Analysis of the forward speed effects on the radiation forces on a Fast Ferry

Ocean Engineering 60 (2013) 136–148

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Analysis of the forward speed effects on the radiation forces on a Fast Ferry R. Datta, N. Fonseca, C. Guedes Soares n Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Te´cnico, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisbon, Portugal

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 June 2012 Accepted 25 November 2012 Available online 26 January 2013

The paper presents an analysis of the radiation forces on a ship advancing with forced heave and pitch motions. This analysis is based on experimental data obtained with model tests and on numerical simulations with two codes, one based on 3D time domain Greens function method and the other is a 2D linear strip theory method. The case study consists of a hull shape from a Fast Ferry and several speeds are investigated between 0 and 40 knot. The objective is to quantify the forward speed effects on the radiation forces and to verify the skill of the conventional strip theory and of a time domain panel methods in dealing with large forward speeds.The consequences of applying simplifications on the linear boundary conditions of the common seakeeping codes are investigated. It is concluded that forward speed has a large effect on the radiation forces, especially on the coupling terms, and thus it is important to consider the full linear interactions between the steady and unsteady flows in the numerical calculations as the speed of the ship increases and the time domain panel code is able to represent these effects. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Boundary integral method Panel code Strip theory Radiation forces Effect of speed Experiments

1. Introduction Predicting hydrodynamic loads and motions of ships when they progress through a wave field is of fundamental interest. The study of wave induced loads and the related motions calculation is important because it is needed for the safe and efficient design of ship hulls. Therefore, for many decades, prediction of wave loads and motions is one of the main research areas in the field of marine hydrodynamics. Although this problem has been studied for many decades, a robust, efficient and complete solution is yet to be devised because of the complexity of the theory and of the numerical solution. In the current literature, many two-dimensional (2D) and three-dimensional (3D) theories are available for the solution of the seakeeping problem. Among the 2D theories, the frequency domain strip theory developed by Salvesen et al. (1970) is very popular in the industry. Fonseca and Guedes Soares (1998) developed a time domain solution based on the strip theory of Salvesen et al. (1970) with the objective of including some nonlinear effects. They studied the well-known S175 hull and obtained good results in large amplitude waves (Fonseca and Guedes Soares, 2002, 2004a, b). It is interesting that this theory was stretched to deal with very large abnormal waves and performed well as shown by various comparisons with experimental results (Guedes Soares et al., 2008; Fonseca et al., 2010),

n

Corresponding author. Tel.: þ351 218417957. E-mail address: [email protected] (C. Guedes Soares).

0029-8018/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2012.11.010

but had problems to deal with high speeds (Fonseca and Guedes Soares, 2004c; Fonseca et al., 2005a). The problem of wave induced loads in high speed ships was studied in a EU project that considered three different types of hulls and used various codes to compare the results as summarised by Schellin and Guedes Soares (2004). In this project experiments were made on a high-speed ferry, a fast monohull, and a containership. The high-speed ferry and the monohull represented large single hull fast ships and the containership served as a reference case for the hydrodynamic theories. The experiments and the calculations for the fast monohull (Maro´n et al., 2004; Fonseca and Guedes Soares, 2004c) and for the Fast Ferry (Fonseca et al., 2005a) showed problems in the performance of the code for high speeds. To understand the origin of the problems of the strip theory code several studies have been made looking at the assessment of the hydrodynamic coefficients and the motion of simplified models (Fonseca et al., 1995, 1997, 2005b) and the present study is a follow-up, analysing in detail the radiation forces in the Fast Ferry and comparing them also with the predictions of a time domain panel code that is able to deal with this effect. Theoretically, the strip theory is simple and relatively easy to implement. Although it gives very consistent and numerically efficient results for many conditions, the method still have some problems due to the simplifications in boundary conditions. Strip theory is mainly a low speed and high frequency theory. It is more adequate for slender ships. Hence for the computation of the non-slender ship with high speed and low frequency situation, a more sophisticated tool is required. The panel method is more

R. Datta et al. / Ocean Engineering 60 (2013) 136–148

sophisticated since it accounts for the 3D effects on the free surface and body boundary conditions. The 3D panel method became popular since early 80’s after the improvement of computer speed and memory power because it demands high memory requirements. Depending on the choice of the Green’s function, 3D panel methods can be classified in three basic types, one is the solution in the frequency domain using a zero or a forward speed Green’s function (Guevel and Bougis, 1982), another is the solution using a Rankine Green’s function (Nakos and Sclavounos, 1990), and third one is the solution in time domain using a transient free surface Green’s function (King, 1987). Frequency domain panel method with zero speed Green’s function has gained popularity in the offshore industry. The first commercial code using this approach is probably that by Garrison (1978), while one of the most widely used is probably WAMIT (Korsmeyer et al., 1988). These codes have been subsequently enhanced to include second order mean drift and slowly varying forces. The time domain panel method is adequate when the seakeeping of ships is concerned. This is mainly because computations of frequency domain Green’s function become extremely difficult when forward speed is considered. Also it is not easy to implement non-linear effects in a frequency domain formulation. Therefore the time domain formulation is useful in such classes of problems. There are few significant development in this domain as follows: Liapis and Beck (1985), who introduced the time domain Green function based solution method for the 3D linear forward speed problem, while King (1987), Lin and Yue (1990), Bingham et al. (1994) and Korsmeyer and Bingham (1998), among others, pursued variants of the same method for different classes of 3D forward speed problems. Although the time domain panel theory and the calculation of wave induced ship motions and loads have been well developed for the last couple of decades, a robust solution and implementation of the panel method approach is still a great challenge. Most of the theories worked very well for simple structures like an hemisphere or the Wigley hull, but they do not work well for the complicated ship hulls of fishing vessels. Till date, to the best of the authors’ knowledge, no such convincing evidence is reported except in Datta et al. (2011, 2012a, b), which was able to deal with short and not very slender fishing vessel hulls. This is one of the reasons why strip theory is still widely used by the industry, since the solution procedure is easy compared to other more sophisticated theories. Adding to this, it is known to provide good results even for some hull shapes that do not completely cope with its restrictive assumptions. Datta and Sen (2007) developed a higher order time domain solution which worked well for simplistic hull shapes, but failed to perform when more complicated hulls are considered as for example from fishing vessels. Datta et al. (2011a) studied this inconsistency and proposed a possible solution to this problem. In this approach, they found that the lower order method is more adequate for realistic complicated hull shapes. They also proposed a technique to refine the mesh in specific areas of the hulls. After incorporating such changes, they showed that the time domain algorithm, with such changes, gives effective results for the fishing vessels in head seas. Later they studied the ship motion problem more rigorously with a variety of ships with different heading angles and speeds and obtained similar results (Datta et al., 2012a,b). One of the strengths of this time domain panel method is the consistent treatment of the free surface boundary condition and body kinematic boundary condition. In fact, besides the linearization, the method assumes no further simplifications on these conditions. For this reason the method solves the complete linear solution of the seakeeping problem. In contrast, the existing

