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OCR for page 539
On the Numerical Solution of the Turbulent Flow-Field past
Double Ship Hulls at I,ow and High Reynolds Numbers
G. D. Tzabiras and T. A. Loukakis
National Technical University of Athens
Athens, Greece
ABS=ACr
Turbulent flow calculations have been carried out
for SSPA 720 double model at a low (Sx106) and a
high (Sx108) Reynolds number. The partially parabolic
algorithm was adopted to solve the complete
momentum equations and k-e model was used for the
Reynolds stress modeBing. At the low Reynolds
number, results for a whole field solution are
compared to those obtained by experimental input
amidships. Comparisons are also made between low
and high Reynolds calculations and conclusions
concermug scaling laws are derived.
1. Introduction
Advanced numerical methods, developed during the
last few years [1] have been applied with encouraging
results for the calculation of the turbulent flow-field
past the stern of double ship models. Most of them
are based on the simultaneous solution of the velocity
and the pressure field (the latter being essential at
the thick boundary layer region) and use zero, one or
two-equation turbulence models. Although the next
step seems to be the development of methods which
take into account the free surface effect and the
presence of a propeller, there is still a lot of useful
numerical investigation to be made on double hulls.
Two of the most important problems in this
investigation are the simultaneous solution of the
whole flow-field past a ship hull as well as the
behaviour of the numerical solution at high Reynolds
numbers, which is the case with real practical interest.
In the present work both of the above problems
have been worked out for the case of the SSPA 720
model, for which extended experimental information is
available [2], [3].The model has been numerically
tested at a low Reynolds number of SxlOb, which
corresponds to the test conditions and at a high
Reynolds number of Sx108, which corresponds to a
full size ship At the low Re No numerics results at
the stern region, obtained by either using input
conditions amidships from experimental data or
solving the complete flow field around the ship hull
have been compared to experimental results.
Moreover, the pressure coefficients at the bow region
predicted by the viscous solver have been compared
to measurements and to predictions from a potential
flow solution. The purpose of these exercises is to
provide some insight with regard to the applicability
of the method to the actual problem of ship design,
for which no experimental input data is available.
The high Re No numerical experiment is obviously
the more important from the practical point of view.
Although experiments at such high Re Nos. do not
exist, comparisons of the velocity profiles, skin
friction and pressure coefficients to those predicted at
the low Reynolds number can illustrate the trends of
the differences between the two solutions. In this
respect it is important to see if some flow
phenomena(such as the strong cross flow reversal or
the rapid changes in the velocity profiles around a
stern frame) which occur at low Reynolds numbers
are substantially lessintense at a high Reynolds
number. Moreover comparisons can be made between
the resistance components (viscous pressure and skin
friction) in order to test various assumptions
concerning the scale effect, when extrapolating model
test data
The method applied in the present work is
basically the fully elliptic algorithm reported in [4],
which solves the complete Reynolds equations on the
physical 3-D space using a sequence of locally
orthogonal Unilinear coordinate systems. The
Reynolds stresses are modelled in this code by the
standard two-equation k-e turbulence model.
2. The Numerical Method
2.1 Governing Equations
For the numerical solution of the transport
equations around the ship hull the computational
domain is covered by a sequence of 2D orthogonal
cunilinear grids as described in [5]. The latter are
generated on transverse sections by the conformal
mapping method [6] taking into acount the local
non-orthogonality at the intersection of the section
contour and the waterplane [71. Interpolated
sections, needed for grid refinement along the ship,
can be easily generated by cubic interpolation between
the transformation coefficients of adjacent sections.
Representative 2D grids used in the present
calculations are shown in Fig. 1 for some sections
along the model. In this figure the origin X=0 of the
longitudinal axis coincides with the midship section.
539
OCR for page 540
2x / L= -.8
`~i\3,43
zx/L= .9
zx/ L=.987
Re 5X10
Re 5X108
Fig. 1 Orthogonal curvi linear grids at various sections
540
OCR for page 541
A local 3D orthogonal curvilinear co-ordinate
system corresponds to each 2D grid generated as
above, whose two co-ordinate lines coincide with the
grid lines and the third one is normal to the section
plane. In a general orthogonal collinear system
(xi,xj,xl) with metrics (hi,hj,hl), where the indices i, i,
1 are in cyclic permutation, the complete Ui
momentum (Reynolds) equation is written as
C(ui) =- P hi aX + u3K3i+u~k,i-uiu3Ki3
uiu~Ki~ + (ail 033)K3i + (ail oD)K,i
+ coil (2Ki3 + Keg) + ',i~(2K 3~ + K 3~) + hi ~
+ hi Dx, + hl axl (l)
where C(ui) shows the convection terms of the Ui
velocity component, that is
C(ui)= h h h [ + ~ + 7~ ]
The stress tensor components appearing on the
right hand side of equation (l) are defined as
rsIi = tic 2eii= He 2 [ham+ hih, aX, + hIh~ ax,]
<,i3=~ e =p [ ]~( ,)+ hi a (ui)] (2)
where the effective viscosity He is modelled according
to the isotropic eddy viscosity concept, i.e.
