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Viscous Flow Past a Ship In a Cross Current
V.C. Patel, S. Ju, I.M. Lew
(Iowa Institute of Hydraulic Research, The University of Iowa, USA)
ABSTRACT
A numerical method for the solution of the Reynolds-
averaged Navier-Stokes equations is used to calculate the
viscous flow over the stern of a ship in a cross current,
i.e., a ship in yaw. The solutions are started with assumed
initial conditions downstream of the bow. The numerical
results are compared with the limited data that are avail-
able. Although the calculations are successful in describ-
ing the port-starboard flow asymmetry and vortex forma-
tion, the solutions indicate a need for a better resolution of
the bow flow.
INTRODUCTION
Patel, Chen and Ju (1988, 1990) have recently devel-
oped a numerical method for the solution of the Reynolds-
averaged Navier-Stokes (RANS) equations and applied it
to study the flow around ship hulls under the assumptions
that the ship is symmetric about the vertical centerplane,
and is advancing in a straight course in a calm sea. To the
authors' knowledge, all viscous-flow calculations for ship
hulls performed over the years with different methods have
been made with these restrictions (see Patel, 1988, for a
review). In this paper, extension and application of the
numerical method of Patel et al. to asymmetric flow around
the stern of a symmetric ship advancing in a straight course
in a cross current, as depicted in Figure 1, are considered.
This problem is equivalent to a ship in a uniform stream at
an angle of yaw. From a basic fluid-flow perspective, this
situation is more general than that of a body of revolution
at incidence, which has been studied experimentally and
numerically in many investigations (see, for example, Patel
and Back, 1985~. We now have a more complex body at
incidence. The resulting flow also bears some resem-
blance to the flow around a turning ship. A study of the
flow around a ship in a cross current is thus of practical
interest and also of value in further developing the capa-
bilities of modern numerical methods.
Before describing the extension of the method, it is use-
ful to make two important observations with regard to the
present work. First, as in the previous work, we will be
concerned with the flow downstream of some section in
the middle body of the ship. This implies that there is
some uncertainty in the establishment of proper initial
conditions which reflect the flow over the bow. Secondly,
experimental information on the viscous flow past a ship in
yaw is limited and, therefore, the success of the method
cannot be ascertained with confidence.
CALCULATION OF ASYMMETRIC FLOWS
A detailed description of the basic numerical method and
its applications to symmetric flows about double models,
i.e., with the water and keel planes treated as planes of
symmetry, is given in Patel, Chen and Ju (1988, 1990~.
The modifications in the method and the associated com-
puter programs to calculate the flow around a ship double
model at an angle of yaw are relatively minor. The main
changes are concerned with the size and shape of the com-
putational domain, the initial conditions, and the boundary
conditions. These are outlined below.
As shown in Figure 1, a positive value of yaw angle or
is used to represent a cross current from the port side. A
cylindrical (x, r, 0) coordinate system in the physical plane
Is used for the velocity components. For the symmetric-
flow calculations presented in Patel et al. (1988, 1990), the
solution domain in the transverse plane was (0° < ~ <
90°), i.e., it extended from the keel plane, ~ = 0°, to the
water plane, ~ = 90°. Plane-of-symmetry conditions were
applied on these planes.
For the current application, the required solution domain
is (-90° < ~ ~ 90°), and plane-of-symmetry conditions are
applied on the water plane at ~ = -90° and ~ = 90°, on the
port and starboard (or, windward and lee) sides, respec-
tively. The outer boundary of the solution domain is
located farther than before, at r = 2.0, i.e., it is a cylindri-
cal boundary, two ship lengths in radius. The number of
grid points in the radial direction is increased to 40 to
accommodate the more severe variations of flow quantities
in that direction. The locations of the transverse sections
(x= constant) and the number of grid points in the axial
direction are the same as before (= 501. The number of
grid lines in the circumferential direction was increased to
27. Thus, the present calculations have been performed
with a 50 x 40 x 27 grid, which is close to the finest that
could be accommodated on a CRAY XMP/48 supercom-
puter.
At the outer boundary of the solution domain, uniform-
stream conditions, i.e., Ux = UO cos a, Ur = UO sin a sin
9, US - UO sin a cos 0, are imposed. The calculations
were started downstream from the bow, with the most
upstream transverse section located at x = 0.3. The pro-
f~les used at this section were generated in a similar manner
to that described in Patel, Chen and Ju (1990~. Briefly, it
involves prescription of a girthwise distribution of the
boundary-layer thickness 8, the friction coefficient Cf (=
~ Presently Manager of Technical Research Center, Daewoo Shipbuilding & Heavy Machinery Ltd., Korea.
