρa − ρ˜i ) ρ˜(x, z) = ρ˜i + c˜(x, z)(˜
(2)
with ρ˜i and ρ˜a representing the interstitial and ambient fluid densities and ρ˜i < ρ˜a . (It is possible to instead consider ρ˜a < ρ˜i . However, in this case, even after all particles settle out, the dense current would remain as a purely ground hugging intrusion making this situation similar to the flows studied in [5].) As in [2], particle quantities will be treated in an Eulerian manner. The particle ˜ (x, y) velocity field and the fluid velocity field will be denoted as u ˜ p (x, y) and u respectively. The particles we consider have a density which is much greater than that of the fluid they are immersed in and so the dominant flow force on ˜ in which F ˜ = 3π μ u −˜ up ). an individual particle is the Stokes drag force, F, ˜d˜p (˜ Note that in this study we make the assumption that the dynamic viscosity, μ ˜, is held constant and is the same for both the ambient and interstitial fluids. Particles of negligible inertia move with a velocity that is equal to the sum of the local fluid velocity and a settling speed, u ˜s (x, y). Since we consider variable density fluid, the settling speed of a particle will depend locally upon ρp − ρ˜(x, y))˜ g / (18˜ μ) with g˜ denoting the density, in particular u ˜s (x, y) = d˜2p (˜ ˜ p = 0 and the gravitational acceleration. Unlike what is assumed in [2], ∇ · u so accumulation of particles is a feature of these flows. The incompressible Navier Stokes equations for variable density flow will be solved with a forcing term that accounts for two-way coupling, i.e. the effect of the particles on the fluid. In dimensional form, with eg = (0, −1) the momentum equation is
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˜ ∂u ˜ + ρ˜g˜eg ˜ · ∇˜ + ρ˜s u ˜ − c˜p F u = −∇˜ p+μ ˜ ∇2 u ∂t
(3)
ρ˜s
where p˜(x, y) denotes the pressure and c˜p (x, y) the particle number density. Furthermore, we solve separate convection-diffusion equations for the particle number density ∂˜ cp ˜ 2 c˜p + (˜ u+u ˜s eg ) · ∇˜ cp + c˜p (∇ · u ˜s eg ) = K∇ ∂t
(4)
and the fluid concentration (which with equation (2) may be written in terms of fluid density) ∂ ρ˜ ˜ 2 ρ˜ . +u ˜ · ∇˜ ρ = K∇ (5) ∂t In the above we have made a simplification by assigning the same molecular ˜ in equations (4) and (5). To nondimensionalize our equations diffusivity, K, ˜ as our length scale while the starting we use the height of the container, H, bulk density in the left compartment, ρ˜0s , serves as the characteristic density. ˜ 1/2 , in which g˜ g H) Velocities are scaled by the buoyancy velocity u ˜b = (˜ denotes the reduced gravity, which is related to g˜ by g˜ = g˜ ρ˜0s − ρ˜a /ρ0s = ρ0s < 1, i.e. γ is the ratio of the bulk densities in the g˜(1 − γ), where γ = ρ˜a /˜ starting compartments. A characteristic pressure is given by u ˜2b ρ˜0s . We thus arrive at the following set of four dimensionless equations governing the four fundamental quantities, u, ρ, s and p, ∇·u=0 1 2 1 1 ∂u + ρs u · ∇u = −∇p + ∇ u+ ρeg + s(D − ρ)eg ∂t Re 1−γ 1−γ 1 2 ∂ρ + u · ∇ρ = ∇ ρ ∂t Pe St 1 2 St ∂s + u+ 1 − D−1 ρ eg · ∇s − ∇ s s∇ · D−1 ρeg = ∂t F r2 F r2 Pe ρs
(6) (7) (8) (9)
ρ0s . In equaHere the dimensionless mass loading D is defined as D = ρ˜p /˜ tion (9) the ratio of Stokes number, St, to the square of the Froude number, F r2 , represents the ratio of the characteristic flow time to the settling time of the particles ( FSt r2 =
d˜2p ρ˜p g ˜ 18˜ μu ˜b ).
The Reynolds number, Re, and the Peclet num˜ ν and P e = u ˜ K. ˜ They are related by ber, P e, are defined as Re = u ˜b H/˜ ˜b H/ ˜ so that P e = ReSc. For most pairs of gases, the Schmidt number Sc = ν˜/K, the Schmidt number lies in the narrow range between 0.2 and 5. We employ Sc = 1 throughout this study and so P e = Re. For us then, it is necessary to assign a value to the four parameters, St/F r2 , Re, γ, and D in each simulation. For the purpose of comparison with [2], in all of our simulations we take a Reynolds number of 2236. In terms of the above parameters the difference in ambient and interstitial fluid density, ρa − ρi , is γ − (1 − so D) (1 − s0 ) with s0 being the initial particle volume fraction.
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3 Computational Approach We briefly outline our computational approach, with a more detailed account of our technique along with its validation given in [5]. For the purpose of numerical simulations, we recast equations (6) - (9) into the vorticitystreamfunction formulation. In a manner that is identical to the one used in our earlier investigation of density-driven gravity currents, spectral Galerkin methods are used in representing the streamwise dependence of the streamfunction and vorticity fields. Vertical derivatives are approximated on the basis of compact finite difference stencils. At interior points, sixth-order spatially accurate stencils are used, with third- and fourth-order accurate stencils employed at the boundaries. A difference in our computational approach from [5] is that we now additionally solve equation (9) during each time step. Derivatives of the fluid density field and the particle volume fraction are computed from compact finite differences in both directions. As in [5], to integrate equations (7) - (9) in time we use a third order Runge-Kutta scheme. At the top and bottom of the computational domain, no-slip walls are prescribed, that is u = 0 at z = ±0.5 while the upstream and downstream boundaries will be considered stress free. Consequently we assign the streamfunction to be identically zero along all boundaries and the vorticity to be zero at both the upstream and downstream boundaries. To enforce zero diffusive mass flux, the fluid density satisfies Neumann boundary conditions along all of the computational boundaries. With the exception of the bottom wall, s also satisfies Neumann boundary conditions. Along the bottom wall, we model the sedimentation of particles through the convective boundary condition, ∂s/∂t = us ∂s/∂z, in which the settling speed is taken as the characteristic velocity. Because of the settling velocity of the particles, the no-flux boundary condition on s takes the following special form along the top wall us s −
1 ∂s =0 Re ∂z
at
z = 0.5 .
(10)
The flow field is initialized with the fluid at rest, i.e., u = 0 everywhere. For the Reynolds number applied in this investigation, successive grid refinements proved that we achieve mesh independent results that adequately resolve the boundary layer at a grid size of Δx = Δz = .0039. So that a smoothly transitioning profile is achieved, both ρ and s are initially given error function profiles in such a way that ρ varies between ρi and ρa while s varies between s0 and 0 from left to right across the midplane. Unlike purely density-driven gravity currents, for a particle-driven gravity current there are two different density contrasts involved. A strong density contrast may occur either 1) in the bulk density through assignment of γ to a value much less than unity and/or 2) in the fluid density if ρa >> ρi . Next, we consider each of these two cases in turn.
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4 non-Boussinesq bulk density contrast
Fig. 1. Grey scale contours of ρ at identical times for St/F r2 value a) 0 (i.e. densitydriven); b) .02; c).12; and d) .32. The starting bulk density ratio, γ, is 0.4. Since the bottom three simulations are particle-driven, in these ρa − ρi is also assigned, but given a small value here
We first consider the case of a strong starting contrast in the bulk density and so we assign γ = 0.4. At the same time, we impose only a moderate contrast in the underlying fluid density by assigning ρa − ρi = 0.1. Figure 1 shows contours of ρ at the time of t = 5.3 for four different simulations where in each simulation a different settling speed was applied. Figure 1(a) gives the result for a purely density-driven situation [5] while figure 1(b) shows the particle-driven simulation for St/F r2 = .02. Although the settling speed and the initial volume fraction of particles is extremely small (s0 = .005), a comparison with figure 1(a) shows that the inclusion of a particle phase has an altering effect upon the flow structure, particularly the vortices along the interface. In figure 1(c), for an increased settling situation of St/F r2 = 0.12, the rightward moving intrusion has commenced reverse buoyancy at the time displayed. In reverse buoyancy, enough particles settle out to make the dense intrusion buoyant. For us, we classify reverse buoyancy as occurring when the velocity of the bulk front becomes zero. In one way, the effect of reverse buoyancy is made apparent in figure 1(c) through the enlarged vortices along
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the interface. The slowing of the dense current during the reverse buoyancy process also has the effect of producing a more flattened front on the dense head. Figure 1(d) is for the exact same time as shown in each of figures 1a-c, however St/F r2 = 0.32. Now particles settle out much more quickly and a reverse buoyancy plume, emerging off of the dense intrusion, is fully evident. A similar anvil-like buoyant plume is observed in the experiments of [4]. The light intrusion is also altered from the form it took in the density-driven situation. In particular, the tip of the light front is positioned at a much lower lateral location (xr = −1.02, z r = 0.15). There emerges an additional, rightward moving, intrusion current positioned above the starting light front. The explanation for a rightward top-hugging current is that by this time enough particles have settled out of the top portion of the left compartment to produce an increase in bulk density from left to right along the top boundary.
5 non-Boussinesq fluid density contrast We next consider the case of moderate bulk density contrast by assigning γ = .9 while creating strictly non-Boussinesq conditions in the fluid density field by taking ρa − ρi = 0.5. The results of this simulation illustrate how, through a fine-tuning of the parameters, a momentary enhanced upward particle dispersion may be produced. Figure 2 shows a grey scale contour of s at an early time of t = 1.7 for a simulation with St/F r2 = 0.22. Contours of the fluid density field are overlaid. Now with such a large starting fluid density contrast, a buoyant plume very quickly emerges. It emerges in back of the dense head creating a counterrotating vortex which, in figure 2, is located near x = −0.25 and z = 0.2. The rising plume pinches off the leftward moving light intrusion
Fig. 2. Grey scale contours of the particle volume fraction at t = 1.7 for γ = 0.9 but strong fluid density contrast. Note the two strands of particles lifted laterally. Overlayed contour line are of the fluid density. Under these non-Boussinesq fluid density conditions, the buoyant plume quickly emerges and it interferes with the leftward moving, light intrusion
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(seen in figure 2 with its tip location at xr = −0.64, z r = 0.34) and, since the leftward intrusion is now more dense than its surroundings it subsequently descends. Particles have not yet settled beyond the influence of the leftward intrusion and the buoyant plume; these each pull a strand of particles laterally. As we described in figure 1(d) again the development of a rightward moving light intrusion along the top boundary is noted. The streamwise location for the tip of this reversed light intrusion eventually extends beyond that of the ground hugging current. In figure 2, the nose location of the ground hugging intrusion is approaching x = 0.5. The front of the head is somewhat flattened, however, the dense current has the typical rounded interface at its top. Figure 3 shows the temporal development of the bulk front velocity from this simulation. As particles settle out, the dense current loses its mass and gradually begins to slow, with zero front velocity occurring at t = 1.91. After t = 1.91, the ground hugging front propagates leftward, eventually proceeding back across the midplane. This flow is ultimately dominated by the top-hugging, rightward moving intrusion. 0.6 0.4
uf
0.2 0
-0.2 -0.4 -0.6
0
1
2
3
t Fig. 3. Bulk density front velocity for γ = 0.9 and ρa − ρi = 0.5. The front velocity becomes zero at t = 1.91 indicating that the dense intrusion has become buoyant
In [3] a study is made of the streamwise liftoff location of particle-driven gravity currents modeled by means of the box model approach. An underlying assumption of all box models is that the settling velocity is very small relative to the characteristic velocity of the gravity current. In many models nondimensional settling velocities only as large as .005 are considered. We are able to extend the results of [3] to look at cases for which the settling velocity of the particles is not small. Similar to their graphs, in figure 4 we plot the streamwise liftoff location as a function of our choice of the starting density contrast. The smaller the settling velocity the longer the distance before liftoff occurs, with this dependence being most pronounced for small density contrast.
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xoff
7
6
5
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
(pa − pi)/(p0s− pi) Fig. 4. Fluid density contrast versus streamwise liftoff location for γ = 0.9 and − St/F r2 = .02; ◦ − St/F r2 = .12; − St/F r2 = .22; − St/F r2 = .32.
6 Conclusion This study has begun our exploration of the early differences which exist between density-driven and particle-driven gravity currents. Future studies will examine particle deposit profiles and these same flows when the container is inclined. Moreover an examination of the energy balance should prove interesting, as the light intrusion in density-driven currents has been found to be always energy preserving [5], while here we have observed occasions with significant vortex production in the rightward-moving, top hugging intrusion.
References 1. Sparks RSJ, Bursik MI, Carey SN, Gilbert JS, Glaze LS, Sigurdsson H, Woods AW (1997) Volcanic Plumes. John Wiley & Sons 2. Necker F, H¨ artel C, Kleiser L, Meiburg E (2002) High-resolution simulations of particle-driven gravity currents. Int. J. Multiphase Flow 28:279–300 3. Hogg AJ, Huppert HE, Hallworth MA (1999) Reversing buoyancy of particledriven gravity currents. Phys. Fluids A 11:2891–2900 4. Sparks RSJ, Bonnecaze RT, Huppert HE, Lister JR, Hallworth MA, Mader H, Phillips, J (1993) Sediment-laden gravity currents with reversing buoyancy. Earth and Planetary Science Letters 114:243–257 5. Birman VK, Martin JE, Meiburg E. (2005) The non-Boussinesq lock-exchange problem. Part 2. High resolution simulations. J. Fluid Mech 537:125–144
LES of the jet in low Mach variable density conditions Artur Tyliszczak and Andrzej Boguslawski Institute of Thermal Machinery, Czestochowa University of Technology Al. Armii Krajowej 21, 42-200 Czestochowa, Poland. [email protected], [email protected] Summary. The paper presents results of LES simulation of isothermal and nonisothermal low Mach number jets. To overcome difficulties related to the time-step restrictions the numerical code uses the low Mach number approximation of the governing equations. The influence of the density ratio between jets and ambient fluid on a flow field evolution has been analysed numerically and compared with experimental data. Additionally we studied influence of an external excitation added to the inlet velocity profile which results in bifurcating jets. We found that bifurcating jets occurs both in isothermal/constant density jets as well as in non-isothermal low Mach number conditions. However, in both cases the excitation parameters required to create bifurcation are different. Key words: Large Eddy Simulation, Low Mach number flow, Bifurcating jet
1 Introduction In the natural, non-excited jets the instability mechanism and inlet turbulence intensity determine the jet mixing efficiency and jet spreading rate. The combustion, aerodynamic noise and evaporization are examples of the physical processes very much dependent on these parameters. In non-isothermal conditions the density differences may considerably change the flow characteristics. An example of such situation is the absolute instability phenomenon [15, 14] occurring when the density ratio between jet and ambient fluid is less than 0.7 approximately. In these particular conditions the jets are characterized by self-excited oscillations leading to increased spreading rate, higher turbulence level and enhanced mixing. The experimental works have shown that absolute instability phenomenon is extremely sensitive to the level and profile of turbulence intensity at the jet nozzle exit - high inlet turbulence level may totally suppress occurrence of the phenomenon. Therefore, from the practical point of view it does not seem proper to develop devices expecting that in various working conditions (various density ratio, various turbulence level, external disturbances, etc.) the natural mechanism will lead to the necessary and desired mixing conditions. The solution of this problem is the active jet control first reported by Crow and Champagne [2]. They observed that
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the mixing and spreading rate of the jet increase when forcing frequency is close to frequency corresponding to Strouhal number equal to 0.3, that corresponds to the so called preffered mode. A very interesting phenomena occurring under particular excitation conditions are the bifurcating jets which split into two separate well defined streams [10, 16, 17]. The necessary conditions to obtain bifurcating jets were formulated as: fa /fh = 2, fa = 0.3 div0.7, where (fa ) and (fh ) are the frequencies of an axial and helical (called also: orbital, flapping) excitations. Numerical simulations performed with direct [4, 6, 7] and large eddy simulations [20, 3] confirmed many experimental results. However, these works deal with isothermal cases only and therefore it is hard to guess whether the additional excitation will promote or prevent enhancement of the jet spreading rate and increase mixing and turbulence intensity in non-isothermal conditions. In other words it is unknown whether (or to what extent) the occurrence of bifurcating jets is affected by density/temperature differences between jet and ambient fluid. In this work we focus on this aspect and therefore we performed LES study of excited jets with axial and helical excitations superimposed on the inlet velocity profile for various density ratios. The influence of excitation frequencies is analysed and compared with constant density case.
2 Governing equations and numerical method In this work we applied the so-called low Mach number expansion [1, 12] which allows for an efficient solution in low Mach number conditions. The Navier-Stokes and energy equations for compressible flows become singular in such conditions what makes their solution very difficult from the numerical point of view. The idea of the low Mach number expansion is based on the assumption that for low speed flows each flow variable may be expressed as the power series of small parameter = γM a2 1 according to the following formula: ρ = ρ(0) + ρ(1) + 2 ρ(2) + . . . . The symbol γ is the specific heat ratio and M a is the Mach number. Expressing the remaining variables in the same way and introducing these expansions into the governing equations results in a set of equations for which the numerical time step is independent of Mach number. The subgrid terms which appear after performing LES filtering are modeled using filtered structure function model [5, 13]. Third order low-storage Runge-Kutta method is used to solve equations in time. Within each Runge-Kutta step the projection method is applied to determine pressure and velocity fields. The spatial discretization is performed with V I th order compact scheme [11] in direction of the jet axis and Fourier approximation in plane perpendicular to the jet. The details of the discretization method and projection algorithm applied may be found in [19]. The computational domain is a rectangular box 10D × 10D × 16D where D is the jet diameter. The periodic boundary conditions are assumed on the lateral walls while the inlet boundary conditions are specified in terms of instantaneous velocity, density and temperature. At the outlet of the computational domain we applied the convective type boundary conditions. For each time step the instantaneous axial velocity is generally defined as: u(x, t) = umean (x) + unoise (x, t) + uexcit (x, t)
(1)
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where the mean velocity umean (x) is defined by hyperbolic-tangent profile [3, 19].The momentum thickness θ of the initial shear layer is characterized by the ratio D/θ = 40. The fluctuating component of velocity unoise (x, t) is the random Gaussian noise adjusted [3] to have turbulence level equal to 5% in the vicinity of the shear layer (0.8 < r/R < 1.2) and 1% in the region where r ≤ 0.8. The same level of turbulence was assumed for the remaining velocity components. The forcing component of the axial velocity is defined as:
uexcit (x, t) = Aa sin 2πSta
U1 U1 π t + Ah sin 2πSth t + D D 4
sin
πx R
(2)
which is a superposition of axial forcing (the first term) and helical/flapping forcing (the second term). The forcing amplitudes are Aa and Ah , the Strouhal numbers are defined as Sta = fa D/U1 and Sth = fh D/U1 where U1 is the jet centerline velocity. The profile of the density was obtained from Busemann-Crocco relation and the temperature was computed from the equation of state.
3 Results The computations were performed for the Reynolds number equal to Re = 20000. The computational mesh consisted of 128 × 128 × 160 nodes, to increase accuracy of the solution in the jet region the mesh was additionally stretched in lateral directions by a hyperbolic-tangent function. Application of such type of mesh may be questionable for the axisymetric jet computations - different grid resolution in particular directions may cause that the jet spreads into some preferential directions. To check the results obtained on stretched mesh we performed additional computations (for selected cases) using uniform mesh with the same number of nodes and also using uniform mesh consisting of 256 × 256 × 256 nodes. The results obtained on the uniform mesh with 256 × 256 × 256 nodes were closer to that obtained on stretched mesh with 128 × 128 × 160 nodes.
3.1 The isothermal jet To verify the numerical code and correctness of excitations we first focused on the solution of isothermal jet for which experimental and numerical data were available. Figure (1) shows comparison of the mean axial velocity and its fluctuation at the jet axis obtained in our computations with literature data. The results are presented for non-excited case (Fig.(1) on the left side) and for the case with axial excitation (Fig.(1) on the right side). The results for the non-excited jet are in good agreement with experimental works [2, 21] and also with numerical one [3] in which the LES method was used together with numerical algorithm similar to that applied in the present work. The results obtained for the case with axial excitation were obtained with the Strouhal number assumed equal to Sta = 0.5 and the amplitude equal to Aa = 0.05U1 . In this case, the results in our opinion are also correct, the differences may originate from the level and distribution of turbulence imposed at the inlet. We note that turbulence imposed in our computations does not have correct time/space
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Fig. 1. Mean axial velocity and its fluctuating component along the jet axis: without excitation (left figure) and with axial excitation (right figure).
Fig. 2. Left figure: mean axial velocity and its fluctuating component along the jet axis. Right figure: mean axial velocity in bifurcating plane along the jet radius obtained with superposition of axial and helical excitations.
correlations, we did not pay attention to this important [8] aspect and we just imposed random Gaussian noise. The next computations were performed for helical excitations (the second term in Eq.(2)) and also for superposition of axial and helical excitations. The results obtained are presented in Fig.(2), the mean axial velocity and its fluctuating component along the jet axis for various excitation types are shown in the left figure. The amplitudes of helical excitations were set equal to 0.05U1 and 0.15U1 , the Sth was equal to 0.5. In these computations we wanted to verify whether it was possible to obtain bifurcating jet with helical excitation only by applying the sufficiently high amplitude of excitation [4]. Unfortunately we could not confirm this observation in our computations. The bifurcating jet was obtained when parameters of excitation
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were set according to [17], i.e. Sta /Sth = 2.0 and Sta = 0.5, the amplitudes of both axial and helical excitations were equal to 0.05U1 . In this case the evident decrease in axial velocity was observed starting from z/D = 3.5, and further z/D = 6.0 the jet practically disappeared. The profile of fluctuations showed two peaks which seem to be related to the region of very fast mean velocity decay. The comparison with experimental data is presented in Fig.(2) on the right hand side, the mean velocity profile along the jet radius in bifurcating plane is compared with data from [10] for two localizations: z/D = 6.5 and z/D = 8.0. Despite the differences in Reynolds number and amplitudes of excitations (Re = 4300 and A = 0.17U1 in [10]), the results obtained are not far from experimental data and they show the proper jet behavior proving correctness of the numerical method applied.
3.2 The non-isothermal jet The results for non-isothermal conditions for density ratios S = 0.8 and S = 0.6 are presented in Fig.(3). The results on the left hand side concern computations without excitation and they are compared with experimental data from [18] for density ratios S = 0.75 and S = 0.5. The length of the potential core in our results is approximately equal to 4.5D regardless on the density ratio, while in results from [18] the potential core is shorter when density ratio decreases. The same effect was observed in [14]. However, we note that in [18] it was shown that the shortening of the potential core is very much dependent on the inlet parameters, i.e. D/θ and turbulence level. The higher was the turbulence level - the less pronounced was the influence of density ratio on the potential core length. In our computations turbulence level was relatively high and moreover the turbulence did not reveal correct space/time correlations. Probably for these reasons we did not observe the effect reported in [18, 14]. However, both in our results as well as in [18], downstream the potential core, the axial velocity decays faster when the density ratio decreases. The maxima of fluctuations increased and moved closer to the jet inlet when the density ratio decreased - this is consistent with observations made in [9] for helium jets. The results obtained for the cases with excitation are presented in Fig.(3) on the right hand side. The parameters of excitations were the same as in isothermal conditions, i.e. for axial excitations: Aa = 0.05U1 , Sta = 0.5; for axial-helical excitations: Aa = Ah = 0.05U1 , Sta = 0.5, Sta /Sth = 2.0. Unlike as in isothermal cases there were no considerable differences between the results obtained for different type of excitations. The effect of axial-helical excitation is relatively high in the mean velocity profile for S = 0.6. However, for all cases both the mean velocity profile and profile of fluctuations are rather typical for axial excitation - this suggests that helical excitation with assumed parameters plays a minor role. Knowing that the frequency corresponding to preffered mode may be different for variable density jets we performed series of computations varying the Strouhal number of axial excitations in range 0.2 − 1.0 while maintaining Aa = Ah = 0.05U1 , Sta /Sth = 2.0 constant. At the same time we performed spectral analysis of the results for the cases without excitations and we observed peaks in the range of higher values of Strouhal number [19] both for S = 0.8 and S = 0.6. Indeed, we found that increasing Sta the jet bifurcates in the same way as in isothermal conditions. Sample results for S = 0.6 and Sta = 1.0 are presented in Fig.(4) where the bifurcating jet is well seen both in 3D view and in presented cross-sections. The mean velocity profile and its fluctuating component along the jet axis are also shown in Fig.(4) in upper
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Fig. 3. Mean axial velocity and its fluctuating component along the jet axis: without excitation (left figure) and with axial and axial-helical excitations (right figure).
Fig. 4. Left and right down figures: instantaneous results for the excited jet for the density ratio 0.6: 3D view shows isosurface of Q-criterion (Q = 0.1), temperature contours in x − z cross-section through the jet axis and vorticity modulus in z − y cross-section. Right up figure: mean and fluctuating component of axial velocity along the jet axis.
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right corner. Comparing to the profile obtained for constant density bifurcating jet (see Fig.(2)) it is easy to see that in both cases profiles present qualitatively the same behavior.
4 Conclusions The results presented in this work show ability of LES method to predict correctly the constant and varying density jets with excitations added to the inlet velocity profile. The results obtained are in good agreement with experimental data for both cases: without excitations (constant and variable density jet) and with axial/axial-helical excitations (constant density jet). Comparing variable and constant density jets we found that when density of the jet was lower than the ambient fluid density the bifurcating jets existed only for excitations with higher Strouhal numbers. However, this conclusion needs experimental confirmation. The support for the research was provided within statutory funds BS-1-103-301/2004/P and the EU FAR-Wake Project No. AST4-CT-2005-012238. The authors are grateful to the TASK Computing Center in Gdansk (Poland) for access to the computing resources on Holk PC Cluster.
References 1. Cook W.C. and Riley J.J. Direct numerical simulation of a turbulent reactive plume on a parallel computer. Journal of Computational Physics, 129:263–283, 1996. 2. Crow S.C. and Champagne F.H. Orderly structure in jet turbulence. Journal of Fluid Mechanics, 48:547–691, 1971. 3. da Silva C.B. and Metais O. Vortex control of bifurcating jets: A numerical study. Physics of Fluids, 14(11):3798–3819, 2002. 4. Danaila I. and Boersma B.J. Direct numerical simulation of bifurcating jets. Physics of Fluids, 12(5):1255–1257, 1999. 5. Ducros F., Comte P., and Lesieur M. Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a flat plate. Journal of Fluid Mechanics, 326:1–36, 1996. 6. Freund J.B. and Moin P. Mixing enhancement in jet exhaust using fluidic actuators: direct numerical simulations. ASME: FEDSM98-5235, 1998. 7. Freund J.B. and Moin P. Jet mixing enhancement by high amplitude fluidic actuation. AIAA Journal, 38(10):1863, 2000. 8. Klein M., Sadiki A., and Janicka J. A digital filter based generation of inflow data for spatially developing direct numerical and large eddy simulations. Journal of Computational Physics, 186:652–665, 2003. 9. Kyle D.M. and Sreenivasan K.R. The instability and breakdown of a round variable density jet. Journal of Fluid Mechanics, 249:619–664, 1993. 10. Lee M. and Reynolds W.C. Bifurcating and blooming jets. Technical Report TF-22, Stanford University, 1985.
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11. Lele S.K. Compact finite difference with spectral-like resolution. Journal of Computational Physics, 103:16–42, 1992. 12. Majda A. and Sethian J. The derivation of the numerical solution of the equations for zero Mach number Combustion. Combust. Sci. and Tech., 42:185–205, 1985. 13. Metais O., Lesieur M., and Comte P. Transition, turbulence and combustion modelling, chapter Large-eddy simulations of incompressible and compressible turbulence. KLUWER Academic Publisher, 1999. 14. Monkewitz P.A., Bechert D.W., Bariskow B., and Lehmann B. Self-excited oscillations and mixing in a heated round jet. Journal of Fluid Mechanics, 213:611–639, 1990. 15. Monkewitz P.A. and Sohn K.D. Absolute instability in hot jets. AIAA Journal, 26(8):911–916, 1988. 16. Parekh D., Leonard A., and Reynolds W.C. Bifurcating jets at high Reynolds number. Technical Report TF-35, Stanford University, 1988. 17. Reynolds W.C., Parekh D.E., Juvet P.J.D., and Lee M.J.D. Bifurcating and blooming jets. Annual Review of Fluid Mechanics, 35:295–315, 2003. 18. Russ S. and Strykowski P.J. Turbulent structure and entrainment in heated jets: The effect of initial conditions. Physics of Fluids, 5(12):3216–3225, 1993. 19. Tyliszczak A. and Boguslawski A. Parametric study of the jet in variable density conditions. In Conference Proceedings of Symposium on Complex Effects in Large Eddy Simulations, Limassol, 2005. 20. Urbin G. and Metais O. Direct and Large Eddy Simulations II, chapter Largeeddy simulations of three-dimensional spatially developing round jets. Kluwer Academic Publishers, 1997. 21. Zaman K.B.M.Q. and Hussain A.K.M.F. Vortex pairing in a circular jet under controlled excitation. Journal of Fluid Mechanics, 101:449–491, 1980.
LES of Spatially-developing Stably Stratified Turbulent Boundary Layers Tetsuro Tamura and Kohei Mori Department of Environmental Science and Technology, Tokyo Institute of Technology, G5-7, 4259 Nagatuda, Midori-ku, Yokohama 226-8502, Japan [email protected], [email protected] Summary. In this research, we carry out LES analysis of the spatially-developing stably stratified turbulent boundary layers and investigate the streamwise variation of turbulence structures. To realize it without enormous computational burden, we employ the pseudo periodic condition using the rescale technique for the velocity field in the streamwise direction. Later for the main computational domain, the temperature field is solved and the thermal effects are introduced into the turbulent flow field. Stratification effects are examined in the wide range including the strongly stable condition. Also, through the comparison with the turbulent channel flows by the previous studies, interesting features of a spatially developing stably stratified boundary layer are clarified. Key words: LES, Thermal stratification, Spatially-developing TBL
1 Introduction The turbulence structure and transport process in thermally stratified turbulent boundary layers have not been completely clarified yet because of the complexity of flow characteristics deformed by the buoyancy effect. Over the last three decades, LES has been proved to be an effective tool to study the details of turbulence in the stratified turbulent boundary layer (e.g. Deardorff 1974, Nieuwstadt et al. 1991, Armenio and Sarkar 2002). One problem occurring in LES is how to impose properly the inflow and outflow boundary conditions of the computational domain, when the boundary layer develops downstream. Without a solution for this problem, it becomes difficult to guarantee physical meaning for solutions of the top inversion layer. Meanwhile, many studies have been carried out till now mainly in order to investigate the turbulence structures in the near-ground region of the boundary layer. The detailed research of turbulence structure in the upper part of the turbulent boundary layer is rare. We can just find Fedrovich’s work (2001), in which the shear and buoyancy effects on the spatially-developing convective turbulent
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boundary layer was studied by using the fluctuation data generated statistically as inflow condition. However we are afraid that turbulence structures in the simulated boundary layer might be influenced by the imposed inflow condition in some way. For this problem, we proposed a new model extended from Lund’s method (1998) to generate the sequential data of inflow turbulence for stratified turbulent boundary layer (Tamura et al. 2003) and carried out DNS/LES analysis of stably/unstably stratified turbulent boundary layer. As a result, it was shown that the unstably stratified boundary layer can be simulated with good accuracy by LES and the inversion layer at the top can be discussed. However, in the case of stably stratified boundary layer, turbulence structures are not so simple. Turbulence is sometimes stabilized by the numerical dissipation without the appropriate numerical discretization. So, previously only DNS with sufficient resolution can be successfully performed for the stably stratified case. Armenio and Sarkar (2002) carried out LES by using the mixed dynamic subgrid scale model for a stably stratified turbulent channel flow where they can use the periodic condition in the streamwise direction and do not need to suffer from formulating the special boundary condition. They clarify the stratification effects on turbulence characteristics and investigate the internal wave structures in the core of the channel. In this research, we firstly provide the numerical model with rescaling technique for the spatially developing stably stratified turbulent boundary layer. By imposing inflow turbulence obtained by this model, we carry out LES analysis of the spatially-developing boundary layers using the dynamic subgrid model. Turbulence structures are investigated from LES data obtained here and the occurrence of the internal wave is discussed. Through the comparison with the turbulent channel flows by Armenio and Sarkar(2002), interesting features of a spatially developing stably stratified boundary layer are clarified. Finally we estimate the turbulent Prandtl number and its relation to the local gradient Richardson number is discussed.
2 Problem Formulation 2.1 Governing Equations In this study, we carry out large-eddy simulation (LES) analysis of spatiallydeveloping stably stratified turbulent boundary layers. Under the Boussinesq approximation, the filtered governing equations consist of the Navier-Stokes equations, the continuity equation and the temperature equation for threedimensional incompressible stratified flow as follows: ∂ui uj ∂ui ∂τij 1 ∂p 1 ∂ ∂uj ∂ui + − =− + + + Riτ θδi3 ∂t ∂xj & ∂xj Reτ ∂xj ∂xj ∂xi ∂xj ∂ui =0, ∂xi
∂θ ∂uj θ ∂ + = ∂t ∂xj ∂xj
1 1 ∂θ Re τ Pr ∂xj
−
∂hj ∂xj
(1)
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where t, ui , p, θ, Reτ = uτ δ/ν, Riτ = gβΔθδ/u2τ and Pr = ν/α = 0.71 (i.e. thermally stratified air) denote time, velocity, pressure, temperature, the Reynolds number, the bulk Richardson number and the Prandtl number (uτ :friction velocity, δ:boundary layer depth, β:thermal expansion coefficient). ui = (u, v, w) are the filtered components of the velocity vector. The quantities τij and hj are the subgrid-scale (SGS) stress and heat flux, respectively as follows: τij = ui uj − ui uj ,
hj = uj θ − uj θ
(2)
2.2 Subgrid Scale Modeling In this study, we use a dynamic Smagorinsky model (Germano et al.1991). The SGS closure is performed by eddy viscosity concept (νt : turbulent viscocity, αt : turbulent thermal diffusivity) as follows: 1 2 (3) τij − δij τkk = −2νt S ij = −2CΔ S S ij 3 ∂θ 2 ∂θ = −Cθ Δ S (4) ∂xj ∂xj 1/3 where C, Cθ , Sij , Δ = Δx Δy Δz denote the model coefficients for SGS modeling of velocity and temperature fields, the strain rate tensor and grid-filter width, respectively. In the dynamic Smagorinsky model, the model coefficients C and Cθ are evaluated dynamically using the Germano identity as follows. 1 2 S δ u − u u = T − τ , T − T = −2C Δ (5) Lij = u S ij i j i j ij ij ij ij kk 3 When using the least-squares procedure (Lilly, 1967), C is given by 2 Lij Mij 2 where Mij = −2 Δ S S ij − Δ S S ij (6) C=− Mij Mij hj = −αt
Similar to the momentum equations, the coefficient Cθ is given by Cθ = −
Pj Qj Qj Qj
(7)
⎞ ∂ θ ∂θ 2 S ⎠ Qj = ⎝−Δ + Δ S ∂xj ∂xj ⎛
where
− j , Pj = θu θu j
2
(8)
The SGS Prandtl number(Prsgs ) can be estimated by the ratio of C and denote the grid and test filtered quantities respectively, Cθ . Here () and () denotes an appropriate ensemble average (here, average in the spanwise direction). In order to avoid numerical instabilities, the dynamic coefficients are set to zero when they become negative.
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2.3 Numerical Method The numerical method used is based on a fractional-step method. For time advancing, the Adams-Bashforth method for convection terms and the CrankNicolson method for diffusion terms are employed. All of the spatial derivatives are approximated by the second-order central difference. The boundary conditions are assumed to be as follows: Bottom surface: No-slip condition for velocity, Dirichlet condition for temperature, Top surface: ∂u/∂z = 0 and w = U∞ dδ ∗ /dx condition (δ ∗ is the displacement thickness) for velocity, Neumann condition for temperature, Spanwise: Periodic condition for velocity and temperature, Outflow boundary: Convective boundary condition of the form ∂/∂t + c∂/∂x = 0 is applied for velocity and temperature, where c is taken to be the bulk velocity. Inflow condition: Spatially-developed turbulent inflow data imposed.
3 Numerical Model for Inflow Turbulence
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Figure 1 shows a numerical model for the simulation of thermally stratified boundary layers. Large eddy simulation of zero-pressure-gradient spatially developing thermally stratified boundary flows is formulated utilizing the technique of the quasi-periodic boundary condition (Lund et al. 1998). In this technique the mid-section variables are rescaled according to the development ratio of the boundary layer thickness along the streamwise direction and reintroduced to the inflow. This process allows the generation of spatially developing neutral turbulent boundary layer with a small computational domain. The velocity is decomposed into a mean and a fluctuating part and applied the appropriate scaling laws to each component separately. The mean streamwise velocity is rescaled according to the law of the wall in the inner region and defect law in the outer region. The generated inflow data of velocities at the recycle station and the temperature data with the assumed profile shown in Fig.2 are introduced into the main computational region, where temperature starts to be solved taking into consideration buoyancy effects. The
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Fig. 1. Numerical model.
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driver computational region is set to be 10δ0 ∗ 3δ0 ∗ 0.5πδ0 (δ0 is the boundary layer thickness at inflow.), with corresponding grid numbers of 100, 45 and 64 in the streamwise (x), wall-normal (z) and spanwise (y) directions. The mesh widths are (Δx+ , Δz + , Δy + ) = (22, 0.6 ∼ 33, 5.5).
4 Stably Stratified Turbulent Boundary Layer 4.1 Numerical Model The main computational region is set to be 30δ0 ∗ 3δ0 ∗ 0.5πδ0 in x, z and y directions, with the same resolution as the driver region. 4.2 Streamwise Variation of Turbulence Structures In order to investigate the streamwise variation of turbulence structures, we visualized flow field by using LES data. Figure 3 illustrates the contours of vertical instantaneous velocities at different time steps for stably stratified turbulent boundary layer, together with the neutral case. We try to evaluate the Reynolds number (Reτ ) and the bulk Richardson number (Riτ ) defined by the friction velocity and boundary thickness at downstream position of 3.7δ(A), 18.1δ(B) and 28.0δ(C). These parameters at each position are as follows: (A) Reτ = 209, Riτ = 64; (B) Reτ = 161, Riτ = 108; (C) Reτ = 144, Riτ = 150. The velocity fluctuation is gradually damped by the stratification effect toward the downstream direction with increasing Riτ . Especially, the turbulence structures are weakened and almost laminarized near the most downstream position (C). In the neutral case, the turbulence is not changed downstream and fully maintained even at the most downstream position. The internal wave-like structures, which can be shown by contours with plus and minus signs repeated downstream, are recognized at relatively high position Stably stratified turbulent boundary layer at time step 1.
(A) Stably stratified turbulent boundary layer at time step 2.
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Fig. 3. Contours of vertical velocity for turbulent boundary layer.
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Tetsuro Tamura and Kohei Mori Armenio Reτ=180 R iτ=0 Armenio Reτ=180 R iτ=9 Armenio Reτ=180 R iτ=60 Armenio Reτ=180 R iτ=120 present Reτ=209 R iτ=64 present Reτ=161 R iτ=108 present Reτ=144 R iτ=155
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present Reτ=209 R iτ=64 present Reτ=161 R iτ=108 present Reτ=144 R iτ=155
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Armenio Reτ=180 R iτ=0 Armenio Reτ=180 R iτ=9 Armenio Reτ=180 R iτ=60 Armenio Reτ=180 R iτ=120 present Reτ=209 R iτ=64 present Reτ=161 R iτ=108 present Reτ=144 R iτ=155
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Fig. 6. Vertical profile of mean temperature.
(around top of boundary layer) shown by a surrounded broken line for the stably stratified boundary layer at time step 2. While, at time step 1, much more damped structures can be seen at downstream position, the internal wave cannot be clearly found anywhere. This means that the internal wavelike structures appear intermittently at top of the turbulent boundary when the boundary layer becomes thickened. So it cannot be expected that the special aspect of internal wave appears in the statistical data such as mean or rms values. Figure 4 shows the vertical profiles of the local gradient Richardson 2 number(Rig = gβ (∂ θ /∂z) / (∂ u /∂z) ). Rig increases with higher location and becomes more than 0.25 at z/δ = 0.4. This value satisfies the condition for the existence of internal wave at higher position of the boundary layer. 4.3 Comparison with Turbulent Channel Flows Next, we compare a spatially developing stably stratified turbulent boundary layer with a stably stratified turbulent channel flow. The LES data shown here for channel flows were obtained by Armenio and Sarkar(2002). Figure 5 and 6 show vertical profiles of mean values of streamwise velocity and temperature at each position, including the previous LES results of turbulent channel flows. The mean velocity at the top increases due to the buoyancy effect. Inner-region velocity is in good agreement with the previous LES data. According to Fig. 6, it can be recognized that the shape of the mean temperature profile is kept as a linear type, so there is a capping inversion part with higher temperature gradient a little below the top of the boundary layer. It is well known that the internal gravity wave can be generated by the presence of high temperature gradient in a capping inversion layer. Figure 7 displays the vertical profiles of various turbulence statistics. The streamwise velocity fluctuation decreases as the location moves downstream and stratification becomes strong. Generally the fluctuations decrease for strong stratification and especially are much more reduced in the lower region compared to the upper region. In the case of channel flow, the temperature fluctuation monotonically increases with height and shows a maximum at the center. Meanwhile, in the case of the boundary layer
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Armenio Reτ=180 Riτ=0 Armenio Reτ=180 Riτ=9 Armenio Reτ=180 Riτ=60 Armenio Reτ=180 Riτ=120 present Reτ=209 Riτ=64 present Reτ=161 Riτ=108 present Reτ=144 Riτ=155
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(a)
Armenio Reτ=180 Riτ=0 Armenio Reτ=180 Riτ=9 Armenio Reτ=180 Riτ=60 Armenio Reτ=180 Riτ=120 present Reτ=209 Riτ=64 present Reτ=161 Riτ=108 present Reτ=144 Riτ=155
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Fig. 7. Vertical profile of turbulence statistics: (a) streamwise velocity fluctuation; (b) vertical velocity fluctuation; (c) temperature fluctuation.
the temperature fluctuation has a peak at lower position which corresponds to the center of the capping inversion layer. Also, the case of channel flow shows that the peak of temperature fluctuation suddenly has jumped up at higher Riτ (=120), because the internal gravity wave occurs in the capping inversion layer. This jump can be recognized for the vertical velocity fluctuations of the channel flows. However, in the case of the spatially developing turbulent boundary layer, the jump cannot be seen clearly because the occurrence of internal waves is intermittent. 4.4 Estimation of the Turbulent Prandtl Number Finally we try to estimate the turbulent Prandtl number by analyzing the turbulence statistics obtained by LES for spatially-developing stably stratified turbulent boundary layers. Figure 8 shows the effects of the gradient Richardson number on the turbulent Prandtl number (Prt = u w (∂ θ /∂z) / θ w / (∂ u /∂z)) and the flux Richardson number (Rif = −gβ θ w / (∂ u /∂z) / u w ), with the previous numerical data (Andren 1995, Brost et al. 1978). Armenio Reτ=180 Riτ=0 Armenio Reτ=180 Riτ=9 Armenio Reτ=180 Riτ=120 Armenio Reτ=180 Riτ=240
present Reτ=162 Riτ=117 present Reτ=159 Riτ=125 present Reτ=157 Riτ=129 present Reτ=153 Riτ=137
6
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(a)
present Reτ=162 R iτ=117 present Reτ=160 R iτ=121 present Reτ=158 R iτ=125 present Reτ=157 R iτ=129 present Reτ=155 R iτ=133
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Fig. 9. Depedence of the Fig. 8. Effect of Rig : (a) the turbulent Prandtl num- SGS Prandtl number on Rig . ber; (b) the flux Richardson number.
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According to Armenio et al.(2002), both Prt and Rif have two types depending on the tendency for stratification level. In the weakly stable case, Prt is nearly equal to 1 and increases slightly with increasing Rig . In the strongly stable case, Prt is coincident with the weak case for Rig < 0.2, but for Rig > 0.2, Prt increases rapidly in a superlinear manner, indicated by Armenio(2002), with increasing Rig . However the present results show that the critical value of Rig from linear to superlinear state is not fixed and decreases for strong stratification. Figure 9 shows the dependence of the SGS Prandtl number on the gradient Richardson number. Prsgs values are scattered around 0.75 and show a slight increase for Rig .
5 Conclusions In this study, we carry out large-eddy simulation(LES) analysis of spatiallydeveloping stably stratified turbulent boundary layers by using dynamic SGS model and the pseudo periodic condition. Focusing on the turbulence statistics in the near wall region, numerical results with thermal effects are validated in comparison with the previous channel flow data. Internal wave can be intermittently recognized by the flow visualization, but statistical data do not show clearly the recovery of the fluctuating part of temperature and vertical velocity. The turbulent Prandtl number rapidly increases as Rig exceeds some value which decreases for strong stratification. The SGS Prandtl number is estimated to be approximately 0.7–1.0.
References 1. Deardorff, J.W. (1974). Three dimensional numerical study of turbulence in an entraining mixed layer,Boundary-Layer Met., 7:199–226 2. Nieuwstadt, F.T.M. et al. (1991). Large-Eddy Simulation of the Convective Boundary Layer, 8th Symp. Turbulent Shear Flows 3. Fedorovich, E. et al. (2001). Numerical and Laboratory Study of a Horizontally Evolving Convective Boundary Layer. Part 1, J. Atmos. Sci., 58:70–86 4. Lund, T.S. et al. (1998). Generation of turbulent inflow data for spatiallydeveloping boundary layer simulations , J. Comput. Phys., 140:233–258 5. Tamura, et al. (2003). LES of spatially-developing stable/unstable stratified turbulent boundary layers, DLES 5 6. Armenio, V. and Sarkar, S. (2002). An investigation of stably stratified turbulent channel flow using large-eddy simulation, J. Fluid. Mech., 459:1–42 7. Germano et al. (1991). A dynamic subgrid-scale eddy viscosity model, Phys. Fluids A, 3, 7:1760–1765 8. Andren, A. (1995). The structure of stable stratified atmospheric boundary layers, Q. J. R. Meteorol. Soc., 121:961–985 9. Brost, R.A. et al. (1978). A model study of the stably stratified planetary boundary layer, J. Atmos. Sci., 35:1427–1440
Numerical Investigation on the Formation of Streamwise Vortices in a Stably Stratified Temporal Mixing Layer Denise Maria Varella Martinez1,2 , Edith Beatriz Camano Schettini2 , and Jorge Hugo Silvestrini3 1
2
3
Funda¸c˜ ao Universidade Federal do Rio Grande - FURG, Rio grande, RS, Brasil. [email protected] Universidade Federal do Rio Grande do Sul - PPGRHSA - IPH, Av. Bento Gon¸calves, 9500 - 91501-970 - Porto Alegre, RS, Brasil. [email protected] Departamento de Engenharia Mecˆ anica e Mecatrˆ onica, Pontif´ıcia Universidade Cat´ olica do Rio Grande do Sul, Av. Ipiranga, 6681 - 90619-900 - Porto Alegre, RS, Brasil. [email protected]
1 Introduction The transition to turbulence in a stably stratified flow is a complex process with great importance for geophysical flows and engineering. The transition is controlled by the competition between the buoyancy and inertial forces. The buoyancy effects act reducing the rate of perturbation growth and delaying the transition to turbulence, while the shear supplies kinetic energy to the flow. The present work investigates the nature of the transition to turbulence in a stably stratified temporal mixing layer through direct numerical simulation (DNS). The purpose of the investigation is to analyze the effect of buoyancy on the formation of streamwise vortices, known as “ribs” vortices. Miles [10] and Howard [5], based on a linear stability, showed that for the occurrence of the Kelvin-Helmholtz (KH) instability in stratified mixing layer from an infinitely small disturbance, the Richardson number (Ri) should be less than 0.25 somewhere within the flow. This first stage of transition is due to the inflectional nature of the velocity profile [9]: the vortex sheet initially created is linearly unstable and rolls up to form the two-dimensional billows of KH. The second stage occurs due to the formation of streamwise vortices, which are developed after the saturation of the primary billows of KH. The translative instability is well known in literature as being responsible for the beginning of the three-dimensionality in the unstratified mixing layer and, consequently, for the formation of the streamwise vortices [11].
