J Mar Sci Technol (1998) 3:82-93
Journal of
Marine Science and Technology 9 SNAJ 1998
Finite-volume simulation of flows about a ship in maneuvering motion TAKUYA OHMORI Ship and Marine Technology Department, Research Institute, Ishikawajima-Harima Heavy Industries Co., Ltd., 1 Shin-Nakahara-cho, Isogo-ku, Yokohama 235-8501, Japan
Abstract: A finite-volume method of computing the viscous flow field about a ship in maneuvering motion was developed. The time-dependent Navier-Stokes equation discretized in the generalized boundary-fitted curvilinear coordinate system is solved numerically. A third-order upwind differencing scheme, a marker and cell (MAC)-type explicit time marching solution algorithm and a simplified subgrid scale (SGS) turbulence model are adopted. The simulation method is formulated, including the movement of a computational grid fitted to the body boundary that allows computation of the flow field around a body under unsteady motion. To estimate the maneuvering ability of a ship, the accurate prediction of the hydrodynamic forces and moments of the hull is important. Therefore, experimental methods of finding the hydrodynamic forces of a ship in maneuvering motion, such as the oblique towing test, the circular motion test (CMT) and planar motion mechanism (PMM) test, were established. Numerical simulation methods for those captive model experiments were developed introducing computational fluid dynamics (CFD). First, numerical methods for steady oblique tow and steady turn simulation were developed and then extended to unsteady forced motion. Simulations were conducted about several realistic hulls, and the results were verified by comparisons with measured results obtained in model experiments. Hydrodynamic forces and the moment, the longitudinal distribution of the hydrodynamic lateral force, and the pressure distribution on the hull surface showed good agreement.
Key words: computational fluid dynamics, maneuvering motion, unsteady flow, hydrodynamic force
Address correspondence to: T. Ohmori
Received for publication on May 12, 1997; accepted on June 15, 1998
Introduction
In recent years, the importance of navigational safety has b e c o m e much greater; accordingly the maneuverability of a ship should comply with accepted m a n e u v e r ability standards. An accurate estimation of a ship's maneuvering ability at the design stage is essential to m e e t these requirements. Therefore, the establishment of a prediction m e t h o d for hydrodynamic forces and m o m e n t s relating to the maneuvering motion, particularly those acting on the main hull of a ship, is indispensable. T h e r e exist several theoretical or numerical methods of estimating the hydrodynamic sway force and yawing m o m e n t acting on a maneuvering ship's hull, which are mainly based on the slender body assumption 1 or the nonlinear lifting surface theory. 2 However, another approach based on the surface panel m e t h o d was proposed recently? These methods concentrated on simplicity in use and an approximate estimation of the size of the hydrodynamic force rather than on the detailed structure of the flow field. However, they also require m a n y empirical parameters, such as the determination of the point of flow separation or the strength of the free vortex. Further, these methods can only express the complicated hull form of real ships in a simplified m a n n e r such as the Lewis-form approximation. Therefore, the only accurate way to obtain the hydrodynamic forces and m o m e n t s acting on a hull has been time-consuming model experiments, and a tool that can resolve the details of the flow field is felt to be necessary for the further a d v a n c e m e n t of ship design techniques. On the other hand, numerical simulation methods for viscous flow fields, by solving the Navier-Stokes equation, have developed rapidly in line with recent advances in c o m p u t e r hardware. Numerical methods have the advantage of requiring no linear approximations, so they can deal with nonlinear flow p h e n o m e n a . Numeri-
T. Ohmori: Finite-volume flows about a ship cal methods can also deal accurately with complex hull forms. Therefore, numerical methods are thought to be suitable for the analysis of viscous flow fields with strong nonlinearity, such as the flow field around a maneuvering ship. The aim of the present study is the application of the technique of computational fluid dynamics (CFD) to ship maneuverability studies by developing the numerical simulation technique of well-established and popular captive model experiments. Three typical captive model tests, namely, the oblique towing test, the circular motion test (CMT), and the planar motion mechanism (PMM) test, were chosen, and a suitable computational technique was adopted for each experiment. The outline of the computational procedure is given in the next section. The oblique towing test--uniform motion--was the primary step. The simulation technique is described first and computations are then made for several real hulls. The validity of the computational method is discussed and compared with experimental results. In the following section, steady turning mot i o n - u n i f o r m l y accelerated m o t i o n - - a s in the CMT, is dealt with in the same manner. The PMM combined motion test--unsteady forced motion--is dealt with in the subsequent section, and brief conclusions are given in the last section.
