urans simulations of static and dynamic maneuvering for ......maneuvering simulation is possible but...
TRANSCRIPT
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ORIGINAL ARTICLE
URANS simulations of static and dynamic maneuveringfor surface combatant: part 1. Verification and validationfor forces, moment, and hydrodynamic derivatives
Nobuaki Sakamoto • Pablo M. Carrica •
Frederick Stern
Received: 18 August 2010 / Accepted: 3 May 2012 / Published online: 16 June 2012
� JASNAOE 2012
Abstract Part 1 of this two-part paper presents the verifi-
cation and validation results of forces and moment coeffi-
cients, hydrodynamic derivatives, and reconstructions of
forces and moment coefficients from resultant hydrody-
namic derivatives for a surface combatant Model 5415 bare
hull under static and dynamic planar motion mechanism
simulations. Unsteady Reynolds averaged Navier–Stokes
(URANS) computations are carried out by a general purpose
URANS/detached eddy simulation research code CFDShip-
Iowa Ver. 4. The objective of this research is to investigate
the capability of the code in regards to the computational
fluid dynamics based maneuvering prediction method. In the
current study, the ship is subjected to static drift, steady turn,
pure sway, pure yaw, and combined yaw and drift motions at
Froude number 0.28. The results are analyzed in view of: (1)
the verification for iterative, grid, and time-step convergence
along with assessment of overall numerical uncertainty; and
(2) validations for forces and moment coefficients, hydro-
dynamic derivatives, and reconstruction of forces and
moment coefficients from resultant hydrodynamic deriva-
tives together with the available experimental data. Part 2
provides the validation for flow features with the experi-
mental data as well as investigations for flow physics, e.g.,
flow separation, three dimensional vortical structure, and
reconstructed local flows.
Keywords URANS � PMM � Verification and validation
1 Introduction
In recognition of the importance of ship maneuverability as
a major factor for navigational safety the International
Maritime Organization (IMO) has developed Standards for
Ship Maneuverability [1]. Meeting these standards has
placed greater emphasis on maneuvering prediction meth-
ods, which historically have been more empirical than those
developed for resistance, propulsion, and seakeeping [2].
Among several methods for maneuvering prediction, static
and dynamic planar motion mechanism (PMM) tests are
one of the most commonly used approaches. They provide
hydrodynamic derivatives by focusing on the creation of a
mathematical model. The PMM tests can be feasible in a
conventional towing tank equipped with a PMM motion
generator or a basin with rotating arm capability. However,
the tests contain several disadvantages; (1) expensive test
facilities and complexity in the experimental settings; (2)
considerable scale effect arising from the impossibility in
practice to achieve Froude number (Fn) and Reynolds
number (Rn) similarities simultaneously; and (3) limitations
in obtaining physical understanding of flow fields around a
ship in maneuvering motions.
Computational fluid dynamics (CFD) based maneuvering
prediction methods significantly contribute to resolve these
disadvantages. Since the viscous effects are very important
for accurate maneuvering prediction, unsteady Reynolds
averaged Navier–Stokes (URANS) simulation and detached
eddy simulation (DES) have been considered to be the most
promising approach rather than inviscid approaches. The
URANS/DES simulations replace the static and dynamic
PMM experiments to obtain hydrodynamic derivatives and
N. Sakamoto � P. M. Carrica � F. Stern (&)IIHR-Hydroscience and Engineering, C. Maxwell Stanley
Hydraulics Laboratory, The University of Iowa, Iowa,
IA 52242-1585, USA
e-mail: [email protected]
Present Address:N. Sakamoto
National Maritime Research Institute, 6-38-1 Shinkawa,
Mitaka, Tokyo 181-0004, Japan
123
J Mar Sci Technol (2012) 17:422–445
DOI 10.1007/s00773-012-0178-x
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provide detail local flow physics around the hull under
maneuvering motions.
The objective of this research is to investigate the
capability of general purpose URANS/DES research code
CFDShip-Iowa Ver. 4 [3–5] simulating surface combatant
Model 5415 under static and dynamic PMM tests. Part 1 of
this two-part paper presents the verification and validation
(V&V) results of forces and moment coefficients, valida-
tion for hydrodynamic derivatives, and reconstructions of
forces and moment coefficients from resultant hydrody-
namic derivatives. Overall results in the present study are
extensive [6], and, thus, the most important outcomes
are presented herein and at SIMMAN 2008 [7] by IIHR-
Hydroscience and Engineering [8] for the CFD-based
method. Part 2 provides the detailed validation for flow fea-
tures with the experimental data [9] as well as investigations
for flow physics, e.g., flow separation, three dimensional
vortical structure and reconstructed local flows [10].
1.1 CFD-based maneuvering prediction
at SIMMAN 2008
For KVLCC1&2 tankers, Broglia et al. [11] perform
dynamic PMM simulations with steering rudder and body-
force propeller. They figure out that the stern region is
more effective in producing lateral hydrodynamic force.
Toxopeus and Lee [12] and Cura Hochbaum et al. [13]
show that the hydrodynamic derivatives determined from
URANS simulations are able to predict ship trajectories
with enough accuracy when they are compared with the
free sailing data. Carrica and Stern [14] demonstrate the
capability of the URANS/DES method with moving rudder
and discretized rotating propeller to simulate full time
domain maneuvers.
For a KCS container ship, Simonsen and Stern [15]
perform a V&V study for the forces and moment coeffi-
cients for pure yaw motion. Due to the relative small grid
refinement ratio and having two degrees of freedom (heave
and pitch), they have difficulties in applying the verifica-
tion method [16] to the time series of forces and moment
coefficients.
For a Model 5415 naval surface combatant, Sakamoto
et al. [8] and Guilmineau et al. [17] perform static and
dynamic PMM simulations for the bare hull. Sakamoto
et al. [8] identify the vortical structures around the hull, and
Guilmineau al. [17] show better resolution of local flow
around vortex cores. Miller [18] uses both bare and fully
appended hulls for static/dynamic PMM simulations, and
shows that the errors of forces and moment coefficients in
fully appended hull is greater than the bare hull results.
Carrica et al. [19] perform full time domain URANS/DES
maneuvering simulations in calm water and in waves with
moving rudder and body-force propeller, showing detail
vortical structures around appendages during the
maneuvers.
1.2 Conclusion from past research
Reviews for the SIMMAN 2008 [7, 20] lists several pre-
liminary conclusions and issues for the CFD-based
maneuvering prediction method: (1) number of study for
surface combatant is much less than commercial type
ships; (2) grid, turbulence model, and inclusion of free
surface may play important roles to predict forces,
moment, and local flow quantities; (3) blockage effect may
not be negligible for ships at larger amplitude PMM tests;
(4) hydrodynamic derivatives are usually not computed
from resultant forces and moment coefficients, although a
few cases show that URANS methods can accurately pre-
dict linear hydrodynamic derivatives; (5) attention is not
paid to evaluation of non-linear and cross-coupling deriv-
atives in most of the cases; (6) local flow physics are
analyzed in limited cases, and no systematic validations
together with the experimental fluid dynamics (EFD) data
are made, and; (7) full time domain URANS/DES
maneuvering simulation is possible but still challenging,
thus, the practical approach is to combine viscous CFD
simulations of static/dynamic PMM tests with systems-
based maneuvering simulation.
2 Test overviews
2.1 Geometry
The geometry used in the current study is the David Taylor
Model Basin (DTMB) Model 5512 (the length between
perpendicular Lpp = 3.048 m), which is the preliminary
design for a surface combatant ca. 1980 and a geosym of
the larger Model 5415 (Lpp = 5.73 m). Model 5415 have
been chosen as one of the benchmark hulls by the Inter-
national Towing Tank Conference (ITTC) Resistance
Committee [21, 22] and Maneuvering Committee [2]. It
has been used in several ship hydrodynamics workshops
[23, 24] and is also adopted in SIMMAN 2008. The model
used in the current study does not have appendages but
with fitted bilge keels at port and starboard.
2.2 Static and dynamic PMM tests
Figure 1 describes the coordinate system utilized for cur-
rent static and dynamic PMM simulations. In the figures,
xE and yE denote the XY-plane in the earth-fixed system,
and xs and ys denote the XY-plane in the ship-fixed system.
The rest of the nomenclature is defined in a later part of this
section. The coordinate system is different from what is
J Mar Sci Technol (2012) 17:422–445 423
123
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usually leveraged in the maneuvering field in that the
positive direction of x is from forward perpendicular (FP)
to aft perpendicular (AP). This is due to the fact that the
positive direction of xE is set to be identical to the direction
of free-stream incoming velocity to the ship. Positive
direction of y is pointing from port to starboard, and thus
the direction of rotation is counter-clockwise in accordance
with the right-handed coordinate system.
