A stable unstructured finite volume method with arbitrary Lagrangian-Eulerian formulation for the numerical simulation of insect flight

An arbitrary Lagrangian-Eulerian (ALE) method based on the side-centered unstructured finite volume method is described for large-scale simulation of moving boundary problems in a fully coupled form. The numerical method based on side-centered finite volume method where the velocity vector components are defined at the mid-point of each cell face while the pressure is defined at the element centroid. The present arrangement of the primitive variables leads to a stable numerical scheme and it does not require any ad-hoc modifications in order to enhance pressure coupling. The continuity equation is satisfied within each element exactly and the summation of the continuity equations can be exactly reduced to the domain boundary, which is important for the global mass conservation. A special attention is also given to satisfy the geometric conservation law (GCL) at discrete level. The interpolation method for the convective fluxes is based on either the least square interpolation or the divergence-free constrained linear reconstruction. The mesh deformation is achieved by using an algebraic approach based on the minimum distance function at each time level while avoiding remeshing in order to enhance numerical robustness. Although the fully couple multigrid methods are very efficient for the computation of the steady-state solutions, they are not suitable for the time-dependent calculations since the advection-diffusion operator is highly diagonally dominant and is well conditioned. Therefore, a matrix factorization is introduced similar to that of the projection method for the whole coupled system and we use two-cycle of BoomerAMG solver for the scaled discrete Laplacian provided by the HYPRE library, a high performance preconditioning package developed at Lawrence Livermore National Laboratory, which we access through the PETSc library. The present numerical algorithm is initially validated for the Kovasznay flow, the 2D/3D lid-driven cavity flow problem, the flow past an oscillating circular cylinder in a channel, the flow around a plunging NACA0012 airfoil and the flow induced by an oscillating sphere in a cubic cavity. Then the numerical algorithm is applied to the numerical simulation of flow field around a pair of flapping Drosophila wing in hover flight. The computed wake structures are in accord with the the experimental observations in the literature.