Decay of a three–dimensional vortex in an enclosure

M. D. Deshpande

Abstract

Consider viscous, incompressible flow in a cubic container, generated by the uniform, linear motion of the top wall. Depending on the Reynolds number Re the flow could be laminar, transitional or turbulent. If the motion of the top wall is now suddenly stopped, the fluid motion will begin to decay. Direct numerical simulations show that no matter what the initial conditions are, the final stages of decay follow an exponential law, i.e. the maximum of the Y-component of velocity Vm is given by Vm = Vm0 exp (−τd/Re), where the decay constant d is about 62. In order to explain this interesting behaviour of three-dimensional vortex motion, exact decaying, axisymmetric solutions of the Navier-Stokes equations were sought. It is shown that in cylindrical coordinates the field (ur,uϕ,uz)[2πJ1(λr)cos(2πz),kJ1(λr)sin(2πz),λJ0(λr)sin(2πz)]ekvtrepresents a swirling decaying vortex and a corresponding field is found for the spherical case. These new exact solutions to the Navier-Stokes equations are used to identify and quantify the different regimes of vortex decay and scaling in the cubic cavity.