spectral les of turbulent flows over bluff bodies: from the ... - BBAA VI

Minguez Matthieu, Pasquetti Richard and Serre Ericclassical experimental and LES works [13, 20]. The influence of the near wall correction, detailedin Minguez et al. [16], is quantified through the turbulent boundary layer characteristicsas well as the aerodynamics properties of the flow. Finally, the study is extended to the morecomplex configuration of the flow around the car model.2 NUMERICAL MODEL2.1 SPACE AND TIME APPROXIMATIONSThe numerical method is based on a multi-domain Chebyshev-Fourier approximation. Inthe streamwise x-direction, the computational domain is decomposed in non overlapping subdomainsof different lengths, depending on the flow region. The continuity of the solution atthe sub-domain interfaces is ensured by using an influence matrix technique, as in Sabbah andPasquetti [22]. In each sub-domain, a collocation Chebyshev method is used in the y-verticaland streamwise directions, whereas a Fourier-Galerkin method is used in the z-spanwise periodicdirection. In order to refine the Gauss-Lobatto-Chebyshev (GLC) mesh around the bluffbody, we use a mapping to accumulate grid-points at the bluff body, especially in y-direction.The discretization in time is based on a fractional step method. Globally 2 nd order accurate, itmakes use of the following three steps, as detailed in Cousin and Pasquetti [4]:- An explicit transport step, based on an Operator Integration Factor (OIF) semi Lagrangianmethod and a 4 th order Runge-Kutta scheme to handle the non-linear convective term.- An implicit diffusion step, to handle the linear viscous term. Time derivatives of the velocitycomponents are approximated at the resolution time t n+1 with second order backwarddifferences and the pressure is expressed at time t n , using the so-called Goda scheme, toobtain a provisional velocity.- A projection step, to obtain a divergence free velocity field. It is based on an unique grid“P N −P N−2 ” approximation, so that no boundary conditions are required for the pressure.It is indeed a Darcy type problem, rather than a Poisson one, which is solved at this step.In our implementation, the P N − P N−2 approximation, where the polynomial degree forthe pressure is two degrees less than for the velocity components, has been extended to thecase of our multi-domain approximation: Unknown values of the pressure are consideredat all the inner grid-points, including the sub-domain interface points.2.2 SPECTRAL VANISHING VISCOSITY BASED LESFirst introduced by Tadmor [24] to solve non-linear hyperbolic equations, typically the Burgersequation, using standard Fourier spectral methods, the SVV method for the LES (SVV-LES)of incompressible turbulent flows has already shown its effectiveness through different works,see e.g. Karamanos and Karniadakis [9], Kirby and Karniadakis [10], Pasquetti [18] and morerecently Severac et al. [23] and Minguez et al. [16].The SVV stabilization is introduced in the Navier-Stokes equations through a SVV modifiedviscous term ∆ SV V , so that the numerical approximation of the velocity and pressure fields,say (u N , p N−2 ), solves (in the collocation Chebyshev-Galerkin Fourier sense) the semi-discrete3