During the static drift test, the model is towed in a
conventional towing tank at a constant velocity U0 with the
initial drift angle b relative to the ship’s axis.During the steady turn test, a yaw angular velocity r is
imposed on the model by fixing it to the end of a radial arm and
rotating the arm with its length R about a vertical axis fixed in
the tank [25]. The yaw angular velocity r is given by
r ¼ U0R
ð1Þ
During the pure sway test, the ship axis is always parallel
to the tank centerline and the model is given sway position
y, sway velocity v and sway acceleration _v as a function oftime
yv_v
2435 ¼
�ymax sinðxtÞ�vmax cosðxtÞ
_vmax sinðxtÞ
24
35 ð2Þ
where x; is the angular frequency of sway motion, vmax ismaximum sway velocity, and _vmax is maximum sway
acceleration. The corresponding drift angle bcorr. of flowrelative to the ship is defined as
bcorr: ¼ tan�1v
U0
� �: ð3Þ
During the pure yaw test, the ship is towed down the tank
with the ship axis always tangent to its path. The model is
given not only sway position, velocity, and acceleration by
Eq. (2), but also yaw angle w, yaw angular velocity r, andyaw angular acceleration _r as a function of time
wr_r
24
35 ¼
�wmax cosðxtÞ_wmax sinðxtÞ€wmax cosðxtÞ
24
35 ð4Þ
where x; is the angular frequency of yaw motion which isequal to the angular frequency of the sway motion, wmax is
maximum yaw amplitude, _wmax is maximum yaw angular
velocity, and €wmax is maximum yaw angular acceleration.During the combined yaw and drift test, the ship is given
r and _r as a function of time by Eq. (4) with constant drift
angle b thus the ship axis is not always tangent to itstowing path.
The computational results of surge force X, sway force
Y, and yaw moment N are subjected to the validation with
the available experimental data [7, 9].
3 Computational method
3.1 Modeling
The CFD solver utilizes an absolute/relative inertial coor-
dinate system and a non-inertial ship-fixed coordinate
system to describe prescribed/predicted ship motions [5].
The flow field is solved in the absolute/relative inertial
coordinate system while the ship motions are solved in the
non-inertial ship-fixed coordinate system. The code solves
an incompressible URANS equation with a single-phase
level-set method as a free surface modeling, and isotropic
blended k - e/k - x (BKW) model or BKW-based alge-braic Reynolds stress (ARS) model with DES option as a
turbulence modeling [3, 26].
All governing equations are made non-dimensional by
U0, Lpp, fluid density q, gravitational acceleration g, andthe dynamic viscosity l which yield the definitions in Fn ¼U0� ffiffiffiffiffiffiffiffiffi
gLppp
and in Rn ¼ qU0Lpp�l. This provides the fol-
lowing non-dimensionalization in v; _v; r; _r, X, Y, and N as
v0
_v0
� �¼
vU0_v
U0
� �ð5Þ
Fig. 1 Coordinate system of the static and dynamic PMM simula-tions (notice that the positive direction in x is from FP to AP)
424 J Mar Sci Technol (2012) 17:422–445
123
-
r0
_r0
� �¼
rLppU0
rLppU0
� 224
35 ð6Þ
X0
Y0
N0
24
35 ¼
X0:5qU2
0TmLpp
Y0:5qU2
0TmLpp
N0:5qU2
0TmL2pp
2664
3775 ð7Þ
where Tm is a draft of a model ship in full-load condition.
3.2 Numerical methods and high-performance
computing
A second-order Euler backward difference is used for a
temporal discretization of all variables. The finite-differ-
ence method is utilized for a spatial discretization, e.g., a
second-order upwind scheme (FD2) or second-order total
variation diminishing with ‘‘superbee’’ (TVD2S) scheme
[27] in momentum convection, a first-order upwind scheme
in turbulence convection, and ahybrid first and second-
order upwind scheme in the level-set convection. The
viscous terms in momentum and turbulence equations are
computed using a second-order central difference scheme.
The pressure implicit split operator (PISO) algorithm is
used to couple the momentum and continuity equations.
The code is made parallel using message passing interface
(MPI) with a domain decomposition technique.
Overset grid technique is adopted to simulate dynamic
ship motions and local grid refinements [28]. Figure 1a, b
show the typical overset grid arrangement in the current
study. The external software SUGGAR is used to obtain
the grid connectivity between overlapping grids. It runs as
a separate process from the flow solver, and is called every
time when the ship motions are prescribed in time to pro-
vide the interpolation information between the overset
grids to the flow solver [4]. Another preprocessing software
USURP [29] provides weights to the active points on the
overlapped region of no-slip surfaces, e.g., between the
hull grid and the bilge-keel grid in the present study. This
avoids counting the same area in space more than once,
and, thus, the flow solver is able to calculate the correct
area, forces, and moments.
4 Simulation design
4.1 Test cases
Table 1 summarizes the test cases presented in this article.
In all the cases Rn is 4.67 9 106 and Fn is 0.28. The hull
configuration are either fixed at even-keel (FX0) or fixed at
sunk and trimmed (FXrs). The CoG is set to (x/Lpp, y/Lpp,
z/Lpp) = (0.5, 0, -0.004) for static PMM simulations
(e.g., static drift and steady turn) and (x/Lpp, y/Lpp, z/Lpp) =
(0.50515, 0, -0.004) for dynamic PMM simulations (e.g.,
pure sway, pure yaw and combined yaw. and drift),
respectively. In both static and dynamic PMM simulations,
the center of rotation (CoR) is set to (x/Lpp, y/Lpp, z/Lpp) =
(0.5, 0, -0.00208) where the yaw moment around the
z-axis is computed at the location. At first, all the simula-
tions except steady turn are performed with side walls since
it is considered to be important to accurately reproduce the
experimental condition. However, the time histories of
forces and moment coefficients for the simulations with
side walls show very large and slowly damped oscillation
due to the spurious waves which are partially reflected by
the upstream, downstream, and side wall boundaries which
yields slow convergence [6]. Therefore, the simulations
without walls are carried out.
4.2 Grid, domain size and time step
Figure 2a–d present the overview of the computational
grids, their domain size and boundary conditions. Table 2
summarizes the size of the fine grids. The commercial
software GRIDGEN� with hyperbolic extrusion for the
curvilinear grids is used to generate all the grids. At the
solid surfaces the first grid point is set at yþ\1 as requiredby the k - e/k - x turbulence model. The grid 1 is ini-tially designed to include the side walls of the IIHR towing
tank. The grid 10 is prepared due to the necessity ofincreasing boundary layer and free surface resolution fol-
lowed by the result of the straight ahead simulation with
grid 1 [6], still maintaining the same domain size as grid 1.
To exclude the side walls, the grid 1NW/grid 10NW is
designed whose base structure is the same as grid 1/grid 10
but the distance from the centerline to side wall boundaries
is 20 times larger. The medium and coarse grids for veri-
fication study are coarsened from fine grids using non-
integer refinement ratioffiffiffi2p
. Prescribed sway/yaw motions
are applied to all the blocks except the outer boundary
which remains stationary during the dynamic PMM
simulations.
For the static PMM simulations, the non-dimensional
time step is set to Dt ¼ 0:01. For dynamic PMM simula-tions, Dt2 ¼ 0:00979 is used which allows 384 time stepsper one sway/yaw period. For the verification study, sys-
tematically refined time steps with refinement ratio 2 are
used, resulting in Dt1 ¼ 0:00489 and Dt3 ¼ 0:01957.
4.3 Boundary conditions
The boundary conditions utilized in the current study are
inlet, outlet, no-slip, and far-field conditions for which their
mathematical descriptions can be found in Carrica et al. [3]
J Mar Sci Technol (2012) 17:422–445 425
123
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Ta
ble
1T
est
mat
rix
for
the
stat
ican
dd
yn
amic
PM
Msi
mu
lati
on
s
#G
rid
Co
nd
itio
nP
resc
.6
DO
FC
on
v.
sch
m.
turb
.m
od
el
EF
Dd
ata
Pu
rpo
se
Par
am.
Val
ue
Sta
tic
dri
ft
1.1
10 ,
20 ,
30
b1
0�
–F
D2
-BK
WX0 ,
Y0 ,
N0
V&
Vw
.w
alls
1.2
10 N
W,
20 N
W,
30 N
WF
D2
-BK
WV
&V
w.o
.w
alls
1.3
30 N
Wb
0�,
2�,
6�,
9�,
10
�,1
1�,
12�,
16
�,2
0�
TV
D2
S-A
RS
X0 ,
Y0 ,
N0 ,
Xvv,
Yv,
Yvvv,
Nv,
Nvvv
Fo
rces
and
mo
men
tco
effi
cien
ts,
hy
dro
dy
nam
icd
eriv
ativ
es
Ste
ady
turn
(Ris
no
n-d
imen
sio
nal
by
Lpp.)
2.1
30 N
WR
1.6
7,
3.3
3,
6.6
7S
urg
e,sw
ay,
yaw
TV
D2
S-A
RS
X0 ,
Y0 ,
N0 ,
Xrr
,Y
r,Y
rrr,
Nr,
Nrr
r
Fo
rces
and
mo
men
tco
effi
cien
ts,
hy
dro
dy
nam
icd
eriv
ativ
es
Pu
resw
ay
3.1
30 N
Wb
max
2�,
4�,
10
�S
way
TV
D2
S-A
RS
X0 ,
Y0 ,
N0 ,
Xvv,
Y_v,
Yv,
Yvvv,
N_ v,
Nv,
Nvvv
Fo
rces
and
mo
men
tco
effi
cien
ts,
hy
dro
dy
nam
icd
eriv
ativ
es
Pu
rey
aw
4.1
1,
2,
3r0 m
ax0
.3S
way
,y
awF
D2
-BK
WX0 ,
Y0 ,
N0
V&
Vw
.w
alls
4.2
10 N
W,
20 N
W,
30 N
W0
.15
,0
.3,
0.6
V&
Vw
.o.
wal
ls
4.3
30 N
WT
VD
2S
-AR
SX0 ,
Y0 ,
N0 ,
Xrr
,Y
_ r,Y
r,Y
rrr,
N_ r,
Nr,
Nrr
r
Fo
rces
and
mo
men
tco
effi
cien
ts,
hy
dro
dy
nam
icd
eriv
ativ
es
Co
mb
ined
yaw
and
dri
ft
5.1
30 N
Wb
9�,
10
�,1
1�
Dri
ft,
sway
,y
awT
VD
2S
-AR
SX0 ,
Y0 ,
N0 ,
Xvr,
Yvrr
,Y
rvv,
Nvrr
,N
rvv
Fo
rces
and
mo
men
tco
effi
cien
ts,
hy
dro
dy
nam
icd
eriv
ativ
es
426 J Mar Sci Technol (2012) 17:422–445
123
-
and Paterson et al. [30]. In the absolute inertial coordinate
system for steady turn simulation, the ship’s surge and
sway coordinate values are prescribed with a constant time
interval as well as the velocity components on the no-slip
surface. Since the ship moves with the prescribed velocity,
the velocity components at the inlet boundary are all zero.
