Third-order discrete unified gas kinetic scheme for continuum and rarefied flows: Low-speed isothermal case

An efficient third-order discrete unified gas kinetic scheme (DUGKS) is presented in this paper for simulating continuum and rarefied flows. By employing a two-stage time-stepping scheme and the high-order DUGKS flux reconstruction strategy, third order of accuracy in both time and space can be achieved in the present method. It is also analytically proven that the second-order DUGKS is a special case of the present method. Compared with the high-order lattice Boltzmann equation-based methods, the present method is capable to deal with the rarefied flows by adopting the Newton-Cotes quadrature to approximate the integrals of moments. Instead of being constrained by the second order (or lower order) of accuracy in the time-splitting scheme as in the conventional high-order Runge-Kutta-based kinetic methods, the present method solves the original Boltzmann equation, which overcomes the limitation in time accuracy. Typical benchmark tests are carried out for comprehensive evaluation of the present method. It is observed in the tests that the present method is advantageous over the original DUGKS in accuracy and capturing delicate flow structures. Moreover, the efficiency of the present third-order method is also shown in simulating rarefied flows.

Figure 1

Parameters $P_n,P^{{}^{\prime }}$ and $P_{n+1}$ with $a$.

Figure 2

Available region for ${\widehat{P}}_n$ with $a$ and $\mathrm{\Delta }t/\tau$.

Figure 3

Timeline of DUGKS-T3S3.

Figure 4

Velocity profiles on $50\times 50$ uniform mesh at different times.

Figure 5

Contours of vorticity at $T=0.8s$ and $T=1.0s$. (a) DUGKS-T2S2 on the mesh size of $128\times 128$; (b) DUGKS-T3S3 on the mesh size of $128\times 128$; (c) DUGKS-T2S2 on the mesh size of $256\times 256$.

Figure 6

Velocity profiles at $\text{Re}=400,1000$ and 3200 on $65\times 65$ uniform mesh. ($\circ$) is reference data.

Figure 7

Pressure contours at $\text{Re}=3200$ on $65\times 65$ uniform mesh. (a) is obtained by DUGKS-T2S2 method. (b) is obtained by DUGKS-T3S3 method.

Figure 8

The maximum values of $\mathrm{\Delta }t/\tau$ for stable computations of the LDF on $25\times 25$ and $50\times 50$ meshes.

Figure 9

Velocity profiles along the centerlines at $\text{Kn}=0.075,1,2$ and 8 on $40\times 40$ uniform mesh. ($\circ$) is reference data.

Figure 10

The velocity profiles of DUGKS-T2S2 and DUGKS-T3S3 methods at $\text{Kn}=10$ on $60\times 60$ uniform mesh with $CFL=0.25$.

Figure 11

Error history of DUGKS-T2S2 and DUGKS-T3S3 methods at $Kn=10$ on $60\times 60$ uniform mesh with $81\times 81$ points particle velocities. ($CFL=0.25$)

Figure 12

The maximum time-step size for stable computations of DUGKS-T2S2 and DUGKS-T3S3.