Time domain three-dimensional fully nonlinear ... - CiteSeerX

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Ocean Engineering 34 (2007) 776–789 www.elsevier.com/locate/oceaneng

Time domain three-dimensional fully nonlinear computations of steady body–wave interaction problem Fuat Kara, Chun Quan Tang, Dracos Vassalos Department of Naval Architecture and Marine Engineering, The Ship Stability Research Centre, The Universities of Glasgow and Strathclyde, Scotland, UK Received 14 July 2005; accepted 19 April 2006 Available online 28 September 2006

Abstract Three-dimensional fully nonlinear waves generated by moving disturbances with steady forward speed without motions are solved using a mixed Eulerian–Lagrangian method in terms of an indirect boundary integral method and a Runge–Kutta time marching approach which integrates the fully nonlinear free surface boundary conditions with respect to time. A moving computational window is used in the computations by truncating the fluid domain (the free surface) into a computational domain. The computational window maintains the computational domain and tracks the free surface profile by a node-shifting scheme applied within it. An implicit implement of far field condition is enforced automatically at the truncation boundary of the computational window. Numerical computations are applied to free surface waves generated by Wigley and Series 60 hulls for the steady problem. The present numerical results are presented and compared with existing linear theory, experimental measurements, and other numerical nonlinear computations. The comparisons show satisfactory agreements for these hydrodynamic problems. r 2006 Elsevier Ltd. All rights reserved. Keywords: Time domain; Fully nonlinear formulation; Wave-making resistance; Computational window

1. Introduction Fully nonlinear body–wave interaction problems can be calculated using a variety of solution methods. In the recent decade, the nonlinear free surface problem was solved in the time domain by the semi-Lagrangian approach. Longuet-Higgins and Cokelet (1976) first developed the mixed Eulerian–Lagrangian method for twodimensional free-surface waves, which requires at each time step first solve a boundary value problem in an Eulerian frame and then update the free surface by integrating the fully nonlinear kinematic and dynamic free-surface boundary conditions with respect to time. Faltinsen (1977) applied this method to compute the interaction of waves with a floating body. Following these pioneering works, variations of this approach have been applied to a variety

Corresponding author. Tel.: +44 141 548 4776; fax: +44 141 548 4784.

E-mail address: [email protected] (F. Kara). 0029-8018/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2006.04.015

of nonlinear free surface problems in two dimensions like Baker et al. (1982), Vinje and Brevig (1981). More recently, the approach has been extended for three-dimensional nonlinear wave problems. Jensen et al. (1986) solved the steady nonlinear ship wave problem using a simple source distribution above the free surface. Dommermuth and Yue (1987, 1988) extended this method to fully three-dimensional nonlinear unsteady waves to solve several axisymmetric problems using spectral expansion procedure that is limited to periodic problems without bodies. Zhou and Gu (1990) tackled the problem of diffraction of nonlinear wave on a surface piercing body. Xu and Yue (1992) successfully solved the problem of overturning waves in three dimensions by using a boundary element method with bi-quadratic isoparametric curvilinear elements. Tang (2005) solved the nonlinear body– wave interaction problems of steady forward speed for both underwater and surface disturbances using fully nonlinear free surface boundary conditions. Their model is based on a first-order panel method with a fourth-order

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Runge–Kutta scheme for time marching. No incoming waves were accounted for in the computations. For three-dimensional nonlinear radiation problems, Cao et al. (1991, 1990) used this time marching solution approach with a desingularization boundary integral method for a preliminary study of the three-dimensional nonlinear unsteady waves caused by simple disturbances (a source–sink pair and a spheroid) moving under the free surface. Beck (1998) and Scorpio (1997) also used this way to calculate wave patterns caused by a Wigley hull for a series of Froude numbers. The successful application of the mixed Eulerian– Lagrangian approach requires not only a numerically stable time marching scheme but also an effective solution method to solve the boundary value problem since it requires most of the computation cost. Among the methods to solve boundary value problems, the boundary integral method is powerful because the integrals are discretized by distributing singularities of the fundamental solution over the domain boundary only, which makes the equations that must be solved satisfy only on the domain surface rather than the entire flow field. Hess and Smith (1964) presented the first panel method that utilizes a distribution of source density on the surfaces of the computational domain and solves for the distribution necessary to meet the specific boundary conditions. Once the source density distribution is known, the flow velocities both on and off the domain surface may be calculated. In the following sections, we describe the problem formulation in Section 2 and the solution procedure and methods in Section 3. A detailed discussion on the numerical implementation and methods are presented in Section 4. Finally we present the numerical results for two computation cases wave resistance generated by a Wigley hull and a Series 60 hull advancing at constant forward speed without motions, respectively. 2. Mathematical formulation of the problem 2.1. Initial boundary value problem To study the body–wave interaction problem with steady forward speed, we consider an irrotational, incompressible * flow in an ideal fluid below the free surface given by F ðx ; tÞ,

777

*

where x ðx; y; zÞ is a right-handed coordinate system with z positive upwards and the origin located at the still free surface. The fluid domain O is bounded by the following surfaces, the free surface Sf, the body surface Sb and the surface at infinity SN. The coordinate system and the problem definition of the fluid domain are shown in Fig. 1, which also presents the discretization of the free surface and the standard Wigley hull. The boundary value problem is formulated by using potential flow theory. The Laplace equation is the governing equation for the velocity potential f in the domain: r2 f ¼ 0

in O.