In the relative inertial coordinate system for static drift and
all the dynamic PMM simulations, the surge motion is not
imposed to the ship thus the velocity components at inlet
are ðU;V ;WÞ ¼ ð1; 0; 0Þ. For straight ahead and static driftsimulations, no motions are prescribed thus the velocity
components at no-slip surface are ðU;V ;WÞ ¼ ð0; 0; 0Þ.For the dynamic PMM simulations, the ship has prescribed
lateral velocity by sway motion and linear components of
axial and lateral velocity due to yaw motion. They are
brought into the U and V-components of no-slip condition.
4.4 Analysis method
4.4.1 Fourier analysis
For the dynamic PMM tests, the sway and yaw motions are
prescribed by sine and cosine functions, thus, the response
of forces and moment coefficients are assumed to be
reconstructed as a Fourier series (FS) with the non-
dimensional angular frequency of sway/yaw motion
xð¼ 2pLpp�
TPMMU0Þ, TPMM is a dimensional period ofprescribed sway/yaw motion period, as
Fig. 2 Grid and boundary conditions: a Grid capable for dynamic motions, b Grid for the ship fixed at sunk and trimmed, c Boundary conditionswith walls, d Boundary conditions without walls
J Mar Sci Technol (2012) 17:422–445 427
123
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FðtÞ ¼ a0 þX1n¼1
an cosðnxtÞþX1n¼1
bn sinðnxtÞ ð8Þ
where F(t) represents the time series of forces and
moment coefficients, an and bn is nth-order Fourier sine
and cosine coefficients, respectively. Equation (5) can be
re-written as
FðtÞ ¼ a0 þX1n¼1
A0 cosðnxt þ unÞ
with An ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2n þ b2n
q; un ¼ tan�1ð�an=bnÞ
ð9Þ
where An; is the nth-order Fourier cosine harmonics and unis the phase angle. Equation (9) is used to evaluate the
iterative error for the harmonics of the forces and moment
coefficients from the dynamic PMM simulations by
marching harmonic analysis [9].
4.4.2 Hydrodynamic derivatives
The hydrodynamic derivatives from static and dynamic
PMM tests are calculated based on the Abkowitz-type
mathematical model [31] with three degrees of freedom
(3DOF), e.g., surge, sway, and yaw. The model describes
the forces and moment coefficients at ship-fixed coordinate
system with bare hull condition as
The hydrodynamic derivatives in Eq. (10) are of the
interest in the current study. Taking the results of static
drift and pure sway cases as an example, the
hydrodynamic derivatives can be calculated as follows.
Notice that the LHS and independent variables in the RHS
are all non-dimensional as defined in [8, 9], and thus the
resultant hydrodynamic derivatives are non-dimensional
as well.
For the static drift results, the forces and moment
coefficients are the function of v thus the right hand side
(RHS) of Eq. (10) are simplified as
X0
Y0
N0
264
375¼
AþBv02
Cv0 þDv03
Ev0 þFv03
264
375
with A¼ X�;B¼ Xvv;C ¼ Yv;D¼ Yvvv;E ¼ Nv;F ¼ Nvvv:ð11Þ
The hydrodynamic derivatives shown in Eq. (11) are
obtained as polynomial coefficients by the least-square
curve fitting method.
For the pure sway results, the resultant forces and
moment coefficients are the function of both v’ and _v0 thus,the RHS of Eq. (11) are simplified as
X0
Y0
N0
24
35 ¼
Aþ Bv02G _v
0 þ Cv0 þ Dv03H _v
0 þ Ev0 þ Fv03
24
35 with G ¼ Y _v; H ¼ N _v
ð12Þ
Using v0 and _v0
representation as Eqs. (2) and (5) to
Eq. (12) results in the FS representation of the forces and
moment coefficients as
Table 2 Description of fine grids
Block name Block type Fine 1 Fine 1NW Fine 10 Fine 10NW
Blocks Grid points Blocks Grid points Blocks Grid points Blocks Grid points
Boundary layer Body-fitted 4 0.51 M 4 0.51 M 12 1.52 M 16 4.30 M
Bilge keel Body-fitted 4 0.51 M 4 0.51 M 4 0.51 M 8 1.44 M
1st refinement Orthogonal 8 1.0 M 8 1.03 M 16 1.95 M 24 5.52 M
2nd refinement Body-fitted 12 1.49 M 12 1.49 M – – – –
2nd refinement Orthogonal – – – – – – – –
Background Orthogonal 4 0.51 M 8 1.01 M 8 1.49 M 36 8.74 M
Total 32 4.02 M 36 4.55 M 40 5.47 M 84 20 M
Domain size -8.6 B x/Lpp B 8.6
-0.5 B y/Lpp B 0.5
-1.0 B z/Lpp B 0.25
-8.6 B x/Lpp B 8.6
-10.0 B y/Lpp B 10.0
-1.0 B z/Lpp B 0.25
Same as fine 1 Same as fine 1NW
X0
Y0
N0
24
35 ¼
X� þ Xvvv02 þ Xrrr
02 þ Xvrv0r0
Y _r _r0 þ Y _v _v
0 þ Yvv0 þ Yvvvv
03 þ Yvrrv0r02 þ Yrr
0 þ Yrrrr03 þ Yrvvr
0v02
N _r _r0 þ N _v _v
0 þ Nv _v0 þ Nvvvv
03 þ Nvrrv0r02 þ Nrr
0 þ Nrrrr03 þ Nrvvr
0v02
24
35: ð10Þ
428 J Mar Sci Technol (2012) 17:422–445
123
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Fourier sine and cosine coefficients are associated with
Eq. (13) as
X0X2;cos
� �¼ � 1
12
v02max
0 12
v02max
� �AB
� �ð14Þ
Y1;sinN1;sin
� �¼ � G
H
� �_v0
max ð15Þ
Y1;cosN1;cos
� �¼ � C
34
DE 3
4F
� �v0max
v03max
� �ð16Þ
Y3;cosN3;cos
� �¼ � 1
4
DF
� �v03max: ð17Þ
To calculate hydrodynamic derivatives from the results
of pure sway tests, there are two approaches, e.g., (1) linear
and non-linear curve fitting methods (LCF and NLCF,
respectively), and (2) single run method (SR). For the CF
methods, Fourier sine and cosine coefficients of forces and
moment coefficients are obtained from multiple pure sway
tests, and the polynomial functions with respect to v0max or
_v0
max are used to calculate the hydrodynamic derivatives
with least-square curve fitting. For the SR method, solving
Eqs. (14–17) algebraically, hydrodynamic derivatives
respect to v0 and _v0 are calculated from a single result ofa pure sway test.
For the rest of hydrodynamic derivatives from the other
static and dynamic PMM tests, they are calculated fol-
lowing the similar manner as explained above [6].
4.4.3 Reconstruction of forces and moment coefficients
The current ‘‘reconstruction’’ approach evaluates how
well the mathematical model with hydrodynamic
derivatives can reproduce originally computed forces
and moment coefficients instead of performing trajec-
tory simulations by resultant hydrodynamic derivatives
[32, 33]. The ‘‘reconstruction’’procedure is given as
follows taking static drift and the pure sway cases as
examples.
For the static drift, Eq. (9) can reproduce the forces and
moment coefficients with the hydrodynamic derivatives.
Reconstructed computational results are termed SR. Sim-
ilarly for the pure sway, the time history of the forces and
moment coefficients over 1 sway period can be repro-
duced by Eq. (10) with the hydrodynamic derivatives.
Reconstructed computational results are instantaneous,
and they are termed SiR. The comparison error between
the experimental data and SR or SiR is the only available
one, and for the current cases, the most acceptable mea-
sure to evaluate the quality of the hydrodynamic deriva-
tives from the CFD simulations.
4.4.4 Definition of comparison error
The comparison error E between the computational results
S and the experimental data D is defined as
Eð%DÞ ¼ D� SD� 100: ð18Þ
The error definition by Eq. (18) is used when the
computational results of forces and moment coefficients
from static drift tests and hydrodynamic derivatives from
all the cases are compared with the experimental data. For
the forces and moment coefficients from dynamic PMM
tests, it is difficult to apply Eq. (18) since the profiles of Y0
and N0 cross 0 at certain planar motion phases. To avoid
this, the average comparison error EX;Y ;N over 1 planar
motion period is defined as
EX;Y ;Nð% Dj jÞ ¼1N
PNi¼1 Di � Sij j
1N
PNi¼1 Dij j
� 100 ð19Þ
where N is the total number of experimental data points,
Di is the instantaneous experimental data, and Si is
the instantaneous computational results. To match the
instantaneous time between the simulation and the
experiment, the computational results are subjected to
cubic spline interpolation before the error is calculated.
