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OCR for page 71
Grict Generation and Flow Computation for Practical Ship Hull Forms
and Propellers Using the Geometrical Method ant! the lAF Scheme
Y. Kodama
Ship Research Institute
Tokyo, Japan
Abstract
Grid generation for ship hulls and a propeller blade
was made using the geometrical method, where an ini
tial grid is modified iteratively under several geomet
rical requirements such as orthogonality, smoothness,
and clustering.
Using the generated grid, the computation of the
incompressible NavierStokes equations was made for
flows past a flat plate and four different ship hull forms
using the IAF scheme and the BaldwinLomax zeros
equation turbulence model. The hull forms chosen are a
Wigley hull and Series 60 (Cb=0.6, 0.7, 0.8) hulls. Prior
to the ship flow computation, the flow around a flat
plate was computed, where the agreement of the com
puted result with experiment was very good in terms
of the wall shear stress, displacement thickness, shape
factor, and velocity distribution. The computed flow
past a Wigley hull, which is not very different from the
flat plate result because of its fine hull form, showed
good agreement with experiments. The computed flows
past Series 60 (Cb=0.6, 0.7, and 0.8) hull forms showed
systematic change with the Cb block coefficient values.
The computed Cb=0.6 result showed reasonable agree
ment with experiments. However, the agreement be
came poorer with increase in the Cb values (=0.7, 0.8~.
1. Introduction
Prediction of flow past a ship hull has been an im
portant subject in ship hydrodynamics because of its
practical importance. If one succeeds in accurate pre
diction of the flow, one can get a resistance value to be
used in powering, and the wake field at the propeller
plane to be used in propeller performance estimation.
A classical approach to the above problem is the
bo~ndarylayer method t14. But recently, aided by
the rapid development of computer hardware, methods
called NS solvers, in which the governing equations of
71
the flow are discretized and computed, are becoming
increasingly popular t2],(3],[43. The author previously
computed the flow past a Wigley hull using the non
conservation form of the NS solver t53. The present work
is an extension of the previous work. The major change
is in the use of the conservation form. The scheme
is called the IAF scheme [6], which is widely used for
computing compressible flows. Pseudocompressil>ility
is introduced in the continuity equation of the incom
pressible flow, in order to make the system of equations
hyperbolic. A conservative 2ndorder central differenc
ings are used for convection and diffusion terms, and
the 4th order numerical dissipation terms are e~cplcitly
added to the equations to damp out high frequency wig
gles.
The degree of accuracy of computed results of NS
solvers is affected by the quality of the grid used. There
fore, it is important to be able to generate grids of high
quality, in order to obtain good computed results. In
the present paper, a grid generation method called the
geometrical method is used. It is an extension of the
method used by the author previously t73. Using the
method, the grid on a propeller blade and the grids
around ship hulls are generated.
Although the use of NS solvers greatly reduces the
need for experimental informations, a turbulence model
is still needed for computing high Reynolds number
flows such as those past ship hulls. The turbulence mocl
els widely used today in engineering applications are
the CebeciSmith (CS) zeroequation turbulence model
t8], and the ke twoequation model t93. In the present
paper the BaldwinLoma.x (BL) zeroequation t~rbu
lence model, derived by making a modification to the
CS model, is used. There, in contrast to the original
CS model, the necessity for finding the edge of flee
boundarylayer is avoided. The model is widely vised
for computing compressble flows in the field of aerody
namics, and is said to be accurate for ~~nseparater1 floes s,
OCR for page 71
lout poor for separated flows. One of the objectives of
the present work is to test the validity of the tubulence
model for incompressible flows past ship hulls, and to
find the range of applicability of the model by com
puting flows past ship hulls with various degree of full
ness. Although further modification to the BL model
was published recently t10], the original BE model is
used here.
In Chapter 2, the NS solver is described, with dis where
cussions on the conservation property, boundary con ~ F = IF + (YG + (ZH
ditions, and the turbulence model. In Chapter 3, the ~ J J J
geometrical grid generation method is described, where ~ G = 92 F + ~9 G + my H
the use of bicubic splines for representing a body sur ~ J J J
face geometry is explained. In Chapter 4, computed re
sults of flows past a flat plate, a Wigley hull, and Series
60 Cb=0.6, 0.7, and 0.8 hulls are shown. In Chapter 5,
conclusions are drawn. The flow past a propeller, which
was originally planned, is not included in the present
paper.
2. NS Solver
2.1 Governing Equations
The governing equations are the combination of the
incompressible NavierStokes equations and the conti
nuity equation. They are, in conservation form,
q' + Fm + Gy + Hz = 0 (2.1a)
where
q= ~ ~ F=
U +p~q. UV~y ~
uv—An G = v + p—MY
uw—~xz vw—Tyz
flu Jv
~ uw—Id
VW—fyz
w + p—fizz
jaw
where all the subscrits except those with ~ denote par
tial derivatives. The 1st, 2nd, and 3rd components are
x, y, and z momentum equations. In the 4th compo
nent, which is the continuity equation, bp/0t is artifi
cially added to give pseudocompressibility,thus making
the sytem of equations hyperbolic. ,B in the equation is
a positive constant. The shearstress terms ~ are ex
pressed as follows.