2 Visiting Professor from Chungnam National University, Daejeon, Korea.
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GUM, where U.; is the friction velocity), and the velocity at
the edge of the boundary layer Us. These are used,
together with the law of the wall and the law of the wake,
to generate the profiles of the longitudinal velocity Ux
inside the boundary layer, and the reduction from Ups to
unity in the inviscid flow is assumed to take place as r2.
Uniform stream conditions are specified beyond a distance
of five boundary-layer thickness from the hull. In the first
instance, Ur and US components are determined by
interpolation in the boundary-layer profiles and the known
conditions at the wall and in the uniform stream, and k and
£ are obtained from correlations for a flat-plate boundary
layer. As the solution progresses, the values of Ur, Us, k
and £ within the boundary layer are updated by scaling
those calculated at the first downstream station. This
process is continued only for the first 20 global sweeps,
and then the initial profiles are fixed. This procedure for
the generation of the initial conditions does not affect the
principal quantity, i.e., the axial velocity profile.
However, it ensures that the subsequent solution is carried
out with initial profiles of transverse velocity components
and turbulence parameters which are compatible with the
governing equations. In the present calculations, the
girthwise distribution of the boundary-layer thickness and
friction velocity were assumed to be the same as those in
the symmetric (zero yaw) case but the velocity at the edge
of the boundary layer was prescribed from an inviscid
solution. This is obviously an approximation and the
resulting initial conditions do not properly reflect all of the
flow phenomena that may occur over the bow, particularly
at an angle of yaw.
In the calculations at nonzero yaw angles, some diffi-
culties were encountered in the application of the boundary
conditions in the ship centerplane and at the exit plane. As
noted in Patel, Chen and Ju (1988), for the symmetric-
flow calculations with a= 0, the ship centerplane was
extended as a false wake plane, and the condition of sym-
metry, Us= 0, was applied on it. Also, along the wake
centerline, r = 0, the conditions Ur = Ue = 0 were implic-
itly imposed. In the present application, the false wake
plane is no longer a plane of flow symmetry although it is
a plane of symmetry for the numerical and. Also, Ur and
US do not vanish along r = 0. In fact, Ur and Ue change
rapidly near the singular point r = 0 in the grid. These
points are no longer calculated directly in the extended
method. Because we are now considering a half plane of
-90° < ~ < 90°, there are no explicit boundary conditions
to be satisfied at the false wake plane. However, because
of the rapid changes occurring in the velocity components
in cylindrical coordinates near the singular point r = 0,
increased accuracy and stability of the solution procedure
was obtained by interpolating in these components in
Cartesian coordinates, and then transforming back to the
cylindrical coordinates.
Finally, at the downstream (exit) boundary, the condi-
tions Px' = 0 and (Ux, Ur, Us, k, £)Xx' = 0 are imposed,
where x' denotes the direction of the uniform stream. This
requires complicated interpolations near the exit plane as x'
is different from the ship axis x. All values required in the
exit plane were extrapolated linearly in the x direction first
and then shifted by the amount of (Xf - Xf 1) tan a in the r
direction, where Xf and Xf 1 denote, respectively, the x
coordinates at the last and the next to last axial stations.
RESULTS
With the modifications described above, the numerical
method was employed to calculate the flow around double
models of the HSVA Tanker (Wieghardt, 1982) and the
SR107 Bulk Carrier (Okajima et al., 1985~. The hull
shapes are shown in Figure 2. Comparisons between
calculations and experiments for both hulls with symmetric
flow, at zero yaw, were presented and discussed in Patel et
al. (1988, 1990) There is no experimental information
with asymmetric flow around the HSVA Tanker but some
measurements at a = 5° and 10° have been reported by
Nishio et al. (1988) for the SR107. In the presentation of
the results all quantities are made dimensionless by the
reference velocity UO and the ship length L.
Figures 3 and 4 provide an overview of the effects of
yaw angle on the HSVA Tanker at Re = 5 x 106. The
limiting, or wall streamlines determined from the calculated
friction vectors on the hull are shown in Figure 3. The
port and starboard asymmetry is very clearly seen. The
regions of strong streamline convergence are those where
the pressure and the friction coefficients assume low
values. These are also the regions which are readily iden-
tified in surface flow-visualizations using such techniques
as oil flow, wool tufts, and dye injection.
Figure 4 shows the velocity field at a section near the
stern, x/L = 0.9. In general, the effect of the cross current
is to drive the flow from the port to the starboard side.