(1)

The fully nonlinear kinematic condition on the instantaneous free surface:   * Dx f q * * þ v r x f ¼ rf on Sf .  (2) qt Dt The fully nonlinear dynamic condition on the instantaneous free surface: Df 1  2 1 ¼ gz þ rf ¼ gz þ rf  rf on S f . (3) Dt 2 2 The non-piercing condition applied on the instantaneous body wetted surface: * * * qf * *  rf  n ðx b ; tÞ ¼ V  n ðx b ; tÞ on Sb . (4) qn The far-field condition on the surface at infinity, which means that the fluid disturbance must vanish at infinity:

rf ! 0

*

ðas x ! 1Þ,

* xf

(5)

¼ ðxf ; yf ; zf Þ is the position vector of a fluid where particle on the free surface, D/Dt is the substantial * derivative following the fluid particle, x b ¼ ðxb ; yb ; zb Þ is * the position vector of a point on the body surface, n is the unit normal vector of the body surface pointing into the * fluid, q/qn is the normal derivative on Sb, and V is the given velocity of the body surface. For free surface flow problems, initial conditions which correspond to starting the body from rest are specified such that the velocity potential on the free surface and the position of the free surface are set equal to those on the calm free surface. In order to investigate the wave motion

Fig. 1. Problem definitions of the fluid domain and the discretization of the free surface and the Wigley hull (half domain presented).

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generated by disturbances starting from rest, we specify that both the velocity potential and the free surface elevation are zero-valued at time t ¼ 0. f  0; qf  0; qt Z  0;

t ¼ 0, t ¼ 0,

ð6Þ

t ¼ 0.

(7)

3. Solution procedure and methods 3.1. Solution procedure The initial boundary value problem (1)–(5) for the body–wave problem is solved using the mixed Euler– Lagrange method with associated boundary conditions. The solution procedure is divided into two parts. In the Eulerean phase the Laplace equation with appropriate boundary conditions is solved in terms of boundary integral equations to obtain fluid velocity over body and free surfaces. In the Lagrangean phase these velocities are used to integrate the fully nonlinear free surface boundary conditions with respect to time to calculate the potential and to follow the points in the free surface. These two parts are taken place alternately at each iteration to solve the problem in time. To solve the appropriate boundary value problem using the mixed Eulerian–Lagrangian method, we specify a Dirichlet condition on the free surface and a Neumann condition on the body surface for the Laplace equation governing a domain whose boundaries change in time. The resulting boundary value problem is solved in the Eulerian frame at each time step: r2 f ¼ 0

in O;

(8)

f ¼ f0

on S f ,

(9)

qf * ( * ¼ V  n ðx b ; tÞ qn rf ! 0

This solution procedure is then repeated for each time step during the simulation time. As time goes on, it makes the computation advance in time domain.

*

on S b ,

as x ! 1, * xf

(10)

(11)

are known from the previous time step. where f0 and After solving the boundary value problem, the velocities of the fluid particles constructing the instantaneous free surface can be obtained. These velocities are then used to update the potentials and positions on the free surface by integrating nonlinear free surface conditions (2) and (3) with respected to time in the Lagrangian form. The free surface is then re-constructed on the new position. The new values of the position and potential are used as boundary input for the next time step. An effective time marching procedure is essential to advance the simulation in the Lagrangian phase.

3.2. Boundary integral method There are many methods to solve (8)–(11). The problem of solving the Laplace equation can be reduced to that of solving an integral equation. This kind of method is the boundary integral method, which reformulates the boundary value problem into a boundary integral equation. There are two versions of the method: direct and indirect. In many cases, both types of integral equations can be solved in an approximate manner. In the indirect method, the integral equation comes from potential flow theory and the solution is constructed by integrating a distribution of some fundamental solutions, namely a source distribution, over the boundary surfaces. The integral equation for the distribution is obtained by satisfying the boundary conditions on the problem boundaries. They are classically used to formulate fundamental Dirichlet, Neumann and Robin boundary value problem. In the direct method, the integral equation is obtained from Green’s second identity and the unknown values are evaluated on the boundary by the primary physical variables: potential and flux, displacement and traction vectors y and their practical applicability is not restricted to specific classes of boundary conditions. By means of indirect boundary integral method, the boundary value problem (8)–(11) can be reformulated into a boundary integral equation by a distribution of sources of constant strength over the boundary surfaces. Using a simple distribution of isolated Rankine sources, the potential at any point in the fluid domain is given by ZZ 1 * * fðx Þ ¼ (12) sðx s Þ * *  dS,  x x s  S where S is the integration surfaces of the domain * * boundary, x s is the integration point on surfaces, x is * the control point, and the sðx s Þ is the unknown source strength distributed over the surfaces. For the problem considered in this work, we construct the solution using a constant-strength source distribution over the integration surface Sf and a constant-strength source distribution over the integration surface Sb: ZZ ZZ 1 1 * * * sf ðx s Þ * *  dS þ sb ðx s Þ * *  dS. fðx Þ ¼  x x s   x x s  S S f

b

(13) The integral equations for the distribution are obtained by satisfying the boundary conditions on the problem boundaries. By applying the boundary conditions (9) and (10), we obtain boundary integral equations for the * * unknown strength of the singularities, sf ðx s Þ and sb ðx s Þ,

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respectively. ZZ 1 *  dS sf ðx s Þ * *  x  x  f s Sf ZZ 1 * *  dS ¼ f0 ðx f Þ þ sb ðx s Þ * *   x f  x s S

ðon Sf Þ

ð14Þ

b

and ZZ q 1 *  dS sf ðx s Þ *  qn  x  * x  b s Sf ZZ * * q 1 *  dS ¼ V  n þ sb ðx s Þ *  qn  x  * x s b S