lte = ~ + lit (3)
where ~ is the fluid viscosity and lit the eddy (or
turbulent) viscosity.
The curvature terms Kij are expressed as
1 phi
K =~
If the xl axis of the adopted co-ordinate system is
parallel to the ship symmetry axis (i.e. normal to the
ship sections) the following simplifications are valid:
hi = 1, Kl2 = K2l = K3l = Kl3 = 0 (4)
As already mentioned, in the present investigation
the standard k-e turbulence model [8] is employed for
the modelling of the Reynolds stresses, that is the
turbulent viscosity ,ut is calculated as
~ t PC D e (5)
where k is the turbulence kinetic energy, ~ its
dissipation rate and CD a constant equal to 0.09.
Two more differential equations have to be solved in
order to determine k and e. These equations can be
cast in the following common form:
div[ pOc - ~ t gradO ] = SO (6)
m
where~=kor e, ok= lam= l.3,Sk=G-Qc,
S = 1 44 G £ _ 1 92 p £
and the generation tell11 G is expressed as
G = 2pt[e2 + e jj + e2 + 1/ 2 (en + Hi + e il )]
The complete equations (l) and (6) are solved
numerically for all tranverse sections following the
finite volume approximation. A staggered grid is
employed and the differential transport equations are
integrated in the corresponding control volume of each
variable Op. resulting in an algebraic equation of the
general form
Apes = ANON + ASKS + ACHE + AW~W
FADED +AUOu + ~ (8)
where the notation P. N. S. E, W. D, U corresponds
to grid points shown in Fig.2. Coefficients Ap, AN....
take into account the combined effect of convection
and diffusion terms modelled according to the hybrid
scheme of Spalding [9]. Equations (8) form a system
of algebraic equations which is solved by succesive
applications of the tndiagonal matrix algorithm.
IN
~X2// 1 D
WE
3
W P. ,
S
Pi g .2 Speci fi cati on of gri d poi nts
541
OCR for page 542
2.2 Boundary Conditions
The calculation domain around the ship can be
divided in two sub-domains surrounding, respectively,
the front and the rear part as shown in Fig. 3. In
the front part domain (I), corresponding to the thin
boundary layer region, relatively coarse grids can be
used to model the viscous flow. Besides, high
convergence rates of the numerical solution can be
achieved due to the strong upstream convective
influence and the existence of favorable pressure
gradients over the major part of the body surface.
The aft part calculation domain (II) covers the thick
boundary layer region around the stem of the ship
and extends in the near wake. The flow there is
characterized by complex phenomena such as vortex
formation, interaction between the boundary layer and
the wake or adverse pressure gradients and finer
grids must be applied in order to obtain reliable
numerical results.
Pi g .3 Def i n i ti on of sub-domai ns
following conditions are valid:
waterplane:
The solution of the elliptic-type algebraic equations
(8) requires specification of boundary conditions at ship symmetry plane:
each bounty of the two sub-domains, that is at the
inlet planes U. the external boundaries N. the exit
planes D, the solid surface S and the syrrunetry
planes of the ship (Fig. 3).
The input boundary values for the velocity
components at the inlet plane UI of the front part
domain, located upstream the ship's bow, are
calculated from the potential flow solution. The latter
is obtained by the application of the classical Hess
and Smith method [10] around the actual shape of the
ship. The values of k and ~ at the same boundary
are assumed to be equal to zero. The corresponding
input boundary conditions for the velocity components
and turbulence quantities at the inlet plane UII of the
second calculation domain are determined from the
front part flow solution by linear interpolation. At
the same plane, input conditions can also be
calculated by empirical formulae, whenever
experimental data are available.
At the exit planes D of the domains the flow is
assumed to be fully developed, corresponding to the
application of Neummann conditions for each variable,
except the pressure. The latter is calculated by linear
extrapolation from the computed values at the
previous sections.
