The 24th Computational Fluid Dynamics Symposium
Paper No.255

Numerical Simulation of Asymmetric Flow Past a Full Hull Form Under Strong Wall
Effecto

Yoshiaki Kodama, National Maritime Research Institute, Tokyo, kodama@nmri.go.jp
Takanori Hino, National Maritime Research Institute, Tokyo, hino@nmri.go.jp
Kunihide Ohashi, National Maritime Research Institute, Tokyo, k-ohashi@nmri.go.jp

The stern flow of a 6m-long bulk carrier model placed in a tunnel of 2m x 0.88m rectangular cross section,
became asymmetric due to strong wall effect, but recovered symmetry by setting a pair of flow liners at thecorners of the section and thus reducing the wall effect. The k- SST turbulence model can reproduce thephenomena, while the Spalart-Allmaras model and he k- BSL model cannot. It has been confirmed that theresult is not affected by grid resolution. A simple oblique towing method with an oblique angle of 0.1 degrees,
whose magnitude is not important because it works as a trigger, has been found useful for testing thesusceptibility or resistance of the flow to asymmetry.

1. Introduction
In order to improve loading efficiency, the hull forms ofdisplacement-type ships such as tankers and bulk carriers are
becoming more and more like a rectangular box. The block
coefficient C B defined byCB [displacement][length]  [width]  [water depth](1)

Therefore, it is a standard procedure (1) to measure the wake
distribution at the propeller disk in the towing tank, (2)to estimate
the full-scale wake distribution using a suitable estimation method(1)such as the Sasajima-Tanaka's method , and to make the wake
distribution measured in the tunnel as close as possible to theestimated distribution by changing the size and the streamwise
location of a pair of flow liners fixed at the lower corners of the test
section in the stern area, as shown in Fig.1(d).

, which indicates the boxiness or fullness of a hull form, takes
slightly over 0.8 with conventional tankers, but exceeds 0.9 with
newly sought Post-Panamax hull forms, which will be used in theenlarged Panama Canal.

W. L.

(a)Towing tank

The stern flow of hull forms of very large C B tends to beunstable due to asymmetry of the flow, alternating sides in time.
This phenomenon restricts the applicability of the hull form.
Therefore, being able to predict flow asymmetry has practicalimportance.
The cavitation tunnel of NMRI has a rectangular test section
whose dimensions are 8.0m length, 2.0m width, and 0.88m depth,
in which a ship model of length up to 6m can be fixed to the top
wall, so that cavitation tests can be carried out for a propeller
operating in the wake of the model. However, since the wall effectis non-negligible, the wake at the propeller plane is not the sameas that in the towing tank.
Fig.1(a) schematically shows a ship model slowly towed in a
towing tank so that the wave effect is negligible. In case the modelis installed in a rectangular test section after cutting it slightly above
the waterline, as shown in Fig.1(b), the wall effect accelerates theflow past the model near the bow, and decelerates near the stern,
resulting in thinning and thickening of the boundary layer on themodel in the front half and rear half, respectively.
As shown in Fig.1(c), if an ideal flow liner, whose shape isequal to a stream tube surrounding the model in the towing tank,is installed in the test section, one should be able to cancel out thewall effect at least theoretically. Since the final goal is to test thepropeller in the wake of a full-scale ship, whose length exceeds
300m, the thinning effect in the front half is useful because thefull-scale boundary layer is thinner than that of the model scale.

top wallConverging flow

Diverging flow(b) Cavitation tunnel (without flow liner)(c) Cavitation tunnel (with an “ideal” flow liner)

Converging flow
Flow liner(d) Cavitation tunnel (with a real flow liner)
Fig.1 Wall effect in the cavitation tunnel and its reduction using apair of flow liners.
In case the wall effect in the cavitation tunnel is strong, thestern flow sometimes becomes asymmetric, in spite of thegeometrical symmetry of the model and the tunnel wall. Then theuse of flow liners works to recover the flow symmetry. In reality, theCopyright © 2010 by JSFM