ðon S b Þ,

ð15Þ

b

*

*

where x s is the integration point on surfaces Sf and Sb, x f * is the control point on surfaces Sf, x b is the control point on surface Sb. The integral Eqs. (14) and (15) are then solved by using the panel method from Hess and Smith (1964) in terms of the plane quadrilateral elements. In this approach, the integral equations for the unknown distribution and the boundary conditions are evaluated on the so-called null points, which is the point where the element itself has no effect on the tangential velocity, i.e., the point where the quadrilateral gives rise to no velocity in its own plane. * * After sf ðx s Þ and sb ðx s Þ are determined, the velocities of the fluid particles on the free surface can be calculated. These velocities are then used in the time marching scheme to update the potential and position on the free surface. 3.3. Time marching scheme The updated values of the potential and position of the free surface are determined in Lagrangian stage by integrating the fully nonlinear free surface boundary conditions with respect to time to make the solution advanced in time step by step. On the free surface, the kinematic boundary condition (2) is used to determine the free surface movement and the dynamic boundary condition (3) is used to update the potential. These fully nonlinear free surface boundary conditions can be rewritten in a simple form: Dxf qf þ U 0, ¼ qx Dt

(16)

Dyf qf , ¼ qy Dt

(17)

Dzf qf , ¼ qz Dt

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hand sides of Eqs. (16)–(19) are the derivatives of potential and position of the moving particles with respect to time. The quantities on the right hand sides are all known: the spatial gradient of the potential are determined analytically after solving the boundary integration equations and free surface position is not changed since the last time step. Thus the new potential f and position ðxf ; yf ; zf Þ on the free surface can be easily determined by integrating the above equations with respect to time. This approach has an advantage that no other spatial derivative is required on the free surface, which helps reduce numerical reflection from the truncated boundary. Many different approaches are possible to integrate these free surface conditions with respect to time to update the values on free surface. Among these time-marching methods, explicit Runge–Kutta method is the traditional method for time-accurate calculations. In this work, the evolution of the free surface and the potential on the free surface is computed by a fourth-order Runge– Kutta method. In the Runge–Kutta method, each of the evolution Eqs. (16)–(19) is regarded as an ordinary differential equation. 3.4. Pressure and force calculation The pressure on the body surface is given by Bernoulli’s equation: p qf 1 ¼   gz  rf  rf r qt 2     * df 1 þ gz þ rf  V  rf , ¼  dt 2

ð20Þ

where rf ¼ ðfx ; fy ; fz Þ is the spatial derivation of the *

potential on the body, and df=dt ¼ ðq=qt þ V rÞf is the material derivation of the potential at fixed points on the body surface and calculated using a two-point backward difference scheme. The second form is more useful when following points fixed on the body moving with *

velocity V . The forces and moments acting on the body are calculated by integrating the pressure over the instantaneous wetted surface: ZZ * * F¼ p n ds (21) Sb

and *

(18)

Df 1 ¼ rf  rf  gz, (19) Dt 2 where U0 is the velocity of the forward speed and D=Dt ¼ q=qt þ rf  r is the usual material derivation. The left

ZZ

*

*

pð r  n Þ ds,

M¼

(22)

Sb *

where n is the generalized unit normal to body surface * (out of fluid) and r is the position vector of the body surface point to a reference point (usually the center of the body).

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4. Numerical implementation and methods 4.1. Discretized panel method The integral Eqs. (14) and (15) are solved by a panel method in which the integrals are discretized to form a system of linear equations. The discretization is for both the boundary surfaces and the singularity element distribution, in which the surfaces are divided into plane quadrilateral elements and isolated sources of constant strength are distributed on each element. The boundary conditions are then satisfied at collocation or node points on the elements, while the sources are distributed on the source points on the elements. In the approach presented here, the collocation or node points and source points on each element are determined by the same way in Hess and Smith (1964). Using this discretization, the integral equations reduce to a simple summation of the influence of each isolated Rankine source. Thus, the discretized forms of integral Eqs. (14) and (15) which satisfy the boundary conditions at the each node of the free surface and body surface are NF X j¼1

NB X sFj sBj *F  F    ¼ f0 ðx ci Þ þ F F B *  *  * *  x ci  x sj  j¼1  x ci  x sj 

ði ¼ 1; . . . ; N F Þ (23)

and NF X j¼1

0 sFj

1

q B 1 C @*B *F A þ qni  x  x  ci sj

* *B

*

¼ V ðx ci Þ  n

NB X

0 sBj

j¼1

1

q B 1 C @*B *B A qni  x  x  ci sj

ði ¼ 1; . . . ; N B Þ;

ð24Þ

where NF is the number of the isolated sources of the free surface; sFj is the unknown strength of the jth free surface source at location xFsj . Similarly, NB, xBsj and sBj are the number, location and strength of the sources of the body. *F x ci

*B

and x ci are the node points of the free surface and body surface, respectively. Eqs. (23) and (24) make up of a system of N ¼ N F þ N B equations for the unknowns source strengths sf and sb. Eqs. (23) and (24) make up a system of equations for the boundary value problem. These linear systems of equations are often solved by iterative methods due to their reasonable computational and storage demands. In this work, we adopt the Generalized Minimal Residual algorithm (Saad and Schultz, 1986), one of the most widely used Krylov subspace iterative methods. The GMRES method minimizes residual norms at each iteration step.

