Dissipation and breakdown of a wing-tip vortex

Abstract

The solutions of the incompressible quasi-cylindrical momentum-integral equations describing the flow in the viscous core of a wing-tip vortex are obtained in a closed form and are shown to have two distinct branches. The discontinuities of these solutions have infinite axial gradients and therefore, following Hall, are assumed to signal the inception of the vortex breakdown. Benjamin's finite transition, with its excess flow force dissipated, is shown to give results equivalent to a sudden cross-over, upstream of the discontinuity, from one branch solution to another. The critical point of such a cross-over is downstream from the cross-over, at the discontinuity. Sarpkaya's experimental data, and the nature of the solutions ahead of the discontinuity, suggest that the physical manifestation of the discontinuity is the spiral breakdown, whereas the cross-over seems to be related to the rapidly expanding and subsequently contracting axisymmetric bubble. This therefore implies that the beginning of the spiral breakdown is the all important disturbance which triggers off not only the downstream asymmetric departure of the flow from its quasi-cylindrical form but also the formation of the upstream axisymmetric cross-over bubble. Solutions for the turbulent flow downstream from the spiral breakdown indicate that the wing-tip vortex breakdown can result in an appreciable reduction of the maximum circumferential velocity and should thus lessen the danger that trailing vortices present to following aircraft.

Solutions of the Navier-Stokes equations for vortex breakdown

“…Reynolds numbers large enough for comparison, it was not possible to conclusively demonstrate that the failure of the quasi-cylindrical approximation, which occurs in the analysis of Mager (1972) for e1 greater than a particular value e 1 independent of…”

“…He thus identified with vortex breakdown both the failure of the quasicylindrical approximation and the appearance of the critical state. Mager (1972), following the work of Gartshore (1963) and Bossel (1967Bossel ( ,1971, showed that the solutions of the quasi-cylindrical momentum-integral equations have two distinct branches and, with the appropriate upstream conditions, discontinuities which represent the appearance of infinite axial gradients and are assumed to signal the inception of vortex breakdown. He then demonstrated that Benjamin's finite transition could be explained as a sudden cross-over, upstream of the discontinuity, from one branch of the solution to the other, and suggested that the physical manifestation of the discontinuity might be the spiral breakdown while the cross-over, if it occurred, might appear as the axisymmetric bubble upstream.…”

“…Mager (1972) provided a very precise criterion for the appearance of discontinuities, which he assumed were the results of such failures, in his integral solutions of the quasi-cylindrical equations. He defined a parameter e which is the invariant flux of axial momentum deficiency in the core plus a term dependent on the circulation around the core plus a third term whose value changes only behind the breakdown.…”

Vortex breakdown incipience: Theoretical considerations

“…(i) vortex breakdown is similar to the separation of a two-dimensional boundary layer (Hall [2,3], Mager [4]);…”

Numerical Simulation of Vortex Breakdown

“…An extensive and comparative analysis has been carried out by Hall (1972) and Leibovitch (1978). In the first theory, due to Gartshore (1963), Hall (1972), andMager (1972), vortex breakdown is interpreted as corresponding to the failure of the quasi-cylindrical _approximation~of_the~Na"Rier-=Stokes_equations_._l'he_theor-y-is~ana-logous-t;o'-----------boundary layer theory. In the second theory, due to Ludwig (1970), vortex breakdown is explained as a consequence of an instability of the flow to helical disturbances.…”