Axisymmetric swirling flow around a vortex breakdown point

The structure of an axisymmetric and inviscid swirling flow around a vortex breakdown point is analysed. The model assumes that a free axisymmetric bubble surface is developed in the flow with a stagnation point at its nose. The classical Squire-Long equation for the stream function psi(x, y) (where y = r(2)/2) is transformed into a free boundary problem for the solution of y(x, psi). The development of the flow is studied in three regions: the approaching flow ahead of the bubble, around the bubble nose and around the separated bubble surface. Asymptotic expansions are constructed to describe the flow ahead of and behind the stagnation point in terms of the radial distance from the vortex axis and from the bubble surface, respectively. In the intermediate region around the stagnation point, the flow is approximated by an asymptotic series of similarity terms that match the expansions in the other regions. The analysis results in two possible matching processes. Analytical expressions are given for the leading term of the intermediate expansion for each of these processes. The first solution describes a swirling flow around a constant-pressure bubble surface, over which the flow is stagnant. The second solution represents a swirling flow around a pressure-varying bubble surface, where the flow expands along the bubble nose. In both solutions, the bubble nose has a parabolic shape, and both exist only when H' > 0 (where H' is the derivative at the vortex centre of the total head H with the stream function psi, and can be determined from the inlet conditions). This result is shown to be equivalent to Brown & Lopez's (1990) criterion for vortex breakdown. Good agreement is found in the region around the stagnation point between the pressure-varying bubble solution and available experimental data for axisymmetric vortex breakdown.