Hydrodynamics of offshore structures with specific focus on wind energy applications

: Three independent wave celerity data sets, measured on natural beaches, are compared with linear, bore, solitary, and hyperbolic wave theories. In the range of relative water depths and wave heights tested, hyperbolic wave theory, which is an asymptotic form of cnoidal theory in shallow water, agreed most closely with measured wave celerities. Linear wave theory also gave satisfactory results; but bore and solitary wave theories overestimated the observed wave speeds. It is concluded that the observed waves are weakly dispersive in amplitude and that care must be taken to apply the theories only in their regime of validity.

This paper presents the development of a "hybrid" method for computing design wave force and moment loading on offshore structures; the method is illustrated by several-examples. The "hybrid" character of the method is due to the combination of linear and nonlinear wave theories to determine wave loading which incorporates the most significant features of a realistic sea system in nature, 1.e., the non1inearities and the directional spectrum. Wave loading due to nonlinear waves with energy present over a continuum of frequencies and directions is represented as the product of a nonlinear wave force and a linearized force transfer coefficient, the latter representing the effect of the directional spectrum. Loading on a single or multiple-pile platform is first calculated from one of a number of nonlinear theories which are valid-for a wave of single fundamental frequency propagating in a single direction. The force transfer function is based on linear wave theory and, in principle, can represent any distribution of the wave energy over frequency and direction. Examples presented include loading on single and multiple pile structures for idealized directional spectra. For the case of a platform supported by four piling on a square array of 60 ft dimensions in plan and spectra for which the princip1e directions associated with each frequency differ substantially, it is found that the linearized force transfer function can result in significantly lower forces (77% of the forces due to unidirectional waves). This case of highly directional waves may be realistic near the center of hurricanes where considerable variability in wind direction occurs over fairly small areas. INTRODUCTION The problem of computing design wave forces on single or multiple piling 11:1 complicated due to the nonlinearity of the waves and the complexity of the random and directional sea surface. Two essentially different approaches have developed in an attempt to establish realistic design wave loading. One approach is to attempt to represent the nonlinearities of the motion for a single fundamental wave characterized by a period and its higher harmonics. A number of such theories have been developed for waves without currents and Dalrymple has extended the stream function theory to include linear and bilinear shear currents. Although more data for wave theory evaluation are always welcome, it has been demonstrated that nonlinear theories are now available which satisfy fairly accurately the mathematical formulation. This approach accounts for the nonlinearities but avoids the random and directional character of the sea surface. The second approach is to exploit the utility of superposition of linear systems by considering the wave system to be composed of an infinite number of random linear waves which, in the general case, can be characterized by a directional spectrum; that is the energy is distributed over a continuum of frequencies and also at each frequency, spread over a range of directions. It is recognized that in design, elements of randomness, directionality and nonlinearity can all be significant.

Using a critical value for Yt> = H./ h. as a wave breaking criterion, where Hb and hb are respectively the wave breaker height and depth, applying Airy wave theory, and assuming conservation of the wave energy flux, one obtains 1/5 2 2/5 Hb = k g (TH. ) relating Hb to the wave period T and to the deep-water wave height H^ . Three sets of laboratory data and one set of field data yield k = 0.39 for the dimensionless coefficient. The relationship, based on Airy wave theory and empirically fitted to the data, is much more successful in predicting wave breaker heights than is the commonly used equation of Munk, based on solitary wave theory. In addition, the relationship is applicable over the entire practical range of wave steepness values.

