J Mar Sci Technol (2013) 18:310–323
DOI 10.1007/s00773-012-0209-7
ORIGINAL ARTICLE
Computational fluid dynamics (CFD) prediction of bank effects
including verification and validation
Lu Zou • Lars Larsson
Received: 5 March 2012 / Accepted: 13 December 2012 / Published online: 18 January 2013
JASNAOE 2013
Abstract Restricted waters impose significant effects on
ship navigation. In particular, with the presence of a side
bank in the vicinity of the hull, the flow is greatly complicated. Additional hydrodynamic forces and moments act
on the hull, thus changing the ship’s maneuverability. In
this paper, computational fluid dynamics methods are utilized for investigating the bank effects on a tanker hull. The
tanker moves straight ahead at a low speed in two canals,
characterized by surface piercing and sloping banks. For
varying water depth and ship-to-bank distance, the sinkage
and trim, as well as the viscous hydrodynamic forces on the
hull, are predicted by a steady state Reynolds averaged
Navier–Stokes solver with the double model approximation
to simulate the flat free surface. A potential flow method is
also applied to evaluate the effect of waves and viscosity
on the solutions. The focus is placed on verification and
validation based on a grid convergence study and comparisons with experimental data. There is also an exploration of the modeling errors in the numerical method.
Keywords Bank effects Reynolds averaged Navier–
Stokes method Hydrodynamic forces and moments Sinkage and trim Verification and validation
1 Introduction
When a ship is traveling in a canal or a narrow channel, the
flow becomes highly complex. Interactions occur between
the ship and the side banks, as additional hydrodynamic
L. Zou (&) L. Larsson
Department of Shipping and Marine Technology, Chalmers
University of Technology, 41296 Gothenburg, Sweden
e-mail: zlu714@gmail.com
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forces and moments generated by the vicinity of the banks
act on the hull and influence ship motion. This phenomenon is named bank effects. When the distance between the
hull and the canal boundary is reduced, the flow is accelerated and the pressure is accordingly decreased, which has
an effect on the hydrodynamic characteristics. The produced hydrodynamic forces, especially in extremely shallow canals, may considerably affect the maneuvering
performance of the ship, making it difficult to steer. The
ship may collide with the side bank and/or run aground due
to the ‘‘squat’’ phenomenon. Bank effects are thus extremely important for ship navigation. In the past few decades, many investigations on bank effects have been
carried out, both experimentally and numerically. A notable event was the International Conference on Ship
Maneuvering in Shallow and Confined Water: Bank Effects
[1], at which 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, which normally treat
the bank effect as a function involving hull-bank distance,
water depth, ship speed, hull form, bank geometry, propeller
performance, etc. During the 1970s, Norrbin at SSPA,
Sweden, carried out experiments and proposed empirical
formulae to estimate the hydrodynamic forces for flooded
[2], vertical [3] and sloping [3] banks. Li et al. [4] continued
Norrbin’s investigations and tested bank effects in extreme
conditions for three different hull forms (tanker, ferry and
catamaran). The influence of ship speed, propeller loading
and bank inclination was evaluated. Ch’ng et al. [5] conducted a series of model tests and developed an empirical
formula to estimate the bank-induced sway force and yaw
moment for a ship handling simulator. In recent years,
comprehensive model tests in a towing tank have been
J Mar Sci Technol (2013) 18:310–323
carried out at the Flanders Hydraulics Research (FHR),
Belgium, to build up mathematical models for bank-effect
investigations and to provide data for computation validation. Vantorre et al. [6] discussed the influence of water
depth, lateral distance, forward speed and propulsion on the
hydrodynamic forces and moments based on a systematic
captive model test program for three ship models moving
along a vertical surface-piercing bank. They also proposed
empirical formulae for the prediction of ship–bank interaction forces. From extensive tests, Lataire et al. [7] developed
a mathematical model for the estimation of the hydrodynamic forces, moments and motions taking into consideration ship speed, propulsion and ship/bank geometry. In
addition, two parameters defining the distance to a bank of
irregular geometry were proposed.
Although experimental tools and empirical formulae are
widely used for bank-effects prediction, they have their
shortcomings. For example, empirical formulae are suitable
only for cases with similar hull forms and conditions.
Otherwise, the prediction is barely reliable. To establish a
mathematical model, many systematic and expensive model
tests are always required. However, the most important
weakness of these tools is their inability to provide detailed
information on the flow field, which can help explain the flow
mechanism behind the bank effects. In view of this,
researchers resort to using numerical (i.e., computational
fluid dynamics, CFD) methods to deal with the phenomena
of bank effects. Among existing CFD methods, the potential
flow method is the most common. Newman [8] applied a
Green function to predict the interaction force between a ship
and an adjacent rectangular canal wall. Miao et al. [9] studied
the case of a ship travelling in a rectangular channel and
estimated the lateral force, yaw moment and wave pattern
based on Dawson’s method. It was shown that the applied
potential flow method was able to predict reasonable
hydrodynamic forces for a water depth to ship draught ratio
larger than 1.5, but it failed for smaller ratios. Lee and Lee
[10] applied the potential flow method to estimate the
hydrodynamic forces as a function of the water depth and the
spacing between the ship and a wedge-shaped bank. Viscous
flow computations were carried out by Lo et al. [11], who
studied the bank effect on a container ship model using CFD
software based on the Navier–Stokes equations. The effect of
vessel speed and distance to the bank on the magnitude and
temporal variation of the yaw angle and sway force were
reported. Some details of the predicted flow field are available from this study. Wang et al. [12] recently studied the
bank effects using a Reynolds averaged Navier–Stokes
(RANS) method to predict viscous hydrodynamic forces on a
Series 60 hull at varying water depth ratios (1.5, 3.0 and 10.0)
and ship–bank distances.
Available literature indicates that although empirical
formulae and model tests have played an important role in
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bank-effect investigations, the mechanism of the effect is
still not fully explained, as the model test normally only
produces global data, like forces and moments, but not
details of the flow. Viscous methods, like the most widely
used RANS method, can provide more details, which can
enhance the understanding of the mechanisms. However,
their application is still limited; only a few reports referring
to this type of computation have been presented, yet some
of them indicate promising results. In the framework of an
ongoing project aiming at extensive CFD investigations of
the hydrodynamic forces on the hull in restricted waterways, the present work intends to study the bank effects in
the case of a ship moving in a shallow canal. By means of
numerical methods, quantitative predictions of the most
interesting hydrodynamic quantities, such as the sinkage
and trim and the viscous forces on the hull are obtained.
