Free-surface effects on interaction of multiple ships moving at different speeds

1

FREE-SURFACE EFFECTS ON MULTIPLE SHIPS

TRAVELLING WITH DIFFERENT SPEEDS

Zhi-Ming Yuan*, †_{, Liang Li}† _{and Ronald W. Yeung}‡,* 3

*_{Jiangsu University of Science and Technology, China.} 4

†_{University of Strathclyde, UK} 5

‡_{University of California at Berkeley, USA} 6

*Corresponding author, E-mail: [email protected], Tel.: +1 (510) 642-7

8347 8

9 10

ABSTRACT 11

Ships often have to pass each other in proximity in harbor area and waterways 12

in dense shipping traffic environment. Hydrodynamic interaction occurs when 13

a ship is overtaking (or being overtaken) or encountering other ships. Such an 14

interactive effect could be magnified in confined waterways, e.g. shallow and 15

narrow rivers. Since Yeung (1978) published his initial work on ship-interaction 16

in shallow water, progress on unsteady interaction among multiple ships has 17

been slow though steady over the following decades. With some exceptions, 18

nearly all the published studies on ship-to-ship problem neglected free-surface 19

effects, and a rigid wall condition has often been applied on the water surface 20

as the boundary condition. When the speed of the ships is low, this assumption 21

is reasonably accurate, as the hydrodynamic interaction is mainly induced by 22

near-field disturbances. However, in many maneuvering operations, the en-23

countering or overtaking speeds are actually moderately high (Froude number 24

Fn>0.2, where 𝐹𝑛 ≡ 𝑈/√𝑔𝐿, U is ship speed, g the gravitational acceleration and

25

L the ship length), especially when the lateral separation between ships is the 26

order of ship length. Here, the far-field effects arising from ship waves can be 27

important. The hydrodynamic interaction model must take into account of the 28

surface-wave effects. 29

Classical potential-flow formulation is only able to deal with the boundary 30

value problem (BVP) when there is only one speed involved in the free-surface 31

boundary condition. For multiple ships travelling with different speeds, it is not 32

possible to express the free-surface boundary condition by a single velocity po-33

tential. Instead, a superposition method can be applied to account for the veloc-34

ity field induced by each vessel with its own and unique speed. The main objec-35

tive of the present paper is to propose a rational superposition method to handle 36

the unsteady free-surface boundary condition containing two or more speed 37

terms, and validate its feasibility in predicting the hydrodynamic behaviour of 38

the ships during overtaking or encountering operations. The solution methodol-39

ogy used in the present paper is a three-dimensional boundary-element method 40

(BEM) based on a Rankine-type (infinite-space) source function, initiated intro-41

duced in Bai & Yeung (1974). The numerical simulations are conducted by using 42

an in-house developed multi-body hydrodynamic interaction program "MHy-43

dro". Waves generated and forces (or moments) are calculated when ships are 44

encountering or passing each other. Published model-test results are used to 45

validate our calculations and very good agreement has been observed. The nu-46

merical results show that free-surface effects need to be taken into account for 47

2

Keywords: Free-surface effect; ship-to-ship problem; hydrodynamic interaction;

49

encountering and overtaking operation; ship manoeuvring. 50

51 52

1 INTRODUCTION

53

The interaction between two or more ships involved in encountering or overtak-54

ing manoeuvring is a classical hydrodynamic problem. Due to the interaction 55

forces, a ship may deviate from its original course and collide with the other 56

ships. The interaction effects are aggravated when the ships are manoeuvring 57

in confined waterways, or when the ships are travelling with high speed. 58

Ship-to-ship problem has been widely studied over the last few decades. No mat-59

ter which kind of methods are used, at least one or more of the following im-60

portant assumptions are often adopted to simplify the problem: 61

1) The fluid is ideal and the viscous effects are neglected. 62

2) The speed is low and the free-surface effects are negligible (rigid-wall 63

free-surface is applicable). 64

3) The ships are slender. 65

4) The shedding of cross-flow vorticity is either ignored, or idealized in a 66

manner similar to thin-wing theory. 67

During1960s-1990s, the slender-body theory has been widely popular to predict 68

the hydrodynamic interaction between multiple ships (Collatz, 1963; Dand, 69

1975; Kijima and Yasukawa, 1985; Tuck, 1966; Tuck and Newman, 1974; 70

Varyani et al., 1998; Yeung, 1978). All of the assumptions mentioned above are 71

adopted in these studies. These assumptions significantly simplified the math-72

ematical model and led to a high-efficiency numerical calculation. For conven-73

tional ships travelling at relatively low Froude numbers, the numerical calcula-74

tions based on strip theory showed a fairly good prediction of the sway force and 75

yaw moment on ships during overtaking or meeting operations. To account for 76

the three-dimensional effects and remove the geometrical idealization described 77

above (Assumption 3)), Korsmeyer et al. (1993) adopted a three-dimensional 78

panel method, which is applicable to any number of arbitrary shaped bodies in 79

arbitrary motions. Pinkster (Pinkster, 2004) extended Korsmeyer’s method with 80

implementation of a model to account for the free-surface effects partially. His 81

model was restricted to simulating the effect of a passing ship on a moored ship. 82

Only the low frequency seiche or solitary waves were taken into account, while 83

the more important far-field waves or so-called Kelvin waves were neglected. Therefore, 84

his conclusions on free-surface effects could not cover the general ship-to-ship 85

operations. More recently, the three-dimensional panel method has been more 86

commonly used (Söding and Conrad, 2005; Xiang and Faltinsen, 2010; Xu et al., 87

2016; Zhou et al., 2012). However, no effort has yet been made to investigate the 88

effects of unsteady free-surface waves on interaction forces. The general conclu-89

sion drawn from these studies is that the potential flow solver could provide a 90

good prediction of interaction forces on ships travelling at relatively low Froude 91

numbers. Benefitted from improving CFD (Computational Fluid Dynamics) 92

technology, the viscous effects on ship-to-ship problem have been investigated 93

by various turbulence models (Jin et al., 2016; Sian et al., 2016; Zou and 94

Larsson, 2013b). In these studies, the free-surface effects are either neglected 95

(Zou and Larsson, 2013b) or treated simply as a steady problem (Jin et al., 2016; 96

3

free-surface waves produced by two or more ships moving with different speeds. 98

Mousaviraad et al. (2016b) analyzed the ship-ship interaction experiments both 99

in calm water and waves. They also run the URANS simulations, in which the 100

free-surface boundary condition was considered (Mousaviraad et al., 2016a). 101

However, these studies focus more on the hydrodynamic forces. The result of 102

free-surface elevation was neither measured in the model tests nor presented in 103

the CFD simulations. The demand in computational power when more than one 104

ship is passing can be the bottleneck if real-time applications should be needed. 105

