Submitted to the School of Mechanical Engineering, Chalmers University of Technology in partial fulfillment of the requirements for the degree
of Doctor of Philosophy.
DEPARTMENT OF NAVAL ARCHITECTURE AND OCEAN ENGINEERING DIVISION OF HYDROMECHANICS
CHALMERS UNIVERSITY OF TECHNOLOGY S-412 96 GOTEBORG, SWEDEN
Goteborg 1990
CHALMERS UNIVERSITY OF TECHNOLOGY
NUMERICAL COMPUTATION OF
SHIP STERN/PROPELLER FLOW
by
Chalmers University of Technology Goteborg, Sweden, 1990
DISSERTATION
This dissertation is based on the following four appended papers:
Paper A: Zhang, D-H, 1988, "Body Fitted Coordinate Systems for Ship Stern Flow Calculations". Report No 73, Department of Marine Hydrodynamics, Chalmers University of Technology, Gothenburg, Sweden
Paper B: Broberg, L. and Zhang, D-H, 1988, "Numerical Solution of the Reynolds-Averaged Navier-Stokes Equations for Ship Stern Flow", SSPA Report 2803-1, SSPA Maritime Consulting AB, Gothenburg, Sweden.
Paper C: Zhang, D-H, 1990, "Numerical Computation of Propeller/Hull Interaction in Viscous Flow, Part I: Rotationally Symmetric Flow", Report No.90:8, Department of Naval Architecture and Ocean Engineering, Chalmers
University of Technology, Gothenburg, Sweden.
Paper D: Zhang, D-H, 1990, "Numerical Computation of Propeller/Hull Interaction in
Viscous Flow, Part II: Three-dimensional Flow", Report No.90:9,
Department of Naval Architecture and Ocean Engineering, Chalmers
ABSTRACT
A numerical method for calculating ship stern/propeller flow is presented. The method is based on the Reynolds-Averaged Navier-Stokes equations, which are completely transformed to a body-fitted coordinate system. Physical velocity components are used. The coordinate system is generated by numerically solving a set of Poisson equations. In this way, the body may be accurately represented also in regions of rapidly changing geometry. The flow equations, with a k-E model for the turbulence, are discretized using a finite-analytic scheme, and the velocity-pressure coupling is treated by the SIMPLER algorithm.
The effect of the propeller is simulated by an actuator disk with distributed body forces. When passing the propeller disk the flow is accelerated in both the axial and tangential directions in a similar way as for a propeller with an infinite number of blades. With
given geometry a lifting-line approach based upon the Kappa method is employed to determine the circulation distribution of the propeller.
To test the grid generator, it was applied to several different bodies including four typical ship hull forms. It turned out that the generator is flexible and capable of generating body-fitted coordinate systems for ship sterns of different complexity.
The Navier-Stokes method was first applied to a fine-formed cargo ship, the S SPA 720 model, for the case without propeller. Both the complete Reynolds-Averaged Navier-Stokes equations and the partially parabolized ones were solved numerically. Good results were obtained comparing with the measured data. The grid dependence and the dependence of the location of the outer edge boundary were also studied.
Thereafter, the effect of the propeller was introduced, and the interactive flow between the stern and the propeller could be computed. An open water propeller was chosen as a first test case. From the calculation it was found that the turbulence generated inthe boundary layer on the propeller blades is important for creating the necessary mixing and diffusion of the shear layer at the edge of the propeller slipstream.The second case was an axisymmetric body with or without an operating propeller. The results indicated that better resolution in the near-wall region is obtained by a two-layer model than by a two-point wall-function approach. Good agreement with the measured data was found. Finally, a practical ship hull form with or without a propeller was computed. The method predicted the characteristics of the interactive flow between the ship stern and the propeller. The computed results were in reasonable agreement with the experimental data.
Keywords
body-fitted coordinate system, body-force, grid generation, lifting-line theory,
numerical calculation, propeller, propeller-hull interaction, Reynolds-Averaged
ACKNOWLEDGEMENTS
I wish to express my sincere gratitude to professor L. Larsson, my thesis advisor, for
his inspiring guidance and constant encouragement during the course of my studies. I am also grateful to professor G. Dyne for his enthusiastic guidance and support.
Grateful thanks are also given to professor V.C. Patel and professor J. Lunde for their excellent courses and helpful discussions, and to professor Sv.Aa. Harvald for his kind care during my stay at DHT, Denmark.
It was a great pleasure to work with Dr. L. Broberg on Paper B. I am grateful to him for sharing his extensive knowledge of mathematics and fluid mechanics.
I am greatly indebted to Wuhan University of Water Transportation Engineering, especially to professor Xiu Heng Wu and Shi Mu Li for their support, encouragement
and deep understanding in every aspect over these past six years.
Further I wish to express my thanks to all my colleagues at Chalmers and SSPA,
particularly to professor C-0 Larsson, Mr. C-A Johnsson, Dr. N. Non-bin, Mr. J. Olofsson, Miss E. Samuelsson, Mr. T. Thorstensson, Mr.J.Schoon, Mr. C-E Janson
and Dr. K-J Kim for all their help in many aspects.
Mrs S. Bemander, Miss B. Engrell and Mr. P. Lindell have assisted me in typewriting my thesis. I am sincerely grateful to them all.
The work was funded by the Swedish Board for Technical Development, financial support for living expenses being provided by P R China for the first two years and thereafter by SSPA.