The 24th Computational Fluid Dynamics Symposium
Paper No.255
model geometry and the upstream flow are not symmetricbecause of manufacturing and setting errors, and somehow it isalways the starboard side (the right side if seen from downstream)
where the stern flow is slower in the cavitation tunnel of NMRI.
(2)
In this paper, the SR174B4 ship model , a research hull form
modeling a bulk carrier is used as a test case. It has the length
Lpp=6.0m, the width B=0.92m, and the block coefficient CB=0.824.
The wake of the model is symmetric in the towing tank,asymmetric in the cavitation tunnel without flow liners, and(2)symmetric again with the use of flow liners .
It is a challenging test for CFD whether it can simulate thinningand thickening of the boundary layer and the formation ofasymmetrical flow past a geometrically symmetrical body, bothdue to the wall effect. As to the thinning and thickening(3)
phenomena, Kodama et al. computed flows past the same ship
(4)
model using the Modified Spalart-Allmaras turbulence modeland a single-side grid assuming symmetry, and obtained goodcorrelation between measured and computed wake distributions.
The formation of asymmetrical flow past a geometricallysymmetrical body is more challenging to CFD. it should be noted
that, in actual computations, although body geometry can bemade symmetric with machine accuracy, solution-updating using
a relaxation method can trigger flow asymmetry throughdirectional sweeps and insufficient convergence at each time step.
2. CFD solver(7)
The CFD solver used is SURF (version 6.43), an
incompressible flow solver for ship flows, based on thepseudo-compressibility approach on unstructured grids.
Incompressibility of the computed flow is satisfied only atconvergence. The solver uses the point Gauss-Seidel relaxation
method for updating the solution.
All the computations were started from the zero-flow state
everywhere, then increasing the u-component uniformly at eachtime step until the 100th time step. The pseudo-compressibility
parameter  was 1.0. The local time-stepping was used with thelocal Courant number CFL=5.0, but sometimes the CFL was
reduced to 2.0 to get convergence. No sub-iteration was applied
at each time step. Ref.(3) used the same solver with the same
parameters.
3. Turbulence models
Four kinds of turbulence models were used.
(4)(1)Spalart-Allmaras model (SA)The model is based on the transport equation for the
kinematic eddy viscosity t (more precisely ~ , a variant of t),which is tuned for free shear layers and wall boundary layers inlaminar and turbulent regimes. In the boundary layer in adverse
pressure gradient, it gives larger t. than experiment, which is also
the case with conventional turbulence models such as theCebeci-Smith model and the standard k- model.
(5)(2)Modified Spalart-Allmaras model (MSA)
In the model, the production term fort is made smaller in aregion where, as in the vortex core, the strain rate tensor is smaller
than the vorticity. The model is known to produce good results inthe ship stern flow where there is a pair of strong longitudinalvortices.

(6)(3) k- BSL model (KOBSL)In order to avoid strong sensitivity of the original k- model on
the free-stream value, Menter combined the original k- modeland the standard k- model, in which the original k- model isused in the inner layer while the standard k- model is used in theouter layer, so that the unwanted sensitivity is avoided.
(4) k- SST model (KOSST) (5)Menter further modified the k- BSL model to develop the k-
SST model, in which t satisfies the Bradshaw's assumption thatthe shear stress in a boundary layer is proportional to the turbulent
kinetic energy k. This modification results in substantiallydecreased values of t in adverse pressure gradient zones.
The actual equation forms are as follows, with  being the
Reynolds shear stress and  being the absolute value of vorticity.
The original form of the kinematic eddy viscosity is

t 

k ,

(2)



The Bradshaw's assumption is in the form
 a1k ,
(3)

whereas, in two-equation models, the Reynolds shear stress isexpressed as t .
(4)

In the k- SST model, the kinematic eddy viscosity is defined as

t 

a1k
max(a1 ; F2 )(5)

F2  tanh(arg22 )

(6)


k
arg 2  max 2
0
.
09
y


500  .
y 2 

(7)