All the above mentioned studies adopted the assumption that the encountering 106

or overtaking speed is low. Therefore, the unsteady free-surface wave effect is 107

not essential. This assumption significantly reduces the complexity of unsteady 108

ship-to-ship problem. But in the real manoeuvring practice, the encounter speed 109

is not always low. The importance of the free-surface effect is determined by 110

whether the far-field waves generated by a ship could propagate to the other 111

ships. At low Froude number, the amplitude of the far-field waves is very small. 112

These far-field waves are dissipated before they propagate to the far-field, as 113

shown in Fig. 1(a). Fig. 1 (b) shows the sketch of flow passing the gap between 114

two ships. The flow is compressed to pass through the narrow gaps between two 115

ships with relative higher velocity. According to Bernoulli's principle, the accel-116

erated fluid velocity could result in a decrease in pressure distribution in the 117

gap, therefore inducing hydrodynamic interaction forces (or moments). In this 118

low-speed case, the free-surface elevation and the hydrodynamic interaction are 119

mainly determined by the near-field disturbance. As the speed increases, the 120

far-field waves can be observed apparently. The far-field wave patterns gener-121

ated by two pressure disturbances moving towards opposite direction are shown 122

in Fig. 2 (a). The encounter process of these two disturbances is time-dependent. 123

It can be anticipated when a disturbance is in the other’s wake region, the hy-124

drodynamic interaction will be unavoidable. In the port or inland waterways, 125

the hydrodynamic interaction between three-dimensional vessels is also con-126

ceivable by the propagation of the far-field waves. The wave elevation reflects 127

the pressure distribution on water surface. The interaction occurs when the 128

waves produced by a ship strike the others, therefore modifying the pressure 129

distribution over their immersed body surfaces. Thus, the hydrodynamic inter-130

action can be apparently observed by wave interference on free water surface. 131

Benefit from satellite and imaging technology, we can observe the wave inter-132

ference phenomenon by analyzing high resolution satellite images. The encoun-133

tering and overtaking process of two real ships are shown in Fig. 2 (b) and (c) 134

respectively. These images show the far-field wave interference, which indicates 135

the ship-to-ship operation is not only limited at low Froude number. Even 136

though the transverse separation between the ships is large, the wave interfer-137

ence effect can still result in strong hydrodynamic interaction. The rigid free-138

surface is not capable to predict the hydrodynamic interactions induced by far-139

field waves. A new methodology should be proposed to deal with the free-surface 140

boundary condition.

141

The main challenge of imposing a non-rigid free-surface condition arises 142

from the speed term in the body boundary condition (See.Eq. (16) later). For 143

multiple ships travelling with various speeds, it is not possible to express the 144

free-surface boundary condition by a single velocity potential (unless one uses 145

an earth-fixed coordinate system as in Yeung (1975)). A superposition method, 146

however, can be applied to account for the velocity potentials induced by each 147

4

appearing in free-surface boundary condition, Yuan et al. (2015) proposed an 149

uncoupled method based on the superposition principle. Therein, the speed dif-150

ference of two ships was assumed to be small. Thus, the free-surface condition 151

could be treated (arguably) as steady-state problem. This method is not appli-152

cable to predict the interaction forces when ships’ speeds are not the same, or 153

when two ships are moving towards each other. In these cases, the unsteady 154

effect becomes essential and the time-dependent terms must be taken into ac-155

count. In the present study, we will extend Yuan’s method to the time domain 156

and discuss the importance of free-surface effects on multi-ship problem. 157

Fig. 1. (a), wave patterns produced by two ships travelling at low Froude number (Fn

158

= 0.043) (Yuan et al., 2015). (b), sketch of flow passing the gap between two ships.

159

d_l/ L

dt

/

L

1 0.5 -0 -0.5 -1 -1.5 -2

-0.5 0 0.5

1 Wave height (mm): -2.0 -1.5 -1.1 -0.6 -0.1 0.3 0.8 1.3

Ship 2

y2

Ship 1

x2

o2

y1

x1

o1

U1

U2

(a) (a)

5

Fig. 2. (a), sketch showing the transverse and divergent waves generating by two

160

pressure disturbances moving towards opposite direction. (b) and (c), satellite image of

161

ship wakes taken from the Google Earth database. (b), two ships encountering at

162

Dordtse Kil, The Netherlands

163

(https://www.google.com/maps/@51.7519406,4.6291446,652a,35y,75.15h,8.71t/data=!3

164

m1!1e3?hl=en). (c), two ships overtaking at Lek, Netherlands

165

(https://www.google.com/maps/@51.9953387,5.0694056,298a,35y,325.35h/data=!3m1!1

166

e3?hl=en). The Froude number of the vessels in the lower part of (b) and upper part of

167

(c) is Fn ≈ 0.15.

168

2 THE BOUNDARY VALUE PROBLEM

169

170

Fig. 3. Coordinate systems. 171

Consider N ships moving at speeds Uj (j = 1, 2, …, N) with respect to a space-172

fixed reference frame 𝐱 = (𝑥, 𝑦, 𝑧) in an inviscid fluid of depth h as shown in Fig. 173

3. A right-handed Cartesian coordinate system 𝐱_𝐣= (𝑥_𝑗, 𝑦_𝑗, 𝑧_𝑗) (j = 1, 2, …, N) is 174

fixed to each ship with its positive xj-axis pointing towards the bow, positive z-175

axis pointing upwards and zj = 0 being the undisturbed free-surface. Let Φ (x, 176

t) be the velocity potential describing the disturbances generated by the forward 177

40 m

40 m

x2 O2

y2

z2

T

2

y1

z1

x1

O1

U1

yn zn

xn On

U2

Un y

z

O _x

dl

dt

B₁

B_N

B2

(b)

6

motion of the ships and ζ (x, y, t) be the free-surface wave elevation. In the fluid 178

domain, the total velocity potential Φ satisfies the Laplace equation 179

2

( , )t 0

  x  (1)

180

The fluid pressure, 𝑝(𝐱, 𝑡), is given by Bernoulli’s equation 181

0

1 ( , )

2

p t gz p

t



  _     _ 

 

x (2)

182

where ρ is the fluid density, p0 is the atmospheric pressure, which is used as a 183

reference pressure and assumed to be constant. Assuming there is no overturn-184

ing and breaking waves on the free-surface, we can use this Eulerian description 185

of the flow to describe the free-surface motion. The free-surface elevation is 186

given by z = ζ (x, y, t). A fluid particle on the free-surface is assumed to stay on 187

the free-surface, which leads to the following kinematic free-surface boundary 188

condition: 189





D z

Dt    , on z = ζ (3)

190

The material derivative in Eq. (3) is given by: 191

D

Dt t



   

 (4)

192

The dynamic free-surface condition is that the fluid pressure equals the con-193

stant atmospheric pressure p0 on the surface, since the position of the free-194

surface is unknown. According to Bernoulli’s equation Eq. (2), the dynamic free-195

surface boundary condition can be written as 196

1

0

2 gz

t

_{    }

 , on z = ζ (5)

