Water Waves and Ship Hydrodynamics

This chapter contains the formulation of boundary and initial boundary value problems in water waves. The basic equations here are the Euler equations and the equation of continuity for a non-viscous incompressible fluid moving under gravity. Throughout the book, in most considerations the motion is assumed to be irrotational and hence the existence of a velocity potential function is ensured in simply connected regions. In this case the equation of continuity for the velocity of the fluid is then reduced to the familiar Laplace equation for the velocity potential function.

Water waves are created normally by the presence of a free surface along which the pressure is constant. For the irrotational motion, on the free surface one than obtains the non-linear Bernoulli equation for the velocity potential function from the Euler equation. Based on small amplitude waves, linearised problems for the velocity potential function and for the free surface elevation are formulated.

At first we follow the derivation as can be found in J.J. Stoker (Water Waves, Interscience, 1957) and J.V. Wehausen and E.V. Laitone in (Encyclopedia of Physics, vol. 9, Springer, pp. 446–814, 1960) to obtain equations for the wave potential in still water and as a superposition on a constant parallel flow potential. The coefficients in the free surface equations are constant. Then we derive linear equation for the superposition of small amplitude waves on a flow disturbed by some three dimensional object. If we consider the magnitude of the steady velocity vector to be small, we obtain for the time-dependent wave potential function a linear equation with non-constant coefficients.

Of particular importance is the asymptotic behaviour of the free surface elevation for large values of relevant spaces and for time variables. This behaviour can be best obtained by the method of stationary phase (see Sect. 9.1). In this connection, the method of characteristics for treating first-order non-linear partial differential equations for the phase function is employed. Hence a brief summary of the concept of characteristics is included in Sect. 9.2.

A systematic derivation of oscillatory source singularity functions is presented for the disturbance below the free surface with and without current in Sects. 2.3 and 2.7.2. In Sect. 2.4 we derive for the steady case the field for a pressure disturbance at the free surface and for a point source below the free surface in Sect. 2.7.1. These source functions are often called Green functions and are used in numerical codes. One may derive different formulations for the functions as is shown.

In the field of ship hydrodynamics the application of integral equations to solve the linear ship motion problem is widely used. For linear problems harmonic in time there are different ways to formulate an integral equation. A popular formulation, described in this chapter, is the one in the frequency domain. A less frequently used approach is a formulation in the time domain. The advantage of the latter approach is that the source function is rather simple and that it can be extended to the non linear case with some minor effort. For the steady ship wave problem a non-linear approach is the currently most used. In this chapter we give an introduction to these formulations. These methods consist of a description by means of regular Fredholm integral equations. In this chapter we present the method based on Green’s theorem. To outline the method and the difficulties encountered we treat a simpler case, namely the diffraction of an acoustic wave by a smooth object. Hence we first consider the Helmholz equation for the scattering of acoustic waves, instead of the free surface problem.

A short introduction is given to the linear ship motion problem. The structure of the equations of motion is explained by the treatment of an object that is free to move in one degree of freedom, namely the vertical heave motion. The coefficients equations of motion of the general problem, with six degrees of freedom, can be determined by means of the numerical solution of the integral equations for the potential. In the same way the exciting forces may be computed. For the zero forward speed case this approach is generally used. Extension to the forward speed case is possible if the steady flow field around the ship does not differ much from the uniform velocity. Here we describe a way to apply a slow speed approximation.

For some applications it is worthwhile to consider second-order wave effects. Especially the influence of low frequency second-order forces on systems moored in waves may cause resonant behaviour, resulting in large oscillating motions at the resonant frequency of the moored system. To describe this phenomenon we first derive the second-order waves in the case that we have a simple spectrum of waves consisting of two monochromatic plane waves with frequencies close to each other. Next a derivation of the second-order low frequency drift forces by means of a local expansion and far-field expansions is given. As a classroom example we consider the forces on a vertical wall caused by the reflection of such a wave system. This serves as an introduction to formulation of the low frequency motion of a moored system.

