Time domain prediction of seakeeping behaviour of catamarans

Abstract

The numerical predictions of the seakeeping characteristics of catamarans with and without forward speed are presented using a direct time domain approximation. Boundary-Integral Equation Method (BIEM) with three-dimensional transient free surface Green function and Neumman–Kelvin linearization is used for the solution of the hydrodynamic interactions in the gap between catamaran hulls and solved as impulsive velocity potential. The numerical results have shown that the response of the fluid between multi-hulls can affect seakeeping performance and influence the hydrodynamic forces and the motion amplitude operators of the floating multi-bodies. It is also shown numerically if the critical reduced frequency is smaller than the ratio of vessel length to hull separation, the wave interactions between hulls are very significant due to trapped waves in the gap of twin-hull. A hemi-spheroid catamaran, Wigley catamaran, and DUT high speed catamaran hull forms are used for the numerical analyses and the comparisons of the present ITU-WAVE numerical results for added-mass, damping coefficients, exciting force, and response amplitude operators show satisfactory agreement with existing numerical and experimental results.

Keywords

Seakeeping multi-hull vessels catamaran time domain transient free-surface wave Green function boundary integral equation method Neumman–Kelvin linearization

1. Introduction

The accurate predictions of wave loads and motion characteristics are of critically important for the design of multi-hull floating structures which are in sufficiently close proximity to experience significant hydrodynamic interactions. The oscillation of each body radiates waves assuming that other bodies are not present. Some of these radiated waves interact with other bodies which can be considered as incident waves and wave diffraction occurs as the consequences while some others radiate to infinity. The response of the fluid between multi-hulls can affect both manoeuvring and seakeeping performance as well as influence the hydrodynamic forces and the motions of the floating multi-bodies.

The hydrodynamic interactions are very significant for many floating bodies including cargo transfer operations among multiple vessels in a seaway, replenishment of two ships, arrays of wave energy converters, multi-hull ships and catamarans, offshore platforms with multiple columns, towing of a ship. The resonance occurs due to hydrodynamic interaction in the wave motion between two hulls when the hulls are forced to oscillate on the free surface. The motions of the fluid between hulls are strongly excited at frequencies corresponding to standing waves. An occurrence of complete reflection or complete transmission of incident waves is possible at standing wave frequencies where wave motion between the hulls is resonant [8,38].

The hydrodynamic interactions of array of vertical axisymmetric cylinders are taken into account exactly in some of the early studies [35,48] while approximate analysis for non-axisymmetric bodies are also studied by many researchers assuming hulls are separated by sufficiently large distances. Mainly the multi-body interaction problem is considered as many radiation diffraction problems and solved independently for each body. The wave interaction problem is then combined analytically using the separately solved problem [12,18,34,38,42,45,49].

The numerical tools to predict hydrodynamic loads over floating bodies are simplified by using two-dimensional strip theory approach by integrating hydrodynamic forces longitudinally by treating the body surface as an ensemble of two-dimensional transverse sections. Because of the computational simplicity and the satisfactory approximation of the body motion of conventional ships, strip theory is still in use to date. The strip theory was used to predict motion of catamarans neglecting the interactions (which are a correct assumption when the radiated waves are swept completely downstream) between the hulls which results in poor prediction. Some early researchers had shown that catamaran motions were much different than that of monohulls. This is mainly due to hydrodynamic interactions which result from forward speed, hull spacing, encounter frequency and the critical reduced frequencies where the standing waves are trapped between hulls [10,17,54–56]. Besides, hydrodynamic interactions are considered as two-dimensional while three-dimensional energy dissipation is required in reality when the vessel has forward speed as the radiated wave are swept downstream.

In the case of multi-hulled vessels, although the flow can be regarded as two-dimensional at zero speed and infinitely long demi-hulls, the flow characteristics around catamarans with finite length and forward speed are three-dimensional. In this case, although strip theory can be used for prediction of wave loads at low forward speed, there would be interactions between two hulls that need to be taken into account. On the other hand, the radiated waves from one hull would not reach the other hull at high forward speed which implies strip theory can be used as there would not be hydrodynamic interactions between hulls. However, the strip theory is not accurate at high forward speeds over Froude number ( $Fn$ ) = 0.3–0.4 which are considered as the upper limit. Furthermore, the interaction between the unsteady and steady potentials could play significant role which is not taken into account in strip theory.

The main limitation of the strip theory due to low frequency and three-dimensional effects due to dissipation of wave energy reflecting between hulls was overcomed by the use of Newman’s [39] unified theory extending for multi-hulls. The inner flow is regarded as a two-dimensional problem combined with a three-dimensional far field solution using asymptotic methods and singularities distributed along the longitudinal axis of the body was developed in the unified theory. The source strength and dipole-moment distributions are used to find the longitudinal distribution of the wave amplitude. At each section of one body, the far-field disturbance of the other body is approximated by a plane wave propagating perpendicularly to its axis. These developed unified theories [4,28,44] are significant improvement over the strip theories. However, as the forward speed effects are not taken into account fully, these unified theories would not predict the motions and hydrodynamic loads over multi-hulls at higher speeds.

In order to remove the restriction of the strip theory for higher forward speed and to maintain the computational efficiency, the new 21/2-dimensional strip theory [9,43,60,61] was developed to predict the response of high forward speed vessels considering the interaction between the steady and unsteady flow fields around the body using the two-dimensional velocity potential, which satisfies the linearized three-dimensional free surface boundary condition. As the diverging waves are dominant at higher speeds, the effects of transverse waves are not taken into account as well as that of hydrodynamic interaction between the hulls. However, this is acceptable assumption as there will be little or no hydrodynamic interactions at high speeds. The solution is started at the bow and stepped toward the stern section-by-section. This implicitly means that waves would not travel forward and Froude number needs to have a lower limit which is around $Fn = 0.4$ . However, this lower limit on Froude number is the one of the disadvantage as the approach cannot be used for lower Froude numbers.

The three-dimensional hydrodynamic interactions can be taken into account automatically by the use of the complete solution of Navier–Stokes equations using computational fluid dynamics methods [6]. Another approach for three-dimensional non-linear flow field due to incident waves is the use of a viscous solution (i.e., computational fluid dynamics methods) in the near field and an inviscid solution (i.e., potential flow methods) in the far field. However, the required computational time to solve these kinds of problems is not suitable for practical purposes yet. An alternative approach to a viscous solution is the three-dimensional potential flow approximation to solve the hydrodynamic interactions as three-dimensional effects play a significant role in the dissipation of wave energy between hulls. The hydrodynamic interactions effects are automatically taken into account as each discretized panel would have its influence on all other panels in three-dimensional numerical models. As a result, due to interactions between two hulls at forward speed, the flow characteristics are inherently three-dimensional and three-dimensional numerical codes needs to be used for accurate prediction of the wave loads and motions over multi-hulls.