197

By applying Taylor series expanded about z = 0 and only keeping the linear 198

terms, the dynamic and kinetic free-surface conditions can be linearized as 199

0

t z



 _ _

  , on z = 0 (6)

200

0

g

t 

_{ }

 , on z = 0 (7)

201

Combining Eq. (6) and (7), we obtain the free-surface boundary condition: 202

2

2 g 0

t z

 _ _

  , on z = 0 (8)

203

It should be noted the free-surface ζ can be found from Eq. (7) when the velocity 204

potential Φ is known. On the wetted body surface, the no-flux boundary condi-205

tions are used, and the following ‘exact’ boundary condition can be formulated: 206

 

j x j

U n

n

_

 , on Bj, where j = 1, 2, …, N (9)

207

where ∂/∂n is the derivative along the normal vector 𝐧 = (𝑛_𝑥, 𝑛_𝑦, 𝑛_𝑧) into the 208

7

Assuming the disturbance of the fluid is small, we represent the total velocity 210

potential produced by the presence of all ships in the fluid domain in a space-211

fixed frame to satisfy the following superposition principle: 212

1

( , ) ( , )

N

j j

t t



 x 



213

where Φj (x, t) is the velocity potential produced by the presence of ship j moving 214

with Uj, while the remaining ships are momentarily stationary in this frame. 215

For the linear problem, the body-fixed coordinate system 𝐱𝐣= (𝑥𝑗, 𝑦𝑗, 𝑧𝑗) (j = 1, 2,

216

…, N) is used to solve the BVP for N vessels in concurrent motion. The relation 217

between the body- and space-fixed coordinate system is straightforward, viz. 218

j j

x  x U t, j = 1, 2, …, N (11) 219

Let ϕj (xj, t) represents Φj (x, t) in the body-fixed coordinate system, the following 220

relation can be obtained 221

j

j j

j

d

U

dt t x



 

 _  _ 

 

_{ } 

  (12)

222

The velocity potential ϕj satisfies the Laplace equation and body ‘exact’ bound-223

ary condition: 224

2

( , ) 0

j t



 x_j  , j = 1, 2, …, N (13)

225

 

j

ijUj nx j

n





 

 , on Bi, where i, j = 1, 2, …, N (14) 226

The Kronecker delta δij is the quantity defined by 227

1

0

ij

i j i j

 _{ }  

 (15)

228

Substituting Eq. (12) into the linearized free-surface condition in Eq. (8), we ob-229

tain the linearized free-surface condition in the body-fixed coordinate system 230

2 2 2

2

2 2 2 0

j j j j

j j

j j j

U U g

t x t x z









   

   

     , on z = 0 (16)

231

The boundary condition on the sea bottom and side walls, if any, can be ex-232

pressed as 233

0 j

n



 

 (17)

234

Besides, a radiation condition is imposed on the control surface to ensure that 235

waves vanish at upstream infinity 236

2 2

0, 0

j j as xj yj

       (18)

237

where ζj is the wave elevation as seen in the j-th body-fixed frame and is given 238

8

3 NUMERICAL SOLUTIONS

240

Eqs. (13) - (18) define a complete set of BVP. Each one of BVP is time-dependent 241

but can be solved individually and independently; only a single speed of ship j 242

appears in the free-surface condition in Eq.(16). The coupled problem is decou-243

pled into N independent BVPs. At each time instant, the BVP in Eqs. (13) - (18) 244

can be solved numerically. Following the work of Hess & Smith (1964), the 245

boundaries are discretized into a number of quadrilateral panels with constant 246

source density σ(ξj), where 𝝃_𝒋= (𝜉_𝑗, 𝜂_𝑗, 𝜍_𝑗) is a position vector on the boundaries

247

in the j-th body-fixed frame and the free-surface (Bai & Yeung, 1974). Let 𝐱𝐣=

248

(𝑥_𝑗, 𝑦_𝑗, 𝑧_𝑗) denote a point inside the fluid domain or on the boundary surface, the 249

velocity potential ϕ can be expressed by a source distribution on the boundary 250

of the fluid domain 251

, j = 1, 2, …, N (19) 252

where G=1/r is the Rankine-type source function, with r being the distance be-253

tween ξj and xj. More detailed numerical implementation on the solution of BVP

254

can be found in Yuan et al. (2014b). The same in-house developed program MHy-255

dro is deployed in the present study as the framework to investigate ship hydro-256

dynamics in restricted waterways. Special care should be taken to implement a 257

suitable open boundary condition to satisfy Eq. (18). In numerical calculations, 258

the computational domain is always truncated at a distance away from the ship 259

hull. In general, waves will be reflected from the truncated boundaries and con-260

taminate the flow in the computational domain. In the present study, a second-261

order upwind difference scheme is applied on the free-surface to obtain the time 262

and spatial derivatives: 263

264

 

 



 





 





 





 



 

 

2

2 2 _{4 3 2 1}

1 1 11 9

2 6

4 2 4

j

j j j j j

k k k k k k

x x



 _  _  _  _ 

 _{ }

 _     _

 xj   xj xj xj xj xj 

265

(20) 266

Here k refers to the panel number. According to Bunnik (1999) and Kim et al. 267

(2005), Eq. (18) can be satisfied consequently by applying Eq.(20). It should be 268

noted that the 2nd _{order upwind difference scheme is applied at each body-fixed} 269

frame locally. This is essential to deal with ships travelling towards opposite 270

directions. 271

For each individual velocity potential ϕj, the BVP is unsteady due to the time-272

dependent terms in Eq. (16). In previous studies on ship-to-ship interaction 273

problem (Xu et al., 2016; Yeung, 1978; Yeung and Tan, 1980), within the frame-274

work of potential-flow theory, the BVP was not in time domain as the free-sur-275

face condition was assumed to be rigid. It was solved independently at each in-276

dividual time step. The unsteady effect was only considered in the pressure cal-277

culations in Eq. (27). The unsteady interaction forces calculated in these studies 278

are not exactly ‘unsteady’, since the velocity potential at each time step is not 279

time dependent. The velocity potential obtained at tn is not related to that ob-280

tained at tn-1, and it will also not determine that at tn+1. In the present study, 281

the unsteady BVP will be solved in time domain by an iteration scheme. The 282

essential steps are: 283

1

( ) ( ) ( , )

N f c b j

j

S S S

G ds

 



 

 



j j j j

9

1. Determine the initial condition. We assume that at the initial stage of 284

ship-to-ship operation, the moving ships are sufficiently far apart so that 285

their interactions are negligible. Thus, the time dependent terms are re-286

moved from the free-surface condition in Eq. (16), and we have 287

 

 

  (21)