The average reconstruction error for the experimental data
EREFD is defined following a similar manner, where Si; is
replaced by DR, the instantaneous and reconstructed
experimental data. The average comparison error between
the reconstructed computational results and the original
experimental data ERCFD is also defined using Eqs. (18) or
(19) where S or Si is replaced by SR or SiR. The total
average comparison error EAve: for all forces and moment
coefficients is defined as
EAve: ¼EX þ Ey þ EN
3: ð20Þ
X0
Y0
N0
24
35 ¼
Aþ 12
Bv02max þ 12 Bv
02max cosð2xtÞ
G _vmax sinðxtÞ � Cv0max þ 34 Dv
03max
�cosðxtÞ � 1
4Dv
03max cosð3xtÞ
H _vmax sinðxtÞ � Ev0
max þ 34 Fv03max
�cosðxtÞ � 1
4Fv
03max cosð3xtÞ
264
375: ð13Þ
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5 Uncertainty analysis for forces and moment
coefficients
5.1 Verification and validation procedure
Uncertainty analysis is performed using the V&V method
following the procedure by Stern et al. [16] with improved
factor of safety [34]. Verification procedures identify the
most important numerical error sources such as iterative
error dI, grid size error dG and time-step error dT andprovide error and estimates of simulation numerical
uncertainty USN.
The forces and moment coefficients are subjected to the
uncertainty analysis in the current study. The iterative
uncertainty UI is estimated for all the cases. The grid
uncertainty UG is estimated for the static drift cases at
b = 10� with and without walls. Both the UG and the timestep uncertainty UT is estimated for the pure yaw cases at
rmax = 0.3 with and without walls. For the static drift and
the pure yaw cases, the USN and the validation uncertainty
UV is also estimated utilizing the experimental uncertainty
UD.
5.2 Iterative convergence
5.2.1 Dependency for inner iteration in pure yaw
The pure yaw is selected to study the dI depending on thenumber of iterations to couple the non-linear terms in
turbulence and momentum equations (termed inner iter-
ation hereafter). The error is evaluated by performing
three simulations using medium grid (grid 20) with med-ium time step (Dt2), and changing the number of inneriterations from 3 to 4 to 6. Using the mean of longitudinal
force coefficient (X0) and most dominant harmonics (X2,
Y1 and N1), the solution changes of harmonics (DFS)based on the solution with inner iteration 6 are computed.
The DFS for all the harmonic amplitudes are at least oneorder of magnitude smaller than the UI which will be
discussed in the next section, and thus the iterative error
depending on the number of inner iteration is considered
to be negligible. In all the cases, the number of inner
iteration is 4.
5.2.2 Solution iterative convergence
5.2.2.1 Static drift and steady turn Two quantities are
extracted from the time history of the forces and moment
coefficients to study the statistical convergence, e.g., (1)
the running mean (RM), and (2) the magnitude of root
mean square of organized oscillation (RMSo) defined as
RMSo ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi¼1 R
2i
N
sð21Þ
where N is a total number of data points and Ri is the
instantaneous forces and moment coefficients. The average
of maximum and minimum RM is considered UI. Notice
that the quantities of RMSo and UI referenced in this sec-
tion are extracted from Sakamoto [6].
The RMSo of static drift simulations with walls are up to
about 29 %M where M is a mean value of forces and
moment coefficients in time. It indicates relatively large
damped oscillation due to the spurious free surface waves
which are partially reflected by the upstream, downstream,
and side wall boundaries. When the wide external domains
are used in order to leverage the numerical dissipation
associated with large grid spacing at inlet/exit/side
boundaries, the levels of RMSo are at most four times
smaller than those of the results with walls. This ensures
the faster statistical convergence and smaller UI (less than
0.7 %M). Statistical convergence for steady turn case is
similar to the static drift cases with walls.
5.2.2.2 Pure sway, pure yaw and combined yaw and
drift In the dynamic PMM cases, the RMSo responds
mainly to the imposed sway/yaw motion. Once the simu-
lation reaches periodic, the amplitude of forces and
moment coefficients should be constant and independent
from the number of iteration. In order to quantify the UI of
the harmonics, the RM of time histories of firstly- and
secondary-dominant harmonic amplitudes are utilized.
Overall, the iterative convergence in X0, Y1 and N1 is
achieved evidenced by the UI less than 4 %M for all the
dynamic PMM cases while it is difficult to achieve iterative
convergence in X2 in pure sway/yaw and X1 in combined
yaw and drift. In pure sway/yaw, although the Y3 is more
than 20 %Y1, small UI in Y1 automatically ensures the
statistical convergence in Y3 (The same discussion can be
applied between N1 and N3).
5.3 Grid and time step convergence
5.3.1 Static drift
Table 3 shows the results of verification in forces and
moment coefficients. The X0, Y0 and N0 are separated intofriction and pressure components described with the sub-
script f and p, respectively.
The UI for X0p has the relative highest value since the
oscillation in the pressure field decay is much slower than
velocity fluctuations [35], thus a sufficient number of
iterations is required to reduce the fluctuations in the
pressure component. The solution change between fine grid
430 J Mar Sci Technol (2012) 17:422–445
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and medium grid eG21 is at least one order of magnitudelarger than most of the UI except X
0p, indicating that the
effect of iteration is almost negligible compared to the
effect of grid refinement. In the b = 10� results, the con-vergence ratio RG shows that only X
0f , X
0p and X
0 are
monotonically converged (MC) while the rest of coeffi-
cients are either oscillatory converged (OC), monotonically
or oscillatory diverged (MD and OD, respectively). The
grid utilized in the current verification study is not likely to
be fine enough since Bhushan et al. [36] who utilize up to
250 M show MC/OC in X0 and Y0 with the acceptable levelof UG.
5.3.2 Pure yaw
Table 4 summarizes the results of the grid and time step
convergence study for X0, X2, Y1 and N1.
5.3.2.1 Grid convergence A general trend shows that X0,
X2 and Y1 are relatively sensitive to the grid resolution
which is evidenced by the eG21 up to 10 %S1 where the S1is the fine grid solution. In contrast, N1 is fairly insensitive
to the grid resolution since the maximum eG21 is up to3.8 %S1. The UI for X0, Y1 and N1 are 2–20 times smaller
than eG21 while the UI for X2 is nearly the same or some-times larger than eG21. As a result, the effect of iteration isalmost negligible compared to grid refinement in X0, Y1 and
N1, while it is not in X2. The RG values show that it is
difficult to achieve MC in most of the harmonics.
5.3.2.2 Time step convergence A general trend shows
that X0, Y0 and N0 are all very sensitive to the size of time
step which is evidenced by the eT21 up to 57 %S1. The eT21for the most dominant harmonics is up to 10 %S1 and
57 %S1 for the case with and without walls, respectively.
As well as the effect of grid refinement to the iteration, the
UI in X0, Y1 and N1 is one order of magnitude smaller than
eT21 while it is not in X2. In consequence, the effect ofiteration is almost negligible in X0, Y1 and N1 compared to
the size of time step. Opposite to the results in the grid
convergence study, most of the dominant harmonics are
MC but with large UT in X2 with/without walls up to
230 %S1. In addition to the time step convergence study for
the dominant harmonics, the UT is computed as a function
of time over 1 yaw motion period, and Table 8 summarizes
the locally time-averaged UT excluding unacceptably
spiked UT. The level of UT varies in between 2.6 %S1 to
7.7 %S1.
5.4 Validation
Table 5 summarizes the validation results of the static
drift and the pure yaw cases. For the static drift, due to
poor grid convergence in Y0 and N0, only X0 is of theinterest for the validation. The |E| is slightly smaller than
UV for the case with walls and thus X0 is validated at
15.2 %D interval, while it is not for the case without
walls. For the pure yaw, it is difficult to use the FS
decomposed forces and moment coefficients for valida-
tion due to the poor grid convergence, thus locally time-
averaged UD and USN are adopted to compute UV [6].
Notice that the large UD in Y0 is due to the electronic
noise from the AC servomotor at the PMM carriage [9].
The results show that X0 is not validated whereas Y0 and
Table 3 Grid convergence forforces and moment coefficients
for static drift b = 10�w/w.o.walls
UI (%S1) |e21(%S1)| |e32 (%S1)| RG pG P Convergence UG (%S1)
w. walls
X0f 1.87e-3 1.68 1.73 0.97 0.16 0.08 MC 145.42
X0p 1.36 4.78 7.79 0.61 2.82 1.41 MC 63.10
X0 0.23 2.92 4.16 0.70 2.03 1.02 MC 12.87
Y 0p 0.046 9.93 0.09 115.46 – – MD –
Y0 0.053 9.60 0.13 74.30 – – MD –
N 0p 0.051 2.19 1.42 1.54 – – MD –
N0 0.043 2.15 1.43 1.50 – – MD –
w.o. walls
X0f 0.08 2.27 0.37 6.14 – – OD –
X0p 0.24 13.33 11.61 -1.15 – – OD –
X0 0.26 2.82 4.04 -0.70 – – OC 2.1
Y 0p 0.014 0.16 0.02 -7.92 – – OD –
Y0 0.015 0.19 0.06 -3.02 – – OD –
N 0p 0.020 0.98 0.84 -1.16 – – OD –
N0 0.020 0.97 0.91 -1.06 – – OD –
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N0 are validated at 37 %D and 11 %D interval,respectively.
