Generalized vortex methods for free-surface flow problems
TL;DR: In this paper, the motion of free surfaces in incompressible, irrotational, inviscid layered flows is studied by evolution equations for the position of the free surfaces and appropriate dipole (vortex) and source strengths.
Abstract: The motion of free surfaces in incompressible, irrotational, inviscid layered flows is studied by evolution equations for the position of the free surfaces and appropriate dipole (vortex) and source strengths. The resulting Fredholm integral equations of the second kind may be solved efficiently in both storage and work by iteration in both two and three dimensions. Applications to breaking water waves over finite-bottom topography and interacting triads of surface and interfacial waves are given.
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TL;DR: In this paper, the authors consider the formation of droplet clouds or sprays that subsequently burn in combustion chambers, which is caused by interfacial instabilities, such as the Kelvin-Helmholtz instability.
Abstract: The numerical simulation of flows with interfaces and free-surface flows is a vast topic, with applications to domains as varied as environment, geophysics, engineering, and fundamental physics. In engineering, as well as in other disciplines, the study of liquid-gas interfaces is important in combustion problems with liquid and gas reagents. The formation of droplet clouds or sprays that subsequently burn in combustion chambers originates in interfacial instabilities, such as the Kelvin-Helmholtz instability. What can numerical simulations do to improve our understanding of these phenomena? The limitations of numerical techniques make it impossible to consider more than a few droplets or bubbles. They also force us to stay at low Reynolds or Weber numbers, which prevent us from finding a direct solution to the breakup problem. However, these methods are potentially important. First, the continuous improvement of computational power (or, what amounts to the same, the drop in megaflop price) continuously extends the range of affordable problems. Second, and more importantly, the phenomena we consider often happen on scales of space and time where experimental visualization is difficult or impossible. In such cases, numerical simulation may be a useful prod to the intuition of the physicist, the engineer, or the mathematician. A typical example of interfacial flow is the collision between two liquid droplets. Finding the flow involves the study not only of hydrodynamic fields in the air and water phases but also of the air-water interface. This latter part
1,949 citations
Cites methods from "Generalized vortex methods for free..."
...Many other formulations and numerical methods have been proposed (Baker et al 1982, 1984; O˜guz & Prosperetti 1990)....
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TL;DR: In this article, Zhou et al. presented the initial condition dependence of Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) mixing layers, and introduced parameters that are used to evaluate the level of mixedness and mixed mass within the layers.
606 citations
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07 Oct 2011TL;DR: In this paper, a review of the state-of-the-art numerical methods used for direct numerical simulations of multiphase flows, with a particular emphasis on methods that use the so-called "one-field" formulation of the governing equations, is presented.
Abstract: Direct numerical simulations of bubbly flows are reviewed and recent progress is discussed. Simulations, of homogeneous bubble distribution in fully periodic domains at relatively low Reynolds numbers have already yielded considerable insight into the dynamics of such flows. Many aspects of the evolution converge rapidly with the size of the systems and results for the rise velocity, the velocity fluctuations, as well as the average relative orientation of bubble pairs have been obtained. The challenge now is to examine bubbles at higher Reynolds numbers, bubbles in channels and confined geometry, and bubble interactions with turbulent flows. We briefly review numerical methods used for direct numerical simulations of multiphase flows, with a particular emphasis on methods that use the so-called "one-field" formulation of the governing equations, and then discuss studies of bubbles in periodic domains, along with recent work on wobbly bubbles, bubbles in laminar and turbulent channel flows, and bubble formation in boiling.
584 citations
TL;DR: In this article, a boundary integral time integration method is presented for computing the motion of fluid interfaces with surface tension in two-dimensional, irrotational, and incompressible fluids.
532 citations
Cites background from "Generalized vortex methods for free..."
...The integral equation (101) is a Fredholm integral equation of the second kind and it is well known that it is invertible and has a bounded inverse on an open and periodic geometry if z is smooth in α (see [6])....
