Quantities which define conically self-similar free-vortex solutions to the Navier–Stokes equations uniquely

Abstract

It is proved that if, in addition to the opening angle of the bounding conical streamsurface and the circulation thereon, one of the radial velocity, the radial tangential stress or the pressure on the bounding streamsurface is given, then a conically self-similar free-vortex solution is uniquely determined in the entire conical domain. In addition, it is shown that for ows inside a cone the same conclusion holds for the Yih et al. (1982) parameter T, but for exterior flows it is shown numerically that non-uniqueness may occur. For given values of the opening angle of the bounding conical streamsurface and the circulation thereon the asymptotic analysis of Shtern & Hussain (1996) is applied to obtain asymptotic formulae which interrelate the opening angle of the cone along which the jet fans out and the radial tangential stress on the bounding surface. A striking property of these formulae is that the opening angle of the cone along which the jet fans out is independent of the value of the viscosity as long as it is small enough for the first-order asymptotic expressions to apply. However, these formulae are shown to be inaccurate for moderate values of the ratio of the circulation at the bounding surface and the viscosity. To amend this shortcoming, an alternative, more accurate, asymptotic analysis is developed to derive second-order correction terms, which considerably improve the accuracy.