Finite Constant Depth Research Articles

The vertical mode method in the problems of flexural-gravity waves diffracted by a vertical cylinder

The linear three-dimensional problem of ice loads acting on a vertical circular cylinder frozen in an ice cover of infinite extent is studied. The loads are caused by an uni-directional hydroelastic wave propagating in the ice cover towards the cylinder mounted to the see bottom in water of constant depth. There are no open water surfaces in this problem. The deflection of the ice cover is described by the Bernoulli–Euler equation of a thin elastic plate of constant thickness. At the contact line between the ice cover and the surface of the cylinder, some edge conditions are imposed. In this study, the edge of the ice plate is either clamped to the cylinder or has no contact with the cylinder surface, with the plate edge being free of stresses and shear forces. The water is of finite constant depth, inviscid and incompressible. The problem is solved by both the vertical mode method and using the Weber integral transform in the radial coordinate. Each vertical mode corresponds to a root of the dispersion relation for flexural-gravity waves. It is proved that these two solutions are identical for the clamped edge conditions. This result is non-trivial because the vertical modes are non-orthogonal in a standard sense, they are linearly dependent, the roots of the dispersion relation can be double and even triple, and the set of the modes could be incomplete. A general solution of the wave-cylinder interaction problem is derived by the method of vertical modes and applied to different edge conditions on the contact line. There are three conditions of solvability in this problem. It is shown that these conditions are satisfied for any parameters of the problem.

Effect of a Floating Elastic Plate/Membrane on the Motion Due to a Ring Source in Water With Porous Bed

The velocity potentials due to the presence of a horizontal circular ring of wave sources of timedependent strength in water of finite constant depth with a floating elastic plate or a floating membrane are determined. The uniform bottom is composed of non-dissipative porous medium. The problems are formulated as the initial value problems and the Laplace transform method is used to solve these. For time-harmonic source strength, the steady-state analysis of the potentials reveals the existence of outgoing progressive waves. Graphs for the surface profiles are presented for different values of the tension parameter for the membrane, flexural rigidity of ice and the porous-effect parameter.

Nonlocal Symmetries and Consistent Riccati Expansions of the (2+1)-Dimensional Dispersive Long Wave Equation

Abstract In this article, the (2+1)-dimensional dispersive long wave equation (DLWE) is investigated, which is derived in the context of a water wave propagating in narrow infinitely long channels of finite constant depth. By using of the truncated Painlevé expansion, we construct its nonlocal symmetry and Bäcklund transformation. After implanting the equation into an enlarged one, then the residual symmetry is localised. Meanwhile, the symmetry group transformation can be computed from the prolonged system. Furthermore, the equation is verified to be consistent Riccati expansion (CRE) solvable. Outing from the CRE, the soliton-cnoidal wave interaction solution in terms of Jacobi elliptic functions and the third type of incomplete elliptic integral are studied, respectively.

HOS-ocean: Open-source solver for nonlinear waves in open ocean based on High-Order Spectral method

HOS-ocean is an efficient High-Order Spectral code developed to solve the deterministic propagation of nonlinear wavefields in open ocean. HOS-ocean is released as open-source, developed and distributed under the terms of GNU General Public License (GPLv3). Along with the source code, a documentation under wiki format is available which makes easy the compilation and execution of the source files. The code has been shown to be accurate and efficient. Program summaryProgram title: HOS-oceanCatalogue identifier: AEZS_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEZS_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 30332No. of bytes in distributed program, including test data, etc.: 2656194Distribution format: tar.gzProgramming language: Fortran.Computer: Tested on Intel Xeon E5504 and Intel Core i7.Operating system: Any system with a Fortran compiler: tested on Linux, OS X and Windows 7.RAM: From several MB up to several GB, depending on problem (512×512, M=3: 385 MB and 256×256, M=3: 99 MB)Classification: 4.12.External routines: FFTW 3.3.4 [1], LAPACK [2] and makedepf90 (linux.die.net/man/1/makedepf90)Nature of problem:HOS-ocean has been developed to study the propagation of highly nonlinear sea-states over large domains and long duration.Solution method:HOS-ocean is an implementation of the High-Order Spectral method, which solves the problem formulated on the free surface by means of a pseudo-spectral method.Restrictions:HOS-ocean is dedicated to the propagation of wave fields in infinite and finite constant depth, the evolution over variable bathymetry is not treated. Furthermore, simulations are restricted to non-breaking waves.Running time:2D simulation of irregular wavefield with Nx=1024 modes and an HOS order M=5: t≅2.010−1s per wave period3D simulation of irregular wavefield with Nx=256, Ny=256 modes, an HOS order M=3: t≅10s per wave period.

