Dependence of Time-periodic Vortex Sheets with Surface Tension on Mean Vortex Sheet Strength

AbstractIn this paper, we show that the only solution of the vortex sheet equation, either stationary or uniformly rotating with negative angular velocity $$\Omega $$ Ω , such that it has positive vorticity and is concentrated in a finite disjoint union of smooth curves with finite length is the trivial one: constant vorticity amplitude supported on a union of nested, concentric circles. The proof follows a desingularization argument and a calculus of variations flavor.

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Collapse of n Point Vortices, Formation of the Vortex Sheets and Transport of Passive Markers

Energies ◽

10.3390/en14040943 ◽

2021 ◽

Vol 14 (4) ◽

pp. 943

Author(s):

Henryk Kudela

Keyword(s):

Differential Equations ◽

Point Vortex ◽

Optimization Procedure ◽

Vortex Sheet ◽

Point Vortices ◽

Vortex Sheets ◽

Impermeable Barrier ◽

Initial Location ◽

Passive Tracers ◽

Initial Iterate

In this paper, the motion of the n-vortex system as it collapses to a point in finite time is studied. The motion of vortices is described by the set of ordinary differential equations that we are able to solve analytically. The explicit formula for the solution demands the initial location of collapsing vortices. To find the collapsing locations of vortices, the algebraic, nonlinear system of equations was built. The solution of that algebraic system was obtained using Newton’s procedure. A good initial iterate needs to be provided to succeed in the application of Newton’s procedure. An unconstrained Leverber–Marquart optimization procedure was used to find such a good initial iterate. The numerical studies were conducted, and numerical evidence was presented that if in a collapsing system n=50 point vortices include a few vortices with much greater intensities than the others in the set, the vortices with weaker intensities organize themselves onto the vortex sheet. The collapsing locations depend on the value of the Hamiltonian. By changing the Hamiltonian values in a specific interval, the collapsing curves can be obtained. All points on the collapse curves with the same Hamiltonian value represent one collapsing system of vortices. To show the properties of vortex sheets created by vortices, the passive tracers were used. Advection of tracers by the velocity induced by vortices was calculated by solving the proper differential equations. The vortex sheets are an impermeable barrier to inward and outward fluxes of tracers. Arising vortex structures are able to transport the passive tracers. In this paper, several examples showing the diversity of collapsing structures with the vortex sheet are presented. The collapsing phenomenon of many vortices, their ability to self organize and the transportation of the passive tracers are novelties in the context of point vortex dynamics.

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Traveling waves from the arclength parameterization: Vortex sheets with surface tension

Interfaces and Free Boundaries ◽

10.4171/ifb/306 ◽

2013 ◽

Vol 15 (3) ◽

pp. 359-380 ◽

Cited By ~ 8

Author(s):

Benjamin Akers ◽

David Ambrose ◽

J. Douglas Wright

Keyword(s):

Surface Tension ◽

Traveling Waves ◽

Vortex Sheets

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Confined vortex surface and irreversibility. 1. Properties of exact solution

International Journal of Modern Physics A ◽

10.1142/s0217751x21500974 ◽

2021 ◽

pp. 2150097

Author(s):

Alexander Migdal

Keyword(s):

Strain Tensor ◽

Special Kind ◽

Local Strain ◽

Vortex Sheet ◽

Exceptional Case ◽

Surface Integral ◽

Time Reversibility ◽

Navier Stokes Equation ◽

Vortex Sheets ◽

Asymmetric Vortex

We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (Migdal, 2021). These surfaces avoid the Kelvin–Helmholtz instability by adjusting their discontinuity and shape. The vorticity collapses to the sheet only in an exceptional case considered long ago by Burgers and Townsend, where it decays as a Gaussian on both sides of the sheet. In generic asymmetric vortex sheets (Shariff, 2021), vorticity leaks to one side or another, making such sheets inadequate for vortex sheet statistics and anomalous dissipation. We conjecture that the vorticity in a turbulent flow collapses on a special kind of surface (confined vortex surface or CVS), satisfying some equations involving the tangent components of the local strain tensor. The most important qualitative observation is that the inequality needed for this solution’s stability breaks the Euler dynamics’ time reversibility. We interpret this as dynamic irreversibility. We have also represented the enstrophy as a surface integral, conserved in the Navier–Stokes equation in the turbulent limit, with vortex stretching and viscous diffusion terms exactly canceling each other on the CVS surfaces. We have studied the CVS equations for the cylindrical vortex surface for an arbitrary constant background strain with two different eigenvalues. This equation reduces to a particular version of the stationary Birkhoff–Rott equation for the 2D flow with an extra nonanalytic term. We study some general properties of this equation and reduce its solution to a fixed point of a map on a sphere, guaranteed to exist by the Brouwer theorem.