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frequency domain panel methods need to introduce some simplification on the linear boundary conditions if reliable numerical results are to be obtained for realistic hull shapes. This need is related to the interactions between the steady flow associated with the ships’ forward speed and the oscillatory flow due to waves and ship motions. Such interactions are relatively simple to consider in the time domain, but very complex in the frequency domain. The simpler frequency domain panel methods consider the forward speed effects in a way similar to that used by consistent strip theories (Papanikolaou and Schellin, 1992). A relatively high forward speed case is ideal to test the advantages of the present time domain panel method. Therefore, in the present paper, one Fast Ferry is considered and experimental results from model tests are compared with time domain code developed by Datta et al. (2011a). The Froude numbers tested go from 0 up to 0.6, therefore clearly outside the range of ‘‘small Froude number’’ assumption. Furthermore, the Fast Ferry hull shape is a challenge for the robustness of the numerical method. This point is discussed further ahead in the text. In this work the major part of the analysis is focused on the hydrodynamics of the ship’s forced motion problem, meaning on the hydrodynamic radiation forces with forward speed. The advantages and limitations of the seakeeping formulations are much better assessed by analysing partial results, like the radiation forces, than analysing global results like the oscillatory ship motions. Global motions may compare well with experiments even when the radiation (or diffraction) forces are inaccurately calculated. Therefore, instead of computing only oscillatory motions in waves, due to the availability of the experimental data, radiation force results are also compared for various speeds. Additionally, the influence of the steady hydrodynamic forces due to forward speed is discussed. This is again one benefit over the frequency domain solution as in frequency domain solution the influence of steady forces is usually ignored in the calculation of the forced motion problem. The comparison with experimental results show the efficiency of the present time domain formulation even for the complicated ship hulls, which is encouraging. The experimental data and time domain panel method results are compared with strip theory predictions based on the method of Salvesen et al. (1970). This is of interest to demonstrate the consequences of simplifying the forward speed effects on the boundary conditions, as well as of assuming high frequency on the free surface condition. Maro´n et al. (2004) describe the experimental programme in which the radiation forces presented in the present paper were measured. The experimental setup and the procedures for analysis of the signals and estimation of the added mass and damping coefficients are presented in detail by Maro´n et al. (2004). Then their analysis of experimental data is focused on: (a) The nonlinear effects on the radiation forces, namely the mean values and the higher harmonic content of the measured signals. (b) The steady forward speed effects, namely the mean radiation forces and the quasi-steady radiation forces. The analysis of the experimental data described in the present paper is new and consists of determining the linear components of the radiation force in: heave due to heave, heave due to pitch, pitch due to heave and pitch due to pitch, all for four Froude numbers. The paper from Maro´n et al. (2004) presents only one illustrative example which is the heave radiation force due to heave for one Froude number and the objective was to show that the nonlinear effects on the first harmonic amplitudes were small. The new contributions from this paper are: (a) New set of the first harmonic radiation forces in the experimental data for a Fast Ferry with different forward speeds of up to 40 kn.

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(b) A demonstration, by comparison with experimental data, of the consequences of simplifying the forward speed effects on the free surface and body boundary conditions on the seakeeping formulations. Although this topic has been discussed before, even since several decades ago, there are only a few 3D codes where the linear seakeeping problem is fully solved (without simplifications in the linear boundary conditions) and comparisons with the results from this type of codes with measured radiation forces is scarce. Furthermore there are no comparisons in the literature for the type of realistic fast hull presented here.

2. Mathematical formulation and numerical implementation of the panel code In the mathematical formulation, only a brief description of the general formulation is given, as a detailed discussion is available in other sources (Lin and Yue, 1990). The whole formulation is based on earth fixed co-ordinate system. Let (O, x, y, z) be the earth fixed coordinate system where origin is placed in mean water surface with z positive upward and (O, x0 , y0 , z0 ) be the body fixed system where the origin of the body fixed system is situated at the waterline (at z¼ 0) above CG of the ship. If the vessel is moving with the velocity U in the direction of positive x axis, then both the co-ordinate system is related by the following relation: x ¼ x0 þ Ut;

