Water Wave Problem Research Articles

Real-coded genetic algorithm optimization in reduction of wave drift forces on an array of truncated cylinders

To reduce wave drift forces acting on an array of truncated cylinders, the optimization of geometrical dimensions of the structure using a genetic algorithm is presented in this paper. Wave drift forces are always the major challenge for offshore floating structures facing to waves. Recently, a cloaking phenomenon has attracted the attention of researchers, which can reduce the scattered wave energy of offshore structures even to zero. In a diffraction problem of water waves, the term cloaking means that there are no scattered waves existing outside the structures. The configuration that a truncated inner cylinder surrounded by four truncated cylinders is considered as an effective way to realize this phenomenon. To accurately calculate the hydrodynamic characteristics of the truncated cylinder array, the Wave Interaction Theory combining with a higher-order boundary element method is adopted. Based on the above numerical methods, three parameters including the radius and the draft of the outer cylinders and the distance between the inner cylinder and outer cylinders are optimized to minimize the scattered wave energy and the wave drift force for a certain wavenumber. It is found that the effect of the draft of outer cylinders on the scattered wave energy is insignificant. Therefore, the optimization with two parameters is carried out additionally without consideration of the draft of outer cylinders. Moreover, the relationship between the wave drift force and the wave distribution around the inner cylinder is discussed.

A Mathematical Justification of the Isobe–Kakinuma Model for Water Waves with and without Bottom Topography

We consider the Isobe–Kakinuma model for water waves in both cases of the flat and the variable bottoms. The Isobe–Kakinuma model is a system of Euler–Lagrange equations for an approximate Lagrangian which is derived from Luke’s Lagrangian for water waves by approximating the velocity potential in the Lagrangian appropriately. The Isobe–Kakinuma model consists of $$(N+1)$$ second order and a first order partial differential equations, where N is a nonnegative integer. We justify rigorously the Isobe–Kakinuma model as a higher order shallow water approximation in the strongly nonlinear regime by giving an error estimate between the solutions of the Isobe–Kakinuma model and of the full water wave problem in terms of the small nondimensional parameter $$\delta $$ , which is the ratio of the mean depth to the typical wavelength. It turns out that the error is of order $$O(\delta ^{4N+2})$$ in the case of the flat bottom and of order $$O(\delta ^{4[N/2]+2})$$ in the case of variable bottoms.

Scattering of water waves by an inclined plate in a two-layer fluid

The problem of water wave scattering by an inclined thin plate submerged in the lower layer of a two-layer fluid is examined here using linear theory. In a two-layer fluid, corresponding to a particular frequency wave propagates with two different wavenumbers. Thus we determine the reflection and the transmission coefficients and the hydrodynamic force for both the wavenumbers. This leads to two separate problems. These problems are solved using a hypersingular integral equation method. Several numerical results for the physical quantities varying the inclination, depth and length of the plate are presented. Making suitable adjustment of the parameters published results for a vertical plate submerged in a single layer fluid are recovered.

Numerical simulations of water wave problems with current effect

Two-dimensional Rankine source model is developed to examine wave–current problems. The resulting wave–current interaction problem is separated into steady and unsteady potential problems. The former problem accounts for the current passing a fixed body, whereas the latter describes the wave interaction with the current through the modification of the free surface boundary conditions. Both steady and unsteady problems are solved using a boundary element model, incorporating a continuous Rankine source panel. The proposed method results are compared with the published data and a reasonably good agreement is obtained. Numerical results show that the current has noticeable effects on the global responses and pressure distribution, and these effects vary in different aspects for both radiation and diffraction problems.

Well-posedness and analyticity of solutions to a water wave problem with viscosity

The water wave problem models the free-surface evolution of an ideal fluid under the influence of gravity and surface tension. The governing equations are a central model in the study of open ocean wave propagation, but they possess a surprisingly difficult and subtle well-posedness theory. In this paper we establish the existence and uniqueness of spatially periodic solutions to the water wave equations augmented with physically inspired viscosity suggested in the recent work of Dias et al. (2008) [16]. As we show, this viscosity (which can be arbitrarily weak) not only delivers an enormously simplified well-posedness theory for the governing equations, but also justifies a greatly stabilized numerical scheme for use in studying solutions of the water wave problem.

Unsteady free surface flow above a moving circular cylinder

A problem on non-stationary free surface flow of infinitely deep ideal fluid generated by the motion of a submerged body is considered. The water wave problem is reduced to the integral–differential system of equations for the functions defining free surface shape, normal, and tangential velocity components on the free boundary. Small-time asymptotic solution is constructed for the case of circular cylinder that moves with constant acceleration from rest. The role of non-linearity is clarified by analysis of this approximate solution which describes the formation of added mass layers, splash jets, and finite amplitude surface waves.