during the simulation. The purpose of adopting the computational window is to concentrate on the free surface wave generated only around the body no matter where the body goes. Since the computational window moves with the body, the position of the body within it is fixed. If we set up a reasonable computational window for the free surface, we are able to obtain the typical behavior of the wave at downstream as well as around the body within the computational window. In fact, the computational window functions as the computational domain for the simulation and the far-field boundary condition can be imposed on the upstream and downstream boundary of the computational window. In the simulation with forward speed, from the point of view of the body, the fluid particles have a tendency to drift downstream in the stream-wise direction. New particles with zero value of potential and elevation drift into the window at upstream while fluid particles drift out of the window at downstream at certain time steps. In order to maintain the same number of the fluid particles in the computational window, we specify the same number of fluid particles in and out. As the time goes on, more and more new fluid particles are introduced into the computational window, while more and more particle are discarded out of the window. There is a special treatment in this process: only the fluid particle itself drifted away, but the values of position and potential of the drifted particle are retained at the original position in the computational window. It is this mechanism that makes it possible to keep the wave behavior and free surface profile in the computational window. This process is called nodes shifting (e.g. Tang, 2005) since the actual process takes place on the nodes or collocation points of the free surface. It is not simply to move the drifted fluid particles back to their original position, but shift the value of the particle to a particle at upstream in the stream-wise direction. Fig. 2 illustrates the process of node shifting. The last four particles in the computational window at time t are discarded while four new particles are introduced into the computational window at time t+Dt. For the other

4.2. Computational window and node shifting scheme For long time simulations with a reasonable-sized fluid domain, we use a computational window moving with the disturbances and maintaining the computational domain

Fig. 2. The process of the nodes shifting. The domain in the black rectangle is the computational window at time t; the domain in the red rectangle is the computational window at time t+Dt; the black particles belong to the computational window at time t; the red particles belong to the computational window at time t+Dt.

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particles, the black particles in the computational window at time t shift its values to the nearby red particles in the computational window at time t+Dt. So the computational window at time t+Dt is able to have the same value as the computational window at time t at the same position. Meanwhile, the computational window at time t+Dt takes in new fluid particles of zero values at it upstream boundary. This process of node shifting is actually consistent with the far-field boundary condition (25) at truncation boundary so that the far field condition can be implemented implicitly in the node shifting scheme. 4.3. Far-field boundary condition Generally speaking, in the initial boundary value problem, fluid motion generated by disturbances goes to zero at a spatial infinity in finite time due to the resting initial conditions. So the infinity condition at the infinity boundary may be written as rf ! 0;

*

x ! 1.

(25)

Eq. (25) is the far-field condition meaning that the fluid disturbance must vanish at infinity. In the computations for the steady forward speed problem, to make the problem well posed, a simple numerical technique is used to implicitly implement the far-field boundary condition (Tang, 2005): at each iteration, the computational domain takes in new nodes with zero values at the upstream boundary and releases nodes at the downstream boundary without consider the values on them. We assume that the nodes consisting of the computation domain convect downstream during the progress of free surface profile because the computation domain follows the body that is moving upstream. It is also assumed that the values of potential and elevation on the nodes beyond the upstream boundary are both zero initially on the ideal undisturbed free surface. So we can consider these nodes as new zero-valued upstream boundary nodes and replace the old ones at each iteration. At the downstream boundary we simply discard the nodes there because they move out the computation domain. So there is no boundary condition explicitly imposed on the downstream boundary. On the side boundaries of the computation domain there are also no special treatments imposed, since we do not need to consider wave reflection to take place because the free surface we considered is open and the reflected waves are outside the domain of interest. 4.4. Free surface grid regeneration For the boundary to track the free surface, a new grid must be generated at each time step. At the end of each time step a grid regeneration scheme is arranged on the free surface to determine the approximation of the actual free surface for the next time step.

781

One of the difficulties associated with determining the free surface grid is that the free surface is a moving boundary and not known in advance, and yet an exact formulation requires the free surface boundary conditions are satisfied on it. The standard method to deal with this challenge is to adopt a grid regeneration method whereby interpolation is used to approximate the actual free surface. In the current grid regeneration algorithm, newborn free surface nodes are produced from the time integration of kinematic boundary condition. After node shifting, these nodes in the computational window are suitable to approximate new free surface by interpolation. We describe our algorithm that it takes as input an unorganized set of points on or near an unknown surface S, and produces as output a simple surface that approximates S. Neither the topology, the presence of boundaries, nor the geometry of S is assumed to be known in advance—all are inferred automatically from the input data. This approach requires that interpolation is carried out on a finite set of scatter points, and the constructed surface passed through the given points. For surface-piercing problem there is intersection part between the Dirichlet and Neumann boundaries, i.e. between the free surface and body surface. To track the moving intersection line, the panel vertices at the edges of the surfaces are used to describe the intersections between the surfaces. The panels from each side are mutually connected since they have these panel vertices in common along the intersection line. The resolution of the panel vertices from both sides matches at the intersection line which implies that panels indeed connect on the intersection line. 5. Numerical computations and results The actual fluid domain of the problem is defined by the free surface and a hull surface of interest, with infinite depth. To be numerically computed, the physical domain has to be converted into a computational domain, which is truncated from the infinite physical domain in a distance from where it has as little as negative effect to the computation. The actual computational domain is then represented by a computational window, which includes a rectangular area truncated from the calm free surface, and the body/hull surface of interest at a suitable location within the window. The whole computational domain is assumed to be symmetry about the x–z plane and we take half domain to save the computational cost. The truncated free surface is discretized evenly both in x and y directions to form N f ¼ N fx  N fy uniform panels. The hull is discretized into N b ¼ N bx  N by quadrilateral panels on its wetted surface. Discretization of the surfaces and the surface grids is shown in Fig. 1. There are totally N ¼ N f þ N b panels on the surfaces of the domain considered and the same number of Rankine sources of constant strength distributed over the surfaces, each at a collocation point of the Nf panels on the free surface and Nb panels on the body

