Below are the first 10 and last 10 pages of uncorrected machineread text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapterrepresentative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 789
On the Numerical Solution of the Total Ship Resistance
Problem under a Predetermined Free Surface
G. Tzabiras, T. Loukakis, G. Garofallidis
(National Technical University of Athens, Greece)
ABSTRACT
The free surface around a ship model
moving at constant speed was determined
experimentally. The resistance components of
the model were computed numerically by solving
the Reynolds equations beneath the predetermined
free surface. The calculations were made using
the finite volume approach and the partially
parabolic procedure. The standard ke turbulence
model was used for the Reynolds stresses.
Calculated and measured values for the total
resistance have been compared and the
applicability of the method is discussed.
NOMENCLATURE
Al finite difference coefficients
C(~) convection term of
Cp pressure coefficient
Cf skin friction coefficient
G generation term of k
g gravitational acceleration
hi metrics
h vertical distance
kij curvature tensor
k turbulence kinetic energy
deformation tensor
 1J
P pressure
PI pressure (P+ggh)
RF frictional resistance
Rp pressure resistance
RT total resistance
velocity components
collinear coordinates
ui
xi
S.`:, source terms
Greek symbols
dissipation of k
~ fluid viscosity
lit eddy viscosity
he effective viscosity
Q fluid density
oil stress tensor
variable
Dept. of NAME, NTUA, 42, 28th Oktovriou Str., Athens 10682, GREECE
789
INTRODUCT ION
The degree of accuracy at which a physical
problem should be solved depends, obviously, on
the application of the solution. In this respect, in
ship design and construction, the most important
problem pertaining to hydrodynamics is the
accurate prediction of the ship speed and the
corresponding propeller revolutions and shaft
horsepower. This is because these quantities are
specified in the contract of a newbuilding and the
shipyard has to pay penalties if the ship
propulsion performance is found inferior during
the ship delivery trials.
Therefore, the efforts of untold numbers of
marine hydrodynamicists and mathematicians
during the last century or so to solve analytically
the ship propulsion problem have been well
directed. However, and from the practical point
of view, these efforts have been largely
unsuccessful. Thus, in current engineering
practice, only the prediction of the propeller
performance in a prescribed wake field is thought
to be trustworthy enough to be actually used on
a routine basis. The prediction of pressures and
shear stresses around the hull of a ship, moving
at constant speed in calm water, remains an
elusive goal. The same is tree for the wake
field behind the hull and for the propeller hull
form interactions. This state of the art is
especially bothersome if the availability of
virtually unlimited computational power is taken
into account.
On the other hand, more difficult but less
important problems, from the point of view of
contractual obligations, such as the dynamic
behaviour of the ship in waves can be and are
successfully treated using simple theories of
moderate accuracy.
The simplest of the yet unsolved problems in
ship hydrodynamics is that of the ship resistance
in calm water and at constant speed. In this case
it is the belief of the authors, that for the vast
majority of practical applications this problem
should be treated from the beginning as a viscous
flow problem. That is, it is believed that the
incomplete modeling of the flow as inviscid, in
order to obtain the wavemaking resistance
separately, will not lead to a successful solution
of the real life problem.
OCR for page 789
In recent years the availability of computer
codes solving the complete Reynolds equations
and of powerful computers has yielded promising
results in the case of computing the flow field
around three dimensional, shiplike bodies in the
absense of a free surface. A recent survey of
this area of research is presented by Patel (11.
But even for this simplifed case some problems
seem to remain as can be deduced from the
scarcity of computations for the high Reynolds
numbers of the ship scale (2~. Very recently,
results have been presented for the solution of
the complete problem of a ship moving in a
viscous fluid, e.g. Hino (3~. Although the results
presented seem reasonable when compared to
experimental values, no result for the ship
resistance is given. Moreover, the method does
not seem at this stage to include the computation
of the equilibrium position of the ship under way.
That is the ship is assumed to move at its static
equilibrium position, which is a common
assumption when trying to solve the ship
resistance problem. However, as it is well
known from experimental results, the dynamic
equilibrium position is an important factor for the
determination of the ship resistance, for a ship
moving at constant speed and weight.
