CHALMERS TEKNISKA HOGSKOLA
CHALMERS UNIVERSITY OF TECHNOLOGY
GOTEBORG
SWEDEN
NUMERICAL CALCULATIONS OF SHIP FLOWS,
WITH SPECIAL EMPHASIS ON THE FREE
SURFACE POTENTIAL FLOW
Fei Xia
Fei Xia
Division of Marine Hydrodynamics Goteborg, Sweden
includes:
The development of a method for solving the linearized free surface potential flow problem. The method is based on the Hess-Smith-Dawson theory. Rankine sources are distributed
on the hull surface and on the local undisturbed free
sur-face. The source distributions are determined so as to satisfy the exact body boundary condition and the free surface condition linearized about the double-model solu-tion and imposed on the undisturbed free surface. Applica-tions to five different hull forms show that the method is efficient for evaluating the flow field, wave pattern and wave resistance of practical ship forms.
The development of a method capable of treating the circu-latory flow around ship configurations in the presence of a free surface. A dipole distribution is added to the source and sink distributions on each lifting section and its wake. In addition to the usual body and free surface conditions, a Kutta condition is imposed at the trailing edge of the lift-ing part. The interaction between the liftlift-ing sections and the free surface can thus be investigated. The method has been successfully applied to two sailing yachts.
The development of a method for solving the fully non-linear ship wave problem. Rankine sources are located on the body surface and on the real free surface. The exact solution is obtained through iterations. In each iteration the source distributions and the free surface location are adjusted simultaneously to satisfy the boundary conditions. Excellent results are obtained when the procedure converges. Since
* A numerical study on the flow around a high speed hull. A
special version of the linear non-lifting method applicable to hulls with a transom stern has been extended to include working propellers. This method is combined with an existing boundary layer method to investigate in detail the resistance
and flow components of a high speed hull, the Athena. Com-parisons with measurements available are satisfactory.
Keywords:
ship flow, potential flow, boundary layer, free surface, numerical calculation, panel method, Rankine source, Kutta condition, wave resistance, wake
The development of a method for solving the linearized free surface potential flow problem. The method is based on the Hess-Smith-Dawson theory. Rankine sources are distributed on the hull surface and on the local undisturbed free sur-face. The source distributions are determined so as to satisfy the exact body boundary condition and the free surface condition linearized about the double-model solu-tion and imposed on the undisturbed free surface. Applica-tions to five different hull forms show that the method is efficient for evaluating the flow field, wave pattern and wave resistance of practical ship forms.
The development of a method capable of treating the circu-latory flow around ship configurations in the presence of a free surface. A dipole distribution is added to the source and sink distributions on each lifting section and its wake. In addition to the usual body and free surface conditions, a Kutta condition is imposed at the trailing edge of the lifting part. The interaction between the
lifting sections and the free surface can thus be investi-gated. The method has been successfully applied to two sailing yachts.
The development of a method for solving the fully non-linear ship wave problem. Rankine sources are located on the body surface and on the real free surface. The exact solution is obtained through iterations. In each iteration the source distributions and the free surface location are adjusted simultaneously to satisfy the boundary conditions. Excellent results are obtained when the procedure converges. Since convergence is not always achieved, alternative solu-tions with lower accuracy have been tried. Further investi-gations are therefore necessary to improve the convergence.
A numerical study on the flow around a high speed hull. A special version of the linear non-lifting method applicable to hulls with a transom stern has been extended to include working propellers. This method is combined with an
exist-ing boundary layer method to investigate in detail the resistance and flow components of a high speed hull, the Athena. Comparisons with measurements available are
satis-factory.
Keywords
ship flow, potential flow, boundary layer, free surface, numerical calculation, panel method, Rankine source, Kutta condition, wave resistance, wake
Fei Xia
Doctoral thesis to be publicly defended on Friday, May 30th, 1986 at 09.15 at SB-salen, Horsalsvagen 1, CTH, Giiteborg for the Degree of Doctor of Philosophy.
The thesis will be defended in English
Division of Marine Hydrodynamics GOteborg, Sweden
organizations and people made it possible. I am indebted to all of them.
I would like to express my sincere thanks to my supervisor Professor L Larsson for his continual guidance and encourage-ment during the course of my studies. He has always been
there to give valuable advice and answers to my questions.
Thanks are also due to Professor G Dyne for his encouragement and for the excellent course on propulsion, and to
Professor C Johnson of the Mathematics Department for several discussions during the course of the work. I am also grateful to Dr J L Hess of the Douglas Aircraft Co, USA, for important suggestions on the second part of the work.
Further, I would like to thank my colleagues at Chalmers and SSPA who have given me kind help in many respects. Especially,
I thank Miss E Samuelsson for her programming advice.
Financial support for the work was provided by the Swedish Board for Technical Development and the Naval Material Department of the Defence Material Administration of Sweden. My living expenses in Sweden were paid by the Chinese Government during the first two years and thereafter by SSPA. All are gratefully acknowledged
here.
Finally, I would like to thank my wife and my family. They have suffered very much because of my absence from home.
For many years the only practicable way to evaluate ship
perfor-mance was through experiments. Theoretical methods were often
under the restrictions of greatly simplified models. During the recent decades, the development of the high speed computer
opened up a new field in naval architecture, computational ship hydrodynamics. Numerical predictions of ship performance became an important part of the design procedures.
The present thesis is part of the continuing study of the numerical calculation of the flow around ships being carried out at SSPA. Special attention has been paid to the potential flow with a free surface.
The free surface problem is perhaps the most classical one in ship hydrodynamics. Ever since Michell's pioneering work at the end of the 19th century many of the most brilliant scientists in the field have devoted their efforts to the development of calculation methods to be used for design purposes. However, the success of their methods was limited for many years due to the necessary linearization of the problem. Although useful information could be extracted in optimization procedures, sometimes with empirical corrections, the underlying theory must be considered too inaccurate for ships of practical dimen-sions, particularly the beam.
During the two latest decades research in this area has pro-ceeded in different directions. An excellent recent survey is
given by Yeung, [ 1 1. The state of the art can also be judged
from the two workshops, held at DTNSRDC in 1979, [ 2], and 1983,
[ 3], where a large number of methods were tested on several well-specified test cases. A number of interesting papers on free surface flows were also presented at the International
Seminar on Wave Resistance in Tokyo in 1976, [ 4], and at the
four Conferences on Numerical Ship Hydrodynamics held in
Washington in 1975, [ 51, Berkeley in 1977, [ 61, Paris in 1981,
In the present work one particular class of methods is of special
interest. The first method of this kind was presented by Gadd,
[ 9], in 1976, with later improvements in 1981, [10]. Rather than
covering the hull or centerplane with the traditional Kelvin sources, which automatically satisfy the linearized free surface boundary condition, Gadd introduced simple Rankine sources that
covered the hull and part of the undisturbed free surface. The
source strengths are determined in such a way that the free sur-face and body boundary conditions are satisfied. In 1977 Dawson, [11], refined Gadd's intuitive idea and gave an elegant formula for the free surface condition. In Dawson's approach the hull
boundary condition is satisfied exactly, while the free surface
condition is linearly satisfied in a way similar to that in the
low speed theory, suggested by Baba in 1976, [12]. The lineari-zation is thus made about the double-model solution rather than
the free stream as in earlier linearized methods.
