Computational fluid dynamics (CFD) prediction of bank effects including verification and validation

DOI 10.1007/S00773-012-0209-7 O R I G I N A L A R T I C L E

Computational fluid dynamics (CFD) prediction of bank effects

including verification and validation

L u Z o u • L a r s L a r s s o n

Received: 5 March 2012/Accepted: 13 December 2012/Published online: 18 January 2013 © JASNAOE 2013

A b s t r a c t Restricted waters impose significant effects on ship navigation. I n particular, w i t h the presence o f a side bank i n the v i c i n i t y of the h u l l , the flow is greatly c o m -plicated. A d d i t i o n a l hydrodynamic forces and moments act on the h u l l , thus changing the ship's maneuverability. I n this paper, computational fluid dynamics methods are u t i -lized f o r investigating the bank effects on a tanker h u l l . The tanker moves straight ahead at a l o w speed i n t w o canals, characterized by surface piercing and sloping banks. For v a r y i n g water depth and ship-to-bank distance, the sinkage and t r i m , as w e l l as the viscous hydrodynamic forces on the h u l l , are predicted b y a steady state Reynolds averaged Navier-Stokes solver w i t h the double model approximation to simulate the flat free surface. A potential flow method is also applied to evaluate the effect o f waves and viscosity o n the solutions. The focus, is placed o n verification and validation based on a grid convergence study and com-parisons w i t h experimental data. There is also an explo-ration o f the m o d e l i n g eiTors i n the numerical method.

K e y w o r d s Bank effects • Reynolds averaged N a v i e r -Stokes method • Hydrodynamic forces and moments • Sinicage and t r i m • V e r i f i c a t i o n and validation

1 Introduction

W h e n a ship is traveling i n a canal or a narrow channel, the flow becomes h i g h l y complex. Interactions occur between the ship and the side banks, as additional liydrodynamic

L. Zou (El) • L. Larsson

Department of Shipping and Marine Technology, Chalmers University of Technology, 41296 Gothenburg, Sweden e-mail: zlu714@gmail,com

forces and moments generated by the v i c i n i t y of the banks act on the h u l l and influence ship m o t i o n . This phenome-non is named bank effects. W h e n the distance between the h u l l and the canal boundary is reduced, the flow is accel-erated and the pressure is accordingly decreased, w h i c h has an effect on the hydrodynamic characteristics. The pro-duced hydrodynamic forces, especially i n extremely shal-l o w canashal-ls, may considerabshal-ly affect the maneuvering performance o f the ship, m a k i n g i t d i f f i c u l t to steer. The ship may collide w i t h the side bank and/or run aground due to the "squat" phenomenon. Bank effects are thus extre-mely important f o r ship navigation. I n the past f e w dec-ades, many investigations on banlc effects have been canied out, both experimentally and numerically. A nota-ble event was the International Conference on Ship Maneuvering i n Shallow and Conflned Water: B a n k E f f e c t s [1], at w h i c h the participants expressed concern about this problem and presented many interesting, papers.

Historical investigations about bank effects have mostly relied on experimental tools, such as model tests and empirical or semi-empirical formulae, w h i c h n o r m a l l y treat the bank effect as a f u n c t i o n i n v o l v i n g hull-bank distance, water depth, ship speed, h u l l f o r m , bank geometry, propeller performance, etc. D u r i n g the 1970s, N o r r b i n at SSPA, Sweden, carried out experiments and proposed empirical formulae to estimate the hydrodynamic forces f o r flooded [ 2 ] , vertical [3] and sloping [3] banks. L i et al. [4] continued N o r r b i n ' s investigations and tested bank effects i n extreme conditions f o r three d i f f e r e n t h u l l f o r m s (tanker, feixy and catamaran). The influence of ship speed, propeller loading and bank inclination was evaluated. C h ' n g et al. [5] con-ducted a series o f m o d e l tests and developed an empirical f o r m u l a to estimate the bank-induced sway force and y a w moment f o r a ship handling simulator. I n recent years, comprehensive m o d e l tests i n a t o w i n g tanic have been

carried out at the Flanders Hydraulics Research ( F H R ) , B e l g i u m , to b u i l d up mathematical models f o r bank-effect investigations and to provide data f o r computation valida-tion. Vantorre et al. [6] discussed the influence o f vi'ater depth, lateral distance, f o r w a r d speed and propulsion on the hydrodynamic forces and moments based on a systematic captive model test program f o r three ship models m o v i n g along a vertical surface-piercing bank. They also proposed empirical formulae f o r the prediction o f ship-bank interac-tion forces. F r o m extensive tests, Lataire et al. [7] developed a mathematical model f o r the estimation o f the hydrody-namic forces, moments and motions taking into consider-ation ship speed, propulsion and ship/bank geometry. I n addition, two parameters defining the distance to a bank o f irregular geometry were proposed.

A l t h o u g h experimental tools and empirical formulae are w i d e l y used f o r bank-effects prediction, they have their shortcomings. For example, empirical formulae are suitable only f o r cases w i t h similar h u l l forms and conditions. Otherwise, the prediction is barely reliable. T o establish a mathematical model, many systematic and expensive model tests are always required. However, the most important wealoiess of these tools is their inability to provide deitailed i n f o r m a t i o n on the flow field, w h i c h can help explain t h è flow mechanism behind the bank effects. I n v i e w o f this, researchers resort to using numerical (i.e., computational fluid dynamics, C F D ) methods to deal w i t h the phenomena o f bank effects. A m o n g existing C F D methods, the potential flow method is the most c o m m o n . N e w m a n [8] applied a Green f u n c t i o n to predict the interaction force between a ship and an adjacentrectangularcanal w a l l . M i a o et al. [9] studied the case o f a ship travelling i n a rectangular channel and estimated the lateral force, yaw moment and wave pattern based on Dawson's method. I t was shown that the applied potential flow method was able to predict reasonable hydrodynamic forces f o r a water depth to ship draught ratio larger than 1.5, but i t f a i l e d f o r smaller ratios. Lee and Lee [10] applied the potential flow method to estimate the hydrodynamic forces as a f u n c t i o n o f the water depth and the spacing between the ship and a wedge-shaped bank. Viscous flow computations were c a n i e d out by L o et al. [ I I ] , w h o studied the bank effect on a container ship model using C F D software based on the Navier-Stokes equations. The e f f e c t o f vessel speed and distance to the bank on the magnitude and temporal variation of the yaw angle and sway force were reported. Some details o f the predicted flow field are avail-able f r o m this study. W a n g et al. [12] recentiy studied the bank effects using a Reynolds averaged Navier-Stokes ( R A N S ) method to predict viscous hydrodynamic forces on a Series 60 h u l l at varying water depth ratios (1.5,3.0 and 10.0) and ship-banic distances.

