Hydrodynamic interactions of cnoidal waves with a vertical cylinder

Diffraction and refraction of nonlinear shallow water waves due to uneven bathymetry is studied numerically in two and three dimensions. The numerical tank consists of a wavemaker at the upwave side of the domain, the submerged obstacles in the middle of the domain, and a numerical wave absorber on the downwave of the domain. The numerical wavemaker is capable of generating solitary and cnoidal waves as solutions of the Green-Naghdi (GN) equations. The nonlinear wave refraction and diffraction is studied by use of the Level I GN equations. The system of equations are solved numerically in time domain by use of a second-order finite difference approach, and in a boundary-fitted coordinate system. Various forms of three-dimensional bathymetry with large slopes, including flat and curved ramps from deep to shallow regions are considered. Results include solitary and cnoidal wave surface elevation and particle velocities and are compared with the existing solutions where possible. Overall very good agreement is observed. Discussion is provided on the nonlinearity and dispersion effects on the wave diffraction and refraction, as well as on the performance of the GN equations in solving these problems.

Propagation and transformation of periodic nonlinear shallow-water waves in basins with selected breakwater systems

Porous structures have the property of diminishing wave loads, so it is important to study the diffraction effects of water waves over porous structures. In the paper, the problems of wave interaction with a porous vertical circular cylinder caused by nonlinear water waves, including cnoidal wave, solitary wave and Stokes second order wave, are discussed in detail, the analytical solutions are derived by using the eigenfunction approach, surface source distribution method and the double Bessel series expansion approach to determine nonlinear diffracted wave potentials for a bottom-mounted surface-piercing porous vertical circular cylinder.

The analytical solutions for the diffraction of the short-crested wave, the cnoidal wave, and the solitary wave incident on a composite bucket foundation of an offshore wind turbine are derived in this paper. The analytical expressions for the pressure, wave run-up, and wave force on the composite bucket foundation are obtained. Based on the numerical results, it is found that the short-crested wave run-up is affected by the wave phase, the cnoidal wave run-up is affected by the wave parameter λ, and the solitary wave run-up is affected by the wave parameter χ. For a composite bucket foundation, the long-crested wave force is larger than the short-crested wave force and the solitary wave force is larger than cnoidal wave force in certain shallow water conditions. Moreover, the short-crested wave force may first increase and then decrease with the increase in the water depth, while the cnoidal wave force and the solitary wave force monotonically increase with the increase in the water depth. In the design of the offshore wind turbine with composite bucket foundation, it will be more conservative if the formula derived from the plane waves is used as design criteria in a short-crested sea, and the wave force in shallow water should be determined by the solitary wave and the cnoidal wave together.

Fluid resonance in the narrow gap of a box-wall system under cnoidal wave action

Internal waves in a two-layer fluid with rotation are considered within the framework of Helfrich’s $f$ -plane extension of the Miyata–Maltseva–Choi–Camassa model. We develop simultaneous asymptotic expansions for the evolving mean fields and deviations from them to describe a large class of uni-directional waves via the Ostrovsky equation, which fully decouples from mean-field variations. The latter generate additive inertial oscillations in the shear and in the phase of both the interfacial displacement and shear. Unlike conventional derivations leading to the Ostrovsky equation, our formulation does not impose the zero-mean constraints on the initial conditions of any variable. Using the constructed solutions, we model the evolution of quasi-periodic initial conditions close to the cnoidal wave solutions of the Korteweg–de Vries (KdV) equation but with local defects, both with and without rotation. We show that rotation leads to the emergence of bursts of internal waves and shear currents, qualitatively similar to the wavepackets generated from solitons and modulated cnoidal waves in earlier studies, but emerging much faster. We also show that cnoidal waves with expansion defects discussed in this work are generalised travelling waves of the KdV equation: they satisfy all conservation laws of the KdV equation (appropriately understood), as well as the Weirstrass–Erdmann corner condition for broken extremals of the associated variational problem and a natural weak formulation. Being smoothed in numerical simulations, they behave, in the absence of rotation, as long-lived states with no visible evolution, while rotation leads to the emergence of strong bursts.

Wave force analysis of the two vertical cylinders by boundary element method

A three-dimensional numerical flume is developed to study cnoidal wave interaction with multiple arranged perforated quasi-ellipse caissons. The continuity equation and the Navier-Stokes equations are used as the governing equation, and the VOF method is adopted to capture the free surface elevation. The equations are discretized on staggered cells and then solved using a finite difference method. The generation and propagation of cnoidal waves in the numerical flume are tested first. And the ability of the present model to simulate interactions between waves and structures is verified by known experimental results. Then cnoidal waves with varying incident wave height and period are generated and interact with multiple quasi-ellipse caissons with and without perforation. It is found that the perforation plays an effective role in reducing wave runup/rundown and wave forces on the caissons. The wave forces on caissons reduce with the decreasing incident wave period. The influence of the transverse distance of multiple caissons on wave forces is also investigated. A closer transverse distance between caissons can produce larger wave forces. But when relative adjacent distanceL/D(Lis the transverse distance andDis the width of the quasi-ellipse caisson) is larger than 3, the effect of adjacent distance is limited.