CONTENTS Page DISSERTATION 1 ABSTRACT 2 ACKNOWLEDGEMENTS 4 CONTENTS 5 LIST OF SYMBOLS 6 I. INTRODUCTION 8
II. THEORETICAL BASIS OF THE CALCULATION METHOD 11
III. NUMERICAL TECHNIQUES 17
IV. BOUNDARY CONDITIONS 19
V. VALIDATION 21
V.1 The Grid Generator 21
V.2 Stern Flow without Propeller 21
V.3 Interactive Flow between a Stem and an Operating Propeller 21
V.3.1 Propeller in Open Water 22
V.3.2 Axisymmetric Body with or without Propeller 22 V.3.3 Series 60 CB = 0.6 Hull with or without Propeller 22
VI. CONCLUSIONS 23
VII. FUTURE APPLICATIONS AND DEVELOPMENTS 25 VIII. SUMMARY OF THE APPENDED PAPERS 26
Paper A: Body Fitted Coordinate Systems for Ship Stem Flow
Calculations 26
Paper B: Numerical Solution of the Reynolds-Averaged Navier-Stokes
Equations for Ship Stern Flow 27 Paper C: Numerical Computation of Propeller/Hull Interaction in
Viscous Flow. Part I: Rotationally Symmetric Flow 28 Paper D: Numerical Computation of Propeller/Hull Interaction in
Viscous Flow. Part II: Three-dimensional Flow 30
REFERENCES 31
APPENDED PAPERS (in full edition of thesis) PAPER A
PAPER B PAPER C PAPER D
LIST OF SYMBOLS
Alphabetical Symbols
AO area of the propeller disk
Ao, Bo, Do, Eo convection coefficients in the transport equation for k or E constant in wall function equation (9)
length of propeller blade section
CL lift coefficient
CLO lift coefficient from two-dimensional profile theory
CT thrust loading coefficient (=2T/pUO2nRp2) ci constant in equation (15)
cp., ed., ce2 turbulence model coefficients propeller diameter
Fi body force term in Reynolds equations (2) empirical function in equation (30) fb , fb
x 0 body force per unit volume
ft
control functions in Poisson equations (1) fm camber of propeller blade sectionturbulence generation term
GL dimensionless circulation (=1-7/cU0pp)
covariant and contravariant general metric tensors for coordinates
hw
length scales ((gii)1/2)(1) Jacobian; (2) advance ratio (=Uf/nDp)
ICQ torque coefficient (=Q/pn2DpS)
turbulent kinetic energy
kf camber correction in equation (29) turbulent kinetic energy of propeller 14, lc length scales defined in equation (14)
revolutions per second of propeller propeller pitch
pressure
Qijk geometrical quantity in equation (2) magnitude of velocity
radius
Re Reynolds number propeller radius
turbulence Reynolds number defined in equation (16) dimensionless radius (=R/Rp)
So source term in transport equations thickness of blade section
Up free-stream velocity
ui physical velocity components
ILE friction velocity (...(twipu02)1/2)
up, UT axial and tangential propeller induced velocity at propeller plane uTh tangential velocity induced by the hull
upm, uTm the mean value of the axial and tangential induced velocity of a propeller with infinite number of blades
V resulting velocity relative to a blade section VA propeller advance velocity
xi orthogonal reference coordinate system normal distance to hull surface
dimensionless distance normal to the wall (=yuilv) number of propeller blades
Greek Symbols
a
the angle of attack13i the hydrodynamic pitch angle
rijk- Christoffel symbol of the second kind circulation
A pressure gradient AT stress gradient
AxP non-dimensional propeller disk thickness 8 boundary layer thickness of the propeller blade
rate of turbulent energy dissipation the Kappa value
Von Karman's constant vE effective viscosity vt turbulent eddy-viscosity
general curvilinear coordinates
Gic, GE constant in turbulence model
tw wall shear stress
Other Symbols
ai partial derivative with respect to
I. INTRODUCTION
In ship hydrodynamics, resistance and propulsion is a very old and important subject. It is always the desire of a ship designer to design hull forms having low resistance and high propulsion efficiency , and to design propellers which would at certain speed and a certain number of revolutions absorb a minimum amount of power. Further, it is requisite that the propeller is free from harmful cavitation, and that the pressure fluctuations on the ship hull are reasonable thus avoiding vibration. In order to fulfill these requirements he must be able to predict the resistance and propulsion performance of the designed ship, and he must know the velocities and pressurevariations around the ship stern and propeller, especially the effective inflow to the propeller disk.
Traditionally towing tanks have been an important tool for ship design. In order to get the whole picture of the ship stem/propeller flow a series of experiments has to be carried out, including bare hull resistance tests, open water propeller tests, self-propulsion tests, stream line tests, pressure measurements on the hull surface, wake
measurements and measurements in waves. A comprehensive analysis is then needed from all these results to get the useful information for the ship designer, such as the resistance, propulsion efficiency, thrust deduction, effective wake, relative rotative efficiency, pressure distribution and so on. Since the first towing tank was built in the last century many tanks have been built in the world and a tremendous number of experiments have been carried out for ship design and for series research in ship hydrodynamics. The experiments are not only time consuming and expensive but also subject to scale effects. It is impossible to fulfill the complete similarity between the model and ship flows, giving rise to uncertainties in the interpretation of the test data.
