Pseudoentropic derivation of the regularized lattice Boltzmann method

The lattice Boltzmann method (LBM) facilitates efficient simulations of fluid turbulence based on advection and collision of local particle distribution functions. To ensure stable simulations on underresolved grids, the collision operator must prevent drastic deviations from local equilibrium. This can be achieved by various methods, such as the multirelaxation time, entropic, quasiequilibrium, regularized, and cumulant schemes. Complementing a part of a unified theoretical framework of these schemes, the present work presents a derivation of the regularized lattice Boltzmann method (RLBM), which follows a recently introduced entropic multirelaxation time LBM by Karlin, Bösch, and Chikatamarla (KBC). It is shown that both methods can be derived by locally maximizing a quadratic Taylor expansion of the entropy function. While KBC expands around the local equilibrium distribution, the RLBM is recovered by expanding entropy around a global equilibrium. Numerical tests were performed to elucidate the role of pseudoentropy maximization in these models. Simulations of a two-dimensional shear layer show that the RLBM successfully reproduces the largest eddies even on a $16\times 16$ grid, while the conventional LBM becomes unstable for grid resolutions of $128\times 128$ and lower. The RLBM suppresses spurious vortices more effectively than KBC. In contrast, simulations of the three-dimensional Taylor-Green and Kida vortices show that KBC performs better in resolving small scale vortices, outperforming the RLBM by a factor of 1.8 in terms of the effective Reynolds number.

Figure 1

Vorticity magnitude for shear-layer simulations using the regularized scheme for resolutions with 16 (a), 32 (b), 64 (c), 128 (d), and 256 (e) grid points per direction. The snapshots were taken at $t/t_c=1.25$ and $t/t_c=2.5$ for the upper and lower row, respectively. None of the snapshots shows spurious vortices of the kind investigated by Minion and Brown [56]. The same simulations with the BGK scheme were unstable for all but the highest resolution. The color scale was cut off at 0.7 to emphasize the structure of the vortices.

Figure 2

Decay of kinetic energy (a) and enstrophy (b) in simulations of a doubly periodic shear layer with both, the regularized and the BGK collision model, compared to the reference by Bösch et al. [50]. Thick lines and points represent normalized kinetic energy and enstrophy, thin lines and points represent their relative standard deviations. The results are shown for two resolutions. The BGK simulations for the $128\times 128$ resolution were unstable.

Figure 3

Dissipation of kinetic energy in simulations with the BGK method and the regularized method for Reynolds numbers $\text{Re}=100,200$, and 400 on an ${80}^3$ lattice. The dissipation rates were calculated via finite differences ($-dk/dt$). The results are compared to the reference [58].

Figure 4

Dissipation of kinetic energy in simulations with the BGK method and the RLBM on an ${80}^3$ lattice for $\text{Re}=200$, $\text{Re}=800$, and $\text{Re}=1600$. The dissipation rates were calculated via finite differences ($-dk/dt$) and via the enstrophy ($\nu \mathcal{E}$). The results are compared to Ref. [58].

Figure 5

Dissipation of kinetic energy and effective Reynolds number in simulations with the regularized method for $\text{Re}=15000$ on an ${80}^3$ lattice. The dissipation rates were calculated via finite differences ($-dk/dt$) and via the enstrophy ($\nu \mathcal{E}$). The results are compared to reference results for $\text{Re}=5000,$ as presented in Ref. [64].

Figure 6

Time-averaged effective Reynolds numbers for the BGK scheme and the regularized scheme on an ${80}^3$ lattice. The BGK simulations were unstable for Reynolds numbers beyond 800.

Figure 7

Principle of KBC A, C, and D: while the regularized post collision state places itself at $\gamma \approx 1$, the KBC post collision state fluctuates around the BGK collision state at $\gamma =2$ and varies in space and time along the ${\mathrm{\Pi }}_{xy}=\text{const}$ level.

Figure 8

Dissipation of kinetic energy and effective Reynolds number in simulations with the regularized method and KBC for $\text{Re}=1600$ (a) and $\text{Re}=15000$ (b) on an ${80}^3$ lattice. The dissipation rates were calculated via finite differences ($-dk/dt$) and via the enstrophy ($\nu \mathcal{E}$). The results of (b) are compared to reference results for $\text{Re}=5000,$ as presented in Ref. [64].

Figure 9

Spectrum of the kinetic energy $E\left(k\right)$ in terms of the wave number $k$ for $\text{Re}=1600$ at $t=10$.

Figure 10

Time-averaged effective Reynolds numbers for the regularized scheme and the KBC scheme on an ${80}^3$ lattice.

Figure 11

Entropy estimates used by KBC and RLBM compared to $H\left(f\right)={\sum }_i{f}_{i}ln(f_i/w_i)$ on a D1Q3 lattice for a constant density, $\rho =1$, and two different velocities, $u=0$ (a) and $u=0.05$ (b). For a better comparison, the functions in the right plot were shifted along the $y$ axis to assume their maximum at zero.

Figure 12

Dissipation of kinetic energy and scaled enstrophy in simulations with the RLBM and KBC for $\text{Re}=2000$ (a) and $\text{Re}=5000$ (b) on a ${160}^3$ lattice. The dissipation rates were calculated via finite differences ($-dk/dt$) and via the enstrophy ($\nu \mathcal{E}$). The results are compared to reference results as presented in Ref. [81].

Figure 13

Time-averaged effective Reynolds numbers of the regularized scheme and the KBC scheme for the Kida vortex on a ${160}^3$ lattice.