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surface of interest. The potential f due to the unknown sources can be expressed as a summation of: (1) The sum of Nf strength-unknown sources of the free surface; (2) The sum of Nb strength-unknown sources of the body surface; and (3) The negative images of (2) above the calm free surface, if we do add negative images of the body about the calm free surface to improve the far-field effect to the computations. The initial boundary conditions of zero potential f ¼ 0 and zero wave elevation Z ¼ 0 are prescribed on the z ¼ 0 plane of the calm free surface at t ¼ 0. In the computations, the Wigley and Series 60 hull starting from rest move in the pffiffiffiffiffiffi x direction with a final speed, V ðtÞ ¼ Fn  gL, where Fn is the Froude number. The initial time increment for the time-marching step is set according to the Froude number. The pressure is integrated over the instantaneous wetted surface in (21) and (22). The usual steady problem coefficients CD, CL, CM and the wave resistance components of the Wigley hull are nondimensionalized by 1/(0.5rU2S), where S is the area of the instantaneous wetted surface. 5.1. Numerical computations on standard Wigley hull The standard Wigley hull is of mathematical hull form with its geometric surface defined as "  2 #  z 2 B 2x 1 yðx; zÞ ¼  , (26) 1 2 L T

where L is the hull length, B the full hull beam and T the hull draft. For the standard Wigley hull used in this computation, the length-to-beam ratio L/B is 10 and the beam-to-draft ratio B/T is 1.6. 5.2. Wave resistance at mean hull surface In this computation case, the Wigley hull has been fixed in position and never changed the wetted hull surface. So the results given here are based on a model-fixed Wigley hull at the mean hull surface. In the computations, 75  24 nodes are used on the free surface and 32  4 nodes on the hull wetted surface. The computational domain extends 2.75 ship lengths downstream, 1.0 ship lengths upstream, and 1.2 ship lengths in the y-direction. To overcome the difficulty that free-surface nodes cross the hull surface between time steps, we have the free-surface nodes around the hull surface re-positioned in y-direction before the next time step. The computed wave patterns for the standard Wigley hull at Froude number 0.316 (for other Froude numbers see Tang, 2005) are shown in Fig. 3. These are threedimensional views of the free surface profiles and the elevation of the free surface is multiplying by a factor of 16 in z-direction. The views are taken at different angles from the air, side and back, respectively. There are no wave reflections at the side- and downstream-boundaries since they are open boundaries for the computational domain. Fig. 4 presents the comparisons of the wave elevations along the hull between the experimental measurements and the numerical results from the current nonlinear solutions

Fig. 3. Wave patterns generated by a Wigley hull advancing at Froude numbers Fn ¼ 0.316.

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Experiment

Nonlinear

Experiment

0.4

0.3

0.2 0.1 0.1

0.3

0.5

0.7

0.9

1.1

Wave Elevation

Wave Elevation

0.3

0.2 0.1

0 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 -0.1

0.1

-0.2

-0.2

-0.3 X/L

-0.3 X/L

Experiment

Nonlinear

Experiment

Fn = 0.289 0.4 0.3

0.3

0.2

0.2

0.1

0 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 -0.1

0.1

0.3

0.5

0.7

0.9

1.1

0.7

0.9

1.1

Nonlinear

Fn = 0.316 0.4

0.3

0.5

0.7

0.9

1.1

Wave Elevation

Wave Elevation

Nonlinear

Fn = 0.267 0.4

Fn = 0.250 0.5

0 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 -0.1

783

0.1

0 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 -0.1

0.1

-0.2

-0.2

-0.3 X/L

-0.3 X/L

0.3

0.5

Fig. 4. Comparisons of wave elevation along Wigley hull with experimental measurements s at Fn ¼ 0.250, 0.267, 0.289 and 0.316, respectively. Experimental measurements are from Kajitani et al. (1983) for model-fixed hull; nonlinear values are from the current nonlinear solutions.

for Froude numbers 0.250, 0.267, 0.289 and 0.316, respectively. The experimental measurements are for model-fixed Wigley hull and given in Kajitani et al. (1983). The nonlinear numerical values agree well with the experimental measurements over the series of Froude numbers considered. The nonlinear calculations indicate significant approximation to the actual free surface elevations along the hull and the wave profiles are quite smooth. The wave elevations from the current solutions are a little lower near the bow than the experimental measurements, in particular at Froude number Fn ¼ 0.250. The rest of the wave elevations along the hull have satisfactory agreement with the experimental measurements, in particular at Froude number Fn ¼ 0.250, where the wave elevation completely approaches to the experimental data. A comparison of wave resistance coefficients with the published results is given in Fig. 5. The wave resistance coefficients are of the form C f ¼ F 1 ð0:5rU 2 SÞ. The published results compared here are from Beck (1998) and Scorpio (1997), which presented four wave resistance coefficients: (1) Cpr is the experimental data obtained

by pressure integration over the hull wetted surface, (2) Cw is the estimation for residual resistance from the resistance test, (3) Cwp is from wave pattern analysis, and (4) CUM-DELTA is the numerical results computed using the UM-DELTA method. As can be seen in the figure, the coefficients from the current nonlinear computations Cpresent approach Cpr for lower Froude numbers and fall into the range of the experimental measurements as Froude number increases, at last approach Cw. The current results also agree with the results of nonlinear computations from the UM-DELTA. 5.3. Wave resistance at instantaneous hull surface In this computation case, the form of the Wigley hull will no longer keep fixed and its wetted surface changes corresponding to the free surface wave elevation at each time step. The exact body boundary condition should be applied on the instantaneous submerged body surface and satisfied instantaneously. The boundary surfaces of the computational fluid domain are therefore iteratively updated by the Lagrangian markers on the free surface

ARTICLE IN PRESS F. Kara et al. / Ocean Engineering 34 (2007) 776–789

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0.002

Coef

0.0018 0.0016

Cpr

0.0014

Cw Cwp

0.0012

Cum_delta 0.001

Cpresent

0.0008 0.0006 0.0004 0.24

0.26

0.28

0.3

0.32

0.34

Fn Fig. 5. Comparisons of the wave resistance coefficients vs. Froude numbers.