6 Validation of forces and moment coefficients,
hydrodynamic derivatives and reconstruction
6.1 Static PMM tests
6.1.1 Static drift
Figure 3 shows the experimental and computational results
of forces and moment coefficients, as well as reconstructed
computational results. The figure also includes the friction-
pressure ratio (Rf/Rp) for X0, Y0 and N0. Table 6 summarizes
the hydrodynamic derivatives and Table 7 presents the
averaged and maximum values of E;ERCFD and EREFD .
6.1.1.1 Validation of forces, moment and hydrodynamic
derivatives The computational results show overall
agreement to the experimental data with E up to 10 %D at
b B 12�, and then they tend to become larger than theexperimental results. At 0� B b B 20�, the Y0 and N0,which are dominated by the pressure force, agree better to
the experimental data than X0 which is dominated by theviscous force. For the Rf/Rp, as the ship encounters stronger
cross flow at larger b, the X0p becomes significant, and at
b = 20� it almost balances the X0f. In Y0 and N0 the pressurecomponent is almost two and three orders of magnitude
larger, respectively, than the friction over the entire b.
The computational results of the linear derivatives agree
very well to the experiment within E�� �� of 5 %D as well as
the length of the de-stabilizing arm Nv=Yv, but the non-
linear derivatives show relatively large E�� ��. The linear
derivatives are likely to be independent from the range of bwhile the non-linear derivatives are not.
Table 5 Validation of forces and moment coefficients for static driftb = 10� w/w.o. walls and for pure yaw rmax = 0.3 w. walls along 1yaw motion period
|E| (%D) UV (%D) UD (%D) USN (%D)
Static drift
With walls
X0 14.3 15.2 3.6 14.8
Y0 0.7 – 5.4 –
N0 4.2 – 2.6 –
Without walls
X0 4.4 4.1 3.6 2.0
Y0 10.0 – 5.4 –
N0 2.0 – 2.6 –
Pure yaw
With walls
E (%D)a UV (%D)a UD (%D)
a USN (%D)a
X0 17.08 9.96 6.4 7.64
Y0 34.03 37.43 14.6 34.46
N0 8.75 10.90 4.1 10.11
a %PN
i Dij j�
N:
Table 4 Verification fordominant harmonics of forces
and moment coefficients for
pure yaw rmax = 0.3 w/w.o.walls
# Grid/time
step
Installation FS UI(%)
|ek21/S1| 9 100
Rk pk P Convergence Uk(%S1)
Grid convergence
4.1 Grid 1, 2, 3
with Dt2FX0, w.
walls
X0 1.7 5.57 -0.61 – – OC 1.79
X2 6.3 4.15 -1.75 – – OD –
Y1 2.2 6.29 -2.09 – – OD –
N1 0.6 2.10 -0.92 – – OC 0.09
4.2 Grid 1NW,
2NW, 3NWwith Dt2
FX0, w.o.
walls
X0 0.3 5.73 -0.61 – – OC 1.80
X2 25.5 9.55 0.05 8.38 4.19 MC 29.84
Y1 0.5 6.58 -2.36 – – OD –
N1 0.4 1.69 -0.79 – – OC 0.23
Time-step convergence
4.1 Grid 1, 2, 3
with Dt2FX0, w.
walls
X0 1.1 2.50 0.32 1.66 0.83 MC 2.01
X2 33.3 8.51 0.25 1.97 0.99 MC 4.68
Y1 1.1 9.89 0.57 0.82 0.41 MC 27.27
N1 1.8 3.85 0.45 1.16 0.57 MC 6.14
4.2 Grid 1NW,
2NW, 3NWwith Dt2
FX0, w.o.
walls
X0 0.2 2.59 0.37 1.43 0.72 MC 2.81
X2 25.7 56.59 -0.12 – – OC 230.45
Y1 0.9 10.49 0.57 0.82 0.41 MC 28.71
N1 2.9 3.74 0.46 1.13 0.57 MC 6.16
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6.1.1.2 Reconstruction The computational and the exper-
imental results show that the derivatives obtained from the
NLCF at 0� B b B 20� give the smallest EREFD in both EAve:�� ��
and Emax�� ��. It indicates that the extrapolation should be
avoided. In the EAve:�� ��, the ERCFD is slightly larger than the
EREFD but still in the same order of magnitude. This implies that
the current CFD simulation for the static drift up to b = 20�may able to be a replacement of the experiment provided that
the Emax�� �� of the computational results, especially in X0 and Y0,
decreases to the similar level for the experiment.
6.1.2 Steady turn
Figure 4 shows the experimental and computational results
of forces and moment coefficients, as well as reconstructed
computational results. The figure also includes the Rf/Rp
Fig. 3 Original/reconstructed forces and moment coefficients for static drift at different drift angles (left) and pressure-friction ratio (right)
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for X0, Y0 and N0. Table 8 summarizes the hydrodynamicderivatives and Table 9 presents the averaged and maxi-
mum values of E;ERCFD and EREFD .
6.1.2.1 Validation of forces, moment and hydrodynamic
derivatives Due to the limited experimental data, it is
difficult to discuss the trend of the computational results for
the experiment. Yet the computational results of the Y0 and N0
give better agreement than X0 within the E of 10 %D. For theRf/Rp, as the ship encounters more cross flow due to the larger
yaw rate the X0p becomes significant, and at r = 0.6 it almost
balances to the X0f . In Y0 and N0 the pressure component is
almost three orders of magnitude larger than the friction over
the entire r0.
The computational results of the linear derivatives show
fair agreement to the experiment within E�� �� of 14 %D as well
as the length of the stabilizing arm Nr � mxG=Yr � mð Þ, butthe non-linear derivatives show relatively large E
�� ��. Althoughthe experimental and the computational results utilize the
same number of data points to calculate derivatives, the range
of r is different between the two which makes the systematic
comparison difficult. Together with the Nv/Yv from the static
drift results, the Nr � mxG=Yr � mð Þ � Nv=Yv is approxi-mately -0.3 which indicates that the ship is naturally course
unstable [25].
6.1.2.2 Reconstruction The computational results show
that the derivatives obtained from the NLCF give almost
Table 6 Hydrodynamic derivatives from static drift tests
Static drift (#1.3) E�� ��
LCF NLCF NLCF
0� B b B 2� 0� B b B 10� 0� B b B 20�
CFD EFD E CFD EFD E CFD EFD E
Xvv – – – -0.130 -0.095 -36.8 -0.148 -0.102 -45.1 41.0
Yv -0.280 -0.264 -6.1 -0.281 -0.271 -3.7 -0.312 -0.297 -5.1 4.9
Yvvv – – – -2.612 -2.023 -29.1 -1.537 -1.292 -19.0 24.0
Nv -0.145 -0.138 -3.6 -0.144 -0.149 3.4 -0.151 -0.161 6.2 4.4
Nvvv – – – -0.507 -0.494 -2.6 -0.234 -0.117 -100.0 51.3
Nv/Yv 0.518 0.523 1.0 0.512 0.550 6.9 0.484 0.542 10.7 6.2
E, E�� �� (%D)
Table 7 Average and maximum comparison error of forces and moment coefficients between original EFD/CFD and reconstructed EFD/EFD
#1.3 Original Hydrodynamic derivatives used for reconstruction
LCF NLCF NLCF
0� B b B 20� 0� B b B 2� 0� B b B 10� 0� B b B 20�
E ERCFD EREFD ERCFD EREFD ERCFD EREFD
Average error (%D)
EX�� �� 5.67 – – 3.60 1.14 6.09 0.63EY�� �� 6.63 16.53 21.53 10.60 4.65 6.90 2.30EN�� �� 3.10 10.50 15.10 4.68 4.18 2.56 2.55EAve:�� �� 5.13 13.52 18.32 6.29 3.32 5.18 1.83
Maximum error (%D)
Emax,X -14.72 – – -7.41 3.29 -14.83 1.32
Emax,Y -9.83 37.39 41.03 -31.13 -13.61 -13.46 -7.47
Emax,N 7.29 16.57 20.74 -17.09 18.87 4.57 -6.16
Emax�� �� 10.61 26.98 30.89 -18.54 11.92 -10.95 4.98
R reconstructed
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identical ERCFD in Y in both EAve: (6 %D) and Emax (9 %D)
compared to the results from the LCF. Since it is opposite
to the conclusion obtained from the static drift and the
experiment, diagnostics for the experimental data and more
CFD simulations with the same range of r to the experi-
ment would be necessary.
6.2 Dynamic PMM tests
6.2.1 Pure sway
Figure 5 shows the experimental and computational results
of the forces and moment coefficients at three different
Fig. 4 Original/reconstructed forces and moment coefficients for steady turn at different yaw rates (left) and pressure-friction ratio (right)
J Mar Sci Technol (2012) 17:422–445 435
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bmax in one sway motion period. The figure also includesthe reconstructed computational results of the forces and
moment coefficients using the hydrodynamic derivatives
obtained from the static drift and current pure sway sim-
ulations. Table 10 summarizes the experimental and com-
putational results of the hydrodynamic derivatives, and
Table 11 presents the E; EAve:; ERCFD and EREFD .