A v a i l a b l e literature indicates that although empirical formulae and model tests have played an important role i n

bank-effect investigations, the mechanism o f the effect is stih not f u h y explained, as the model test normally only produces global data, like forces and moments, but not details o f the flow. Viscous methods, l i k e the most w i d e l y used R A N S method, can provide more details, w h i c h can enhance the understanding o f the mechanisms. However, their application is still l i m i t e d ; only a f e w reports r e f e n i n g to this type o f computation have been presented, yet some o f them indicate p r o m i s i n g results. I n the f r a m e w o r k o f an ongoing project a i m i n g at extensive C F D investigations o f the hydrodynamic forces on the h u l l i n restricted water-ways, the present w o r k intends to study the bank effects i n the case o f a ship m o v i n g i n a shallow canal. B y means o f numerical methods, quantitative predictions o f the most interesting hydrodynamic quantities, such as the sinkage and t r i m and the viscous forces on the h u l l are obtained. Since this is a validation study, emphasis is placed on f o r m a l verification and validation (VcfeV) and an investi-gation o f modehng enors.

2 Computational method

Generally i n a R A N S method, the fluid m o t i o n around a ship is governed by a system o f equations consisting o f the Navier-Stokes equations (1) and the continuity equation (2), describing the conservation o f m o m e n t u m and o f mass. Assuming the fluid to be incompressible, the governing equations given i n a Cartesian coordinate system read [13]: 9 , , 1 8 » d \ diii 'dt p dxi dui 0 (2)

where M,^) represents velocity components, x,-,^) denotes coordinates, p is the pressure, v is the kinematic viscosity,

Fj represents the body force (such as gravity) and p is the

fluid density. For a three dimensional flow, i, j = 1, 2, 3. A s f o u r variables («,(,), p) are present i n the three-dimen-sional equations above, the Navier-Stokes equations combined w i t h the continuity equation, establish a closure o f the system o f equations.

T i m e averaging Eqs. 1 and 2 yields:

8w/ 'dt düi 6 ^ d_ : 0 / - - N 1 9P 1 9 / -(3) (4)

where M / Q - ) , ^ and <Ty denote the average velocity, pressure and stress; Rji = R/j = -pu'iii'j a symmetric quantity.

termed "Reynolds stress"; u'.fj-^ represents the fluctuating velocity i n time. As seen i n Eqs. 3 and 4, the fluctuating values are a l l removed during the time-averaging, but new u n k n o w n variables, the Reynolds stresses Rji, are introduced. The Rji needs to be modeled to close the system o f Eqs. 3 and 4, w h i c h then y i e l d a l l the mean flow properties.

3 C F D solver

A C F D solver f o r Ship Hydrodynamics, S H I P F L O W [14], is utilized i n the present w o r k to implement the C F D compu-tations. I t includes several modules w i t h different, but associated functions. T w o o f the modules are used i n the present work: X P A N , w h i c h is a solver f o r the potential flow based o n a surface singularity panel method; and X C H A P , a finite volume method f o r solving the R A N S equations. The X C H A P can compute complex geometries, o w i n g to the capability o f handling overlapping grids, e.g., rudder, shafts, brackets and vortex generators. I t can also deal w i t h the grids imported f r o m external g r i d generators. The code contains several turbulence models and i n this paper, the explicit algebraic stress model ( E A S M ) [15] and the Menter shear stress transport (SST) k-m m o d e l [16] are adopted. The discretization o f convective terms is implemented by a Roe scheme [17] and f o r the d i f f u s i v e fluxes central differences are applied. A flux coiTection [18] is adopted to approach second order accuracy. A n alternating direction i m p l i c i t scheme ( A D I ) is utilized to solve the discrete equations. As f o r the boundary conditions, the available options are: i n f l o w , outflow, no-slip, slip and interior conditions. Their descriptions are as f o l l o w s , (a) I n f l o w condition specifies a fixed velocity equal to the ship speed and estimated values o f

k, CO at an inlet plane; sets a zero pressure gradient normal to

the i n l e t plane, (b) O u t f l o w condition describes zero normal gradients o f velocity, k, QJ and a fixed pressure at a down-stream outlet plane, (c) No-slip condition simulates a solid w a l l boundary (e.g., a h u l l suiface) by designating zero value to velocity components, k, normal pressure gradient, and treating co f o h o w i n g [ 1 9 ] . (d) Slip condition specifies the normal velocity component and normal gradient o f a l l other flow quantities (e.g., pressure) as zero. I t simulates a sym-metry condition on flat boundaries, (e) Interior condition describes the boundary data interpolated f r o m another grid.

systematic investigations, a grid convergence study is always suggested. Its purpose is to provide some l o i o w l -edge and understanding o f the numerical error or uncer-tainty i n the computation. I t also helps to determine the m i n i m u m g r i d density (computing expense) f o r an acceptable level o f accuracy i n the computations. As a preliminary study, grid convergence computations were carried out f o r a model o f the 2nd variant o f the K R I S O Very Large Crude-oil Carrier ( K V L C C 2 ) [ 2 0 ] . Its scale ratio was 1/45.714. Thus the principal dimensions o f this model h u l l were: length between perpendiculars Lpp = 7.0 m , beam B = 1.269 m , draught T = 0.455 m . N o appendage was attached to the model. The model h u l l m o v e d straight ahead at a speed U = 0.530 m/s (corre-sponding a Froude number Fr = 0.064 and a Reynolds number Re = 3.697 x lO'') i n a canal w i t h t w o surface-piercing vertical side banks. The computed condition was quite extreme; a water depth to ship draught ratio

h/T — 1.12 and a non-dimensional ship-to-bank distance ys = 0.65.