~m = (R + ut)2u~, boxy = (R + ~~) (fly + vm)
My = (R +~2vy ~ me = (R +~) two +uz) (2.2)
''ZZ = (R Jr 7~2wz ~ adz = ('R + ut) (vz + wy)
where u~ is the kinematic eddy viscosity.
72
Coordinate transformation from the (x,y,z) Carte
sian coordinate system to the bodyfitted ((,r~,() coordi
nate system is made to the governing equation eq.~2. ] ).
The transformation is assumed timeindependent. The
resulting equation is, again in conservation form
Hi ~ J ~ + Fit + Go + H; = 0 (2.3~)
(2.3b)
tH=~JF+(JYG+(JZH
J is the Jacobian, and all the x, y, and z derivatives of
g", A, and ¢ are expanded using the chain rule.
Numerical dissipation terms are added to the above
equation to enhance numerical stability.
~ ~ J )+F~ +Gn +H: +~(q6466 +q0000+q<~¢) = 0 (2.~)
where cat is a positive constant. In order that computed
results obtained by solving eq.~2.4) satisfy the origi
nal equation with accuracy, these added terms must be
small.
2.2 Discretization
First, the time derivative is replaced by the time
differencing. The Pade time differencing form is used
here.
~q 1
_ = qn
Bt fit l + c4/\
where i\qn En+ _ qn (2 5)
where qn denotes q at timestep n. ~ is a constant which
(2.1.b) takes the value of either 0 (Euler explicit), 0.5 (Trape
zoidal), or 1.0 (Euler implicit). Here ~—1.0 is adopted.
The nonlinear flux terms OF, AG, /\H are locally lin
earized into the form, for example,
i\F = A/\q + A6/\qf. + A~\q,, + A¢~\q¢ (~2.6a)
where A = ~' , At = ~ , An = ~ , At = `9 (2.66)
Then the governing equation becomes
/\qn+67\tt J~(` Ai\q + A6/\q~ + A'7/\qr~ + A¢Aq¢)
+ J~,,(~ Bi\q + BfAq~. + Br77\q,7 + B
OCR for page 71
The mixed derivative terms such as 0(A~^q~/~:
are lagged to the timestep (n1) and evaluated time
explicitly. Then the above equation can be approxi
mately factored into A, A, and ¢sweeps.
(sweep
{I + 9 At [J ,~: (A + Af ~ ~ + a, ~ in } A*
=—At [ J(F~ + Gn + H: ~ + w f qua + q + qua< )]n
—§^tJ[(A~^q~ + A(Aq`~4 + (B(Aq~ + B
0.4 Turbulence Model
The turbulence model used is the BaldwinLomax
(BL) zeroeq~ation algebraic model, whose original
form is the CebeciSmith (CS) model. The kinematic
eddy viscosity ~` is evaluated in the inner and outer lay
ers separately.
In the inner region, the kinematic eddy viscosity Eli has
the same form with the CS model.
with
I'd = 12 ~~ (2. 15)
I = knrL1—exp(<—n+ /A+ )] (2.16)
n+ = nRe ~/~ (2.17)
where cat is the vorticity and n is the normal distance
from the wall. In the present computation, as shown
in Fig. 23, the normal distance n is determined by
projecting a vector connecting the point in concern with
the root point on the same ¢direction line, onto the
normal vector defined at the root point.
In the outer region, the necessity of finding the edge
of the boundarylayer, which existed in the CS model,
is removed in the BL model. The eddy viscosity RIO has
the following form.
lJ`o = KCCp Fkiebnmam Fma2 ~ 2. 1 S)
or
Jto = KCcpFElebCwkUdiff F. (2.19)
mar:
with K=0.0168, Ccp =1.6, and Cask =0.25. The smaller
value of the above two equations is taken. The quanti
ties Fma~ and nma.r are determined from the function
F = n ~~ t1—exp(<—n+/A+~] (~.20)
The quantity Fma2 is the maximum value of F that
occurs in a velocity profile, and nmna is the value of n
at that point. Uaif f is the difference btweeen maximum
and minimum velocity in the profile.
Ujif f = USA — Umin ~ 2.21 )
where Umin is taken to be zero except in wakes. The
intermittence factor FKieb is given by
1 ~ 5.5(C~`ebn/nma~6 (2.22)
with C~;'eb= 0. 3.
3. Grid Generation
The grid generation method used is the geometrical
method. Since the method is described in detail in t11i,
descriptions are limited here on the modifications made
thereafter.