288

Here (𝜙_𝑗𝑘₎∗ _{is the time-independent velocity potential at the time step k.}

289

The computational domain and the corresponding panel distribution at 290

each time step k can be constructed and the steady BVP in Eqs. (13) to 291

(15), (21), (17) and (18) can be solved straightforwardly by using the Ran-292

kine source panel method. The time-independent velocity potential (𝜙_𝑗𝑘₎∗

293

can be obtained, which will be used as the initial guess to calculate the 294

time derivatives of unsteady velocity potential 𝜙_𝑗𝑘 _{in Eq. (22).}

295 296

2. By applying the second-order backward difference scheme, the time de-297

rivatives in Eq. (16) can be calculated according to the following formulas 298

   

 

     

 





* ₁ * ₂ *

2

* ₁ * ₂ * _{* 3} *

2 2

1 3 1

2

2 2

1

2 5 4

k

j k k k

j j j

k

j k k k k

j j j j j

t t t t

















 

3. Substituting Eq. (22) into Eq. (16), the following time-domain free-sur-300

face condition can be obtained 301

2 2

2

2 2 2 0

k k k k k

j j j j j

j j

j j j

U U g

t t x x z











    

    

     (23)

302

Solving the unsteady BVP in Eqs. (13) to (15), (23), (17) and (18), we can 303

obtain the unsteady velocity potential 𝜙_𝑗𝑘_{. Residual errors of time}

deriv-304

atives of |(𝜙_𝑗𝑘₎∗_{− 𝜙}

𝑗𝑘| can be evaluated. If |(𝜙𝑗𝑘)∗− 𝜙𝑗𝑘| < 𝜀, the iteration

305

stops and 𝜙_𝑗𝑘 _{will be used to calculated the pressure and wave elevation.}

306

Otherwise, (𝜙_𝑗𝑘₎∗ _{in Eq. (22) will be replaced by the newly obtained 𝜙} 𝑗𝑘,

307

and the iteration continues until |(𝜙_𝑗𝑘₎∗_{− 𝜙}

𝑗𝑘| < 𝜀. It is known that the

308

iterative scheme has advantages of high accuracy and good numerical 309

stability. 310

Once the unknown potential ϕj is solved on the plane z = 0 and on the body Bj, 311

the unsteady pressure components under its individual coordinate system can 312

be obtained from linearized Bernoulli’s equation 313 j j j j j p U t x         _  _     

j j j

x x x , j = 1, 2, …, N (24)

314

We should point out that because of the first unsteady term in Eq. (24), the total 315

pressure p in coordinate system 𝐱_𝐣 cannot be expressed directly as the sum of 316

all the pressure components in their local frames. To transfer the pressure from 317

coordinate system 𝐱𝐢 to 𝐱𝐣, the following relation needs to be observed

10





i

j i i

i

d

U U

dt t x



_

_{ } 

 

j j

x x , i, j = 1, 2, …, N (25)

319

It should be noted that the partial derivative symbol of the first term in Eq. (24) 320

is retained to make it consistent with Eq. (12) where the potential is expressed 321

in the body-fixed coordinate system 𝐱_𝐣. But here the body-fixed coordinate sys-322

tem 𝐱𝐣 turns out to be in the reference frame for the other body-fixed coordinate

323

system 𝐱𝐢. Therefore, 𝜕∅_𝜕𝑡𝑗 is actually calculated as a total derivative by using Eq.

324

(25). The unsteady pressure in coordinate system 𝐱_𝐢 (i = 1, 2, …, N, i ≠ j ) can 325

then be ‘transferred’ to 𝐱_𝐣 as 326





i j i i i j i

i i i

p U U U U

t x x t x

             _   _  _  _  _            

j j j j

x x x x , i, j = 1, 2, …, N

327

(26) 328

Note the subtle differences between Eq.(24) and (26). The total pressure p in 329

coordinate system 𝐱𝐣 can be written as

330

1 1

N N

i j i

i i i

p p U

t x            _  _    





j j j

x x x , i, j = 1, 2, …, N (27)

331

Integrating the pressure over the hull surface, we can express the forces (or 332

moments) on the i-th hull induced by the j-th ship as: 333 334 j j k k S

F 



335

where k = 1, 2, …, 6, representing the force in surge, sway, heave, roll, pitch and 336

yaw directions, and 337

, 1, 2,3

, 4,5,6

k k n k     _{ }  n

x n (29)

338

The free-surface elevation can be obtained from dynamic free-surface boundary 339

condition in Eq. (7). Similar to the pressure expression, the unsteady wave ele-340

vation in coordinate system 𝐱𝐢 ( i = 1, 2, …, N, i ≠ j ) can be transferred to 𝐱𝐣 as

341

1

i j i

i

U

g t x

   _    _

 

 

j j

x x , i, j = 1, 2, …, N (30)

342

The total wave elevation in coordinate system 𝐱𝐣 can be written as

343 1 1 N j i i i U

g t x

         _  _    



x x , i, j = 1, 2, …, N (31)

344

We note that we have not imposed a Kutta condition at the stern, as a first 345

approximation, or equivalently, the stern is pointed. 346

4 VALIDATIONS OF NUMERICAL MODEL

347

The convergence study is carried out on two identical Wigley III hulls in head-348

on encounter. We calculate the lateral force and wave elevation to exam the 349

11

panel size to ship length ratio at each Froude number is fixed at Δx/L=1/κ. The 351

time then can be non-dimensionalized by 352

1

' /

n

L

t x U

F g



   (32)

353

In the present study, κ=60 was found adequate to obtain a convergent result. 354

The results shown in Fig. 4 confirm the convergence of the present superposition 355

method by reducing the time-stepping. It should be noted that the convergence 356

becomes slower as the encounter speed increases. 357

358

Fig. 4. Convergence study on two identical Wigley III hulls (Journee, 1992) in head-on 359

encounter with dt/B=2, dt being the lateral separation between two ships (a) Sway force;

360

(b) wave profile at the center line between two ships at the moment of side-by-side

con-361

figuration (dl=0). The black, red and blue cures correspond to Fn=0.1, 0.2 and 0.3

respec-362

tively. CY and C_ζ is non-dimensionalized by 1

2𝜌𝐵𝑇|𝑈1𝑈2| and by 2𝜋|𝑈1𝑈2|/𝑔 respectively.

363

Model-test data on ship-to-ship interaction with different speeds as a parameter 364

is rather rare. To run the tests, an auxiliary carriage must be installed, in ad-365

dition to the main tow carriage. Therefore, the encountering tests were not in-366

cluded in Oltmann (1970). In the present study, as another check, the bench-367

mark data published by Vantorre, et al. (2002) is used to validate the numerical 368

results of encountering cases. Two ship models with scale factor 1/75 are used 369

for encountering or overtaking tests (referred as Model D and Model E). The 370

main particulars of Model D (j=2) and Model E (j=1) in model scale can be found 371

in Table 1. In the model test, Model E was towed by the main carriage along the 372

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

-1 0 1

CY

Repulsion

Attraction ∆t=t' ∆t=2t' ∆t=4t'

-0.02 -0.01 0 0.01 0.02

-1 0 1

Cζ

x/L

dl/L

(a)

12

center line (y = 0) of the tank, while Model D was towed by an auxiliary carriage. 373

The transverse separation is dt = BD + 0.5BE and the water depth d is 0.248m. 374

Table 1. Main particulars of Model D and Model E in Vantorre, et al. (2002).