A schematic diagram o f the canal configuration is given i n F i g . 1, indicating that the h u l l is m o v i n g close to the bank o n the port side o f the canal. T o s i m p l i f y the com-putation, effects o f waves and sinkage and t r i m were assumed negligible. The speed was very l o w . Hence, the double model approximation was adopted and a flat free surface considered at z — 0. The computational domain was defined by seven boundaries: inlet plane ( i n f l o w ) , outlet plane (outflow), h u l l suiface, flat free surface, seabed boundary, as w e l l as two side banks. The i n f l o w plane was located at l.OLpp i n f r o n t o f the h u l l and the o u t f l o w plane at L5Lpp behind the h u l l . The distance between t w o side walls was fixed as lOB to exclude the influence f r o m the far side o f the canal on the h u l l . As f o r boundary conditions, the no-slip condition was satisfied on the h u l l surface (no w a l l f u n c t i o n was introduced and y'^ < 1 was employed); the i n f l o w / o u t f l o w condition was set at the respective inlet/ outlet boundary plane; the slip condition was set at the flat free suiface {z — 0), the seabed and the side walls. For closing the R A N S equations, the E A S M turbulence model was adopted. Figure 2 illustrates the b o d y - f i x e d and right-handed Cartesian coordinate system used i n the computa-tions. The axes x, y, z are directed towards the b o w , to starboard and downwards, respectively. Figure 3 presents a general view o f the grid distribution; the grids are coars-ened f o r better illustration. As can be seen, the overlapping

4 G r i d convergence study

I n this paper, a w i d e range o f test conditions are to be considered and computed. For the assessment o f the

accuracy i n numerical computations, especially i n recent Fig. 1 Canal configuration

Fig. 2 Computational domain

Fig. 3 Grid distribution (coarse grids)

grids establislied by a cylindrical body-fitted H - 0 grid and a rectilinear H H background grid cover the whole c o m -putational domain. M o r e details.of the grid generation w i l l be given i n the next section.

h i the study, the estimation o f numerical errors and uncertainties f o l l o w e d the procedure by Ega et al. [ 2 1 , 2 2 ] . A u n i f o r m grid refinement ratio r = \/2 was utilized i n gen-erating the grid series. S i x systematically refined grids were created to enable the curve fit by the least squares root method, so as to minintize the impact of scatter on the determination o f grid convergence. F r o m the finest ( G r i d l ) to the coarsest (Grid6) density, the number o f grid points are given i n Table 1, where the grid refinement ratio is denoted b y hi/hi. hi is the grid spacing o f the ith grid, w h i l e hi is the grid spacing o f the finest grid. Cp, Cpy and X', y , K', N' are introduced i n the error and uncertainty estimation. Cp, Cpy represent the f r i c t i o n a l and viscous pressure resistance coefficients; X\ Y, K', N' stand f o r the non-dimensional longitudinal and sway force, r o l l and y a w moment, respec-tively. I n most cases, these forces and moments are essential quantities i n the prediction o f ship maneuverability. The quantities are defined as f o l l o w s :

_ RF ^ ^ Rpv

^ O.SpU^S^ ' 0 . 5 p f / 2 5 w '

Y' ^ ^ Y' = = (51

0.5pU^LppT' O.SpU^LppT' ^ '

0.5p(72LppT2' O.SpU^l^pT where S^,y is the wetted h u l l surface area.

Table 1 Grid sizes in grid convergence study

No. Grid points /!,//; 1 (;• = 1, 2, .. .,6)

Gridl 7943783 1.0 Grid2 4593870 1.189 Grids 2815261 1.414 Grid4 1707270 1.682 Grid5 1003110 2.0 Grid6 605898 2.378

G r i d convergence tendencies o f Cp, Cpy and X', Y, K',

N' are presented i n F i g . 4 a - f together w i t h fitted curves

(including the fits f r o m a theoretical order o f accuracy

Pill = 2.0) and extrapolated solutions SQ. Note that the coarsest

Grid6 is dropped f r o m the curve fit, as i t is too coarse and contaminates the results. As can be seen, the resistance coef-ficients converge w e l l w i t h an increasing number o f grid points. The fitted curves go ttaough all the solutions. The convergence o f Cp is most reasonable (see F i g . 4a). Its con-vergence r a t e p (=3.55) is closer to the theoretical one and is only about half o f the convergence rate f o r Cpy. The larger

p value f o r Cpy is probably a combined effect o f scatter and

grid independence i n solutions o f G r i d l - G r i d 4 .

The X', Y' forces and K' moment are more scattered (see F i g . 4c-e). The X' and K' solutions apparently oscillate around the fitted curves (oscillatory convergence), and are almost independent o f the grid density, producing fitted curves as straight lines (p ^ 0). The Y solutions present a slightiy better convergence trend, but still very slow. The most satisfactory grid convergence appears i n the N' solutions, as the points are a l l on the fitted curve, and most importantly, the observed order o f accuracy is identical to the theoretical one, p = ptii = 2.0.

The estimated numerical uncertainties are listed i n Table 2 f o r G r i d I , Grid2 and Grid3. Oscillatory convergence i n Cpy and X', K' is demonstrated i n the uncertainty esti-mation, since C / S N fluctuates slightiy between G r i d l , G r i d 2 and Grid3. Uncertainties i n the other three quantities Cp and y , N' tend to be converged. Y presents a slower convergence i n {7sN. l i k e the solution tendency shown i n F i g . 4d. USN i n Y drops about 18 % f r o m Grid3 to G r i d l , w h i l e the reduction i n Cp and N' is 37 and 47 %, respectively.

F r o m this study, i t seems very d i f f i c u l t to obtain grid convergence i n bank-effects computations, at least f o r extreme eases, as the predicted hydrodynamic quantities display either fluctuating or slow convergence, and even a fine g r i d discretization (i.e., approximately, 8 m i l l i o n grid points) does not help. However, there is always a concern about the computing expenses. The computing time and the numerical accuracy have to be balanced, and this suggests that a density similar to that o f Grid3 can be adopted i n further systematic computations o f bank effects.

Table 2 Numerical uncertainties of Cp, Cpy and X', y , N' Cp Cpv X' r K' N' p 3.55 7.12 0.0009 0.56 0.004 2.00 IC/sN%Sli 7.69 14.53 3.32 24.45 4.95 4.54 \UsN%S\2 9.29 14.59 3.34 26.21 4.91 6.67 12.17 14.45 3.33 29.91 4.90 8.54 5 Systematic computations

The test case f o r bank-effects (ship-bank interaction) i n the present paper was set up according to available captive m o d e l test data f o r a 1/75 scale model o f the same h u l l as above, the K V L C C 2 , fitted w i t h a horn-type rudder. The

principal dimensions o f the model were: Lpp — 4.267 m , B = 0.773 m , r = 0.277 m ; the body plan represented by solid lines (dashed lines denote a d i f f e r e n t but related h u l l f o r m , K V L C C l ) and the h u l l geometry w i t h a rudder are presented i n F i g . 5. The tests were p e r f o r m e d i n the shal-l o w water t o w i n g tank at the F H R , i n cooperation w i t h the M a r i t i m e Technology D i v i s i o n o f Ghent U n i v e r s i t y , B e l -g i u m [ 2 3 ] . I n the t o w i n -g tank, strai-ght-line tests (6 knots f u l l scale) o f the K V L C C 2 m o d e l were conducted at three d i f f e r e n t under keel clearances ( U K C ) , namely 50, 35 and 10 % o f the draught, at d i f f e r e n t lateral positions and at a zero and a non-zero propeller rate w i t h a fitted propeller. Here we w i l l o n l y use the zero propeller rate data.

