Similarity models for viscous vortex cores

Abstract

Results are presented for a new class of selfsimilar solutions to the steady, axisymmetric Navier-Stokes equations, modelling the flows in vortices whose viscous cores grow proportionally to an arbitrary power of the axial coordinate. Effects of Reynolds number, growth rate, free-stream Mach number and vortex strength are investigated. Results for incompressible vortices are compared to experimental results for leading-edge vortices and to the inviscid limiting case. It is shown that the core of an incompressible vortex should, in absence of an axial pressure gradient, grow with the onehalf power of the axial coordinate. The total pressure level on the axis of the vortex is seen to 6e independent of Reynolds number and assumed growth rate, although the distribution of total pressure in the core is dependent on both. Large axial velocity surpluses are seen in the core as well as strong pressure gradients. Results for the viscous compressible vortex are compared to the inviscid potential and rotational model flows. Near-zero densities and pressures and very low temperatures are seen on the axis for reasonable freestream Mach numbers and vortex strengths. The presence of viscosity prevents the density from actually reaching zero, unlike with the inviscid models. Limiting velocity concepts are found to restrict the possible vortex strengths for a given freestream Mach number. Compressibility is shown to have a significant influence upon the distribution of vorticity in the core of the vortex.