The velocity components and the pressure at the
external boundaries N are calculated from the
potential flow solution, whereas for k and ~ the
normal to the boundary derivatives vanish
(Neummann condition).
The turbulent flow near the solid boundary is
modelled according to the standard wall function
method [11] assuming that the velocity in the adjacent
to the wall cells follows the logarithmic law
u+ = 1/x In (Ey+) (9)
where =0.42, E=9.79, y+ the non-dimensional
distance from the wall and u+ the non-dimensional
velocity parallel to it. Relation (9) is implicitly
introduced in the momentum and k-e equations
leading to a simplified set of boundary conditions for
the corresponding variables.
Finally at the two flow symmetry planes the
U3 =0 , ant =0 , An= Ul'U2'P'~6
U2 = 0 , Ax' = 0 , ~ = Mu flak
2.3 The Solution Algorithm
The existence of a dominant flow direction along
the co-ordinate axis xl, which is parallel to the
symmetry axis of the ship allows for a marching
solution of the governing transport equations, known
as the partially parabolic algorithm [12]. The method
has been ordinally developed to solve the parabolized
Navier-Stokes equations [13] but it can also be
applied to the solution of the complete form of
equations (1) and (6).
According to the partially parabolic algorithm, a
local numerical solution is performed in each
transverse section of the calculation domain. Firstly
the U3, u2 and u1 momentum equations are solved
and then the pressure field and the velocity
components are corrected to satisfy the continuity
equation. Then the k-e equations are solved and the
eddy viscosities are updated using relation (5). In this
local solution two-dimensional in-core storage is
essentially needed for various geometrical and flow
parameters, permitting the use of fine grids even with
conventional computers. After solution for every
section of the domain is performed, a sweep is
completed and calculations start again. Several sweeps
are needed until both the velocity and pressure fields
converge.
The most crucial point in the application of the
partially parabolic method is the treatment of the
pressure field. In the present work the SIMPLE [14]
algorithm has been adopted for the local correction
of the pressure and the velocities. The application of
this algorithm requires underrelaxation of variables
during the iterative solution procedure, that is the
updated value ~ of a variable is calculated as linear
combination of its previous value NO and the solution
En Of system (8), through relation
~ = r En + (1-r) To
where r is the underrelaxation factor which is
constant for every grid node of a transverse section.
542
OCR for page 543
Although SIMPLE and partially parabolic algorithms
form the basis of the convergence procedure, it has
been found that at the front part calculation domain
(Fig.3) they can be combined in a different way than
at the rear part domain. While at the stern region
several iteration SIMPLE steps are needed to achieve
local convergence, the existence of a thin boundary
layer over the bow and the middle body of the ship
allow a single step local solution at the front part
domain. The latter results in a decrease of the
computational cost by a factor of 30.
A new approach has also been applied for the
numerical solution at high Reynolds numbers. The
high grid densities, required to model the turbulent
flow near the wall at the above numbers, lead to the
generation of computational cells having their normal
to the wall dimension substantially lower that the
other two dimensions. This geometrical property is
quite unfavorable for the pressure correction methods
applied in this case, especially at the stern region
where steep longitudinal and transverse pressure
gradients occur. Moreover, at the same region, it is
difficult to obtain the necessary grid clustering near
the wall by global grid generation methods.
To overcome the aforementioned problems a special
near wall treatment has been developed, as shown in
Fig.4. The near wall computational cells
corresponding to the initial mesh generation, can be
automatically subdivided to any desired number of
sub-cells and a second computational domain is
created. Two different solutions are applied in the
resulting internal and external domains. For the
internal solution the pressure values within the normal
to the wall generated cells is assumed to be equal
to the "external" value at node N. This assumption
valid near the solid boundary, is quite beneficial for
the solution procedure followed: the US and u1
momentum and k-e equations are solved in the
sub-domain as in the external domain, while the u2
component (normal to the wall) is calculated explicitly
from the integrated continuity equation. For the
external solution the SIMPLE procedure is followed.
At the common boundary (B) of the two domains the
boundary conditions for various variables are updated
according to the adopted finite difference formulation
for the convection and diffusion terms. Convergence
in a transverse section is achieved after several
successive internal and external solutions.
~ 4 IN ~
,,,,,, ;:
Fig.4 Near wal 1 treatment
543
3. The Numerical Tests
As already mentioned in the introduction,
calculations with the described methods were carried
out for SSPA-720 double model.
