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OCR for page 268
Towards the Simulation of Seakeeping and Manoeuvring based
on the Computation of the Free Surface Viscous Ship Flow
A. Cura Hochbaum and M. Vogt (Hamburg Ship Model Basin, Germany)
ABSTRACT
The simulation of ship motions due to rud-
der manoeuvres and/or to incoming waves based on
free surface viscous flow computations will be one
of the most fascinating challenges in ship hydrody-
namics for the coming years. The present paper
deals with the steps we are taking to achieve this
long-term objective. The numerical method used
is described, predicted forces and moments on the
hull of a ship in a variety of cases involving ba-
sic manoeuvring flow situations are compared with
measurements and results of simulations for ships
moving against harmonic waves coming from ahead
are presented.
INTRODUCTION
There have been research activities in this
field since the 21th ONR Symposium in Trond-
heim 1996 and the 7th Int. Conf. on Numerical Ship
Hydrodynamics in Nantes 1999. McDonald et al.
[6] and Takada et al. [12] simulated manoeuvring
based on viscous flow computations. Wilson et al.
t16], Kinoshita et al. t5] and Sato et al. [9] simulated
ship motions in waves coming from ahead. Miyake
et al. t7] also treated oblique incoming waves. They
all generated regular waves prescribing velocities at
the inlet according to the potential wave theory.
Because the techniques used in the publica-
tions mentioned above have shown to be eEective
and promising, we extended our numerical method
in a similar way in order to simulate the motions of
a ship model moving straight ahead against incom-
ing waves. Due to the assumed symmetry, the ship
Ship motions and loads are traditionally cat- only performs surge, heave and pitch motions. An
culated using methods based on the strip theory or accurate computation of forces and moments on the
with panel methods. These methods rely on po- hull during steady oblique and turning motions, is
tential flow theory and take viscous effects into ac- certainly a prior requirement for a reliable simula-
count in a very coarse manner. For this reason,
motions with a significant viscosity influence, e.g.
surge and roll, can hardly be predicted well. More-
over, most of the methods are not able to deal with
steep waves nor with breaking waves, especially be-
cause they linearise the motions and/or the water
free surface, impeding the treatment of cases where
slamming and green water occur.
The long-term objective of this work is to
achieve numerical simulations of ship motions based
on the calculation of the free surface viscous flow
around a manoeuvring ship in calm water or in
ocean waves. For most tasks concerning seakeeping
and manoeuvring, conventional methods will still
be used in future. However, the present method
may become useful for the special cases mentioned
above, because it takes all relevant aspects involved
into account. For the present first step however, we
have focussed on much simpler cases than the really
interesting problems in practice.
tion of motions of a manoeuvring ship. In order to
carry on the validation of our code for these cases,
extensive work is being performed parallel to the
simulation of motions in waves. This includes the
computation and comparison of results with model
test measurements for a large combination of yaw
rate, drift angle, rudder angle and heel angle. Re-
sults for zero rudder and heel angle are presented
at the end of this paper.
METHOD
Coordinatesystem
In order to describe the motion of the ship
we define the earth fixed coordinates X, Y. Z and
the ship fixed coordinates x, y, z as well as the Eu-
ler angles ,o, I, ¢.
The origin of the ship fixed coordinate sys-
tem is positioned mid ship, on the symmetry plane
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Representative terms from entire chapter:
head waves
X,I x,~
f Y ~ Projection of x
~ ~ ~
b_\ -Y,J
Z k ~
, _
v
Z K
m[(u+w2w—TV) — XG (W2 +W3)
—YG (W3—~1~2) + ZG (~2 + ~1 W3)] = Fit
m [(v—~1 W + W3 U) + XG (W3 + w1 ¢~2)
—YG (~1 + ~3) — ZG (~1 —~2 W3)] = FY
m [(w + ~1 v—~2 U) — XG (~2—~1 W3)
+ YG (W1 + ~2 ~3) — ZG (W1 + ~2)] = FZ
IX ~1 - (IY—IN ) W2 ~3—IXY (W2—~1 W3)
—IT (~2 _ ~3 )—IXZ (~3 + ~1 W2)
+m[yG(W+~tlV—MU)
Figure 1: Coordinate systems —ZG (V—~1 W + (1)3 U) ] = K
at the undisturbed free surface. The hybrid base i,
r', u and the projection of the x-axis on the X' O Y'
plane serve to define the rotations and Euler angles
cp (roll angle), ~ (pitch angle) and ~6 (yaw angle).
