Numerical computation of solitary waves in a two-layer fluid

2011 ◽  
Vol 688 ◽  
pp. 528-550 ◽  
Author(s):  
H. C. Woolfenden ◽  
E. I. Pǎrǎu
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Froude Number ◽  
Solitary Waves ◽  
Integral Formula ◽  
Density Ratio ◽  
Bond Number ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Damped Oscillations

AbstractWe consider steady two-dimensional flow in a two-layer fluid under the effects of gravity and surface tension. The upper fluid is bounded above by a free surface and the lower fluid is bounded below by a rigid bottom. We assume the fluids to be inviscid and the flow to be irrotational in each layer. Solitary wave solutions are found to the fully nonlinear problem using a boundary integral method based on the Cauchy integral formula. The behaviour of the solitary waves on the interface and free surface is determined by the density ratio of the two fluids, the fluid depth ratio, the Froude number and the Bond numbers. The dispersion relation obtained for the linearized equations demonstrates the presence of two modes: a ‘slow’ mode and a ‘fast’ mode. When a sufficiently strong surface tension is present only on the free surface, there is a region, or ‘gap’, between the two modes where no linear periodic waves are found. In-phase and out-of-phase solitary waves are computed in this spectral gap. Damped oscillations appear in the tails of the solitary waves when the value of the free-surface Bond number is either sufficiently small or large. The out-of-phase waves broaden as the Froude number tends towards a critical value. When surface tension is present on both surfaces, out-of-phase solitary waves are computed. Damped oscillations occur in the tails of the waves when the interfacial Bond number is sufficiently small. Oppositely oriented solitary waves are shown to coexist for identical parameter values.

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

scholarly journals Do true elevation gravity–capillary solitary waves exist? A numerical investigation

2002 ◽  
Vol 454 ◽  
pp. 403-417 ◽  
Author(s):  
A. R. CHAMPNEYS ◽  
J.-M. VANDEN-BROECK ◽  
G. J. LORD
Keyword(s):  
Solitary Waves ◽  
Finite Depth ◽  
Bond Number ◽  
Boundary Integral ◽  
Numerical Results ◽  
Boundary Integral Method ◽  
Small Function ◽  
Asymptotic Results ◽  
Water Wave Problem ◽  
Strong Conclusion

This paper extends the numerical results of Hunter & Vanden-Broeck (1983) and Vanden-Broeck (1991) which were concerned with studies of solitary waves on the surface of fluids of finite depth under the action of gravity and surface tension. The aim of this paper is to answer the question of whether small-amplitude elevation solitary waves exist. Several analytical results have proved that bifurcating from Froude number F = 1, for Bond number τ between 0 and 1/3, there are families of ‘generalized’ solitary waves with periodic tails whose minimum amplitude is an exponentially small function of F−1. An open problem (which, for τ sufficiently close to 1/3, was recently proved by Sun 1999 to be false) is whether this amplitude can ever be zero, which would give a truly localized solitary wave.The problem is first addressed in terms of model equations taking the form of generalized fifth-order KdV equations, where it is demonstrated that if such a zero-tail-amplitude solution occurs, it does so along codimension-one lines in the parameter plane. Moreover, along solution paths of generalized solitary waves a topological distinction is found between cases where the tail does vanish and those where it does not. This motivates a new set of numerical results for the full problem, formulated using a boundary integral method, namely to probe the size of the tail amplitude as τ varies for fixed F > 1. The strong conclusion from the numerical results is that true solitary waves of elevation do not exist for the steady gravity–capillary water wave problem, at least for 9/50 < τ < 1=3. This finding confirms and explains previous asymptotic results by Yang & Akylas.

Bow and stern flows with constant vorticity

1999 ◽  
Vol 399 ◽  
pp. 277-300 ◽  
Author(s):  
SCOTT W. McCUE ◽  
LAWRENCE K. FORBES
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Radiation Condition ◽  
Finite Depth ◽  
Free Surface Flows ◽  
Boundary Integral ◽  
The Body ◽  
Boundary Integral Method ◽  
Infinite Body ◽  
Constant Vorticity

Free surface flows of a rotational fluid past a two-dimensional semi-infinite body are considered. The fluid is assumed to be inviscid, incompressible, and of finite depth. A boundary integral method is used to solve the problem for the case where the free surface meets the body at a stagnation point. Supercritical solutions which satisfy the radiation condition are found for various values of the Froude number and the dimensionless vorticity. Subcritical solutions are also found; however these solutions violate the radiation condition and are characterized by a train of waves upstream. It is shown numerically that the amplitude of these waves increases as each of the Froude number, vorticity and height of the body above the bottom increases.

