Viscous Rankine vortices

A generalized version of the Rankine vortex is taken as initial state in an infinite fluid domain in two dimensions. The inner rigid-body motion of the Rankine vortex is surrounded by an outer irrotational vortex, where the inner flow and outer flow are allowed to have different amplitudes. We study the time evolution in a viscous fluid, where the kinematic viscosity scales the unit of dimensionless time. By an inverse Hankel transform, the Rankine vortex appears as an integrated continuous spectrum of axisymmetric Fourier–Bessel modes with radial wave numbers k. Each of these velocity modes is represented by a Bessel function $J1(kr)$ where r is the radial distance from the vortex center. The amplitude of each individual mode will decay exponentially at a rate of $νk2$⁠, where ν is the kinematic viscosity. The decaying continuous Fourier–Bessel spectrum is calculated analytically, while the evolution of the flow field is found by numerical evaluation of an inverse Hankel transform. We also consider the presence of a free surface in the gravity field, where pressure distribution in the vertical direction is hydrostatic. The far field remains irrotational during the viscous decay.