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Figure 7 Time evolution of normalized turbulence kinetic energy 2k/(3u2, yf), and root mean square values of —-— den- sity \/(p'p’) -3/ (prep Maz), —--— pressure \/(p'p’) -3/(prepyMa;) , and ---- temperature \/(T’T’) -3/ (Tres (y — 1)Ma?), for implicit LES of forced isotropic turbulence with 16? cells.

Figure 7 Time evolution of normalized turbulence kinetic energy 2k/(3u2, yf), and root mean square values of —-— den- sity \/(p'p’) -3/ (prep Maz), —--— pressure \/(p'p’) -3/(prepyMa;) , and ---- temperature \/(T’T’) -3/ (Tres (y — 1)Ma?), for implicit LES of forced isotropic turbulence with 16? cells.

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Table I Model parameters of ALDM for compressible turbulent flows with shocks.
Table I Model parameters of ALDM for compressible turbulent flows with shocks.
scalar mixing?!© in the limit of Ma — 0, we define the dissipation matrix as Stiga we ee ee ee eS Se Me, Se ey oe mire | Ren Sense Me A: Sete MR, ee The free parameters {a,y, a} in the solution-adaptive stencil-selection scheme and the numerical flux function car be used to adjust the spatial truncation error of the discretization. For implicit SGS modeling, the values of these free parameters are chosen in such a way that the truncation error of the discretization method acts as a physically consistent SGS model. The deconvolution parameters a and ¥y should result in an optimum spectral representatior of turbulence structures. It is generally accepted, that for most turbulent flows of practical interest, compressibility effects are mainly due to variations of the mean density and transport properties, whereas the direct effect of density fluctuations on turbulence structures is weak as long as the root mean square (r.m.s.) density fluctuation is smal compared to the mean density (Morkovin’s hypothesis). We therefore use values for a and y that were determined by Hickel et al. for incompressible homogeneous turbulence. The parameter in the flux function, o, can be used to adjust the broadband plateau of the spectral eddy viscosity and diffusivity?®. Isentropic flow requires that o? and o?° have identical values. We use the same value as for passive scalars, which has been calibrated by Hickel et al. !© for isotropic turbulence and later confirmed by Remmler and Hickel® for stratified turbulence. o?“ has been re-calibrated using the same procedure as Hickel and Larsson!” to account for differences between the incompressible and compressible flux function. All parameter values are given in Table I. Tt qa * y 7 7 Pa) 1,71 o + n r +, ye + aw
scalar mixing?!© in the limit of Ma — 0, we define the dissipation matrix as Stiga we ee ee ee eS Se Me, Se ey oe mire | Ren Sense Me A: Sete MR, ee The free parameters {a,y, a} in the solution-adaptive stencil-selection scheme and the numerical flux function car be used to adjust the spatial truncation error of the discretization. For implicit SGS modeling, the values of these free parameters are chosen in such a way that the truncation error of the discretization method acts as a physically consistent SGS model. The deconvolution parameters a and ¥y should result in an optimum spectral representatior of turbulence structures. It is generally accepted, that for most turbulent flows of practical interest, compressibility effects are mainly due to variations of the mean density and transport properties, whereas the direct effect of density fluctuations on turbulence structures is weak as long as the root mean square (r.m.s.) density fluctuation is smal compared to the mean density (Morkovin’s hypothesis). We therefore use values for a and y that were determined by Hickel et al. for incompressible homogeneous turbulence. The parameter in the flux function, o, can be used to adjust the broadband plateau of the spectral eddy viscosity and diffusivity?®. Isentropic flow requires that o? and o?° have identical values. We use the same value as for passive scalars, which has been calibrated by Hickel et al. !© for isotropic turbulence and later confirmed by Remmler and Hickel® for stratified turbulence. o?“ has been re-calibrated using the same procedure as Hickel and Larsson!” to account for differences between the incompressible and compressible flux function. All parameter values are given in Table I. Tt qa * y 7 7 Pa) 1,71 o + n r +, ye + aw
Figure 1 Results for the shock tube problem of Toro *! with 200 cells. solver, ---- ALDM optimized for turbulence without shock capturing, exact solution computed with an iterative Riemann ALDM with shock capturing, Eqs. (36,37).
