Quasi-equilibrium states for helical vortices with swirl

We present a family of exact equilibrium solutions of the Euler equations consisting of a single helical vortex with an axial flow component along the vortex core. Relations between vorticity, velocity and streamfunction are analytically derived in the inviscid framework and constitute a generalization of relations valid for vortex rings (Batchelor, 1967 An Introduction to Fluid Dynamics. Cambridge University Press). Through Navier-Stokes simulations, it is shown that these relations hold for quasi-steady viscous solutions and become independent of the Reynolds number when sufficiently large. We also elaborate a procedure which generates a quasi-equilibrium with prescribed characteristics (circulation, helix radius, helical pitch, vortex core size, swirl level) and compare the obtained state with the results of an asymptotic theory. Finally, we illustrate how a strong axial flow jeopardizes such an evolution towards a quasi-equilibrium.

Automated summary

We present a family of exact equilibrium solutions of the Euler equations consisting of a single helical vortex with an axial flow component. The relations between vorticity, velocity and streamfunction are analytically derived and shown to hold in viscous solutions and become independent of the Reynolds number for sufficiently large values. A procedure to generate a quasi-equilibrium with prescribed characteristics is elaborated and compared with asymptotic theory. It is illustrated how a strong axial flow can jeopardize the evolution towards a quasi-equilibrium.