For the numerical computations, the test conditions were determined f r o m the model tests, w i t h the ship m o v i n g along one side o f the t o w i n g tank at d i f f e r e n t distances to the bank.

In addition, three water depths, and two canal configurations w i t h different side-bank geometries were included. The general plan was to investigate the influence o f the water depth, ship-bank distance and bank geometry on hydrody-namic quantities. The canal configurations i n the computa-tions are shown i n F i g . 6a and b, where the h u l l is m o v i n g close to the vertical bank ( i n Canal A ) and the bank w i t h slope 1:1 ( i n Canal B) at its starboard side. The non-dimen-sional ship-bank distance is defined as shown below, f o l l o w i n g the proposal by, e.g., C h ' n g et al. [ 5 ] :

±4fi + l) (6)

}'B 2 \yp ysj

yp and ys represent the distance f r o m the ship center-plane

to the toe o f the bank at the port and starboard side, respectively. This description thus takes both side banks into consideration, w h i c h is important due to the non-uiti-f o r m bank geometries. 4.40 m (a) Canal A w w ^ w w w w w.-^ .-• w w w w w ^ ^ ^ 1:1 0 1:1 1:1 (b) Canal B

Fig. 6 Cross-sections of Canal A and Canal B, seen in the direction of motion

Table 3 Matrix of test conditions

The computed combinations o f ship-bank distance yp, and water depth hIT i n Canal A and Canal B are presented i n Table 3. As can be seen, there are six conditions i n each canal, some o f w h i c h are rather extreme, w h i c h puts the computational tools to a severe test. N o waves were con-sidered, since the tanker m o v e d at a l o w speed:

IJ = 0.356 m/s. The corresponding Froude number Fr = 0.055 and the Reynolds number i?e = 1.513 x 10*^.

The double model approximation was adopted, and no sinkage and t r i m was allowed. Since a propeller was fitted even i n the resistance tests, the drag o f the fixed propeller has been deducted f r o m the measured resistance.

Systematic computations were flrst c a n i e d out f o r Canal A . Thereafter, an analysis was made o f the possible m o d -eling enors, and finally computations f o r the more d i f f i c u l t Canal B were c a n i e d out u t i t i z i n g the lessons leamt f r o m Canal A .

5.1 Computational settings

Due to the asymmetry o f the geometry and the flow field, the computational domain had to cover the w h o l e h u l l i n the canal. The computational settings f o r Canal A and Canal B were essentially the same. A sketch o f the com-putational domain and the coordinate system is g i v e n i n F i g . 7a. A s seen i n the figure, the computational d o m a i n is made up by seven boundaries l i k e i n the g r i d convergence study part. The flat free surface is considered at z = 0, w h i l e the seabed and the t w o side banks are placed at specific locations conesponding to the test conditions i n Table 3. The coordinate system and the applied boundary conditions are defined i n the grid convergence study. I t should be mentioned that both the E A S M and the Menter

k-a> SST turbulence model were used f o r the closure o f

R A N S equations at different stages o f the computations. Slightly more comphcated overlapping grids were adopted here. As illustrated i n Fig. 7b, the overlapping grid is b u i l t up by three components: a c y l i n d r i c a l H - O grid (fitted to the h u l l ) , a curvilinear 0 - 0 g r i d (fitted to the rudder) and a rectilinear H - H grid (fitted to the canal boundaries). The f o r m e r two grids are immersed into the latter. The body-fitted H - O g r i d covers the m a i n flow fleld around the h u l l , and two clusters o f g r i d points are

MT yB Canal A . Canal B 1.180 1.316 1.961 2.431 0.758 0.909 1.632 2.173 1.50 (UKC = 50%r) O O 1.35 (UKC = 35%r) O O O O O O O O 1.10 (UKC = io%r) O O ,

(a) Computational domain and coordinate system

(b) Grid distribution

Fig. 7 Computational domain, coordinate system and grid distribu-tion of Canal A and Canal B

concentrated aromtd the b o w and stern regions so as to resolve the flow field more precisely. A smaU outer radius (0.5Lpp) o f the cylindrical g r i d is used f o r Canal A to save grid points and an even smaller radius (0.12Lpp) o f the c y h n d r i c a l H - O grid is applied f o r Canal B .

5.2 Results

I n this section, results f r o m varying water depths and s h i p -bank distances i n Canal A and Canal B are presented. I n addition to hydrodynamic forces X', Y and moments K!, N', the mean sinkage and t r i m are reported. These t w o vari-ables are o f great engineering interest w i t h respect to the U K C . The mean sinkage cr and t r i m T are defined as:

a = Z/pgA,,

where sinkage is positive downwards and t r i m positive bow-up. Z is the sinking force, M the t r i m moment, A,v the water plane area and the l o n g i t u d i n a l moment o f inertia o f the water-plane area about the center of floatation.

I n addition to the R A N S computations, the steady state potential flow panel method X P A N i n S H I P F L O W was used to compute the sinkage and t r i m (without rudder) f o r Canal A . The m o t i v a t i o n was m a i n l y to study the influence o f neglecting the wave effects i n the R A N S method. I n X P A N non-linear free surface boundary conditions are satisfied [ 1 4 ] . I t is also o f interest to compare the accuracy i n the bank-effects simulation o f two completely d i f f e r e n t numerical methods.

A l l the predicted hydrodynamic quantities have been compared w i t h the test data f r o m F H R . I t should be noticed that the measured data were obtained i n a confidential project, so no quantitative values o f the data and computed results can be presented i n the f o l l o w i n g figures. Instead, the results f o r all and h/T variations are expressed i n percent o f the largest measured value o f a l l variations. Specifically, the predicted sinkage and t r i m are normalized b y the m a x i m u m measured sinkage and t r i m , respectively, the X', Y forces by the largest \Y\, and the K', N' moments b y the largest \K'\. Absolute values o f a l l m a x i m u m quan-tities are thus used i n the normalization. I n addition, the measured negative thrust f r o m the non-rotating propeller is subtracted f r o m the total longitudinal force to enable a straightforward comparison between computations and measurements.