Minguez Matthieu, Pasquetti Richard and Serre Ericwhere C is a penalization constant and χ(x) the characteristic function of the obstacle, equalto 1 inside the bluff body and 0 elsewhere. A modified penalization method, named pseudopenalization(see Pasquetti et al. [18]) because the penalization is implicitly involved within thetime scheme, is in fact applied here to improve the numerical stability.Commonly, for this kind of modelization and more particularly when spectral methods areconcerned, one has to smooth the characteristic function in order to avoid Gibbs phenomena.However, despite the fact such a smoothing is carefully discussed in [18], as detailed in thepresent paper our best SVV-LES results have been obtained without smoothing. Tests haveequally shown that no Gibbs phenomena occurred, certainly due to the SVV high frequencydissipation term.2.4 NEAR WALL TREATMENTIn parallel to the near wall treatments [19] developed for finite volume approximations andso inapplicable in our SVV-LES methodology, we make use of an original SVV near wallcorrection, as presented in Minguez et al. [16]. In order to capture at the best the turbulenceproduction phenomena localized in the near-wall region, keeping the global stability of thesolution, we simply relax the SVV stabilization within the boundary layers by increasing theactivation parameter m N . Consequently, one diminishes the range of frequencies on which theSVV acts. This near-wall correction is introduced with a new characteristic function χ BL (x),equal to 1 in the near wall (NW) region and to 0 outside. The stabilized Navier-Stokes equationsare then reformulated as :∂ t u N + u N · ∇u N = −∇p N−2 + ν∆ SV V u N − Cχu N + f BL (8)with BL for Boundary Layer and where :f BL = χ BL ν(∆ BLSV V u N − ∆ SV V u N ) . (9)The operator ∆ BLSV V is defined like ∆ SV V but makes use of a greater value of m N . Note thata smaller value of ɛ N is also possible to modify the influence of the SVV terms, but wouldbe more drastic. Moreover, in order to take into account the strong anisotropy of the flowin this region and the distribution of grid-points in the three directions, the operator ∆ BLSV V isanisotropic. Thus, the values of the parameter m N differ, depending on the direction (x, y, z).In practice, its values are increased till preserving the numerical stability of the computation.3 SQUARE CYLINDER TURBULENT FLOWThe flow over the square cylinder is very well documented and so allows the validation ofLES of separated turbulent flows. The ability of our SVV-LES approach to predict such acomplex flow is shown by comparing our results to former ones, obtained both by LES [20]and experimental measurements [13]. We especially point out the efficiency of the near walltreatment previously described. To this end, we compare results obtained with the standardSVV-LES methodology, i.e., with a smoothing of the bluff body characteristic function andwithout near wall treatment, and the hereafter so-called SVV-NW methodology, i.e., withoutsmoothing and with near wall treatment (SVV-LES + NW).5

Minguez Matthieu, Pasquetti Richard and Serre Eric3.2.1 Mean velocity and stress tensor components profilesThe mean profiles of the streamwise velocity and components of the stress tensor have beenaveraged both in time and in the z-spanwise homogeneous direction. Results obtained with thestandard SVV-LES and the SVV-NW approaches are plotted in Figure 2 and 3, together withthose of the experimental and numerical reference studies [13, 20].21.81.6Exp LynLES Rodi UK2SVV−LES with smoothingSVV−LES + NW21.81.6Exp LynLES Rodi UK2SVV−LES with smoothingSVV−LES + NW1.41.4y1.2y1.2110.80.80.60.6a0.4−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Ub0.4−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6U21.81.6Exp LynLES Rodi UK2SVV−LES with smoothingSVV−LES + NW21.81.6Exp LynLES Rodi UK2SVV−LES with smoothingSVV−LES + NW1.41.4y1.2y1.2110.80.80.60.6c0.4−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Ud0.4−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6UFigure 2: y-profiles of the mean streamwise velocity U =< u x > in the boundary layer of the square cylinder atx = −0.375D (a), x = −0.125D (b), x = 0.25D (c) and x = 0.5D(d), obtained with the standard and improvedSVV-LES approaches. Comparisons with the LES of Rodi et al. [20]) and with the experiments of Lyn [13].Present results are globally in good agreement with the experimental data despite of a coarserresolution than in Rodi et al. [20]. The NW treatment considerably improves the present LESresults.In near-wall region, the SVV-NW simulations give a satisfactory estimation of the extremumof the mean streamwise velocity U as well as its location, within the boundary layer, see Figure2. The results are even better than those provided, 15 years ago, by Rodi et al. [20] whichtend to overestimate the maximum of U. The recirculation zone on the cylinder is particularlywell described, with a maximal relative error of about 10% with respect to the first experimentalmeasurement point from the wall in Figure 2b). Within the shear layer generated by the juxtapositionof the recirculation zone and the external velocity field (at y ≃ 0.8), all the LES resultsoverestimate the streamwise velocity in an intermediate region of height ∆y ≃ 0.5. This ten-7

