Arbitrary Lagrangian–Eulerian formulation of lattice Boltzmann model for compressible flows on unstructured moving meshes

Abstract

We propose the application of the arbitrary Lagrangian-Eulerian (ALE) technique to a compressible lattice Boltzmann model for the simulation of moving boundary problems on unstructured meshes. To that end, the kinetic equations are mapped from a moving physical domain into a fixed computational domain. The resulting equations in the computational domain are then numerically solved using the second-order accurate finite element reconstruction on an unstructured mesh. It is shown that the problem regarding the geometric conservation law (GCL), which needs a special treatment in the ALE Navier-Stokes solvers, does not appear here and the model satisfies the GCL exactly. The model is validated with sets of simulations including uniform flow preservation and compressible flow past airfoil with plunging and pitching motions at different Mach numbers. It is demonstrated that the results are in good agreement with the experimental and other available numerical results in the literature. Finally, in order to show the capability of the proposed solver in simulating high-speed flows, transonic flow over pitching airfoil is investigated. It is shown that the proposed model is able to capture the complex characteristics of this flow which involves multiple weak shock waves interacting with the boundary and shear layers.

Consistency study of Lattice-Boltzmann schemes macroscopic limit

“…where and are the discretized probability density functions corresponding to their BE counterpart from Eqs. (9,10). Accordingly the BE is also discretized such that we should now solve BEs, one for each discrete velocities ,…”

“…(ii) Second, time and space are discretized, as in most computational uid dynamics (CFD) methods. Albeit initially limited to low-Mach athermal ows, the range of applicability has been steadily growing, to encompass compressible ows [7][8][9][10] , multiphase ows [11][12][13][14][15][16] and combustion [17][18][19] .…”

Lattice Boltzmann method for fluid-structure interaction in compressible flow

“…In this section we briefly review the ALE formulation for LBM as proposed in [40]. To that end, we consider the Boltzmann equation,…”

“…Another avenue are semi-Lagrangian methods [33][34][35][36][37], which start from a characteristic equation to resolve these issues. Promising results have been shown for wall-bounded turbulent flows [38] and even compressible flows [37,39,40].…”

“…In our previous works [39,40], it was demonstrated that by using an arbitrary Eulerian-Lagrangian (ALE) formulation in combination with a dual population LB formulation, highspeed compressible flows with moving geometries can be captured accurately and efficiently. However, in those works, only a single body was considered and the motion was prescribed.…”