The prediction of three-dimensional effects for multihulls can be obtained using three-dimensional frequency and time domain approaches and two kinds of formulations were used for this purpose. These are Green’s function formulation [1,16,19,29,31–33] or Rankine type source distribution [2,30,36,37,52,58,59]. The former satisfies the free surface boundary condition and condition at infinity automatically, and only the body surface needs to be discretized with panels, while in the latter source and dipole singularities are distributed discretizing both the body surface and a portion of the free surface. The main disadvantage of Rankine type source distribution is the stability problem for the numerical implementation, since the radiation condition or condition at infinity is not satisfied exactly. The requirement of the discretization of some portion of the free surface using quadrilateral or triangular elements increases the computational time substantially. The time domain and frequency domain results are related by the Fourier Transform in the context of the linear theory.

It appears that in both time domain and frequency domain it is an advantage to use the Green’s function approach for computational and practical purposes and computationally efficient numerical codes are developed recently [13,47]. The extension of the time domain approach to more general cases, such as non-constant forward speed case, large amplitude body motion, water on deck, unsteady manoeuvres of the body surface, non-linear cable forces etc., is much easier than the frequency domain approach.

In the present paper, the fluid boundaries are described by the use of Boundary Integral Equation Methods (BIEM) with Neumann–Kelvin linearization. The exact initial boundary value problem is then linearized using the free stream as a basis flow and replaced by the boundary integral equation applying Green theorem over three-dimensional transient free surface Green function [19–27]. The resultant boundary integral equation is discretized using quadrilateral panels over which the value of the potential is assumed to be constant and solved using the trapezoidal rule to integrate the memory part of the transient free surface Green function in time. The free surface and body boundary conditions are linearized on the discretized collocation points over each quadrilateral element to obtain algebraic equation. The accuracy of the present method is assessed by comparing the results with the available numerical and experimental results [4,46,53].

2. Linearized initial-boundary value problem

A right-handed coordinate system is used to define the fluid action and a Cartesian coordinate system $\vec{x} = (x, y, z)$ is fixed to the body which is used for the solution of the linearized problem in the time domain Fig. 1. Positive x-direction is towards the bow, positive z-direction points upwards, and the $z = 0$ plane (or $x y$ plane) is coincident with calm water. The body is translating through an incident wave field with velocity $U_{0}$ while it undergoes oscillatory motion about its mean body position. The origin of the body-fixed coordinate system $\vec{x} = (x, y, z)$ is located at the centre of the $x y$ plane. The solution domain consists of the fluid bounded by the free surface $S_{f} (t)$ , the body surface $S_{b} (t)$ , and the boundary surface at infinity $S_{\infty}$ Fig. 1 [19].

Fig. 1.

Coordinate system and surface of the problem.

The following assumptions are taken into account in order to solve the physical problem. If the fluid is unbounded (except for the submerged portion of the body on the free surface), ideal (inviscid and incompressible), and its flow is irrotational (no fluid separation and lifting effect), the principle of mass conservation dictates the total disturbance velocity potential $Φ (\vec{x}, t)$ . This velocity potential is harmonic in the fluid domain and is governed by Laplace equation everywhere in the fluid domain as $\nabla^{2} Φ (\vec{x}, t) = 0$ and the disturbance flow velocity field $\vec{V} (\vec{x}, t)$ may then be described as the gradient of the potential $Φ (\vec{x}, t)$ (e.g. $\vec{V} (\vec{x}, t) = \nabla Φ (\vec{x}, t)$ ).

In the present paper, it is assumed that the fluid disturbances due to steady forward motion and unsteady oscillations of the floating body are small and may be separated into individual parts for the linearized problem. In addition to the separation of the fluid disturbance into steady and unsteady part, the free surface boundary condition, body boundary condition, and Bernoulli’s equation may be linearized. In the steady problem, the body starts its motion at rest and then suddenly takes a constant velocity $U_{0}$ parallel to free surface. After some oscillation all transients are allowed to decay to zero for the steady problem which gives rise to the calculation of the steady resistance, sinkage force, and trim moment [19,26]. Then the unsteady problem [19–25], which consists of radiation and diffraction analyses, is solved, when the body is in its equilibrium position. Because of the small disturbance of the fluid, the total velocity potential produced by the presence of the floating body in the fluid domain may be separated into three different parts Eq. (1) $\begin{array}{rcl} Φ (\vec{x}, t) & = & φ_{basis} (\vec{x}) + φ_{steady} (\vec{x}) + φ_{I} (\vec{x}, t) \\ (1) & + φ_{S} (\vec{x}, t) + \sum_{k = 1}^{6} φ_{k} (\vec{x}, t) . \end{array}$

The steady problem is the combination of $φ_{basis} (\vec{x})$ and $φ_{steady} (\vec{x})$ potentials due to the steady translation of the floating body at forward speed $U_{0}$ . The incident potential $φ_{I} (\vec{x}, t)$ is produced when the steadily translating floating body meets with an incident wave field. If the incident wave is reflected by the floating body, the resultant potential is the scattering potential $φ_{S} (\vec{x}, t)$ and comprises the diffraction potential. The solution of the incident wave potential and diffraction potential is called diffraction problem. When the steadily translating floating body is forced to oscillate at any of its rigid-body modes k, the floating body produces the unsteady effects of the radiation potential $φ_{k} (\vec{x}, t)$ , the solution of which comprises the radiation problem. This kind of decomposition is given by [14] and gives rise to the linearization of the governing equations including the free surface boundary condition, body boundary condition, and Bernoulli’s equation for the fluid pressure field. Physically, this kind of decomposition Eq. (1) ignores the interaction of the waves produced by the individual components.

The traditional selection for the basis flow is the double body flow and free stream flow. The latter is used in the present paper which is given as $φ_{basis} (\vec{x}) = - U_{0} x$ and it is far from the body and assumed its contribution is much bigger than the remaining potentials. This kind of selection of the basis flow results in the Neumann–Kelvin linearization of pressure, free surface boundary condition and body boundary condition and eliminates the interaction between the various potentials except for the interaction of the steady flow with the body boundary conditions.

For the free surface boundary condition, the Eulerian description of the flow is used. Thus, no overturning and breaking waves are allowed to exist. The linearized free surface boundary condition about the mean positions of the floating body in the moving coordinate system may be written as $\begin{matrix} (2) & {(\frac{\partial}{\partial t} - U_{0} \frac{\partial}{\partial x})}^{2} φ + g \frac{\partial φ}{\partial z} = 0 on z = 0, \end{matrix}$ where φ is used for all the perturbation potentials. The unsteady problem is coupled to the steady problem due to the presence of the steady velocity vector $\vec{W} = \nabla (- U_{0} x + φ_{steady})$ on the body boundary condition. By assuming that the bodies of interest are slender, the perturbation of the steady flow field due to the floating body is neglected giving $\vec{W} \approx - U_{0} i$ which is used in the present paper.