Theoretical studies on the resistance and propulsion characteristics of ships in a viscous flow have progressed relatively slowly because of the difficulties in the theory. It is natural that first the boundary-layer calculation method, which has been quite
successful and widely used in aerodynamics, was applied in hydrodynamics to predict the ship stern flow. Several ship boundary layer calculation methods were presented at the SSPA-TITC Workshop on Ship Boundary Layers held in Gothenburg 1980,
Larsson [1]. It turned out that almost all the methods failed in the stern and wake region due to a variety of factors related to the simplification of the governing equations,
deficiencies of the physical models and the numerical solution techniques. None of the methods could accurately predict the near wake and viscous/inviscid interaction. The general features of ship stern and wake flow was reviewed by Patel [2]. By examining the available experimental information on thick three dimensional boundary layers on bodies of revolution and ship forms, he conjectured that the thin boundary layer theory could not be properly used to describe the ship stern flow, and simple extensions of boundary layer type methods were unlikely to succeed in capturing the essential features of stern and wake flows. It is clear that more general equations of motion are required for describing such flow.
Several approaches for calculating ship stern and wake flow using the complete Reynolds-Averaged Navier-Stokes equations or the somewhat less general partially-parabolic equations were developed during 1980's, as is evidenced by the many
publications, for example, Broberg [3], Chen and Patel [4], [5], [6], Hoekstra and Raven [7], [8], Huang and Zhou [9], Janson and Larsson [10], Kodama [11],[12],
Raven and Hoekstra [13], Tzabiras [14], and Piquet and Visonneau [15]. The methods of several investigators differ in several respects. The principal differences are reviewed by Patel [16].
Computational Fluid Dynamics (CFD) has been a major research area at Chalmers University of Technology and SSPA. The development of a method for predicting ship
stern flow was started in 1980. First, a boundary-layer method was developed by Broberg and Larsson [17]. The method is based on the thin boundary layer equations cast in an analytic body-fitted coordinate system. Later on Broberg [18] developed a method based on the partially-parabolic Reynolds-Averaged Navier-Stokes equations. The governing equations are transformed to a body-fitted coordinate system using physical velocity components. The body-fitted coordinate system is generated as a numerical solution of a set of Poisson equations. This is the work described in Paper A. Furthermore the method was extended to fully elliptic by Broberg and the author, see Paper B. A systematical study on the grid independence and the location of the outer edge boundary was carried out also in this work.
All the methods mentioned above are restricted to the so called double-model in which the free surface is assumed to be flat and treated as a symmetry plane. Also the effect of the propeller is excluded. As mentioned at the beginning, the ship hull and propeller as a whole dynamic system can not be separated. It is very important for a ship designer to be able to predict the interactive viscous flow between the ship stern and the propeller. The knowledge of this interactive flow is quite useful and necessary as guidance to the designer to design a more efficient ship-propeller dynamic system, i.e. lower resistance of the hull form and higher propulsion efficiency.
Theoretical methods for predicting the effective inflow to a propeller were developed at the beginning for simple geometry bodies such as axisymmetrical bodies. Huang et al [19] and Dyne [20] have proposed methods to estimate the effective inflow by
correlating with measured wake distributions in the presence of a propeller for axisymmetric bodies. Zhou and Yuan [21] developed a streamline curvature method based on the basic equation of turbulent flow. With the development of viscous flow theory Stern et al [22], [23] calculated the interactive flow between axisymmetric bodies and propellers by combining the stern flow calculation method of Chen and Patel [5] and Kerwin's lifting-surface method for calculating propeller performance [24].
In Paper C a method for calculating the interactive viscous flow around an
axisymmetric body in the presence of an operating propeller is presented. Broberg 's stem flow calculation method is combined with a simple lifting-line propeller analysis method based on the Kappa theory. In this method the turbulence production at the propeller blades is taken into account in the calculations. The prediction of the propeller slipstream is improved by introducing the turbulent kinetic energy generated by the propeller blades.
The final object of the thesis is to develop a method for calculating the interactive viscous flow around practical ship hulls and propellers. Due to the difficulties in many
aspects, especially in generating a grid which is suitable for both ship stem and propeller calculations, there seems to be no successful procedure which can predict such complex flow in the open literature. Some investigators, for instance Masuko and Ogiwara [25] have made calculations considering the effect of a propeller on the ship stem flow by using simple pressure jump propeller model. The pressure jump is assumed uniform in the propeller disk and its value is derived from the measured thrust of the self-propulsion test. There is no interaction between the propeller and stern flow, and also the propeller is not rotative.
In the last part of the present thesis, Paper D, the method described in Paper C is extended to calculate the ship stem/propeller flow. The extension is not straight forward. Since the propeller introduces asymmetric flow the vertical centreplane of the ship is no longer a symmetry plane of the flow even though it is still a symmetry plane of the grid. The calculation has to consider the full plane problem, i.e. both port and starboard side. In order to avoid the geometrical grid singularity an interactive procedure between port and starboard is introduced. A grid including the hub of the propeller is generated. The body forces of the propeller are calculated at the grid points
within the propeller disk by a lifting-line method based on the Kappa theory. The ship hull form of Series 60, CB = 0.6 is chosen to be a test case for validating the method. The experimental data of the complete flow field both with and without propeller for the hull form is available in Toda et al [26].
where i
u'. =a .0 +
Qj 1u1 J . =aa = hol)h(orii -8ithol)aih(i) 110) = h-1ki
_h i k,vt" (k)u ,k (i)g 'IcP
ak(Jh(ki)uk)= 0
( no summation ) i
h-1 gi juk)a (k)° ,j' k E
a .ui gj
kni ul
kr1 .uiE k \4k1 ,j k
An index inside a parenthesis is excluded from the summation convention. J is the Jacobian, gii and gii are the contravariant and covariant components of the metric tensor respectively. The effective viscosity vE is defined as
1
VE = R Vt ( 4 )
IL THE THEORETICAL BASIS OF THE CALCULATION METHOD
A body-fitted coordinate system is generated by numerically solving the following set of inverted Poisson equations with the Cartesian coordinate system xi as a reference frame
fkakx, = aia,x, ( 1 )
The control functions fk are used for stretching the coordinate surfaces in three directions to concentrate the grid near the surface of the hull and in the stern region where the rapid variation of the velocities and pressure call for a better resolution.
