A Model Equation to Study the Effects of Nonlinearity,Surface Tension and Viscosity in Water Waves

The formation of short capillary waves on long, finite amplitude gravity waves is studied by solving numerically a non-linear partial differential equation which models effects of surface tension, viscosity, unsteadiness and finite amplitude.     

Two-dimensional travelling waves over a moving bottom

Two-dimensional travelling waves on an ideal fluid with gravity and surface tension over a periodically moving bottom with a small amplitude are studied. The bottom and the wave travel with a same speed. The exact Euler equations are formulated as a spatial dynamic system by using the stream function. A manifold reduction technique is applied to reduce the system into one of ordinary differential equations with finite dimensions. A homoclinic solution to the normal form of this reduced system persists when higher-order terms are added, which gives a generalized solitary wave—the homoclinic solution connecting a periodic solution.     

Flexural‐Gravity Wave Scattering by a Circular‐Arc‐Shaped Porous Plate

The interaction of flexural‐gravity waves with a thin circular‐arc‐shaped permeable plate submerged beneath the ice‐covered surface of water with uniform finite depth is considered under the assumption of linear theory. The problem is reduced to a second kind hypersingular integral equation for the potential difference across the plate which is solved approximately by an expansion–collocation method. Utilizing the solution, the reflection and the transmission coefficients and the hydrodynamic forces are evaluated numerically. The focus of the paper is to illustrate the effect of a porous curved plate submerged in finite depth water with an ice‐cover on the normally incident waves. Numerical results for a circular‐arc‐shaped plate for different configurations are derived and represented graphically. Also, by choosing an appropriate set of parameters, the known results for a circular‐arc‐shaped rigid plate submerged in deep water and a semicircular porous plate submerged in finite depth water with a free surface are recovered as special cases.     

Influence of Finite Amplitude Disturbances on the Nonstationary Modes of a Compressible Boundary Layer Flow

A weakly nonlinear stability analysis is performed to search for the effects of compressibility on a mode of instability of the three-dimensional boundary layer flow due to a rotating disk. The motivation is to extend the stationary work of [ 1 ] (hereafter referred to as S90) to incorporate into the nonstationary mode so that it will be investigated whether the finite amplitude destabilization of the boundary layer is owing to this mode or the mode of S90. Therefore, the basic compressible flow obtained in the large Reynolds number limit is perturbed by disturbances that are nonlinear and also time dependent. In this connection, the effects of nonlinearity are explored allowing the finite amplitude growth of a disturbance close to the neutral location and thus, a finite amplitude equation governing the evolution of the nonlinear lower branch modes is obtained. The coefficients of this evolution equation clearly demonstrate that the nonlinearity is destabilizing for all the modes, the effect of which is higher for the nonstationary waves as compared to the stationary waves. Some modes particularly having positive frequency, regardless of the adiabatic or wall heating/cooling conditions, are always found to be unstable, which are apparently more important than those stationary modes determined in S90. The solution of the asymptotic amplitude equation reveals that compressibility as the local Mach number increases, has the influence of stabilization by requiring smaller initial amplitude of the disturbance for the laminar rotating disk boundary layer flow to become unstable. Apart from the already unstable positive frequency waves, perturbations with positive frequency are always seen to compete to lead the solution to unstable state before the negative frequency waves do. Also, cooling the surface of the disk will be apparently ineffective to suppress the instability mechanisms operating in this boundary layer flow.     

Drop formation from a wettable nozzle

The process of drop formation from a nozzle can be seen in many natural systems and engineering applications. However, previous research focuses on the pinch-off mechanism of drops from a non-wettable nozzle. Here we investigate the formation of a liquid droplet from a wettable nozzle. In the experiments, drops forming from a wettable nozzle initially climb the outer walls of the nozzle due to surface tension. Then, when the weight of the drops gradually increases, they eventually fall due to gravity. By changing the parameters such as the nozzle size and fluid flow rate, we have observed different behaviors of the droplets. Such oscillatory behavior is characterized by an equation that consists of capillary force, viscous drag, and gravity. Two asymptotic solutions in the initial and later stages of drop formation are obtained and show good agreement with the experimental observations.     

