From a boundary integral formulation to a vortex method for viscous flows

Abstract

For the numerical solution of flow problems past a solid body it is worth to consider boundary integral techniques for their inherent capability to manage efficiently the far-field boundary conditions as well as the approximation of the solid body contour. However, for the analysis of large Reynolds number flows, of major interest in the applications, several computational difficulties appear when using the integral representation for the velocity or for the vorticity field in its classical form with interpolating functions (BEM). In particular, the evaluation of the volume integrals is a serious drawback while the steepness of their kernel introduces artificial diffusion in the calculation. To satisfy the opposite requirements of the advective and of the diffusive part of the Navier-Stokes equations, we adopt an operator splitting scheme according to the Chorin-Marsden product formula (Chorin et al. 1978), together with a proper vorticity generation scheme at the solid boundary. A solution procedure based on the approximation of the vorticity field by a finite number of point vortices (PVM) follows as a natural evolution of the boundary integral formulation.

The numerical results given by the two methods for the merging of two like-signed vortices in free space reveal the excessive numerical diffusion of BEM. The better accuracy of PVM is also established through the evaluation of some first integrals of motion. Several results are also reported for flows in presence of solid boundaries where the vorticity generation is crucial. In this case accurate solutions are only obtained with PVM, while BEM is even less satisfactory than in free space. Finally, the proposed vortex-like method (PVM) is tested on the classical problem of the wake behind a cylinder, in comparison with other well established techniques.