In this chapter we consider the two-dimensional interaction of an incident wave with a flexible floating dock or very large floating platform (VLFP) with finite draft. The water depth is finite. The case of a rigid dock is a classical problem. For instance Mei and Black (J. Fluid Mech. 38: 499–511, 1969) have solved the rigid problem, by means of a variational approach. They considered a fixed bottom and fixed free surface obstacle, so they also covered the case of small draft. After splitting the problem in a symmetric and an antisymmetric one, the method consists of matching of eigenfunction expansions of the velocity potential and its normal derivative at the boundaries of two regions. In principle, their method can be extended to the flexible platform case. Recently we derived a simpler method for both the moving rigid and the flexible dock (Hermans in J. Eng. Math. 45: 39–53, 2003). However we considered objects with zero draft only. In this chapter we present our approach for the case of finite, but small, draft. The draft is small compared to the length of the platform to be sure that we may use as a model, for the elastic plate, the thin plate theory, while the water pressure at the plate is applied at finite depth. The method is based on a direct application of Green’s theorem, combined with an appropriate choice of expansion functions for the potential in the fluid region outside the platform and the deflection of the plate. The integral equation obtained by the Green’s theorem is transformed into an integral-differential equation by making use of the equation for the elastic plate deflection. One must be careful in choosing the appropriate Green’s function. It is crucial to use a formulation of the Green’s function consisting of an integral expression only. In Sect. 9.4 we derive such a Green’s function for the two-dimensional case. One may derive an expression as can be found in the article of Wehausen and Laitone (Encyclopedia of Physics, vol. 9, Springer, pp. 446–814, 1960) after application of Cauchy’s residue lemma. In the three-dimensional case one also may derive such an expression. The advantage of this version of the source function is that one may work out the integration with respect to the space coordinate first and apply the residue lemma afterwards. In the case of a zero draft platform this approach resulted in a dispersion relation in the plate region and an algebraic set of equations for the coefficients of the deflection only. Here we derive a coupled algebraic set of equations for the expansion coefficients of the potential in the fluid region and the deflection.

The surface waves of the sea are almost always random in the sense that detailed configuration of the surface varies in an irregular manner in both space and time. Section 7.1 contains a brief description of the Wiener spectrum in connection with the generalised Fourier representations for the surface waves (S. Bochner, Vorlesungen über Fouriersche Integrale, Chelsea, 1948 and N. Wiener, The Fourier Integral and certain of Its Applications, Dover, 1933). In this way we see how one may represent the surface elevation by a superposition of harmonic waves with amplitudes being a stochastic process.

The remaining sections in the chapter are devoted to non-linear waves. In Sect. 7.2 we give a systematic derivation of the shallow water theory from the exact hydrodynamical equations as the approximation of lowest order in a perturbation procedure. Here the relevant small parameter is the ratio of the depth of water to some characteristic length associated with the horizontal direction such as the wave length; the water is considered shallow when this parameter is small. It is a different kind of approximation from the previous linear theory for waves of small amplitude. The resulting equations here are quasi-linear and are exactly analogous to the ones in gas dynamics. Second order approximations are included in the last Sect. 7.3. In particular, an asymptotic theory will be developed for slowly varying wave trains, which may be considered as nearly uniform in the regions of order of magnitude of a small number of wave lengths and periods. Some non-linear dispersive wave phenomena will be discussed and more details can be found in H.W. Hoogstraten (Thesis, Delft University of Technology, 1968).

In this chapter we consider slender ships in shallow water; we discuss three related topics. We are, among others, interested in the influence of the bottom on the vertical motion of the ship. It turns out that a ship in shallow water experiences a certain sinkage and trim due to the bottom effect. This is of importance if one wishes to determine the required depth of harbours in such a way that a ship may enter safely. It is also of importance to know the wall effects for ships travelling in a shallow channel. Here we wish to know sinkage and trim but also the force and moment due to the interaction of the walls of the channel. The third topic we shall consider is the interaction among ships.

The topics have been investigated by E.O. Tuck (J. Fluid Mech. 26: 81–95, 1966), R.F. Beck (J. Ship Res. 2, 1977) and R.W. Yeung (Proc. Sympos. Aspects of Navigability, 1978 and J. Fluid Mech. 85:143–159, 1978). The first two are steady flow problems, while the third one is unsteady due to the difference in forward speed of the two ships. The method we use to solve these problems is related to the well-known method of matched asymptotic expansions. To obtain insight into this method, we apply it to the unsteady flow around a two-dimensional airfoil, where the introduction of the inner expansion is not commonly done. The incoming flow is in principle nearly parallel to the airfoil while the fluid domain is extended toward infinity. Thereafter, we treat the case of a crosscurrent flow around a slender body wedged between two parallel plates. The notion blockage will be introduced in this way. After the preliminary investigations, we shall pay attention to the three problems mentioned above.

In this chapter we present the derivations of some mathematical tools used in this book. No proofs of validity are given. We have used the method of stationary phase and the method of characteristics singular integral equations, for instance. Here we give a short introduction to these methods. We also give the derivation of a two-dimensional Green’s function and a simplification of a set of algebraic equations used in Chap. 6.