Minguez Matthieu, Pasquetti Richard and Serre Eric21.8Exp LynLES Rodi UK2SVV−LES with smoothingSVV−LES + NW21.8Exp LynLES Rodi UK2SVV−LES with smoothingSVV−LES + NW1.61.61.41.4y1.2y1.2110.80.80.60.6a0.40 0.02 0.04 0.06 0.08 0.1 0.12 0.14< u’ 2 >b0.40 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45< u’ 2 >21.8Exp LynLES Rodi UK2SVV−LES with smoothingSVV−LES + NW21.8Exp LynLES Rodi UK2SVV−LES with smoothingSVV−LES + NW1.61.61.41.4y1.2y1.2110.80.80.60.6c0.40 0.1 0.2 0.3 0.4 0.5 0.6 0.7< u’ 2 >d0.40 0.1 0.2 0.3 0.4 0.5 0.6 0.7< u’ 2 >Figure 3: y-profiles of the streamwise stress term < u ′2 > in the boundary layer of the square cylinder at x =−0.375D (a), x = −0.125D (b), x = 0.25D (c) and x = 0.5D (d), obtained with the standard and improvedSVV-LES approaches. Comparisons with the LES of Rodi et al. [20]) and with the experiments of Lyn [13].dency seems even enforced for the SVV-NW results, although the NW treatment is not activein this flow region. However, the difference becomes weaker when x increases.As expected, the agreement on the streamwise stress tensor term < u ′2 >, see Figure 3, isnot so good. At the different locations in the x-streamwise direction, all the LES results overestimatethe maximum values even if their locations are well predicted, within the shear layer atabout y ≃ 0.8. Nevertheless, SVV-NW results are closer to the experiments of reference [13]and so provide a realistic level of turbulence.In order to estimate the impact of the SVV-NW on the local turbulence level, a comparison ofthe turbulent local Reynolds number is presented in Figure 4 between SVV-LES and SVV-NWresults. The local Reynolds number is defined as Re = u τ ∆y p /ν, with ∆y p the distance of thefirst mesh point from the wall, and u τ the friction velocity estimated from the simplified onedimensional integrated boundary layer equation [25],ν dUdy − < u′ v ′ >≃ u 2 τ (10)The results show that the viscous contribution νdU/dy remains globally equal in a) and b) while,as expected, the turbulent contribution − < u ′ v ′ > is largely increased with the SVV-NW simulations.The stability constraints and turbulence considerations on which the NW treatment8

Minguez Matthieu, Pasquetti Richard and Serre Eric2520h.(nu d yU) 1/2 / nuh.(−< u’v’ >) 1/2 / nuh.(nu d yU − < u’v’ >) 1/2 / nu6050h.(nu d yU) 1/2 / nuh.(− < u’v’ >) 1/2 / nuh.(nu d yU − < u’v’ >) 1/2 / nu4015Re turbRe turb301020510a0−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5xb0−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5Figure 4: Contribution of the different terms of the boundary layer equation (10) computed from SVV-LES withsmoothing of the characteristic function and without SVV-NW (a) and without smoothing of the characteristicfunction but with SVV-NW (b); in legend nu ≡ ν, h ≡ ∆y p and Re turb ≡ Re τ .xwas built are thus verified. Consequently, it appears that although the boundary layer is notcompletely resolved by the mesh, the SVV-NW approach provides a good estimation of theturbulent production.3.2.2 Mean aerodynamic coefficientThe mean drag coefficient, < C d >, and its rms fluctuations, C drms , have been computed.They are compared with reference values in Table 1.Table 1: Aerodynamic features.< C d > C drms < C dp > StExp[13] 1.9 − 2.2 0.1 − 0.2 0.132LES[20] 2.2 0.14 0.138SV V − LES 2.31 0.131 2.26 0.141SV V − LES + NW 2.18 0.116 2.17 0.141SVV-NW provides a good estimation of the drag coefficient < C d >. Without NW treatment,the agreement is not so good and the drag coefficient is overestimated. In order to understandwhy the drag coefficient is larger without NW treatment, we have computed the pressure contributionsto the total drag, see column < C dp > in Table 1. The delay of the detachment onthe wall of the cylinder (cf subsection 3.2.1, Figure 2), shifts downstream the recirculation zonealong the cylinder and behind in the near wake. Consequently, the pressure repartition and thetotal drag are modified.As in the former LES results of Rodi et al. [20], the present SVV-LES overestimate the Strouhalnumber (+7%) with respect to the experiments.9