For an incompressible flow the dimensionless Reynolds-Averaged Navier-Stokes equations and the continuity equation transformed to a body-fitted coordinate system using physical velocity components ui can be written as:
where Re is the Reynolds number and vt is the eddy viscosity, which is related to the turbulent kinetic energy k and its rate of dissipation E by
k2
V =C
I
iie
By using the standard k-E model, the dimensionless transport equations for k and E are as follows
V
-1E
atk + hwu aik = Ja.a kj - j k] +
1gija.v
a.k + G-1 j E
k
( 5 )
( 6 )
-1 VE j-.1 2
a E + h(i)u' aiE = [glja.a.E -galC] +
1g
a. a.e + c-c
( 7 )j
j E
El ks2k
where the generation term G is
-1 -1 m i ni -1 -1 m i
G=v[h h.u.0 +g g .hh,,u trn]
t (m) (1) ,1 ,m m j (m) kj) ,1 ,The constants c
cei
, cE2, ak
and oe have the values of 0.09, 1.44, 1.92, 1.0 and 1.3 respectively. The free-stream velocity Up, reference length L and the density p are used for non-dimensionalization.Two approaches of near-wall treatment are employed in the thesis. One is the two-point wall-function of Chen and Patel [5], in which it is assumed that at least the first two near-wall grid points lie within the fully turbulent layer, and the velocity components at these two grid points are determined by the logarithmic wall-function. A generalized law of the wall given by Patel [5] is used.
1 + A y+ - 1
q .1011[4
+2[11
+ Aty+ - 1]) + B + 3.7AA
+ Ay+ + 1
The boundary values for k and E are obtained from the equilibrium relations
( 8 )
The other approach is the two-layer method [27] in which the calculation domain is divided into two regions: the near-wall region and the outer region. The near-wall region includes the sublayer, the buffer layer and a part of the fully turbulent layer. In this region the dissipation equation is not solved, and E is specified by
( 10 )
y is the normal distance from the wall, and Re is the Reynolds number. The standard k-E model is used in the outer region, and the match boundary between two regions is chosen along a grid line where the minimum Ry is of the order of 250.
3/2 k
E = ( 12 )
1
E
The eddy-viscosity vt is obtained from
V =c
tjc 1
p. 0, where1 = cly[l
-exp(-R.Y/A )] P. il ( 13 ) lE = ciy[l - exp(-R/AE)] ci is given by -3/4 CI = KC 11.and AE= 2 c1 A 70. Ry is the turbulence Reynolds number defined by
( ( 14 ) 15 )
R = R lic y
( 16 ) Y eThe propeller is replaced by a body-force actuator disk of constant axial thickness Axe. which has the effect of accelerating the flow passing the propeller disk in both the axial and tangential directions in the similar way as a propeller with an infinite number of blades. If GL = FinDpUo is the circulation at radius r and angular position 0, the body forces can be calculated by the Kutta-Joukowslcy theorem.
ZG,
-fbx(r'e) =
AxU r
ZG, VA
UAfb (r,0) =
+Ax r
Uo Uowhere Z is the number of blades, J is the propeller advance ratio, uA and LIT are the propeller induced axial and tangential velocities respectively and uTH is the hull induced tangential velocity. The body forces obtained above have to be transformed to the body-fitted coordinate system to get the body force components Fi and imposed in the momentum equations (2) just at the grid elements in the propeller disk. The thrust loading coefficient CT and torque coefficient K.Q of the propeller can be calculated by integrating the body forces over the propeller disk as
2 AAx,,
--413x(r,0)ds
0 Ao J2Ax--a fb (r,e)ds
Q D2p Ao 0where Ao is the area of the propeller disk. The circulation GL is calculated by a lifting-line propeller analysis program based on the simple Kappa theory [28]. According to the lifting-line theory a propeller with an infinite number of blades and its slipstream can be replaced by an infinite number of elementary vortex systems. The tangential induced velocity uTm of the propeller at the radius R can be calculated through the
circulation F(= GL7cDpUo) as
UTM =
Zr
4.7rIZwhere Z is the number of blades.
In the Kappa theory the tangential induced velocity of a finite number of blades propeller UT at radius R can be obtained by uTm and a K value
( 17 )
( 18 )
( 19 )
( 20 )
where K is the Kappa value, a function of the number of blades, radius and hydrodynamic pitch angle pi.
Furthermore the theory assumes that the "condition of normality" exists, which means that the axial induced velocity uA of a finite number of blades propeller can be
calculated from UT andr3i as
UT
UA = tgpi
And again the axial induced velocity of an infinite number of blades propeller uAm is calculated using the K value as
uAM = KuA ( 24 )
Thus the advance velocity of a finite number of blades propeller can be obtained by modifying the total axial velocity (VA + uAm) calculated from the flow calculation program as
VA = (VA + uAm)- KUA ( 25 )
The resulting velocity at a blade section of a propeller with finite number of blades is then obtained
V = (27an + uTH - UT)2 + (VA + UA)2
The circulation GL is calculated from the lift coefficient CLas
nD U
p0
27c U0 Dpwhere C is the length of the blade section and Dp is the diameter of the propeller.