Numerical method The computational m e t h o d in this study is based on a finite-volume method WISDAM-V. 4 The timedependent Reynolds averaged Navier-Stokes equation for incompressible fluid in arbitrary Lagrangian and Eulerian ( A L E ) form is solved by a MAC-type algorithm. The computational method includes a technique to deal with the moving boundary problem by way of a moving grid system which is based on that of Rosenfeld and Kwak. 5
Computationalgrid The formulation is conducted in a boundary-fitted generalized curvilinear coordinate system. Flow variables are arranged in a staggered manner. Scalar variables such as pressure are defined at the center of a control volume, and components of vector variables such as velocity are defined at the center of each surface of the control volume (Fig. 1). A boundary-fitted coordinate (BFC) system is suitable to compute the viscous flow around a ship. The choice of the most appropriate grid topology is of great importance when BFC system is employed. In this study, an O - H - t y p e grid system is employed: the ~l-axis
83
Fig. 1. Definition of the variable arrangement
is the longitudinal direction of the ship, the ~2-axis is in the radial direction, and the ~3-axis is in the girth direction. The computational domain includes the whole ship in order to deal with the nonsymmetrical flow as the ship maneuvers. The area vector S of each cell surface is calculated as the sum of two divided triangular sections. The Jacobian J which corresponds to the cell volume is divided into three tetrahedrons and calculated as the sum of three sections. Although W I S D A M - V uses threedimensional generalized curvilinear coordinates, vectors have Cartesian components.
Governingequations The governing equations of an incompressible fluid, the continuity equation, and the Reynolds averaged Navier-Stokes equation are discretized by a finitevolume method. The conservation form of the Navier-Stokes equation is expressed as 0u = divT 0t
(1)
where u is velocity vector of the fluid. The stress tensor T, in the A L E expression, is written as
T=-pi-(u-v)u+l ~gradu+(gradu)T}-u'u"
(2)
Ke L
where i is a unit tensor, p is pressure divided by density, and v is the velocity of the grid movement. The stress term which comes from the turbulent model, is included in the diffusion term. The treatment of the
u'u',
84
T. Ohmori: Finite-volume flows about a ship
(j.)l,,+ll
__s,l.+lt pi.+ll,
At
-S i(n) . (u(") -- v(n))u (") +J( n<) {1/ R ~+ vr}V~u ('+1)
(7)
The stress term which comes from the turbulence model is incorporated in the diffusion term as the eddy viscosity, Yr. The pressure term is separated here since a MAC-type solution algorythm is employed in WISDAM-V. Applying the differential operator, Eq. (7) is rewritten as e (s~,pa,k).f,
at
a~J
Fig. 2. The convective momentum flux caused by grid deformation
-RT+Vr
2e0)
(8)
where the strain tensor is moving grid system in this method is to calculate convection terms using the relative velocity between the grid and the fluid. Consequently, the volume flux, which comes from the volume swept by the surface of the control volume, is included in the momentum flux (Fig. 2). The control volume itself changes at every time step and must be considered in the computation.