Since this is a validation study, emphasis is placed on
formal verification and validation (V&V) and an investigation of modeling errors.
2 Computational method
Generally in a RANS method, the fluid motion around a
ship is governed by a system of equations consisting of
the Navier–Stokes equations (1) and the continuity
equation (2), describing the conservation of momentum
and of mass. Assuming the fluid to be incompressible,
the governing equations given in a Cartesian coordinate
system read [13]:
oui
o
1 op
o2 ui
þ
ðuj ui Þ ¼ þ Fi þ m
ð1Þ
q oxi
ot oxj
oxj oxj
oui
¼0
ð2Þ
oxi
where ui(j) represents velocity components, xi(j) denotes
coordinates, p is the pressure, v is the kinematic viscosity,
Fi represents the body force (such as gravity) and q is the
fluid density. For a three dimensional flow, i, j = 1, 2, 3.
As four variables (ui(j), p) are present in the three-dimensional equations above, the Navier–Stokes equations
combined with the continuity equation, establish a closure
of the system of equations.
Time averaging Eqs. 1 and 2 yields:
oui
o
1 op
1 o
þ
ðuj ui Þ ¼ þ Fi þ
ð
rji þ Rji Þ
q oxi
q oxj
ot oxj
ð3Þ
oui
¼0
oxi
ð4Þ
ij denote the average velocity, pressure
where uiðjÞ ; p and r
and stress; Rji ¼ Rij ¼ qu0i u0j a symmetric quantity,
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312
termed ‘‘Reynolds stress’’; u0iðjÞ represents the fluctuating
velocity in time. As seen in Eqs. 3 and 4, the fluctuating
values are all removed during the time-averaging, but
new unknown variables, the Reynolds stresses Rji, are
introduced. The Rji needs to be modeled to close the
system of Eqs. 3 and 4, which then yield all the mean
flow properties.
3 CFD solver
A CFD solver for Ship Hydrodynamics, SHIPFLOW [14], is
utilized in the present work to implement the CFD computations. It includes several modules with different, but
associated functions. Two of the modules are used in the
present work: XPAN, which is a solver for the potential flow
based on a surface singularity panel method; and XCHAP, a
finite volume method for solving the RANS equations. The
XCHAP can compute complex geometries, owing to the
capability of handling overlapping grids, e.g., rudder, shafts,
brackets and vortex generators. It can also deal with the grids
imported from external grid generators. The code contains
several turbulence models and in this paper, the explicit
algebraic stress model (EASM) [15] and the Menter shear
stress transport (SST) k–x model [16] are adopted. The
discretization of convective terms is implemented by a Roe
scheme [17] and for the diffusive fluxes central differences
are applied. A flux correction [18] is adopted to approach
second order accuracy. An alternating direction implicit
scheme (ADI) is utilized to solve the discrete equations. As
for the boundary conditions, the available options are:
inflow, outflow, no-slip, slip and interior conditions. Their
descriptions are as follows. (a) Inflow condition specifies a
fixed velocity equal to the ship speed and estimated values of
k, x at an inlet plane; sets a zero pressure gradient normal to
the inlet plane. (b) Outflow condition describes zero normal
gradients of velocity, k, x and a fixed pressure at a downstream outlet plane. (c) No-slip condition simulates a solid
wall boundary (e.g., a hull surface) by designating zero value
to velocity components, k, normal pressure gradient, and
treating x following [19]. (d) Slip condition specifies the
normal velocity component and normal gradient of all other
flow quantities (e.g., pressure) as zero. It simulates a symmetry condition on flat boundaries. (e) Interior condition
describes the boundary data interpolated from another grid.
J Mar Sci Technol (2013) 18:310–323
systematic investigations, a grid convergence study is
always suggested. Its purpose is to provide some knowledge and understanding of the numerical error or uncertainty in the computation. It also helps to determine the
minimum grid density (computing expense) for an
acceptable level of accuracy in the computations. As a
preliminary study, grid convergence computations were
carried out for a model of the 2nd variant of the KRISO
Very Large Crude-oil Carrier (KVLCC2) [20]. Its scale
ratio was 1/45.714. Thus the principal dimensions of this
model hull were: length between perpendiculars
LPP = 7.0 m, beam B = 1.269 m, draught T = 0.455 m.
No appendage was attached to the model. The model hull
moved straight ahead at a speed U = 0.530 m/s (corresponding a Froude number Fr = 0.064 and a Reynolds
number Re = 3.697 9 106) in a canal with two surfacepiercing 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
yB = 0.6B.
A schematic diagram of the canal configuration is given
in Fig. 1, indicating that the hull is moving close to the
bank on the port side of the canal. To simplify the computation, effects of waves and sinkage and trim were
assumed negligible. The speed was very low. 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 (inflow),
outlet plane (outflow), hull surface, flat free surface, seabed
boundary, as well as two side banks. The inflow plane was
located at 1.0LPP in front of the hull and the outflow plane
at 1.5LPP behind the hull. The distance between two side
walls was fixed as 10B to exclude the influence from the far
side of the canal on the hull. As for boundary conditions,
the no-slip condition was satisfied on the hull surface (no
wall function was introduced and y? \ 1 was employed);
the inflow/outflow condition was set at the respective inlet/
outlet boundary plane; the slip condition was set at the flat
free surface (z = 0), the seabed and the side walls. For
closing the RANS equations, the EASM turbulence model
was adopted. Figure 2 illustrates the body-fixed and righthanded Cartesian coordinate system used in the computations. The axes x, y, z are directed towards the bow, to
starboard and downwards, respectively. Figure 3 presents a
general view of the grid distribution; the grids are coarsened for better illustration. As can be seen, the overlapping
4 Grid convergence study
In this paper, a wide range of test conditions are to be
considered and computed. For the assessment of the
accuracy in numerical computations, especially in recent
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Fig. 1 Canal configuration
J Mar Sci Technol (2013) 18:310–323
313
Table 1 Grid sizes in grid convergence study
No.