The unit vector ~ is defined as the product of K
and ~ for K x ~ 76 0, see Fig. 1. With this def-
inition of the rotations the position of the ship is
unique, independent of the order of rotations.
The components of the angular velocity in
the ship fixed coordinate system are:
we = ~—~ sine
. .
~2 = %6cos~sin~+§cos~ (1)
. .
w3 = ~ Cost COsy'—~ sing
The ship fixed and the earth fixed coordi-
nates are related to each other as follows:
(, Y \) = (\ YO \\) + T ~ y ~ (2)
where XO, YO, ZO are the coordinates of O and T
is the transformation matrix. The elements of the
rows of T are the components of the earth fixed
unit vectors expressed in the ship fixed system.
Motion equations of the ship
The motion of the assumed rigid ship is de-
scribed by the momentum and the moment of mo-
mentum equations. These equations are written in
the ship fixed coordinate system:
(3)
IY ~2 - (\IZ—IX ~ Wit ~3 —IXY (
Fat = For + mg cosoe1 + Fop
Fy = Fyh + mg costar + Fyp
Fz = Fzh + mg COS(X3 + Fzp
= Kh + mg (COSC\3 YG—costly ZG) + KP
(5)
M = Mh+ mg (cosa1 ZG—COSCX3 XG) + Mp (6)
N = Nh + mg (cosot2 XG—COSOY1 YG) + Np
The directional cosine of the gravity force, in
the ship fixed coordinate system, costar, i = 1, 2, 3
coincide with the components of the earth fixed ba-
sis vector K and as such, with the elements of the
third row of the transformation matrix T. By inte-
grating the pressure and shear stresses on the hull
we get the hydrodynamic forces Fish, Fyh' Fzh and
moments Kh, Mh, Nh in Eqn. (5) and (6) where
9 is the acceleration due to gravity. Fop, Fyp' Fzp
and Up, Mp, Np are the propeller contributions to
the forces and moments and are here approximated
with a simple model.
Equations of motion of the fluid
The fluid flow around a moving ship is con-
sidered to be a flow of one fluid with two immisci-
ble phases (water and air). The interface between
both phases represents the free surface. Because
each phase is incompressible, the global flow can as
such be handled as incompressible. The governing
equations are the RANSE and continuity equation.
Reynolds stresses are approximated with an eddy
viscosity modell. In the Cartesian ship fixed coor-
dinate system xi=x, x2=y, X3=Z the conservative
form of the governing equations can be written:
dvi + 0(vivj) _ 1 0(p + 2/3 r: k) (7'
lit {Jxi rat Ski
+ F 2 + Ski [( Rn ( fJ /; ~ )
—(~i + (ijk (l)j Ok) — (ijk DJj~km7' Am X
—2 (ijk Wj ok — f ijk JO X
~ i = 0 (8)
All variables have been nondimensionalised
by the ship speed Uo, the ship length L and the wa-
ter density Pw vi is the relative velocity component
in the Cartesian direction i, xi the corresponding
Cartesian coordinate and t the time. Fin = To//
is the Ffoude number and Rn = Uo L/Z/W is the
Reynolds number. p is the mean pressure and ~
the turbulent kinetic energy. The eddy viscosity
zig = k/w where w is the specific dissipation rate of
~ is determined by using the k—w turbulence model
from Wilcox t154. \Vall functions were used in all
computations presented in this paper. r, = P/Pw
and r2 = ',/Z'w are the ratios between the local den-
sity and kinematic viscosity and the corresponding
values for water. These ratios have constant values
P'/Pw and i1/uw in air, pi and i'` being the density
and viscosity of air, and both equal one in water.
f ijk iS the permutation symbol. Surface tension has
not been taken into account in Eq. (7~.
The terms in the third and forth row of Eq.
(7) are the inertial forces stemming from the ac-
celeration and rotation of the coordinate system
used. The first term represents the acceleration
of the origin, the second and third the centrifugal
and the Coriolis forces respectively. The last term
is identical to zero for steady motions e.g. steady
oblique motions. Different from our earlier work,
where the motion of the ship was given, the inertial
forces now have to be updated in the course of the
simulation.