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In stratified mixing layers, the three-dimensionality process is more complex than in unstratified mixing layers. This fact is due to the greater number of secondary instabilities that are developed in the flow. The instabilities that may be developed in a three-dimensional (3D) stably stratified mixing layer should be divided in two groups: one that grows within the vortex core and the other that develops in the region between the cores (the braids). Within the cores two types of instabilities are found: the one discovered by Pierrehumbert and Widnall [11] that does not depend on buoyancy effects, and the gravitational convective instability that is driven by the buoyancy force. The gravitational convective instability makes unstable the sub layers of density generated during the roll-up of KH billows. The other instability that grows in the braid region between two Kelvin-Helmholtz vortices, was predicted theoretically by Klaassen and Peltier [6] and verified by Schowalter et al. [12] in laboratory experiments, is called secondary shear instability [1]. The gravitational convective instability and the secondary shear instability that are restricted to a stratified mixing layer, are caused by the streamwise density gradient imposed by the buoyancy force. The presence of secondary shear instability in a stratified mixing layer is due to baroclinic vorticity generation given by a source term (−g/ρ0 ∂ρ/∂x), which concentrates the vorticity in the baroclinic layer [2, 3]. How the streamwise vortices are formed in a stably stratified mixing layer and how much the stratification affects this formation are issues that still have to be explored. In this paper, we investigate the influence of the spanwise size of the domain and the initial condition in the development of the streamwise vortices.
2 Mathematical model and numerical method The temporal mixing layer with periodic conditions in the streamwise (x) and spanwise (y) directions and free-slip boundary condition in the vertical (z) direction is considered. The basic governing equations in dimensionless form for mass conservation, Navier-Stokes in the Boussinesq approximation and energy equation, in a Cartesian frame of reference Re = (0; x, y, z), are: ∇·u=0
(1)
1 2 ∂u = −∇P − ω × u − Ri ρ iz + ∇ u (2) ∂t Re 1 ∂ρ + u · ∇ρ = ∇2 ρ (3) ∂t ReP r where ρ(x, y, z, t) is density or active scalar, u(x, y, z, t) is the velocity field, P (x, y, z, t) is the modified pressure field and ω is the vorticity field. There are two dimensionless relevant parameters; the Reynolds number Re = U δi /ν based on the half velocity difference across the shear layer and on the initial vorticity thickness, defined by δi = 2U/(du/dz)max , and the Richardson
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number Ri = gΔρRδi /ρ0 U 2 (where ΔρR is density scale and R is the ratio of initial vorticity thickness and the density thickness). Here the thickness of the initial velocity profile is approximately 80% of the thickness of the initial density profile. The time is made dimensionless using the advective scale δi /U . We choose the units of length, velocity and density such that δi = 1, U = 1 and Δρ = 1/R. In this manner, Re = 1/ν and Ri = g/ρ0 . The initial profiles of velocity and density are √ √ 1 πz πRz ρ(z, t = 0) = − erf . (4) u(z, t = 0) = U erf δi R δi In the present case, no density fluctuation is superposed upon ρ(z) at t = 0. A two-dimensional sinusoidal perturbation is superimposed on the basic velocity profile. This perturbation is composed by two waves corresponding to the most amplified wave number αa and its first sub-harmonic αa /2. The most unstable associated wavelength given by linear stability theory is approximately λa = 7δi (the most amplified wave number is αa = 2π/λa being 0.889δ1− 1 [9]). These perturbations promote the development of the KelvinHelmholtz instability and vortex pairing, respectively. In such case, the side of the computational domain is taken equal to Lx = 7N δi in order to obtain N vortices in the streamwise direction. Numerical tests are done with different kinds of initial conditions for the spanwise velocity fluctuation (v ), while a forced condition for the other two components of speed fluctuation (u , w ) is considered. In particular, spanwise length effects are tested by calculating with different configurations while keeping the same streamwise and vertical dimensions. Equations (1-3) are solved numerically using a sixth-order compact finite difference scheme [8] to compute the spatial derivatives, while the integration in time is performed with a third-order low-storage Runge-Kutta method [14]. The incompressibility condition, Eq.(1), is ensured with a fractional step method via resolution of the Poisson equation for the pressure (more details about the numerical code can be found in [7, 13]).
3 Code Validation - Amplification Rate In order to validate the numerical code, the evolution of a small disturbance was considered in a two-dimensional domain. The results were compared with linear stability theory, where the disturbance is described by the TaylorGoldstein equation [4]. The computational domain used is a square of side L = 7δi . The Reynolds number is 300, the Prandtl number is 1 and the Richardson numbers tested are 0, 0.1 and 0.2. The initial amplitude of the perturbation was 10−6 U . Tests with different computational grids of nx × nz points were done (see Tab.1). As it is expected, the grid size has a great influence on the amplification rate In test GII it was noticed that the streamwise resolution affects the evolution of the wave amplitude (stratified case), when
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comparing with grid GI (see Tab.1). Thus, for the stratified case (Ri = 0.1) the grid of the test GII showed a diminution of the amplification rate due to the growth of the vertical resolution while for Ri = 0.2 there is an increase in the amplification rate. Probably this occurs because the streamwise density gradient is not being well solved. Comparison of the simulation GIII with the numerical results of [4], shown in Tab.1, gives some accord for the temporal growth rates of stably stratified mixing layers, for Ri = 0 and 0.1 while for Ri = 0.2, an even finer grid should be used.
4 Three-dimensional visualizations - streamwise vortices in a stably stratified mixing layer The computational parameters for the simulations are given in Table 2. The names in the first column refer to simulation nomenclature in accord with the initial velocity conditions. Thus, “3DF ” for three-dimensional with forced condition, “3DF R” for combined forcing-random condition. The parameter εi denotes the amplitude of the perturbation superimposed on the basic velocity profile for the fundamental (i = 1), the first sub-harmonic (i = 2) and for the spanwise mode (i = 3), respectively. The random component of the initial condition for the spanwise velocity fluctuation (v ) is composed of white noise. The Reynolds number is 200 for all numerical tests. 4.1 Short spanwise domain results Figure 1 shows the results of the simulation “3DFI” at two characteristic times and in a domain with a spanwise length (Ly ) of about 2/3 of the streamwise extension (Lx ). The choice of the spanwise length was done to force the most Table 1. Comparison of amplification rate with reference value for different grids. Ri
GI 64 × 65 0 0.1873 0.1 0.1949 0.2 0.1359
GII 64 × 129 0.1863 0.1705 0.1653
GIII 128 × 129 0.1861 0.1650 0.1388
Ref.Value Hazel,1972 0.1867 0.1594 0.1259
Table 2. Physical and numerical parameters. Simulation
Ri
3DFI 3DFII 3DFRI 3DFRII 3DFIII
0; 0.1; 0.2 0; 0.1; 0.2 0; 0.1; 0.2 0; 0.1; 0.2 0.2
Domain (Lx , Ly , Lz ) (14, 10.5, 14) (14, 14, 14) (14, 14, 14) (14, 42, 14) (14, 42, 14)
Grid Forcing nx × ny × nz (ε1 ,ε2 ,ε3 ) 128 × 96 × 129 (1%U, 0.1%U, 0.1%U ) 128 × 128 × 129 (1%U, 0.1%U, 0.1%U ) 128 × 128 × 129 (1%U, 0.1%U, random) 128 × 384 × 129 (1%U, 0.001%U, random) 128 × 384 × 129 (1%U, 0.1%U, 0.1%U )
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amplified mode in the spanwise direction, predicted by Pierrehumbert and Widnall[11]. In the unstratified case, at t = 76, two pairs of counter-rotating streamwise vortices arise. The streamwise vortices show a high degree of coherence and extend with nearly the same vortical intensity over the complete braid region. The picture of the isosufaces Q for Ri = 0.1 shows the translative instability acting on the resultant vortex of the pairing process, while for Ri = 0.2 the flow is still two-dimensional. The three-dimensionalization of the
Ri = 0
Ri = 0.1
Ri = 0.1
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Ri = 0.2
Fig. 1. Q-criterium isosurfaces for 3DFI test at Re = 200, t = 76 (above) and t = 104.5 (below).
mixing layer is dominated by the initial forcing condition of the sub-harmonic and spanwise modes. As the sub-harmonic mode grows before the spanwise mode, the formation of the streamwise vortices is delayed. The stratification also delays the pairing process and the development of the translative instability. This fact can be observed when comparing the unstratified case with the stratified one in Fig. 1. Some time later, at t = 104.5, Fig. 1 shows intense streamwise vortices for the two stratified cases considered. The streamwise vortices are now seen to be less developed over the complete braid region and are confined in a vertical length shorter than in the unstratified case. Figure 2 shows results obtained for simulation “3DFRI”, in a domain with a spanwise extension of two times the fundamental streamwise wavelength (λx = λa = 7) using a small initial white noise condition for the perturbation of the velocity field in the spanwise direction while in the other direction it is forced (Tab. 2). The pictures show that the separation between streamwise vortices is close to λx , higher than the classical value for unstratified case (2/3λx ). It is observed that the streamwise vortices are well visible just
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Ri = 0
Ri = 0.1
Ri = 0.2
Fig. 2. Q-criterium isosurfaces and spanwise cross-sectional plots (x = 6) of vorticity modulus for 3DFRI test at Re = 200, t = 114(left) and t = 162 (middle and right).
Ri = 0
Ri = 0.1
Ri = 0.2
Fig. 3. Streamwise cross-sectional plots (y = 4) of vorticity modulus for 3DFRI test at t = 19 and Re = 200.
above and below the of the vortex at Ri = 0.1. The streamwise vortices extend over/under the core of the main vortex forming the upward/downward classical mushroom-like vortices. These vortices are part of one continuous hairpin (Ri = 0 and Ri = 0.1). These hairpins continue to extend around the core, though their most visible part is the mushroom structure. The vorticity is more amplified in the braids and directly above and below the core (Fig. 2), the stretching in these regions being much stronger than within the core. This is due to the streamwise density gradient which decreases the level of vorticity in the KH billows but increases it in the braid region. This effect can be observed in Fig. 3, which is easier to be seen in highly stratified flow, Ri = 0.2, than in “moderately” stratified flow, Ri = 0.1. Thus a strong stable stratification has a stabilizing effect on the growth of the primary KH instability. This results in a reduction of the gravitational convective and translative instability. Figure 4 shows the simulation “3DFII” using a forced condition for the spanwise velocity fluctuation. The initial conditions strongly dictate the time at which the flow takes place three-dimensional, with the 2D and 3D instabilities competing for dominance. In the spanwise cross-sectional plots of Fig. 2 and Fig. 4 the effect of the initial condition for unstratified and stratified cases can be observed. If the spanwise mode is forced, the flow is more coherent than in the absence of the forced spanwise mode. Thus, major differences
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Ri = 0
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Fig. 4. Spanwise cross-sectional plots (x = 6) of vorticity modulus for 3DFII test at t = 104.5 and Re = 200.
in the formation of the streamwise vortices may be expected depending on the initial condition. In this way, the formation of the longitudinal structure changes due to the stratification and the type of the imposed condition. This is apparent from the development of the vertical extent of the mixing layer for different Ri, which is progressively suppressed with increasing stratification. The transversal length of the domain also modifies the longitudinal structure, what can be observed comparing Fig. 1 (3DFI) with Fig. 2 (3DFII). 4.2 Large spanwise domain results In test 3DFRII, the spanwise length (Ly ) of the domain is six times the fundamental streamwise wavelength. The results are show in Fig. 5. For Ri = 0 strong tubes are created by amplification of the spanwise mode and seven pairs of counter-rotating streamwise vortices are observed. In this case the wavelength associated with the streamwise vortices is appreciably smaller than the wavelength of the KH billows (λx ). For Ri = 0.1, on the other hand, a deformation in the structure of the streamwise vortices is visible. This deformation, probably caused by baroclinic effects, can be traced back to the translative instability occurring in the region between the billows of KH and not in its core, as in the unstratified case. This may indicate that other mechanisms are triggered when a large spanwise computational domain is used. Also, in Fig. 5 is observed that the separation between the streamwise vortices is higher than 1λx for Ri = 0.1. The high three-dimensional flow distortion at Ri = 0.1 is very different in relation to the cases in Fig. 1 and Fig. 2, where the classical mushroom-like vortices are observed.
t = 76
t = 95.5
t = 76
t = 95.5
t = 152.5
Fig. 5. Spanwise cross-sectional plots (x = 6) of vorticity modulus for 3DFRII test at Re = 200, Ri = 0 (above) and Ri = 0.1 (below).
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t = 95.5
t = 114.5
Fig. 6. Spanwise cross-sectional plots (x = 6) of vorticity modulus for 3DFIII test at Re = 200 and Ri = 0.2.
The formation of the longitudinal structures is attenuated by high stratification. In test 3DFIII, for Ri = 0.2, streamwise vortices appear and the separation between them is higher than 1λx . However, for combined forcingrandom initial condition the stratification suppresses the formation of the streamwise vortices and the flow behaves as a two-dimensional one. Further insight is necessary to verify if the absence of the streamwise vortices is a consequence of the stratification or of the low resolution used. The present results show that the spanwise length affects the number of streamwise vortices, which are formed in the mixing layer. In summary, for the unstratified case (Ri = 0) the results indicate that when the spanwise length of the domain is 6λa = 42 (3DFRII test) the spanwise wavelength (λy ) is close to the one predicted theoretically by the Pierrehumbert and Widnall[11](λy ∼ 2/3λx ). However, if the spanwise extension of the domain is 2λa = 14 (3DFRI and 3DFII tests) the spanwise wavelength is 1λx . For the stratified case (Ri = 0.1) the spanwise wavelength is the same as that observed for Ri = 0 if Ly = 2λa = 14 is used (3DFRI test). Otherwise, for a large-spanwise domain Ly = 6λa = 42 the spanwise wavelength is higher than 1λx , and as a consequence the number of vortex pairs is smaller than observed for the unstratified case.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Caulfield C.P., Peltier W.R. (1994) Phys. Fluids 6: 3803–3805 Caulfield C.P., Peltier W.R. (2000) J. Fluid Mech. 413: 1–47 Cortesi A.B., Yadigaroglu G., Banerjee S. (1998) Phys. Fluids 10: 1449–1473 Hazel P. (1972) J. Fluid Mech. 51: 39–61 Howard L.N. (1961) J. Fluid Mech. 10: 509–512 Klaassen G.P., Peltier W.R. (1991) J. Fluid Mech. 227 : 71–106 Lardeau S., Lamballais E., Bonnet J.P. (2002) J. Turbulence 3: 002 Lele S.K. (1992) J. Comp. Phys. 103: 16–42 Michalke A. (1964) J. Fluid Mech. 19: 693–704 Miles J.W. (1961) J. Fluid Mech. 10: 496–508 Pierrehumbert R.T., Widnall S.E. (1982) J. Fluid Mech. 114: 59–82 Showalter D.G., Van Atta C.W., Lasheras J.C. (1994) J. Fluid Mech. 281: 247– 291 13. Silvestrini J.H., Lamballais E. (2002) Int. J. Comp. Fluid Dyn. 16 : 305–314 14. Williamson J.H. (1980) J. Comp. Phys. 35: 48
Part XIII
Inflow/Initial conditions
Interfacing Stereoscopic PIV measurements to Large Eddy Simulations via Low Order Dynamical Systems L. Perret1,2 , J. Delville1 , R. Manceau1 , and J.P. Bonnet1 1
2
Laboratoire d’Etudes A´erodynamiques - UMR CNRS 6609 Universit´e de Poitiers - ENSMA, C.E.A.T., 43, route de l’A´erodrome F-86036 Poitiers - France Present address: Laboratoire de M´ecanique de Lille, UMR CNRS 8107, Cit´e scientifique, F-59655, Villeneuve d’Ascq Cedex - France. [email protected]
Summary. An original method consisting in interfacing dual-time Stereoscopic PIV measurements to a numerical code is proposed to provide inflow data to numerical simulations. It relies on the use of the Proper Orthogonal Decomposition to model the temporal dynamics of the coherent structures of the flow via a low order model and adapt the experimental mesh to the numerical one. LES of a turbulent plane mixing layer is performed to demonstrate the viability of the method.
Key words: Inflow conditions, LES, SPIV, dynamical system, mixing layer
1 Introduction When performing unsteady simulations of spatially developing flows, the prescription of correct inlet conditions is crucial to obtain a realistic dowstream development of the flow. To address this issue, several methods have been designed. The simplest solution to generate inflow data is to superimpose onto a mean flow profile random fluctuations, the level of which being of very small amplitude to let the flow evolve to a turbulent state, or adjusted to match targetted spatial or spectral distribution of energy [8]. To avoid the appearance of a transient region near the domain inlet resulting from the fact that the randomly generated fluctuations are not solutions of the filtered NavierStokes equations and that true turbulent fluctuations must be regenerated by the code, more sophisticated methods have been developed [7]. These methods consist in running auxiliary calculations to provide inflow data to the main computation [9], recycling downstream instantaneous velocity profiles as inlet conditions after a rescaling operation [10], or generating synthetic eddies, as for instance in [6, 15]. In order to provide more realistic turbulent inflow
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conditions, attemps were recently made to couple an experimental database obtained via hot-wire measurements to a Large Eddy Simulation (LES)[3]. Perret et al. [12] extended this concept of experiment-simulation interfacing to the use of databases acquired by Stereoscopic Particle Image Velocimetry (SPIV). In this study, synthetic random time series were employed to model the temporal dynamics of the flow in the measurement section. It is now of evidence [7] that inflow conditions must feature realistic temporal dynamics and account for the presence of coherent structures [3]. Thus, a method is proposed in the present paper to generate velocity fields from SPIV by modelling the dynamics of the coherent structures of the flow with a low order dynamical system (LODS). Such a model will provide the correct phase information between the velocity signals. This method consists of two main steps. Firstly, the spatial mesh of the SPIV data must be adapted to the numerical one and the generated velocity fields must satisfy the other boundary conditions of the simulation. Secondly, the temporal dynamics of the flow in the inlet section must be modelled for SPIV technique cannot offer acquisition rate compatible with the time-step constraint of the simulation. To address these two different issues, the Proper Orthogonal Decomposition (POD) is used to decompose the velocity field into spatial and temporal modes: ui (x, t) =
Ns
an (t)Φni (x)
(1)
n=1
where Φni (x) and an (t) are the n-th spatial eigenvector and temporal eigenmode respectively. Consequently, the spatial processing is directly performed on each spatial eigenvector while modelling the temporal eigenmodes enables the reproduction of the dynamics of the flow. This temporal modelling relies on the joint use of a LODS, directly derived from samples of an (t) and their temporal derivatives a˙ n (t), to model the behaviour of the large scale structures, and synthetic random time series to model higher order POD modes.
2 Flow configuration The present method to generate inflow conditions was tested in a plane subsonic turbulent mixing layer configuration. Figure 1 presents the experimental set-up. The mean velocity was Um = 29.5 m/s and the velocity ratio 0.67. Measurements were performed in a cross-section normal to the mean flow, located at x0 = 300 mm downstream of the trailing-edge. At this location, the Reynolds number based on Um and on the vorticity thickness δω0 = 18.7 mm is Re = Um δω0 /ν 36,000. To measure both the velocity fields and their temporal derivatives, a dual-time SPIV (DT-SPIV) arrangement composed of two synchronized in time SPIV systems was implemented. Such a set-up provides pairs of velocity fields measured at the same location, the two velocity fields within a pair being separated in time by an adjustable time-delay τ .
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Double pulsed Nd-YagLaser B Mirror x0
Light sheets
Ua CCD Cameras Ub
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Fig. 1. Experimental set up.
Thus, temporal derivatives can be computed by using a second-order finite difference scheme between time t and t + τ . Ns = 2,000 pairs of velocity fields were acquired at a rate of 1 Hz, τ being 80μs. More details on the test-flow and the experimental set-up can be found in Perret [11].
3 Coupling method 3.1 Proper Orthogonal Decomposition In the present work, the implemented POD technique is the version proposed by Sirovich [16], called the snapshot POD, adapted to a database composed of a limited number of samples of velocity fields that are well-resolved in space. The main features of this decomposition are recalled here. Eigenmodes an (t) result from the integral problem: C(t, t )an (t )dt = λn an (t) (2) T
where C(t, t ) is the two-time correlation tensor, and λn the n-th eigenvalue associated to the n-th eigenmode. By definition, eigenmodes an (t) are uncorrelated in time. Moreover, in the present study, they are rescaled so that an (t), am (t) = λn δnm , where , is the ensemble average operator. Spatial eigenvectors Φn (x) are computed by: Φni (x) = an (t)ui (x, t)dt (3) T
The spatial domain considered here being the y − z plane, Φn (x) are functions of y and z: Φn (y, z), i = 1, 2, 3.
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3.2 Modelling of the temporal dynamics In the present study, temporal evolution of the velocity fluctuations are modelled through the POD eigenmodes an (t). To ensure that the generated velocity fields present the correct amount of energy, all the POD modes are taken into account. First POD modes representing the large scales structures are modelled by a LODS while higher order modes corresponding to the finer scales are modelled by synthetic time-series [12]. The method of the LODS derivation is briefly presented here, readers being referred to Perret et al [13] for more details. It relies on the postulate that the dynamics of the temporal POD coefficients an (t) is driven by a set of ordinary differential equations (ODE’s) of the same nature as those that can be obtained via a POD-Galerkin approach, namely polynomial ODE’s of order at most cubic: a˙ i = Di + Lij aj + Qijk aj ak + Cijkl aj ak al
(4)
with implicit summation over repeated indices, i ∈ [1 : Ntr ] and 1 ≤ i ≤ j ≤ k ≤ l ≤ Ntr , Ntr being the number of POD modes retained to develop the LODS. In this approach, Di , Lij , Qijk and Cijkl are the unknown parameters that must be indentified. From the DT-SPIV database, Ns pairs of POD coefficients {an (t), an (t + τ )} can be computed by solving the eigenvalue problem (2). Thus, time derivatives a˙ n (t) of POD coefficients an (t) can be computed by finite difference between two samples separated by a time-delay τ . To keep the samples of a˙ n and an simultaneous, temporal coefficients are evaluated as an (t + τ /2) = (an (t + τ ) + an (t))/2 [13]. From the Ns samples of both an (t) and a˙ n (t), the LODS parameters D, L, Q and C can be estimated by a least square approach. Once these parameters obtained, a 4-th order Runge-Kutta algorithm is used to integrate in time equation (4) to compute the temporal evolution of modelled eigenmodes a ˜n (t). Due to the influence of the SPIV measurement noise, only Ntr = 12 POD modes are retained to develop the LODS, representing almost 40% of the total kinetic turbulent energy in the whole section. This appears to be sufficient to model the large scale structures of the mixing layer [11]. Higher order POD modes (n > Ntr ), representing the remaining 60% of the energy, are modelled by random time series bn (t) to which a spectral transfer function is applied in order to enforce a realistic spectral energy repartition (a modified Von Karman spectrum as proposed by Pao [5]) and the condition bn (t), bn (t) = λn [12]. 3.3 Spatial adaptation Spatial adaptation of the measured velocity fields to the mesh of the simulation is directly performed on the spatial eigenvectors Φn (y, z) to take into account the simulation requirements, namely periodic boundary condition in the homogeneous spanwise direction and free-slip condition on the upper and
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lower horizontal boudaries of the computational domain. In the z-direction, extrapolating the velocity fields and imposing a periodic condition is processed via the selective deconvolution algorithm proposed by Franke [4]. Based on the method proposed by Druault and Delville [2], a one-dimensional POD approach is then performed to extrapolate each Φn (y, z) in the y-direction. ˜n (y, z) are obtained on the numerical Hence, extrapolated spatial modes Φ mesh. These two steps of the spatial processing are described in details in Perret et al [12]. Eventually, the velocity fluctuations are reconstructed in the whole inlet section by: u ˜i (y, z, t) =
Ntr
˜n (y, z) + a ˜n Φ i
n=1
Ns
bn Φ˜ni (y, z)
(5)
n=Ntr +1
4 Inflow data characteristics The synthetized distribution of Reynolds stresses (figure 2) across the shearlayer compares well with the original experimental data, satisfying the anisotropy of the flow and the shear-stress level. Spectra of the modelled field (figure 3) show that the combination of a LODS and synthetic random time series accounts for the spectral content of both the large scale structures (corresponding to a Strouhal number f δω0 /Um 0.3) and the finer scales at higher frequencies contrary to purely random eigenmodes as in [12]. The proposed technique, based on the proper orthogonal decomposition of the velocity field and modelling techniques satisfying statistics of the temporal POD coefficients an (t), enables the generation of data that satisfy also the
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Fig. 4. Two-point correlation Rii (y = 0, y ) (left) and Rii (y = 0, δz ) (right) of the original and the modelled velocity fields obtained at y = 0.
two-point correlation tensor of the original flow (figures 4). Both the level and the shape of the correlation, in both direction y and z, are correctly reproduced.
5 LES results Generated data were utilized as inflow conditions to perform a LES of a plane mixing-layer with the unstructured collocated finite volume solver Code Saturne developed at EDF [1] using 8 processors of a PC cluster. The Piomelli and Liu [14] subgrid scale model is used to solve the filtered NavierStokes equations. Space discretization is based on a collocation of all the variables at the center of gravity of the cells, using central differencing. Time discretization is based on a second order Crank-Nicolson/Adams-Bashforth method. The velocity/pressure system is solved by a SIMPLEC algorithm, with the Rhie and Chow interpolation. The Poisson equation is solved using a conjugate gradient method, with diagonal preconditioning. The Reynolds number of the simulation, based on the parameters of the flow in the inlet section, is the same as the experiment. The spatial domain corresponds to Lx × Ly × Lz = 30 δω0 × 10 δω0 × 11 δω0 and the number of points used is Nx × Ny × Nz = 112 × 53 × 65.
Interfacing SPIV measurements to LES via low order dynamical system
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Fig. 5. Left: Longitudinal evolution of the vorticity thickness δω ; — experimental results; ◦: present study; ×: Perret et al [12]. Right: Longitudinal evolution of the turbulent kinetic energy k normalized by the value obtained experimentally in the self-similarity region: : experimental results; ◦: present study; ×: Perret et al [12].
The growth rate of the simulated mean flow (figure 5 left) is found to be in good agreement with experimental results. Moreover, compared to the results obtained with only random temporal POD eigenmodes [12], the use of a LODS to model the dynamics of the most energetic structures improves slightly the prediction of the growth rate of the shear layer. Streamwise evolution of the turbulent kinetic energy k = 12 (u 2 + v 2 + w 2 ) (figure 5 right) shows good agreement with that obtained via hot-wire measurements. Despite the presence of a region where a decrease of k can be seen, the part of the velocity fields corresponding to the LODS modelling improves the prediction of the level of energy and then reduces the length of the region where the code must regenerate realistic turbulence. These results confirm the importance of the realistic spatio-temporal dynamics of the large scale structures for the correct development of the simulated flow. Furthermore, even if differences still exist between the simulation and the experimental data, the obtained improvement demonstrates the viability of the proposed approach.
6 Conclusion A novel method to generate inflow conditions for LES of spatially developing flow has been presented. It relies on the interfacing of experimental data acquired by dual-time SPIV to a computational code. Based on the POD, the proposed method allows us to perform separately the spatial adaptation of the experimental mesh to the numerical one and the temporal modelling. A low order dynamical system is derived from the experimental database to model the temporal behaviour of the 12 first POD modes, thus reproducing the dynamics of the coherent structures. Higher order modes are randomly generated. Consequently, the present technique enables the generation of inflow data that respect one- and two-point spatial statistics in the whole section, realistic turbulent spectra as well as realistic spatio-temporal dynamics of the coherent structures of the original flow. LES of a plane mixing layer performed
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with such inflow data has demonstrated the viability of such an approach and the improvement due to the coherent dynamics of the inflow velocity field compared to the use of purely random temporal modes.
References 1. Archambeau F., Mehitoua N., and Sakiz M. (2004) Code Saturne: a Finite Volume Code for the Computation of Turbulent Incompressible flows Int. J. Finite Volumes, Electronical edition: http://averoes.math.univ-paris13.fr/html 2. Druault P. and Delville J. (2000) Representation of the spatial correlation tensor of the velocity in free turbulent flows C.R. Acad. Sci. Paris, S´erie IIb 328: 135– 141 3. Druault P., Lardeau S., Bonnet J. P., Coiffet F., Delville J., Lamballais E., Largeau J. F., and Perret L. (2004) A Methodology for the Generation of Realistic 3D Turbulent Unsteady Inlet Conditions for LES AIAA J. 42: 447–456 4. Franke U. (1987) Selective deconvolution: a new approach to extrapolation and spectral analysis of discrete signals Int. Conf. on Acoustics, Speech Signal Processing IEEE May, pp. 30.3.1-30.3.4 5. Hinze J. O. (1975) Turbulence, 2nd ed, Mc Graw-Hill, New York 6. Jarrin N., Benhamadouche S., Addad Y. and Laurence D. (2003) Synthetic turbulent inflow conditions for Large-Eddy Simulation Proc. 4th Int. Conf. Turb., Heat and Mass Transfer, to appear in Progress in Computational Fluid Dynamics (2005) 7. Keating A., Piomelli U., Balaras E., and Kaltenbach H.-J. (2004) A priori and a posteriori tests of inflow conditions for large-eddy simulation Phys. Fluids 16: 4696–4712 8. Lee S., Lele S. K., Moin P. (1992) Simulation of spatially evolving turbulence and the applicability of Taylor’s hypothesis in compressible flow. Phys. Fluids 4: 1521–1530 9. Li N., Balaras E., and Piomelli U. (2000) Inflow conditions for large-eddy simulations of mixing layers Phys. Fluids 12: 935–938 10. Lund T. S., Wu X., and Squires K. D. (1998) Generation of turbulent inflow data for spatially-developing boundary layer simulations J. Comput. Phys. 140: 233–258 11. Perret L. (2004) Etude du couplage instationnaire calculs-exp´erience en ´ecoulements turbulents Ph.D. Thesis, Poitiers University, France. 12. Perret L., Delville J., Manceau R. and Bonnet J. P. (2005) Generation of turbulent inflow conditions for Large Eddy Simulation from Stereoscopic PIV measurements. 4th International Symposium on Turbulent Shear Flow Phenomena 13. Perret L., Collin E. and Delville J. Polynomial identification of POD based low-order dynamical system. to appear in J. of Turbulence 14. Piomelli U. and Liu J. (1995) Large-eddy simulation of rotating channel flows using a localized dynamic model Phys. Fluids 7: 839–848 15. Sandham N. D., Yao Y. F. and Lawal A. A. (2003) Large-eddy simulation of transonic turbulent flow over a bump Int. J. of Heat and Fluid Flow 24: 584-595 16. Sirovich L. (1987) Turbulence and the dynamics of coherent structures. Part I: coherent structures Quarterly of Applied Mathematics, XLV 3: 561–571
LES of background fluctuations interacting with periodically incoming wakes in a turbine cascade Jan Wissink and Wolfgang Rodi Institute for Hydromechanics, University of Karlsruhe, Kaiserstrasse 12, D-76128 Karlsruhe, Germany. [email protected], [email protected] Summary. Large-Eddy Simulations (LES) of flow in a turbine cascade with and without background fluctuations and incoming wakes have been performed. The results allow a qualitative study of the influence of background turbulence on both periodically impinging wakes and on boundary layer transition. It is found that the interaction of uniformly distributed small-scale background fluctuations with wakes is quite strong, especially as the wakes travel through the passage between blades. Here, background turbulence hinders the production of kinetic energy at the apex of the wake. As a consequence, the impingement of the apex of the wakes on the downstream half of the suction side becomes weaker, which shows up, for instance, as a slight local decrease in the phase-averaged friction velocity.
1 Introduction Generally, three-dimensional flows in turbine cascades are very complex and call for powerful computational tools and models. Because of their computational efficiency and recent improvements in turbulence modelling, Unsteady Reynolds-Averaged Navier-Stokes (URANS) solvers are likely to remain the most popular industrial tool for the simulation of realistic flows in turbine and compressor stages in the foreseeable future [2]. The aerodynamic performance of a turbine blade is largely affected by disturbances carried by incoming wakes generated by the preceding row of blades. The effects of impinging wakes on the blades’ boundary layer are very complex and call for advanced modeling strategies to describe the state of the boundary layer [7]. The streamwise pressure gradient along the suction side of a turbine blade changes from favourable in a large region, somewhat downstream of the leading edge, to adverse in a region immediately upstream of the trailing edge. The favourable streamwise pressure gradient stabilizes the boundary layer, making it very difficult for impinging disturbances to penetrate, while the
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adverse pressure gradient destabilizes the boundary layer such that impinging wakes can easily trigger transition. To study the effect of periodically impinging wakes on the boundary layer flow along turbine blades, several experiments [5, 6, 11, 12] have been performed in the past. Recently, because of advances in computational power, Large Eddy Simulations (LES) and Direct Numerical Simulations (DNS) of such flows in Low-Pressure Turbine (LPT) cascades have become feasible [4, 8, 9, 13, 15]. The main goals of both experiments and simulations are to gain insight into physical mechanisms that play a role in the transition to turbulence of the blades’ boundary layer and to provide data for the development of new models to be used in URANS simulations. Several simulations have been performed with periodically incoming wakes superposed on a steady, uniform flow field [8, 9, 13, 15], while Kalitzin et al. [4] also report first DNS results of flow in a LPT cascade with uniformly distributed free-stream fluctuations superposed on a steady inflow. Especially along the downstream half of the suction side, it is found that impinging wakes are generally far more effective in triggering boundary layer instabilities than small-scale free-stream disturbances. Though - compared to the background fluctuations - the wake carries relatively strong disturbances, it is also subjected to severe stretching as it is wrapped around the leading edge of the blade. Due to this stretching, fluctuations carried by the wake are damped such that the impinging wake only very mildly triggers boundary layer fluctuations immediately downstream of the leading edge. While the wake travels through the passage between blades, it is deformed by the mean flow. At the apex of the wake a significant quantity of fluctuating kinetic energy is produced as the axis of the wake aligns with the direction of compression [10]. When these strong fluctuations impinge on the downstream half of the suction side they usually trigger boundary layer transition far more effectively than small-scale, uniformly distributed free-stream disturbances. The precise interaction of uniformly distributed small-scale free-stream disturbances - also called background fluctuations, which can be found in realistic flows in turbine cascades - with periodically incoming wakes inside the passage between blades is not well understood. Therefore it was decided to perform LES to study this subject. The results obtained are reported in the present paper. 1.1 Computational details Large-eddy simulations of flow in a turbine cascade in the presence of periodically incoming wakes combined with and without uniformly distributed background fluctuations have been performed. The set-up of the simulations is largely in accordance with experiments performed by Liu and Rodi [5, 6]. In the experiments, the cylinders, that generate the wakes, are mounted on a squirrel cage device. The diameter of the cylinders is 4mm, while the cage diameter to cylinder diameter ratio is 650 : 4. Because of this large ratio, there is a significant decay and merging of wakes generated by the upward
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moving cylinders on the upstream half of the squirrel cage. As a result, these wakes only generate background turbulence. The pitch between the blades, mounted in a linear cascade, is 149.5mm, which is rather small compared to the diameter of the squirrel cage. It was therefore decided to ignore the curvature of the cage in the LES and assume that the cylinders move downwards in a vertical plane at a distance of 100mm from the leading edge of the blades. The blades have a chord-length of 230mm and span the 750 mm test section, such that the mean flow at midspan can be regarded as being two-dimensional (2D). The axial chord-lenght of the blades is 150mm and their stagger angle is 40.45o . The inlet angle is 0o and the outlet angle is approximately 69o . The tangential speed of the wake-generating cylinders, finally, is Ucyl = 5.1m/s. Figure 1 shows a (x, y)-slice through the computational domain (a) and the corresponding computational mesh (b) employed in the LES. The Reynolds number of the flow problem, based on the mean inflow velocity U0 and the axial chord-length L, see Figure 1a, is Re = 72 000. Periodic boundary conditions are employed in the y-direction for x/L < 0 and x/L > 1, while on the surface of the blade no-slip boundary conditions are prescribed. The size Location of inlet plane (b)
(a) 1
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Fig. 1. (a): Geometry of the flow problem, (b): computational mesh showing every tenth gridline in the (x, y)-plane.
of the spanwise direction, where periodic boundary conditions are employed, is lz = 0.20L. At the outflow plane a convective boundary condition is prescribed. Instead of using cylinders to generate wakes, at the inflow plane, x/L = −0.5, artificial turbulent wakes and - only in Simulation A - background fluctuations (u , v , w ) are introduced superposed on the steady flow field (u, v, w) = U0 (1, 0, 0). The wake data have been kindly made available by Xiaohua Wu and Paul Durbin of Stanford University (see also Wu and Durbin [15]). The artificial wake data correspond to wakes generated by a row of cylinders, located in the plane x/L = −0.67L, which move in the negative y-direction with speed Ucyl = 1.36U0 . The diameter of the cylinders is
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d = 0.02681L, while the distance between the cylinders and the inlet plane is 6.25d. The pitch between blades equals the axial chord-length L, while the distance between the inlet-wake generating cylinders, Dcyl , is 12 L. The freestream disturbances, used to model the background fluctuations, stem from a separate LES of isotropic turbulence and were made available by Jochen Fr¨ ohlich of the University of Karlsruhe. A description of its properties can be found in Wissink and Rodi [14]. Two simulations, listed in Table 1, were performed. Note that only in Simulation A grid turbulence is added at the inlet. In both simulations, the Table 1. Overview of the Large Eddy Simulations and corresponding experiments, WHW is the Wake Half-Width and MVD is its Mean Velocity-Deficit. T umin is the minimum turbulence level of the phase-averaged flow in the plane x/L = −0.20. Sim. Grid A B
614 × 294 × 64 614 × 294 × 64
WHW
MVD
Dcyl
T umin
0.029L 0.029L
30% 30%
1 L 2 1 L 2
3.4% 0.8%
period of the flow, P , was determined by the time for the cylinders, shown in Figure 1a, to perform a vertical sweep over a distance 12 L, that is P = 1 2 L/(1.36U0 ) = 0.3676L/U0 . The Turbulence level, T u, is defined by 5 Tu =
1 uu φ + vv φ + ww φ , 3 u 2φ + v 2φ
(1)
where uu φ , vv φ and ww φ are the phase-averaged velocity fluctuations. Since - because of spanwise homogeneity - w φ will tend to zero when the sampling time of the phase-averaged statistics goes to infinity, it is omitted from the equation. Note that in Simulation B, owing to diffusion of fluctuations carried by the wakes, at x/L = −0.20 a minimum turbulence level of T umin = 0.8% was obtained even though no uniformly distributed background fluctuations were added at the inlet. To resolve the flow, a mesh containing 11.55 × 106 grid points is employed. The mesh, illustrated in Figure 1(b), is optimised using experience gained in earlier DNS and LES of periodic unsteady flow in a T106A cascade [8, 13] and provides a fair resolution of both the suction side boundary layer and the pressure side boundary layer. The distance between the wall and the wallnearest grid-point in wall units, Δy + , is mostly bounded between 2 and 4 and the cell-sizes Δx+ and Δz + are mostly between 40−60 and 10−20 wall units, respectively. The simulations were performed using a second-order accurate central finite-volume discretization of the three-dimensional, incompressible NavierStokes equations in space, combined with a three-stage Runge-Kutta method
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for the time-integration. In both simulations, the sub-grid-scale stresses were modelled using the dynamical Smagorinsky approach of Germano et al. [1]. The calculations were performed on the Hitachi SR8000-F1 in Stuttgart using 64 processors. After letting the flow develop for 10 periods, phase-averaging was performed for 15 further periods. In the phase-averaging procedure, each period was subdivided into 64 equal phases, φ. In the phase-averaging procedure, the statistical quantities were averaged in the homogeneous spanwise direction. For each of the phases 15 spanwise-averaged (x, y)-fields containing statistics were stored.
2 Results Figure 2 shows a comparison between both simulations of the evolution of the phase-averaged kinetic energy at two phases φ = 0 and 4/8. As also observed by Wu and Durbin [15], incoming wakes in a turbine cascade are passively convected by the mean flow. As in the T106 cascade [8, 13, 15], the wake clearly bends as it reaches the passage between blades. Along the suction side the wake wraps around the leading edge and impinges downstream at virtually zero angle of attack. At the same time, a considerable amount of fluctuating kinetic energy is produced at the apex of the wake. The same mechanism, explained by Rogers [10] (case C), was previously observed in the T106 cascade by Wu and Durbin [15] and Wissink [13]. As shown in the latter paper, the same wake is able to impinge at two locations simultaneously on the suction side boundary layer: The most upstream impingement by the part of the wake that wraps around the region close to the leading edge is quite weak. Further downstream, for s/s0 = 0.4, the lower part of the apex of the wake with its strong fluctuations impinges and triggers strong fluctuations in the suction side boundary layer. For s/s0 > 0.65 the streamwise pressure gradient turns adverse, which makes the suction side boundary layer more unstable such that it eventually becomes mildly turbulent just upstream of the trailing edge. Along the pressure side, the wakes mostly impinge at a non-zero angle of attack. This angle of attack gradually decreases until it is virtually zero immediately upstream of the trailing edge. The production of k at the apex of the deformed wake in the passage between blades appears to be larger in Simulation B than in Simulation A. A further quantification is given below. Figure 3 compares the (maximum) magnitude of k at the apex of the wake for Simulations A & B, respectively, as a function of the streamwise location of this apex. In Simulation B, the increase of k is considerably larger than the increase found in Simulation A. In both simulations a maximum is obtained at an apex location of x/L = 0.8. In Simulation A this maximum k is 0.049, while in Simulation B it is 0.079. As witnessed in Figure 2, the phase-averaged kinetic energy level inside the wake, upstream of the leading edge, is virtually the same in both simulations. Hence, it can be concluded that the background fluctuations that are uniformly distributed at the inlet in
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Fig. 2. Development of the phase-averaged kinetic energy, k , in the vane of the turbine cascade. Left: Simulation A, Right: Simulation B. k has been made dimensionless using U 2 .
max. k in apex wake
0.1
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Fig. 3. Maximum phase-averaged fluctuating kinetic energy obtained inside the apex of the wake as it travels through the passage between blades as a function of the axial coordinate. The markers correspond to the 16 phases φ = 0, 1/8, ΣΣΣ, 16/8, 17/8 where the maximums are determined.
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Simulation A, hinder the production of kinetic energy inside the apex of the wake. Most likely, the background fluctuations partly destroy the coherence of the wake by the breakdown of its large-scale vortical structures, leading to an increased dissipation. Figure 4 shows the phase-averaged friction velocity, uτ φ for φ = 0, 1/8, . . . , 7/8 along both the suction side and the pressure side of the blade. In the suction side plot, the thick, dashed line corresponds to the approximate path of the wake as it impinges on the blade’s boundary layer. The path of the wake is easily identified by locally increased levels of uτ φ , which are caused by the triggering of boundary layer fluctuations by the impinging wake. Compared to Simulation A, at most phases the wakes from Simulation B are found to be more effective in increasing the friction velocity upon impingement. Somewhat upstream of the location of impingement, however, the explicitly added background fluctuations in Simulation A perpetually slightly disturb the boundary layer flow and hence induce a slightly higher uτ φ -level. The fact that uτ φ is found to remain positive for all phases indicates that the suction side boundary layer always remains attached. Along the pressure side, the levels of uτ φ obtained in Simulations A & B are virtually identical. For all phases the same pattern is observed: After the initial decline upstream of s/s0 = 0.15, uτ φ gradually increases downstream of s/s0 = 0.15 up until the trailing edge. The main reason for this gradual increase in the uτ φ -level is the presence of accelerating streamwise flow along the pressure side boundary, induced by the favourable pressure gradient in this region, which also casuses the entire pressure side boundary layer to remain laminar. 0.6
0.6 Simulation A Simulation B
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Fig. 4. Simulations A, B: Comparison of the phase-averaged friction velocity uτ φ . The thick dashed line corresponds to the approximate path of the wakes.
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3 Conclusions Large eddy simulations of flow around a MTU turbine blade with incoming wakes and with and without background turbulence have been performed. It was shown that in Simulation A - with incoming wakes and background fluctuations - the impingement of the uniformly distributed background turbulence on the suction side of the blade is very weak compared to the impingement of the lower part of the apex of the wake. Though the direct effect of the freestream disturbances on the suction side boundary layer transition is negligible, the free-stream disturbances do hinder the production of kinetic energy inside the apex of the wake as large-scale vortical structures present in the wakes are largely destroyed. A direct comparison of the maximum levels of phaseaveraged kinetic energy at the bow-apex of the wake as it travels through the passage between blades shows that k in Simulation B - without background fluctuations - reaches a maximum that is more than 50% larger than the maximum obtained in Simulation A. Hence, in Simulation A the impingement of the lower part of the apex of the wake on the downstream half of the suction side boundary layer is weaker than the corresponding impingement in Simulation B.
Acknowledgements The authors would like to thank the Steering Committee of the Computing Centre in Stuttgart for granting computing time on the Hitachi SR8000-F1.
