Time-Domain Calculations of Drift Forces and Moments. Henk Prins

Transcription

1 Time-Domain Calculations of Drift Forces and Moments Henk Prins

2 Time-Domain Calculations of Drift Forces and Moments Tijdsdomein Berekeningen van Drift Krachten en Momenten PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus Prof. ir. K.F. Wakker, in het openbaar te verdedigen ten overstaan van een commissie, door het College van Dekanen aangewezen, op vrijdag 31 maart 1995 te 16. uur door Henk Jurriën PRINS, wiskundig ingenieur, geboren te Amsterdam.

3 Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. A.J. Hermans Samenstelling promotiecommissie: Prof. dr. O.M. Faltinsen (Norwegian Institute of Technology) Prof. dr. ir. A.W. Heemink (TU Delft) Prof. dr. J.N. Newman (MIT) Prof. dr. ir. J.A. Pinkster (TU Delft) Prof. dr. ir. P.J. Zandbergen (U Twente) Dr. ir. J.E.W. Wichers (Marin, Wageningen) Copyright c 1995 by H.J. Prins CIP DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG Prins, Henk Jurriën Time-domain calculations of drift forces and moments / Henk Jurriën Prins. - [S.l. : s.n.]. - Ill. Thesis Technische Universiteit Delft. - With index, ref. - With summary in Dutch. ISBN NUGI 811 Subject headings: ship-hydrodynamics / time-domain simulations / drift forces.

4 Contents Introduction 5 List of symbols 11 1 Problem Definition and Mathematical Model Physical problem Mathematical model Linearization of the boundary conditions Free-surface condition Body boundary condition Forces and moments Perturbation series Hydrodynamic coefficients and equation of motion Drift forces and moments Double-body potential Numerical Method Introduction Boundary-integral method Discretization of the boundary conditions Time integration Algorithm for zero speed New algorithm for forward speeds Suggestion concerning time integration of non-linear equations Matrix solver Two-dimensional test problem: cylinder of infinite length Introduction Simplified model Numerical aspects Convergence and stability

5 2 Contents Effectiveness of absorbing boundary condition Results Comparison of results found by using the doublebody potential and the undisturbed-flow potential Conclusions Three-dimensional test problem: floating sphere Introduction Mathematical model Absorbing boundary condition Numerical aspects Results Conclusions Results for a 2kDWT tanker Introduction Mathematical model Numerical aspects Numerical differentiation on the hull Double-body potential Results Hydrodynamic coefficients Improvement of roll damping Drift forces and moments Conclusions and recommendations General time signals Introduction General absorbing-boundary condition Step-response functions Fitting of the step-response function using Laguerre polynomials Results Forward speed effects Slow-drift forces Results for infinite depth Results for finite depth Conclusions Conclusions 119

6 Contents 3 A Asymptotic analysis of the Green s function 123 A.1 Green s function A.2 Asymptotic analyses of the function Ψ A.2.1 Complex integration A.2.2 Method of stationary phase A.2.3 First correction B Error estimation of the absorbing-boundary condition 133 B.1 Introduction B.2 Error in the potential B.3 Error in the normal derivative B.4 Integral over the Green s function B.5 Total error Summary 139 Samenvatting 141 Acknowledgements 145 Curriculum Vitae 147 Bibliography 151 Index 152

7 Introduction When a ship is sailing on an ocean, it is exposed to several forces. These forces are due to wind, waves, current, and speed of the vessel. Especially the first three may cause the ship to loose its course, i.e. to drift away. Drifting forces can be very large, in particular the force due to the incoming waves. In this thesis this part of the drifting effect is studied, and the effect of forward speed of the vessel is taken into account. The drifting of a ship due to waves may be surprising. In the case of harmonic waves, the motion of the ship may be expected to be harmonic as well, thus no drifting would be expected. However, drifting does occurs, even in harmonic waves. This may be illustrated by two practical examples. When kids play football, it is very common, especially in Holland, that the ball ends up in a ditch or pool. A very practical way to get the ball out of the water, is throwing stones or dirt at the ball. The stones generate waves, which cause the ball to drift to the shore. This drifting even occurs when the waves are small. A second example is a ship lying at anchor in waves. If the waves do not come head on, the ship wants to drift away into the direction of wave propagation. Because this is prevented by the anchor, the ship will rotate until it lies in head waves. The examples show that drifting effects can be large and practical. They also show that the effect of drifting is important for both moored systems and freely floating objects. In the case of moored systems the importance of drifting may seem to lessen, when the system lies in head waves, as the example illustrates. However, the drift force is not a steady force, but may also include slowly varying components. The frequency of these components may be close to the eigen-frequency of the mooring system, thus possibly yielding severe damage to the system. This damage might for instance be breaking of anchor lines or the destruction of the mooring system. In the case of a freely floating ship a major contribution to the drifting effect is caused by incoming waves. However, from measurements it appears that the forward speed of the ship increases the drift forces considerably. Thus the forward speed has to be taken into account, together with existing