A Cliffed Cape Does Not Trap Water Waves in the Sea of Constant Depth

The water wave problem is considered for a class of semi-infinite domains each having its shore shaped as a cliffed cape. In particular, the free surface of a water domain is supposed to be an infinite sector whose vertex angle is greater than π, whereas the water layer lying under the free surface is of constant depth. Under these assumptions it is shown that there are no trapped mode solutions of the problem for all values of a non-dimensional spectral parameter; in other words, no point eigenvalues are embedded in the continuous spectrum of the problem.

Spectral gaps and non-Bragg resonances in a water channel

In this paper the essential spectrum of the linear problem of water-waves on a 3d-channel with gently periodic bottom will be studied. We show that under a certain geometric condition on the bottom profile the essential spectrum has spectral gaps. In classical analysis of waveguides it is known that the Bragg resonances at the edges of the Brillouin zones create band gaps in the spectrum. Here we demonstrate that the band gaps can be opened also in the frequency range far from the Bragg resonances. The position and the length of the gaps are found out by applying an asymptotic analysis to the model problem in the periodicity cell.

Water wave scattering by two surface-piercing and one submerged thin vertical barriers

The problem of water wave scattering by three thin vertical barriers present in infinitely deep water is investigated assuming linear theory. Out of the three, two outer barriers are partially immersed and the inner one is fully submerged and extends infinitely downwards. Havelock’s expansion of water wave potential along with inversion formulae is employed to reduce the problem into a set of linear first-kind integral equations which are solved approximately by using single-term Galerkin approximation technique. Very accurate numerical estimates for the reflection and transmission coefficients are then obtained. The numerical results obtained for various arrangements of the three vertical barriers are depicted graphically in several figures against the wavenumber. These figures exhibit that the reflection coefficient vanishes at discrete wavenumbers only when the two outer barriers are identical. Few known results of a single submerged wall with a gap, single fully submerged barrier extending infinitely downwards, and two barriers partially immersed up to the same depth in deep water are recovered as special cases. This establishes the correctness of the method employed here.

Oblique Water Wave Scattering by a Trench in a Two-Layer Fluid

The problem of oblique water wave scattering and trapping by a submarine trench in a two-layer fluid is analyzed within the context of the linearized water wave theory. Under the potential flow approach, the solution of the associated boundary value problem is derived by the matched eigenfunction expansion method. Numerical results for the reflection and transmission coefficients in surface and internal waves for several physical parameters are obtained. The reflection and transmission coefficients are computed and analyzed in various configurations. The depth and width of the trench affect the reflection and transmission coefficients in both modes. For a symmetric trench, the periodicity of zero and the minimum reflection values increase as the width of the trench increases. The angle of incidence significantly affects the reflection and transmission coefficients at surface mode. In addition to the structural properties of the submarine trench, wave reflection and transmission in surface and internal modes are strongly dependent on the fluid density ratio and the interface location. The numerical results are compared with cases in the specialized literature and very good agreement is achieved. In order to verify the computational results, energy relations are derived and the numerical results are found to satisfy these relations accurately.

An improved Rankine source panel method for three dimensional water wave problems

An improved three dimensional Rankine source method is developed to solve numerically water wave problems in time domain. The free surface and body surface are both represented by continuous panels rather than a discretization by isolated points. The integral of Rankine source 1/r on free surface panel is calculated analytically instead of numerical approximation. Due to the exact algorithm of Rankine source integral applied on the free surface and body surface, a space increment free surface source distribution method is developed and much smaller amount of source panels are required to cover the fluid domain surface than other numerical approximation methods. The proposed method shows a higher accuracy and efficiency compared to other numerical methods for various water wave problems.

Mathematical methods via the nonlinear two-dimensional water waves of Olver dynamical equation and its exact solitary wave solutions

The problem formulations of the nonlinear for the small-long amplitude two-dimensional water waves propagation with free surface are studied. The water wave problem leads to the nonlinear Olver dynamical equation. By applying the extended mapping method, We derive the solitary wave solutions of the nonlinear Olver dynamical equation. These solutions for the nonlinear Olver dynamical equation are obtained efficiency and precisely of the method can be demonstrated. The movement role of the waves by making the graphs of the exact solutions and the stability of these solutions are analyzed and discussed. All solutions are stable and exact.

Existence and amplitude bounds for irrotational water waves in finite depth.

