Waves Generated by a Moving Source in a Two-Layer Ocean of Finite Depth

The velocity potentials of a point source moving at a constant velocity in the upper layer of a two-layer fluid are obtained in a form amenable to numerical integration. Each fluid layer is of finite depth, and the density difference between the two layers is not necessarily small. The far-field asymptotic behavior of the surface waves and internal waves are also derived using the method of stationary phase. They show that the wave system at the free surface or at the interface each contains contributions from two different modes: a surface-wave mode and an internal-wave mode. When the density difference between the two layers is small or the depth of the upper layer is large, the surface-wave mode mainly affects the surface waves while the internal-wave mode mainly affects the internal waves. However, for large density difference, both modes contribute to the surface wave or internal wave system. For each mode, both divergent and transverse waves are present if the total depth Froude number is less than a certain critical Froude number which is mode-dependent. For depth Froude number greater than the critical Froude number, only divergent waves exist for that mode. This classification is similar to that of a uniform fluid of finite depth, where the critical Froude number is simply unity. The surface waves and internal waves are also calculated using the full expressions of the source potentials. They further confirm and illustrate the features observed in the asymptotic analysis.     

Subcritical free-surface flow caused by a line source in a fluid of finite depth

An integral equation is derived and solved numerically to compute the flow and the free surface shape generated when water flows from a line source into a fluid of finite depth. At very low values of the Froude number, stagnation point solutions are found to exist over a continuous range in the parameter space. For each value of the source submergence depth to free stream depth ratio, an upper bound on the existence of stagnation point solutions is found. These results are compared with existing known solutions. A second integral equation formulation is discussed which investigates the hypothesis that these upper bounds correspond to the formation of waves on the free surface. No waves are found, however, and the results of the first method are confirmed.     

On subsonic, supersonic and hypersonic inflectional-wave/vortex interaction

An unstable inflection point developing in an oncoming two-dimensional boundary layer can give rise to nonlinear three-dimensional inflectional-wave/vortex interaction as described in recent papers by Hall and Smith [1], Brown et al. [2], and Smith et al. [3]. In the current study on the compressible range the flow is examined theoretically just downstream of the linear neutral position, in order to understand how the interaction may be initiated. The research addresses both moderately and strongly compressible regimes. In the latter regime the vorticity mode, the most dangerous one, is taken as the wave part, causing the hypersonic interaction to become concentrated in a thin temperature-adjustment layer lying at the outer edge of the boundary layer, just below the free stream. In both regimes, the result is a nonlinear integro-differential equation for the wave-pressure which implies four different types of downstream behaviour for the interaction-a far-downstream saturation, a finite-distance singularity, exponentially decaying waves (leaving pure vortex motion) or periodicity. In a principal finding of the study, the coefficients of the equation are worked out explicitly for hypersonic flow, and in particular for the case of unit Prandtl number and a Chapman fluid, where it is shown that for sufficiently high wall temperatures the wave angle of propagation must lie between 45° and 90° relative to the free-stream direction and also no periodic solutions may occur then. The theory applies also to wake flows and others. Connections with experimental findings are noted.     

A BEM for the prediction of free surface effects on cavitating hydrofoils

 In this paper, a boundary element method (BEM) for cavitating hydrofoils moving steadily under a free surface is presented and its performance is assessed through systematic convergence studies, comparisons with other methods, and existing measurements. The cavitating hydrofoil part and the free surface part of the problem are solved separately, with the effects of one on the other being accounted for in an iterative manner. Both the cavitating hydrofoil surface and the free surface are modeled by a low-order potential based panel method using constant strength dipole and source panels. The induced potential by the cavitating hydrofoil on the free surface and by the free surface on the hydrofoil are determined in an iterative sense and considered on the right hand side of the discretized integral equations. The source strengths on the free surface are expressed by applying the linearized free surface conditions. In order to prevent upstream waves the source strengths from some distance in front of the hydrofoil to the end of the truncated upstream boundary are enforced to be equal to zero. No radiation condition is enforced at the downstream boundary or at the transverse boundary for the three-dimensional case. First, the BEM is validated in the case of a point vortex and some convergence studies are done. Second, the BEM is applied to 2-D hydrofoil geometry both in fully wetted and in cavitating flow conditions and the predictions are compared to those of other methods and of the measurements in the literature. The effects of Froude number, the cavitation number and the submergence depth of the hydrofoil from free surface are discussed. Then, the BEM is validated in the case of a 3-D point source. The effects of grid and of the truncated domain size on the results are investigated. Lastly, the BEM is applied to a 3-D rectangular cavitating hydrofoil and the effect of number of iterations and the effect of Froude number on the results are discussed. Received 6 November 2000     

