The free-ALE method : for unsteady incompressible flow in deforming geometries

Abstract

An algorithm has been developed for the solution of the Navier-Stokes equations, applicable to unsteady incompressible flows. The algorithm is based around an unstructured mesh of flexible topology, intended to facilitate flow modelling in complex and deforming geometries. The unstructured mesh consists of Voronoi finite volumes. The algorithm has been implemented in two and three dimensions in a set of computer programs named "Free-ALE", reflecting the free topology of the mesh and the Arbitrary Lagrangian Eulerian (ALE) approximation used to account for motion of the finite volumes. The algorithm is an adaptation of MAC-like explicit methods to a collocated unstructured Voronoi mesh. Rhie-Chow like interpolation is used to avoid the problem of spurious pressure oscillations. Second order accurate in space, flux limited upwind advection approximations based upon the ULTRAQUICK and ULTRAQUICKEST schemes have been adapted to the Voronoi mesh. An automatic interior node motion algorithm has been developed, which maintains homogeneity of the Voronoi mesh in the presence of prescribed boundary deformations. The duality of the Delaunay and Voronoi geometric structures is exploited to maintain the Voronoi properties of the finite volumes in the presence of mesh deformation. Lawson's local transformation procedure has been used in the two-dimensional version of the program for this purpose, while in three dimensions a more sophisticated local transformation procedure has been derived from Joe's method for generating three dimensional Delaunay triangulations. A feature of this algorithm is its ability to cope with improper triangulations which occur frequently in the deforming three dimensional case. The algorithms have been applied to a number of test problems to validate the implementation and assess accuracy. Second order spatial accuracy has been demonstrated on a static mesh three dimensional unsteady problem having an analytic solution. The code has been applied to the physical problem of modelling blood flow in an artificial ventricle (the "Spiral Vortex VAD", Nugent and Bertram, 1992). This pulsatile flow device undergoes severe deformation and shape change during its pumping cycle and thus provides a challenging and interesting application for the method.