The linearization of the complete normal body boundary condition on the instantaneous body surface to the mean position of the floating body in the moving coordinate system is given by Timman and Newman [50], and Newman [39] and may be written as for rigid body problem $\begin{array}{l} (3) & \frac{\partial φ_{steady}}{\partial n} = U_{0} n_{1} on S_{0}, \\ (4) & \frac{\partial φ_{I}}{\partial n} = - \frac{\partial φ_{S}}{\partial n} on S_{0}, \\ (5) & \frac{\partial φ_{k}}{\partial n} = n_{k} {\dot{x}}_{k} (t) + m_{k} x_{k} (t) on S_{0}, \\ (6) & m_{k} = (0, 0, 0, 0, U_{0} n_{3}, - U_{0} n_{2}), \end{array}$ where $S_{0}$ is the mean position of the floating body, $n_{k}$ is the generalized unit normal vectors, $x_{k}$ is the amplitude of the unsteady motion, and the $m_{k}$ -terms on the body boundary condition Eq. (5) for the radiation problem Eq. (6) implies that the steady and unsteady potentials are coupled through the presence of these $m_{k}$ -terms and are the gradient of the steady velocities in the normal direction. Equations (3)–(5) represent the steady body boundary condition, diffraction body boundary condition, and radiation body boundary condition, respectively. The linearized Bernoulli’s equation for the fluid pressure field can be written in the body fixed coordinate system as $\begin{matrix} (7) & p (\vec{x}, t) = - ρ (\frac{\partial φ}{\partial t} - U_{0} \frac{\partial φ}{\partial x}) . \end{matrix}$

Two initial conditions are required, since the free surface condition Eq. (2) is second order; $φ = φ_{t} = 0$ on $z = 0$ and $t < 0$ for the radiation problem and $φ = φ_{t} = 0$ on $z = 0$ and $t < - \infty$ for the diffraction problem. Since an initial boundary value problem is being solved, the gradient of the velocity potential must vanish ( $\nabla φ \to 0$ when $\vec{x} \to \infty$ ) at a spatial infinity for all finite time. This kind of formulation is the linearized description of the physical problem of a body starting at rest and reaching a uniform speed in the presence of an incident wave field. The more detailed discussion of the initial boundary value problem is presented by Wehausen and Laitone [57].

3. Solution of boundary integral equation

The initial boundary value problem consisting of initial, free surface and body boundary conditions for the solution may be represented as an integral equation using a transient free surface Green’s function [57]. This integral equation is derived by applying Green’s theorem over the transient free surface Green function which satisfies the initial boundary value problem without a body [11]. Integrating Green’s theorem in terms of time from $- \infty$ to ∞ using the properties of transient free surface Green’s function and potential theory, the integral equation for the potential approximation on the body surface may be written as [19]. $\begin{array}{l} φ (P, t) + \frac{1}{2 π} \int \int_{S_{b (t)}} d S_{Q} φ (Q, t) \frac{\partial}{\partial n_{Q}} (\frac{1}{r} - \frac{1}{r^{'}}) \\ = \frac{1}{2 π} \int \int_{S_{b (t)}} d S_{Q} (\frac{1}{r} - \frac{1}{r^{'}}) \frac{\partial}{\partial n_{Q}} φ (Q, t) \\ - \frac{1}{2 π} \int_{t_{0}}^{t} d τ \int \int_{S_{b (τ)}} d S_{Q} {φ (Q, τ) \frac{\partial}{\partial n_{Q}} \tilde{G} (P, Q, t - τ) \\ - \tilde{G} (P, Q, t - τ) \frac{\partial}{\partial n_{Q}} φ (Q, τ)} \\ - \frac{U_{0}^{2}}{2 π g} \int_{t_{0}}^{t} d τ \oint_{Γ (τ)} d η {φ (Q, τ) \frac{\partial}{\partial ξ} \tilde{G} (P, Q, t - τ) \\ - \tilde{G} (P, Q, t - τ) \frac{\partial}{\partial ξ} φ (Q, τ)} \\ - \frac{U_{0}}{2 π g} \int_{t_{0}}^{t} d τ \oint_{Γ (τ)} d η {φ (Q, τ) \frac{\partial}{\partial τ} \tilde{G} (P, Q, t - τ) \\ (8) & - \tilde{G} (P, Q, t - τ) \frac{\partial}{\partial τ} φ (Q, τ)}, \end{array}$ where $\begin{matrix} (9) & \begin{matrix} \tilde{G} (P, Q, t, τ) = 2 \int_{0}^{\infty} d k \sqrt{k g} sin (\sqrt{k g} (t - τ)) e^{k (z + ς)} J_{0} (k R), \\ R = \sqrt{{(x - ξ + U_{0} t)}^{2} + {(y - η)}^{2}}, \\ r = \sqrt{{(x - ξ)}^{2} + {(y - η)}^{2} + {(z - ς)}^{2}}, \\ r^{'} = \sqrt{{(x - ξ)}^{2} + {(y - η)}^{2} + {(z + ς)}^{2}}, \end{matrix} \end{matrix}$ where $Γ (t)$ is the intersection between the body surface and the free surface, $\tilde{G} (P, Q, t, τ)$ the memory part of the transient free surface source potential, $P (x (t), y (t), z (t))$ the field point, $Q (ξ (t), η (t), ς (t))$ the source point, r the distance between field and source point and represent the Rankine part of source potential, $r^{'}$ the distance between field point and image point over free surface, $J_{0}$ the Bessel function of zero order. The Green function $\tilde{G} (P, Q, t, τ)$ represents the potential at the field point $P (x (t), y (t), z (t))$ and time t due to an impulsive disturbance at source point $Q (ξ (t), η (t), ς (t))$ and time τ.

The solution of the integral equation Eq. (8) is done using time marching scheme. The form of the equation Eq. (8) is the same for both the radiation and the diffraction potentials so that the same approach may be used for all potentials. Since the transient free surface Green function $\tilde{G} (P, Q, t, τ)$ satisfies free surface boundary condition and condition at infinity automatically, in this case only the underwater surface of the body needs to be discretized using quadrilateral or triangular elements. The resultant boundary integral equation Eq. (8) in the present paper is discretized using quadrilateral elements over which the value of the potential is assumed to be constant and solved using the trapezoidal rule to integrate the memory or convolution part in time. This discretization reduces the continuous singularity distribution to a finite number of unknown potentials. The integral equation Eq. (8) is then satisfied at collocation points located at the null points of each panel. This gives a system of algebraic equations which are solved for the unknown potentials. At each time step the new value of the potentials is determined on each quadrilateral panel.

The evaluation of the Rankine source type terms (e.g. $1 / r$ , $1 / r^{'}$ ) in Eq. (8) is analytically integrated over quadrilateral panels using the method and formulas of Hess and Smith [15]. For small values of r the integrals are done exactly whilst for intermediate values of r a multi-pole expansion is used. For large values of r a simple monopole expansion is used. The surface and line integrals over each quadrilateral element involving the wave term of the transient free surface Green function $\tilde{G} (P, Q, t, τ)$ are solved analytically ([19,29,31]) and then integrated numerically using a coordinate mapping onto a standard region and Gaussian quadrature. For surface elements the arbitrary quadrilateral element is first mapped into a unit square. A two-dimensional 2 × 2 Gaussian quadrature formula is then used to numerically evaluate the surface integrals and 16 points Gaussian quadrature for line integrals. The line integral is evaluated by subdividing $Γ (t)$ into a series of straight line segments. The source strength $σ (t)$ on a line segment is assumed equal to the source strength of the panel underneath it.

