HYSTERESIS IN SWIRLING JETS

This paper explains hysteretic transitions in swirling jets and models external flows of vortex suction devices. Toward this goal, the stead y rotationally symmetric motion of a viscous incompressible fluid abov e an infinite conical stream surface of a half-angle theta(c) is studi ed. The flows analysed are generalizations of Long's vortex. They corr espond to the conically similar solutions of the Navier-Stokes equatio ns and are characterized by circulation Gamma(c) given at the surface and axial flow force J(1). Asymptotic analysis and numerical calculati ons show that four (for theta(c) less than or equal to 90 degrees) or five (for theta(c) > 90 degrees) solutions exist in some range of Gamm a(c) and J(1). The solution branches form hysteresis loops which are r elated to jump transitions between various flow regimes. Four kinds of jump are found: (i) vortex breakdown which transforms a near-axis jet into a two-cell flow with a reverse flow near the axis and an annular jet fanning out along conical surface theta = theta(s) < theta(c); (i i) vortex consolidation causing a reversal of (i), (iii) jump flow sep aration from surface theta = theta(c); and (iv) jump attachment of the swirling jet to the surface. As Gamma(c) and/or J(1) decrease, the hy steresis loops disappear through a cusp catastrophe. The physical reas ons for the solution non-uniqueness are revealed and the results are d iscussed in the context of vortex breakdown theories. Vortex breakdown is viewed as a fold catastrophe. Two new striking effects are found: (i) there is a pressure peak of O(Gamma(c)(2)) inside the annular swir ling jet; and (ii) a consolidated swirling jet forms with a reversed ( 'anti-rocket') flow force.