Conservative discrete-velocity method for the ellipsoidal Fokker-Planck equation in gas-kinetic theory

A conservative discrete velocity method (DVM) is developed for the ellipsoidal Fokker-Planck (ES-FP) equation in prediction of nonequilibrium neutral gas flows in this paper. The ES-FP collision operator is solved in discrete velocity space in a concise and quick finite difference framework. The conservation problem of the discrete ES-FP collision operator is solved by multiplying each term in it by extra conservative coefficients whose values are very close to unity. Their differences to unity are in the same order of the numerical error in approximating the ES-FP operator in discrete velocity space. All the macroscopic conservative variables (mass, momentum, and energy) are conserved in the present modified discrete ES-FP collision operator. Since the conservation property in a discrete element of physical space is very important for the numerical scheme when discontinuity and a large gradient exist in the flow field, a finite volume framework is adopted for the transport term of the ES-FP equation. For $nD-3V$ ($n<3$) cases, a $nD$-quasi $nV$ reduction is specifically proposed for the ES-FP equation and the corresponding FP-DVM method, which can greatly reduce the computational cost. The validity and accuracy of both the ES-FP equation and FP-DVM method are examined using a series of $0D-3V$ homogenous relaxation cases and $1D-3V$ shock structure cases with different Mach numbers, in which $1D-3V$ cases are reduced to $1D$-quasi $1V$ cases. Both the predictions of $0D-3V$ and $1D-3V$ cases match well with the benchmark results such as the analytical Boltzmann solution, direct full-Boltzmann numerical solution, and DSMC result. Especially, the FP-DVM predictions match well with the DSMC results in the Mach 8.0 shock structure case, which is in high nonequilibrium, and is a challenge case of the model Boltzmann equation and the corresponding numerical methods.

Figure 1

Stable Maxwellian distribution computed by FP-DVM on ${50}^3$ and ${100}^3$ uniform meshes.

Figure 2

The relaxation process of distribution with initial anisotropic temperatures, only the region with distribution function greater than $5\times {10}^{-3}$ is plotted, and the contour is on the plane ${\xi }_3=0$. The evolution time $\tau$ is (a) 0, (b) 1, (c) 2, and (d) 10, respectively, in clockwise from the top-left figure.

Figure 3

The relaxation process of anisotropic temperatures with initial values $T_{11}=2T_{22}=2T_{33}$.

Figure 4

The relaxation process of bi-model distribution, only the region with distribution function greater than (a)–(c) $5\times {10}^{-5}$ or (d) $5\times {10}^{-4}$ is plotted, and the contour is on the plane ${\xi }_3=0$. The evolution time $\tau$ is (a) 0, (b) 1, (c) 2, and (d) 10, respectively, in clockwise from the top-left figure.

Figure 5

The relaxation process of anisotropic temperatures and heat flux of initial bi-model distribution. The left figure is the (a) anisotropic temperatures, the right figure is the (b) heat flux.

Figure 6

The relaxation process of discontinuous distribution, only the region with distribution function greater than $5\times {10}^{-3}$ is plotted, and the contour is on the plane ${\xi }_3=0$. The evolution time $\tau$ is (a) 0, (b) 1, (c) 2, and (d) 10, respectively, in clockwise from the top-left figure.

Figure 7

The relaxation process of anisotropic temperatures and heat flux of initial discontinuous distribution. The left figure is the (a) anisotropic temperatures, the right figure is the (b) heat flux.

Figure 8

The structures of density, temperature, stress, and heat in $\mathrm{Ma}=1.2$ shock wave. The left figure is the (a) density and temperature, the right figure is the (b) stress and heat flux.

Figure 9

The structures of density, temperature, stress, and heat flux in $\mathrm{Ma}=3.0$ shock wave. The left figure is the (a) density and temperature, the right figure is the (b) stress and heat flux.

Figure 10

The structures of density, temperature, stress, and heat flux in $\mathrm{Ma}=8.0$ Argon shock wave. The left figure is the (a) density and temperature, the right figure is the (b) stress and heat flux.

Figure 11

The structures of density, temperature, stress, and heat flux in $\mathrm{Ma}=8.0$ Argon shock wave predicted by FP-DVM with different amount of discrete velocity points in ${\xi }_1$ direction. The left figure is the (a) density and temperature, the right figure is the (b) stress and heat flux.

Figure 12

The distribution function at different $x_1$ locations inside a $\mathrm{Ma}=8.0$ Argon shock wave predicted using 300 and 50 discrete velocity points in ${\xi }_1$ direction, respectively. The left figure is the (a) number distribution $F$, the right figure is the (b) energy distribution $G$.