Difference between revisions of "Category:Time-Dependent Linear Water Waves"

Generally the focus of research is on the Frequency Domain Problem. The time-domain problem can be solved by Generalized Eigenfunction Expansion for Water Waves or by an Expansion over the Resonances or using Memory Effect Function.

Introduction

Analytical solutions of the linearised time-domain equations for a coupled-motion or forced-motion problems are very rare. In fact, only a small number of analytic time-domain solutions have been obtained (see the papers by Kennard 1949 and McIver 1994) for structures with simple geometries. Therefore, a variety of numerical methods have been developed to solve initial-value time-domain wave-structure interaction problems. The Fourier transform enables the time and frequency domain quantities to be related and the frequency domain forces and potentials have important roles in the development of the numerical time-domain solution methods. This can be partly attributed to the fact that, in the past, analytical or semi-analytical solutions to linearised water-wave problems were more easily obtained in the frequency domain given that the assumption of time-harmonic motion results in a significant simplification of the interaction equations. Therefore, it was natural to develop time-domain solutions based on frequency domain results and the Fourier transform relation between the domains.

Equations of motion in the Time Domain

Two Dimensional Equations for fixed bodies in the time domain

We consider a two-dimensional fluid domain of constant depth, which contains a finite number of fixed bodies of arbitrary geometry. We denote the fluid domain by [math]\displaystyle{ \Omega }[/math], the boundary of the fluid domain which touches the fixed bodies by [math]\displaystyle{ \partial\Omega }[/math], and the free surface by [math]\displaystyle{ F. }[/math] The [math]\displaystyle{ x }[/math] and [math]\displaystyle{ z }[/math] coordinates are such that [math]\displaystyle{ x }[/math] is pointing in the horizontal direction and [math]\displaystyle{ z }[/math] is pointing in the vertical upwards direction (we denote [math]\displaystyle{ \mathbf{x}=\left( x,z\right) ). }[/math] The free surface is at [math]\displaystyle{ z=0 }[/math] and the sea floor is at [math]\displaystyle{ z=-h }[/math]. The fluid motion is described by a velocity potential [math]\displaystyle{ \Phi }[/math] and free surface by [math]\displaystyle{ \zeta }[/math].

The equations of motion in the time domain are Laplace's equation through out the fluid

At the bottom surface we have no flow

At the free surface we have the kinematic condition

and the dynamic condition (the linearized Bernoulli equation)

The body boundary condition for a fixed body is

The initial conditions are

Two dimensional equations for a floating body

We now consider the equations for a floating two-dimensional structure. The equations of motion in the time domain are Laplace's equation through out the fluid

At the bottom surface we have no flow

At the free surface we have the kinematic condition

and the dynamic condition (the linearized Bernoulli equation)

The body boundary condition for a fixed body is given in terms of the 3 rigid body motions, namely surge, heave and pitch which are indexed as [math]\displaystyle{ \mu=1,3,5 }[/math] in order to be consistent with the three-dimensional problem. We have a kinematic condition

where [math]\displaystyle{ \xi_{\nu} }[/math] is the motion of the [math]\displaystyle{ \mu }[/math]th mode and [math]\displaystyle{ \mathbf{n}_{\nu} }[/math] is the normal associated with this mode. The dynamic condition is the equation of motion for the structure:

where summation over repeated indices is assumed. In this equation, [math]\displaystyle{ M_{\mu\nu} }[/math] are the elements of the mass matrix

for the structure and [math]\displaystyle{ c_{\mu\nu} }[/math] are the elements of the buoyancy matrix