Generation of water waves and bores by impulsive bottom flux

The inviscid free-surface flow due to an impulsive bottom flux on constant depth is investigated analytically and numerically. The following classes of two-dimensional flow are considered: an upwelling flow which is uniform over a half-plane, a line source/sink, and a dipole aligned along the bottom. The bottom flux is turned on impulsively and may decay with time. The fully nonlinear problem is solved numerically. A small-time asymptotic expansion to third order is found for the nonlinear problem. An asymptotic large-time solution is found for the linearized problem. A steady source will generate a pair of symmetric bores, and their breaking is investigated. A steady sink generates a depression wave if it is weak, and dip instability if it is strong. Wave breaking will occur for intermediate sink strengths. A decaying source emits solitary waves.     

The shallow water equations

Summary In this paper a set of exact nonlinear equations is derived for gravity flows. By assuming the flow to be shallow and assuming the vertical acceleration to be small these equations reduce to the classical equations for long waves in shallow water. If only shallowness is assumed a set of equations results, which admits in the steady case periodic solutions for Froude numbers smaller than 1 and laminar jumps for Froude numbers larger than 1. In the last section potential flows are discussed.Supported, in part, by the National Science Foundation (GP-6632).     

Analytical Solution for Nonlinear Schrödinger Vortex Reconnection

Analysis of the nonlinear Schrödinger vortex reconnection is given in terms of coordinate-time power series. The lowest order terms in these series correspond to a solution of the linear Schrödinger equation and provide several interesting properties of the reconnection process, in particular the nonsingular character of reconnections, the anti-parallel configuration of vortex filaments and a square-root law of approach just before/after reconnections. The complete infinite power series represents a fully nonlinear analytic solution in a finite volume which includes the reconnection point, and is valid for finite time provided the initial condition is an analytic function. These series solutions are free from the periodicity artifacts and discretization error of the direct computational approaches and they are easy to analyze using a computer algebra program.     

A model equation for steady surface waves over a bump

The objective of this paper is to study the solutions of a model equation for steady surface waves on an ideal fluid over a semicircular or semielliptical bump. For upstream Froude number F>1, we show that the numerical solution of the equation has two branches and there is a cut-off value of F below which no solution exists. For F<1, the problem is reformulated to overcome the so-called infinite-mass dilemma. A branch of solutions and a cut-off value of F, above which no solution exists, are found. Furthermore, we also obtain a branch of hydraulic-fall solutions which decrease monotonically from upstream to downstream.     

Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis

The effect of inertia on the Yih–Marangoni instability of the interface between two liquid layers in the presence of an insoluble surfactant is assessed for shear-driven channel flow by a normal-mode linear stability analysis. The Orr–Sommerfeld equation describing the growth of small perturbations is solved numerically subject to interfacial conditions that allow for the Marangoni traction. For general Reynolds numbers and arbitrary wave numbers, the surfactant is found to either provoke instability or significantly lower the rate of decay of infinitesimal perturbations, while inertial effects act to widen the range of unstable wave numbers. The nonlinear evolution of growing interfacial waves consisting of a special pair of normal modes yielding an initially flat interface is analysed numerically by a finite-difference method. The results of the simulations are consistent with the predictions of the linear theory and reveal that the interfacial waves steepen and eventually overturn under the influence of the shear flow.     

Nonlinear solution for an applied overpressure on a moving stream

Summary A boundary-integral method is given for the numerical solution of the exact equations for steady two-dimensional potential flow past a fixed pressure distribution on the free surface of a fluid of infinite depth. The variation in wave-resistance coefficient with overpressure and Froude number is presented. A drag-free nonlinear profile is obtained.     