The effect of surfactant on bursting gas bubbles

1995 ◽  
Vol 302 ◽  
pp. 231-257 ◽  
Author(s):  
Jeremy M. Boulton-Stone
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Stress Condition ◽  
Numerical Technique ◽  
Gas Bubbles ◽  
Free Surface Flows ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Surface Shear ◽  
Surface Compression

A numerical technique, based on the boundary integral method, is developed to allow the modelling of unsteady free-surface flows at large Reynolds numbers in cases where the surface is contaminated by some surface-active compound. This requires the method to take account of the tangential stress condition at the interface and is achieved through a boundary-layer analysis. The constitutive relation that forms the surface stress condition is assumed to be of the Boussinesq type and allows the incorporation of surface shear and dilatational viscous forces as well as Marangoni effects due to gradients in surface tension. Sorption kinetics can be included in the model, allowing calculations for both soluble and insolube surfactants. Application of the numerical model to the problem of bursting gas bubbles at a free surface shows the greatest effect to be due to surface dilatational viscosity which drastically reduces the amount of surface compression and can slow and even prevent the information of a liquid jet. Surface tension gradients give dilatational elasticity to the surface and thus also significantly prevent surface compression. Surface shear viscosity has a smaller effect on the interface motion but results in initially increased surface concentrations due to the sweeping up of surface particles ahead of the inward-moving surface wave.

Numerical studies of singularity formation at free surfaces and fluid interfaces in two-dimensional Stokes flow

1997 ◽  
Vol 331 ◽  
pp. 145-167 ◽  
Author(s):  
C. POZRIKIDIS
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Numerical Study ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Analytic Structure ◽  
Two Dimensional ◽  
Thin Film Lubrication ◽  
Transient Deformation ◽  
Lubrication Equation

We consider the analytic structure of interfaces in several families of steady and unsteady two-dimensional Stokes flows, focusing on the formation of corners and cusps. Previous experimental and theoretical studies have suggested that, without surface tension, the interfaces spontaneously develop such singular points. We investigate whether and how corners and cusps actually develop in a time-dependent flow, and assess the stability of stationary cusped shapes predicted by previous authors. The motion of the interfaces is computed with high resolution using a boundary integral method for three families of flows. In the case of a bubble that is subjected to the family of straining flows devised by Antanovskii, we find that a stationary cusped shape is not likely to occur as the asymptotic limit of a transient deformation. Instead, the pointed ends of the bubble disintegrate in a process that is reminiscent of tip streaming. In the case of the flow due to an array of point-source dipoles immersed beneath a free surface, which is the periodic version of a flow proposed by Jeong & Moffatt, we find evidence that a cusped shape indeed arises as the result of a transient deformation. In the third part of the numerical study, we show that, under certain conditions, the free surface of a liquid film that is levelling under the action of gravity on a horizontal or slightly inclined surface develops an evolving corner or cusp. In certain cases, the film engulfs a small air bubble of ambient fluid to obtain a composite shape. The structure of a corner or a cusp in an unsteady flow does not have a unique shape, as it does at steady state. In all cases, a small amount of surface tension is able to prevent the formation of a singularity, but replacing the inviscid gas with a viscous liquid does not have a smoothing effect. The ability of the thin-film lubrication equation to produce mathematical singularities at the free surface of a levelling film is also discussed.

A note on the instabilities of a horizontal shear flow with a free surface

2000 ◽  
Vol 406 ◽  
pp. 337-346 ◽  
Author(s):  
L. ENGEVIK
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Shear Flow ◽  
Velocity Profile ◽  
Froude Number ◽  
The Other ◽  
Algebraic Equations ◽  
Horizontal Shear ◽  
Analytical Treatment ◽  
Unstable Region