In the inner layer, arg2 becomes very large, and F2 becomes unity.
Then eq.(5) reduces to

t 

a1k ,


(8)

which produces eq.(3) by substituting into eq.(4).
In the outer layer, arg2 goes to zero, and so is F2. Then eq.(5)reduces to eq.(2), i.e. the original form.
(8)
Hino computed flows past a tanker form using SURF withthe k- SST turbulence model.
4. Grid(1)Grid topology and resolution
Grids have the H-O topology, covering both sides of the ship
hull. The grid coordinates are normalized by the ship length Lpp(length between perpendiculars) with the origin at midship center.
The bottom boundary consists of the front cut, the hull surface,
and the rear cut. The top boundary is the outer boundary, which
consists of a half cylinder of radius 1.5, or the cavitation tunnel wall,or the tunnel wall with a pair of flow liners as a part of it. In the latter
Copyright © 2010 by JSFM

two cases, the grid was clustered toward the top boundary as well,with the minimum spacing being 0.001 and the free-slip boundary
condition being imposed. The left and right boundaries consist of
the z=0 symmetry plane, simulating the free surface or the top wallof the tunnel. The upstream and downstream boundaries arelocated at x=-1.5 and x=1.5. The Dirchlet boundary condition forpressure, i.e. p=0.0, was imposed at the downstream boundary.
Most computations have been carried out using the standard
grid (StdG), whose number of grid points are 131*89*61 in theI(upstream-downstream), J (girthwise), and K (bottom-top)-6
directions. The minimum spacing is 0.35*10 .
For the grid resolution study, the fine grid (FineG), whose
number of grid points is 191*133*91, has been used. The grid
resolution has been made 1.5 times in all the three directions in
the rear half, while keeping the minimum spacing unchanged.
(2)
The flow liners used are FL2 , which have the cylindricalparallel part of radius 0.44m and length 0.6m with front (1.1mlength) and rear (0.5m length) conical parts. In the computation,
the rear conical part has been made three times longer than real,in order to stabilize the computation.
Fig.2 shows the surface grids. The propeller disk is at
x=0.4888 with the radius 0.0205.

The 24th Computational Fluid Dynamics Symposium
Paper No.255
shifted accordingly. The grid points on the top boundary areunchanged. The grid points between the two boundaries are
shifted by linear interpolation based on the distance. The rotation
angle  is 0.1 or 0.2 degrees. The rotation  =0.1 deg. causes they-shift of +1cm at the stern end of a 6m-long ship model.
yOuter boundary
Original grid point
New grid point

Original hull surface

Upstream boundary

Downstream boundary

Fig.3 Hull surface grid rotation by angle 
4. Towing tank (rad1.5)
The model with turbulence stimulators at the bow was towed
in a towing tank, whose width is 18m and the water depth is 7m.
The towing speed was 1.35m/s, resulting in the Froude number
6
Fn=0.176 and the Reynolds number Re=8.28*10 .
Fig.4 shows the measured velocity distribution at the propeller(2)plane (x=0.4888) , looked from behind, where the u-component
is normalized by the towing speed. Since there is a pair of
longitudinal vortices of opposite sign in the wake, the contour lines
exhibit a so-called "hook" shape. The bulge of the slow speedzone at the center is caused not only by the induced velocities ofthe longitudinal vortices but also by the shaft housing for the model
propeller.
0.5

0.3

0.3

0.2

0.5

0.7

0.7

(a)Hull surface and tunnel wall grids with flow liners (StdG)

x



0.9

0.9
0.2

Fig.4 Measured towing tank wake at propeller plane(b)Stern grid with the propeller disk(FineG).
Fig.2 Surface grids(2)Hull surface grid rotation
We have found out that computing the flow by rotating the hull
surface only slightly about the hull center is a good test for thesusceptibility of the flow to asymmetry. As shown in Fig.3, thebottom surface grids are shifted first. The hull surface grid isrotated by the angle , then the grid points on the two cuts are