74
3.1 Body Surface Grid
In the grid generation method, first tile surface
grids at inner and outer boundaries are generated. The
inner boundary usually corresponds to a body surface.
The body surface geometry is represented using para
metric bicubic splines. Then the grid points on the sur
face can be shifted along the surface according to four
geometrical requirements, i.e., orthogonality, smooth
ing, clustering, and minimum spacing. Fig. 31 shows
a spline surface on a Series 60 (Cb=0.6) hull, which
has been determined from given offset points. Fig. 39
shows a surface grid on the same hull. Clustering has
been imposed toward bow and stern edges, as well as
toward bottom and waterline edges. Orthogonality has
been imposed at all the four edges.
The same method has been applied to the surface
grid on a propeller blade. Fig. 33 (a), (b) show surface
grids on a propeller blade before and after mo~lif~ca
tion. Clustering is made toward leading, baling, and
tip edges, with various degree of clustering. Here all
the points on the surface are allowed to move except
for th\ose at the four corners, and therefore highly fle~c
ible grid generation is possible.
3.2 Total Grid
After the surface grid is generated, intermediate
grid points, which are the points between the inner and
outer boundaries, are generated, by simply connecting
the corresponding points on the inner and outer bound
aries with a straight line, and disributing points on it.
Then the initial grid is modified iteratively under the
same geometrical requirements as those used in the sur
face grid modification. The grid points on the bound
aries are kept fixed.
The most important change since ref.~11] is in the
combination of the grid point modifications. In the orig
inal method, grid points were shifted separately dice to
each modification requirement, whereas in the present
method the grid pains are shifted only once in each
iteration as a result of the combined shifts. The com
bination is made with weights which change locally ac
cording to the local grid geometry. The parameter vised
to represent local grid geometry is the aspect ratio of
a grid cell. The value of the parameter varies greatly
depending on the location, since the grid is highly cl~s
tered near the solid wall surface. The minimum spacing
requirement has been modified. In the old version, if a
spacing between two neighboring points in a certain di
rection is smaller than a given value, that spacing is
made larger to the given value in that direction by re
distributing points along that direction with the total
length kept unchanged. This requirement was found
to be too "active" in some cases, especially near the
solid wall boundary where the grid is highly clustered
in the normal direction. There the redi.ctribution in
the alongthewall direction sometimes results in kinks
in the other direction. Therefore in the present version
OCR for page 71
the requirement has been changed to a "passive" one,
where, if a grid spacing is found to be smaller than a
given value, the modification which makes the spacing
even smaller is made zero.
After the grid is modified according to the foure
requirements, the final grid for high Reynolds num
ber flow computation is obtained by reclustering.
Reclustering is the process in which the grid is re
distributed, in a single sweep, in the normaltowall
direction according to a given minimumm spacing ad
jacent to the solid wall and a new given number of grid
points in that direction, making the information inher
ent in the original grid point distribution reflected in
the new distribution. This process makes possible to
use smaller number of grid points and much larger min
imum spacing near the solid wall in the iterative mod
ification stage, thus making the grid generation much
faster and easier. No change has been made to the re
clustering process. Fig. 34 (a),~b),tc) shows the gener
ated grid around a Series 60 (Cb=0.6) hull. The i=38
section shown in (b) is approximately at midship, and
the i=60 section is at the stern edge (see Table 41~.
4. Computed Results
4.1 Tested Cases and Parameters
Computations were made for seven cases with five
different bodies. The bodies vary in the degree of full
ness, from a flat plate to a Series 60 (Cb=0.8) hull. Ta
ble 41 shows the cases and the parameters. There im,
jm, and km are the number of grid points in ~ (stream
wise), ~ (girthwise) and ~ (normaltc~wall) directions.
The experiments with Froude number values (Fn) were
made in cirulating water channels with free surface, and
all the others were made using double models in wind
tunnels. Followings are the parameters common to all
the computational cases:
ifp=16 iap=60 Xupstream end = ~0 5
Xfp = 0.0 Cap = 1.0 Xdo~vnstream end = 2 0
Outer boundary radius = 1.0 ,l? = 0.1
where ifp and lap are the numberings in i at bow and
stern (or leading and trailing for a flat plate) edges. As
shown in Fig. 41, the minimum spacing of grid points
adjacent to the inner boundary (solid shall or symmetry
plane) /\min is determined as
/\mi77~ = /\min14
= /\ mini
= /\ mini
0.05
/\ m ~ 77 ~ = ~
v Re
(x < 0.1)
(0.1 < x < 1.0)
(1.0 < x)
(my)
(4.1)
Computations were made using the Stellar GS
1000 graphic workstation with a rector processor unit
and 3~) SIB main memory. The NS solver took approx
imately 40 sees (Cases .5,6,7) CPU time per timestep,
7s
using a highly vectorized code. /\t varied from 0.005
NTU (Nondimensional Time Unit) for the Wigley hull
case, to 0.001 NTU for the Series 60 Cb=0.8 case. It
took approximately 4 to 5 NTU to reach the stea
turbulent thereafter, to be consistent with the experi
ment, where the studs were pla.cecl at the same location.
