Semi-Lagrangian lattice Boltzmann method for compressible flows

This work thoroughly investigates a semi-Lagrangian lattice Boltzmann (SLLBM) solver for compressible flows. In contrast to other LBM for compressible flows, the vertices are organized in cells, and interpolation polynomials up to fourth order are used to attain the off-vertex distribution function values. Differing from the recently introduced Particles on Demand (PoD) method [Dorschner, Bösch, and Karlin, Phys. Rev. Lett. 121, 130602 (2018)], the method operates in a static, nonmoving reference frame. Yet the SLLBM in the present formulation grants supersonic flows and exhibits a high degree of Galilean invariance. The SLLBM solver allows for an independent time step size due to the integration along characteristics and for the use of unusual velocity sets, like the D2Q25, which is constructed by the roots of the fifth-order Hermite polynomial. The properties of the present model are shown in diverse example simulations of a two-dimensional Taylor-Green vortex, a Sod shock tube, a two-dimensional Riemann problem, and a shock-vortex interaction. It is shown that the cell-based interpolation and the use of Gauss-Lobatto-Chebyshev support points allow for spatially high-order solutions and minimize the mass loss caused by the interpolation. Transformed grids in the shock-vortex interaction show the general applicability to nonuniform grids.

Figure 1

D2Q25 velocity set based on Hermite polynomials of order 5. Although the velocity can be scaled, matching the regular grid is not possible.

Figure 2

Distribution of Gauß-Lobatto-Chebyshev support points on a two-dimensional reference cell with $p=4$.

Figure 3

Numerical errors of incompressible 2D Taylor-Green vortex benchmarks with horizontal Mach numbers ${\mathrm{Ma}}_h=0$ (a) and ${\mathrm{Ma}}_h=0.05$ (b).

Figure 4

Numerical errors of incompressible 2D Taylor-Green vortex benchmarks over horizontal Mach numbers ${\mathrm{Ma}}_h$ with the D2Q25 velocity set for three different time step sizes. The resolution was 64 grid points per direction.

Figure 5

Sod shock tube with $p=4,N=800\left(200\mathrm{cells}\right),\delta t=0.002$ after 735 iterations showing (a) density $\rho$, (b) velocity $u$, and (c) temperature $T$. The SLLBM is compared with Sod's reference results [58].

Figure 6

Stable and unstable Sod shock tube simulations of the on-lattice Boltzmann solver detailed in Sec. 2g.

Figure 7

Stable Sod shock tube simulations of the SLLBM at three different time step sizes, where ${\delta }_t^{\prime }$ is the time step size of the on-lattice method in Fig. 6.

Figure 8

Outline of 45 equally spaced density contours of the 2D Riemann problem with $N=512\times 512$ in the interval $\rho \in [0.412,1.753]$. The order of finite elements was $p=4$. The simulation end time was reached in 2091 iterations. Reference solutions obtained by two different solvers are available in [61] and [62].

Figure 9

Pressure, velocity, and temperature curves of the two-dimensional Riemann problem, shown in Fig. 8 along the $y=x$ diagonal $d$.

Figure 10

Comparison of the pressure $P$ with Gauß-Lobatto-Chebyshev nodes and conventional equidistant nodes at order of finite elements of $p=4$ after 200 time steps. The test run with equidistant nodes provoked heavy oscillations, which resulted in a crash shortly afterwards.

Figure 11

Schematic grid for the shock vortex interaction. The actual grid used in the simulation was further refined by splitting each cell in half along each axis. Each cell contains $5\times 5$ support points, whereby the outer support points are shared with the neighboring cells.

Figure 12

Density contours of the shock-vortex interaction with ${\mathrm{Ma}}_v=0.5$ and $\mathrm{Re}=400$ in the range $\rho \in [0.92,1.55]$ in 119 steps.

Figure 13

Sound pressure $\mathrm{\Delta }P=(P-P_B)/P_B$ of the shock-vortex interaction with $\mathrm{Re}=800,{\mathrm{Ma}}_v=0.25$, and ${\mathrm{Ma}}_A=1.2$ at times $t^{\prime }=6,t^{\prime }=8,$ and $t^{\prime }=10$. The measurements were taken from the center of the vortex along the radius with $-{45}^{\circ }$ with respect to the $x$ axis. DNS from [63].

Figure 14

Relative mass $m/m_0$ for polynomial orders $p=4$ and $p=2$ over the number of iterations for the test case shown in Fig. 12.