Entropic lattice Boltzmann representations required to recover Navier-Stokes flows

There are two disparate formulations of the entropic lattice Boltzmann scheme: one of these theories revolves around the analog of the discrete Boltzmann $H$ function of standard extensive statistical mechanics, while the other revolves around the nonextensive Tsallis entropy. It is shown here that it is the nonenforcement of the pressure tensor moment constraints that lead to extremizations of entropy resulting in Tsallis-like forms. However, with the imposition of the pressure tensor moment constraint, as is fundamentally necessary for the recovery of the Navier-Stokes equations, it is proved that the entropy function must be of the discrete Boltzmann form. Three-dimensional simulations are performed which illustrate some of the differences between standard lattice Boltzmann and entropic lattice Boltzmann schemes, as well as the role played by the number of phase-space velocities used in the discretization.

Figure 1

(Color online) The decay of the (a) normalized kinetic energy and the (b) normalized enstrophy for the viscosity run $\nu =2\times {10}^{-3}$ using the LB and ELB algorithms for D3Q15, D3Q19, and D3Q27. Time is in units of LB iterations, and the spatial grid is ${512}^3$. These simulations are fully resolved at $\mathrm{Re}=2037$, with integral to dissipation length scales $L∕{\ell }_D\approx {\mathrm{Re}}^{3∕4}\approx 300$. The kinetic energy decay is insensitive to whether the simulation uses the LB or ELB algorithm as well as the number of velocity bits in the model. However, the enstrophy decay shows dependence on both the algorithm and bit model used in the simulation. Since the ELB viscosities are somewhat higher than the corresponding LB bare viscosities, the LB enstrophy maxima (which are controlled by the viscosity) are greater than for ELB. The rapid rise in the enstrophy for early times is due to vortex stretching, which is independent of the viscosity.

Figure 2

(Color online) The decay of the (a) normalized kinetic energy and the (b) normalized enstrophy for the lower viscosity run $\nu =2\times {10}^{-4}$. At this viscosity the standard LB algorithm rapidly becomes numerically unstable. The ELB algorithm is unconditionally stable for all viscosities. These runs at $\mathrm{Re}=20370$ are somewhat under-resolved $(L∕{\ell }_D\approx 1700)$. Both the energy and enstrophy evolution now show little variation with bit model. The enstrophy maxima are now enhanced by over a factor of 3 from those in Fig. 1 due to the reduced viscosity.

Figure 3

(Color online) The time evolution of $\widehat{\omega }\left(t\right)={\max }_{\mathbf{x}}{\omega }_x(\mathbf{x},t)$ where ${\omega }_x=\widehat{\mathbf{x}}\bullet (\mathbf{\nabla }\times \mathbf{u})$ is the $x$ component of vorticity for the various velocity bit models: D3Q27, D3Q19, and D3Q15 models. The spatial grid is ${512}^3$. (a) LB at $\nu =2\times {10}^{-3}$, (b) ELB at $\nu =2\times {10}^{-3}$, and (c) ELB at $\nu =2\times {10}^{-4}$. For early times there is strong vortex stretching that is independent of viscosity. There are some bit-model variations in both the LB and ELB runs at $\nu =2\times {10}^{-3}$, but these are not present for the ELB simulation at the lower viscosity $\nu =2\times {10}^{-4}$.

Figure 4

(Color online) The evolution of the isosurface of ${\tilde{\omega }}_x(\mathbf{x},t)=0.15{\omega }_x(\mathbf{x},t)$ at (a) $t=0$, (b) $t=2.5\mathrm{K}$, (c) $t=5\mathrm{K}$, (d) $t=7.5\mathrm{K}$, (e) $t=10\mathrm{K}$, and (f) $t=15\mathrm{K}$, for the ELB D3Q27 model at viscosity $\nu =2\times {10}^{-3}$. Some of the symmetries of the initial Kida velocity profile are evident in the 15% isosurfaces at $t=0$ (a). The vortex stretching is seen in (b), followed by the breakup of these vortex structures in (c)–(e).

Figure 5

(Color online) The evolution of the isosurface of ${\tilde{\omega }}_x(\mathbf{x},t)=0.15{\omega }_x(\mathbf{x},t)$ at (a) $t=0$, (b) $t=2.5\mathrm{K}$, (c) $t=5\mathrm{K}$, (d) $t=7.5\mathrm{K}$, (e) $t=10\mathrm{K}$, and (f) $t=15\mathrm{K}$, for the ELB D3Q27 model at viscosity $\nu =2\times {10}^{-4}$. Following the vortex stretching there is a more rapid breakup into smaller vortex clusters due to the higher Reynolds number than in Fig. 4.

Figure 6

(Color online) The effect of the choice of the relaxation distribution function on the normalized enstrophy for different Reynolds and Mach numbers for the D3Q27 model. There is little difference between the use of the standard polynomial relaxation distribution function [including $O\left(u^3\right)$ terms], Eq. (6), to the entropic form, Eq. (13), even at high Reynolds. This difference is nearly totally reduced when the Mach number is decreased from (a) $\mathrm{Ma}=0.086$ to (b) $\mathrm{Ma}=0.035$. The number of time iterations scales as $U_0^{-1}$ so that lower-Mach-number simulations required longer-time runs.