Grid points
hi/h1 (i = 1, 2, …, 6)
Grid1
7943783
1.0
Grid2
4593870
1.189
Grid3
2815261
1.414
Grid4
1707270
1.682
Grid5
1003110
Grid6
605898
2.0
2.378
Fig. 2 Computational domain
Fig. 3 Grid distribution (coarse grids)
grids established by a cylindrical body-fitted H–O grid and
a rectilinear H–H background grid cover the whole computational domain. More details of the grid generation will
be given in the next section.
In the study, the estimation of numerical errors and
uncertainties followed the procedure by Eça et al. [21, 22]. A
pffiffiffi
uniform grid refinement ratio r ¼ 4 2 was utilized in generating the grid series. Six systematically refined grids were
created to enable the curve fit by the least squares root
method, so as to minimize the impact of scatter on the
determination of grid convergence. From the finest (Grid1)
to the coarsest (Grid6) density, the number of grid points are
given in Table 1, where the grid refinement ratio is denoted
by hi/h1. hi is the grid spacing of the ith grid, while h1 is the
grid spacing of the finest grid. CF, CPV and X0 , Y0 , K0 , N0 are
introduced in the error and uncertainty estimation. CF, CPV
represent the frictional and viscous pressure resistance
coefficients; X0 , Y0 , K0 , N0 stand for the non-dimensional
longitudinal and sway force, roll and yaw moment, respectively. In most cases, these forces and moments are essential
quantities in the prediction of ship maneuverability. The
quantities are defined as follows:
RF
RPV
; CPV ¼
;
0:5qU 2 SW
0:5qU 2 SW
X
Y
; Y0 ¼
;
X0 ¼
2
0:5qU LPP T
0:5qU 2 LPP T
K
N
K0 ¼
; N0 ¼
0:5qU 2 LPP T 2
0:5qU 2 L2PP T
CF ¼
where SW is the wetted hull surface area.
ð5Þ
Grid convergence tendencies of CF, CPV and X0 , Y0 , K0 ,
N0 are presented in Fig. 4a–f, together with fitted curves
(including the fits from a theoretical order of accuracy
pth = 2.0) and extrapolated solutions S0. Note that the coarsest
Grid6 is dropped from the curve fit, as it is too coarse and
contaminates the results. As can be seen, the resistance coefficients converge well with an increasing number of grid
points. The fitted curves go through all the solutions. The
convergence of CF is most reasonable (see Fig. 4a). Its convergence rate p (=3.55) is closer to the theoretical one and is
only about half of the convergence rate for CPV. The larger
p value for CPV is probably a combined effect of scatter and
grid independence in solutions of Grid1–Grid4.
The X0 , Y0 forces and K0 moment are more scattered (see
Fig. 4c–e). The X0 and K0 solutions apparently oscillate
around the fitted curves (oscillatory convergence), and are
almost independent of the grid density, producing fitted
curves as straight lines (p ? 0). The Y0 solutions present a
slightly better convergence trend, but still very slow. The
most satisfactory grid convergence appears in the N0
solutions, as the points are all on the fitted curve, and most
importantly, the observed order of accuracy is identical to
the theoretical one, p = pth = 2.0.
The estimated numerical uncertainties are listed in
Table 2 for Grid1, Grid2 and Grid3. Oscillatory convergence
in CPV and X0 , K0 is demonstrated in the uncertainty estimation, since USN fluctuates slightly between Grid1, Grid2
and Grid3. Uncertainties in the other three quantities CF and
Y0 , N0 tend to be converged. Y0 presents a slower convergence
in USN, like the solution tendency shown in Fig. 4d. USN in Y0
drops about 18 % from Grid3 to Grid1, while the reduction in
CF and N0 is 37 and 47 %, respectively.
From this study, it seems very difficult to obtain grid
convergence in bank-effects computations, at least for
extreme cases, as the predicted hydrodynamic quantities
display either fluctuating or slow convergence, and even a
fine grid discretization (i.e., approximately, 8 million 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 of Grid3 can be adopted in
further systematic computations of bank effects.
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314
J Mar Sci Technol (2013) 18:310–323
Fig. 4 Grid convergences of
CF, CPV and X0 , Y0 , K0 , N0
5.5
11.5
CPV (bank)
fitted curve (p = 7.12)
fitted curve (pth= 2.0)
11.0
S0
10.5
CPV ×103
CF ×103
5.0
4.5
4.0
3.5
0.0
-0.052
-0.054
1.0
1.5
9.5
9.0
CF (bank)
fitted curve (p = 3.55)
fitted curve (pth= 2.0)
0.5
10.0
S0
8.5
2.0
2.5
8.0
0.0
3.0
0.5
1.0
1.5
hi /h1
hi /h1
(a) CF
(b) CPV
2.0
2.5
3.0
-0.040
X' (bank)
fitted curve (p →0)
fitted curve (pth= 2.0)
-0.044
Y'
X'
-0.048
-0.056
-0.058
-0.056
Y' (bank)
fitted curve (p =0.56)
fitted curve (pth= 2.0)
-0.060 S0
-0.060
-0.062
0.0
-0.052
0.5
1.0
1.5
2.0
2.5
-0.064
0.0
3.0
0.5
1.0
N'
K'
0.022
-0.058
0.5
1.0
1.5
(e) K'
Table 2 Numerical uncertainties of CF, CPV and X0 , Y0 , K0 , N0
X
0.0009
0
K
N
Y
0
0
p
3.55
7.12
0.56
0.004
2.00
|USN%S|1
7.69
14.53
3.32
24.45
4.95
4.54
|USN%S|2
|USN%S|3
9.29
12.17
14.59
14.45
3.34
3.33
26.21
29.91
4.91
4.90
6.67
8.54
5 Systematic computations
The test case for bank-effects (ship–bank interaction) in the
present paper was set up according to available captive
model test data for a 1/75 scale model of the same hull as
above, the KVLCC2, fitted with a horn-type rudder. The
123
2.0
2.5
3.0
N' (bank)
fitted curve (p =2.00)
fitted curve (pth= 2.0)
0.020
S0
hi /h1
CPV
3.0
0.018
-0.060
CF
2.5
0.024
K' (bank)
fitted curve (p → 0)
fitted curve (pth= 2.0)
-0.056
0
2.0
(d) Y'
-0.054
-0.062
0.0
1.5
hi /h1
hi /h1
(c) X'
2.0
2.5
3.0
0.016
0.0
0.5
1.0
1.5
hi /h1
(f)
N'
principal dimensions of the model were: LPP = 4.267 m,
B = 0.773 m, T = 0.277 m; the body plan represented by
solid lines (dashed lines denote a different but related hull
form, KVLCC1) and the hull geometry with a rudder are
presented in Fig. 5. The tests were performed in the shallow water towing tank at the FHR, in cooperation with the
Maritime Technology Division of Ghent University, Belgium [23]. In the towing tank, straight-line tests (6 knots
full scale) of the KVLCC2 model were conducted at three
different under keel clearances (UKC), namely 50, 35 and
10 % of the draught, at different lateral positions and at a
zero and a non-zero propeller rate with a fitted propeller.
