Numerical Study of the Viscous Lamb Dipole

The Lamb dipole is a steady propagating solution of the inviscid fluid equations with opposite-signed vorticity in a circular disk. As known from previous studies, in the presence of viscosity the dipole grows in size, the propagation velocity decreases, and a thin tail of low-level vorticity is left behind as the head propagates forward. To clarify the roles of convection and diffusion, we compare finite-difference solutions of the Navier-Stokes equation (NSE) and the linear diffusion equation (LDE) using the inviscid Lamb dipole as the initial condition. We find that the maximum core vorticity decreases at the same rate for the NSE and LDE, for Reynolds numbers in the range from 125 to 1000. However as the Reynolds number increases, the total positive circulation decreases slightly faster for the NSE than the LDE, showing that convection enhances the cancellation of opposite-signed vorticity in the dipole. Other details of the flow are examined including the structure of the tail and entrainment of initially irrotational fluid in the head.