The memory part of the Green function is given as $\tilde{G} (P, Q, t - τ) = \sqrt{g / {r^{'}}^{3}} \tilde{G} (μ, β)$ where $\tilde{G} (μ, β) = 2 \int_{0}^{\infty} d λ \sqrt{λ} sin (β \sqrt{λ}) e^{- λ μ} J_{0} (λ \sqrt{1 - μ^{2}})$ where $λ = k r^{'}$ , $μ = - (z + ς) / r^{'}$ , and $β = \sqrt{g / r^{'}} (t - τ)$ . λ is the relative position coordinate between field and source points. The non-dimensional parameter μ is the relative non-dimensional vertical coordinates and varies from zero to one. The non-dimensional parameter β depends on time and represents the phase of the generated waves. The evaluation of the memory part of the transient free surface Green function and its derivatives with an efficient and accurate method is one of the most important elements of the present study. Depending on the values of $(μ, β)$ , and t, the following five different methods are used to evaluate memory part $\tilde{G} (μ, β)$ ; power series expansion, an asymptotic expansion, a Filon integration quadrature, Bessel function, and asymptotic expansion of complex error function.

4. Hemi-spheroid catamaran with zero forward speed

A hemi-spheroid catamaran hull form with zero forward speed is used for numerical analysis as a first test case. This hemi-spheroid catamaran has the length to beam ratio of $L / B = 8.0$ and hull separation to beam ratio of $H / B = 2.0 .$ It is assumed that twin-hull is free for heave and pitch modes and fixed for other modes and is studied to predict heave and pitch radiation impulse response functions in time and added-mass and damping coefficients in frequency domain. The present ITU-WAVE numerical results for heave and pitch added-mass and damping coefficients at $Fn = 0.0$ and head seas $β = 180 °$ are compared with the other three-dimensional numerical results that were provided in Breit et al. [5] which are presented in Breit and Sclavounos [4].

The generalized radiation force acting on the body surface in the jth direction due to an arbitrary motion in the kth mode may be written in a form which is essentially proposed by Cummins [7]. $\begin{array}{rcl} F_{j k} (P, t) & = & - a_{j k} (P) {\ddot{x}}_{k} (t) - b_{j k} (P) {\dot{x}}_{k} (t) - c_{j k} (P) x_{k} (t) \\ (10) & - \int_{0}^{t} d τ K_{j k} (Q, t - τ) {\dot{x}}_{k} (τ), \end{array}$ where $\begin{array}{l} (11) & a_{j k} (P) = ρ \int \int_{S_{0}} d S_{Q} ψ_{1 k} (Q) n_{j}, \\ (12) & b_{j k} (P) = ρ \int \int_{S_{0}} d S_{Q} (ψ_{1 k} (Q) m_{j} - ψ_{2 k} (Q) n_{j}), \\ (13) & c_{j k} (P) = - ρ \int \int_{S_{0}} d S_{Q} ψ_{2 k} (Q) m_{j}, \\ (14) & K_{j k} (P, t) = ρ \int \int_{S_{0}} d S_{Q} {\frac{\partial}{\partial t} χ_{k} (Q, t) n_{j} - χ_{k} (Q, t) m_{j}} . \end{array}$

The displacement of the floating bodies from its mean position in each of its rigid-body modes is given $x_{k} (t)$ in Eq. (10) and the overdots indicate differentiation with respect to time. The time dependent radiation force Eq. (10) is composed of the time independent hydrodynamic coefficients and time dependent impulse response functions. The hydrodynamic coefficients in Eq. (10) $a_{j k}$ , $b_{j k}$ , and $c_{j k}$ account for the instantaneous forces proportional to the acceleration, velocity, and displacement, respectively. The coefficient $a_{j k}$ is the time and frequency independent constant and depends on the body geometry and is related to added mass. The coefficients $b_{j k}$ and $c_{j k}$ are the time and frequency independent constants and depend on the body geometry and forward speed and are related to damping and hydrostatic restoring coefficient, respectively.

The radiation impulse response (or memory) function $K_{j k} (t)$ is the force on the body in jth direction due to an impulsive velocity in kth direction. The memory function $K_{j k} (t)$ accounts for the free surface effects which persist after the motion occurs and $K_{j k} (t)$ is the time dependent part and depends on body geometry, forward speed, and time. It contains the memory effect of the fluid response. The convolution integral in Eq. (10), whose kernel is a product of the radiation impulse response function $K_{j k} (t)$ and velocity of the floating body ${\dot{x}}_{k} (t)$ , is a consequence of the radiated wave of the floating body. When this wave is generated, it affects the floating body at each successive time step [40].

The instantaneous potential $ψ_{1 k} (P)$ represents the instantaneous fluid response to the motion of the body. If the body moves and suddenly stops, the entire fluid motion associated with the $ψ_{1 k} (P)$ potential stops. The time independent impulsive potential $ψ_{2 k} (P)$ represents the potential due to the steady displacements. In other words, if the body is given a unit impulsive velocity in kth mode, the floating body will have a unit displacement in that mode. The time dependent memory potential $χ_{k} (t)$ represents the transient potential, which results from the effect of the free surface. In the case of the transient problem, all motions die out after a reasonable time and all displacements approach zero asymptotically. In other words, the transient potential $χ_{k} (t)$ is the velocity potential of the motion which results from the impulse of the floating body velocity at time $t = 0$ . The time independent impulsive potentials $ψ_{1 k} (P)$ and $ψ_{2 k} (P)$ provide initial conditions on the potentials which describe the transient motion $χ_{k} (t)$ [40].

Figure 2 shows the convergence test of radiation impulse response functions for heave and pitch modes. As hemi-spheroid twin hull form is symmetric in terms of $x z$ -coordinate plane of the reference coordinate system, only single hull form is discretized for numerical analysis. Numerical experience showed that numerical results are not very sensitive in terms of non-dimensional time step size $t * \sqrt{g / L}$ of 0.01, 0.03, and 0.05 over the range of panel numbers of 96, 150, 216, 294 on single hull of hemi-spheroid twin-hull form whilst the numerical results are quite sensitive in terms of panel numbers as can be seen in Fig. 2 and the results at panel number 216 on single hull form is converged and used for the present ITU-WAVE numerical calculations for both semi-spheroid mono-hull and twin-hull forms with the non-dimensional time step size of 0.05.

Fig. 2.

Hemi-spheroid catamaran with $L / B = 8$ and $H / B = 2$ , non-dimensional radiation heave-heave and pitch-pitch impulse response functions at $Fn = 0.0$ .