8 6 Introduction Figure I.1: Wave pattern generated at the stern. ocean currents. But also in harbour circumstances the drift forces may be very important. Although the waves are in general not very high inside harbours, drift forces become large due to shallow water. Thus besides speed and current, bottom effects have to be considered. Before studying the combination of waves and forward speed, it is important to understand the forces due to waves and forward speed separately. The forces due to waves only are a special case of the problem we are going to consider, i.e. the forward speed is zero. Thus insight in these forces will be gained within this thesis. The forces in the case when no waves are present, however, are not part of this study, but knowledge about these forces is assumed. Therefore a short explanation will be given about the forces solely due to the forward speed of the ship. When a ship sails through otherwise calm water, it will generate a steady ship-fixed wave pattern. The generation of the waves leads to a power loss of the ship: the wave resistance. At low speeds, these waves have two distinct sources: the bow and the stern. The bow waves travel in a V-like shape, the so-called Kelvin-wave pattern. These waves can be observed when watching a water-bird swimming. This Kelvin-wave can be predicted very well using analytical methods. The second kind of waves are generated at the stern, see Figure I.1. They are very pronounced for sailing-boats at reasonably high speeds. These stern waves can not be predicted very easily; even numerical techniques experience difficulties trying to calculate these waves. To compute the wave resistance at finite speed, it is shown by Raven [25] that the complete non-linear free surface has to be taken into account. The results obtained by his method RAPID are very promising. For an extensive study of the wave resistance, the reader is referred to Wehausen [32].

9 Introduction 7 Figure I.2: Double-body flow. Unfortunately, the drift forces we want to study are strongly influenced by the total fluid flow due to the forward speed of the ship (i.e. waves generated at both bow and stern, and the flow around the hull). Thus it may seem important to know the stationary wave field. As mentioned before, this wave field is difficult to calculate. Therefore one has the choice either to solve an extra problem for these waves, or to make an assumption about the influence of these waves. In this study we will make an assumption, because we are interested in forces acting on the ship, and not in the waves themselves. The most important part of the fluid flow to be taken into account, is the fluid velocity around the ship. This velocity can be approximated using the double-body flow: the flow past the double body, see Figure I.2. The ship and the fluid domain are mirrored in the undisturbed water surface. Thus note that Figure I.2 still represents a side-view of the geometry. Up to the stern the double-body fluid velocity is a good approximation of the general flow which included wavemaking. Because no steady wave pattern is taken into account, the flow is much easier to calculate than the general flow including waves. The double-body flow is therefore a practical approximation of the general flow. If the speed of the ship is constant, the wave-resistance problem is a timeindependent problem. The drift-force problem, however, is time-dependent because the waves and thus the motion of the ship will be a function of time. This complicates the problem considerably. In the past the only possible way to solve these time-dependent equations was to assume the waves and all other quantities to be perfectly harmonic. This assumption reduces the equations to much simpler equations in the so-called frequency domain. At first this may seem a reasonable assumption, because most waves indeed look rather harmonic. However, at full sea a ship may encounter rather

10 8 Introduction strange and large waves. These waves are definitely not harmonic. Until recently, the only way of predicting the ship s behaviour under these severe seastates was by performing model tests. With the recent development of large computers, however, it became possible to study the time-dependent equations themselves. The waves are then allowed to be non-harmonic and more realistic sea-states can be simulated. Thus much attention to the time-dependent equations has been paid in recent years, and is still being paid, with the ultimate goal of making expensive model tests superfluous. Besides the time-dependency of the hydrodynamic equations, another major problem arises when studying ship motions. The equations are highly non-linear, even if forward speed is absent. The equation which has to be solved is the Laplace-equation, which is linear itself. The necessary boundary conditions, however, are non-linear. On the hull of the ship, for example, we have that the velocity of the fluid should match the velocity of the ship. In other words, the ship should always stay in contact with the ocean. Unfortunately, the velocity of the ship is not known beforehand: it is part of the solution. Besides, the position of the hull is also unknown, thus it is not known where exactly the condition should be applied. A second non-linearity arises from the water surface. The position of the surface is unknown, and the condition to be imposed is non-linear as well. In recent years some attention has been paid to solving the complete non-linear equations. First attempts have been made by for instance Romate [26], Broeze [2] and Van Daalen [4]. The first two of these studies were focussed on the propagation and breaking of water waves, without the presence of an object. The most recent study by Van Daalen concerned the two-dimensional water flow induced by the motion of an object. However, no incoming waves are simulated and current or forward speed are not included in the mathematical model. Because it is not known yet, what the influence of forward speed is on the behaviour of the equations and on the numerical algorithm to be used, we chose to study a linear problem including forward speed. As a next step the non-linear approach and the knowledge of forward-speed effects can be combined in order to allow numerical simulation of the non-linear motion of sailing ships. To obtain a linear problem, we linearize the boundary conditions of the differential equation. We will assume that all waves and motions arising in the simulations are small. Note that this assumption is also made for the stationary waveheight due to the double-body flow; this waveheight is therefore neglected. Then it is allowed to linearize the water surface around the undisturbed surface, which is a flat plane. The non-linear condition which

