A High-Order Flux Reconstruction Method for 2-D Vorticity Transport

A compact high-order finite difference method on unstructured meshes is developed for discretization of the unsteady vorticity transport equations (VTE) for 2-D incompressible flow. The algorithm is based on the Flux Reconstruction Method of Huynh [1, 2], extended to evaluate a Poisson equation for the streamfunction to enforce the kinematic relationship between the velocity and vorticity fields while satisfying the continuity equation. Unlike other finite difference methods for the VTE, where the wall vorticity is approximated by finite differencing the second wall-normal derivative of the streamfunction, the new method applies a Neumann boundary condition for the diffusion of vorticity such that it cancels the slip velocity resulting from the solution of the Poisson equation for the streamfunction. This yields a wall vorticity with order of accuracy consistent with that of the overall solution. In this paper, the high-order VTE solver is formulated and results presented to demonstrate the accuracy and convergence rate of the Poisson solution, as well as the VTE solver using benchmark problems of 2-D flow in lid-driven cavity and backward facing step channel at various Reynolds numbers.