JOURNAL OF COMPUTATIONAL PHYSICS 123, 341–353 (1996) ARTICLE NO. 0028 An Impulse-Based Approximation of Fluid Motion due to Boundary Forces RICARDO CORTEZ Department of Mathematics and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720 October 4, 1994; revised June 19, 1995 order blobs and in which the dipole strengths are updated with the appropriate equation. The motion of an incompressible, inviscid fluid in a region sur- rounded by a massless, elastic membrane can be approximated by McCracken and Peskin [19] applied a vortex-grid hybrid transmitting the effect of the boundary forces to the fluid through method to the study of blood flow through heart valves. vortex dipoles. We present a Lagrangian numerical method for ap- This method made use of a simple vortex layer to introduce proximating this motion based on the impulse (a.k.a. magnetization) the effects of normal boundary forces over each time step variables introduced by Buttke. In particular, we explain the corre- and a layer of discrete vortex dipoles to introduce the spondence between impulse variables and vortex dipoles with a prescribed dipole moment. Numerical examples that illustrate the effects of tangential forces. By a discrete vortex dipole application of impulse variables in this context are given.  1996 we mean a pair of vortex blobs with equal but opposite Academic Press, Inc. strengths a small distance apart. Our method for introducing the effects of elastic forces on the fluid is based on impulse rather than vorticity. This 1. INTRODUCTION feature is attractive since the boundary forces naturally interact with the fluid by imparting impulse and these There are many interesting situations in which a fluid forces are easily accounted for in the method. Furthermore, flows in a region bounded by elastic membranes. Examples the use of a discrete approximation of dipoles is no longer of such situations are air flow inside the lungs, blood flow necessary because the impulse variables in fact represent through the heart, and water sloshing inside a balloon. A dipoles. key feature of these examples is the interaction between Other uses of vortex dipoles or simple vortex layers on the elastic forces that arise on the boundary as the mem- free surfaces are found in [2–4, 14, 16, 24]. Baker, Meiron, brane stretches, and the fluid inside. When the fluid is and Orszag [3, 4] modeled the motion of a periodic free incompressible, these forces immediately affect the motion surface between two incompressible, inviscid fluids of dif- of the entire fluid; in turn, this motion changes the configu- ferent densities using vortex dipoles on the free surface. ration of the boundary which determines the forces. Our No surface forces were included in their application. Their goal is to solve problems with this type of force–fluid inter- method required the formulation of an evolution equation action. for the dipole strengths which was derived from Bernoulli’s One approach for the numerical treatment of the interac- equation. The impusle field in our method equals the dipole tion between boundary forces and fluid uses the forces to strength multiplied by a unit vector normal to the surface, generate vorticity on the boundary and lets the vorticity and so the equation of motion for impulse updates the induce the motion of the fluid (see [20, 19]). Typically the dipole strengths appropriately. The problems illustrated in boundary is idealized as infinitely thin. This gives rise to the present paper feature a single fluid (no density jumps) a situation in which forces are singular since they act on and surface forces. We point out that the methods found a set whose dimension is lower than the spatial dimension in [3, 4, 14, 24] use desingularizing techniques to evaluate of the problem. The singularity of the force field has been the principal value integrals for the velocity at the interface a source of instabilities (see [19]). Mendez [20] tracked the and the vorticity (or dipole) strength. The method adopted motion of an elastic ellipse immersed in an inviscid fluid here is to use blobs in the discretization of the integrals. by introducing pairs of vortex blobs along the boundary Buttke [7] recently proposed a Lagrangian numerical to approximate dipoles. The blobs used were of low order method for incompressible Euler flow which uses a compu- and questions regarding changes of the dipole strengths tational variable that he called velicity. The method is remained unresolved. In the present paper we improve on these ideas by presenting a method which uses higher- based on the Hamiltonian formulation proposed by Osele- 341 0021-9991/96 $18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.