~(fvadV)=IvdivTdV
(3)
By applying the Gaussian theorem, the right-hand side of Eq. 5 is rewritten as (4)
( ~ 0xj i +-~7 eq= !2~
(9)
The continuity equation for an incompressible fluid is expressed as
dt +~(,)(a-v).dS=O
(10)
where v is the velocity of the grid point movement and V is the volume of a cell. These variables are required to satisfy the geometric conservation law
dV dt
~s v.dS= (,)
0
(11)
Using this equation, the mass conservation law Eq. (10) is rewritten as
-~(t)u dS =
0
(12)
where S is the area vector defined at the face of the control volume. In the Cartesian coordinate system, T has three components defined in those coordinates and can be discretized as follows:
Finally, it is seen that the continuity equation is in the same form as for a fixed grid system. Discretized by finite-volume method, it is written as
JdivT -
J a~J
k
1 e (5)
Then the discretized form of Eq. 4 can be derived as at
- ~S'T i
Thus
(6)
~,~ )
(13)
Differencing scheme A third-order upwind differencing scheme 6 which can cope with a nonuniform computational mesh is employed for the convection term. Following the fluxsplitting method, convection term C is represented as
T. Ohmori: Finite-volume flows about a ship
C-
OcF
85 (14)
J where F is the inviscid momentum flux. F is defined at the cell face surrounding the definition point of each component of C and the velocity. The cell around the velocity point (secondary cell) shifts from the cell around the pressure point by half of the cell size. The momentum flux is written as 8~F = F+_I - F , 2
(16)
u ("+1) = 0u(") - AtV("+l)p("+1) + 0o)i
(17)
where 0 = Jr and the pressure increment between the mth iteration step to the (m + 1)th step is defined as
which is the volume flux caused by the grid movement (Fig. 2) subtracted from the mass flux. The momentum flux is given by
F] = I (u/ - U/ )q{ + I (U/ + U] )q7
(18)
where
6p(m) = p(m+l) _ p(m)
V2p(n+l) = 0V.{//(n) -t- Iff~}
(25)
(26)
At
t2
q[ = q~+! + O~&/~+z +O56q,
(24)
and 6fi is the velocity increment which comes from all the terms except the pressure term. Taking the divergence of Eq. (24) and applying it to Eq. (23),
qi q. 1 + O;6qi-t + O~&li =
(23)
V (n+l) " U (n+l) = 0
Also, from Eq. (8), the velocity at the new time level is
where U is the volume flux, and is defined as
U=S~(u k -v k)
The MAC-type solution algorithm is employed in this study, and time advancement is explicitly conducted. After calculating the terms of the stress tensor T, the pressure is solved. First, the velocity divergence at time step (n + 1) is assumed to be zero from Eq. (13)
(15) 2
fJ = U~q
Solution algorithm
(19)
2
The Laplacian operator in generalized coordinates is c)2p 1 0 r ,,jl 0p VZp= g m' O~mO~j +7~-~[ Jg ] cg~j
(27)
and 6q~ = q,+12- q,_l
(20)
The coefficients O~, ~ , O~, and O~ are given as functions of grid spacing as follows:
Turbulence model Three types of turbulence model are available in the WISDAM-V code. These are the subgrid-scale (SGS) model, the Baldwin-Lomax model, and a hybrid of these two? The SGS model was the main one selected in this study. A simplified subgrid-scale turbulence model in the form of the eddy viscosity of the subgrid-scale turbulence model, v,, following Smagorinsky's definition of the eddy viscosity is employed.
hihi+l
hi(hi_a+hi) hihi+l ~=(h i d-hi+, +hi+2)(hi§ +hi+2) hi+l(hi+l+hi+z)
while gO is the contravariant metric tensor. Equations (27) and (26) are iteratively solved by the successive over relaxation (SOR) procedure.
(21)
*~=(h i -t-hi+l-~-hi+2)(hi 4- hi+l)
vs -- L 2 ~ = L] 09
where hi is the conjugate length, and is defined as (22)
(28)
As the length scale L,, the definition of Takakura et a12 is introduced and L, is set at half of the minimum grid spacing of the cell. The Van-Driest-type damping function is introduced in the neighborhood of the wall,
86
T. Ohmori: Finite-volume flows about a ship Simulation
L,=L,
1-exp -
of oblique
towing
(29)
Simulation technique where y+ is the viscosity unit. In addition, a wall function is introduced. Spalding's universal model of the wall s is used when y+ > 1.0.
Boundary conditions The computational domain has six types of b o u n d a r y - the inflow boundary, the side boundary, the outflow boundary, the water surface boundary, the body boundary, and the center plane boundary. The inflow and side boundary conditions are implemented in the Dirichlet manner: the pressure is set at zero and the velocity is set at a uniform velocity. However, the outflow boundary condition is the zero-normal-gradient condition. The free-surface motion is not considered in this study and the flow field is assumed to be a double-model flow. Thus a mirror-image condition is given at the water surface boundary. Since the flow field around a whole ship is computed, the flow variables are continuously computed across the center plane. As for the body boundary, a no-slip condition is imposed for the velocity. The time-dependent body boundary condition for pressure is required to compute the flow about a body in unsteady motion. Pressure on the body surface is extrapolated from the neighboring point using the acceleration of the body surface. The pressure gradient in a direction normal to the body surface is estimated from the Navier-Stokes equation together with the no-slip condition,
8P = On
--
n. VP = -n.--
& Ot
(30)
Here n is the vector normal to the body. This expression coincides with the zero-normal-gradient condition in case of steady motion.