Figure 1 Results for the shock tube problem of Toro *! with 200 cells. solver, ---- ALDM optimized for turbulence without shock capturing, exact solution computed with an iterative Riemann ALDM with shock capturing, Eqs. (36,37).
Figure 2 Results for the double expansion (1-2-3 problem) with 200 cells. exact analytical solution, ---- ALDM optimized for turbulence with shock capturing deactivated, ALDM with shock capturing, Eq. (37), and shock sensor Eq. (36), —-— fifth-order WENO reconstruction with Rusanov flux working on characteristic variables, —--— HLLC flu working on fifth-order WENO reconstruction of primitive variables.
Figure 2 Results for the double expansion (1-2-3 problem) with 200 cells. exact analytical solution, ---- ALDM optimized for turbulence with shock capturing deactivated, ALDM with shock capturing, Eq. (37), and shock sensor Eq. (36), —-— fifth-order WENO reconstruction with Rusanov flux working on characteristic variables, —--— HLLC flu working on fifth-order WENO reconstruction of primitive variables.
Figure 3 Results for the Noh problem at t = 0.6. Density profiles (a) along grid lines (b) diagonal to grid lines. analytical solution (42), ALDM with shock capturing, Eqs. (36,37), ---- Results of Johnsen et al.°° obtained with a HyBRIr method, which is essentially identical to WENO5-Rusanov for this case.
Figure 3 Results for the Noh problem at t = 0.6. Density profiles (a) along grid lines (b) diagonal to grid lines. analytical solution (42), ALDM with shock capturing, Eqs. (36,37), ---- Results of Johnsen et al.°° obtained with a HyBRIr method, which is essentially identical to WENO5-Rusanov for this case.
Figure 5 Profiles of the entropy measure, Eq. (44), for the Shu-Osher problem®® on (a) 200 cells and (b) 400 cells. eri converged reference solution, ALDM with 200 cells, ---- characteristic WENO5-Rusanov with 200 cells, —-— ALD with 400 cells, —--— characteristic WENO5-Rusanov with 400 cells.
Figure 5 Profiles of the entropy measure, Eq. (44), for the Shu-Osher problem®® on (a) 200 cells and (b) 400 cells. eri converged reference solution, ALDM with 200 cells, ---- characteristic WENO5-Rusanov with 200 cells, —-— ALD with 400 cells, —--— characteristic WENO5-Rusanov with 400 cells.
Figure 6 Time averaged spectra of turbulence kinetic energy k= s0;07 , and variances of —-— density 66*, —--— pres- sure pp", and ---- temperature TT", for implicit LES of forced isotropic turbulence with 16° and 32° cells. Symbols denote the respective asymptotic scalings*®: © €~5/3, 0 €-3, A €-3, and 7 €-°°.
Figure 6 Time averaged spectra of turbulence kinetic energy k= s0;07 , and variances of —-— density 66*, —--— pres- sure pp", and ---- temperature TT", for implicit LES of forced isotropic turbulence with 16° and 32° cells. Symbols denote the respective asymptotic scalings*®: © €~5/3, 0 €-3, A €-3, and 7 €-°°.
relations: the turbulence kinetic energy is scaled by 2/(3u?,,), the r.m.s. density fluctuation by 3/(PrefMa?), the r.m.s. pressure fluctuation by 3/(prefyMa;z), and the r.m.s. temperature fluctuation by 3/(Tre¢(y — 1)Ma?). The reference quantities are
relations: the turbulence kinetic energy is scaled by 2/(3u?,,), the r.m.s. density fluctuation by 3/(PrefMa?), the r.m.s. pressure fluctuation by 3/(prefyMa;z), and the r.m.s. temperature fluctuation by 3/(Tre¢(y — 1)Ma?). The reference quantities are
Figure 9 Instantaneous 3-D spectra of kinetic energy and vorticity at t/Teaay = {0.5, 3.5, 6.5, 10} for implicit LES of decayin; compressible isotropic turbulence at Ma; = 0.3 (left) and Ma; = 0.6 (right). LES with 64% cells, ---- DNS with 256 cells. Note that DNS and LES are different realizations with different random initial data.