Here we will only use the zero propeller rate data.
For the numerical computations, the test conditions were
determined from the model tests, with the ship moving along
one side of the towing tank at different distances to the bank.
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315
In addition, three water depths, and two canal configurations
with different side-bank geometries were included. The
general plan was to investigate the influence of the water
depth, ship–bank distance and bank geometry on hydrodynamic quantities. The canal configurations in the computations are shown in Fig. 6a and b, where the hull is moving
close to the vertical bank (in Canal A) and the bank with
slope 1:1 (in Canal B) at its starboard side. The non-dimensional ship–bank distance yB is defined as shown below,
following the proposal by, e.g., Ch’ng et al. [5]:
1
B 1
1
¼
þ
ð6Þ
yB 2 yP yS
yP and yS represent the distance from the ship center-plane
to the toe of the bank at the port and starboard side,
respectively. This description thus takes both side banks
into consideration, which is important due to the non-uniform bank geometries.
The computed combinations of ship–bank distance yB
and water depth h/T in Canal A and Canal B are presented
in Table 3. As can be seen, there are six conditions in each
canal, some of which are rather extreme, which puts the
computational tools to a severe test. No waves were considered, since the tanker moved at a low speed:
U = 0.356 m/s. The corresponding Froude number
Fr = 0.055 and the Reynolds number Re = 1.513 9 106.
The double model approximation was adopted, and no
sinkage and trim was allowed. Since a propeller was fitted
even in the resistance tests, the drag of the fixed propeller
has been deducted from the measured resistance.
Systematic computations were first carried out for Canal
A. Thereafter, an analysis was made of the possible modeling errors, and finally computations for the more difficult
Canal B were carried out utilizing the lessons learnt from
Canal A.
5.1 Computational settings
Fig. 5 KVLCC2 geometry with a rudder
(a) Canal A
(b) Canal B
Fig. 6 Cross-sections of Canal A and Canal B, seen in the direction
of motion
Due to the asymmetry of the geometry and the flow field,
the computational domain had to cover the whole hull in
the canal. The computational settings for Canal A and
Canal B were essentially the same. A sketch of the computational domain and the coordinate system is given in
Fig. 7a. As seen in the figure, the computational domain is
made up by seven boundaries like in the grid convergence
study part. The flat free surface is considered at z = 0,
while the seabed and the two side banks are placed at
specific locations corresponding to the test conditions in
Table 3. The coordinate system and the applied boundary
conditions are defined in the grid convergence study. It
should be mentioned that both the EASM and the Menter
k–x SST turbulence model were used for the closure of
RANS equations at different stages of the computations.
Slightly more complicated overlapping grids were
adopted here. As illustrated in Fig. 7b, the overlapping grid
is built up by three components: a cylindrical H–O grid
(fitted to the hull), a curvilinear O–O grid (fitted to the
rudder) and a rectilinear H–H grid (fitted to the canal
boundaries). The former two grids are immersed into the
latter. The body-fitted H–O grid covers the main flow field
around the hull, and two clusters of grid points are
Table 3 Matrix of test conditions
h/T
yB
Canal A
1.180
1.10 (UKC = 10%T)
1.316
1.961
2.431
0.758
s
1.50 (UKC = 50%T)
1.35 (UKC = 35%T)
Canal B
s
s
s
0.909
1.632
2.173
s
s
s
s
s
s
s
s
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316
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r ¼ Z=qgAw
s ¼ M=qgIw
(a) Computational domain and coordinate system
(b) Grid distribution
Fig. 7 Computational domain, coordinate system and grid distribution of Canal A and Canal B
concentrated around the bow and stern regions so as to
resolve the flow field more precisely. A small outer radius
(0.5LPP) of the cylindrical grid is used for Canal A to save
grid points and an even smaller radius (0.12LPP) of the
cylindrical H–O grid is applied for Canal B.
5.2 Results
In this section, results from varying water depths and ship–
bank distances in Canal A and Canal B are presented. In
addition to hydrodynamic forces X0 , Y0 and moments K0 , N0 ,
the mean sinkage and trim are reported. These two variables are of great engineering interest with respect to the
UKC. The mean sinkage r and trim s are defined as:
123
;
ð7Þ
where sinkage is positive downwards and trim positive
bow-up. Z is the sinking force, M the trim moment, Aw the
water plane area and Iw the longitudinal moment of inertia
of the water-plane area about the center of floatation.
In addition to the RANS computations, the steady state
potential flow panel method XPAN in SHIPFLOW was
used to compute the sinkage and trim (without rudder) for
Canal A. The motivation was mainly to study the influence
of neglecting the wave effects in the RANS method. In
XPAN non-linear free surface boundary conditions are
satisfied [14]. It is also of interest to compare the accuracy
in the bank-effects simulation of two completely different
numerical methods.
All the predicted hydrodynamic quantities have been
compared with the test data from FHR. It should be noticed
that the measured data were obtained in a confidential
project, so no quantitative values of the data and computed
results can be presented in the following figures. Instead,
the results for all yB and h/T variations are expressed in
percent of the largest measured value of all variations.