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...Xt = U n̂ + T ŝ (26) γt − ∂α ((T − W · ŝ) γ/sα) = −2Aρ ( sαWt · ŝ + 1 8 ∂α (γ/sα) 2 + gyα − (T − W · ŝ)Wα · ŝ/sα ) + Sκα (27) where Aρ = (ρ1 − ρ2)/(ρ1 + ρ2) is the Atwood ratio of densities and S = 2τ/(ρ1 + ρ2) is a rescaled surface tension parameter (see [6] and [45])....
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...This representation is well known; see [6] and [51, 27] for some applications to inertial and Hele-Shaw flows, respectively....
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...(20) is a Fredholm integral equation of the second kind for γ, and is, in general, uniquely solveable (see [6])....
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...This approach is based on the boundary integral formulation [6] and applies more generally, even to problems beyond the fluid mechanical context....
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TL;DR: In this paper, the authors present a new numerical method for studying the evolution of free and bound waves on the nonlinear ocean surface using a slope expansion of the velocity potential at the free surface and not an expansion about a reference surface.
Abstract: We present a new numerical method for studying the evolution of free and bound waves on the nonlinear ocean surface. The technique, based on a representation due to Watson and West (1975), uses a slope expansion of the velocity potential at the free surface and not an expansion about a reference surface. The numerical scheme is applied to a number of wave and wave train configurations including longwave-shortwave interactions and the three-dimensional instability of waves with finite slope. The results are consistent with those obtained in other studies. One strength of the technique is that it can be applied to a variety of wave train and spectral configurations without modifying the code.
455 citations
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TL;DR: In this paper, the authors present a method for following the time-history of space-periodic irrotational surface waves, where the only independent variables are the coordinates and velocity potential of marked particles at the free surface at each time step an integral equation is solved for the new normal component of velocity.
Abstract: Plunging breakers are beyond the reach of all known analytical approximations Previous numerical computations have succeeded only in integrating the equations of motion up to the instant when the surface becomes vertical In this paper we present a new method for following the time-history of space-periodic irrotational surface waves The only independent variables are the coordinates and velocity potential of marked particles at the free surface At each time-step an integral equation is solved for the new normal component of velocity The method is faster and more accurate than previous methods based on a two dimensional grid It has also the advantage that the marked particles become concentrated near regions of sharp curvature Viscosity and surface tension are both neglected The method is tested on a free, steady wave of finite amplitude, and is found to give excellent agreement with independent calculations based on Stokes’s series It is then applied to unsteady waves, produced by initially applying an asymmetric distribution of pressure to a symmetric, progressive wave The freely running wave then steepens and overturns It is demonstrated that the surface remains rounded till well after the overturning takes place
1,151 citations
TL;DR: In this article, the authors present a similar method, but with the exception that the problem is solved in the physical plane and finite depth is introduced, whereas in this paper, the same problem is stated in the same way, except that certain other effects can be included without much modification of the program.
216 citations
TL;DR: In this article, the energy transfer due to weak nonlinear interactions in random wave fields is reinterpreted in terms of a hypothetical ensemble of interacting particles, antiparticles, and virtual particles.
Abstract: The energy transfer due to weak nonlinear interactions in random wave fields is reinterpreted in terms of a hypothetical ensemble of interacting particles, antiparticles, and virtual particles. In the particle picture, the interactions can be conveniently described by Feynman diagrams, which may be regarded either as branch diagrams of the perturbation expansion or as collision diagrams. The derivation of the transfer expressions can then be reduced to a few general rules for the construction of the diagrams and the associated collision cross sections. The representation follows closely the standard treatment of nonlinear lattice vibrations, but the particle picture differs from the usual phonon interpretation of lattice waves. It has the unrealistic property that the energies and number densities of antiparticles are negative. This is offset by simpler interaction rules and a closer correspondence between the perturbation graphs and collision diagrams. The method is illustrated for scattering processes in the oceanic wave guide involving surface and internal gravity waves, horizontal currents (turbulence), seismic waves, and bottom irregularities.
202 citations
TL;DR: In this paper, a vortex technique capable of calculating the Rayleigh-Taylor instability to large amplitudes in inviscid, incompressible, layered flows is introduced, whose results show the formation of a steady-state bubble at large times, whose velocity is in agreement with the theory of Birkhoff and Carter.
Abstract: 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.
194 citations