The transport equations for the turbulent
quantities k and ~ for a incompressible fluid with
two immiscible phases in Cartesian coordinates are:
0t + fJ2;i foci [(Rn 0~£ ]
+ (0Vi + i9Vj) Levi _ id* k
0t + deli = hi [(Rn +a Gil (10)
~ (49~i + Ski) If - jB,-4,2
a* = ~ = 0.5 it* = 0.09 ,3 = 0.075 By = 0.555
The transport equations for k and ~ Eqn.
(9) and (10) and the continuity equation Eq. (8)
remain without changes valid in the ship fixed co-
ordinate system.
These equations and the momentum equa-
tions Eq. (7) are solved in the whole computa-
tional domain. Special treatment of the two phases
and boundary conditions on the free surface is not
needed. The inertial forces in Eq. (7) are calcu-
lated with the velocities pi and wi and their time
derivatives obtained from Eqn. (3) and (4~.
Level Set Method
The free surface is taken as the interface be-
tween the two phases water and air. This is cap-
tured with the Level Set method see e.g. Osher
and Sethian t8] and Sussman et al. t114. Among
the first ship hydrodynamics related applications
of the Level Set method was reported in t133.
The interface can be expressed by the equa-
tion Phi, t) = 0. Since the function o is always zero
in all points on the interface its material deriva-
tive is zero there. This also applies for compli-
cated topologies e.g. breaking waves. In the Level
Set technique, a scalar field ~ having a zero level
set ~ = 0 coinciding with the interface is defined
throughout the computational domain. We have
air where ~ is negative and water where o is posi-
tive. Extending the condition ~ = 0 to all points
in the domain leads to the transport equation for
the level set function:
do + 0(Vi ¢) o (11)
In every point (cell center) of the compu-
tational domain, the initial value of o is chosen
to be the distance (with sign) to the initial posi-
tion of the interface. According to Eq. (11), o is
exclusively transported by convection and always
remains positive in one phase and negative in the
other.
For the case of water and air the density is
P = Pw (water) for ~ > 0 and p = pi (air) for 0 < 0.
Also the molecular viscosity of the fluid changes
its value suddenly when ~ changes sign. In order
to avoid numerical problems, these discontinuities
are somewhat smoothed and the density and the
viscosity are determined as follows:
D
Pw
V
LAW
= (1—c) Pi + c
Pw
1
o
2 [1 + sir (26)~
> or (water)
<—or (air)
—of _ 0 < ~
The thickness of the transition region can so,
be prescribed with the parameter or. As long as ~
is the nondimensional distance from the considered
point to the interface, this thickness is 2 or, as can
be seen in Eq. (12~. We choose or to be roughly the
vertical grid spacing near the waterline of the ship.
During the time marching procedure, first
the momentum equations, then the pressure-cor-
rection equation (mass conservation) followed by
the transport equations for the turbulence param-
eters are solved assuming a known o distribution.
After that a new approximation of ~ is computed.
COMPUTATIONAL GRID
The quality of the numerical grid used is cru-
cial for the accuracy of the results and for the con-
vergence behaviour of the method. While the loss
of accuracy due to a poor grid can (in principle)
be reduced increasing the grid resolution, conver-
gence problems mostly remain. Usual grid require-
ments are smoothness and the proper resolution
to capture all interesting aspects of the flow. The
latter usually only depends on the Reynolds num-
ber (boundary layer, wake) and the Froude num-
ber (free surface). However, in order to calculate
ship motions using a ship fixed grid, the expected
motions themselves have to be taken into account
when generating the grid. The free surface (includ-
ing incoming waves) changes the position in the
course of the simulation. Even small pitch motions
for instance, lead to large vertical shifts of the free
surface in the grid somewhat away from the ship.
To avoid related numerical problems, without dras-
tically reducing the time step of the time integra-
tion, we simply generate the grid in such a way that
the waves are always inside regions foreseen for this
purpose in front of and behind the ship.
Grids with satisfactory quality have in the
past been generated with our own elliptical grid
generator. The grid for the KRISO container ship
of the CFD Workshop Gothenburg 2000, [4], and
for the Series 60 ship in this paper, Figs. 10 and
11, were constructed with this tool. However, due
to the poor user interface, the grid generation takes
several days using our program. Therefore, we now
have begun to use a commercial grid generator, re-
ducing this time considerably, at acceptable costs
regarding grid quality. The grid for the container
ship C-Box in this paper was generated in this way.