Specifically, the predicted sinkage and trim are normalized
by the maximum measured sinkage and trim, respectively,
the X0 , Y0 forces by the largest |Y0 |, and the K0 , N0 moments
by the largest |K0 |. Absolute values of all maximum quantities are thus used in the normalization. In addition, the
measured negative thrust from the non-rotating propeller is
subtracted from 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 trim in Canal A from the
potential flow method and the RANS method (with the
EASM model) are plotted in Figs. 8 and 9, coupled with
the experimental data. Dashed lines indicate the zero level.
Figure 8 shows the sinkage and trim versus the water depth
ratio h/T at a fixed small ship-to-bank distance yB = 1.316.
The sinkage and trim, in general, rise with 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 yB at a fixed water depth h/
T = 1.35 are presented in Fig. 9. Similar tendencies are
displayed, indicating a blockage from the bank: when the
hull moves closer to the bank, the sinkage and trim
increase.
It is shown in Figs. 8 and 9 that the RANS method
works very well in predicting the sinkage and trim, while
the potential flow method fails to obtain a solution at the
closest ship–bank distance and under-predicts the trim
J Mar Sci Technol (2013) 18:310–323
317
(%) 160
σ (RANS_EASM)
σ (potential)
σ (EFD)
σ %|σmax| , τ %|τmax|
120
80
40
0
-40
-80
τ (RANS_EASM)
τ (potential)
τ (EFD)
-120
-160
1.0
Canal A: yB=1.316
1.1
1.2
1.3
1.4
1.5
1.6
h/T
Fig. 8 Sinkage r and trim s versus h/T in Canal A
(%) 160
σ (RANS_EASM)
σ (potential)
σ (EFD)
Canal A: h / T=1.35
σ %|σmax| , τ %|τmax|
120
80
Fig. 10 RANS predicted axial velocity uX/U contour around the
tanker in the horizontal plane (from top to bottom: z/LPP = 0, -0.032,
-0.06) (Canal A: h/T = 1.1, yB = 1.316)
40
0
-40
-80
τ (RANS_EASM)
τ (potential)
τ (EFD)
-120
-160
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
yB
Fig. 9 Sinkage r and trim s versus yB in Canal A
considerably at all other conditions. The predicted trim is
only half the measured value. This may be due to the fact
that there is significant flow separation along the hull,
which influences the pressure at the stern. Thus, a large
pressure difference between the bow and stern is produced.
A pronounced flow separation around the stern at h/
T = 1.1 and yB = 1.316 is indeed predicted by the RANS
method, as indicated in the non-dimensional axial velocity
uX/U contours in Fig. 10. The potential flow method,
however, is unable to simulate this effect, as verified by the
predicted pressure distribution on the hull surface. Normalizing the pressure by 0.5qU2, pressure coefficient distributions from potential flow and RANS methods are
displayed in Fig. 11. While the potential flow pressure is
relatively symmetrical fore-and-aft, the suction under the
bow has no correspondence under the stern in the RANS
results. A much larger trim by the bow can thus be
expected for this case.
5.2.2 Hydrodynamic forces and moments in Canal A
The predicted forces and moments for Canal A by the
RANS method are shown in the following figures, where
Fig. 11 Pressure distribution on the hull surface (Canal A: h/T = 1.1,
yB = 1.316). (Bottom view) upper potential flow, lower RANS
only the zero level is given. Note that no result from the
potential flow method is presented here, as it is indicated
from the predicted sinkage and trim, as well as the pressure
distribution in Figs. 8-11 that viscous effects cannot be
neglected. Results for the X0 , Y0 forces and the K0 , N0
moments at a specific ship–bank distance, yB = 1.316,
against the water depth ratio are shown in Fig. 12a and b.
Results at a specific water depth ratio h/T = 1.35 against
the ship–bank distance are shown in Fig. 13a and b.
Compared with measurements, the tendencies of
hydrodynamic forces and moments are captured well. As
seen in Fig. 12a and b, when the hull bottom approaches
the seabed, the X0 force and K0 moment increase, while the
Y0 force (a suction force towards the bank) behaves in a
different way. It increases slightly between h/T = 1.5 and
1.35, but drops rapidly between h/T = 1.35 and 1.1. The N0
moment shows a monotonic increase for diminishing water
depth as well. However, magnitudes of the yaw moment
are very small. In Fig. 13a and b, with the varying ship-tobank distance at h/T = 1.35, the hull is attracted to the
bank but the bow is pushed away.
