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.

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.

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.

Generation and Propagation of Long-Crested Finite-Depth Surface Waves: Comparison Between the Corresponding Results From Numerical Models and a Wave Tank

2008 ◽  
Author(s):  
M. Hasanat Zaman ◽  
Wade Parsons ◽  
Okey Nwogu ◽  
Wooyoung Choi ◽  
R. Emile Baddour ◽  
...  
Keyword(s):  
Surface Waves ◽  
Numerical Solutions ◽  
Numerical Models ◽  
Finite Depth ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Surface Elevation ◽  
Ocean Engineering ◽  
Measured Time ◽  
Wave Maker

The evolution of long-crested surface waves subject to side-band perturbations is investigated with two different numerical models: a direct solver for the Euler equations using a non-orthogonal boundary-fitted curvilinear coordinate system and an FFT-accelerated boundary integral method. The numerical solutions are then validated with laboratory experiments performed in the NRC-IOT Ocean Engineering Basin with a segmented wave-maker operating in piston mode. The numerical models are forced by a point measurement of the free surface elevation at a wave probe close to the wave-maker and the numerical solutions are compared with the measured time-series of the surface elevation at a few wave probe locations downstream.

scholarly journals Numerical study of interfacial solitary waves propagating under an elastic sheet

2014 ◽  
Vol 470 (2168) ◽  
pp. 20140111 ◽  
Author(s):  
Zhan Wang ◽  
Emilian I. Părău ◽  
Paul A. Milewski ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Wave Packet ◽  
Solitary Waves ◽  
Large Range ◽  
Numerical Study ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Surface Modes ◽  
Elastic Sheet ◽  
Pure Gravity ◽  
Two Fluid

Steady solitary and generalized solitary waves of a two-fluid problem where the upper layer is under a flexible elastic sheet are considered as a model for internal waves under an ice-covered ocean. The fluid consists of two layers of constant densities, separated by an interface. The elastic sheet resists bending forces and is mathematically described by a fully nonlinear thin shell model. Fully localized solitary waves are computed via a boundary integral method. Progression along the various branches of solutions shows that barotropic (i.e. surface modes) wave-packet solitary wave branches end with the free surface approaching the interface. On the other hand, the limiting configurations of long baroclinic (i.e. internal) solitary waves are characterized by an infinite broadening in the horizontal direction. Baroclinic wave-packet modes also exist for a large range of amplitudes and generalized solitary waves are computed in a case of a long internal mode in resonance with surface modes. In contrast to the pure gravity case (i.e without an elastic cover), these generalized solitary waves exhibit new Wilton-ripple-like periodic trains in the far field.

Numerical studies of cusp formation at fluid interfaces in Stokes flow

1998 ◽  
Vol 357 ◽  
pp. 29-57 ◽  
Author(s):  
C. POZRIKIDIS
Keyword(s):  
Surface Tension ◽  
Stokes Flow ◽  
Boundary Integral ◽  
Numerical Results ◽  
Boundary Integral Method ◽  
Fluid Interfaces ◽  
Two Dimensional ◽  
Numerical Studies ◽  
Insoluble Surfactant ◽  
Cusp Formation

Numerical studies are performed addressing the development of regions of high curvature and the spontaneous occurrence of cusped interfacial shapes in two-dimensional and axisymmetric Stokes flow. In the numerical simulations, the velocity field is computed using a boundary-integral method, and the evolution of the concentration of an insoluble surfactant over an evolving interface is computed using an implicit finite-volume method. Three configurations are considered in detail, and the results are used to elucidate three different aspects of cusp formation. In the first series, the deformation of a two-dimensional bubble immersed in a family of straining flows devised by Antanovskii, and of an axisymmetric bubble immersed in an analogous family of flows devised by Sherwood, are examined. The numerical results indicate that highly elongated and cusped two-dimensional shapes, and pointed or cusped axisymmetric shapes, are unstable and should not be expected to occur in practice. In the second series of studies, the role of an insoluble surfactant on the transient deformation of bubbles subject to the Antanovskii or Sherwood flow is investigated. Under certain conditions, the reduced surface tension at the tips raises the local curvature to high values and causes the ejection of a sheet or column of gas by means of tip streaming. In the third series of studies, the coalescence of a polygonal formation of five viscous columns of a fluid placed in an arrangement that differs only slightly from one proposed recently by Richardson is examined. The numerical results confirm Richardson's predictions that transient cusps may occur at a finite time in the presence of surface tension. The underlying physical mechanism is discussed on the basis of reversibility of surface-driven Stokes flow and with reference to the regularity of the motion driven by negative surface tension. Replacing the inviscid ambient gas with a slightly viscous fluid whose viscosity is as low as one hundredth the viscosity of the cylinders suppresses the cusp formation.

