An implicit edge-based ALE method for the incompressible Navier-Stokes equations

A new finite volume method for the incompressible Navier-Stokes equations, expressed in arbitrary Lagrangian-Eulerian (ALE) form, is presented. The method uses a staggered storage arrangement for the pressure and velocity variables and adopts an edge-based data structure and assembly procedure which is valid for arbitrary n-sided polygonal meshes. Edge formulas are presented for assembling the ALE form of the momentum and pressure equations. An implicit multi-stage time integrator is constructed that is geometrically conservative to the precision of the arithmetic used in the computation. The method is shown to be second-order-accurate in time and space for general time-dependent polygonal meshes. The method is first evaluated using several well-known unsteady incompressible Navier-Stokes problems before being applied to a periodically forced aeroelastic problem and a transient free surface problem.