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

1998

Math. Comp.

Abstract. Boundary integral methods to simulate interfacial flows are very sensitive to numerical instabilities. In addition, surface tension introduces nonlinear terms with high order spatial derivatives into the interface dynamics. This makes the spatial discretization even more difficult and, at the same time, imposes a severe time step constraint for stable explicit time integration methods.A proof of the convergence of a reformulated boundary integral method for two-density fluid interfaces with surface tension is presented. The method is based on a scheme introduced by Hou, Lowengrub and Shelley [ J. Comp. Phys. 114 (1994), pp. 312-338] to remove the high order stability constraint or stiffness. Some numerical filtering is applied carefully at certain places in the discretization to guarantee stability. The key of the proof is to identify the most singular terms of the method and to show, through energy estimates, that these terms balance one another.The analysis is at a time continuous-space discrete level but a fully discrete case for a simple Hele-Shaw interface is also studied. The time discrete analysis shows that the high order stiffness is removed and also provides an estimate of how the CFL constraint depends on the curvature and regularity of the solution.The robustness of the method is illustrated with several numerical examples. A numerical simulation of an unstably stratified two-density interfacial flow shows the roll-up of the interface; the computations proceed up to a time where the interface is about to pinch off and trapped bubbles of fluid are formed. The method remains stable even in the full nonlinear regime of motion. Another application of the method shows the process of drop formation in a falling single fluid.

“…E T can be computed with spectral accuracy [4]. In our computations, we monitor the number of digits in the fractional change of energy, that is,…”

Section: Numerical Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convergence of a non-stiff boundary integral method for interfacial flows with surface tension

Ceniceros

1998

Math. Comp.

“…21 The convergence of this method has been established by Ceniceros and Hou 22 for a general two-fluid interface. The stability of boundary integral methods for vortex sheets with surface tension has also been investigated by Baker and Nachbin 23 based on Fourier analysis near equilibrium solutions.…”

Section: Introductionmentioning

confidence: 99%

Dynamic generation of capillary waves

Ceniceros

1999

Physics of Fluids

We investigate the dynamic generation of capillary waves in two-dimensional, inviscid, and irrotational water waves with surface tension. It is well known that short capillary waves appear in the forward front of steep water waves. Although various experimental and analytical studies have contributed to the understanding of this physical phenomenon, the precise mechanism that generates the dynamic formation of capillary waves is still not well understood. Using a numerically stable and spectrally accurate boundary integral method, we perform a systematic study of the time evolution of breaking waves in the presence of surface tension. We find that the capillary waves originate near the crest in a neighborhood, where both the curvature and its derivative are maximum. For fixed but small surface tension, the maximum of curvature increases in time and the interface develops an oscillatory train of capillary waves in the forward front of the crest. Our numerical experiments also show that, as time increases, the interface tends to a possible formation of trapped bubbles through self-intersection. On the other hand, for a fixed time, as the surface tension coefficient is reduced, both the capillary wavelength and its amplitude decrease nonlinearly. The interface solutions approach the ϭ0 profile. At the onset of the capillaries, the derivative of the convection is comparable to that of the gravity term in the dynamic boundary condition and the surface tension becomes appreciable with respect to these two terms. We find that, based on the ϭ0 wave, it is possible to estimate a threshold value 0 such that if р 0 then no capillary waves arise. On the other hand, for sufficiently large, breaking is inhibited and pure capillary motion is observed. The limiting behavior is very similar to that in the classical KdV equation. We also investigate the effect of viscosity on the generation of capillary waves. We find that the capillary waves still persist as long as the viscosity is not significantly greater than surface tension.

“…As a consequence, this avoids the difficulty of differentiating discontinuous fluid quantities across the fluid interface, and makes it possible to design high-order discretizations for the governing equations. On the other hand, it is also well known that boundary integral methods are very sensitive to numerical instabilities [10,5,2,3,7].…”

Section: Introductionmentioning

confidence: 99%

A new stabilizing technique for boundary integral methods for water waves

Zhang

2000

Math. Comp.

Abstract. Boundary integral methods to compute interfacial flows are very sensitive to numerical instabilities. A previous stability analysis by Beale, Hou and Lowengrub reveals that a very delicate balance among terms with singular integrals and derivatives must be preserved at the discrete level in order to maintain numerical stability. Such balance can be preserved by applying suitable numerical filtering at certain places of the discretization. While this filtering technique is effective for two-dimensional (2-D) periodic fluid interfaces, it does not apply to nonperiodic fluid interfaces. Moreover, using the filtering technique alone does not seem to be sufficient to stabilize 3-D fluid interfaces.Here we introduce a new stabilizing technique for boundary integral methods for water waves which applies to nonperiodic and 3-D interfaces. A stabilizing term is added to the boundary integral method which exactly cancels the destabilizing term produced by the point vortex method approximation to the leading order. This modified boundary integral method still has the same order of accuracy as the point vortex method. A detailed stability analysis is presented for the point vortex method for 2-D water waves. The effect of various stabilizing terms is illustrated through careful numerical experiments.