Semi-Lagrangian lattice Boltzmann model for compressible flows on unstructured meshes

Compressible lattice Boltzmann model on standard lattices [M. H. Saadat, F. Bösch, and I. V. Karlin, Phys. Rev. E 99, 013306 (2019).] is extended to deal with complex flows on unstructured grid. Semi-Lagrangian propagation [A. Krämer et al., Phys. Rev. E 95, 023305 (2017).] is performed on an unstructured second-order accurate finite-element mesh and a consistent wall boundary condition is implemented which makes it possible to simulate compressible flows over complex geometries. The model is validated through simulation of Sod shock tube, subsonic and supersonic flow over NACA0012 airfoil and shock-vortex interaction in Schardin's problem. Numerical results demonstrate that the present model on standard lattices is able to simulate compressible flows involving shock waves on unstructured meshes with good accuracy and without using any artificial dissipation or limiter.

Figure 1

Schematic of a second-order finite-element mesh, the semi-Lagrangian propagation along the discrete velocity ${\mathit{v}}_i$ and mapping from the global coordinate $(x,y)$ to local coordinate $(\xi ,\eta )$.

Figure 2

Schematic representation of the wall boundary condition implementation.

Figure 3

Sod shock tube simulation results at nondimensional time $t^{*}=0.2$: (a) density and (b) reduced velocity. Symbols: present model; line: exact solution.

Figure 4

Second-order finite-element mesh (Mesh-1) used for the simulation of subsonic flow over NACA0012 airfoil. Bottom is the zoom near leading edge of the airfoil.

Figure 5

Mach contour for subsonic flow over NACA0012 airfoil at $\mathrm{Ma}=0.5, \mathrm{Re}=5000$, and $\alpha =0^{\circ }$. Bottom figure shows streamlines near trailing edge.

Figure 6

Distribution of (a) pressure coefficient and (b) skin friction coefficient on the NACA0012 airfoil surface for subsonic flow at $\mathrm{Ma}=0.5, \mathrm{Re}=5000$, and $\alpha =0^{\circ }$. Line: present model; symbols: DG solver [33].

Figure 7

Distribution of (a) pressure coefficient and (b) skin friction coefficient on the NACA0012 airfoil surface for subsonic flow at $\mathrm{Ma}=0.5, \mathrm{Re}=5000$, and $\alpha =2^{\circ }$. Lines: present model; symbols: reference solution [34].

Figure 8

Mesh-2 used for the simulation of subsonic flow over NACA0012 airfoil.

Figure 9

Temperature (top) and Mach number (bottom) contours for supersonic flow over NACA0012 airfoil at $\mathrm{Ma}=1.5, \mathrm{Re}=10000$.

Figure 10

Pressure coefficient upstream, downstream and on the NACA0012 airfoil surface for supersonic flow at $\mathrm{Ma}=1.5, \mathrm{Re}=10000$. Line: present model with D2Q9 lattice; symbols: ELBM solution with $D2Q49$ lattice; dashed line: results reported in Ref. [37].

Figure 11

Evolution of the density for the Schardin's problem at shock Mach number ${\mathrm{Ma}}_s=1.34$. $T_1$ and $T_2$ are triple points.

Figure 12

Comparison of the position of triple points $T_1$ and $T_2$ for the Schardin's problem. Squares: present model with D2Q9 lattice; dashed line: ELBM solution with $D2Q49$ lattice [14]; circles: experiment [38].