On the stability of lumps and wave collapse in water waves

In the classical water-wave problem, fully localized nonlinear waves of permanent form, commonly referred to as lumps, are possible only if both gravity and surface tension are present. While much attention has been paid to shallow-water lumps, which are generalizations of Korteweg-de Vries solitary waves, the present study is concerned with a distinct class of gravity-capillary lumps recently found on water of finite or infinite depth. In the near linear limit, these lumps resemble locally confined wave packets with envelope and wave crests moving at the same speed, and they can be approximated in terms of a particular steady solution (ground state) of an elliptic equation system of the Benney-Roskes-Davey-Stewartson (BRDS) type, which governs the coupled evolution of the envelope along with the induced mean flow. According to the BRDS equations, however, initial conditions above a certain threshold develop a singularity in finite time, known as wave collapse, due to nonlinear focusing; the ground state, in fact, being exactly at the threshold for collapse suggests that the newly discovered lumps are unstable. In an effort to understand the role of this singularity in the dynamics of lumps, here we consider the fifth-order Kadomtsev-Petviashvili equation, a model for weakly nonlinear gravity-capillary waves on water of finite depth when the Bond number is close to one-third, which also admits lumps of the wave packet type. It is found that an exchange of stability occurs at a certain finite wave steepness, lumps being unstable below but stable above this critical value. As a result, a small-amplitude lump, which is linearly unstable and according to the BRDS equations would be prone to wave collapse, depending on the perturbation, either decays into dispersive waves or evolves into an oscillatory state near a finite-amplitude stable lump.     

Steady two-layer flows over an obstacle

Nonlinear waves in a forced channel flow of two contiguous homogeneous fluids of different densities are considered. Each fluid layer is of finite depth. The forcing is due to an obstruction lying on the bottom. The study is restricted to steady flows. First a weakly nonlinear analysis is performed. At leading order the problem reduces to a forced Korteweg-de Vries equation, except near a critical value of the ratio of layer depths which leads to the vanishing of the nonlinear term. The weakly nonlinear results obtained by integrating the forced Korteweg-de Vries equation are validated by comparison with numerical results obtained by solving the full governing equations. The numerical method is based on boundary integral equation techniques. Although the problem of two-layer flows over an obstacle is a classical problem, several branches of solutions which have never been computed before are obtained.     

A boundary integral method for gravitational fluid flow over a semi-circular obstruction using an interpolative technique

A boundary integral technique is developed to study the behaviour of a steady, two-dimensional free surface flow of an incompressible, irrotational and inviscid fluid over a submerged semi-circular obstruction in the presence of gravity. The solution technique is different to that employed by many of the previous research workers since it involves the application of the Riemann–Hilbert problem in the derivation of the nonlinear boundary integral–differential equations. The boundary integral equations are solved using piecewise constant and linear interpolative techniques for the fluid velocity on both the solid boundary and the free surface for various values of the upstream Froude number and the radius of the semi-circular obstruction. An investigation into the numerical accuracy of the interpolation techniques is employed. It is found that it is difficult to obtain a solution when the non-dimensional radius of the semi-circular obstruction is large and hence a hybrid technique is developed which is capable of computing the free surface profiles for all values of the radius of the semi-circular obstruction. Also by considering the local Froude number we have found that the fluid flow can become subcritical, i.e. the local Froude number is less than unity, in the vicinity just above the obstacle but no waves are found to be present on the free surface.     

Viscosity and Surface-Tension Effects on Wave Generation by a Translating Body

Various theoretical and experimental studies have been carried out to examine the generation of waves ahead of a translating body. Not all issues pertaining to this wave-motion problem are, however, fully resolved. In particular, mechanisms pertaining to generation of white-water instability and inception of vortices in the bow region are not fully understood. In this paper, the two-dimensional, unsteady, nonlinear, viscous-flow problem associated with a translating surface-piercing body is solved by means of finite-difference algorithm based on boundary-fitted coordinates. Effects of surface tension and surfactants are examined. Results of this work resolve certain classic issues pertaining to bow flows. A continuous generation of short and steepening bow waves is observed at low (draft) Froude number, a nonlinear phenomenon uncovered recently in the case of inviscid fluid also. This indicates that, steady-state nonlinear bow-flow solution may not exist, even at low speed. It is postulated that these short bow waves are responsible for the white-water instability commonly observed ahead of a full-scale ship. The amplitudes of these short bow waves are suppressed by surface tension, which is, possibly, the reason why white-water instability is not distinctly observed in laboratory-scale experiments. The presence of surfactants on the free surface is found to intensify the generation of free-surface vorticity, thus resulting in the formation of bow vortices. The accumulation of surface-active contaminants at the bow is hence responsible for the generation of bow vortices observed in laboratory experiments at low Froude number. At high Froude number, an impulsive starting motion of the body results in the generation of a jet-like splash at the bow and a gentle start an overturning bow wave, as previously observed in the case of inviscid bow flow.     