From this discussion, one might conclude
that, although many steps have been made
towards the prediction of the ship resistance, no
practical solution of the problem is in sight. This
conclusion is strenghtened by the fact that no
model of turbulence, required for the Reynolds
equations, has been derived with free surface
flows in mind.
In view of the above it was decided that a
meaningful intermediate step, in the long route
necessary before ship resistance can be
analytically computed, was to remove as many
uncertainties as possible and to treat a simpler
case with the method described in (4~. Thus, a
three meter model of a liner ship was tested in a
Towing Tank and its dynamic equilibrium position 9~nx4.65mx3.0m.
as well as the wave pattern around it were
measured together with its total resistance. In
this manner, and looking at the ship from below,
the actual solid and liquid boundaries for the flow
were determined. It was thed straightforward to
run the NTUA viscous flow computer code for
this prescribed fluid region, applying appropriate
conditions at the boundaries and obtain the total
ship resistance, as the sum of the pressure and
the wall shear stress forces.
The viscous flow computer code of NTUA
uses the standard ke turbulence model which has
been applied with relative success but it is
unknown if it is valid near a free surface. For
axisymmetric fully submerged bodies, the method
gives good pressure predictions and a small
overestimation of the velocities (5~. The
integrated results, that is the force predictions,
are good. For double hull ship forms, the
method gives again good results for pressures
and wall shear stresses although it overestimates
the velocities in some areas (2~. However, when
the method was used to obtain the integrated
resistance force for the case of a landing ship
running at a low Froude number, the computed
results seemed to overpredict the measured
resistance by almost 7% (6~.
790
The results of the present research effort,
which is modest in scope as it is unsponsored,
can be of value since they focus on the
capabilities of the numerical solution of the
viscous flow problem in a predetermined domain.
As will be seen in the sequel, the analytical
prediction of the total ship resistance, although
reasonable, does not compare well with the
experimental value. Therefore, more research is
necessary as will be explained in the conclusions.
For this reason, the idea of using the
predetermined free surface to compute the ship
resistance for the Reynolds number of the full
size ship was abandoned as premature.
DESCRIPTION OF THE EXPERIMENTS
The 1:50 scale model of a liner ship was
selected for the purposes of the investigation.
This model had been tested previously at the
Towing Tank of N.T.U.A. For the same ship,
another model at a 1:30 scale had been tested at
N.T.U.A. and a third model, at a 1:22.43 scale at
the Bulgarian Ship Hydrodynamics Center. Thus,
enough data were available for the determination
of the form factor according to the lllC
methods and definitions.
The principal chacracteristics of the 1:50
model are shown below and the body plan of the
model is shown In Fig.1.
Length on Waterline LWL= 3.083 m
Beam B = 0.429 m
Draught T = 0.179 m
Block Coefficient CB = 0.575
Prismatic Coefficient Cp = 0.601
Midship section Coefficient CM = 0.956
Wetted Surface S = 1.694 m
Displacement ~ = 133 kp
The dunmensions of the Towing Tank are
Figure 1. The body plan of the model.
OCR for page 789
Resistance Measurements
The model was attached to a
dynamometerheave rodpitch bearing assembly,
which can measure the model resistance, parallel
sinkage and running trim as the model is towed
at constant speed.
The model was tested repeatedly at a speed
of 1.346 m/sec., which corresponds to a Froude
Number of 0.245. At this speed and for a water
temperature of 22°C, the measured resistance and
parallel sinkage were:
Speed Resistance Parallel Sinkage
1.346 (mls) 0.682 (kp) 0.6 (cm)
The measured valued of running trim was
practically zero.
With regard to the resistance measurements
in particular and the test conditions in general,
two remarks are in order. Firstly, the blockage
effect on this model is negligible, about 0.4%
when expressed as a speed correction and
secondly that, although the model speed is quite
constant, the resistance force is not. This can be
explained by the fact that resistance
measurements are obtained by measuring the
deflection of an elastic member connecting the
model to the towing carriage. During the run this
spring  mass system can be excited to oscilate
along the longitudinal axis of the model. For the
particular model, an analysis of the time history
of the resistance force during a typical 30 seconds
run showed an oscillation of approximately +10%
about the mean value. However, since the actual
logitudinal deflections of the system are extremely
small, the oscillation of the model is not expected
to affect the steady wave pattern.