;: ~ ~
'in
Figure 2: Partial view of the coarse grid at the cen-
ter plane of the C-Box container ship
The grid has roughly 1 million cells and 34
blocks. Because of the symmetry of the case con-
sidered, only the starboard side of the ship was
meshed. Most of the simulation results presented
in this paper were obtained on a coarse version of
this grid, in which every other grid line was skiped.
Fig.2 shows a cut at the center plane of the ship.
Because of the very intensive computational time,
fine grid computations have been restricted to the
model fixed condition till now.
NUMERICAL METHOD
The mathematical model described in the
previous section was implemented by the authors
in the RANSE code Neptun in the course of several
research projects. The implementation details are
certainly crucial for the accuracy of the resulting
code. Most of these details are common to many
other codes and/or have been described in previous
publications e.g. A. Therefore, the overall numer-
ical method is briefly outlined here and we focus
only on a few particular aspects of our method.
The conservation equations of momentum
(7), mass (8) and turbulence parameters (9), (10)
are discretised with a Finite Volume Method. Con-
vective terms are approximated with the Linear
Upwind Differencing Scheme (LUDS) and diffusive
terms with the Central Differencing Scheme (CDS).
The SIMPLE method is used to iteratively solve
the resulting set of equations in each step of the
time marching procedure used. The convection
equation of the Level Set Function Eq. (11) is also
discretised with LUDS in a finite volume manner
and explicitely integrated in order to get the new
position of the free surface. Then the derivatives of
the ship velocities (u, v, w) and of the angular ve-
locities Owl, w2, ~3) are eliminated in the equations
of motion of the ship Eq. (3) and Eq. (4) and ex-
plicitely integrated in order to get the position and
velocities for the next time step. Hereby, known
velocities and angular velocities from the previous
time step (or initial condition) are used together
with the forces and moments on the hull, deter-
mined when solving the fluid equations.
Note that a more sophisticated time integra-
tion, e.g. as recommended by Soeding [10], was not
necessary in the cases considered, because the aris-
ing accelerations were relatively small. Because the
integration of the ship motion equations does not
take much computational time, significant time is
not saved performing the integration with a larger
time step than for the fluid equations. Thus, as
long as the time step needed when solving the flow
is smaller than the time step needed for stability
reasons during the explicit integration of the ship
motion equations, it is not worthwhile changing to
an implicit integration, which demands more com-
putational effort.
Unbalance of Discretised Terms
Because all variables are only stored for the
cell centers, pressures and velocities at cell sides are
interpolated linearly along grid lines to determine
fluxes at the cell faces. If the mass term coso~i/F2
in Eq. (7) is treated in non-divergence form, while
the pressure term is treated in divergence form, an
unbalance between the discretised expressions of
these terms can arise on curvilinear grids, leading
to large numerical disturbances.
L==~
Figure 3: Velocity field at a cross section of the
KRISO container ship at rest. Usual algorithm
(left) and improved algorithm (right)
In case of an uniform flow, with hydrostatic
pressure distribution away from the ship, both terms
should cancel. Fig.3 illustrates this effect at the
cross section of the KRISO container ship at rest.
The exact velocity distribution vi = 0 is strongly
disturbed after some hundred iterations on the left
hand side of the figure. A remedy of this problem
consists in transforming the mass term in diver-
gence form before integrating this term over the
considered finite volume, determining the needed
coordinates at the centers of the cell faces through
linear interpolation like the pressures. The result-
ing expression is then identical to the pressure term
in case of a hydrostatic pressure distribution, avoid-
ing the flow disturbancies almost completely, as
shown on the right hand side of Fig.3.
A similar effect could arise from an unbal-
ance between the discretised forms of the convec-
tive terms and the inertial terms in Eq. (7~. In
a steady turning motion for instance, these terms
should cancel each other, as well as the mass and
pressure terms, in the undisturbed regions away
from the ship. Despite no attempt having been
performed to ensure the balance in this case, no
disturbances have been detected in all cases anal-
ysed untill now, because the angular velocities dur-
ing ship motions are quite small.