Solution of a singular integral equation involving two intervals arising in the theory of water waves

Surface gravity wave interaction with a floating and submerged elastic plate system is analyzed under the assumption of small-amplitude surface water wave theory and structural response. The plane progressive wave solution associated with the plate system is analyzed to understand the characteristics of the flexural gravity waves in different modes. Further, linearized long-wave equations associated with the wave interaction with the elastic plate system are derived. The dispersion relations are derived based on small-amplitude wave theory and shallow-water approximation and are compared to ensure the correctness of the mathematical formulation. To deal with various types of problems associated with gravity wave interaction with a floating and submerged flexible plate system, Fourier-type expansion formulae are derived in the cases of water of both finite and infinite depths in two dimensions. Certain characteristics of the eigensystems of the developed expansion formulae are derived. Source potentials for surface wave interaction with a floating flexible structure in the presence of a submerged flexible structure are derived and used in Green’s identity to obtain the expansion formulae for flexural gravity wavemaker problems in the presence of submerged flexible plates. The utility of the expansion formulae and associated orthogonal mode-coupling relations is demonstrated by investigating the diffraction of surface waves by floating and submerged flexible structures of two different configurations. The accuracy of the computational results is checked using appropriate energy relations. The present study is likely to provide fruitful solutions to problems associated with floating and submerged flexible plate systems of various configurations and geometries arising in ocean engineering and other branches of mathematical physics and engineering including acoustic structure interaction problems.

Determining the impact of external excitations on a dynamic marine object such as ship hull in a seaway is the main goal of simulations. Now such simulations is most often based on approximate mathematical models that use results of the theory of small amplitude waves. The most complicated software for marine objects behavior simulation LAMP IV (Large amplitude motion program) uses numerical solution of traditional hydrodynamic problem without often used approximations but on the basis of theory of small amplitude waves. For efficiency reasons these simulations can be based on autoregressive model to generate real wave surface. Such a surface possesses all the hydrodynamic characteristics of sea waves, preserves dispersion relation and also shows superior performance compared to other wind wave models. Naturally, the known surface can be used to compute velocity field and in turn to determine pressures in any point under sea surface. The resulting computational algorithm can be used to determine pressures without use of theory of small-amplitude waves.

ABSTRACTResearch on the response of random wave on offshore structures has received great deal of attention of many researchers and engineers in the design of marine structures. Most previous investigations have been limited to the regular waves. In this paper, based on Longuet–Higgins random wave theory and finite element method, a numerical model for random wave-induced seabed response is established. The seabed is treated as poroelastic medium and characterized by Biot’s partly dynamic equations (u–p model). The JONSWAP spectrum is adopted in Longuet–Higgins model, which is based on the cumulative superposition of linear diffraction solution. Based on the numerical results, the effects of random wave on seabed response are investigated by comparing with the corresponding Stokes wave and cnoidal wave. Then, a parametric study is conducted to examine the effect of wave and soil characteristic on the seabed.

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.

This paper uses an analytical method to examine the motion of a Tension Leg Fish Cage (TLFC) in regular and random waves. TLFC is a conceptual design of a fish cage based on the Tension Leg Platform (TLP) working principle that is usually used in deep water offshore oil and gas exploration. The idea of providing a safe environment to combine ecotourism and fish farming in a single platform led us to perform an analytical calculation to assess the possibility of using the TLP concept in fish farming. A preliminary conceptual design of TLFC using an HDPE floater with steel cable tendon is presented. The analytical calculation of the response amplitude operator for surge and heave motion is presented using linear airy wave theory with head seas encountering angle. This paper also presents the calculation of TLFC surge and heave motion under random wave loads. The random wave spectra used in this paper are JONSWAP and ISSC spectra. The result shows that the surge and heave motion response of TLFC is relatively small and, therefore, can be analyzed further with more detailed consideration. It is admitted that HDPE is a brittle material that cannot sustain any long period of constant tension. Hence the optimum tendon-floater connection for the structure is subject to further research.