Domain-adaptive finite difference methods for collapsing annular liquid jets

A domain-adaptive technique which maps a time-dependent, curvilinear geometry into a unit square is used to determine the steady state mass absorption rate and the collapse of annular liquid jets. A method of lines is used to solve the one-dimensional fluid dynamics equations written in weak conservation-law form, and upwind differences are employed to evaluate the axial convective fluxes. The unknown, time-dependent, axial location of the downstream boundary is determined from the solution of an ordinary differential equation which is nonlinearly coupled to the fluid dynamics and gas concentration equations. The equation for the gas concentration in the annular liquid jet is written in strong conservation-law form and solved by means of a method of lines at high Peclet numbers and a line Gauss-Seidel method at low Peclet numbers. The effects of the number of grid points along and across the annular jet, time step, and discretization of the radial convective fluxes on both the steady state mass absorption rate and the jet's collapse rate have been analyzed on staggered and non-staggered grids. The steady state mass absorption rate and the collapse of annular liquid jets are determined as a function of the Froude, Peclet and Weber numbers, annular jet's thickness-to-radius ratio at the nozzle exit, initial pressure difference across the annular jet, nozzle exit angle, temperature of the gas enclosed by the annular jet, pressure of the gas surrounding the jet, solubilities at the inner and outer interfaces of the annular jet, and gas concentration at the nozzle exit. It is shown that the steady state mass absorption rate is proportional to the inverse square root of the Peclet number except for low values of this parameter, and that the possible mathematical incompatibilities in the concentration field at the nozzle exit exert a great influence on the steady state mass absorption rate and on the jet collapse. It is also shown that the steady state mass absorption rate increases as the Weber number, nozzle exit angle, gas concentration at the nozzle exit, and temperature of the gases enclosed by the annular liquid jet are increased, but it decreases as the Froude and Peclet numbers, and annular liquid jet's thickness-to-radius ratio at the nozzle exit are increased. It is also shown that the annular liquid jet's collapse rate increases as the Weber number, nozzle exit angle, temperature of the gases enclosed by the annular liquid jet, and pressure of the gases which surround the jet are increased, but decreases as the Froude and Peclet numbers, and annular liquid jet's thickness-toradius ratio at the nozzle exit are increased. It is also shown that both the ratio of the initial pressure of the gas enclosed by the jet to the pressure of the gas surrounding the jet and the ratio of solubilities at the annular liquid jet's inner and outer interfaces play an important role on both the steady state mass absorption rate and the jet collapse. If the product of these ratios is greater or less than one, both the pressure and the mass of the gas enclosed by the annular liquid jet decrease or increase, respectively, with time. It is also shown that the numerical results obtained with the conservative, domain-adaptive method of lines technique presented in this paper are in excellent agreement with those of a domain-adaptive, iterative, non-conservative, block-bidiagonal, finite difference method which uncouples the solution of the fluid dynamics equations from that of the convergence length.     

Transient evolution of weakly nonlinear sloshing waves: an analytical and numerical comparison

The problem of water waves generated in a horizontally oscillating basin is considered, with specific emphasis on the transient evolution of the wave amplitude. A third-order amplitude evolution equation is solved analytically in terms of Jacobian elliptic functions. The solution explicitly determines the maximum amplitude and nonlinear beating period of the resonated wave. An observed bifurcation in the amplitude response is shown to correspond to the elliptic modulus approaching unity and the beating period of the interaction approaching infinity. The theoretical predictions compare favorably to fully nonlinear simulations of the sloshing process. Due to the omission of damping, the consideration of only a single mode, and the weakly nonlinear framework, the analytical solution applies only to finite-depth, non-breaking waves. The inviscid numerical simulations are similarly limited to finite depth.     