This applies to all the cases with ship hulls.
Fig. 4S shows pressure contours on the hull sur
fa.ce, (xz) symmetry plane, and (xy) symmetry plane.
The flow is from left to right. The contour lines are
wigglefree. In the previous computation [5], where the
nonconservation form was used, there were wiggles at
fore and aft ends of the hulll.
Fig. 49 (a),tb),~c),td) show wake contours at
x=constant sections, from midship (~=0.5) to down
strea.m (x=1.5~. The contours at an x=constant section
is obtained by interpolating the values at grid point lo
cations using parametric tricubic splines, which is an
extension of bicubic splines to threedimensions. The
computed values show good agreement with the mea
sured values. Fig. ~10 shows the kinematic eddy vis
cosity u~ contours at i=53 section (x~ 0.95~. The step
wise change of z'` in the girthwise direction occurs be
ca.use the location of FmaX in (direction is determined
at either of the grid points, which are widely spaced in
the outer region.
Fig. ~ , ~ is,
rameters on the wake contour at x=1.0. The figure (a)
shows the result with ~ = 5.0 (Case 3~. The difference
from the ~ = 1.0 result is small, which implies that
the value ~ = 1.0 in Case 2 is small enough such that
the added numerical dissipation terms do not affect the
computed result. The figure (b) shows the result where
the number of grid points in ¢direction is doubled. The
difference is again small, which implies that the number
of grid points used in Case 2 is large enough.
Fig. 412 (a),~b),tc) show the wall shear stress Id
distributions at z/D= 0.2, 0.5, and 0.8 sections, where
D is the depth of the hull. The empirical curves are
the same as those in the flat plate case. The agreement
with the measured values are generally good. The id
values agree well with the flat plate values in most of
the regions.
Fig. ~13 (a),tb),tc) show the displacement thick
ness {~ distributions at z/D= 0.9, 0.5, and 0.8 sections.
The agreement is again generally good. The b~ values
deviate considerably from the empirical flat plate values
shown as solid lines in the figures.
Fig. 414 shows the shape factor H at z/D= 0.2,
0.5, and 0.8 sections. They show tendencies similar to
those of flat plate results, except near the stern.
Fig. 415 (a),tb),~c) show logarithmic plot of ve
locity at three i=consta.nt sections, i.e. i=25 (x~ 0.07),
i=38 (x=0.5), and i=53 (x~ 0.95~. It can be seen
that logarithmic law holds in every case. Fig. ~16
(a),tb),tc) shows the distribution of the kinematic eddy
viscosity id, the locations of which correspond to those
in Fig. 415. The tedency is similar to that of the flat
plate.
411 (a).(b) show the effect of chancing oa
4.3 Series :60 (Cb=0.6, 0.7, 0.8) Hulls
In this section, the measured data. shown are taken
76
from [15] (Cb=0.6), [16] (Cb=0.7), and [17] (Cb=0.8).
Fig. 417 (a),tb),~c) shows computed pressure contours
for Series 60 Cb=0.6, 0.7, and 0.8 hulls. They are
plotted at the same pressure values as those for the
Wigley case (Fig. ~8~. There are slight pressure oscil
lations just below the points of mapping singularity in
the wake.
Fig. 418 shows comparison of the computed pres
sure distribution near the stern region on the C1.~=0.6
hull with the measurement [1S]. The agreement is good,
though there is systematic deviation which increases
toward midship, and the adverse pressure gradient is
greater in the computed result.
Fig. 419 (a) to. (h) show the wake contours. The
agreement with the measurements are good in general,
except in the wake. There the downward movement of
the low speed region near the (xz) symmetry plane is
not captured in the computation. There are two possi
ble reasons for this failure. One is that the intensity of
the downwash due to the pair of longitudial vortices is
insufficient. The other is in the way the eddy viscosity
is determined. In the wake, the normal distance n+ is
ta.ken from the (xz) symmetry plane. However, when
the flow near the stern is highly threedimensional, the
(xz) symmetry plane is not necessarily suital:)le for this
purpose, in contrast to the situation with the Wigley
hull. There the longitudinal vortices are not strong,
and the flow in the wake remains similar to that of a
flat plate.
Figs. 420,~21,~22 show the wall shear stress tow,
the displacement thickness 5~, and the shape factor H
at three z=constant locations. There at z/D=0.8, the
separation occurs near the stern.
Fig. 423 (a),~b),tc) show the logarithmic plot of
velocity at three streamwise locations. It is seen that all
the points adjacent to the solid wall are well within the
viscous sublayer as the turbulence model used demands,
and that all the velocity profiles follow the logarithmic
distribution law in the inner layer. They suggest that
the BaldwinLomax turbulence model, a simple zero
equation model, can be used for this type of flow. Fig.