6.2.1.1 Validation of forces, moment and hydrodynamic
derivatives The overall trend shows that the computa-
tional results agree well to the experimental data within
EAve: of 10 %D. In both the experimental and the compu-
tational results, the dominant harmonic is 2nd in X0 and 1stin Y0 and N0 which agree to the Abkowitz’s approximation.The Y0 shows phase lead by about 25� with respect toimposed sway motion, and a similar trend is observed for
KVLCC1&2 under the same pure sway motion [11]. The
virtual mass force is proportional to the acceleration which
has a maximum at the maximum y position of the ship
(t/T = 0.25 and 0.75), while all other forces peak at the
maximum velocity point at t/T = 0 and 1, i.e., at the center
of the towing tank. This causes Y0 to peak approximately att/T = 0.9. Different from Y0, N0 is in phase with the swaymotion. Since the yaw moment is mostly caused by lateral
forces acting through the CoR, a symmetric N0 should havebeen caused by a symmetric Y0. The simulation shows that,after the peak in Y0, N0 is increasing while Y0 is decreasing.This is most likely due to a local decrease of the lateral
force near the stern which causes an increase of the yaw
moment.
The computational and the experimental results show
that the CF and the SR methods give consistent value in Yvbut not in N _v. Since the N _v is coupled added inertia and is
nearly zero as long as the ship is geometrically symmetric
between port and starboard, it is difficult for both the
simulation and the experiment to calculate it accurately.
Table 8 Hydrodynamic derivatives from steady turn tests
Hydrodynamic derivative EFDa CFD (#2.1) E (%D) E�� ��(%D)
LCF NLCF LCF NLCF LCF NLCF
0.15 B r0 B 0.3 0.15 B r0 B 0.3 0 B r0 B 0.15 0 B r0 B 0.60
Xa – -0.0162 – -0.0162 – 0.00 0.0
Xrr – -0.0591 – -0.0327 – 44.67 44.7
Yr -0.051 -0.0452 -0.0492 -0.0506 3.53 -11.95 7.7
Yrrr – -0.0925 – -0.0192 – 79.24 79.2
Nr -0.049 -0.0393 -0.0424 -0.0451 13.47 -14.76 14.1
Nrrr – -0.0955 – -0.0201 – 78.95 79.0
(Nr - mxG)/(Yr - m) 0.262 0.211 0.227 0.240 13.36 -13.74 13.6
a Raw data by Bassin d’Essai des Carenes (BEC), http://www.simman2008.dk/
Table 9 Average and maximum comparison error of forces and moment coefficients between original EFD/CFD and reconstructed EFD/EFD
#2.1 Original Hydrodynamic derivatives used for reconstruction
LCF NLCF
0.15 B r0 B 0.3 0 B r0 B 0.15 (CFD), 0.15 B r0 B 0.3 (EFD) 0 B r0 B 0.6 (CFD), 0.15 B r0 B 0.3 (EFD)
E ERCFD EREFD ERCFD EREFD
Average error (%D)
EX�� �� 6.83 – – 7.27 4.74EY�� �� 6.19 6.06 3.17 5.19 0.91EN�� �� 0.50 6.29 7.37 5.36 1.02EAve:�� �� 4.51 6.18 5.27 5.94 2.22
Maximum error (%D)
Emax,X 13.27 – – 9.61 -8.53
Emax,Y 8.02 7.94 -6.29 -8.22 -1.68
Emax,N -0.97 11.64 -12.11 -8.58 -2.01
Emax�� �� 7.42 9.79 9.20 8.80 4.07
R reconstructed
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For linear derivatives, the CF and the SR methods give
consistent values in both Yv and Nv while the trend is
opposite in non-linear derivatives. The current bmax rangeis almost within the linear range in connection to the results
from static drift (see Fig. 2) which makes it difficult to
calculate non-linear derivatives. The length of the stabi-
lizing arm is slightly longer (e.g., Nv/Yv * 0.6) than theresult from the static drift, and it tends to be shorter as bmaxbecomes larger.
6.2.1.2 Reconstruction The computational results show
that the ERCFD is small only when the reconstruction is done
using its own derivatives which is the same conclusion as is
obtained from the experiment. The ERCFD from the NLCF
using pure sway is about 5 %D which is almost the same
level as EREFD . It implies that the CFD simulation of pure
sway test at bmax up to 10� can be a replacement of theexperiment.
6.2.2 Pure yaw
Figure 6 shows the experimental and computational results
of the forces and moment coefficients at three different r0maxin one yaw motion period. The figure also includes the
reconstructed computational results of forces and moment
coefficients using the hydrodynamic derivatives obtained
from the steady turn and the pure yaw simulations.
Table 12 summarizes the experimental and computational
results of the hydrodynamic derivatives, and Table 13
presents the E; EAve:; ERCFD and EREFD .
6.2.2.1 Validation of forces, moment and hydrodynamic
derivatives The overall trend shows that the computa-
tional results show fair agreement to the experimental data
within EAve: of 20 %D. The larger EAve: compared to the
pure sway is mostly due to large E in Y0. In Fig. 6, theexperimental and the computational results show the same
trends about the dominant frequency of forces and moment
coefficients over one yaw motion period to the pure sway
results. The peaks of N0 is likely to appear prior to the peakof r0 (i.e., before t/T = 0.25 and 0.75) since the addedhydrodynamic moment of inertia increases as the yaw rate
becomes larger.
The computational and the experimental results show
that the CF and the SR methods give a different result in Y _rbut a consistent result inN _r. Similar to N _v in the pure sway,
Y _r is coupled added inertia and is a very small quantity and,
thus, it is difficult to calculate. For linear derivatives, the
CF and the SR methods give different Yr but gives con-
sistent Nr. For non-linear derivatives, the CF and the SR
methods give different derivatives. The length of the de-Fig. 5 Original/reconstructed forces and moment coefficients forpure sway at different bmax
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Ta
ble
10
Hy
dro
dy
nam
icd
eriv
ativ
esfr
om
pu
resw
ayte
sts
Pu
resw
ay(#
3.1
)E� �� �
SR
SR
SR
NL
CF
bm
ax
=2
�b m
ax
=4
�b
max
=1
0�
0�
Bb m
ax
B1
0�
CF
DE
FD
Ea
CF
DE
FD
EC
FD
EF
DE
CF
DE
FD
E
Xa
-0
.01
67
-0
.01
61
-3
.83
-0
.01
69
-0
.01
61
-5
.33
-0
.01
90
-0
.01
86
-2
.44
-0
.01
63
-0
.01
62
-1
.07
3.1
7
Xvv,0
––
––
––
––
–-
0.2
40
-0
.07
0-
12
6.7
12
6.7
Xvv,2
0.3
46
-0
.75
31
46
.00
.06
2-
0.0
01
46
50
.0-
0.0
08
0.0
53
11
6.2
-0
.00
60
.05
11
12
.41
25
6.1
5
Yvdot
-0
.10
2-
0.0
92
-1
1.0
-0
.10
4-
0.0
80
-3
0.4
-0
.11
1-
0.1
02
-9
.2-
0.1
10
-0
.09
9-
11
.61
5.5
5
Yv
-0
.23
6-
0.2
44
3.1
-0
.24
3-
0.2
69
9.6
-0
.27
6-
0.2
91
5.0
-0
.25
5-
0.2
68
4.9
4.8
6
/Y
(�)
-1
4.5
3-
12
.75
-1
4.0
-1
4.3
8-
10
.08
-4
2.9
-1
3.5
5-
11
.04
-1
3.5
8–
––
23
.45
Yvvv,1
––
––
––
––
–-
3.8
87
-2
.44
3-
59
.25
9.2
0
Yvvv,3
-1
6.6
94
-1
2.3
60
-3
5.1
-7
.73
5-
3.4
85
-1
21
.9-
2.9
45
-1
.44
2-
10
4.3
-2
.96
7-
1.4
51
-1
04
.49
1.4
3
Nvdot
-0
.01
2-
0.0
12
-0
.6-
0.0
11
-0
.00
7-
58
.3-
0.0
11
-0
.00
8-
38
.90
.01
1-
0.0
08
-3
9.4
34
.30
Nv
-0
.15
0-
0.1
63
0.7
-0
.14
7-
0.1
64
0.7
-0
.14
6-
0.1
59
-1
3.1
-0
.15
4-
0.1
62
5.0
4.8
0
/N
(�)
-7
.50
-6
.89
-8
.9-
7.2
6-
4.1
2-
75
.9-
7.0
7-
4.6
8-
51
.23
––
–4
5.3
4
Nvvv,1
––
––
––
––
–-
0.3
69
-0
.08
3-
34
2.7
34
2.7
0
Nvvv,3
-6
.16
23
.02
1-
10
4.0
-2
.16
6-
0.2
23
-8
33
.9-
0.7
45
-0
.24
64
02
.5-
0.7
54
-0
.24
3-
20
8.1
18
5.8
8
Nv/Y
v0
.63
60
.66
84
.90
.60
50
.61
00
.80
.52
90
.54
63
.20
.60
40
.60
40
.13
.8
aE
=D
-S
(%D
),u
Y¼
tan�
1�
Yv=x
Yv
ðÞ,
sam
efo
r/
N[9
]
438 J Mar Sci Technol (2012) 17:422–445
123
-
stabilizing arm is similar to the solution from the steady
turn, and it tends to be longer as r0max becomes larger. Using
the lengths of the stabilizing and de-stabilizing arm
obtained from the pure sway and the pure yaw, respec-
tively, Nr � mxG=Yr � mð Þ � Nv=Yv is approximately -0.4thus the ship is naturally course unstable which is the same
conclusion obtained from the static PMM tests.
