Stability of the lattice kinetic scheme and choice of the free relaxation parameter

The lattice kinetic scheme (LKS), a modified version of the classical single relaxation time (SRT) lattice Boltzmann method, was initially developed as a suitable numerical approach for non-Newtonian flow simulations and a way to reduce memory consumption of the original SRT approach. The better performances observed for non-Newtonian flows are mainly due to the additional degree of freedom allowing an independent control over the relaxation of higher-order moments, independently from the fluid viscosity. Although widely applied to fluid flow simulations, no theoretical analysis of LKS has been performed. The present work focuses on a systematic von Neumann analysis of the linearized collision operator. Thanks to this analysis, the effects of the modified collision operator on the stability domain and spectral behavior of the scheme are clarified. Results obtained in this study show that correct choices of the “second relaxation coefficient” lead, to a certain extent, to a more consistent dispersion and dissipation for large values of the first relaxation coefficient. Furthermore, appropriate values of this parameter can lead to a larger linear stability domain. At moderate and low values of viscosity, larger values of the free parameter are observed to increase dissipation of kinetic modes, while leaving the acoustic modes untouched and having a less pronounced effect on the convective mode. This increased dissipation leads in general to less pronounced sources of nonlinear instability, thus improving the stability of the LKS.

Figure 1

Effects of the EDF order on the linear stability domain of the SRT collision model.

Figure 2

Stability areas of the SRT ($\mathbf{–}\mathbf{–}\mathbf{–}$) and LKS with EDFs of (a) second, (b) third, and (c) fourth order, for different values of the free parameter: $\eta {\delta }_t/{\delta }_x^2=5(——)$, $1(——)$, $5\times {10}^{-1}(——)$, $1\times {10}^{-1}(——)$, $5\times {10}^{-2}(——)$, $1\times {10}^{-2}(——)$, $5\times {10}^{-3}(——)$, $1\times {10}^{-3}(——)$, $5\times {10}^{-4}(——)$, $1\times {10}^{-4}(——)$, $5\times {10}^{-5}(——)$, $1\times {10}^{-5}(——)$.

Figure 3

Stability maps for EDFs of order (a) 2, (b) 3, and (c) 4 (color codes are the same for all figures).

Figure 4

Effects of the EDF order and of the LKS collision operator on stability domain—the darker gray area represents the gain induced by the order of the EDF (going from second to fourth order), while the lighter gray area corresponds to the LKS collision operator (LKS with fourth-order EDF).

Figure 5

Effects of the first relaxation coefficient on (a) spectral dispersion and (b) dissipation for the SRT collision operator.

Figure 6

Effects of the second relaxation coefficient on spectral dispersion [(a), (b), and (c)] and dissipation [(d), (e), and (f)] for large values of nondimensional viscosity, from left to right: $\nu {\delta }_t/{\delta }_x^2=1$, 0.5, 0.3.

Figure 7

Spectral velocity of the LKS model with $\nu {\delta }_t/{\delta }_x^2=0.3$ and Ma = 0.1 for different choices of the free parameter. Choices of the free parameter and the color coding are the same as in the rightmost cases in Fig. 6.

Figure 8

Effects of the second relaxation coefficient for $\nu {\delta }_t/{\delta }_x^2=5\times {10}^{-4}$ and Mach = 0.1 on (a) dispersion of acoustic and convective modes, (b) dissipation of acoustic and convective modes, and (c) dissipation of hydrodynamic and kinetic modes. The dispersion and dissipation curves of convective and acoustic modes for all values of $\eta {\delta }_t/{\delta }_x^2$ cannot be clearly distinguished as they exactly fall onto each other.

Figure 9

von Neumann stability domain (VN) vs stability domain for the double periodic shear layer (DPSL) test case for the SRT operator and the LKS with maximum linear stability (LKS-1) and $\eta {\delta }_t/{\delta }_x^2=0.05$ (LKS-2).

Figure 10

Energy spectrum for Taylor-Green vortex at $t_c=10$ for different spatial resolutions and values of the free parameter $\lambda$.

Figure 11

Isosurfaces of the $z$ component of vorticity ${\omega }_z=0$ (bottom view, only the upper left quadrant is shown) at $t=10t_c$ obtained using the LKS at three different resolutions (from top to bottom), ${32}^3$, ${64}^3$, and ${128}^3$, with five different values for the free parameter (from left to right), $\lambda =0.515$, 0.53, 0.59, 0.65, and 1.

Figure 12

Isosurfaces of the $z$ component of vorticity ${\omega }_z=0$ (bottom view) at $t=10t_c$ obtained using the SRT at resolutions ${512}^3$.

Figure 13

Effect of the Mach number on propagation speed [(a), (b) and (c)] and dissipation [(d), (e) and (f)] for SRT collision operator with (from left to right): second, third and fourth-order EDFs.