W~ ~ uO
/~`
// ~~\ N\
// ~~N N\
/~> <<
Figure 4: Comparison of the computed (black) and
the undisturbed velocity field (red) in the water-
plane of a twin rudder RoRo ship in steady turning
The computed vector field at the waterplane
of the twin rudder FSG RoRo ship in steady turn-
ing (black) is compared with the exact velocity in
the far field (red) in Fig.4. The non-dimensional Eq. (13) is explicitely integrated in the pseudo
yaw rate, based on the ship speed Uo and ship time A. S(o) = I/ is a smoothed sign
length L, is r' = 0.4. The agreement between both function. We choose ~ to be roughly as large as or.
vector fields is very satisfactory in those regions When the steady state is achieved, the first term in
where they are expected to agree. It is interesting
to note that departures are restricted to the near
neighbourhood of the ship and the wake. This can
be seen more clearly in Fig.5, where the magnitude
of the difference of both vector fields are depicted.
o.' 1
QO9 ~
0.08'
0.07 ~
0.06'
0.05!
0.04 ~
0.03'
0.02'
0.01 ~
o 1
Figure 5: Magnitude of the difference between the
computed and the undisturbed velocity fields in the
waterplane of the same RoRo ship in steady turning
Reinitialization of ~
The convection equation (11) moves the free
surface in the correct way but 0 does not remain a
distance function. Keeping o a distance function is
crucial to assure a constant thickness of the tran-
sition zone between water and air, which is needed
for stability and for limiting mass losses. Thus, the
Level Set function ~ is 'reinitialized' before starting
a new time step, i.e. replaced by a new distribution
¢, which in each point again represents the distance
to the free surface. Because in all points on the in-
terface the distance is zero, the interface remains
unchanged while doing so, i.e. the isosurfaces ~ = 0
and o=0 coincide.
The reinitialization is here performed deter-
mining ~ as the steady state solution of the follow-
ing equation, with the initial condition ¢(xt, 0) =
¢(xi, t):
'9r (
Eq. (13) vanishes. Thus, it must be ~~ = 1. This
is exactly the constraint which must be fulfilled by
o to be a distance function.
Usual convection schemes are not suitable
for the discretisation of Eq. (13~. Here, the 1st
order ENO scheme described in [11] for Cartesian
grids was extended for 3D curvilinear grids. After
integrating Eq. (13) in time, we get:
+i = on—fir S(O) (:D2 + D2 + D2 _ 1) (14)
D denotes a discretisation operator for the
magnitude of the x, y and z derivatives, see below.
The non-dimensional pseudo time step b~ is set to
0.01 - 0.001 times or (smaller values for finer grids).
Using the ENO scheme on a Cartesian grid, the x
derivative writes:
Do = max (max(D2, 0), min(D+, 0~) 0 > 0
Do = max (max(D+, 0),—min(D2 ~ 0~) 55 < 0
D2 = (°i,j,k—(i—I,j,k) /^X
D+ = (°i+l,j,k—°i,j,k) /~X
(15)
The indices i, j, k denote the position of the
grid point considered. DO and D+ are backward
and forward finite differences. The role of the min/
max expressions is mainly to build the derivative
with those points, which are closer to the interface
o = 0. For y and z, analog expressions are valid.
In order to extend the scheme to curvilin-
ear grids, with coordinates (i, we denote the com-
ponents of the contravariant base vectors, e.g. if],
with As. and the Jacobian with J. The backward
x-derivative can then be written:
_ ~ (D-(2)Af +D-(~)A~ +D~ (~)A<) (16)
The upper—(x) in the RHS means, that the
corresponding backward finite difference has to be
built regarding to x. In order to determine how to
proceed, we consider the components al of the first
covariant base vector:
D; ( ) = (¢i,j,k—Oi—I,j,k)/~( for al > 0
Do ( ) = (¢i+l,j,k—(i,j,k)/~( for al < 0
(17)
All other derivatives are built in the same
way. When solving Eq. (14), needed values of v
at boundaries are set equal to the direct neighbour
inside the domain.
Figure 6: Isolines of the Level Set function at a
cross section of the KRISO container ship at the
beginning (left) and end (right) of the computation
Fig.6 shows the result of the reinitialization
at a cross section of the KRISO container ship men-
tioned before, moving straight ahead in calm water.
On the left hand side of the figure, the initial o dis-
tribution is shown. The distribution at the end of
the computation on the right hand side of the figure
agrees satisfactorily with the distance to the free
surface represented by the first isoline from above.
Because o has to be reinitialized in the neighbour-
hood of the free surface only, just 3 to 5 pseudo
time steps solaces when solving Eq. (14~.