Experiment

Mean Position

Instantaneous

Fn = 0.316 0.4

Wave Elevation

0.3 0.2 0.1 0 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 -0.1

0.1

0.3

0.5

0.7

0.9

1.1

-0.2 -0.3 X/L Fig. 6. Comparisons of wave elevation along Wigley hull at the mean position and the instantaneous wetted surface at Froude Numbers Fn ¼ 0.316. The experimental measurements are from Kajitani et al. (1983) for model-fixed hull; the mean position values are for the modelfixed Wigley hull; the instantaneous values are for the instantaneous wetted surface.

and by the instantaneous position of the wetted surface of the hull. Fig. 6 presents the comparisons of wave elevations along the hull generated by the model fixed Wigley hull at the mean position and by the hull with instantaneous wetted surface for Froude number 0.316 (for other Froude numbers see Tang, 2005). The experimental measurements from Kajitani et al. (1983) for model-fixed Wigley hull are also plotted together. The instantaneous wave elevations agree satisfactorily with the mean position results along the hull. The bow wave elevations from the instantaneous cases are still under estimation than these from the experimental measurements, but better than these from mean position cases, especially at low Froude numbers. This indicates that we model the body surface more accurately when we consider the instantaneous wetted surface for the hull. The

instantaneous wetted surface case has greater value at the first trough and the second crest. The reason is likely that we include a relatively large part of the wetted surface at the first crest and exclude a relatively large part of hull surface at the first trough, which may result in significant influences on the downstream part after them. These influences are correlated and have effects on each other. Alternatively, we should note that the experimental measurements used here are for the model-fixed hull only, which implies that the instantaneous experimental measurements should have the same behaviors as the instantaneous numerical solutions. It is obvious from the above discussion that the construction of the instantaneous wetted hull surface should be treated accurately by using a proper approach in order to get better solutions to the wave profile. 5.4. Wave pattern comparisons of linear and nonlinear solutions at far field for mean hull surface In this section, we try to compare the free surface waves at different distances from the centerline of the hull, which are computed using different far-field radiation conditions, and investigate the nonlinearity on the free surface in the computation domain. The nonlinear wave computation uses a simple treatment implemented by the node-shifting scheme. The linear wave computation is a solution (Kara, 2000; Kara and Vassalos, 2003, 2005) of transient outgoing radiated waves, which automatically satisfies the free surface boundary condition and condition at infinity. The comparisons are carried out on a fictitious boundary some distance away between the outer linear domain and the inner non-linear domain. The body used is a model fixed Wigley hull. Fig. 7 illustrates the layout on the free surface and panel discretization. The computed wave patterns generated by a standard Wigley hull of model-fixed at Froude number Fn ¼ 0.30 with linear and nonlinear computations solved together are presented in Fig. 8. These are three-dimensional views of

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Fig. 7. Illustration of the layout of the outer domain and the inner domain and panel discretization on the free surface. The green panels are in the inner domain for the nonlinear solution; the pinks panels are on the outer domain for the linear solution; the yellow panels are the panels on the fictitious boundary.

Fig. 8. Wave patterns generated by a Wigley hull advancing at Froude number Fn ¼ 0.3 with a fictitious boundary. The green panels are in the inner domain for the nonlinear solution; the pinks panels are on the outer domain for the linear solution; the yellow panels are the panels on the fictitious surface.

the free surface profiles and the elevation of the free surface is multiplying by a factor of 16 in z-direction. The views are taken at different angles from the air, side and back,

respectively. In this computation, 32  4 nodes are used on the hull wetted surface and 75  24 nodes on the free surface. The domain parameters and panel spacing on the

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free surface are the same as Fig. 3. The computational domain extends 2.75 ship lengths downstream, 1.0 ship lengths upstream, and 1.2 ship lengths in the y-direction. The fictitious boundaries extend 1.0 ship lengths upstream, and 0.9 ship lengths in the y-direction. There are 820 panels on the outer linear domain, 897 panels on the inner nonlinear domain and 83 panels on the fictitious surface. As we can see from Fig. 8, the wave profiles continue smoothly from the inner nonlinear domain to outer linear domain. The crests and troughs of the wave profiles from the outer and inner domain have the same phases and elevations at the fictitious surface. Fig. 9 shows the wave cuts of the free surface profiles generated by a Wigley hull advancing at Froude number Fn ¼ 0.30. To investigate the influence of the nonlinear effect around the hull, we set the fictitious boundary at different distances from the centerline of the hull and obtain the wave elevations on these fictitious boundaries. These wave cuts are then compared with the wave cuts from the complete nonlinear computation (that is, the computational domain consists of only nonlinear domain) at the same distances from the centerline of the hull. The distances we take are Y =L ¼ 0:225, 0.325, 0.425, 0.525 and 0.625, respectively. Our computation do not converge for the distance less than Y =L ¼ 0:2. The two wave elevations does not agree well for a closer distance such as Y =L ¼ 0:225. The poor simulation of the behaviour on the fictitious boundary indicates that linear solution is not applicable at a closer distance where the nonlinearity is still quite strong. The wave elevations agree quite well for a farther distance at Y =L ¼ 0:625 and the values of the wave elevations approach the same and the downstream wave behaviour is almost similar. These comparisons indicate that the nonlinear effect is greater where closer to the hull and the linear solution cannot get satisfactory results there. In general, the farther the distance, the less nonlinear effect and the better agreement between the two wave elevations. This conclusion is helpful when we choose the distance for the fictitious boundary. 5.5. Numerical computations on Series 60 hull For the Series 60 hull used in this computation, the length-to-beam ratio L/B is 7.61, the draft-to-beam ratio B/T is 0.4 and block coefficient CB is 0.6. In this computation case, the Series 60 hull has been fixed in position and never changes the wetted hull surface. So the results compared with experimental data are based on a model-fixed Series 60 hull at the mean hull surface. In the computations, 90  25 nodes are used on the free surface and 40  5 nodes on the hull wetted surface. The hull length is 3.096 with bow position at 1.524 and stern position at 1.572. The computational domain extends 1.5 ship lengths downstream, 0.75 ship lengths upstream, and 0.6 ship lengths in the y-direction. The computed wave pattern for the Series 60 hull at Froude numbers 0.32 (for other Froude numbers see Tang,