It may be noticed that the magnitude of Impulse Response Functions (IRF) of twin-hull form for heave mode Fig. 2 is approximately twice of IRF of mono-hull form while it is more than double in the case of pitch mode. The other distinctive difference of IRF of mono-hull and twin-hull forms in Fig. 2 is the behaviour of these functions in longer times. In the case of twin-hull, twin-hull IRFs has oscillation over longer times with decreasing amplitude while mono-hull IRF decays to zero just after first oscillation. This behaviour of IRF implicitly means that the energy between twin-hulls is trapped in the gap and only a minor part of the energy is radiated outwards each time the wave is reflected off the hull while all energy is dissipated in the case of mono-hull. It is expected that geometry and forward speed of twin-hull would significantly affects the radiated and trapped waves which result from due to standing waves in the gap.

The time domain radiation force coefficients Eq. (10) are related to the frequency domain force coefficients through Fourier transforms when the motion is considered as a time harmonic motion. The Fourier transform of radiation impulse response functions in time domain gives the frequency dependent added mass and damping coefficients in frequency domain and may be written as: $\begin{array}{l} (15) & A_{j k} (ω) = a_{j k} - \frac{1}{ω} \int_{0}^{t} d τ K_{j k} (τ) sin (ω τ) - \frac{c_{j k}}{ω^{2}}, \\ (16) & B_{j k} (ω) = b_{j k} + \int_{0}^{t} d τ K_{j k} (τ) cos (ω τ), \end{array}$ where the coefficients $A_{j k} (ω)$ and $B_{j k} (ω)$ are the frequency dependent added mass and damping coefficients, respectively and are obtained by Fourier transforms of Eq. (10). Figures 3 and 4 is obtained by the Fourier transform of heave-heave IRF $K_{33} (t)$ and pitch-pitch IRF $K_{55} (t)$ of Fig. 2, respectively.

Fig. 3.

Hemi-spheroid catamaran with $L / B = 8$ and $H / B = 2$ , non-dimensional heave-heave added-mass and damping coefficients at $Fn = 0.0$ .

The validation of ITU-WAVE predicted numerical results for both heave and pitch added-mass and damping coefficients are compared with those of Breit et al. [5] which are presented in Breit and Sclavounos [4]. The numerical solutions of Breit et al. [5] are based on the three-dimensional exact linear boundary integral equation formulation. ITU-WAVE numerical results of twin-hull are very good agreement with those of Breit et al. [5] as can be seen in Figs 3 and 4. In addition to twin-hull added-mas and damping coefficients in Figs 3 and 4, the mono-hull results are presented as the comparison with twin-hull results. It can be seen in Figs 3 and 4 the behaviour of both heave and pitch twin-hull results are significantly different from those of mono-hull due to trapped waves in the gap of twin-hull.

Fig. 4.

Hemi-spheroid catamaran with $L / B = 8$ and $H / B = 2$ , non-dimensional pitch-pitch added-mass and damping coefficients at $Fn = 0.0$ .

The effects of hydrodynamic interactions in both heave and pitch modes are noticed from Figs 3 and 4 in which interactions are effective in the whole frequency range. This interaction effects are even stronger in a limited frequency range which is of interest for multi-hull floating body motions in waves and is around 2 and 4.8 of non-dimensional frequency for heave mode in Fig. 3 and is around 2.2 and 4.5 of non-dimensional frequency for pitch mode in Fig. 4. The pitch-pitch added-mass coefficients become negative and damping coefficients have peak values and close to zero values at these non-dimensional frequencies both for heave and pitch modes. There would not be energy transfer or radiated waves from floating body to sea when the damping coefficients are zero. These two resonance behaviour in damping coefficients for both heave and pitch modes implies that high standing waves occur between the maximum and minimum damping coefficients [41,51]. It may be noticed the peaks are finite at non-dimensional resonance frequencies as some of the wave energy dissipate under the floating body and radiates to the far field.

5. Wigley catamaran with forward speed

A modified Wigley hull form with forward speed which has parabolic sections and waterlines is used for numerical analysis. This Wigley hull form has the length to beam ratio of $L / B = 7.0$ , length to draft ratio of $L / T = 18.0$ and duplicated to form a catamaran configuration, and actual length of Wigley hull form is 2.5 m. Twin Wigley hull form Fig. 5 is symmetric about the $x z$ -coordinate plane at the origin with the hull separation H which is defined as the horizontal distance between the two midship sections at the waterline. The ratio of hull separation to beam of $H / B = 3.14$ with $Fn = 0.15$ , 0.30 which are used in the experimental study of Siregar [46] is used for the validation of the present ITU-WAVE numerical results against experimental results. It is assumed Wigley hull form, which is free for heave and pitch modes and fixed for other modes, is studied to predict hydrodynamic coefficients and motions of Wigley catamaran with head seas $β = 180 °$ . The underwater part of analytically described Wigley hull form is given by the equation. $\begin{matrix} (17) & η = (1 - ς^{2}) (1 - ξ^{2}) (1 + 0.2 ξ^{2}) + ς^{2} (1 - ς^{8}) {(1 - ξ^{2})}^{4}, \end{matrix}$ where $η = 2 y / B$ , $ξ = 2 x / L$ , and $ς = z / T$ and L, B, T are length, beam, and draft of the floating body, respectively. The last term in the equation Eq. (17) is the modification compared to the original Wigley hull form. Siregar [46] used Wigley monohull with forward speed at the vicinity of the tank wall. It is assumed that tank wall would be regarded as the symmetry plane of the Wigley catamaran.

Fig. 5.

Wigley catamaran with $L / B = 7$ , $L / T = 18$ and $H / B = 3.14$ discretized with 432 panels with 36 panels in longitudinal direction and 12 panels along the girth direction on single hull form.

The transient generalized exciting force including Froude–Krylov and diffraction forces in the presence of an incident wave field acting on the body surface in the jth direction may be written in a form which is essentially proposed by King [29]. $\begin{array}{l} F_{j D} (t) = \int_{- \infty}^{\infty} d τ K_{j D} (t - τ) ζ (τ) \\ (18) & = \int_{- \infty}^{\infty} d τ {K_{j S} (t - τ) + K_{j I} (t - τ)} ζ (τ), \\ (19) & K_{j I} (t) = \int \int_{S_{0}} d S_{Q} \hat{p} (t) n_{j}, \\ (20) & K_{j S} (t) = ρ \int \int_{S_{0}} d S_{Q} {- \frac{\partial}{\partial t} {\hat{ϕ}}_{S} (t) n_{j} + {\hat{ϕ}}_{S} (t) m_{j}} . \end{array}$

The term $K_{j D} (t)$ in Eq. (18) has two components representing the exciting forces and moments due to the diffraction and Froude–Krylov forces, respectively. The forces are due to the incident wave elevation $ζ (t)$ which is the arbitrary wave elevation and defined at the origin of the body-fixed coordinate system Fig. 1. The kernel $K_{j D} (t)$ is the diffraction IRFs which are the forces on the body in the jth direction due to a uni-directional impulsive wave elevation with a heading angle β. The kernels $K_{j S} (t)$ and $K_{j I} (t)$ are the IRFs for diffraction (scattering) and Froude–Krylov forces, respectively and are of the form which corresponds to a time-invariant linear system since the reference point of the waves is fixed with respect to the moving floating body. $\hat{p} (t)$ is the IRF for the pressure calculation and $K_{j I} (t)$ is found by direct integration of the $\hat{p} (t)$ over the floating body surface. The scattering (diffraction) perturbation potential ${\hat{ϕ}}_{S} (t)$ represents the diffracted wave potential due to an impulsive incident wave [29].

