### Abstract

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 J_{1}. 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 J_{1}. 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 J_{1} 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_{c}^{2}) inside the annular swirling jet ; and (ii) a consolidated swirling jet forms with a reversed ('anti-rocket') flow force.

Original language | English |
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Pages (from-to) | 1-44 |

Number of pages | 44 |

Journal | Journal of Fluid Mechanics |

Volume | 309 |

DOIs | |

State | Published - Feb 25 1996 |

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## Cite this

*Journal of Fluid Mechanics*,

*309*, 1-44. https://doi.org/10.1017/S0022112096001541