Efficient Adaptive Cartesian Vorticity Transport Solver for Vortex-Dominated Flows

An efficient solver for the velocity―vorticity form of the Navier―Stokes equations on adaptive Cartesian grids is presented. The excessive numerical dissipation common to most grid-based Navier―Stokes solvers is avoided by solving the fluid dynamic equations in vorticity conservation form. Additionally, an adaptive Cartesian solver is employed to efficiently capture and preserve vorticity on-the-fly as the flow develops. For practical purposes, this solver would be used in the wake region and coupled with a full Navier―Stokes solver in the near-body region, thus allowing vorticity to be accurately generated and then convected in the wake region with minimal dissipation. The adaptive Cartesian framework allows for the rapid evaluation of the velocity field using a fast-summation technique based on the Cartesian Treecode method. The implementations of both the solution algorithm and velocity calculation are described in detail. Results are presented for vortex convection applications that show good agreement with the analytical solution, and the accuracy of the scheme is verified numerically using a series of increasingly fine grids. Additionally, fully three-dimensional flow in the presence of a vortex ring is investigated and results are shown to be in very close agreement with published data.