Coupled discrete unified gas kinetic scheme for the thermal compressible flows in all Knudsen number regimes

In this paper, a coupled discrete unified gas kinetic scheme (CDUGKS) with a flexible Prandtl number is developed for the thermal compressible flows in all Knudsen number regimes. Different from the existing thermal discrete unified gas kinetic scheme based on the Shakhov model, the proposed CDUGKS based on the total energy double-distribution-function model can well preserve the nonnegative property of the distribution function, especially for the strong shock in the continuum regime. In the CDUGKS, the velocity distribution function (VDF) is used to recover the compressible continuity and momentum equations, while the energy distribution function (EDF) is used to recover the energy equation. The VDF and EDF are evaluated in a similar way and then coupled via the thermal equation of state. With the un-splitting treatment of the particle transport and collision in the distribution function evolution and the flux evaluation, the time step in CDUGKS is not limited by the particle collision time. Furthermore, the CDUGKS is an asymptotic preserving scheme, in which the Navier-Stokes solution in the hydrodynamic regime and the free transport mechanism in the kinetic regime can be precisely recovered with the second-order accuracy in both space and time. Finally, several numerical experiments, including the weak shock tube and the strong one in the whole Knudsen number flows, as well as the two-dimensional Riemann problem and the Rayleigh-Taylor instability in both hydrodynamic regime and kinetic regimes, are performed to validate the method. Numerical results agree fairly well with other benchmark results in different flow regimes, which demonstrates the current CDUGKS is a reliable and efficient method for multiscale flow problems.

Figure 1

Density (a), velocity (b), temperature (c), and pressure profiles (d) of weak shock tube problem with ${\mu }_r=10$.

Figure 2

Density (a), velocity (b), temperature (c), and pressure profiles (d) of weak shock tube problem with ${\mu }_r=0.1$.

Figure 3

Density(a), velocity(b), temperature (c), and pressure profiles (d) of weak shock tube problem with ${\mu }_r={10}^{-4}$.

Figure 4

Density (a), velocity (b), temperature (c), and plot of distribution function at $x=0.44$ (d) of the strong shock tube with ${\mu }_r=10.$

Figure 5

Density (a), velocity (b), temperature (c), and plot of distribution function at $x=0.52$ (d) of the strong shock tube with ${\mu }_r=0.1$.

Figure 6

Density (a), velocity (b), temperature (c), and plot of distribution function at $x=0.32$ (d) of the strong shock tube with ${\mu }_r={10}^{-4}$.

Figure 7

Density (a) and temperature distributions (b) of 2D Riemann problem with ${\mu }_r={10}^{-6}$.

Figure 8

Density(a) and temperature distributions (b) of 2D Riemann problem with ${\mu }_r=10$. The black dashed lines and the white solid lines are the DUGKS and CDUGKS results, respectively.

Figure 9

Rayleigh-Taylor instability under gravitational field with ${\mu }_r={10}^{-6}$. Density and (a) and temperature (b) at four different times ($t=0.0,0.8,1.4,$ and 2.0) are shown in the four quadrants, starting with the initial data in the upper right corner and progressing clockwise.

Figure 10

Density distribution along the radial direction of Rayleigh-Taylor instability under gravitational field with ${\mu }_r={10}^{-6} t=0.0$ (a), $t=0.8$ (b), $t=1.4$ (c), and $t=2.0$ (d).

Figure 11

Rayleigh-Taylor instability under gravitational field with ${\mu }_r=10$. Density (a) and temperature (b) at four different times (t = 0.0, 0.08, 0.16, and 0.24) are shown in the four quadrants, starting with the initial data in the upper right corner and progressing clockwise.

Figure 12

Density distribution along the radial direction of Rayleigh-Taylor instability under gravitational field with ${\mu }_r=10$ at $t=0.0$ (a), $t=0.08$ (b), $t=0.16$ (c), and $t=0.24$ (d).