The applications published so far have shown significant advan-tages in Dawson's method in simplicity and generality.
Improve-ments, modifications, and numerical studies on themethod have
been conducted by many researchers.
In order to reduce computer time and storage Mori et al, [13],
proposed a modified Rankine source method, where only the free
surface source is considered as unknown and determined
by the
free surface condition, while the double-model solution is used
for the hull surface source distribution. The method is based on
the assumption that the effect of the violation of the body boundary condition by the free surface sources is negligibly small on the final results of wave resistance calculations.
Ogiwara, [14], however, took Mori's type of solution as a first
approximation. The body boundary condition is correccted by
adding sources, which are determined by the normal velocity left
on the body surface from the first approximation. The second
approximation of the free surface source is then obtained from
the free surface condition, taking into account the effect of
Cheng et al,
substantial improvements to its computational techniques. A new method for satisfying the transom boundary condition resulted in a more realistic model of the physical problem.
Dawson's free surface boundary condition has also been studied by Katazawa, [16], Mori, [17], and Baba, [18].
In the present study a computer program has been developed for the linear solution of the potential flow free surface problem.
The program is based on Dawson's approach for treating the free
surface condition and the Hess-Smith method, [19], for the
solu-tion procedure. A special version of the program permits the com-putation to be made for a ship with a transom stern. Applications
and numerical parametric studies are presented in Paper A.
In the case of a lifting free surface potential flow, not only sources and sinks but also vortices or doublets are needed to generate the circulatory flow, if a singularity method is to be used in the solution. For the unbounded flow around a lifting body, the Hess method, [20], is wellknown, where in addition to the conventional body boundary condition a Kutta condition is
imposed at the edge of the lifting surface. A dipole distribution
on the lifting surface and its wake generates the required
circu-lation.
Hess's method is a good representative of existing panel methods used in the aircraft industry. It is linear in the sense that the panels are flat and covered by sources of constant density. There are, however, a number of other methods, linear and
non-linear, in aerodynamics, and a good survey in given by Hunt, [21].
In hydrodynamics slender body theories have frequently been used for calculating side forces and yawing moments, particularly in manoeuvring investigations. A low aspect ratio theory including viscous effects was recently proposed by Gadd, [22], to be used for ships at small angles of yaw.
A common feature of all methods mentioned for generating lift is that the free surface effect is not taken into account. Inspired by the recent interest in sailing hydrodynamics the aeronautical
institute in the Netherlands, NLR, in cooperation with MARIN, developed a panel method with lifting elements and a linearized
free surface boundary condition. The method is mentioned by Slooff in a recent paper, [23], but does not seem to have been presented in the open literature.
A new method has therefore been developed in the present study,
Paper B. This method is capable of treating the flow around ship
configurations with or without lifting surfaces. It combines Dawson's approach for the linear treatment of the free surface with the Hess method for the unbounded lifting flow. The inter-action between the lifting element and the free surface may now be investigated.
Several fully non-linear wave resistance methods for three-dimensional cases have been reported in the literature. Chan &
Chan, [24], developed a finite difference method for solving the
Navier-Stokes equation in the presence of a free surface. °omen's
method, [25], is based on finite elements, and in the very active
attempts to solve the bow wave breaking problem at Tokyo
Univer-sity, Miyata and his co-workers, [26], developed a modified marker and cell method.
Of more immediate interest in connection with the present work
are the methods based on surface singularity distributions. Gadd's
panel method, [ 9], has already been mentioned. Although the free
surface sources are put on the undisturbed level a non-linear, but not exact, boundary condition can be satisfied. Maruo & Ogiwara, [27], proposed a numerical approach, in which the
non-linear effect is taken into account iteratively and two relaxa-tion factors are introduced to improve the stability of the iterative procedure. An attempt has also been made by Daube et al, [28]. An iterative procedure is employed, at each step of which a linear problem is solved. A free surface condition in
condition has been derived on the exact location of the free surface.
Quite interesting is also a paper by Hess, [29], where an itera-tive approach for solving the non-linear two-dimensional free surface problem is described. Panels are distributed on the exact free surface as well as on the (subnerged) body. Several types of panel distributions and iteration techniques are tested and the
most important finding is that, in order to achieve convergence
of the iterative process, the wave height change and the new source distribution must be computed simultaneously, not sequen-tially as in other attempts.
Hess's paper inspired the work described in Paper C of this thesis. This is an attempt to solve the fully non-linear three-dimensional ship wave problem directly by the use of a purely numerical method. The mathematical problem is formulated in such a way that it is solvable by using a panel method with Rankine source distributions. The exact solution is obtained through
iterations, with small perturbation approximations being employed in each iteration. The source distribution both on the hull sur-face and the free sursur-face as well as the wave elevation are
adjusted simultaneously in iterations to satisfy the boundary conditions. The major problem of the procedure is that of con-vergence, which has to be further investigated.
The primary purpose of the research on numerical hydrodynamics at SSPA is to develop methods which can be used with sufficient accuracy for design purposes, thus offering an alternative to the traditional model testing. Several methods for the viscous flow have been developed and reported elsewhere and in this thesis
emphasis has been on inviscid free surface flows. It may thus be of interest to combine these methods to see what accuracy can be
obtained for the viscid and inviscid resistance components and flow properties around the hull.
To the author's knowledge, reports published on such calculations are very scarce. Larsson & Chang, [30], investigated the flow and
resistance components around a mathematical model. The
displace-ment thickness calculated from the boundary layer computation is
added to the hull and wake and the reduction in wave resistance as compared with that for the bare hull is obtained.
von Kerczek et al, [31], performed a computational study on the
resistance characteristics of transom stern hull forms. A slender
body linearized free surface potential flow technique is used to
calculate the wave resistance, while a three-dimensional turbulent
boundary layer momentum integral method is used for the viscous
flow calculations. In a computer system for designing SWATHs,
Salvesen et al, [32], included hydrodynamic performance predictions,
where simplified theories are used to calculate the resistance
com-ponents. A numerical method to estimate the total resistance of a
ship has also been presented by Hinatsu et al, [33]. The
viscid-inviscid interaction is included in the method.
Applying computational methods for the viscous flow, problems are
always encountered at the stern, where the boundary layer grows
thick. Until sufficiently accurate methods for this type of flow
have been developed theoretical methods will not be very useful
for wake flow predictions for ordinary ship forms. There is,
how-ever, one class of ships where the viscous stern flow problem is
completely avoided, namely the high speed transom stern ships.
Further, since the propellers of such ships usually are located
well below the hull, potential flow methods may be used for
ob-taining the wake.
