Conically similar swirling flows at high Reynolds numbers

Abstract

A new class of both inviscid and boundary layer self-similar solutions for conical swirling flows at high Reynolds numbers is analysed. For the case of one-cell solutions, the flow consists of an inviscid, but in general rotational, core with a velocity field, in spherical polar coordinates, of the form u = rm-2V(θ), where m is any real number. Due to the known existence of two integrals to Euler's equations, the vector function V(θ) is obtained by the integration of a second-order ordinary differential equation, containing the two integration constants K and K1 associated with the intensities of the swirl and the meridional motion, respectively. This inviscid flow is, however, singular at the axis and must be regularized through a thin viscous layer, which also has self-similar structure. A variety of flow regimes are obtained for different ranges of m, all of them exhaustively analysed. In particular, for 0 < m < 2, the solution to the near-axis boundary layer equations has the interesting property of losing existence when a certain inviscid swirl parameter, D ≈ K1/K4, is either larger or smaller than a critical value, depending on m. We hypothesize that when this occurs, a two-cell flow structure develops. For 1 < m < 2, we find that the two-cell structure consists of a thin fan-jet separating two inviscid regions; the flow in the outer cell being vortical while that in the inner one is potential. Flows of the two-cell type cannot exist for 0 < m < 1. Transition from a one- to a two-cell solution is discussed with relevance to a simple example of vortex breakdown. In order to meet any given boundary condition on a certain cone surface θ = α (or a plane for α = ½π), another viscous boundary layer is needed near t, which also has self-similar structure. In the most interesting range 0 < m < 2, this boundary layer also regularizes the singular behaviour of the inviscid flow at the cone surface; in this range, the pressure gradient is negligible inside that boundary layer, allowing for an exhaustive two-dimensional phase space analysis. Two different boundary conditions are considered on the cone surface: a no-slip boundary condition, modelling the interaction of general conical vortices (at high Reynolds numbers) with a cone or a plane, and a shear stress varying as rn. For this last boundary condition, three different relations between the powers n and m are obtained for three different inviscid flow regimes. This shear driven flow appears in some instances to model the motion inside so-called Taylor cones for which n = -5/2(m = 10/13).