Figure 9 Instantaneous 3-D spectra of kinetic energy and vorticity at t/Teaay = {0.5, 3.5, 6.5, 10} for implicit LES of decayin; compressible isotropic turbulence at Ma; = 0.3 (left) and Ma; = 0.6 (right). LES with 64% cells, ---- DNS with 256 cells. Note that DNS and LES are different realizations with different random initial data.
Table II Computational grids for LES of a turbulent boundary layer at Max = 2 at a common point and viscous heating lets them slowly diverge during the decay. We observe excellent agreement of LES and DNS, and emphasize that ALDM correctly predicts the distinctly different decay rates of kinetic energy and the thermodynamic fluctuations for both Mach numbers.
Table II Computational grids for LES of a turbulent boundary layer at Max = 2 at a common point and viscous heating lets them slowly diverge during the decay. We observe excellent agreement of LES and DNS, and emphasize that ALDM correctly predicts the distinctly different decay rates of kinetic energy and the thermodynamic fluctuations for both Mach numbers.
Figure 11 Grid convergence study for a supersonic turbulent boundary layer at Ma. = 2 and Re, ~ 360. (a) van-Driest transformed mean velocity profile: —--— coarse LES grid, ---- medium LES grid, fine LES grid, law of the wall ut = 4In(yt)+C? with « = 0.41 and C* = 5.25, e reference DNS by Pirozzoli et al.°°. (b) Reynolds stresses: —--— coarse LES grid, ---- medium LES grid, —— fine LES grid, e reference DNS by Pirozzoli et al.%°.
Figure 11 Grid convergence study for a supersonic turbulent boundary layer at Ma. = 2 and Re, ~ 360. (a) van-Driest transformed mean velocity profile: —--— coarse LES grid, ---- medium LES grid, fine LES grid, law of the wall ut = 4In(yt)+C? with « = 0.41 and C* = 5.25, e reference DNS by Pirozzoli et al.°°. (b) Reynolds stresses: —--— coarse LES grid, ---- medium LES grid, —— fine LES grid, e reference DNS by Pirozzoli et al.%°.
Figure 12 Computational grid for LES of a 25° compression-expansion ramp flow at Maso = 2.88 and Res = 132840. This picture shows not all but only a few grid lines to illustrate the grid topology and the ramp geometry.
Figure 12 Computational grid for LES of a 25° compression-expansion ramp flow at Maso = 2.88 and Res = 132840. This picture shows not all but only a few grid lines to illustrate the grid topology and the ramp geometry.
Figure 13 Compression-expansion ramp flow at Maw = 2.88 and Res = 132840. Lines are LES results and symbols are experimental data by Zheltovodov et al.°° for (from left to right) ---- A mean density, © mean streamwise velocity, see eon mean temperature, and —-— © mean Mach number at x/do = —8.05.
Figure 13 Compression-expansion ramp flow at Maw = 2.88 and Res = 132840. Lines are LES results and symbols are experimental data by Zheltovodov et al.°° for (from left to right) ---- A mean density, © mean streamwise velocity, see eon mean temperature, and —-— © mean Mach number at x/do = —8.05.
Figure 14 Mean turbulence kinetic energy for LES of a 25° compression-expansion ramp flow at Mas = 2.88 and Res = 132840.
Figure 14 Mean turbulence kinetic energy for LES of a 25° compression-expansion ramp flow at Mas = 2.88 and Res = 132840.