Minguez Matthieu, Pasquetti Richard and Serre Eric4 AHMED BODY TURBULENT FLOWThe SVV-LES methodology has also been applied to a much more challenging benchmark:the flow over a simplified car model, the “Ahmed body”, at Reynolds number Re = 768000.The SVV-LES results have been compared to the experiments of Lienhart et al. [12] and to thenumerical ones of Guilmineau [7] and Hinterberger et al. [8], using RANS or LES approaches,respectively.4.1 TEST CASE DESCRIPTION AND NUMERICAL DETAILSThe geometry was proposed in the ERCOFTAC benchmark [15] and corresponds to a slantincidence angle of 25 o . The dimensions and the shape of the car model are the ones of Lienhartet al. [12], see figure 5. The bluff body, of length L = 1044 mm, height h = 288 mm and widthw = 389 mm, is positioned at d = 50 mm from the ground. The (x,y,z) directions correspondto the streamwise, vertical and spanwise directions, respectively. The channel is of rectangularsection, 1370 mm × 1000 mm, in the (x,y) directions, and spreads on 4 lengths L of the bluffbody, which is located at the distance L from the inlet.The computational domain is decomposed in eight sub-domains in the x-streamwise direction.Figure 5: Geometry of the Ahmed body.Three are devoted to the Ahmed body and one entirely for the slant part. Each sub-domain isdiscretized using N x = 40, N y = 190 and N z = 340, so that one has about 21.10 6 mesh points.Despite using a mapping in the y-vertical direction to accumulate grid points on the upper panel,the first grid point is located at ∆y + = 350 from the body roof.At the inlet, a bulk velocity U e = 40m/s is imposed and a stiff variation is used to match theno-slip condition at the ground. The Reynolds number, Re=768000, is based on this velocityand on the height of the car h.The SVV parameters have been chosen anisotropic as (m Nx , m Ny , m Nz ) = (2 √ N x , 5 √ √N y , 4 N z /2)and ɛ N = 1/N in the boundary layers whereas in the complete domain (m N , ɛ N ) = ( √ N, 1/N).The simulations have been performed on the NEC-SX8 and have been required 18 GB of memoryand 500 CPU hours to get convergence.It would be noticed that the present SVV-LES results have been obtained with the NW treatmentpreviously justified for the square cylinder wake flow.10

Minguez Matthieu, Pasquetti Richard and Serre Eric4.2 TURBULENCE STATISTICSThe statistics have been converged over the 40 time units, h/U e , corresponding to about0.29 s (physical time).4.2.1 Mean flow and stress tensor components profilesThe topology of the mean flow, fully discussed in Minguez et al.[16], is visualized in Figure6 by the three-dimensional streamlines. As expected from the experiments of Ahmed et al. [1],two strong contra-rotative trailing vortices coming from the slant side edge develop and spreadfarther in the wake. Moreover, a partial recirculation bubble, between the streamwise vorticalstructures, is confined on the upper half part of the slant. Downstream the recirculation zonebehind the bluff body, two large contra-rotative vortices develop in the near wake.Figure 6: Mean three-dimensional streamlines colored by the mean streamwise velocity, view from behind.The profiles of the mean streamwise velocity in the symmetry plane (see Figure 7a) showa rather satisfactory agreement with the experiments. The main experimental characteristicsare recovered, despite a too thick boundary layer arriving on the slant. At the beginning of theslant the fluid detaches and reattaches farther on the middle part of the panel. Consequently, theboundary layer thickness increases along the slant which is in good agreement with the experiments.Classical RANS approaches [7] fail to predict this behavior : the flow does not separateover the slant so that all the main characteristics of the boundary layer get lost, see Figure 7a.In the near wake, the numerical results match relatively well the experimental profiles whilefarther in the wake the SVV-LES results are worse due to a less fine local resolution.Profiles of turbulence kinetic energy k (cf. Figure 7b) exhibit the same behavior. At the junctionbetween the end of the upper panel and the beginning of the slant, the turbulent kineticenergy experimentally measured in the shear layer is not well captured. The maximum of turbulence,within the shear layer, measured by Lienhart et al. [12] at the beginning of the slantis here shifted farther in its middle part. As already mentioned in Minguez et al. [16], this11