375

Model E (j=1) Model D (j=2)

Length (m) LE = 3.824 LD = 3.864

Breadth (m) BE = 0.624 BD = 0.55

Draft (m) TE = 0.207 TD = 0.18

Block coefficient CBE = 0.816 CBD = 0.588 376

377

Fig. 5. Panel distribution on partial computational domain. There are 9,950 panels 378

distributed on the entire computational domain: 1,900 on the wetted body surface of

379

Model E, and 2,170 on the wetted body surface of Model D, 5,880 on the free-surface.

380

The free-surface is truncated at 2LE upstream and 2LE downstream with regard to

381

body-fixed frame on Model E.

382

Fig. 5 is the mesh distribution on the partial computational domain when Model 383

E encounters Model D. It should be noted that the side walls of the tank are not 384

modelled. In order to minimize the panel number, the free surface is truncated 385

at 0.27LE and 0.42LE laterally with regard to Model D and Model E respectively. 386

In calm water test, it has been proved by Yuan and Incecik (2016b) that the side 387

wall effects are negligible at dsb / L > 0.25 and Fn < 0.25. It should also be noted 388

that in the encountering simulation, the longitudinal separation dl is measured 389

in body-fixed frame on Model E. The longitudinal separation between two ships 390

at the moment shown in Fig. 5 has a positive sign. The time step ∆t in the nu-391

merical calculation is 0.18s. The numerical results, as well as the experimental 392

measurements, are shown in Fig. 6-Fig. 9. 393

Fig. 6 shows the interactions forces on Model E at Fn=0 passed by Model D at 394

Fn =0.078. In engineering practice, this case study aims to investigate the moor-395

ing forces induced by a passing ship in the harbor areas or inland waterways. 396

Fig. 7 and Fig. 8 shows the interaction forces on Model E at Fn=0.039 and 397

Fn=0.078 encountered by Model D at Fn=0.078. These two case studies aim to 398

validate the feasibility of the present superposition method on simulating the 399

ships moving towards opposite directions. Fig. 9 shows the interactions forces 400

on Model E at Fn=0.078 overtaken by Model D at Fn =0.117. In all of these four 401

cases, the forces on both ships are calculated numerically. However, only the 402

forces on Model E, which was towed by the main carriage, were measured in 403

model tests. Generally, the agreement between present potential flow solver 404

(MHydro) and experimental measurement is very good. It indicates the poten-405

tial flow method is applicable to predict the hydrodynamic interactions between 406

two ships with different forward speeds. 407

X

13

It is found from Fig. 6-Fig. 9.(a) that the resistance (F1) is overestimated by the 408

present potential flow solver, even though the viscous effect is not taken into 409

account. It indicates the hydrodynamic interaction force plays a dominate role 410

in total resistance, and the frictional component due to the viscosity is negligi-411

ble. The negative values shown in Fig. 6-Fig. 9.(a) represent the resistance that 412

is opposite to the moving direction, while the positive values represents a thrust 413

which is the same as the moving direction. An interesting finding is that a very 414

large thrust force is observed at dl / LE = -0.5 during the passing and encounter-415

ing maneuvering. Physically it can be explained that before passing and encoun-416

tering (0 < dl / LE < 1), the presence of the other moving vessel stops the water 417

from spreading evenly into the surrounding field. As a result, the pressure dis-418

tributed over ship bow increases. At the same time, the pressure distributed 419

over ship stern retains the same level. An increased resistance is expected by 420

pressure integral. After encountering (-1 < dl / LE < 0), the high pressure area 421

transfers to the ship stern, which will correspondingly lead to a propulsion force. 422

But in overtaking maneuvering shown in Fig. 9.(a), the thrust force is observed 423

at dl / LE = 0.5, where the bow of Model D approaches the midship of Model E 424

longitudinally. It can be explained that before overtaking (-1 < dl / LE < 0), the 425

presence of faster ship (Model D) accelerates the fluid velocity around the stern 426

area of Model E. As a result, the pressure distributed over ship stern decreases. 427

At the same time, the pressure distributed over ship bow retains the same level. 428

An increased resistance is expected by pressure integral over the hull surface of 429

Model E. After overtaking (0 < dl / LE < 1), the high pressure area transfers to 430

the ship bow, which will correspondingly lead to a propulsion force. 431

During the passing, encountering and overtaking process, the symmetry of the 432

flow in the starboard and port side is violated by the presence of the other vessel. 433

The maximum asymmetric flow is observed when the midships of the two ships 434

are aligned (dl / LE ≈ 0, as shown in Fig. 1), and the suction force reaches its peak 435

value (see Fig. 6-Fig. 9.(b)). The pressure distribution is not only asymmetric in 436

port side and starboard, but also in bow and stern. Consequently, a yaw moment 437

will be induced, as shown in Fig. 6-Fig. 9.(c). Generally, there are four peaks of 438

yaw moment during passing and encountering maneuvering, which appear at 439

dl / LE ≈ -0.6, dl / LE ≈ -0.1, dl / LE ≈ 0.4 and dl / LE ≈ 0.9. But in overtaking 440

process, only three peaks are observed at dl / LE ≈ -0.8, dl / LE ≈ -0.1 and dl / LE 441

≈ 0.5. Based on these peaks, some empirical formulas were established to model 442

the interaction moment (Lataire et al., 2012; Vantorre et al., 2002; Varyani et 443

al., 2002). However, as the numbers of the peaks are not predictable, the ap-444

plicability of those empirical formulas is very limited. It should be noted that in 445

ship-bank and ship-lock problem, potential flow method fails to predict the sign 446

of yaw moment due to the weak lifting force caused by the cross-flow in the stern 447

(Yuan and Incecik, 2016a). However, in ship-ship problem, the hydrodynamic 448

interaction is much more important than cross-flow effects. The predictions of 449

yaw moment by a potential flow solver are therefore reliable. 450

It is also found from Fig. 6-Fig. 9 that the interaction forces on the ship with 451

lower speed is larger than that with higher speed. In passing operation (see Fig. 452

6), the hydrodynamic interaction on Model E is significant, even though Model 453

E is stationary without forward speed. On the contrary, the interaction is neg-454

ligible on the moving ship (Model D) during passing operation. It indicates that 455

the slower ship is more likely to lose its maneuverability during passing, en-456

14

Fig. 6. (a) The resistance, (b) the sway force and (c) the yaw moment on Model E

(j=1) at Fn=0 passed by Model D (j=2) at

Fn =0.078. The positive dl values denote

that Model D is in the upstream side of Model E. As Model D moves to the down-stream side, dl becomes negative. EFD

results are published by Vantorre et al. (2002).