This article provides a succinct overview of potential water waves and the water wave instabilities. Its aim is to provide both an easy‐to‐use reference and to show how the often used models and concepts give an insight into particular aspects of wave motions and, crucially, what assumptions are employed. Starting with the Euler equations, the potential theory of water waves is formulated along with the underlying assumptions. A very brief overview of existing numerical approaches of how to simulate the nonlinear potential Euler equations is provided. Linear theory of potential waves is given in more detail: the key concepts of dispersion relation, phase and group velocity, wave packets, wave energy and momentum, wave refraction, shoaling, and wave action conservation are introduced. Nonlinear effects are first discussed for steady waves (the Stokes waves), where they manifest in the change of wave shape and speed, and in a new phenomenon of bound waves . Then the Benjamin–Feir instability is explained within the framework of the nonlinear Schrödinger equation. Also on its basis, the envelope soliton is demonstrated. The second fundamental instability, the crest instability of steep waves leading to the crest overturning and breaking, is described very briefly. To complete the overview of the key nonlinear phenomena, basics of waves in shallow water, such as solitary wave , are considered within the framework of the Korteweg–de Vries (KdV) equation. A cursory exposition of wave resonant interactions for waters of arbitrary depth and the Hasselmann or the kinetic equation that describes spectral evolution of random wind wave fields and is the basis of all modern wave forecasting concludes the article.

Detailed comparisons are made between the predictions of Benjamin's weakly nonlinear theory for internal solitary waves in fluids of great depth, with observational data on solitary wave-type disturbances in the lower atmosphere associated with the “morning glory” phenomenon. It is shown that, while the theory is not wholly unreasonable, neither is it completely satisfactory. In particular, although the calculated wave speeds are generally close to those observed, they are no improvement on those based on linear long wave theory; at the same time the predicted wave half-widths are too large by a factor of two to three. The limitations of the theory appear to be associated with the requirement that wave half-widths are much less than the total fluid depth, a condition not satisfied in the atmospheric case. However, the alternative theory for shallow fluids, based on the Korteweg-de Vries equation is found to be even more unsuitable. Our analyses highlight some of the problems in comparing theory w...

A self-consistent theory of solitary hole structures in coasting beams is presented. These phase space vortices are known from particle simulations and appear, e.g., due to a resistive wall instability. The analysis reveals new intrinsic nonlinear modes which owe their existence to a deficiency of particles trapped in the potential well, showing up as notches in the thermal range of the distribution function, where linear wave theory would predict strong Landau damping. This sheds light on the spectrum of small amplitude perturbations proving the incompleteness of linear and associated nonlinear wave theories in the kinetic regime and offers a new interpretation of recent synchrotron experiments. {copyright} {ital 1997} {ital The American Physical Society}

The primary aim of the analysis presented herein is to consolidate the ideas of the ‘conjugate-flow’ theory, which proposes that vortex breakdown is fundamentally a transition from a uniform state of swirling flow to one featuring stationary waves of finite amplitude. The original flow is assumed to be supercritical (i.e. incapable of bearing infinitesimal stationary waves), and the mechanism of the transition is explained on the basis of physical principles that are well established in relation to the analogous supercritical-flow phenomenon of the hydraulic jump or bore. In previous presentations of the theory the existence of appropriately descriptive solutions to the full equations of motion has only been inferred from these general principles, but here the solutions are demonstrated explicitly by means of a perturbation analysis. This has basically much in common with the classical theory of solitary and cnoidal waves, which is known to explain well the essential properties of weak bores.In § 2 the basic equations of the problem are set out and the leading results of the original theoretical treatment are recalled. The new developments are mainly presented in § 3, where an analysis of finite-amplitude waves is completed by two different methods, each serving to illustrate points of interest. The effects of small energy losses and of small flow-force reductions (i.e. wave-resistance effects) are considered, and the analysis leads to a general classification of possible phenomena accompanying such changes of integral properties in either slightly supercritical or slightly subcritical vortex flows. The application to vortex breakdown remains the focus of attention, however, and § 3 includes a careful appraisal of some experimental observations on the phenomenon. In § 4 a summary is given of a variant on the previous methods which is required when the radial boundary of the flow is taken to infinity. The main analysis is developed without restriction to particular flow models, but in § 5 the results are applied to a specific example.