References 1. Germano M., Piomelli U., Moin P., Cabot W.H. (1991) Physics of Fluids A 3(7):1760-1765. 2. Hummel F. (2002) ASME J. of Turbomachinery, 124:69-76 3. Jeong J., Hussain F. (1995) J. Fluid. Mech. 285:69-94 4. Kalitzin G., Wu X., Durbin P.A. (2003) Int. J. of Heat and Fluid Flow 24:636645. 5. Liu X., Rodi W. (1994) Exp. in Fluids 17:45-58. 6. Liu X., Rodi W. (1994) Exp. in Fluids 17:171-178. 7. Mayle R.E. (1991) ASME Paper 91-GT-261. 8. Michelassi V., Wissink J.G., Fr¨ ohlich J., Rodi W. (2003) AIAA J. 41(11):21432156. 9. Raverdy B., Mary I., Sagaut P., Liamis N. (2003) AIAA J. 41(3):390-397. 10. Rogers M.M. (2002) J. Fluid Mech. 463:53-120. 11. Schulte W., Hodson H.P. (1998) ASME J. of Turbomachinery 120:28-35. 12. Stadtm¨ uller P., Fottner L. (2001) ASME Paper 2001-GT-311. 13. Wissink J.G. (2003) Int. J. of Heat and Fluid Flow 24:626-635. 14. Wissink J.G., Rodi W. (2004) DNS of a laminar separation bubble affected by free-stream disturbances. In: Proceedings of the workshop Direct and LargeEddy Simulation V. 15. Wu X., Durbin P.A. (2001) J. Fluid Mech. 446:199-228.
Large Eddy Simulations Of Spatially Developing Flows Using Inlet Conditions Dimokratis G.E. Grigoriadis NCSR ”DEMOKRITOS” Research Reactor Lab Institute of Nuclear Technology & Radiation Protection Aghia Paraskevi Attikis 15310, Athens, Greece [email protected] Summary. A synchronous inflow condition generation technique for Large Eddy Simulations of incompressible turbulent flows is presented and validated. The method is a modification of the inflow method of Lund et al. [1]. A scaling approach is used to extract only the fluctuating component of the boundary layer (BL hereafter) further downstream which is subsequently reintroduced at the domain’s inlet. The technique has been extensively validated using DNS [2, 3] and experimental [4] data for : a) equilibrium BL at zero pressure gradient for Reθ = 300 − 1410, b) separated BL under the influence of an adverse to favorable pressure gradient due to suction-blowing and c) the flow over a wall mounted rectangular obstacle with line passive scalar injection inside the wake of the obstacle according to the experimental study of Vincont et al.
1 Introduction The generation of realistic turbulent inflow conditions is a crucial aspect of large eddy and direct simulations of turbulent flows. Fully developed turbulent flows are normally treated by imposing periodic boundary conditions in the streamwise direction thus eliminating the need to specify inlet conditions. In space developing flows however one should supply the inlet of the domain with a field which should not be far from a realistic turbulent flow, both in terms of mean properties and the energy content of various scales coexisting in the flow. That translates to the consistent prescription of a turbulent inflow condition that provides realistic time dependent flow properties at the domain’s inlet section for each computational step. Spalart [2, 5] performed DNS computations of a BL using suitably computed sink terms to account for the BL growth. Alternatively, the simplest and least accurate approach consists of the random perturbation [6] or triggering [7] of the oncoming BL. However a purely random inlet signal severely lacks in spatial and temporal coherency of the entering turbulent structures and development sections as large as 50 BL thicknesses δ are needed before the flow stabilises to a realistic BL flow [1]. In most simulated cases the extra energy associated to the superposition is rapidly
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dissipated only a few BL thicknesses downstream [8, 9]. A more accurate variant of that approach was presented by Le & Moin [10] who used a random inlet method such that a prescribed space and time energy spectra were satisfied at the inlet. However their method requires a priori knowledge of time and space frequency spectra for the simulated flow. Therefore the majority of the researchers follow the auxiliary or parallel simulation approaches [11, 12] where another simulation is used to sequentially feed the inlet of the domain. The simulation generating the inlet profiles can be performed either before the main simulation or simultaneously i.e. in an asynchronous or a synchronous mode. In that case, the main disadvantage is the associated additional computing cost of the parallel simulation. That can be a restricting parameter especially in single node computations where due to simultaneous execution the computational effort of the parallel simulation limits the resources for the main simulation. Additionally, in asynchronous mode the two simulations have to be matched in terms of domain dimensions and time stepping and one has to account for the physical disk space storage. In the present work a synchronous inlet generation technique is presented which does not imply significant additional computing power or memory costs and can be easily incorporated in existing flow solvers.
2 Inflow generation The inflow generation presented does only requires the mean inflow profile as an input which extends its predictive capability to flow cases were the exact turbulent characteristics of the simulated inflow are not known a priori. The method is similar to the modified Spalart approach of Lund et al. [1] and it is based on the self preserving extend of the BL. The coordinate transformation of Spalart [2] is used which recovers streamwise homogeneity along wall normal coordinate lines η such that, z m (0.001z + z m ) + z m (z/δ) 5.0 with m = (1) η= 2 z2m + z m log 10 (z3 /z1 ) where z1+ = 15, z2 = (z1 z3 )1/2 and z3 = 0.3δ. The basic idea is the utilisation of multiple scales to minimise inhomogeneity i.e. using the viscous scale ν/uτ close to the wall, and the outer scale z/δ in the outer region. The above coordinate transformation is applied only at two planes perpendicular to the flow direction namely at the inlet and recycle stations. The velocity profile from a recycle position is first decomposed into a mean and a fluctuating part. The transformation (1) is then applied and only the fluctuating part of the velocity components from the recycle station is re-introduced accordingly at the inlet, while the inlet mean flow properties remain unaltered during the simulation. In doing so the dynamic characteristics of the inlet BL such as physical, displacement and momentum thicknesses are fixed at the inlet of the domain through the definition of the inlet profile. Additionally, since only the mean part of the velocity field is needed as an input, one can easily simulate and compare with other numerical or experimental studies, with prerequisite mean inflow, allowing the flow to self-adjust to a realistic BL at the inlet. Starting from a recycle station at a distance xrec from the domain’s inlet, the fluctuating part ui of each velocity component Ui is estimated,
LES Of Spatially Developing Flows ui (xrec , y, z) = Ui (xrec , y, z)− < Ui > (xrec , z)
619 (2)
where < Ui > (xrec , z) is the space and time averaged velocity component at the recycling station. Following Lund et al. [1], such space and time averaged quantities < Φs > are computed at each computational step n + 1 according to, < Φs >n+1 =
Δt n Δt Φs + 1 − Ts Ts
< Φs >n
(3) n
which was also used for the average values of uτ , δ, δ ∗ , and θ 1 . The terms Φs correspond to the space averaged values along the homogeneous direction y. The produced spaced averaged velocity profile at the recycle station and the mean inlet velocity profile, are then subjected to a coordinate transformation from z to η coordinates (1). The final velocity field at the inlet is then simply the sum of the mean inlet field and the reintroduced fluctuations part,
Ui (xinl , y, η) =< Ui > (xinl , η) +
uτ,inl uτ,rec
ui (xrec , y, η)
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where the mean inlet velocity < Ui > (xinl , η) is the predefined input. The fluctuating component ui (xrec , y, η) is produced by simple second order interpolations so that it is added at the same η location at the inlet.
3 Numerical method The full set of the Navier-Stokes equations are solved for an incompressible fluid, using a fractional step approach [13] with pressure correction. Details of the numerical procedures can be found in [9, 14]. Due to the utilisation of the immersed boundary concept for the inclusion of complicated geometrical boundaries, the solution of the derived Poisson’s equation is achieved by a direct pressure solver. Time advancement was based on the Adams-Bashforth scheme, with a variable time step which is dynamically computed according to the convection (CF L) and viscous time scale (V SL) criteria: CF L < 0.2 and V SL < 0.05. The resulting computer code has excellent parallel efficiency, requires 120M b of physical memory per million nodes and reaches performances of 0.9μsec/node/iter on personal computers (WinXP, [email protected]).
4 Results For all the cases presented in the following sections periodic boundary conditions have been used along the homogeneous direction y. The initial field for all simulations was a uniform field produced from the inlet profile on which divergence-free random 1
The characteristic time scale Ts in (3) adjusts the contribution of each realisation to the overall average and it was set equal to Ts = 12 (xrec /U∞ ). Increasing this time unit significantly was found to lead to delayed statistical stationarity. In practice, using half or double of that time scale value lead to insignificant changes in the simulation’s quality.
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perturbations of intensity 2%−3% of the local value have been imposed. Calculations have been performed with both the Smagorinsky model with Cs = 0.1, and the Filtered Structure Function (FSF) model [16].
4.1 Flat plate BL at zero pressure gradient at Reθ = 300 − 1410 A series of flat plate BL test cases were first computed using a computational domain equal to (Lx , Ly , Lz ) = (25δinl , πδinl /2, 3δinl ) for the Reθ = 300 case and (20δinl , πδinl /2, 3δinl ) for Reθ = 670 and 1410 cases, where δinl corresponds to the inlet BL thickness. The mean inflow profile consisted of the DNS data of Spalart [2] for each Reθ case. The recycling station was defined at the middle of each computational domain. Due to the available numerical resolution the near wall dynamics were not computed explicitly and a modified Werner & Wengle [15] approximate wall boundary condition [9] was used for all the simulations presented here. Fig. 1 shows the predicted mean and fluctuation field for the three computed cases. The agreement with the reference data is excellent for the lower Reθ cases and slightly deteriorates for lower numerical resolutions or higher Reθ . In all cases the mean streamwise velocity is computed accurately and differences can be only found very close to the wall surface for the lower resolutions. It should be noted that the effect of the recycle station proved to have a negligible effect on the simulation’s quality as long as the station was located in the interval xrec = 30% − 90% Lx .
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4.2 Flat plate BL under an adverse to favorable pressure gradient The validation of the method for flows under the influence of mean pressure gradients was based on the DNS study of Na & Moin [3, 17] who promoted separation by imposing a suction-blowing profile at the upper surface of the domain as shown in Fig. 2. LES computations were performed with respect to the physical momentum thickness δ, on a computational domain with dimensions (Li + Lx , Ly , Lz ) = (80.0δinl , 9.88δinl , 12.65δinl ). An inlet section Li = 10δinl was added just before the domain where an inlet profile from the DNS data of Spalart at Reθ = 300 was imposed. Approximate wall boundary conditions have been used along the wall surface.
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The predicted size of the recirculation region extended from x ≈ 40δinl to x ≈ 60δinl and the mean location of detachment-reattachment were correctly captured along the streamwise direction. Slight deviations can only be observed along the normal direction of the recirculation area where coarser grids lead to an overestimated size of the bubble in the normal direction and increased turbulent kinetic energy in that region (Fig. 3). For higher resolutions though, both the size and the magnitude of turbulent fluctuations where found in very good agreement with the reference data.
4.3 Flow over an obstacle with line passive scalar injection The last flow case used to evaluate the inflow procedure concerned the flow over a wall mounted two dimensional square obstacle. A passive scalar was line-injected inside its wake according to the experimental study of Vincont et al. [4] as shown in Fig. 4. The Reh number based on the obstacle’s dimension h and the free stream velocity U∞ was 740. The oncoming BL had a thickness δ = 7h while the Schmidt number of the injected dye was Sc = 2500. LES calculations were performed on a computational domain (Li +Lx , Ly , Lz ) = (5δinl + 4δinl , 3δinl , 3δinl ). Preliminary tests with coarser grids revealed that a satisfactory development of the oncoming BL could be realised even if such a marginally short inlet section of Li = 5δinl was used. With the flow parameters considered this inlet section extended to x+ ≈ 3320, while the recycle station was located at x+ ≈ 2730 i.e. at a distance x = 6.0h before the obstacle. Approximate wall boundary conditions were used along the solid surfaces [9] and the mean velocity profile from the DNS of Spalart at Reθ = 670 was prescribed at the inlet. Fig. 5 shows the recirculation patterns formed before and after the obstacle. A large recirculation region of length 7.2h is clearly formed, and a secondary smaller one just after the body. The size of the latter is clearly determined by fluid injection in the wake. Simulations performed without injection lead to a larger secondary bubble of length 2h instead of 1.0h. The extend and the statistical turbulent characteristics of the wake are also correctly captured as shown in Fig. 6.
Fig. 4. Computational domain for the flow over an obstacle. A passive scalar is injected normally through a slot of width h/7 at a distance h after the body with an injection speed Ud /U∞ = 4%.
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5 Conclusions Analysis of the examined cases suggests that very short development sections were adequate for the production of a realistic turbulent inflow while the location of the recycling station was found to have a negligible effect on the simulation’s quality. The characteristics of the incoming BL and the produced location and size of the recirculation regions were also found in very good agreement with the reference data. The computed cases suggest that the inflow method is capable of generating realistic inlet conditions appropriate for inflow-outflow type LES of nominally two dimensional space developing flows.
Acknowledgements The author is grateful to Dr. G. Hoffman, Dr. E. Simons and Dr. E. Balaras for fruitful discussions, helpful suggestions and support. Financial support by the European Commission and the Hellenic Ministry of National Education and Religious Affairs under the program ”Pythagoras II” is also greatly appreciated.
References 1. Lund T.S., Wu X., Squires K.D. (1998) J. of Computational Physics 140(2):233– 258. 2. Spalart P.R. (1988) J. of Fluid Mechanics 187:61–98.
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3. Na Y., Moin P. (1998) J. of Fluid Mechanics 374:379–405. 4. Vincont J.Y., Simoens S., Ayrault M., Wallace J.M. (2000) J. of Fluid Mechanics 424:127–167. 5. Spalart P.R. (1986) J. of Fluid Mechanics 172:307–328. 6. Neto A.S., Grand D., M´etais O., Lesieur M. (1993) J. of Fluid Mechanics 256:1– 25. 7. Voke P.R., Gao S., Leslie D. (1995) Int. J. for Numerical Methods in Engineering 38:489–507. 8. Keating A., Piomelli U., Balaras E., Kaltenback H.J. (2004) Physics Of Fluids A 16(12):4696–4712. 9. Grigoriadis D.G.E., Bartzis J.G., Goulas A. (2004) Computers and Fluids 33(2):201–222. 10. Le H., Moin P. (1994) Direct numerical simulation of turbulent flow over a backward-facing step. Technical Report TF-58, Thermoscience division, Stanford University, Stanford, California. 11. Simons E. (2000) Efficient multi-domain approach to Large Eddy Simulation of incompressible turbulent flows in complex geometries. Phd Thesis, The Von Karman Institute for Fluid Mechanics, Brussels, Belgium. 12. Balaras E., Piomelli U., Wallace M. (2000) Physics of Fluids 12:935–938. 13. Le H., Moin P. (1991) J. of Computational Physics 92:369–79. 14. Grigoriadis D.G.E., Bartzis J.G., Goulas A. (2003) Int. J. for Numerical Methods in Fluids 41(6):615–632. 15. Werner H., Wengle H. (1993) Large-eddy simulation of turbulent flow over and around a cube in a plate channel. In 8th Symp. on Turb. Shear Flows. 16. Ducros F., Comte P., Lesieur M. (1996) J. of Fluid Mechanics 326:1–36. 17. Na Y., Moin P. (1998) J. of Fluid Mechanics 377:347–373.
Impact of Initial Flow Parameters on a Temporal Mixing Layer Evolution M. Fathali1 , J. Meyers1 , C. Lacor2 , and M. Baelmans1 1
2
Department of Mechanical Engineering, Katholieke Universiteit Leuven [email protected] Department of Fluid Mechanics, V.U.B., Pleinlaan 2, 1050 Brussel, Belgium.
Summary. This paper aims at determining the most influential initial flow field parameters on the evolution of turbulent mixing phenomena. To this end a stochastic method is developed to generate a divergence-free random velocity field with a prescribed energy distribution in physical and wave-number space. In addition, predetermined integral length scales are established. Eight direct numerical simulations of a temporally evolving turbulent mixing layer are examined in detail. Simulation results show a large disparity in mean and instantaneous turbulent quantities, mainly effected by the energy spectrum, the integral length scale and the divergence freeness of the initial field.
1 Introduction The specification of suitable inlet boundary conditions for mixing layers and jets is of crucial importance to achieve reliable behaviour of their downstream flow evolution. A computationally demanding solution is to foresee a turbulent flow simulation of the upstream components in order to obtain realistic conditions at the initial development of a shear layer. Of course, less time consuming methods are favorable as long as the main features of the upstream flow are captured. This task is hampered by the fact that turbulent mixing phenomena are, due to their chaotic behavior, very sensitive to small changes in operating conditions, as was shown both in experimental and numerical studies [1, 2, 3]. In order to construct appropriate but compact boundary conditions, the effective parameters should be distinctly identified. Various investigations have already been carried out to reveal the influence of different parameters in inflow conditions, such as turbulent kinetic energy level, type of turbulence, computational domain size and the mean flow profile shape. However the impact of the energy spectrum, the turbulent integral time and length-scales and the flow-field kinematic parameters including, e.g., the divergence freeness
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of the velocity field are not yet fully explored. The objective of the present study is to complement this previous work by establishing possible sensitivity of the following inflow characteristics: 1 –the shape of the velocity spectrum, 2—the distribution of the velocity fluctuations in physical space, 3—the total turbulent kinetic energy, 4— the integral length scale and 5—the divergence of the initial velocity field ∇ · u which must be zero if the constructed inflow or initial conditions are to satisfy the incompressible continuity condition. Furthermore, the turbulent flow structures initially excited with a continuous spectrum are compared with those resulting from the most unstable modes of the linearized Navier-Stokes equations. The effects of the above-mentioned parameters are studied in detail by performing different direct numerical simulations (DNS) of a temporally evolving turbulent mixing layer. Hence, the inlet boundary condition in a spatial framework is transformed—using a Taylor hypothesis—into an initial condition in the temporal setting.
2 Computational setup and initial conditions In the current research, the compressible Navier-Stokes equations are solved for a temporal mixing layer at Mach 0.2 and a Reynolds number of 50 based on half the vorticity thickness δ0 /2 and half the velocity difference ΔU . The computational box is a cube with side L, which corresponds to four times the wave length of the most unstable mode with periodic boundary condition in streamwise and spanwise directions. Convective and viscous fluxes are discretized on a uniform mesh with 963 grid cells using a second-order accurate finite-volume discretization. Computations are initialized with a tangent hyperbolic mean velocity profile on which velocity perturbations are superimposed and with constant initial pressure and temperature. Velocity perturbations are generated either based on Linear Stability Theory (LST) or on a stochastic method. In the first method, the velocity perturbations are unstable eigenmodes of the linearized Navier-Stokes equations. The stochastic method generates a divergence-free random velocity in a box with two homogenous directions. This field has a prescribed two-dimensional energy spectrum in the homogeneous planes and a prescribed energy distribution in the non-homogenous direction. It can be further processed to generate a divergence-free velocity field with prescribed 2L , integral length scale [4] by applying a gaussian filter with filter width Δ = √ π where L is the integral length scale. Table 1 displays the main properties of different initial conditions used for the simulations. As a reference case a divergence free velocity field with a white noise spectrum is chosen with a medium energy level (Eref ) of 2, 230.3ΔU 2 and a medium variance of the gaussian energy distribution in normal direction (σref ) of 5.8692θ0 . To investigate their influence, their values are respectively
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doubled for the cases Ehigh σref and Eref σhigh . Case LST and S are performed to address the effects of the initial energy spectrum shape. Hereby the first one represents a discrete spectrum with energy in the first nine most unstable modes and the latter one a -5/3rd power spectrum. Furthermore, cases Eref Llow and Eref Lhigh allow to investigate the effects of respectively lower and higher values for the initial integral length scale. These test cases have initially constant isotropic integral length scales through the computational box. Finally, simulation D examines the effects of the initial divergence freeness of the velocity field with respect to the reference case. Test Energy σ Spectrum ∇ · V Integral case E(k) Length scale Eref σref ref. ref. constant 0 – Ehigh σref high ref. constant 0 – Eref σhigh ref. high constant 0 – LST ref. – – ≈0 – 5 S ref. ref. Ak− 3 0 – Eref Llow ref. – – 0 low Eref Lhigh ref. – – 0 high D ref. ref. constant = 0 – Table 1. Properties of the random-perturbed initial fields used for the mixing layer
3 Results and discussion In this section, the effects of initial-condition parameters on the evolution the momentum thickness are first qualitatively assessed. It is shown that the amount of energy contained in the large scales is a crucial parameter. Subsequently, in order to quantitatively assess the effects, the instantaneous coherent structures are studied. Hereby large differences for the different cases are detected, especially after the first roll-up. 3.1 Momentum thickness evolution Figure 1b presents the evolution of the momentum thickness θ defined by +∞ < ρux > < ρux > + 1 dz, (1) <ρ> 1− θ= <ρ> <ρ> −∞ where, throughout this paper, < . > denotes an averaging over the plane of homogeneity. After an initial transition period, an approximately linear growth region, characterizing a self-similar behavior of the turbulent mixing layer, is observed in all cases. Furthermore, the figure clearly reveals that both
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the transition period as well as the linear growth rate are largely affected by the different initial condition parameters. A large effect of the initial energy spectrum is retrieved when comparing test cases LST and S with Eref σref . It can be seen that the more energy contained in the large scales, the shorter the transition period is at one hand and the faster the linear growth of the momentum thickness is on the other. Also the effect of different integral length scales can be understood in this context (cases Eref Lhigh and Eref Llow ). Indeed, the evolutions of the momentum thicknesses for both cases are very close to the one with the -5/3rd spectrum (case S). However a larger integral length scale slightly fastens the development of the mixing layer. This can again be explained by the fact that larger integral length scales result in more energy in the larger scales. The effect of the distribution of the velocity fluctuations in physical space can be observed by considering the test case Eref σhigh . As can be seen in figure 1a,b, this parameter has a minor effect in comparison with other parameters: more concentrated energy in physical space only slightly shortens the transition period without effecting the linear growth rate. Case Ehigh σref reveals that higher initial turbulent kinetic energy clearly produces faster transients, but does not change the linear growth rate of the momentum thickness. Finally, comparison of the non-divergence free test case D with the reference case Eref σref shows that the non-divergence free velocity field gives rise to a shorter transition. On the other hand, the linear growth after transition is almost the same for both cases. It can be concluded that all investigated parameters do affect the transition period. For the linear growth rate, it is mainly the initial energy spectrum that has a major influence. 3.2 Rollup and pairing time In this section different aspects of the flow field evolution are examined at first, second, and third rollup time. Formation times for 4, 2 and 1 rollers are listed in table 2 for the different cases based on the criteria proposed by Moser & Rogers [3]. From figure 1a it can be seen that in all simulations, except for the LST case, four eddies appear at approximately the same momentum thickness. On the other hand, figure 1b shows that the momentum thickness at which the two rollers appear after the first pairing varies somewhat more for the different simulations. In particular it can be seen that case D shows a smaller and LST and Eref Lhigh show a higher momentum thickness in comparison to the reference simulation. Moreover, after second pairing, it can be seen from figure 1b that the disparity in momentum thickness for the different cases is remained in the further development of the mixing layer. Figures 1c and d show the two-dimensional energy spectrum at the mixing layer centerline at appearance times of four and two eddies. It can be seen that the effects of the initial conditions diminish faster in the small scales than in the large ones.
Impact of Initial Parameters on a Temporal Mixing Layer case LST Eref σref Ehigh σref Eref σhigh 4 eddies 20 40 30 40 2 eddies 40 95 85 95 1 eddies 80 150 140 150
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Figure 2 presents the contour plots of the instantaneous spanwise vorticity ωy , after first pairing. Using this visualization, the simulations can be classified into two categories, i.e.: 1—the LST , Eref σhigh , Eref σref and Ehigh σref cases; and 2—the Eref Lhigh , Eref Llow , S and D simulations. The first class of simulations shows well-pronounced spanwise rollers, whereas the second group shows more chaotic structures with significantly twisted rollers. Spanwise velocity variance is a good measure of the three-dimensionality of the flow. It is shown in figure 3. The lower level of < v v > for LST supports the observation of well-organized coherent structures in this case, whereas the three-dimensional structure of the large-scale eddies can be related to the high level of < v v > for cases S and D only.
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To give a concise overview of the main initial condition effects on the evolution of the coherent structures, we will next address the effects of different parameters. First, the effect of the initial shape is studied by comparing cases LST and S with Eref σref in figures 2 a,c,e, 3, and table 2. It can be observed that the uniform spectrum ‘hinders’ the pairing times. Furthermore, the 5/3rd spectrum gives rise to more chaotic structures, whereas LST generates more organized coherent structures. The effects of different integral length scales
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are revealed by comparing the Eref Lhigh , Eref Llow cases with Eref σref with initially zero integral length scale. It can be observed that initial non zero length scales fasten the pairing process, whereas the pairing time does not show sensitivity to the variation of this parameter. Also, in contrast to the reference case Eref Lhigh and Eref Llow generate more chaotic coherent structures and Kelvin-Helmholtz rollers probably do not emerge due to the early presence of coherent three-dimensional eddies [1]. Next, an increase in initial total energy, as investigated by case Ehigh σref , seems to fasten the pairing process without influencing the coherent structures and three-dimensionality of the flow (see figures 2 b,c, and 3). Furthermore, the spatial distribution of energy (case Eref σhigh ) in figures 2d, and 3 does not show an important influence neither on pairing time and coherent structures nor on the threedimensionality of the flow. Finally, from case D in figures 2f one can observe that an initial non-divergence-free velocity field fastens the pairing process and produces less organized coherent structures, whereas it only slightly enhances the three-dimensionality of the flow. Comparison of the divergence of the velocity field for this case with some other cases in figure 4 reveals that ∇ · u only converges to a realistic value in between its first and second pairing (approx. at tΔU/δ ≈ 80). To summarize the above observations: coherent structures and their formation time are largely affected by the initial condition under investigation, except for the spatial energy distribution. On the other hand, the presence of well-pronounced two-dimensional rollers is largely affected by the initial shape of the energy spectrum, the divergence freeness of the initial field and the integral length scale.
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4 Conclusion In the present paper, a stochastic method for the generation of turbulent initial condition is presented and used to examine DNS of a temporally evolving, turbulent mixing layer. The initial conditions for these simulations are designed to study the following effects on the flow evolution: 1—the energy spectrum, 2—the spatial energy distribution, 3—the total turbulent kinetic energy, 4— the integral length scale and 5—the divergence of the initial velocity field ∇ · u. A transition period followed by a linear growth region for the momentum thickness evolution is observed for all cases. The transition period is found to be sensitive to all investigated parameters. However, the linear growth rate is mainly affected by the initial shape of the energy spectrum and the imposed integral length scale. The Kelvin-Helmholtz instability generates large vortical structures in most cases. Their formation time and the presence of well-pronounced 2-dimensional rollers is mainly affected by the initial shape of the spectrum and the presence of integral length scales, and the divergence of the velocity field. As such the conclusions of this work confirm experimental and theoretical evidences that the initial or inlet conditions have a profound effect on the development of turbulent mixing phenomena. Therefore, in order to reconstruct these conditions in numerical simulations without altering the main characteristics of the mixing layer development, at least the key parameters as listed above should be incorporated in the numerical procedure for initial or inlet condition.
5 Acknowledgement This research was performed in the framework of FWO-project G.0130.02. The authors acknowledge the financial support by FWO-Vlaanderen.
References 1. E. Balaras, U. Piomelli, and J. M. Wallace. Self-similar states in turbulent mixing layers. Journal of Fluid Mechanics, 446:1–24, 2001. 2. A. K. M. F. Hussain and M. F. Zedan. Effects of the initial condition on the axisymmetric free shear layer: Effects of the initial momentum thickness. Physics of Fluids, 21(7):1100–1112, 1978. 3. R. D. Moser and M. M. Rogers. The three-dimensional evolution of a plane mixing layer: pairing and transition to turbulence. Journal of Fluid Mechanics, 247:275–320, 1993. 4. M. Klein and A. Sadiki and J. Janicka. A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations. Journal of Computational Physics,186:652–665, 2003.
Part XIV
Separated/Reattached Flows
A Comparative Study of the Turbulent Flow Over a Periodic Arrangement of Smoothly Contoured Hills Michael Breuer, Benoit Jaffr´ezic, Nikolaus Peller, Michael Manhart, Jochen Fr¨ ohlich, Christoph Hinterberger, Wolfgang Rodi, Ganbo Deng, Oussama Chikhaoui, Sanjin uSari´c, and Suad Jakirli´c Large–Eddy Simulation of Complex Flows – A French-German Research Group [email protected]/[email protected] Summary. The paper presents DNS and LES predictions for the flow over smoothly contoured constrictions in a plane channel, a demanding benchmark case for separated flows. Reynolds numbers in the range 700 ≤ Re ≤ 10, 595 are investigated in order to assess the dependence of the flow on this parameter. New interesting features of the flow were found. Additionally, different codes and discretization schemes, curvilinear and Cartesian, are compared focussing on simulations with about 1 Mio. grid points. The effect of grid refinement is studied as well as the impact of wall modeling at the upper plane wall on the flow. These investigations are performed using a new highly resolved LES prediction with about 13 Mio. grid points. Key words: DNS, LES, periodic hill flow, benchmark case, separated flow
1 Introduction The French-German research group ’Large-Eddy Simulation of Complex Flows’ has been established to improve the current state-of-the-art of large-eddy simulation (LES) in complex flow situations such as separation and reattachment at high Reynolds numbers. As one of the common test beds, the flow over a periodic arrangement of smoothly contoured hills placed in a channel proposed by Mellen et al. [10] has been selected (see Fig. 1). This case is especially useful for basic investigations of the performance of turbulence models – not only subgrid-scale but also statistical models – because of its well defined flow state which is independent of inflow conditions. In the near future this configuration will be investigated experimentally at TU Munich. Length and height of the channel are Lx = 9.0 h and Ly = 3.036 h, respectively, where h denotes the hill height. The flow is assumed to be periodic in the streamwise direction. Based on the investigations in [10] a spanwise extension of the computational domain of Lz = 4.5 h is used in all computations presented.
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A number of comparative studies for this flow case have already been performed, including statistical models [8, 9] and LES [12, 13, 6]. For LES the influence of grid resolution, SGS models and wall models has been assessed with two different curvilinear codes at a Reynolds number of Re = UB h/ν = 10, 595 based on the hill height h, the bulk velocity UB taken at the crest of the first hill, and the viscosity ν. The current work presents a new, extremely fine resolved LES [4] for this Reynolds number using a grid even finer than employed in the highly resolved simulations by [6]. That allowed to resolve both, + ≈ 1.2) and the lower wall (y + ≤ 0.45 except at the windward side: ymax + the upper wall (y ≤ 0.95) of the channel. Furthermore, an extensive study based on direct numerical simulations (DNS) and LES was carried out in the range 700 ≤ Re ≤ 5600 allowing to assess the Reynolds number dependence of the flow. Also, different codes, numerical schemes and physical models are compared for the same Reynolds number. Re Geometry 10,595 BJ, DC, HF, SJ
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DC = Deng & Chikhaoui HF = Hinterberger, Fröhlich & Rodi BJ = Breuer & Jaffrezic PM = Peller & Manhart SJ = Saric & Jakirlic TL = Temmerman & Leschziner [12]
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Fig. 1. Overview over the simulations discussed in the present paper.
2 Simulations performed In order to assess the Reynolds number effect on the flow, simulations at five different Reynolds numbers are performed at Re = 700, 1400, 2800, 5600, and 10, 595. The Reynolds numbers Re = 2800 and 5600 are investigated by DNS using a second-order finite-volume code (MGLET) based on a Cartesian grid in which the geometry of the hill is represented by an immersed boundary method [11]. The grid resolution of these two DNS was selected carefully according to criteria based on the Kolmogorov scale in the inner flow and on the wall shear stress. This leads to numbers of grid cells of 48 × 106 and 233 × 106 , respectively. Furthermore, at the three lowest Reynolds numbers up to Re = 2800 additional DNS predictions were performed with the curvilinear finite-volume code LESOCC [2, 3] using 12.4 × 106 grid cells. At the two larger Reynolds
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numbers (5600 and 10, 595) well-resolved LES were carried out with the same code and grid. In contrast to the simulations published in [6] these new simulations resolve not only the lower wall (the hills) in more detail but also resolve the upper wall, thus being able to establish the influence of the resolution of the upper wall on the results. To reduce statistical errors due to insufficient sampling to a reasonable minimum, the data were averaged over about 140 flow-through times and in the spanwise direction. Thus the prediction at Re = 10, 595 represents a new reference data set for the hill flow test case [4]. Additionally, LES predictions on coarser grids (about 1.0 × 106 ) under a variety of different conditions were carried out using LESOCC. A similar code, LESOCC2, developed in [7] was used to perform a coarsegrid simulation with about 1.15 × 106 grid points on a different grid than employed for the previous computations with this code in [13]. At Re = 10, 595, several coarse-grid LES predictions were also performed with a second curvilinear collocated code (ISIS) using the WALE model [5]. Fig. 1 shows an overview of the present simulations.
3 Results 3.1 Reynolds Number Effects First, the influence of the Reynolds number is investigated by taking the most accurate predictions into account. These are on the one hand the two LES predictions (Re = 5600 and 10, 595) based on the dynamic Smagorinsky model performed on the very fine curvilinear grid using LESOCC. Because the grid was designed for the high-Re case, it is also sufficient for the low-Re cases. Especially, at Re = 2800, 1400 and 700 the grid resolution is appropriate for DNS which are also considered here. On the other hand the two DNS predictions carried out on the extremely fine Cartesian grids using MGLET are taken into account. Fig. 2(a) displays the distribution of the wall shear stress on the lower wall for all Reynolds numbers, and Fig. 2(b) shows a direct comparison of the results of both codes at two Re. Based on this comparison of the DNS and the well-resolved LES predictions, we are not only able to show the Reynolds number effect on the results, but also to validate the new reference solution at Re = 10, 595. Comparing the wall shear stresses (Fig. 2(b)) shows on the one hand an excellent agreement between the results of both codes at Re = 2800 and Re = 5600 except at the windward side of the hill. On the other hand the dependence of the wall shear stress on the Reynolds number can be clearly appreciated in Fig. 2(a). Based on this series of simulations, some peculiar features of the flow field become evident. With increasing Re the main recirculation region first becomes larger with a maximum at Re = 2800 and then gets smaller again. The corresponding values of the reattachment lengths are xreat /h = 5.14, 5.19, 5.41, 5.09, and 4.69 for Re = 700 − 10, 595, respectively. A similar behavior of the separation length with increasing Re has been observed for the backward-facing step flow by Armaly et al. [1]. The undulations
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Fig. 2. (a) Comparison of wall shear stress at five different Reynolds numbers between Re = 700 and 10, 595; (b) comparison of the results of two codes.
in the wall shear stress at the beginning of the main recirculation bubble seem to be geometry related and not affected by Reynolds number effects. There is a small recirculation zone just before the hill starts at x/h ≈ 7.0, which forms in all simulations. At the hill crest, just before the main separation, a small recirculation zone forms at the highest Re. It is unlikely that this is an artifact because a clear trend of the wall shear stress with increasing Reynolds number definitely supports this observation. It should also be noted that right on the hill crest the boundary is flat. The peak value of the wall shear stress appearing shortly before the hill crest strongly increases with decreasing Re. Fig. 3 shows exemplarily the distribution of the mean velocity U/UB , the Reynolds shear stress u v /UB2 as well as the turbulent kinetic energy k/UB2 . The left column displays the profiles at x/h = 2 which is located at the beginning of the flat floor and hence within the main recirculation region, whereas the right column shows the same quantities at x/h = 6 located behind the recirculation bubble. The results predicted at different Re using the same code (LESOCC) are compared. It is visible that on both walls the velocity gradients of U increase with increasing Re and owing to a fixed mass flow rate the maximum U velocity is decreasing. Furthermore, the back flow velocity increases with increasing Re and overall the recirculation bubble becomes slightly thinner. A clear trend is also obvious for V (not shown here). At x/h = 2 the peak value of u v appearing in the free shear layer is approximately independent of Re. However, with increasing Re the position of the maximum is shifted closer to the lower wall which is in accordance with the thinner recirculation region. The shear layer broadens with increasing Re. That is also obvious in the distribution of the turbulent kinetic energy k at x/h = 2. Furthermore, both peak values of k , in the vicinity of the upper wall and in the free shear layer, increase with increasing Re and move closer to the walls. However, the maximum of k only changes about 14% in the entire Re range considered. Behind the recirculation bubble at x/h = 6 the situation is different. Here
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Fig. 3. Comparison of mean velocity profiles U/UB , Reynolds shear stresses 2 2 and turbulent kinetic energy k/UB at five different Reynolds numbers
u v /UB between Re = 700 and 10, 595 at x/h = 2 (left) and x/h = 6 (right).
the peak values of u v increase strongly with decreasing Re. Contrarily to x/h = 2 the peak values of k at x/h = 6 also increase with decreasing Re. 3.2 Investigations for Re = 10, 595 The results in the previous section clearly show that the LES prediction on the fine curvilinear grid [4] can serve as a new reference case. Compared to the prediction by Temmerman et al. [13] the total number of grid points is about three times larger which allows to resolve the upper wall, too. More important is, however, that the resolution at the hill crest was improved, e.g., Δxcrest /h = 0.026 vs. 0.032 and Δycrest /h = 2.0 × 10−3 vs. 3.3 × 10−3 .
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Fig. 4 shows a direct comparison of both LES predictions at x/h = 2. Regarding the time-averaged velocity profiles both results are in close agreement. Small deviations are visible near the upper wall since this region was modeled with a wall function by Temmerman et al. [13]. The same applies to the Reynolds stresses which agree well. The trend is confirmed at other locations in the flow field except in those regions where the wall model was applied in [13]. For comparison two other results obtained with LESOCC on a coarse grid with about 1 Mio. grid points using the no-slip boundary condition and either the fixed parameter (Cs = 0.1) or the dynamic Smagrinsky model are included in Fig. 4. Regarding the mean velocity the agreement is excellent. With respect to the Reynolds shear stress the peak value is either slightly overpredicted or underpredicted, respectively. Additionally, Fig. 5 depicts three other results in comparison to the new reference solution. The first is an LES prediction using MGLET, a Cartesian grid with about 3.8 × 106 points and a dynamic Smagorinsky model. The second represents LES results based on ISIS using a coarse curvilinear grid with about 1.45 × 106 points and the WALE SGS model [5]. Finally, the third is also based on a curvilinear grid of comparable size (1.15 × 106 points) using the Smagorinsky model with Cs = 0.065 and LESOCC2. Additionally, in this last case a wall function based on profiles from DNS data is applied [7]. The grid size for MGLET was chosen in order to guarantee approximately the same computational effort than spent for the curvilinear codes. The deviations observed for the mean axial velocity U are small except for MGLET. Evaluating the shear stress u v , differences of the same order of magnitude as visible in Fig. 4 are obvious. A more critical position (not shown here) for the comparison of the Reynolds stresses is at x/h = 0.5 located shortly after the separation line and through the strong shear layer. Compared with the reference solution larger deviations can be observed here for all three cases, especially in the free shear layer. Generally, the Reynolds stresses are overpredicted by the simulations using coarser grid resolutions. Note that for the immersed boundary method used in MGLET it is by construction more difficult to refine the grid near a curved wall. Thus, the grid resolution of 3.8 × 106 points corresponds to a lower near-wall resolution compared to the curvilinear grids using about 1 Mio. points. Using finer Cartesian grids (e.g. 23.5 × 106 points, see Fig. 1) for LES which are primarily improved in wallnormal direction thus leading to larger aspect ratios of the cells was found to be a critical issue for the immersed boundary technique. The aspect ratio has to be carefully chosen when generating Cartesian grids for LES predictions.
4 Conclusions The paper presents a large number of different simulations for the turbulent flow over a periodic arrangement of hills [10] at varying Reynolds numbers. They support the physical interpretations of an earlier study [6] at
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Fig. 4. Comparison of mean velocity profiles U/UB and Reynolds shear stresses 2 at Re = 10, 595 and x/h = 2 using different methods and grids.
u v /UB
Fig. 5. Comparison of mean velocity profiles U/UB and Reynolds shear stresses 2 at Re = 10, 595 and x/h = 2 for different methods at coarse grids.
u v /UB
Re = 10, 595 and shed new light on the flow by a variation of the Reynolds number. In particular, the existence of a tiny recirculation at the foot of the windward face of the hill was confirmed for Re = 10, 595 and a recirculation on the hill crest was found which has not been discussed before. This was possible due to a new LES prediction with increased resolution [4] supported by a series of DNS at lower Reynolds numbers. Overall the flow does not strongly change within the Re range investigated. Nevertheless, clear trends in the distributions of the mean velocities, Reynolds stresses and the wall shear stresses were found. The comparison of results obtained by different codes, models and grid resolutions at Re = 10, 595 reveals the following conclusions: (i) as long as the grid resolution is finer than 1−2 wall units based on the maximum wall shear stress, results with the standard Smagorinsky, the dynamic Smagorinsky and the WALE model are comparable; (ii) thus with about 1 Mio. CVs, reasonable results can be achieved for that configuration and Reynolds number; (iii) if the wall region is poorly resolved (y + ≥ 5), then the wall model has a strong influence on the results (partially results not shown in the paper); (iv) if Cartesian grids combined with the immersed boundary technique are
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applied special diligence has to be dispensed in choosing the aspect ratio of the cells. Acknowledgements: The present research is supported by DFG and CNRS through the French-German programme ’LES for complex flows’ (FOR 507). The authors thank CIRM, Marseille for its hospitality during CEMRACS 2005.
References 1. Armaly, B. F., Durst, F., Pereira, J. C .F. and Sch¨ onung, B. Experimental and theoretical investigation of backward-facing step flow. J. Fluid Mechanics, 127:473–496, 1983. 2. Breuer, M., Rodi, W. Large-eddy simulation of complex turbulent flows of practical interest. In: Flow Simulation with High–Performance Computers II, ed. E.H. Hirschel. Notes on Numerical Fluid Mechanics, 52:258–274, 1996. 3. Breuer, M. Large-eddy simulation of the sub-critical flow past a circular cylinder: numerical and modeling aspects. Int. J. for Numerical Methods in Fluids, 28:1281–1302, 1998. 4. Breuer, M. New reference data for the hill flow test case. http://www.hy.bv.tum.de/DFG-CNRS/, 2005. 5. Ducros, F., Nicoud, F., Poinsot, T. Wall-adapting local eddy-viscosity models for simulations in complex geometries. In: 6th ICFD Conf. on Numerical Methods for Fluid Dynamics. pp. 293–299, 1998. 6. Fr¨ ohlich, J., Mellen, C. P., Rodi, W., Temmerman, L., and Leschziner, M.A. Highly resolved large-eddy simulation of separated flow in a channel with streamwise periodic constrictions. J. Fluid Mechanics, 526:19–66, 2005. 7. Hinterberger, C. Dreidimensionale und tiefengemittelte Large–Eddy–Simulation von Flachwasserstr¨ omungen, PhD Thesis, University of Karlsruhe, 2004. 8. Jakirli´c, S., Jester–Z¨ urker, R., Tropea, C. (eds.) 9th ERCOFTAC/IAHR/COST Workshop on Refined Flow Modeling, Darmstadt University of Technology, Germany, Oct. 4–5, 2001. 9. Manceau, R., Bonnet, J.-P., Leschziner, M.A., Menter, F. (eds.) 10th Joint ERCOFTAC(SIG–15)/IAHR/QNET–CFD Workshop on Refined Flow Modeling, Universit´e de Poitiers, France, Oct. 10–11, 2002. 10. Mellen, C. P., Fr¨ ohlich, J., Rodi, W. Large-eddy simulation of the flow over periodic hills, Proc. of 16th IMACS World Congress, Lausanne, Switzerland, Deville M. and Owens R. (eds.), CD–ROM, 2000. 11. Peller, N., Le Duc, A., Tremblay, F., and Manhart, M. High-order stable interpolations for immersed boundary methods. Int. J. for Numerical Methods in Fluids, subm. 12. Temmerman, L., Leschziner, M. A. Large-eddy simulation of separated flow in a streamwise periodic channel constriction. In Lindborg, Johansson, Eaton, Humphrey, Kasagi, Leschziner, Sommerfeld (eds.), Turbulence and Shear Flow Phenomena. 2nd Int. Symp., pp. 399–404, Stockholm, June 27–29, 2001. 13. Temmerman, L., Leschziner, M. A., Mellen, C. P., Fr¨ ohlich, J. Investigation of wall-function approximations and subgrid-scale models in large-eddy simulation of separated flow in a channel with periodic constrictions. Int. J. Heat Fluid Flow, 24:157–180, 2003.
Large-Eddy Simulation of a Subsonic Cavity Flow Including Asymmetric Three-Dimensional Effects Lionel Larchevˆeque1 , Odile Labb´e2 , and Pierre Sagaut3 1
2
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IUSTI, Aix-Marseille I university, UMR CNRS 6595, 5 rue Enrico Fermi, F-13453 Marseille cedex 13, France. [email protected] ONERA, CFD and Aeroacoustics department, 29 avenue de la division Leclerc, F-92322 Chˆ atillon, France. [email protected] LMM, Paris VI University, UMR CNRS 7607, boˆıte 162, 4 place Jussieu, F-75252 Paris cedex 5, France. [email protected]
Summary. Large-Eddy Simulations of a subsonic turbulent cavity flow which experimentally exhibits a streamwise non-symmetrical mean flowfield have been carried out. Both MiLES and LES based on the selective mixed scale model match well the experimental data and include the spanwise modulation of the flow. Moreover, by comparing a priori statistically identical computations, it is found that a symmetrybreaking bifurcation process is responsible for the streamwise structuration of the flow. By means of auxiliary computations, it is demonstrated that the bifurcation is due to the inviscid confining effects originating in the lateral walls of the wind tunnel. Nonetheless, a simulation with spanwise periodic boundary conditions shows evidence of an underlying streamwise modulation of the flow.
1 Introduction Subsonic flows over shallow open cavities are most often dominated by strong self-sustained oscillations due to an aeroacoustic coupling between vortical structures and pressure waves[1]. Numerous experimental, analytical and numerical works have been devoted to the study of these oscillations. However secondary phenomena like the three-dimensionality of the flow have received little attention, although in incompressible and laminar regime such phenomena have been clearly identified[2, 3]. Nevertheless, recent experimental studies dedicated to oscillating M = 0.8 and ReL = 8.6 × 105 flows over a deep[4] and a shallow[5] cavities of the same length and width have highlighted noticeable three-dimensional effects. As the deep cavity flowfield, away from the lateral walls, was been found almost homogeneous in the spanwise direction, the mean velocity field inside the
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shallow cavity exhibited a marked asymmetrical structuration despite the fully symmetrical configuration of the wind-tunnel. A Large-Eddy Simulation (LES) of the flow over the deep cavity has been successfully carried out[6] jointly with the experiments. Beyond validation purposes, results from the simulation have been used so as to explain some features of the flow already highlighted by the experimental data. The aim of the present study is to extend this coupled experiment-LES approach to the shallow cavity case. It is expected that computations will be able to reproduce the three-dimensionality of the flow so as to help clarifying the origin of the intriguing asymmetrical pattern.