2005) are shown in Fig. 10. These are three-dimensional views of the free surface profiles and the elevation of the free surface is multiplying by a factor of 8 in z-direction. The views are taken at different angles from the air, side and back, respectively. The experimental measures and the current numerical computations of the wave elevations along the hull for Froude numbers 0.28, 0.30, 0.32 and 0.35 are shown in Fig. 11. The wave elevation calculated by the extended SWIFT approach from Kim and Lucas (1991) is also plotted together. Note that the experimental data determined by Kim and Jenkins (1981) are for model-fixed Series 60 hull only. The comparison of wave elevation along the hull between computed and measured is generally satisfied. Our non-linear solutions significantly agree with the wave profiles at the bow and the first trough, which have better estimates of wave profile than the SWIFT solution. The agreement of the wave profiles improves significantly as the Froude number increases. In particular at Froude number 0.35, the wave elevation from our nonlinear solution significantly agrees with the experiment measures. A comparison of nonlinear wave resistance coefficients against Froude numbers with the experimental data is given in Fig. 12. The wave-making resistance coefficients

are of the form C w ¼ F 1 ð0:5rU 2 SÞ. The experimental results for Series 60 hull with CB ¼ 0.6 are available online (https://pronet.wsatkins.co.uk/marnet/) and cover a range of Froude number from 0.0847 to 0.3691. The nonlinear results are calculated for Froude numbers at 0.20, 0.22, 0.25, 0.28, 0.30, 0.32, 0.35 and 0.40. The nonlinear results show very close agreement with the experimental data for Froude numbers from 0.20 to 0.30. But from 0.30 the nonlinear results are deviated from the experimental data. It is likely that the aberrance results from the incompatibility of the fixed hull surface with the increased Froude number. The wave elevation along the body surface changes a lot at higher Froude number than at lower Froude number. But the nonlinear computations always use the same surface-fixed Series 60 hull to obtain the wave resistance.

6. Conclusions A solution to three-dimensional nonlinear body–wave interaction problems of steady forward speed without motions has been constructed using the mixed Euler– Lagrange method. The initial boundary value problem is solved in terms of indirect boundary integral method and Runge–Kutta approach which integrates the fully nonlinear free surface boundary conditions with respect to time. This method has been applied to a variety of computational cases and compared with experimental measurements and other numerical computations, which shows satisfactory agreement and accuracy. The three-dimensional wave

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Wave Elevation

Y/L = 0.225

-2

-1.5

-1

-0.5

0.01 0.008 0.006 0.004 0.002 0 -0.002 0 -0.004 -0.006 -0.008 -0.01

Nonlinear Linear

0.5

1

X/L

Wave Elevation

Y/L = 0.325

-2

-1.5

-1

-0.5

0.01 0.008 0.006 0.004 0.002 0 -0.002 0 -0.004 -0.006 -0.008 -0.01

Nonlinear Linear

0.5

1

X/L

Wave Elevation

Y/L = 0.425

-2

-1.5

-1

-0.5

0.01 0.008 0.006 0.004 0.002 0 -0.002 0 -0.004 -0.006 -0.008 -0.01

Nonlinear Linear

0.5

1

X/L

Wave Elevation

Y/L = 0.525

-2

-1.5

-1

-0.5

0.01 0.008 0.006 0.004 0.002 0 -0.002 0 -0.004 -0.006 -0.008 -0.01

Nonlinear Linear

0.5

1

X/L

Wave Elevation

Y/L = 0.625

-2

-1.5

-1

-0.5

0.01 0.008 0.006 0.004 0.002 0 -0.002 0 -0.004 -0.006 -0.008 -0.01

Nonlinear Linear

0.5

1

X/L

Fig. 9. Comparisons of wave elevations on the fictitious boundary at different distances from the centerline of the hull at Froude number Fn ¼ 0.3. The linear values come from the linear computations; the nonlinear values come from the complete nonlinear computations.

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Fig. 10. Wave patterns generated by a series 60 hull advancing at Froude numbers Fn ¼ 0.32.

SWIFT

0.3

0.3

0.2

0.2

0 -0.3 -0.1 -0.1

0.1

0.3

0.5

0.7

0.9

1.1

Wave Elevation

Fn = 0.30 0.4

0.1

-1.1 -0.9 -0.7 -0.5

SWIFT

Fn = 0.28 0.4

-1.1 -0.9 -0.7 -0.5

0 -0.3 -0.1 -0.1

0.1

-0.2

-0.3 X/L

-0.3 X/L

SWIFT

Present

Experiment

Fn = 0.35 0.4

0.3

0.3

0.2

0.2

0 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 -0.1

0.1

0.3

0.5

0.7

0.9

1.1

0.3

SWIFT

Fn = 0.32 0.4

0.1

Present

0.1

-0.2

Experiment

Wave Elevation

Experiment

Present

Wave Elevation

Wave Elevation

Experiment

0.5

0.7

0.9

1.1

0.9

1.1

Present

0.1 0 -1.1 -0.9 -0.7 -0.5 -0.3 -0.1 -0.1

0.1

-0.2

-0.2

-0.3 X/L

-0.3 X/L

0.3

0.5

0.7

Fig. 11. Wave elevation comparison at hull surface series 60, CB ¼ 0.6. The experimental values for model-fixed were determined by Kim and Jenkins (1981); the SWIFT approach was extended by Kim and Lucas (1991); the present values are from the current computations for model-fixed hull.