3.1 Low Reynolds Number Computations
For the front part calculations, a 32x20x61 grid was
used where 32 is the number of grid nodes along the
girth, 20 the number of nodes along the normal
direction to the section contour and 61 the number of
transverse sections. The inlet plane of the calculation
domain was placed at 2X/I'-1.2 and the exit plane
at 2X/I'0.30. Convergence of the numerical solution
was achieved in 80 single-step sweeps of the domain
and constant underrelaxation factors equal to 0.4
were used for each variable. The values of y+ in the
adjacent to the wall cells ranged between 30 and 50,
that is within the suggested region for the application
of wall functions (30~150). In Fig. 6 results for the
predicted Cp coefficient are compared to experiments
as well as to potential flow calculations. The latter
were obtained using 673 quadrilateral elements on the
model surface.
For the rear part calculation domain a 32x30x44 grid
was used starting at 2X/I'O. 1 and extending up to
2X/L=1.4. Both types of input boundary conditions
were tested, the first one corresponding to a whole
field solution and the second to an experimental
input. In the second case the velocities within the
boundary layer were calculated according to the 1/n
power law using the experimental data of Larsson [2],
while the initial values for k and ~ were estimated by
empirical formulae [5]. A total number of 35 sweeps
was needed to obtain convergence in either case. An
initial number of 15 iterative steps was required for
local convergence in a transverse section, which
reduced to one up to 5 steps during the last sweeps.
The values of the underrelaxation factors at the rear
part were constant and equal to 0.5 for every
veriable. The values of y+ ranged between 30 and
170.
In Fig.7 computational results for the streamwise
(IJ/Ue) and crosswise (W/Ue) velocity components are
compared to experiments for points 11 tol7 of
station 2X/I'O.9 shown in Fig.5. In this Figure the
vertical axis refers to the non-dimensionalized normal
distance from the body surface with respect to the
experimental [2] boundary thickness be and the
horizontal axis to the non-dimensionalized velocities
with respect to the velocity at the edge of the
boundary layer. A special output program has been
developed to compute the necessary variables along
normal' to the body surface by linear interpolation
among the stored values. It should be noted here
that the experimental results were subject to
blockage effects while no such effect has been
accounted for in the calculations.
In Fig. 8 the calculated, non-dimensionalized by the
free stream velocity, CF coefficient is compared to
the experimental data around the girth of the
previous section. Results for the pressure distribution
are also presented.
3.2 High Reynolds Number Calculations
The same transverse sections and girthwise points
as in the case of the low Reynolds number
OCR for page 544
calculations, have been used for the high Reynolds
number tests. The grid density was different only
along the normal direction, as shown in Fig. 1. A
32x30x61 grid was employed for the front part
calculations and convergence of both the velocity and
pressure fields was obtained in 150 sweeps. Constant
underrelaxation factors equal to 0.3 were used for
each variable. The values of y+ ranged between 100
and 300.
A 32x30x44 grid was used for the stern part
calculations. The underrelaxation factors were equal to
0.5 for all variables and convergence was achieved
after 25 sweeps. Two grid types have been tested,
that is a coarse near the wall grid with y+ varying
from 100 to 1400 and a fine grid according to the
method previously described. The latter was created
by dividing the initial near wall cells in 10 sub-cells,
that is a total of 40 grid nodes along the normal
direction was used. Starting with the coarse grid
solution, 15 more sweeps were needed to obtain
convergence with the fine grid. The corresponding
values of ye ranged between 15 and 150.
Comparisons between the calculated results by the
two grids as well as with the results for the low
Reynolds number are presented in Figs. 9 to 12.
/15
/13
112
~9
'119
11
Fig.S Distribution of calculation points
at 2X/L=O.9
4. Discussion of the Results
4.1 Low Reynolds Number
The calculated pressures along the girth of venous
sections of the fron part of the ship compare well
with the experiment values, Figs. 6 a to 6 d, when
corrected for blockage effects as proposed by Larsson
[15]. The agreement is good both for the Viscous
and the potential flow pressure calculations, with the
exception of the lower part of the stern section with
2X/I'-0.93. In this case it is believed that the
discrepancy is due to the insuffient accuracy of
classical Hess and Smith method near the extremeties
of the body.
Cp
.1 ~
.1
d
of'
~ ~ 2SC/L=.93
''/
/ ~ pot. CDIC.