Form Factor Determination
During previous tests with the 1:50 and 1:30
scale models, it has been determined that the
value of the form factor for  this hull form is
0.14. That is the total viscous resistance is
CV=1.14CF, where CF is the frictional resistance
of a flat plate according to the 1TTC 1957
formulation.
The same value for the 1:22.43 scale model
is 0.16. Needless to say that the ITT C method
for deterring these form factors is
approximate, as one tries to establish Usually at
what speed the wavemaking resistance practically
disappears and what is the corresponding model
resistance, at a low speed region around Fr.No =
0.12, where the experimental results show no
negligible scatter.
Measurements of the Steady Wave Pattern
The wavy free surface necessary for the
numerical calculations was obtained in a mixed
manner. Different methods were used for the
intersection of the free surface and the hull
surface, for the wave region near the hull surface
and for the wave region away from the hull
surface.
Measarements of the Free SurfaceHull
Intersection
An auxiliary grid was painted on the surface
of the model about the waterline. The dimensions
of the grid were lem for the waterlines and 2cm
for the transverse sections. It was then tried to
determine the intersection photographically. The
resolution of this procedure was not satisfactory,
in particular near the bow where the
measurements are most important. The required
intersection was finally obtained by scratching
several points at the side of the model during the
run and then taking the model out of the water
and drawing a faire d line through these points.
Measurement of the Wave Region Near
the Hull Surface
This region was defined to extend from the
centerline of the model to a distance of 38.9cm
sideways. It is reminded here that the maximum
half breath of the model is 21.45cm. For this
region the wave pattern was obtained using
photogrammetric methods. The methodology
applied and the results obtained have been
described in (7) and will only brieflfy discussed
here.
Two nonmetric motor driven cameras,
Hasselblad ELM with normal angle lenses of
8~nm principal distance, were used. The cameras
were positioned on a rigid base, which could slide
along a specially constructed guiding rail bolted
on the towing bridge. In this way the relative
position of the cameras remained undisturbed and
the coverage of the ship model with stereoscopic
models became possible. The two cameras were
electronically synchronized and a powerful flash
was attached to the system in order to take care
of the poor lighting conditions under the bridge.
The cameras were positioned approximately
200mm apart, at a distance of lm from the
waterline and with an inclination of approximately
35 grad In this way a favourable
basetodistance ratio was ensured, while at the
same time maximum possible coverage of the
object was obtained.
The problem of providing detail points on
the water surface was solved by spraying, just
before the shutters were fired, yellow papertape
punch of lmm diameter on the water surfing.
For the basic control of the orientations, an
aluminum bar bearing two retrotargets at a
distance of 283mm was hung on the model. The
stereoscopic models were levelled, by taking
two pairs of pictures, one at rest and one
underway for each case. Nine pairs of
stereoscopic models were taken in order to cover
the whole legth of the model and the necessary
area behind it.
The photogrammetric processing of the
photography was carried out on a ZEISS
Stereocard G2 connected via a DIREC 1 unit to a
desk top computer. Analytical processing of the
stereocard data was used. An accuracy of about
lmm for the wave heights can be obtained using
this method.
791
OCR for page 789
Measurement of the We're Region Away
from the Hull Surface
The wave region extending from a distance
of 38.9cm from the centerline of the model to a
distance of 218.9cm was covered by taking 95
longitudinal cuts of the wave surface. The
longitudinal cuts were obtained by the repeated
use of commercially available wave monitors of
the resistance type. A specially constructed
overhang beam was used for the attachment of
five wave monitors at predetermined distances
from the centerline of the tank. The probes were
stationary and they were recording the wave
elevation as the model was passing by. One
probe was always positioned at a distance of
28.9cm from the centerline and it was used to
"allign" the other probes longitudinally.
Thus, four longitudinal cuts were obtained
per run and 24 runs were necessary to obtain 96
longitudinal cuts. The adignement of the wave cuts
was based on the crest of the first wave of the
28.9 cm cut. The same point was used to allign
the whole system of the longitudinal cuts to the
model using the results of the stereoscopic
model of the bow region. An additional cut at a
distance of 33.9 cm was used to check the results
of the photogrammetric procedure. The details of
the longitudinal cut measurements are given in
(8).
The accuracy of the wire probe measurements
is of the order of one milimiter, on the basis of
their static calibration curves. How this accuracy
is affected during dynamic measurements is not
known.