This produces a thinning of the boundary layer on the port
side and a thickening on the starboard side. The secondary
motion at this section reveals reversals in direction that are
characteristic of converging limiting streamlines and vortex
formation. They are also associated with a local decrease
in the wall shear stress and thickening of the viscous layer.
These flow features are particularly difficult to handle in
space-marching numerical methods usually employed for
the solution of the three-dimensional boundary-layer equa-
tions. Here they are predicted without any special treat
ment.
The general features of these and other results presented
below are qualitatively similar to those observed on a body
of revolution at angles of attack. However, subtle differ-
ences arise in the shapes of the distributions and in the
magnitudes of the pressure and friction coefficients due to
the present three-dimensional geometry and, in particular,
due to the rapid changes in this geometry at the stern.
The remaining results pertain to the SR107 hull. In this
case, comparisons are made between the calculations and
experimental data at a = 10° and Re = 2.7 x 106. We
examine the velocity field as well as the contours of the
axial component of vorticity, fox, in the format presented
by Nishio et al . (1988~. Figures 5, 6 and 7, show the
results at three sections, x/L = 0.5, 0.7 and 0.9, respec-
tively.
From Figure 5 it is clear that the effect of the cross cur-
rent is to thin the boundary layer on the port side and
thicken it on the starboard side. A vortex is observed
around the turn of the bilge, particularly in the experiment.
In general, at this section the calculations reproduce the
overall trends but differ in intensity. For example, the
measured secondary motion and axial vorticity are both
stronger than predicted, and Figure 5(c) indicates some
difference in the location of the vortex. It is possible that
better agreement between measurements and calculations
can be secured, at least in some respects, by adjustment of
the initial conditions at x/L = 0.3. An alternative, of
course, is to include the bow in the calculations. We shall
return to this later.
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At the next axial station, x/L = 0.7, the results are
similar to those at the previous section. Figure 6(b) again
indicates that the secondary motion observed in the
experiment was considerably stronger than is calculated
although there is good correspondence in the extent over
which axial vorticity is found. The calculations fail to
reproduce a detached core of high axial vorticity.
At the last section where measurements were made, x/L
= 0.9, Figure 7 shows continued qualitative agreement
between the calculations and data but somewhat greater
differences in the details. The dual vortex structure indi-
cated by the experiments is perplexing because it gives the
appearance of vortex shedding and yet such a phe-
nomenon, even if it were present, could not have been
captured by the simple pressure probes employed for the
measurements. The calculations also do not produce the
high levels of axial vorticity observed in the experiments.
Be that as it may, Figure 7(c) shows that the calculations
are successful in identifying the region over which there is
. . ~ . . .
slgmilcant axle . vortlclty.
Although the calculations were continued well into the
wake, there is no data to gauge their success. An interest-
ing feature of the calculated wake is that the longitudinal
vorticity (secondary motion) is destroyed rather rapidly
although a measurable axial velocity defect persists for
large distances.
DISCUSSION
This first attempt to calculate the viscous flow over a
ship in a cross current is regarded only as a partial success
because it has raised a number of issues which need fur-
ther investigation. Among these, the most important is the
problem of specifying realistic initial conditions at the
upstream station. Numerical experiments performed with
different initial conditions during the course of this work
indicated that the strength and location of the vortices on
the lee side depended rather markedly on the initial condi-
tions. For this reason, it was decided to present results
obtained with the same initial conditions as those used ear-
lier and found satisfactory for the zero-yaw case. Of
course, calculations can be performed for the inviscid flow
and the boundary layer over the bow to determine these
conditions in a systematic way. But, it is possible that
even these will fail if, as the results from the present calcu-
lations indicate, a vortex is formed well ahead of midships.
In fact, the likelihood of vortices arising near the bow is
greatly increased for a full hull form, and with a bulbous
bow, at angles of yaw. Thus, it appears that the problem
of initial conditions may have to be addressed, in the final
analysis, by extending the full Navier-Stokes type of solu-
tions to include the complete ship. At model scale, this
would also require a careful treatment of transition or trip-
ping devices.
The failure of the calculations to reproduce the observed
high levels of axial vorticity in well defined cores is
another feature brought forth by this investigation. The
differences between calculations and experiment arise as
early as midships and continue over the stern. It is possi-
ble that this is due, at least in part, to the use of the wall-
functions approach in the turbulence model. In this
approach, the flow near the wall is not explicitly calcu-
lated, and therefore, flow features arising from a roll up of
the near-wall layer, such as vortex formation, are unlikely
to be resolved in detail. A near-wall turbulence model,
along with integration of the equations of motion up to the
wall, are needed to obtain an adequate resolution of these
features.