11 Introduction 9 has to be applied on the water surface is linearized using the fact that the waves are small. The condition on the hull is linearized around the averaged position of the ship, instead of imposing it on the actual position of the ship. This linearization process is described in chapter 1. The non-linear equations are summarized briefly, without going into too much detail. (For a more thorough discussion of the principles of water waves and ship motions the reader is referred to Newman [15]. ) Then a perturbation series is used to derive the equations for the forces and moments and the equation of motion. This derivation is an extension of the derivation given by Pinkster [2]. The last part of chapter 1 will concern the approximation of the stationary potential by the double-body potential. To solve the mathematical model a numerical algorithm is needed. This numerical algorithm is discussed in chapter 2. The three-dimensional problem will be reduced to two dimensions by means of the boundary-element method. The resulting integral equation is discretized assuming constant quantities on each element. Then it will be shown that the numerical algorithm suggested in literature is unstable when forward speed is included, and thus unsuitable for our numerical simulations. A new algorithm will be derived which is stable for all simulations carried out in this study. A suggestion is given how to extend our algorithm to the non-linear problem. The numerical algorithm is first tested for the case of an infinitely long cylinder, see chapter 3. The integral equation can then be reduced to only one dimension. Convergence of the time-integration will be shown, and the results will be compared with results given by Vugts [31] and Zhao [36]. Further testing is done in chapter 4. The three-dimensional case of a floating hemisphere is studied, thus eliminating rotations and moments. An absorbing-boundary condition will be derived by an asymptotical analysis of the Green s function satisfying the Neumann-Kelvin free-surface condition. Results will be presented for the hydrodynamic coefficients and the drift forces. The results will be compared with Pinkster [2] for zero forward speed, and with Nossen [17] for the coupling coefficients when forward speed is included. After these two tests, the numerical method is used in chapter 5 to calculate the hydrodynamic coefficients and the drift forces and moments on a commercial super tanker. The results for zero forward speed will be compared with Pinkster [21]; the results for non-zero forward speed are new and can not be compared with literature. In the final chapter, chapter 6, the numerical model is extended to general

12 1 Introduction time signals. This enables us to calculate the hydrodynamic coefficients in one single simulation. Furthermore it becomes possible to calculate the slow-drift motion of an object, in deep and shallow water. The calculations are performed for the case of the infinitely long cylinder. To ensure accuracy of the simulations an improved absorbing boundary condition, developed and programmed by Sierevogel [28], will be used. Finally, conclusions will be drawn and recommendations will be made for improvements of the numerical algorithm and the computer programs.

13 List of symbols A ij : added-mass coefficients [kg] A : asymptotic added mass [kg] A wl : water-line area [m 2 ] B ij : added-damping coefficients [kg s 1 ] B : asymptotic added-damping [kg s 1 ] C ij : restoring-force coefficients [kg s 2 ] C : asymptotic restoring-force [kg s 2 ] F : force [N] F i : i-th order force [N] Fn : Froude number [ ] Fn h : grid Froude number [ ] G : Green s function [m 1 ] H : momentary wetted hull surface [m 2 ] H : averaged wetted hull surface [m 2 ] K ij : step-response functions [kg s 2 ] L p : Laguerre polynomials of the order p [ ] M : moment [N m] M i : i-th order moment [N m] M ij : mass matrix [kg] R : radius [m] T : transfer function [kg m 1 s 2 ] U : forward speed [m s 1 ] V : momentary velocity of the hull [m s 1 ] V : displacement volume [m 3 ] X : translation displacement [m] c i : wave velocity [m s 1 ] d : diameter [m] g : gravitational constant [m s 2 ]

14 12 List of symbols h : fluid depth [m ] k : wave number [m 1 ] n : normal vector [ ] p : pressure [Pa] p : atmospheric pressure [Pa] t : time [s] u : local fluid velocity [m s 1 ] x : coordinates [m] (x, y, z) : coordinates [m] x b : point of buoyancy [m] x g : point of gravity [m] Φ : total potential [m 2 s 1 ] Ω : rotational displacement [ ] α : total displacement [m] β : angle of incidence [ ] ε : small parameter [ ] ζ : wave height [m] ζ a : wave height of incoming wave [m] λ : wave length [m] ξ : field coordinate in Green s function [m] ρ : density [kg m 3 ] τ : non-dimension parameter [ ] φ : time-dependent potential [m 2 s 1 ] φ : stationary potential [m 2 s 1 ] ω : frequency of encounter [s 1 ] : scaled frequency of encounter [ ] ω : frequency [s 1 ] : scaled frequency [ ] t : time step [s] x : grid size [m] : mean submerged volume of the hull [m 3 ]

15 Chapter 1 Problem Definition and Mathematical Model In this chapter the physical problem will be defined and formulated in a mathematical model. Then the non-linear boundary conditions arising, will be linearized by assuming that the quantities involved are small. Finally, equations for the forces and moments will be derived using a perturbation series for both the position of and the pressure on the hull. 1.1 Physical problem In sailing on oceans, a lot of physical processes influence the behaviour of a ship. Some of these are waves, wind (especially under stormy conditions) and currents. Also the speed of the ship itself is of great importance. The problem we will consider, is the interaction between a sailing ship and the ocean it is sailing at. Especially the influence of waves and current on the behaviour of the ship is considered in detail. The influence of current and the speed of the ship, is evident. They will cause a resistance, which has to be overcome by the ships engines. Side currents will of course make the vessel drift aside, but this effect can be calculated rather easily, or measured in towing tanks, i.e. the equivalence of wind tunnels. Model tests show that the influence of waves can be very large. It may cause drifting of the ship, i.e. loosing its course. This effect is widely known and used by kids, trying to get a football out of the water. Throwing stones into the water generates waves which cause the ball to drift slowly into the direction of wave propagation. This effect is measured in model tests, performed in large water basins. Although difficult to perform, these measurements are reliable.