In the present work, Paper D, calculations have been carried out
for the high speed hull Athena, which has been extensively
inves-tigated experimentally. Boundary layer methods, developed earlier
at SSPA, [34], [35], are combined with the potential flow method
of part A of this theses. Since the influence of the working
pro-peller is of interest, the latter method has been extended to take
this effect into account. All resistance and flow components may
2. THE MATHEMATICAL DESCRIPTION OF THE FREE SURFACE POTENTIAL FLOW PROBLEM
Using a Cartesian coordinate system oxyz, fixed in the ship, with x parallel to the centreplane and directed from the bow to the stern, z vertically upwards, and the origin at the intersection of the planes of the undisturbed free surface and the midship section, a steady potential flow past the ship can be described by a velocity potential Ox, y, z), which satisfies the Laplace equation
V2CP = 0 (1)
in the fluid. On the free surface z =
c(x, y),
where is thewave elevation, the velocity potential needs to satisfy the pressure condition
gc + 1(4)2 + (1)2 + (1)2
u2)
0X
y
zand the kinematic condition
(1)xx (1)yCy- (Pz = °
where U is the speed of the free stream.
Eliminating Eqs (2) and (3) reduce to
i[(1) (4)2 + 4)2 + 4)2) +
+ + 4)2) ] + 0
xx
yyx
yzy
The boundary condition on the wetted ship surface, y=n(x,z) is for z < c(x,y),
= (5)
where n denotes the outward normal to the hull surface. The function cb approaches the uniform stream potential at infinity. Symbolically,
=
According to Dawson, [11], the free surface boundary condition, Eq (4), can be linearized in terms of the double-model solution as follows.
The velocity potential (I) is expressed as the sum of two
poten-tial functions cl) and (1)1 as
= (I) + (1)1
(7)
where (I) is the double-model potential and
(1)1 is the perturbation
potential of the free surface wave. Inserting Eq (7) into Eq (4),
neglecting the non-linear terms of (1)1 and assuming that Eq (4)
holds on z = 0 instead of on the real free surface, then is
obtained
lEcI)x4)X +(
:Y
x(1)17;)
"
4:";y
2 (I'x"x(1)X + `13y(l)yxi)+ 21)y(cDx(!)3(' +0y(1)171)y + (I)x' (<1)x2 +(1)y2)x + (1)((l)x2 + cl)y2)y] +
Oz' =
0which can be written as
(1)1(1)11 4.
"111)1 4-
gc/pZ = °(6)
where 1 denotes differentiation along a streamline of the
double-model flow on the symmetry plane z = 0.
iUx
cos 8 + Uy sin 8
Ux cos 8 + Uy sin 8
8 is the yaw angle and vanishes in the case of a symmetric flow. It is defined as positive when the y-component of U is positive. Finally, a radiation condition should be imposed on the free surface wave, namely that there are no upstream waves.
3. THE LINEAR SOLUTION OF THE FREE SURFACE POTENTIAL FLOW PROBLEM
The method presented in Paper B is an extension of both the
Dawson method, [11], for satisfying the free surface condition and
of the Hess method, [2d, for computing unbounded circulatory flows.
Replacing q)1 by (1)
- c,
Eq (9) becomes(4)1Y1
g(Pz = 2 cl)(1)11 (10)The wave elevation is given by
1 2 2
= ( U + (I)1 - 2 cr)1cp 1)
In solving the problem a panel method is used. Both the hull sur-face and a local portion of the free sursur-face are covered with source panels. The source density is supposed to be constant on each panel and determined by solving a linear equation system
relevant to the boundary conditions, Eqs (5) and (10). Obtaining
the source distribution, the flow field is determined. The hydro-dynamic forces are calculated by integrating the pressure on the wetted hull surface. A special version of the computer program allows the calculations for ships with a transom stern to be made.
Test computations have been run for the Wigley hull, the Series 60, CB = 0.60, the high speed hull Athena, a tanker designed at HSVA and the sailing yacht Antiope, see Paper A. Results are given for the wave profile, the wave resistance coefficient and the trim and sinkage. Parametric studies were also conducted. The method has been shown to be efficient for evaluating the wave-making characteristics of practical ship forms. The calculated result depends to a certain extent on the numerical parameters, e g the discretization of the hull and the extent of the free surface. It has also been shown that the non-linear effect may be import-ant in some regions along the hull, especially when the Froude number is high.
4. THE LINEAR SOLUTION OF THE FREE SURFACE POTENTIAL FLOW PROBLEM WITH LIFT
The ship configuration considered now may have lifting surfaces. In addition to the other boundary conditions, a Kutta condition at the trailing edge of the lifting surface is imposed. The Kutta
condition, which states that the pressure on both sides of the
lifting surface adjacent to the trailing edge should be equal, is
satisfied by the determination of a dipile distribution, which
generates the required circulation, on the lifting surface and its wake.
In solving the problem by a panel method, the ship configuration
is divided into lifting and non-lifting sections. The panels on a
lifting section are specially organized into a number of lifting
strips. Each lifting strip is associated with a circulatory onset flow. The potential in question, (1), is now written in a linear
combination form
(k) (k)
(1) = B
k=1
where 0. denotes the uniform onset flow and k the circulatory flow
for each lifting strip. L is the number of lifting strips. The
(k)
combination constants B are unknown and to be determined by
applying the Kutta condition. Accordingly, Eq (10) becomes
WW-)
B(k)(I)(k)) ] + g(q)( co) +E B(k)(1)(k)) = 2 (I)2 1 1 1 1 11 k=1 k=1 It can be rewritten asB(k)["14k))1
gcl)Z1c)] = 2 1)1(1)11 k=1Eq (14) is satisfied for any B(k) if the following equation system
is satisfied
(4)2(1)(c"))
+c")
= 2 1)21)1 1 1 1 11
(k) (k)
(c1)11 )1 glpz = °
The same procedure is applied to Eq (5)
(12)
(15)
(0.)
(k)
(k)_
=+kI1B
cicinIt may be resolved as
(w) cPn = 0 0k = 1,
LThe boundary conditions can be summarized as follows
,(k)
*n
(00,k)
yhn = 0
on the hull
2 4)20
for cP('') on z
= 0
1 11
(w,k)
(4)1cp1 ) 1 + gcl)(k)
0for
cl)on z
= 0
k = 1,
L(16)
Test computations showed a significant free surface grid
depend-ence of the result. The reason was considered to be the inadequate
representation of the free surface by the streamline grid, which
is inconvenient in the bow and stern regions, especially in yawed
cases. The difficulties were circumvented by the use of a
body-fitted grid, created using a set of hyperbolas in the transverse
direction and a number of quasi-streamlines in the longitudinal
direction. The quasi-streamlines are located between the edge of
the free surface and the hull contour and change shape gradually
between these two extremes. The resolution of the free surface
may be improved very much in this way. This is, however, gained
at the expense of a more complicated numerical solution procedure.