2 Numerical models The numerical methods retained for the present simulations are mostly similar to the ones used for the aforementioned successful LES of the deep cavity configuration[6]. Namely two classes of computation have been carried out. The first one is based on an implicit subgrid modeling through a upwind optimized AUSM-type scheme[7] and is hereafter classically denoted MiLES. The second one relies on a traditional LES with a 2nd order centered scheme coupled to the mixed-scale model on its standard[8] or selective[9] form. A hybridization with the previous upwind scheme is performed through an oscillation detector[7] to avoid point-to-point oscillations induced by purely centered schemes. The second column of Tab. 1 lists the subgrid modeling strategies adopted in each of the simulations described in this paper. Time integration is a carried out by means of a 3rd order compact Runge-Kutta scheme. Label MIL MS SMS MILl MILr MILs MILp
subgrid spanwise Forcing Bifurcation model boundary condition MiLES Wall function No Left Mixed scales Wall function No Left Selective mixed scales Wall function No Left MiLES Wall function Right Right MiLES Wall function Left Left MiLES Slip wall No Right MiLES Periodicity No No Table 1. Main characteristics of the computations
For cavity flow simulations, non-reflective boundary conditions are highly desirable to prevent an alteration of the aeroacoustic resonant loop. Moreover, realistic velocity profiles at the upstream corner of the cavity are required as a key parameter to accurately predict the growth of the mixing layer. A characteristic-based inflow condition coupled with a synthetic turbulence generated through the addition of space-filtered and time-correlated random
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fluctuations[10] is retained to fulfill such requirements. Both first and second order statistical moments of velocity are set accordingly to the experimental data. A subsonic characteristics-based condition is used at the outflow plane. Finally, wall modeling is adopted to make the computations feasible despite the low thickness of the boundary layer and the large dimensions of the wind tunnel having to be discretized. It relies on a two-layer instantaneous wall function computed in a reference frame related to the local friction line. The dimensions of the computational domain are shown in Fig. 1. The mesh is built using similar cells dimensions as for the finer mesh described in the deep cavity study by Larchevequeet al.[6]. It includes the upper wall of the wind tunnel. The full width of the wind tunnel and its lateral boundary layers are also taken into account because of the spanwise inhomogeneity of the flow. This yields a nearly 6 millions cells mesh the characteristics of which are described in Tab. 2. The number of cells in the initial vorticity thickness is of special importance because of its influence on the accuracy of the computation in the receptivity region of the mixing layer. For the present simulation, this number is slightly increased compared to the already large value of the deep cavity computations[6] because of a slightly thicker incoming boundary layer. For all the computations, statistical data have been averaged over more than 50 periods of the first mode of oscillation, resulting in CPU times ranging from 300 h to 450 h on one processor of ONERA’s NEC SX-5.
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Fig. 1. Dimensions of the computational domain. The black dot located on the upstream wall of the cavity marks the location of the pressure sensor for both the experiment and the simulations.
Table 2. Mesh parameters including typical dimensions of the boundary layer cells in local wall units. Cells 39750 × 144 (spanwise) = 5.724 × 106
Δx+ Δy+ Δz+ Cells (streamwise) (normal) (spanwise) within δω0 80 ∼ 300
70 ∼ 90
10 ∼ 20
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3 Validation Since experimental velocity databases of cavity flows are rare, corresponding simulations are usually evaluated by means of pressure spectra. Such spectra are plotted in Fig. 2 which shows that MIL and SMS spectra are in remarkable agreement with the experiment. The global shape is correctly predicted up to 30 kHz. Moreover the levels and the frequencies of the peaks related to the selfsustained oscillations process differ from the experimental ones by less than 5 dB and 3% respectively. It demonstrates that the MIL and SMS simulations are able to accurately reproduce the aeroacoustic loop responsible for the oscillations of the flow. On the contrary, the MS simulation yields inaccurate pressure spectra with multiple spurious harmonics of the main peaks. 160
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Fig. 2. Power density spectra of pressure on the upstream wall at the location highlighted in Fig. 1. The frequency in Hz (horizontal axis) is plotted against the pressure level in dB scale with the experimental data labeled as “Exp.”.
A similar superiority of the MIL and SMS simulations over the MS one is found in the velocity profiles plotted in Fig. 3. The left part of the figure, dedicated to the mean streamwise velocity, shows that the growth rate of the shear layer is underestimated in the MS simulation whereas MIL and SMS computations are in good agreement with the measurements, except for a slight upper shift of the mixing layer. The turbulent kinetic energy (TKE) profiles plotted on the right part of Fig. 3 are also in good agreement with the experiment for the inner part of the mixing layer. However levels are underpredicted inside the cavity between one-third of its length and the vicinity of the downstream wall. On the basis of the theoretical velocity fluctuation profiles of a Stuart vortex, it is conjectured that the discrepancies originate from a fragmentation of the unique large vortex involved in the aeroacoustic loop into multiple intermingled smaller vortices. Indeed such a lower coherence due to a higher three-dimensionality
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0.5
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of the vortex has been found in the deep cavity computations[6]. Nevertheless, numerical schlieren pictures clearly demonstrate the presence of the structure inside the cavity. One can therefore conclude from figures 2 and 3 that both
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Fig. 3. Streamwise velocity (left) and 2D turbulent kinetic energy (right) profiles on the half-span vertical plane. ◦: experiment; − − −: MIL computation; · · · · · · : MS computation; −−−−−: SMS computation.
MIL and SMS computations are able to described the structure and dynamics of the flow in the half-span plane. However the ability of the computations to reproduce the spanwise non-symmetric patterns remains to be checked. Streamwise velocity and 2D turbulent kinetic energy maps of the horizontal aperture plane of the cavity are drawn in Fig. 4 for this purpose. They show that all the computations capture the complex highly non-symmetrical features of the flow though in an inverted form. Note that the lower streamwise velocity values in the computations are due to the already mentioned slight shift of the mixing layer in the upper direction. The turbulent kinetic energy maps plotted in the right part of Fig. 4 show a similar inversion of the asymmetrical structure between the experiments and the three computations. One can remark that low mean velocity regions also exhibit high TKE levels. This suggests that these areas correspond to a vertical extension of the recirculation zone beyond the dimension of the cavity. Also note that the MS results exhibit a spanwise structure which is quite similar to that the other two computations despite marked differences in terms of pressure spectrum. This seems to prove that the asymmetric flowfield is not directly related to the aeroacoustic loop.
4 Bifurcation of the mean flow It has been demonstrated in the previous section that the present computations are able to reproduce the asymmetry of the experimental flowfield. Consequently this feature is not due to an imperfection of the wind tunnel and remains to be explained. Bearing in mind this aim, it has to be specified that an initial MiLES computation yielded the same orientation of the asymmetry as for the experiments. Moreover, when comparing the statistics
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Fig. 4. Streamwise velocity (left figures) and 2D turbulent kinetic energy (right figures) maps in the horizontal aperture plane of the cavity. Experiment (a), MIL computation (b), MS computation (c) and SMS computation (d).
coming from this computation and the ones issued from MIL, it was found that the two resulting flowfields were clearly enantiomorphic. One must emphasize that the preliminary simulation and the MIL one were based on the same numerical parameters, including the mesh, schemes and boundary conditions with the exception of the turbulent fluctuations added to the mean inflow velocity profile. The fluctuations were differing by their time series although first and second-order moments of these series were identical. Therefore the two computations were equivalent in a statistical sense. As a matter of fact, statistically equivalent systems resulting in mirror image pairs are the distinctive sign of a symmetry-breaking fork bifurcation. Auxiliary computations are carried out to identify some of the characteristics of the newly-identified bifurcation. Because the MIL simulation is in similar agreement with the experimental measurements as the SMS one, all the computations described hereafter are carried out using the MiLES model in order to save CPU time. Note that all simulations rely on the same time series of turbulent fluctuations as the MIL one for consistency reasons. First the influence of the initial condition on the direction of the bifurcation is tested. The initial velocity profile in the channel above the cavity is unsymmetrized by adding 1% sinusoidal variations of the streamwise velocity component in the spanwise direction. Two computations are carried out starting from two anti-symmetric sets of perturbations. They are stopped after about 20 periods once it is obvious from the statistics that the same left variant of the bifurcation is obtained from both of them.
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The characteristics of the four remaining computations are summarized in Tab. 1. The MILr / MILl simulations rely on a right / left permanent alteration of the mean inflow velocity profiles using the same 1% sinusoid modulation as for the cases of the initial condition forcing. Comparisons between Figs. 5(a) and 5(b) show that the flowfields resulting from MILr and MILl computation are anti-symmetrical. Consequently it demonstrates that the constant low amplitude asymmetry of the velocity allows selecting the orientation of the bifurcation. This contributes to explain how only the right variant of the bifurcation was found in the experiment because of small permanent inhomogeneities in the wind tunnel.
Fig. 5. Streamwise velocity (left part) and 2D turbulent kinetic energy (right part) maps in the horizontal aperture plane of the cavity. MILr (a), MILl computation (b), MILs computation (c) and MILp (d).
Moreover, it is seen from the eduction of the mean vortical structures plotted in Fig. 6 that the flow is fully bifurcated inside the cavity yielding a complex vortex pattern. Outside of the cavity, the modulated elevation of the mixing layer is seen from the difference in altitude of the two lateral wall vortices originating from the upstream corners of the cavity. Next, the lateral wall boundary condition is changed to a slip condition in computation MILs to check if the lateral boundary layers are involved in the bifurcation. Figure 5(c) demonstrates little influence on the bifurcation, except in selecting the opposite variant as for MIL despite the same inflow time series. Nonetheless, the bifurcation process seems to be essentially inviscid. The last computation, denoted MILp is based on span periodicity. For such a boundary condition, the bifurcation of course disappears but underlying spanwise modulations of the flow are still seen in Fig. 5(d).
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Fig. 6. Isosurfaces of Q = 0.75 L∞ 2 computed from the mean velocity field. Left and right plots are respectively related to MILr and MILl computations.
5 Conclusion The combination of experimental measurement and LES data of a wallbounded subsonic cavity flow has highlighted a bifurcation being responsible for asymmetrical effects. The bifurcation has been found to be inviscid in nature and to be sensitive to an unsymmetrical forcing of low amplitude. Future work will focus on the analysis of the complex space and time fluctuations of the flow using the unsteady and three-dimensional database generated from the MILr and MILl simulations.
References 1. Rockwell, D., Naudascher, E.: Self-sustained oscillations of impinging free shear layer. Annu. Rev. Fluid Mech. 11 (1979) 67–94 2. Maull, D.J., East, L.F.: Three-dimensional flow in cavities. J. Fluid Mech. 16 (1963) 620–632 3. Rockwell, D., Knisely, C.: Observations of the three-dimensional nature of unstable flow past a cavity. Phys. Fluids 23 (1980) 425–431 4. Forestier, N., Jacquin, L., Geffroy, P.: The mixing layer over a deep cavity at high-subsonic speed. J. Fluid Mech. 475 (2003) 101–145 ´ 5. Forestier, N., Geffroy, P., Jacquin, L.: Etude exp´erimentale des propri´et´es instationnaires d’une couche de m´elange compressible sur une cavit´e : cas d’une cavit´e ouverte peu profonde. Technical Report RT 22/00153 DAFE, ONERA (2000) In French. 6. Larchevˆeque, L., Sagaut, P., Mary, I., Labb´e, O., Comte, P.: Large-Eddy Simulation of a compressible flow past a deep cavity. Phys. Fluids 15 (2003) 193–210 7. Mary, I., Sagaut, P.: LES of a flow around an airfoil near stall. AIAA J. 40 (2002) 1139–1145 8. Sagaut, P.: Large-eddy simulation for incompressible flows - An introduction. Scientific Computation. Springer-Verlag, Berlin (2005) 3rd edition. 9. Lenormand, E., Sagaut, P., Ta Phuoc, L., Comte, P.: Subgrid-scale models for Large-Eddy Simulation of compressible wall bounded flows. AIAA J. 38 (2000) 1340–1350 10. Sagaut, P., Garnier, E., Tromeur, E., Larchevˆeque, L., Labourasse, E.: Turbulent inflow conditions for large-eddy simulation of supersonic and subsonic wall flows. AIAA J, 42 (2004) 469–477
Numerical Simulation of the Flow Around a Sphere Using the Immersed Boundary Method for Low Reynolds Numbers Campregher, R.1 , Mansur, S.S.2 , and Silveira-Neto, A.1 1
2
Mechanical Engineering College (FEMEC), Federal University of Uberlˆ andia (UFU), Av. Jo˜ ao Naves de Avila 2.160, Bloco 1M, Campus Santa Mˆ onica, 38400-902 Uberlˆ andia, MG, Brazil [email protected] [email protected] S˜ ao Paulo State University (UNESP) - Dept. of Mechanical Engineering (FEIS), Cx. Postal 31, 15385-000 Ilha Solteira, SP, Brazil [email protected]
Key words: Immersed Boundary Method, Flow around a Sphere, Numerical Method, Low Reynolds Numbers Summary. In the immersed boundary methodology, the effect of a fluid/solid interface inside a flow can be modeled by a body force term added to the discretized Navier-Stokes Equations. Moreover, the immersed body and the fluid don’t share the same domain. In fact, one uses a Cartesian mesh based computational domain to solve the flow equations and a Lagrangian mesh to define the interface of the body. This approach allows to move and/or to deform the body surface without remeshing the fluid grid since the meshes overlap each other without interference. In the present work, numerical simulations of the flow around a sphere were performed employing an in-house immersed boundary methodology named Virtual Physical Model [1]. The sphere boundary was represented by a Finite Element mesh and the flow domain was discretized by the Finite Volume Method. Parallel processing techniques were used to solve the system of equations originated from an implicit second-order time and space discretization algorithm. The presented results were compared against those from literature.
1 Introduction Numerical simulations of flows around complex deforming and/or moving geometries have been a very complex challenge to the researchers. The class of problems that fit into this category is the simulation of two-phase flows and fluid-structure interactions, i.e., two issues of great scientific and technological relevance. It is possible to tackle them by employing body-fitted meshes
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that need to be rebuilt constantly. On the other hand, the Immersed Boundary (IB) Method allows one to work on those problems with no remeshing requirements. The Immersed Boundary Method was due to Peskin [2] and collaborators in order to simulate the blood flow through heart valves back in 70’s. Later on, several methodologies have been developed following this approach. Basically, the body/flow interface, represented by a Lagrangian framework, are simulated by adding a force field into the discretized Navier-Stokes equation in the Eulerian domain. The way how such force fields are evaluated differentiates the methodologies among them. In the work of [2], the force term was calculated using stress forces originating from the boundary strain rate. Lai [4] used higher-order discretization to improve the numerical stability dependence on the time-step and [5] added adaptive meshes and a new interpolation function. Two phase-flows in both two and three-dimensional domains have been simulated by [6], where the force term was evaluated considering the interfacial tension. Furthermore, in [6] the bubbles in the flow were represented by Finite Element-like meshes and the Eulerian domain was discretised by a Finite Difference mesh in a staggered arrangement. Later on, [7] proposed a function that could relate the flow velocity at interface to the interface velocity itself. This function was adjusted by ad hoc constants where one of them produces a natural oscillation frequency while the other one dumps the response oscillations. Due to the fact that these constants were adjusted according to the response of the flow when they were introduced, the method was known as feedback forcing method. Mohd-Yusof [8] calculated the force field by a momentum balancing over the portion of flow adjacent to the Lagrangian interface. The method based on that approach was called direct forcing method and requires no constants to be adjusted. However, it demands some efforts to locate the geometry and several B-splines interpolation procedures to evaluate the flow properties close to the interface. Kim et al. [10], have run experiments using the methodology from [8] in domains discretised by the Finite Volume technique. The authors have applied linear and bilinear interpolations to the velocity field and have introduced mass sources and sinks into the flow cells aiming to improve the accuracy. More recently, [1] have proposed a model that has no ad hoc constants, by evaluating the force term from the momentum balancing over a flow particle close to the interface, and uses very simple interpolation procedures. Due to the fact that the momentum equations is employed to force the no-slip conditions on the body surface, the method was named Virtual Physical Model (VPM). Once the method has been tested for several flows in two-dimensional problems, the present work presents an extension to three-dimensional domains choosing, as a first test case, the flow around a single sphere at Reynolds numbers ranging from 100 to 1000. The Eulerian domain was discretized by the Finite Volume method with second order accuracy in space and time and the Lagrangian domain was built from a Finite Element mesh using triangular
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elements. The numerical code was written in Fortran 90 and has run in an in-house Beowulf cluster.
2 Mathematical Modelling 2.1 The Eulerian Domain The flow was discretized by the Finite Volume Method over a non-uniform mesh, the grids points of which were more concentrated on the sphere region. The grid expansion and contraction ratios follow a linear function. In Fig. 1 the computational domain characteristics are depicted. The numbers above and below the arrows represent, respectively, the length and the number of grid points in each region. In its integral form, the incompressible Navier-Stokes equations without body force, can be written as: ∂ ρui dΩ + ρui v · ndS = (τ · n)i dS + fi dΩ (1) ∂t Ω S S Ω where fi is the force field term associated with the volume element i, τ is the stress tensor, and ρ is the fluid density. The time derivative was approximated by the second-order three-time level [11] and spatial derivatives by the CentralDifference Scheme (CDS). Other body forces than the force field can also be added to the model. The pressure-velocity coupling was solved by the SIMPLEC method with no relaxation in the momentum equations. A co-located arrangement of the variables was employed along with the Rhie-Chow [13] interpolation method to avoid numerical oscillation due to eventual pressure-velocity decoupling.
Fig. 1. Cartesian domain used in all simulations.
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As the Cartesian mesh was not fine enough to keep the solution stable when using CDS interpolation in the advective terms for all Reynolds numbers, a deferred-correction algorithm was implemented. The linear system generated by the velocity component discretization was solved by the SOR (Successive Over-Relaxation) algorithm whereas the SIP (Strongly Implicit Procedure) algorithm was employed to solve the pressure correction system of equations. Since the problem has required very high memory resources to be simulated accurately, the domain decomposition technique was employed using an in-house Beowulf cluster. The way the problem was partitioned, for every simulation presented in this work, can be seen in Fig. 2.
Fig. 2. Eulerian domain subdivided into subdomains for parallel computing.
The second-order spatial discretization requires one neighbor cell along each coordinate direction. Thus, the overlapping between two adjacent processors was done by a computational halo of one volume width. Higher order would require more overlapping sub-domain layers, increasing substantially the communication overhead. The data communication among the processors was done by the MPI (Message Passage Interface) library that was chosen, mainly, due to its better performance on clusters of homogeneous machines. 2.2 The Virtual Physical Model The fluid-solid interface is a surface discretized by a Finite Element mesh. As previously stated, the Virtual Physical Model evaluates the force field by
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applying the momentum balance over the Lagrangian fluid particle which must be non-slipping on the boundary. Then, let k be a Lagrangian point, placed at − → (see Fig. 3) that was chosen to be at the triangular finite element centroid. x k The index i means each of the Eulerian coordinate system directions. Applying the momentum balance equation over this Lagrangian point (see [1]), one has: ∂(ρukj uki ) ∂uki ∂pk ρuki − ρuf ki ∂ + + −μ (2) Fki = Δt ∂xj ∂xj ∂xj ∂xi where Fki is the force, along the direction i, needed to change the velocity of the fluid particles uf ki adjacent to the Lagrangian point k, in order to attain the wall velocity uki , i.e., performing the no-slip condition on the solid wall. The discretization of the eq. 2 is done by constructing a reference threedimensional axis with origin at each point k, as can be seen in Figure 3. Later on, a Lagrangian polynomial is used to obtain the space derivatives along each direction.
Fig. 3. Lagrangian point k and auxiliary points.
Once Fki has been evaluated in the Lagrangian domain, it is interpolated back to the Eulerian domain via Eq. 3: (Fki )Dm ΔAk ΔSk (3) fi = k
where ΔAk and ΔSk are the triangular element area and average side length, respectively. The Dm is the distribution function detailed in [1].
3 Results and Discussion The flow around a sphere was tested for Reynolds numbers, based on the sphere diameter D = 0.04m and inflow velocity u∞ , running from 100 to 1000
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in steps of 100. The center of the sphere was placed in (x,y,z )=(0.38,0.34,0.34). The boundary conditions for the Eulerian domain were set as free slip conditions at the side walls (along the y and z axis), v = w = 0 and u = u∞ at the i inflow, and ∂u ∂xi = 0 at the outflow. The result for u-velocity component profile across the sphere equator at for Re = 200 is depicted in Figure 4a. One can see that its value goes approximately to zero at the solid/fluid interface. This is a result from the l2 norm that was kept in the range 10−4 ≤ l2 ≤ 10−3 for all Reynolds numbers simulated. In Figure 4b the l2 norm time evolution can be seen for Re = 200. For lower Reynolds numbers, the flow presents a mainly bi-dimensional pattern. As the Reynolds number reaches the range 270 ≤ Re ≤ 285, it presents a three-dimensional configuration but keeps its planar symmetry. For Reynolds numbers above approximately 420, the planar symmetry is completely lost and a fully three-dimensional pattern is established. Some flow patterns can be seen in Figure 5. An interesting point to be remarked here is that the evaluation of the total force that acts on the body surface is a natural issue
Fig. 4. a) Velocity profile across the sphere equator and b) l2 norm time evolution for component u at Re = 200. Table 1. Comparison of drag coefficients for all Reynolds numbers simulated. Re
This work Correlations Ref. [3] Ref. [9] Ref. [10]
100 200 300 400 500 600 700 800 900 1000
1.178 0.815 0.675 0.594 0.520 0.530 0.505 0.495 0.485 0.478
1.087 0.776 0.653 0.594 0.555 0.528 0.508 0.493 0.481 0.471
1.085 0.768 0.482 0.319
1.079 0.757 0.476 0.321
1.087 0.657 -
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Fig. 5. Streamline patterns for a) Re = 200, b) Re = 300, c) Re = 500, and d) Re = 1000.
for the Virtual Physical Model. So, the drag and lift coefficients are easily obtained, and the former are listed in Table 1 and compared against those from literature and analytical correlations.
4 Conclusions Numerical simulations of the flow around a sphere were done for Reynolds numbers ranging from 100 to 1000. The results for flow pattern and drag coefficients show good agreement with those from the literature. The transition phenomena were also captured with good accuracy.
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Acknowledgements The authors wish to thank the Brazilians’s CNPq Council and CAPES Foundation for the financial support of this work.
References 1. A.L.F. Lima e Silva, A. Silveira-Neto, and J.J.R. Damasceno, Numerical simulation of two dimensional flows over a circular cylinder using the immersed boundary method, J. Comp. Phys. 189, 351 (2003) 2. C.S. Peskin, Numerical analysis of the blood flow in the heart, J. Comp. Phys. 25, 220 (1977) 3. B. Fornberg, Steady viscous flow past a sphere at high reynolds number, J. Fluid Mech. 190, 471 (1988) 4. M.C. Lai, Simulations of the flow past an array of circular cylinders as a test of the Immersed Boundary Method, Ph.D. Dissertation (New York University, NY, 1998) 5. A. Roma, C.S. Peskin, M. Berger, An adaptative version of the immersed boundary method, J. of Comp. Phys. 153, 509 (1999) 6. S. Unverdi, G. Tryggvason, A front-tracking method for viscous, incompressible multi-flui flows, J. Comp. Phys. 100, 25 (1992) 7. D. Goldstein, R. Handler, and L. Sirovich, Modeling a no-slip flow boundary with an external force field, J. Comp. Phys. 150, 354 (1993) 8. J. Mohd-Yusof, Combined immersed boudaries/B-splines methods for simulations in complex geometries, CTR Annual Research Briefs, NASA Ames/Stanford University, 1997. 9. E. Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusof, Combined immersedboundary finite-difference methods for three-dimensional complex flow simulations, J. Comp. Phys. 161, 35 (2000) 10. J. Kim, D. Kim, and H. Choi, An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. Comp. Phys. 173, 636 (2001) 11. J. Ferziger, M. Peric, Computational methods for fluid dynamics 3th ed., 2002, Springer-Verlag, New-York, USA. 12. A.G. Tomboulides and S.A. Orszag, Numerical inestigation of transitional and weak turbulent flow past a sphere, J. Fluid Mech. 416, 45 (2000) 13. C. Rhie and W. Chow, Numerical study of turbulent flow past an airfoil with trailing edge separation, AIAA Journal 21, 1525 (1983)
LES studies on the correspondence between the interaction of shear layers in post-reattachment recovery and in a plane turbulent wall jet A. Dejoan and M. A. Leschziner Aeronautics Department, Imperial College London, London SW7 2AZ, UK [email protected], [email protected] Summary. Turbulence budgets, extracted from highly-resolved LES, are used to contrast the turbulence characteristics of the flow behind a backward-facing step with those of two wall jets, one bounded by a real wall and the other by a frictionless wall, thus allowing wall-blocking effects to be isolated from those of near-wall shear. The objective is to identify common features in the turbulence processes associated with the interaction between the near-wall region and the outer layer of the separated flow and the wall jet, and thus isolate mechanisms possibly responsible for the poor representation of the post-reattachment recovery returned by most of RANS closures. The budgets show that certain regions of the separated flow present similarities with the wall-shear-free jet, while others share features with the real wall jet. Turbulent transport by third moments and pressure-velocity correlations are shown to be important processes in the interaction between the near-wall region and the outer layer.
1 Introduction The correct prediction of separation over a broad range of conditions continues to elude turbulence models, at whatever level of closure [1], [2]. Although the overall features and dimensions of the recirculation zone are returned correctly by some turbulence models, its detailed features are almost always misrepresented. One common defect which occurs almost invariably, whatever the predictive quality of the recirculation zone itself may be, is an insufficient rate of post-reattachment recovery. The ubiquity of this defect suggests that the problem may lie in the complex interaction between the outer shear layer, originally above the recirculation zone, and the wall over which the wake recovers. The effect of the wall is two-fold: first, it acts to block the wall-normal velocity fluctuations and to reflect pressure fluctuations, thus modifying the turbulence field away from the wall in the shear layer; and second, it causes the formation of a sheared boundary layer, with its own scales and turbulence characteristics, which interacts with the outer shear layer, thus affecting its
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evolution. It is this two-pronged interaction that is the focus of the present study. Its principal objective is to gain insight into the peculiar characteristics of the turbulence processes within the composite shear layer in the wake recovering from separation and reattachment, and to assess the implications for its modelling. In this context, a flow that appears to be highly pertinent to postreattachment recovery is the wall jet. This flow also consists of two interacting shear layers, one a separated outer layer and the other a boundary layer. The turbulence processes in this flow have recently been the subject of three studies by the present authors [3], [4], [5], in which data derived from large eddy simulations have been analysed. These studies focus, respectively, on the turbulence structure of the wall jet, on the discrimination of effects arising from wall blocking and wall shear and on a-priori studies of elements of secondmoment RANS closures. Some important features that the simulations have brought out include the significant contribution of triple-correlations-driven stress diffusion in the interaction region, the surprisingly high level of pressure diffusion, the strong influence of wall blocking on the redistribution mechanism and the major effects on the turbulence structure brought about by the addition of near-wall shear to wall blocking. An outcome of the a-priori RANS studies [5] has been the illumination of some of the reasons for the poor representation of the above-noted features by popular second-moment-closure models. The present study contrasts the turbulence characteristics of the recovery region of the separated flow behind a backward-facing step against those of two wall jets, one with and the other without wall shear, the comparative analysis being based, principally, on Reynolds-stress budgets. The rationale is to identify parallels in the turbulence mechanisms and to understand the extent to which the supposition of the equivalence of the processes in the recovering wake and the jets is valid. The back-step-flow simulations were undertaken by the present authors in the context of a study on the control of separation by means of a synthetic jet [6] which perturbs the separated shear layer. Here, the data for the non-perturbed flow are exploited.
2 The flow configurations and their simulation The flow configurations studied are shown in Fig. 1. The back-step flow is at a Reynolds number Re = Uc h/ν = 3700, at conditions identical to those of the experiments of Yoshioka et al. [7]. The computational domain extends from 4h upstream of the step to 18h downstream. The spanwise direction is statistically homogeneous and 4π/3 deep. A two-block grid of 2.106 nodes is used, with the maximum ratio of the grid scale, Δ, to the Kolmogorov scale, η, not exceeding 7 throughout the domain. The maximum value of y + = yuτ /ν at both the lower and upper walls is less than 1, and the cell-aspect ratio is, typically, Δy + /Δx+ /Δz + = 1.1/18/10 at the wall and 2.5/18/10 in the shear
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layer. The inlet conditions, prescribed at 4h upstream the step, were derived from a fully-developed channel-flow precursor computation. The wall jet is at a Reynolds number Re = Uo b/ν = 9600, Uo being the inlet velocity and b the slot height. Two cases were simulated, one in which the jet develops along a real wall and the other along a frictionless, impermeable boundary. Here, the intention is to permit effects arising from wall-blocking and near-wall shear to be distinguished or separated within the whole interaction process between the outer shear layer and the wall. The real jet simulated corresponds to one studied experimentally by Eriksson et al. [8]. In both jet flows, the domain extends from the wall to 10b above it and to 22b in the streamwise direction. The grid is composed of 8.106 computational cells, and the ratio Δ/η is typically 5 − 10. For the real jet, the near-wall cell-aspect ratio is, typically, Δy + /Δx+ /Δz + = 1.2/24/24, the wall-nearest grid node located at y + = 0.6. More details can be found in [3], [4]. Computational tests, including the use of different grid-resolution levels and subgrid-scale models, have shown the back-step-flow simulation to be insensitive to subgrid-scale modelling, and the standard (damped) Smagorinsky model was therefore adopted. A complete description is given in [6]. The transitional character of the wall jet, albeit only within a few slot heights downstream of the discharge, made the use of Germano’s dynamic Smagorinsky model more appropriate for this flow, yielding good agreement between the simulation and the experiments (see [3]). 8 6
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All simulations were performed with a general multiblock finite-volume scheme incorporating second-order discretisation in time and space. Timemarching was based on a fractional-step method. The code is fully parallelised and was run on 128 to 256 ORIGIN 3800 processors.
3 Results 3.1 Preliminaries A few results for mean-flow quantities and second moments are presented first, but the emphasis is on budgets. For the back-step flow, properties are normalised by the step height h and the maximum velocity in the inlet channel Uc . Results for the jet flows are given only at the far-field location, x = 20b,
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where the flow approaches a state of self-similarity. Here, the flow properties are normalised with the maximum velocity, Umax , and y1/2 , the distance from the wall to the half-of-maximum-velocity point (see Fig. 1). Using this scaling, the outer layers of the two jets display a self-similar behaviour (see [4]). 3.2 Mean flow and turbulence properties Figure 2 presents profiles of the mean-velocity and Reynolds shear stress for all three flows considered. For the back-step flow, the profiles are given at two locations ahead of the reattachment location (xr = 7h) and one, well downstream, in the recovery region. The data are compared with two sets of experimental results, one by Yoshioka et al. [7] and the other by Kasagi et al. [9], as well as with the DNS results of Le et al. [10], the last two being for a Reynolds number of 5100 – that is, not far above the present value. However, the DNS data relate to a geometry that does not include an upper wall. Considering the differences, the comparisons show fair agreement among the results. It is noted that Yoshioka et al. report a shorter reattachment length, xr = 5.5h, than the present value of 7, this discrepancy being most likely due to 3D contamination in the experiments, an issue discussed in [6]. A complete comparison of the full turbulence statistics can also be found in [6]. 5
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For the real wall jet, extensive comparisons with the experiments are presented and discussed in [3], while a detailed discussion on the discrimination of wall-blocking effects from wall-shear effects is given in [4]. The main differences between the two wall jets arise from the non-vanishing values of the
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streamwise and wall-parallel stresses at the wall of the wall-shear-free jet. In the absence of wall-shear, large eddies originating from the outer layer impinge on the wall without damping by viscous action, thus resulting in a high finite value of turbulence energy at the wall. The near-wall structural features of the two jets also differ greatly. In particular, the near-wall shear gives rise to small-scale, elongated eddies, while in the absence of shear, the eddies are more “isotropic”. This implies different redistribution mechanisms of turbulence energy among the normal stresses in the near-wall region, discussed in more detail in [4]. 3.3 Budgets Space constraints only permit some representative budgets to be included, allowing features common to both wall jets and the back-step flow to be highlighted. For the former, budgets are reported for x/b = 20, while for the latter flow, they are given for two locations, one within the recirculation zone, x/h = 4, and the other well downstream of the reattachment point, x/h = 16. The terms included in the budgets terms are those constituting the transport equations that govern the evolution of the stresses: ∂ui uj + U k ui uj ,k = −uk uj Ui,k − uk ui U j,k ∂t "# $ ! "# $ ! Pij
Cij
+ νui uj ,kk −ui uj uk ,k − ! "# $ ! "# $ ! Dν
T T Tij
(ui p,j + uj p,i ) ρ "# $
(1)
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+
εij !"#$ dissipation
The turbulence-energy budget is deduced from the sum of the normal-stress equations. Here, we point out that the dissipation is deduced from the balance of Eq. (1), which implies that εij represents the total level of dissipation, including the viscous and subgrid-scale contributions. The budgets for the turbulence energy are shown in Fig. 3. The main feature linking the recirculation region of the backward-facing step (x/h = 4) to the zero-wall-shear jet is the low near-wall shear production. The low production in the latter flow results from a small streamwise contribution while in the recirculation region, the production, slightly negative, reflects the sign reversal of the velocity profile in this region. With the near-wall shear being small, it is observed that the main input of turbulence energy into the near-wall layer is caused by a high level of turbulent transport from the outer shear layer towards the wall, in particular in the respective interaction layers 0.05 < y/h < 0.25 and 0.02 < y/y1/2 < 0.2, although convection is also significant, because of the slip at the wall in the zero-wall-shear jet. Very near the wall, both energy budgets indicate a balance mainly between turbulent transport (combining triple correlation and pressure diffusion) and
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Fig. 3. Budgets for the turbulence energy : P : production; C: convection; Dν : viscous diffusion; T T T : transport by Triple correlations; Π: velocity-pressure correlations (identical to pressure diffusion for k); ε: dissipation. See Eq. (1) for definition of budget terms.
dissipation. Features similar to those in the budget for x/h = 4, were also observed in budgets at other locations within the wall region of the whole recirculation region. Downstream of reattachment, as the flows recovers, a new boundary layer develops and mean-shear production close to the wall becomes a major contributor to the turbulence energy. Here, a balance akin to that in a standard boundary layer is observed, wherein production is mainly balanced by dissipation in the near-wall region. However, a major difference to a standard boundary layer is the large input of energy by turbulent transport from the outer shear layer towards the wall, in particular in the interaction region (0.15 < y/h < 0.3). While in the recirculation zone the turbulent transport provides the main input of energy, this process contributes equally with production in the interaction region of the back-step recovery region. This combined influence is also observed in the wall jet, although the contribution of diffusion is more pronounced as the production is low in the interaction region (0.08 < y/y1/2 < 0.25). Not included herein, because of space limitations, are plots of production-to-dissipation and length-scales distributions. These show that the high input of turbulent transport from the outer shear layer towards the wall, in both the wall jet and the back-step recovery region, results in strong departures from equilibrium in the interaction regions – stronger in
Comparative LES studies on flows with interacting shear layers
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the wall jet than in the back-step flow, because production is smaller in the former than in the latter. Budgets for the shear stress are presented in Fig. 4. All show significant levels of transport by triple correlations in the near-wall region. However, for the back-step flow, this transport is negative, while it is positive in both jet flows, this difference being associated with the opposite signs in the respective strain rates (see Fig. 2). Both in the recirculation region of the back-step flow and in the zero-wall-shear jet, the production is low, and the main balance occurs between turbulent transport and pressure-velocity correlation. On the other hand, in the back-step recovery region and in the real wall jet, production is high, and this is counterbalanced mainly by pressure-velocity correlation. However, the contribution of transport by triple correlations is still significant. uv budget
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Fig. 4. Budgets of the shear stress: see caption of Fig. 3 and Eq. 1
Again because of space constraints, only budgets for the wall-normal stress are included herein. These are also the most interesting ones, because the wallnormal intensity is intimately linked to the shear stress. On the whole, the budgets of uu are similar to those of the turbulence energy, and the budgets of ww do not differ greatly among the three flows examined herein, all being broadly similar to those found in a standard boundary layer. Figures 5 shows that the budgets for the back-step flow within the separation region and in the recovering wake have features similar to those of the zero-wall-shear and the real wall jet, respectively. Thus, the pair of budgets on the left-hand side of Fig. 5 both feature a high level of transport by triple correlations from the outer layer towards the wall, which is compensated by a high negative contribution of the pressure-velocity correlation in
A. Dejoan and M. A. Leschziner
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Fig. 5. Budgets of the wall-normal stress: see caption of Fig. 3 and Eq. 1
the near-wall region, this latter contribution being, in absolute terms, higher than dissipation. However, in the back-step flow, dissipation tends to become more important as the flow recovers. As regards the budget for real wall jet, the stress vv is also elevated by a large gain through turbulent transport, but a lower and slightly negative pressure-velocity correlation arises near the wall, the dissipation being here the dominant process. It is important to point out that the vv budgets for both the back-step flow in the recovery region and the real wall jet differ strongly from that for the standard boundary layer, in which turbulent transport is low, the main balance being between a large gain by pressure-velocity correlation and loss by dissipation in the near-wall region. This implies that the large gain of turbulent transport modifies the energy-redistribution process in the present wall flows, when compared to a standard boundary layer. This will be demonstrated below by reference the non-deviatoric part of the velocity-pressure correlation. 3.4 Redistribution of turbulence energy Figure 6 shows profiles of the pressure-strain term for the normal-stress components. The pressure-strain – obtained as the difference between the velocity-pressure correlations and pressure diffusion, and defined as Φij = p ρ (ui,j + uj,i ) – is trace-free and thus responsible for the redistribution of the energy among the normal-stress components. One particularly interesting feature is the similarity of the energy-redistribution processes in the recirculation region and the zero-wall-shear jet. In both, this redistribution is from vv to
Comparative LES studies on flows with interacting shear layers
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both uu and ww. As the flow recovers, the energy-redistribution process becomes similar to the real wall jet, wherein the energy is transferred from vv and from uu to ww. In the latter case, a large level of uu is produced by near-wall shear, and energy has to be removed from uu, so that all stresses are made to vanish at the wall (see [4] for more details). These similarities suggest strongly that inside the recirculation region of the back-step flow wall-blocking effects dominate over the wall-shear effects. In contrast, as the flow recovers, wallshear effects predominate. As demonstrated by the present authors for the wall jets [4], the energy-redistribution is closely linked to structural features close to the wall, a main difference between the two jets being the formation of streaky structures in presence of wall shear, which inhibit the `ısplattingˆı of large-scale eddies originating in the outer shear layer onto the wall itself. In contrast, in the absence of wall shear and hence ¨eshielding´ı by the small-scale eddies in the near-wall layer, the large-scale eddies impinge freely on the wall (or surface). Φuu Φvv Φww Φuu Φvv Φww
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Fig. 6. Turbulence energy redistribution through the pressure-strain Φii .
4 Conclusions The present study has demonstrated, mainly by reference to budgets, that turbulence mechanisms effective in different regions of a separated flow relate to those in two wall jets, one evolving along a real wall and the other along a slip boundary. The latter jet only includes wall-blocking effects in its near-wall region, while the former combines wall-blocking with near-wall shear effects. An interesting finding is the similarity between the wall regions in the recirculation zone of the back-step flow and the zero-wall-shear jet, which suggests a predominance of wall-blocking effects inside the recirculation zone. In this region, mean-shear production is low, and the main contributor is transport by triple correlations from the free shear layer towards the wall, which is counterbalanced by the pressure-velocity-correlation terms. In the near-wall layer of the back-step recovery region, mean-shear production becomes predominant, and the budgets for the back-step flow contain features that are
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close those in the corresponding budgets of the wall jet. However, transport by triple correlation transport is also important, especially in the interaction region. In fact, in all flow regions considered herein, the turbulent transport provides a significant, if not major, input of energy in the wall-normal stress, a process that is compensated by pressure-velocity correlation and dissipation. This type of interaction differs substantially from that in standard near-wall flows close to equilibrium, for which the wall-normal stress is mainly governed by pressure-velocity correlation and dissipation, the transport by triple correlation being small. The redistribution of the turbulence energy among the normal stresses through the pressure-strain process reinforces the similarities between, on the one hand, the reverse-flow layer and the zero-wall-shear jet, in which energy is transferred from the wall-normal component to the both wall-parallel components, and, on the other hand, between the post-recovery region and the real jet, in which shear-produced energy has to be removed from the streamwise-stress component to the spanwise component. The above different energy-redistribution scenarios, applicable to the recirculation and the post-reattachment-recovery regions of the back-step flow, suggest that the importance of wall-blocking effects depends strongly on the streamwise evolution and local structure of the flow, a fact that presents a significant challenge to statistical closures, second-moment models in particular. Acknowledgements This study was supported financially by the UK Science and Engineering Research Council (EPSRC). The computations were performed on the Origin 3800 computer at the national CSAR service in Manchester, using resources provided as part of the EPSRC grant.
References 1. Jang Y-J., Leschziner M. A., Abe K. and Temmerman L. (2002) Flow, Turbulence and Combustion, 69:161–203 2. Wang C., Jang Y-J. and Leschziner M. A. (2004) Int. J. Heat and Fluid Flow, 25:499–512 3. Dejoan A. and Leschziner M. A. (2005) Phys. Fluids, 17, Issue 2–025102 4. Dejoan A. and Leschziner M. A. (2005) Proc. TSFP4, Williamsburg, June 27–20, 2:401–406 5. Dejoan A. and Leschziner M. A. (2005) Proc. ETMM6, Sardinia, May 23–25:97– 106 6. Dejoan A. and Leschziner M. A. (2004) Int. J. Heat and Fluid Flow, 25:581–592 7. Yoshioka S., Obi S. and Masuda S. (2001) Int. J. Heat and Fluid Flow, 22:393– 401 8. Eriksson J. G., Karlsson R. I. and Persson J. (1998) Expt. in Fluids, 25:50–60 9. Kasagi N. and Matsunaga A. (1995) Int. J. Heat and Fluid Flow, 16:477–485 10. Le H., Moin P. and Kim J. (1997) J. Fluid Mech. 342:119–139
Three Dimensional Wake Structure of Free Planar Shear Flow Around Horizontal Cylinder Marcelo A. Vitola1 , Edith B. C. Schettini1 , and Jorge H. Silvestrini2 1
2
Universidade Federal do Rio Grande do Sul - Av. Bento Gon¸calves, 4500 94501-970 - Porto Alegre, RS, Brasil [email protected];[email protected] Pontif´ıcia Universidade Cat´ olica do Rio Grande do Sul - Av. Ipiranga, 6681 90619-900 - Porto Aegre, RS, Brasil [email protected]
1 Introduction The transition to turbulence and the wake formation behind a circular cylinder have been the subject of a large number of researches, mainly the uniform flow case, due to the practical application. Indeed, in the last decades, some new insights have been found. The previous differences of experimental results have been associated with various factors including the surface roughness, free-stream turbulence, small aspect ratio, blockage and end effects [10, 16]. Nowadays, it has been confirmed, both experimentally and analytically, that as the Reynolds number increases a sequence of transitions events takes place in the wake of the cylinder which leads to turbulence. The first bifurcation occurs at Reynolds number 47 leading to vortex shedding. The second bifurcation occurs at Reynolds number 188, when the wake first becomes threedimensional (mode A). This is characterized by regular streamwise vortices appearing in the wake, with spanwise wavelength of approximately 4 cylinder diameters. At higher Reynolds number, i.e. 230, a second mode (mode B) appears, consisting of fine-scale streamwise vortices with a spanwise wavelength of about 1 cylinder diameter [16]. In the last two decades, a great effort has been made to reproduce numerically these three-dimensional results. The main problem is the significant computational resources required to resolve the flow structures properly. Since the Karniadakis and Triantafyllow work [3], many other studies have been published confirming the existence of modes A and B [9]. Although threedimensional numerical simulations are now available, little attention has been given to the influence of three-dimensional structures on the flow-induced forces [10, 12]. Another important flow configuration for many practical applications, is the non-uniform flow past a cylinder. In this kind of flow, the vortex shedding behaviour and the hydrodynamic forces acting upon the cylinder are affected
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by the shear of the oncoming flow. Examples include a cylinder in the wake of another one, a cylinder near a plane wall and a cylinder in a channel flow. The simplest case of non-uniform flow is the free shear flow. In this case the mean streamwise velocity has a linear distribution in the transversal direction. Two distinct configurations exist depending on the cylinder orientation: i) axial shear flow and ii) planar shear flow Most of the previous investigations which handle free planar shear flow have been developed using two-dimensional numerical simulations [2, 7]. Only in the last decades, some experimental studies have been developed [4, 13] due to the difficulty in generating a uniform free planar shear flow in the laboratory. The main point of divergence observed in literature for this kind of flow is the occurrence of vortex shedding suppression. This phenomenon was observed experimentally by Kiya et al. [4] for low Reynolds numbers and high shear parameter; however, it has not been observed in any other study. Recent results presented by Vitola et al. [14], for the same range of Reynolds number and shear parameter as used by Kiya et al. [4], have not shown the vortex shedding suppression. According to that work, this divergence can be the consequence of the high blockage and/or low aspect ratio used in the experimental study or some three-dimensional processes that can not be reproduced by the two-dimensional simulation. In this paper, some preliminary results of three-dimensional Direct Numerical Simulations of free planar shear flow around a horizontal cylinder are presented. The purpose of this study is to analyse the three-dimensional aspect of the wake in this kind of flow. It will be checked if the vortex shedding suppression occurs in the range of shear parameter β = 0 − 0.25 (β - defined below) and the influence of three-dimensional vortical structures on the flow-induced forces will be shown.
2 Flow configuration and parameters The uniform and free planar shear flow over a horizontal circular cylinder is considered in a Cartesian frame where the cylinder axis is oriented along the spanwise direction z and normal to the xy plane (see Fig. 1). At the inflow boundary, a uniform or free planar shear flow is imposed with the mean shear aligned with the z-direction. The shear extends over a zone −Ly /2 < y < Ly /2. Outside this interval, two streams of constant velocities U1 and U2 are imposed, with U1 > U2 . The mean velocity profile at the inflow is given by ⎧ ⎫ Ly ⎬ ⎨ 6 cosh[ D (y + 2 )] U1 − U2 D U1 + U2 + u(y) = ln (1) 2 12 Ly ⎩ cosh[ 6 (y − Ly )] ⎭ D
2
A weak white noise (≈ 10−3 ) was superposed in all directions to accelerate the transition process. For all simulations, free-slip boundary conditions are
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Fig. 1. Schematic view of the flow configuration.
applied to y = ±Ly /2 while periodicity is imposed in the z-direction. At the outflow boundary condition, an advection equation is prescribed to ensure that the disturbance created by vortex shedding leaves the computational domain without reflection. Two non-dimensional parameters are fundamental for free planar shear flow, the Reynolds number, Re = Uc D/ν, and the shear parameter,β = −D/Uc dU/dy, where Uc = (U1 + U2 )/2 is the mean velocity of the inflow profile.