Fig.5 shows the computed flow field using StdG and KOSST.
The water surface was approximated as the symmetry plane,
because the corresponding Froude number is low enough toneglect the free-surface wave effect. The cylindrical outerboundary is at the radius 1.5 (Rad1.5), which assures the wall
6
effect being negligibly small. The Reynolds number is Re=8.28*10 ,the k- SST turbulence model, and the standard grid.
Fig.5(a) shows the contour lines in case the rotation angle =0.0 deg.. They are symmetrical and show reasonable agreementwith the measurement, except that the u=0.9 line is significantly
wider and that the slow speed bulge is smaller. It should be notedthat the grid resolution is not sufficient and the shaft housing is nottaken into account in the computation.
Fig.5(b) shows the rotation angle  =0.1 deg. case. The low
Copyright © 2010 by JSFM

The 24th Computational Fluid Dynamics Symposium
Paper No.255
speed bulge is slightly shifted toward left.
Fig.5(c) shows the hull surface pressure at the stern. It also
shows the u=0 line, denoting the separation zone.

1.0E‐03
5.0E‐04Cy

0.0E+00‐5.0E‐04‐1.0E‐03
0

1000

2000

3000

4000

5000

6000

7000

8000

time stepFig.7 Time history of Cy in the cavitation tunnel (no FL, StdG, =0.0 deg., MSA)(a)Wake at propeller plane ( =0.0 deg.)(b)Wake at propeller plane ( =0.1 deg.)
x=0.4

0.45

0.5

0.05

p=0.0

0.1
u=0.0

Figs.8(a),(b),(c) show the wake distribution at the propeller
plane (x=0.4888) at  =0.0, 0.1, and 0.2 deg. The distributions arevery similar to each other and asymmetric, the left side being
slower, opposite to the measured result. Fig.8(d) shows the
distribution at  = - 0.1 deg., which is anti-symmetric with theprevious three. These results suggest that the stern flow is highly
susceptible to asymmetry and that only a small bias triggers theasymmetry.
Figs.8(e),(f) show the surface pressure distribution and theseparation zone (u<0.0) at the stern. At first the surface pressure islower than that of the unbounded case, because of the flow
acceleration up to midship, but the recovery toward the stern end
is more rapid. In agreement with the flow asymmetry, thedistributions are asymmetric.

‐0.05
propeller disk(c)Stern surface pressure
0.1

Fig.5 Computed flow field (Rad1.5, StdG,  =0.0 deg., KOSST)
5. Cavitation tunnel without flow liners (no FL)
The same ship model was fixed to the top wall of thecavitation tunnel, after cutting it slightly above the water plane. The
average flow speed in the test section was 3.8m/s corresponding7
to Re=1.97*10 . Fig.6 shows the measured u-contours at thepropeller plane, normalized by the maximum value inside thepropeller disk. The pattern shows strong asymmetry, the right side
being slower.
0.5

0.3

u=1.0

(a)Wake at propeller plane ( =0.0 deg.)

0.3
0.5

0.7

0.1

(b)Wake at propeller plane ( =0.1 deg.)
0.7

0.9
0.9
0.3

Fig.6 Measured cavitation tunnel wake at propeller plane (no FL)(c)Wake at propeller plane ( =0.2 deg.)

In case the modified Spalart-Allmaras turbulence model wasused, the computed flow in the  =0.0 deg. case showed sternflow separation on alternating sides, and did not converge. Fig.7
shows the time history of the side force Cy.
Copyright © 2010 by JSFM

The 24th Computational Fluid Dynamics Symposium
Paper No.255(d)Wake at propeller plane ( =-0.1 deg.)
x=0.4
‐0.1