424 (a),~b),~c) show the kinematic eddy viscosity ~~`
distributions at the same three strea.mwise locations as
in the logarithmic velocity plot. The clistributions are
similar to those in the Wigley hull case shown in Fig.
416.
Finally, Fig. 425 (a),tb) show the wake contours
for the Cb=0.7 case (Case 6 in Table 41), and Fig. ~1
26 shows the wake contours for the Cb=0.8 case scene
7~. In the Cb=0.7 case, the computed wakes still show
reasonable agreement with the measurements in those
streamwise locations. However, in the Cb=0.8 case, the
agreement becomes poor. By looking at the three com
puted wake contours, i.e. the cases Cb=0.6, Cb=0.7,
and Cb=0.8, a systematic trend is observed. In the
stern region, the wake near the bottom becomes thin
ner than the measured one, and the wake in the mid
de;pth region becomes very thick, as the fullness of the
OCR for page 71
ship hull (Cl:) becomes greater. Clearly the tubulence
remodel needs n~c~clification there (see Supplement.
5. Conclusions
Flows past ship hulls were computed and compared
with measurements. The NS solver used is the IAF
scheme, where the pseudocompressibility is introduced
in the continuity equation, in order to make the system
of equations hyperbolic. The accuracy and convergence
of the computed results were tested by computing flows
rising different numl:>er of grid points, or using different
amount of the added numerical dissipation terms.
The BaldwinLomax zeroequation turbulence
model was used. The validity and limitation of the tur
bulence model was tested by computing flows past five
different bodies. They are a flat plate, a Wigley hull,
Series 60 Cb=06, 0.7, and 0.8 hulls. They vary in the de
gree of fullness, from complete flatness of the flat plate
to high fullness of the Series 60 C=0.8 hull. By com
paring the computed results with measurements, the
turl:'ulence model was found to be useful for fine hull
forms, such as a flat plate, a Wigley hull, and a Se
ries 60 Cb=0.6 hull. However, the agreement between
the computed and measured data was not satisfactory
for the Series 60 Cb=0.7 or 0.8 hulls. This suggests
that the turbulence model needs modification for such
flows where strong adverse pressure gradient exists and
threedi~nensionality becomes important.
The grid generation method called the geometrical
method was used to generate grid around those ship
hulls mentioned above. The method was also applied
to generate a surface grid on a propeller blade. The
body surface was represented using parametric bicubic
splines, and the grid points on the body surface were
allowed to move along the surface, in order to meet
the requirements imposed, i.e. orthogonality, cluster
ing, smoothing, and minimum spacing.
Sato,T. et al."FiniteDi~erence Simulation Method
for Wave and Viscous Flows al:'out a Ship", J. of
SNAJ vol. 160, (Dec. 1986).
4] Masuko, A. et al."Numerical Simulation of Viscous
Flow around a Series of Mathematical Ship Mod
els", J. of SNAJ vol. 162, pp.110 (Dec. 1987).
5] Kodama, Y."Computation of High Reynolds Num
ber Flows Past a Ship Hull Using the IAF Scheme"
J. of SNAJ vol. 161, pp.2534 (1987~.
6] Beam, R.M. and Warming, R.F."An Implicit Fac
tored Scheme for the Compressible NavierStokes
Equations", AIAA Journal, Vol.16, No.4, (April
1978).
7] Kodama, Y."ThreeDimensional Grid Genera
tion around a Ship Hull Using the Geometrical
Method", J. of SNAJ vol. 164, pp.916 (1988).
8] Cebeci, T. and Smith, A.M.O."A FiniteDifference
Method for Calculating Compressible Laminar and
Turbulent Boundary Layers", J. of Basic Engi neer
ing, Trans. of the ASME, pp.523535 (Sept. 1970~.
9] Rodi, W."Turbulence Models and Their Applica
tion in Hydraulics", IAHR (1980~.
t10; Stock, H.W. and Haase, W." Determination of
Length Scales in Algebraic Turbulence Models for
NavierStokes Methods", AIAA Journal, Vol.97,
No.1, (January 1989~.
[11] Kodama, Y."ThreeDimensional Grid Genera
tion around a Ship Hull Using the Geometrical
Method", J. of SNAJ vol. 164, pp.916 (Dec.
1988~.
[12]
Acknowledgements [13]
The author would like to thank Profs. I. Tanaka
and T. Suzuki of Osaka University, Prof. T. Okuno of
University of Osaka Prefecture, and Prof. V.C. Patel of
University of Iowa for providing references and material
on the measured data used in the present work. The au
t.hor also thanks the members of the CFD group at the
Ship Research Institute for many valuable discussions.
References
1] Himeno? Y."Calculation Method of the Twos
Dimensional Turbulent BoundaryLayers", Pros
ceedings of the Symposium on Viscous Resistance,
SNAJ, pp.5993 (1973) (In Japanese).