y ¼ y0 ; z ¼ z0

ð1Þ

For the formulation of the hydrodynamic problem, assume that a linear (sinusoidal) wave is approaching an angle a with the positive x axis. Then the fluid motion can be defined by the velocity potential !  !  !  fT X ; t ¼ fI X ; t þ f X ; t ð2Þ !  where fI X ; t is denoted as incident wave potential and !  !  !  f X ; t ¼ fT X ; t fI X ; t is called total disturbed potential. !  The total disturbed potential f X ; t satisfies the following governing equations, boundary condition, radiation conditions and initial conditions: !  ! r2 f X ; t ¼ 0; X A O ð3Þ @2 f @f ¼0 þg @z @t 2

on z ¼ 0

ð4Þ

@f @fI ¼ V n @n @n

on S0

ð5Þ

The disturbed potential can be obtained by following two integral equations: 8 2 > ZZ Z t ZZ 1 < 6 o fðp,tÞ ¼  sðq,tÞG ðp,qÞdS þ dt4 sðq,tÞGft ðp,q; ttÞdS 4p > 0 : S0 ðtÞ

S0 ðtÞ

39 > Z = 1  sðq,tÞGft ðp,q; ttÞV N V n dL7 5 > g GðtÞ ;

ð9Þ

Differentiating (9) with respect to n gives 8 > ZZ @fðp; t Þ 1 < @G0 ðp,qÞ ¼ sðq,tÞ dS @np 4p > @np : S0 ðtÞ 2 Z t ZZ @Gf ðp,q; ttÞ 6 þ dt4 sðq,tÞ t dS @np 0 S0 ðtÞ

1  g

Z

39 > = @Gft ðp,q; ttÞ 7 sðq,tÞ V N V n dL5 > @np GðtÞ ;

ð10Þ

Eq. (9) represents the total disturbed potential f at any point p at any instant of time t in terms of the distribution of sources over the body surface and waterline contour. Here, q¼q(x,Z,z) is the location of a point source on the body surface S0, s(q,t) is the associated source strength at retarded time t, and Go(p,q) and Gf(p,q;t t) are the Rankine and regular parts of the transient free surface Green’s function G(p,t,q,t): Gðp,t,q, tÞ ¼ G0 þ Gf with p a q, t 4 t

ð11Þ

G(t)represents the curve defined by the instantaneous intersection of the hull surface with the z¼0 plane, and VN corresponds to the two dimensional normal velocity in the z¼0 plane of a point on G. VN and  !! Vn are related by V N ¼ V n = N n . For the numerical solution of the above problem, a lower order panel method approach is taken. The details solution process are given by Sen (2002), Datta and Sen (2007) and Datta et al. (2011), and hence not repeated here. The method proposed by Datta and Sen (2007) faced some numerical instability for the hydrodynamic solution when the motion is studied for complicated hulls like fishing vessels. Datta et al. (2011) proposed a solution scheme to recover from the numerical instability for the hydrodynamic solution. The solution is given in terms of modification of the mesh near free surface. In the present paper, the same scheme is used and hence the description of the scheme is not repeated here.

rf-0 as RH -1, on z ¼ 0

ð6Þ

3. Linear frequency domain solutions

f, rf-0 as t-0

ð7Þ

While the time domain method presented in the previous section is formulated in an earth fixed co-ordinate system, the linear frequency domain seakeeping problem is usually formulated in an inertial reference system which advances with the ship constant forward speed. In the scope of the present section only, this moving reference system is defined as (O,x,y,z). The reference system is parallel to the earth fixed reference system and also parallel to the body fixed reference system when the ship has no oscillatory motions. There is another important difference between the time domain and the frequency domain formulations: while the first solves the boundary value problem directly for the total linear perturbation potential, within the frequency domain approach the perturbation potential is decomposed into steady, f , plus oscillatory potentials and the second is further decomposed into

In the above expressions, O represents the fluid domain, S0 is the mean wetted body surface, Vn is the normal velocity of the body surface, g is the acceleration due to gravity. For the radiation problem, the boundary condition (5) is modified as follows: 9 @f = @n ¼ Un1 þ Lðt Þnk for k ¼ 1,2,3 on S0 ð8Þ @f ¼ Un1 þ Lðt Þðr  nÞk3 for k ¼ 4,5,6 ; @n

In Eq. (8), n is the unit normal vector and L(t) represents the time harmonic impulsive function. The solution of the above initial boundary value problem is based on the transient Green’s function approach of Lin and Yue (1990).

R. Datta et al. / Ocean Engineering 60 (2013) 136–148

radiation, fR and diffraction, fD, potentials: R

D

f ¼ f þf þf

ð12Þ

The incident waves are assumed harmonic and, within the linear approach, the oscillatory potential and all ship related responses are harmonic as well. The linear boundary value problem, equivalent to the time domain one presented by Eqs. (3)–(7) which is established by: !  ! r2 f X ; t ¼ 0; X A O ð13Þ   R,D @ 2 R,D @f ¼ 0, on z ¼ 0 f þg ioU @x @z

ð14Þ

R

@fj ¼ ionj þUmj , j ¼ 1,. . .,6, on S0 @n D

ð15Þ

I

@f @f ¼ , on S0 @n @n

ð16Þ

Radiation condition as RH -1, on z ¼ 0

ð17Þ

where i is the imaginary unit number, o is the frequency of oscillation, n~ is the generalised outward unit vector perpendicular to the hull surface S0 and the radiation condition is imposed so that the waves generated by the body radiate away to infinity (RH is the ~ vector represents, in radial distance with respect to the body). The m a condensed way, the linear interactions between the steady and oscillatory flows in the body radiation boundary condition ( nk for k ¼ 1,2,3 n~ ¼ ð18Þ ðr  nÞk3 for k ¼ 4,5,6 8 ! > < ðm1 ,m2 ,m3 Þ ¼  ð n rÞrf U ~ ¼ m ! ! > : ðm ,m ,m Þ ¼  ð n rÞð r rfÞ 5 4 6 U