We prove the existence of solutions to the irrotational water-wave problem in finite depth and derive an explicit upper bound on the amplitude of the nonlinear solutions in terms of the wavenumber, the total hydraulic head, the wave speed and the relative mass flux. Our approach relies upon a reformulation of the water-wave problem as a one-dimensional pseudo-differential equation and the Newton-Kantorovich iteration for Banach spaces.This article is part of the theme issue 'Nonlinear water waves'.

A quasi-cloaking phenomenon to reduce the wave drift force on an array of adjacent floating bodies

The mooring of offshore floating structures, such as offshore platforms, in large waves against drift forces and rotational moments is a challenging problem in offshore engineering. To accurately investigate such problems, called positioning problems, the time-averaged steady forces of the second order known as the wave drift forces must be taken into account. Fortunately, a cloaking phenomenon occurs under certain conditions and dramatically reduces the wave drift force acting on such a floating body, as previously reported by several researchers. In the diffraction problem of water waves, cloaking refers to the condition where there is no scattering in the form of radial outgoing waves. The reduction of wave drift force on a truncated cylinder with the occurrence of cloaking phenomenon has been numerically and experimentally confirmed. In this paper, the arrangement of several small circular cylinders at regular intervals in a circle concentric with a fixed floating body is considered as an effective means of reducing the wave drift force. Using a combination of a higher-order boundary element method (HOBEM) and wave interaction theory, the influences of the geometric parameters of the outer surrounding cylinders on the wave drift force and the total scattered-wave energy are systematically investigated and discussed. A quasi-cloaking phenomenon is first found and reported in the present study, which is beneficial and flexible for application in practical engineering. More than one quasi-cloaking trigger (where a trigger is an occurrence condition) can be found simply by varying the distance between the inner and outer floating bodies.

A Variational Reduction and the Existence of a Fully Localised Solitary Wave for the Three-Dimensional Water-Wave Problem with Weak Surface Tension

Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence has been predicted on the basis of numerical simulations and model equations (in which context they are usually referred to as `lumps'), and a mathematically rigorous existence theory for strong surface tension (Bond number $\beta$ greater than $\frac{1}{3}$) has recently been given. In this article we present an existence theory for the physically more realistic case $0<\beta<\frac{1}{3}$. A classical variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the functional associated with the Davey-Stewartson equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.

A numerical continuation approach for computing water waves of large wave height

We analyse an algorithm for the calculation of travelling water waves in flows with constant and variable vorticity. The algorithm is based on numerical continuation techniques, which are suitably adapted to the water wave problem. Numerical examples illustrate the performance of the algorithm for flows of constant vorticity, where the results are compared with the literature. We observe agreement with already existing results, but we also have some new qualitative and quantitative results considering the characteristics of the water waves both for constant and variable vorticity.

On a Special Type of Water Wave Problem Arising in Oceanography

On a Special Type of Water Wave Problem Arising in Oceanography

Two-dimensional wave radiation and diffraction problems in a flat or sloping seabed environment

Two-dimensional water wave problems are investigated in an environment with a flat or sloping rigid seabed adopting a continuous Rankine source method. All the fluid domain surfaces, that is the free, body and seabed surfaces, are discretized using continuous panels. These panels are positioned exactly on the fluid boundary surfaces and no desingularization technique is required. A new seabed source panel distribution method is developed to accommodate both symmetric and asymmetric seabed profiles. To validate the numerical model comparisons are made with published findings from other mathematical models and experimental data. The presence of a sloped seabed alters the symmetry of the fluid domain, causing wave reflection and shoaling, and therefore, significantly affects the hydrodynamic characteristics of water wave problems. The influence of these topographies on the responses in all three degrees of freedom (heave, sway and roll) of a rigid floating body are investigated and discussed accounting for wave radiation and diffraction problems.

Boundary observability of gravity water waves

Consider a three-dimensional fluid in a rectangular tank, bounded by a flat bottom, vertical walls and a free surface evolving under the influence of gravity. We prove that one can estimate its energy by looking only at the motion of the points of contact between the free surface and the vertical walls. The proof relies on the multiplier technique, the Craig–Sulem–Zakharov formulation of the water-wave problem, a Pohozaev identity for the Dirichlet to Neumann operator, previous results about the Cauchy problem and computations inspired by the analysis done by Benjamin and Olver of the conservation laws for water waves.

Elliptic estimates for the Dirichlet–Neumann operator on a corner domain

We consider the elliptic estimates for Dirichlet-Neumann operator related to the water-wave problem on a two-dimensional corner domain in this paper. Due to the singularity of the boundary, there will be singular parts in the solution of the elliptic problem for D-N operator. To begin with, we study elliptic problems with mixed boundary condition to derive singularity decompositions and estimates. Based on the analysis, we present the estimates for both D-N operator and its shape derivative with the existence of singular parts.