Analytical Determination

Cuts of the Ware Pattern
of Transverse
curvilinear orthogonal grid is created using the
method of singular distributions, as described in
the sequel. In Fig. 2 eight orthogonal
curvilinear meshes are shown at various sections
along the ship model and the wake. The
boundary S (Fig. 2a) is the model section contour
and the boundary W is the free surface
intersection with the corresponding transverse
plane.
The generation of an orthogonal grid in the
2D domain defined by the boundaries N,S, E, W
shown in Fig. 2a is based on the incompressible
potential flow solution (9~. A singularity
distribution on the four boundaries is assumed,
i.e., a source distribution on boundaries N. S and
an eddy distribution on E. and W. Using
rectilinear elements, the unknown distributions are
calculated to satisfy the boundary conditions of a
potential function A, that is
at ~ =o
an N,S
at
1 =0
as E,W
whre n is the normal direction on N or S
contours and s the direction tangential to E or W
boundaries. After the element source or eddy
distributions are computed, the grid nodes can be
specified as intersections of equipotential and
equistream function lines, following the iterative
procedure described in (9~.
The velocity components and the other flow
variables refer to local orthogonal curvilinear
coordinate systems coinciding in two dimensions
with the grid lines x1 = const and x2 = const,
while their third direction X3 is always parallel to
the ship longitudinal axis. These systems vary, in
general, along the ship as the geometry of the
frames and the free surface changes. It should
be noticed here that the coordinate system is
always orthogonal, while the numerical grid is
nonorthogonal in the X3  direction.
Based on the model speed and the frequency
at which the probe signals were digitized, a
minimum distance of 1.034 cm between successive
transverse cuts could be used to determine the
free surface in the outer field. Since the The Governing Equations
measured points near the hull were located at
random, an interpolation procedure was firstly
applied to estimate the wave elevation in the
inner field, on the transverse cuts determined at
the outer field. Actually, all points within a
bandwidth of 1.034 cm were used to determine
the wave contour on the midplane of the band.
Finally, a second order smoothing method was
used to generate the waveform in the combined ulmomentum
inner and outer field domain of each transverse
cut, as shown in Fig.2. In this manner 300
transverse cuts were generated along the ship
length, of which 150 were used for the numerical
computations.
DESCRIPTION OF THE NUMERICAL
METHOD
The coordinate system
In a local orthogonal cirvilinear coordinate
system, described as above, with metrics hi, h2,
ho end curvatures k12' k21, the time averaged
NavierStokes (Reynolds) equations can be written
as in (101:
Crud) =  h aX + punks  pu~u2k,2 +
~ ~t 622)k2~+2~2k~2+h .0X +h~ ~ ~2 +ht i3
u2momentum
The transport equations describing the flow C(u)=h aX +pUlkl2pulu2k2l+t022 all) 12
around the ship are solved numerically in the ~ 9~ aft
physical space. The calculation domain consists of ~ 22 1 ~2 ~ 23
transverse sections and on each section a +2k2l~l2 + hi ~x2 + h Dxl ha dx3
792
OCR for page 789
w "~!
(I?.P) a
.
c
the Main. Wear gods along
793
b
OCR for page 789
art
g
re 2 (conffnued)
794
e
l
OCR for page 789
US momentum
in*
C(U 3) =  h OX +kl2(~13 + 23)
1 ~ 33 1 ~ 23 1  13
+ + ~
ha DX 3 ha DX 2 hi dX1
where C(ui) shows the convection terms, i.e.:
(1)
C(Ui )= h P [I i + ~ ~ i] + p;_ (2)
and the stress tensor cij includes the viscous
stresses and the double velocity correlations. The
components of cij are expressed as:
Ail =2ple[h SEX + U2k~2]= he Eli
22 2~e[h tax + Ulk21]= He e22
Ou3
=2~e0X = Pee33
C7~2 Em 1~ OXEN + h2 OX2 U2k21 Ulkl2]steely
3=~e[h Ox + Ox ]=Reel3
623=~e~ i~x3 + dX2]=~ee23 (3)
*
The value of p appearing on the right
hand side of momentum equations (1) is equal to
p+ggh, where h is the vertical distance from a
fixed level.
