Some similarity solutions for three-dimensional boundary layers

Abstract

A similarity solution of a three-dimensional boundary layer is investigated. The outer flow is given by U  = ( −  xz , −  yz , z 2 ), corresponding to an axisymmetric poloidal circulation with constant potential vorticity. This flow is an exact solution of the Navier–Stokes. A wall is introduced at y  = 0 along which a boundary layer develops towards x  = 0. We show that a similarity reduction to a system of ODEs is possible. Two distinct solutions are found, one of them through numerical path-continuation, and their stability is investigated. A second three-dimensional solution is also identified for two-dimensional outer flow. The solutions are generalized for outer flows scaling with different powers of z and similar results are found. This behaviour is related to the non-uniqueness of the Falkner–Skan flows in a three-dimensional sense, with a transverse wall-jet.