Oblique wave-free potentials for water waves in constant finite depth

In this paper, a method to construct oblique wave-free potentials in the linearised theory of water waves for water with uniform finite depth is presented in a systematic manner. The water has either a free surface or an ice-cover modelled as a thin elastic plate. For the case of free surface, the effect of surface tension may be neglected or taken into account. Here, the wave-free potentials are singular solutions of the modified Helmholtz equation, having singularity at a point in the fluid region and they satisfy the conditions at the upper surface and the bottom of water region and decay rapidly away from the point of singularity. These are useful in obtaining solutions to oblique water wave problems involving bodies with circular cross-sections such as long horizontal cylinders submerged or half-immersed in water of uniform finite depth with a free surface or an ice-cover modelled as a floating elastic plate. Finally, the forms of the upper surface related to the wave-free potentials constructed here are depicted graphically in a number of figures to visualize the wave motion. The results for non-oblique wave-free potentials and the upper surface wave-free potentials are obtained. The wave-free potentials constructed here will be useful in the mathematical study of water wave problems involving infinitely long horizontal cylinders, either half-immersed or completely immersed in water.

Initial surface disturbance on a shear current: The Cauchy-Poisson problem with a twist

We solve for the first time the classical linear Cauchy-Poisson problem—the time evolution of an initial surface disturbance—when a shear current of uniform vorticity is present beneath the surface. The solution is general, including the effects of gravity, surface tension, and constant finite depth. The particular case of an initially Gaussian disturbance of width b is studied for different values of three system parameters: a “shear Froude number” \documentclass[12pt]{minimal}\begin{document}$S\sqrt{b/g}$\end{document}Sb/g (S is the vorticity), the Bond number and the depth relative to the initial perturbation width. Different phase and group velocity in different directions yield very different wave patterns in different parameter regimes when the shear is strong, and the well-known pattern of diverging ring waves in the absence of shear can take on very different qualitative behaviours. For a given shear Froude number, both finite depth and nonzero capillary effects are found to weaken the influence of the shear on the resulting wave pattern. The various patterns are analysed and explained in light of the shear-modified dispersion relation.

Nonlinear free surface flows past a semi-infinite flat plate in water of finite depth

We consider the steady free surface two-dimensional flow past a semi-infinite flat plate in water of a constant finite depth. The fluid is assumed to be inviscid, incompressible and the flow is irrotational; surface tension at the free surface is neglected. Our concern is with the periodic waves generated downstream of the plate edge. These can be characterized by a depth-based Froude number F and the depth d (draft) of the depressed plate. Previous analytical studies have been restricted to small values of d and subcritical flows F<1, when linearized theory can be used. Here, our main concern here is with the critical regime, F≈1, where we use a weakly nonlinear long-wave analysis, For finite values of d, we solve the fully nonlinear problem numerically using a boundary integral equation method. Our results confirm previous studies and extend these to the critical regime. As d increases, our numerical results demonstrate that the downstream waves approach the highest possible wave. We also find some wave-free solutions, which when compared to the weakly nonlinear results are essentially just one-half of a solitary wave.

Green Function of Steady Motion in Finite Water Depth

The Green function associated with a steady translating source on a straight horizontal course in water with finite constant depth and infinite horizontal extent satisfying the classical free surface condition is studied by decomposing it into three parts: an array of Rankine singularities A, local disturbance D, and downstream wave part S. Each of the three parts is studied by several methods. This is used to verify the numerical scheme and find the most time-efficient procedure. The method of repeated averaging of partial sums for oscillating series is efficiently used to evaluate the infinite sum of Rankine singularities and the downstream wave part. The local disturbance needed in vertical force and pitch moment calculations is most demanding. The Green function is used in combination with thin ship theory to calculate wave resistance, vertical force, pitch moment, and far-field wash for a Wigley hull. The results are compared with Tuck's (1966) slender body theory for shallow water and experimental and theoretical results of wave resistance by Everest and Hogben (1970). The agreement is satisfactory. A shallow water wave resistance ratio r expressing the ratio between wave resistance in finite depth and infinite depth is introduced as an indirect way to minimize wash. It is demonstrated that a large influence of critical depth Froude number requires the ratio between fluid depth and ship length to be small.