The effective viscosity Be in expressions (3
is calculated according to the standard ke
turbulence model (11) as follows:
2
~e= it+ ~t=~+0~09pk /£ (4)
where ,ut is the eddy viscocity, k the turbulence
kinetic energy and ~ its dissipation rate. The
values of k and ~ are determined by solving two
more differential equations, which in the
orthogonal curvilinear system under consideration
are written as:
kequation
C(k)= h h [aX <~th1 Act ) Ala th2 Aft ]
+ 00 (~c Bilk )+G pe
e equation
)= x ~ h x )+ x ~h x
~ 1 £ 1 1 2 £ 2 2 ]
t t}x (~, fix )+1.44G k1.92p (5)
where o~=1.3 and the generation term G is
expanded as:
G =2 p~ he 1~ + e222 + e333 + 2 (e212 + e223 + e223~3
The Reynolds equations (1) as well as the
turbulence model equations (5) are discretized
according to the finite volume approach using a
staggered node arrangement (10). The resulting
algebraic equations have the general form
6
Ap4>p = ~Aid~i + S `:,, (6)
1
where dip stands for the velocity components, the
turbulence kinetic energy and its discipation rate
and Hi are the values at the neighbouring
nodes of P. Central differences are used to
model Al along x1 and x2 directions while the
corresponding coefficients on upstream ala
downstream planes are calculated by the hybrid
scheme (12).
Boundary Conditions
The boundaries of the calculation domain
shown in Figs. 2a and 3 are the inlet U and
outlet D planes, the external boundary N. the
solid surface S. the free surface W and the flow
symmetry plane E. The elliptic form of equations
(6) requires specification of boundary conditions
(Dirichlet or Neummann type) on each of these
boundaries.
[. ~
Figure 3. Definition of boundaries.
795
OCR for page 789
At the inlet plane U and external boundary
N the values of the velocity components u1, u2
U3 and the pressure are calculated by the
potential flow solution under the predetermined
free surface. The latter is performed by the
classical Hess and Smith method (131. The ship
hull and the free surface are covered by
quadrilateral panels. Once the wave elevations are
a priori known, the source distribution on each
panel is calculated by satisfying the unique
boundary condition un=O, where n is the normal
to the hull or the free surface. The panel
arrangement on the free surface region which has
been used for the computations is shown in Fig.4.
The external NP and the upstream UP boundaries
of this region are located in the undisturbed free
surface part in order to avoid as far as possible
errors due to end effects (14~. It should be
noticed here that the external boundary N for the
viscous flow calculations is almost five times
closer to the ship hull than NP and, therefore, it
is expected that the calculation of the velocity
components at N will not be practically affected
by the aforementioned effects. A total of 1400
panels has been used to model the ship hull and
2000 panels to model the free surface. These
numbers refer to the one half of the whole
domain, since the flow has one symmetry plane.
At the same boundaries U and N the values of k
and ~ are assumed to be equal to zero. For
viscous flow computations the inlet plane was
placed at x = 0.2 m and the external boundary
almost 25 cm (in the mean) apart from the solid
surface.
Neglecting surface tension, the dynamic
boundary condition on the free surface can be
written as onp~c=0, where on and ~ are the
nonnal and the shear stresses respectively. It is
easy to show that these conditions, together with
the elimination of the convective terms on the
free surface (due to the kinematic condition)
result in the application of Neummann type
boundary conditions for the u2 and U3 velocity
components on the Wboundary (Fig.5~. The
same conditions are assumed to hold for k and
(i.e. ~klDn~/0n=0), while the normal to the
surface ulcomponent is set equal to zero.
L/8 1 _ ~ Unto L
~1~
U2= · U3
Figure 5. The free surface boundary.
The wall function method (10), (11) has been
employed to model the flow characteristics near
the solid boundary. The values of y+ ranged
between 30 and 180 in any case.
On the symmetry plane the following
conditions are valid:
u = 0 ~ =0 ~ =U1,U3,~ ~
Finally, at the outlet plane D the flow is
assumed to be fully developed, corresponding to
the application of Neummann conditions for each
variable. This plane was placed at x = 3.6 m.
