Conserved discrete unified gas-kinetic scheme with unstructured discrete velocity space

Discrete unified gas-kinetic scheme (DUGKS) is a multiscale numerical method for flows from continuum limit to free molecular limit, and is especially suitable for the simulation of multiscale flows, benefiting from its multiscale property. To reduce integration error of the DUGKS and ensure the conservation property of the collision term in isothermal flow simulations, a conserved-DUGKS (C-DUGKS) is proposed. On the other hand, both DUGKS and C-DUGKS adopt Cartesian-type discrete velocity space, in which Gaussian and Newton-Cotes numerical quadrature are used for calculating the macroscopic physical variables in low-speed and high-speed flows, respectively. However, the Cartesian-type discrete velocity space leads to huge computational cost and memory demand. In this paper, the isothermal C-DUGKS is extended to the nonisothermal case by adopting coupled mass and inertial energy distribution functions. Moreover, since the unstructured mesh, such as the triangular mesh in the two-dimensional case, is more flexible than the structured Cartesian mesh, it is introduced to the discrete velocity space of C-DUGKS, such that more discrete velocity points can be arranged in the velocity regions that enclose a large number of molecules, and only a few discrete velocity points need to be arranged in the velocity regions with a small amount of molecules in it. By using the unstructured discrete velocity space, the computational efficiency of C-DUGKS is significantly increased. A series of numerical tests in a wide range of Knudsen numbers, such as the Couette flow, lid-driven cavity flow, two-dimensional rarefied Riemann problem, and the supersonic cylinder flows, are carried out to examine the validity and efficiency of the present method.

Figure 1

Sketch of two neighboring cells and the characteristic line.

Figure 2

Flow chart of C-DUGKS.

Figure 3

Discrete velocity spaces. (a) Structured velocity mesh. (b) Unstructured velocity mesh.

Figure 4

Geometry of the Couette flow.

Figure 5

Unstructured velocity mesh for Couette flow with 1882 points.

Figure 6

U-velocity profile in vertical direction predicted by UGKS and C-DUGKS with unstructured velocity mesh [C-DUGKS(U)] and structured velocity mesh [C-DUGKS(S)].

Figure 7

Geometry of the cavity flow.

Figure 8

Spatial mesh (a) and unstructured velocity mesh (b) for cavity flow.

Figure 9

Simulation results of cavity flow at $\text{Kn}=0.075$. (a) Temperature (UGKS, the background and white solid lines; C-DUGKS with unstructured velocity mesh, black dashed lines; C-DUGKS with structured velocity mesh, red dashed lines). (b) Heat flux (UGKS, blue solid lines; C-DUGKS with unstructured velocity mesh, black solid lines; C-DUGKS with structured velocity mesh, red solid lines). (c) $U$ velocity along the central vertical lines and $V$ velocity along the central horizontal lines (C-DUGKS with unstructured velocity mesh [C-DUGKS(U)], black solid lines; C-DUGKS with structured velocity mesh [C-DUGKS(S)], blue dashed lines; UGKS, red circle symbol).

Figure 10

Simulation results of cavity flow at $\text{Kn}=10$. (a) Temperature (UGKS, the background and white solid lines; C-DUGKS with unstructured velocity mesh, black dashed lines; C-DUGKS with structured velocity mesh, red dashed lines). (b) Heat flux (UGKS, blue solid lines; C-DUGKS with unstructured velocity mesh, black solid lines; C-DUGKS with structured velocity mesh, red solid lines). (c) $U$ velocity along the central vertical lines and $V$ velocity along the central horizontal lines (C-DUGKS with unstructured velocity mesh [C-DUGKS(U)], black solid lines; C-DUGKS with structured velocity mesh [C-DUGKS(S)], blue dashed lines; UGKS, red circle symbol).

Figure 11

Simulation results of cavity flow at $\text{Re}=1000$. (a) Velocity streamline. (b) $U$-velocity alone central vertical line and $V$-velocity alone central horizontal line.

Figure 12

Unstructured velocity mesh for 2D rarefied Riemann problem.

Figure 13

The contours of the density (a), temperature (b), velocity magnitude (c), and the streamlines (d) for the 2D rarefied Riemann problem at $t=0.15$. In (a), (b), and (c), the background and white solid lines are from the collisionless Boltzmann equation, and the black dashed lines and red dashed lines are the results of C-DUGKS with unstructured and structured velocity mesh, respectively. In (d), the blue solid lines are the solutions of the collisionless Boltzmann equation, and the black solid lines and red solid lines are the results of C-DUGKS with unstructured and structured velocity mesh, respectively.

Figure 14

Unstructured velocity mesh for C-DUGKS.

Figure 15

Flow variables along the stagnation line for the case of $\text{Kn}=0.1$. (a) Density, (b) pressure, (c) temperature, and (d) horizontal velocity. C-DUGKS(U) and C-DUGKS(S) represent C-DUGKS with unstructured and structured velocity mesh, respectively.

Figure 16

Flow variables along the surface of the cylinder for the case of $\text{Kn}=0.1$. (a) Heat flux, (b) pressure, and (c) shear stress. C-DUGKS(U) and C-DUGKS(S) represent C-DUGKS with unstructured and structured velocity mesh, respectively.

Figure 17

Flow variables along the stagnation line for the case of $\text{Kn}=1.0$. (a) Density, (b) pressure, (c) temperature, and (d) horizontal velocity. C-DUGKS(U) and C-DUGKS(S) represent C-DUGKS with unstructured and structured velocity mesh, respectively.

Figure 18

Flow variables along the surface of the cylinder for the case of $\text{Kn}=1.0$. (a) Heat flux, (b) pressure, and (c) shear stress. C-DUGKS(U) and C-DUGKS(S) represent C-DUGKS with unstructured and structured velocity mesh, respectively.