Two topics in computational fluid dynamics

1. The Lamb dipole is a steady propagating solution of the inviscid fluid equations with opposite-signed vorticity in a circular disk. We compare finite-difference solutions of the Navier-Stokes equation (NSE) and the linear diffusion equation (LDE) using the Lamb dipole as the initial condition. We find some expected and some unexpected results; among the latter is that the maximum core vorticity decreases at the same rate for the NSE and LDE, but at higher Reynolds numbers, convection enhances the viscous cancellation of opposite-signed vorticity.
(This is joint work with Ling Xu.)

2. We discuss a new implementation of the vortex method for the incompressible Euler equations. The vorticity is carried by Lagrangian particles and the velocity is recovered by a regularized Biot-Savart integral. The new work employs remeshing and adaptive refinement to resolve small-scale features in the vorticity as well as a treecode for efficiency. The method is demonstrated for vortex dynamics on a rotating sphere (with Peter Bosler) and axisymmetrization of an elliptical vortex (with Ling Xu).