A new technique for vortex methods in computational fluid dynamics

Abstract

The random vortex method is a Lagrangian technique which involves solving the stream function-vorticity formulation of the Navier-Stokes equations. Discrete vortex particles, which are created at the surface to satisfy zero slip, are convected in the inviscid flow, while diffusion is modelled as a random walk. The interaction of the discrete vortices can be computed more efficiently by use of a cloud-in-cell technique. Here the vorticity of the particles is spread onto a fixed Eulerian mesh, providing a source term for the Poisson equation which can be easily solved to provide the stream function (and hence velocity) field on the mesh. The cloud-in-cell technique has traditionally been used with regular polar or Cartesian grids in two dimensions, thus requiring conformal transformation and/or multi-grid methods for application to complex geometries. A new method, referred to herein as the cloud-in-element method, is presented which allows a single unstructured mesh to be used for multiple bodies and arbitrary geometries in two dimensions. The technique requires a computationally efficient location of point vortices in an unstructured mesh, which has been made possible by use of reference matrices. A finite element method is used to solve the velocity field on the unstructured mesh. The method is shown to be computationally efficient, with a process time which varies linearly with the number of point vortices used in the simulation. A preliminary study of accuracy and mesh dependency has shown the method to be stable, with increasing accuracy for finer discretisations in space and time. The algorithm is described, and results for the standard case of impulsively started flow over a circular cylinder are presented for a range of Reynolds numbers from 40 to 9500. Good quantitative agreement for Strouhal number and total drag has been achieved. Qualitative agreement of the instantaneous streamline pattern and flow structure was also observed. Vorticity plots are presented for flow over two cylinders, showing the application of the method to multiple bodies.