On the ‘δ-equations’ for vortex sheet evolution (original) (raw)

The long-time motion of vortex sheets with surface tension

Physics of Fluids, 1997

We study numerically the simplest model of two incompressible, immiscible fluids shearing past one another. The fluids are two-dimensional, inviscid, irrotational, density matched, and separated by a sharp interface under a surface tension. The nonlinear growth and evolution of this interface is governed by only the competing effects of the Kelvin-Helmholtz instability and the dispersion due to surface tension. We have developed new and highly accurate numerical methods designed to treat the difficulties associated with the presence of surface tension. This allows us to accurately simulate the evolution of the interface over much longer times than has been done previously. A surprisingly rich variety of behavior is found. For small Weber numbers, where there are no unstable length-scales, the flow is dispersively dominated and oscillatory behavior is observed. For intermediate Weber numbers, where there are only a few unstable length-scales, the interface forms elongating and interpenetrating fingers of fluid. At larger Weber numbers, where there are many unstable scales, the interface rolls-up into a ''Kelvin-Helmholtz'' spiral with its late evolution terminated by the collision of the interface with itself, forming at that instant bubbles of fluid at the core of the spiral. Using locally refined grids, this singular event ͑a ''topological'' or ''pinching'' singularity͒ is studied carefully. Our computations suggest at least a partial conformance to a local self-similar scaling. For fixed initial data, the pinching singularity times decrease as the surface tension is reduced, apparently towards the singularity time associated with the zero surface tension problem, as studied by Moore and others. Simulations from more complicated, multi-modal initial data show the evolution as a combination of these fingers, spirals, and pinches.

The collapse of an axi-symmetric, swirling vortex sheet

Nonlinearity, 1993

An ax-symmetric and swirling vortex sheet is investigated as the simplest flow in which there is non-trivial vortex stretching and as a possible setting for studying vortex cancellation and singularity formation. Rayleigh's criterion indicates linear stabiliry of a single sheet but instability for other configurations of sheets. Due to the simplicity of vortex sheet problems, the linear modes and growth rates (or frequencies) can be explicitly expressed. Subsequent nonlinear evolution is numerically simulated using a vortex method. The numerical results for an ai-symmetric swirling sheet with a vortex line along the axis of symmetry show detlchment of a vortex ring from the sheet into the outer fluid, and Follapse of the sheet onto t 2?c vortex line at some points. Vortex cancellation, which in the presence of viscosity would likely lead to vortex line reconnection. seems to occur in both of these phenomena. The evolution of two co-axial, axi-symmetric, swirling vortex sheets is similar.

Singular Solutions and Ill-Posedness for the Evolution of Vortex Sheets

SIAM Journal on Mathematical Analysis, 1989

The evolution of a planar vortex sheet is described by the Birkhott-Rott equation. Duchon and Robert [C.R. Acad. Sci. Paris, 302 (1986), pp. 183-186], [Comm. Partial Ditterential Equations, 13 (1988), pp. 1265-1295 have constructed exact solutions of this equation that are analytic for all < 0 but have a possible singularity in the curvature of the sheet at 0. This shows that smooth initial data for a vortex sheet can lead to singularity formation at a finite time, in agreement with the results of numerical computation [We present an independent construction of these solutions and use these results to infer that the vortex sheet problem is ill-posed in Sobolev class Hn with n > 3/2.

Stable Methods for Vortex Sheet Motion in the Presence of Surface Tension

SIAM Journal on Scientific Computing, 1998

Boundary integral techniques provide a convenient way to study the evolution of an interface between inviscid liquids. Several studies have revealed that standard numerical approximations tend to lead to unstable methods, and various remedies have been introduced and tested. In this paper, we conduct a stability analysis of the linearized equations with a particular objective in mind | the determination of how the discrete system fails to capture the physical dispersion relation precisely for the available discrete modes. We discover two reasons for the typical failure in numerical discretizations: one is the inability of the mesh to represent the vorticity created by surface tension e ects on the nest scale; and the other is the inaccuracies in the evaluation of the boundary integral for the velocity. With the insight gained from our linear analysis, we propose a new method that is spectrally accurate and linearly stable. Further, the exact dispersion relation is obtained for all the available discrete modes. Numerical tests suggest that the method is also stable in the nonlinear regime. However, our method runs into di culties generic to methods based on Lagrangian motion. The markers accumulate near a stagnation point on the interface, forcing us to use an ever decreasing timestep in our explicit method. We introduce a redistribution of markers to overcome this di culty. When we redistribute according to equal arclength, we nd excellent agreement with a method based on preserving equal spacing in arclength.

Geometry and dynamics of vortex sheets in 3 dimension

Theoretical and Applied Mechanics, 2002

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 the evolution of a singular vortex patch in a two-dimensional incompressible fluid flow

Computer Physics Communications, 1999

The angle evolution of a tangent-slope discontinuity on a singular vortex patch, i.e. a patch of vorticity with tangent discontinuities in its boundary, is studied from a numerical and theoretical point of view. Different numerical examples show that the angle shrinks for an initial angle less than 90 • , the angle widens when the initial angle is greater than 90 • or is approximately preserved for 90 • for small time evolution. An asymptotic expansion of the initial velocity field near a singularity is performed for a class of singular vortex patches to reinforce analytically these results.

On the formation of small-time curvature singularities in vortex sheets

IMA Journal of Applied Mathematics

The Kelvin-Helmholtz model for the evolution of an infinitesimally thin vortex sheet in an inviscid fluid is mathematically ill-posed for general classes of initial conditions. However, if the initial data, say imposed at t = 0, is in a certain class of analytic functions then the problem is well-posed for a finite time until a singularity forms, say at t = t s , on the vortex-sheet interface, e.g. as illustrated by Moore (1979). However, if the problem is analytically continued into the complex plane, then the singularity, or singularities, exist for t < t s away from the physical real axis. More specifically, Cowley et al. (1999) found that for a class of analytic initial conditions, singularities can form in the complex plane at t = 0+. They posed asymptotic expansions in the neighbourhood of these singularities for 0 < t 1, and found numerical solutions to the governing similarity differential equations. In this paper we obtain new exact solutions to these equations, show that the singularities always correspond to local 3 2-power singularities, and determine both the number and precise locations of all branch points. Further, our analytical approach can be extended to a more general class of initial conditions. These new exact solutions can assist in resolving the small-time behaviour for the numerical solution of the Birkhoff-Rott equations.

Analytic structure of vortex sheet dynamics. Part 1. Kelvin–Helmholtz instability

Journal of Fluid Mechanics, 1982

The instabilit,y of an initially flat vortex sheet to a sinusoidal perturbation of the vorticity is studied by means of high-order Taylor series in time t. All finite-amplitude corrections are retained at each order in t. Our analysis indicates that the sheet develops a curvature singularity at t = t, < 00. The variation oft, with the amplitude a of the perturbation vorticity is in good agreement with the asymptotic results of Moore. When a is O(I), the Fourier coefficient of order n decays slightly faster than predicted by Moore. Extensions of the present prototype of Kelvin-Helmholtz instability to other layered flows, such as Rayleigh-Taylor instability, are indicated.

Kaplanski, F.; Rudi, U. Evolution of a viscous vortex ring

Evolution of a viscous vortex ring, 2001

The velocity field inside a viscous vortex ring and its overall velocity are obtained on the basis of an existing solution of the Stokes equation for the vorticity in a moving coordinate system. It is shown that these characteristics of the motion relate the results obtained earlier for limiting times and are confirmed by the experimental data. The effect of the initial Reynolds number on the development of the vortex ring is analyzed by constructing entrainment diagrams.