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J. Fluid Mech. (1996), vol. 309, pp. 1-44 Copyright © 1996 Cambridge University Press Hysteresis in swirling jets By VLADIMIR SHTERN AND FAZLE HUSSAIN Department of Mechanical Engineering, University of Houston, Houston, TX 77204-4792, USA (Received 4 October 1994 and in revised form 16 September 1995) 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 6. 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 I’, given at the surface and axial flow force J,. Asymptotic analysis and numerical calculations show that four (for 6, < 90°) or five (for 4, > 90°) solutions exist in some range of I’, and J,. 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 0 = 6, < 6,; (ii) vortex consolidation causing a reversal of (i); (iii) jump flow separation from surface 6 = 6,; and (iv) jump attachment of the swirling jet to the surface. As I, and/or J, 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(T?) inside the annular swirling jet; and (ii) a consolidated swirling jet forms with a reversed (‘anti-rocket’) flow force. CONTENTS 1. Introduction . Observations of bi-stability in swirling flows . Models of vortex suction devices . Flow force as a control parameter . Limitations of Long’s model . Hysteresis loops 2. Problem formulation 2.1. Reduction of NSE to ODE 2.2. Boundary conditions 2.3. The flow force 3. Asymptotic analysis 3.1. Reduced Euler equations 3.2. Two-cell flow 3.3. Regime with near-surface outflow V. Shtern and F. Hussain 3.4. Outer flow of the near-axis jet 3.5. Near-axis jet 3.6. Summary of the asymptotic analysis . Swirling flows in cones 4.1. Flow regimes inside the 0, = 45° cone 4.2. Flow regimes outside the 45° cone . A flow driven by a half-line vortex 5.1. Modification of the flow force 5.2. Hysteresis 5.3. Cusp . Pressure peak in swirling annular jets . Discussion 7.1. The physical nature of the jumps 7.2. Dynamics of stagnation points 7.3. Folds in the context of vortex breakdown theories . Conclusions . Introduction 1.1. Observations of bi-stability in swirling flows Swirling flows in nature and technology have many striking and even enigmatic features that need thorough analysis and interpretation. In addition to the well-known examples of intriguing phenomena such as the Ranque-Hilsch effect and vortex breakdown, several new ones are revealed here: jump separation and attachment of near-surface swirling flows; a strong peak of pressure inside annular swirling jets; and a paradoxical, upstream-directed flow force in a consolidated swirling jet. These intriguing features are not only of intrinsic scientific interest but also very important for technological applications. Perhaps the most problematic feature of swirling flows is their bi- or multi-stability which can lead to abrupt transitions between different states, occurring at the same values of control parameters. Similar bi-stability is observed in tornadoes (Burggraf & Foster 1977; Shtern & Hussain 1993, hereinafter referred to as SH), vortices above delta wings (Schmucker & Gersten 1988) as well as in the inner and outer flow domains of vortex devices (Goldshtik 1990; Spotar’ & Terekhov 1987). These abrupt or jump transitions are most dangerous for aircraft because the resultant changes in lift and drag can lead to a loss of flight control. This has been the main practical motivation for numerous studies of vortex breakdown since its discovery by Peckham & Atkinson 1957). Although not necessarily as dangerous, several highly undesirable sequences of transitions between flow states occur in various vortex devices. Figure 1 (a) shows a schematic of a ‘focused’ suction device (Boguslavskii & Ivanskii 1987) to remove hazardous fumes or aerosols generated at a distant spot. Fresh air pumped through guidevane | forms swirling outflow 2, turns around near stagnation point 3, collects hazardous fumes from pollution source 4, and is sucked out through axial pipe 5.