The terms [math]\displaystyle{ I^b_{11} }[/math], [math]\displaystyle{ I^b_{33} }[/math] are the moments of inertia of the body about the [math]\displaystyle{ x }[/math] and [math]\displaystyle{ z }[/math] axes and the terms [math]\displaystyle{ I_1^{A} }[/math], [math]\displaystyle{ I^{A}_{11} }[/math] are the first and second moments of the waterplane (the waterplane area is denoted [math]\displaystyle{ W }[/math]) about the [math]\displaystyle{ x }[/math]-axis (see Chapter 7, Mei 1983). In addition, [math]\displaystyle{ (x^c,z^c) }[/math] and [math]\displaystyle{ (X^R,Z^R) }[/math] are the positions of the centre of mass and centre of rotation of the body and [math]\displaystyle{ I^{V}_{3} }[/math] is [math]\displaystyle{ z }[/math]-component centre of buoyancy of the structure. Thus, the coupled equations of motion for a floating structure have been derived. (N.B. if is assumed that the centre of rotation and the centre of mass of the structure coincide, i.e. if it is assumed that the body is semi-submerged, the mass and buoyancy matrices become diagonal). Any wave incidence is assumed to be propagating in the positive [math]\displaystyle{ x }[/math] direction.) The scattering and radiation problems are simpler than the coupled problem because the motion of the the structure is then prescribed.

The initial conditions are

and The initial generalised displacements [math]\displaystyle{ \xi_{\mu} }[/math] and velocities [math]\displaystyle{ \dot{\xi}_{\mu} }[/math] of the body must be specified for all modes [math]\displaystyle{ \mu }[/math]

Floating body constrained to move in heave

The simplest type of floating body problem concerns the motion of a body constrained to move in heave in two-dimensions. Apart from the boundary condition on the structure surface which becomes

where [math]\displaystyle{ \xi_{3} }[/math] is the heave displacement of the structure, the equations governing the motion of the fluid remain the same. The equation of motion for a body constrained to move in heave only is

which is a significant simplification compared to the general case - the mass matrix and buoyancy matrix are replaced simply by the mass [math]\displaystyle{ M }[/math] and the hydrostatic term [math]\displaystyle{ \rho g W }[/math]. The initial conditions for the fluid and the structure ([math]\displaystyle{ \xi_{3}(0) }[/math], [math]\displaystyle{ \partial_t{\xi}_{3}(0) }[/math]) must also be prescribed to complete the problem specification.

Frequency domain definitions

Linearised water-wave problems were first investigated in the frequency domain wherein all motions are assumed to be time-harmonic and to have begun from [math]\displaystyle{ t= -\infty }[/math] . The time-dependence can be removed in a straightforward manner by substituting for the potential and any structural motions using

and this results in a considerable simplification of the problem. The complex amplitudes of the dynamic quantities vary with frequency and by removing the time-dependence the governing equations and boundary conditions will feature these terms only. However, \citeasnoun{mciver2003} have shown that the frequency-domain potential can also be defined using the following operations on the time-domain solution:

• take a Laplace transform of the time-domain potential

• apply the change of variables [math]\displaystyle{ s=-i\omega }[/math] so that

• noting that [math]\displaystyle{ \phi(\mathbf{x},-\omega)=\bar{\phi}(\mathbf{x},\omega) }[/math], the inverse Fourier transform is given by

where the path of integration must pass over any singularities of [math]\displaystyle{ \phi }[/math] that lie on the real axis and it has been assumed that there is no motion prior to [math]\displaystyle{ t=0 }[/math], i.e. [math]\displaystyle{ \Phi(\mathbf{x},t)=0 }[/math] for [math]\displaystyle{ t\lt 0 }[/math].