The Solution Procedure
The solution of the transport equations (6)
together with the determination of the pressure
are made according to the partially parabolic
algorithm (151. An initial guess of the pressure
field is made, based on the calculated pressure
values at the external boundary N. Then the
solution proceeds by solving the momentum,
pressure correction and ke equations in each
transverse section successively. The pressure is
corrected according to the SIMPLE (16) algorithm
so that continuity is satisfied in each cell of the
domain. Once the free surface is known, a
Dirichlettype boundary condition for the pressure
correction cannot be applied at this boundary,
since it leads to overdetermination of the
problem and prevents the satisfaction of the
continuity equation in the adjacent to the surface
cells.
1 L/4
Figure 4. The free surface panel arrangement.
796
OCR for page 789
During the application of the partially parabolic
algorithm only twodimensional incore storage is
essentially needed for the various geometrical
and flow parameters and, therefore, fine grids
can be used. The upstream and downstream
values of different variables are constant when
calculations are performed at a certain station.
After the solution for every section of the
domain is obtained, a sweep is completed and the
calculations start again. Several sweeps are
needed until both the velocity and the pressure
field converge.
The use of an orthogonal grid in two
dimensions proved to be quite successful with
respect to convergence. This is due to the rather
simple way that velocity corrections are coupled
to changes of the corresponding pressure
gradients, the latter being of crucial importance
when the SIMPLE approach is followed.
It has been found that relatively high
underrelaxation factors can be used for
the solution of the momentum and ke equations
even if only one SIMPLE step is performed in
each station. Underrelaxation, necessary to
obtain convergent solutions, is applied for every
variable as:
~ bran + (I  redo
where r is the underrelaxation factor (constant
throughout the calculation domain), En the
solution of (6) and NO the previous value of the
variable.
A 178 x 32 x 31 grid was used for the
computations in any case, where 178 is the
number of transverse sections, 32 the nodes
girthwise and 31 along a normal. Constant
underrelaxation factors equal to 0.5 were applied
for all variables exept the pressure correction for
which the value of r = 0.3 was adopted.
Convergence was achieved in 400 singlestep
sweeps of the domain.
RESULTS AND DISCUSSION
As mentioned earlier, the predetermined
boundary used for the computer runs consisted of
the measured wave pattern plus the actual wetted
surface of the hull. In this case the model
sustained a parallel sinkage of 6 mm but no
running trim. As a first result it should be
mentioned that the numerical calculations under
the aforementioned boundary exhibited good
behaviour, i.e. they converged always. The
running time for the computations was about 24
hours on a 2.6 Plops workstation for the grid
described in the previous section. The
corresponding time
calculations was one
the same machine.
In order to gain some more insight in the
relative magnitudes of the different components
of ship resistance, it was decided to obtain similar
results for a double model of the tested hull,
even keeled but with the draught increased by
6 mm (3.35%~. This reasonable chaise turned out
to be very meaningful! from the point of view of
the wetted surface. That is whereas the static
for the potential flow
hour for 3500 elements on
equilibrium wetted surface of the model was 1.694
me, the wetted surface underway was 1.738 ma,
that is increased by 2,6%, and the wetted surface
of the double model was 1.734 ma, very close to
the actual wetted surface. In addition, the
pressure resistance of the potential flow for the
actual wetted surface and the measured wave
pattern was computed.
Because of the differences in the wetted
surface, it was decided to present all calculated
and measured resistance values in terms of force
(lip), and not in the form of nondimensional
coefficients. The results of calculations and
measurements are shown in Table 1, whereas
the nondimensional, integrated resistance force
along the length of the ship are shown in Figs 6
and 7. In both the Table and the Figures the
subscripts T. F & P correspond to the total
force, the shear stress component and the
pressure component respectively. This rule does
not apply to the RF value computed from the
experimental results, which is thought to contain
both the shear stress and the viscous pressure
component of the resistance. This value of RF
was computed on the basis of the llTC friction
line and a form factor of 1.14 determined from
the experiments.
Ache
_~_
Calculated
for actual
free surface
Calculated
for double
model
Calculated
for potential
flow & actual
free surface
Measured R T
& Calculated
Radon the basis
of form factor
method
R
(kp~
0.766
R F
(lip)
0.604
Rp
(kp)
0.162
0.694
0.572
0.122
0.682
0.596
0.128
Table 1. Comparison of resistance componets.