Fig. 7. (a) The resistance, (b) the sway force and (c) the yaw moment on Model E

(j=1) at Fn=0.039 encountered by Model D

(j=2) at Fn =0.078. The positive dl values

denote that Model D is in the upstream side of Model E. As Model D moves to the downstream side, dl becomes negative.

EFD results are published by Vantorre et al. (2002).

458

-6 -4 -2 0 2 4 6

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F1

(N

)

E (MHydro) D (MHydro) E (Exp.)

-6 -4 -2 0 2 4 6

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F1

(N

)

-10 -5 0 5 10 15 20 25

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F2

(N

)

-10 -5 0 5 10 15 20 25

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F2

(N

)

-15 -10 -5 0 5 10 15

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F6

(N∙m

)

d_l/L_E

-15 -10 -5 0 5 10 15

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F6

(N∙m

)

d_l/L_E

-6 -4 -2 0 2 4 6

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F1

(N

)

E (MHydro) D (MHydro) E (Exp.)

-15 -10 -5 0 5 10 15

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F1

(N

)

(a)

(b)

(c)

(a)

(b)

(c)

15

Fig. 8. (a) The resistance, (b) the sway force and (c) the yaw moment on Model E

(j=1) at Fn=0.078 encountered by Model D

(j=2) at Fn =0.078. The positive dl values

denote that Model D is in the upstream side of Model E. As Model D moves to the downstream side, dl becomes negative.

EFD results are published by Vantorre et al. (2002).

Fig. 9. (a) The resistance, (b) the sway force and (c) the yaw moment on Model E

(j=1) at Fn=0.078 overtaken by Model D

(j=2) at Fn =0.117. The negative dl values

denote that Model D is in the down-stream side of Model E. As Model D moves to the upstream side, dl becomes

positive. EFD results are published by Vantorre et al. (2002).

5 DISCUSSIONS ON FREE-SURFACE EFFECTS

459

The predictions of the yaw moment by a potential-flow solver are therefore reli-460

able. After the aforementioned validations against physical model tests, the pre-461

sent superposition method was extended to investigate the free-surface effects. 462

Here, we study the interactions between two identical Wigley III hulls in head-463

on encounter. The geometry of the hull can be found in Journee (1992). Fig. 10 464

shows the panels distributed on the partial computational domain. The panel 465

number per ship length κ=60. Δt=2t’ is applied in all of the numerical simula-466

tions reported below. We computed the interaction forces in 6DoF (6 Degrees of 467

Freedom), as well as the total wave elevation. 468

469

Fig. 10. Panel distribution on the computational domain of two identical Wigley III 470

hulls in head-on encounter with Fn=0.3, dt/B=2, and dl/L=1. There are 17,760 panels

471

distributed on the entire computational domain: 600 on the wetted body surface of each

472

-30 -20 -10 0 10 20 30

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F2

(N

)

-30 -20 -10 0 10 20 30

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F2

(N

)

-15 -10 -5 0 5 10 15

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F6

(N∙m

)

dl/LE

-60 -40 -20 0 20 40

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

F6

(N∙m

)

d_l/L_E

(b)

(c)

(b)

16

hull and 16,560 on the free-surface. The computational domain is truncated at 2L

up-473

stream, 2L downstream and 0.5L laterally with regard to the body-fixed reference

474

frame.

475

5.1 The effect of near-field disturbance and far-field waves 476

Fig. 11 shows the results of the lateral (sway) forces on two identical Wigley III 477

hulls in head-on encounter with dt/B=2. Here we compare the results obtained 478

by using three different methods. In the first method, the encountering problem 479

is treated as a steady problem, although the steady linearized free-surface con-480

dition is applied. Mathematically, in the pressure calculation, the first term in 481

Eq. (27) is neglected. Meanwhile, the first two time-dependent terms in Eq. (16) 482

are also not taken into account. It has been approved to be an efficient method 483

to deal with the steady problems, e.g., interactions between two ships travelling 484

with the same speed (Yuan et al., 2015), or between the hulls of a catamaran or 485

trimaran (Shahjada Tarafder and Suzuki, 2007). In the second method, the en-486

countering is treated as an unsteady problem, while the rigid condition is ap-487

plied on the free-surface. Mathematically, the free-surface condition in Eq. (16) 488

is replaced by an impermeable boundary condition. The BVP therefore is solved 489

as a steady problem. The only unsteady effects are reflected by the time-depend-490

ent term in Eq. (27). Nearly all the published studies on ship-to-ship problem 491

are based on this partially unsteady method (Korsmeyer et al., 1993; Xu et al., 492

2016; Yeung, 1978; Zhou et al., 2012). The advantage of this rigid-free-surface 493

method is obvious. As the image method can be applied on the free-surface, it 494

doesn’t require panels to be distributed on the free surface. It significantly re-495

duces the panel number, herein reducing the calculation time to solve the BVP. 496

However, this method is only applicable when the speed of the ships is low. The 497

third method proposed by the present study takes all the unsteady effects into 498

account. The time derivatives in both Eq. (16) and Eq. (27) are considered. The 499

BVP is solved in the time domain by using an iteration scheme. The advantage 500

of this fully unsteady method is that it can predict the hydrodynamic interaction 501

induced by the ship-generated waves. But the panels must be distributed on the 502

free-surface, which not only increases the total mesh number, but only add dif-503

ficulties to construct the computational domain at each time step. To deal with 504

this issue, we developed a dynamic meshing technique to generate the mesh 505

automatically at each time step. With regard to the computational time, the 506

third method takes longer than the other two methods. But within the frame-507

work of potential flow theory, the computational time is still very satisfactory. 508

Most of the computational efforts are spent on generating the so-called coeffi-509

cient matrix (Hess and Smith, 1964) Even though it involves time iteration, the 510

coefficient matrix retains unchanged. The time scale to solve the unsteady BVP 511

for each time step is few minutes. 512

The results shown in Fig. 11 clearly demonstrate the effects of unsteady pres-513

sure and unsteady free-surface. Here, we note that the unsteady pressure term 514

in Eq. (27) is very important at all the range of encountering speeds, while the 515

free-surface effect is only important when the encounter speed is moderate or 516

high. Ignoring the unsteady pressure term in Eq. (27) will lead to mis-estima-517

tion of the interaction force. At Fn = 0.1, the free-surface elevation and hydrody-518

namic interaction are mainly determined by the near-field (non-wave-like) dis-519

turbances. The rigid free-surface condition (RFC) is adequate to predict the in-520

teraction forces, as shown in Fig. 11(a). As the Froude number increases to Fn 521

=0.2, the far-field waves become manifest, and the interaction force oscillates 522