Spectral Stability of Deep Two‐Dimensional Gravity‐Capillary Water Waves

In this contribution we study the spectral stability problem for periodic traveling gravity‐capillary waves on a two‐dimensional fluid of infinite depth. We use a perturbative approach that computes the spectrum of the linearized water wave operator as an analytic function of the wave amplitude/slope. We extend the highly accurate method of Transformed Field Expansions to address surface tension in the presence of both simple and repeated eigenvalues, then numerically simulate the evolution of the spectrum as the wave amplitude is increased. We also calculate explicitly the first nonzero correction to the flat‐water spectrum, which we observe to accurately predict the stability (or instability) for all amplitudes within the disk of analyticity of the spectrum. With this observation in mind, the disk of analyticity of the flat state spectrum is numerically estimated as a function of the Bond number and the Bloch parameter, and compared to the value of the wave slope at the first finite amplitude eigenvalue collision.     

On stable composite capillary-gravitational waves of finite amplitude on the surface of a fluid of finite depth : PMM vol. 39, n 6, 1975, pp. 1023–1031

The problem of stable plane capillary-gravitational waves of finite amplitude on the surface of a perfect incompressible fluid stream of finite depth is considered. It is assumed that the waves are induced by pressure periodically distributed along the free surface, and that these, unlike induced waves, do not vanish when the pressure becomes constant, are transformed into free waves. Such waves are called composite; they exist similarly to free waves, for particular values of velocity of the stream.The problem, which is rigorously stated, reduces to solving a system of four nonlinear equations for two functions and two constants. One of the equations is integral and the remaining are transcendental. Pressure on the surface is defined by an infinite trigonometric series whose coefficients are proportional to integral powers of some dimensionless small parameter; these powers are by two units greater than the numbers of coefficients.The theorem of existence and uniqueness of solution is established, and the method of its proof is indicated. The derivation of solution in any approximation is presented in the form of series in powers of the indicated small parameter. Computation of the first three approximations is carried out to the end, and an approximate equation of the wave profile is presented.Composite capillary-gravitational waves in the case of fluid of infinite depth were considered by the author in [1].     

Numerical stability of the KdV equation with a negative forcing

The waves at the free surface waves of an incompressible and inviscid fluid in a two dimensional domain with horizontal rigid flat bottom with a small obstruction are considered. A time dependent KdV equation with a negative forcing is derived and studied both theoretically and numerically. The existence of a negative solitary-wave-like solution of the equation near the Froude number is proved and the numerical stability of the solution is also studied. The numerical stability of the positive both symmetric and unsymmetric solitary-wave-like solutions are also studied. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Three-dimensional simulation of the runup of nonlinear surface gravity waves

The runup of nonlinear surface gravity waves is numerically simulated in two and three dimensions on the basis of the Navier-Stokes equations. The three-dimensional problem is formulated, and the boundary and initial conditions are described. The splitting method over physical processes is used to construct a discrete model taking into account the cell occupation coefficient. The runup of nonlinear surface gravity waves is simulated in two dimensions for slopes of various geometries, and the numerical results are analyzed. The structural features of the simulated three-dimensional basin are described. Three-dimensional models for the staged runup of nonlinear surface gravity waves breaking on coastal slopes in shallow water areas are considered.     

The Cauchy-Poisson problem in an inviscid stratified liquid

Based upon the Boussinesq approximation, an initial value investigation is made of the axisymmetric free surface response of a nonrotating inviscid stratified liquid of finite or infinite depth to the initial displacement of the free surface. The asymptotic analysis of the integral solution is carried out by the stationary phase method to describe the solution for large time and distance from the origin of disturbance. It is shown that the asymptotic solution consists of the classical free surface gravity waves and the internal waves.     

