Stable methods for vortex sheet motion in the presence of su

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 int

STABLE METHODS FOR VORTEX SHEET MOTION IN THE PRESENCE OF SURFACE TENSIONGREGORY BAKERy AND ANDRE NACHBINz 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.

Abstract.

Key words. Vortex Sheets, Surface Tension, Stable Numerical Methods. AMS subject classi cations. Subject Classi cation: 65M99,65R20,76C10

1. Introduction.. Boundary integral techniques have gained wide popularity as an e cient approach to studying the evolution of interfaces between incompressible liquids. In particular for inviscid ows, methods based on representing the interface by a dipole or vortex sheet have been used to study Rayleigh{Taylor instabilities 7, 21, 26, 33, 36]; the motion of bubbles or drops 9, 35]; and the motion of water waves and internal waves 8, 12, 13, 18, 26, 27, 28]. Although there are di erences in these methods, they all use markers to represent the interface and simple approximations to the boundary integrals to determine the velocity of the markers. Bernoulli's equation, or variants of it, is used to update the velocity potential, dipole sheet strength, or the vortex sheet strength along the interface. When the interface lies between a liquid and vacuum (the special limit of a liquid with vanishing density), the evidence from most numerical simulations is that the motion is reasonably well behaved 6]. In contrast, the typical motion of an interface between inviscid liquids of nonvanishing density su ers from the rapid formation of curvature singularities in the absence of surface tension e

ects. The Kelvin{Helmholtz instability is the underlying mechanism which causes such singularities to form 20, 29, 25, 6]. As the interface moves, there will be regions where the liquids ow with di erent speeds on either side of the interface. On a local level, these regions appear as vortex sheets with almost uniform strength. In the absence of stabilizing e ects, these regions su er from Kelvin{Helmholtz instability, and form curvature singularities in nite time.This work was partially supported by the National Science Foundation under contract DMS9005932, and by grant of computing resources at the Ohio Supercomputer Center y Department of Mathematics, The Ohio State University, Columbus, Ohio, USA. z IMPA, Rio de Janeiro, Brazil. 1

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 int

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Gregory Baker and Andre Nachbin

Several asymptotic studies 23, 14] provide an explanation for the origin of these singularities on a periodic vortex sheet, while others 25, 6] show clearly the connection between curvature singularities in interfacial ows with those in vortex sheets. Numerical studies 20, 29] con rm the ideas behind the asymptotic predictions. In these numerical studies, the behavior of the singularity is determined by a study of the Fourier spectrum of the location of the vortex sheet, which requires the evolution of the Fourier spectrum to be calculated very accurately. Typically, time steps in the range 10?3{10?5 are used with fourth order Runge{Kutta or predictor-corrector methods. Presumably, the inclusion of surface tension e ects will prevent the formation of curvature singularities. Historically, this belief is based on the behavior of in nitesimal perturbations to a at interface between two liquids streaming past each other 15]. Above a critical wavenumber, surface tension prevents the growth of sinusoidal perturbations, causing them to oscillate instead. The next step in understanding the e ects of weak surface tension is the study of the long time evolution of the interface as the low modes grow into signi cant perturbations. Two recent numerical calculations 27, 18] show that instead of a curvature singularity, the vortex sheet starts to twist and create arms of a spiral. In 18], the arms show oscillations, while in 27] they do not, but there is overwhelming evidence in 18] that their calculations are more accurate. Only a few arms are created, however, before the innermost arms of the spiral appear to touch each o …… 此处隐藏：49197字，全部文档内容请下载后查看。喜欢就下载吧 ……