Fig. 6.

Wigley catamaran with $L / B = 7$ , $L / T = 18$ and $H / B = 3.14$ , non-dimensional exciting heave and pitch impulse response functions at $Fn = 0.30$ and $β = 180 °$ .

The excitation of the floating body is provided by the incident wave $ζ (t)$ , which is the arbitrary wave elevation at the body-fixed coordinate system and measured at the origin of the coordinate system Fig. 1. The incident wave potential which is known and given as [29] $\begin{matrix} (21) & φ_{I} (\vec{x}, t) = \frac{i g}{ω} e^{k (z - i ϖ)} e^{i ω_{e} t}, \end{matrix}$ where the encounter frequency is given as $ω_{e} = ω - U_{0} k cos (β)$ , ω the absolute frequency of the linear system, β the angle of the wave propagation direction with the positive x-direction, k the wave number and is related to the absolute frequency ω (in the case of infinite depth) by $k = ω^{2} / g$ , and $ϖ = x cos (β) + y sin (β)$ which is the total distance in the wave direction. It is assumed that the incident wave potential Eq. (21) is a uni-directional wave system which contains all frequencies, and it describes a wave elevation which is Dirac delta function $δ (t)$ in time when it is viewed from the origin of the body-fixed coordinate system Fig. 1.

Figures 6 and 7 shows the convergence test of the exciting and radiation IRFs for heave and pitch modes, respectively. As twin Wigley hull form is symmetric in terms of $x z$ -coordinate plane of the reference coordinate system, only single hull form is discretized for numerical analysis, and symmetry condition is applied while solving the numerical problem. Numerical experience showed that numerical results are not very sensitive in terms of non-dimensional time step size $t * \sqrt{g / L}$ of 0.01, 0.03, and 0.05 over the range of panel numbers of 96, 150, 216, 294 on single hull of twin Wigley hull form whilst the numerical results are quite sensitive in terms of panel numbers as in shown in Fig. 6 for exciting IRFs. However, this is not the case for radiation IRFs Fig. 7 and convergence is achieved even with smallest panel number of 96. As the exciting force IRFs Fig. 6 are converged at panel number of 216 on single hull form, this panel number is used for the present calculations for both mono-hull and twin-hull exciting and radiation IRFs with the non-dimensional time step size of 0.05.

Fig. 7.

Wigley catamaran with $L / B = 7$ , $L / T = 18$ and $H / B = 3.14$ , non-dimensional radiation heave-heave and pitch-pitch impulse response functions at $Fn = 0.30$ .

It may be noticed that the magnitude of IRF of twin-hull for both heave and pitch modes Fig. 6 is quite big when these IRFs are compared to IRFs of mono-hull and the magnitude of the trapped waves is rapidly decaying to zero in both heave and pitch modes. Implicitly this means there would not be energy trap over longer times between hulls. Unlike the radiation IRFs, the exciting IRFs (including Froude–Krylov and diffraction IRFs which are not shown in Fig. 6) are nonzero at time $t < 0$ as can be seen from Fig. 6. This is the result of the dispersion of the free surface waves. The impulse is the wave elevation at the body-fixed coordinate system at time $t = 0$ and the response is the fluid velocity or pressure due to this incident wave elevation at the origin.

As in hemi-spheroid, it may be noticed that the magnitude of IRF of twin Wigley hull form with forward speed for both heave and pitch modes is quite big compared to IRF of mono-hull in both modes. As we expected the energy loss in the gap between hulls is much larger due to divergence of the waves when the floating bodies have forward speed. This effect can be observed clearly when zero speed case in Fig. 2 is compared with forward speed case in Fig. 7. The magnitude of the trapped waves is rapidly decaying to zero in heave mode in forward speed case.

The numerical solution of the memory part of the transient discretized boundary integral equation Eq. (8) shows an oscillatory behaviour when the impulse response function approaches a zero value for the radiation problem. The amplitude of the oscillation persists indefinitely in time at zero forward speed. The oscillatory amplitude decreases when the forward speed increases Fig. 7. This kind of behaviour of the time domain approach is related to irregular frequencies in the frequency domain formulation. Decreasing the time step size and increasing the number of elements over the body surface, the oscillatory amplitude may be removed.

The other distinctive difference between the zero and non-zero forward speed cases is the behaviour of the IRFs. In the case of zero forward speed, the IRFs decay to zero value and it is possible to obtain the Fourier transform of this kind of functions to express the results in the frequency domain. In the case of non-zero forward speed, some of the IRFs (e.g. pitch IRF in Fig. 7) do not decay to zero value in the steady state limit. This is mainly due to solution method that is used to predict IRFs. The solution of boundary integral equation Eq. (8) in time domain can be found with impulsive displacement, impulsive velocity, or impulsive acceleration memory potentials. The IRFs from the impulsive displacement memory potential approach the zero at large times with and without forward speed cases for all modes while IRFs of pitch and yaw modes from the impulsive velocity memory potential (which is used in the present study) does not approach the zero at large times in the case of forward speed (e.g. pitch IRF in Fig. 7). The impulsive displacement IRFs are the time derivative of impulsive velocity IRFs $\partial K_{j k} (t) / \partial t = K_{j k}^{d} (t)$ . Where $K_{j k} (t)$ is the impulsive velocity IRFs. $K_{j k}^{d} (t)$ is the impulsive displacement IRFs which always decays to zero value and can be used for a Fourier transform [3].

The validation of ITU-WAVE predicted numerical results for both heave and pitch motion amplitudes and phase angles at $Fn = 0.15$ and head seas $β = 180 °$ are compared with experimental results of Siregar [46]. ITU-WAVE results of twin-hull are in satisfactory agreement with those of Siregar [46] which are presented in Figs 8 and 9. In addition to twin-hull numerical and experimental results in Figs 8 and 9, ITU-WAVE numerical mono-hull results are also presented as the comparison with twin-hull results. It can be seen in Figs 8 and 9 that the behaviour of both heave and pitch twin-hull results are significantly different from those of mono-hull due to trapped waves and hydrodynamic interactions in the gap of twin-hull.

Fig. 8.

Wigley catamaran with $L / B = 7$ , $L / T = 18$ and $H / B = 3.14$ , non-dimensional heave motion amplitude and phase angle at $Fn = 0.15$ and $β = 180 °$ .