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318
J Mar Sci Technol (2013) 18:310–323
(%) 120
(%) 125
Y' (RANS_EASM)
Y' (EFD)
75
80
50
60
25
0
-25
-50
-75
-100
-125
1.0
X' (RANS_EASM)
X' (EFD)
Canal A: yB =1.316
1.1
1.2
Y' (RANS_EASM)
Y' (EFD)
100
X'%|Y'max| , Y'%|Y'max|
X'%|Y'max| , Y'%|Y'max|
100
1.3
1.4
1.5
40
20
0
-20
-40
-60
-80
-100
1.0
1.6
h/T
(a) X', Y' force
X' (RANS_EASM)
X' (EFD)
Canal A: h /T=1.35
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
yB
(a) X', Y' force
(%) 150
Canal A: yB=1.316
K' (RANS_EASM)
K' (EFD)
(%) 60
75
50
25
0
-25
N' (RANS_EASM)
N' (EFD)
-50
-75
1.0
1.1
1.2
1.3
1.4
1.5
1.6
h/T
(b) K', N' moment
Fig. 12 X0 , Y0 force and K0 , N0 moment versus h/T in Canal A
K' (RANS_EASM)
K' (EFD)
N' (RANS_EASM)
N' (EFD)
50
100
K'%|K'max| , N'%|K'max|
K'%|K'max| , N'%|K'max|
125
40
30
20
10
0
-10
Canal A: h /T=1.35
-20
1.0
1.2
1.4
1.6
1.8
yB
2.0
2.2
2.4
2.6
(b) K', N' moment
Fig. 13 X0 , Y0 force and K0 , N0 moment versus yB in Canal A
While tendencies of the predicted force and moment
correspond well with those of the measured data, the
absolute level is not well predicted in some cases. The most
obvious difference between computations and measurements is seen in the Y0 force, for which there is a more or
less constant shift to lower predicted values. This tendency
is seen clearly in Figs. 12a and 13a. To more quantitatively
investigate the absolute accuracy, a formal validation study
was made for hydrodynamic variables in an extreme condition: h/T = 1.1 and yB = 1.316. The validation follows
the procedure proposed by the ASME V&V 20 2009
standard [24], in which the concepts of a validation
uncertainty Uval (U2val & U2num ? U2D) and a comparison
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 in the measurements. Assuming that the roundoff error in a computation is negligible, the numerical
uncertainty mainly consists of iterative uncertainty UI due
to the lack of convergence in the iteration process and grid
discretization uncertainty USN caused by the limited grid
resolution. According to the procedure, the validation
result is to be interpreted as follows: if |E| B Uval, the
modeling error is within the ‘‘noise level’’ imposed by the
numerical and experimental uncertainty, and not much can
123
be concluded about the source of the error; but if
|E| Uval, the sign and magnitude of E could be used as
to improve the modeling. In the present work, the iterative
uncertainty is quantified by the standard deviation of the
force in per cent of average force over the last 10 % iterations and kept below 0.2 %, which is a negligible contribution to the numerical uncertainty Unum. Therefore,
Unum is approximated as the grid discretization uncertainty
USN in the grid convergence study; while the experimental
data and data uncertainty are available from FHR. Note
that UD is estimated as twice the standard deviation of
measured variables in repeated captive shallow water
model tests at FHR for the SIMMAN 2012 Workshop [25].
In this way, the precision error is accounted for, but not the
bias error.
The estimated uncertainties and comparison errors for
hydrodynamics quantities X0 , Y0 , K0 , N0 are presented in
Table 4, where the measured data (D) are used for normalization. It is seen that three quantities Y0 , K0 and N0 ,
exhibit a larger comparison error than the validation
uncertainty, implying that there are significant modeling
errors, in computations and/or measurements. The neglected bias error in the measured uncertainty corresponds to a
modeling error and has not been investigated in the present
J Mar Sci Technol (2013) 18:310–323
319
There are several potential sources of modeling errors in
the bank-effect computations for Canal A. Examples
include the computational domain size, neglect of free
surface, non-free sinkage and trim, turbulence modeling,
absence of propeller, and boundary condition on the bank/
seabed.
First of all, the selected computational domain is composed of the boundaries of seabed, side banks, flat water
surface, hull surface, inflow and outflow. The boundaries of
flat water surface and hull surface are fixed, while the
positions of seabed and side banks are decided according to
the test conditions. In theory, the inflow and outflow
boundaries should be kept far from the hull to diminish the
effect on the solutions. In the computations presented
above, their locations were defined from early experiences
with 1.0LPP in front of and 1.5LPP behind the hull,
respectively, yielding a total longitudinal size of 3.5LPP. If
these two boundaries are not sufficiently far away from the
hull, they may introduce some error to the results. To
investigate the influence of the domain size, we computed
the shallowest water case (h/T = 1.1 and yB = 1.316) as an
example like in the validation study. With larger longitudinal size (1.5 times of the previous one), the X0 , Y0 forces
and K0 , N0 moments reduced about 2.8%D. Recalling
Fig. 12 and Table 4 of the previous results, the forces and
moments are all under-predicted. The decreased results do
not improve the validation, thus the domain size appears to
be a not so important source of modeling error. Moreover,
a large domain size also increases the grid number and
consumes longer computing time, which is important for a
series of systematic computations. Given these factors, the
previous domain size was considered acceptable.
As mentioned above, the wave effect is neglected in the
RANS computations. Its influence on hydrodynamic
quantities should be investigated. As shown in Figs. 8 and
9, the computation of sinkage and trim in Canal A indicates
that without considering the waves in the RANS
Table 4 Validation results of X0 , Y0 , K0 , N0
X0
Y0
K0
N0
|UD%D|
4.40
18.80
7.10
2.78
|USN%D|
3.27
8.38
3.57
6.92
|Uval%D|
5.48
20.58
7.95
7.46
|E%D|
1.85
71.99
27.19
18.94
(%) 140
X'%|Y'max| , Y'%|Y'max|
5.2.3 Investigation of modeling errors
computations, the predicted sinkage and trim correspond
very well with measurements. This shows that it is
acceptable to neglect the wave effect when predicting the
sinkage and trim at the present low speed and at relatively
close ship–bank distances. However, the effect of the
sinkage and trim on the other forces and moments, especially at small water depths, needs to be investigated. Thus,
further computations with given sinkage and trim from
prior RANS computations were carried out for the case
with a variation of water depth in Canal A at yB = 1.316.
Results including the initial sinkage and trim (r&s) are
indicated by symbols and dotted lines in Fig. 14a and b,
also containing the previous predictions (symbols and solid
lines). From the comparisons, it is interesting to see that the
influence of sinkage and trim is generally very small.
However, it cannot be neglected for the K0 and N0 moments
at the very shallow water depth (h/T = 1.1), where visible
differences in the results are noted. Returning to Table 4, a
quantitative comparison can also be made. The correction
from sinkage and trim increases |Uval%D| for N0 to 8.34 %
and reduces the |E%D| to 7.92 %, while for other quantities
the improvement is not large enough to reduce E to the Uval
level, so there must be other significant modeling issues.
As for the turbulence modeling, the EASM model was
used in the systematic computations, as this turbulence
120
100
80
60
40
20
0
-20
-40
-60
-80
-100
1.0
Y' (RANS_EASM)
Y' (RANS_EASM_ σ&τ)
Y' (EFD)
X' (RANS_EASM)
X' (RANS_EASM_ σ&τ)
X' (EFD)
Canal A: yB=1.316
1.1
1.2
1.3
1.4
1.5
1.6
h/T
(a) X', Y' force
(%) 140
K'%|K'max| , N'%|K'max|
work, and neither has the absence of the side bank in the
precision measurements. However, systematic investigations of the modeling in the numerical computations can be
carried out. This is the topic of the next section.