The buoyancy-driven motion of a train of viscous drops within a cylindrical tube

1992 ◽  
Vol 237 ◽  
pp. 627-648 ◽  
Author(s):  
C. Pozrikidis
Keyword(s):  
Cylindrical Tube ◽  
Terminal Velocity ◽  
Bond Number ◽  
Boundary Integral ◽  
Effective Radius ◽  
Creeping Flow ◽  
Boundary Integral Method ◽  
Main Body ◽  
Viscous Drops ◽  
Asymptotic Analyses

The buoyancy-driven motion of a train of viscous drops settling or rising along the axis of a vertical cylindrical tube is investigated. Under the assumption of creeping flow, the evolution of the drops is computed numerically using a boundary integral method that employs the axisymmetric periodic Green's function for flow in a cylindrical tube. Given the drop volume and assuming that the viscosity of the drops is equal to that of the suspending fluid, the motion is studied as a function of the radius of the tube, the separation between the drops, and the Bond number. Two classes of drops are considered: compact drops whose effective radius is smaller than the radius of the tube, and elongated drops whose effective radius is larger than the radius of the tube. It is found that compact drops may have a variety of steady shapes including prolate and oblate, dimpled tops, and shapes containing pockets of entrained ambient fluid. When the surface tension is sufficiently small, compact drops become unstable, evolving to prolate rings with elongated tails. The terminal velocity of compact drops is discussed and compared with that predicted by previous asymptotic analyses for spherical drops. Steady elongated drops assume the shape of long axisymmetric fingers consisting of a nearly cylindrical main body and two curved ends. Relationships between the terminal velocity of elongated drops, the gap between the drops and the wall of the tube, and the Bond number are established. The results are discussed with reference to previous analyses and laboratory measurements for inviscid bubbles.

scholarly journals Head-on collision of two solitary waves and residual falling jet formation

2009 ◽  
Vol 16 (1) ◽  
pp. 111-122 ◽  
Author(s):  
J. Chambarel ◽  
C. Kharif ◽  
J. Touboul
Keyword(s):  
Residence Time ◽  
Solitary Waves ◽  
Vertical Wall ◽  
Integral Equation Method ◽  
Threshold Value ◽  
Finite Depth ◽  
Boundary Integral ◽  
Jet Formation ◽  
Fully Nonlinear Equations ◽  
Time Calculation

Abstract. The head-on collision of two equal and two unequal steep solitary waves is investigated numerically. The former case is equivalent to the reflection of one solitary wave by a vertical wall when viscosity is neglected. We have performed a series of numerical simulations based on a Boundary Integral Equation Method (BIEM) on finite depth to investigate during the collision the maximum runup, phase shift, wall residence time and acceleration field for arbitrary values of the non-linearity parameter a/h, where a is the amplitude of initial solitary waves and h the constant water depth. The initial solitary waves are calculated numerically from the fully nonlinear equations. The present work extends previous results on the runup and wall residence time calculation to the collision of very steep counter propagating solitary waves. Furthermore, a new phenomenon corresponding to the occurrence of a residual jet is found for wave amplitudes larger than a threshold value.

scholarly journals The boundary integral method for the D.C. geoelectric problem in the 3-layered earth with a prismoid inhomogeneity in the second layer

2012 ◽  
Vol 42 (4) ◽  
pp. 313-343 ◽  
Author(s):  
Milan Hvoždara
Keyword(s):  
Integral Method ◽  
Apparent Resistivity ◽  
Boundary Integral Equations ◽  
Sedimentary Basins ◽  
Boundary Integral ◽  
Numerical Results ◽  
Boundary Integral Method ◽  
Layered Earth ◽  
The Earth ◽  
Schlumberger Array

Abstract The paper presents algorithm and numerical results for the boundary integral equations (BIE) method of the forward D.C. geoelectric problem for the three-layered earth which contains the prismoidal body with sloped faces in the second layer. This situation occurs in the sedimentary basins. Although the numerical calculations are more complicated in comparison with faces orthogonal to the x, y, z axes, the generalization to the sloped faces enables treatment of the anomalous fields for the bodies of more general shapes as rectangular prisms. The graphs with numerical results present isoline maps of the perturbing potential as well as the resistivity profiles when the source field is due to the pair of D.C. electrodes at the surface of the earth. Also presented apparent resistivity curves for the Schlumberger array AMNB sounding.

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.

On asymmetric gravity–capillary solitary waves

1997 ◽  
Vol 330 ◽  
pp. 215-232 ◽  
Author(s):  
T.-S. YANG ◽  
T. R. AKYLAS
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Solitary Wave Solution ◽  
Kdv Equation ◽  
Phase Speed ◽  
Capillary Waves ◽  
Asymptotic Results ◽  
Water Wave Problem ◽  
Fifth Order

Symme tric gravity–capillary solitary waves with decaying oscillatory tails are known to bifurcate from infinitesimal periodic waves at the minimum value of the phase speed where the group velocity is equal to the phase speed. In the small-amplitude limit, these solitary waves may be interpreted as envelope solitons with stationary crests and are described by the nonlinear Schrödinger (NLS) equation to leading order. In line with this interpretation, it would appear that one may also co nstruct asymmetric solitary waves by shifting the carrier oscillations relative to the envelope of a symmetric solitary wave. This possibility is examined here on the basis of the fifth-order Korteweg–de Vries (KdV) equation, a model for g ravity–capillary waves on water of finite depth when the Bond number is close to 1/3. Using techniques of exponential asymptotics beyond all orders of the NLS theory, it is shown that asymmetric solitary waves of the form suggested by the NLS theory in fact are not possible. On the other hand, an infinity of symmetric and asymmetric solitary-wave solution families comprising two or more NLS solitary wavepackets bifurcate at finite values of the amplitude parameter. The asymptotic results are consistent with numerical solutions of the fifth-order KdV equation. Moreover, the asymptotic theory suggests that such multi-packet gravity–capillary solitary waves also exist in the full water-wave problem near the minimum of t he phase speed.

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