16 14 Chapter 1. Problem Definition and Mathematical Model However, it appears from measurement that the coupling between waves and current causes a considerable increase in the drift effects. Unfortunately, the drift measurements become even more difficult when current is involved. Therefore this effect is subject of mathematical modelling and calculations in this thesis. Drifting effects are not only important for ships sailing at the ocean, but also for ships manoeuvring close to or in a harbour. Although waves are generally not very high inside harbours, the drift forces may still be very large due to bottom effects. The influence of the bottom on the drift forces will therefore be studied for a commercial super tanker. Ocean waves can in general be modelled reasonably well by a sine function of one or two frequencies. If two frequencies are involved, the object (ship or a floating platform) will not only react to those frequencies, but also responses to the sum and difference frequencies. Especially this difference frequency may be in the order of magnitude of the resonance frequency, and may therefore cause significant damage to the structure. This effect should also be included in the mathematical model. For the convenience of readers who are not familiar with naval hydrodynamics, a short summary of the most important jargon will be given: surge : horizontal translational motion of the ship in the sailing direction, i.e. in the direction of the length of the ship. sway : horizontal translational motion of the ship, perpendicular to surge. heave : vertical translational motion of the ship. roll : rotational motion round the longitudinal axis, or the surge -axis pitch : rotational motion round the sway -axis. yaw : rotational motion round the heave -axis. point of gravity : point of gravity of the whole ship, including load and structures above the water-line. point of buoyancy : mathematical point of gravity of the hull shape; point of application of the buoyancy force. drifting : being carried along by current and waves; also used for moments to indicate that these moments have the same physical cause as the forces which cause the actual drifting.

17 1.2. Mathematical model Mathematical model We will consider an object of general shape in three dimensions, sailing at sea in the presence of waves and current. The object is free to move in all directions or to rotate round any of its axis. The most obvious choice for a coordinate system would be a system which is fixed relatively to the earth. However, this will cause the object to sail out of the coordinate system. Because we are only interested in processes which act on the object, we would like to have our object within the frame of our coordinate system. Therefore we define a coordinate system moving with the average speed of the body, such that the undisturbed free surface coincides with the plane z = and that the centre of gravity of the body is on the z-axis. The z-axis points upwards, which implies that the fluid domain is in the half space with negative z-values. Note that this coordinate system moves relatively to the earth. This changes the frequency of the waves as described in the model. Therefore we will distinguish between the frequency in the space-fixed reference frame, ω, and the frequency of encounter, ω. The forward speed is translated into a constant undisturbed horizontal velocity U of the fluid the object is floating in. The fluid depth is supposed to be constant, h. In order to study drift effects, we assume that regular incoming waves are travelling in the water-surface in a direction which makes an angle β with the positive x-direction, see also Figure 1.1. y U O x β Figure 1.1: Aerial view of the geometry. The goal of this study, as stated in section 1.1, is to calculate the interaction between the ocean and an object. This can be done by calculating the forces and moments acting on the object. The force is given by F = p nds, H

18 16 Chapter 1. Problem Definition and Mathematical Model with n the normal on the objects surface H. The pressure p can be calculated using the Navier-Stokes equations or Bernoulli s equation, which couple the local fluid velocity with the pressure. Because the fluid under consideration is water, we assume our fluid to be incompressible. If we furthermore assume the flow to be irrotational at all times, we can introduce a velocity potential Φ given by Φ = u. (1.1) Note that the latter assumption implies that we neglect the effects of viscosity. Using this velocity potential, the Euler equations can be integrated into the equation of Bernoulli: p p ρ = Φ t Φ Φ + gz + const. (1.2) This equation will be used calculating the pressure distribution on the object and the wave heights at the water surface. In order to use Bernoulli s equation, we need a set of differential equations for the potential including proper boundary conditions. The potential has to satisfy the Laplace equation: 2 Φ =, (1.3) which can be derived easily from the continuity equation. This equation represents the physical principle of conservation of mass. At the water surface we have two boundary conditions: 1. the pressure should equal the atmospheric pressure (p ), and 2. a fluid particle can not leave the surface. In mathematical form these conditions are given by and Φ t Φ Φ + gζ + const =, ζ t + Φ ζ = Φ z, both at z = ζ, the unknown momentary position of the water surface, which is given by ζ = 1 ( Φ g t + 1 ) Φ 2 Φ + const.

19 1.2. Mathematical model 17 These conditions are well known as the dynamic and kinematic conditions. Because there are no exterior restrictions to the water surface, this boundary will be referred to as the free surface. To obtain one condition, the above two are combined, leading to 2 Φ t 2 +2 Φ Φ t + 1 Φ 2 ( Φ Φ )+g Φ = at z = ζ.(1.4) z The floating object is assumed to be impermeable, so no fluid particles can cross this boundary. The normal velocity of the fluid should therefore equal the normal velocity of the object: Φ n = V n on H. (1.5) Note that slip may occur along the hull, due to the assumption of an inviscid fluid. The velocity V is the objects velocity in the coordinate system fixed to the average position of the object. In the earth-fixed coordinate system, this velocity has to be superimposed on the undisturbed velocity U. On the bottom of the fluid domain we have, like on the boundary of the floating object, that no fluid particles may cross this boundary. As the bottom is fixed, we have Φ n = at z = h. (1.6) As our free-surface boundary condition is a second-order differential equation in time, we have to provide two initial conditions. We assume that at t = the fluid is undisturbed, i.e. and Φ( x,) =, (1.7) Φ ( x,) =. (1.8) t To make the solution of our mathematical model unique, we have to impose an extra condition, called the radiation condition. This condition states that waves can only be generated by the body, except for possible incoming waves, and that they should travel away from the body. When studying the equations analytically, this condition can be imposed at infinity. In numerical studies however, this is not possible due to limited computer time and memory. Therefore this condition has to be given on an artificial boundary. There are several possibilities for the radiation condition to be imposed, extensively reviewed by Romate [26]. Most commonly used are a damping

