Singularity formation in three-dimensional vortex sheets

We study singularity formation of three-dimensional (3-D) vortex sheets without surface tension using a new approach. First, we derive a leading order approximation to the boundary integral equation governing the 3-D vortex sheet. This leading order equation captures the most singular contributions of the integral equation. By introducing an appropriate change of variables, we show that the leading order vortex sheet equation degenerates to a two-dimensional vortex sheet equation in the direction of the tangential velocity jump. This change of variables is guided by a careful analysis based on properties of certain singular integral operators, and is crucial in identifying the leading order singular behavior. Our result confirms that the tangential velocity jump is the physical driving force of the vortex sheet singularities. We also show that the singularity type of the three-dimensional problem is similar to that of the two-dimensional problem. Moreover, we introduce a model equation for 3-D vortex sheets. This model equation captures the leading order singularity structure of the full 3-D vortex sheet equation, and it can be computed efficiently using fast Fourier transform. This enables us to perform well-resolved calculations to study the generic type of 3-D vortex sheet singularities. We will provide detailed numerical results to support the analytic prediction, and to reveal the generic form of the vortex sheet singularity

Fully nonlinear interfacial waves in a bounded two-fluid system

We study the nonlinear flow which results when two immiscible inviscid incompressible fluids of different densities and separated by an interface which is free to move and which supports surface tension, are caused to flow in a straight infinite channel. Gravity is taken into consideration and the velocities of each phase can be different, thus giving rise to the Kelvin-Helmholtz instability. Our objective is to study the competing effects of the Kelvin-Helmholtz instability coupled with a stably or unstably stratified fluid system (Rayleigh-Taylor instability) when surface tension is present to regularize the dynamics. Our approach involves the derivation of two-and three-dimensional model evolution equations using long-wave asymptotics and the ensuing analysis and computation of these models. In addition, we derive the appropriate Birkhoff-Rott integro-differential equation for two-phase inviscid flows in channels of arbitrary aspect ratios. A long wave asymptotic analysis is undertaken to develop a theory for fully nonlinear interfacial waves allowing amplitudes as large as the channel thickness. The result is a set of evolution equations for the interfacial shape and the velocity jump across the interface. Linear stability analysis reveals that capillary forces stabilize short-wave disturbances in a dispersive manner and we study their effect on the fully nonlinear dynamics described by our models. In the case of two-dimensional interfacial deflections, traveling waves of permanent form are constructed and it is shown that solitary waves are possible for a range of physical parameters. All solitary waves are expressed implicitly in terms of incomplete elliptic integrals of the third kind. When the upper layer has zero density, two explicit solitary-wave solutions have been found whose amplitudes are equal to h/4 or h/9 where 2h is the channel thickness. In the absence of gravity, solitary waves are not possible but periodic ones are. Numerically constructed traveling and solitary waves are given for representative physical parameters. The initial value problem for the partial differential equations is also addressed numerically in periodic domains, and the regularizing effect of surface tension is investigated. In particular, when surface tension is absent it is shown that the system of governing evolution equations terminates in a singularity after a finite time. This is achieved by studying a 2 x 2 system of nonlinear conservation laws in the complex plane and by numerical solution of the evolution equations. The analysis shows that a sinusoidal perturbation of the flat interface and a cosine perturbation to the unit velocity jump across the interface, develop a singularity at time tc = ln 1/ε+0 (ln(ln 1/ε)) where ε is the initial amplitude of the disturbances. This result is asymptotic for small ε and is derived by studying the asymptotic form of the flow characteristics in the complex plane. We also derive the analogous three-dimensional evolution equations by assuming that the wavelengths in the principal horizontal directions are large compared to the channel thickness. Surface tension is again incorporated to regularize short-wave Kelvin-Helmholtz instabilities and the equations are solved numerically subject to periodic boundary conditions. Evidence of singularity formation is found. In particular, we observe that singularities occur at isolated points starting from general initial conditions. This finding is consistent with numerical studies of unbounded three-dimensional vortex sheets (see Introduction for a discussion and references). In the final part of this work we consider the vortex-sheet formulation of the exact nonlinear two-dimensional flow of a vortex sheet which is bounded in a channel. We derive a Birkhoff-Rott type integro-differential evolution equation for the velocity of the interface in terms of the vorticity as well as the evolution equation for the unnormalized vortex sheet strength. For the case of a spatially periodic vortex sheet, this Birkhoff-Rott type equation is written in terms of Jacobi\u27s functions. The equation is shown to recover the limits of unbounded and non-periodic flows which are known in the literature

