Deformation and Regeneration of the Computational Grid for CFD with Moving Boundaries

Abstract

A general method for computational fluid dynamics with boundaries moving in any prescribed fashion, is presented. The method adapts the boundary fitted mesh to the changing spatial domain by deforming it and, when necessary due to grid quality requirements, regenerating it. Finite volumes are used for the discretisation of Navier-Stokes equations, the mesh is regenerated by the advancing front method and an elliptic differential equation, or an analytical expression, is used to compute the mesh motion. Results are presented for two model problems. I. Introduction In many industrial application it is necessary to compute the flow field in a domain whose boundary is moving. One large class of such problem contains a rotating part, with applications to turbo-machinery and ship and air craft propellers. Another special problem type is piston motion in engines and hydraulic, pneumatic systems. There are also, of course, many problems which have moving boundaries and do not fall into the two mentioned restricted classes. We mention only the movement of rudders on a ship or an air plane and flows in blood vessels but the examples are countless. In this article we describe an implementation of a general method, for incompressible flow, which is applicable, in principle, to all of the problems mentioned above. The computational problem is formulated precisely in section II. The basis for the method is a finite volume approximation with an unstructured, boundary fitted, moving grid, as described in Demirdzic and Peric. 4 The new development is to dynamically determine where and when the mesh motion causes too large decrease in the mesh quality, and then regenerate the mesh for an appropriate region, interpolate the flow variables and continue the solution. The main components of the method are: The finite volume discretisation, the advancing front meshing algorithm, the method to deform the mesh and the interpolation scheme used at the mesh switching/regeneration. These steps are described in section III. The method is applied to two model problems for which we present results in section IV. We have performed a grid refinement study which indicates the appropriate convergence. We have also compared different methods for grid deformation. The first computes the grid motion by the solution of an elliptic partial differential equation and the second bases the node displacements on a weighting using the distance to the moving boundary. The results indicate that the elliptic operator, with a certain form of the coefficient function, gives slowest decrease in mesh quality. The coefficient is chosen to be large at the moving boundary, see section IV for the exact expression. This counter-acts the well-known tendency of the grid to deteriorate in this region. Throughout this article, we use the terms grid and mesh interchangeably. Another commonly used possibility for problems with rotating parts is to let the moving grid “slide” along the fixed surrounding grid, see e.g. Demirdzic, Muzaferija, Peric and Schreck. 3 The method presented here however has several advantages compared to this. First all cells in the grid has matching interfaces, there are no “hanging nodes”. Sliding meshes also necessitates book-keeping concerning which faces overlap at the sliding interface. Furthermore, the method presented here can be applied to problems involving more general boundary motion than rotation. The drawbacks with our method compared to sliding meshes are that we must regenerate the grid, and we need an algorithm for the mesh motion. Thus it is a trade-off which method to prefer. Finally, we mention the approach using overlapping (Chimera) grids as an alternative for the class of problems considered here, see e.g. Chesshire and Henshaw. 2