17

still dominated by the near-field disturbance. The contribution of the force in-524

duced by far-field waves is smaller than that induced by the near-field disturb-525

ance. The fluctuations due to the far-field waves will not deviate from the near-526

field induced forces. The interaction force predicted by rigid free-surface condi-527

tion is symmetric with respect to dl/L=0. But this symmetry property is violated 528

by the presence of the far-field waves. As the far-field waves could not propagate 529

ahead of the ship, the free-surface effect cannot be observed before the encoun-530

tering taken place (dl/L>1). As the encountering ships are maneuvering to each 531

other’s wake region, the free-surface effect then can be observed, and some fluc-532

tuations can be observed at dl/L<1 correspondingly. These fluctuations will not 533

disappear (but the amplitude will decrease) after the encountering operation. 534

The relationship between the near- and far-field induced force is very similar to 535

that between low- and wave-frequency surge or sway motions of a floating struc-536

ture in irregular waves (Yuan et al., 2014a). The free-surface effect becomes 537

even more significant at Fn = 0.3. The force amplitude induced by the far-field 538

waves is larger than that induced by the near-field disturbance, as shown in 539

Fig. 11(c). There are only three peaks induced by near-field disturbance. How-540

ever, the peaks altered by the far-field waves are unpredictable. Therefore, the 541

empirical formulas based on low speed model (Lataire et al., 2012; Vantorre et 542

al., 2002; Varyani et al., 2002) is not applicable to predict the interaction forces 543

when the free-surface effect becomes important. It can be concluded that the 544

free-surface effects must be taken into account at Fn > 0.2. 545

-0.1 -0.05 0 0.05 0.1 0.15

-2 -1 0 1 2

CY

dl/L

Unsteady + LFC Unsteady + RFC Steady + LFC

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

-2 -1 0 1 2

CY

dl/L

λ/L

(a)

18

Fig. 11. Sway force acting on two identical Wigley III hulls in head-on encounter with 546

dt/B=2. (a) Fn=0.1; (b) Fn=0.2; (c) Fn=0.3. dl/L=0 corresponds to the moment t=ts, when

547

the midships of the two ships are aligned. dl/L>0 corresponds to t<ts, dl/L<0 corresponds to 548

t>ts. CY is non-dimensionalized by 1

2𝜌𝐵𝑇|𝑈1𝑈2|. LFC indicates that the linearized

free-549

surface condition is used; RFC indicates that the rigid-wall free-surface condition is

550

used.

551

Fig. 12 shows the wave profile at the moment when the midships of two Wigley 552

hulls are aligned. The ‘Steady’ indicates the first two terms in Eq. (16) are ig-553

nored, while ‘Unsteady’ indicates the BVP is solved fully in the time domain by 554

using an iteration scheme. At low Froude number Fn=0.1, the unsteady effect 555

on free-surface condition is not essential. As the wave elevation is dominant by 556

the near-field disturbance, the wave-like fluctuations can hardly be observed at 557

low forward speed. At moderate Froude number, the unsteady effect becomes 558

manifest, especially at the gap between two aligned ships (-0.5<x/L>0.5). as the 559

Froude number increases to Fn=0.3, the difference between ‘Steady’ and ‘Un-560

steady’ can be observed in a wider range of x/L, especially at the bow (x/L=0.5) 561

and stern (x/L=-0.5) areas. Fig. 13 shows the wave elevation components ob-562

tained by the present superposition principle. It should be noted that the total 563

wave elevation presented in Fig. 13(c) is not the simple superposition of the 564

waves produced by two individual hulls without considering the presence of the 565

other one. When we calculate the wave elevation produced by B1, the presence 566

of B2 is also considered, treated as an obstacle, by being momentarily stationary 567

in the body-fixed frame of B1. Therefore, the diffraction and reflection by B2 is 568

considered in the present study. These reflected waves can be seen clearly from 569

Fig. 13(a) and (b). 570

571

Fig. 12. Wave profiles at the center line between two identical Wigley III hulls in head-572

on encounter with dt/B=2, dl/L =0 and Fn=0.3. The black, red and blue curves

corre-573

spond to Fn=0.1, 0.2 and 0.3 respectively.

574

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

-2 -1 0 1 2

CY

d_l/L

-0.02 -0.01 0 0.01 0.02

-1.5 -1 -0.5 0 0.5 1 1.5

Cζ

x/L

Steady Unsteady

(c) Attraction

19 575

Fig. 13. Waves produced by two Wigley III hulls in head-on encounter with dt/B=2,

576

dl=0 and Fn=0.3. (a)Cζ1, the waves produced by B₁ moving at Fn=0.3 while B₂is

mo-577

mentarily stationary in the body-fixed frame of B1; (b)Cζ2, the waves produced by B2 578

moving at U2 while B₁is momentarily stationary in the body-fixed frame of B₂; (c)Cζ,

579

the total waves superposing Cζ1 and Cζ2; (d) Wave profile at the centre line between

580

two hulls shown in (a), (b) and (c).x in the abscissa of (d) refers to the

midship-to-mid-581

ship distance between left-moving ship and the encountered ship.

582

5.2 The effect of divergent and transverse waves 583

Fig. 14 shows the encountering process of two ships in the body-fixed frame of 584

B2. The contour only shows the wave patterns generated by B2 at Fn=0.3 in an 585

unconfined waterway. For a 3D ship, its far-field wave patter includes two wave 586

systems: bow wave and stern wave. Each wave system has two wave compo-587

nents: divergent wave and transverse wave. In the body-fixed frame of B2, B1 588

approaches B2 from its upstream side to its downstream side. Theoretically, B2 589

will experience 6 stages over the entire encountering process: i, non-interference 590

→ ii, local wave disturbance → iii, divergent bow wave disturbance → iv, trans-591

verse bow wave disturbance → v, divergent stern wave disturbance → vi, trans-592

verse wave disturbance. The non-interference stage can only be observed when 593

two ships are sufficient far apart from each other. The transverse bow waves 594

are always interfered with the divergent stern waves. The disturbance in the 595

stage iii, iv and v is supposed to be violate and unpredictable. In the present 596

study, stage iii, iv and v are categorized as a combined stage, namely divergent 597

disturbance. Finally, the interference is divided into three regions: I: t < t1, B1 598

is in the local wave disturbance region of B2; II: t1 < t < t2, B1 is in the divergent 599

wave disturbance region of B2; III: B1 is in the transverse wave disturbance re-600

gion of B2. Here t1 refers to the moment when the bow of B1 reaches the Kelvin 601

envelope of the waves generated by B2, and t2 refers to the moment when the 602

stern of B1 leaves the divergent stern waves generated by B2. 603

C

:

-0.02 -0.01 0 0.01 0.02

-1.5 -1 -0.5 0 0.5 1 1.5

Cζ

x/L

ζ1 ζ2 ζ

(a) (c)

(b) (d)

20 604

Fig. 14. Sketch showing the encountering process of two ships in the body-fixed frame 605

of B2. The bow and stern of the ships act like two sources (or sinks). The blue and red 606

curves represent bow and stern wave patterns respectively.

