Lattice Boltzmann model capable of mesoscopic vorticity computation

It is well known that standard lattice Boltzmann (LB) models allow the strain-rate components to be computed mesoscopically (i.e., through the local particle distributions) and as such possess a second-order accuracy in strain rate. This is one of the appealing features of the lattice Boltzmann method (LBM) which is of only second-order accuracy in hydrodynamic velocity itself. However, no known LB model can provide the same quality for vorticity and pressure gradients. In this paper, we design a multiple-relaxation time LB model on a three-dimensional 27-discrete-velocity (D3Q27) lattice. A detailed Chapman-Enskog analysis is presented to illustrate all the necessary constraints in reproducing the isothermal Navier-Stokes equations. The remaining degrees of freedom are carefully analyzed to derive a model that accommodates mesoscopic computation of all the velocity and pressure gradients from the nonequilibrium moments. This way of vorticity calculation naturally ensures a second-order accuracy, which is also proven through an asymptotic analysis. We thus show, with enough degrees of freedom and appropriate modifications, the mesoscopic vorticity computation can be achieved in LBM. The resulting model is then validated in simulations of a three-dimensional decaying Taylor-Green flow, a lid-driven cavity flow, and a uniform flow passing a fixed sphere. Furthermore, it is shown that the mesoscopic vorticity computation can be realized even with single relaxation parameter.

Figure 1

Sketch of D3Q27 lattice grid.

Figure 2

The statistics in a decaying 3D Taylor-Green flow with ${\mathrm{Re}}_0=600\pi$: (a) kinetic energy, (b) dissipation rate, (c) skewness, and (d) flatness. $k_0$ and $D_0$ are the kinetic energy and dissipation rate at the initial time, respectively.

Figure 3

The kinetic energy results of a decaying 3D Taylor-Green flow with different shear viscosities.

Figure 4

Contours of vorticity in $x$ direction at $t^{*}=4$ and $t^{*}=8$ with $\mathrm{Re}=300$ using different (a) FD, $t^{*}=4$, (b) ME, $t^{*}=4$, (c) PSM, $t^{*}=4$, (d) FD, $t^{*}=8$, (e) ME, $t^{*}=8$, and (f) PSM, $t^{*}=8$.

Figure 5

Contours of vorticity in $y$ direction at $t^{*}=4$ and $t^{*}=8$ with $\mathrm{Re}=300$ using different (a) FD, $t^{*}=4$, (b) ME, $t^{*}=4$, (c) PSM, $t^{*}=4$, (d) FD, $t^{*}=8$, (e) ME, $t^{*}=8$, and (f) PSM, $t^{*}=8$.

Figure 6

Contours of vorticity in the $z$ direction at $t^{*}=4$ and $t^{*}=8$ with $\mathrm{Re}=300$ using different (a) FD, $t^{*}=4$, (b) ME, $t^{*}=4$, (c) PSM, $t^{*}=4$, (d) FD, $t^{*}=8$, (e) ME, $t^{*}=8$, and (f) PSM, $t^{*}=8$.

Figure 7

Sketch of the 3D lid-driven cavity flow.

Figure 8

Contours of vorticity in a lid-driven cavity flow: (a) FD ${\omega }_x$ at $n_x/2$ plane; (b) ME, same as (a); (c) FD, ${\omega }_y$ at $n_y/2$ plane; (d) ME, same as (c); (e) FD, ${\omega }_z$ at $n_z/2$ plane; (f) ME, same as (e).

Figure 9

Sketch of a uniform flow passing a fixed sphere.

Figure 10

Vorticity contours in a plane cutting across the sphere center for a uniform flow passing a sphere: (a) standard MRT ${\mathrm{Re}}_p=50$; (b) standard MRT ${\mathrm{Re}}_p=100$; (c) present MRT, ${\mathrm{Re}}_p=50$, with finite-difference approximation; (d) present MRT ${\mathrm{Re}}_p=100$, with finite-difference approximation; (e) present MRT, ${\mathrm{Re}}_p=50$, with moment equations; and (f) present MRT ${\mathrm{Re}}_p=100$, with moment equations.