Self-consistent force scheme in the spectral multiple-relaxation-time lattice Boltzmann model

In the present work, the force term is first derived in the spectral multiple-relaxation-time high-order lattice Boltzmann model. The force term in the Boltzmann equation is expanded in the Hermite temperature rescaled central moment space (RCM), instead of the Hermite raw moment space (RM). The contribution of nonequilibrium RCM moments beyond second order are neglected. For the collision operator in the RCM space, each order of the force term can be incorporated directly. Through the transformation between the RCM space and the RM space, the force term for practical numerical implementation in the RM space can be derived. It can be demonstrated that the present force scheme is self-consistent for the isothermal flow and compressible thermal flow with adjustable Prandtl number via the numerical experiments.

Figure 1

Numerical (symbols; resolution: $64\times 64$) and analytical (solid line) results of 2D Taylor-Green flow at different times $t=nT$ with $n=1,2,4$ and $T=\text{ln}2/\left(2\nu k_1^2\right)$.

Figure 2

Convergence test: six resolutions are tested ($16\times 16,32\times 32,64\times 64,128\times 128,256\times 256,\text{and}384\times 384$), the solid dot line is of slope 2, and the convergence order of the present numerical results is between first and second order.

Figure 3

Numerical (symbols) and analytical (solid line) results of Womersley flow at different times $t=nT/16$ with (a) $n=4,6,8,10$ and (b) $n=0,2,12,14$.

Figure 4

In the $\text{Pr}=0.2$ case, the effects of different terms in the third-order relaxation, Eq. (50d), are investigated. “RM collision” denotes only the relaxation of the nonequilibrium RM is retained; “RM collision $+$ collision correction” denotes that the Galilean invariance is considered with the collision correction term; “RM collision $+$ collision correction $+$ 3rd-order force” denotes the third-order force is considered; “RCM collision” denotes the complete collision form with the force correction term.

Figure 5

In the $\text{Pr}=2$ case, the effects of different terms in the third-order relaxation, Eq. (50d), are investigated. “RM collision” denotes only the relaxation of the nonequilibrium RM is retained; “RM collision $+$ collision correction” denotes that the Galilean invariance is considered with the collision correction term; “RM collision $+$ collision correction $+$ 3rd-order force” denotes the third-order force is considered; “RCM collision” denotes the complete collision form with the force correction term.

Figure 6

In the $\text{Pr}=1$ case, the effects of different terms in the third-order relaxation, Eq. (50d), are investigated. “RM collision” denotes only the relaxation of the nonequilibrium RM is retained; “RM collision $+$ collision correction” denotes that the Galilean invariance is considered with the collision correction term; “RM collision $+$ collision correction $+$ 3rd-order force” denotes the third-order force is considered; “RCM collision” denotes the complete collision form with the force correction term.