Grid refinement in the three-dimensional hybrid recursive regularized lattice Boltzmann method for compressible aerodynamics

Grid refinement techniques are of paramount importance for computational fluid dynamics approaches relying on the use of Cartesian grids. This is especially true of solvers dedicated to aerodynamics, in which the capture of thin shear layers require the use of small cells. In this paper, a three-dimensional grid refinement technique is developed within the framework of hybrid recursive regularized lattice Boltzmann method (HRR-LBM) for compressible high-speed flows, which is an efficient collide-stream-type method on a compact D3Q19 stencil. The proposed method is successfully assessed considering several test cases, namely, an isentropic vortex propagating through transition interface, shock-vortex interaction with intersection between grid refinement interface and shock corrugation, and transonic flows over three-dimensional DLR-M6 wing with seven levels of grid refinement.

Figure 1

Visualization of the grid arrangement around transition in resolution; the nodes in $◯$ are located on the coarse grid and the nodes in $•$ are located on the fine grid. The nodes in ($•+◯$) are in collocated transition nodes, the nodes in ($•+\square$) are transition fine nodes located on the edge-center of coarse grid, and the nodes in ($•+\diamond$) are transition fine nodes located on the face-center of coarse grid.

Figure 2

Temporal interpolation along time line. The nodes in ($•+◯$) are in collocated transition nodes. (a) Odd time step, (b) even time step.

Figure 3

1D view along transition edge. The nodes in ($•+◯$) are in collocated transition nodes, the nodes in ($•+\square$) are transition fine nodes located on the edge-center of coarse grid.

Figure 4

2D spatial interpolation in $x\text{−}y$ plane. The nodes in ($•+\square$) are transition fine nodes located on the edge-center of coarse grid. ($•+\diamond$) are transition fine nodes located on the face-center of coarse grid.

Figure 5

Computed density field at different time of 19.2T (a) and 19.3T (b). The grid-refined subdomain is marked by the square in gray.

Figure 6

Computed streamwise velocity ($x$-direction) field at different time of 19.2T (a) and 19.3T (b). The grid-refined subdomain is marked by the square in gray.

Figure 7

Computed temperature field at different time of 19.2T (a) and 19.3T (b). The grid-refined subdomain is marked by the square in gray.

Figure 8

Distributions of density, velocity, and temperature at midline at $t=20\mathrm{T}$. The refined domain is located between the two dotted lines.

Figure 9

L2 error obtained by the LBM with the present grid refinement approach. $N=10/\mathrm{\Delta }{x}_{\text{min}}$.

Figure 10

Sound pressure field obtained by the present HRR-LBM with grid refinement on Grid-a configuration at $t=1$ (a), and $t=6$ (b), respectively. The grid-refined domain is inside the square marked by red dotted line.

Figure 11

Sound pressure field obtained by the present HRR-LBM with grid refinement on Grid-b configuration at $t=1$ (a), and $t=6$ (b), respectively. The grid-refined domain is inside the square marked by red dotted line.

Figure 12

Distributions of the sound pressure. top: radial distribution at the angle $\theta =-{45}^{\circ }$; bottom: circumferential distribution at $t=6$. The reference data in Ref. [58] are denoted by symbols.

Figure 13

Density field at $t=8, M_v=0.5,\text{Re}=400$. The contour levels are from 0.92 to 1.55 with an increment of 0.0053.

Figure 14

Instantaneous Ma number (a) and vorticity fields (b) at time $t=16.36$. Grid refinement interfaces are denoted by solid lines.

Figure 15

Pressure coefficients on the NACA0012 airfoil at $t=16.36$.

Figure 16

Sketch of grid refinement for inviscid flows past ONERA-M6. There are approximately 800 nodes in chord length at the finest level.

Figure 17

Pressure coefficient obtained by the present LB with grid refinement for inviscid flows past ONERA-M6.

Figure 18

Pressure coefficient at $y=20\%$ span, $y=44\%$ span, $y=65\%$ span, and $y=90\%$ span (from top to bottom) obtained by the present LB with grid refinement for inviscid flows past ONERA-M6.