‐0.05

0.45
0.47
p=0.0

(b) =0.1 deg.
0.0

0.5
0.05
u=0.0

0.5
0.5

0.7

0.3

0.7

0.1
propeller disk

0.9
0.3

(e)Stern surface pressure (left side,  =0.0 deg.)
u/u_max=0.9(c) =0.0 deg.(umax=0.98)(f)Stern surface pressure (right side,  =0.0 deg.)
Fig.8 Computed flow field in the cavitation tunnel (no FL, StdG,
KOSST)
Figs.9(a),(b) show the wake distributions at the propeller planein case the fine grid (FineG) is used. By comparing Fig.8(b) and
Fig.9(b), it is seen that the grid refinement makes the slow region
even slower and wider, while maintaining the overall character. As
shown in Fig.9(a), the t  =0.0 case of the fine grid gives the slow
region on the right, in contrast to the standard grid case. Insummary, it may be stated that the  =0.0 case gives the
distribution very close to either the  =0.1 case or the  = - 0.1
case, which are anti-symmetric.
Fig.9(c) shows the wake distribution at the propeller plane (=0.0), normalized by the maximum value in the propeller disk
(umax=0.98). By comparing the figure with Fig.6, it is seen that thecomputed result shows reasonable agreement with themeasurement.
Fig.9(d) shows the wake distribution plus the velocity vectors
on the plane. A pair of longitudinal vortices are clearly seen withthe center of the left one being lower than the right one, causing
the asymmetric wake distribution.

(d) =0.0 deg.(u_max=0.98, vectors)
Fig.9 Computed cavitation tunnel wake at propeller plane (no FL,
FineG, KOSST)
Figs.10(a),(b) show the wake distributions at the propeller
plane in case the Spalart-Allmaras turbulence model is used. Incase  =0.0 deg., the converged flow is highly symmetric, which isin contrast to the k- SST turbulence model case. As shown in
Fig.10(b), the rotation by 0.1 degrees causes asymmetry only
slightly.

0.0

0.1
0.9

0.1
0.9

u=1.0

u=1.0

(a) =0.0 deg.

(a) =0.0 deg.

Copyright © 2010 by JSFM

The 24th Computational Fluid Dynamics Symposium
Paper No.255
hull

hull

0.3e‐4‐0.1(b) =0.1 deg.

u=1.0
0.1e‐4

Wake
Fig.10 Computed cavitation tunnel wake at propeller plane (no Fl,
StdG, SA )t(a)x=0.47

hull

hull
0.3e‐4

The results using the k- BSL turbulence model, shown in
Fig.11(a),(b) are similar to that of SA, i.e. the model fails to predictflow asymmetry. By comparing it with those of the k- SST
turbulence model shown in Fig.8, it is amazing that such a small
modification as eq.(5) significantly improves the capability of the
turbulence model.

0.3e‐4

propeller
disk

propeller
disk

0.4e‐40.1

0.6e‐40.9e‐4

u=1.0

0.1e‐4Waket(b)x=0.4888

Fig.12 Computed contours(StdG, no FL,  =0.0 deg., KOSST)
0.2

u=1.0

hull

hull
0.4e‐4(a) =0.0 deg.
0.0

u=1.0
0.1e‐4Wakehull(b) =0.1 deg.

hull

0.5e‐4

max=5.2e‐4

propeller
disk

propeller
disk

Fig.11 Computed cavitation tunnel wake at propeller plane (no Fl,
StdG, KOBSL )Figs.12, 13, 14 show the comparison of the wake and thekinematic eddy viscosity t distributions using the KOSST, the
KOBSL or the SA turbulence model at x=0.47 and
x=0.4888(propeller plane). At x=0.47, where there is flow
separation, the t distribution is slightly asymmetric with KOSST,and symmetric with KOBSL and SA, in agreement with thetendency in the wake. In general, the t value is the smallest with
KOSST. With SA, the t distribution is more confined toward thehull surface than the other two.
At x=0.4888 (propeller plane), the t distribution is highly
asymmetric with KOSST, and almost symmetric or symmetric with
KOBSL and SA. With KOSST, the large t zone corresponds tothe high wake zone. There are sharp peaks on the symmetry
plane in the distribution of KOBSL. In the computation, the residual
showed occasional bursts, and therefore they are regarded as
local numerical oscillation. SA gives the most symmetrical
distributions.