The lift coefficient CL is approximated by
aCL
CL = CLO + --a
a a GL= FCL V C
( 22 ) ( 23 ) ( 26 ) ( 27 ) ( 28 )Cu) is calculated from two-dimensional profile theory and corrected for lifting surface
effects. The lift coefficient slope aglacc is approximated by acL
= 27tkf
aa
where kf is the camber correction caused by lifting-surface effects [28].
When the flow passes the propeller disk the turbulent kinetic energy will be increased due to both the large mean velocity gradient and the turbulent production on the propeller blades. This increased turbulent kinetic energy will increase the diffusion, so
the shear layer at the edge of the propeller slipstream will grow more rapidly. It is assumed that the turbulent kinetic energy generated by the propeller blade is produced in the boundary layer of the blade. The energy can be estimated, as proposed by
Klebanoff and Bradshaw, see [29], from
k = fV2 ( 30 )
where V is the resulting total velocity relative to the propeller blade, and f is a function of the distance from the blade surface determined from Klebanoff's flat plate data
f = 0.008214 - 0.05888(y/8) for y/8 0.05
( 31 )
f = 0.00555(1 - y/8) for 0.05 <y/8 1.0
where 6 is the boundary layer thickness at the trailing edge of the propeller blade and y is the normal distance to the blade. The turbulent kinetic energy of the actuator disk at a certain radius generated by the corresponding propeller can be estimated by integrating the equation (30) across the boundary layer at the trailing edge of the blade and added for all blades from both sides of each blade. Averaging in the circumferential direction yields
k(r) =
Zif V2d(y/S)/Er ( 32 )where Z is the number of propeller blades. Extra terms uiaikp and uiaikpe/k will then be added to the k and E transport equations, equation (6) and (7), respectivelyjust at the grid points of the actuator disk.
III. NUMERICAL TECHNIQUES
The governing equations (2), (6) and (7) for u1, k and c can be written in the form
g11a1a14) g22a2a24) g33a3a30=
2A3
4)a+ 282 DO,Dao+Eao+s
1 twhere 4) = ui, k or E.
The finite-analytical discretization scheme of Chen and Patel [5] is used in the V-V plane and a second-order central and first-order upwind difference approximation is used for the second and first derivative in the 1-direction respectively. This results in a twelve-point algebraic equation, including nine nodes in the V-V plane, one node upstream, one downstream and one for the previous time step.
The pressure-velocity coupling is treated with the SIMPLER-algorithm of Patankar [30]. Pressure correction and pressure equations are derived from the continuity equation (3) using a staggered-grid system. The solution is found as the stationary solution of a transient problem started from a initial guess of the velocities, turbulent quantities and the pressure. In each time step the momentum equations (2) are first solved successively for the pressure from the previous time step. The corresponding algebraic equations are solved by a line-by-line sweeping technique, using a fixed number of iterations. After solving the momentum equations the velocity field is corrected to become divergence free, and thereafter the pressure is calculated. Finally, before the next time step, the transport equations for the turbulent quantities k and E, equation (6) and (7), are solved using the same technique as for the momentum equations.
For the partially-parabolic approximation, equation (33) is made parabolic in the V-direction, the dominant flow V-direction, by neglecting the term g11a1a10, but remains elliptic in the transverse V-V plane. An eleven-point, in stead of twelve-point, algebraic equation is obtained by leaving out the downstream point. Therefore a marching technique in the V- direction can be used to solve the parabolic equations. The pressure correction equation is also made parabolic by assuming the downstream pressure to be known . But the pressure equation is kept fully-elliptic. For a known pressure field from the previous time step (in the first sweep the pressure is assumed zero everywhere) the momentum equations and the transport equations for the turbulent kinetic energy k and its rate of dissipation c are solved section by section from the inlet to the outlet plane. At each marching step the velocities just obtained are corrected by the pressure correction equation .After a complete marching sweep the pressure field is
updated through the use of the pressure equation based on the corrected velocities. The procedure goes back to the inlet plane and continues the next time step. The resulting algebraic equation systems for velocities, turbulent kinetic energy and its rate of dissipation and pressure are solved using a fixed number of line by line sweeps with a
tri-diagonal matrix algorithm.
For calculating the interactive flow between a stern and a propeller another iterative procedure between the flow calculation program and the propeller analysis program is introduced. For a guessed velocity field, or the velocity field from the previous iteration, the circulation and body forces are calculated by the propeller analysis program. The procedure then calls the flow calculation program to calculate the stern flow with the effects of the propeller. The velocity field is then modified due to the effects of the propeller. The procedure goes to next iteration and stops when the body force change is within a certain limit.
IV. BOUNDARY CONDITIONS
The appropriate boundary conditions are summarized as follows.
Inflow Condition
Standard boundary layer profiles for 10, k and E are specified at the inlet plane (u2=u3=0). Outside the boundary layer, k and c are set to be zero, and the velocity is computed from a potential flow solution.
Outflow Condition
In principle, only apial = o is required, but in the present thesis the zero gradient conditions ao/aV = o, (4) =ui, k, E) are also specified for all other quantities, since the coefficients in the momentum equations are based on central approximations of the velocities and their gradients.
Outer edge surface
For a large calculation domain, the uniform-flow conditions are applied, i.e. p = 0, ul = 1, u3 = 0,ak/a42=
aa42
= o , and u2 is calculated from the local continuity. Otherwise the potential flow solutions are specified.Free Water Surface
The double-model assumption is used, so the free water surface is treated as a
symmetry plane, i.e. u3 = 0, a4)/a43 = 0, = u1,u2,k or c).