2] Chen, H. C. and Patel, V.C." Calculation of
TrailingEdge, Stern and Wake Flows by a Time
Marching Solution of the PartiallyParabolic Equa
tions", IIHR Report No. 285, IIHR, University of
Iowa (198.5).
77
Schlichting, H."BoundaryLayer Theory", 6th Edi
tion, McGrawHill (1968).
Cebeci, T. and Bradshaw, P." Momentum Trans
fer in Boundary Layers", Hemisphere Publishing,
p. 168 ( 1977).
[14] Sarda, O.P."Turbulent Flow Past Ship Hulls An
Experimental and Computational Study", Ph.D.
Thesis, Univ. of Iowa (Aug. 1986).
[15] Toda, Y. et al."MeanFlow Measurements in the
Boundary Layer and Wake of a Series 60 Cb=0.6
Model Ship With and without Propeller", IIHR
Report No.326, Iowa Institute of Hydraulic Re
search, Univ. of Iowa (Nov. 1988~.
[16] Okuno,T." Study on Threedimensional Boundary
Layers on Ship Hulls", Ph.D. Thesis, Univ. of Os
aka Prefecture (In Japanese) (Nov. 1980~.
t17] Fukuda, K. and Fujii, A."Turbulence Measure
ments in Three Dimensional Boundary Layer and
Wake around a Ship Model", J. of SNAJ, Vol.1.50
pp.8598 (In Japanese) (Dec. 1981~.
OCR for page 71
Fig. 21 Area covered by
the discretized governing
equations.
( xy ) Symmetry Plane ~
Fig. 32 Surface grid on a Series 60 (Cb=0.6) hull
BY
n
(b) Body section (c) Wake region
Fig. 02 Grid topology and boundaries
n ~ = from body surface
~ Bay Surface
Mapping
Singularity
Fig. ~>3 Norma1 distance
Fig. 31 Spline surface on a Series 60 (Cb=0.6) hull
(a) Init,ia,1 grid (b) Modified grid
Fig. 33 Surface grid on a propeller blade
(a) Perspective view
~/~/~N
(b) i=.3(S section (midship) (c) i=60 section (stern)
Fig. 3~1 Gricl around a Series 60 (Cb=0.6) hull
78
OCR for page 71
0.005
Iw
o
O.
Case
1
2
4
5
6
7
n
~ \
~ u
0 0.1 1.0 ~ x
Tal~le 41 Computation paran~etei~
Body
.
f l a t p l a t e
Wi g I ey
Wi g l ey
Wi g ley
S. 60 Cb=O. 6
S. 60 Cb=O. 7
S. 60 Cb=O. 8
. _
i~
_
8
. 8
8
8
8
. 8l
_ 81
km _
_
3l l.
3l 1.
31 5.
61 1.
3l l.
31 1.
_
31 1.
.
Re
4. Ox106
4. Ox106
.
4. OX106
4. OxlO6
4. Ox106
1. 7X106
.
2. 1x106 .
E xper i men t
Re=1. 7X106 to 18xlO6
Re=4. 5x106
Re=4. 5X106
Re=4. 5x106
Re=3. 94X106' Pn=O. 16
Re=1. 7x106, Fn=O. 21
Re=2. 1X106
Fig. 41 Minimum sDacine ~ ~
Fig. 42 Flat plate grid
'1 ' ' ' ' ' ' ' ' ' 1
O Computed
Empirical
—v v ~ >
Fig. 43 NVall shea.r stress ~w on a flat plate
~1
o
0
1
30
1 u
0 Computed
_ Empirical O ^~V
1,~. ~
O x 1
Fig. 44 Displacement thickness 51 on a flat plate
1 . O _
H
O
o
_ Ref. I
_ ,
[12]
[14]
1
[14]
[14
[15]
[~
_ [17] 1
30L
J~
~ .
oOO~OOOOOOOOOOOOO
0
O Computed
~v , ~
Empirical 
2 n+ 104 1o6
(a) x=0.07
I ~ r ~ r ~ ~ I
i ' ~' ~
y~U~ O Computed
Empirlcal
2 n+ 104
(~)) ~=0.5
' ',~ 1
T _ '~ 1
_ /0
_ , . . . .
1 102 n+ 104 1o6
(c) X=0.953
0 Computed
 Empirical
Fig. 46 Log plot of velocity on a flat plate
~ooo° O 0 0 0 0 0 0 Oooomcmn"
<' Computed
Fig. 45 Shape fa.ctor H on a flat plate
~ x=O.953
O /\
O
10 15
Fig. 4/ Kinema.t.ic eddy visco~ity ~t on a flat pla.te
1
79
OCR for page 71
   t~P 1
Fig. 48 Pressure contours of a. \\Tigley hull
Computed ~ Computed
u=1.0,0.9,0.8, ~ u=1.0, 0.95,0.