The method makes it possible to investigate the interaction
be-tween the lifting elements and the free surface - a phenomenon
which is of much interest in sailing hydrodynamics. In Paper B
5. A NUMERICAL METHOD FOR SOLVING THE FULLY NON-LINEAR FREE SURFACE POTENTIAL FLOW PROBLEM
The fully non-linear free surface potential flow problem is
attacked iteratively. Using a grid generated on the undisturbed
free surface, z = 0, the wavy free surface is represented by
points which are located vertically above the centroid of each
element of the grid. cx and c in Eq (3) are calculated as
=a x1Cn -a
= -a
y1
Cn + act
y2where n and t denote respectively the transverse and longitudinal
directions of the flat surface grid. The coefficients axi, ax2,
ayl
and
ay2 are calculated from the grid geometry.Starting from a known solution (I) and a known free surface z
=
co(x,y)
the boundary conditions for the unknown solution can be written as follows1)n =
on the hull
axl(i)xn - a q)
co
- a (1) co + a cl)co + axlxAcn
-x2 x t y1 y n y2 y t
- ax2(I)xAct - ay1(I)yAcn
+ ay2yAct -z = 0 on
the free surfaceAc = (1, (1) (4)2 ÷ +2 2) ]
g x'x 174'17 z z x y z
on the free surface Cx
In Eqs (20) the relations are based on the small perturbation
assumption, i e the non-linear terms are small and may be dropped
in each iteration. Upon convergence, they will vanish also in the
exact expression. The wave elevation change, Ac, has been
intro-duced. It is to be determined along with the surface source
dis-tribution, both on the hull surface and on the free surface
sim-ultaneously in each iteration. In each iteration the free surface
are parallel to the undisturbed free surface and pass the points which define the known wavy surface. Each free surface panel is
determined by its projection on the undisturbed free surface which is the element of the flat grid.
The preliminary results show good promise for the method. The major problem encountered has been the problem of convergence which indicates that further investigations into the numerical
procedure are necessary. A few attempts and alternative solutions are presented in Paper C.
6. APPLICATIONS OF THE COMPUTATIONAL METHODS TO A HIGH SPEED HULL, THE ATHENA
An example of the numerical study of the flow around a ship including the viscosity is given in Paper D.
For a high speed ship with a transom stern the flow is assumed to clear the stern completely. The pressure at the edge of the tran-som is atmospheric and the velocity there can be determined by the Bernoulli equation. In the free surface potential flow calcu-lation, a boundary condition which has the same form as that any-where else on the free surface and makes use of the known velocity at the transom edge is imposed on the free surface behind the
transom stern.
The working propeller has been introduced into the flow computa-tion as a sink disk with a uniform sink distribucomputa-tion. The sink density, a, is determined by the propeller thrust coefficient, CTh' as
-1+/1+CTh
-27
To implement the flow calculation the boundary layer development is investigated by employing Larsson's method, [34], and a method applicable to the laminar and transitional flow, [35]. The free surface effect may now be taken into account in the boundary layer calculation.
Since the flow clears the transom, the boundary layer never grows thick, so the problems connected with the stern flow of a low speed ship are never encountered. Further, the propellers are positioned below the hull and essentially outside the boundary layer, so the wake can be computed using the potential flow method.
The computation provides resistance components, the nominal, and effective wake distribution, the wave profile, the sinkage and trim, boundary layer velocity profiles, and viscous and potential flow streamlines. Comparisons with the measurements confirm a satisfactory accuracy.
7. CONCLUSIONS
A numerical method based on the Hess-Smith-Dawson theory has been developed for solving the linearized problem of
ship potential flow with a free surface. The method is
efficient for evaluating the flow field, wave pattern and
wave resistance of practical ship forms.
A method has also been developed for treating the potential
flow around ship configurations with or without lifting surfaces in the presence of a free surface. The method
pro-vides a possibility of observing the interaction between
the lifting elements and the free surface.
The fully non-linear ship wave problem has been attacked
directly by the use of a purely numerical method. The method
proposed is promising, but further investigations on the
numerical procedure should be carried out.
A numerical study on the flow around a high speed hull has
been carried out using the first method above, extended to
include the transom stern case and the working propeller. This method is combined with existing boundary layer methods
and the results show reasonable correspondence with measured
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SUZUKI, A, MIYATA, H, KAJITANI, H, and KANAI, M: "Numerical Analysis of Free Surface Shock Waves around Bow by Modified MAC-Method". Journal of the Society of Naval Architects of Japan, Vol 150, 1981
OGIWARA, S, and MARUO, H: "A Numerical Method of Non-Linear Solution for Steady Waves around Ships". Journal of the Society of Naval Architects of Japan, Vol 157, 1985
DAUBE, 0, and DULIEU, A: "A Numerical Approach of the Non-linear Wave Resistance Problem". Proceedings of the Third International Conference on Numerical Ship Hydrodynamics,
1981
HESS, J L: "Progress in the Calculation of Nonlinear Free-Surface Problems by Free-Surface-Singularity Techniques".
Proceedings of the Second International Conference on
LARSSON, L, and CHANG, M S: "Numerical Viscous and Wave Resistance Calculations Including Interaction". Proceedings of the Thirteenth Symposium on Naval Hydrodynamics, 1980
VON KERCZEK, C H, STERN, F, SCRAGG, C A, and SANDBERG, W: "The Use of Theoretical and Computational Methods for
Evaluating the Resistance of Appended Destroyer Hull Forms".
SAIC Report, 1983
SALVESEN, N, VON KERCZEK, C H, SCRAGG, C A, CRESSY, C P,
and MEINHOLD, M J: "Hydro-Numeric Design of SWATH Ships". SAIC-85/1673, 1985
HINATSU, M, and TAKESHI, H: "A Calculation Method for
Resistance Prediction Including Viscid-Inviscid Interaction".
Proceedings of the Second International Symposium on Ship
Viscous Resistance, 1985
LARSSON, L: "A Calculation Method for Three-Dimensional Turbulent Boundary Layers on Ship-Like Bodies". Proceedings of the First International Conference on Numerical Ship Hydrodynamics, 1975
XIA, F, JOHANSSON, L E, and LARSSON, L: "Experimental and Theoretical Studies of Boundary Layer Transition on Ship Models". Proceedings of the Second International Symposium on Ship Viscous Resistance, 1985
and
SSPA Report No 2912-1
Calculation of Potential Flow with a Free Surface
by
Fei Xia
Goteborg 1986-04-28 (amended version)
CONTENTS Page Abstract 1 Introduction 2 Fundamental formulation 3 Numerical computation 5
Results and discussions 13
4.1 The Wigley hull 13
4.2 Series 60, CB = 0.60 17
4.3 The high speed hull Athena 18
4.4 The sailing yacht Antiope 20
4.5 The HSVA tanker 23
Conclusions 24
Acknowledgements 26
ABSTRACT
The wavy potential flow around a ship is calculated with a method developed from the Hess-Smith-Dawson
theory. Test cases have been run and reults are given for
the Wigley hull, theSeries 60, CB = 0.60, the high speed
hull Athena, a tanker designed at HSVA and the sailing yacht Antiope. The comparison between the calculation and the measurements available shows that the present method is efficient for the evaluation of practical
INTRODUCTION
Ship waves and wave resistance are inter-related
sub-jects of great importance. Scientists and engineers
have followed a variety of ways to deal with the problems
in that field, hoping that a continuous effort might
ultimately enable satisfactory predictions of wave
re-sistance and optimizations of ship wavemaking
character-istics. Although the success is so far limited, the
future is bright.