Vertical Center Plane
Without a propeller, the vertical center plane is a symmetry plane, i.e. above the grid
singularity at the keel level u2 = 0, avav = 0, = u1,u3,k or E), but below that
u3 = o, ao/av = 0,
= u1,u2,k or E).With a propeller, in principle, the vertical center plane is not a boundary, but it is treated as a boundary plane since an iterative procedure between the port and starboard sides is adopted. The boundary values of ui, p, k and c for one side are obtained from the corresponding grid points at the other side. But u2 above the grid singularity at the vertical center plane are calculated through the local continuity and u3 below that is obtained by interpolation between the sides.
(6) Hull Surface
No-slip conditions are applied on the hull surface. For the two-point wall-function approach, u2 = 0, and u1 and u3 are calculated from the wall-function (9) interactively at the grid point 2 and 3. The boundary values for k and e are obtained from equation (10) and (11). For the two-layer approach, all the velocity components and the
turbulent kinetic energy are equal to zero on the hull surface, i.e. u1 = u2 = u3 = k = 0.
For an axisymmetric body, conditions (4) and (5) are as follows: In the no propeller case u3 = 0,
aotav
= 0, = u1, u2, k, e), while with an operating propeller the flow gets rotationally symmetric, i.e.agov
= o, = u1, k, E).IV. BOUNDARY CONDITIONS
The appropriate boundary conditions are summarized as follows.
Inflow Condition
Standard boundary layer profiles for ul, k and E are specified at the inlet plane (u2=u34). Outside the boundary layer, k and E are set to be zero, and the velocity is computed from a potential flow solution.
Outflow Condition
In principle, only ap/a41= 0 is required, but in the present thesis the zero gradient
conditions 4/w = o,
= ui, k, E) are also specified for all other quantities, since the coefficients in the momentum equations are based on central approximations of the velocities and their gradients.Outer edge surface
For a large calculation domain, the uniform-flow conditions are applied, i.e. p = 0,
ul
= 1, u3 = 0, ak/W = ao42 = o , and u2 is calculated from the local continuity.Otherwise the potential flow solutions are specified.
Free Water Surface
The double-model assumption is used, so the free water surface is treated as a
symmetry plane, i.e. u3 = 0, a0/a43 = 0, = u1,u2,Ic or E). Vertical Center Plane
Without a propeller, the vertical center plane is a symmetry plane, i.e. above the grid singularity at the keel level u2 = 0, 4/(342 = 0, (4) = u1,u3,k or E), but below that
u3 = o, a0A3
= 0, = u1,u2,k or E).With a propeller, in principle, the vertical center plane is not a boundary, but it is treated as a boundary plane since an iterative procedure between the port and starboard sides is adopted. The boundary values of ui, p, k and E for one side are obtained from the corresponding grid points at the other side. But u2 above the grid singularity at the vertical center plane are calculated through the local continuity and u3 below that is obtained by interpolation between the sides.
(6) Hull Surface
No-slip conditions are applied on the hull surface. For the two-point wall-function approach, u2 = 0, and u1 and u3 are calculated from the wall-function (9) interactively at the grid point 2 and 3. The boundary values for k and E are obtained from equation (10) and (11). For the two-layer approach, all the velocity components and the
turbulent kinetic energy are equal to zero on the hull surface, i.e. u1 = u2 = u3 = k = 0.
For an axisymmetric body, conditions (4) and (5) are as follows: In the no propeller
case u3 = 0, a403 = 0, (0 =
ul, u2, k, e), while with an operating propeller the flowV. VALIDATION
The method summarized in the previous Chapters has been validated at different stages during the course of development.
V.1 The Grid Generator
The grid generator, the Poisson solver, was applied to several different bodies
including four typical ship hull forms, i.e. the SSPA 720 model, the HSVA tanker, the Series 60 CB = 0.6 hull and the Wigley hull. It was evident that the generator is very flexible and capable of generating body-fitted coordinate systems for ship hulls of different complexity.
V.2 Stern Flow without Propeller
The method was developed first for the case without propeller in two levels, i.e. the fully-elliptic Reynolds-Averaged Navier-Stokes equations and the partially-parabolic approximation. A fine-formed cargo ship, the SSPA 720 model, was chosen for validation, since detailed measured data by Larsson [32] are available. The comparison of the calculated results from the fully-elliptic approach and the partially-parabolic approximation indicated that no noticeable difference can be seen. This is not surprising for a stern flow without separation. The second derivative term neglected in the
partially-parabolic approximation is of significant importance only for bluff stern flows with separation, but not for slender sterns, like the one of the SSPA 720 model. This means that the partially-parabolic assumptions are valid, at least for slender ships.
The grid dependence and the location of the outer edge boundary were also studied at this stage. The computed results showed remarkably slight sensitivity to the grid variations made in the numerical tests. The location of the outer edge boundary can be placed as close as 0.125L from the ship axis if the boundary conditions are calculated from a potential flow solution. The calculation results are in very good agreement with the experiments.
V.3 Interactive Flow between a Stern and an Operating Propeller
The method was further developed for calculating the interactive flow between a stern and a propeller. Only the partially-parabolic Reynolds-Averaged Navier-Stokes equations were used, since the operation of the propeller reinforces the parabolic assumptions.
The method was validated for three different cases of increasing complexity, starting with an open water case with only a propeller and a shaft. Somewhat more complicated
is the second case, where the flow around an axisymmetric body with a propeller was computed. Finally, the fully three-dimensional flow around a ship hull with an operating propeller was investigated.