ð19Þ

In Eqs. (12) and (19) f represents the steady potential related to the ship constant forward speed, and this is further decomposed into f ¼ Un1 þ fS where Un1 is the potential due to the uniform incident flow and fs represents the modification of the uniform flow due to the ship’s hull. The equations above show that two boundary value problems need to be solved to obtain the unknown radiation and diffraction potentials. The radiation problem is related to the hull advancing with constant speed and forced harmonic motions in otherwise calm water (no incident waves). The diffraction problem is related to the hull advancing through the incident harmonic wave field and restrained in the mean position (no oscillatory motions). There are two important differences between the boundary value problems formulated under Sections 2 and 3: (a) the time domain formulation considers the steady effects due to forward speed directly in the disturbed velocity potential and therefore in the body boundary and free surface boundary conditions, which means that the steady and oscillatory problems are solved simultaneously, while the frequency domain approach decomposes the potential into steady and oscillatory potentials and the boundary value problem is formulated for the oscillatory potential only; (b) the diffraction effects in the time domain formulation are coupled to the oscillatory motions of the hull, while the frequency domain approach assumes that the radiation and diffraction problems are decoupled meaning that the diffraction forces are independent of the ship oscillatory motions. Some authors solved the linear problem in the frequency domain as presented in Eqs. (12)–(17), both applying the Green’s function panel methods (as for example Inglis and Price, 1981a,b) and the Rankine panel methods (Nakos et al., 1993).

139

Although the problem stated above is fully linear, the numerical solution is still difficult to obtain and for this reason many of the existing seakeeping codes which are used for practical applications simplify further the boundary conditions. The objective is to simplify the interactions between the steady and oscillatory flows and the three-dimensional effects of the flow. The simplifications are introduced by imposing restrictions on some parameters governing the solution, namely: the speed of the ship, the slenderness of the hull and the frequency of oscillation of the boundaries. The selection and combination of simplifications to introduce distinguish the different seakeeping theories. The body boundary condition can be further simplified by assuming that the ship is (very) slender and the steady potential is simplified to f ¼ Un1 , meaning that the modification of the uniform flow due to the hull can be neglected. With this assumption the gradients of the steady potential are very simple to calculate and the Eq. (19) is simplified to: ( ðm1 ,m2 ,m3 Þ ¼ ð0,0,0Þ ~ ¼ m ð20Þ ðm4 ,m5 ,m6 Þ ¼ ð0,n3 ,n2 Þ This approach was used by Chang (1977), who first solved the seakeeping problem applying the panel method and three dimensional Green’s functions. Several other authors followed a similar approach. In order to simplify the longitudinal gradients of the oscillatory potentials in the free surface boundary condition (14) a strip theory assumption is often used, namely the high frequency assumption. The resulting condition is. R,D

o2 f

R,D

þg

@f @z

¼ 0, on z ¼ 0

ð21Þ

The potential is still 3D, however the forward speed effects have been removed from the free surface condition by assuming the frequency is high (and the ship speed is low). This solution has been applied, for example, by Papanikolaou and Schellin (1992), Schellin and Rathje (1995) and Chan (1993). There is a small step between the boundary conditions (18) and (19) and the ones used by consistent strip theory methods (as for example the one from Salvesen et al., 1970), which is the assumption that the unit normal vector to the hull surface, which is at this point already assumed as (very) slender, is two-dimensional, therefore the longitudinal component can be neglected. The limitations of the panel methods which use Eqs. (18) and (19) and strip theories are similar and this is the reason why the predicted motions and global structural loads in waves are similar as well. The most simplified level of accuracy consists on using the ! frequency of encounter approach, meaning that the m vector of Eq. (18) is zero in the body boundary conditions and the free surface condition is given by (19). Tuitman and Malenica (2009) applied this approach, together with a 3D panel method and zero speed Green’s function, to calculate the coupled seakeeping and whipping responses of ships. In the next sections a strip theory code based on the formulation from Salvesen et al. (1970) is used to compute the heave and pitch radiation forces and compare with the experimental data and with the time domain panel code results. The objective is to demonstrate the limitations of introducing the simplifications on the linear boundary conditions, namely those related to the interactions between the steady and the unsteady potentials and to verify how the 3D panel code deals with the problem. 4. Experimental programme with a Fast Ferry model The experimental programme was carried out at the El Pardo Model Basin (CEHIPAR) and it included forced motion tests to

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R. Datta et al. / Ocean Engineering 60 (2013) 136–148

20 19 -0.25 -0.5 -0.25 -0.5

18

waterline 10 9 4 3

2

1

17 16

87 6 5

15 14 13 12 11 10

0.5 0

Length overall (m) Length between perpendiculars (m) Length waterline (m) Beam max (m) Beam waterline (m) Draught (m) Trim (m)

125.0 110.0 110.0 18.7 14.7 2.80 0.00

Fig. 1. Fast Ferry bodylines.