Figure 15 Time averaged normalized wall pressure and friction coefficient for compression-expansion ramp flow at Mas = 2.88. Lines denote LES results and symbols denote experimental data by Zheltovodov et al. °°.
Figure 15 Time averaged normalized wall pressure and friction coefficient for compression-expansion ramp flow at Mas = 2.88. Lines denote LES results and symbols denote experimental data by Zheltovodov et al. °°.
Figure 16 Streamwise mean density profile for LES of canonical shock-turbulence interaction with N = 128 x 64 x 64 cells: top-hat filtered DNS1°8, ---- ALDM, -------- HyBrip-DSM-W5!!", —-— Hyprip-DSM-W5xX!”, —--— Hysrip-DSM- Ww7xt??,
Figure 16 Streamwise mean density profile for LES of canonical shock-turbulence interaction with N = 128 x 64 x 64 cells: top-hat filtered DNS1°8, ---- ALDM, -------- HyBrip-DSM-W5!!", —-— Hyprip-DSM-W5xX!”, —--— Hysrip-DSM- Ww7xt??,
Figure 17 Results for LES of canonical shock-turbulence interaction with N = 128 x 64 x 64 cells. Streamwise evolution of (a) streamwise Reynolds stress, (b) transverse Reynolds stress, (c) anisotropy of the Reynolds stress tensor, and (d) density variance for top-hat filtered DNS!°5, ---- ALDM, -------- Hysrip-DSM-W5!"7, —-— Hysrip-DSM-W5X!””, —--— Hysrip- DSM-W7X"??.
Figure 17 Results for LES of canonical shock-turbulence interaction with N = 128 x 64 x 64 cells. Streamwise evolution of (a) streamwise Reynolds stress, (b) transverse Reynolds stress, (c) anisotropy of the Reynolds stress tensor, and (d) density variance for top-hat filtered DNS!°5, ---- ALDM, -------- Hysrip-DSM-W5!"7, —-— Hysrip-DSM-W5X!””, —--— Hysrip- DSM-W7X"??.
Figure 18 2-D spectra of (a,c,e) streamwise velocity and (b,d,f) density variance for implicit LES of canonical shock-turbulence interaction with N = 128 x 64 x 64 cells.; (a,b) grid plane at (a — xg)&) = —6, (c,d) grid plane at (a — xg3)&) = +6, and (e,f) grid plane at (x — xs) = +30; top-hat filtered DNS1°8, ---- ALDM, -------- Hysrip-DSM-W5!!*, —-— Hysrip- DSM-W5X'!?, —--— HyBrip-DSM-W7X'”.
Figure 18 2-D spectra of (a,c,e) streamwise velocity and (b,d,f) density variance for implicit LES of canonical shock-turbulence interaction with N = 128 x 64 x 64 cells.; (a,b) grid plane at (a — xg)&) = —6, (c,d) grid plane at (a — xg3)&) = +6, and (e,f) grid plane at (x — xs) = +30; top-hat filtered DNS1°8, ---- ALDM, -------- Hysrip-DSM-W5!!*, —-— Hysrip- DSM-W5X'!?, —--— HyBrip-DSM-W7X'”.
Figure 19 Grid convergence for implicit LES of canonical shock-turbulence interaction; (a) mean density, (b) density variance (c) streamwise Reynolds stress, (d) transverse Reynolds stress, (e) turbulence kinetic energy, and (f) anisotropy of the Reynolds stress tensor. DNS! with N, x N. = 1024?,---- LES with N, x N, = 32?,---- LES with N, x Nz = 48°, ++ LES with Ny, x N, = 647, —-— LES with N, x N. = 96, —--— LES with N, x Nz = 128?.