The global artificial boundary conditions for numerical simulations of the flow around a submerged body

We consider the numerical approximations of the two-dimensional steady potential flow around a body moving in a liquid of finite-constant depth at constant speed and distance below a free surface. One vertical segment is introduced as the upstream artificial boundary and two vertical segments are introduced as the downstream artificial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then, the original problem is reduced to a problem defined on a finite-computational domain, which is equivalent to a variational problem. After solving the variational problem by the finite-element method, we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.

Axisymmetric gravity waves in two-layered fluids with the upper fluid having a free surface

The forced axisymmetric motion, produced by a vertical cylindrical porous wave maker immersed vertically in two-layered fluids with the upper fluid having a free surface, is considered. Analytical solutions are obtained for two cases when the lower fluid is of infinite depth and when it is of finite constant depth. Also the scattering of harmonic waves incident normally to a fixed cylindrical wall is investigated. Wave trapping phenomenon is discussed. The results are plotted for different values of porosity, different values of the radius of the cylinder and different values of density ratio.

Numerical simulations of non‐linear wave radiation in inviscid fluid with a free surface

AbstractThe paper deals with the numerical simulation of non‐linear wave radiation by vertical free‐surface piercing structures moving horizontally in water of finite constant depth. A time‐stepping approach combined with the finite element method is used to simulate the potential flows due to high‐frequency oscillations of the structure. The evolution equations for the wave elevation and velocity potential at the free surface, exact up to second order, are derived using Stokes perturbation expansion. The boundary value problems up to second order at each time‐step are solved using the Eulerian description and a fixed mesh of finite elements. The results of simulations are presented for vertical cylinders of circular cross section, and for sloshing waves in annular tanks. Copyright © 2003 John Wiley & Sons, Ltd.

Interaction of Surface Waves in a Basin with Floating Broken Ice

The multiscale method is used to obtain asymptotic expansions up to the quantities of the third order for the elevations of the surface of the basin and the velocity potential of motion of liquid particles in the wave disturbances formed in the process of nonlinear interaction of periodic running waves of the first and second harmonics in a homogeneous ideal incompressible liquid of constant finite depth covered with broken ice. The dependences of the amplitude-phase structure of disturbances on the ice thickness, depth of the basin, and the parameters of interacting harmonics are investigated. We estimate the error of evaluation of the characteristics of the formed vertical displacement of the surface of the basin and nonlinear mass transfer introduced by neglecting the curvature of the wave profile in the expression for the velocity potential in deducing the kinematic and dynamic surface boundary conditions for nonlinear approximations.

Effects of viscosity on linear gravity waves due to surface disturbances in water of finite depth

AbstractThe three‐dimensional problem of waves due to an arbitrary initial time‐dependent surface pressure together with an elevation of the surface in a viscous fluid of constant finite depth h is examined. It is shown that the multiple‐integral expression for the surface displacement ζ is reducible to one which is correct to terms of order O(ν'), ν' = ν/(4gh3)1/2, for small coefficient of viscosity ν, under certain conditions. When the Laplace inversion is completed in ζ we arrive at new results which differ significantly from those obtained in an earlier analysis of the problem by Nikitin and Potetyunko [1].

On the diffraction of Poincaré waves

AbstractThe diffraction of tidal waves (Poincaré waves) by islands and barriers on water of constant finite depth is governed by the two‐dimensional Helmholtz equation. One effect of the Earth's rotation is to complicate the boundary condition on rigid boundaries: a linear combination of the normal and tangential derivatives is prescribed. (This would be an oblique derivative if the coefficients were real.) Corresponding boundary‐value problems are treated here using layer potentials, generalizing the usual approach for the standard exterior boundary‐value problems of acoustics. Singular integral equations are obtained for islands (scatterers with non‐empty interiors) whereas hypersingular integral equations are obtained for thin barriers. Copyright © 2001 John Wiley & Sons, Ltd.