The critical reduced frequency τ, which is the ratio of forward speed of the vessel $U_{0}$ to the wave velocity $U_{w}$ , may be used to predict the position of the waves whether the waves are generated from one hull would reach to the other hull or not [54]. $\begin{matrix} (22) & τ = \frac{U_{0}}{U_{w}} = \frac{U_{0}}{λ / T} = \frac{U_{0} k}{ω_{e}} = \frac{U_{0} ω_{e}}{g} . \end{matrix}$

The wave interaction between hulls can be determined by comparing the critical reduced frequency τ with ratio of vessel length to hull separation $L / H$ . It would be assumed there would be no wave interactions between hulls as a wave generated at the bow section would not reach the aft section of the other hull if the reduced frequency τ is bigger than the ratio of length to beam $L / H = 0.796$ .

It can be observed from Fig. 8 there are no any differences between mono-hull and twin-hull results at very low and very high frequencies while the motion amplitude results for mono-hull and twin-hull results show very different behaviour between the non-dimensional frequencies of $k L = 3$ and $k L = 6$ . This different behaviour for both heave and pitch modes are due to the hydrodynamic interactions and trapped energy in the gap between hulls. The hydrodynamic interactions are significant or not can be determined using Eq. (22) which gives numerical value of 0.279 and 0.661 for non-dimensional frequencies of $k L = 3$ and $k L = 6$ at $Fn = 0.15$ , respectively. Both values are less than $L / H = 0.796$ which implicitly means that hydrodynamic interactions between the twin hulls are significant as it can be observed in Figs 8 and 9.

Fig. 9.

Wigley catamaran with $L / B = 7$ , $L / T = 18$ and $H / B = 3.14$ , non-dimensional pitch motion amplitude and phase angle at $Fn = 0.15$ and $β = 180 °$ .

It is possible to have the reduction in motion amplitude and a quite small shift in resonance frequency due to nonlinear effects which results from changing hull geometry during the oscillatory motions. Both added-mass and damping coefficients can be affected due to this type of non-linear effects which are directly related to whether the floating bodies have wall-sided cross-sections or not around waterline. As the change in waterplane area and moment is quite small due to the form of Wigley hull which has wall-sided cross-sections around waterline, it is expected the nonlinear effects in the geometry during in both heaving and pitching oscillation is not significant. However, ITU-WAVE numerical results show a small shift in resonance frequency in motion amplitudes towards higher frequencies at low Froude number $Fn = 0.15$ for both heave mode in Fig. 8 while there are no any shift at high Froude number $Fn = 0.30$ in Figs 10 and 11.

Figures 10 and 11 show the comparison of ITU-WAVE numerical results with experimental results of Siregar [46] for both heave and pitch motion amplitudes and phase angles at $Fn = 0.30$ and head seas $β = 180 °$ . ITU-WAVE numerical results of twin-hull are in satisfactory agreement with those of Siregar [46].

Fig. 10.

Wigley catamaran with $L / B = 7$ , $L / T = 18$ and $H / B = 3.14$ , non-dimensional heave motion amplitude and phase angle at $Fn = 0.30$ and $β = 180 °$ .

In addition to twin-hull numerical and experimental results in Figs 10 and 11, ITU-WAVE numerical mono-hull results are also presented as the comparison with twin-hull results. It can be seen in Figs 10 and 11 that there are no any difference between twin-hull and mono-hull results as the hydrodynamic interactions are not significant at this Froude number $Fn = 0.30$ and the energy loss in the gap is also much larger at Froude number $Fn = 0.30$ compared to low Froude number $Fn = 0.15$ .

Fig. 11.

Wigley catamaran with $L / B = 7$ , $L / T = 18$ and $H / B = 3.14$ , non-dimensional pitch motion amplitude and phase angle at $Fn = 0.30$ and $β = 180 °$ .

It can be confirmed by the use of Eq. (22) that the hydrodynamic interactions have much less effects in the case of $Fn = 0.30$ . Equation (22) gives numerical value of 0.661 and 1.734 for non-dimensional frequencies of $k L = 3$ and $k L = 6$ at $Fn = 0.30$ , respectively. Although critical reduced frequency 0.661 is smaller than hull separation ratio $L / H = 0.796$ in the lower boundary at non-dimensional frequency of $k L = 3$ , critical reduced frequency 1.734 is quite big in the higher boundary at non-dimensional frequency of $k L = 6$ . This implicitly means that the hydrodynamic interactions between the twin hulls are not as significant as in the case of $Fn = 0.15$ . This can be observed in Figs 10 and 11 for $Fn = 0.30$ and in Figs 8 and 9 for $Fn = 0.15$ when the mono-hull and twin-hull results are compared for heave and pitch modes.

6. Delft University of Technology (DUT) high-speed catamaran

DUT high-speed catamaran hull form which is designed at Delft University of Technology is used for numerical predictions of seakeeping variables. DUT catamaran has the length to beam ratio of $L / B = 12.5$ , length to draft ratio of $L / T = 20 .$ The mono-hull DUT form is duplicated to form a catamaran configuration and actual length of DUT catamaran hull form is 3.0 m. The twin DUT catamaran is symmetric about the $x z$ -coordinate plane at the origin with the hull separation H which is defined as the horizontal distance between the two midship sections at the waterline. The hull separation to beam ratio of $H / B = 2.917$ with $Fn = 0.75$ which is used in the experimental study of van’t Veer [53] is used for the validation of the present ITU-WAVE numerical results against experimental results. The experimental study is conducted assuming that hull is free for heave and pitch modes and fixed for other modes is studied to measure hydrodynamic coefficients, exciting forces, and motions of DUT catamaran. The present ITU-WAVE numerical results of added-mass, damping coefficients, exciting forces, and vertical motions including phase angles in heave and pitch modes are validated against the experimental results of van’t Veer [53] at $Fn = 0.75$ and head seas $β = 180 °$ . The present ITU-WAVE numerical results are obtained by the use of 196 panels on one hull of the catamaran Fig. 12.

Fig. 12.

DUT high-speed catamaran with $L / B = 12.5$ , $L / T = 20$ and $H / B = 2.917$ discretized with 196 panels on single hull form.

The non-dimensional heave-heave added-mass and damping coefficients are presented in Fig. 13 in which ITU-WAVE numerical results are compared with experimental results of van’t Veer [53]. As the contribution of the trim and sinkage to added-mass and damping coefficients is quite significant at high Froude number $Fn = 0.75$ , experiment was conducted to take into account these effects. The experimental results with trim and sinkage correction together with zero trim and sinkage are compared with present ITU-WAVE results in Fig. 13.

Fig. 13.

DUT catamaran with $L / B = 12.5$ , $L / T = 20$ and $H / B = 2.917$ , non-dimensional heave-heave added-mass and damping coefficients at $Fn = 0.75$ .

It can be observed from Fig. 13 the present ITU-WAVE predicted numerical heave-heave added-mass and damping coefficient results are in a good agreement with experimental results of van’t Veer [53]. The distinctive difference between with and without trim and sinkage correction in experimental results is the trend of heave-heave damping coefficients as can be observed from Fig. 13. ITU-WAVE numerical results are in a good agreement with experimental results with sinkage and trim corrections Fig. 13.