The diffraction of waves by three bottom fixed vertical circular cylinders is investigated by using the boundary element method. This method has been successfully applied to the isolated vertical circular cylinder and now is used to study the interaction between waves and multiple vertical cylinders. In this paper, a numerical analysis by the boundary element method is developed by the linear potential theory. The numerical analysis by the boundary element method is based on Green's second theorem and introduced to an integral equation for the fluid velocity potential around the vertical circular cylinders. To verify this method, the results obtained in present study are compared with the results computed by the multiple scattering method. The results of the comparisons show strong agreement. Also in this paper, several numerical examples are given to illustrate the effects of various parameters on the wave exciting force such are the separation distance, the wave number and the incident wave angle. This numerical computation method might be used broadly for the design of various offshore structures to be constructed in the future.

Enhancement of wave energy harvesting performance by wave concentration with vertical cylinders

Asymptotic solutions representing slowly varying wavetrains are obtained for two kinds of non-linear shallow water waves, i.e. Korteweg — de Vries waves (cnoidal waves) and Boussinesq waves. Differential equations for slowly varying parameters are derived and it is shown that some of these equations can also be obtained by an averaging technique applied to the conservation equations of the problem. After an asymptotic expansion with respect to the small amplitude/depth ratio equations are given that determine the slow variations of amplitude, wavenumber, frequency, mean waveheight, etc. It is shown that these equations are hyperbolic. A transformation of these equations into their characteristic form shows that-two equations for wavenumber and energy density uncouple from the other ones. An explicit method of solution is indicated.

Chen, Y.-H.; Chu, T., and Wang, K.-H., 2019. Analytical and experimental investigation of waves propagating through thin, porous walls for coastal protection applications. Journal of Coastal Research, 35(6), 1294–1306. Coconut Creek (Florida), ISSN 0749-0208.This paper presents a combined analytical and experimental investigation of the interaction between either a linear monochromatic or a nonlinear solitary wave and a thin, porous wall as a breakwater for coastal protection. This study will determine with validation a unique porous-wall property defined as a material constant in Darcy's law based on theoretical formulation of the porous flow condition using the laboratory data and will evaluate the effectiveness of the porous walls on the transmitted waves through the analytical solutions derived for both the monochromatic and solitary wave conditions. A series of wave tank experiments were performed to record wave elevations in front of and behind the porous walls. Monochromatic waves with various incident wave conditions were first generated to test eight thin, porous walls with varying pore sizes and porosities. The porous-wall properties determined and validated with monochromatic wave data are found to be dependent on the porosity and pore size. The transmitted wave height is reduced as the porosities of the walls decrease. When comparing two porous walls with the same porosity, less incident wave energy gets transmitted through the wall with a smaller pore diameter. Additionally, those porous walls were applied in laboratory tests with inputs of incident solitary waves of varying wave heights. The derived analytical solutions with the determined porous-wall material constants for the cases of solitary waves are verified with the results, similar to the monochromatic wave cases, showing good matches between the predicted and measured wave profiles. Overall, this study provides an important analytical approach with confirmed experimental verifications to be used for coastal engineering application analyzing waves interacting with a thin, porous wall.

The second-order wave force is analysed for diffraction of monochromatic water waves by a vertical cylinder. The force is evaluated directly from pressure integration over the cylinder, and the second-order potential is derived by Weber transformation of the corresponding forcing function on the free surface. This forcing function is reduced to a form which involves a simple factor inversely proportional to the radial coordinate plus an oscillatory function which decays more rapidly in the far field. This feature alleviates the slow rate of convergence involved in capturing the far-field effect. Benchmark computations are obtained and compared with other works. Asymptotic approximations are derived for long and short wavelengths. The analysis and results are primarily for the case of infinite fluid depth, but the finite-depth case is also considered to facilitate comparison with other computations and to illustrate the importance of finite-depth effects in the long-wavelength asymptotic regime.

The diffraction of water waves by a vertical circular cylinder is considered in the regime where the wave amplitude A and cylinder radius a are of the same order, and both are small compared to the wavelength. The wave slope is small, and a conventional linear analysis applies in the outer domain far from the cylinder. Significant nonlinear effects exist in the complementary inner domain close to the cylinder, associated with the free-surface boundary condition. Using inner coordinates scaled with respect to a, it is shown that the leading-order nonlinear contribution to the velocity potential includes terms proportional to both A2a and A3. The wave load which acts on the cylinder near the free surface includes second- and third-harmonic components which are proportional respectively to A2a2 and A3a. In a conventional perturbation analysis, where A [Lt ] a, these components would be ordered in magnitude corresponding to the different powers of A, but here they are of the same order. The second- and third-order components of the total force are of comparable magnitude for practical values of the wave slope.

This study investigates the Mach reflection of random waves by numerical simulations and with experimental observations, especially for the difference and similarity between periodic waves and random waves. As the incident angle becomes large, the normalized significant wave height in front of a reflecting wall becomes large. This tendency is the same as the Mach reflection due to the Stokes wave or the cnoidal wave. The wave height distribution in the normal direction to the wall, that is, the width of stem wave is almost the same even if the nonlinearity of incident waves becomes large. This is a different point between the periodic waves and random waves.