More than twenty years ago John L Hess & A M 0 Smith [1] presented their remarkable paper on the calculation of the potential flow about arbitrary three-dimensional
bodies. A Rankine source panel method was introduced.
Several years later C W Dawson [2] made a modification
of Hess & Smith's method taking the free surface effect
into account. The hull
surface and
a local portionof the undisturbed free surface are geometrically
re-presented by quadrilateral panels. The source density
is determined so as to satisfy the exact boundary
con-dition on the hull and a linearized boundary condition
on the free surface. Since then substantial improvements,
practical use.
A FORTRAN program has been developed based on
Hess-Smith-Dawson theory at SSPA. The program calculates the local flow field, wave resistance and wave pattern for ships moving at a constant speed. The hydrodynamic sinkage force and trim moment are also calculated and
can be used to reposition the ship as if it was free
to sink and trim. In that case a complete re-calculation
gives the results for the sunk and trimmed hull.
A special version of the program allows calculations for ships with a transom stern to be made. In this case the flow is assumed to clear the transom completely, so a new boundary condition is introduced around the transom edge.
2. FUNDAMENTAL FORMULATION
Using a Cartesian coordinate system oxyz,Fig 1,fixed in the ship, with x in the direction of the uniform onset
flow, z vertically upwards, and the origin at the inter-section of the planes of the undisturbed free surface and the midship section, a steady flow past the ship can be described by a velocity potential cp(x,y,z), which satisfies the Laplace equation
V2cp(x,y,z) = 0 (1)
in the fluid. On the free surface z = (x,y), where
is the wave elevation, the velocity potential needs to satisfy the pressure condition
liA
2 qb 2 4. 2u2)
0(1)xx cl)Y17 q)2 = 0
(3)
where
u
is the speed of the uniform onset flow.Eliminating C Eqs (2) and (3) reduce to
1[CI)X(4)2(1)
2 ( 2))x
(1) (CPX2 (1)y2z2)}
I-+gz = 0
(4)The boundary condition on the wetted ship surface, where z <C (x,y) is
(Pn =
(5)
where n denotes the outward normal to the hull surface.
Finally, it is also required that the function qb
ap-proaches the uniform stream potential at infinity and
that there are no waves upstream from the ship.
Symbolically
ux
+ 0(1) X < 0 0 = Ux + (-1)X >
0 (6) ro 1as r
= (X2+ y
2+ z2)
2 0Since the free surface condition given in Eq (4) is
highly non-linear, a linearization procedure is
necessary to obtain the solution if any numerical
ap-proach is applied. The total velocity potential kp is
then expressed as the sum of two potential functions
(I) and ci)' as
In the present method (I) is the double-model solution,
effect of the free surface wave. Substituting Eq (7) into Eq (4), the latter can be linearized about the
double-model solution by neglecting the non-linear
terms of (1)' and assuming that Eq (4) holds on z = 0
instead of on the free surface, that is
+ 24)y(4)x(1)xI 4.(1) 1(4, 2 y x + 4, ) 2 y x y x (..p I) 4. (1,
I(
Y Y Y X X + (1) 2) ] + acp ' = 0 Y Y + (I) 2) + 2D OD 1) + Y Y X X X y y xFor any function F it can be verified that
F +F =1F1
(9)x x Y Y
where the subscript 1 denotes differentiation along a streamline of the double-model potential on the symmetry
plane z = 0. Thus Eq (8) can be written as
'151211 ( 124)1')1
"Z' =
(10)
Replacing c15' with (1)-4), Eq (10) becomes
°121)1)1 = 2(1)1211
The corresponding wave elevation may be given by
1 2
+ CD2
-21cp1) (12)
3. NUMERICAL COMPUTATION
The total velocity potential may be generated by a
source distribution on a double-model of the ship and a local portion of the undisturbed free surface, and be represented as
s r(P'cl)
where a is the source density, r is the distance from
the integration point
q(x',y',z')
on S to the fieldpoint P(x,y,z) where the potential is being evaluated.
The integration domain S consists of the double-model surface and the undisturbed local free surface. The form
of shown in Eq (13) automatically satisfies Laplace's
equation and the infinity condition for any function o.
Thus the function c must be determined so as to satisfy the boundary conditions on the hull and on the free sur-face. It might be understood that an appropriate method
is needed to handle the wave radiation condition.
The numerical solution begins with the discretization of the integration domain S so that integration is re-placed by summation. Both the double-model surface and the undisturbed local free surface are divided into
quadrilateral panels. Each panel is characterized by its null point, normal vector and various geometry properties
such as surface area and second moment. The source density
over each panel is supposed to be constant. The velocity components at the null point of the i-th panel may be obtained from 4)xi = U + E a.X. 7)=1 r. E o.Y.. (14) j=1 3 13 4) = Z1 o.Z j=1 3 1J
where M is th number of panels, X.., Y. and Z. are the
ij ij
velocity components induced at the null point of the i-th panel by a unit source density on the j-th panel. They
(15)
the i-th panel to the centroid point of the j-th panel.
Their complete sets of values for all i and j compose
threeinducedvelocitymatrices.G.is the unknown
value of the source density on the various panels.The boundary condition will be satisfied at the null point of each panel. On the hull surface the boundary
condition Eq (5) at the i-th null point is
=
43ni +43xiNxi cPyiNyi 43ziNzi =
M = UN . + E (X..N . + Y..N . + Z. .N Jo. = 0
xi .13 xi
13 yi
13 zi 3 3=1 or (X. .N . + Y. .N +Z. .N
.)u.
= -UN
. j=.113 xi
13 yi iD zi xiwhere
(N,N .,N
.)
is the unit normal vector to thexi yi zi
i-th panel.
On the free surface the panels are constructed by using the streamlines of the double-model solution as the
transverse coordinate of a two-dimensional grid. In the longitudinal direction, a hyperbola is used.
(X
x0)2
y2 a02 (16)where xo and ao may be chosen to create a suitable grid.
For the panels on the free surface the normal direction
is replaced by the 1 direction. The i direction vector
L.
-yiL. =0
zi where = U + 7 G . X. . Xij=1
03lj
Ojsolution. At the free surface 0 . has a value
0]
of zero.
Now for each panel on the free surface the terms in
Eq (11) may be expressed as 1 2 -1i = X1
.L.+.L.=((li
xi yi yi xi2 Yi . ,1") 2 ) (19) 4)1i ==.L.
4)x
i xi 4)yi Lyi , .yi
(1,2. xi yi j=1oj
1]
and a . are obtained from the double-model
= UL . + xi (X.. L . + Y. . L .)
a.