Minguez Matthieu, Pasquetti Richard and Serre Eric[3] G. Bosch, Experimentelle und theoretische Untersuchung der instationren Strmung umzylindrische Strukturen, Dissertation, University of Karlsruhe, 1995.[4] L. Cousin and R. Pasquetti, ”High-order methods for the simulation of transitional toturbulent wakes”, Advances in Scientific Computing and applications, Y. Lu, W. Sun andT. Tang Eds, Science Press Beijing/New York, 133-143, 2004.[5] E. Fares, Unsteady flow simulation of the Ahmed reference body using a lattice Boltzmannapproach, Computers & Fluids, 35 (8-9), 940, 2006.[6] R. Franke, W. Rodi, Calculation of a vortex shedding past a square cylinder with variousturbulence models, in: U. Schumann et al (Eds.), Turbulent Shear Flows, 8, Springer,Berlin, 1993.[7] E. Guilmineau, Computational study of flow around a simplified car body, J. WindEng.Ind.Aerodyn.,doi:10.1016/j.jweia.2007.06.041, 2007.[8] M. Hinterberger, M. Garcia-Villalba and W. Rodi, Large Eddy Simulation of flow aroundthe Ahmed body, In Lecture Notes in Applied and Computational Mechanics / The Aerodynamicsof Heavy Vehicles: Trucks, Buses, and Trains, McCallen, R., Browand, F., Ross(Eds.), J., Springer Verlag, 2004.[9] G.S. Karamanos and G.E. Karniadakis, A spectral vanishing viscosity method for largeeddysimulation, J. Comput. Phys., 163, 22, 2000.[10] R.M. Kirby and G.E. Karniadakis, Coarse resolution turbulence simulations with spectralvanishing viscosity - Large Eddy Simulation (SVV-LES), J. Fluids Engineering, 124 (4),886-891, 2002.[11] S. Krajnovic and L. Davidson, Flow around a simplified car, Journal of Fluids Engineering.127, 907-918 et 919-928, 2004.[12] H. Lienhart, C. Stoots and S. Becker, Flow and turbulence structures in the wake of asimplified car model (Ahmed Body). New Results in Numerical and Experimental FluidMechanics, 15-17 Nov., 2002.[13] D.A. Lyn, S. Einav, W. Rodi and J.H. Park, A laser-Doppler velocimetry study ofensemble-averaged characteristics of the turbulent wake of a square cylinder, J. FluidMech., 304, 285 - 319.[14] Y. Maday, S.M.O. Kaber and E. Tadmor, Legendre pseudo-spectral viscosity method fornonlinear conservation laws, SIAM J. Numer. Anal., 30 (2), 321-342 1993.[15] R. Manceau and J.-P. Bonnet, 10th joint ERCOFTAC (SIG-15)/IAHR/QNET-CFD Workshopon Refined Turbulence Modelling, Poitiers, 2000.[16] M. Minguez, R. Pasquetti and E. Serre, High-order Large Eddy Simualtion of flow overthe ”Ahmed body” car model, Phys. of Fluids, (accepted).[17] R. Pasquetti, Spectral vanishing viscosity method for large-eddy simulation of turbulentflows, J. Sci. Comp., 27 (1-3), 365, 2006.14

Minguez Matthieu, Pasquetti Richard and Serre Eric[18] R. Pasquetti, R. Bwemba and L. Cousin, A pseudo-penalization method for high Reynoldsnumber unsteady flows, Applied Numerical Mathematics, under press, 2006.[19] U. Piomelli and E. Balaras, Wall-layer models for large-eddy simulations, Annu. Rev. FluidMech., 34, 349-374, 2002.[20] W. Rodi, Comparison of LES and RANS calculations of the flow around bluff bodies, J.Wind Eng. and Ind. Aerodyn., 46-47, 3-19, 1993.[21] W. Rodi, J.H. Ferziger, M. Breuer, M. Pourquie, Status of Large-eddy simulation : Resultsof a workshop, J. Fluids Eng., 119, 248-262, 1997[22] C. Sabbah and R. Pasquetti, A divergence-free multidomain spectral solver of the Navier-Stokes equations in geometries of high aspect ratio, Comput. Phys., 139, 359, 1998.[23] E. Severac and E. Serre, A Spectral Viscosity LES for the simulation of turbulent flowswithin rotating cavities, Comput. Phys. 226, (2), 1234, 2007.[24] E. Tadmor, Convergence of spectral methods for non linear conservation laws, SIAM J.Numer. Anal., 26(1),30, 1989.[25] H. Tennekes and J. Lumley, A first course in Turbulence, The MIT Press, 1972.[26] P. Voke, 2 n d ERCOFTAC Workshop on Direct and Large-Eddy Simulation, Grenoble,September 1996.15