BOUNDARY CONDITIONS
The boundary conditions on the boundaries
of the computational domain have remained almost
unchanged compared to previous work, see e.g. [1],
the only difference being that they now have to be
expressed in a moving coordinate system. On the
hull the no slip condition is enforced. Boundary
conditions on the free suface are not necessary in
conection with the Level Set method for two-phase
flows. The level set function and the velocities are
given at the inlet. At all other boundaries the
OCR for page 275
are still extrapolated by LUDS. Where the fluid
enters the domain, all velocities are given as usual.
On the outlet the pressure is given. For free surface
computations it is set to the hydrostatic pressure
relative to the undisturbed free surface:
.o.o~s
.o.o,
-0.005
0.005
0.01
0.015
Poutiet = Fn2 (ZO + tat x + t32 y + t33 z) (18) 0.02
t3i, i = 1, 2, 3 are the components of the third row of
the transformation matrix T. Due to ship motions
and incoming waves, the boundary conditions in
a point (x, y, z) at the inlet are in our coordinate
system:
(>irliet = Zo + tat x + t32 y + t33 z + (w (19)
=
=
V3 inlet =
—(A—~3y+W2Z)
+ (ill US + t21 VW + t31 WW)
—(v + w3 x—w1 z) (20)
+ (tl2Uw+t22Vw +t32WW)
—(w —w2x+wly)
+ (tl3Uw +t23vw +t33WW)
(w is the wave elevation relative to the earth
fixed coordinate system. Uw, Vw and Ww are the
earth fixed velocity components in the generated
incoming waves. The values at the inlet boundary
consist of two parts, the contribution from the rigid
motion of the ship fixed coordinate system and the
absolut values in the far field. According to the
potential flow theory we have in deep water:
N
(w = —Mew sin(kjX+=jt+olj) (21)
j=1
N
Uw = —~Wj~we-kiz sin(kjX+wjt+o~j)
j=
VW = 0
(22)
N
Ww = —~ wj (w e kj Z cost; X + Wj t + A)
j=1
N ist the number of superimposed harmonic
waves. w; is the frequency, (w the amplitude, kj
the wave number and off the phase of the wave
component j. Each individual wave is assumed to
run in negative earth fixed direction X. In order
to completely express the inlet values in terms of
t\,'>~:
1 0.75 0.5 0.25
~ All- r -f TV,- - -I- - -it -A - - ~ - - - -
~ \ ~ / 1~ ~ g:
;~ ~ _ _ jet _ I_ ~ ~ V~ ~ '~ ~ 2
-0.25 -0.5 -0.75 - -1.25 -1.5 1.75
x/L
Figure 7: Profile and velocities of a harmonic wave
with amplitude (w/L = 0.01 and length A/L = 1.
Computed profile (red) and potential theory (blue)
o
0.05
0.1
n 1.~
0.2 _
x/L
Figure 8: Profile and streamlines of a harmonic
wave with amplitude (w/L= 0.01 and length A/L= 1
ship fixed variables, the earth fixed coordinates X
and Z of the boundary point considered have to be
transformed according to Eq. 2.
Using this technique, the waves are consid-
ered to be fully developed already at the inlet.
They can be seen as having been generated far in
front of the grid and pass without disturbance the
inlet into the computational domain. The main
problem when generating waves with a finite vol-
ume method is to avoid an excessive damping due
to numerical dissipation. The implemented Level
Set technique seems to do a good job in this regard.
In Fig. 7 a snapshot of a computed harmonic
wave (in red j superimposed to an uniform flow is
compared with the corresponding profit and veloc-
ity distribution of the potential theory (in blue).
The given non-dimensional wave lenght and ampli-
tude were (w/L = 0.01 and A/L = 1 in this case.
The computational domain extends from x/L = 1
to x/L = -5, but a strong cell stretching was used
starting from x/L = - 2 in order to damp out the
waves toward the outlet. Only 100 cells were lo-
cated in the shown region of the Cartesian grid used,
in order to simulate a situation, which can be af-
forded when performing ship motion simulations.