Large Amplitude Internal Waves of Permanent Form

In this paper the weakly nonlinear theory of long internal gravity waves propagating in stratified media is extended to the fully nonlinear case by treating Long's nonlinear partial differential equation for steady inviscid flows without restriction to small amplitudes and long wavelengths. The existence of finite amplitude solutions of “permanent form” is established analytically for a large class of stratification profiles, and properties are calculated numerically for the case of a hyperbolic tangent density profile in a large range of fluid depths. The numerical results agree well with the experimental data of Davis and Acrivos over the full range of wave amplitudes measured; such agreement is not obtainable with existing weakly nonlinear theories.     

Oblique wave scattering by an impermeable ocean-bed of variable depth in a two-layer fluid with ice-cover

Within the framework of linearized theory, obliquely incident water wave scattering by an uneven ocean-bed in the form of a small bottom undulation in a two-layer fluid, where the upper layer has a thin ice-cover while the lower one has the undulation, is investigated here. In such a two-layer fluid, there exist two modes of time-harmonic waves—the one with lower wave number propagating just below the ice-cover and the one with higher wave number along the interface. An incident wave of a particular mode gets reflected and transmitted by the bottom undulations into waves of both the modes. Assuming irrotational motion, a perturbation technique is employed to solve the first-order corrections to the velocity potentials in the two-layer fluid by using Fourier transform appropriately and also to calculate the reflection and transmission coefficients in terms of integrals involving the shape function representing the bottom undulation. For a sinusoidal bottom topography, these coefficients are depicted graphically against the wave number. It is observed that when the oblique wave is incident on the ice-cover surface, we always find energy transfer to the interface, but for interfacial oblique incident waves, there are parameter ranges for which no energy transfer to the ice-cover surface is possible.     

Resonant Interactions between Vortical Flows and Water Waves. Part II: Shallow Water

Any weak, steady vortical flow is a solution, to leading order, of the inviscid fluid equations with a free surface, so long as this flow has horizontal streamlines coinciding with the undisturbed free surface. This work considers the propagation of long irrotational surface gravity waves when such a vortical flow is present. In particular, when the vortical flow and the irrotational surface waves are both periodic and have comparable length scales, resonant interactions can occur between the various components of the flow. The interaction is described by two coupled Korteweg-de Vries equations and a two-dimensional streamfunction equation.     

Three‐dimensional internal gravity‐capillary waves in finite depth

We consider three‐dimensional inviscid‐irrotational flow in a two‐layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite‐dimensional Hamiltonian system, where an unbounded spatial direction x is considered as a time‐like coordinate. In addition, we consider wave motions that are periodic in another direction z. By analyzing the dispersion relation, we detect several bifurcation scenarios, two of which we study further: a type of 00(is)(iκ₀) resonance and a Hamiltonian Hopf bifurcation. The bifurcations are investigated by performing a center‐manifold reduction, which yields a finite‐dimensional Hamiltonian system. For this finite‐dimensional system, we establish the existence of periodic and homoclinic orbits, which correspond to, respectively, doubly periodic travelling waves and oblique travelling waves with a dark or bright solitary wave profile in the x direction. The former are obtained using a variational Lyapunov‐Schmidt reduction and the latter by first applying a normal form transformation and then studying the resulting canonical system of equations.     

Propagation of kelvin waves from a channel into a semibounded tank

The problem of propagation of Kelvin waves from a channel into a semibounded tank is considered. An exact solution of the problem is constructed using the Wiener — Hopf method. The solution is analyzed asymptotically and numerically. The Wiener — Hopf method was used in /1,2/ to solve the problem of diffraction of the Kelvin waves in tanks bounded by the infinite and semiinfinite parallel walls. Below a generalized method of matching /3/ is used to solve the problem of diffraction of Kelvin waves in the case when the walls confining the fluid meet at the right angles.     