20 18 Chapter 1. Problem Definition and Mathematical Model zone in the free surface or partial differential equations on the artificial boundaries. A damping zone has as advantages that it is easy to implement, and that it has good reflection properties for a wide range of frequencies. The disadvantage is that a large domain is needed to have good absorption. Partial differential equations, however, can be applied closer to the body. The radiation condition would then read N ( ) Φ t + c Φ i =, n i=1 with c i the local phase velocity of the wave to be absorbed and N the number of different waves in the problem. Romate showed that this condition leaves us with a well posed problem, or at worst a weakly-ill posed one. A disadvantage of these equations is that it absorbs only those waves whose wave velocity is included. Other waves will be partly absorbed, partly reflected. Recently, a new boundary condition has been developed by Sierevogel [28], which has good reflection properties for a wide range of frequencies and can still be chosen very close to the object, thus combining the advantages of the previously mentioned methods. In major parts of this thesis, partial differential equations will be used, because in these problems only one wave has to be absorbed at the boundary. In the last chapter, concerning the investigation of general time signals, the new absorbing boundary condition developed by Sierevogel will be applied. 1.3 Linearization of the boundary conditions The problem formulated in the previous section contains several non-linearities. Unfortunately it is not possible yet to solve these equations, especially in the case of an object in current or an object with a forward speed. In the case of zero current, or no forward speed, first attempts have been made by Romate [26], Broeze [2] and van Daalen [4]. However, major problems arise in solving these non-linear equations, even without forward speed. Including forward speed probably worsens these problems. Therefore we will linearize the boundary conditions, in order to study the problems arising from introducing the forward speed. Our results and experiences may then be used solving the non-linear problem including forward speed or current. Linearizing the boundary conditions restricts our results to small amplitudes of both the motions and the incoming wave field.

21 1.3. Linearization of the boundary conditions Free-surface condition The non-linear free-surface condition is given by (1.4): 2 Φ t Φ Φ t + 1 Φ 2 ( ) Φ Φ + g Φ z = at z = ζ. This condition is non-linear in two ways. The first obvious non-linearity is in the potential Φ itself. The second non-linearity is hidden in the fact that the position of the boundary, ζ, is unknown and a function of the potential Φ. To linearize this equation we assume the potential to be a linear combination of a time-independent part and a time-dependent part: Φ( x,t) = φ( x) + φ( x,t). (1.9) Furthermore we assume that both the time-dependent and the time-independent potential are small. We assume that for φ linear terms are sufficient, and for φ quadratic terms. Using this in (1.4), we get 2 φ t φ φ t + 1 φ 2 ( ) φ φ + φ ( ) φ φ + g φ z + g φ = z at z = ζ. (1.1) For ζ we linearize Bernoulli: ζ = 1 ( φ g t + 1 φ 2 φ + φ φ 1 ) 2 U2 at z =. (1.11) Because at infinity there will be no wave elevation, the constant in the equation of Bernoulli is given by 1 2 U2. To overcome the second non-linearity, the unknown position of the free surface, we expand the condition into a Taylor series around z =. Thus we assume ζ to be small, which is in accordance with both φ and φ being small. We get for the time-dependent problem: 2 φ t φ φ t + 1 φ 2 ( ) φ φ + φ ( ) φ φ + g φ ( φ φ ) ( ) U 2 2 φ z ( 2 φ z 2 3 φ 2 t z z 2 1 g φ φ + φ ) = at z =. (1.12) t For the time-independent problem we have, up to the second order of the stationary potential, g φ z = at z =. (1.13) +

22 2 Chapter 1. Problem Definition and Mathematical Model Body boundary condition On the hull we have, as stated earlier, that the normal velocity of the fluid should equal the normal velocity of the body. Therefore we have Φ n = V n. Timman [29] showed that this condition can be linearized into and φ n =, (1.14) φ n = α [( ) ( t n + φ α α ) ] φ n. (1.15) Both conditions now hold on the average position of the hull. Here, α is the total displacement vector, given by α = X + Ω ( x x g ), (1.16) with X the translational, and Ω the rotational motion of the body relative to the centre of gravity x g. 1.4 Forces and moments The goal of this thesis as stated in the Physical problem, is to calculate the interaction between an object and the presence of waves and current. To calculate this interaction, we have to calculate the forces and moments acting on the object due to incoming waves and current. They can be calculated using Bernoulli s equation. The pressure on the hull is given by ( Φ p = ρ t + 1 Φ 2 Φ + gz 1 ) 2 U2 on H. (1.17) The forces and moments then follow from integrating the pressure over the body s wetted surface: F = p nds, (1.18) and H M = p {( x x g ) n}ds. (1.19) H