(20) =11]
xiij
y1
3 _ kli
UL . 1). = (Xij Lxi 4-11 X1 11j=1
+ Y. . L ) 0.ijyi
(21)A four point, upstream, finite difference operator is used to obtain (1)11 and (I)
1 )1 so that 1 (17) (18)
= CA..
11-1 + CB. . + CC. 11.-2 + CD.lii
1 li-3 (22)(
'
4) ) . = CA. . UL + -(x.. L .+ Y.
. L.)(.7.]
11 11
1 Jxi
li
.=1 13xi
13yi
3 CB.li_1
+ H
(X. . L . + 1li-1
.1-13
x1-1
7=1 Y. . L .)(7.]
1-13
y1-1
CC.ULxi-2 +
_ (X. . L 1j=
11-2
j xi-2
Y. . L .)0.]
1-23
yi-2
CD11-3UL
+ IT(Xi-3j Lxi-3 +
i
-3
xi-3
li-3
j=1
Y. . L .)c.]
1-33
y1-3
= CA.1 11UL .+ CB.
xi
1ULxi-1
CC.UL .
+ CD. . UL . 1i-2
x1-2
111-3
x1-3
I [CA.1 '.t'2li
(X.L
ij
xi
+ Y.
1j
Lyi
.)
+j=1
CB.t2
-(Xij
Li
x-1
+ Y.
. L 11i-1
_l
. )1-13
y1-1
CCt2
11. (X. . L .+ Y.
. L . ) +i
-2
1-2j
x1-2
1-2j
y1-2
CD.(X.
L 1qi-3
1-3j
xi
-3
yi-3yi-3]Gj
j L
wherecA.,c13.,=,aricicty,the coefficients of the four
1 1 1 1point operator, are approximately calculated from
CD.=1-
12 (1 + 1 )(21
+ 12)/Di
1 1 CC.= -1,1
(11 + 1, + 13)
(12 + 13)
(211 + 12 + 13)/Di
CB.1= 13
(11 + 1
)-
(11 + 12 + 13)2
(211 + 2 12 + 13 )/D.
1 CA.= -(CB. + CC.
+ CD.)
(24)
1 1 1 1(23)
(25) Di = 11 12 13 (11 + 12) (1 + 13) (11 + 12 + 13) (3 1 + 2 12 + 13) 1 1
1= = - [(xni - xni-1)2 + (yni - yni-1)]
12 = - [(xni-1
2 112
xni-2' 'Yni-1 Yni-2
13 = - [(xni-2 - x 2
n. 3) (Yni-2 - Yni-3) 12
where (xni, y)
ni,
(xni_., v-ni
),(x.
(xni-2' ,y. ),(x (xni-3'. ) are the null point coordinates of the i-th,
Yni-3
(i-1)-th, (i-2)-th and (i-3)-th panels in a same
longi-tudinal set, and i increases in the downstream direction.
Near the upstream end of each set, a smaller number of
points is used in the finite difference operators. For
the first panel a is set equal to zero.
It is known that
,4) . =
-27
a.(26)
zi 1
Thus the boundary condition, Eq (11), at the i-th null
normalized as M1
C.
. AS1-1
pi xi
i
2 A S. 1 i=1 (27) (29) SSPA & CTH 11 [CA. (X. .L . 'Y..L ) + CB
1 lj X1 13 yi ij=1
-1i-1(Xi_li Lxi-1 + Yi_ii Lyi-1)
4- CC .';'2 CD.
;2.
i
li-2
(xi-2j Lxi-2 Y.Lyi-2)
1 11-3(X.
.L
. + Y. . L . )]Jj-2-IT G.1-3j x1-3 1-3] y1-3 1
(CA. CB. - + CC :it . + CD.
1
li
1 11-1 i1 11-2 -3)- (CA. UL xi + CE. ULxi-1 + CCi ULxi-2
+ CD. 2 UL )
1 11-3 xi-3
The complete sets of Eqs (15) and (27) compose a system of M equations in M unknown values of c that is solved
by Gaussian Elimination. Iterative procedures may not be used, since the matrix of coefficients is not diagonally dominant.
Once the values of the source density on all panels have been determined, the full flow velocities at each hull point are calculated by inserting the oj values into Eq(14)The pressure coefficient is, however, calculated
using the linearized formula,ie
12
C = 1 - D2 [(Dx2+ 2 + z 2(4, (C
i
$1'Z
$1z)] P X x Y Y (28)The wave resistance is calculated by summing the x-compo-nents of the pressure forces acting on the hull panels. The wave resistance coefficient is thus calculated and
where
M1
is the number of panels on the hull and A Si
is the area of each panel. Ideally this coefficient should be zero for the double model case, but due to the discre-tization error, a prior - zero value is frequently obtained. This value is subtracted from the wave resistance.
The hydrodynamic sinking force and trim moment are calculated
in the similar way as that adopt in the wave resistance
calculation. The coefficients are obtained as follows
. C N . AS . 1=1
p1 Z11
Cz -1 ASii=1
M ,-1i=1
Cpi (NI' . :z . - 1\1 . . xrii) AS.xi ni xi i Cm = M 1 AV. i=1 ( 31)
where Y:
.and z
. are the coordinates of each null pointni ni and AV. = AS. . . z . N . 1 ni zi
It might be noticed that it is not always necessary to
represent the whole double-model surface and the free surface portion directly. The double-model possesses at
least one symmetry plan, namely the undisturbed free surface.
Usually it possesses two symmetry plans, the other one being
the vertical ship centerplane.On the other hand, the free
surface portion always possesses one symmetry plan less than its correlating double-model. These facts may be noted
The first test case was a mathematical hull with parabolic waterlines and sections, usually referred to as the Wigley
hull. Employing the coordinate system defined in Fig 1, the
equation for the hull surface reads
1
TO- '
I-
(12
-
-6-4- )A body plan is shown in Fig 2.
Since one of the most important questions in connection with
the present method is the dependence of the solution on the
panel distribution, particularly on the free surface, a
systematic variation was carried out. In table 1 the number
of panels on the hull and on the surface, the extent of the
(32)
in the input to the computer program and only the non-redundant portion needstobe specified by input points. The other portions are automatically taken into account.
In the case of a ship with a transom stern the free surface behind the stern is treated as a special section. It is known that a depression in the free surface occurs behind a transom stern hull and the flow will separate cleanly
from the stern if the operating speed is sufficiently high. Accordingly the pressure at the edge of the transom has to be atmospheric and the velocity there is completely deter-mined by the Froude Number and the stern geometry. The boundary condition applied in that section has the same form as that anywhere else on the free surface, but for
the panels next to the transom a two point finite difference operator is used to obtain
-11' assuming that the known
velocity at the transom may be put equal to
4.
RESULTS AND DTscusTnNS
panelled part of the free surface in the x- and v-directions,
the Froude number and the wave resistance coefficient are
given. The latter is also plotted in Fig 3 and compared
with a number of measurements, reported by the Resistance
Committe of the 17th ITTC [6]. (In the following,
compari-sons will frequently be made with data from this report,
based on a comprehensive cooperative research project, where a large number of different organization
partici-pated).