3 Numerical Method The numerical model integrates the time-dependent, incompressible, NavierStokes equations in primitive variables form, normalised by the cylinder diameter D and the mean streamwise velocity Uc . These equations are discretized using sixth-order compact centred difference schemes [8] for all spatial derivatives. The time integration is performed with an explicit third-order lowstorage Runge-Kutta method [15]. The non-slip condition at the cylinder surface is imposed via an immersed boundary method based on the feedback force proposed by Goldstein et al. [1]. For more details about the numerical code, see Lardeau et al. [6] and for the immersed boundary method see Lamballais and Silvestrini [5].
4 Validation Results Before proceeding with the numerical investigation of the three-dimensional free planar shear flow, intensive tests have been carried out in order to analyse the influence of computational domain parameters. For this purpose, we use a two-dimensional version of the computational code to compute the uniform flow around a circular cylinder for Reynolds number 300. Our results were compared with those obtained by Mittal and Balanchander [9]. The size of a computational domain is defined by three dimensions: the length Xc , which locates the cylinder in relation to the inflow boundary, the length Lx which defines the total horizontal domain length and Ly which
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defines the vertical domain length. Other parameters that influence the results are the grid resolution (Δ) and the method used to apply the feedback force (F.V.). The influence of the numerical parameters in the flow characteristics (Strouhal number, drag and lift coefficients) are summarised in Tab. 1. The cases I, II and III describe the blockage effect (Ly /D). The results indicate that as the vertical domain size increases the drag and lift decreases as expected. The same behaviour was not observed for the Strouhal number. This difference is a consequence of the low resolution used. This was confirmed in the cases V and V I. The location of the cylinder has a similar behaviour as the blockage, as it is shown in cases I, IV and V . Another parameter that demonstrated to be important was the method used for the application of the virtual force (cases V I, V II and V III). The best result obtained in case V III was a consequence of the recirculation induced inside the cylinder by the feedback force, which reduced the discontinuity at the geometry boundary. Similar results were obtained by Parnaudeau et al. [11] using a direct setting approach. Table 1. Validation results for the 2D inform flow case at Re = 300. Case
Lx
Ly
Xc
Δ
F.V.
< CD >
CLrms
St
I II III IV V VI VII VIII IX Ref.
19 19 19 19 19 19 19 19 45 30
12 18 24 12 12 12 12 12 30 30
6 6 6 8 10 8 8 8 15 15
0.055 0.055 0.055 0.055 0.055 0.028 0.028 0.028 0.028 -
1 1 1 1 1 1 2 3 1 -
1.719 1.704 1.702 1.690 1.676 1.522 1.511 1.476 1.420 1.380
0.807 0.798 0.797 0.788 0.769 0.723 0.713 0.694 0.665 0.650
0.195 0.194 0.194 0.192 0.192 0.203 0.204 0.210 0.205 0.213
1 - solid; 2- Gaussian; 3-half Gaussian
After all these tests, the drag and lit values obtained were higher than those presented by Mittal and Balanchander [9], what leads us to repeat some tests in a similar domain as presented by them (Case IX). In this case, the difference between our results and those presented by Mittal and Balanchander have been reduced. This confirms the importance of using a large domain and locating the cylinder far from the inlet boundary as pointed out in cases I −V.
5 Free planar shear flow results Three DNS for free planar shear flow at Reynolds number 300 are presented. For all cases the cylinder is located at Xc = 8 and due to computational
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limitation they were computed in a small domain with a lower resolution than the one used in the two-dimensional tests. In each case it is possible to define a local Reynolds number Rel (y) = U (y)D/ν associated to the inflow velocity profile. The minimum, mean and maximum values are presented in Tab. 2. However, near the cylinder the local Reynolds number has a smaller variation from Re = 276 − 324 in case II and from Re = 263 − 339 in the case III. In all cases the minimum value is greater than that necessary for mode B to occur. Table 2. Flow configuration and simulation parameters for 3D simulation Case I II III
(Lx , Ly , Lz )
(nx , ny , nz )
Remin
Re
Remax
β
(19,4,12) (19,4,12) (19,4,12)
(343,64,217) (343,64,217) (343,64,217)
300 165 75
300 300 300
300 435 525
0.00 0.15 0.25
Instantaneous isosurfaces for the second invariant of the velocity gradient tensor (Q - criterium) coloured with the streamwise vorticity are shown in Fig. 2. These pictures were taken at the final time of each calculation (T ≈ 124). The plots presented clearly demonstrate the complexity of this flow. In all cases the von K´ arm´an vortices can be observed behind the cylinder even for the highest shear parameter (Fig. 2c). Another common feature for all cases is the formation of longitudinal vortices between the von K´ arm´an vortices. The mean spanwise wavelength of these vortices in the near wake is approximately 1. As already pointed out, the local Reynolds number for all cases is greater than the critical value for mode B, what leads us to identify these longitudinal vortices as being this mode. The animations of the fields show the influence of the shear parameter. Due to the velocity profile asymmetry, the vortex in the higher velocity side (V 1) has a faster convection velocity than the lower one (V2), resulting in a reduction of the distance between them (Fig. 2). This difference of convection velocities also induces an increase of the stretching of longitudinal vortices as the shear parameter increases (see Fig. 4).
Hydrodynamic forces The spanwise variation of mean drag and lift forces is presented in Fig. 3, for the three cases. The length of the time series used is T ≈ 50. The mean value of the drag force seems to be not affect by the increase of the shear parameter. The mean lift force has a different behaviour. It increases in module due to the shear parameter. The negative sign is due to the asymmetry of the incoming velocity profile, which dislocates the stagnation
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Fig. 2. Views showing isosurfaces for the second invariant tensor of velocity gradient (Q = 0.20 for T ≈ 124) coloured with streamwise vorticity (dark grey - positive; light grey - negative). Top, side and bottom views from top to bottom.
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point towards the high velocity side creating an asymmetry in the pressure field with respect to the horizontal axis (x). This results in a vertical force from the high velocity side towards the low velocity side. Similar results were presented by Vitola et al. [14] for two-dimensional simulation. The mean drag (Fig. 3a) shows a periodic-like behaviour in the spanwise direction. The mean spanwise wavelength is 0.75 − 1D which is very similar to the wavelength observed for mode B. This makes us associate the spanwise oscillation of the mean drag with the formation of the streamwise vortices of mode B. This relation is more evident in Fig. 4 where the instantaneous variation of the drag and lift forces are compared with the isosurfaces of the Q-criterium at the same time. The lift force has also a periodic-like behaviour in the spanwise direction, however the fluctuations are smaller than observed for the drag force and the mean spanwise wavelength is ≈ 0.5D. This is confirmed in Fig. 4 where a
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2
z
1 0 −1 −2 2
z
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Fig. 4. Spanwise variation for (a) drag force; (b) lift force and (c) isosurface for the second invariant of the velocity gradient tensor (top - β = 0.00; bottom -β = 0.25).
more uniform behaviour is observed for the lift force compared with the drag. These results indicate that the lift force oscillations are influenced principally by the primary vortices of von K´ arm´an street. The streamwise vortices have a secondary influence, inducing a small variation of the lift in the spanwise direction. Frequency analysis Generally, hot-wire measurements are used to identify the vortex shedding. However, in nonuniform flows, different results could be obtained depending on the probe location. An alternative way is to use the temporal variation of the aerodynamic forces acting on the cylinder to define the vortex shedding frequency [7]. Figures 5a1 and 5b1 show spatio-temporal variations of the drag and lift forces along the cylinder span, while Figs. 5a2 and 5b2 show the spanwise variation of the frequency spectra of those forces at some sections. For all cases the lift force has a constant distribution along the cylinder (Fig. 5b1), which is confirmed by a constant peak of frequency in the spectra for different sections (Fig. 5b2). This confirms the idea that a lift force has a two-dimensional behaviour and that primary vortices of von K´ arm´ an street control the lift behaviour. Another important observation is that the shear parameter has a small effect on the selected frequency of the lift force and the vortex shedding, as it can be observed in Fig. 5b2. The drag force has a completely different behaviour (see Fig. 5a1 and a2). For all cases the peak frequency varies along the cylinder span and in some cases it has more than one peak. In all cases a peak of frequency twice the vortex shedding frequency defined by the lift force analyses is observed, what is associated with the primary vortex structures. However, they have small energy. The origin of the other peaks seems to be related to with the location of the longitudinal vortices as argued in the previous section, although further insight is necessary to clarify this relation.
Vitola M., Schettini, E. B. C., Silvestrini, J.
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Fig. 5. Spanwise spatio-temporal distribution and frequency for (a) drag and (b) lift forces (top - β = 0.00; middle - β = 0.15; bottom - β = 0.2).
Acknowledgments This study was supported by the CNPq.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Goldstein, D., Handler, R., Sirovich, L. (1993) J. Comp. Phys. 105:354–366. Jordan, S., and Fromm, J. (1972) Phys. Fluids 15:972–976. Karniadakis, G., and Triantafyllou, G. (1992) J. Fluid Mech. 238:1–30. Kiya, M., Tamura, H., Arie, M. (1992) J. Fluid Mech. 141:721–735. Lamballais, E., and Silvestrini, J. (2002) J. Turbulence 3:28. Lardeau, S., Lamballais, E., Bonnet, J. (2002) J. Turbulence 3:2. Lei, C., Cheng, L., Armfield, S., Kavanagh, K. (2000) Ocean Eng. 27:1109–1127. Lele, S. J. Comp. Phys. 103:16–42. Mittal, R., and Balachandar, S. (1997) In: FEDSM97, 1–10. Mittal, S. Phys. Fluids 1:177–191. Parnaudeau, P., Lamballais, E., Heitz, E., Silvestrini, J. (2003) In: DLES5 (Munich - Germany). So, R., Liu, Y., Cui, Z., Zang, C., Wang, X. J. (2005) Fluids Struct. 20:373–402. Sumner, D., and Akosile, O. J. (2003) Fluids Struct. 18:441–454. Vitola, M., Schettini, E., Silvestrini, J. (2004) In: X-ENCIT (Rio de Janeiro, Brasil). Williamson, J. (1980) J. Comp. Phys. 35:48–56. Williamson, C. (1996) Annu. Rev. Fluid Mech. 28:477–539.
Part XV
Hybrid RANS-LES Approach
Coupling from LES to RANS using eddy-viscosity models G. Nolin1 , I. Mary1 , L. Ta-Phuoc1 , C. Hinterberger2 , J. Fr¨ ohlich3 1
2
3
ONERA, 29 Avenue de la Division Leclerc, 92322 Chˆ atillon Cedex, France [email protected] Institute for Hydromechanics, University of Karlsruhe, 76128 Karlsruhe, Germany Institute for Technical Chemistry and Polymer Chemistry, University of Karlsruhe, 76128 Karlsruhe, Germany
Summary. The paper proposes two methods to couple an LES zone with a downstream RANS zone. Both employ eddy viscosity models. One is based on a filtering– enrichment procedure, the other on an intermediate damping zone. Simulations of developed plane channel flow at Reτ = 395 are carried out to assess the methods.
1 Introduction LES and RANS both have their respective merits for the computation of turbulent flows. RANS is substantially less costly than LES and will for this reason be applied to the majority of industrial flows for several decades to come. It is, however, not well suited for some important problems of applied aerodynamics such as noise prediction, flow control, transitional flows, etc. Indeed, (U)RANS can only predict the low frequency part of an unsteady signal and suffers from a lack of accuracy in case of separated and/or transitional flows. LES is relatively free of these problems, but leads to high computational cost. Often, however, acoustic sources, separation regions or transition zones, etc., are confined to a comparatively small and well-localized region within the computational domain. A hybrid LES/RANS method hence seems a suitable approach to treat these problems accurately and cost effectively. Several LES/RANS methods have been proposed during the last decade. They can be divided into two categories. The first one is based on a model which is able to switch gradually from RANS to LES model, like DES. While this approach seems favourable for massively separated flow, a critical issue is the “grey zone” where the blending between the two models takes place. This is grid-dependent and may deteriorate the accuracy for weaker flow separation. A second type of methods comprises the zonal approaches. Here, classical LES and RANS models are used in pre-defined sub-zones of the computational domain connected by explicit interfaces which appears more flexible for
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the location of the LES zone and more rigorous from a theoretical point of view. LES can then be confined to critical regions, e.g., those where acoustic sources are generated through large-scale vortices. The goal is to obtain accurate simulations with high frequency information at lower computational cost than full LES. The main difficulty of this technique resides in the exchange of information at the LES/RANS interface between two solutions with very different frequency spectrum. The present paper addresses the situation where the interface between an LES and a RANS zone is located at the downstream end of the LES zone. The first issue is to propagate the information concerning average pressure and velocity from the RANS zone adequately upstream into the LES zone. Depending on the discretization scheme spurious oscillations can occur as a consequence of the coupling. The second issue is to take advantage of the potentially improved solution in the LES part to improve the solution also in the downstream RANS zone in order to correct the lack of accuracy of these models.
2 Turbulence modelling The filtered Navier-Stokes equations and RANS equations are used in the LES and RANS domain, respectively. The Boussinesq hypothesis is employed for both the RANS and the LES models. The former uses the Spalart-Allmaras (SA) model determining μt by a transport equation for a modified νt . For the latter, the selective mixed scale model is used by ONERA, whereas IFH employed the Smagorinsky model. With these choices the coupling through the turbulent viscosity is natural. Note, however, that in the RANS domain the turbulent viscosity represents the totality of turbulent fluctuations whereas only the fine-scale contributions are represented by the eddy viscosity in the LES domain.
3 Coupling strategies 3.1 Overview The present work results from an ongoing collaboration between ONERA and the University of Karlsruhe on zonal LES/RANS methods. Two independent codes were used, both cell-centered second order Finite Volume codes for curvilinear coordinates, one solving the compressible (ONERA) the other solving the incompressible equations (IFH). In both codes different blocks of the multi-block grid can be defined as LES or RANS blocks. Different strategies have been developed in both groups to couple RANS and LES domains. Three issues must be addressed in this respect: the unsteady LES solution has to provide steady inflow data for the downstream RANS domain, the RANS domain must be able to influence the upstream LES domain, and
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finally upstream boundary conditions must be provided for the SA transport equation. The developed strategies are conceived for arbitrary geometries. For simplicity they are described here using the two-dimensional test case chosen for assessment (Fig. 1). 3.2 Explicit filtering at the block interface The approach used by ONERA relies on explicit filtering and enrichment at the block interfaces between the LES domain and the RANS domain (see [5]). Different grids are employed in both types of domains according to the respective resolution requirements. For LES the traditional rules are followed, in the present case to obtain a wall-resolving grid, whereas the RANS grid is coarser in the streamwise and spanwise direction and two-dimensional (2D) since the geometry is 2D. At the LES/RANS interface, different techniques based on space and/or time filtering have been assessed in [5]. It appears that both, space and time filtering must be used simultaneously to filter the LES solution in a satisfactory way to generate inflow conditions for the RANS domain. Outflow conditions are implemented for the LES domain in order to couple the steady-state solution in the 2D RANS domain to the unsteady three-dimensional (3D) LES solution as follows. A first order extrapolation is used to generate high frequency information in the ghost cells of the LES domain. This leads to the 0 1 ghost 1 following definition of Qghost LES as QLES = QLES + C (QLES − QLES ) where Q1LES represents the value of QLES in the first row of the LES grid and C is a
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scalar coefficient. QLES is determined by a second order interpolation of the RANS solution and the filtered LES solution. The basic idea of this procedure is to add to the RANS solution QRAN S spectral information similar to the solution in the LES block. Two values of C have been assessed in the present study: C = 0 (no enrichment) and C = 0.95. Further details are described in [3]. The second issue is the turbulence modelling. The 2D transport equation for the turbulent viscosity needs to be provided with suitable inflow conditions. It turned out that for reasons of implementation it is easiest to solve the SA transport equation on a 2D grid covering the 3D domain as well (Fig. 1a). Here, the filtered LES field described above is used in the convection term of the 2D SA equation. The computed viscosity does not impact on the LES, its only purpose is to provide a suitable value at the LES/RANS interface. Hence, νt = 0 was used at the upstream boundary of this domain. Based on the idea that the LES solution is more reliable than the RANS solution in the same part of the computational domain, it was tried to reconstruct a turbulent RANS-type viscosity from the LES without using the SA model and to provide this value for the SA model at the LES/RANS interface. This was done deducing a RANS-type equation by averaging the LES data as described above and solving it for the eddy viscosity, denoted μrec t . More details are given in [5]. 3.3 Coupling by a three-dimensional damping zone At IFH, the LESOCC2 code developed at this institute [1] was enhanced by grid interfaces allowing different resolution in the blocks on either side. To ensure simplicity of the method it was decided to avoid any explicit averaging operations in the LES domain or at interfaces. Instead, a 3D URANS damping zone between the 3D LES and the 2D RANS zone was used in a straightforward manner (Fig. 1b). In this intermediate block, the only change with respect to the LES zone is the use of a RANS viscosity instead of the SGS expression. This turbulent viscosity is obtained from a 3D computation of the SA transport equation using the LES velocity in the convection term and hence running in unsteady mode. The higher viscosity in the 3D URANS zone damps the 3D unsteady fluctuations and at the downstream end provides a steady state solution which can be coupled to the 2D RANS solution. Reversely, the average pressure and velocity of the 2D RANS zone is communicated to the upstream LES. For simplicity the SA equation is also solved in the inflow generator and the main LES zone where the average velocity, determined by incremental averaging during the simulation, is used in the SAequation. Since an additional 3D transport equation is used here and in the main LES zone as well as a 3D damping zone, this approach is more costly. The coding requirements, on the other hand are negligible.
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4 Channel flow test case The plane channel flow at a Reynolds number of Reτ = 395, was chosen to test the methods described. The Reynolds number is based on the friction velocity uτ and the half hight h of the channel. To allow best comparison computational domain and grid were identical in all simulations. The domain features three zones as displayed in Fig. 1. The first one is a classical LES zone with periodic conditions in streamwise and spanwise direction which is used as inflow generator. Its size is 2πh × 2h × πh in the streamwise (x), wall-normal (y) and spanwise (z) direction, respectively. Here, 80 × 100 × 80 points are used for discretization. The resulting resolution of y1+ = 1.4, Δ+ x = 32, Δ+ z = 16 is relatively coarse for a wall-resolving LES [2] and was selected to enhance possible differences. The second domain is the principal LES zone, with inflow and outflow conditions in streamwise direction using the same grid. It extends from x/h = 2π to x/h = 4π. The third zone, computed with a RANS model, extends from x/h = 2π to x/h = 8π. This zone is a two-dimensional zone in ONERA simulations, whereas IFH has decomposed this part into a 3D RANS domain (from x/h = 4π to x/h = 5π) and a 2D RANS domain (from x/h = 5π to x/h = 8π). Here, the grid in x is gradually stretched. The streamwise velocity component is denoted u. The present flow is fully developed. Hence, any gradient of the solution in x−direction results from physical or numerical modelling which makes the configuration a very severe and good test for LES/RANS coupling.
5 Analysis of results 5.1 Periodic channel To check that both codes give similar results for the canonical case, the solutions in the streamwise periodic inflow zone are compared. Mean and rms
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Fig. 3. Streamwise evolution of: a) average streamwise velocity component u in the center of the channel (since streamwise changes are very small, curves have been normalized to u = 1 at x = 0); b) urms in the center of the channel; c) wall shear stress coefficient cf ; d) mean turbulent viscosity at y = 0.1, i.e. y + ≈ 40.
values of the streamwise velocity component are plotted in Fig. 2 and compared with the DNS of [4]. The results of IFH are in very good agreement with the DNS, whereas those of ONERA are slightly less accurate for the mean profile. These differences result from different discretization schemes for the convective term, the use of compressible versus incompressible Navier-Stokes equations and different SGS models and remain very small. Hence, an assessment of the coupling methods described above can be carried out with confidence. 5.2 Spatially evolving flow and coupling to downstream RANS Since the mesh resolution and the domain size in the principal LES zone are the same as in the periodic zone used as inflow condition generator, the statistical quantities (velocity and Reynolds stresses) should remain unchanged in
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streamwise direction if the coupling of the conservative variables works well. Such data is presented in Fig. 3. The variations of u in Fig. 3a are insignificant since they are less than 0.5%. The small discontinuity at the inflow plane of the LES domain in the ONERA computation is due to a non-reflective condition at the inflow plane (compressible solver). The streamwise evolution of urms is plotted in Fig. 3b along the center line. All simulations give satisfactory results, except those of ONERA without enrichment at the outflow of the LES domain, C=0. In this case, the level of turbulent fluctuations slightly decreases just upstream of the LES/RANS interface over a distance of the order of h. Results obtained by IFH with the 3D URANS domain and ONERA with the enrichment, C=0.95, are indistinguishable in the plot, which shows that both methods are accurate in this respect. Downstream of the interface in the RANS domain, the results from the two methods differ. The filtering applied to the LES field in the ONERA approach yields resolved turbulent fluctuations close to zero, as expected for the simulation of a channel flow with a RANS model. In the IFH results, the turbulent fluctuations, which rapidly decrease in the 3D RANS domain due to the dissipation of the SA viscosity, level off at a small but non-zero value in the downstream 2D RANS domain. At the interface between the 3D and the 2D RANS zone where both domains are coupled without further measures such as the enrichment procedure in the ONERA approach, small wiggles are observed. These are however not of importance as they occur far from the principal LES domain. Differences are larger for the friction coefficient displayed in Fig. 3c. It is proportional to ux at the first grid point from the wall. The computation of ONERA with explicit filtering at the interface shows a fairly constant value in the LES zones. Towards the RANS zone a substantial increase by about 12% is observed which is due to the difference between the LES and the RANS model. The solution in the LES domain is affected by this jump only at the very end. In the IFH simulation, the jump of cf at the LES/RANS interface is about twice as large. It occurs mainly in the 3D RANS domain and may be attributed to the fact that the increased viscosity moves the flow out of equilibrium. The immediate switch from the LES viscosity to the higher RANS viscosity introduces wiggles near the downstream end of the LES domain which can presumably be remedied by a more gradual increase. A similar phenomenon appears at the transition from the 3D URANS zone to the 2D RANS zone. These are acceptable since they occur remote from the LES zone. The streamwise evolution of the turbulent viscosity is presented in Fig. 3d. The ONERA result reflects the viscosity treatment. If the SA equation is solved in the main LES zone with the ONERA approach, the turbulent viscosity does almost not depend on the streamwise location. If it is determined by reconstruction a discontinuity appears in this quantity at the LES/RANS interface. In both cases a slight kink is observed in the RANS zone in Fig. 3d related to an adjustment of the velocity profile (see Fig. 3a,c). This is accomplished after about 2h. In the IFH data, the curves of μt do not extend
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into the LES zone for technical reasons. At the LES/RANS interface, a jump in μt is generated close to the wall (Fig. 3d). In the center of the channel a smooth variation is observed (not shown here). The eddy viscosity is larger in the 3D URANS zone as the production term in the SA-equation is determined from the gradients of the instantaneous flow and not the average flow. In the RANS zone, a higher value of the eddy viscosity persists than in the ONERA simulation which may also result from the fact that a compressible and an incompressible simulation are compared.
6 Conclusions Different zonal LES/RANS coupling techniques have been assessed for a plane channel flow, where a RANS zone is located downstream of an LES zone. All methods give satisfactory results in the LES domain, showing that the simulation is weakly affected by the use of the different outflow conditions tested in this study. The explicit filtering is somewhat more rigorous while the 3D damping zone is appealing due to its simplicity. In the downstream RANS region, results are less satisfactory (Fig. 3c), since the accuracy improvement related to the use of the LES does not propagate far downstream in the RANS domain. Relaxation to the pure RANS solution takes place over a few h since the wall-normal direction dominates the behaviour of the RANS solution. Hence, an embedded LES in the upstream part can improve the solution in the LES part of the domain but does not necessarily improve the solution in a downstream RANS zone. The current fully developed flow, however, is a very severe test case in this respect. If, e.g., the LES zone leads to an improved determination of the average flow at the interface compared to a pure RANS computation, the downstream RANS solution will be improved, not by improved turbulence quantities but through a better mean flow at the interface. Acknowledgements: This work was funded through the DFG-CNRS programme LES for Complex Flows and ONERA.
References 1. C. Hinterberger, Dreidimensionale und tiefengemittelte Large–Eddy–Simulation von Flachwasserstr¨ omungen, PhD thesis, University of Karlsruhe, 2004. 2. J. Fr¨ ohlich, W. Rodi, Introduction to Large Eddy Simulation of turbulent flows, Chapter 8 in B. Lauder, N. Sandham (eds), Closure Strategies for Turbulent and Transitional Flows, Cambridge University Press, 2000. 3. I. Mary, LES of vortex breakdown behind a delta wing, Int. J. Heat Fluid Flow, 24(4):596–605, 2003. 4. R. D. Moser, J. Kim, N. N. Mansour, DNS of Turbulent Channel Flow up to Reτ = 590, Phys. Fluids, 11:943–945, 1999. 5. G. Nolin, I. Mary, L. Ta-Phuoc, RANS eddy viscosity reconstruction from LES flow field, AIAA Paper 2005-4998, 2005.
SAS Turbulence Modelling of Technical Flows F.R. Menter and Y. Egorov ANSYS Germany GmbH, Staudenfeldweg 12, D-83624 Otterfing, Germany [email protected], [email protected]
Summary. Scale-Adaptive Simulation (SAS) allows the simulation of unsteady flows with both RANS and LES content in a single model environment. As SAS formulations use the von Karman length scale as a second external scale, they can automatically adjust to resolved features in the flow. As a result, SAS develops LESlike solutions in unsteady regions, without a resort to the local grid spacing. In the present article, a definition of the SAS modelling concept will be given. In addition, issues of the numerical treatment of the source terms will be discussed. Results will be shown for the flow in a 3D cavity and for a combustion chamber.
1 Introduction The concept of Scale-Adaptive Simulation (SAS) [1, 2, 3, 4] allows the simulation of unsteady turbulent flows without the limitations of most Unsteady RANS (URANS) models. Contrary to standard URANS, SAS provides two independent scales to the source terms of the underlying two-equation model. In addition to the standard input in form of the velocity gradient tensor, ∂Ui /∂xj , SAS models compute a second scale from the second derivative of the velocity field. The resulting length scale is the well known von Karman length scale LvK . The introduction of LvK allows the model to react more dynamically to resolved scales in the flow field which cannot be handled by standard URANS models. As a result, SAS offers a single framework, which covers steady state RANS as well as LES regions, without an explicit switch in the model formulation. SAS therefore offers an attractive framework for many “multi-scale” flow problems encountered in industrial CFD. The functionality of SAS is similar to Detached Eddy Simulation (DES) [5, 6]. It provides a steady state (or mildly unsteady) solution in stable flow regions (like boundary layers), and unsteady structures in unsteady regions within a single model framework. The difference is that the LES activity in DES is enforced by the grid limiter, whereas SAS allows a breakdown of the large unsteady structures by adapting the turbulence model to the locally
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resolved length scale. In order of avoiding multiple definitions and naming conventions, as observed in DES, the following definition is given for SAS models. 1. SAS modeling is based on the use of a second mechanical scale in the source/sink terms of the underlying turbulence model. In addition to the standard input from the momentum equations in the form of first velocity gradients (strain rate tensor, vorticity tensor, . . . ) SAS models rely on a second scale, in the form of higher velocity derivatives (second derivatives). 2. SAS models satisfy the following requirements: a) Provide proper RANS performance in stable flow regions. b) Allow the break-up of large unsteady structures into a turbulent spectrum. c) Provide proper damping of resolved turbulence at the high wave number end of the spectrum (resolution limit of the grid). 3. Functions 2a and 2b are achieved without an explicit grid or time step dependency in the model. Naturally, function 2c has to be based on information on the grid spacing, other information concerning the resolution limit (dynamic LES model, etc.), or the numerical method (MILES damping etc.). In the following, two different √ physical formulations of the SAS model will be listed. The first is the k − kL model, which is the natural two-equation model environment, for which the SAS concept was developed [2]. The second formulation is for the SST model, which was augmented by Menter and Egorov √ [3] by the transformation of the SAS terms from the k − kL model. The numerical treatment of the source terms will then be discussed for the SST model. It should be noted that SAS modeling is a relatively new concept. The following formulations reflect our current best understanding of an optimal model. They are the subject of intense research and will most likely undergo some changes in the future.
2 The k −
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The details of the derivation of the model are given in [2] and are therefore not repeated here. The basic idea behind the model is to start from Rotta’s k − kL model [7] and to re-evaluate the critical terms, which distinguish this model from standard two-equation models. It was shown in [2] that some of the assumptions made by Rotta in the derivation of the k − kL model were overly restrictive and can be relaxed. This results in the appearance of the second derivative of the velocity field in the length-scale equation, instead of the third √ as in Rotta’s model. The current version is formulated as √ derivative a k − kL (Φ = kL) model:
SAS Turbulence Modelling of Technical Flows
μt k2 ∂ρk + ∇ · ρUk = Pk − c3/4 +∇· ∇k μ ρ ∂t Φ σk Φ ∂ρΦ L μt + ∇ · ρUΦ = Pk ζ1 − ζ2 κ − ζ3 ρk + ∇ · ∇Φ ∂t k LvK σΦ ) S = 2 · Sij Sij ; Sij = ((∇U)ij + (∇U)ji ) /2 νt = c1/4 μ Φ; LvK = κ · S/U ; U = ∇2 U
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where S is the invariant of the strain rate tensor and Pk is the production rate of the turbulent kinetic energy. The model constants are given in [2]. The SAS-relevant term in the equation for Φ is the term featuring the von Karman length scale LvK . As a result of this term, the predicted turbulent length-scale L is largely proportional to the von Karman length-scale: L ∼ LvK = κ · S/U (5) √ The transformation of the k − kL model, with some modifications to leave the SST model undisturbed in boundary layers (model function 2a in Introduction), results in an additional term, which is simply added to the right hand side of the ω-equation of the SST model [3, 8]: 2 2 2k |∇ω| |∇k| 2 L ˜ , 0 (6) − · max , QSST −SAS = ρFSAS · max ζ2 κS LvK σΦ ω2 k2 The constants are taken from the k −
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3 Discretisation in CFX Solver The SST-SAS model has been implemented in the commercial CFD code CFX-10. Its solver is based on a finite volume formulation for structured and unstructured grids. In the case of hybrid RANS/LES or SAS simulations, the numerical switch between a second order upwind and a second order central scheme as proposed by Strelets [6] is applied for the advection terms. Switching to the non-dissipative scheme is required to let the SAS model properly resolve turbulent structures down to the grid limit (function 2b). Satisfying the function 2c by the current implementation of the SAS model essentially depends on the way of discretising the source terms of the turbulence model. One of the main issues here is the numerical treatment of the strain rate S, which appears in the production of the turbulent kinetic energy Pk and in the SAS term QSST −SAS . (Calibration results for the SST-SAS model for decaying isotropic turbulence can be found in [3]). The infrastructure of the CFX solver provides different schemes for the velocity gradients, with three of them being considered for S: the standard nodal values SN , an average of the ) surrounding elements S (element averaged values), and a quadratic average
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The above treatment of the strain rate provides function 2c (Introduction). Assuming source term equilibrium conditions, the eddy viscosity resulting from the SST-SAS model is given by [3]: 2 β 1 −α · ρL2vK S (9) μt = FSAS κζ˜2 cμ This makes clear, that the high wave number damping depends on the specific implementation of S. For a 1D velocity field U = U (y) the two different discretisations SN and S of the strain rate S give: Ui+1 − Ui−1 ; S = 1 Ui+1 − Ui + Ui − Ui−1 (10) SN = 2Δy 2 Δy Δy Clearly both formulations are second order in space and are nearly equivalent in a smoothly resolved flow field. However in the extreme case of the odd-even grid oscillation Ui = ±U0 one gets: SN = 0 −→ LvK = 0
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4 Industrial Applications 4.1 3-D Acoustic Cavity Air flow past a 3-D rectangular shallow cavity is calculated in this test, with the cavity geometry and flow conditions corresponding to the M219 experimental test case of [9]. The geometry dimensions of the cavity are L × W × D = 5 × 1 × 1 (length, width, and depth), with the depth of 4 (all sizes in inches). The side boundaries are treated as symmetry planes, the top boundary is a far-field boundary, and all the solid surfaces are adiabatic non-slip walls. The amount of ambient space, included into the computational domain, is: 31 from the inlet to the cavity leading edge, 21 from the cavity trailing edge to the outlet, 68 from the cavity opening level to the top boundary. Space equal to a half of the cavity width is left between each of the side boundaries and the correspondent side edge of the cavity. The inlet Mach number is 0.85. The grid consists of 1.05 · 106 hexahedral elements. The time step for the simulation is 2 · 10−5 s, which is 18 times less than the hydrodynamic time scale based on the inlet velocity and the cavity depth. 104 time steps have been computed. Figure 2 shows the turbulent structures, produced by the SST-SAS model. Figure 3 shows the power spectral density of the transient pressure signal, calculated and measured at the K29 sensor location on the cavity bottom near the downstream wall. Despite the relatively coarse grid and short calculated physical time, the main acoustic modes are predicted in good agreement with the experiment. The simulation is currently running on a finer grid. 4.2 ITS Combustion Chamber The instabilities in gas turbine combustion chambers are of strong technical interest, as they can produce noise and also compromise the structural Pressure PSD, Pa 2 106 105 104 103
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integrity of the chamber. The unsteadiness can be caused by different mechanisms like flow instabilities introduced by the high swirl in the burner or thermo-acoustic instabilities from combustion itself. The SST-SAS model is applied to the ITS test rig with a single burner. This burner is typical for industrial gas turbine combustion systems. The test rig was built as a rectangular combustion chamber. This configuration was investigated experimentally by Schildmacher and Koch [10]. A lean preheated methane-air mixture is supplied through a ring inlet with the external diameter of 120 mm, which encircles an additional axial inlet of the preheated dilution air. A partially premixed combustion model [11] available in CFX is used for this simulation. The grid consists of 3.6 · 106 tetrahedral elements, corresponding to 6 · 105 control volumes of the dual mesh. As the experimental data are only available in the central part of the chamber, the ITS combustion chamber serves mainly as a test for the LEScapabilities of the SAS approach. It should be noted however, that industrial combustion chambers are very complex, making a complete LES simulation impractical. The flow structures for the cold and the hot flow simulation at a given instance in time are shown in Figures 4 and 5. Experience shows that RANS models are not reliable in predicting the change in flow topology indicated by that figure. This can be seen in more detail in Figures 6–10 showing the radial distributions of the statistically averaged velocity and temperature at the distance from the inlet, approximately equal to one ring diameter (note that the SST and the k − models are virtually identical for free shear flows). Superior accuracy of SAS results relative to the RANS simulation confirms, that SAS is a viable method for such a complex flow.
Fig. 4. ITS combustion chamber, nonreacting. Isosurface Ω 2 − S 2 = 105 s−2 .
Fig. 5. ITS combustion chamber, reacting. Isosurface Ω 2 − S 2 = 105 s−2 .
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5 Acknowledgment The current work was partially supported by the EU within the research projects DESIDER (Detached Eddy Simulation for Industrial Aerodynamics) under contract No. AST3-CT-200-502842 (http://cfd.me.umist.ac.uk/desider) and PRECCINSTA (ENK5-CT-2000-00060).
References 1. Menter F. R., Kuntz M., Bender R. (2003) A scale-adaptive simulation model for turbulent flow predictions. AIAA Paper 2003-0767 2. Menter F. R., Egorov Y. (2005) Re-visiting the turbulent scale equation. In: Heinemann H.-J. (ed) Proc. IUTAM Symposium “One hundred years of boundary layer research”. G¨ ottingen, Germany, 12-14 August, 2004. Kluwer Academic Publishers (in press) 3. Menter F. R., Egorov Y. (2005) A Scale-adaptive simulation model using twoequation models. AIAA paper 2005–1095 4. Menter F. R., Egorov Y. (2005) Turbulence models based on the length-scale equation. In: Humphrey J .A. C., Gatski T. B., Eaton J. K., Friedrich R., Kasagi N., Leschziner M. A. (eds) Proc. Fourth International Symposium on Turbulence and Shear Flow Phenomena. Williamsburg, VA USA, 27–29 June, 2005. 5. Spalart P. R. (2000) Strategies for turbulence modelling and simulations. Int J Heat Fluid Flow 21:252–263 6. Strelets M. (2001) Detached eddy simulation of massively separated flows. AIAA paper 2001-0879 7. Rotta J. C. (1972) Turbulente Str¨ omungen. Teubner Verlag, Stuttgart 8. Menter F. R. (1994) Two-equation eddy-viscosity turbulence models for engineering applications. AIAA Journal 32(8):269–289 9. Henshaw M. J. de C. (2000) M219 cavity case. In: Verification and validation data for computational unsteady aerodynamics, Tech. Rep. RTO-TR-26, AC/323/(AVT)TP/19:453–472 10. Schildmacher K.-U., Koch R., Wittig S., Krebs W., Hoffmann S. (2000) Experimental Investigations of the Temporal Air-Fuel Mixing Fluctuations and Cold Flow Instabilities of a Premixing Gas Turbine Burner. ASME Paper 2000-GT0084 11. Zimont V. L., Biagioli F., Syed K. (2001) Modelling turbulent premixed combustion in the inter-mediate steady propagation regime. Progress in Computational Fluid Dynamics 1:14–28
Zonal LES/RANS modelling of separated flow around a three-dimensional hill F. Tessicini, N. Li and M. A. Leschziner Department of Aeronautics, Imperial College, London, UK [email protected]
Summary. A complex three-dimensional flow, separating from a hill-shaped circular obstacle in a duct, is studied numerically with variations of large eddy simulation. Two zonal near-wall approximations, one based on simple log laws and the other involving the solution of parabolized Navier-Stokes equations, are applied and contrasted against a fine-grid LES solution as well as experiments. The wall models, although very simple, are shown to yield satisfactory results.
1 Introduction The ability of computational schemes to predict three-dimensional separation from curved surfaces is of much practical interest, because of the frequency of its occurrence and its operational impact in a wide range of applications in external aerodynamics, turbomachinery and ship hydrodynamics. Unlike separation from a sharp edge, that from a curved surface is always characterised by an intermittent collection of separated and attached patches on the surface, which form, move and disappear rapidly in time and space over a substantial proportion of the surface. Moreover, large vortical structure are ejected intermittently from the separation area, and the boundary layer merging into the separation zone is subject to strong skewing and normal straining, with consequent major changes to its turbulence structure, relative to a standard boundary layer. In such circumstances, turbulence is distinctly non-local and out of equilibrium, and its dynamics are important. The flow around a hill-shaped obstacle that is placed in a duct, as shown in Fig. 1, is a generic configuration representative of the conditions described above. This flow, at hill-height Reynolds number of 130,000, has been the subject of extensive experimental studies by Simpson et al. [7] and Byun & Simpson [3]. In a time-averaged sense, separation from the hill is in the form of a pair of strong vortices ejected from the leeward surface, between which a thin recirculation bubble is nesting. There have been a number of recent attempts to compute this flow with statistical closures. The study by Wang et al [13] is, arguably, the best
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representative of such attempts, for it involved the careful application of several non-linear eddy-viscosity and second-moment-closure models to this flow. In none of the applications reported did RANS models yield solutions in satisfactory agreement with the experimental observations. Typically, separation is predicted to occur far too early, reattachment occurs much too far downstream, and recovery is seriously delayed. All these defects suggest that the dynamics associated with large-scale turbulent motions are highly influential, and these cannot be captured effectively within the RANS framework. The above suggests that LES is the appropriate approach to adopt. However, LES faces a serious resource challenge, especially at high Reynolds numbers, because of the importance of resolving accurately the near-wall processes, likely to be influential in any flow separating from a continuous surface. This challenge is brought to light in several recent attempts to compute with LES the present flow (Benhamadouche et al [2], Li et al [5], Davidson [4]). It thus appears that RANS-LES methods need to be exploited if the enormous resource requirements of wall-resolving LES, estimated to require 30-50 million nodes for the present flow, are to be avoided. Two such ’zonal’ approaches are investigated herein. In previous papers, Temmerman et al [9] and Tessicini et al [12] report on the development and application to several flows of two distinctly different approximate approaches to near-wall modelling: a hybrid RANS-LES method and a zonal two-layer method, both pursued in preference to the well-known DES strategy of Spalart et al [8] for reasons given in the above papers. One of these - the zonal variant - is applied herein to the 3D-hill flow, alongside log-law-based wall functions. The performance of the approximate methods, applied over a mesh of 3.5 million nodes, is contrasted against a pure LES computation, performed on a mesh of 9.6 million nodes. Y Z
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2 The Zonal Two-layer Approach The objective of the zonal strategy is to provide the LES region with the wall-shear stress, extracted from a separate modelling process applied to the near-wall layer. The wall-shear stress can be determined from an algebraic law-of-the-wall model or from differential equations solved on a near-walllayer grid refined in the wall-normal direction - an approach referred to as ”two-layer wall modelling”. The method, shown schematically in Fig. 2, was
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originally proposed by Balaras and Benocci [1]. At solid boundaries, the LES equations are solved up to a near-wall node, which is located, typically, at y + = 50. From this node to the wall, a refined mesh is embedded into the main flow, and the following simplified turbulent boundary-layer equations are solved: ˜i U ˜i ˜j ˜i ∂ρU dP˜ ∂ ∂U ∂ρU + [(μ + μt ) ] i = 1, 3 + = ∂t ∂xj dxi ∂y ∂y
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where y denotes the direction normal to the wall and i identify the wallparallel directions (1 and 3). In the present study, the left-hand-side terms have been nullified, yielding a 1D form of equation (1). The effects of including the remaining terms are being investigated and will be reported in a future paper. The eddy viscosity μt is here obtained from a mixing-length model with near-wall damping, as done by Wang and Moin [14]: + μt + = κyw (1 − e−yw /A )2 μ
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Fig. 2. Two-layer zonal model.
3 The Computational Framework The computational framework is a block-structured finite-volume method with non-orthogonal-mesh capabilities allowing the mesh to be body-fitted. The scheme is second-order accurate in space, based on central differencing for advection and diffusion. Time-marching is based on a fractional-step method, with the time derivative being discretized by a second-order backward-biased approximation. The flux terms are advanced explicitly using the AdamsBashforth method. The provisional velocity field is then corrected via the pressure gradient by a projection onto a divergence-free velocity field. To this end, the pressure is computed as a solution to the pressure-Poisson equation by means of a V-cycle multigrid algorithm operating in conjunction with a successive line over-relaxation scheme. The code is fully parallelised using MPI and was run on several multi-processor computers with up to 128 processors.
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4 Application to 3D Hill 4.1 Computational representation The 3D circular hill, of height-to-base ratio of 4, is located on one wall of a duct, as shown in Fig. 1. The computational domain is 16H ×3.205H ×11.67H, encompassing the whole duct, H being the hill height. The hill-crest is 4H downstream of the inlet plane. The mean inlet flow, containing a boundary layer of thickness 0.5H, was taken from a RANS simulation that accurately matched the experimental conditions [13]. The spectral content was then generated separately by superposing onto the mean profile fluctuations taken from a separate precursor boundary-layer simulation, performed with the quasiperiodic recycling method at Reθ = 1700, and rescaling this for Reθ = 7000. The fluctuations only roughly match the experimental conditions at the inlet. However, specifying reasonably realistic spectral representation proved to be decisively superior to simply using uncorrelated fluctuations, even if the latter could be matched better to the experimental profile of the turbulence energy. Because the upper and side walls of the domain were far away from the hill, the spectral state of the boundary layers along these walls was ignored. y + of interface or near-wall-node location Under-resolved LES 448 × 112 × 192 Dynamic 5-10 Under-resolved LES 448 × 112 × 192 Smagorinsky 5-10 Log-law Wall Function 192 × 96 × 192 Dynamic 20-40 Two-layer Zonal 192 × 96 × 192 Dynamic 20-40 Case
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Table 1. Grids, modelling practices and near-wall distances. The Smagorinsky model is combined with van Driest damping. The y + values for the under-resolved LES identify the wall distance of the wall-nearest nodes.
