Vorticity and circulation decay in the viscous Lamb dipole

The Lamb dipole is a steady translating structure in 2D ideal fluid flow with opposite-sign vorticity of compact support in a circular disk. Previous studies have shown that when viscosity is present, the resulting viscous Lamb dipole develops a head-tail structure in which the head expands in size, while a tail of low amplitude vorticity is left behind as the head moves forward; in addition, the maximum vorticity and total circulation on each side of the dipole decay in time. Here we examine these decay properties by comparing numerical solutions of the Navier-Stokes equation (NSE) and diffusion equation (DE) in the Reynolds number range using the inviscid Lamb dipole as initial condition; this enables us to compare the combined effects of convection and diffusion in the NSE with the sole effect of diffusion in the DE. The results show that for a given Re, the vortex core size, shape, and maximum vorticity are nearly the same for the NSE and DE, indicating that convection has little effect on these properties. Nonetheless, compared to the DE, convection in the NSE inhibits circulation decay at low Re, while it enhances circulation decay at high Re, and the lateral separation of the vortex cores is a critical factor in this transition.