V.3.1 Propeller in Open Water
Before applying the method to more complicated geometries it was considered
appropriate to apply it to a simple case, so a propeller-shaft configuration was selected for this purpose. An experiment with such a configuration, i.e. an open water propeller
test, was made by Voigt [33] in 1933. It was found from this study that by introducing the turbulent kinetic energy generated in the boundary layers at the propeller blades a more realistic growth of the mixing layer at the edge of the propeller slipstream was obtained. The method proposed for estimating the propeller generated turbulent kinetic
energy seems to be proper. The calculated velocity profiles are in very good agreement with the experiments.
V.3.2 Axisymmetric Body with or without Propeller
Although the propeller-shaft configuration is an example of propeller/hull interaction the influence of the shaft on the propeller is rather small. The boundary layer of the shaft is very thin and most of the propeller works outside the boundary layer. Therefore the axisymmetric Afterbody 1 of Huang et al and a DTNSRDC seven-blade propeller was used for further validation, for which experimental data are available [34],[35]. For the case without propeller, two near-wall treatments in the turbulence model was tested, and it was found that the two-layer model gives better agreement with the experimental results than the two-point wall-function approach. For the case with propeller, the method predicted the interactive flow well. The calculated axial velocity profiles, especially the effective inflow velocity at the propeller plane, are in excellent agreement
with the measurements. Also the propeller loading coefficient CT and the torque coefficient 1{.Q, are very well predicted.
V.3.3 Series 60 CB = 0.6 Hull with or without Propeller
The final test case for the method was the well-known Series 60 CB = 0.6 hull, extensively investigated within the rr-rc Cooperative Experimental Program. Of particular interest is the work carried by Toda et al [26], who measured the detailed flow field around the hull in a towing tank with and without an operating propeller. From the calculations it was seen that most characteristics of the ship stem/propeller interactive turbulent flow are well predicted by the present method, such as the suction of the flow ahead of the propeller, the thinner boundary layer thickness near the stem, a lower pressure on the hull surface at the aftermost part of the hull, the rotational flow in
VI. CONCLUSIONS
The numerical method for calculating ship stem/propeller flow presented in the present thesis contains the development of a method for generating body-fitted coordinate systems, a Navier-Stokes method for stern flow calculations including the effect of a propeller, and applications of the method first to a ship hull without propeller, then to axisymmetric bodies and a practical ship hull form with or without propeller. The following conclusions may be drawn.
The quality of the grid is important in the numerical solution of the partial differential equations. Ship hull forms are usually quite complex, especially the afterbody. It is desirable to employ body-fitted coordinate systems so that the boundary conditions can be accurately represented and the flow in the boundary layer and stem region can be accurately resolved with a reasonable number of grid points. The method of numerically solving a set of Poisson equations for
generating such body-fitted coordinate systems is found to be flexible and powerful. The method is capable of generating nearly orthogonal and smooth coordinates with desirable stretching in all three directions for different kinds of
bodies with different geometrical complexity.
Stem flows can be calculated with reasonable accuracy and in detail by the Navier-Stokes method. Two levels of such a method, i.e. the fully-elliptic and the
partially-parabolic approximation, are studied. It is seen that the partially-parabolic assumptions are valid at least for slender ships. The near wall treatment seems to be important. Even though the two-point wall-function is good enough for most calculations, the results from an axisymmetric body, Afterbody 1, indicate that a more accurate resolution of the flow in the near-wall region is obtained by the two-layer model than by the two-point wall-function approach.
The Navier-Stokes method can be readily extended to include a propeller, rudder or other appendages. The effect of an operating propeller is introduced into the stem flow by replacing the propeller with a force actuator disk. The body-force propeller model is a good representation of the propeller in the numerical solution of the propeller/hull interaction problem and it is easy to implement in the Navier-Stokes method. The relatively simple lifting-line propeller analysis method based upon the Kappa theory for calculating the circulation distribution of a propeller blade with specified propeller geometry, operating condition and effective inflow condition is found to be accurate enough in combination with the Navier-Stokes method.
o An operating propeller increases the turbulent kinetic energy in the slipstream of the propeller due to both the large mean velocity gradient and the turbulent production on the propeller blades. This affects the turbulent diffusion in the slipstream, so that the shear layer at the edge of the propeller slipstream is mixed and grows more rapidly, resulting in a more smooth velocity profile. Therefore, in order to accurately predict the propeller slipstream, the turbulence caused by the propeller should be considered. The method proposed in the present thesis for
estimating the turbulent kinetic energy generated by the propeller is based on the flat plate turbulent boundary layer theory. The propeller slipstream is more accurately predicted by adding an extra term in the k and E transport equations for introducing the turbulence caused by the propeller.
o The method is capable of calculating the three-dimensional viscous flow around a practical ship hull form with an operating propeller. The calculated results of the effective inflow to the propeller, the lower pressure at the stern due to suction of the propeller, and the rotative slipstream in the wake are in reasonable agreement with the experiment. For a single right hand screw propeller operating behind a ship, with water flowing upward into the propeller disk from underneath, a larger portion of the thrust is developed on the starboard side, where the blades are
moving downward. The asymmetric loading introduces asymmetric flow in front of and behind the propeller, and gives rise to a side force and a turning moment on the ship. This is the so called Hovgaard effect. It may be predicted by the present method. The axial velocity is more accelerated on the starboard side than on the port side, and the center of the rotative slipstream is shifted to the port side.
VII. FUTURE APPLICATIONS AND DEVELOPMENTS
In order to test the present method, flows around bodies with different geometrical complexity need to be calculated, especially for different ship hull forms. Experimental data for such bodies are required for the validation of the calculations.
With regard to the further development of the method more attention should be paid to the grid generation. A better grid is certainly required for better description of the flow around the stern and the propeller. More grid points are required for better resolution in the propeller tip region where the shear layer is developed. From the view of
representation of the propeller, fairly well distributed grid points on the propeller disk are preferred. Multi-block grids are worth investigating with finer grid in the stern and propeller region.