From the contents of Table 1, it can be
concluded that the method used overpedicts the
resistance. This is obvious from the double
model calculations which yield a total resistance
value slightly higher than the experimental one,
which contains a wavemaking resistance
component. The same trend has been noticed in
previous calculations for another ship shown in
(6~. In the present case the overprediction for
the total resistance is 12.5%, which renders the
use of the results doubtful for practical
applications.
797
OCR for page 789
There are however some useful conclusions
to be drawn from the contents of Table 1 and
Figs 6 and 7. Firstly, if the difference between
the calculated total resistance for the ship and the
double model is taken to represent the
wavemaking resistance, its value is Rw =
0.766  0.694 = 0.072 lip. This is significantly
smaller than the wavemaking (or pressure)
solution. If one then adds this value to the
experimentally determined RF of 0.596, a value
of RT = 0.668 is obtained, which is very close to
the experimental value. Secondly, the value of
RF along the length of the model is slightly
different for the actual free surface and the
double model, as the shape of the free surface
seems to increase the shear flow resistance.
And, finally, that the accuracy of the
computations at the aftermost part of the ship
for the pressure component of the resistance is
very important due to the large slope of the
curves, just before the final result is obtained.
From the point of view now of possible
improvements of the calculation procedure, there
are at least two areas which can lower the
predicted value of the resistance. The first such
area is connected with an artificial blockage
effect, which is inserted to the procedure by
imposing the potential flow velocities as boundary
conditions relatively close to the surface of the
body. Obviously, this is done to reduce the time
of computations and our experience a, shows
that if this effect is eliminated, the predicted
value of the resistance will be reduced by 23%.
The second area of possible improvements is of
a more fundamental nature as it questions the
accuracy of the wall function approach in turbulent
flow calculations. As it has been shown in (17)
and (18), if a direct solution of the Reynolds
equations is used all the way up to the solid
boundary, better values for the wall shear
stresses are predicted. Unpublished results of
NTUA indicate that in this case a reduction of the
frictional resistance by 5,5% is obtained for the
aft part of a tanker hull. Unfortunately, this
improvement is accompanied by a 25% increase in
computing time.
Nevertheless the up to now discussion of the
results should also be seen in the light of the
unavoidable shortcomings of any such
experimentalnumerical investigation. In this
respect, the authors cannot guarantee the degree
of accuracy of the free surface measurements and
subsequent interpolation. Also, no attempt was
made to achieve a grid independent solution,
although the grid is fine enough according to our
experience, but not necessarily so in the
aftermost part of the hull surface.
Finally, we recall the discussion made in the
introduction about the shortcomings of the ke
turbulence model, which need to be further
investigated.
ACKNOWLEDGMENTS
The authors would like to thank Dr. D.
Lyrides for the careful measurements of the free
surface, while he was carrying out his Diploma
Thesis. Many thanks to our colleagues from the
Leboratory of Photogrammetry for their
companion measurements of the free surface near
the model. Finally, we appreciate the help of
Dr. S. Voutsinas and Dr. Y. Glekas for the
analytical determination of the free surface.
REFERENCES
1. Patel, V.C., "Ship Stern and Wake Flows:
Status of Experiment and Theory",
Proceedings of 17th ONR Symposium on
Naval Hydrodynamics. The Hague, 1988.
2. Tzabiras, G.D. and Loukakis, T.A., "On the
Numerical Solution of the Turbulent
FlowField past Double Ship Hulls at Low
and ~ Reynolds Numbers", Proceedings of
5th International Conference on Numerical
Ship Hydrodynamics. Hiroshima, 1989, pp.
395408.
3. Hino, T., "Computation of a Free Surface
around an Advancing Ship by the
NavierStokes Equations", Proceecings of 5th
International Conference on Numerical Ship
Hydrodynamics, Hiroshima, 1989, pp. 6983.
4. Tzabiras, G.D., "On the Calculation of the
3D Reynolds Stress Tensor by two
Algorithms", Proceedings of 2nd International
Symposium on Ship Viscous Resistance.
Goteborg, 1985.
5. Tzabiras, G., Hytopoulos, F. and Nassos
G., "On the Numerical Calculation of the
Turbulent FlowField around Bodies of
Revolution at Zero Incidence", Proceedings
of Marine and Offshore Computer
Applications Conference, Southampton, 1988,
pp. 3147.