Figure 19 Grid convergence for implicit LES of canonical shock-turbulence interaction; (a) mean density, (b) density variance (c) streamwise Reynolds stress, (d) transverse Reynolds stress, (e) turbulence kinetic energy, and (f) anisotropy of the Reynolds stress tensor. DNS! with N, x N. = 1024?,---- LES with N, x N, = 32?,---- LES with N, x Nz = 48°, ++ LES with Ny, x N, = 647, —-— LES with N, x N. = 96, —--— LES with N, x Nz = 128?.
Figure 22 shows the error in the post-shock streamwise and transverse Reynolds stress defined as
Figure 22 shows the error in the post-shock streamwise and transverse Reynolds stress defined as
Figure 20 Grid convergence of 2-D spectra of (a,c,e) turbulence kinetic energy and (b,d,f) density variance for implicit LES of canonical shock-turbulence interaction; (a,b) grid plane at (« — xs) = —6, (c,d) grid plane at (a — xs5)& = +6, and (e,f) grid plane at (2 — xs)& = +30; DNS!°8 with N, x Nz = 10247, ----- LES with N, x Nz = 32°, ---- LES with Ny x Nz = 487, 0 LES with N, x N. = 64°, —-— LES with Ny, x Nz = 96’, —--— LES with N, x Nz = 128°.
Figure 20 Grid convergence of 2-D spectra of (a,c,e) turbulence kinetic energy and (b,d,f) density variance for implicit LES of canonical shock-turbulence interaction; (a,b) grid plane at (« — xs) = —6, (c,d) grid plane at (a — xs5)& = +6, and (e,f) grid plane at (2 — xs)& = +30; DNS!°8 with N, x Nz = 10247, ----- LES with N, x Nz = 32°, ---- LES with Ny x Nz = 487, 0 LES with N, x N. = 64°, —-— LES with Ny, x Nz = 96’, —--— LES with N, x Nz = 128°.
Figure 21 Shock-normal grid convergence of (a) streamwise and (b) transverse Reynolds stress for implicit LES of canonical shock-turbulence interaction; Ny = Nz = 64, Ng =7 +++ 32,---- 48,-+-+++-- 64, —-— 96, —--— 128, «+ 256, 512.
Figure 21 Shock-normal grid convergence of (a) streamwise and (b) transverse Reynolds stress for implicit LES of canonical shock-turbulence interaction; Ny = Nz = 64, Ng =7 +++ 32,---- 48,-+-+++-- 64, —-— 96, —--— 128, «+ 256, 512.
Figure 22 Error in the post-shock streamwise and transverse Reynolds stress for implicit LES of canonical shock-turbulence interaction; © €11, X €22.
Figure 22 Error in the post-shock streamwise and transverse Reynolds stress for implicit LES of canonical shock-turbulence interaction; © €11, X €22.
Table III Average computational performance of several numerical methods for a realistic compressible flow problem on an Intel® i7 core. Nate that one time step amounts to three flux evaluations.
Table III Average computational performance of several numerical methods for a realistic compressible flow problem on an Intel® i7 core. Nate that one time step amounts to three flux evaluations.