Finally, it should be noted that the present calculations
were carried out for a ship in a uniform stream (in unre-
stricted waters) whereas the experiments on the SR107
were conducted on a double model in a wind tunnel. The
effects of tunnel blockage, which increases when the
model is mounted at an angle of yaw, remain to be
explored.
CONCLUDING REMARKS
Calculations of the flow past a ship in a cross current,
presented here, have served to illustrate not only the
capabilities of the numerical method employed but also the
additional complexities that arise in real-life situations.
The asymmetries in the stern flow, resulting from asym-
metric stern shapes, cross currents, or maneuvers, are
obviously of interest in the design of propulsors and
appendages, and in the prediction of hull vibration. The
present calculations represent a first step in the
development of numerical methods capable of addressing
these issues in a comprehensive and realistic manner.
ACKNOWLEDGEMENTS
This research was partially sponsored by the Office of
Naval Research, first under the Accelerated Research
Initiative (Special Focus) Program in Ship
Hydrodynamics, Contract N00014-83-K-0136, and then
under Contract N00014-88-K-0001. The calculations
reported here were performed on the CRAY X/MP-24
supercomputer of the Naval Research Laboratory, and on
the CRAY X/MP-48 machine of the National Center for
Supercomputing Applications at Urbana-Champaign.
REFERENCES
Nishio, S., Tanaka, I., and Ueda, H. (1988), "Study on
Separated Flow around Ships at Incidence," J. Kansai
Soc. Nav. Architects, Japan, Vol. 210, pp. 9-17.
Okajima, R., Toda, Y., and Suzuki, T. (1985), "On a
Stern Flow Field with Bildge Vortices," J. Kansai Soc.
Nav. Architects, Japan, Vol. 197, pp. 87-95.
Patel, V.C. (1988), "Ship Stern and Wake Flows: Status
of Experiment and Theory," Proc. 17th ONR Sym. Naval
Hydrodyn., The Hague, The Netherlands, pp. 217-240.
Patel, V.C. and Baek, J.H. (1985), "Boundary Layers and
Separation on a Spheroid at Incidence," AIAA Journal,
Vol. 23, pp. 55-63.
Patel, V.C., Chen, H.C. and Ju, S. (1988), "Ship Stern
and Wake Flows: Solutions of the Fully-Elliptic
Reynolds-Averaged Navier-Stokes Equations and
Comparisons with Experiments," Iowa Inst. Hydraulic
Research, Uni. Iowa, IIHR Report No. 323.
Patel, V.C, Chen, ~G, and Ju, S. (1990), "Computa-
tions of Ship Stern and Wake Flow and Comparisons with
Experiment," J. Ship Research, to appear.
Wieghardt, K. (1982), "Kinematics of Ship Wake Flow,
The Seventh David W. Taylor Lecture," DTNSRDC-
81/093. See also Z. Flugwiss Weltraumforsch., Vol. 7,
pp. 149-158.
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z
43 =- 90° 11 ~ - 90°
Port\ Starboard
a
e= 0
(a) transverse section A-A
A ~
(' ~_.
Ct:~ _ .
/ UOA ~
(b) waterplane
(a) HSVA Tanker
lLLL:~Sl
(b) SR107 Bulk Carrier
Figure 1. Ship in a cross current; notation Figure 2. Hull shapes
a = 10°
(a) port
(by starboard
Figure 3. Surface streamlines, HSVA Tanker
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whoa
-. 1_
.~2
_ no
-~ od
- or
_ 17
-. od
-. OB
_ , ~
- n2
aft
(a) Axial velocity
(b) Transverse velocity
Figure 4. Velocity field on HSVA tanker
at transverse section x/L = 0.9
0.7
1~
'1 --
Experiment
Calculation
(~1 Axial velocity
us
(b) Transverse velocity
28 24 20 16 .
(c) Axial vorticity
\ \l1 6
Figure 5. Velocity field and axial vorticity
on SR107 at x/L = 0.5
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Experiment
Calculation
AIL = 0.7
(a) Axial velocity
~ ~o.s
_ :~_
0.809
0.6
GIL = 0.9 At 0.6
(a) Axial velocity
(b) Transverse velocity
TIC
(c) Axial vorticity
=~84
16 12
Figure 6. Velocity field and axial vorticity
on SR107 at x/L = 0.7
(b) Transverse velocity
(c) Axial vorticity
24 2016
Figure 7. Velocity field and axial vorticity
on SR107 at x/L = 0.9
726