t(a)x=0.47

0.2

0.8e‐4

u=1.0
0.1e‐4

Wake(b)x=0.4888

t

Fig.13 Computed contours(StdG, no FL,  =0.0 deg., KOBSL)
hull

hull

0.4e‐40.0

0.1e‐4u=1.0

Wake(a)x=0.47

t

Copyright © 2010 by JSFM

The 24th Computational Fluid Dynamics Symposium
Paper No.255
hull

hull

0.5e‐4
propeller
disk

propeller
disk
0.4e‐40.6e‐40.1

0.1e‐4(b) Wake at propeller plane =0.1 deg.)

u=1.0

Wake

t

(b)x=0.4888

x=0.4

0.45

Fig.14 Computed contours(StdG, no FL,  =0.0 deg., SA)
6. Cavitation tunnel with flow liners
Fig.15 shows the measured wake distribution at the propeller
plane in case the flow liners are installed in the cavitation tunnel.
The distribution is normalized by the maximum u-value in thepropeller disk. The distribution has regained symmetry, using theflow liners. It should be noted that the wake region is thinner thanthat in the towing tank, because it simulates the estimated fullscale wake.

p=‐0.2

‐0.15

‐0.05

‐0.05

u=0.0
‐0.25

‐0.3
propeller disk

‐0.35(c)Left stern surface pressure( =0.0 deg.)

0.5 0.3 0.3 0.5
0.7

0.5
‐0.1

0.1
0.5

0.7

0.3

0.7
0.9

0.9

0.2

Fig.15 Measured cavitation tunnel wake at propeller plane (FL2).
Normalized by the maximum value.
Figs.16(a),(b) show the computed wake distribution at thepropeller plane using the standard grid and the k- SST
turbulence model. In agreement with the measurement, the flow issymmetric at  =0.0 deg.. The flow is highly resistant to asymmetry,as the  =0.1 deg. case shows.
Fig.16(c) shows the hull surface pressure distribution at the left
stern. The pressure recovery in the downstream direction is slow
because of the flow acceleration caused by the flow liners. The
flow acceleration also causes a low pressure zone along the hullbottom. Due to this, the wake distribution shows a sharp peak atthe bottom. The separation zone is hardly affected by the flow
liners.
Fig.16(d) shows the wake distribution normalized by the
maximum value in the propeller disk. The distribution agrees wellwith the measurement.

u/umax=0.90.2(d)Wake at propeller plane( =0.0 deg., umax=1.113)
Fig.16 Computed flow field in the cavitation tunnel(FL2, StdG,
KOSST)
Fig.17(a) shows the distributions of the maximum
non-dimensional pressure as a function of x. In the no-FL and FL2cases, due to the wall effect, the hull surface pressure in theparallel part is significantly lower than the Rad1.5 case. Also,
Fig.17(c) shows that the boundary layer at x=0.0 is significantly
thinner.
In the stern part, the no FL case gives steeper adverse
pressure gradient and larger pressure rise than the Rad1.5 case.
That is perhaps why the flow asymmetry occurs in the no FL case.
In the FL2 case, the flow acceleration due to the flow liners retards
the pressure rise. Once the pressure rise starts, it is gradual in the
beginning, followed by an even steeper one than that of the no FL
case. But it does not last long, and the resultant pressure rise issmaller than that of the no FL case. These two effects might be
the reason why the flow asymmetry disappears by using the flow
liners.

1.0
u=1.1

0.3

(a) Wake at propeller plane =0.0 deg.)
Copyright © 2010 by JSFM

0 x 0.1 0.2 0.3(a)pmax(x) on hull surface-0.6 -0.5 -0.4 -0.3 -0.2 -0.1

x=‐0.6

p=‐0.3

‐0.1

0.5

‐0.2

x=0.0

‐0.2

‐0.2

0.4

0.1

p_max(x)

0.5
0.4
0.3
0.2
0.1
0
‐0.1
‐0.2
‐0.3

rad1.5
no FL
FL2

The 24th Computational Fluid Dynamics Symposium
Paper No.255
Association for giving permission to use the SR174B4 hull form,
and Dr. Yoshitaka Ukon of NMRI for providing the measurements.