Nevertheless, the computed wave elevation and ve-
locities agree well with the theory. The streamlines
-0.01 5
-0.01
-0 ~5
n
0.005
0.01
0.015
, _ , I , , it, I , I , ~
tar - - - r - -~/~A,~ a- _ _ ~ _ _ _
~ ~ ~ ~ ~ 5, \ ~ I'm ~t i-~ ~ r ~ ~
. >~ /~ ' \ ~ ~ \ ' ~12~ ~ ' ~ z~ IN ~ \
0.02 -~,wk~l~l/,<~~,~]l&~-,~l/~',-,l~-
Figure 11: Two snapshots of the simulation of the
flow around the Series 60 ship in head waves
german ship yard FSG, in order to exploit available
force measurements for validation purpose. Simu-
lations were done at Fr' = 0.23 and Rr' = 8.2106,
A
again with (WIL = 0.01 and A/L = 1.
In Fig. 12 the wave patterns computed on
the coarse grid and on the fine grid are compared
for a given instant. Due to the relatively low Troupe
number, the wave system of the ship can be seen
less clearly now. Nevertheless, it is interesting that
on the coarse grid having roughly 125 000 cells, in-
stead of 1 million as for the fine grid, almost all
wave features are captured. In Fig. 13 the cor-
responding wall shear stresses obtained on both
grids are compared. Now, the poorer quality of
the coarse grid results is more obvious.
Fig. 14 compares the computed and mea-
sured longitudinal force acting on the hull of the
model. Because of the very high computational
time, simulations on the fine grid could not be suf-
ficiently continued to compare the obtained time
history with experiments. Therefore, the predicted
force on the coarse grid is shown instead. The
agreement is surprinsingly good, taking into ac-
count the grid resolution. The encountering pe-
riod has been captured almost exactly and even the
amplitude is well predicted. The computed mean
value of the longitudinal force departes roughly 10%
from the measured value, but confirmes the very
low added resistance compared to the calm water
value when all degrees of freedom are suppressed.
A snapshot from the simulation on the fine
grid at a moment when a wave crest is at the bow is
compared with a photo from experiment in Fig.17.
Due to non- hydrodynamical considerations, this
fine gnd
Hi_
coarse grid
Figure 12: Wave patterns in head waves computed
in the coarse grid (bottom) and in the fine grid
(top). C-Box container ship at En = 0.23
_ ~ ~ ; , ~ ~ I, , ' ~ ~ ' ~ ~ ~ ', ',' ~ ~ ~ ~ ~ ~ ~ ~ ~ , 'I , = 'a ; ~
_<,3,~ ~~ . it. . .~ ... 5 ~ 5 ~ ~-3——__= ~ + ~,;~
~-.3-~ A= -= ~ I =,
—;, ..~ ~ I.= ~ ~~ :::— .—:::,—~ .—: _ ~=.~ :-~ >3 += — zest
(~ ~~:-~,-:~
.~P~:~
Figure 13: Wall shear stress field at the hull of the
C-Box container ship in head waves computed on
the coarse and on the fine grid
container ship has got a relatively large block co-
eflicient CB = 0.74 and was not provided with a
bulbous bow. As a result, pronounced wave break-
ing occur at the bow. This is captured, at least
qualitatively, by the simulation. Unfortunately, the
side walls of the ship were simply vertically pro-
longed when generating the grid, regardless of the
real form of the back. This seems to have an im-
portant influence on the wave breaking process.
The above computations were then repeated
for a condition where the ship model is free to pitch,
heave and surge. During the simulation the pro-
peller thrust was kept constant at the value corre-
sponding to the calm water situation. Again the
simulation started from the calm water solution
without taking dynamic sinkage and trim into ac-
count. Therefore, when the wave generation be-
gins, the initial conditions does not represent an
equilibrium situation. This explains why the ship
carries out small heave and pitch motions already
.i,IN]
-1 60
-120
-80
.dn
40
80
on
Figure 14: Comparison of the computed (solid red
line) and the measured (dashed blue line) longitudi-
nal force at the C-Box container ship in model-fixed
condition in head waves
before the waves reach the hull. Moreover, the gen-
erated waves are not harmonic at the beginning,
neither in the towing tank nor in the simulation.
For these reasons, a comparison of computed and
measured complete time histories is not possible. A
selected interval of the computed pitch angle time
history is compared with experiment in Fig. 16.
Again, the period is well predicted, but the am-
plitude is strongly underestimated. The reason for
this discrepancy most probably relies on a stronger
influence of the coarse grid in this case then for the
model-fixed condition, leading to an increased wave
damping.
In Fig. 15 the computed axial velocity in the
propeller plane is compared with the results in calm
water. The axial velocity field shows much higher
values now. The chosen instant corresponds to the
situation where a wave trough is at the stern, lead-
ing to a strong acceleration of the flow there.