The instabilities of a free surface shear flow are considered, with special emphasis on the shear flow with the velocity profile U* = U*0sech2 (by*). This velocity profile, which is found to model very well the shear flow in the wake of a hydrofoil, has been focused on in previous studies, for instance by Dimas & Triantyfallou who made a purely numerical investigation of this problem, and by Longuet-Higgins who simplified the problem by approximating the velocity profile with a piecewise-linear profile to make it amenable to an analytical treatment. However, none has so far recognized that this problem in fact has a very simple solution which can be found analytically; that is, the stability boundaries, i.e. the boundaries between the stable and the unstable regions in the wavenumber (k)–Froude number (F)-plane, are given by simple algebraic equations in k and F. This applies also when surface tension is included. With no surface tension present there exist two distinct regimes of unstable waves for all values of the Froude number F > 0. If 0 < F [Lt ] 1, then one of the regimes is given by 0 < k < (1 − F2/6), the other by F−2 < k < 9F−2, which is a very extended region on the k-axis. When F [Gt ] 1 there is one small unstable region close to k = 0, i.e. 0 < k < 9/(4F2), the other unstable region being (3/2)1/2F−1 < k < 2 + 27/(8F2). When surface tension is included there may be one, two or even three distinct regimes of unstable modes depending on the value of the Froude number. For small F there is only one instability region, for intermediate values of F there are two regimes of unstable modes, and when F is large enough there are three distinct instability regions.

Atmospheric solitary waves: some applications to the morning glory of the Gulf of Carpentaria

1996 ◽  
Vol 321 ◽  
pp. 137-155 ◽  
Author(s):  
Lawrence K. Forbes ◽  
Shaun R. Belward
Keyword(s):  
Solitary Waves ◽  
Lower Layer ◽  
Sea Breeze ◽  
Boundary Integral ◽  
Morning Glory ◽  
Boundary Integral Method ◽  
Far North ◽  
Constant Vorticity ◽  
Shallow Layer ◽  
Meteorological Phenomenon

A mathematical model is proposed to describe atmospheric solitary waves at the interface between a ‘shallow’ layer of fluid near the ground and a stationary upper layer of compressible air. The lower layer is in motion relative to the ground, perhaps as a result of a distant thunderstorm or a sea breeze, and possesses constant vorticity. The upper fluid is compressible and isothermal, so that its density and pressure both decrease exponentially with height. The profile and speed of the solitary wave are determined, for a wave of given amplitude, using a boundary-integral method. Results are discussed in relation to the ‘morning glory’, which is a remarkable meteorological phenomenon evident in the far north of Australia.

Theory and experiment on the low-Reynolds-number expansion and contraction of a bubble pinned at a submerged tube tip

1998 ◽  
Vol 356 ◽  
pp. 93-124 ◽  
Author(s):  
HARRIS WONG ◽  
DAVID RUMSCHITZKI ◽  
CHARLES MALDARELLI
Keyword(s):  
Surface Tension ◽  
Reynolds Number ◽  
Numerical Solutions ◽  
Full Range ◽  
Boundary Integral ◽  
Gas Pressure ◽  
Boundary Integral Method ◽  
Liquid Density ◽  
Gravitational Acceleration ◽  
Dynamic Surface

The expansion and contraction of a bubble pinned at a submerged tube tip and driven by constant gas flow rate Q are studied both theoretically and experimentally for Reynolds number Re[Lt ]1. Bubble shape, gas pressure, surface velocities, and extrapolated detached bubble volume are determined by a boundary integral method for various Bond (Bo=ρga2/σ) and capillary (Ca=μQ/σa2) numbers, where a is the capillary radius, ρ and μ are the liquid density and viscosity, σ is the surface tension, and g is the gravitational acceleration.Bubble expansion from a flat interface to near detachment is simulated for a full range of Ca (0.01–100) and Bo (0.01–0.5). The maximum gas pressure is found to vary almost linearly with Ca for 0.01[les ]Ca[les ]100. This correlation allows the maximum bubble pressure method for measuring dynamic surface tension to be extended to viscous liquids. Simulated detached bubble volumes approach static values for Ca[Lt ]1, and asymptote as Q3/4 for Ca[Gt ]1, in agreement with analytic predictions. In the limit Ca→0, two singular time domains are identified near the beginning and the end of bubble growth during which viscous and capillary forces become comparable.Expansion and contraction experiments were conducted using a viscous silicone oil. Digitized video images of deforming bubbles compare well with numerical solutions. It is observed that a bubble contracting at high Ca snaps off.

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