For ship hydrodynamics, the CFD technique has been developed mainly in the fields of resistance and propulsion. The oblique towing test is similar to the resistance test, since the motion is uniform. Therefore the simulation of steady oblique towing is right as the first step toward an application to the field of maneuverability. The oblique towing condition is given by simply setting the inflow boundary condition 9 as shown in Fig. 3, u,',o. = u
cos/3
Uinflow2 m U
sinfl
(31)
where U= is the uniform flow velocity and fl is the angle of oblique towing. Flow acceleration at the start of computation is also given in the oblique direction.
Conditions of computation Two full-size ships, SR221A and SR221B, were computed. These ships had the same principal particulars (Table 1) and the fore part had the same lines. However, the ships differed in the frame line of the aft part, i.e., SR221A was V-type and SR221B was U-type. The computed drift angles were 0, 3, 6, and 9 degrees for each ship. The computed Reynolds number was set to 2.835 x 106, as in the experimental conditions. The number of grid points was 95625 (85 x 25 • 45). The inflow boundary is set as half the ship's length from the fore parpendicular (FP), the outflow boundary is set at one ship's length after the after parpendicular (AP), and the side boundary is set at 0.75 of the ship's length from the centerline. The minimum grid spacing in the longitudinal direction at bow and stern is set at 5 x 10-3 of the ship's length, and that in radial direction at midship is 7 x 10 ~ of the ship's length, with 6 x 10-4 at AP and FP. The computation is continued until the flow field reaches the steady state for T = 2.0 or 2.5, where T
Fig. 3. The oblique tow simulation
T. Ohmori: Finite-volume flows about a ship
87
is nondimensionalized time. The computation was fairly stable, and the convergence of the pressure iteration was good.
Results and discussion A concise description is given here, and detailed results about oblique towing and steady turning can be found elsewhere.10-12
Table 1. Principal data for the models used in the experiments Length Breadth Depth Draft Displacement volume Block coefficient Prismatic coefficient Prismatic coefficient Prismatic coefficient Midship coefficient LCB
(L) (B) (D) (d)
(m) (m) (m) (m) (m 3) (CB) (Cp)
(Cpa) (CpF) (CM) (%L) SR221A
0.08 0.07 0.06 0.05
-
'
'
i -:: :
'
I
9 [] o A
:t
'
'
Y' Y' Y' Y'
'
I
'
'
'
I
SR221B
3.5000 0.6344 0.3281 0.2111 0.37719 0.8045 0.8084 0.7504 0.8611 0.9952 -2.4500
3.5000 0.6344 0.3281 0.2111 0.37595 0.8018 0.8057 0.7557 0.8611 0.9952 -2.6100
'
(32)
where U is the uniform velocity, L is the length between parpendiculars, and d is the draft. A definition of the positive direction is shown in Fig. 3. C o m p a r i n g the two ships, SR221A, with a V-type frame line, shows a smaller Y' and a greater N ' than SR221B with a U-type f r a m e line. This tendency is also qualitatively evident in the c o m p u t e d results. H o w e v e r , quantative agreement becomes worse as/3 increases.
'
'
'
'
'
I ................................................. I i I ........... i. . . . . . i _ I z~
I
i
Y'
0.08
'
i
0.07 0.06
................
i ..................
i ................
0.05 0.04
0.03
..........~............... =
~ ....~.......... ~ 1i......i..1.....i...............~..o. ; ; ~.....
0.03
- ......... ~ - ~ , . . ~ .
0.02
0.02
A
~
..........
0.01
, , , i,'=o,j~, i . . . . . . . . . .
0 -2
0
2
4 beta (deg)
SR221A 0.03
i
,
J
i
J
i
i
~
,
i
6
'
................
i
" ..................
" .........
i
,
0
2
4
beta (deg)
6
8
10
,
,
0.03
I ~
,
,
,
,
~
,
,
,
,
i
i
,
i
i
i
~
i
J
r'=O
::r'=0
0.02
1
...... ~ . . . . . . . i...........Q
~
........... i .........
SR221 B N'
'
"i .... "r'~
i
v
-2
1
::
i
i .-,.,.=oi
................
0.01 0
8
............. J ..................................................... i .................. i ......
