Arbitrary Lagrangian-Eulerian-type discrete unified gas kinetic scheme for low-speed continuum and rarefied flow simulations with moving boundaries

In this paper, the original discrete unified gas kinetic scheme (DUGKS) is extended to the arbitrary Lagrangian–Eulerian (ALE) framework to enable simulation of low-speed continuum and rarefied flows with moving boundaries. For the proposed ALE-type DUGKS, the mesh motion velocity is introduced in the Boltzmann–BGK equation and a remapping-free scheme is used to discretize the governing equation. Under this coupling framework, the complex rezoning and remapping phases implemented in the traditional ALE method are avoided. In some application areas, large discretization errors are introduced in the simulation if the geometric conservation law (GCL) is not guaranteed. Therefore, three GCL-compliant approaches are discussed, and a uniform flow test case is conducted to validate these schemes. Further, to illustrate the performance of the proposed method, four test cases are simulated, including the continuum flow around an oscillating circular cylinder, the continuum flow around a pitching NACA0012 airfoil, a moving piston driven by a rarefied gas, and the rarefied flow caused by a plate oscillating in the normal direction. Finally, an extended test case considering the rarefied flow over an oscillating circular cylinder is also studied, as this condition is not sufficiently researched. Consistent and good results obtained from the above test cases demonstrate the capability of the proposed ALE-type DUGKS to simulate moving boundary problems in different flow regimes.

Figure 1

Schematics of (a) an unstructured mesh used in the DUGKS and (b) flux calculation for a cell interface.

Figure 2

Mesh for a uniform flow: (a) initial mesh and (b) instantaneous mesh.

Figure 3

(a) Pressure and (b) velocity contours of uniform flow obtained with a DGCL-violated scheme.

Figure 4

Mesh for flow around an oscillating circular cylinder: (a) full domain and (b) near the circular cylinder surface.

Figure 5

Dimensionless inline force $F_x$ at different $U_{\text{max}}$ ($\mathrm{\Delta }t=0.05, \mathrm{\Delta }x=d/256$): (a) the full oscillating period $T$ and (b) part of the enlarged view.

Figure 6

Pressure isolines at phases (a) $0^{\circ }$ and (b) ${288}^{\circ }$, Dütsch et al. [36] (left) and present (right) results.

Figure 7

Velocity profiles at four vertical cross sections $\overline{x}=-0.6,0,0.6,1.2$ for phase positions (a) ${180}^{\circ }$, (b) ${210}^{\circ }$, and (c) ${330}^{\circ }$. The present results are compared with the computational and experimental values of Dütsch et al. [36].

Figure 8

Comparison of inline force $F_x$ with the result of Dütsch et al. [36] and empirical result obtained by Morison equation [38] [Eq. (50)].

Figure 9

Mesh for flow around a pitching NACA0012 airfoil: (a) full domain and (b), (c), (d) parts of the enlarged view near the airfoil.

Figure 10

Time evolutions of drag coefficient at different reduced frequencies $k$ for $d=2^{\circ }$.

Figure 11

Mean thrust coefficients ${\overline{C}}_T$ at different reduced frequencies $k$ for (a) $d=2^{\circ }$ and (b) $d=4^{\circ }$.

Figure 12

Vortex shedding patterns at different reduced frequencies $k$ for $d=2^{\circ }$.

Figure 13

Vortex shedding patterns at different reduced frequencies $k$ for $d=4^{\circ }$.

Figure 14

Schematics of a moving piston driven by a rarefied gas: (a) initial stage and (b) equilibrium stage.

Figure 15

Time histories of piston position at (a) $\text{Kn}=0.031$ and (b) $\text{Kn}=0.31$ (the black dash and dot lines represent the theoretical solutions).

Figure 16

Time histories of density fluctuation at the two walls of the piston for (a) $\text{Kn}=0.031$ and (b) $\text{Kn}=0.31$ (the red dash and dot lines represent the theoretical solutions).

Figure 17

Time histories of pressure fluctuation at the two walls of the piston for (a) $\text{Kn}=0.031$ and (b) $\text{Kn}=0.31$ (the red dash and dot lines represent the theoretical solutions).

Figure 18

Convergence tests based on different number of abscissas at (a) $\text{Kn}=0.031$ and (b) $\text{Kn}=0.31$ (the black dash and dot line represent the theoretical solution). 28GH: 28-abscissas Gauss–Hermit quadrature rule.

Figure 19

Comparisons of time history of piston position at different masses for (a) $\text{Kn}=0.031$ and (b) $\text{Kn}=0.31$ (the black dash and dot line represent the theoretical solution).

Figure 20

Schematic of the rarefied flow caused by a moving plate oscillating in its normal direction.

Figure 21

Profiles of density at (a) $t/2\pi =0.1,\cdots ,0.5$ and (b) $t/2\pi =0.6,\cdots ,1.0$ and at two $K$ numbers.

Figure 22

Profiles of velocity at (a) $t/2\pi =0.1,\cdots ,0.5$ and (b) $t/2\pi =0.6,\cdots ,1.0$ and at two $K$ numbers.

Figure 23

Profiles of temperature at (a) $t/2\pi =0.1,\cdots ,0.5$ and (b) $t/2\pi =0.6,\cdots ,1.0$ and at two $K$ numbers.

Figure 24

The (a) maximum lift coefficient $C_l^{\prime }$ and (b) mean drag coefficient ${\overline{C}}_d$ at two Reynolds numbers in a continuum flow regime.

Figure 25

The (a) maximum lift coefficient $C_l^{\prime }$ and (b) mean drag coefficient ${\overline{C}}_d$ at two Kn numbers and at $\text{Re}=10$.

Figure 26

Convergence tests for the lift coefficient $C_l$ and the drag coefficient $C_d$ at (a) $\text{Kn}=0.01$ and (b) $\text{Kn}=0.05$. Single $f$: only $g$ is solved. Double $f$: both $g$ and $h$ are solved. 16GH: 16-abscissas Gauss–Hermit quadrature rule.