Mathematics for 2d Interfaces

We present here a survey of recent results concerning the mathematical analysis of instabilities of the interface between two incompressible, non viscous, fluids of constant density and vorticity concentrated on the interface. This configuration includes the so-called Kelvin-Helmholtz (the two densities are equal), Rayleigh-Taylor (two different, nonzero, densities) and the water waves (one of the densities is zero) problems. After a brief review of results concerning strong and weak solutions of the Euler equation, we derive interface equations (such as the Birkhoff-Rott equation) that describe the motion of the interface. A linear analysis allows us to exhibit the main features of these equations (such as ellipticity properties); the consequences for the full, non linear, equations are then described. In particular, the solutions of the Kelvin-Helmholtz and Rayleigh-Taylor problems are necessarily analytic if they are above a certain threshold of regularity (a consequence is the illposedness of the initial value problem in a non analytic framework). We also say a few words on the phenomena that may occur below this regularity threshold. Finally, special attention is given to the water waves problem, which is much more stable than the Kelvin-Helmholtz and Rayleigh-Taylor configurations. Most of the results presented here are in 2d (the interface has dimension one), but we give a brief description of similarities and differences in the 3d case.Comment: Survey. To appear in Panorama et Synth\`ese

Singularity formation in vortex sheets and interfaces

ISBN: 1-4020-1950-5One of the paradigms of nonlinear science is that patterns result from instability and bifurcation. However, another pathway is possible: self-similar evolution, singularity formation, and form. One example of this process is the formation of spherical drops through the pinch off of a cylindrical thread of liquid. Other example is given by the evolution of a vortex sheet, which from an initial regular shape, develops a finite time singularity of the curvature, resulting in the generation of a spiraling vortex. We investigate some simple systems, a stretched vortex sheet, the free surface of a perfect fluid driven by a vortex dipole, and the splash produced by a convergent capillary wave, in order to illustrate some specific scenarios to the appearance of a “form'' through a singularit

A dam-break driven by a moving source: a simple model for a powder snow avalanche

We study the two-dimensional, irrotational flow of an inviscid, incompressible fluid injected from a line source moving at constant speed along a horizontal boundary, into a second, immiscible, inviscid fluid of lower density. A semi-infinite, horizontal layer sustained by the moving source has previously been studied as a simple model for a powder snow avalanche, Caroll et al. (2012). We show that with fluids of unequal densities, in a frame of reference moving with the source, no steady solution exists, and formulate an initial/boundary value problem that allows us to study the evolution of the flow. After considering the limit of small density difference, we study the fully nonlinear initial/boundary value problem and find that the flow at the head of the layer is effectively a dam-break for the initial conditions that we have used. We study the dynamics of this in detail for small times using the method of matched asymptotic expansions. Finally, we solve the fully nonlinear free boundary problem numerically using an adaptive vortex blob method, after regularising the flow by modifying the initial interface to include a thin layer of the denser fluid that extends to infinity ahead of the source. We find that the disturbance of the interface in the linear theory develops into a dispersive shock in the fully nonlinear initial/boundary value problem and overturns. For sufficiently large Richardson number, overturning can also occur at the head of the layer

Vortex simulations of the Rayleigh–Taylor instability

A vortex technique capable of calculating the Rayleigh–Taylor instability to large amplitudes in inviscid, incompressible, layered flows is introduced. The results show the formation of a steady‐state bubble at large times, whose velocity is in agreement with the theory of Birkhoff and Carter. It is shown that the spike acceleration can exceed free fall, as suggested recently by Menikoff and Zemach. Results are also presented for instability at various Atwood ratios and for fluids having several layers

Complex singularity analysis for vortex layer flows

We study the evolution of a 2D vortex layer at high Reynolds number. Vortex layer flows are characterized by intense vorticity concentrated around a curve. In addition to their intrinsic interest, vortex layers are relevant configurations because they are regularizations of vortex sheets. In this paper, we consider vortex layers whose thickness is proportional to the square-root of the viscosity. We investigate the typical roll-up process, showing that crucial phases in the initial flow evolution are the formation of stagnation points and recirculation regions. Stretching and folding characterizes the following stage of the dynamics, and we relate these events to the growth of the palinstrophy. The formation of an inner vorticity core, with vorticity intensity growing to infinity for larger Reynolds number, is the final phase of the dynamics. We display the inner core's self-similar structure, with the scale factor depending on the Reynolds number. We reveal the presence of complex singularities in the solutions of Navier-Stokes equations; these singularities approach the real axis with increasing Reynolds number. The comparison between these singularities and the Birkhoff-Rott singularity seems to suggest that vortex layers, in the limit, behave differently from vortex sheets

On Tensionless Strings in 3+13+1 Dimensions

We argue for the existence of phase transitions in $3+1$ dimensions associated with the appearance of tensionless strings. The massless spectrum of this theory does not contain a graviton: it consists of one $N=2$ vector multiplet and one linear multiplet, in agreement with the light-cone analysis of the Green-Schwarz string in $3+1$ dimensions. In M-theory the string decoupled from gravity arises when two 5-branes intersect over a three-dimensional hyperplane. The two 5-branes may be connected by a 2-brane, whose boundary becomes a tensionless string with $N=2$ supersymmetry in $3+1$ dimensions. Non-critical strings on the intersection may also come from dynamical 5-branes intersecting the two 5-branes over a string and wrapped over a four-torus. The near-extremal entropy of the intersecting 5-branes is explained by the non-critical strings originating from the wrapped 5-branes.Comment: latex, 16 pages; version to appear in Nucl. Phys.