The present ITU-WAVE numerical results of the cross-coupling added-mass and damping coefficients between heave-pitch and pitch-heave modes are presented in Fig. 14 together with experimental results of van’t Veer [53] with and without trim and sinkage correction. As in Fig. 13, the numerical and experimental results are in a very good agreement in Fig. 14 in which only zero trim and sinkage results are presented for cross-coupling heave-pitch and pitch-heave added-mass results as there were no any differences between with and without trim and sinkage correction of experimental results.

Fig. 14.

DUT catamaran with $L / B = 12.5$ , $L / T = 20$ and $H / B = 2.917$ , non-dimensional cross-coupling heave-pitch and pitch-heave added-mass and damping coefficients at $Fn = 0.75$ .

Pitch-pitch added-mass and damping coefficients are presented in Fig. 15 in which the present ITU-WAVE numerical results are compared with experimental results of van’t Veer [53] and shows satisfactory agreement. It may be noticed that in Fig. 15 the experimental pitch-pitch damping coefficient with and without trim and sinkage corrections shows the same trend as opposite to heave-heave damping coefficient.

Fig. 15.

DUT catamaran with $L / B = 12.5$ , $L / T = 20$ and $H / B = 2.917$ , non-dimensional pitch-pitch added-mass and damping coefficients at $Fn = 0.75$ .

Similar to the radiation problem, the time domain exciting force is related to the frequency domain exciting force via Fourier transform. $\begin{matrix} (23) & X_{j} (ω_{e}) = \int_{- \infty}^{\infty} d τ [K_{j I} (τ) + K_{j S} (τ)] e^{- i ω τ}, \end{matrix}$ where $X_{j} (ω_{e})$ is the complex exciting force. The absolute value of Eq. (23) is the amplitude of the exciting force, while argument of Eq. (23) is the phase angle of the exciting force in the frequency domain. Thus, exciting force in the frequency domain can be determined from Fourier transform of the impulse response functions of Froude–Krylov and diffraction forces based on impulsive wave elevation.

Fig. 16.

DUT catamaran with $L / B = 12.5$ , $L / T = 20$ and $H / B = 2.917$ , non-dimensional heave exciting force amplitude and phase angle at $Fn = 0.75$ and $β = 180 °$ .

The present ITU-WAVE numerical results of the heave and pitch exciting force amplitudes and phase angles which result from the superposition of Froude–Krylov and diffraction forces are compared with experimental results of van’t Veer [53] in Figs 16 and 17, respectively. ITU-WAVE numerical results are in an acceptable agreement with experimental results of van’t Veer [53] both in amplitude and in phase at $Fn = 0.75$ although there is discrepancy in heave phase angle in Fig. 16.

Fig. 17.

DUT catamaran with $L / B = 12.5$ , $L / T = 20$ and $H / B = 2.917$ , non-dimensional pitch exciting force amplitude and phase angle at $Fn = 0.75$ and $β = 180 °$ .

The present ITU-WAVE heave and pitch response amplitude operators and phase angles are compared with experimental results of van’t Veer [53] which shows satisfactory agreement with ITU-WAVE numerical results although there are some discrepancies around resonance frequency in both heave and pitch modes for response amplitudes in Figs 18 and 19. It may be noticed the experimental results at $Fn = 0.75$ shows dependency on incident wave amplitude especially in pitch motion amplitude Fig. 19. This implicitly means that the non-linear effects would be significant around the resonance frequency.

Fig. 18.

DUT catamaran with $L / B = 12.5$ , $L / T = 20$ and $H / B = 2.917$ , non-dimensional heave motion amplitude and phase angle at $Fn = 0.75$ and $β = 180 °$ .

Fig. 19.

DUT catamaran with $L / B = 12.5$ , $L / T = 20$ and $H / B = 2.917$ , non-dimensional pitch motion amplitude and phase angle at $Fn = 0.75$ and $β = 180 °$ .

As the inertia force is balanced by restoring force for heave and pitch modes, the motion amplitudes are controlled by the damping forces at the resonance frequency while at zero frequency as there would not be velocity and acceleration, damping and inertia would not affect the motion which are controlled by restoring forces. On the other hand, at high frequencies, the response amplitude falls toward zero where the oscillation is so rapid that the floating system has insufficient time to respond appreciably.

As DUT catamaran hull has transom stern, there would be flow separation due to immersion of transom stern at high Froude numbers $Fn = 0.75$ . The flow separation significantly influences the viscous effects which would affect the response amplitude at resonance frequency as can be observed in Figs 18 and 19. In the case of the contribution from viscosity increase, the damping of the floating system would increase which result in the reduction of the motion amplitude. It is expected the potential theory based numerical approximations overpredict the motion amplitudes as the potential theory ignore the viscous effects. As ITU-WAVE numerical code depends on three-dimensional potential formulation for arbitrary multi-hull bodies, the overprediction of wave amplitudes can be observed clearly for heave and pitch modes in Figs 18 and 19 while the phase angles for both heave and pitch modes are better agreement with experimental results in Figs 18 and 19. ITU-WAVE numerical results also show a small shift against experimental results in motion amplitudes towards higher frequencies in pitch mode Fig. 19.

7. Conclusions

A three-dimensional transient wave-body interaction computer code (ITU-WAVE) with the Boundary-Integral Equation Method (BIEM) and Neumann–Kelvin linearization was developed for the time domain prediction of the motions and the first-order unsteady hydrodynamic forces including radiation, diffraction and Froude–Krylov forces of the mono-hull and multi-hull floating bodies.

It was shown that the behaviour of both heave and pitch twin-hull results are significantly different from those of mono-hull due to trapped waves in the gap of twin-hull at zero and low forward speeds. It was also shown numerically the hydrodynamics interactions are effective in the whole frequency range and are even stronger in a limited frequency range which is of interest for floating body motions in waves. The numerical experience also shows the energy loss in the gap between hulls is much larger due to divergence of the waves when the floating multi-bodies have higher forward speed. Implicitly this means there would not energy trap over longer times between hulls.

The critical reduced frequency, which is defined as the ratio of forward speed of the vessel to the wave velocity, is used to predict the position of the waves whether the waves are generated from one hull would reach to the other hull or not. The position of the waves is then used to determine hydrodynamic interactions between the hulls to predict the effectiveness of wave interactions on motion and hydrodynamic forces.

Numerical results related to radiation and exciting (including Froude–Krylov and diffraction) forces in frequency domain are obtained by the Fourier transform of radiation and exciting time domain impulse response functions, respectively in the present paper. The present ITU-WAVE numerical results for heave and pitch RAOs, added-mass and damping coefficients, exciting forces and phase angles with and without forward speed for multi-hull floating bodies of hemi-spheroid catamaran, Wigley catamaran, and DUT high-speed catamaran shows satisfactory agreement with both other numerical and experimental results.

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