Breaking probabilities for dominant surface waves on water of finite constant depth

This paper extends our previous study of the breaking probability of dominant deep water gravity surface waves into the finite water depth environment. It reports a unified behavior of the mean breaking statistics once the effects of finite water depth are taken into account. The shallow water wave data that form the basis of this study were acquired at a field experiment site at Lake George, New South Wales, Australia. The breaking events were detected through visual observation of videotaped records of the wave field in combination with acoustic signatures of the breaking waves from a collocated hydrophone. Following Banner et al. [2000], we argue that when constant finite depth bottom influence is operative, nonlinear hydrodynamical effects associated with energy and momentum fluxes within deforming wave groups remain the primary determinant of breaking onset. This underpins our proposed finite depth water parameterization for the environmental dependence of dominant wave breaking probability, given by the average number of breakers passing a fixed point per dominant wave period. The additional influence of bottom interaction with the wind drift current shear and wind forcing are also included in our finite constant depth formulation. This is a natural extension of our recently proposed deep water dependence and reduces to it as the significant wave height becomes much smaller than the water depth. In common with the deep water case we propose that there exists a threshold of the significant peak steepness below which negligible dominant wave breaking occurs. The available data show encouraging agreement with our proposed dependence, with a correlation coefficient approaching 0.9.

The Local Artificial Boundary Conditions for Numerical Simulations of the Flow Around a Submerged Body

We consider in this work the numerical approximations of the two-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a f...

The artificial boundary conditions for computing the flow around a submerged body

In this paper we consider the numerical simulations of the steady linearized potential flow around a body which moves in a liquid of finite constant depth at constant speed and distance below a free surface. Two vertical segments are introduced as the artificial boundary to limit the unbounded physical domain to a bounded computational domain. A series of approximate artificial boundary conditions at the given artificial boundary are designed by imposing the continuity of the solution and its normal derivative. Then the original problem is reduced to a problem defined in a bounded computational domain, which is solved numerically by the finite element method. The numerical examples show that our artificial boundary conditions are very effective.

Incidence of flexural gravitational waves on the line of contact of two floating ice plates of different thickness

We analyze the influence of a rift in an ice field on the propagation of flexural gravitational waves in a basin of finite constant depth. The ice cover is simulated by two floating semiinfinite elastic plates of different thickness. We studied the dependence of the amplitude coefficients of reflection and transmission of waves incident on the rift on the frequency of running waves, the thickness of ice on both sides of the rift, and the type of contact boundary conditions at the rift.

A fast iterative method to compute the flow around a submerged body

We develop an efficient iterative method for computing the steady linearized potential flow around a submerged body moving in a liquid of finite constant depth. In this paper we restrict the presentation to the two-dimensional problem, but the method is readily generalizable to the three-dimensional case, i.e., the flow in a canal. The problem is indefinite, which makes the convergence of most iterative methods unstable. To circumvent this difficulty, we decompose the problem into two more easily solvable subproblems and form a Schwarz–type iteration to solve the original problem. The first subproblem is definite and can therefore be solved by standard iterative methods. The second subproblem is indefinite but has no body. It is therefore easily and efficiently solvable by separation of variables. We prove that the iteration converges for sufficiently small Froude numbers. In addition, we present numerical results for a second-order accurate discretization of the problem. We demonstrate that the iterative method converges rapidly, and that the convergence rate improves when the Froude number decreases. We also verify numerically that the convergence rate is essentially independent of the grid size.

Evolution of near-inertial waves

The three-dimensional evolution of near-inertial internal gravity waves is investigated for the case of a laterally unbounded fluid layer of constant finite depth. A general Green's function formulation is derived which can be used to solve initial value problems or study the effect of forcing. The Green's function is expanded in vertical normal modes, and is very singular. Convolutions with finite-sized initial conditions lead however to well-behaved solutions. Expansions in similarity solutions of the diffusion equation are shown to be an alternative for finding exact solutions to initial value problems, with respect to one normal mode. For the case of constant buoyancy frequency normal modes expansions are shown to be equivalent to expansions in an alternative series of which the first term is the response on the infinite domain, all the others being corrections to account for the no-flux boundary condition on the upper and lower boundaries.