Long's Vortex Revisited
Abstract
The conical selfsimilar vortex solution of Long (1961) is reconsidered, with a view toward understanding what, if any, relationship exists between Long's solution and the morerecent similarity solutions of Mayer and Powell (1992), which are a rotationalflow analogue of the FalknerSkan boundarylayer flows, describing a selfsimilar axisymmetric vortex embedded in an external stream whose axial velocity varies as a power law in the axial (z) coordinate, with phi=r/z^n being the radial similarity coordinate and n the core growth rate parameter. We show that, when certain ostensible differences in the formulations and radial scalings are properly accounted for, the Long and MayerPowell flows in fact satisfy the same system of coupled ordinary differential equations, subject to different kinds of outerboundary conditions, and with Long's equations a special case corresponding to conical vortex core growth, n=1 with outer axial velocity field decelerating in a 1/z fashion, which implies a severe adverse pressure gradient. For pressure gradients this adverse Mayer and Powell were unable to find any leadingedgetype vortex flow solutions which satisfy a basic physicality criterion based on monotonicity of the totalpressure profile of the flow, and it is shown that Long's solutions also violate this criterion, in an extreme fashion. Despite their apparent nonphysicality, the fact that Long's solutions fit into a more general similarity framework means that nonconical analogues of these flows should exist. The farfield asymptotics of these generalized solutions are derived and used as the basis for a hybrid spectralnumerical solution of the generalized similarity equations, which reveal the existence of solutions for more modestly adverse pressure gradients than those in Long's case, and which do satisfy the above physicality criterion.
 Publication:

arXiv eprints
 Pub Date:
 March 2013
 arXiv:
 arXiv:1303.1212
 Bibcode:
 2013arXiv1303.1212L
 Keywords:

 Physics  Fluid Dynamics;
 76U05
 EPrint:
 30 pages, including 16 figures