Hysteresis in swirling jets

This paper explains hysteretic transitions in swirling jets and models external flows of vortex suction devices. Toward this goal, the steady rotationally symmetric motion of a viscous incompressible fluid above an infinite conical stream surface of a half-angle θ_c is studied. The flows analysed are generalizations of Long's vortex. They correspond to the conically similar solutions of the Navier-Stokes equations and are characterized by circulation Γ_c given at the surface and axial flow force J₁. Asymptotic analysis and numerical calculations show that four (for θ_c ≤ 90°) or five (for θ_c > 90°) solutions exist in some range of Γ_c and J₁. The solution branches form hysteresis loops which are related 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 θ = θ_s < θ_c ; (ii) vortex consolidation causing a reversal of (i) ; (iii) jump flow separation from surface θ = θ_c; and (iv) jump attachment of the swirling jet to the surface. As Γ_c and/or J₁ decrease, the hysteresis loops disappear through a cusp catastrophe. The physical reasons for the solution non-uniqueness are revealed and the results are discussed 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(Γ_c²) inside the annular swirling jet ; and (ii) a consolidated swirling jet forms with a reversed ('anti-rocket') flow force.