Conically similar swirling flows at high Reynolds numbers

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 o f one-cell solutions, the flow consists of an inviscid, but in general rota tional, core with a velocity field, in spherical polar coordinates, of the form u = r(m-2)V(theta), where m is any real number. Due to the known exist ence of two integrals to Euler's equations, the vector function V(theta) is obtained by the integration of a second-order ordinary differential equati on, containing the two integration constants K and K-1 associated with the intensities of the swirl and the meridional motion, respectively. This invi scid 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 rn, all of them exhaustiv ely analysed. In particular, for 0 < m < 2, the solution to the near-axis b oundary layer equations has the interesting property of losing existence wh en a certain inviscid swirl parameter, D similar to K-1/K-4, is either larg er or smaller than a critical value, depending on m. We hypothesize that wh en this occurs, a two-cell Bow structure develops. For 1 < m < 2, we find t hat the two-cell structure consists of a thin fan-jet separating two invisc id regions; the Bow in the outer cell being vortical while that in the inne r 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 t o a simple example of vortex breakdown. In order to meet any given boundary condition on a certain cone surface theta = alpha (or a plane for alpha = 1/2 pi), another viscous boundary layer is needed near it, which also has s elf-similar structure. In the most interesting range 0 < m < 2, this bounda ry layer also regularizes the singular behaviour of the inviscid Bow at the cone surface; in this range, the pressure gradient is negligible inside th at boundary layer, allowing for an exhaustive two-dimensional phase space a nalysis. Two different boundary conditions are considered on the cone surfa ce: a no-slip boundary condition, modelling the interaction of general coni cal vortices (at high Reynolds numbers) with a cone or a plane, and a shear stress varying as r(n). For this last boundary condition, three different relations between the powers n and m are obtained for three different invis cid Row regimes. This sheer driven Flow appears in some instances to model the motion inside so-called Taylor cones for which n = 5/2 (m = 10/13).