Finally, we show some snapshots of the sim-
ulation performed for the C-Box container ship in
harmonic head waves. The sequence depicted in
Fig. 18 covers a periode of the surge, heave and
pitch motions, starting on the top of the figure from
a slightly positiv trimed position. The wave break-
ing mentioned above and observed in the model
tests in both, the model-fixed and the model-free
condition can certainly not be predicted very real-
istically on the coarse grid used, but are at least
present.
~ o.s
3 0.8
0-7
0.6
0.5
0.4
0-3
0.2
0.1
1 o
- :~
Figure 15: Computed axial velocity in the propeller
plane of the C-Box container ship in calm water
(top) and in head waves (bottom)
Results for the FSG RoRo ship
Computations for many combinations of the
non-dimensional yaw rate r, = rL/Uo and the drift
angle ~ at the main section were performed for a
modern RoRo ship of the FSG. The Froude number
En = 0.20 and the Reynolds number Rn = 1.53 107
were determined with a suitable speed for steady
turnings. The predictions of forces and moments
on the hull of this twin screw, twin rudder ship
model are compared with experimental data ob-
tained with the Computarized Planar Motion Car-
riage (CPMC) in the towing tank of the Hamburg
Ship Model Basin.
The ship motions were prescribed and the
free surface was not taken into account. Because
of the low Fioude number, we do not expect this
to have a significant influence on the results. The
computations were performed on a non-matching
block-structured grid with roughly 1 million cells.
Between 1000 and 2000 SIMPLE iterations were
necessary to achieve a good convergence, demand-
ing a CPU time of 7 to 14 hours on a PC.
16
1R
Figure 16: Comparison of computed (solid red line)
and Reassured pitch motion (dashed blue line) in
head waves. C-Box container ship at Fit = 0.23
__
~ = 1—·1 1 _
- _ - ~ 51 1 _
__ ~ , ~ ~ _
- _e ~ 1
-
-
i
-
Figure 17: C-Box container ship in model fixed con-
dition in head waves. Comparison of a snapshot of
the simulation on the fine grid with a photo of the
model test
The flow features at the fore body of the
RoRo ship turning with constant yaw rate r'=0.4
and drift angle p=10°, which represents a realis-
tic situation, look very similar to those of a ship in
straight motion. Because the bow shows towards
the center of the circle, the local drift angle at the
bow is practically zero. This yields a very sym-
metric pressure distribution at the bulbous bow as
shown in Fig.19. Contrary, on the rear part of the
ship both the contributions of r' and ,B to the lo-
cal drift angle yield a large cross flow which can be
discern on the pressure distribution on the skeg.
The computed non-dimensional side force Y'
Ye/ Uo2 L2) is compared with measurements in
Fig.20. Although the overall trends are captured
well, discrepancies are somewhat larger than in ear-
-
-
Figure 18: Sequence of the simulation of the flow
around the C-Box container ship in head waves,
coarse grid
tier cases. The latter especially applies to the com-
parison of the non-dimensional yaw moment N' =
N/~2 Uo2 L3) with experiments shown in Fig. 21.
These larger discrepancies are probably dun
to the propellers, shafts and V-brackets present in
the ship model, Fig. 22, which were not taken into
account in our computations. To check this and
to improve our predictions we now are going to in-
clude shafts and brackets in the computational grid
and repeat the computations. Next, computations
will include rudder deflections and heel angles.
ACKNOWLEDGEMENT
This work was supported by the German
Ministry of Education and Research.
4
2
4 ,
-6
Figure 19: Computed pressure coefficient on the
hull of the FSG RoRo ship at steady turning with
r' = 0.4 and ~ = 10°
-
-10
. , ,_____,_
...... , i,
,~ : _1 ___ _'
- ~ - ~ .......... ... ~r'=0.4~ ~
.................................................. I I
...................... , ~ '^? .,
. ~,
/~ 1 2.
.~ 1 '
............................................ 1 -.
~ : ~ - 1
3
1.5
1 5
o
-A
t
Figure 21: Comparison of measured and computed
non-dimensional yaw moment on the FSG RoRo
ship for different yaw/drift combinations
Figure 20: Comparison of measured and computed
non-dimensional side force on the FSG RoRo ship
for different yaw/drift combinations
Figure 22: Stern of the model of the FSG RoRo
ship with rudders and propellers, shafts and V-
brackets
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