0.6

x=0.6 ‐0.1

‐0.4

‐0.3

0.0

(b)Pressure on hull surface and top boundary (right half, FL2)

Rad1.5

u=1.0

no FL

0.9

u=1.2

1.0

(c)u-contours (x=0.0)
Fig.17 Effects of tunnel wall and flow liners on the surfacepressure and u-contours (StdG,  =0.0 deg., KOSST)
7.Conclusions
The stern flow of a 6m-long bulk carrier model placed in atunnel of 2m x 0.88m rectangular cross section, became
asymmetric due to strong wall effect, but recovered symmetry by
setting a pair of flow liners at the corners of the section and thus
reducing the wall effect.
The use of the modified Spalart-Allmaras model did not show
convergence for the case without flow liners.
The k- SST turbulence model can reproduce thephenomena (flow symmetry or asymmetry), while the
Spalart-Allmaras turbulence model and the k- BSL turbulencemodel always produce symmetrical flows. The clear distinction
between the results by the k- BSL and the k- SST models
shows that the modification based on the idea that shear stress ina boundary layer is proportional to the turbulent kinetic energy k isessential to simulating the flow asymmetry. It has been confirmedthat this result is not affected by the grid resolution.
The combination of the k- SST turbulence model and asimple oblique towing method, in which the hull form is rotated bya small angle  (typically =0.1 degrees), has been found usefulfor testing the susceptibility or the resistance of the flow toasymmetry. In case the wake distribution is asymmetric at =0.0
deg., it agrees either with the =0.1 deg. case or with the = - 0.1
deg. case. Doubling the rotation angle to =0.2 deg. hardly affects
the wake distribution, which suggests that rotating a hull by a small
angle works as a trigger to flow asymmetry, and the magnitude of
the rotation angle is unimportant. In case the wake distribution issymmetric at =0.0 deg., the case =0.1 deg. shows the degree ofsusceptibility or resistance of the flow to asymmetry.

Bibliography(1) Tanaka, I., "Scale Effects on Wake Distribution and Viscous
Pressure Resistance of Ships," J. of the Society of Naval
Architects of Japan, Vol. 146 (1979), pp. 53-60.
(2) Ukon, Y., Fujisawa, J., Kodama, Y., Hinatsu, M., and Kurobe,Y., "Wake Simulation and Cavitation Experiment on a Full
Ship Model with Flow Liners," Conference Proceedings of
the Japan Society of Naval Architects and Ocean Engineers,Vol. 9E (2009), pp. 107-110 (in Japanese).
(3) Kodama, Y., Hino, T., Ukon, Y., and Hinatsu, M., "Numerical
Simulation of Wall Effect Reduction Using Flow Liners in a
Cavitation Tunnel," J. of the Japan Society of Naval
Architects and Ocean Engineers, Vol. 11 (2010), pp. 81-90(in Japanese).
(4) Spalart, P. R. and Allmaras, S. R.," A One-Equation
Turbulence Model for Aerodynamics Flows," La Recherche
Aerospatiale, 1994, No.1, pp.5-21.
(5) Hirata, N. and Hino, T.: A Comparative Study of Zero- and
One-Equation Turbulence Models for Ship Flows, J. of the
Kansai Society of Naval Architects Japan, Vol. 234 (2000),
pp.17-24.
(6) Menter, F.R., "Two-Equation Eddy-Viscosity Turbulence
Models for Engineering Applications," AIAA Journal, vol.32,
No.8 (1994), pp.1598-1605.
(7) Hino T., "A 3D Unstructured Grid Method for Incompressible
Viscous Flows," J. of the Soc. Naval Archit. Japan, Vol.182(1997), pp.9 - 15.
(8) Hino, T., "Navier-Stokes Computations of Ship Flows on
Unstructured Grids," The Proceedings of the 22nd
Symposium on Naval Hydrodynamics, Washington, D.C.
(1998), pp.463-475.

Acknowledgments
The authors thank the Japan Ship Technology Research
Copyright © 2010 by JSFM