Recursive regularization step for high-order lattice Boltzmann methods

A lattice Boltzmann method (LBM) with enhanced stability and accuracy is presented for various Hermite tensor-based lattice structures. The collision operator relies on a regularization step, which is here improved through a recursive computation of nonequilibrium Hermite polynomial coefficients. In addition to the reduced computational cost of this procedure with respect to the standard one, the recursive step allows to considerably enhance the stability and accuracy of the numerical scheme by properly filtering out second- (and higher-) order nonhydrodynamic contributions in under-resolved conditions. This is first shown in the isothermal case where the simulation of the doubly periodic shear layer is performed with a Reynolds number ranging from ${10}^4$ to ${10}^6$, and where a thorough analysis of the case at $\text{Re}=3\times {10}^4$ is conducted. In the latter, results obtained using both regularization steps are compared against the Bhatnagar-Gross-Krook LBM for standard (D2Q9) and high-order (D2V17 and D2V37) lattice structures, confirming the tremendous increase of stability range of the proposed approach. Further comparisons on thermal and fully compressible flows, using the general extension of this procedure, are then conducted through the numerical simulation of Sod shock tubes with the D2V37 lattice. They confirm the stability increase induced by the recursive approach as compared with the standard one.

Figure 1

Rollup of the double shear layer at $M_0=0.2$ and $\text{Re}=3\times {10}^4$. Visualization of the dimensionless vorticity field at $t_c=L/u_0=1$, using $L=512$ grid points in each directions, with the D2Q9 and the BGK collision model.

Figure 2

Double shear layer at $M_0=0.2$ and $\text{Re}=3\times {10}^4$ for the D2Q9. From top to bottom: dimensionless mean kinetic energy, standard deviation of the kinetic energy, mean enstrophy, and standard deviation of the enstrophy. All quantities are spatially averaged over all the simulation domain. The projection-based (PR) and the recursive (RR) regularizations, at order $N=2$, 3, and 4, are compared against the standard collision model (BGK). The first two columns are the results obtained using a $L\times L$ mesh with $L=128$ and 256, respectively. The last column illustrates the mesh convergence of the fourth-order recursive regularization (RR 4), where the reference solution was obtained using the BGK collision operator with $L=2048$. The characteristic time $t_c$ is defined as $t_c=L/u_0$, while the characteristic speed is $u_0=M_0{c}_s$.

Figure 3

Double shear layer at $M_0=0.35$ and $\text{Re}=3\times {10}^4$ for the D2V17. From top to bottom: dimensionless mean kinetic energy, standard deviation of the kinetic energy, mean enstrophy, and standard deviation of the enstrophy. All quantities are spatially averaged over all the simulation domain. The projection-based (PR) and the recursive (RR) regularizations, at order $N=2$ and 3, are compared against the standard collision model (BGK). The first two columns are the results obtained using a $L\times L$ mesh with $L=128$ and 256, respectively. The last column illustrates the mesh convergence of the third-order recursive regularization (RR 3), where the reference solution was obtained using the BGK collision operator with $L=2048$. The characteristic time $t_c$ is defined as $t_c=L/u_0$, while the characteristic speed is $u_0=M_0{c}_s$.

Figure 4

Double shear layer at $M_0=0.57$ and $\text{Re}=3\times {10}^4$ for the D2V37. From top to bottom: dimensionless mean kinetic energy, standard deviation of the kinetic energy, mean enstrophy, and standard deviation of the enstrophy. All quantities are spatially averaged over all the simulation domain. The projection-based (PR) and the recursive (RR) regularizations, at order $N=2$, 3, and 4, are compared against the standard collision model (BGK). The first two columns are the results obtained using a $L\times L$ mesh with $L=128$ and 256, respectively. The last column illustrates the mesh convergence of the fourth-order recursive regularization (RR 4), where the reference solution was obtained using the BGK collision operator with $L=2048$. The characteristic time $t_c$ is defined as $t_c=L/u_0$, while the characteristic speed is $u_0=M_0{c}_s$.

Figure 5

Stability range of the double shear layer simulation using $L=128$, for $t\le 2t_c$, and with the characteristic time $t_c=L/M_0{c}_s$. The standard PR (filled symbols) and the most stable RR (open symbols) procedures are compared for every LBM. The RR approach always shows a higher stability range regarding $M_0^{\text{Max}}$, where the level of accuracy of $M_0^{\text{Max}}$ is $\mathrm{\Delta }{M}_0=0.01$.

Figure 6

1D Riemann problem: $[P_L/P_R,{\rho }_L/{\rho }_R]=[10,8]$ (left) and $[10,2]$ (right) with $L_x=400$ grid points, and a relaxation time of $\tau =0.595$ and $0.760$, respectively. From top to bottom: dimensionless pressure, density, temperature, and velocity profiles. Results obtained using the fourth-order PR and RR steps are compared to the reference solution (dashed line) for a specific heat ratio $\gamma =2$. They are plotted at time $t/t_c=0.2$ (left) and $t/t_c=0.1$ (right), with the characteristic time $t_c=L_x/\sqrt{\gamma T_R}$.