/ viscous calc.
experiments
a
*
fit
. ~
b
2X / L=-.7
if\
__ ~ .._ ~ .~
* *
~< ~ ~
2x/ L=-.4
. . .
_ .1
In_ _ _ ~ _ _ ~ _ _ _ ~ _ _ a_ _K
2X/L- O
L GIRTH %
o
1 0 0
Fig.6 Comparisons of pressure coeffici ents at
the fron part for Re=5xiO 6
544
OCR for page 545
lo
Ax
1 14~,
a)
alc- =4~
can *
CL
o
n
*
a,
,,_, C
~ _
Q ~
. _ ~
X X
1
~ ~ -.
1
1 ~
3 .
.U ~ . ~ ~ L_.
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a) `
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~ -
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U'
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_ *-
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545
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OCR for page 546
Therefore it may be concluded that the proposed
Viscous flow calculations for the front part of the ship
can be conveniently used in conduction with the same
calculation method, which is widely used for the aft ·1
end of the ship. The advantages of such calculations
are that they do not require any assumptions with
regard to the upstream input boundary conditions,
while they automatically generate input conditions for
the aft part solution. Needless to say that the method
is more expensive to run than simpler approaches.
The calculated velocity profiles, using both the whole
field solution and the aft part solution based on
experimental input, are compared to measured values
along the girth of section with 2X/L=O.9, Fig.7.
The calculated results agree in general well with the -1
measurement for both velocity components. The
results based on experimental input are somewhat
better but, as it will be pointed out later, the overall
difference in the total resistance prediction between
the two methods is very small. The overprediction of
the streamwise velocities near the surface at points
19 and 9 can be explained by observing from f~.7
that in this region the geometry of the hull surface is
rapidly changing, a situation for which the k-e 4
turbulence model is known to overpredict [1], [16],
[17]. It should be noted here that the comparison
between measured and calculated velocity profiles is
somewhat indirect because the measured results are
affected by blockage effects. However numerical
calculations for the same hull [4], taking into account
blockage effects, have shown that the non-dimensional
velaecity profiles remain practically the same.
The predicted by both aforementioned methods
values for CF along the girth of the same as above
section are compared to experimental values in fig.8.
The agreement is particularly good for both methods.
The corresponding values for Cp, predicted by boath
methods shown in the same Figure, are in close
agreement. No experimental pressure values exist for
this section.
Finally, although there exist no measured values for
the total resistance of the ship to be used for
comparison purposes, it is interesting to note that
the predicted by the whole field solution total
resistance is 2.5% higher than the one predicted using
experimental input for the stern part solution.
4.2 High Revnolds Number
The results for the velocity profiles at station
2X/L=O.9 shown in fig. 9, allow us to conclude that
the local and refinement produces no noticable effect,
although it requires approximately 30% more
computer time. The same conclusion is reached by
observing Fig.10, where the girthwise results for Cp
and CF at the same station are shown. This is a
remarkable result showing that at high Reynolds
numbers the wall function method is valid for a wide
range of y+. However more numerical experiments
should be made for various hull forms to establish
this behaviour.
The overall difference in the prediction of the total
ship resistance using both methods is of the order of
1%.
/~ GIRTH %
~ v / ~ ~ ~ . ~ ~ ~ ~ .
it, . . . . . . . .
Re 5~106
2 L
546
~ *
*I\\
\\\
\
I\
Cf~l00
whole field
- exp. input
experiments
I , , GIRTH %
Fig.8 Comparisons of pressure and friction
coeffi ci ents at 2X/L=0.9
4.3 Comparison of the Low and High ReYnolds
Number Cases
100
The profiles of the streamwise and crosswise velocity
components are shown in Fig. 11 for the station with
2X/I'O.9. All calculations were performed using the
whole field solution. The streamwise component is
higher for the high Re. No., as expected, with larger
differences near the keel. A more interesting
conclusion is reached by observing the crosswise
velocity profiles. At the high Re.No. the S shape of
this profile, which exist at the low Re. No., is lost or
is reduced. Consequently, the hull form is less prone
to vortex formation at the high Re. No., a fact which
has also been experimentally verified.
Finally the girthwise distribution of the Cp and CF
distribution for the same station are shown in Fig. 12
for both Re. Nos. In the same Figure the calculated
values for pressure using potential flow are plotted.
As expected the potential flow solution yields larger
values than the viscous flow solutions, with the
differences diminishing with increasing Re.No.