Measured
u=0.9,0.8,0.7  Measured (a) z/D = 0.2
(a.) x=0.5
 km=31 ~
'1
(a) ~ values (b) Number of grid points.
411 Effect of changing parameters on wake at
O Computed
~ ~< Measured (z/D=0.25)
t~q~O O `3 n ~  
tO.8rO.7, .. ~
u=0~95~0~9~0~8
,0.7 r O. . 6
(b) x=O.9
1 1
1 1
1 1
l I
1 1
~ 1
\ ~
Computed ~ \~
u=0.9,0.8,0.7 \\
rOe6' ~ ~ ~ \
Measured
u=0.9,0.8,0.7,0.6
(c) x=1.0
Computed ~ \~\\
u=1.0,0.95, ~ `\~\\
0.9,0.85,0.8 \ \\\
Measured
u=0.95,0.9,0.85,0.8 1
(d) x=1.5
Fia. 49 Wake contours of a TViale~ hull
Fig. 410 Ixinematic eddy viscosity ~` of a Wigley hull
at i=53 (x~0.95)
80
Iw
0.005
Iw
(c) z/D = 0.8
Fig. 412 Wall shear stress ~w on a \Vigley hull
0 x
0 Computed
r ~ ~ _ _ · ~ _
o
O x 1
(b) z/D = 0.5
J
r
O Computed
Measured (z/D=0.80)
1
OCR for page 71
2
1
H
o
Fig. 413 Displacement thickness 5~ on a Wigley hull 1
_ O ~ 00 Q ~ ~ ~ Q ~ Q ~ 0 ~Q
~ O z/D=0.2 _
_ ~ z/D=0.5
~ z/D=0.8
. . . . . . . . .
O x
Fig. 414 Shape factor H on a Wigley h~ll
0.005 1 ~ I I ' ' ' ' ' ' O 30 ~ .
/,,, ~
C Computed O  + _ ~ _
~ Measured (z/D=0.25) Oi u ~ O j=13 (z/D=0.22)
61 ~ O 1 ~ ~ ~ j=9 (z/D=0.51) 1
O ~ ~ ~ j=5 (z/D=0, 79) ~
1 1o2 n+ 104 1o6
~ 0 _ (a) i = 25 (x~ 0.07)
~ ~ O , , , , , , 1 30 ~ , , ~ ~
O ~ 1 _ ~
(a) z/D = 0.2 u+ ~ o j=14 (z/D=O.l9)
°°°51 ' ' ' ' ' ' ' ' ' ~ ~ ~ ~ j=10 (z/D=0.5
j=6 (z/D=0.7E
Computed ~ O , , ,
Measured (z/D=0.54) ~ 1 102 n 104 1C 6
~ (b) i = 38 (x~ 0.5)
61 1 ^~' 1 4OI . . , . . —
~=~ u+~ ~ ~
0 1 ~ O j=13 (z/D=0.22
O x 1 ~ ~ j=9 (z/D=0.51) _
(b) z/D= 0.5 ~ 0 j=5 (z/D=0.79) 1
o
0.0~ 1 1o2 n+ 104 10
_ (c) i = 53 (x~ 0.95)
~ Computed
Fig. 415 Log plot of velocity on a Wigley hull
1 1 ~ Measured (z/D=0.80) l 1 ~ ~
0: ~: ~ L; , 0~
O x 1 0 5 n/62 10 15
(c) z/D = 0.8
(a) i = 25 (x~ 0.07)
vt ~~ ~ ~ ao
uw6
0 1
0
1
;~ ~~o~
0 0 ~

o
5 n/62 10 15
(b) i = 38 (x~ 0.5)
81
OCR for page 71
1
At
uw62
r
Sopor on on
Lo
0
o
n/62 1° 15
(c) i = 53 (x~ 0.95)
t16 Kinematic eddy viscosity u~ on a Wigley hull
Computed
Measured
u=1.0,0.9,0.8,...
(a) x = 0.5
;
(b) Cb=0.7
/
. . .
(C) C13=0.8
Fig. 417 Pressure contours on Series 60 hulls
x=o .8 P~'O O 9 PO .05 1 0
(A ~'~\~\L'L§LL~
Computed
Measured
Fig. 418 Pressure contours at the stern of a Series
60 (Cb=0.6) hull
Computed
 Measured
u=1.0,0.9,0.8,...
(b) x = 0.7
Computed
Measured
u=0.9,0.8,0.7,.,
(d) x = o.g
Computed
Measured
u=0.9,0.8,0.7,.
(e) x = 0.95
 Computed i:'
Measured
u=0.9,0.8,0.7,..1.
(g) X = 1.05
 Computed
Measured
u=0.9,0.8,0.
