Parallel finite-volume discrete Boltzmann method for inviscid compressible flows on unstructured grids

In this paper, a finite-volume discrete Boltzmann method based on a cell-centered scheme for inviscid compressible flows on unstructured grids is presented. In the new method, the equilibrium distribution functions are obtained from the circle function in two-dimensions (2D) and the spherical function in three-dimensions (3D). Moreover, the advective fluxes are evaluated by Roe's flux-difference splitting scheme, the gradients of the density and total energy distribution functions are computed with a least-squares method, and the Venkatakrishnan limiter is employed to prevent oscillations. To parallelize the method we use a graph-based partitioning approach that also guarantees the load balancing. The method is validated by seven benchmark problems: (a) a 2D flow pasting a bump, (b) a 2D Riemann problem, (c) a 2D flow passing the RAE2822 airfoil, (d) flows passing the NACA0012 airfoil, (e) 2D supersonic flows around a cylinder, (f) an explosion in a 3D box, and (g) a 3D flow around the ONERA M6 wing. The benchmark tests show that the results obtained by the proposed method match well with the published results, and the parallel numerical experiments show that the proposed parallel implementation has close to linear strong scalability, and parallel efficiencies of $95.31\%$ and $94.56\%$ are achieved for 2D and 3D problems on a supercomputer with up to 4800 processor cores, respectively.

Figure 1

Lattice velocity model in 2D and 3D. (a) D2Q13 and (b) D3Q25.

Figure 2

Cell-centered scheme on unstructured grids in 2D (a) and 3D (b).

Figure 3

Illustration of the ghost cell method.

Figure 4

An example of the unstructured grid (a) for the flow simulation around the NACA0012 airfoil and its partition (b) for parallel computing. Different colors denote different subdomains.

Figure 5

The setup of the flow simulation in the channel with a circular bump, $h=0.1$ is the height of the bump.

Figure 6

Mach number contours ($M_{\infty }=0.675$).

Figure 7

Mach number along the top wall. WENO in the figure refers to the result from Ref. [57].

Figure 8

Density contours of the 2D Riemann problem, $t=0.2$. (a) our result, (b) result in Ref. [58].

Figure 9

The computational domain and boundary condition of a flow around the RAE2822 airfoil.

Figure 10

Pressure contours of flow around the RAE2822 airfoil ($M_{\infty }=0.75$, $\alpha =3^{\circ }$).

Figure 11

$C_p$ of the flow around the RAE2822 airfoil ($M_{\infty }=0.75$, $\alpha =3^{\circ }$). In the figure, Meister refers to the result published in Ref. [59].

Figure 12

Pressure contours of the flow around the NACA0012 airfoil ($M_{\infty }=0.63$, $\alpha =2^{\circ }$).

Figure 13

$C_p$ of the flow around the NACA0012 airfoil ($M_{\infty }=0.63$, $\alpha =2^{\circ }$).

Figure 14

Pressure contours of the flow around the NACA0012 airfoil ($M_{\infty }=0.85$, $\alpha =1^{\circ }$).

Figure 15

$C_p$ of the flow around the NACA0012 airfoil ($M_{\infty }=0.85$, $\alpha =1^{\circ }$).

Figure 16

Pressure contours of the flow around the NACA0012 airfoil ($M_{\infty }=0.8$, $\alpha =1.{25}^{\circ }$).

Figure 17

$C_p$ of the flow around the NACA0012 airfoil ($M_{\infty }=0.8$, $\alpha =1.{25}^{\circ }$).

Figure 18

Pressure contours of supersonic flow past a cylinder. (a) $M_{\infty }=3$ and (b) $M_{\infty }=5$.

Figure 19

Pressure coefficient profile along the central line. (a) $M_{\infty }=3$ and (b) $M_{\infty }=5$. C6F8-ROE refers to the results in Visbal and Gaitonde [62].

Figure 20

Configuration of the explosion in a 3D box.

Figure 21

Density isosurfaces of the explosion in a 3D box at (a) $t=0.25$, (c) $t=0.375$, and (e) $t=0.5$. Panels (b), (d), and (f) are the corresponding results in Ref. [63].

Figure 22

Density contours of the explosion in a 3D box at $z=0.4$ and $t=0.5$. (a) Our result and (b) result in Ref. [63].

Figure 23

Comparison of the pressure coefficient distribution at different sections on the OMERA M6 wing, $\eta$ refers to the spanwise location, and the $x$ directions are nondimensionalized by local chord. (a) $\eta =0.2$, (b) $\eta =0.44$, and (c) $\eta =0.65$.

Figure 24

The speedup (a) and parallel efficiency (b) of the proposed method for the 2D NACA0012 case. The grid has $137523200$ quadrangular cells in this test.

Figure 25

The speedup (a) and parallel efficiency (b) of the proposed method for the 3D ONERA M6 case. The grid has $55362624$ tetrahedron cells in this test.