5.2.1 Sinkage and trim in Canal A

The predicted sinkage and t r i m i n Canal A f r o m the potential flow method and the R A N S method ( w i t h the E A S M model) are plotted i n Figs. 8 and 9, coupled w i t h the experimental data. Dashed lines indicate the zero level. Figure 8 shows the sinkage and t r i m versus the water depth ratio li/Tat a fixed smaU ship-to-bank distance yp = L 3 I 6 . The sinkage and t r i m , i n general, rise w i t h decreasing water depth, especially at a water depth less than h/T ~ 1.35, revealing a significant shallow water effect. The results against the ship-bank distance y^ at a fixed water depth /?/

T= 1.35 are presented i n F i g . 9. S i m i l a r tendencies are

displayed, indicating a blockage f r o m the bank: when the h u l l moves closer to the bank, the sinkage and t r i m increase.

It is shown i n Figs. 8 and 9 that the R A N S method works very w e l l i n predicting the sinkage and t r i m , while the potential flow method fails to obtain a solution at the closest ship-bank distance and under-predicts the t r i m

(%) 160 120 SO Ï ^° 0 " -80 -120 -160 - B - c r ( R A N S _ E A S M ) (7 (potential) • cr ( E F D ) T ( R A N S _ E A S M ) 0--"-r(pötenlial) Canal A:)^1.316| ^ ^^^^^p^ 1.0 1.1 1.2 1.3 hIT 1.4 1.5 1.6

Fig. 8 Sinkage a and trim T versus hIT in Canal A

( % ) 160 120 80 40 0 -40 b -80 -120 4 6 0

: [Canal A:/i/r=1.35] a (RANS_EASivl) cr (potential) .-cr (EFD) - D — g _ _ _ _ o a (RANS_EASivl) cr (potential) .-cr (EFD) A Ö - T ( R A N S _ E A S M ) • •r(pötential) , , , TiWP} 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 yB

Fig. 9 Sinkage a and trim T versus in Canal A

considerably at all other conditions. The predicted t r i m is only half the measured value. This may be due to the fact that there is significant flow separation along the h u l l , w h i c h influences the pressure at the stern. Thus, a large pressure difference between the b o w and stern is produced. A pronounced flow separation around the stern at hi

T = 1.1 and ) ' B = 1.316 is indeed predicted by the R A N S

method, as indicated i n the non-dimensional axial velocity

Ux/U contours i n F i g . 10. The potential flow method,

however, is unable to simulate this effect, as verified b y the predicted pressure distribution on the h u l l surface. N o r -m a l i z i n g the pressure by 0.5pU^, pressure c o e f f i c i e n t dis-tributions f r o m potential flow and R A N S methods are displayed i n F i g . 11. W h i l e the potential flow pressure is relatively symmetrical fore-and-aft, the suction under the b o w has no correspondence under the stern i n the R A N S results. A much larger t r i m by the b o w can thus be expected f o r this case.

5 . 2 . 2 Hydrodynamic forces and moments in Canal A

The predicted forces and moments f o r Canal A b y the R A N S method are shown i n the f o l l o w i n g figures, where

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 10 RANS predicted axial velocity iixlU contour around the tanker in the horizontal plane (from top to bottom: zILpp = 0, -0.032, -0.06) (Canal A: lilT = 1.1, = 1.316)

Pressure: -0.5 -0.4-0.3 -0.2-0.1 0 0.1 0.2 0.3 0.4 0.5

Fig. 11 Pressure distribution on the hull surface (Canal A: hIT = 1 . 1 , J'B = 1.316). {Bottom view) upper potential flow, lower RANS

only the zero level is given. Note that no result f r o m the potential flow method is presented here, as i t is indicated f r o m the predicted sinkage and t r i m , as w e l l as the pressure distribution i n Figs. 8-11 that viscous' effects cannot be neglected. Results f o r the X ' , Y forces and the K', N' moments at a speciflc ship-bank distance, y^ = 1.316, against the water depth ratio are shown i n F i g . 12a and b . Results at a specific water depth ratio hIT = 1.35 against the ship-bank distance are shown i n F i g . 13a and b .

Compared w i t h measurements, the tendencies o f hydrodynamic forces and moments are captured w e l l . A s seen i n F i g . 12a and b , when the hull bottom approaches the seabed, the X' force and K' moment increase, w h i l e the y force (a suction force towards the bank) behaves i n a different way. I t increases slightly between hIT — 1.5 and 1.35, but drops rapidly between hIT =• 1.35 and 1.1. The N' moment shows a monotonic increase f o r d i m i n i s h i n g water depth as w e l l . However, magnitudes o f the y a w m o m e n t are very small. I n F i g . I 3 a and b , w i t h the v a r y i n g ship-to-bank distance at hIT = 1.35, the h u f l is attracted to the bank but the b o w is pushed away.

(%) 125 r 100 : 75 : ~ l _{50 :} 25 -0 ~a -25 r 50 --75 r -100 i--125 E-T ê ^ r ( R A N S _ E A S M ) • y ' ( E F D ) (%) 150 125 ~% ^ ° 75 50 - | 25 0 S< -25 -50 -75 ;|CanaIA:)'j=1.3I6 | K' ( R A N S _ E A S M ) r— • • K ' C E F D ) r « w . ^—- • - # • A»'(RANS_EASM) • A''(EFD) 1.0 1.2 1.3 1.4 1.6 h/T (b) K', N' moment

Fig. 12 X', Y force and K*, A'' moment versus h/T in Canal A W h i l e tendencies o f the predicted force and moment correspond w e l l w i t h those o f the measured data, the absolute level is not w e l l predicted i n some cases. The most obvious difference between computations and measure-ments is seen i n the Y' force, f o r w h i c h there is a more or less constant s h i f t to l o w e r predicted values. This tendency is seen clearly i n Figs. 12a and 13a. T o more quantitatively investigate the absolute accuracy, a f o r m a l validation study was made f o r hydrodynamic variables i n an extreme con-d i t i o n : h/T = 1.1 ancon-d ye = 1-316. The valicon-dation f o l l o w s the procedure proposed by the A S M E V & V 20 2009 standard [24], i n w h i c h the concepts of a validation

^num + UD) and a comparison uncertainty Uy^i (C/vai

error \E\ = \S — D\ are introduced. Here .S" and D represent the simulated solution and experimental data, respectively.