In run No 1, only approximately 1/4 of the half ship length,
was used for the free surface extent in the y-direction.
The culculation gave an unreasonably high wave resistance
at Fn = 0.266 and broke down at Fn = 0.45. In runs of No
3 and 4, the free surface region was enlarged and divided
into 5, 10, and 15 strips respectively. At Fn = 0.266, the
resistance changed significantly from run
to run, but at
Fn = 0.45 there was no change between the last two runs.
The free surface extent in the x-direction was tested in runs
No 3, 5, 6, 7 and 8. From No 3 to No 5 the free surface
region was extended by 1/4 1. The wave resistance did not
change very much, but as the free surface region was
enlarged by 1/2 1 from run No 6 to No 7, the wave resis-tance changed, particularly for the highest Froude numbers.
From run No 7 to run No 8, a part of the free surface
region was moved from the bow to the stern. The wave
re-sistance did not change at Fn = 0.266, whereas it was
increased by 10 % at Fn = 0.45.
There were no systematic changes in the panel distribution
on the hull, but two slightly different hull panel
distri-butions were used in the successive tests. This should not
cause large changes in the final results. The considerable
difference in the wave resistance between runs No 3 and No 6
The calculated values of trim and sinkage are given in on the free surface. More even panels were placed on the
free surface for run No 6 than those for case No 3.
The results obtained imply that the choice of the panels depends on the Froude number to be evaluated. Smaller panels should be used for low Froude numbers, while a larger free surface portion is required for high Froude numbers. It is wise to vary the panel arrangement for different Froude numbers.
For the further tests a panel distribution was chosen,
which is shown in Fig 4 and presented as run No 9 in Table 1. Considering the fact that test runs would be carried out
in a Froude number range from 0.20 to 0.50, a fairly large
free surface portion and relatively small panels were used.
The detailed results of this test are given in Table 2
and in Figs 5-10.
The calculated wave profiles along the hull for different Froude numbers are shown in Fig 5.
As in all the other cases presented below, the wave
pro-files were taken from the free surface elevation at the
panels next to the hull, not on the actual ship surface, which resulted in some error especially near the bow. The comparisons between the calculated wave profiles and measurements were made for Fn = 0.22 and 0.31, which are shown in Figs 6 and 7. Thr, agreement is fairly good. In
the case of Fn = 0.31, the phases are almost the same
but the magnitude of the first crest an6 though has been
under-predicted. In the case of Fn = 0.22, there is no large
difference in the magnitude but a phase shift seems to exist. There is however a fairly large discrepancy between the two sets of measurement.
Table 2. The definitions adopted in this report are as follows
Trim (T)
= "bow - AZstern
)Sinkage (h) =
1
(AZ b
ow AZstern)
(34)
where AZbow and AZstern are the changes in water level
measured at the bow and the stern respectively, and L is the ship length.
The calculated trim has been plotted in Fig 8 and it is seen that the plots follow the measured curves quite well. Good results are obtained also for the calculated sinkage at the bow in Fig 9.
The calculated wave resistance coefficients are also given
in Table 3. A comparison with the measurements is displayed in Fig 10. The numerical prediction follows the experimental curve very well but the predicted magnitudes are generally somewhat larger than the measured ones.
It should be mentioned that the wave resistance coefficients
presented in Table 1 were calculated using the full velocity.
In the later calculations the linearized formula was
used, according to Eq (28). To show the difference, the
wave resistance coefficients calculated using the linearized
formula for run No 9 are also presented in Table 1 within
brackets.
/L
The second test case was the series 60 hull with CB = 0.60. A body plan is shown in Fig 11. 21 x 11 panels were used
for representing the main body and 1 x 6 panels were used
for the stern. 41 x 10 panels were placed on a free surface portion with -1.50< x_ 2.00, -0.775 < y< 0. The
_
panel arrangement on the free surface is shown in Fig 12.
The test runs were mainly carried out for the model fixed condition, but for comparison runs were also carried out
for the model free condition at Fn = 0.25 and 0.30. Fig 13
presents the calculated wave profiles as wellas the measured ones [7], They are in good agreement. The agreement on
the middle part of the hull is better than that at the two ends. The prediction always underestimates the magnitude
of the first crest and overestimates that of the last
crest. As already ponted out, a possible reason for the first effect is the fact that the wave height is computed slightly outside the hull. To investigate this, the wave height was extrapolated linearly to the hull, using the values at the two closest points. The result for Fn = 0.30
is presented in Fig 14. The agreement near the bow is
improved but there is still a discrepancy at the extreme
end of the bow. The calculated wave profile did not change
very much for the most part of the hull. The fact that
the
predicted stern wave is too large is most probably due to
the neglect of viscoisty, i e the boundary layer
displace-ment effect.
In Fig 15 the calculated wave profiles for both the model fixed condition and the model free condition are plotted together. It can be seen that the difference between these two cases is very small. The phases are the same but the
amplitude for the free model case is always slightly bigger
than that for the fixed model case.
Table 3 gives the calculated trim and sinkage and Fig 16
shows the comparison of the calculated and the measured
[6] trim. It displays a discrepancy since the calculated
values are smaller than the measured ones. Comparison
of the calculated and the measured [6] bow sinkage is shown
in Fig 17. The agreement seems fairly good.
In Fig 18 the calculated wave resistance coefficients are compared with the measured values presented by DTNSRDC,
[7]. The prediction follows the experimental curve very
well, but the predicted magnitudes around Fn = 0.30 are
considerably larger than the measured ones. Calculations
for the model free condition carried out at two Froude
numbers, 0.25 and 0.30, are also plotted in the figure.
As can be seen the magnitude of the change seems to be
about right.
In Fig 19 the calculated results are also compared with those
obtained in the ITTC experiments [6] . The calculated values
are presented in Table 3.
4.3 The high speed hull Athena
The third test case was the high speed hull. Athena. A body
plan is shown in Fig 20. 25 x 9 panels were used for repre-senting the hull. As mentioned earlier, the Athena
calcula-tion requires a special section of free surface behind
the transom. 10 x 6 panels were placed on that section and
40 x 8 panels were used for the main free surface portion,
Fig 21 shows the panel distribution on the free surface.
The calculated wave profiles along the hull are presented in
Fig 22 and the corresponding stern wave profiles, taken
same figure. In Fig 23 the stern wave profiles are reproduced in transverse planesfor Fn = 0.36 and 0.48.
A comparison in hull wave profile between calculations and measurements is shown in Figs 24 and 25, where a discrepancy
is noticible especially for the lower Froude number. The predicted wave profiles are lower than the measured ones. The present results are in good agreement with Dawson's except on the bow quarter where Dawson-s results show a dip which was not observed either in present calculation or in the experiment.
