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. Introduction Vortex flows occur in nearly every area of continuum fluid mechanics from the stirred fluid in a teacup to the flows in tornado funnels and hurricanes. An important instance of vortices in aerodynamics is the flow about a slender delta wing at angle of attack, in which the rolling up of the vortex sheets over the leading edges produces a vortex pair. These leading-edge'vortices often have a roughly axisymmetric core region where viscous effects dominate the flow, and velocities and pressure gradients may be large. Experimental studies of incompressible flows past delta wings have shown some remarkable features, including core velocities that are four to five times the freestream values, accompanied by high gradients in the static and total pressures [l]. Experimental studies of compressible flows past delta wings have shown similar velocity and pressure gradients, accompanied by near-vacuum values of density on the core axis [Z]. Analytical and numerical studies of incompressible leading-edge vortices have duplicated, at least qualitatively, many of these results. The incompressible models of Hall [3], Stewartson and Hall [4] and Powell and Murmaii [5 ] give axial velocity and static pressure values that agree fairly well with the experimental results of Earnshaw [I]. Theoretical studies of compressible vortices have given less satisfactory results. The compressible model of Brown [6] for an inviscid rotational flow shows that compressibility has a boundary-layer effect on the velocity field, but exhibits some odd behavior at higher Mach numbers. Mack [7]studied a viscous heat-conducting compressible vortex, but the flow field is a rather contrived one, with density becoming unbounded at the surface of a rotating cylinder in 'Doctord Candidate, ALAA Member fAssirtant Professor, AIAA Member