0.02
S,-.o.2
....... ~ - - - i
N' P
::
0
0.01
.............i..................i...
~
i
-
-
0 -0.01
-0.01
I
*
N'
i I .........................................
[] o A
N' Exp. (r'=0) It N' Exp. (r'=0.2) I1 N' Exp. (r'=0.4) I~
$ -0.02
/
SR221B
'
0.04
0.01
Y 1/2pUZLd N N" 1/2pU2LZd y
Y'
'
Cal. Exp. (r'=0) Exp. (r'=0.2) Exp. (r'=0.4)
SR221A
First, a comparison of the c o m p u t e d and measured 1~ nondimensional hydrodynamic lateral force Y' and the yawing m o m e n t N' is shown in Fig. 4, which also incorporates the case of steady turning motion. The oblique towing case is shown as r' = 0, where r' is nondimensional angular velocity. The c o m p u t e d value is obtained by integrating the hydrodynamic pressure over the ship's surface. Y" and N' are nondimensionalized as follows:
-0.03
i
-2
t
=
~
,
0
,
,
,
2
,
,
.
.
.
.
.
4 beta (deg)
.
.
.
.
6
Cal.
.
.
.
.
I~
-0.03
.
8
10
i'=0-2
:
-0.02 t
-2
=
i
i
0
i
i
i
2
r
i
=
4
=
beta (deg)
i
=
6
Fig. 4. Comparison of the computed and measured hydrodynamic Lateral force Y' and yawing moment N'
i
~
m
8
i
t
88
T. Ohmori: Finite-volume flows about a ship
Z~Y' 0.3
I
~.
I
I
..+. "~
~
I
SR221A Cal. SR221B Cal. SR221A Exp. SR221B Exp.
~
-- .... -o---+ --
0.2
0.1
-0.1 -0.6
F.P.
I
I
-0.4
-0.2
0 X/Lpp
I
I
0,2
0.4
A.P.
0.6
Fig. 5. Longitudinal distribution of lateral force in oblique tow conditions when fl = 9 ~
Second, the longitudinal distribution of the computed and measured hydrodynamic lateral force for both models in the case of/3 = 9 ~ is shown in Fig. 5, where the horizontal axis is the nondimensionalized coordinate in the longitudinal direction: -0.5 at FP and 0.5 at AP. AY" is the lateral force per unit length, and is nondimensionalized as follows: AY A Y ' = 1/2pU2 d
(33)
and AY is obtained by integrating the surface pressure along the girth line. The measured value is obtained from the pressure measurements at more than 400 points in each ship. The simulated and measured lateral force distributions agree well, and show that a large amount of positive lateral force is generated in the bow region, a relatively small amount in the parallel section, and a negative amount in the aft part which then returns to a positive amount at the aft end. Since the fore part of the two models are identical, the distribution in the fore part is almost the same. However, there are significant differences in the aft part. In the case of SR221A, the lateral force has a peak negative value at SS(square station)2, and gradually approaches zero towards the aft end, while in the case of SR221B it has a smaller negative peak at SS2.5 and becomes positive again near the aft end. All these features are clearly simulated by the numerical computation, which means that this method can predict the differences in the hydrodynamic lateral force caused by the change of frame line. In addition, the relation be-
tween the hull form, flow field, and hydrodynamic forces might be shown by examining the simulated flow field. Such a study, which requires a huge amount of experimental work, is an ideal use of the CFD technique. However, some points still need improvement. Computed values disagree with experimental values in the bow section, which might be caused either by the inadequacy of the grid system in the bow region, or by the free surface effect, which was neglected in the computation.