6.2.2.2 Reconstruction The computational results show
that the ERCFD is small only when the reconstruction is done
using its own derivatives. The ERCFD from the NLCF using
pure yaw is at least two times larger than EREFD , thus it is
still inconclusive to state that the CFD simulation of pure
yaw test at r0max up to 0.6 can be a replacement of the
experiment. The ERCFD from the NLCF with the steady turn
results is two times smaller than the ERCFD from the NLCF
with the pure yaw results.
6.2.3 Combined yaw and drift
Figure 7 shows the experimental and computational results of
the forces and moment coefficients at three different b withconstant r0max in one yaw motion period. The figure also
includes the reconstructed computational results of forces and
moment coefficients using the hydrodynamic derivatives
obtained from the static drift, pure sway, and pure yaw sim-
ulations. Table 14 summarizes the experimental and compu-
tational results of the hydrodynamic derivatives, and Table 15
presents the E; EAve:; ERCFD and EREFD .
6.2.3.1 Validation for forces, moment and hydrodynamic
derivatives The overall trend shows that the computa-
tional results show fair agreement to the experimental data
within EAve: of 16 %D regardless of the different installa-
tion condition between the experiment (FX0) and the
simulation (FXrs). Relatively large EAve: is mostly due to
large E in X0. According to the towing path of the com-bined yaw and drift test, at 0 \ t/T \ 0.5 the direction ofthe yaw rotation is towards the leeward side which reduces
the cross flow to the ship induced by the given b. Duringthis period the apparent b become smaller than the given b.At 0.5 \ t/T \ 1, the yaw rotation is towards the windwardside which increases the cross flow towards the ship.
During this motion period the instantaneous b is up to 21.2�at t/T = 0.75 when the given b = 11�, thus, the E in X0becomes larger. In Fig. 7, both the experiment and the
Table 11 Average comparison error over 1 sway motion period between original and reconstructed forces and moment coefficients from puresway tests
E (%D) Original pure sway Hydrodynamic derivatives used for reconstruction UD (%D)
Static drifta (#1.3) Pure sway (#3.1)
NLCF SR SR SR NLCF
0� B b B 20� bmax = 2� bmax = 4� bmax = 10� 0� B bmax B 10�
bmax E ERCFD EREFD ERCFD EREFD ERCFD EREFD ERCFD EREFD ERCFD EREFD UD
2�EX 6.93 5.77 3.7 4.16 4.7 4.13 4.7 15.34 12.4 3.50 4.4 –
EY 6.56 20.98 15.1 6.17 1.6 7.17 9.0 12.85 12.8 10.45 5.8 –
EN 3.72 5.71 4.8 3.28 1.7 7.11 5.9 8.71 4.7 3.66 5.0 –
EAve: 5.74 10.82 7.9 4.54 2.7 6.14 6.5 12.30 10.0 5.87 5.1 –
4�EX 9.05 6.04 7.3 6.60 12.5 6.99 5.3 18.58 14.8 6.95 5.5 –
EY 13.99 18.61 11.9 14.90 8.3 13.38 1.7 17.01 12.3 16.99 10.9 –
EN 6.41 7.79 1.2 8.94 8.6 6.13 0.8 9.69 2.8 5.75 1.2 –
EAve: 9.82 10.81 6.8 10.15 9.8 8.83 2.6 15.09 10.0 9.90 5.9 –
10�EX 7.86 11.33 3.1 35.76 55.5 10.06 9.5 7.86 6.2 14.37 13.4 6.8
EY 7.01 6.97 3.2 70.02 49.8 23.16 14.0 7.09 2.8 7.41 5.9 5.2
EN 3.22 5.67 3.3 68.62 38.7 18.43 3.6 3.54 1.1 3.49 3.0 4.9
EAve: 6.03 7.99 3.2 58.13 48.0 17.22 9.0 6.16 3.4 8.42 7.4 5.6
a Y _v and N _v are taken from curve fit results of pure sway
J Mar Sci Technol (2012) 17:422–445 439
123
-
computational results show that the 1st-harmonic is the
most dominant in X0, Y0, and N0 which is different trendfrom the pure yaw cases, but it mathematically agrees with
the Abkowitz’s approximation [6].
The computational results show that the CF and the SR
methods give a different result in all the cross-coupling
derivatives which is the similar trend for the experiment,
with the exception in Yvrr and Nvrr.
6.2.3.2 Reconstruction The computational results show
that the ERCFD from the NLCF is about 1.2 times larger than
ERCFD from the SR, but it is nearly the same level of EREFDfrom the NLCF. This implies that the CFD simulation of
combined yaw and drift test at rmax 0.3 with b up to 11�may able to be a replacement of the experiment, although
more diagnostics are necessary for the pure yaw results
which provide the hydrodynamic derivatives (Yr, Yrrr, Nr,
Nrrr) for the reconstruction.
7 Conclusions
Static and dynamic PMM simulations of a surface com-
batant Model 5415 are performed using a viscous CFD
solver with dynamic overset interface. The objective of this
research is to investigate the capability of the current solver
for CFD-based maneuvering prediction.
The V&V study is performed for forces and moment
coefficients in the static drift at b = 10� and the pure yawwith r0max = 0.3. The use of wide background domain is
strongly recommended in static drift simulation in order to
dissipate spurious free surface waves so that faster statis-
tical convergence can be achieved, although the systematic
quantification of the blockage effect to the forces and
moment [37] would be suggested as one for future work. In
the pure yaw (and the other dynamic PMM cases), the
statistical convergence of the forces and moment coeffi-
cients in terms of their firstly and secondary dominant
harmonics is mostly ensured. For grid and time step con-
vergence, the current static drift results show difficulties in
obtaining grid convergence in most of the forces and
moment coefficients. As reported by Bhushan et al. [36]
who utilize the DES simulation with finer grid (up to
250 M grid points) to simulate Model 5512 in static drift
(b = 10� and 20�), the resolution of vortical structure andits unsteadiness is the key for better grid and time-step
convergence. Thus, such simulations would be suggested
as future work to achieve grid convergence with acceptable
USN in X0 and Y0. In the pure yaw, the effect of grid is likely
to have a stronger influence to the forces and moment
coefficients than the effect of the size of time steps.Fig. 6 Original/reconstructed forces and moment coefficients forpure yaw at different r0max
440 J Mar Sci Technol (2012) 17:422–445
123
-
Table 12 Hydrodynamic derivatives from pure yaw tests
Pure yaw (#4.3) E�� ��
SR SR SR NLCF
r0max = 0.15 r0max = 0.30 r
0max = 0.60 0 B r
0max B 0.60
CFD EFD Ea CFD EFD E CFD EFD E CFD EFD E
Xa -0.0162 -0.0166 2.21 -0.0173 -0.0179 3.07 -0.0183 -0.0210 12.73 -0.0159 -0.0157 -1.7 4.92
Xrr,0 -0.074 0.092 180.4 -0.037 0.010 453.6 -0.029 -0.021 -39.2 -0.0293 -0.0289 -1.4 168.65
Xrr,2 -0.024 0.091 126.6 0.0003 0.039 99.1 -0.013 0.004 445.2 0.0129 -0.006 311.5 245.60
Yrdot -0.006 -0.002 -189.6 -0.009 -0.001 -549.3 -0.013 -0.005 -188.2 -0.008 -0.006 -36.0 240.78
Yr -0.038 -0.050 24.8 -0.030 -0.047 35.0 -0.009 -0.023 60.9 -0.042 -0.053 20.0 35.18
/Y (�) 74.94 85.87 12.7 63.09 87.09 27.55 17.96 67.31 73.30 – – – 37.85
Yrrr,1 – – – – – – – – – -0.019 -0.021 9.8 9.80
Yrrr,3 -0.348 -0.076 -363.8 -0.211 -0.119 -76.8 -0.144 -0.130 -11.4 -0.145 -0.130 -12.2 116.05
Nrdot -0.008 -0.007 -18.4 -0.008 -0.006 -21.3 -0.008 -0.005 -39.1 -0.008 -0.006 -36.0 28.70
Nr -0.041 -0.043 5.1 -0.042 -0.045 6.6 -0.041 -0.047 11.6 -0.042 -0.046 7.4 7.68
/N (�) 71.45 74.97 4.69 72.66 76.48 4.99 72.96 75.98 3.9 – – – 4.53
Nrrr,1 – – – – – – – – – -0.032 -0.036 12.0 12.00
Nrrr,3 -0.161 -0.037 76.9 -0.043 -0.050 13.9 -0.0360 -0.0328 -9.6 -0.0361 -0.0332 -8.9 27.33
(Nr - mxG)/
(Yr - m)
0.233 0.229 -1.89 0.251 0.244 -2.85 0.280 0.294 4.48 0.234 0.241 3.15 3.09
a E = D - S (%D), uY ¼ tan�1 �Y _r=xY _rð Þ, same for uN : [9]
Table 13 Average comparison error over 1 yaw motion period between original and reconstructed forces and moment coefficients from pureyaw tests