Generalised critical free-surface flows

Nonlinear waves in a forced channel flow are considered. The forcing is due to a bottom obstruction. The study is restricted to steady flows. A weakly nonlinear analysis shows that for a given obstruction, there are two important values of the Froude number, which is the ratio of the upstream uniform velocity to the critical speed of shallow water waves, F _C>1 and F _L<1 such that: (i) when F<F _L, there is a unique downstream cnoidal wave matched with the upstream (subcritical) uniform flow; (ii) when F=F _L, the period of the cnoidal wave extends to infinity and the solution becomes a hydraulic fall (conjugate flow solution) – the flow is subcritical upstream and supercritical downstream; (iii) when F>F _C, there are two symmetric solitary waves sustained over the site of forcing, and at F=F _C the two solitary waves merge into one; (iv) when F>F _C, there is also a one-parameter family of solutions matching the upstream (supercritical) uniform flow with a cnoidal wave downstream; (v) for a particular value of F>F _C, the downstream wave can be eliminated and the solution becomes a reversed hydraulic fall (it is the same as solution (ii), except that the flow is reversed!). Flows of type (iv), including the hydraulic fall (v) as a special case, are computed here using the full Euler equations. The problem is solved numerically by a boundary-integral-equation method due to Forbes and Schwartz. It is confirmed that there is a three-parameter family of solutions with a train of waves downstream. The three parameters can be chosen as the Froude number, the obstruction size and the wavelength of the downstream waves. This three-parameter family differs from the classical two-parameter family of subcritical flows (i) but includes as a particular case the hydraulic falls (ii) or equivalently (v) computed by Forbes.     

Mild-Slope Approximation for Long Waves Generated by Short Waves

The mild-slope approximation has become a popular basis for calculating infinitesimal surface waves on slowly varying depth. It is less restrictive hence more advantageous than the ray and parabolic approximations for describing diffraction and refraction by bathymetry and/or by complex coastlines. Since its computation involves only two horizontal coordinates, the mild-slope equation is also numerically less demanding than the solution of fully three- dimensional equations for a horizontal area with sides much greater than the typical wavelength. By consideration of nonlinear effects of the second order, the mild-slope approximation for long waves over slowly varying depth is derived here, in order to provide a convenient basis for predicting long-period resonance in a large harbor by short-period wind waves.     

Fully nonlinear gravity-capillary solitary waves in a two-fluid system of finite depth

Large-amplitude waves at the interface between two laminar immisible inviscid streams of different densities and velocites, bounded together in a straight infinite channel are studied, when surface tension and gravity are both present. A long-wave approximation is used to develop a theory for fully nonlinear interfacial waves allowing amplitudes as large as the channel thickness. The result is a set of evolution equations for the interfacial shape and the velocity jump across it. Traveling waves of permanent form are studied and it is shown that solitary waves are possible for a range of physical parameters. All solitary waves can be expressed implicitly in terms of incomplete elliptic integrals of the third kind. When the upper layer has zero density, two explicit solitary-wave solutions have been found whose amplitudes are equal to h/4 or h/9, where 2h is the channel thickness. In the absence of gravity solitary waves are not possible but periodic ones are. Numerically constructed solitary waves are given for representative physical parameters.     

Withdrawal from a fluid of finite depth through a line sink, including surface-tension effects

The steady withdrawal of an inviscid fluid of finite depth into a line sink is considered for the case in which surface tension is acting on the free surface. The problem is solved numerically by use of a boundary-integral-equation method. It is shown that the flow depends on the Froude number, F _B=m(gH ³ _B)^–1/2, and the nondimensional sink depth =H _S/H _B, where m is the sink strength, g the acceleration of gravity, H _B is the total depth upstream, H _S is the height of the sink, and on the surface tension, T. Solutions are obtained in which the free surface has a stagnation point above the sink, and it is found that these exist for almost all Froude numbers less than unity. A train of steady waves is found on the free surface for very small values of the surface tension, while for larger values of surface tension the waves disappear, leaving a waveless free surface. It the sink is a long way off the bottom, the solutions break down at a Froude number which appears to be bounded by a region containing solutions with a cusp in the surface. For certain values of the parameters, two solutions can be obtained.