120
100
80
60
40
20
0
-20
-40
-60
-80
1.0
K' (RANS_EASM)
K' (RANS_EASM_ σ&τ)
K' (EFD)
Canal A: yB=1.316
N' (RANS_EASM)
N' (RANS_EASM_ σ&τ)
N' (EFD)
1.1
1.2
1.3
1.4
1.5
1.6
h/T
(b) K', N' moment
Fig. 14 X0 , Y0 force and K0 , N0 moment versus h/T in Canal A
123
320
J Mar Sci Technol (2013) 18:310–323
model is known to predict more accurate wake profiles in
deep water, see e.g. [26], but in the present case, hydrodynamic quantities are under-predicted with this model.
Therefore, the Menter k–x SST model was applied to
evaluate the influence of the turbulence modeling. The
focus was placed on computations with varying bank distance at the same water depth h/T = 1.35. Results are
presented by dotted lines in Fig. 15a and b. There is a slight
improvement in the prediction of the sway force Y0 (the
maximum decrease in |E%D| is around 8 %), but the
improvement is very small compared to the difference
between computed and measured results. It should be
pointed out that the massive separation mentioned above—
near the stern on the bank side of the hull—may be influenced by the turbulence modeling. The prediction of this
separation is crucial for the prediction of hydrodynamic
forces.
Another modeling error is the absence of the nonrotating propeller that was fitted to the hull in the measurements. In the previous assessment of the computed
longitudinal force X0 the measured thrust was deducted. So,
the influence of the propeller blades on the flow field was
neglected. To look into this, a non-rotating propeller, fixed
behind the hull was investigated. The study was carried out
for the second water depth h/T = 1.35 and closest ship–
X'%|Y'max| , Y'%|Y'max|
(%) 120
100
80
60
40
20
0
-20
-40
-60
-80
-100 Canal A: h/T=1.35
-120
1.0
1.2
1.4
1.6
Y' (RANS_EASM)
Y' (RANS_k-ω SST)
Y' (EFD)
bank distance yB = 1.180 in Canal A. Considering the
improvement in the turbulence modeling above, the k–x
SST model was adopted for these computations.
The propeller was designed by the Maritime and Ocean
Engineering Research Institute (MOERI) in Korea. Its data
are given in Table 5 and the geometry is illustrated in
Fig. 16. However, the angular position of the propeller
blades in the experiment is unknown, so it is impossible to
exactly capture the flow disturbance resulting from the
blades in the experiment. Therefore, four designated positions were considered in the investigation. Taking the ship
center-plane as a reference, the propeller reference line
originally coincided with the ship center-plane (0), and
then its blades were turned clockwise 30, 45 and 60
from this position (see Fig. 17). New computations with
these blades were then performed.
Changes in forces and moments due to the presence of
the propeller are presented in Table 6. It is seen that the
measured negative thrust of the fixed propeller in 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 PR/DR (0.7R)
0.721
Expanded area ratio AE/A0
0.431
Rotation
Hub ratio
Right hand
0.155
Skew ()
21.15
Rake ()
0.0
X' (RANS_EASM)
X' (RANS_k-ω SST)
X' (EFD)
1.8
yB
2.0
2.2
2.4
2.6
(a) X', Y' force
(%) 70
K' (RANS_EASM)
K' (RANS_k-ω SST)
K' (EFD)
N' (RANS_EASM)
N' (RANS_k-ω SST)
N' (EFD)
K'%|K'max| , N'%|K'max|
60
50
40
30
20
10
0
-10
Canal A: h/T=1.35
-20
1.0
1.2
1.4
1.6
1.8
yB
2.0
2.2
2.4
2.6
(b) K', N' moment
Fig. 15 X0 , Y0 force and K0 , N0 moment versus yB in Canal A
123
Fig. 16 Propeller geometry (ship side and back view)
J Mar Sci Technol (2013) 18:310–323
321
Table 6 X0 , Y0 , K0 , N0 changes in experiment and computations
(without and with a propeller)
h/T = 1.35,
yB = 1.180
Dprop
Sw/o
14.49
–
S0
S30
S45
S60
DX0
w/o ? prop.
(%)
(S - D) %D
–
16.47
0.67
0.97
0.66
0.88
2.41
2.72
2.40
2.62
DY0
w/o ? prop.
(%)
Fig. 17 Variation of blade position (0 ? 30 ? 45 ? 60)
–
-2.49
-2.71
-2.64
-2.60
–
–
-2.88
-2.89
-3.12
-2.68
–
–
27.82
34.83
33.22
26.30
DK0
w/o ? prop.
(%)
experiment was 14.49 % of the resistance. However, the
resistance increases in the computations with different
blade positions are all no more than 1 %. This could mean
either that the flow separation at the stern is over-predicted
in the computations, or that the interaction between the
propeller and the hull causes a thrust on the hull, which is
almost as large as the propeller drag. What speaks in favor
of the latter explanation is the fact that X0 without the
propeller is over-predicted by 16.47 %, compared with the
data, if the measured negative propeller thrust is subtracted
from the measured X0 , while the predicted X0 with the
propeller is very close to the measured value without
subtraction of the thrust. The error is then only 2–3 %. On
the other hand, the errors in Y0 , K0 and N0 are much larger
than this, as seen previously, so the first explanation is
more likely: the separation is over-predicted at the stern.
This causes the drag of the propeller to be much smaller
than in the measurements. As seen in Table 6, the different
blade positions cause very similar drag increases. The
increase/decrease of X0 , Y0 and K0 are similar in magnitude,
but the increase in N0 is much larger: about 10 times of the
change in X0 , Y0 or K0 . Referring to Fig. 15b, the small
value of N0 (close to zero) may explain the larger percentage increase.
The last modeling error stated here is that the slip
condition is satisfied at the walls (seabed, side bank) in
computations, but when the hull moves over/along the
seabed/side bank, a boundary layer is developed on these
boundaries due to the disturbance velocity from the hull.
Therefore, the slip condition might be inadequate, and a
moving no-slip condition could be more suitable. Preliminary computations have indicated some effect, but this
will be investigated further.