23 1.4. Forces and moments 21 Note that the moments are calculated relative to the centre of gravity, not relative to the origin, see also (1.16). Unfortunately, the actual position of the body, H, is not known, and due to the linearization all quantities are known on the average position. Besides the position of the hull, the size of the integration domain, the wetted part of the hull, is also unknown. Thus some kind of an approximation has to be made to evaluate the above integrals. We will derive approximate formulae extending the method outlined by Pinkster [2]. He used the same kind of perturbation series as we will in the following, but forgot to expand one of the variables. This forced him to correct his expressions along the way, which will not be necessary in the correct derivation. Furthermore an extra term will appear in the drift forces and moments, which was overseen by Pinkster. We previously assumed that the time-dependent potential is small. Because this potential causes the movements of the hull, the latter is assumed to be small as well. The pressure at the actual position of the body can therefore be expanded into a Taylor series around the average position, H : p H = p H + α p H ( α ) 2 ph + O ( α 3). (1.2) Furthermore the size of the actual wetted surface can be estimated by the sum of the average wetted surface and an oscillatory disturbance of this average surface. We therefore have f( x)ds = f( x)ds + f( x)ds. H H Hosc The integral over the oscillatory wetted surface can be approximated by Hosc f( x)ds ζ f( x)dzdl. (1.21) wl α 3 Combining these two approximations results in an estimate of the total forces and moments. As may be seen from the Taylor expansion of the pressure, the forces and moments will be proportional to powers of the movement of the object. However, the dependency is more complicated than suggested by this expansion. The pressure on the hull is dependent on the potentials, which in its turn are dependent on the displacement, see equations (1.17) and (1.15). The displacements are on the other hand dependent on the pressure through the forces and moments. Because the movements and the potentials were assumed to be small, the obvious way to proceed is introducing a perturbation series.

24 22 Chapter 1. Problem Definition and Mathematical Model Perturbation series To evaluate the integrals resulting from the estimates made above, we assume that all quantities can be written in a perturbation series. The small parameter involved in these series could for instance be the wave height of the incoming wave. This wave height is indeed small, because we assumed that the time-dependent potential is small as well. The perturbation series for the quantities involved are: ( ζ = ζ + εζ (1) + ε 2 ζ (2) + O ε 3), ( Φ = φ + εφ (1) + ε 2 φ (2) + O ε 3), ( p H = p + εp (1) + ε 2 p (2) + O ε 3), X = εx ( (1) + ε 2 X (2) + O ε 3), Ω = εω ( (1) + ε 2 Ω (2) + O ε 3), x x g = x x g + εω ) (1) ( x x g + ) ε 2 Ω (2) ( x x g n = n + ε Ω (1) n + ε 2 Ω (2) n + O ( ε 3). ( + O ε 3), The first terms in the perturbation series are time independent, so ζ = 1 2 φ φ 1 2 U2 is the stationary wave-height. These stationary contributions are assumed to be small, in order to be able to linearize the free-surface condition as has been done in equation (1.12). For the same reason the time-dependent parts of the series are assumed to be small as well. Note, however, that these assumptions are independent from each other and that no difference in order is assumed between the stationary and first-order contributions. Substituting these series into the Taylor series of the pressure on the actual wetted surface, (1.2), and collecting equal powers of ε, we get: p H = p, p (1) { H = p (1) + X (1) + Ω )} (1) ( x x g p,

25 1.4. Forces and moments 23 p (2) { H = p (2) + X (2) + Ω ) (2) ( x x g + Ω (1) [ Ω (1) ( x x g )]} p { + X (1) + Ω )} (1) ( x x g p (1) + 1 {[ X (1) + Ω 2 )] (1) 2 ( x x g } p. From Bernoulli and the perturbation series for the potential we get for the components of the pressure series on the average wetted surface: p = ρ (gz φ φ 1 ) 2 U2, ( ) p (1) = ρ φ φ (1) + φ(1), t ( φ p (2) (2) = ρ + 1 ) t 2 φ(1) φ (1) + φ (2) φ The floating bodies of interest for our calculations sail at low speed or float in moderate currents. Therefore, terms of O ( U 2) in p will be neglected in the calculations of the forces and moments. We substitute these expansions of the pressure into the equations for the force and moment, (1.18) and (1.19). This results in a power series of the forces and moment in ε. The stationary contribution is then given by: and F = p H nds = H M = H p H ρgv, (1.22) ) } {( x x g n ds = ρgv y b x b.. (1.23) Equation (1.22) represents Archimedes law. The stationary resistance of the object is of the order U 2, and has therefore been neglected. At the end of this chapter, the stationary potential will be approximated by the double-body potential. For this potential, the stationary resistance is zero due to the paradox of d Alembert, as shown for instance by Meyer [12]. The zero-forward-speed moment, equation (1.23), is zero if the buoyancy point and the point of gravity of the body are on the same vertical axis. If this is not the case, the object will rotate until this requirement has been fulfilled. Therefore we assume the object to be in equilibrium as far as buoyancy is concerned.