Table 1 summarizes the simulations performed. As the table shows, the mesh of 9.6 million nodes used for the LES computation is still insufficient for an adequate near-wall resolution. Also, the near-wall cell-aspect ratio is very high. Thus, although the simulation is found to capture the major flow features reasonably well, its quality is questionable. In particular, an undesirably strong sensitivity to subgrid-scale modelling is observed, as will be demonstrated below. For both the log-law and zonal implementations, the near-wall nodes are at around y + = 20 − 40, thus resulting in a much lower cell-aspect ratio of the LES grid near the wall. The above range of y + values may appear low. The lowest y + values arise predominantly in the leeward region of the hill, where the velocity is low, but where the turbulence level is high. Thus, here, y + , when defined by reference to the wall shear stress, is misleadingly low. In fact, the near-wall flow is far more turbulent than suggested by y + = O(20), and use of the turbulence energy as the velocity scale implies a far higher value. As a consequence of the reduced mesh size - 36% of that used by the pure LES - a saving of some 80% of the CPU time arose. A mesh of 1.5 million nodes has also been used, and this fails to resolve any
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separation, unless combined with the wall approximations. Results for this mesh are reported in Tessicini et al. [11] 4.2 Results Fig. 3 - 6 show, respectively, pressure-coefficient distributions, velocity vector fields across the hill/duct symmetry plane, flow topology in the leeward side of the hill and velocity profiles on a downstream plane at x/H = 3.63. The results are compared with experimental data of Simpson et al. [7], Byun & Simpson [3] and Ma & Simpson [6]. The LES solutions are time-averaged over 80 non-dimensional time units, corresponding to 5 flow-through times. The flow accelerates on the windward side and decelerates on the leeward side of the hill, resulting in the corresponding sharp decrease of pressure before the hill-crest and then increase after it, shown in Fig. 3(a). The experiments uncover the presence of a small and thin recirculation zone across the midplane on the leeward side, with separation occuring at x/H = 0.96 and reattaches around the foot of the hill, at x/H = 2.0. This separation is reflected by an inflexion in the pressure-coefficient distribution around x/H = 1.0. Both wall-model approximations and the full LES with the dynamic SGS model capture the variation in the pressure recovery associated with the weak separation. The full LES with the Smagorinsky model, on the other hand, performs distinctly less well, predicting a separation that is excessively large and strong, so that the pressure recovers much too slowly. The sensitivity of SGS modelling is normally weak in highly-resolved simulations in which the model contribution of the subgrid-scale viscosity is relatively small. However, when the model contribution is significant, as it is with an under-resolving mesh, it is preferable to use the dynamic model, as this represents better the wall-asymptotic variation of the turbulent viscosity. Fig. 4 gives comparisons between the computed and experimental velocityvector fields across the hill centre-plane. The lines identify the edges of the reverse-flow zones. Apart from the full LES with the Smagorinsky model, the simulations predict the separation region reasonably well, and this is remark0.4
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Fig. 5 demonstrates that the simulations give a broadly faithful representation of the flow topology. The experiment shows a pair of counter-rotating vortices detaching from the leeward surface of the hill and approximately centred at x/H = 1.2 and z/H = ±0.7. This feature is well captured by the simulations, especially by those involving wall models. Consistent with the previous results, the separation and reattachment lines (the latter only clearly visible in the plots relating to the two wall models) can be identified through the limiting streaklines that are approximately normal to the x -axis (these streaklines were constructed with the wall-parallel velocity components closest to the wall, as also done in the experiment). Because of the strong spatial and temporal variations of the locations of the detaching vortices, a longer integration period is required to obtain smoother patterns. This is obviously especially costly in the case of the fine-grid simulation. Finally, Fig. 6 presents wall-normal streamwise- and spanwise-velocity profiles at x/H = 3.63. At this streamwise location, the flow is attached and features a pair of counter-rotating vortices which redistribute streamwise momentum by transverse advection. The U -velocity profiles contain three sets of experimental data, two obtained with LDA on either side of the symmetry plane and one determined with HWA. Some significant differences between the LDA and HWA data are noticeable, perhaps due to the inevitable disturbances caused by the physical nature of the HWA probes. The computed U -velocity profiles are in fairly good agreement with the experiments, especially with the HWA data, although close to the wall, the simulated flow generally appears to recover more quickly than the measured one, reflecting the slightly early reattachment. Away from the wall, in the range y/H = 0.2 − 0.4, all velocity profiles feature inflexion regions, reflecting the wake associated with
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Fig. 5. Flow topology on the leeward side. The streakline patterns are extracted from the velocity field at the nodes closest to the hill surfaces. Exp. (LDV) Exp. (LDV) Log Law Two-layer Zonal LES (Dynamic) LES (Smag.) Exp. (HWA)
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the upstream separation, as well as the upward transport of slow fluid by the transverse vortical motion, which is clearly implied by the W -profiles. However, further away from the wall, beyond y/H = 0.4, the simulations fail to reproduced the distinctive low-velocity region measured with the LDA. Nor is this feature returned by the HWA data, a fact that makes it difficult to judge the quality of the computational profiles. As regards the W -profiles, the best results are obtained with the zonal treatment, while the pure LES results significantly under-estimate the transverse near-wall motion at the outer spanwise location. The latter variance reflects, in part, defects in the separation process, as seen in the left-most plot of Fig. 5, but may also partly be a consequence of insufficient convergence of the related 9.6-million-grid solutions. Notwithstanding these differences, all the results of the present simulations are far superior to those obtained for this flow in earlier RANS studies.
5 Conclusions The simulations reported herein demonstrate that separation from curved surfaces in, supposedly, highly challenging high Reynolds-number conditions
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can be resolved well with relatively simple near-wall approximations intended to allow the use of relatively coarse near-wall meshes. In the present 3D-hill flow, a wall-resolving mesh would need to contain about 30 million nodes, yet the use of a much coarser mesh of 3.5 million nodes, in combination with a simple zonal scheme, has been found to reproduce well all essential aspects of the separation and reattachment processes. Although the approach offers the advantage of using relatively modest cell-aspect ratios near the wall, the close agreement achieved with the experimental data is a surprising outcome, in view of the expected role of near-wall turbulence in the separation process, and current studies aim to establish the sensitivity of the solution to refinements in the zonal model and to the use of alternative hybrid RANS-LES strategies.
6 Acknowledgement This work was undertaken, in part, within the DESider project (Detached Eddy Simulation for Industrial Aerodynamics). The project is funded by the European Union and administrated by the CEC, Research DirectorateGeneral, Growth Programme, under Contract No. AST3-CT-2003-502842.
References 1. Balaras E. and Benocci C. (1994) In: Applications of Direct and Large Eddy Simulation, AGARD. pp. 2-1-2-6. 2. Benhamadouche S., Uribe J., Jarrin N. and Laurence D. (2005) In: 4th International Symposium on Turbulence and Shear Flow Phenomena (TSFP4), Williamsburg. pp. 325-330. 3. Byun G., and Simpson R.L. (2005) AIAA Paper, 2005-0113. 4. Davidson L. (2005) In: 11th ERCOFTAC/IAHR Workshop on Refined Turbulence Modelling, Gothenburg. 5. Li N., Wang C., Avdis A., Leschziner M.A. and Temmerman L. (2005) In: 4th International Symposium on Turbulence and Shear Flow Phenomena (TSFP4), Williamsburg. pp. 331-336. 6. Ma R., and Simpson R.L. (2005) In: 4th International Symposium on Turbulence and Shear Flow Phenomena (TSFP4), Williamsburg. pp. 1171-1176. 7. Simpson R.L., Long C.H. and Byun G. (2002) Int. J. of Heat and Fluid Flow, 140(2):233-258. 8. Spalart P.R., Jou W.-H., Strelets M. and Allmaras S.R. (1997) In: Advances in DNS/LES, 1st AFOSR Int. Conf. on DNS/LES, Greyden Press, pp. 137-148. 9. Temmerman L., Hadziabli´c M., Leschziner M.A. and Hanjali´c K. (2005) Int. J. of Heat and Fluid Flow, 26(2):173-190. 10. Temmerman L., Wang C. and Leschziner M.A., (2004) In: 4th European Congress on Computational Methods in Applied Sciences and Engineering. 11. Tessicini F., Li N. and Leschziner M.A. (2005) In: Complex Effects in Large Eddy Simulation, Limassol, Cyprus. 12. Tessicini F., Temmerman L. and Leschziner M.A. (2005) In: 6th Engineering Turbulence Modelling and Measurements (ETMM6), Sardinia. 13. Wang C., Jang Y. and Leschziner M.A. (2004) Int. J. of Heat and Fluid Flow, 25(3):499-512. 14. Wang M. and Moin P. (2002) Physics of Fluids, 14(7):2043-2051.
Progress In Subgrid-Scale Transport Modeling Using Partial Integration Method For LES Of Developing Turbulent Flows Bruno Chaouat1 and Roland Schiestel2 1 2
ONERA, 92322 Chˆ atillon, France [email protected] IRPHE, Chˆ ateau-Gombert, Marseille 13384, France [email protected]
Summary. A partially integrated transport modeling method (PITM) for subgridscale turbulent stresses including a dissipation rate equation is applied for LES of developing turbulent flows. The present PITM method allows to perform LES simulations on coarse grids, in particular when the spectral cutoff happens to be located before the inertial zone of the energy spectrum. This method guaranties compatibility with the two extreme limits that are the full statistical Reynolds stress modeling and the direct numerical simulation and is viewed as a continuous approach of hybrid RANS/LES methods with seamless coupling. Two LES simulations of the fully turbulent channel flow are performed, for both coarse and medium meshes. This model allows to control the sharing out of turbulence energy among the subgrid and resolved turbulence scales depending on the filter width. Then, the application to the channel flow with wall mass injection which undergoes the development of natural unsteadiness with a transition process from laminar to turbulent regime is considered for illustrating the potential of the PITM model.
1 Introduction Due to the progress in powerful computers, large-eddy simulation (LES) is a promising route toward the calculations of turbulent flows which has been now largely developed. New trends in LES of turbulence have been proposed in the past decade [1], such as for instance the dynamic model [2] or the structure model of M´etais and Lesieur [3]. Another route was initiated by Deardorff [4] who has developed a model which lies on the transport equations of the subgrid scale turbulent stresses with an algebraic relation for the length scale. This approach allows to determine the subgrid scale stresses independently by solving their individual equations. Despite improvements in the more advanced approaches, several modeling problems remain. These problems are more acute when considering coarse meshes and non-equilibrium situations. Among these problems, for instance, the filter width may no longer be a good estimate of the characteristic subgrid-scale turbulence length when the filter cutoff is located at a wave number below the inertial range in non-equilibrium flows.
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To overcome this problem, Schiestel and Dejoan [5] have developed a partial integrated model (PITM) based on two transport equations for the subgrid-scale energy and the dissipation rate equation for calculating the length scale without referring directly to the mesh size. These authors successfully simulated the unsteady pulsed channel flow showing the potential of the model in comparison with the Smagorinsky model. Recently, Chaouat and Schiestel [6] have developed a more sophisticated PITM model which is based on all stress transport equations of the subgrid-scale turbulence including the dissipation rate equation. This modeling strategy is motivated by the idea that the recognized advantages of usual second order closures (RSM) [7] are worth to be transposed to subgrid-scale modeling when the SGS part is not small compared to the resolved part. Due to the presence of the subgrid-scale pressure-strain correlation term in the transport equations, this model allows a more realistic description of the flow anisotropy than eddy viscosity models. This subgridscale model complies with the two extreme limits that are DNS and full statistical Reynolds stress model in a continuous way that bridges URANS and LES methods with “seamless coupling ”. In this study, we present simulations of the fully turbulent channel flows performed on two different grids. A particular attention is devoted to the sharing out of turbulence energy among the subgrid and resolved turbulence scales when the filter width is changed. The application to the channel flow with wall mass injection which undergoes the development of natural unsteadiness with a transition process from laminar to turbulent regime is then considered. This case is of central interest for engineering applications in solid rocket motors (SRM) [8].
2 Governing Equations As in the usual treatment of turbulence, the flow variable ξ is decomposed into a filtered part including mean value and large-scale fluctuation ξ and a subgrid-scale fluctuating part ξ such that ξ = ξ + ξ where the quantity ξ is defined by the filter function GΔ as: ξ=
3
GΔi (xi , xi ) ξ(x1 , x2 , x3 ) dx1 dx2 dx3
(1)
i=1
where Δi is the filter width in the i-direction. The Reynolds statistical average of ξ is denoted by < ξ > so that the large scale fluctuation is ξ− < ξ >. In the present case, the Favre averaging is used for compressible flows. In that definition, the variable ξ can be written as ξ = ξ˜+ ξ where ξ˜ = ρξ/ρ. Closure of the filtered equations is nec essary for the turbulent stress ρ u i uj and the turbulent heat flux h ui . The filtered flow equations are solved using a centered numerical scheme of second or fourthorder accuracy in space discretization. The transport model equations need some degree of upwinding to be solved without any spurious oscillations. The precision requirements for the model are not so stringent than for the momentum equations [6].
3 Partial Integrated Transport Model The derivation of the PITM model results from the analysis of the turbulent processes in the spectral space for homogeneous turbulence [6, 9]. In this framework, the
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cutoff wave number κc featuring the filter width is introduced in the medium range of eddies. The PITM method involves partial integration of the spectral equations of the turbulence field in the range [κc , ∞[. Thus transport equations are obtained for the subfilter energies together with a new form of the dissipation rate equation. These transport equations look formally like the corresponding RANS model but the coefficients are no longer constants. Using the total turbulent kinetic energy denoted k and the dissipation rate , we show that these coefficients are functions of the non-dimensional cutoff κc k3/2 / , which can be interpreted as the ratio of the characteristic length scales LRAN S /LLES . The details of the derivation can be found in reference [5] for a two-equation framework. In the present paper, we have transposed the Launder-Shima model to PITM formalism [6]. To do that, we define the subfilter tensor (τij )sgs = u i uj which can be formally obtained by integrating the spherical mean ϕij (κ) of the Fourier transform of the two-point ∞ velocity fluctuations by the relation (τij )sgs = κ ϕij (κ)dκ. The turbulent transport c equations of the subfilter tensor (τij )sgs is then modeled in the physical space after extension to inhomogeneous flows by inclusion of the diffusion terms [6]:
∂ ∂ ρ (τij )sgs u ˜k = Pij − ρ ij + Φ1ij + Φ2ij + Φw (ρ (τij )sgs ) + ij + Jij ∂t ∂xk
(2)
˜j /∂xk − ρ(τjk )sgs ∂ u˜i /∂xk , ij = 23 δij . The terms on where Pij = −ρ(τik )sgs ∂ u the right-hand side of equation (2) are identified as the production by the filtered velocity Pij , the turbulent viscous dissipation ij , the redistribution of the subgridscale turbulent kinetic energy among the stress components Φij , and the turbulent diffusion due to the fluctuating velocities and pressure with the molecular diffusion, together denoted Jij . The redistribution terms of the pressure-strain subgrid-scale fluctuating correlations are modeled in the range [κc , κd ] where the wave number κd is located at the end of the inertial range of the spectrum after the transfer zone: Φ1ij = −csgs1 ρ
ksgs
(τij )sgs −
1 Pmm δij 3 takes the expression: Φ2ij = −c2 Pij −
where the coefficient csgs1
2 ksgs δij 3
csgs1 =
1 + αN Nc2 c1 1 + Nc2
(3)
(4)
(5)
In this expression, Nc = κc L is the dimensionless wave number computed by means 1 of the filter width κc = π/Δ, where Δ = (Δ1 Δ2 Δ3 ) 3 and the normal distance to arm´ an constant. The coefficient αN is set to the wall L = Kx3 where K is the Von K´ 1.5. Like the Launder and Shima model, the function c1 depends on the second and third subgrid-scale invariants A2 = aij aji , A3 = aij ajk aki and the flatness coefficient parameter A = 1 − 98 (A2 − A3 ) where aij = ((τij )sgs − 23 ksgs δij )/ksgs . According to the classical physics of turbulence, the coefficient csgs1 increases with the parameter Nc in order to increase the return to isotropy. This behavior is reproduced by equation (5), the value of the coefficient csgs1 ranging from c1 to αN c1 throughout the spectrum. The term Φij takes into account the wall reflection effect of the pressure fluctuations:
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ρ ((τkl )sgs nk nl δij − 32 (τki )sgs nk nj − 32 (τkj )sgs nk ni ) fw ksgs
2 3 2 3 2 + cw 2 Φkl nk nl δij − 2 Φik nk nj − 2 Φjk nk ni fw
(6)
where ni is the normal unit vector to the wall and fw is a near wall damping function. The diffusion process Jij is modeled assuming a gradient law: Jij =
∂ ∂xk
μ
∂(τij )sgs ∂(τij )sgs ksgs + cs ρ (τkl )sgs ∂xk ∂xl
(7)
where cs is a numerical coefficient which takes the value 0.22. As a result of modeling, the transport equation of the dissipation rate takes the following form:
Pmm ∂ ∂ ˜ ρ ˜ uj = c1 + J (ρ ) + − csgs2 ρ ∂t ∂xj ksgs 2 ksgs
(8)
where the diffusion process J is modeled assuming a tensorial gradient law: J =
)
∂ ∂xj
and ˜ = −2ν ∂ ksgs /∂xn (8) is modeled as [6]:
2
μ
∂ ∂ ksgs + c ρ (τjm )sgs ∂xj ∂xm
(9)
. For LES simulation, the coefficient csgs2 in equation
csgs2 = c1 +
c2 − c1
(10) 2/3 1 + βN Nc The value of the dissipation rate remains the same as the one used in statistical modeling but the production, destruction and diffusion terms that appear in its transport equation (8) need to be modified. In particular, the production of the subgrid turbulent energy Pmm is expressed by a tensorial product of the subgridstresses by the filtered velocities. Both production and destruction terms are now functions of the subgrid turbulent kinetic energy ksgs instead of the total turbulent kinetic energy as usually obtained in RANS modeling. Physically, These differences in turbulence modeling are necessary because in LES, just a part of the spectrum is modeled. Equation (10) allows the turbulence model to “see ”the grid and then to determine the part of the spectrum that has to be modeled. The optimized value βN = 0.5 is chosen. The values of the numerical coefficients are the w following: c1 = 1.45, c2 = 1.9 and c = 0.18. The functions c1 ,c2 ,cw 1 ,c2 and fw are listed in reference [6].
4 LES of Fully Developed Turbulent Channel Flow Numerical simulations are performed on coarse and medium meshes requiring 16 × 32 × 64 grids (LES 1) and 32 × 64 × 84 grids (LES 2) respectively with different spacings Δi . This choice is motivated by the necessity of checking numerically the grid independence of the statistics of the solution as well as the consistency of the subgrid scale model when the filter width is changed. The present LES are compared with direct numerical simulation of Moser et al. [10] for a Reynolds number Rτ = ρτ uτ δ/2μ = 395, based on the averaged friction density ρτ , the averaged friction velocity uτ and the channel half width δ/2. The total
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Reynolds stress τij is calculated as the sum of the subgrid and large scale parts τij = (τij )sgs + (τij )les where the resolved scales are computed using the statistical ˜i u ˜j > − < u ˜i >< u ˜j >. Figure 1 shows the ratios of the reaveraging (τij )les =< u solved energy scale to the total energy kles /k, respectively for the coarse and medium meshes. As a results of interest, one can observe that these ratios are close to zero near the walls. When moving to the center of the channel, they increase going to unity. This means that the PITM model behaves like a RANS model near the walls and continuously evolves toward a DNS in the center of the channel since all the 1.2
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scales are resolved. This remark demonstrates the usefulness of the present PITM model that takes into account the flow anisotropy, specially in the wall regions, contrary to viscosity models. Note that the evolutions of the ratio ksgs /k result from equation (10) which governs the proportion of the turbulence distributed in the subgrid part and large scale part through the parameter Nc = πL/Δ. As expected, comparisons of these ratios also reveal that the part of modeled scales for the coarse mesh if of higher intensity than those of the medium mesh, specially near the walls. Figure 2 shows the variations of the total stresses, in the streamwise, spanwise and normal direction, for the medium mesh. Although the sharing out between the SGS and LES scale parts is quite different as previously observed in Figure 1, excellent agreement with the DNS data is observed for each normal stress. This confirms that the total energy remains nicely preserved.
5 LES of Channel Flow with Wall Injection The application of channel flow with appreciable fluid injection through a permeable wall as sketched in Fig. 3 is considered. Modeling such flow is a challenging task since the flow evolves from a laminar to a turbulent regime due to the transition of the mean axial direction [8]. The objective is to simulate the flow which develops in the specific experimental setup VECLA made at ONERA. Values of the length, height and width of the channel are respectively L1 = 58.1 cm, L2 = 6 cm and δ = 1.03 cm. The present LES results are compared to the experimental data [11]. The numerical 3d simulation is performed on a mesh composed of 400 × 44 × 80 grid points in the streamwise, spanwise and normal directions for a computational domain [L1 , 2δ, δ]. This mesh can be compared with the refined mesh of the LES of Apte and Yang [12] using a dynamical Smagorinsky model [2] for a quasi-similar computational domain. In the present case, the present hybrid method which can be applied to coarse grids in contrast to conventional LES computation [12] allows to get satisfactory results while reducing the grid points by 64 %. Figure 3 shows the instantaneous spanwise
δ
x3
U1
x2 U3
x
1
Fig. 3. Schematic of channel flow with fluid injection of Vecla setup
filtered vorticity ω ˜ i = ijk ∂ u ˜k /∂xj in different planes of the channel and provides the detail of the flow structures subjected to mass injection as well as the location of the transition which occurs at x1 /δ ≈ 35. In the downstream transition location, the flow is then characterized by the presence of roll-up vortex structures of large magnitude of vorticity in the spanwise direction. Because of the injection, it is found that these structures are squeezed upwards in the normal direction to the axial flow (see Fig.4a) as previously observed by Apte and Yang [12]. Compared to usual channel
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(a)
(b) 4 −1
Fig. 4. Snapshots of spanwise instantaneous filtered vorticity. |˜ ω | < 4. 10 s , Δω = 2000 s−1 . (a) mid-plane (x1 , x3 ), x2 /δ = 1; (b) plane (x1 , x2 ), x3 /δ ≈ 0. x1 ∈ [0, L1 ], x2 ∈ [0, 2δ], x3 ∈ [0, δ].
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Fig. 5. Turbulent shear stresses in the cross section x1 = 57 cm. (a)
:< (τ13 )sgs > /u2m , : < (τ13 )les > /u2m ; (b) +: < τ13 > /u2m , •: experiment.
flows, the slope of these expanding eddies is fare more pronounced. The simulation provides turbulent stresses that agree well with the experimental data (see also reference [6]). However, it is of interest to mention that the spanwise direction plays an essential role in the vortex-stretching mechanism and therefore in the prediction of the turbulence intensity. Indeed, previous simulations taking into account less than 40 planes in the spanwise direction have underpredicted the turbulence intensity of the flow. But the flow intensity remains unchanged when increasing the spanwise mesh resolution. In the region where the flow is fully turbulent, one can determine the part of modeled turbulent scales compared to the part of the resolved scales. Figure 5a shows the LES and SGS shear stresses normalized by the bulk velocity in the last section at x1 = 57 cm. As a result of interest, it appears that the intensity of the large scale part remains greater than those of the subgrid-scale part in the core flow whereas the contribution of the subrid-scale is of higher intensity near the walls. This result which is depending on grid refinement, is similar to the previous
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observation made in Fig. 1 for the fully turbulent channel flow. In addition, Fig. 5b indicates that the total turbulent shear stress agrees fairly well with the experimental data.
6 Conclusion Progress in subgrid-scale transport modeling using partial integration method for LES of developing turbulent flows have been made. In particular, the concept of turbulent viscosity is no longer necessary. This PITM model has been developed for LES simulations of practical engineering applications which require coarse meshes. As a result of interest, it is shown that the total turbulent stresses agree with the DNS or experimental data whatever the sharing out of turbulence energy among the subgrid and resolved scales which is governed by mesh size. The present PITM model can be easily implemented in RANS codes using statistical Reynolds stress models for performing 3d LES simulations, provided the numerical schemes are precise enough.
References 1. Lesieur M, Metais O (1996) Ann. Rev. Journal of Fluid Mechanics 28:45–82 2. Germano M, Piomelli U, Moin P, Cabot WH (1992) Physics of Fluids 3:1760– 1765 3. M´etais O, Lesieur M (1992) Journal of Fluid Mechanics 239:157–194 4. Deardorff J (1970) Journal of Fluid Mechanics 41:433–480 5. Schiestel R, Dejoan A (2005) Theoret. Comput. Fluid Dynamics 18:443-468 6. Chaouat B, Schiestel R (2005) Physics of Fluids 17 7. Chaouat B (2001) Journal of Fluid Engineering, ASME 123:2-10 8. Chaouat B, Schiestel R (2002) Journal of Turbulence 3:1–15 9. Schiestel R (1986) Physics of Fluids 30:722–731 10. Moser R, Kim D, Mansour N (1999) Physics of Fluids 11:943–945 11. Avalon G, Casalis G, Griffond J (1998) AIAA paper 3218 12. Apte SA, Yang V (2003) Journal of Fluid Mechanics 477:215–225
Part XVI
Compressible Flows
Towards Large Eddy Simulations of Scramjet Flows Christer Fureby1 and Magnus Berglund2,3 1
2
3
FOI, Swedish Defence Research Agency, SE-164 90 Stockholm, Sweden [email protected] FOI, Swedish Defence Research Agency, SE-164 90 Stockholm, Sweden [email protected] Dept. of Fluid Mechanics, Lund Institute of Technology, SE-221 00 Lund, Sweden
Summary Large Eddy Simulation (LES) has been used to examine supersonic flow in a model scramjet combustor. The LES model is based on an unstructured finite volume discretization, using total variation diminishing (TVD) flux reconstruction, and a TVD Runge–Kutta time integration scheme. The configuration used is similar to the laboratory scramjet at DLR and consists of a one-sided divergent channel with a wedge-shaped flameholder at the base of which hydrogen is injected. Two cases are investigated: (i) supersonic flow and (ii) supersonic flow with hydrogen injection. For the purpose of validation, the LES results are compared with experimental data for velocity and pressure. In addition, qualitative comparisons are also made between predicted and measured schlieren fields. The LES computations are capable of predicting both the flow and the large-scale mixing well.
1 Introduction Turbulent mixing of fuel and oxidizer and the subsequent reactions in the supersonic flow inside a scramjet combustor are of considerable interest due to the renewed interest in hypersonic airbreathing propulsion. As experimental studies in the Mach number range of interest, Ma ≈ 5 − 15, mostly suffer from run-time limitations, numerical simulations appear to be a cost-effective and practical candidate for prediction of the performance of hypersonic propulsion systems despite the complexity and the large computational resources required. However, only a few computational studies have been reported to date, the most noteworthy being for a scramjet engine model developed and experimentally studied at DLR in Germany [1, 2, 3, 4]. In [5], results from two-dimensional steady-state Reynolds Averaged Navier–Stokes (RANS) calculations using an adaptive mesh together with a flamelet combustion model
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was presented. More recently, LES was used by G´enin et al. [6] and Berglund, Wikstr¨ om, and Fureby [7]. In [6] an eddy dissipation model and a linear eddy model were applied whereas in [7] a flamelet model was used. In the latter two studies, the computational domain encompassed three jets (instead of 15 as in the experimental studies) and periodic boundary conditions were used in the spanwise direction. For both RANS and LES, reasonable results were obtained, but the LES results are in better qualitative and quantitative agreement with the measurement data, since they are more realistic in reproducing the unsteady turbulent flow and the associated unsteady turbulent mixing. In order to simulate the reacting flow in the scramjet engine model, a detailed understanding of the flow and the mixing process is required. To accomplish this we use LES in order to investigate the flow and the supersonic fuel-air mixing process in the aforementioned scramjet engine model.
2 Theoretical Formulation and Numerical Procedures The mathematical model employed consists of the conservation and balance equations of mass, momentum and energy for a linear viscous fluid with Fourier heat conduction, i.e. the compressible Navier–Stokes equations [8]. Furthermore, the fluid is assumed to be an ideal gas with constant specific heat capacities. Since the flow under consideration involves shock and rarefaction waves it is important to ensure the correct Rankine–Hugoniot jump conditions, and therefore a conservative formulation of the energy equation, in terms of the total energy is employed. By low pass-filtering and Favr´e-averaging the governing equations, we obtain the corresponding conservative LES equations [9], that contain a number of unresolved transport, or subgrid stress tensor and flux vector, terms. These quantities hold information about the unresolved flow and need to be modeled in order to close the LES equations. The modeling employed here is essentially the same as used in [9], and consists of a scale similarity term, which essentially is an integral part of the LES equations [10], and a subgrid viscosity model, employing the constant coefficient Smagorinsky model. In order to alleviate some of the requirements for resolving the near wall boundary layer, a subgrid wall-model, based on the logarithmic law-of-the-wall [11] has been employed. The practical effect of this modeling is a modified subgrid viscosity such that the flow locally satisfies a logarithmic velocity distribution in the near wall-region. The flow code is based on an unstructured finite volume discretization with a colocated cell-centered variable arrangement, TVD flux reconstruction of the convective fluxes, and central differencing for the viscous fluxes. The time integration is fully explicit, using a second order accurate TVD Runge– Kutta scheme by Gottlieb and Shu [12].
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3 Experimental Setup In the scramjet engine model, Fig. 1, preheated air enters the combustor section through a connected Laval nozzle. When hydrogen is injected, this is done through an array of 15 holes, each with a diameter of 1.0 mm and separated by 2.4 mm, at the base of the wedge. Typical mass flows in the experiments were varied between 1.0 kg/s and 1.5 kg/s for the air and between 1.5 g/s and 4.0 g/s for hydrogen. A wide range of measurements were conducted in this rig, including LDV and PIV measurements of the velocity, side-wall pressure measurements, CARS measurements of the temperature, OH-LIF for mapping regions of combustion as well as more conventional schlieren and shadow photography in order to characterize the flow and combustion dynamics.
Fig. 1. Schematic of the supersonic combustor
In the present study we have focused on the non-reacting experiments without and with hydrogen injection and with the inflow conditions uair = 730 m/s, uH2 = 1200 m/s, Tair = 340 K, TH2 = 250 K and p = 1.0 atm. These values correspond to mass flow rates of about 1.0 kg/s and 1.0 g/s and Mach numbers of 2.0 and 1.0 for air and hydrogen, respectively.
4 Computational Details The computational configuration is simplified in the sense that instead of modeling the full width of the combustor, including the 15 injector holes and the side walls, a smaller domain with three injector holes and enforced periodicity in the spanwise (or z-) direction was chosen. The reason for this simplification is that we need to reduce the computational cost as much as possible, but still retaining as much as possible of the relevant physics of the scramjet combustor, including the jet-to-jet interactions. Two meshes have been used, consisting of 1.6 and 3.2 million hexahedral control volumes. The meshes are block-structured, using 12 blocks per jet. The first grid point is located at a normalized wall distance of y + ≈ 50 in order to make maximum use of the aforementioned subgrid wall model. Dirichlet boundary conditions are used for all variables at the air inlet and at the hydrogen jet inlets, according to the values given above whereas
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extrapolated values are used at the (supersonic) outlet. Periodic boundary conditions are used in the spanwise direction whereas at the upper, lower, and wedge walls, adiabatic no-slip conditions are applied. The computations for the case without hydrogen injection were initialized with the state of the incoming air and time-averaging was carried out over four flow through-times. The simulations with hydrogen injection were started from results obtained in the former case in order to save computer time. Again, time-averaging was carried out over four flow through-times.
5 Results and Discussion Here we will focus on the results from the fine grid, since the spatial resolution is imperative for capturing the dynamics of the shocks, expansion fans, shear layers, and their mutual interactions. 5.1 Supersonic Flow without Hydrogen Injection The first case considered is without hydrogen injection and is hereafter referred to as Case I. Its principle purpose is to aid in characterizing the flow in the combustion chamber, and since experimental data is available it will also aid in validating the LES model and the supersonic flow code used. The measurements carried out for Case I are however limited to schlieren photographs and time-averaged pressure data at the bottom wall.
Fig. 2. Results from Case I: numerical schlieren image
Schlieren imaging measures the deflection of the optical path-length, l = S n(x(s)) ds, where n is the refractive index and x = x(s) is the path in parametric form, for a light ray transmitted through an inhomogeneous medium. For a gaseous mixture, n can be estimated by the Lorentz–Lorenz formula (n2 + 1)ρ i Yi Ai = Mmix (n2 − 1), where Mmix is the molecular weight of the mixture and Ai the molar refractivity for species i. In the experiments the schlieren images are obtained by using a horizontal knife-edge that cuts off light propagating in the vertical (i.e. the y-) direction, and for small deviations along the optical path the intensity variations in the schlieren L image are ΔI ∝ 0 z (∇y n) dz. Here, we use the LES data to calculate the re) fractive index from the Lorentz–Lorenz formula as n = (1 + 2α)/(1 − α),
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where α = p i (Yi Ai )/(R0 T ), from which numerical schlieren images can be constructed and subsequently compared with images from the experiments. Figure 2 shows a numerical schlieren image of the scramjet engine flow. Oblique shocks are formed at the tip of the wedge and then reflected by the upper and lower walls downstream of the small triangular-shaped recirculation bubble formed just behind the wedge. This causes a characteristic shock pattern in the downstream region. At the upper and lower walls, the boundary layer is strongly affected, at least locally, by the reflected oblique shocks. These local modifications involve thickening of the boundary layer, higher rmspressure fluctuations, and elevated wall temperatures. The boundary layer on the wedge surface separates at the base and a shear layer is formed. This shear layer is naturally unstable and is therefore prone to break-up and develop into Kelvin–Helmholtz (KH) structures. However, before these structures fully develop, the reflected oblique shocks from the tip of the wedge interact with the shear layer, causing it to bend, distort, and form undulating large-scale structures. Strong expansion fans originating at the upper and lower edges of the wedge rapidly deflect the flow towards the centerline. After some distance the flow in the wake of the wedge is accelerated back to supersonic speed and the subsequent shocks (reflecting off the walls) passes through the accelerating wake. Moreover, the resemblance between Fig. 2 and the experimental schlieren image (Fig. 4a in [3]) is good. 5.2 Supersonic Flow with Hydrogen Injection The second case considered is with hydrogen injection, which adds significant complexity to the flow due to the mixing of gases with different molar masses. This case is hereafter referred to as Case II. Figure 3 shows a perspective view and a numerical schlieren image of the scramjet engine flow. With hydrogen injection the flow field is similar to that of Case I but with a few differences: (i) the expansion fans coming off the base of the wedge are not as strong due to the presence of hydrogen that affects the thermodynamic properties of the fluid, (ii) the recirculation region is now much larger with a somewhat higher base pressure, (iii) the curved recompression shocks of Case I are here replaced with straighter recompression shocks, (iv) the shear layers, originating at the upper and lower edges of the wedge, are now clearly visible due to the different thermodynamic properties of the wake in Case II as compared to Case I. The hydrogen-rich stream is clearly visible, in particular where the shock crosses the core flow and where the shock angle is increased. For the shear layers, we find that the strength of the gradient between the hydrogen and the air decreases with distance from the wedge due to the effects of the mixing. The shear layers are unstable due to the KH instabilities. Because of the one-sided divergent channel the upper reflecting shock hits the hydrogen filled wake further downstream than the lower shock, causing an asymmetric flow field in which the KH modes are amplified. This in turn results in shedding and periodically bent structures
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whereby air and hydrogen are mixed by advection. It is important also to notice that compressibility leads to reduced shear layer growth, affecting the formation and subsequent break-up of large coherent structures. In addition, the reflected shock waves are deflected by the hydrogen jets. The agreement with the experimental schlieren image (Fig. 4b in [3]) is good, suggesting that LES can differentiate between the curved recompression shocks observed in Case I and the straighter recompression shocks found in Case II.
Fig. 3. Results from Case II: perspective view from the rear of the simulated flow (left) and numerical schlieren image (right)
5.3 Time-Averaged Results In Fig. 4 we compare the predicted and experimentally measured pressure at the lower wall and the axial velocity distribution and its rms fluctuations at four cross-sections downstream of the wedge. For the time-averaged pressure, p , we find reasonable qualitative agreement; the reflection of the leading edge shock is well predicted both with respect to position and amplitude and the reflection of the recompression shock and the reflected leading edge shock are clearly separated in the LES computations, whereas in the experiments they partially overlap. Furthermore, the amplitude of p is somewhat under-predicted and the reason for this is believed to be insufficient spatial resolution in this region. For the time-averaged axial velocity, ˜ v x , the overall agreement is good and, in particular, we find that the recovery of the subsonic hydrogen jet forming the wake is predicted reasonably well. Just after the jet exits the wedge it is accelerated in the low-pressure recirculation region followed by a deceleration by a shock wave. After further acceleration the hydrogen jet is well mixed with the ambient air and has reached a nearly constant velocity of 760 m/s, which is in good agreement with the experimental data. Moreover, the influence of the hydrogen jet on the flow is limited to the region in the immediate vicinity of the jets and only very small differences between Case I and Case II can be detected in the velocity field.) vx − ˜ vx
2 , are found to The axial rms-velocity fluctuations, vxrms = ˜ be large in the wake and around the jets, but decrease with downstream distance. Also, these fluctuations are found to be large in the vicinity of the upper and lower walls, which is associated with the near-wall production of
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turbulence emulated by the wall model. Comparison with experimental data at x/h = 22.54 and 28.36 shows good agreement.
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Fig. 4. Results from Case I and II: (a) Time-averaged pressure at the lower wall (y = 0), (b) time-averaged axial velocity and (c) axial rms-velocity fluctuations at four cross-sections downstream of the wedge at x/h = 13.00, 20.83, 26.16, and 38.83. Legend: Dashed lines are used for Case I, solid lines for Case II, and symbols for experimental data. Note also that profiles are shifted 300 units in the horizontal direction.
6 Conclusions In the present work LES has been used to investigate supersonic flow and hydrogen-air mixing in a scramjet model under realistic operating conditions. The configuration is similar to the laboratory scramjet developed and experimentally investigated at DLR in Germany, with the difference that periodicity is invoked in the spanwise direction (encompassing three hydrogen jets instead of 15) to reduce the computational cost of the simulations. Qualitative and quantitative comparisons were made with experimental data for both velocity and pressure for non-reacting flow without and with hydrogen injection. These comparisons are supplemented with comparisons of schlieren images. In summary, we find that the LES computations are capable of predicting the non-reacting flowfields reasonably well. This study also served the additional
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purpose of laying the foundations for LES computations of scramjets with chemical reactions. These will be presented elsewhere. Acknowledgment Mr. H.G. Weller is acknowledged for the development of the C++ class library FOAM (Field Operation And Manipulation), version 1.9.2β, used in this study. The authors also wish to acknowledge Dr. Torgny Carlsson for valuable discussions concerning the interpretation and the numerical modeling of the light deflection based experimental images as well as Mr. Niklas Wikstr¨ om for excellent work on the computational meshes.
References 1. Oschwald M, Guerra R, Waidmann W (1993) Investigation of the Flowfield of a Scramjet Combustor with Parallel H2 -Injection through a Strut by Particle Image Displacement Velocimetry. In: Int Symp on Special Topics in Chem Prop, pp 498–503, May 10–14, Scheveningen, NL 2. Waidmann W, Alff F, Brummund U, B¨ ohm M, Clauss W, Oschwald M (1994) Experimental Investigation of the Combustion Process in a Supersonic Combustion Ramjet (SCRAMJET), In: DGLR Jahrestagung, pp 629–638, Erlangen 3. Waidmann W, Brummund U, Nuding J (1995) Experimental Investigation of Supersonic Ramjet Combustion (SCRAMJET). In: 8th Int Symp on Transp Phenom In Comb, pp 1473–1484, July 16–20, San Francisco, US 4. Waidmann W, Alff F, Brummund U, B¨ ohm M, Clauss W, Oschwald M (1995) Space Tech 15:421–429 5. Oevermann M (2000) Aerosp Sci Tech 4:463–480 6. G´enin F, Chernyavsky B, Menon S (2003) AIAA Paper 2003-7035 7. Berglund M, Wiksr¨ om N, Fureby C (2005) FOI-R--1650--SE, FOI Technical Report 8. Hutter K, J¨ ohnk K (2004) Continuum Methods of Physical Modeling. Springer, Berlin Heidelberg New York 9. Erlebacher G, Hussaini MY, Speziale CG, Zang TA (1990) ICASE Report No 90-76. NASA Langley 10. Fureby C, Bensow R, Persson T (2005) In: Turbulence and Shear Flow Phenomena 4, pp 1077–1082, June 27–29, Williamsburg, US 11. Fureby C, Alin N, Wikstr¨ om N, Menon S, Persson L, Svanstedt N (2004) AIAA J 42:457–468 12. Gottlieb S, Shu C-W (1998) Math Comp 67:73–85
DNS and LES of compressible turbulent pipe flow with isothermal wall S. Ghosh, J. Sesterhenn, and R. Friedrich Fachgebiet Stroemungsmechanik,Technische Universitaet Muenchen, Boltzmannstr. 15, 85748, Garching, Germany. [email protected]
Summary : Compressible turbulent flow through a pipe at Mach numbers 0.3 and 1.5 is studied by means of DNS and LES. The focus is on compressibility effects on turbulence statistics. The effectiveness of explicit filtering based on approximate deconvolution for LES is demonstrated by comparison with DNS results.
1 Introduction Wall-bounded compressible turbulent flows have been the subject of numerical studies by different groups in the recent past with the aim of finding suitable scalings to account for compressiblity effects in such flows. Coleman et al. [2] performed DNS of channel flow with isothermal walls up to M = 3.0 and found that Morkovin’s hypothesis holds in these flows i.e. the major compressibility effect comes through mean property variations. Huang et al. [5] introduced a ’semi-local’ scaling instead of the conventional wall units to account for mean property variations. Foysi et al. [4] showed that a reduction in the pressure strain correlation at higher Mach numbers is the result of mean density variations and causes changes in Reynolds stress anisotropy. A recent study of channel flow between adiabatic and isothermal walls by Morinishi et al. [10] examines the applicability of Morkovin’s hypothesis to turbulence statistics and the validity of strong Reynolds analogy relations. They showed that Morkovin’s hypothesis is not applicable to near-wall asymptotic behaviour of wall-normal turbulence intensity. In this work we aim at studying compressibility effects in turbulent pipe flow with an isothermal, impermeable wall. We chose to study pipe flow with constant cross-section as a first test case for computations in cylindrical coordinates. An isothermal condition at the wall is a natural choice in the supersonic case since an adiabatic wall will lead to ’choking’.
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2 Numerical Method 2.1 Discretization schemes and LES approach We solve the governing Navier-Stokes equations in a special non-conservative pressure-velocity-entropy form [11] and use compact 5th order upwind schemes of Adams et al. [1] for the convective terms and 6th order compact central schemes of Lele [7] for the molecular transport terms. The flowfield is advanced in time using a 3rd order low-storage Runge-Kutta scheme of Williamson [15]. Lechner et al. [6] and Foysi et al. [4] have used the same space and time integration schemes for computing turbulent, supersonic channel flows with isothermal walls and their results are in excellent agreement with the spectral results of Coleman et al. [2] The flow is driven by a homogeneous body force balancing the wall shear stress allowing for periodicity of the flow variables in streamwise direction. This avoids the use of inflow and outflow conditions which are still sensitive issues in compressible turbulent flow simulations. The computations are done in a cylindrical coordinate system which requires a special treatment at the axis. We use the method of Mohseni et al. [9] in which the radial derivative is computed without using any boundary schemes at the axis and without placing a gridpoint directly on the axis. To avoid unnecessary timestep restrictions near the axis due to the present cylindrical coordinate system, a high wavenumber cut-off filter is applied to the flow variables after every time-step in a small region (0 < r/R < 0.05) around the axis. Thus timestep restrictions come only from the region near the wall. The code uses MPI routines for parallelisation in streamwise and azimuthal directions. A two-way algorithm [14] is used for faster computation of the radial derivative across the axis. The computations were performed on the Hitachi SR8000-F1 high-performance computer at the Leibniz Rechenzentrum in Munich with a performance of 1.8 Gflops per node (8 processors). The DNS in our case typically uses 32 processors on SR8000-F1. LES studies have also been performed for the pipe flow configuration in order to test the effectiveness of our LES method. The explicit filtering approach of Mathew et al. [8], based on approximate deconvolution (Stolz and Adams [13]), is used to treat the interaction between resolved and unresolved scales. In this approach the Navier-Stokes equations are integrated on the LES grid and explicit filtering of the solution is done after every timestep with a composite filter Q ∗ G, where Q is the approximate inverse of a low-pass filter G. For G, we use one-parametric, implicit Pad´e filters in the homogeneous directions and an explicit filter [12] in the wall-normal direction. A ’secondary’ filtering is done in the two homogeneous directions following [8] which leads to an equivalent filter (Q ∗ G)2 in these directions. Since the LES are performed at the same low Reynolds numbers as the DNS we can afford to resolve the wall layer and need not use any wall models. As shown in [8] the LES results in the case of a compressible channel flow are in good agreement with DNS results. Hence, we expect similar good LES results for the pipe.
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2.2 Simulation parameters We present results from DNS and LES with parameters given in Table 1. It may be noted that in order to focus on compressibility effects we kept the Reynolds number nearly constant in our computations. The computational domain is a pipe of diameter 2R and length 10R. It is discretised in the DNS by 256 × 128 × 91 points in streamwise, circumferential and radial directions. In both LES 64 × 64 × 50 points are used. The resolution has been chosen in accordance with that in [3]. We use uniform grids in the streamwise and circumferential directions. In the wall-normal direction the points are clustered using tanh functions. M is a global Mach number based on the speed of sound at constant R wall temperature and the bulk velocity, uav = 2 0 ux Rr d Rr . The friction ) Reynolds number, Reτ , is based on friction velocity uτ = τw /ρw , pipe radius and kinematic viscosity νw (Tw ). Reτ is a result of the computation. The mesh size has been non-dimensionalised by uτ and νw . Table 1. Flow and computation parameters Case
+ + Δx+ rΔφ+ max Δrmin Δrmax M
Reτ
DNSM0.3 DNSM1.5 LESM0.3 LESM1.5
8.3 9.5 33.2 38
214 245 216 244
10.5 12.0 19.2 21.2
1.18 1.3 0.9 2.5
3.26 3.73 6.1 6.79
0.3 1.5 0.3 1.5
3 Results 3.1 Effects of compressibility derived from DNS We can expect our near-wall results to be similar to those obtained from compressible, turbulent channel flow (Reτ = 221, M = 1.5) [6], [4], since the Reynolds numbers, Reτ , are similar. The mean streamwise velocity (Fig. 1a) shows a higher value in the core region of the pipe than in the channel, because the pipe flow is ’less turbulent’ due to the effect of the wall. The nearwall values of the rms fluctuations normalized with the local mean velocity (Fig. 1b) are close to the limiting values found in DNS and experiments of incompressible pipe flow [3] which indicates negligible compressibility effects near the wall. The rms pressure and total temperature fluctuations, normalized by their mean values, are negligibly small at least up to Mach numbers of 1.5. In our isothermal case, because of heat transfer to the wall, we have relatively
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Fig. 1. (a) Mean streamwise velocity at M = 1.5 : — DNS pipe, - - -, DNS channel (b) Rms velocity fluctuations, symbols as in (a).
high total temperature fluctuations. Due to lack of space these results are not presented here. The mean density and mean temperature profiles (Fig 2a) have sharp gradients near the wall due to wall-cooling. This variation in mean properties is seen as the most important Mach-number effect in wall-bounded turbulent flows along isothermal walls. The Van Driest transformation, u+ ) u+ = ρ/ρw du VD 0
brings the mean velocity profile of the supersonic case closer to the incompressible log law (Fig. 6a). The remaining deviation is due to combined effects of mean viscosity and density in the buffer layer. Huang et al. [5] suggested a ’semi-local’, ’inner’ scaling for the rms of velocity fluctuations e.g. in order to take into account the mean property ) variation in the near-wall region. Here y ∗ ≡ (R−r)∗u∗τ /ν, where u∗τ = τw /ρ. As shown in Fig. 2b, this change in normalization of the coordinate gives better results than the use of the coordinate y + ≡ (R − r) ∗ uτ /νw . The peaks in the buffer layer show distinctly better collapse especially for the streamwise velocity fluctuations. Fig. 3 contains the streamwise and shear Reynolds stresses, scaled with τw , and plotted against the radial coordinate. This is the ’outer’ scaling as used in incompressible flows. As can be seen the profiles show remarkable collapse in the outer layer (core region). These diagrams also show a comparison between DNS and LES results, which will be discussed below. Concerning effects of compressibility we note an increase in streamwise Reynolds stress with increase in Mach number. At the same time all the remaining stresses are reduced (only the shear stress is shown here). Hence, an increasing Mach number changes the Reynolds stress anisotropy and the reason for this is the reduction in pressure strain correlations. An ’inner’ scaling, following Foysi et al. [4], τw2 /μ is used for the terms in the transport equation for the streamwise Reynolds stress, R11 (Fig. 4) which
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Fig. 2. (a) Mean temperature and density profiles (b) Rms velocity fluctuations.