(I) x= 1.0
Computed
 Measured
u=0.9,0.8,0.7
(h) x = 1.1
Fig. 419 Wake contours on a Series 60 (Cb=O.G) hull
82
OCR for page 71
o . 408
~w
O z/D=0.2
z/D=0.5
$~8096~^ ·:
o
O x 1
Fig. 420 Wall shear stress ~w on a Series 60 (Cb=0.6)
hull
o.oo
~1
H
+
u
O x=0.507 z/D=0.240
x=0.501 z/D=0.495
x=0.496 z/D=0.834
(b) i = 38
0 z/D=0.2
z/D=0.5
0 z/D=0.8
~ ^0 /\
O x 1
Fig. 421 Displacement thickness 5~ on a Series 60
(Cb=0.6) hull
I l ,
o
Q
~ oo
O
o
/\ o
,!\ o
'\ o
~ _
'` ° 8
o

~ _
~4o~
0 z/D=0.2
~ z/D=0.5
O z/D=0.8
Fig. 422 Shape factor H on a Series 60 (Cb=O.~) hull
o
L:
x=0.073 z/D=0.221
x=0.074 z/D=0.489
x=0.074 z/D=0.800
. . .
1 1o2 n+ 104 1`'
(a) i = 05
Jo6
 ,L/~~
_
. . . .
~ _
0 o° /
~/
O x=0.942 z/D=0.184
x=0.935 z/D=0.493 _
0 x=0.935 z/D=0.805
1 ~ I I
1o2 n+ 104 1o6
(c) i = 53
Fig. 423 Log plot of velocity on a Series 60 (Cb=0.6)
hull
vt
uw62
83
ol
 /0 tP dP ~o
/B ~'
0 5 n/62 1
(a) i = _5
0 x=0.073 z/D=0.221
x=0.074 z/D=0.489 ~
x=0.074 z/D=0.800 _
=0 ~
_0 15
,1 . . . o
0 5 n/62 10 15
_ i
/ 0 x=0.507 z/D=0.240
iO 0 0 ~ O ~3 x 0.501 z/D=0.495_
~ x=0.496 z/D=0.834
(b) i = 38
OCR for page 71
1
at
uwd2~ ~
5 n/62 10
_ r
^~^ a A
o,~08 V @^
O o ~ AL—U . ~ ;) J I/ L,—I . OU:)
Cal x=0.942 z/D=0.184
~ x=0.935 z/D=0.493 
~ .~_^ ~ ~ ~ ~ /~_m O^C
o
o
~ O ~ A
(c) i = 53
Fig. 494 Kinematic eddy viscosity zig on a Series 60
(Cb=0.6) hull
, ,, \`
Computed
u=l.O,O.9, . .
Computed ~ ~
~~ ~ n)
_
u=l.O,O.9, . . .
Measured ~1.O  Measured 1.
u=0.9,0.8,... \
(a) x = 0.9
u=0.9,0.8,...
(b) x= 0.95
Fig. 425 Wake contours on a Series 60 (Cb=0.7) hull
Computed ~1
u=1.0,0.9,...1.0
 Measured
u=0.9,0.8,...
Fig. 496 Wa.ke contours on a Series 60 (Cb=0.8) hull.
x= 0.95
Supplement
After the present paper was submitted for the pre
sentation at the Conference, the following disagreement
between the two measured results t16],(17] on the wake
contours of the Series 60 (Cb=O.~) model has been
found out. In Fig. A1 the computed wake is compared
u=o.9
with the ~~easure~ent obtained by Okuno t164? while
in Fig. 426 it is compared with the the measured re
sult ol:'tainecl l:>,y Ful;uda. and Fujii [174. The co~nputecl
wake in Fig. A1 shows reasonable agreement with the
Ol;uno's measurement, whereas it shows significant dis
crepa.ncy from the Fukuda's measurement in Fig. 426.
This clearly shows that measurements must be carefully
valiclat.ed before they are used in the validation of com
5 licit. a,tio~.
it
Computed 1 0
u=l.O,O.9, . . .
Measured[16]
1
~4
u=0.9,0.8,...
Fig. A1 Wake contours on a Series 60 (Cb=0.8) hull.
x=0.95
OCR for page 71
DISCUSSION
by C.M. Lee
I know that a few people in the past have
shown the NSSolver results for the DTRC body
forms. I am wondering if further progresses in
computational techniques for a fully appended
submerged body with propeller have been made.
From the paper of Dr. Fu jii this morning,
the CDF people in aeronautics seem to have
progressed to the stage that they can compute
the f low about a fully appended airplane. Have
you tried to include the sail and stern
control surfaces in your computations?
Author's Reolv
I have tried to compute a shipstern f low
with propeller effect, using the body force
method. Though I got a converged result, there
was pressure oscillation, which is due to the
use of central differencing. I have not yet
tried to compute f lows with appendages. In
order to do that, I think a multiblock ap
proach should be used.
85
OCR for page 71