Three-Dimensional Vortex Simulation of Time Dependent Incompressible Internal Viscous Flows

Abstract

A hybrid random vortex-boundary element method is developed for the solution of time-dependent incompressible three-dimensional internal flow problems. The numerical scheme is grid-free within the flow domain and is based on a combination of the Lagrangian vortex method to capture the convection and stretch of the vortical field, the random walk method to describe the diffusion process, and the boundary element method to superimpose a potential flow on the vortical field such that the normal flux boundary condition is satisfied. The no-slip boundary condition is satisfied by generating vorticity tiles on solid boundaries, which are subsequently diffused and convected into the flow interior. Additionally, a boundary condition is devised for the application of fully developed flow properties at the exit plane. In this paper, the formulation and the numerical scheme are presented, followed by a parametric study of the accuracy of the method using the model problem of the flow in a duct with square cross section at Re= 100. We show that the method converges to the analytical solution of the problem as the resolution of the time integration and the discretization are improved, and we discuss the impact of each resolution parameter on the accuracy. In addition, selected results from the simulation of an impulsively started flow over a cube at Re= 100 are presented. We use the results of this test case to demonstrate that the method captures the effect of sharp edges, parallel and normal to the streamwise flow direction, on the flow dynamics.