+
Fig. 3. Reynolds stresses (a) streamwise (b) shear stress.
brings the profiles close to each other in the near-wall region. It is apparent from Fig. 4b that the pressure strain correlation term goes down appreciably in the buffer layer when the Mach number is increased. This has already been noted in previous studies and the cause has been attributed to mean density variations with Mach number [4]. A reduced streamwise pressurestrain correlation implies less energy transfer to the other components and hence allows for an increase in the peak value of R11 . 3.2 Evaluation of LES method In this section we compare our LES results with those from DNS. The mean temperature profile shows differences with the DNS (Fig. 5a) since the unfiltered DNS data are used for comparison. The rms temperature fluctuations, however, agree very well with the DNS results in both the computations (Fig. 5b). The Van Driest transformed mean streamwise velocity profile is very close to the DNS profile (Fig. 6a). The LES makes reasonably good predictions of
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Fig. 4. Terms in R11 transport equation: lines, DNSM1.5; symbols, DNSM0.3.
Fig. 5. (a) Mean temperature (b) Rms temperature fluctuations.
the streamwise Reynolds stress and the shear stress (Fig. 3a, 6b).The fact that the LES result for the streamwise Reynolds stress (Fig. 3a) overshoots the DNS result in the buffer region, is due to lack of resolution. Such a result is also typically found in an underresolved DNS. A look at the terms in the R11 transport equation shows that the LES predicts those terms remarkably well, that mainly depend on the large turbulent scales, like production and pressure strain correlation. The agreement between DNS and LES is not as good for the turbulent transport term and for terms associated with the dissipative scales (Fig. 7a,b).
4 Conclusions DNS and LES of compressible, turbulent pipe flow have been performed in order to investigate compressibility effects on various turbulence statistics. The major compressibility effect is due to variations in mean properties with
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Fig. 6. (a) Mean streamwise velocity (b) Shear stress.
Fig. 7. Terms in R11 transport equation: symbols DNSM1.5, lines LESM1.5.
increasing Mach number as noted in previous studies. Morkovin’s hypothesis holds for some turbulence statistics and as a consequence Van Driest’s transformation applies to mean streamwise velocity. ’Semi-local’ scalings incorporating mean property variations work better than scalings using wall values. The various terms contributing to the R11 transport equation have been scaled using local mean viscosity and wall shear stress. The difference between the subsonic and supersonic cases finally lies primarily in the pressure strain correlation term. Results from the LES of supersonic pipe flow are compared with the DNS results and the agreement is found to be good. Hence, this shows the effectiveness of the explicit filtering approach as a viable and easy-to-implement option for subgrid-scale modelling in LES of this particular class of compressible flows.
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References 1. N. A. Adams and K. Shariff. A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems. Journal of Computational Physics, 127:27–51, 1996. 2. G. N. Coleman, J. Kim, and R. D. Moser. A numerical study of turbulent supersonic isothermal-wall channel flow. Journal of Fluid Mechanics, 305:159– 183, 1995. 3. J. G. M. Eggels, F. Unger, M. H. Weiss, J. Westerweel, R. J. Adrian, R. Friedrich, and F. T. M. Nieuwstadt. Fully developed turbulent pipe flow: A comparison between direct numerical simulation and experiment. Journal of Fluid Mechanics, 268:175–209, 1994. 4. H. Foysi, S. Sarkar, and R. Friedrich. Compressibility effects and turbulence scalings in supersonic channel flow. Journal of Fluid Mechanics, 509:207–216, 2004. 5. P. G. Huang, G. N. Coleman, and P. Bradshaw. Compressible turbulent channel flows: DNS results and modelling. Journal of Fluid Mechanics, 305:185–218, 1995. 6. R. Lechner, J. Sesterhenn, and R. Friedrich. Compressibility effects and turbulence scalings in supersonic channel flow. J. of Turbulence, 2:1–25, 2001. 7. S. K. Lele. Compact finite difference schemes with spectral-like resolution. Journal of Computational Physics, 103:16–42, 1992. 8. J. Mathew, R. Lechner, H. Foysi, J. Sesterhenn, and R. Friedrich. An explicit filtering method for large eddy simulation of compressible flows. Physics of Fluids, 15:2279–2289, 2003. 9. K. Mohseni and T. Colonius. Numerical treatment of polar coordinate singularities. Journal of Computational Physics, 157:787–795, 2000. 10. Y. Morinishi, S. Tamano, and K. Nakabayashi. Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls. Journal of Fluid Mechanics, 502:273–308, 2004. 11. J. Sesterhenn. A characteristic-type formulation of the Navier-Stokes equations for high order upwind schemes. Computers and Fluids, 30:37–67, 2001. 12. S. Stolz. Large eddy simulation of complex shear flows using an approximate deconvolution model. PhD thesis, ETH, Zurich, 2000. 13. S. Stolz and N. A. Adams. An approximate deconvolution procedure for largeeddy simulation. Phys. Fluids, 11:1699–1701, 1999. 14. C. Walshaw and S. J. Farr. A two-way parallel partition method for solving tridiagonal systems. Technical Report 93.25, School of computer studies, Univ. of Leeds, 1993. 15. J. K. Williamson. Low-storage Runge-Kutta schemes. Journal of Computational Physics, 35:48–56, 1980.
Large-eddy simulation of separated flow along a compression ramp at high Reynolds number M. S. Loginov1,2 , N. A. Adams1 , and A. A. Zheltovodov2 1
2
Institute of Aerodynamics, Technical University of Munich, Garching 85747, Germany [email protected] Institute of Theoretical and Applied Mechanics, Novosibirsk 630090, Russia
1 Introduction Shock-wave / turbulent-boundarylayer interaction for compression ramp flow is a canonical test case for turbulence modeling. In fig. 1 basic flow features are sketched [1]. The undisturbed incoming turbulent boundary layer interacts with the shock wave, for sufficiently large deflection angles resulting in a separation region near the compression corner and a λ-shock system containing the separation region. The separa- Fig. 1. Essential flow phenomena in comtion and reattachment positions are pression ramp flows indicated by S and R respectively. Subsequently the disturbed boundary layer passes through the Prandtl–Meyer expansion near the decompression corner. Turbulence is amplified by interaction with a rapid compression within the boundary layer (inset 1) and by direct interaction with the shock in the external flow, item 2. Item 3 points to the damping of turbulent fluctuation by the interaction with the expansion waves at the expansion corner. After reattachment at the deflected part of the compression ramp a turbulent boundary layer is reestablished, item 4. Experimental and computational results, which are discussed in section 2, support the existence of pairs of large counterrotating streamwise vortices in the reattachment region, item 5. Within the area of flow separation the reverse mean flow has the character of a wall jet which exhibits indications of relaminarization, item 6.
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Large-eddy simulation (LES) nowadays can be considered as the most appropriate numerical tool for the numerical analysis of complex turbulent flows. Although different subgrid-scale models have been extensively tested, most LES computations are still limited to Reynolds numbers which are significantly lower then their experimental counterparts. Specifically, there is to our knowledge no direct comparison with a compression-ramp experiment so far. The objective of the current numerical investigation is a direct comparison with an available experiment, along with a detailed investigation of the instantaneous and the averaged flow structure. For this purpose all flow parameters and the flow geometry are matched to the reference experiment [2, 3]. The free-stream Mach number is M ∞ = 2.95, the Reynolds number based on the incoming boundary-layer thickness is Re δ0 = 63560, the ramp deflection angle is β = 25◦ . For our LES we employ the Approximate Deconvolution Model (ADM) [4] for modeling the sub-grid scales. The conservation equations for the filtered density, momentum and total energy are solved in curvilinear coordinates. A 6-th order compact finite-difference scheme is used for spatial discretization and an explicit low-storage 3-rd order Runge-Kutta scheme is applied for time advancement. Boundary conditions are applied as follows: periodic conditions in the spanwise direction, sponge technique at the outflow [5], non-reflecting condition with sponge layer at the upper boundary, and isothermal condition at the wall. The wall temperature is uniform in spanwise, distribution along the streamwise direction is taken from the experiment. Further numerical details can be found in reference [6]. A well-resolved LES of the entire compression-decompression ramp configuration requires about 30 million gridpoints, that is too large for completing the computations in reasonable time, so we performed this simulation in two steps. At the first step only the flow in the compression corner was considered. This simulation is completed and results are presented in section 2. Then a separate simulation of the decompression corner is continued using data from first simulation. This step is still in progress and a preliminary analysis, based on limited statistical data, is provided in section 3.
2 Compression-corner flow The thickness of the undisturbed boundary layer δ0 is used as reference length for our simulations. The compression-corner domain with an extent of 25.8δ0 × 4δ0 × 4δ0 was split into 701 × 132 × 201 gridpoints in streamwise (x1 ), spanwise (x2 ), and wall-normal (x3 ) direction respectively. The inflowboundary conditions have been generated by a separate flat-plate boundarylayer simulation using the rescaling and recycling procedure [7]. After an initial transient from artificially composed initial data, the simulation was continued for 703.45 characteristic time scales of the incoming boundary layer δ0 /U∞ .
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During this time interval the flow field was sampled 1272 times for statistical analysis. The meanflow flow structure is shown in fig. 2. The computational-domain boundaries are indicated by thin black lines, crossflow-planes are colored with local mean density. A translucent isosurface of mean pressure p = 0.1 represents the mean forward shock. Despite the fact that the flow geometry is nominally two-dimensional, the interaction breaks the spanwise translational symmetry. A rake of 10 colored mean streamlines identifies the recirculating flow in the separation region. Furthermore, a non-planar motion in the separation zone and a rotational motion after reattachment are evident.
Fig. 2. Thee-dimensional meanflow.
Two pairs of counterrotating streamwise vortices can be identified in the reattaching shear layer from isosurfaces of streamwise vorticity. Positive and negative rotation is indicated by the different colors. Vortex pair width is about 2δ0 , which is consistent with experimentally observed values [3]. These streamwise vortices affect turbulence structure and the properties of the mean flow significantly. It is obvious that this influence should be taken into account when comparing computational data with the experiment. To assess the agreement of our computation with the experiment we compare skin-friction coefficient and surface pressure in fig. 3. The experimental data were obtained near the model centerline, but the exact position, with respect to streamwise vortices is unknown. The computed Cf , averaged in time
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Cf 3×10
5 -3
4 2×10
1×10
-3
〈 pw〉
-3
2
0
-1×10
-3
1 S .
-20
3
-10
P .
R .
0 (a)
x1
S .
10
20
-20
-10
P .
R .
0
10
20
x1 (b)
Fig. 3. Averaged skin-friction coefficient (a) and wall-pressure (b) distributions , current LES averin the streamwise direction. ◦ , reference experiment; , current LES averaged in time aged in time and over spanwise direction; only, min and max values over spanwise. The leftmost dashed vertical line indicates the compression-corner position, the middle dashed vertical line the decompressioncorner of the experiment and the right-most dashed vertical line the beginning of the sponge-zone at outflow of the computation.
and in the spanwise direction, are in very good agreement with the experimental data, fig. 3(a). Deviations of experimental data from computed spanwise averaged values are in between of min Cf and max Cf denoted by dotted lines. x2
x2
Whereas Cf in the undisturbed boundary layer varies by a magnitude of approximately ±0.48 · 10−3 , this variation increases to about ±1.38 · 10−3 after reattachment R. A less strong spanwise variation is observed for the surface pressure, shown in fig. 3(b) normalized by the surface pressure of the incoming boundary layer value. The surface pressure exhibits a pressure plateau with an inflection point P as indicated in the figure. Again, a very good agreement between computational and experimental results is found. The solution inside the sponge layer at outflow has no physical meaning. An impression of the instantaneous shock-wave structure can be obtained from a computed Schlieren-type visualization in fig. 4. The shock (2) is strong enough to separate the incoming boundary layer (1), forming a detached shear layer (4) which contains the reverse flow region (3). The rear compression shock (5) appears as a converging set of compression waves originating from the reattachment region. Instantaneous data show, however, that the rearward shock in the λ-configuration is in fact created by highly unsteady compression waves and shocklets. An animation of a time-series of shock visualizations for our computation clearly shows the unsteady motion of the shock system and the shedding of compression waves behind the forward shock. We find in our simulations small-scale shock motion [8, 9], along with a large scale shock motion [10]. Our simulation covers roughly two large-scale shock-motion cycles of about 340δ0 /U∞ each, corresponding to 1.26ms using reference data
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(a)
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(b)
Fig. 4. Schlieren-type visualization at two time instants: simulation, computed as density gradient ∇ρ averaged in spanwise direction (a, b).
from the experiment. These data are consistent with a value of about 1ms for the experiments at Re δ0 = 1400000 [11]. We also observe that the rearward shock is highly unsteady and becomes invisible at irregular time intervals. The compression waves, indicated as 5 in fig. 4, travel downstream with a speed of about 0.1U∞ to 0.4U∞ . For verification of the shocklet character of the stronger compression waves, we have confirmed that the change of the flow state across the shocklets satisfies the Rankine-Hugoniot conditions relative to the ambient flow behind the separation shock. The shocklet Mach number varies between 1.2 and 2.2. The stronger shocklets with larger Mach number have lower absolute velocity and belong to the unsteady second stem of the λ-shock. We believe that the presence of traveling compression waves and shocklets in the wake of the compression shock explains the high level of turbulent fluctuations in the external flow between the separation shock and the detached shear layer which was observed earlier in experiments by hot-wire measurements [3]. It was shown experimentally that the acoustic mode is prevalent in this region which is consistent with the existence of weak shocklets. The observed phenomenon provides evidence for an additional mechanism which enhances the level of turbulent fluctuations in this flow region along with direct interaction of shock and turbulence. A characteristic shape of the standard-deviation distribution of wallpressure fluctuations is found experimentally [10], fig. 5. When normalized by the local mean wall pressure, our computational results agree quantitatively with the experimental results, although for these results the Reynolds numbers are one to two orders of magnitude larger. A global maximum at more than 20% of the local mean wall pressure is observed for the LES in the separation region at x1 ≈ −4.5δ0 which corresponds to the mean separation position S, fig. 5(b). The maximum value agrees well with the experimental data at larger Reynolds number. The observed large variance of pressure fluctuations is a consequence of the shock-foot motion. Characteristic for the standard-deviation distribution is a second peak which can be found near the
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0.2
0.2
σpw pw
σpw pw
0.1
0 −10
S .
−5
P .
0.1
R .
0
x1
(a)
5
10
0 −10
S .
−5
R .
0
x1
5
10
(b)
Fig. 5. Standard deviation of wall pressure fluctuation (a) computation and (b) , Re δ0 = 780000; , Re δ0 = 1400000. Vertical lines experiment [10]. have the same meaning as in fig. 3.
reattachment position. Its value is at about 9% of the local mean wall pressure for both, computation and experiment.
3 Preliminary results on decompression corner The domain for the decompression-corner calculation has an extent of 14.5δ0 × 4δ0 × (4 div5)δ0 , and is discretized with 401 × 132 × 201 gridpoints. The combined domain of the two simulations is shown in fig. 6. The decompressioncorner domain is indicated by the thick line, the overlapping part of the compression corner domain is not shown. The cross sections are colored by instantaneous density. The incoming boundary of the second part is matched to the cross-section of compression corner grid, where all conserved quantities were saved 5173 times during a time interval of 226δ0 /U∞ . These data are used as inflow boundary conditions. Currently statistical data from 50 samples are collected during a period of 30δ0 /U∞ . This amount is not enough for conclusive analysis, only a preliminary comparison of experimental meanflow and numerical data can be made. A good agreement in term of mean wall pressure is evident from figure 7. Data were averaged in spanwise direction in order to improve statistical sampling, data from the compression corner simulation in the overlapping part of the domain (x1 ≥ 4.3) are not shown. One of the interesting questions which this simulation will answer is how the shocklets and the streamwise vortices are affected by the expansion waves. As suggested by experimental observation, a high level of external fluctuations behind the expansion fan has an influence on heat transfer. The observed shocklets in the outer flow can sustain a high level of external fluctuations.
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Fig. 6. Instantaneous flowfield in combined domain. 5
4
3 (pw)
2
1 S .
-20
-10
P .
R .
0
10
20
x1
Fig. 7. Spanwise averaged mean wall-pressure distributions in the streamwise direction. Legend is the same as in fig. 3.
4 Conclusions A Large-Eddy Simulation which was performed for flow parameters matching a reference experiment proves the possibility of a correct numerical prediction of shock-wave / turbulent-boundary-layer interaction at compressiondecompression ramps. Its analysis shows a good agreement with a reference experiment. The existence of 3D G¨ortler-like vortices and large-scale shock motion is confirmed numerically. For the successive compression-decompression corner a good agreement of mean-flow data between experiment and simulation
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is obtained. Further analysis of the decompression-corner flow is in progress.
References 1. A. A. Zheltovodov. Peculiarities of development and modeling possibilities of supersonic turbulent separated flows. In V. V. Kozlov and A. V. Dovgal, editors, Separated Flows and Jets: IUTAM Symposium, pages 225–236, 1991. Novosibirsk, USSR July 9 - 13, 1990. 2. A. A. Zheltovodov, E. Sch¨ ulein, and V. N. Yakovlev. Development of turbulent boundary layer under conditions of mixed interaction with shock and expansion waves. Preprint 28–83 ITAM, USSR Academy of Sciences, Siberian Branch, Novosibirsk, 1983. (in Russian). 3. A. A. Zheltovodov and V. N. Yakovlev. Stages of development, flowfield structure and turbulence characteristics of compressible separated flows in the vicinity of 2-D obstacles. Preprint 27–86 ITAM, USSR Academy of Sciences, Siberian Branch, Novosibirsk, 1986. (in Russian). 4. S. Stolz, N. A. Adams, and L. Kleiser. The approximate deconvolution model for large-eddy simulation of compressible flows and its application to shockturbulent-boundary-layer interaction. Phys. Fluids, 13:2985–3001, 2001. 5. N. A. Adams. Direct numerical simulation of turbulent compression corner flow. Theor. Comp. Fluid Dyn., 12:109–129, 1998. 6. M. S. Loginov, N. A. Adams, and A. A. Zheltovodov. LES of shock wave / turbulent boundary layer interaction. In E. Krause, W. J¨ ager, and M. Resch, editors, High Performance Computing in Science and Engineering 04, pages 177–188, 2004. 7. S. Stolz and N. A. Adams. Large-eddy simulation of high-Reynolds-number supersonic boundary layers using the approximate deconvolution model and a rescaling and recycling technique. Phys. Fluids, 15:2398–2412, 2003. 8. J. Andreopoulos and K. C. Muck. Some new aspects of the shockwave/boundary-layer interaction in compression-ramp flows. J. Fluid Mech., 180:405–428, 1987. 9. N. A. Adams. Direct simulation of the turbulent boundary layer along a compression ramp at M = 3 and Reθ = 1685. J. Fluid Mech., 420:47–83, 2000. 10. D. S. Dolling and M. T. Murphy. Unsteadiness of the separation shock wave structure in a supersonic compression ramp flowfield. AIAA J., 12:1628–1634, 1983. 11. D. S. Dolling and C. T. Or. Unsteadiness of the shock wave structure in attached and separated compression ramp flows. Exp. in Fluids, 3:24–32, 1985.
DNS of Transitional Transonic Flow about a Supercritical BAC3-11 Airfoil using High-Order Shock Capturing Schemes Igor Klioutchnikov and Josef Ballmann Mechanics Department, RWTH Aachen University, Templergraben 64, D-52062 Aachen, Germany [email protected]
1 Numerical Method The present work is a DNS investigation of transitional transonic threedimensional flow about a rectangular wing segment with the supercritical BAC 3-11 airfoil [1] in the (x,y)-plane. Periodic boundary conditions are assumed in z-direction. Flow conditions are laminar inflow at Mach number M∞ = 0.75, Reynolds number based on chord-length Rec = 2.11 · 106 and two fixed angles of attack α = 0o and α = 4o . The Navier-Stokes equations for unsteady three-dimensional compressible fluid flow are transformed from Cartesian to general coordinates ξ(x, y), η(x, y), ζ = z and read U,t +
1 1 1 1 F,ξ + G,η + H,z = Fv,ξ + Gv,η + Hv,z . J J J J
(1)
Here U is the solution vector of conserved variables, F, G, H and Fv , Gv , Hv represent the inviscid and viscous fluxes, J is the Jacobian of the transformation. FCT Method The numerical method developed in [2] is of arbitrary even order N = 2, 4, 6, 8, 10... in space and employs central differences for spatial discretisation together with an explicit time integration using a twostep Richtmyer scheme of second order with modified first step: n+1/2
Uir +1/2 = Lr Unir +1/2 − 0.5λr Anrir +1/2 (Rr Unir +1/2 ) Un+1 = Un• − •
3
n+1/2
n+1/2
λr {(Frir +1/2 − Frir −1/2 )
r=1
vn n n − (Fvn rir +1/2 − Frir −1/2 ) − (Sr Uir +1/2 − Sr Uir −1/2 )},
(F1 , F2 , F3 ) = (F, G, H),
(λ1 , λ2 , λ3 ) = (
Δt Δt Δt , , ). J• Δξ J• Δη Δz
(2)
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Here upper indices n and n + 1/2 correspond to time levels and lower indices ir , r = 1, 2, 3 are running in the space directions r = 1, 2, 3 from ir = 1 to ir = irmax . Ar is the flux Jacobian, and Lr , Rr , Sr are special difference operators of order N for spatial discretisation and artificial viscosity. These are defined as
N/2
Lr Uir +1/2 =
(−1)m+1 am (Uir +m + Uir −m+1 ),
(3)
(−1)m+1 bm (Uir +m − Uir −m+1 ),
(4)
m=1 N/2
Rr Uir +1/2 =
m=1 N/2
Sr Uir +1/2 =
1 max Λrir +1/2 (−1)m+1 dm (Uir +m − Uir −m+1 ), N 2 m=1
(5)
where Λmax rir +1/2 , r = 1, 2, 3, are the maximum eigenvalues of the flux Jacobian matrices. The viscous fluxes Fvr are calculated using the difference operators Rr of order N in the final time step (n + 1). The coefficients am , bm and dm are determined analytically for every order N [2]. For simulating flows with steep gradients or strong shocks, an FCT (Flux Corrected Transport) method has been introduced to modify the two-step Richtmyer scheme and the scheme order [2], which has been tested for DNS of airfoil flows [3]. The inviscid fluxes Fr are calculated with the high-order N on the one hand (upper index (h) in Eq.(6)), and a low-order (upper index (l)) on the other hand, after which they are used for calculating an anti-diffusion term. The equations for the anti-diffusive flux (index (ad)) and the corrected flux (index (cor)) read (ad)
(h)n+1/2
(l)n+1/2
Frir +1/2 = Frir +1/2 − Frir +1/2 ,
(cor)n+1/2
Frir +1/2
(ad)
= cir +1/2 Frir +1/2 , (6)
with the limiting function cir +1/2 . WENO Method The WENO (Weighted Essentially Non-Oscillatory) finite difference scheme [4] in this work has been modified regarding the NavierStokes equations in general coordinates (1) and combined with central differences of order (N + 1) for the viscous fluxes. The method is of arbitrary odd order N = 3, 5, 7, 9, 11... in space and employs p sub-stencils with operators for spatial discretisation of order N = (2p − 1) of the scheme. For the time integration an expicit Runge-Kutta TVD-scheme of third order is used that reads U∗• = Un• − Qn• , = U∗∗ Un+1 • • − Q• =
3 r=1
1 ∗ n ∗ U∗∗ • = U• − {−3Q• + Q• }, 4
1 {−Qn• − Q∗• + 8Q∗∗ • }, 12
λr {(Frir +1/2 − Frir −1/2 ) − (Fvrir +1/2 − Fvrir −1/2 )}.
(7)
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For the computing the viscous fluxes Fvr the central high-order discretisation uses the difference operators Rr (eq.(4)) of the FCT method as well but with even order (N + 1). The inviscid fluxes Fr are calculated using the WENO operators of odd high-order N = (2p−1) employing possible p sub-stencils. For monotonicity reasons of spatial discretisation the nonlinear Roe flux-difference splitting or the Lax-Friedrichs flux-vector splitting are used. The Roe flux-difference splitting reads + Fir +1/2 = 0.5{[F(U− ir +1/2 ) − F(Uir +1/2 )]
U± ir +1/2
+ −|Arir +1/2 |(U− ir +1/2 − Uir +1/2 )}, p p ± ± = αl± P± , P = s± l ml Uml , lir +1/2
(8)
m=1
l=1
where αl are nonlinear weigth coefficients and Pl are sub-stencils with coefficients sml . The Jacobian matrix is transformed to: |Arir +1/2 | = Rrir +1/2 |Λrir +1/2 |R−1 rir +1/2 ,
(9)
where R−1 rir +1/2 , Rrir +1/2 are matrices of the left and right eigenvectors and Λrir +1/2 is the diagonal matrix of the eigenvalues. − The Lax-Friedrichs flux-vector splitting Fir +1/2 = F+ ir +1/2 + Fir +1/2 with U = (u1 , u2 , ..., uM ) (for the Navier-Stokes equations M = 5) reads Fir +1/2 = φr Fir +1/2 +
M
m+ [−θN (R−1 mrir +1/2 ΔFir −N/2+1 , ...
m=1 m+ −1 m− ..., R−1 mrir +1/2 ΔFir −N/2+N −1 ) + θN (Rmrir +1/2 ΔFir +N/2 , ... m− ..., R−1 mrir +1/2 ΔFir +N/2−N +2 )]Rmrir +1/2 ,
m± m± ΔFm± ir +1/2 = Fir +1 − Fir ,
m(max)
m Fm± ir = 0.5(Fir ± Λir
(10)
Uir ),
where φr is the central difference operator of even order (N − 1). R−1 mrir +1/2 , Rmrir +1/2 are m-th left and right eigenvectors of the Jacobian in general coorm(max) dinates and Λir are the maximum eigenvalues (see eq.(5)). The nonlinear weight operators θN (a1 , a2 , ..., aN −1 ) are defined using smoothness estimators for every order N . Application Three-dimensional flow simulation was done on the CH type grid with 1280x130x120 grid points on a Sun Fire cluster with 20 CPUs. The computational domain is Lx = 40c, Ly = 40c and Lz = 1c. The numerical results shown here were obtained using the FCT method of order N = 8 in space and the two-step Richtmyer scheme of second order in time. Computation using the WENO method is still in progress.
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a)
b)
c)
d)
e)
f)
Fig. 1. DNS of 3D flow about the BAC3-11 airfoil for M∞ = 0.75, α = 0o . Instantaneous distributions of a)density by DNS and b)Schlieren picture taken during experiment [5]; DNS: c),d)Mach number and e),f)vortical structures (iso-surfaces of λ = −10) near the airfoil.
DNS of Transonic Flow about BAC3-11 Airfoil using High-Order Schemes
a)
b)
c)
d)
e)
f)
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Fig. 2. DNS of 3D flow about the BAC3-11 airfoil for M∞ = 0.75, α = 0o . From a) to f): Instantaneous distributions of Mach number in the plane z = 0 at different times: a)to , b)to + 0.23ms, c)to + 0.82ms, d)to + 1.50ms, e)to + 3.26ms, f)to + 3.76ms.
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a)
b)
c)
d)
e)
f)
Fig. 3. DNS of 3D flow about the BAC3-11 airfoil for M∞ = 0.75, α = 4o . From a) to f): Instantaneous distributions of Mach number in the plane z = 0 at different times: a)to , b)to + 0.19ms, c)to + 0.24ms, d)to + 0.38ms, e)to + 0.51ms, f)to + 1.70ms.
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1.5 2
1
1.5
1
-c p [-]
-c p [-]
0.5
0.5
0
0 o
α=0 , Ma=0 .75
-0.5
-1
a)
0
0.2
0.4
x/l [-]
0.6
0.8
al=4 o , Ma=0 .75
-0.5
6
D NS (R e=2 .11⋅10 ) Q UADF LO W (Re= 2.11⋅10 6 ) 6 E xp. [KR G cas e 4] (Re= 4.46⋅10 )
6
D NS (R e=2 .11⋅10 ) 6 Q UADF LO W (Re= 2.11⋅10 ) 6 E xp. [KR G cas e 60] (Re= 4.58⋅10 )
-1
1
0
b)
0.2
0.4
x/l [-]
0.6
0.8
1
Fig. 4. DNS of 3D flow about the BAC3-11 airfoil for M∞ = 0.75. Instantaneous distributions of pressure coefficient in the plane z = 0 at two different angles of attack α = 0o (a) and α = 4o (b) in compare with mean distributions: RANS [6] and experiments [6].
2 Results An experimental result for the supercritical BAC3-11 airfoil has been chosen from [5]. The time resolved shadowgraph visualisations and time resolved pressure measurements show the shock-wave structures in the flow field about the BAC3-11 airfoil in a certain Mach and Reynolds number range where the supersonic domain is not yet well established. The mechanism which generates the waves as well as their interactions with the boundary layer in the transitional regime is not yet fully understood. The present work is a numerical investigation of unsteady shock-waves structures in transonic transitional flow about the BAC3-11 airfoil. Fig.1 presents a DNS result at M∞ = 0.75, Rec = 2.11 · 106 , α = 0o for an instantaneous density distribution (a) and an experimental Schlieren picture at M∞ = 0.7, Rec = 3.3 · 106 , α = 0o (b) for comparison. The results are in good phenomenological agreement and show the presence of a local supersonic region closed by the family of shock waves in form of λ-shocks which are evolving along the airfoil surface. Transition inception to turbulent flow may be located on the upper side in the supersonic domain manifesting itself in strong vortex separation and interaction with the shocks (Fig.1 c,e). On the lower side the transition region begins to form behind the leading edge in the region where an inflection point of the profile geometry is present (Fig.1 d,f). Fig.2a-f show the development of Mach number distributions in the mean plane of the flow field at different times. Complex shock-vortex- and boundary-layer-interactions cause new shock waves which propagate upstream in the subsonic region and interact with λ-shocks in the supersonic region. During that time a new λ-shock is formed. This happens periodically. Fig.3a-f show the development of Mach number distributions in the mean plane of the flow field at different times after increase of the angle
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of attack up to α = 4o at M∞ = 0.75, Rec = 2.11 · 106 . Instead of family of shocks in this case a big λ-shock is formed (see [3], too). The vortices originated in the transition region interact with the normal part of the λ-shock and develop further to strong turbulent flow separation behind the shock. This causes new shock waves which steepen on the upstream side while propagating in y-direction along the main part of the λ-shock. They are diffracted round the supersonic region being weakened, finally they disappear. The oblique part of the λ-shock moves upstream and disappears at about x = 0.2. During that time a new oblique shock is formed. This happens periodically. On the lower side in this case the flow field stays laminar without vortical structures. Fig.4a,b show the DNS-result of instantaneous distributions of the pressure coefficient on the airfoil at two different angles of attack α = 0o (a) and α = 4o (b) in comparison with RANS-results (QUADFLOW, [6]) and experimental results [6] of mean distributions. The DNS-results show pressure fluctuations which correspond to vortex-boundary layer interactions on the airfoil. Conclusion Two high order finite difference schemes- the one based on FCT, the other on WENO techniques- which are appropriate for DNS of unsteady compressible fluid flow with shocks and shock-boundary layer interaction have been developed for application to transitional transonic flow about supercritical BAC3-11 airfoil. The high order in space enables to reproduce well vortical structures and internal dynamics of weak shock-structures in the transitional region to turbulent flow. The numerical results are in good agreement with experimental data. The support of Deutsche Forschungsgemeinschaft within the Collaborative Research Center SFB 401 “Flow Modulation and Fluid-Structure Interaction” is greatfully acknowledged.
References 1. Moir I.R.M. (1994) Measurements on a two-dimensional airfoil with high lift devices, AGARD-AR-303 2. Klioutchnikov I. (1998) Direct Numerical Simulation of Turbulent Compressible Fluid Flow. Habilitation Thesis, Russian Akademy of Sciences (in Russian) 3. Klioutchnikov I., Ballmann J. (2004) Direct Numerical Simulation of Transonic Flow about a Supercritical Airfoil. DLES V, 2003, Kluwer Academic Publishers, 9, 223-230 4. Jiang G.S., Shu C.W. (1996) Efficient implementation of weighted ENO schemes. J. of Comp. Physics, 126, N1, 202-228 5. Olivier H., Reichel T., Zechner M. (2003) Flow visualisation and pressure measurements on an airfoil in high Reynolds number transonic flow. AIAA J., 41, N8, 1405-1412 6. Ray S.R. (2005) Numerische Vergleichsrechnungen mit QUADFLOW zu den KRG-Versuhen am DLR Goettingen. Interner Bericht des TP A3 im SFB 401, LFM, RWTH Aachen
Part XVII
Numerical Techniques and POD
A Preconditioned LES Method for Nearly Incompressible Flows N. A. Alkishriwi, M. Meinke, and W. Schr¨ oder Aerodynamisches Institut, RWTH Aachen University, W¨ ullnerstraße zw. 5 und 7, D-52062 Aachen, Germany [email protected] Summary. A procedure for large-eddy simulations (LES) of a continuous tundish flow using the viscous conservation equations for compressible fluids is described. The numerical computations are performed by solving Navier–Stokes equations for compressible flow using an implicit dual time stepping scheme combined with low Mach number preconditioning and a multigrid acceleration technique. The impact of jet spreading, jet impingement on the wall, and wall jets on the flow field and steel quality is investigated. The characteristics of the flow field in a one-strand tundish such as the time-dependent turbulent flow structure and vortex dynamics is analyzed. The resolution of these phenomena is of importance to investigate the inclusion of nonmetallic material.
1 Introduction Large-eddy simulation (LES) has emerged as a viable and powerful tool to investigate the physics of unsteady compressible and incompressible turbulent flows. However, the LES of low Mach number flows based on compressible solvers has not been adequately discussed in the literature. At low Mach numbers the performance of LES codes developed for the conservation equations of compressible fluids deteriorates in terms of speed and accuracy due to the presence of two different time scales associated with acoustic and convective waves. In many subsonic turbulent flows low Mach number regions exist, which require large integration times until a fully developed flow is established. Different methods have been proposed to solve such mixed flow problems by modifying the existing compressible flow solvers. One of the most popular approaches is to use low Mach number preconditioning methods for compressible codes (Turkel, (1999)). The basic idea of this approach is to modify the time marching behavior of the system of equations without altering the steady state solution. This is, however, only useful when the steady state solution is sought. A straightforward extension of the preconditioning approach to unsteady flow problems is achieved when it is combined with a dual time stepping technique.
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In this paper, issues related to the development of a new LES method for low Mach number flows are discussed. A highly efficient large-eddy simulation method is presented based on an implicit dual time stepping scheme combined with preconditioning and multigrid. This method is validated by well-known case studies in Alkishriwi et al. (2004, 2005a, 2005b). The preconditioned large-eddy simulation method is used here to simulate the flow field in a continuous casting tundish. The main function of a continuous casting tundish is to act as distributor of molten steel from the ladle to the moulds. The tundish plays a critical role in determining the final steel quality. In continuous casting of steel, the tundish enables to remove nonmetallic inclusions from molten steel and to regulate the flow from individual ladles to the mold. There is two types of flow conditions in a tundish. Steady-state casting where the mass flow rate through the shroud msh is equal to the mass flow rate through the submerged entry nozzle into the mold mSEN , and transient casting in which the mass of steel in a tundish varies in time during the filling or draining stages. The motion of the liquid steel is generated by jets into the tundish and continuously casting mold. The flow regime is mostly turbulent, but some turbulence attenuation can occur far from the inlet. The characteristics of the flow in a tundish include jet spreading, jet impingement on the wall, wall jets, and an important decrease of turbulence intensity in the core region of the tundish far from the jet (Gardin et al. (1999)) and (Alkishriwi et al. (2005b)). In previous studies, a large amount of research has been carried out to understand the physics of the flow in a tundish mainly through numerical simulations based on the Reynolds-averaged Navier–Stokes (RANS) equations plus an appropriate turbulence model. To gain more knowledge about the time-dependent turbulence process, which cannot be achieved via RANS solutions, large-eddy simulations of the tundish flow field are performed.
2 Governing equations The governing equations are the unsteady three-dimensional compressible Navier-Stokes equations written in generalized coordinates ξi , i = 1, 2, 3 ∂Q ∂(Fci − Fvi ) + =0 ∂t ∂ξi
,
(1)
where the quantity Q represents the vector of the conservative variables and Fci , Fvi are inviscid and viscous flux vectors, respectively. As mentioned before, preconditioning is required to provide an efficient and accurate method of solution of the steady Navier–Stokes equations for compressible flow at low Mach numbers. Moreover, when unsteady flows are considered, a dual time stepping technique for time accurate solutions is used. In this approach, the solution at the next physical time step is determined as a steady state problem to which preconditioning, local time stepping and multigrid are applied.
A Preconditioned LES Method for Nearly Incompressible Flows
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Introducing of a pseudo-time τ in (1), the unsteady two-dimensional governing equations with preconditioning read Γ−1
∂Q ∂Q + + R = 0, ∂τ ∂t
(2)
where R represents R=
∂Fc ∂Ev ∂Fv ∂Ec + + + ∂ξ ∂η ∂ξ ∂η
(3)
and Γ−1 is the preconditioning matrix, which is to be defined such that the new eigenvalues of the preconditioned system of equations are of similar magnitude. In this study, a preconditioning technique from Turkel (1999) has been implemented. It is clear that only the pseudo-time terms in (2) are altered by the preconditioning, while the physical time and space derivatives retain their original form. Convergence of the pseudo-time within each physical time step is necessary for accurate unsteady solutions. This means, the acceleration techniques such as local time stepping and multigrid can be immediately utilized to speed up the convergence within each physical time step to obtain an accurate solution for unsteady flows. The derivatives with respect to the physical time t are discretized using a three-point backward difference scheme that results in an implicit scheme, which is second-order accurate in time ∂Q = RHS ∂τ RHS = −Γ
3Qn+1 − 4Qn + Qn−1 n+1 + R(Q ) 2Δt
(4)
.
To advance the solution of the inner pseudo-time iteration, a 5-stage RungeKutta method in conjunction with local time stepping and multigrid is used. n+1 For stability reasons the term 3Q 2Δt is treated implicitly within the RungeKutta stages yielding the following formulation for the lth stage Q0 = Qn .. . −1 3Δτ Ql = Q0 + αl Δτ I + αl Γ RHS 2Δt
(5)
.. . Qn+1 = Q5 . The additional term means that in smooth flows the development in pseudotime is proportional to the evolution in t.
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N. A. Alkishriwi, M. Meinke, and W. Schr¨ oder
3 Numerical Procedure The governing equations are the Navier–Stokes equations filtered by a lowpass filter of width Δ, which corresponds to the local average in each cell volume. The monotone integrated large-eddy simulations (MILES) approach is used to implicitly model the small scale motions through the numerical scheme. The approximation of the convective terms of the conservation equations is based on a modified second-order accurate AUSM scheme using a centered 5-point low dissipation stencil (Meinke et al. (2002)) to compute the pressure derivative in the convective fluxes. The viscous stresses are discretized to second-order accuracy using central differences, i.e., the overall spatial approximation is second-order accurate. A dual time stepping technique is used for the temporal integration. In this approach, the solution at the next physical time step is determined as a steady state problem to which preconditioning, local time stepping and multigrid are applied. A second-oder 5-stage Runge-Kutta method is used to propagate the solution from time level $n$ to $n+1$. The Runge-Kutta co6 4 9 12 24 , 24 , 24 , 24 , 24 ) are optimized for maximum stability of a efficients αl = ( 24 centrally discretized scheme. The physical time derivative is discretized by a backward difference formula of second-order accuracy. The method is formulated for multi-block structured curvilinear grids and implemented on vector and parallel computers.
4 Tundish Flow In the following the numerical setup of the tundish flow, the computational domain, and the boundary conditions are briefly described. Figure 1 shows the geometry of the tundish and its main dimensions. Furthermore, the notation of the boundaries is given. The numerical simulations consist of two simultaneously performed computations. On the one hand, the pipe flow is calculated to provide timedependent inflow data for the jet into the tundish and on the other hand, the flow field within the tundish is computed. The geometrical values and the flow parameters of the tundish are given in Table 1. The computational domain is discretized by 12 million grid points, 5 million of which are located in the jet domain to resolve the essential turbulent structures. Since the jet possesses the major impact on the flow characteristics in the tundish, it is a must to determine in great detail the interaction between the jet and the tundish flow. The boundary conditions consist of no-slip conditions on solid walls. At the free surface, the normal velocity components and the normal derivatives of all remaining variables are set zero. An LES of the impinging jet requires
A Preconditioned LES Method for Nearly Incompressible Flows
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Fig. 1. Description of the main parameters of the tundish geometry and integration domain. Table 1. Physical parameters of the tundish flow Parameter
value
tundish length L tundish width B tundish height H inclination of side walls diameter of the shroud dsh diameter of the SEN dSEN diameter of the stopper rod dsr hydraulic diameter dhyd Re based on the jet diameter
1.847 m 0.459 m 0.471 m 7o 0.04 m 0.041 m 0.0747 m 0.6911 m 25000
a prescription of the instantaneous flow variables at the inlet section of the jet. To determine those values a slicing technique based on a simultaneously conducted LES of a fully developed turbulent pipe flow is used.
5 RESULTS The typical structure of the flow field inside the tundish is shown in Figs. 2-5 in different planes. Figs. 2 and 3 visualize the jet flow into the tundish. It is clear that the turbulent flow contains a wide range of length scales. Large eddies
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N. A. Alkishriwi, M. Meinke, and W. Schr¨ oder
Fig. 2. Instantaneous entropy contours and velocity vectors in the center plane of the jet.
Fig. 3. Instantaneous entropy contours and velocity vectors in the X/L= 0 plane.
with a size comparable to the diameter of the pipe occur together with eddies of very small size. The figures evidence the jet spreading, jet impingement on the wall, and wall jets, hence a complicated flow field with secondary flow patterns like recirculating pockets results in the tundish. The flow ejected from the ladle reaches the bottom of the tundish at high velocities, spreads in all directions and then, mainly develops along the side walls of the tundish to generate a dominant double vortex system. This behavior can be seen in Fig. 4. In this figure the turbulent flow in a tundish is visualized by the λ2 criterion. The pronounced vertical flow structure near the shroud is emphasized. To be more precise, to avoid nonmetallic inclusions it is this region of the flow field that has to be optimized. Such a flow pattern leads to nonmetallic inclusions, which are to be avoided. Figs. 5 represents flow field structures in the vertical longitudinal center plane of the tundish. The streamlines in this figure illustrate the vortex dominated flow. It can be seen that the flow includes strong vortices and recirculation regions mainly in the inlet region, which is fully turbulent.
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Fig. 4. Visualizations of the LES of a flow field in the tundish by using λ2 contours shaded by the time averaged velocity; top (side view), center (plan view), bottom (Y-Z plane at x = Lsh ).
6 CONCLUSION An efficient large-eddy simulation method for nearly incompressible flows based on solutions of the governing equations of viscous compressible fluids has been applied. The method uses an implicit time accurate dual timestepping scheme in conjunction with low Mach number preconditioning and multigrid acceleration. A large-eddy simulation of the flow field in a tundish is conducted to analyze the flow structure, which determines to a certain extent the steel quality. The findings evidence many intricate flow details that have not been observed before by customary RANS approaches.
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N. A. Alkishriwi, M. Meinke, and W. Schr¨ oder
Fig. 5. Streamlines in the longitudinal symmetry plane of the tundish.
References 1. Alkishriwi N., Meinke M., Schr¨ oder W. (2004) Efficient Large-eddy simulation of low Mach number flow. Presented in 14th -DGLR Symposium der STAB, Nov. 16-18, 2004, Bremen, Germany. 2. Alkishriwi N., Meinke M., Schr¨ oder W. (2005a) A Large-eddy simulation method for low Mach number flows using preconditioning and multigrid. Computers and Fluids in press. 3. Alkishriwi N., Meinke M., Schr¨ oder W. (2005b) Preconditioned large-eddy simulations for tundish flows. Proc. of turbulence and shear flow phenomena TSFP-4, Williamsburg, June 27-29. 4. Meinke M., Schr¨ oder W., Krause E., Rister T. (2002) A comparison of secondand sixth-order methods for large-eddy simulations. Computers and Fluids, Vol. 31, pp. 695-718. 5. Turkel E. (1999) Preconditioning techniques in computational fluid dynamics. Annual Review of Fluid Mechanics 31:385-416. 6. Gardin P., Brunet M., Domgin J.F., Pericleous K. (1999) An Experimental and numerical CFD study of turbulence in a tundish container. Second International Conference on CFD in the Minerals and Process Industries CSIRO, Melbourne, Australia, 6-8 December.
Particle dispersion in turbulent flows by POD low order model using LES snapshots C. Allery, C. B´eghein, A. Hamdouni and N. Verdon LEPTAB, Universit´e de La Rochelle, Avenue Michel Cr´epeau, 17042 La Rochelle C´edex 1, France [email protected]
Abstract This study deals with the computation of particle dispersion in a lid driven cavity, with a dynamical system obtained by projecting the Navier-Stokes equations onto a POD basis. This POD basis was built using LES. The particle behavior is qualitatively coherent. A substantial decrease of the computaing was observed.
1 Introduction The context of this work is the numerical modeling of indoor air quality, and the long term objective is to control solid particle dispersion in an active way, for realistic configurations encountered in buildings. Using the Lagrangian technique to compute particle dispersion in a turbulent flow requires the knowledge of the instantaneous fluid velocity at the particle’s location. The instantaneous fluid velocity is usually obtained by DNS, LES or RANS coupled with a stochastic model. Instead, in order to treat heterogeneous flows encountered in ventilated rooms, we propose to compute the instantaneous fluid velocity with a low order dynamical system obtained by proper orthogonal decomposition (POD). We already applied this technique to the computation of particle dispersion in a 2D ventilated cavity [1]. In this communication, we will consider particle dispersion in a 3D driven cavity.