Simulation of steady turning Simulation technique CMT is a steady turning motion and the flow field is uniformly accelerated. Therefore the computation is carried out in a body-fixed coordinate system, and the steady turning motion is simulated by adding the centrifugal force and the Coriolis force to the external force term of the Navier-Stokes (NS) equation in the form of body force. A coordinate system o-xtx2x 3 is fixed in a hull which is in steady turning motion, and which is located in the earth's coordinate system O-X~X2X3. The origin 0 is set at the waterline at midship, and the axes x ~, x 2, x 3 are in the backward, starboard, and upward directions, respectively. The hull-fixed coordinate system is assumed to rotate steadily with a radius ro, angular velocity 09, and drift angle fl in an anticlockwise direction. Consider a point P with the positional vector P,
T. Ohmori: Finite-volume flows about a ship dP -
=
-
dt
cox
P
89 (34)
= p V c o 2 r - pV(co, r)co
where ~o is the angular velocity vector, and assume that the X3-axis is the axis of rotation. This can be written as
(35)
c o = eoI 3
w h e r e / 3 is the unit vector in the X 3 direction. F r o m Eq. (34), di m dt
- p V c o x ( cox r) = p V ( cox r) x co
= cox i"
= pVw2r,
(41)
where r,, is a vector from the rotational axis to the location r in the x L x 2 plane. Considering the motion of the origin 0, the accerelation of fluid at the location p(xl, x~, x 3) caused by the centrifugal force is rp cosr + r0 sinfl ~
(36)
where i m is the unit vector in the x 'n direction in the hullfixed coordinates. The time difference in the inertial coordinates of any vector r with components (rl, r2, r3) defined in the hull-fixed coordinates can be written using Eq. (35)
ace.my = O)2trp s i n ~ ; r0 cos]3
(42)
where rp = ~/(x~)g + (x12) e r = tan_l(x~(2 )
(43)
as
di 2 r3 -di- 3 dr _ drl il + dr2 i2 + dr3 i3 + rl -di 1 dt dt dt dt dt + r2 ~ + dt 6r = - - + cox r &
(37)
where
acoriolis =
& - dr1 i 1 + dr2 i 2 + -dr3 -i 3 & dt dt dt
expressed as in Eq. (37), and the acceleration is dZr ~2F ~O) ~l" r + 2cox-~+ cox (cox r) dt 2 ~2t 4-~ x
(39)
6c0 dr Considering the 8t - d t motion equation for an infinitely small volume of fluid at a location r in the hullfixed coordinates is ~2r Sr P V &2 : F - 2 p V c o x - ~ - p V c o x ( c o x r ) dto
-pV--~
x r
--2m
(44)
(38)
is the rate of variation in time of r in the hull-fixed coordinates and o~x r is the variation in time caused by the rotation of the coordinates fixed to the hull. dr. Thus, the velocity ~-~ in the inertial coordinate is
(40)
The second term of the right-hand side of Eq. (40) is the Coriolis force and the third term is the centrifugal force. The fourth term vanishes in the case of steady turning motion. Considering that the axis of rotation is parallel to the x3-axis, the centrifugal force term can be rewritten as
The Coriolis force term is derived as -209/3 x u from Eq. (35). T h e r e f o r e the components of the acceleration term are
where u = (u 1, u 2, u 3) is the fluid velocity in the coordinates fixed to the hull. Conditions o f computation
Simulatiosn were conducted for the models SR221A and SR221B. The selected combination of turning rate and drift angle was (r',/3) = (0.2, 0), (0.2, 9 deg), (0.4, 0). The conditions of the computation were the same as in the case of oblique towing except that the n u m b e r of grid points was increased to about 190000 (101 x 31 x 61), the hybrid-type turbulence model was employed, and thus the minimum grid spacing in the radial direction at midship was set at 1 x 104 of ship's length, with 8 = 104 at A P and 1.5 x 10-3 at FP. The hybrid turbulence model is a very recent development, and it would be desirable to calculate the oblique towing cases twice in order to compare and validate the two models. Since the computational results for the same number of grid points as in the case of oblique towing were not fully satisfactory, the n u m b e r was increased. For the case of steady turning motion, it seems that a higher grid resolution is required than for oblique towing. Although it was suspected that the grid resolution was insufficient, the physical restrictions of the computer hardware did not allow the use of a larger number of grid points.
90
T. Ohmori: Finite-volume flows about a ship
0,2
~y'
0.1 0
" {*'~'! '~(;.... ........ r ~ :..................................................................................... ~ ~ : i a:~ ! '~/*'~ ~ " ~,i'[3t~~
-0.1 -0.2 9g -0,3 -0.4 -0.5
/I /I V
SR221BCal. -..... SR221AExp. --~---SR221BExp...-§
9 (a)
I -0.6 -0.6 F.P. -0.4
r'-0.4
B ~0deg
i
i
I
-0.2
0 X/Lpp
0.2
i
o.4 A.P.o.6
0,2
0
t
. . . . . . . . . . . .
. . . . . . . . . . . . .