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Experiment

Nonlinear

CB=0.6 5

CW∗1000

4 3 2 1 0 0.05

0.15

0.25 Fn

0.35

0.45

Fig. 12. Comparison of the wave resistance coefficients vs. Froude numbers for Series 60, CB ¼ 0.6 at Fn ¼ 0.20, 0.22, 0.25, 0.28, 0.30, 0.32, 0.35 and 0.40.

patterns present a clear and apparent view of the characteristics of free surface wave. We also compare free surface wave patterns generated by computations using different far-field radiation condition schemes. The nonlinear computation uses a simple treatment implemented by the node-shifting scheme. The linear wave computation is a solution of transient outgoing radiated waves, which automatically satisfies the free surface boundary condition and condition at infinity. The comparisons are carried out on a fictitious boundary some distance away between the outer linear domain and the inner nonlinear domain. We get similar wave patterns and smooth wave profile between the inner nonlinear solution and the outer linear solution. The influence of the nonlinear effect around the hull is investigated and we conclude that the closer to the hull, the stronger the nonlinear effect. The results obtained from nonlinear computations are of great interest when considering cases for which the solution exhibits strongly nonlinear characteristics, as accurate solutions for such cases can not be obtained from the linear theory. References Baker, G.R., Meiron, D.I., Orszag, S.A., 1982. A generalized vortex methods for free surface flow problem. Journal of Fluid Mechanic 123, 477–501. Beck, R.F., 1998. Fully nonlinear water wave computations using a desingularized Euler–Lagrange time-domain approach, Nonlinear Water Wave Interaction, International Series on Advances in Fluid Mechanics, UK. Cao, Y., Schultz, W.W., Beck, R.F., 1990. Three-dimensional, unsteady computations of nonlinear waves caused by underwater disturbances. In: Proc. 18th Symposium on Naval Hydrodynamics, pp. 417–425.

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Cao, Y., Schultz, W.W., Beck, R.F., 1991. A three-dimensional desingularized boundary integral method for potential problems. International Journal of Numerical Methods in Fluids 12, 785–803. Dommermuth, D.G., Yue, D.K.P., 1987. Numerical simulations of nonlinear axisymmetric flows with a free surface. Journal of Fluid Mech. 178, 195–219. Dommermuth, D.G., Yue, D.K.P., 1988. The nonlinear three-dimensional waves generated by a moving surface disturbance. In: Proc. of 17th Symposium on Naval Hydrodynamics, The Hague. Faltinsen, O., 1977. Numerical solution of transient nonlinear free surface motion outside or inside moving bodies. In: Proceedings of the 2nd International Conference on Numerical Ship Hydrodynamics, Berkeley, California, C.A., pp. 347–357. Hess, J.L., Smith, A.M.O., 1964. Calculation of nonlifting potential flow about arbitrary three-dimensional bodies. Journal of Ship Research 8 (2), 22–44. Jensen, G., Mi, Z.X., Soding, H., 1986. Rankine source methods for numerical solutions of the steady wave resistance problem. In: Proc. of 16th Symposium on Naval Hydrodynamics. University of California, Berkeley. Kajitani, H., Miyata, H., Ikehata, M., Tanaka, H., Adachi, H., 1983. Summary of the Cooperative Experiment on Wigley Parabolic Model in Japan. In: Proceedings of the Workshop on Ship Wave Resistance Computations, pp. 5–35. Kara, F., 2000. Time domain hydrodynamic & hydroelastic analysis of floating bodies with forward speed, PhD. thesis, Department of Ship and Marine Technology, University of Strathclyde in Glasgow, UK. Kara, F., Vassalos, D., 2003. Time domain prediction of steady and unsteady marine hydrodynamic problem. International Shipbuilding Progress 50 (4), 317–332. Kara, F., Vassalos, D., 2005. Time domain computation of wavemaking resistance of ships. Journal of Ship Research 49 (2), 144–158. Kim, Y.H., Jenkins, D., 1981. Trim and Sinkage Effects on Wave Resistance with Series60, CB ¼ 0.60, DTNSRDC/SPD-1013-01. Kim, Y.H., Lucas, T., 1991. Nonlinear ship waves. Eighteenth Symposium on Naval Hydrodynamics, pp. 439–452. Longuet-Higgins, M.S., Cokelet, C.D., 1976. The deformation of steep surface waves on water: I. A numerical method of computation. In: Proceedings of Royal Society of London, volume A350, pp. 1–26. Saad, Y., Schultz, H., 1986. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal of Scientific Computation 7 (3), 856–869. Scorpio, S.M., 1997. Fully non-linear ship-wave computations using a multipole accelerated, desingularized method, Ph. D. thesis, University of Michigan. Tang, C.Q., 2005. Time Domain Three-Dimensional Fully Nonlinear Computations For body–Wave Interaction Problems In A Dynamic Visualization Architecture. Ph. D. thesis, University of Strathcldye, UK. Vinje, T., Brevig, P., 1981. Nonlinear ship motions. In: Proceedings of the 3rd International Symposium on Numerical Hydrodynamics, Paris, France, pp. 257–268. Xu, H., Yue, D., 1992. Computations of fully nonlinear three-dimensional water waves. 19th ONR Symposium on Naval Hydrodynamics, Seoul, Korea. Zhou, Z., Gu, M., 1990. Numerical research of nonlinear body–wave interactions. 18th ONR Symposium on Naval Hydrodynamics, Ann Arbor, Michigan.