Investigation of an entropic stabilizer for the lattice-Boltzmann method

The lattice-Boltzmann (LB) method is commonly used for the simulation of fluid flows at the hydrodynamic level of description. Due to its kinetic theory origins, the standard LB schemes carry more degrees of freedom than strictly needed, e.g., for the approximation of solutions to the Navier-stokes equation. In particular, there is freedom in the details of the so-called collision operator. This aspect was recently utilized when an entropic stabilizer, based on the principle of maximizing local entropy, was proposed for the LB method [I. V. Karlin, F. Bösch, and S. S. Chikatamarla, Phys. Rev. E 90, 031302(R) (2014)]. The proposed stabilizer can be considered as an add-on or extension to basic LB schemes. Here the entropic stabilizer is investigated numerically using the perturbed double periodic shear layer flow as a benchmark case. The investigation is carried out by comparing numerical results obtained with six distinct LB schemes. The main observation is that the unbounded, and not explicitly controllable, relaxation time for the higher-order moments will directly influence the leading-order error terms. As a consequence, the order of accuracy and, in general, the numerical behavior of LB schemes are substantially altered. Hence, in addition to systematic numerical validation, more detailed theoretical analysis of the entropic stabilizer is still required in order to properly understand its properties.

Figure 1

Numerical solution of the pressure-Poisson equation for $L=512$ and $U_0=1/32$: isolines of the pressure field are presented.

Figure 2

Average density variation for the central moment space LB scheme (Sec. 2b): $L=128$, $U_0=1/16$, and $\nu ={10}^{-4}(Re=8\times {10}^4)$.

Figure 3

Average density variation for the original regularized LB scheme (Sec. 2a): $L=128$, $U_0=1/16$, and $\nu ={10}^{-4}(Re=8\times {10}^4)$.

Figure 4

Average kinetic energy (normalized) for the central moment space LB scheme (Sec. 2b): $L=128$, $U_0=1/16$, and $\nu ={10}^{-4}$ ($Re=8\times {10}^4$). The average values are normalized with ${\rho }_r{U}_0^2/2$, where ${\rho }_r=1$.

Figure 5

Average kinetic energy (normalized) for the original Regularized LB scheme (Sec. 2a): $L=128$, $U_0=1/16$, and $\nu ={10}^{-4}$ ($Re=8\times {10}^4$). The average values are normalized with ${\rho }_r{U}_0^2/2$, where ${\rho }_r=1$.

Figure 6

Decay of average kinetic energy for $Re=3\times {10}^4$: $L=128$, $U_0=1/32$, and $\nu =4/30000$. The average values are normalized with ${\rho }_r{U}_0^2/2$, where ${\rho }_r=1$. The LB-BGK scheme becomes unstable early on, around time $t^{*}=0.3$. The central moment space LB scheme (Sec. 2b) with $\gamma =1/\beta$ experiences instability close to $t^{*}=2.5$. The natural and central moment space LB schemes (Sec. 2b, $\gamma \ne 1/\beta$) as well as both regularized LB schemes remain stable.

Figure 7

Isolines of the vorticity field for $Re=8\times {10}^4$ at $t^{*}=1$: $L=256$, $U_0=1/16$, and $\nu =2\times {10}^{-4}$. The simulation with the LB-BGK scheme is for $Re=3.2\times {10}^4$ ($\nu =5\times {10}^{-4}$) due to stability reasons: instabilities arise from the small secondary vortices. The five other cases remain stable. However, the vorticity fields appear visibly distorted for all three schemes utilizing the entropic stabilizer. Computations with the two LB schemes which filter out the high-order nonequilibrium moments, i.e., the original regularized LB scheme (Sec. 2a) and the central moment space LB scheme (Sec. 2b) with $\gamma =1/\beta$ result in undisturbed fields.

Figure 8

Relative magnitude of the maximum $\left|\gamma \right|$ for the central moment space LB scheme (Sec. 2b): $U_0=1/32$, $\nu$ is varied in order to enforce the prescribed Reynolds number, and the simulations are run until $t^{*}=1$.

Figure 9

Convergence of the numerical error for LB-BGK, the natural and central moment space LB schemes (Sec. 2b), and for the regularized LB schemes; $Re=2\times {10}^4$, $U_0=1/32$, $\nu$ is varied in order to enforce the prescribed Reynolds number, and the errors are measured at $t^{*}=1$.

Figure 10

Convergence of the numerical error for the natural and central moment space LB schemes (Sec. 2b), and for the regularized LB schemes; $Re=4\times {10}^4$, $U_0=1/32$, $\nu$ is varied in order to enforce the prescribed Reynolds number, and the errors are measured at $t^{*}=1$.

Figure 11

Average magnitude of the relaxation parameter for the higher-order moments: the central moment space LB scheme (Sec. 2b) is utilized, $U_0=1/32$, $\nu$ is varied in order to enforce the prescribed Reynolds number, and the simulations are run until $t^{*}=1$.

Figure 12

Relative occurrence of the distribution updates where the higher-order moments are amplified, i.e., $|1-\beta \gamma |>1$: the central moment space LB scheme (Sec. 2b) is utilized, $U_0=1/32$, $\nu$ is varied in order to enforce the prescribed Reynolds number, and the simulations are run until $t^{*}=1$.