obtain the radiation forces and free motion tests in incident regular waves. A description of the setup is given in Maro´n et al. (2004), although in the context of testing another ship hull. Four different speeds were considered for both groups of tests. Some hydrodynamic coefficient results and vertical motions have been presented by Fonseca et al. (2005a). Most of the forced motion tests were performed at the Still Water Tank, namely all those with forward speed. Zero speed tests were carried out at the Laboratory of Ship Dynamics, whose tank is wider and in this way the wall effects can be minimised. Experiments in waves were conducted at the Laboratory of Ship Dynamics as well. The Still Water Tank is 320 m long, 12.5 m wide and 6.5 m deep, while the Laboratory of Ship Dynamics has a tank 145 m long from the face of the wave generator to the intersection of the beach with the water free surface, 30 m wide and 5 m deep. The experiments were carried out with a model of a Fast Ferry. The ship is 125.0 m overall and has service speed of 44 knot. Fig. 1 presents the hull bodylines and Table 1 the ship main particulars. The model was constructed in fibre reinforced plastic (FRP), the length is 4.4 m between perpendiculars (scale of 1:25) and it is made up of four segments connected by a rigid aluminium backbone. The cuts between segments are located in section 5, 10 and 15. Vertical cross sectional loads were measured at these positions. The neutral axis of the ‘‘almost rigid’’ backbone is approximately at the waterline level. One observes that the ship hull has a transom stern, which for ships with forward speed introduces local viscous effects, namely the flow separation at the hard chine. In the present paper the main focus in on potential flow radiation forces, therefore the transom stern viscous effects are not taken into account nor discussed. The forced oscillations were produced by means of two vertical linear actuators. The aft actuator was installed at station 7½, using the usual reference system with station 0 at the aft perpendicular and 20 at the forward perpendicular. This actuator was clamped to the towing carriage so that it remained always

Table 1 Fast Ferry main characteristics. Fast Ferry Length overall Length between perp. Breadth overall Depth Draught Displacement Service speed Longitudinal position of CG Vertical position of CG Pitch radiation of gyr.

Loa(m) L(m) B(m) D(m) T(m) D(ton) V(kn) LCG (m) VCG (m) Kyy/Lpp

125.0 110.0 18.7 16.7 2.80 2267 40  11.47 4.475 0.248

vertical providing the towing force necessary to maintain the model at speed. The forward actuator was located at station 12½. The connection to the carriage was through a hinge allowing some rotation in the longitudinal vertical plane such that the model could pitch. Fig. 2 shows a sketch with the model and the position of the actuators, as well as all the instrumentation used during the tests, namely load cells to measure the forces on the actuators, potentiometers to measure the forced motions and load cells installed at the backbone for the structural loads at the segmented cross sections. For the tests in waves the model was towed by a system connected to the carriage which allows free heave, pitch and roll motions. Only head waves were tested and the motions were measured with an optical system.

5. Results and discussion 5.1. Radiation force for a hemisphere Initially, to check the correctness of the proposed scheme, heave radiation force of a hemisphere is computed for a heave

R. Datta et al. / Ocean Engineering 60 (2013) 136–148

141

Y

X Z

Non Dimensional Force

Fig. 2. Sketch with the instrumentation of the model.

Radiation Force (F33)

6 5 4

TD

3

WAMIT

2 1 0 0

5.2. Radiation force for the Fast Ferry Fig. 3 represents the initial mesh of the Fast Ferry (without introducing the vertical panel) and it can be seen that the hull is not simplistic. The hull is highly unsymmetrical on bow and stern parts, with ‘‘v’’ shaped cross section on the bow which gradually become flatter and shallow draft near the stern (for semiplanning characteristics). The details of the hull are given in Table 1. From the table, one may notice that the longitudinal position of the centre of gravity (LCG) of the ship is 11.5 m aft of midship, which simply confirm the high asymmetry level of the hull. 5.2.1. Steady radiation force Initially, the time history for the forced motion is demonstrated for two different speeds (in this case, 0 and 40 knot which

Nondimensional Heave Force

forcing function (F33) for unit amplitude motion. The force results are computed for a range of frequencies and then compared with WAMIT. The force results and frequencies are nondimensionalised by 9F 33 9=rg D and kR, respectively, where 9F339 represents the radiation force amplitude, D represents the underwater volume, k and R represents the wave number and radius of the hemisphere, respectively. Fig. 4 confirms the excellent agreement between the industrially accepted WAMIT code and the present time domain panel code. From the figure, it may be noted that present scheme is working well and produce very good results for the hemisphere.

4

8

6 kR

10

12

Fig. 4. Heave radiation force for a hemisphere due to heave forcing function.

Heave Force History

0.2 0.15 0.1 0.05

TD time history

0 -0.05

0

10

20

30

40

50

-0.1 -0.15 -0.2 t

Fig. 5. Heave radiation force time history for Fast Ferry due to Heave forcing function. Zero speed case, kL¼2.58.

Nondimensional Pitch Moment

Fig. 3. Mesh of the Fast Ferry.

2

Pitch Moment History

0.015 0.01 0.005

TD time History

0 0

10

20

30

40

50

-0.005 -0.01 -0.015

t

Fig. 6. Pitch radiation moment time history for Fast Ferry due to pitch forcing function. Zero speed case, kL¼2.58.

R. Datta et al. / Ocean Engineering 60 (2013) 136–148

Nondimensional Heave Force

142

Steady heave force history

0.25 0.2 0.15 0.1

Steady heave force

0.05 0 -0.05

0

10

20

30

40

50

-0.1 t Fig. 9. Steady heave force, 40 knot speed.

Fig. 7. Heave radiation force time history for Fast Ferry due to Heave forcing function. 40 knot speed, kL¼2.58.