where p and é are suitable approximations of density and sound speed at the respective cell face. These functionals combine an arithmetic average with coupling terms that provide an upwind-type dissipation. LES of compressible turbulence, see Sec. IV.F, showed that acoustic and entropy fluctuations are excessively damped by the term pe(uy - tu; )/2, which is obviously inconsistent with low Mach number fluid dynamics. We note that an exact Riemann solver (B3) gives almost the same results and is only slightly less dissipative than the linearized version (B4a,b). of the inviscid Riemann problem. The weights {b*, b~ } are nonlinear functions of the solution and additionally depend on the wave structure of the Riemann problem. An exact solution based on an iterative procedure is undesirable, not only because it is computationally expensive, but also because it simply is the right solution to the wrong problem if the difference between the left and right state is of pure numerical origin. In shock-free turbulence, the jump is in most cases a result of reconstruction errors and not a physical feature. Linearization of Eqs. (B3a,b) leads to
where p and é are suitable approximations of density and sound speed at the respective cell face. These functionals combine an arithmetic average with coupling terms that provide an upwind-type dissipation. LES of compressible turbulence, see Sec. IV.F, showed that acoustic and entropy fluctuations are excessively damped by the term pe(uy - tu; )/2, which is obviously inconsistent with low Mach number fluid dynamics. We note that an exact Riemann solver (B3) gives almost the same results and is only slightly less dissipative than the linearized version (B4a,b). of the inviscid Riemann problem. The weights {b*, b~ } are nonlinear functions of the solution and additionally depend on the wave structure of the Riemann problem. An exact solution based on an iterative procedure is undesirable, not only because it is computationally expensive, but also because it simply is the right solution to the wrong problem if the difference between the left and right state is of pure numerical origin. In shock-free turbulence, the jump is in most cases a result of reconstruction errors and not a physical feature. Linearization of Eqs. (B3a,b) leads to
‘igure 23 Effect of the interface velocity on the evolution of the dilatation, see Eq. (B6), in decaying isotropic turbulence. ») reference DNS, final ALDM scheme (B5a,b), ---- fixed-stencil scheme (B2) with Eq. (B5a), —-— naive scheme Bla,b) —--— linearized Riemann solution (B4b) with low Mach number consistent pressure reconstruction (B5a). A low Mach number consistent scheme is obtained by combining a central pressure scheme (Bla) with the approx imate Riemann solution (B4b). This scheme has been successfully applied to flows of compressible liquids but give a slightly higher effective numerical diffusivity for the temperature variance at small wavenumbers than required b turbulence theory. We found that the numerical diffusivity scales correctly if the pressure coupling term is compute: only from the third-order reconstruction polynomials. This seems reasonable, as spurious waves are most likely cause: by the high-order contributions to the reconstructed solution. We therefore propose
‘igure 23 Effect of the interface velocity on the evolution of the dilatation, see Eq. (B6), in decaying isotropic turbulence. ») reference DNS, final ALDM scheme (B5a,b), ---- fixed-stencil scheme (B2) with Eq. (B5a), —-— naive scheme Bla,b) —--— linearized Riemann solution (B4b) with low Mach number consistent pressure reconstruction (B5a). A low Mach number consistent scheme is obtained by combining a central pressure scheme (Bla) with the approx imate Riemann solution (B4b). This scheme has been successfully applied to flows of compressible liquids but give a slightly higher effective numerical diffusivity for the temperature variance at small wavenumbers than required b turbulence theory. We found that the numerical diffusivity scales correctly if the pressure coupling term is compute: only from the third-order reconstruction polynomials. This seems reasonable, as spurious waves are most likely cause: by the high-order contributions to the reconstructed solution. We therefore propose
“igure 24 Modified wavenumber analysis for (a) centered differences of second, fourth and sixth order, (b) compressible ALDM: ~~ AR(E), ---- amplification factor | exp S(E)| . settee exact (spectral scheme),
“igure 24 Modified wavenumber analysis for (a) centered differences of second, fourth and sixth order, (b) compressible ALDM: ~~ AR(E), ---- amplification factor | exp S(E)| . settee exact (spectral scheme),
Figure 25 Modified wavenumber analysis for (a) fifth-order WENO reconstruction with HLLC flux function working on primitive variables, (b) sixth-order WENO-CU6-M1 reconstruction with Rusanov flux working on characteristic variables: -------- exact “ A (spectral scheme), AR(E), amplification factor | exp S(&)| .
Figure 25 Modified wavenumber analysis for (a) fifth-order WENO reconstruction with HLLC flux function working on primitive variables, (b) sixth-order WENO-CU6-M1 reconstruction with Rusanov flux working on characteristic variables: -------- exact “ A (spectral scheme), AR(E), amplification factor | exp S(&)| .
Related topics:
Large Eddy SimulationShock WavesShock Waves Boundary Layer InteractionLES and DNS
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