An Arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations

This paper presents a novel semi-implicit hybrid finite volume / element (FV/FE) scheme for the numerical solution of incompressible and weakly compressible Navier-Stokes equations on moving unstructured meshes using direct Arbitrary-Lagrangian-Eulerian (ALE) formulation. The is based suitable splitting governing partial differential into subsystems employs staggered grid arrangement, where pressure defined primal simplex mesh, while velocity remaining flow quantities are an edge-based dual mesh. key idea presented in this to discretize nonlinear convective viscous terms at aid explicit that space-time divergence form control volumes. For terms, ALE extension Ducros flux introduced, which can be proven kinetic energy preserving stable norm when adding dissipation terms. use closed volumes inside guarantees important geometric conservation law (GCL) Lagrangian schemes verified by construction. Finally, equation system solved new mesh configuration classical continuous method, traditional P1 Lagrange elements. A convergence study confirms second order accurate space. Subsequently, FV/FE method applied several test problems ranging from non-hydrostatic free surface flows over rising bubble oscillating cylinder ellipse. Via simulation circular explosion problem we show able capture also weak shock waves, rarefactions contact discontinuities. We provide evidence shows compared fully scheme, proposed particularly efficient low Mach number limit.