0.05 0 0

10

20

30

40 Without removing steady contribution

-0.05

Removing steady contribution

-0.1 -0.15 -0.2 t

Nondimensional Pitch Moment

Nondimensional Pitch Moment

Steady pitch moment time history Pitch moment history(40 knot)

0.01 0 -0.01 0 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09

20

40

60

Steady Pitch moment

t Fig. 8. Pitch radiation moment time history for Fast Ferry due to pitch forcing function. 40 knot speed, kL¼2.58.

is almost the service speed of the vessel) (Fig.4). Figs. 5 and 6 represent the radiation force history for zero speed, at kL¼2.58, where L is the length between perpendiculars, amplitude of forced heave motion and forced pitch motion is 1 m and 11, respectively. Fig. 7 represents the time history (time axis is given in second scale for all time history graphs) for non-dimensional heave force due to heave forced harmonic function and Fig. 8 presents the time history for the non-dimensional pitch moment due to forced pitch. The force and moments are nondimensionalised by F 33 ðtÞ=rg D and F 55 ðtÞ=rg DL, respectively. From Figs. 5 and 6 it may be noted that for the zero speed case responses are very steady and perfectly symmetrical about the x-axis. Figs. 7 and 8 represent the heave and pitch force and moment for the 40 knot speed (almost service speed) of the vessel. In Figs. 7 and 8, the continuous line represents the radiation force history without removing the effect of steady force component. The dotted line represents the radiation force history after removing the steady linear force component. The steady component is defined here as the hydrodynamic force generated on the hull advancing with constant forward speed without forcing oscillatory motions (and without incident waves as well). Although the hull is advancing, it is restrained at the zero speed static sinkage and trim, meaning that the steady hydrodynamic force represents the effects of the forward speed on the steady pressure distribution on the hull. Mathematically it is calculated by modifying the body boundary condition (8) as follows: @f ¼ Un1 @n

on S0

ð22Þ

It simply means that to calculate the steady forces, the code is run in the absence of any waves or impulse and computes the heave and pitch moment. Figs. 9 and 10 give the steady heave force and pitch moment component using boundary condition (22). The heave force

Fig. 10. Steady pitch moment, 40 knot speed.

and pitch moment are negative, meaning the ship would tend to submerge under speed and rotate the bow up. From the Figs. 7 and 8, it can be noted that the continuous time series is not symmetrical about the x-axis, which means that it is influenced by some steady forces. Compared to the oscillatory amplitude of the radiation forces, one concludes that the steady effects are very pronounced for the ship’s service speed. The dotted lines of Figs. 7 and 8 give the force history after removing the steady component. It can be observed that, after removing the steady effect, the force history asymmetry reduces, but does not disappears. This means that, besides the non-oscillatory hull steady hydrodynamic force component, there is also a steady radiation force induced by forward speed effects. While the time domain method is able to include the steady force contribution, it is very difficult to include the steady contribution in the case of the frequency domain solution. This is one major advantage of using the time domain solution over frequency domain solution.

5.2.2. Amplitude of the radiation forces The results for radiation forces due to heave forcing harmonic function and pitch forcing harmonic function are presented for the different speeds in Figs. 11–27. The forced motion amplitudes are 1.0 m for heave and 1.01 for pitch. The computed results with the strip theory and 3D time domain panel method are compared with experimental data. In the figures, the amplitude of forces and moments are normalised by 9F9=rg D and 9M9/rp gDffiffiffiffiffiffiffi L, ffirespectively. The frequencies are non-dimensionalised by o L=g . In the legends, ‘‘Exp’’ represents the results from experiments, ‘‘TD’’ represents the results from panel code calculations and ‘‘ST’’ is the results from strip theory code. It may also be noted that F33, F53, F35 and F55 represent the following cases:

R. Datta et al. / Ocean Engineering 60 (2013) 136–148

143

Experimental Results (F33) 1.2 Nondimensional Force

1 0.8

Speed = 0knot Speed = 20 knot

0.6

Speed = 30 knot Speed = 40 knot

0.4 0.2 0 0

2 4 6 nondimensional Frequency

8

Experimental Results (F53) Nondimensional Moment

0.045 0.04 0.035 0.03

Speed = 0 knot

0.025

Speed = 20 knot

0.02

Speed = 30 knot Speed = 40 knot

0.015 0.01 0.005 0 0

2 4 6 Nondimensional Frequency

8

Fig. 11. (a) Comparison of radiation force amplitude in different speeds for F33, (b) comparison of radiation moment amplitude in different speeds for F53, (c) comparison of radiation force amplitude in different speeds for F35, (d) comparison of radiation moment amplitude in different speeds for F55.

F33, 0 speed Nondimensional Force Amplitude

1.2 1 0.8 Exp

0.6

TD ST

0.4 0.2 0 0

2 4 6 Nondimensional Frequency

8

Phase, F33, 0 speed 0.0

Phase, F53, 0 speed 150.0

-20.0

Exp TD ST

-30.0 -40.0 -50.0

Angle (degree)

Angle (degree)

-10.0

100.0 50.0

Exp TD ST

0.0 -50.0 -100.0 -150.0 0

-60.0 0

2 4 6 Nondimensional Frequency

8

2 4 6 Nondimensional Frequency

8

Fig. 13. Pitch moment response function and phase angle for Fast Ferry due to heave forcing function, 0 speed.

Fig. 12. Heave force response function and phase angle for Fast Ferry due to heave forcing function, 0 speed.

d) F55: Pitch radiation moment due to pitch forcing function. a) F33: Heave radiation force due to heave forcing function. b) F53: Pitch radiation moment due to heave forcing function. c) F35: Heave radiation force due to pitch forcing function.

