Vortex Rings with Swirl: Axisymmetric Solutions of the Euler Equations with Nonzero Helicity

1989 ◽  
Vol 20 (1) ◽  
pp. 57-73 ◽  
Author(s):  
Bruce Turkington
Keyword(s):  
Euler Equations ◽  
Vortex Rings ◽  
Axisymmetric Solutions

On the existence of localized rotational disturbances which propagate without change of structure in an inviscid fluid

1986 ◽  
Vol 173 ◽  
pp. 289-302 ◽  
Author(s):  
H. K. Moffatt
Keyword(s):  
Euler Equations ◽  
Stream Function ◽  
Wide Class ◽  
Magnetic Relaxation ◽  
Topological Invariant ◽  
Vortex Rings ◽  
Inviscid Fluid ◽  
Arbitrary Signature ◽  
Axisymmetric Solutions ◽  
Uniform Stream

A wide class of solutions of the steady Euler equations, representing localized rotational disturbances imbedded in a uniform stream U0 is inferred by considering the process of magnetic relaxation to analogous magnetostatic equilibria. These solutions, which may be regarded as generalizations of vortex rings, are characterized by their streamline topology, distinct topologies giving rise to distinct solutions.Particular attention is paid to the class of axisymmetric solutions described by Stokes stream function ψ(s, z). It is argued that the appropriate topological ‘invariant’ characterizing the flow is the function Vψ representing the volume inside toroidal surfaces ψ = const, in the region of closed streamlines where ψ > 0. This function is described as the ‘signature’ of the flow, and it is shown that in a certain sense, flows with different signatures are topologically distinct. The approach yields a method by which flows of arbitrary signature V(ψ) may in principle be found, and the corresponding vorticity ωφ = sFψ calculated.

Shock cavity implosion morphologies and vortical projectile generation in axisymmetric shock–spherical fast/slow bubble interactions

1998 ◽  
Vol 362 ◽  
pp. 327-346 ◽  
Author(s):  
N. J. ZABUSKY ◽  
S. M. ZENG
Keyword(s):  
Mach Number ◽  
Euler Equations ◽  
Mach Disk ◽  
Vortex Rings ◽  
Tvd Scheme ◽  
Downstream Side ◽  
Bubble Interactions ◽  
Vortex Layer ◽  
Mesh Resolution ◽  
Order Of Magnitude

Collapsing shock-bounded cavities in fast/slow (F/S) spherical and near-spherical configurations give rise to expelled jets and vortex rings. In this paper, we simulate with the Euler equations planar shocks interacting with an R12 axisymmetric spherical bubble. We visualize and quantify results that show evolving upstream and downstream complex wave patterns and emphasize the appearance of vortex rings. We examine how the magnitude of these structures scales with Mach number. The collapsing shock cavity within the bubble causes secondary shock refractions on the interface and an expelled weak jet at low Mach number. At higher Mach numbers (e.g. M=2.5) ‘vortical projectiles’ (VP) appear on the downstream side of the bubble. The primary VP arises from the delayed conical vortex layer generated at the Mach disk which forms as a result of the interaction of the curved incoming shock waves that collide on the downstream side of the bubble. These rings grow in a self-similar manner and their circulation is a function of the incoming shock Mach number. At M=5.0, it is of the same order of magnitude as the primary negative circulation deposited on the bubble interface. Also at M=2.5 and 5.0 a double vortex layer arises near the apex of the bubble and moves off the interface. It evolves into a VP, an asymmetric diffuse double ring, and moves radially beyond the apex of the bubble. Our simulations of the Euler equations were done with a second-order-accurate Harten–Yee-type upwind TVD scheme with an approximate Riemann Solver on mesh resolution of 803×123 with a bubble of radius 55 zones.

Axisymmetric solutions to the Euler equations

2007 ◽  
Vol 66 (9) ◽  
pp. 1938-1948
Author(s):  
Shen Gang ◽  
Zhu Xiangrong
Keyword(s):  
Euler Equations ◽  
Axisymmetric Solutions
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