The girthwise values of CF are more constant for
the high Re.No., whereas CF has very high values
near the keel for the low Re.No.
OCR for page 547
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air
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air
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r
OCR for page 548
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548
OCR for page 549
cp
[18], for which grid independency of the numerical
solution was achieved.
-41
.1
1\ ~ GIRTH ;:
. . , ~ · ~ ~ ~ . . I
VIellelI.
R4r Axiom
-Cf *1000
- y+=100....1400
ye = 15.. 150
Fi g . to Compari sons of pressure and fri cti on
coeff i ci ents at 2X/L=0 .9
4.4 Scaling Laws for Ship Resistance Prediction
With regard to the independency of the numerical
results to the transverse grid parameters, previous
calculations with a coarser grid [4] have yielded quite
solar results for the low Re No case. Similarly for
the high Re No. case, a more dense grid near the
body surface has produced effectively the same
velocity profiles. Therefore we can assume that the
numerical results are reliable in this respect.
However, no such examination was performed with
respect to other grid parameters, which Night affect
the numerical solution, e.g. the number of the hull
cross sections used for the calculations especially for
the low Re.No. case. Nevertheless, the numerical
results seem to be good enough to allow for an
attempt to demonstrate their impact on practical
procedures for the prediction of the resistance of the
ship. In this endeavour we are encouraged from
similar trends for the case of a body of revolution
As it is well known, ship resistance predictions are
based on model experiments, on a flat plate friction
line and on appropriate scaling laws. We now
consider that the examined low Re.No case represents
model tests with results shown In Table 1, which also
contains the results of a corresponding full scale
experiment. Since the free surface effect has been
neglected, this pair of experiments is thought to be
conducted at a Froude No equal to, say, 0.15.
If we now use both the form factor (K) method
and Froude's method, each in combination with both
the I.T.T.C. and the A.T.T.C. friction lines, we can
derive Table 2.
Then, it can be concluded that none of the above
combinations predicts the ship resistance adequatly aIld
that the form factor method underpredicts, but it is
closer to the calculated ship resistance than the
Froude method, which overpredicts.
Needless to say that the numerical methods
presented herein can be easily used for the direct
prediction of the ship resistance, a fact which
necessitates full scale experiments to validate their
accuracy.
Finally, it should be mentioned that a complete set
of calculations for a hull form require approximately
60 hours of computer time on a MicroVAX II
machine, amrnount which can be reduced to about 2
hours on a modern RISC technology workstation.
, GIRT X, ,
100 REFERENCES
1. Patel, V.C., "Ship stern and wake flows: Status of
experiment and theory", 17th ONR Symposium, The
Hague (1988).
2. Larrson, L., "Boundary layers of ships (lbree-
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,1 ~
4
550
:\` ~ /; GIRTH ;:
_r I L ~ . . I · I ·
1~ , j , ~ ~ . . .
_ ~
*
* * * * * ****
* ~.
- Re- 5~108
Re-5xl°6
1` ~ Pot.flow
\
\
Cf *1000 '~= ~ ~ ~ - _
, , , , , , {;IRT~ X, ,
keel 100
Fi g. 12 Compari sons of pressure and fri cti on
coefficients at 2XiL=O.9
OCR for page 551
Tab l e 1: Cal cul ated resi stance coeff i ci ents
Re = 5X10
CT 4.754xlO ~
Cp 1. 024x10 -3
CF 3.73 ~0-3
Re = 5x10,3
2.560 10 3
0.77 x1o ~ 3
1. 79XlO -3
Table 2: Calculation of ship total resistance coefficient by the form factor and Froude method
.
CTM x 10 3 CFMX10 3 K CRX10 3 CFSX10 3 CTsx 10 ~ CTsxlO 3
computed
form factor
pl us 4.754 3.397 0.399 _ 1.671 2.21 2.56 -13.6
I TTC
fr~ct~on l ~ne
form factor
plus
ATTC
friction 1 ine 4.754 3.294 0.443 1.671 2.32 2.56
Froude s
plhsd 4.754 j .397 1 1 1 35 1 1 671 1 3 OZ8 1 2.5
I TTC f . l .
;roude s T 4.754 1 ,.294 ~ _ ~ 1.46( 1 1.671 1 3~ 13 1 2.5~ :
pl us
ATTC f . l
% Di ff .
- 9.4
+ 17
+ 22
551
OCR for page 552
Representative terms from entire chapter:
ship resistance