Scattering of oblique waves by bottom undulations in a two-layer fluid

Scattering of waves obliquely incident on small cylindrical undulations at the bottom of a two-layer fluid wherein the upper layer has a free surface and the lower layer has an undulating bottom, is investigated here assuming linear theory. There exists two modes of time-harmonic waves propagating at each of the free surface and the interface. Due to an obliquely incident wave of a particular mode, reflected and transmitted waves of both the modes are created in general by the bottom undulations. For small undulations, a simplified perturbation analysis is used to obtain first-order reflection and transmission coefficients of both the modes due to oblique incidence of waves of again both modes, in terms of integrals involving the shape function describing the bottom. For sinusoidal undulations, these coefficients are plotted graphically to illustrate the energy transfer between the waves of different modes induced by the bottom undulations.     

Extreme wave phenomena in down-stream running modulated waves

Modulational, Benjamin-Feir, instability is studied for the down-stream evolution of surface gravity waves. An explicit solution, the soliton on finite background, of the NLS equation in physical space is used to study various phenomena in detail. It is shown that for sufficiently long modulation lengths, at a unique position where the largest waves appear, phase singularities are present in the time signal. These singularities are related to wave dislocations and lead to a discrimination between successive ‘extreme’ waves and much smaller intermittent waves. Energy flow in opposite directions through successive dislocations at which waves merge and split, causes the large amplitude difference. The envelope of the time signal at that point is shown to have a simple phase plane representation, and will be described by a symmetry breaking unfolding of the steady state solutions of NLS. The results are used together with the maximal temporal amplitude MTA, to design a strategy for the generation of extreme (freak, rogue) waves in hydrodynamic laboratories.     

Numerical investigation of impinging gas jets onto deformable liquid layers

Impinging jets over liquid surfaces are a common practice in the metallurgy and chemical industries. This paper presents a numerical study of the fluid dynamics involved in this kind of processes. URANS simulations are performed using the volume of fluid (VOF) method to deal with the multiphase physics. This unsteady approach with the appropriate computational domain allows resolution of the big eddies responsible for the low frequency phenomena. The solver we used is based on the finite volume method and turbulence is modelled with the realisable k-? model. Two different configurations belonging to the dimpling and splashing modes are under consideration. The results are compared with PIV and LeDaR experimental data previously obtained by the authors. Attention is focused on the surroundings of the impingement, where the interaction between jet and liquid film is much stronger. Finally, frequency analysis is carried out to study the flapping motion of the jet and cavity oscillations.     

A ray‐integral solution for the wave‐field in the Sommerfeld model

A ray‐integral solution is presented for the wave‐field in the Sommerfeld model (liquid half‐space over solid half‐space), where a point source is placed in the fluid and the two media (fluid and solid) of contrasting densities and wave speeds are homogeneous. This exact closed form solution is then used to evaluate complete time records of the acoustic pressure (at a point receiver located in the fluid in the vicinity of the penetrable fluid‐solid interface) for two Sommerfeld models: one where the shear wave speed in the solid bottom is lower than the sound speed in the fluid and the other where the shear wave speed is higher. These pressure response curves indicate the relative importance of the various wave‐forms (the critically refracted longitudinal and shear waves, the pseudo‐Rayleigh and Stoneley interface waves, the direct wave, and the totally reflected wave) contributing to the solution and the possibility of utilizing the arrival times of the refracted and interface waves to determine the bottom rigidity. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Group invariant solution for a pre-existing fluid-driven fracture in impermeable rock

The propagation of a two-dimensional fluid-driven fracture in impermeable rock is considered. The fluid flow in the fracture is laminar. By applying lubrication theory a partial differential equation relating the half-width of the fracture to the fluid pressure is derived. To close the model the PKN formulation is adopted in which the fluid pressure is proportional to the half-width of the fracture. By considering a linear combination of the Lie point symmetries of the resulting non-linear diffusion equation the boundary value problem is expressed in a form appropriate for a similarity solution. The boundary value problem is reformulated as two initial value problems which are readily solved numerically. The similarity solution describes a preexisting fracture since both the total volume and length of the fracture are initially finite and non-zero. Applications in which the rate of fluid injection into the fracture and the pressure at the fracture entry are independent of time are considered.