Non-Viscous Vortex Generation due to Buoyancy, an Example of Application of Compulsive Forces in Fluids

Abstract

In the application of the Euler momentum equation to the flow of a two-dimensional incompressible vortex pair, the density of the accompanying fluid of which is greater than the one of the ambient fluid, the generation of a pressure discontinuity, i. e. the generation of a shock, at the density discontinuity is predicted.  However, even weak shock waves are propagating at the infinite speed of sound in an ideal incompressible fluid, which is a good approximation for low subsonic flows of air.  In order to avoid the pressure discontinuity, the condition of pressure continuity is introduced as a compulsive condition.  If this condition is not fulfilled, which up to now is the case in relevant papers, the compulsive condition of pressure continuity has to be introduced.  This results in a supplementary compulsive force, which in the present case is non-conservative, and consequently leads to the generation of circulation in the barotropic, non-viscous fluid accompanying the vortex pair, for which Bjerknes theorem of baroclinic generation of circulation does not hold.  It may be applied to the velocity jump across the density discontinuity, but without the contribution from the compulsive forces does not satisfy the condition of pressure continuity and the cinematic condition at the stagnation points.  Thus, another exception to Kelvin’s theorem of the constancy of circulation is found.  The theory presented is applied to the interference between an aircraft wake and atmospheric instability.  In the case of a latent atmospheric instability the aircraft wake may trigger off instability.