Unum is the numerical uncertainty and UD is the data

uncertainty i n the measurements. Assuming that the round-o f f eiTround-or i n a cround-omputatiround-on is negligible, the numerical uncertainty m a i n l y consists o f iterative uncertainty Ui due to the lack o f convergence i n the iteration process and grid discretization uncertainty f/sN caused by the l i m i t e d g r i d resolution. A c c o r d i n g to the procedure, the validation result is to be interpreted as f o l l o w s : i f \E\ < U^a., the modeling eiTor is w i t h i n the "noise l e v e l " imposed by the numerical and experimental uncertainty, and not m u c h can

(%) 120 100 80 60 40 ^ 20 - „ 0 -20 eS; -40 -60 -100 - Y' (RANS_EASM) T _{T r (EFD)} V T • r ,

•

•

•

Fig. 13 X', f force and K', N' moment versus )'B in Canal A

be concluded about the source o f the error; but i f

\E\ ^ f/vai, the sign and magnitude o f E could be used as

to improve the modeling. I n the present w o r k , the iterative uncertainty is quantified by the standard deviation o f the force i n per cent o f average force over the last 10 % iter-ations and kept below 0.2 %, w h i c h is a neghgible con-tribution to the numerical uncertainty Unum- Therefore,

Unum is approximated as the grid discretization uncertainty UsN i n the g r i d convergence study; w h i l e the experimental

data and data uncertainty are available f r o m F H R . Note that C/d is estimated as twice the standard deviation o f measured variables i n repeated captive shallow water model tests at F H R f o r the S E V I M A N 2012 W o r k s h o p [ 2 5 ] . I n this way, the precision eiror is accounted for, but not the bias error.

T h e estimated uncertainties and comparison errors f o r hydrodynamics quantities X', Y, K', N' are presented i n Table 4, where the measured data (D) are used f o r nor-malization. I t is seen that tbi-ee quanfities Y, K' and N', exhibit a larger comparison error than the v a l i d a t i o n uncertainty, i m p l y i n g that there are significant m o d e l i n g errors, i n computations and/or measurements. The neglec-ted bias en-or i n the measured uncertainty corresponds to a modeling error and has not been investigated i n the present

work, and neither has the absence o f the side bank i n the precision measurements. However, systematic investiga-tions o f the modeling i n the numerical computainvestiga-tions can be canied out. This is the topic o f the next section.

5.2.3 Investigation of modeling errors

There are several potential sources o f m o d e l i n g enors i n the bank-effect computations f o r Canal A . Examples include the computational domain size, neglect o f free surface, non-free sinkage and t r i m , turbulence modeling, absence o f propeller, and boundary condition on the bank/ seabed.

Eirst o f all, the selected computational domain is com-posed of the boundaries o f seabed, side banks, flat water surface, h u l l suiface, i n f l o w and outflow. The boundaries o f flat water suiface and h u l l surface are fixed, w h i l e the positions o f seabed and side banks are decided according to the test conditions. I n theory, the i n f l o w and outflow boundaries should be kept far f r o m the h u l l to diminish the effect on the solutions. I n the computations presented above, their locations were defined f r o m early experiences w i t h l.OLpp i n f r o n t o f and l.SLpp behind the h u l l , respectively, yielding a total longitudinal size o f 3.5Lt)p. I f these t w o boundaries ai^e not sufficiently f a r away f r o m the h u l l , they may introduce some error to the results. T o investigate the influence o f the domain size, w e computed the shallowest water case {hIT = 1 . 1 and yg = 1.316) as an example like i n the vahdation study. W i t h larger longitu-dinal size (1.5 times of the previous one), the X ' , Y forces and K', N' moments reduced about 2.8%D. Recalling Fig. 12 and Table 4 o f the previous results, the forces and moments are a l l under-predicted. The decreased results do not i m p r o v e the validation, thus the domain size appears to be a not so important source o f m o d e l i n g en-or. Moreover, a large domain size also increases the grid number and consumes longer computing time, w h i c h is important f o r a series o f systematic computations. G i v e n these factors, the previous domain size was considered acceptable.

A s mentioned above, the wave effect is neglected i n the R A N S computations. Its influence on hydrodynamic quantities should be investigated. A s shown i n Figs. 8 and 9, the computation o f sinkage and t r i m i n Canal A indicates that without considering the waves i n the R A N S

Table 4 Validation results of X', F, K', N'

X' 1" N'

\UD%D\ 4.40 18.80 7.10 2.78

It/SN%ÖI 3.27 8.38 3.57 6.92

iC/val%öl 5.48 20.58 7.95 7.46

\E%D\ 1.85 71.99 27.19 18.94

computations, the predicted sinkage and t r i m conespond very w e l l w i t h measurements. This shows that i t is acceptable to neglect the wave e f f e c t when predicting the sinkage and t r i m at the present l o w speed and at relatively close ship-bank distances. However, the effect o f the sinkage and t r i m on the other forces and moments, espe-cially at small water depths, needs to be investigated. Thus, further computations w i t h given sinkage and t r i m f r o m prior R A N S computations were carried out f o r the case w i t h a variation o f water depth i n Canal A at J B = 1.316. Results including the initial sinkage and t r i m ( C T & T ) are indicated by symbols and dotted lines i n F i g . 14a and b, also containing the previous predictions (symbols and solid lines). F r o m the comparisons, i t is interesting to see that the influence o f sinkage and t r i m is generally very small. However, i t cannot be neglected f o r the K' and N' moments at the very shallow water depth {h/T = 1.1), where visible differences i n the results are noted. Returning to Table 4, a quantitative comparison can also be made. The correction f r o m sinkage and t r i m increases l/7vai%DI f o r A'' to 8.34 % and reduces the \E%D\ to 7.92 %, w h i l e f o r other quantities the improvement is not large enough to reduce E to the Uy^i level, so there must be other significant m o d e l i n g issues.