.....
between computed and measured values at the bow region, as in the case of oblique towing. As stated in the previous section, the sign of the lateral force caused by the drift is positive in the fore and parallel parts of the ship and negative in the aft part. Conversely, that resulting from the turning motion is negative in the fore part and positive in the parallel and aft parts. Moreover, the lateral force in the aft part of SR221A tends to have a positive value, while that of SR221B tends to be negative. The total lateral force distribution of a turning hull is composed of these elements, and the mechanism of its generation is very complicated. That may be why it was not possible to get a sufficiently accurate prediction using theoretical methods. It is encouraging that the present method succeeded so well in predicting such an intricate flow field and the resulting hydrodynamic forces, since it requires few artificial parameters such as the position or strength of the free vortex.
................-
i
~
~
-0.1
Simulation of unsteady motion
-0.2 ~7: -0.3 -0.6 F.P.-O.4
Simulation technique
(b) r'-0 2 B -0deg -o.2
0 X/Lpp
0.2
0.4 A.P. 0.6
0.2 Ay' 0.1
,4 v
.........................................
-0,1
~
%"-z52 ' 1 ,
~,<+..: . . . . .
~.5 /
.........
(e) r'=0.2, /3=gdao
-0.2 -0.6 F.P.-0.4 I
I
0.2
i
I
0 X/Lpp
0.2
L
0.4 A.P. 0.6
Fig. 6. Comparison between the measured and computed longitudinal distribution of the hydrodynamic lateral force in steady turning motion
T o develop a simulation method for a captive model test which consists of unsteady motion, the planar motion mechanism (PMM) test was selected. The most suitable approach to the problem is to employ the moving grid technique, since the idea of a moving grid system is included in the formulation of WISDAM-V. The combined motion of sway and yaw is expressed in such a way that the heading becomes the cosine function of time and the center of the ship moves uniformly,
/ as shown in Fig. 7. The velocity and acceleration of the body surface m o v e m e n t which are used in the body boundary conditions are also derive.d from this function.
Conditions of computation Results and discussion The computed forces and moments are shown in Fig. 4, and the distribution of the hydrodynamic lateral force z~Y' is shown in Fig. 6. Quantitatively, simulated results agree well with measured results, 11,12which shows that this method can precisely predict the hydrodynamic forces in steady turning motion. However, several problems do exist. Any rapid change in the hydrodynamic force distribution, such as that around SS1 of SR221B (r',/3) = (0.2, 0), is likely to cause an insufficient resolution of the vortical fluid which comes from the grid and the turbulence model. Some discrepancies also exist
The SR221B model was chosen for the test. The number of grid points was about 115000 (91 x 31 x 41). The amplitude of the motion was/30 = 14.5 ~ and the period was T = 4.98 in nondimensional time. The inflow boundary was set at one ship's length from FP, and the radius of the domain was set at twice the ship's length. The grid spacing was the same as for the oblique tow and steady turn.
Results and discussion The computed time-history variation of the lateral force and yawing m o m e n t is compared with the measured
T. Ohmori: Finite-volume flows about a ship
91 0.31 -'
L.
v
0.2
/
9
0.1
-0.1 ( ./
-0.2
y,
__
N'
Beta ...... i
-0.3 ' 0
1
2
3
7
0.3 /" /
0.2
/ / {
0.1
-0.1
-0.2
y,
_ _
Beta -
-~
7'
8
i
9
10
11
12
13
Fig. 8. Time history of the lateral force Y', the yawing moment N', and the yaw angle /3, measured (above) and computed (below) Fig. 7. Computational grids moving together with the ship's motion
results in Fig. 8. The agreement is satisfactory for lateral force, but is not so good for the yawing moment. The cause of that discrepancy is presumed to be partly experimental uncertainty. Because the magnitude and phase difference of the lateral force is satisfactory, the flow field around the hull in motion is considered to be well resolved. The time-history variation of the lateral force distribution for half a period is shown in Fig. 9. The lateral force of the fore section is a very large part of the total lateral force and is in a different phase from that of the aft section, while the lateral force in the parallel section does not make a significant contribution. The detailed flow field at the instant t = 1/8T is shown in Fig. 10. The fore part is moving toward the starboard side and the aft part is moving toward the port side. The hull is fully covered with vortical layers, and the wake is
spread widely in the lateral direction. Such a detailed resolution of this complicated flow field is extremely important in a rigorous evaluation of the forces acting on a hull in maneuvering motion. However, it must be noted that the development of the simulation technique is not yet complete, and some discrepancies still exist between computed and measured results.