In this investigation of modeling errors, several possibilities have been tested and most of them demonstrate an
influence on hydrodynamic forces and moments. Although
the magnitude of the influence—from turbulence modeling,
neglect of free surface, non-free sinkage and trim, absence
of propeller—varies, a combination of these contributions
–
DN0
w/o ? prop.
(%)
may yield more accurate solutions. Note that the influence
on individual hydrodynamic quantities is not always uniform: improving the modeling does not always result in an
improved result. Take for instance one of the conditions: h/
T = 1.35 and yB = 1.180 in Canal A. Applying more
appropriate turbulence modeling (k–x SST model) makes
the comparison error E for N0 drop 9 %, and based on this
turbulence model, including a non-rotating propeller yields
a further reduction of 4 %. However the error E of Y0 is
reduced by 21 % with better turbulence modeling, but is
then increased by 7 % including the non-rotating propeller.
5.2.4 Results in Canal B
The numerical predictions of the ship–bank interaction in
Canal A facilitated the computations for the same tanker
moving in Canal B, since the experience with respect to the
turbulence modeling, the inclusion of the effect of sinkage
and trim, as well as the fixed propeller could be utilized.
The computational settings in Canal B were similar to
those in Canal A, as described in Sect. 5.1, and the computations were carried out for varying ship-to-bank distance (at h/T = 1.35). The k–x SST turbulence model was
used and the non-rotating propeller included, but the initial
sinkage and trim were not considered. As clearly shown in
Fig. 14a and b, their influence on the forces and moments
is negligible at the medium water depth (h/T = 1.35).
The sinkage and trim results with varying bank distance
in Canal B are shown in Fig. 18. As can be seen, a smaller
ship–bank distance leads to larger sinkage and trim, indicating a pronounced blockage when the hull approaches the
bank. Like in the results for Canal A, sinkage and trim are
predicted well by the RANS method, the sinkage in particular, as shown in the figure. Trim values are slightly
123
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J Mar Sci Technol (2013) 18:310–323
σ %|σmax| , τ %|τmax|
(%) 150
125
100
75
50
25
0
-25
-50
-75
-100
-125
-150
0.6
σ (RANS_k-ω SST & prop.)
σ (EFD)
Canal B: h/T=1.35
τ (RANS_ k-ω SST & prop.)
τ (EFD)
0.8
1.0
1.2
1.4
1.6
yB
1.8
2.0
2.2
2.4
included in computations, like here, the measured total X0
shall be used for evaluation. Figure 19 shows the forces X0
and Y0 for varying ship–bank distance, and the computed K0 ,
N0 results are available in Fig. 20. The correspondence
between computed X0 results and measured data is good as
are the trends in Y0 force and K0 , N0 moments. However the
absolute accuracy of the latter three quantities is less satisfactory. A slight improvement relative to Canal A may be
noted, probably due to improved modeling, but there are still
significant differences between the measured and computed
results. Note that Canal B should be a more difficult case to
compute with the sloping wall.
Fig. 18 Sinkage r and trim s versus yB in Canal B
6 Conclusions
(%) 120
X' (RANS_ k-ω SST & prop.)
X' (EFD)
Y' (RANS_ k-ω SST & prop.)
Y' (EFD)
X'%|Y'max| , Y'%|Y'max|
100
80
60
40
20
0
-20
-40
-60
-80
0.6
Canal B: h/T=1.35
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
yB
Fig. 19 X0 and Y0 force versus yB in Canal B
K'%|K'max| , N'%|K'max|
(%) 80
70
K' (RANS_ k-ω SST & prop.)
K' (EFD)
60
N' (RANS_ k-ω SST & prop.)
N' (EFD)
50
40
30
20
10
0
-10
-20
0.6
Canal B: h/T=1.35
0.8
1.0
1.2
1.4
1.6
yB
1.8
2.0
2.2
2.4
Fig. 20 K0 , N0 moment versus yB in Canal B
under-predicted, which may be due to the over-predicted
flow separation, typically at the most extreme condition
(closest ship–bank distance).
Results for forces and moments are given in Figs. 19 and
20. Again, the hydrodynamic variable of the primary interest
is the longitudinal force X0 . As mentioned above, to evaluate
the predicted X0 in the absence of a non-rotating propeller, the
measured propeller force is deducted. When the propeller is
123
Investigating bank effects is a challenging task due to the
importance and complexity of the hydrodynamic problem
itself and the difficulty in modeling the physical problem
by numerical tools. The present work aims at investigating
the accuracy achievable in the prediction of several
hydrodynamic forces and moments of interest in maneuvering. Great emphasis has been placed on uncertainty
assessment and numerical and physical modeling issues.
A grid convergence study was performed first to obtain
the numerical uncertainty in the computations for varying
grid densities. The study illustrated the difficulty in
achieving converged results in bank-effects computations,
but it also indicated a suitable grid density for the problem
at hand.
In a series of systematic computations the sinkage, trim
and hydrodynamic forces and moments for varying water
depths, ship-to-bank distances and bank geometries were
then predicted. Very accurate results for sinkage and trim
were demonstrated using a RANS method, while a potential flow method predicted trim much less accurately. For
other quantities the RANS method predicted the trends
with varying bank distance and bottom clearance quite
well, but the absolute accuracy was less satisfactory. A
formal validation analysis showed that there are modeling
errors in the computations and/or the measurements. Five
types of computational modeling errors, i.e., computational
domain size, neglect of waves, non-free sinkage and trim,
turbulence modeling, and absence of propeller were discussed. It was shown that the applied domain size is
acceptable and that the waves at the low speed of interest
have a negligible influence on the results, while the effect
of sinkage and trim cannot be neglected at the shallowest
water depth. Including the non-rotating propeller improved
the predicted X0 force significantly, but the changes in other
forces/moments were minor. As for the turbulence modeling, the k–x SST model produced slight improvement in
comparison with the EASM model.
J Mar Sci Technol (2013) 18:310–323
In further studies it would be of interest to investigate
the effect of the no-slip condition on the bank walls and
bottom. Also, measurements of the wake flow would be
most valuable. Strong separation of the flow occurs in the
most severe conditions close to the bank and bottom, and it
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 Mr. 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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