Heuristic representation of the swirl velocity in the core of the bidirectional vortex

In this article, we discuss the merits of a heuristic, piecewise representation of the swirl velocity in the core of the bidirectional vortex. This combined vortex representation is based on the notion that a uniform, Couette-like, shear stress distribution may be assumed in the inner vortex region, especially at high Reynolds numbers. At the outset, direct integration of the shear stress enables us to retrieve an expression for the swirl velocity that overcomes the inviscid singularity at the centerline. The solution we obtain must be patched to the outer, free vortex approximation at some intermediate position in the chamber. The resulting piecewise distribution is then used to represent the swirl velocity throughout the chamber. Two Rankine-type patching schemes are explored for this heuristic model. In the first, the core solution, along with its first derivative, are patched to the free vortex at the mantle location of 0.707 found by Vyas and Majdalani (Vyas, A. B., and Majdalani, J., "Exact Solution of the Bidirectional Vortex," AIAA Journal, Vol. 44, No. 10, 2006, pp. 2208-2216). In the second, the patching is performed at a point that is representative of the thickness of the forced vortex core. The more general representation provides the freedom of using either laminar or turbulent models to estimate the thickness of the core boundary layer at a given vortex Reynolds number. The first model that we explore assumes that the outer, annular region of the bidirectional vortex is driven by free vortex motion, whereas the inner region (inside the mantle) is entirely dominated by constant shear. Being purely inviscid and insensitive to the vortex Reynolds number, it is discarded in favor of a more portable model that can be set to mimic laminar or turbulent profiles. The second, more general approximation is subsequently compared to the existing laminar solution derived directly from first principles. Its pressure distribution is calculated and shown to be non-singular even in the purely inviscid case. The versatility of the piecewise solution is illustrated by specifying a constant shear radius that scales with the existing laminar core layer thickness. Other heuristic schemes are discussed and the conclusion is reached that further refinements for high Reynolds number flows must await the advent of sufficient laboratory and numerical experiments.