A s f o r the turbulence modeling, the E A S M m o d e l was used i n the systematic computations, as this turbulence

{%) 140 120 too h -100 ; ^S^r(RANS^EASM) •r(RANS_EASM_ a & T ) • _y'(EFD) . •

fl

model is k n o w n to predict more accurate wake profiles i n deep water, see e.g. [ 2 6 ] , but i n the present case, h y d r o -dynamic quantities are under-predicted w i t h this model. Therefore, • the Menter k-co SST model was apphed to evaluate the influence o f the turbulence modeling. The focus was placed on computations w i t h varying bank dis-tance at the same water depth h/T = 1.35. Results are presented by dotted lines i n Fig. 15a and b. There is a slight improvement i n the prediction o f the sway force Y (the m a x i m u m decrease i n \E%D\ is around 8 % ) , but the improvement is very small compared to the difference between computed and measured results. I t should be pointed out that the massive separation mentioned above— near the stern on the bank side o f the h u l l — m a y be i n f l u -enced by the turbulence modehng. The prediction o f this separation is crucial f o r the prediction o f hydrodynamic forces.

Another modehng error is the absence o f the non-rotating propeller that was fitted to the h u l l i n the mea-surements. I n the previous assessment o f the computed longitudinal force X' the measured thrust was deducted. So, the influence o f the propeller blades on the flow field was neglected. T o l o o k into this, a non-rotating propeller, fixed behind the h u l l was investigated. The study was carried out f o r the second water depth Ji/T = 1.35 and closest s h i p

-(%) 120 100 SO 60 40 20 0 -20 -40 -60 - « - y ( R A N S J A S M ) T _{•O -Y'iRANS i ' - ö S S T )} 7 rCEFD)

•

:| Canal A; Ii/T=l35 \ •• A -X'(RANS k-o) SST) _{• X ' ( E F D )} 1.6 1.; 2.0 2.2 (a)X', r force (%) 3 - E i - A " ( R A N S _ E A S M ) ••<£>•• AT'(RANS_/t-(a S S T ) A S-'fEFDI ; - ^ A ' ' ( R A N S „ E A S M ) • • A . . A ' ' ( R A N S _ / . - - ö S S T ) . • A ' ( E F D ) A • • A . . A ' ' ( R A N S _ / . - - ö S S T ) . • A ' ( E F D ) i -: 1 Canal A-: Wr=1.35 I 2.0 2.2 3'„ (b) K', N' moment

bank distance = 1.180 i n Canal A . Considering the improvement i n the turbulence modeling above, the k-oj SST model was adopted f o r these computations.

The propeller was designed by the M a r i t i m e and Ocean Engineering Research Institute ( M O E R I ) i n Korea. Its data are given i n Table 5 and the geometry is illustrated i n F i g . 16. However, the angular position o f the propeller blades i n the experiment is u n k n o w n , so i t is impossible to exactly capture the flow disturbance resulting f r o m the blades i n the experiment. Therefore, f o u r designated posi-tions were considered i n the investigation. T a k i n g the ship center-plane as a reference, the propeller reference line originally coincided w i t h the ship center-plane (0°), and then its blades were turned clockwise 3 0 ° , 4 5 ° and 6 0 ° f r o m this position (see F i g . 17). N e w computations w i t h these blades were then performed.

Changes i n forces and moments due to the presence o f the propeller are presented i n Table 6. I t is seen that the measured negative thrust o f the fixed propetier i n the

Table 5 Propeller data

Name MOERI KP458

Type Fixed pitch

No. of propeller Single

No. of blades 4

Diameter DR (m) 0.131

Pitch ratio P R / Ö R (0.7R) 0.721

Expanded area ratio AE/AQ 0.431

Rotadon Right hand

Hub ratio 0.155

Skew (°) 21.15

Rake (°) 0.0

Fig. 15 X', F force and K', N' moment versus ys in Canal A Fig. 16 Propeller geometry (ship side and back view)

Table 6 A", Y, K', N' changes in experiment and computations (without and with a propeller)

h/T= 1.35, j'B = 1.180

•^w/o S20' 5450 Seo"

Fig. 17 Variation of blade position (0° -> 30° ^ 45° ^ 60°)

experiment was 14.49 % o f the resistance. However, the resistance increases i n the computations w i t h d i f f e r e n t blade positions are all no more than 1 %. T h i s could mean either that the flow separation at the stern is over-predicted i n the computations, or that the interaction between the propeller and the h u l l causes a thrust on the h u l l , w h i c h is almost as large as the propeller drag. W h a t speaks i n f a v o r o f the latter explanation is the fact that X' without the propeller is over-predicted by 16.47 % , compared w i t h the data, i f the measured negative propeller thrust is subtracted f r o m the measured X', w h i l e t h é predicted X' w i t h the propeller is very close to the measured value without subtraction o f the thrust. The error is then only 2-3 % . On the other hand, the errors mY,K' and A'' are m u c h larger than this, as seen previously, so the first explanation is more l i k e l y : the separation is over-predicted at the stern. This causes the drag o f the propeller to be much smaller than i n the measurements. As seen i n Table 6, the d i f f e r e n t blade positions cause very similar drag increases. The increase/decrease o f X', Y and K' are similar i n magnitude, but the increase i n N' is m u c h larger: about 10 times o f the change i n X', Y or K'. Referring to F i g . 15b, the small value o f A'' (close to zero) may explain the larger per-centage increase.

The last modeling eiTor stated here is that the slip condition is satisfied at the walls (seabed, side bank) i n computations, but when the h u l l moves over/along the seabed/side bank, a boundary layer is developed on these boundaries due to the disturbance velocity f r o m the h u l l . Therefore, the slip condition m i g h t be inadequate, and a m o v i n g no-slip condition could be more suitable. Pre-l i m i n a r y computations have indicated some effect, but this w i l l be investigated further.

I n this investigation of modeling eiTors, several possi-bilities have been tested and most of them demonstrate an influence on hydrodynamic forces and moments. A l t h o u g h the magnitude o f the i n f l u e n c e — f r o m turbulence m o d e l i n g , neglect o f free surface, non-free sinkage and t r i m , absence o f propeller—varies, a combination o f these contributions

AX' w/o -> prop. (%) {S - D) %D AY' w/o -> prop. (%) AK' w/o —> prop. (%) AN' w/o -> prop. (%) 14.49 - 0.67 0.97 0.66 0.88 16.47 2.41 2.72 2.40 2.62 _2.49 -2.71 -2.64 -2.60 -2.88 -2.89 -3.12 -2.68 27.82 34.83 33.22 26.30

may yield more accurate solutions. Note that the influence on i n d i v i d u a l hydrodynamic quantities is not always u n i -f o r m : i m p r o v i n g the m o d e l i n g does not always result i n an improved result. Take f o r instance one o f t h e conditions: h/

T= 1.35 and ys = 1-180 i n Canal A . A p p l y i n g more

appropriate turbulence modeling (k-co SST model) makes the comparison error E f o r A'' drop 9 % , and based on this turbulence model, i n c l u d i n g a non-rotating propeller yields a further reduction o f 4 % . However the error E o f y is reduced by 21 % w i t h better turbulence modeling, but is then increased b y 7 % including the non-rotating propeller.

