Generalized vortex methods for free-surface flow problems

Baker, Greg; Meiron, D. I.; Orszag, Steven A.

doi:10.1017/s0022112082003164

Cited by 344 publications

(315 citation statements)

References 2 publications

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“…If s~ initially satisfies the constraint (6), this choice for T maintains that constraint in time. Here, T(0, t) is simply an arbitrary change of frame that is taken to be 0.…”

Section: Of F~n T(at)=t(ot)+-ej ° Oda'-e O]a~ ' (7)mentioning

confidence: 99%

“…1 2~ d~'=lL(t), s~(g, t) = ~ fo s,,(0~', t) (6) Fig. 1, which shows the simulation of a gas bubble expanding into a Hele-Shaw fluid (see [ 15,16]) over long times.…”

Section: Of F~n T(at)=t(ot)+-ej ° Oda'-e O]a~ ' (7)mentioning

confidence: 99%

“…Due to the presence ofy in the velocity W, Eq. (20) is a Fredholm integral equation of the second kind for 7, and is, in general, uniquely solveable (see [6]). Further, from Eq.…”

Section: Hele-shaw Flowsmentioning

confidence: 99%

“…Thus, our new numerical methods have no high order time step stability constraints that are usually associated with surface tension. This approach is based on the boundary integral formulation [6] and applies more generally, even to problems beyond the fluid mechanical context. Here, they are applied to computing the motion of interfaces in Hele-Shaw flow, which is quasi two-dimensional and viscously dominated, and to computing the motion of free surfaces in inviscid, inertially dominated flows governed by the Euler equations.…”

Section: Introductionmentioning

confidence: 99%

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Removing the stiffness from interfacial flows with surface tension

Hou

Lowengrub

Shelley

1994

Journal of Computational Physics

495

722

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A new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two-dimensional, irrotational, and incompressible fluids. Through the Laplace-Young condition at the interface, surface tension introduces high-order terms, both nonlinear and nonlocal, into the dynamics. This leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult. This new formulation has all the nice properties for time integration methods that are associated with having a linear, constant coefficient, highest order term. That is, using this formulation, we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in Fourier space. The approach is based on a boundary integral formulation and applies more generally, even to problems beyond the fluid mechanical context. Here they are applied to computing with high resolution the motion of interfaces in Hele-Shaw flows and the motion of free surfaces in inviscid flows governed by the Euler equations. One Hele-Shaw computation shows the behavior of an expanding gas bubble over long-time as the interface undergoes successive tip-splittings and finger competition. A second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification. In Euler flows, the computation of a vortex sheet shows its roll-up through the Kelvin-Helmholtz instability. This motion culminates in the late time self-intersection of the interface, creating trapped bubbles of fluid. This is, we believe, a type of singularity formation previously unobserved for such flows in 2D. Finally, computations of falling plumes in an unstably stratified Boussinesq fluid show a very similar behavior.

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“…If s~ initially satisfies the constraint (6), this choice for T maintains that constraint in time. Here, T(0, t) is simply an arbitrary change of frame that is taken to be 0.…”

Section: Of F~n T(at)=t(ot)+-ej ° Oda'-e O]a~ ' (7)mentioning

confidence: 99%

“…1 2~ d~'=lL(t), s~(g, t) = ~ fo s,,(0~', t) (6) Fig. 1, which shows the simulation of a gas bubble expanding into a Hele-Shaw fluid (see [ 15,16]) over long times.…”

Section: Of F~n T(at)=t(ot)+-ej ° Oda'-e O]a~ ' (7)mentioning

confidence: 99%

“…Due to the presence ofy in the velocity W, Eq. (20) is a Fredholm integral equation of the second kind for 7, and is, in general, uniquely solveable (see [6]). Further, from Eq.…”

Section: Hele-shaw Flowsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Removing the stiffness from interfacial flows with surface tension

Hou

Lowengrub

Shelley

1994

Journal of Computational Physics

495

722

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show abstract

“…[17]), boundary integrals (see e.g. [4][5][6]), operator expansions [14,26,27,31], and Fourier series with domain flattening changes of variables [28]. The computations presented here are based on operator expansions [14] and have given us spectrally accurate results within the parameter ranges given in this paper.…”

Section: Equations Of Motionmentioning

confidence: 99%

On hexagonal gravity water waves

Nicholls

2001

Mathematics and Computers in Simulation

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In this paper we produce numerical, genuinely three-dimensional, hexagonal traveling wave solutions of the Euler equations for water waves using a surface integral formulation derived by Craig and Sulem. These calculations are free from the requirements of either long wavelength or two-dimensionality, both of which are crucial to the KdV and KP scaling regimes, and we produce hexagonal traveling waves of not only small but also moderate amplitude.

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On the accuracy of the desingularized boundary integral method in free surface flow problems

Lalli

1997

Int. J. Numer. Meth. Fluids

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Generalized vortex methods for free-surface flow problems

Cited by 344 publications

References 2 publications

Removing the stiffness from interfacial flows with surface tension

Removing the stiffness from interfacial flows with surface tension

On hexagonal gravity water waves

On the accuracy of the desingularized boundary integral method in free surface flow problems

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