2 Low order dynamical POD systems POD, which was introduced in fluid mechanics in 1967 by Lumley [2] consists in finding a physical basis which is optimal in an energetic sense. Thus, we search a deterministic function φ which gives the best representation of the set of flow fields u (assumed random and at real values) in the following sense:
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u, φ )2 = (u
max
ψ∈L2 (Ω)
< (u, ψ )2 >
with
(ψ, ψ) = 1
(1)
where (•, •) denotes the inner product of L2 (Ω) and < • > denotes a statistic average operator. L2 is the space of functions of finite energy in the flow volume Ω. From variational calculus it follows that the above expression is equivalent to the Fredholm integral: 2 φ(x’)dx’ = λφ φ(x) φ R(x, x’)φ (2) Find λ ∈ R and ∈ L (Ω) with Ω
representing an eigenvalue problem for φ. R is the spatial correlation tensor. The eigenfunctions φn are orthogonal and all realizations of the flow u are written: u(x, t) =
+∞
φn (x) in L2 (Ω) sense, with an (t) = (u(x, t), φ n ) an (t)φ
(3)
n=1
In practice, if the sampling of the flow field is obtained by numerical simulation, the evaluation of the tensor R is a very large computational task. In order to reduce the calculation, the snapshots method proposed by Sirovitch [3] can be used. Each eigenvalue λn , taken individually, represents the energy contribution of the corresponding eigenfunction. The eigenvectors φ n are by construction incompressible, satisfy the boundary conditions and can be normalized to form an orthonormal set. POD gives the best approximation of the flow in an energetic sense [4]. The energy contained in the first N POD modes is indeed always greater than the energy contained in any other basis such as the Fourier basis. By applying POD to the fluctuating velocity and keeping the first N more energetic modes, the instantaneous velocity can be written according to N
u(x, t) = u(x) +
φn (x) an (t)φ
(4)
n=1
where u is the mean velocity. The Galerkin projection of the fluctuating Navier-Stokes equations on the spatial structures φ n , taking φ orthonormal and the divergence of φ null, is written: N N N dan = Cnmk am ak + Bnm am + En + Dn for n = 1, ..., N dt m=1 m=1 k=1
with ⎧ ⎪ ⎪ φn , ∇φ φm .φ φk ) ⎨ Cnmk = −(φ ⎪ ⎪ ⎩ Dn = − P φ n ndS Γ
1 φ , Δφ φm − ∇u.φ φm − ∇φ φm .u) (φ Re n 1 φn , −∇P + Δu − ∇u.u) En = (φ Re
Bnm =
(5)
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where n is the outward normal on the domain Ω of boundary Γ . For the driven cavity studied in this paper, since the φ n functions are null on the boundaries, Dn is equal to 0. This technique has been successfully applied to many flow configurations, among these the driven cavity studied by Cazemier et al. [5]. They built a dynamical system using DNS snapshots (at Re = 22000), and this enabled them to predict properly the transition behavior of the flow.
3 Particle model A monodisperse aerosol in an isothermal flow is considered. The particles are solid and spherical. Aerosols in buildings are dilute (for example, the particle number concentration for the particle size range > 1μm in an office building is about 105 particles per m3 [6]). A one-way coupling approach is thus used. The Lagrangian approach is employed to compute the particles’ trajectories. Since the density ratio (particle density versus gas density) is rather high (ρp = 2000kg/m3 , ρ = 1.2kg/m3 ), the particles are mainly submitted to the gravity and drag forces [7]. Other forces such as the buoyancy force, the added mass effect, the Basset force are negligible. In addition, the particles do not exhibit a random motion due to Brownian effects, because the particle diameters in this study (5μm and 10μm, which corresponds to particle relaxation times of respectively 1.85×10−4 s. and 7.41×10−4 s.) are greater than one micron. The particle equation of motion is therefore: ρp
πd3p πd3p dup πd2p = ρp g + ρCd u − up (u − up ) 6 dt 6 8
with: 24 24 if Rep < 1, Cd = 0.44 if Rep > 1000, Cd = Cd = Rep Rep
(6)
2/3
Rep 1+ 6
else
where ρp , dp , up are the density, diameter and velocity of the particle, g is the gravitational acceleration, ρ is the fluid density and u is the instantaneous fluid velocity at the particle’s location. Rep is the particle Reynolds number. The particle position at each time step is obtained by integrating: up =
dxp dt
(7)
Computing the instantaneous fluid velocity at the particle’s location by using the low order dynamical system (5) yields the following coupled problem: da = F (a, t) dt
dup = G(a, up , t) dt
dxp = up dt
where a = (a1 , a2 , . . . , an ). F (resp. G) is defined in Equ. (5) (resp. Equ. (6)). This set of differential equations is solved by the Runge-Kutta method.
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4 Discussion The application case considered is particle dispersion in an isothermal flow in a lid driven cavity filled with air, which dimensions are 1m. × 1m. × 1m.. The fluid motion in the cavity is enhanced by the motion of the upper horizontal wall. The wall velocity is 0.15 m./s., which corresponds to a Reynolds number based on the cavity height of 10000. The turbulent flow was computed with the Smagorinsky LES model. The flow domain is discretized into a non uniform mesh of 58 × 58 × 58 cells. A cloud of particles (density ρp = 2000kg/m3 ) is released in the upper part of the cavity, close to the upper horizontal wall (0.03m. < xp < 0.07m., 0.48m. < yp < 0.49m., −0.05m. < zp < 0.05m.). The initial particles’velocity is equal to the instantaneous fluid velocity. The time step chosen for particle computation is 0.0002s. . Particles are attached to the wall when they contact the surfaces. Since particles are injected in the vicinity
0.4
0
Y
0.2
-0.2
-0.4
-0.4
-0.2
0 X
0.2
0.4
Fig. 1. Mean velocity field in the z = 0 plane.
of the vertical mid plane z = 0 of the cavity we present in figure 1 the mean velocity field u, obtained by averaging the LES flow, in this plane. Dynamical system (DS) obtained by POD Once the LES computation of the turbulent flow was converged, 100 snapshots (M = 100) of fluctuating velocity (defined by u’ = u − u) fields were recorded every 0.1s. to build the POD basis. In table 1 the eigenvalues λn relative to mode n, the cumulative energy Ec contained in the N first POD modes, and a normalized residual showing the ability of the N first POD modes to reproduce the flow velocity are presented. These quantities are defined according to: 9M N : λi λi resN = β ∞ η ∞ (8) Ec = i=1
2
with β =
[u(x, t)− Ω
i=1 N k=1
2
[u(x, t)]2 dx. Here • ∞
2
φk (x)] dx and η = ak (t)φ Ω
is the infinite norm, f (t) ∞ = sup f (t). In table 1 we can notice that one
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Table 1. Eigenvalues, cumulative energy, residuals Mode n
λn
100 × Ec
resn
1 2 3 4
4.613E-06 6.755E-07 9.939E-08 1.899E-08
85.24 97.73 99.56 99.81
2.313E-01 5.331E-02 1.439E-02 3.947E-03
mode contains 85 % of energy and with four modes the fluctuating energy of the signal is almost captured. It also indicates that only four modes are necessary to obtain a residual less than 4 × 10−3 (which corresponds to a relative error of 1.6 × 10−5 ). With four modes the fluctuating velocities are hence well reconstructed. Therefore, the low-order system, used to compute the flow dynamics and particle dispersion, is obtained with four modes. Figure 2 shows the temporal evolution of the two first modes a1 (t) and a2 (t) during the sampling time used for the snapshot method. In this figure 0.2
0.15 POD DS
POD DS
0.1
0.1
temporal coefficient a2
temporal coefficient a1
0.15
0.05 0 -0.05 -0.1
0.05 0 -0.05 -0.1 -0.15
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-0.2 0
1
2
3
4
5 6 time (s)
a) mode 1
7
8
9
10
0
1
2
3
4
5 6 time (s)
7
8
9
10
b) mode 2
Fig. 2. Temporal evolution of the two first modes.
P OD corresponds to the reference solution obtained by projecting the fluctuating LES velocity field onto the spatial structure φ n (equation 3) and DS is the solution obtained by solving the dynamical system1 (equation 5). It can be noticed that the temporal evolution of the reference functions POD are properly recovered from the dynamical system. A similar remark can be made when considering a3 (t) and a4 (t). Figure 3 gives the fluctuating velocity field obtained by LES and by the low order dynamical system at time t = 9s. The velocity vectors show that there is no difference between the reference solution and the reconstructed fluctuating velocity field. But if we examine the velocity magnitude in detail (not shown here) we can see that there are small differences between these 1
In order to account for the effect of the unresolved modes on the resolved ones, a viscosity of 2.7 × 10−5 was added to the kinematic viscosity of (DS).
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two fields. For a quantitative study of particle dispersion, attention should be paid to the influence of the neglected modes as done by Wang and Squires for example [8]. For this communication focused on the ability of POD to predict qualitatively particle dispersion by a low order model, we nevertheless consider that the dynamical behavior of the flow is properly predicted by “our” low order system.
0.25
0.25
0
0
-0.25
-0.25
-0.5 -0.5
-0.25
0 X
0.25
0.5
a) Reference solution LES
Y
0.5
Y
0.5
-0.5 -0.5
-0.25
0 X
0.25
0.5
b) Obtained by solving DS (4 modes)
Fig. 3. Fluctuating velocity vectors at time t = 9s. in the z = 0 plane.
Particle dispersion Although the POD basis was built by using 10s. of snapshots, particle dispersion was computed with the low order dynamical system for a duration of 120s.. In a first series of simulations, a cloud of 5μm diameter particles was released in the upper part of the cavity. Since POD enables to compute the instantaneous flow with a reduced computational effort, it is “easy” to treat various cases of particle’s injection, in comparison with LES or DNS. Indeed, the first step was to vary the number of particles released in the air flow in order to obtain statistically independent results. Figure 4 depicts the percentage of particles stuck on the lower horizontal wall as function of time. It shows that injecting 5000 particles in the flow yields statistically independent results. Figure 5 presents the temporal evolution of the particles’cloud. One can notice that, since particles are driven by the air flow with a strong velocity very close to the wall, some of them are stuck on the floor due to the combined effects of inertia and gravity. Nevertheless, since these particles are light, many of them follow the jet path, go to the small recirculation regions in the lower left corner of the cavity (see figure 1), and are lifted towards the upper part of the cavity. In a second series of simulations, a cloud of 5000 10μm diameter particles was released in the same location as in the previous simulations. In figure
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45
% of attached particles
40
1000 particles 5000 particles 10000 particles
35 30 25 20 15 10 5 0 0
20
40
60
80
100
120
time (s.)
Fig. 4. Percentage of attached particles according to time (diameter=5μm)
Fig. 5. Temporal evolution of the 5μm particles’ cloud in the cavity (Left: black: 20s., white : 64s. – Right: black : 80s., white : 120s since particles’ injection)
Fig. 6. Temporal evolution of the 10μm particles’ cloud in the cavity (black: 20s., white : 64s. since particles’ injection)
6, the temporal evolution of the particles’ cloud in the cavity is given. Since these particles are heavier than in the previous case, they deposit once they reach the lower horizontal wall, in a smaller duration (about 85 seconds).
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Computing time POD enables to significantly decrease the computing time in comparison with LES. If a reduced order system is used with 4 modes, 4 ODE are solved at each time step to compute the instantaneous flow (rather than partial differential equations with LES). Computing the particle dispersion of 5000 particles with one processor yields a computing time of2 48h while we estimate that the computing time with LES and ten processors is more than one week.
5 Conclusion In this communication, we presented a numerical study of particle dispersion in a lid driven cavity. This non physical case (selected to avoid the pressure term in the DS) has shown the ability of POD to qualitatively predict particle dispersion for one way coupling cases with a minimum effort of computing time. The next step of our work is to validate quantitatively this approach and possibly include a model to account for the effect of the non resolved modes on the particles.
References 1. Allery C., Beghein C., Hamdouni A. (2005) Applying proper orthogonal decomposition to the computation of particle dispersion in a two-dimensional ventilated, Com. in Nonlinear Science and Numerical Simulation 10(8):907-920 2. Lumley J.L. (1967) The structure of inhomogeneous turbulent flows, In: Yaglom and Tarasky (eds) Atmospheric turbulence and radio wave propagation. 3. Sirovitch L. (1987) Turbulence and the dynamics of coherent structures, part 1: Coherent structures, Quarterly of Applied Mathematics 45:561–571 4. Berkooz G., Holmes P., Lumley J.L. (1993) The proper orthogonal decomposition in the analysis of turbulent flows, Annual Review of Fluid Mechanics 25:539–575 5. Cazemier W., Verstappen R.W., Veldman A.E.P. (1998) Proper orthogonal decomposition and low-dimensional models for driven cavity flows, Physics of fluids 10(7):1685-1699 6. Fisk W.J., Faulkner D., Sullivan D., Mendell M.J. (2000) Particle concentrations and sizes with normal and high efficiency air filtration in a sealed airconditioned office building, Aerosol Science and Technology 32:527-544 7. Armenio V., Fiorotto V. (2001) The importance of the forces acting on particles in turbulent flows, Physics of Fluids 13(8):2437-2440 8. Wang Q., Squires K. D. (1996) Large eddy simulation of particle deposition in a vertical turbulent channel flow, Int. J. Multiphase Flow 22(4):667–683 2
This time is still large because at each time step the velocity field is computed for the whole domain. It could be reduced if it were solved only for the grids surrounding each particle considered
On the influence of domain size on POD modes in turbulent plane Couette flow Anders Holstad1 , Peter S. Johansson2 , Helge I. Andersson2 and Bjørnar Pettersen1 1
2
Department of Marine Technology, Norwegian University of Science and Technology [email protected] [email protected] Department of Energy and Process Engineering, Norwegian University of Science and Technology [email protected]
Summary. Coherent structures in turbulent plane Couette flow are studied by extracting three-dimensional spatial modes from data from several direct numerical simulations. The spatial modes are extracted using proper orthogonal decomposition. The results show that the first mode can be associated with the mean flow, and that the second mode can be associated with a pair of counter-rotating streamwise vortices. The results also demonstrate that the distribution of energy between the modes is sensitive to the size of the computational domain. As the size of the domain is increased, the turbulent energy becomes more evenly distributed between the modes. The flow thus becomes more complicated in the sense that more modes become energetically important.
1 Introduction Several studies have shown the existence of large-scale structures in numerically generated turbulent plane Couette flow, cf. Lee & Kim [1] and Komminaho et al. [2]. These structures, which appear in the form of pairs of counterrotating streamwise vortices, have however not been observed experimentally. It has therefore been hypothesized (Andersson et al. [3]) that these vortices are a spurious flow phenomenon caused by a combination of a small box size (computational domain) and periodic boundary conditions in the streamwise direction. Andersson et al. [3] examined the effect of the periodic boundary conditions on the vortices by abandoning the conventional periodicity and introducing new boundary conditions. The tendency for streamwise vortices to develop was reduced, but not eliminated, and it was concluded that further studies were needed in order to conclude as to whether the vortices are a real or spurious flow phenomenon. The effect of the box size on the large-scale structures has been examined by Komminaho et al. [2] at a Reynolds number of Re = 750 (based on half the
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velocity difference between the walls and the channel half-height h) by comparing two-point velocity correlations for various box sizes. They concluded that a very large computational domain, indicating lengths of at least 28πh and 8πh in the streamwise and spanwise directions, was necessary in order to obtain proper decay of the two-point correlations. Tsukahara et al. [4] have also studied coherent structures in turbulent plane Couette flow (Re = 750 and Re = 2150) by comparing two-point correlations for various box sizes. They concluded that a box length of 89.6h − 128h was necessary if the largescale structures were to move freely. Their work was extended in Tsukahara et al. [5], where they concluded that a box width of at least 16h was required if the large-scale structures were to move freely also in the spanwise direction. In this paper, coherent structures in turbulent plane Couette flow are studied by extracting three-dimensional spatial modes from data from several direct numerical simulations (DNSs). The objective of the study is to examine the influence of the size of the computational domain on the spatial modes. The computational domain is systematically increased in the streamwise and spanwise directions, while keeping the Reynolds number and the spatial resolution in all directions constant. The distribution of energy between the modes is then compared for the various domain sizes. The spatial modes are extracted using proper orthogonal decomposition (POD). POD is a method for extracting information from a stochastic field or variable, and in the context of turbulence, POD can be used to extract coherent structures from a three-dimensional flow field (Berkooz et al. [6]). Moehlis et al. [7] and Smith et al. [8] also used POD to extract spatial modes from data from a DNS of turbulent plane Couette flow. However, they focused on low-dimensional modelling, and considered flow at Re = 400 in a box with lengths of 4πh and 2πh, [7], and 1.75πh and 1.2πh, [8], in the streamwise and spanwise directions. The latter case corresponds to flow in a minimal flow unit, i.e. a domain whose streamwise and spanwise extent is just sufficient to sustain turbulence. The flow considered here is fully developed turbulent plane Couette flow at Reynolds number Re = Uw h/ν = 1300, where Uw is half the velocity difference between the walls and ν is the kinematic viscosity. The flow geometry and the coordinate system are shown in Figure 1. Lx , Ly and Lz (= 2h) are the streamwise, spanwise and wall-normal lengths, respectively. 2Uw Lz z, w y, v x, u
Ly
Lx
Fig. 1. Flow geometry and coordinate system
POD modes in turbulent plane Couette flow
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2 Proper orthogonal decomposition The POD procedure consists of projecting an ensemble of velocity fields, uk , onto basis functions, ϕ, chosen such that the averaged projection of uk onto ϕ is maximized. This means that the chosen basis functions are optimal in terms of representing the energy of the ensemble of velocity fields. It can be shown (Holmes et al. [9]) that the POD modes are the Fourier modes in the homogeneous directions. Since the plane Couette flow considered here is homogeneous in the streamwise and spanwise directions, the expansion of the POD modes is unknown only in the wall-normal z-direction. The eigenfunctions or the POD modes can then be expressed as ϕqmn (x) = φqmn (z)e2πi(mx/Lx +ny/Ly ) ,
(1)
where m and n are the number of periods in x- and y-direction, and q is the quantum number indexing the wall-normal expansions. The velocity field can be expressed as aqmn (t)ϕqmn (x), (2) u(x, t) = mnq
ϕqmn (x)
are the spatial basis functions and aqmn (t) are the associated where basis coefficients. The basis coefficients can be found by a Galerkin projection of the Navier-Stokes equations onto the POD modes.
3 Database for the POD The DNSs, which produced the data from which the spatial modes were extracted, were carried out using the computer code MGLET (Manhart et al. [10]). The code is based on a finite-volume discretization of the incompressible Navier-Stokes equations. Central differences are used for the spatial discretization and the explicit second-order Adams-Bashforth scheme for the time advancement. Periodic boundary conditions are used in the streamwise and spanwise directions, and no-slip boundary conditions are used at the walls. The computational mesh is inhomogeneous in the wall-normal z-direction. The databases consist of 240 velocity fields (except for Case 15 which has only 200) sampled every half large-eddy-turnover-time after first having reached a statistically stationary state. The large-eddy-turnover-time is defined as tτ = h/uτ , where uτ is the wall friction velocity defined as uτ = (τw /ρ)1/2 . τw is the wall shear stress and ρ is the density of the fluid.
4 Overview of flow cases The present work started with an examination of the POD modes using a computational domain with Lx = 4πh, Ly = 43 πh and Lz = 2h (Case 0). The
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influence of the box size on the POD modes was then examined by systematically increasing the computational domain in the streamwise and spanwise directions, while keeping the Reynolds number Re and the spatial resolution constant. The largest domain had a streamwise length Lx = 16πh and a spanwise length Ly = 16 3 πh (Case 15). Although the box sizes used in the present work are smaller than what Komminaho et al. [2] and Tsukahara et al. [5] recommended in order to assure sufficient decay of the two-point correlations, it is believed that the present box sizes are large enough for a relevant investigation of the effect of the box size on the POD modes. Some details of the simulations and an overview of the different box sizes are shown in Table 1. Table 1. Overview of flow cases. N1 , N2 and N3 are the number of grid points in the streamwise, spanwise and wall-normal directions. Reτ is the Reynolds number based on uτ and h. The relative extent of the domains is indicated in the figure to the right. All domains begin in the lower-left corner. Case 0 1 2 3 4 5 6 8 9 10 12 13 14 15
Lx /h
Ly /h
Lz /h
4π 4π 4π 4π 6π 6π 6π 8π 8π 8π 16π 16π 16π 16π
4π/3 2π 8π/3 16π/3 4π/3 2π 8π/3 4π/3 2π 8π/3 4π/3 2π 8π/3 16π/3
2 2 2 2 2 2 2 2 2 2 2 2 2 2
N1 × N2 × N3 64 64 64 64 96 96 96 128 128 128 256 256 256 256
× × × × × × × × × × × × × ×
64 96 128 256 64 96 128 64 96 128 64 96 128 256
× × × × × × × × × × × × × ×
64 64 64 64 64 64 64 64 64 64 64 64 64 64
12 13 14
Reτ 84.9 84.8 84.7 84.9 84.7 84.6 84.6 84.7 84.8 84.7 84.7 84.6 84.6 84.6
Lx
8 9 10
15
11
4 5
6
7
1
2
3
0
Ly
5 Verification of DNS Figure 2 shows the distribution of shear stress and root-mean-square values of the fluctuating velocities from Case 14. Turbulent plane Couette flow is characterized by a constant shear stress distribution. Here, the shear stress has been normalized by ρu2τ such that the sum of viscous and turbulent shear stress equals 1. The normalization is with respect to the viscous scales, i.e. wall friction velocity uτ defined earlier, viscous length l∗ = ν/uτ and viscous time t∗ = ν/u2τ . The spatial resolution is somewhat coarse, but sufficient for the present purpose as demonstrated in Figure 2 where results from Case 14 and from Bech et al. [11] are compared.
POD modes in turbulent plane Couette flow 1.4
3
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0.2 0 0
0.5
1
1.5
2
0 0
0.5
1
z
1.5
2
z
Fig. 2. Left: Distribution of dimensionless shear stress from Case 14: dU/dz − uw(· · · ); −uw( ); dU/dz( ). Right: Distribution of root-mean-square values ); of fluctuating velocities normalized by uτ from Case 14: urms ( ); vrms ( wrms (· · · ). In both figures, the markers (◦) indicate the corresponding distributions from Bech et al. [11].
6 Results 6.1 Distribution of energy between the modes Table 2 shows the 10 most energetic modes with corresponding quantum numbers (m, n, q), energy density (λ), degeneracy (d) and fraction of turbulent kinetic energy (E%) for Case 0 (left) and Case 14 (right). The degeneracy is a measure of how many complex symmetry modes there exist for each complex POD mode ϕqmn . The (real) POD “flow unit” corresponding to the quantum number combination (m, n, q) consists of the modes ϕqmn , ϕq−mn , ϕqm−n and ϕq−m−n . The total energy in this flow unit is the basis for the energy fractions shown in Table 2. Table 2. The 10 most energetic POD modes of turbulent plane Couette flow with corresponding quantum number combination (m, n, q), energy density λ, degeneracy d and fraction of turbulent kinetic energy E%. Left: Case 0. Right: Case 14. Mode 1 2 3 4 5 6 7 8 9 10
mnq 0 0 1 0 1 1 0 1 2 0
0 1 1 2 2 1 2 2 1 1
1 1 1 1 1 2 2 2 1 2
λ
d
E%
Mode
4.12E-02 1.02E-03 1.40E-04 1.31E-04 6.36E-05 5.74E-05 1.02E-04 4.79E-05 4.17E-05 7.39E-05
1 2 4 2 4 4 2 4 4 2
1.37E-01 3.75E-02 1.76E-02 1.71E-02 1.54E-02 1.37E-02 1.29E-02 1.12E-02 9.92E-03
1 2 3 4 5 6 7 8 9 10
mnq 0 1 0 1 2 0 1 2 1 1
0 2 2 3 2 3 1 3 2 4
1 1 1 1 1 1 1 1 2 1
λ
d
E%
4.23E-02 8.47E-05 1.64E-04 3.30E-05 2.80E-05 5.24E-05 2.40E-05 2.32E-05 2.19E-05 2.08E-05
1 4 2 4 4 2 4 4 4 4
2.32E-02 2.24E-02 9.05E-03 7.67E-03 7.18E-03 6.58E-03 6.37E-03 5.99E-03 5.70E-03
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The results for Case 0 in Table 2 indicate that there is a substantial reduction in turbulent kinetic energy from one mode to the next, in particular from mode 2 to mode 3. Mode 2 contains for instance fourteen times more energy than mode 10. This may be taken as an indication that a few dominant modes carry most of the energy in the flow. However, as the computational domain is increased, the energy distribution between the modes changes. The energy in Case 14, which domain is 4 times longer and 2 times wider than that of Case 0, is far more evenly distributed between the modes. In this case, mode 2 contains only four times more energy than mode 10. It is also noteworthy that the relative importance of a certain mode changes from one case to the other. Mode 2 of Case 0 corresponds to (m, n, q)=(0, 2, 1) of Case 14, i.e. mode 3 in the latter case. Mode 3 of Case 0 corresponds to (m, n, q)=(4, 2, 1) of Case 14, but this mode is not among the ten most energetic modes in the latter case. The distribution of turbulent kinetic energy between the modes has been calculated for all the flow cases, and the results are shown in Figure 3. The results indicate that the distribution of energy between the modes is sensitive to the size of the computational domain. The smaller domains seem to cause a build-up of energy on the second and third mode, whereas the larger domains lead to a more even distribution of energy between the modes. For the largest domain, Case 15, the energy is quite evenly distributed between the modes. It also seems like the energy becomes more evenly distributed as the domain is increased in the streamwise direction than in the spanwise direction, indicating that box length is more important than box width. This sensitivity of the energy distribution on the size of the computational domain has not been found in plane Poiseuille flow. Johansson & Andersson [12] extracted POD modes from a DNS of a plane Poiseuille flow at Reτ = 180 using a computational domain equal to that of Case 0 in the present work, and found the energy to be quite evenly distributed between the modes. 0.16
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Fig. 3. Distribution of turbulent kinetic energy between modes. Left: Streamwise increase of computational domain. Right: Spanwise increase of computational domain. Case 15 is included in both figures as a reference. This case, having the largest domain, is believed to produce the most correct distribution of energy between the modes.
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Fig. 5. Mode 2 (m, n, q) = (0, 1, 1) for Case 0. Left: Wall-normal variation of the ) and wall-normal ( · ) components. Right: Velocities streamwise ( ), spanwise ( in the yz-plane.
6.2 Configuration of the modes The results from the POD show that the first mode (m, n, q)=(0, 0, 1) can be associated with the mean flow, and that the second mode can be associated with a pair of counter-rotating streamwise vortices. The first mode for Case 0 is shown in Figure 4. The mode is seen not only to have a wall-normal variation similar to the characteristic S -shaped mean velocity profile of turbulent plane Couette flow, but also to agree nicely with the statistical mean flow from the DNS. Figure 5 (right) shows a vector plot of the velocities in the yz-plane of the second mode for Case 0. A pair of counter-rotating streamwise vortices is easily seen. The wall-normal variation of the mode is shown in Figure 5 (left), and it is clear that the streamwise component carries appreciably more energy than the spanwise and wall-normal components. The wall-normal variation of the streamwise component displays a z-dependence similar to the turbulent shear stress shown in Figure 2 (left). This may suggest that the second mode contributes significantly to the turbulent shear stress.
7 Concluding remarks POD modes have been extracted from a series of DNSs of turbulent plane Couette flow. The results show that the first mode can be associated with the mean flow, and that the second mode can be associated with a pair of counter-rotating streamwise vortices. The results also demonstrate that the
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distribution of energy between the modes is sensitive to the size of the computational domain. As the size of the domain is increased, the turbulent energy becomes more evenly distributed between the modes. The flow thus becomes more complicated in the sense that more modes become energetically important. This suggests that if a significant portion of the energy is to be carried by the modes in a model for turbulent plane Couette flow, more than just a few modes will have to be included. More work is underway to further investigate the effect of the domain size on the energy decay properties of the POD modes, and to relate these properties to the decay properties of the two-point velocity correlations. This work has received support from The Research Council of Norway (Programme for Supercomputing) through a grant of computing time.
References 1. Lee M. J. & Kim J. The structure of turbulence in a simulated plane Couette flow. In Eight Symposium on Turbulent Shear Flows, 5-3-1–5-3-6, 1991. 2. Komminaho J., Lundbladh A. & Johansson A. V. Very large structures in plane turbulent Couette flow. Journal of Fluid Mechanics, 320, 259–285, 1996. 3. Andersson H. I., Lygren M. & Kristoffersen R. Roll cells in turbulent plane Couette flow: Reality or artifact. In Sixteenth International Conference on Numerical Methods in Fluid Dynamics (ed. Bruneau C.), 117–122, 1998, Springer. 4. Tsukahara T., Kawamura H. & Shingai K. DNS of turbulent Couette flow with respect to a large scale structure in the core region. Advances in Turbulence X (ed. H. I. Andersson & P.-˚ A. Krogstad), 611–614, 2004, CIMNE. 5. Tsukahara T., Kawamura H. & Shingai K. DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region. Journal of Turbulence (in print), 2005. 6. Berkooz G., Holmes P. & Lumley J. The proper orthogonal decomposition in the analysis of turbulent flows. Annual Review of Fluid Mechanics, 25, 539–575, 1993. 7. Moehlis J., Smith T. R., Holmes P. & Faisst H. Models for turbulent plane Couette flow using the proper orthogonal decomposition. Physics of Fluids, 14, 2493–2507, 2002. 8. Smith T. R., Moehlis J. & Holmes P. Low-dimensional models for turbulent plane Couette flow in a minimal flow unit. Journal of Fluid Mechanics, 538, 71–110, 2005. 9. Holmes P., Lumley J. & Berkooz G. Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, 1996. 10. Manhart M., Tremblay F. & Friedrich R. MGLET: a parallel code for efficient DNS and LES of complex geometries. In Parallel Computational Fluid Dynamics - Trends and Applications, 449–456, 2001, Elsevier. 11. Bech K. H., Tillmark N., Alfredsson P. H. & Andersson H. I. An investigation of turbulent plane Couette flow at low Reynolds number. Journal of Fluid Mechanics, 286, 291–325, 1995. 12. Johansson P. S. & Andersson H. I. Generation of inflow data for inhomogeneous turbulence. Theoretical and Computational Fluid Dynamics, 18, 371–389, 2004.
Implicit Time Integration Method for LES of Complex Flows Fr´ed´eric Daude1 , Ivan Mary1 and Pierre Comte1,2 1
2
ONERA 29, Avenue de la Division Leclerc 92322 Chˆ atillon Cedex, France [email protected] CEAT/LEA/ENSMA, 43, route de l’A´erodrome, 86036 Poitiers Cedex, France
Summary. Implicit time integration is studied on the coupled RANS/LES of the transitional boundary layer around an airfoil near stall. A reference solution to compare with the implicit Gear scheme is determined with an explicit three-stage Runge-Kutta scheme. It is first shown that implicit simulation can find again explicit results in the transition region with a reduction factor of 2 for the CPU time. Then, large time steps can be used without threatening the agreement with explicit simulation, with similar reduction of the computational cost.
1 Introduction For high Reynolds number (about one million) flow around airfoils or cavities, Large Eddy Simulation (LES) makes it possible to obtain more precise and more complete results than those provided by Reynolds-Averaged NavierStokes calculations (RANS). For example, Larchevˆeque et al.[3] obtains numerical results in very good agreement with experimental measurements in the case of a cavity flow with Re = 7×106 . This is also the case for the LES of a turbulent boundary layer around an airfoil performed by Mary and Sagaut [6] with similar numerical and modeling techniques. Nevertheless, a limiting factor for LES to be used for practical purposes is its computational cost. Apart from faster computing resources, more sophisticated numerical methods need to be developed for LES in order to further reduce the computational costs. This approach is addressed in Refs. [3, 6] by the use of implicit schemes to make the calculation feasible and to deal with the wide range of scales. Indeed the time step in the explicit scheme is bounded by numerical stability requirements which can lead to a computational time step which is significantly smaller than the physical time scales of turbulent flow. Implicit methods can be used to overcome these time step limitations. Each calculation in Refs. [3, 6] is characterized by a couple of parameters (CFL,N ), in which CFL and N denote the Courant-Friedrichs-Lewy number and the number of the iterations in the inner process, respectively. (CFL,N ) is (20, 5) in [3] and (16, 4) in [6]. The cavity computation [3] needed 5 million points and 200 h CPU time on a single processor of NEC SX5, and
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we are now tackling the simulation of its controlled counterpart [2], in which the wake of a circular cylinder oriented spanwise interferes with the detached mixing layer in the vicinity of the upstream corner of the cavity. This should take between 10 and 20 million grid points and presumably 10 times more CPU time unless substantial numerical efficiency is gained. We are also aiming at the simulation of the buffeting of an airfoil, for which the low-subsonic coupled RANS/LES [7] is a prerequisite. The validation procedure of the implicit schemes to be used is analogous to that of Choi and Moin [1], in which the influence of CFL was investigated in the case of an incompressible turbulent channel flow, showing that spurious relaminarization could be obtained for too high values of the CFL number. After introducing the different modeling and numerical techniques chosen, we show that the explicit prediction of transition can be obtained with implicit simulation with a reduction factor of 2 in CPU time. Then, large CFL numbers can be used without damaging the results quality with similar computational cost reduction.
2 Governing Equations The three-dimensional unsteady filtered Navier-Stokes equations are used for a viscous compressible Newtonian fluid. In the conservative form, these equations can be expressed as: (1) ∂t U + ∇. F (U ) − F ν (U ) = 0 The flow variable vector U , inviscid flux F (U ) and using the Boussinesq eddy viscosity assumption the viscous flux F ν (U ) are given by ⎛ ⎞ ⎛ ⎛ ⎞ ⎞ ρuj ρ 0 ⎠ (2) U = ⎝ ρui ⎠ , F (U ) = ⎝ ρui uj + pδij ⎠ , F ν (U ) = ⎝ σij ρe (ρe + p)uj σlj ul + qj where ρ, e and p denote the density, the specific total energy, the fluid pressure respectively, ui is the flow velocity in the coordinate direction xi and δij is the Kronecker tensor. This set of equations is completed by the addition of p = ρ(γ − 1)(e − 12 ui ui ) σij = (μ + ρνt )(∂xj ui + ∂xi uj − 23 ∂xk u k) qj = κ∂xj T
with
κ=
μCp Pr
+
ρνt Cp P rt
(3)
where γ, T , P r, Cp and P rt are the specific heat ratio, temperature, Prandtl number, the specific heat and turbulent Prandtl number respectively. This corresponds to the macro-temperature closure of the compressible LES equations in conservation form [5]. The molecular viscosity μ is given by the Sutherland formula. The expression of the eddy viscosity νt is computed from the Selective Mixed Scale Model [4]. The same subgrid scale model was used in [3, 6].
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3 Numerical Method 3.1 Temporal and spatial discretization The equations (1) are discretized by a cell-centered finite volume method using structured multi-block meshes. The time integration employed is the implicit Gear scheme leading to a second-order accuracy. |Ωijk |
n+1 n−1 n − 4Uijk + Uijk 3Uijk
2Δt
n+1 + Rijk =0
(4)
where Uijk is the mean state vector, |Ωijk | the volume of the cell ijk, Δt the time step, n the time level, and Rijk the residual of the discretized convective and viscous terms. In order to have a reference numerical solution, an explicit three-stage RungeKutta scheme is used. The advection terms which have an important role in the accuracy of LES are evaluated using a simplified formula of the AUSM+(P) scheme [6] in order to minimize the dissipative error. The viscous fluxes are discretized by a second-order accurate centered scheme. 3.2 Approximate Newton method Given U n and U n−1 in each cell ijk, system (4) is a non-linear fixed-point problem of the form (5) F(U n+1 ) = 0 It is solved by an approximate Newton-Raphson method ⎧ ⎨ ∂U F(U (k) )ΔU (k) = −F(U (k) ) = ⎩
(6) U (k+1) = U (k) + ΔU (k)
where ∈ IR5 is called the Newton residual. In the exact Newton-Raphson method, ∂U F is the exact Jacobian matrix, too costly to evaluate for the method to be of practical use. However, the method can converge with a suitable approximation of it, which uses the spectral radii of the matrices of the inviscid and viscous operators, as in Ref. [8] in an unsteady RANS context. At each iteration of the inner process, the inherent linear system is solved using the Lower-Upper Symmetric Gauss-Seidel (LU-SGS) factorization. The efficiency of this approximate Newton method depends on the cost of one iteration and the convergence speed of the iterative method. We denote N the iteration number in the inner process. U (0) and U (N ) are set to U n and U n+1 , respectively. Furthermore, the solution is fixed at the ghost cells (ΔU (k) = 0).
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4 Results 4.1 Computational Setup Several configurations are under consideration, ranging from analytical test cases such as 2D vortex advection to the controlled transonic cavity flow. We will here focus on the coupled RANS/LES of the transitional boundary layer around an airfoil near stall (see [7] for details). The Reynolds number, based on the upstream velocity (u∞ = 50 m.s−1 ) and the chord (c = 0.6 m) is 2.1 × 106 . The angle of attack and the upstream Mach number are set to 13.3 deg and 0.15, respectively. Due to the strong adverse pressure gradient, transition occurs naturally, ie without external or upstream forcing, with a laminar separation bubble at x/c = 0.09, which leads to turbulent reattachment at x/c = 0.12 (Figure 1). The LES zone (x/c ∈ [0.07; 0.2]) contains the whole of transition zone, and extends in the laminar and the developed turbulent regions in order to minimize the matching problems between the RANS and the LES regions. The meshes of the LES domain contain 6 × 105 points. A first run is performed at (CFL,N )=(17, 4) similar to [6] which corresponds with a time step Δt = 0.25μs. An explicit, and costly, calculation (with CFL = 1) has also been performed in order to provide a reference. The assessment is based on mean-flow, Reynolds-stress profiles and wall pressure spectra at different streamwise positions in the laminar, transitional and turbulent regions. The exploration of the (CFL,N ) parameter plane is performed as follows. CFL = 17 CFL = 34 CFL = 68 N = 4; 8; 12 8; 16 16; 32 The influence of N is first investigated at fixed CFL = 17. To work out a possible optimal way of reducing the computational cost, the simulations are repeated at larger time step, and thus CFL. In the following, we denote “4 it cfl=17” results obtained with the implicit method with the two parameters N = 4 and CFL = 17.
Fig. 1. Pressure fluctuation isosurface in the transitional zone [6]
4.2 Results Analysis Figure 2 shows that the velocity fluctuations decrease smoothly for increasing abscissa beyond x = 0.15c, and that the mean streamwise velocity profile at
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x = 0.17c has apparently recovered from the separation. The region x/c ≥ 0.15 will thus be referred to as ”turbulent”. In contrast, position x/c = 0.09 is visibly transitional, shortly after the closure of the laminar separation bubble. It is in this region that the influence of the numerical parameters (CFL,N ) is most visible. The peaks in the pressure spectra shown in Fig. 3 are linked to the Kelvin-Helmholtz unstable modes that develop in the bubble, in which the velocity profile is inflectional. We can see that the highest peak values are obtained from the explicit simulation, and that they are well reproduced by the implicit simulations as long as N is ≥ 8. With N = 4, almost all the peak levels are lower. It is thus tempting to interpret the influence of the convergence errors in terms of numerical dissipation, but the investigation of the streamwise evolution of the velocity fluctuations shows more complex trends to interpret. We first notice from Fig. 2 (left) that the explicit simulation yields a plateau of the spanwise component wrms , whereas the streamwise component urms exhibits a sharp peak farther upstream, with an undershoot near the maximum of urms . The peak value of urms decreases when N decreases, the peak moving slightly forward, which would support the interpretation of convergence errors in terms of dissipation. However, this is not supported by the spanwise profiles of wrms , which become more peaky at lower N . Figure 4 shows the evolution of wrms with the distance to the wall at x/c = 0.09 and x/c = 0.17. At x/c = 0.17, in the turbulent region, the higher the number of sub steps, the larger the maximum, with convergence toward the explicit reference. However, at x/c = 0.09, in the transitional region, we can see that the reference is quasi two-dimensional, with nearly zero wrms , whereas the convergence errors yield spurious spanwise fluctuations.
explicit 4it cfl=17 8it cfl=17 12it cfl=17
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Let us now look at the computational efficiency by increasing the time step, keeping in mind that the cost of a (CFL,N )=(17, 4) simulation is
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4 times less than that of its explicit counterpart. Of course, doubling N at the same CFL doubles the cost. We will thus consider iso-cost couples (CFL,N )=(17, 4), (34, 8) and (68, 16) with respect to the twice-as-costly counterparts (CFL, N ) = (17, 8), (34, 16) and (68, 32). The rms fluctuations and the pressure spectra obtained with (CFL, N )=(17, 4), (34, 8) and (68, 16) are similar. Figure 5 (left) also shows a good agreement between the (17, 8), (34, 16) and (68, 32) cases, with the same peak values and positions for urms and wrms at y + = 30, and the same asymptotic behavior at increasing x/c. Figure 5 (right) shows the corresponding wall pressure spectra at x/c = 0.09. The peak values and frequencies are quite close not only to each other, but also to the explicit reference. Note however that the main peaks obtained with (CFL, N )=(34, 16) and (68, 32) are significantly narrower than those from the explicit or the (17, 8) simulations, which, at given peak level, would denote a lower power. Anyway, in contrast with the incompressible channel-flow simulations of [1], it seems that large time steps, thus CFL numbers, can be used with-
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out threatening the consistency with explicit simulations, provided N is large enough. This shows that the explicit time step restriction is too restrictive with respect to the physical time scales which justifies the use of implicit scheme on this test case. Since we found that N should increase proportionally to CFL, there is no increased efficiency: there is still a reduction of the CPU time by 2 comparatively to the explicit calculation. However, this requirement depends on the overall solver’s convergence speed over the whole computational domain, for which possibilities of improvement do exist. 120 20
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5 Conclusion and future plans The effect of the couple of numerical parameters (CFL, N ) is studied on the flow around an airfoil in near stall configuration. The prediction of statistics in the transition region shows a higher numerical demand than those in the developed turbulent region [6] or for cavity flows [3] in which the physics are strongly dominated by the coupling between the detached layer and the acoustic reflections. Indeed, even if the ratio CFL/N ≈ 17/4, which is the case in [3] and [6], allows a reduction factor of the CPU time by 4 comparatively to the explicit simulation, the difference with explicit results is significant. Hence, we must double the iteration number N which allows a reduction factor of 2 for the CPU time without damaging the results quality. Furthermore, simulations performed by increasing the CFL number give similar observations: there is a very good agreement between numerical solutions if the ratio CFL/N is kept constant with similar reduction of the CPU time. Efforts are underway to improve on the numerical implementation in order to satisfy the requirement of increased computational efficiency. Two strategies are under consideration. The first one concerns various approximations in the Newton method (for example, Jacobian matrices) in order to accelerate the convergence of the inner process. Concerning this idea, several test have been made. First, an approximation of the Jacobian using the sign of eigenvalues of these matrices like the Flux-Vector Splitting approach [9] was
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tested. Even if the convergence speed of the inner process increased, the cost of one iteration became too expansive to make a gain in CPU time. Then, the first-order representation of the inviscid numerical flux based on a simplified formula of the AUSM+(P) scheme [6] was linearized. Unfortunately, this linearization would lead to a linear system that is not diagonally dominant which is an important requirement for the LU-SGS factorization. So the algorithm diverges. Afterwards, a time extrapolation for the initial solution of the Newton method and a time correction to reduce numerical errors [10] was implemented. Awkwardly, the effect of these two methods is the divergence of the calculation. The second strategy consists in a local determination of N linked to that CFL, which is inherently local. This implies the derivation of (necessarily) empirical relations between the convergence rate, CFL and N . As we can use larger time step if a suitable N is chosen, the use of a local iteration number can reduce appreciably the cost of the simulation. The validation is now performed on accessible test cases. Preliminary results on the test case of the boundary layer over a flat plate are promising and allow a CPU time reduction. The next validation level is the simulation of realistic test cases such as a turbulent boundary layer around an airfoil and as a passively-controlled transonic cavity flow.
References 1. H. Choi and P. Moin, Effects of the Computational Time Step on Numerical Solutions of Turbulent Flow, J. Comput. Phys., 113 (1994), pp. 1–4. 2. H. Illy, P. Geffroy, and L. Jacquin, Control of Flow Oscillations over a Cavity by Means of a Spanwise Cylinder, 21st ICTAM, Warsaw, Poland, (2004). ˆque, P. Sagaut, T.-H. Le ˆ, and P. Comte, Large-eddy simu3. L. Larcheve lation of a compressible flow in three-dimensional open cavity at high Reynolds number, J. Fluid Mech., 516 (2004), pp. 265–301. 4. E. Lenormand, P. Sagaut, L. Ta Phuoc, and P. Comte, Subgrid-Scale Models for Large-Eddy Simulation of Compressible Wall Bounded Flows, AIAA J., 38 (2000), pp. 1340–1350. 5. M. Lesieur and P. Comte, Favre filtering and macro-temperature in LargeEddy Simulation of compressible turbulence, C.R. Acad. Sci. Paris, 329, S´erie IIb (2001), pp. 363–368. 6. I. Mary and P. Sagaut, Large Eddy Simulation of Flow Around an Airfoil Near Stall, AIAA J., 36 (2002), pp. 1139–1145. 7. G. Nolin, I. Mary, and L. Ta Phuoc, RANS eddy viscosity reconstruction from LES flow field, AIAA Paper 2005-4998, Toronto, Canada, (2005). ´chier, Pr´evisions Num´ 8. M. Pe eriques de l’Effet Magnus pour des Configurations de Munitions, PhD thesis, Universit´e de Poitiers, 1999. 9. J. L. Steger and R. F. Warming, Flux Vector Splitting of the Inviscid Gasdynamic Equations with Application to Finite Difference Methods, J. Comput. Phys., 40 (1981), pp. 263–2933. 10. C. Weber, D´eveloppement de m´ethodes implicites pour les ´equations de Navier-Stokes moyenn´ees et la simulaton des grandes ´echelles : Application ` a l’a´erodynamique externe, PhD thesis, INPT, 1998.
ERCOFTAC SERIES 1. 2.
3. 4.
5. 6. 7. 8. 9.
A. Gyr and F.-S. Rys (eds.): Diffusion and Transport of Pollutants in Atmospheric Mesoscale Flow Fields. 1995 ISBN 0-7923-3260-1 M. Hallb¨ack, D.S. Henningson, A.V. Johansson and P.H. Alfredsson (eds.): Turbulence and Transition Modelling. Lecture Notes from the ERCOFTAC/IUTAM Summerschool held in Stockholm. 1996 ISBN 0-7923-4060-4 P. Wesseling (ed.): High Performance Computing in Fluid Dynamics. Proceedings of the Summerschool held in Delft, The Netherlands. 1996 ISBN 0-7923-4063-9 Th. Dracos (ed.): Three-Dimensional Velocity and Vorticity Measuring and Image Analysis Techniques. Lecture Notes from the Short Course held in Z¨urich, Switzerland. 1996 ISBN 0-7923-4256-9 J.-P. Chollet, P.R. Voke and L. Kleiser (eds.): Direct and Large-Eddy Simulation II. Proceedings of the ERCOFTAC Workshop held in Grenoble, France. 1997 ISBN 0-7923-4687-4 A. Hanifi, P.H. Alfredson, A.V. Johansson and D.S. Henningson (eds.): Transition, Turbulence and Combustion Modelling. 1999 ISBN 0-7923-5989-5 P.R. Voke, N.D. Sandham and L. Kleiser (eds.): Direct and Large-Eddy Simulation III. 1999 ISBN 0-7923-5990-9 B.J. Geurts, R. Friedrich and O. M´etais (eds.): Direct and Large-Eddy Simulation IV. 2001 ISBN 1-4020-0177-0 R. Friedrich, B.J. Geurts and O. M´etais (eds.): Direct and Large-Eddy Simulation V. Proceedings of the fifth international ERCOFTAC workshop on direct and large-eddy simulation, held at the Munich University of Technology, August 27-29, 2003. 2004 ISBN 1-4020-2032-5
SPRINGER – DORDRECHT
