The self-induced motion of vortex sheets

1985 ◽  
Vol 150 ◽  
pp. 203-231 ◽  
Author(s):  
J. J. L. Higdon ◽  
C. Pozrikidis
Keyword(s):  
Vortex Sheet ◽  
The Self ◽  
Trigonometric Polynomials ◽  
Infinite Plane ◽  
Vortex Sheets ◽  
Periodic Disturbances ◽  
Circular Arcs ◽  
Special Cases ◽  
Induced Motion ◽  
Circulation Distribution

A method is presented for following the self-induced motion of vortex sheets. In this method, we use a piecewise analytic representation of the sheet consisting of circular arcs with trigonometric polynomials for the circulation. The procedure is used to study the evolution of the motion in two special cases: a circular vortex sheet with sinusoidal circulation distribution and an infinite plane vortex sheet subject to periodic disturbances. The first problem was studied by Baker (1980) as a test of the method of Fink & Soh (1978), while the second has been studied by a number of authors, notably Meiron, Baker & Orszag (1982). In each case, we follow the motion of the sheet up to the appearance of a singularity at a finite time. The singularity takes the form of an exponential spiral with the simultaneous development of singularities in the curvature and in the circulation distribution. In the final stages of the calculations, up to 155 marker points are used to specify the position of the sheet. If it were possible to execute a stable calculation with equally spaced point vortices, approximately 106 points would be required to achieve the same resolution. Problems with instabilities have been reduced, but not entirely eliminated, and prevent a rigorous verification of the results obtained.

Vortex sheet roll-up revisited

2018 ◽  
Vol 855 ◽  
pp. 299-321 ◽  
Author(s):  
A. C. DeVoria ◽  
K. Mohseni
Keyword(s):  
Classical Problem ◽  
Critical Time ◽  
Vortex Sheet ◽  
The Self ◽  
Physical Consideration ◽  
Considerable Difficulty ◽  
Circulation Distribution ◽  
Continuous Object ◽  
Induced Velocity ◽  
Vorticity Transport

The classical problem of roll-up of a two-dimensional free inviscid vortex sheet is reconsidered. The singular governing equation brings with it considerable difficulty in terms of actual calculation of the sheet dynamics. Here, the sheet is discretized into segments that maintain it as a continuous object with curvature. A model for the self-induced velocity of a finite segment is derived based on the physical consideration that the velocity remain bounded. This allows direct integration through the singularity of the Birkhoff–Rott equation. The self-induced velocity of the segments represents the explicit inclusion of stretching of the sheet and thus vorticity transport. The method is applied to two benchmark cases. The first is a finite vortex sheet with an elliptical circulation distribution. It is found that the self-induced velocity is most relevant in regions where the curvature and the sheet strength or its gradient are large. The second is the Kelvin–Helmholtz instability of an infinite vortex sheet. The critical time at which the sheet forms a singularity in curvature is accurately predicted. As observed by others, the vortex sheet strength forms a finite-valued cusp at this time. Here, it is shown that the cusp value rapidly increases after the critical time and is the impetus that initiates the roll-up process.

scholarly journals Geometry and dynamics of vortex sheets in 3 dimension

2002 ◽  
pp. 55-76 ◽  
Author(s):  
D.A. Burton ◽  
R.W. Tucker
Keyword(s):  
Fluid Velocity ◽  
Normal Component ◽  
Vortex Sheet ◽  
The Self ◽  
Vortex Filament ◽  
Vortex Sheets ◽  
Filament Dynamics ◽  
De Rham ◽  
Induced Velocity

We consider the properties and dynamics of vortex sheets from a geometrical, coordinate-free, perspective. Distribution-valued forms (de Rham currents) are used to represent the fluid velocity and vorticity due to the vortex sheets. The smooth velocities on either side of the sheets are solved in terms of the sheet strengths using the language of double forms. The classical results regarding the continuity of the sheet normal component of the velocity and the conservation of vorticity are exposed in this setting. The formalism is then applied to the case of the self-induced velocity of an isolated vortex sheet. We develop a simplified expression for the sheet velocity in terms of representative curves. Its relevance to the classical Localized Induction Approximation (LIA) to vortex filament dynamics is discussed. .

On a generalization of Kaden's problem

1981 ◽  
Vol 104 ◽  
pp. 45-53 ◽  
Author(s):  
D. I. Pullin ◽  
W. R. C. Phillips
Keyword(s):  
Velocity Field ◽  
Power Law ◽  
Vortex Sheet ◽  
Vortex Sheets ◽  
Parabolic Distribution ◽  
General Power ◽  
Circulation Distribution ◽  
Field Singularity

Kaden's problem of the roll-up of an initially planar semi-infinite vortex sheet with a parabolic distribution of circulation is extended to include vortex sheets exhibiting a general power law circulation distribution, resulting in the presence of a power law, and in one case a logarithmic-like, velocity-field singularity. Both semi-infinite and infinite initially plane sheets with this property are considered and the form of their roll-up in the similarity plane, into single and double-branched spirals respectively, is obtained numerically. Estimates of the Betz constant obtained from the solutions are found to be significantly different from values predicted by the Betz approximation.

Bayesian Prediction in Hybrid Conjoint Analysis

2002 ◽  
Vol 39 (2) ◽  
pp. 253-261 ◽  
Author(s):  
Frenkel Ter Hofstede ◽  
Youngchan Kim ◽  
Michel Wedel
Keyword(s):  
Prior Information ◽  
Current Model ◽  
The Self ◽  
Attribute Importance ◽  
Individual Level ◽  
Special Cases ◽  
Full Profile ◽  
Credible Intervals ◽  
Competing Models ◽  
The Individual

The authors propose a general model that includes the effects of discrete and continuous heterogeneity as well as self-stated and derived attribute importance in hybrid conjoint studies. Rather than use the self-stated importances as prior information, as has been done in several previous approaches, the authors consider them data and therefore include them in the formulation of the likelihood, which helps investigate the relationship of self-stated and derived importances at the individual level. The authors formulate several special cases of the model and estimate them using the Gibbs sampler. The authors reanalyze Srinivasan and Park's (1997) data and show that the current model predicts real choices better than competing models do. The posterior credible intervals of the predictions of models with the different heterogeneity specifications overlap, so there is no clear superior specification of heterogeneity. However, when different sources of data are used—that is, full profile evaluations, self-stated importances, or both—clear differences arise in the accuracy of predictions. Moreover, the authors find that including the self-stated importances in the likelihood leads to much better predictions than does considering them prior information.

scholarly journals Remarks on Stationary and Uniformly-rotating Vortex Sheets: Rigidity Results

2021 ◽  
Author(s):  
Javier Gómez-Serrano ◽  
Jaemin Park ◽  
Jia Shi ◽  
Yao Yao
Keyword(s):  
Angular Velocity ◽  
Disjoint Union ◽  
Vortex Sheet ◽  
Vortex Sheets ◽  
Smooth Curves ◽  
Rigidity Results ◽  
Positive Vorticity ◽  
Constant Vorticity ◽  
Concentric Circles ◽  
Finite Disjoint Union

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.

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

Energies ◽  
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.

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

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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