If [math]\displaystyle{ \omega }[/math] is real then [math]\displaystyle{ \phi(\mathbf{x},\omega) }[/math] is just the frequency domain potential. By the principle of causality, which precludes the existence of the response before the cause so that [math]\displaystyle{ \Phi(\mathbf{x},t)=0 }[/math] for [math]\displaystyle{ t\lt 0 }[/math] , no singularities can exist in [math]\displaystyle{ \textrm{Im } \omega\gt 0 }[/math]. Therefore, if it is assumed that no singularities exist on the real axis the inverse transform will integrate the frequency-domain potential over the full range of [math]\displaystyle{ \omega }[/math]. With this approach, the frequency-domain equations are obtained by applying the Fourier transform to the time-dependent equations. However, unlike the equations resulting from the assumptions~(2) of time-harmonic motion from [math]\displaystyle{ t= -\infty }[/math], the initial conditions will be present in the frequency-domain equations arising from the Fourier transformation. Although, the latter equations are more mathematically consistent it is usual to set all initial conditions to zero before attempting to solve the equations even if this is not the case. Nevertheless, in the frequency-domain analysis of resonances in coupled motion problems \citeasnoun{pmciver2005} retains the initial condition terms in the equation of motion for the structure and obtains some important results regarding the resonant behaviour of the motion. Therefore, the retention of the initial condition terms depends on the application of the subsequent equations.

Many analytical and approximate solutions exist for frequency-domain problems involving structures with simple geometries. Thus, in the simplest frequency-domain radiation and diffraction problems it is possible to completely determine the velocity potential analytically. The hydrodynamic forces, which are crucial in the context of marine engineering, can then be calculated directly from the potential. For more general geometries this is not possible and the hydrodynamic forces must be computed using numerical methods. Nevertheless, it is useful to consider the analytical expressions for the various potentials and forces in the frequency-domain problem.

In a linearised coupled motion problem, it is customary to decompose the total velocity potential into a scattering potential [math]\displaystyle{ \phi^{S} }[/math] and a radiation potential [math]\displaystyle{ \phi^{R} }[/math]. Thus, the coupled motion problem will require the solution of the radiation problem and the scattering problem. The radiation potential can be further decomposed into a sum over the six modes of motion so that the total frequency domain potential is

where [math]\displaystyle{ u_{\mu} }[/math] is the complex amplitude of the generalised velocity in the [math]\displaystyle{ \mu }[/math] direction and [math]\displaystyle{ \phi_{\mu} }[/math] describes the fluid response due to the forced oscillations in mode [math]\displaystyle{ \mu }[/math] with unit velocity amplitude. The velocities in each mode will be determined from the frequency domain equation of motion for the body. However, this requires the computation of the radiation and exciting forces on the structure. Therefore, it is necessary to consider the radiation problem prior to the coupled motion equation. It is first important to note that, for a given mode [math]\displaystyle{ \mu }[/math] the boundary condition on the structure will be

because the total velocity is

where [math]\displaystyle{ n_{\mu} }[/math] is the [math]\displaystyle{ \mu }[/math] component of the generalised normal and [math]\displaystyle{ v_{\mu} }[/math] is the component of the generalised complex velocity amplitude in this direction. The hydrodynamic force on the structure in the [math]\displaystyle{ \mu }[/math] direction due to the fluid response to the forced oscillations is

and the force due to the diffraction of an incident wave by the fixed structure is

where the time-dependence has been removed in both cases. By expressing the radiation potential as a sum over the individual modes, the radiation force can written as [math]\displaystyle{ \sum_{\nu} v_{\nu}f_{\nu\mu} }[/math] where

It is conventional to decompose the radiation force into a term featuring the added mass matrix [math]\displaystyle{ a_{\alpha\beta} }[/math] and a term featuring the damping matrix [math]\displaystyle{ b_{\alpha\beta} }[/math] as follows

where the added mass and damping coefficients are in phase with the acceleration and velocity respectively. These real quantities depend only on the geometry of the structure and the oscillation frequency [math]\displaystyle{ \omega }[/math] and describe many of the properties of the structure. In particular, (as will be shown see later) if the variation of these coefficients with frequency is known then we can predict the resonant behaviour of the structure in the time-domain as well as the frequency domain. The computation of these quantities was considered to be very important in marine engineering and naval architecture and so a variety of different computer codes were developed to compute the coefficients efficiently. The boundary element method forms the basis of most of the computer code algorithms, the most widely-used example being WAMIT (see http:[math]\displaystyle{ \backslash\backslash }[/math]www.wamit.com) which determines frequency domain solutions for prescribed structures and structure motions.