607

Fig. 15. Yaw moment acting on two identical Wigley III hulls in head-on encounter at

608

Fn=0.3. (a) dt/B=2; (b) dt/B=5; (c) dt/B=10. CZZ is non-dimensionalized by 1₂𝜌𝐵𝑇𝐿|𝑈1𝑈2|.

609

dl/L>0 corresponds to t<ts, dl/L<0 corresponds to t>ts.

610

Fig. 15 shows the yaw moment on B1 during the encountering process. Different 611

lateral separations are investigated here. As the lateral separation increases, 612

-0.1 -0.05 0 0.05 0.1 0.15

-4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5

t=t₁

t=t₂ t > t₂ Transverse wave

disturb-ance region

C_ZZ

-0.1 -0.05 0 0.05 0.1

-4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5

t=t₁

Divergent wave dis-turbance region t=t₂ t₁ < t < t₂

-0.1 -0.05 0 0.05 0.1

-4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5

dt/L

t=t₁

t < t₁ Local wave

disturb-ance region U₂

U₁ Diverging waves

Transverse waves

t < t₁ Local wave dis-turbance region

t₁ < t < t₂ Divergent wave disturbance region

t > t₂ Transverse wave disturbance region

t=t₁ t=t₂

B₁ B2

(a)

(b)

21

the non-interference region expands and the disturbance region shifts down-613

stream with regards to the body-fixed frame of B2. It agrees with the physical 614

observation of the far-field waves (Kelvin waves) that confines within Kelvin 615

wedge downstream. Before B1 reaches the Kelvin envelope, some interactions 616

are observed at t < t1, which is due to the disturbance caused by the local waves. 617

The typical wave patterns at t < t1 is shown in Fig. 16(a). At t = t1, when the bow 618

of B1 is stroken by the divergent waves produced by B2, a very large yaw moment 619

can be induced. When B1 is partly or completely in the divergent disturbance 620

region (t1 < t < t2), the interaction becomes significant. The bow and stern waves 621

interfere in this region, and the wave energy concetrated in this region is 622

usually high, especially when the ship speed is moderate or high. The typical 623

wave patterns at t1 < t < t2 is shown in Fig. 16(b). When B1 completely leaves the 624

divergent disturbance region and enters into the transverse disturbance region 625

(t > t2), the amplitude of the interaction force decreases with the decay of the 626

transverse waves. The typical wave patterns at t > t2 is shown in Fig. 16(c). It 627

should be noted that at dt/B=10, the forces at the moment t = t2 is not captured 628

in Fig. 15(c). As the lateral separation increases, t2 will shift further 629

downstream. Numerically, to simulate the case with larger lateral separation, 630

the computational domain must be expanded not only to the lateral sides, but 631

also to the downstream direction. Much more computational efforts are required 632

to simulate the entire encountering process when the lateral separation 633

becomes large. It can also be found from Fig. 15 that as the lateral separation 634

increases, the interaction diminishes, but not significantly. The maximum yaw 635

moment at dt/B=10 still accounts for 40% of that at dt/B=2. It indicates that 636

the hydrodynamic interaction induced by the far-field waves is very important 637

at moderate or high speed encountering operation. It must be taken into account 638

even though the lateral separation between ships is large. 639

640

Fig. 16. Wave patterns produced by two identical Wigley III hulls in head-on encounter 641

at Fn=0.3. (a) dt/B =10 and dl/L= -1, corresponding to t < t1 when a ship is in the other

642

ship’s local wave disturbance region; (b) dt/B =5 and dl/L = -1.5, corresponding to t1 < t

643

< t2 when a ship is in the other ship’s divergent wave disturbance region; (a) dt/B=2 and

644

C_ 0.017 0.012

0.007

0.002 -0.003

-0.008

-0.013 -0.018

(a) (b)

22

dl/L = -2.5, corresponding to t > t2 when a ship is in the other ship’s transverse wave

645

disturbance region.

646

6 CONCLUSIONS

647

A linearized free-surface boundary condition was used to solve the BVP involved 648

in multiple bodies travelling with various speeds. Based on superposition prin-649

cipal, the traditional coupled BVP could be decoupled into N (assuming there 650

are N bodies) sets of independent unsteady BVPs, which can be solved individ-651

ually in time domain. The advantage of this decoupled method is that the free-652

surface boundary condition can be taken into consideration for each set of inde-653

pendent BVPs. Thus, the unsteady hydrodynamic interaction problem can be 654

solved in a fully unsteady manner, and the far-field wave effect can be can be 655

accounted for. 656

The present formulation provides an effective way to predict the free-surface 657

effects, with particular application for calculating the lateral interaction force 658

on arbitrary number of ships, each with its own speed. By integrating the pre-659

sent superposition method into a Rankine source (simple-source) panel code, we 660

calculated the unsteady hydrodynamic interaction forces and wave elevation 661

when two ships were under passing, overtaking and encountering operations. 662

Experimental measurements confirm the applicability of the present approach. 663

Numerical results indicate the near-field disturbances are the most important 664

component of the interaction force when the encountering speed is low. As the 665

encountering speed increases, the interaction force induced by the far-field 666

waves becomes manifest gradually. It was found the free-surface effects must 667

be considered at Fn > 0.2 for slender ships. For blunt-body ships, the lower limit 668

of Froude number is smaller. When the encountering speed reaches Fn = 0.3, 669

the free-surface effect becomes the dominant component. The interaction force 670

induced by the divergent waves could reach a very large value, which may cause 671

ship accidents, such as grounding, capsizing or collisions. By increasing the sep-672

aration distance between encountering ships could reduce the interaction am-673

plitude, but not significantly. At high encountering speed, the free-surface must 674

be taken into account even though the lateral separation between ships is large. 675

The superposition method proposed in the present study is not limited to 676

solving the unsteady interaction problem between ships. It can also be applied 677

to predict the hydrodynamic interactions between competitive swimmers in a 678

swimming pool, or between aquatic animals swimming near the free surface. 679

The present approach provides a rational and rapid (real-time level) tool for 680

analyzing and computing interaction effects, without going through a lengthy 681

detailed CFD type computations, which would be prohibitively slow (e.g., Zou 682

and Larsson (2013a)) and have yet reached a state to effectively model unsteady 683

multi-body interaction. 684

7 ACKNOWLEDGEMENTS

685

The first author acknowledges the financially support by a Sir David Anderson 686

Award for his visit to UC Berkeley during which this work was formulated. The 687

third author acknowledges partial support of the American Bureau of Shipping 688

via an endowed chair in Ocean Engineering at UC Berkeley. Discussions with 689

Dr. Lu Wang and Mr. Dongchi Yu of UC Berkeley during the course of this work 690