Conclusions A time-dependent finite-volume simulation method for the viscous flow field around a ship in maneuvering motion has been developed and verified using computations for full ship forms. The method successfully resolved the flow around a hull, and the hydrodynamic forces and moments during steady oblique towing,
92
T. Ohmori: Finite-volume flows about a ship
0.6 Ay' 0.4
1116 - 2/16 ..... 3/16 ...... 4/16 ......... 5/16 ..... 6/16 ..... 7/16 ...... 8/16
~ - ~ ? ,. ~ ! ~ ],., ~
. . . . . .
0.2 ;. . . .
."
,.'
..
--'
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........
,..""".-..
...
"2=,-z'"
......
!.
'.
.
227--,.,,
.
,.,.
.
~,~.,
-0.2
-0.4
-0.6 -0.6
i
I
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I
-0.2
0
I
0.2
0.4 A.P. 0.6
Fig. 9. Time-history variation of the longitudinal distribution of the computed lateral force
iJ
\,,,
s s6',
port ~
S,S.5
&
S.S.5
~ J
.-
~
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) '" j
steady turning, and unsteady further work is needed to this m e t h o d is believed to be a ing the hydrodynamic forces state.
motion. Although some improve the accuracy, valuable tool for evaluaton a ship at the design
Fig. 10. Contours of computed u l ( l e f t ) and ~ ( r i g h t ) in cross-sectional planes at t = 1 / 8 7 , Aul = 0.1, and Ao~ = 20. The vorticity contours of anticlockwise rotation are drawn in bold lines, and those of clockwise rotation are drawn in thin lines
The author wishes to express sincere gratitude to the Shipbuilding Research Association of J a p a n (JSRA). Thanks are also expressed to Prof. H. Miyata, Prof. M. Fujino, and all laboratory m e m b e r s for their assistance in the experiment.
Acknowledgments.
T. O h m o r i : F i n i t e - v o l u m e flows a b o u t a ship
References 1. Fuwa T (1973) Hydrodynamic forces acting on a ship in oblique towing (in Japanese). J Soc Nav Archit Jpn 134:135-192 2. Inoue S, Hirano M, Kijima K (1981) Hydrodynamic derivatives on ship manoeuvring. Int Shipbuild Prog 14:321-328 3. Yang J, Matsui S, Tamashima M et al (1994) On the mututal interaction between hull and propeller-rudder system of an obliquely sailing ship (in Japanese). Trans West Japan Soc Nav Archit 87:61-79 4. Zhu M, Yoshida O, Miyata H et al (1993) Verification of the viscous flow-field simulation for practical hull forms by a finitevolume method. Proceedings of the 6th International Conference on Numerical Ship Hydrodynamics, IA, pp 469-487 5. Rosenfeld M, Kwak D (1991) Time-dependent solutions of viscous incompressible flows in moving co-ordinates. Int J Numer Methods Fluids 13:1311-1328 6. Sawada K, Takanashi S (1987) A numerical investigation on wing/nacelle interfaces of USB configuration. A I A A Paper 870455
93 7. Takakura Y, Ogawa S, lshiguro T (1989) Turbulence models for transsonic viscous flow. A I A A Paper 89-1952CP 8. Spalding DB (1961) A single formula for the "Law of the Wall". J Appl Mech Sept 455-458 9. Ohmori T, Miyata H (1993) Oblique two simulation by a finitevolume method. J Soc Nav Archit Jpn 173:27-34 10. Ohmori T, Fujino M, Miyata H et al (1994) A study on flow field around full ship forms in maneuvering motion (lst report: in oblique two) (in Japanese). J Soc Nav Archit Jpn 176:241250 11. Fujino M, Ohmori T, Usami S e t al (1995) A Study of flow field around full ship forms in maneuvering motion (2nd report: Hydrodynamic forces and pressure distribution on ship's hull in steady turning condition) (in Japanese). J Soc Nav Archit Jpn 177:13-28 12. Ohmori T, Fujino M, Tatsumi K et al (1996) A study on flow field around full ship forms in maneuvering motion (3rd report: Flow field around ship's hull in steady turning condition) (in Japanese). J Soc Nav Archit Jpn 179:125-~138