INTRODUCTION The wave forces on an offshore structure are determined by a wave theory (e.g., Stokes or Stream Function) which relates the water kinematics (velocity and acceleration) to the wave parameters (height and period), and a theory which relates the resulting pressures on the structure to the predicted water kinematics (e.g., Morison Equation or Refraction Theory). Generally, the Morison Equation is used which incorporates two force coefficients-the drag and inertia coefficients. The wave parameters experienced by a structure during a storm are random. Also, inferred values of the force coefficients from field measurements indicate a random scatter from wave to wave due to the random nature of the processes involved and imperfect wave and hydrodynamic theories. Therefore, for offshore structures the prediction of wave forces and ultimately the selection of design criteria involve both the random nature of the wave parameters (e.g., height) and the uncertainty in the force coefficients. Procedures for selecting wave heights for design criteria have received considerable attention and are well established; however, the problem of considering the uncertainty in the force coefficients has received little attention. Currently there is no rational procedure to generally account for coefficient uncertainty except to use arbitrary, and potentially unrealistic, guidelines such as the mean value plus a multiple of the standard deviation. The purpose of this paper is to provide a rational framework for dealing with the uncertainty in force coefficients. This framework is statistical and incorporates into the force statistics the uncertainty of the force coefficients and the random occurrence of the wave parameters. BACKGROUND The wave forces on an offshore structure are generally determined by the Morison equation (Mathematical Equation Available in Full Paper) and QD and Qr are the drag and inertia forces (respectively) per unit length acting normal to a structural element; CD and Cr are the drag and inertia coefficients (i.e., the force coefficients), v and v are the water velocity and acceleration normal to the element, D is the element diameter and P is the mass density of water. As shown in Appendix I, the maximum total horizontal forces on a vertical cylinder during a wave cycle are (Mathematical Equation Available in Full Paper) which are obtained by integrating (1) over the water depth. FD is the total drag force; Fr is the total inertia force; H is the wave height and s is the wave steepness (height/wave length). The assumptions used to derive (2) are deepwater waves and linear wave theory. Similar expressions have been presented in Ref. 1. Except for the properties of the water and cylinder (p,D), (2) depends only on the force coefficients, CD and Cr, and wave height, H, if the steepness is small (exp(2ns)=1) or constant. It is noted that for these conditions the total forces are independent of wave period.

The various aspects of the propagation of long waves onto a shelf (i.e., reflection, transmission and propagation on the shelf) are examined experimentally and theoretically. The results are applied to tsunamis propagating onto the continental shelf. A numerical method of solving the one-dimensional Boussinesq equations for constant depth using finite element techniques is presented. The method is extended to the case of an arbitrary variation in depth (i.e., gradually to abruptly varying depth) in the direction of wave propagation. The scheme is applied to the propagation of solitary waves over a slope onto a shelf and is confirmed by experiments. A theory is developed for the generation in the laboratory of long waves of permanent form, i.e., solitary and cnoidal waves. The theory, which incorporates the nonlinear aspects of the problem, applies to wave generators which consist of a vertical plate which moves horizontally. Experiments have been conducted and the results agree well with the generation theory. In addition, these results are used to compare the shape, celerity and damping characteristics of the generated waves with the long wave theories. The solution of the linear nondispersive theory for harmonic waves of a single frequency propagating over a slope onto a shelf is extended to the case of solitary waves. Comparisons of this analysis with the nonlinear dispersive theory and experiments are presented. Comparisons of experiments with solitary and cnoidal waves with the predictions of the various theories indicate that, apart from propagation, the reflection of waves from a change in depth is a linear process except in extreme cases. However, the transmission and the propagation of both the transmitted and the reflected waves in general are nonlinear processes. Exceptions are waves with heights which are very small compared to the depth. For these waves, the entire process of propagation onto a shelf in the vicinity of the shelf is linear . Tsunamis propagating from the deep ocean onto the continental shelf probably fall in this class.