26 24 Chapter 1. Problem Definition and Mathematical Model Hydrodynamic coefficients and equation of motion When we continue expanding the forces and moments along the way indicated above, we find formulae for the first-order quantities: F (1) = H ρg ( p (1) H n + p H n (1)) ds = X (1) 3 A wl Ω (1) 2 ρ H xds + Ω (1) 1 yds D D ( φ (1) + φ t ) nds φ (1), (1.24) and M (1) = { p (1) H ( x x g ) n ρg H X (1) 3 X (1) 3 D D [ yda + Ω (1) 1 D xda Ω (1) 1 ρ H ) + p HΩ (1) [( x x g ] ]} n ds = y 2 da + (z b z g )V Ω (1) 2 xyda [ D ] xyda + Ω (1) 2 x 2 da + (z b z g )V D D Ω (1) 1 x b Ω (1) 2 y b ( ) φ (1) ( + φ t φ ) x x g nds, (1.25) with D the water-line surface of the object, and A wl the area of D. Note that the integral over the oscillatory wetted surface does not contribute to the first-order quantities. However, if we would have included second-order effects in the velocity in the derivation, the first-order quantities would have been affected. For instance, the first-order force would be corrected with ρg wl α (1) 3 ζ ndl. As stated before, these terms are not important in the applications we are interested in.

27 1.4. Forces and moments 25 As can be seen from both (1.24) and (1.25), only few terms depend on the first-order potential directly. Most terms depend on the first-order motions in combination with stationary quantities. However, these motions can only be calculated when the first-order forces and moments are known. This problem is very well known in calculating the displacement of a mass connected to a string. The internal force of the string is proportional to the displacement of the mass. To overcome this problem, the restoring force is taken as part of the differential equation. Only external forces, independent of the displacement, are considered as forcing terms. Analogously, the terms depending on the first-order motion are considered to be restoring forces and moments in the equations of motion, instead of being terms of the first-order forces and moments. If we assume the object to be symmetrical in the y-plane, the restoring-force coefficients are given by: C 33 C 35 C 44 C 53 C 55 = ρg = ρg D D da, xda, = ρg (z b z g )V + y 2 da, D = ρg xda, D = ρg (z b z g )V + x 2 da. D The other elements of the restoring force matrix C are zero, some due to the symmetry assumption. The restoring coefficients are independent of the forward speed, up to the second order. For ships sailing at higher velocities, these coefficients must be adapted. Unfortunately, the terms depending directly on the first-order potential are not entirely independent of the motions. This may be observed from (1.15), where the potential φ is coupled with the motion α. To calculate the forces on the object these dependencies have to be removed. To achieve this we introduce added mass (A ij ) and added damping (B ij ) coefficients. These coefficients represent the dependency of the force on the motion, or vice versa. For an extensive explanation of the principle of added hydrodynamic coefficients, the reader is referred to Ogilvie [18]. Because the forces and movements depend on each other, we first force the object to oscillate in a given direction. The forces thus calculated depend directly on the motion of the object. For purely sinusoidal motions the

28 26 Chapter 1. Problem Definition and Mathematical Model added mass and damping coefficients are given by fitting the forces to the acceleration and velocity of the body: 2 x j F ij = A ij t 2 B x j ij t. Here j is the direction of the motion, either translational or rotational, and i the direction of the force. This can, of course, also be done for the moments. The coefficients are then still called A ij and B ij but now with i [4,5,6]. When these coefficients are known, we calculate the forces due to the incoming wave field, for a fixed object. Then the motion of the object is calculated using the equations of motion and the hydrodynamic coefficients: (M + A) 2 Y t 2 + B Y ( Finc t + C Y = M inc ), (1.26) with Y = (X 1,X 2,X 3,Ω 1,Ω 2,Ω 3 ) T. The mass matrix M is diagonal and consists of the mass and the relevant moments of inertia. For the calculation of the hydrodynamic coefficients in the case of general time signals, the reader is referred to chapter Drift forces and moments In studying drift effects, only the time-average values of the forces and moments are of interest, not the oscillatory forces themselves. If we assume the incoming wave to be a sine function with zero mean value, it may be seen from (1.24) and (1.25) that the mean value of the first-order quantities are zero. This is, of course, caused by the fact that these forces and moments have a linear dependency on the time-dependent quantities φ, X and Ω. Thus second-order forces and moments will be needed to calculate the drifting of an object. In general sea keeping, however, the incoming wave may be seen as a finite sum of sine functions. Still, the first-order forces and moments will have zero mean value and second-order effects have to be taken into account. By continuing our perturbation series, we arrive at formulae for the mean values of the second-order forces and moments:

29 1.4. Forces and moments 27 F (2) = ρ H ρg ( α (1) ) ( φ (1) ) + φ t (1) φ nds wl ( ζ (1) α (1) 3 ) 2 ndl + Ω (1) M X (1) 1 2 ρ φ (1) φ (1) nds H ρg. (1.27) Ω (1) 1 Ω(1) 3 C 35 Note that several terms with zero mean value have been left out. This formula is equivalent to the one given by Pinkster [2], except for the last vector which is missing in his thesis, and the forward speed contributions. The last vector represent the force generated by the displacement of the point of gravity of the hull deck due to two combined rotations. For the average second-order moment we find M (2) = ρ H 12 ρ φ (1) φ (1) ( x x g ) nds H ( α (1) ) ( ) φ (1) + φ t (1) φ ( x x g ) nds + 1 ( 2 ρg ζ (1) α (1) ) 2 3 ( x xg ) ndl wl + Ω (1) (1) Ω (1) 2 M Ω(1) 3 C 44 Ω ρg Ω (1) 1 Ω(1) 3 C 55. (1.28) Again, the last vector is missing in Pinkster s thesis. As may be seen from the above equations, the formulae for the drift forces and drift moments are almost equivalent. Not regarding the last vector, translational motion is equivalent to rotational motion, and the normal n to ( x x g ) n. Note that the integral over the oscillatory wetted surface does contribute to the second-order quantities, even now that we did not include second-order effects of the velocity. Of course, including those effects would give rise to much more contributions of this integral. However, these terms are not important for objects sailing at low to moderate speed.