E (%D) Original pure yaw Hydrodynamic derivatives used for reconstruction UD (%D)
Steady turn (#2.1) Pure yaw (#4.3)
NLCF SR SR SR NLCF
0 B r0 B 0.60 r0max = 0.15 r0max = 0.30 r
0max = 0.60 0 B r
0max B 0.60
r0max E ERCFD ERCFD EREFD ERCFD EREFD ERCFD EREFD ERCFD EREFD UD
0.15
EX 8.24 8.43 10.18 4.8 11.37 12.1 50.58 34.8 7.76 7.7 –
EY 21.99 9.61 22.92 8.9 44.88 9.8 89.35 52.5 37.35 10.3 –
EN 9.65 4.40 9.77 2.2 7.50 3.1 8.50 5.6 6.75 4.6 –
EAve: 13.29 7.48 14.29 5.3 21.25 8.3 49.48 31.0 17.29 7.5 –
0.3
EX 8.85 14.11 13.26 22.4 8.41 1.1 48.21 29.2 12.86 12.0 6.4
EY 29.21 9.26 21.87 3.6 29.61 2.3 76.47 43.5 39.82 9.0 14.6
EN 8.16 4.23 10.94 11.1 8.24 1.3 9.74 3.5 8.43 2.9 4.1
EAve: 15.41 9.20 15.36 12.4 15.42 1.6 44.81 25.4 20.37 8.0 8.4
0.6
EX 10.57 21.76 16.22 98.5 15.08 46.6 26.83 2.8 19.14 18.5 –
EY 37.20 18.37 117.26 25.7 51.82 39.0 36.92 4.0 40.52 17.7 –
EN 10.68 8.23 13.07 49.6 10.41 6.8 10.86 3.8 11.25 3.8 –
EAve: 19.48 16.21 48.85 57.9 25.77 30.8 24.87 3.5 23.64 13.3 –
J Mar Sci Technol (2012) 17:422–445 441
123
-
Fig. 7 Original/reconstructed forces and moment coefficients forcombined yaw and drift at different b T
ab
le1
4H
yd
rod
yn
amic
der
ivat
ives
fro
mco
mb
ined
yaw
and
dri
ftte
st
#5
.1Y
awan
dd
rift
E� �� �
SR
SR
SR
NL
CF
b=
9�
b=
10�
b=
11�
9�
Bb
B1
1�
CF
DE
FD
EC
FD
EF
DE
CF
DE
FD
EC
FD
EF
DE
Xvr
-0
.03
51
0.0
26
62
52
.0-
0.0
38
90
.02
33
26
6.8
-0
.04
21
0.0
20
43
05
.8-
0.0
95
30
.15
95
15
9.7
52
46
.09
Yvrr
,0-
0.4
96
9-
0.6
63
52
5.1
2-
0.4
95
2-
0.6
02
01
7.7
3-
0.4
83
8-
0.3
47
7-
39
.16
-3
9.9
25
.57
88
15
.16
22
4.2
9
Yvrr
,2-
0.6
52
8-
0.6
21
3-
5.0
6-
0.6
07
4-
0.6
48
56
.35
-0
.56
90
-0
.86
20
33
.99
-0
.60
43
-0
.72
67
16
.84
15
.56
Yrv
v-
0.7
10
1-
0.7
80
38
.99
-0
.87
64
-0
.87
42
-0
.25
-1
.00
64
-0
.90
16
-1
1.6
3-
1.6
14
0-
1.1
48
2-
40
.57
15
.36
Nvrr
,0-
0.1
03
30
.15
20
16
7.9
6-
0.1
09
20
.15
12
17
2.1
6-
0.1
15
50
.24
69
14
6.7
9-
7.6
47
38
.58
21
89
.11
16
9.0
1
Nvrr
,2-
0.1
55
9-
0.1
44
2-
8.1
4-
0.1
45
3-
0.1
29
6-
12
.15
-0
.13
55
-0
.13
66
0.7
6-
0.1
44
2-
0.1
36
3-
5.8
46
.72
Nrv
v-
0.5
87
8-
0.3
56
4-
64
.93
-0
.55
21
-0
.32
23
-7
1.3
0-
0.5
26
1-
0.2
92
3-
80
.01
-0
.39
95
-0
.16
05
-1
48
.94
91
.30
442 J Mar Sci Technol (2012) 17:422–445
123
-
Static drift, steady turn, pure sway, pure yaw, and
combined yaw and drift simulations are performed using
different planar motion parameters, and resultant forces
and moment coefficients are compared with the experi-
mental data. For the static drift, the URANS with relatively
coarse grid (2.4 M) provides satisfactory agreements to the
experimental data up to b = 12�. When the b is larger than12�, the refinement grid must properly be embedded to theregion where the massive flow separation occurs, and the
DES should be utilized rather than the URANS simulation.
These treatments decrease the E at b = 20� down to 5 %D.In the steady turn, the large E that appeared in X0 atr0 = 0.6 is also likely to be improved by the DES withappropriate local refinement. In the pure sway, pure yaw,
and combined yaw and drift, the forces and moment
coefficients over 1 sway/yaw motion period generally
agree well to the experimental data although the reason for
phase difference in Y for the pure yaw at medium and large
rmax has not yet identified.
The acceleration, linear, non-linear, and cross-coupling
hydrodynamic derivatives are calculated from both the com-
putational and the experimental results of the forces and
moment coefficients in the static and dynamic PMM tests. The
predictions for most of the linear derivatives obtained either
from the single-run method or the non-linear curve fitting are
satisfactory, evidenced by the E less than 10 %D. For
acceleration, non-linear, and cross coupling derivatives, the
predictions are fair but not as good as linear derivatives.
Resultant hydrodynamic derivatives are utilized to
reconstruct the forces and moment coefficients. It is com-
mon in all the cases that extrapolation should be avoided,
e.g., it is strongly recommended to use the hydrodynamic
derivatives calculated from the non-linear curve fitting and
not from the single-run method to estimate forces and
moment coefficients in the mathematical model. It is also
common for all the cases that the EREFD is non-negligible
quantity. In view of the ERCFD relative to the EREFD they are
close each other in pure sway, and thus the CFD simulation
can be a replacement of the experiment. In pure yaw, the
ERCFD is larger than EREFD and additional diagnostics and
simulations would be necessary to conclude that the CFD
simulation can be a replacement of the experiment. In
combined yaw and drift, ERCFD is again close to EREFD , and,
thus, it may be able to conclude that the CFD simulation
can be a replacement of the experiment. Yet attention must
be paid since the pure yaw results are utilized for recon-
structing the combined yaw and drift results.
Overall, the current solver is confirmed to have the
capability of handling static and dynamic PMM simula-
tions, and the resultant forces and moment coefficients as
well as hydrodynamic derivatives show general agreement
Table 15 Average comparison error over 1 yaw motion period between original and reconstructed forces and moment coefficients fromcombined yaw and drift tests
# 5.1 Original yaw
and drift
Hydrodynamic derivatives used for reconstruction
Yaw and drift
SR SR SR NLCF
b = 9� b = 10� b = 11� 9�B b B11�
b E ERCFD EREFD ERCFD EREFD ERCFD EREFD ERCFD EREFD
9�EX 18.57 15.51 4.3 15.68 4.3 15.22 4.2 16.01 4.3
EY 8.78 9.74 4.7 8.77 4.5 8.02 4.2 12.40 10.6
EN 6.06 6.53 13.3 6.23 13.3 5.95 14.0 10.83 15.3
EAve: 11.14 10.59 7.4 10.23 7.4 9.73 7.5 13.08 10.1
10�EX 21.64 19.03 1.8 19.66 1.6 19.30 1.6 19.19 1.6
EY 8.92 10.68 3.9 9.70 3.5 8.96 3.3 13.67 10.4
EN 6.76 7.68 12.7 7.38 12.7 7.10 13.3 9.21 14.7
EAve: 12.44 12.46 6.1 12.25 5.9 11.79 6.1 14.02 8.9
11�EX 24.71 22.31 4.1 23.05 4.1 22.21 4.1 22.18 4.1
EY 12.21 14.42 3.8 13.71 3.2 13.23 2.5 15.52 11.9
EN 10.82 11.98 12.0 11.67 12.0 11.39 12.3 13.91 15.1
EAve: 15.91 16.24 6.6 16.14 6.4 15.61 6.3 17.20 10.4
J Mar Sci Technol (2012) 17:422–445 443
123
-
to the experimental data. Although some unsatisfactory
results are found in grid convergence, forces and moment
coefficients at large b, and some non-linear hydrodynamicderivatives, DES with appropriate local refinement is likely
to improve them.
Part 2 provides the detailed validation for flow features
with the experimental data as well as investigations for
flow physics, e.g., flow separation, three dimensional vor-
tical structure and reconstructed local flows.
Acknowledgments This research was sponsored by the US Officeof Naval Research, Contract N00014-01-1-0073 under the adminis-
tration Dr. Patrick Purtell. Computations were performed at the DoD
NAVO MSRC on IBM P4? and P5.
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c.773_2012_Article_178.pdfURANS simulations of static and dynamic maneuvering for surface combatant: part 1. Verification and validation for forces, moment, and hydrodynamic derivativesAbstractIntroductionCFD-based maneuvering prediction at SIMMAN 2008Conclusion from past research
Test overviewsGeometryStatic and dynamic PMM tests
Computational methodModelingNumerical methods and high-performance computing
Simulation designTest casesGrid, domain size and time stepBoundary conditionsAnalysis m