5.2.4 Results in Canal B

The numerical predictions of the ship-banlc interaction i n Canal A facilitated the computations f o r the same tanker m o v i n g i n Canal B , since the experience w i t h respect to the turbulence modeling, the inclusion o f the effect o f sinkage and t r i m , as w e l l as the fixed propeller could be utihzed. The computational settings i n Canal B were s i m i l a r to those i n Canal A , as described i n Sect. 5.1, and the c o m -putations were cairied out f o r varying ship-to-bank dis-tance (at h/T = 1.35). The k-co SST turbulence m o d e l was used and the non-rotating propeller included, but the i n i t i a l sinkage and t r i m were not considered. A s clearly shown i n Fig. 14a and b, their influence on the forces and moments is negligible at the m e d i u m water depth (h/T = 1.35).

The sinkage and t r i m results w i t h v a r y i n g bank distance i n Canal B are shown i n F i g . 18. As can be seen, a smaher shipbank distance leads to larger sinkage and t r i m , i n d i -cating a pronounced blockage when the h u l l approaches the bank. L i k e i n the results f o r Canal A , sinkage and t r i m are predicted w e l l by the R A N S method, the sinkage i n par-ticular, as shown i n the figure. T r i m values are slightly

(%) 150 125 100 1^= 50 25 -25 -50 b -75 -100 -125 -150

— a — cr(RANS_/!r-ö)SST & prop.) • ( T C E F D )

L - f r- H L i _ . . _ _

I

-; |CanalB:/!/r=1.35 1

> —0— T(RANS fcfflSST & prop.)

T. . ^ E F D )

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Fig. 18 Sinkage a and trim T versus )'B in Canal B

(%) 120 100 _ 80 60 40 - „ 20 " # -20 ^ -40 -60 -80 A . .- A

—0— X'(RANS_t-ft) SST & prop.)

O A" ( E F D )

. — B — . -}" (RANS_/t-ai SST & prop.) 4 J " ( E F D )

. A

r _{2 z ^ z ^ , e}

ICanalB: /;/7i=1.35 |

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2 4

Fig. 19 X' and F force versus )'B in Canal B

(%) 80 70 —a 60 50

I

- X"(RANS_/t-£» SST & prop.)

^A— N' (RANS_<:-ffl SST & prop.) • ^ ' ( E F D )

[Canal B: ;i/r=1.35 |

0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Fig. 20 K', N' moment versus }'B in Canal B

under-predicted, w h i c h may be due to the over-predicted flow separation, typically at the most extreme condition (closest ship-bank distance).

Results f o r forces and moments are g i v e n i n Figs. 19 and 20. A g a i n , the hydrodynamic variable o f the primary interest is the longitudinal force Z ' . As mentioned above, to evaluate the predicted X' i n the absence of a non-rotating propeller, the measured propeller force is deducted. W h e n the propeller is

included i n computations, l i k e here, the measured total X' shall be used f o r evaluation. Figure 19 shows the forces X' and Y f o r v a r y i n g ship-bank distance, and the computed Z ' ,

N' results are available i n F i g . 20. The coiTespondence

between computed X' results and measured data is good as are the trends i n Y force and N' moments. H o w e v e r the absolute accuracy o f the latter three quantities is less satis-factory. A slight improvement relative to Canal A may be noted, probably due to i m p r o v e d modeling, but there are s t i l l significant differences between the measured and computed results. Note that Canal B should be a more d i f f i c u l t case to compute w i t h the sloping w a l l .

6 Conclusions

Investigating bank effects is a challenging task due to the importance and c o m p l e x i t y o f the hydrodynamic p r o b l e m itself and the d i f f i c u l t y i n modeling the physical p r o b l e m by numerical tools. The present w o r k aims at investigating the accuracy achievable i n the prediction o f several hydrodynamic forces and moments o f interest i n maneu-vering. Great emphasis has been placed o n uncertainty assessment and numerical and physical m o d e l i n g issues.

A g r i d convergence study was performed first to obtain the numerical uncertainty i n the computations f o r v a r y i n g grid densities. The study illustrated the d i f f i c u l t y i n achieving converged results i n bank-effects computations, but i t also indicated a suitable grid density f o r the p r o b l e m at hand.

I n a series o f systematic computations the sinkage, t r i m and hydrodynamic forces and moments f o r v a r y i n g water depths, ship-to-bank distances and bank geometries were then predicted. V e r y accurate results f o r sinkage and t r i m were demonstrated using a R A N S method, w h i l e a poten-tial flow method predicted t r i m much.less accurately. For other quantities. the R A N S method predicted the trends w i t h varying bank distance and bottom clearance quite w e l l , but the absolute accuracy was less satisfactory. A f o r m a l validation analysis showed that there ai-e m o d e l i n g errors i n the computations and/or the measurements. F i v e types o f computational m o d e l i n g eiTors, i.e., computational domain size, neglect o f waves, non-free sinkage and t r i m , turbulence modeling, and absence o f propeller were dis-cussed. I t was shown that the applied domain size is acceptable and that the waves at the l o w speed o f interest have a negligible influence on the results, w h i l e the e f f e c t o f sinkage and t r i m cannot be neglected at the shallowest water depth. I n c l u d i n g the non-rotating propeller i m p r o v e d the predicted X' f o r c e significantly, but the changes i n other forces/moments were minor. As f o r the turbulence m o d -eling, the k-co SST model produced slight i m p r o v e m e n t i n comparison w i t h the E A S M model.

I n further studies i t w o u l d be of interest to investigate the effect o f the no-slip condition on the bank walls and bottom. Also, measurements of the wake flow w o u l d be most valuable. Strong separation of the flow occurs i n the most severe conditions close to the bank and bottom, and i t is important to predict this flow accurately to obtain the correct forces and moments.

Acknowledgments The present work was funded by Chalmers University of Technology, Sweden and the China Scholarship Council. Computing resources were provided by C3SE, Chalmers Centre for Computational Science and Engineering. The authors thank Mi-. Guillaume Delefortrie (Flanders Hydraulics Research, Belgium) and Mr. Evert Lataire (Maritime Technology Division, Ghent University, Belgium) for providing the measurement data.

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