30 28 Chapter 1. Problem Definition and Mathematical Model 1.5 Double-body potential In the section concerning the linearization of the boundary conditions, section 1.3, the potential has been split up into a stationary and an instationary part: Φ( x,t) = φ( x) + φ( x,t). The stationary potential includes the resistance of the object in calm water and the Kelvin wave pattern; the instationary part includes all timedependent effects. Unfortunately, the stationary potential is difficult to calculate. However, from (1.13) and (1.14) it can be seen that up to the second order in the velocity U, the normal derivatives of the stationary potential equal zero. Therefore, no waves are created up to this order in our linearization. This allows us to approximate the stationary potential with the double-body potential: the potential arising from the case when the whole computational domain is extended with its mirror image in the plane z =. One would expect that taking into account higher-order terms in the velocity U would lead to the well-known Kelvin wave pattern. Unfortunately, this is not the case. To find this stationary wave pattern, the method has to be extended in another way. A first possibility is solving the complete non-linear system. This has been done by Raven [25], using a Dawsonlike approach. However, these calculations are very time-consuming. A faster way has been developed by Sakamoto [27]. He linearized the freesurface condition around the wave-height resulting from the double-body potential. He then found wave-like corrections to this double-body potential. Hermans and Van Gemert [7] showed that Sakamoto s choice of the double-body potential was the only correct one to yield solutions resembling the Kelvin-wave pattern. The disadvantage of this linearization is that it is only applicable to low speeds. In the derivation of our linear free-surface condition we linearized around z =. Thus no wave-like solution can be found for the stationary potential. This may only be correct when these waves are small, i.e. when the speed of the object is low. This is indeed the case for the commercial supertanker for which we will perform calculations in chapter 5. For simplified geometries, the double-body potential can be calculated analytically, which means that the derivatives needed in the boundary condition of the instationary potential are exactly known. This enables us to eliminate the errors arising from numerical differentiation of the stationary potential.

31 Chapter 2 Numerical Method In this chapter a numerical method will be given to solve the equations as given in the previous chapter. The differential equation will be discretized using a boundary-integral method. Then it is shown that the timeintegration algorithm known in literature is unstable for problems including forward speed. A new algorithm will be given, which is stable for all time steps and space discretizations. 2.1 Introduction In the previous chapter, we derived a mathematical model for the interaction between current and waves, and an object. The model consists of a governing equation with linearized boundary conditions. The wanted interactions, i.e. the forces, all follow from integrals over the wetted surface of the object, as soon as the potentials on this object are known. To calculate these potentials, we have to solve the Laplace equation. This equation itself is time-independent. However, the boundary conditions do depend on time, so some sort of time integration has to be carried out. Therefore it will be useful to choose a fast solver of the Laplace equation. There are several ways of solving the Laplace equation. It can be discretized straightforwardly by using a finite-difference or a finite-element method. These methods have the advantage of discretizing the differential equation directly, and generating a sparse matrix. However, since we are only interested in values of the potential or its derivatives on the surface of the object, there is no need to discretize the entire fluid. Still doing so will lead to very large, though sparse matrices. Especially for computations in three dimensions, this can be a big problem. An other method to solve the Laplace equation is using a boundary-integral method. In this method, only values of the potential on the boundary are

32 3 Chapter 2. Numerical Method needed. This reduces the number of dimensions to two, and therewith shrinks the matrices with a factor N, the amount of elements in one direction. Unfortunately, this method has the disadvantage of generating full matrices, which in its turn will increase computer time needed to solve the matrix system. Although the advantages and disadvantages of the methods may be balanced, we chose the boundary-integral method, as has been done by many authors in recent years. A major consideration was, that computer time is less limited than computer memory. The time integration of the equations is a major problem in itself. Until quite recently, all calculations were performed in the frequency domain, thus avoiding the time-integration problem. However, the limitations of the frequency-domain approach forced the investigations to be focussed on time-domain analyses. Most of the recent studies use a rather obvious and straight-forward algorithm to perform the time integration. Romate [26], for instance, used this algorithm to calculate non-linear wave propagation. However, the algorithm used in literature is not suitable for problems including forward speed, as will be shown in this chapter. Thus a new algorithm has to be developed. 2.2 Boundary-integral method The boundary-integral method is based upon Green s second identity. It reformulates the differential equation into an integral over the boundary, thus reducing the number of dimensions of the problem. It can be shown that if the boundary of the fluid domain is twice continuous differentiable, and if the function φ is twice continuous differentiable inside the domain and once continuous differentiable on the boundary of the domain, the following identity holds, see Colton [3]: ( Here G 1 2 φ( x) = dv ) φ ( ( ) G x, ) ξ ξ n ξ ( ) φ ξ ( ξ) G x, n ds. (2.1) ξ x, ξ is the Green s function of the Laplacian, and n the normal pointing out of the fluid domain. Note that in this representation the Green s function should represent a source, not a drain. The Green s function depends on the number of dimensions of the problem, and will therefore be given in the consecutive chapters. Now the problem has been reduced to solving this integral equation together with the boundary conditions of the original differential equation.