Self-similar, slightly compressible, free vortices

Abstract

Exact and numerical similarity solutions for compressible perturbations to an incompressible, two-dimensional, axisymmetric vortex reference flow are presented. The reference flow consists of a set of two-dimensional, self-similar, incompressible vortices. Similarity variables, which give explicit expressions for the decay rates of the velocities and thermodynamic variables in the vortex flows, are used to reduce the governing partial differential equations to a set of ordinary differential equations. The ODEs are solved analytically and numerically for a Taylor vortex reference flow, and numerically for an Oseen vortex reference flow. The solutions are employed to study the dependences of the temperature, density, entropy, dissipation and radial velocity on the Prandtl number. Additionally, several integral relations, which allow one to trace the energy transfer in a slightly compressible vortex, are derived.

Speed of a von Kármán point vortex street in a weakly compressible fluid

“…A surprising circumstance, remarked upon by previous authors [6], is that the study of compressible vortex dynamics has received markedly less attention even though gaining an understanding of compressible vortex flows is of fundamental interest for compressible wakes [7,8] and the theory of vortex sound and aeroacoustics [9][10][11]. There have been a few fundamental studies of compressible vortices [12][13][14][15], but many basic theoretical questions remain to be answered. There is, by now, evidence in the literature [16][17][18][19][20] of the existence of continuous families of shock-free transonic compressible flows with embedded vortices.…”

Flight-Test Measurement of the Aero-Optical Environment of a Helicopter in Hover

“…Since even the fastest helicopters travel at incompressible speeds, the kinds of aero-optical flows that are important for fixed-wing aircraft are generally inconsequential for helicopters. Instead, for helicopters, aero-optical aberrations are expected to originate primarily from the wake and (especially) tip vortices that are shed from the helicopter rotor blades, where flow speeds close to and especially within the cores of the vortices are sufficiently high to produce significant density variations and associated optical aberrations [9,10]. As shown by the analyses described in [11], high-speed flows in the tip-vortex cores persist for some distance along the trajectory of the vortex, so that compressibleflow effects can be expected even after the vortices have been convected by the rotor downwash into the line of sight of an optical system mounted within or under the helicopter fuselage [12][13][14][15].…”