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.

New Exact Solutions of Steady Plane Flows of an In Compressible Fluid of Variable Viscosity

2016 ◽  
Vol 2 (1) ◽  
Author(s):  
Sobia Younus
Keyword(s):  
Exact Solutions ◽  
Stream Function ◽  
Compressible Fluid ◽  
Variable Viscosity ◽  
Plane Motion ◽  
Transformation Technique ◽  
Vorticity Distribution ◽  
New Exact Solutions ◽  
Steady Plane ◽  
Uniform Stream

<span>Some new exact solutions to the equations governing the steady plane motion of an in compressible<span> fluid of variable viscosity for the chosen form of the vorticity distribution are determined by using<span> transformation technique. In this case the vorticity distribution is proportional to the stream function<span> perturbed by the product of a uniform stream and an exponential stream<br /><br class="Apple-interchange-newline" /></span></span></span></span>

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.

On steady vortex rings of small cross-section in an ideal fluid

1970 ◽  
Vol 316 (1524) ◽  
pp. 29-62 ◽  
Keyword(s):  
Cross Section ◽  
Propagation Speed ◽  
Vortex Rings ◽  
Uniform Density ◽  
Inviscid Fluid ◽  
Small Cross Section ◽  
Cross Sectional ◽  
The Core ◽  
Ring Radius ◽  
Explicit Formulae

This paper is concerned with vortex rings, in an unbounded inviscid fluid of uniform density, that move without change of form and with constant velocity when the fluid at infinity is at rest. The work is restricted to rings whose cross-sectional area is small relative to the square of a mean ring radius. An existence theorem is proved for distributions of vorticity in the core that are arbitrary, apart from the condition imposed by the equation of motion and certain smoothness requirements. The method of proof relies on the nearly plane, or two-dimensional, nature of the flow in the neighbourhood of a small cross-section, and leads to approximate but explicit formulae for the propagation speed and shape of the vortex rings in question.

scholarly journals Lie symmetry analysis and invariant solutions of 3D Euler equations for axisymmetric, incompressible, and inviscid flow in the cylindrical coordinates

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
R. Sadat ◽  
Praveen Agarwal ◽  
R. Saleh ◽  
Mohamed R. Ali
Keyword(s):  
Euler Equations ◽  
Inviscid Flow ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Inviscid Fluid ◽  
Analysis Method ◽  
Governing Equations ◽  
Lie Symmetry Analysis ◽  
Cartesian Plane ◽  
Intensive Observation

AbstractThrough the Lie symmetry analysis method, the axisymmetric, incompressible, and inviscid fluid is studied. The governing equations that describe the flow are the Euler equations. Under intensive observation, these equations do not have a certain solution localized in all directions $(r,t,z)$ ( r , t , z ) due to the presence of the term $\frac{1}{r}$ 1 r , which leads to the singularity cases. The researchers avoid this problem by truncating this term or solving the equations in the Cartesian plane. However, the Euler equations have an infinite number of Lie infinitesimals; we utilize the commutative product between these Lie vectors. The specialization process procures a nonlinear system of ODEs. Manual calculations have been done to solve this system. The investigated Lie vectors have been used to generate new solutions for the Euler equations. Some solutions are selected and plotted as two-dimensional plots.

Dynamics of thin vortex rings

2008 ◽  
Vol 609 ◽  
pp. 319-347 ◽  
Author(s):  
IAN S. SULLIVAN ◽  
JOSEPH J. NIEMELA ◽  
ROBERT E. HERSHBERGER ◽  
DIOGO BOLSTER ◽  
RUSSELL J. DONNELLY
Keyword(s):  
Viscous Fluid ◽  
Vortex Ring ◽  
Long Range ◽  
Direct Measurement ◽  
Vortex Rings ◽  
Inviscid Fluid ◽  
Core Size ◽  
Physical Pendulum ◽  
Range Study

As part of a long-range study of vortex rings, their dynamics, interactions with boundaries and with each other, we present the results of experiments on thin core rings generated by a piston gun in water. We characterize the dynamics of these rings by means of the traditional equations for such rings in an inviscid fluid suitably modifying them to be applicable to a viscous fluid. We develop expressions for the radius, core size, circulation and bubble dimensions of these rings. We report the direct measurement of the impulse of a vortex ring by means of a physical pendulum.

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