Fig. 11a–d represents the complete experimental results for the Fast Ferry. Each graph presents the radiation force results for

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F35, 0 speed

0.06 0.05 Exp TD ST

0.04 0.03 0.02 0.01

Nondimensional Force Amplitude

Nondimensional Force Amplitude

F33, 20 knot speed

1.2

0.07

1 0.8 Exp TD ST

0.6 0.4 0.2

0 0

2 4 6 Nondimensional Frequency

0

8

0

Phase, F35, 0 speed 150.0

Exp TD ST

0.0

-50.0 -100.0 -150.0

Angle (degree)

Angle (degree)

50.0

-200.0 2 4 6 Nondimensional Frequency

-40.0 -50.0

F55, 0 speed

0.1 0.08 Exp TD ST

0.06 0.04 0.02 2 4 6 Nondimensional Frequency

8

Exp TD ST

0

Exp TD ST

-30.0 -40.0 -50.0 -60.0 0

2 4 6 Nondimensional Frequency

8

Fig. 15. Pitch moment response function and phase angle for Fast Ferry due to pitch forcing function, 0 speed.

each mode with different speed. The results are clustered in this way just to show the effect of speed on radiation forces. From Fig. 11a, it may be noted that for F33 the variation of radiation force amplitude is minimum when speed is increased. The variation is significantly increased with speed for F55, and for F53 and F35, results largely vary when speed is increased. Figs. 12–15 represent the results for the zero speed case. From the figures, it can be seen that the panel code computation agrees very well with the experimental data and the strip theory results.

Angle (degree)

-10.0 -20.0

8

F53, 20 knot speed

0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

Phase,F55, 0 speed

0.0

2 4 6 Nondimensional Frequency

Fig. 16. Heave force response function and phase angle for Fast Ferry due to heave forcing function, 20 knot speed.

Nondimensional Force Amplitude

Nondimensional Force Amplitude

Exp TD ST

-30.0

0

0

Angle (degree)

-20.0

8

Fig. 14. Heave force response function and phase angle for Fast Ferry due to pitch forcing function, 0 speed.

0

-10.0

-60.0

0

0.12

8

Phase, F33, 20 knot speed

0.0

100.0

2 4 6 Nondimensional Frequency

2 4 6 Nondimensional Frequency

8

Phase, F53, 20 knot speed

45.0 40.0 35.0 30.0 25.0 20.0 15.0 10.0 5.0 0.0

Exp TD ST

0

2 4 6 Nondimensional Frequency

8

Fig. 17. Pitch moment response function and phase for Fast Ferry due to heave forcing function, 20 knot speed.

It is interesting to observe that the strip theory and the panel code results match very well for all four cases in zero speed while have some difference with experiments especially for the computations of F53 and F35. Figs. 16–27 represent the same results for different speeds: Figs. 16–19show the radiation force amplitude results with phases for 20 knot, Figs. 20–23correspond to the results for 30 knot and 24–27 are for 40 knot.

R. Datta et al. / Ocean Engineering 60 (2013) 136–148

Nondimensional Force Amplitude

0.14 0.12 0.1

Eep TD ST

0.08 0.06 0.04 0.02

Nondimensional Force Amplitude

F35, 20 knot speed 0.16

2 4 6 Nondimensional Frequency

8

8

-10.0

-20.0 -40.0 Exp TD ST

-60.0 -80.0

Angle (degree)

0.0

-20.0 Exp TD ST

-30.0 -40.0 -50.0

-100.0

-60.0

-120.0

-70.0 0

-140.0 2 4 6 Nondimensional Frequency

8

Fig. 18. Heave force response function for Fast Ferry due to pitch forcing function, 20 knot speed.

0.12 0.1 0.08 Exp TD ST

0.06 0.04

8

F53, 30 knot speed

0.16

F55, 20 knot speed

2 4 6 Nondimensional Frequency

Fig. 20. Heave force response function and phase angle for Fast Ferry due to heave forcing function, 30 knot speed.

Nondimensional Force Amplitude

0

0.14 0.12 0.1

Exp TD ST

0.08 0.06 0.04 0.02 0 0

0.02 0 2 4 6 Nondimensional Frequency

8

0.0 -10.0 -20.0 Exp TD ST

-30.0 -40.0

50.0 Exp TD ST

40.0 30.0 20.0 10.0 0.0 0

-50.0 -60.0 -70.0 2 4 6 Nondimensional Frequency

8

60.0

Phase, F55, 20 knot speed

0

2 4 6 Nondimensional Frequency Phase, F53, 30 knot speed

70.0 Angle (degree)

0

Angle (degree)

2 4 6 Nondimensional Frequency Phase, F33, 30 knot speed

0.0

Phase, F35, 20 knot speed

Angle (degree)

Exp TD ST

0 0

Nondimensioanl Force Amplitude

F33, 30 knot speed

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

145

8

Fig. 19. Pitch moment response function for Fast Ferry due to pitch forcing function, 20 knot speed.

Starting with the heave radiation forces due to heave forced motion (F33), the experimental data shows an almost negligible influence of the forward speed. There is only a very small increase of the force amplitude for the higher speed and higher frequency of oscillation. The agreement between the numerical and experimental results is very good, even for the strip theory at high speed.

2 4 6 Nondimensional Frequency

8

Fig. 21. Pitch moment response function and phase angle for Fast Ferry due to heave forcing function, 30 knot speed.

Regarding the pitch moment due to forced pitch motion (F55), the experimental results show evidence of significant forward speed effects. In fact the amplitude of the radiation moment increases with the forward speed by around 40–60% between zero and 40 knot. The time domain code is able to represent the forward speed effects quite well. Regarding the strip theory, the predictions are good for high frequencies, but not for the low to medium frequency range. This reflects the limitation of the high

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F35, 30 knot speed

0.2

0.15

Exp TD ST

0.1

Nondimensional Force Amplitude

Nondimensional Force Amplitude

F33, 40 knot speed

1.2

0.25

1 0.8 Exp TD ST

0.6 0.4 0.2 0

0.05

0

2 4 6 Nondimensional Frequency

8

0 0

2 4 6 Nondimensional Frequency

8

-10.0

0.0 -20.0 -40.0 -60.0

Exp TD ST

-80.0 -100.0

Angle (degree)

Phase, F35, 30 knot speed

Angle (degree)

Phase, F33, 40 knot speed

0.0

-20.0 Exp TD ST

-30.0 -40.0 -50.0 -60.0

-120.0

-70.0

-140.0

0

-160.0 0

2 4 6 Nondimansional Frequency

8

2 4 6 Nondimensional Frequency

8

Fig. 24. Heave force response function and phase angle for Fast Ferry due to heave forcing function, 40 knot speed.