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A numerical investigation of the rolling-up of vortex sheets

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1980.0137 ◽

1980 ◽

Vol 373 (1752) ◽

pp. 67-91 ◽

Cited By ~ 32

Keyword(s):

Numerical Integration ◽

Numerical Investigation ◽

Equations Of Motion ◽

Additional Term ◽

Vortex Sheet ◽

Time Step ◽

Vortex Sheets ◽

Numerical Instabilities ◽

Sinusoidal Disturbance ◽

Induced Velocity

In this paper the development of a vortex sheet due to an initially sinusoidal disturbance is calculated. When determining the induced velocity in points of the vortex sheet, it can be represented by concentrated vortices but it is shown that it is analytically more correct to add an additional term that represents the effect of the immediate neighbourhood of the point considered. The equations of motion were integrated by a Runge-Kutta technique to exclude numerical instabilities. The time step was determined by the requirement that a quantity (Hamiltonian) that remains invariant as a result of the equations of motion, should not change more than a certain amount in the numerical integration of the equations of motion. One difficulty is that if a greater number of concentrated vortices are introduced to represent the vortex sheet, the effect of round-off errors becomes more important. The number of figures retained in the computations limits the number of concentrated vortices. Where the round-off errors have been kept sufficiently small, a process of rolling-up of vorticity clearly occurs. There is no point in pursuing the calculations much beyond this point, first because the representation of the vortex sheet by concentrated vortices becomes more and more inaccurate and secondly because viscosity will have the effect of transforming the rolled-up vortex sheet into a region of vorticity.

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Paper 7. Unsteady Flow around a Slender Fish-Like Body

Journal of Mechanical Engineering Science ◽

10.1243/jmes_jour_1972_014_062_02 ◽

1972 ◽

Vol 14 (7) ◽

pp. 43-52 ◽

Cited By ~ 5

Author(s):

Th. Y. Wu ◽

J. N. Newman

Keyword(s):

Unsteady Flow ◽

Theoretical Calculations ◽

Cross Flow ◽

Vortex Sheet ◽

The Body ◽

Dynamic Force ◽

Cross Sectional ◽

Vortex Sheets ◽

Flat Fish ◽

The Cross

This paper attempts to extend some recent theoretical calculations on the unsteady flow generated by body movements of a slender ‘flat’ fish by further including the effect of finite body thickness in the consideration for various configurations of side and caudal fins as major appendages. Based on the slender-body approximation, the cross-flow is determined for different longitudinal body sections which are characterized by a variety of cross-sectional shapes and flow conditions (such as having smooth or fin-edged body contours, with or without vortex sheets alongside the body section). The effect of body thickness is found to arise primarily from its interaction with the vortex sheet already existing in the cross-flow. New results for the transverse hydro-dynamic force acting on the body are obtained, and their physical significances are discussed.

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Fine Structure of Vortex Sheet Rollup by Viscous and Inviscid Simulation

Journal of Fluids Engineering ◽

10.1115/1.2926492 ◽

1991 ◽

Vol 113 (1) ◽

pp. 31-36 ◽

Cited By ~ 44

Author(s):

G. Tryggvason ◽

W. J. A. Dahm ◽

K. Sbeih

Keyword(s):

Reynolds Number ◽

High Reynolds Number ◽

Large Scale ◽

Stokes Equations ◽

Flow Region ◽

Vortex Sheet ◽

Navier Stokes ◽

Vortex Sheets ◽

Limiting Behavior ◽

High Reynolds

Numerical simulations of the large amplitude stage of the Kelvin-Helmholtz instability of a relatively thin vorticity layer are discussed. At high Reynolds number, the effect of viscosity is commonly neglected and the thin layer is modeled as a vortex sheet separating one potential flow region from another. Since such vortex sheets are susceptible to a short wavelength instability, as well as singularity formation, it is necessary to provide an artificial “regularization” for long time calculations. We examine the effect of this regularization by comparing vortex sheet calculations with fully viscous finite difference calculations of the Navier-Stokes equations. In particular, we compare the limiting behavior of the viscous simulations for high Reynolds numbers and small initial layer thickness with the limiting solution for the roll-up of an inviscid vortex sheet. Results show that the inviscid regularization effectively reproduces many of the features associated with the thickness of viscous vorticity layers with increasing Reynolds number, though the simplified dynamics of the inviscid model allows it to accurately simulate only the large scale features of the vorticity field. Our results also show that the limiting solution of zero regularization for the inviscid model and high Reynolds number and zero initial thickness for the viscous simulations appear to be the same.

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