An extension to the extended Beltrami solution method and solutions to the 3D Navier-Stokes equations*

Recirculating flows, such as those seen in vortex separators, swirl combustion chambers, and Ranque-Hilsch vortex tubes can seldom be modelled analytically, and where exact solutions may be found, viscous effects are neglected. In this work we attempt to describe swirling motions, by extending the extended Beltrami solution method for the Navier-Stokes equations to describe 3D flows wherein the flow variables may be written as a function of two spatial co-ordinates. The new formulation emerges when auxiliary functions are added to the general Bragg-Hawethorne equations and reinserted into the governing equations. As is the case with the extended Beltrami approach, solutions are sought by guessing forms of the auxiliary functions, and attempting to solve the system of equations. Using this technique, we find many new solutions. Well known planar flows including Kovasznay flow and Wang's flow have been generalized to three-dimensions. A unique solution in plane polar co-ordinates has been found. A new 3D swirling flow solution which can be considered the angular analogue to Kovasznay flow is found which exhibits many realistic features observed in swirling flow applications.