6. Tzabiras, G.D. and Loukakis, T.A.,
"Reynolds Number Effect on the Resistance
Components of 3D Bodies", Proceedings of
18th I.T.T.C., vol.2, 1987, pp 7~71.
7. Georgopoulos, A., Ioannidis, C., Potsiou, C.
and Badekas, 1., "Photogrammetric Wave
Profile Determination", Proceedings of XVI
ISPRS Congress, vol. 27, part V.II,
commision 5, Qyoto, 1988.
Lyridis, D.B., "Experimental and Numerical
Investigation of the Wave Pattern around a
Ship Moving with Cosntant Speed", Diploma
Thesis, Dept. of N.A.M.E., N.T.U.A., 1987.
Tzabiras, G., Vaftadou, M. and Nassos, G.
"A Numerical Method for the Generation of
2D Orthogonal Curvilinear Grids"
Proceedings of 1st Conference on Numerical
Grid Generation in Computational Fluid
Dynamics. 1986, pp. 183195.
10. Tzabiras, G.D., "Numerical and Experimental
Investigation of the Turbulent FlowField at
the Stern of Double Ship Hulls", Ph.D.
Thesis, N.T.U.A., 1984.
. Launder, B.E. and Spalding, D.B., "The
Numerical Computation of Turbulent Flows",
Computer Methods in Applied Mechanics and
Engineering. vol. 3(3), 1974, pp. 269289.
798
OCR for page 789
12. Spalding, D.B., "A Novel F~teDifference
Formulation for Different Expressions
Evolving both First and Second
Derivatives", Bit. J. of Numerical Methods in
Engineering. vol.4, 1972, pp. 551559.
13. Hess, J.L. ad Smith, A.M.O., "Calculation
of Potential Flow about Arbitrary Bodies",
Progress in Aeronautical Sciences. vol.8,
1966, pp. 1138.
14. Kim, K.J., "Ship Flow Calculations and
Resistance Minimization", Ph. D. Thesis,
(palmers University of Technology, 1989.
15. Pratap, V.S. and Spalding D.B., "Numerical
Computations of the Flow in Cunred Ducts",
Aeronautical Quarterly, vol. 26, 1975, pp.
219232.
0.~30 i
i /
0.0020 ~ /
o.oolo ~ / ~\
// \
/ \
0.0000 


0.0010 ~1
J
3 Distance from F.P. in meters.
~.~.0 ~ i',,,,, ~ ~' i,, , ', ~ ', ~ i,, i,,,,,, i, ',,, i,, ~ i,, ~ ~ ', r, ~ I ~ ~ I,,,, r, ~ ~ I,, rr~T1
 _
\
16. Patankar, S.V. and Spalding, D.B., "A
Calculation Procedure for Heat, Mass and
Momentum Transfer in 3D Parabolic Flows",
Int. 1. of Heat and Mass Transfer vol.15,
1972, pp. 17871806.
~~7. Chen, H.C. and Patel, V.C., "Practical
NearWall Turbulence Models for Complex
Flows Including Separation", Proceedings of
AIAA 19th Fluid Dynamics, Plasma
Dynamics and Lasers Conference Honolulu,
1987.
18. Tzabiras, G.D., "A Numerical Investigation
of the Turbulent FlowField at the Stern of
a Body of Revolution", J. of Applied
Mathematical Modelling, vol.11, 1987, pp.
4561.
a/ .,~ ~
Have / ~\ '
, \< 1
' Double model
_ _ _
_ _ _
~,
Rp
o.no 0.50 1.00 1.50 2.00 2.50 3.00 3.bc)
Figure 6. Nondimensional integrated pressure force.
0.00~ 
O.0030 
0.0020
0.0010 
Caere ~
An,
/'
/
'K' ~
a'
Double model
i ''~ ~
''' Distance from F.P. in meters.
0~3~0 ~rl,, I I r, I,,, I ~ ,,, I I ',,,,,,, I I I I 1,, I,,, I, I,,, . I I ~ I,,,, r~~tTr,, ~ rrr~r
0.00 0.50 1.00 1.50 2.00 2.50
Figure 7. Nondimensional integrated frictional force.
799
3.00 .~.50
OCR for page 789