Fig. 22. Heave force response function and phase angle for Fast Ferry due to pitch forcing function, 30 knot speed.

0.1 Exp TD ST

0.08 0.06 0.04 0.02 0

Angle (degree)

0

2 4 6 Nondimensional Frequency

8

Phase, F55, 30 knot speed

0.0 -10.0 -20.0 -30.0 -40.0 -50.0 -60.0 -70.0 -80.0

2 4 6 Nondimensional Frequency

0.14 0.12 0.1

Exp TD ST

0.08 0.06 0.04 0.02 0

8

Fig. 23. Pitch moment response function and phase angle for Fast Ferry due to pitch forcing function, 30 knot speed.

2 4 6 Nondimensional Frequency

8

Phase, F53, 40 knot speed

90.0

Exp TD ST

0

F53, 40 knot speed

0.16

0

80.0 Angle (degree)

Nondimensional Force Amplitude

0.12

Nondimensional Force Amplitude

F55, 30 knot speed

0.14

70.0 60.0 Exp TD ST

50.0 40.0 30.0 20.0 10.0 0.0

frequency hypothesis assumed to simplify the free surface boundary condition. The experimental coupling radiation force amplitudes (F53 and F35) present a very large dependence of the forward speed. These

0

2 4 6 Nondimensional Frequency

8

Fig. 25. Pitch moment response function and phase angle for Fast Ferry due to heave forcing function, 40 knot speed.

R. Datta et al. / Ocean Engineering 60 (2013) 136–148

F35, 40 knot speed

0.25 0.2 Exp TD ST

0.15 0.1 0.05

0.14 0.12 0.1

Exp TD ST

0.08 0.06 0.04 0.02 0

0

0 0

2 4 6 Nondimensional Frequency

8

2 4 6 Nondimensional Frequency

8

Phase, F55, 40 knot speed

0.0

Phase, F35, 40 knot speed

0.0 -40.0 -60.0

Exp TD ST

-80.0 -100.0 -120.0

Angle ( degree)

-10.0

-20.0 Angle (degree)

F55, 40 knot speed

0.16 Nondimensional Force Amplitude

Nondimensional Force Amplitude

0.3

147

-20.0 -30.0

Exp TD ST

-40.0 -50.0 -60.0 -70.0

-140.0

-80.0

-160.0 0

2 4 6 Nondimensional Frequency

8

Fig. 26. Heave force response function and phase angle for Fast Ferry due to pitch forcing function, 40 knot speed.

forces increase up to 10 times between 0 and 40 knot. Interestingly, the effect of forward speed on the radiation force due to forced pitch motion (F35) is similar along the whole frequency range, while for the pitch moment due to forced heave motion (F53) the effects are much stronger for low frequencies. Regarding the time domain computations it is possible to say that the agreement with experiments is good for all speeds and frequencies. The method is able to predict very well the coupling pitch moment due to heave and reasonably well the heave force due to pitch. The strip theory fails to predict the coupling forces when the forward speed is present. From the figures, it may be concluded that the time domain code predicts all four radiation forces well, there are some difference for coupled forces, but the overall behaviour is very consistent and follow the same pattern as that of the experiments, whereas for the coupled forces, strip theory results are far from the experiments. Therefore it may be concluded that the present time domain method produce good and robust results for the radiation forces.

6. Conclusions The paper investigates the hydrodynamic radiation forces on a Fast Ferry mono-hull with a time domain three dimensional panel method and a strip theory code and compares their predictions with the experimental data of model tests. The time domain formulation accounts for the full linear interaction between the steady flow due to the ship’s forward speed and the oscillatory flow and this is one of the main advantages of the method. On the other hand, the common frequency domain seakeeping theories introduce some simplifications in the former interactions, including the panel methods. In order to

0

2 4 6 Nondimensional Frequency

8

Fig. 27. Pitch moment response function and phase angle for Fast Ferry due to pitch forcing function, 40 knot speed.

assess the consequences of these simplifications the strip theory results and the time domain panel code predictions are compared with the experimental data. The experimental data shows that the heave radiation forces due to forced heave motion are almost independent of the forward speed. In this case both the time domain method and the strip theory give very good predictions of the hydrodynamic forces. Regarding the pitch radiation moment due to forced pitch motion, one observes a significant increase of the moment amplitude with the forward speed between 0 and 40 knot. In this case the time domain results agree very well with the experiments for all speeds and for the whole frequency range, while the agreement with strip theory is poor at the high speeds and low frequency range. The experimental coupling forces show the largest influence from the ship’s speed with amplification of the forces up to 10 times between 0 and 40 knot. The panel code represents these effects well, while the strip theory fails for the higher speeds. As a general conclusion, the present time domain panel code results show very good agreement with experimental results, which shows the efficiency and correctness of the proposed scheme. It is also observed from the time signals that the present time domain code is able to give very stable signals. Therefore it may be concluded that the proposed method is very robust as well. Another interesting aspect of the present time domain code is to calculate the steady contribution automatically from the signals, which is not possible with frequency domain methods. Considering the complexity of the hull, it can be concluded that proposed time domain code is able to produce very good results even for the complicated hull shapes and high speeds.

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