The calculated trim and sinkage are given in Table 4 and
Fig 26 shows the sinkage at the bow and the stern being compared with Dawson's results and some measurements. The present results agree quite well with the measured values at the stern and at the bow when Fn < 0.40. When Fn > 0.40 the calculated bow sinkage is too small.
Table 4 also gives the wave resistance coefficient Cwd,
calculated from the dynamic pressure and plotted in Fig 27.
A ship hull like Athena has its special resistance charac-teristics, essentially different from those of regular ship hulls. Because of the flow detachment the pressure on the
stern becomes equal to zero (atmospheric). A resistance
from the hydrostatic pressure is then added. This can be
calculated as R = if pgz ds
B/2 z(y)
2J f Pcz(y) dzdy o o ,13/2 2 = pg J z (y) dv (35)where S is the transom stern surface below the design water-line. A calculated curve of the hydrostatic resistance
coef-ficient denoted as C is presented in Fig 27. The sum of the
dynamic and the hydrostatic resistances should generally
be compared with the measured wave pattern resistance. Fig 27 shows the comparison between these two. A great discrepancy at the low Froude number end is expected since the flow will not be able to clear the stern completely when the Froude number is lower than 0.30. [8].
Even if the flow is not detached, there will be a flow
sepa-ration at the transom edge giving rise to a large resistance component which is found in the residuary resistance, Cr,
so also plotted in Fig 27. In fact it might be more
approp-riate to compare the calculated resistance with Cr at these Froude numbers since the pressure in the separated area might not be very different from the atmospheric pressure.
In the intermediate Froude number range (0.4-0.65) there is
a relatively good correlaticnbetween the wave pattern resis-tance and the calculated one. An attempt was also made to make a computation at Fn <0.8, but the results were quite bad. A possible reason for this is the very large change in the under-water part of the hull due to the waves, which makes the fixed condition a very unrealistic one. Other possible explanation might be the fact that the free surface panels
were not extended downstream as the wave length increased,
or simply the fact that the theory cannot handle much large disturbances.
4.4 The sailing yacht Antiope
A sailing yacht, Antiope, was also tested. Its body plan is
shown in Fig 28. The hull has been divided into six parts to get a perfect representation. Altogether 186 panels were used. The panel distribution on the free surface is shown in
Fig
29. 33
x 8 panels were used in the region - 1.5 --<x1.5,- 0.775 y<0.
The geometry of a sailing yacht is widely different from that of ordinary ships. Apart from the fact that the keel plays an important part in the flow behavior, the most im-portant feature is that Antiope has very inclined framelines all along the waterline. According to Baba [9], the term
gC I should be included in the free surface condition for
o zz
such hull forms. The calculations were thereforecarried out with and without the term gC
6-zz'
This term appears in the equations as follows:
Substituting Eq(7) into Eq (4) , neglecting the non-linear
terms of (1)1 and assuming that (Dz is one order of magnitude
2
smaller than 1) , , so that D can also be omitted,
x z) z z Eq (4) becomes 1 -2 (cD[c()x21-$ y2+2 ('D 'Di +4) cf` )] + y[(Dx24- y2+2(cD cf +4) )] x X' X V y X
xxyyy
±t'x(x2+$17-)x+1 y(x2+1' 2y
+geDz+q) z) = 0(36)
Eq (36) is supposed to apply on the wavy free surface z
=
Introducing the expression
!
= CO
z'z=C zz Z=0
01)24,1 bis2A
14'11 ''14
Replacing (1) by d?'- (I), yields
(4)2cP ) +
go
aq) = 24,24)
1 1 1 zz z
111
were the condition ,1) = 0 is used. (,0, can now be
Z Z=0
calculated as
12
r=0 =
(u2)
In the present method it turns out to be simple to compute
zlz=co
than
(I)zzlz=0
so the former is used instead of
ozzz=0
. The calculated wave profiles areplotted in Fig 30. Tables 5 and 6 give
the calculated wave resistance, trim and sinkage for the two
cases. The sinkage at the bow and the stern is plotted in
Fig 31 and the wave resistance is displayed in Fig 32. The
measured curves [10] presented in Fig 32 are for the model
free case, but the calculations were carried out for the
model fixed case only. Fetter agreement can be expected if
the calulations are carried out for the model free case.
g°1)zz + gcp'z = 0
where s]o is evaluated on the plan z=0 and
1 = , ()x2 4' (D
2)]
2 g (38)Eq (36) is approximately satisfied on the plane z=0 instead of on z-=.c. As explained previously, the following equation can be obtained
The term a.:o zz did not change the results very much. Its effect varies with the Froude number.
To investigate the effect of the panel distribution calcu-lations were carried out using 406 panels on hull and 333 panels on the free surface. The results are presented in
Table 7 and the calculated wave resistance is plotted in
Fig 32. No large change can be observed.
4.5 The HSVA Tanker
The last test case was a tanker hull designed at HSVA. 347
panels were used in the hull representation. A body plan
is shown in Fig 33. A free surface portion (-1.5 < x< 1.50 and -0.775 y< 0)
_
_
was represented by 41 x 8 panels as shown in Fig 34. The
calculated results are uiven in Table 8 and plotted
in Figs 35 and 36. Fiu 36 reveals that the waves were not
resolved perfectly. A finer arid with 55 x 8 panels was therefore used on the free surface portion and a second
test run was carried out. The results are given in Table
8 and are also plotted in Figs 35 and 36. It appears
that wave profiles were considerably improved but calculated
wave resistance was still significently larger than Dawson-s
[3]. In Dawson-scalculation the hull was treated as two semi-infinite bodies. Separate calculations were made for the bow and stern ends and the results were added. A good agreement was obtained with the experiment Since the
inter-action between the two ends is lost in this procedure, it
was not tried in the present work.
Another attempt was made using panels on the free surface,
at richt angles to the centerline, as shown in Fig 37. In this way a better resolution
couldbe obtained at the bow, but no improvements were observed in the results.
The problems encountered in the HSVA tanker calculation are most probably due to the fact that low Froude numbers require
better resolution of the hull and the free surface. As
re-commended by Dawson [2] , at least 8 to 14 panels should
be used per wave length.
In the present calculations 8 panels Per wave length were
achieved only for the highest Froude numbers. In order to
improve this situation without increasing the number of
panels too much a new scheme is being considered, where the
panels are concentrated near the free surface. The number of stations should be larger in this region than further down on the hull.
5. CONCLUSIONS
The present method has successfully predicted the flow field,
the wave pattern and the wave resistance for five different hulls which represent a wide variety of ship forms. The results agree with the measurements fairly well, especially
in the moderate Froude number range. Problems usually occurred when the Froude number was either too high or too low. The
calculations broke down in certain cases. It also turned out
that good results considerably depend on the representation
of the hull and the free surface.
The failure of the calculation at high Froude numbers is not
surprising, since the underlying assumption of the present
method is a small perturbation theory. When the Froude num-ber is fairly high, the derivatives of ('p' are no longer of
second order. In the low Froude number range the calcula-tion may be improved by reducing the size of panels to