A further extension of the present method is to integrate it with the SHIPFLOW system [31], a suite of computer programs for calculating the flow, resistance and propulsion characteristics of ships based on a zonal approach. The potential flow and the wave resistance is computed using a panel method, while the viscous flow and the
corresponding resistance components and propulsion characteristics are obtained using boundary layer theory for the forebody and the present method for the afterbody and in the wake.
VIII. SUMMARY OF THE APPENDED PAPERS
Paper A: Body Fitted Coordinate Systems for Ship Stern Flow Calculations
An elliptic generation system for generating body-fitted coordinate systems is described in the paper. The emphasis lies on the generation of grids for ship stern flow
calculations. The geometry of ship sterns is usually complex. The resolution of the flow around the stem and near wake requires a reasonable amount of grid points located in the near-wall layers and in the region of rapid change of geometry. A method for numerically solving a set of Poisson equations for generating such body-fitted coordinate systems is presented in the paper. The coordinate surfaces can be stretched in each coordinate direction by the control functions on the right hand side of the Poisson equations. The method for evaluating the control functions numerically from the prescribed distributions of the boundary nodes is explained in the paper. Neuman boundary conditions are specified on all the boundaries of the cross sections in order to
get as closely as possible orthogonal coordinates in each cross sections. The techniques for treating boundary conditions in different complex geometries are discussed in more detail in the paper.
The method has been applied to several different bodies including four typical ship hull forms, i.e. the SSPA model 720, the HSVA tanker, the Series 60, and the Wigley hull. It is concluded that the method is very flexible and capable of generating body-fitted coordinate systems for ship hull forms of different complexity
Paper B: Numerical Solution of the Reynolds-Averaged Navier-Stokes Equations for Ship Stem Flow
This paper describes a numerical solution procedure for the Reynolds-Averaged Navier-Stokes equations with application to ship stem flows. It is an extension of the previous work did by the co-author of the paper [18]. In there a time-marching calculation procedure was developed for the calculation of turbulent three-dimensional partially-parabolic flows. The procedure is extended into the fully-elliptic mode in the present paper by including the second derivatives in the main flow direction, which is neglected in the governing equations due to the partially-parabolic assumptions.
The fully-elliptic Reynolds-Averaged equations and the continuity equation are transformed to a general body-fitted coordinate system using physical velocity
components. A partially finite-analytical scheme is employed in the discretization of the governing equations. The first and second derivatives in the dominant direction are replaced by a first order upwind difference and a second order central approximation respectively. The pressure-velocity coupling is treated with the SIMPLER-algorithm. The SSPA 720 Model, for which the detailed measured data by Larsson [32] are available, is used as a test case. Comparison is made with the partially-parabolic method. No noticeable difference between the solutions can be seen. That means the partially-parabolic assumptions are valid at least for slender ship hulls, such as the
SSPA 720 Model. Of course for full ship hull forms such as tankers where might appear reverse flow the fully elliptic method is required.
The grid dependence has been investigated in the paper. Four grids with the dimensions of LLxMMxNN as 60x19x9, 60x21x15, 30x21x15 and 30x41x15 are used. LL, MM and NN are the number of clusters in the longitudinal, normal and circumferential directions respectively. These systematic studies of grid-dependence indicate that there is some grid sensitivity in the cross section, but not in the longitudinal direction. Special attention must be paid to regions of abrupt change in geometry, such as the stern region and near the keel. Concentration of clusters to these regions may be an adequate alternative to fine grids.
A study of the outer edge boundary location has been done in the paper to investigate the location of the region of viscous-inviscid interaction for the SSPA 720 Model. A series of calculations with a variation of the outer edge boundary location, the greatest distance to the edge measured from the keel ranging from 0.54 L to 0.025 L, was carried out. The potential flow solution was taken as the boundary condition there. The results indicate that the location of the outer edge can be placed as close as 0.125 L from the ship axis if the boundary conditions are calculated from a potential flow solution. This means that with the same number of clusters in the normal direction, better resolutions can be obtained with a smaller calculation domain.
Paper C: Numerical Computation of Propeller/Hull Interaction in Viscous Flow. Part I: Rotationally Symmetric Flow
In this paper the effect of an operating propeller is introduced into the stern flow. The effect of the propeller is simulated by an actuator disk of constant thickness with distributed body forces thus accelerates the flow passing the propeller disk in both the axial and tangential directions in a similar way as a propeller with an infinite number of blades. The body forces depend on the circulation distribution of the propeller, which is calculated by a relatively simple lifting-line propeller analysis programbased on the -Kappa theory with specified inflow conditions and propeller geometry. The body forces obtained are added to the momentum equations just at the actuator disk grid
elements. The inflow to the propeller will be changed due to the operating propeller. So the partially-parabolic Navier-Stokes method for calculating stem flow is combined with the propeller analysis program for calculating the body forces in an interactive manner to predict the interactive flow between the hull and the propeller.
Only axisymmetric bodies are considered in this paper. The method was first applied to a shaft-propeller configuration, an open water propeller. From this test it appears that the turbulence generated by the propeller is important for creating the necessary mixing and diffusion of the shear layer at the edge of the propeller slipstream. A method based on the turbulent flat plate boundary layer theory is proposed in this paper to estimate the turbulent kinetic energy generated by the propeller. Extra terms considering the
propeller generated turbulent kinetic energy are added in the k and c transport
equations. The propeller slipstream is more accurately predicted and in good agreement with the measurements by Voigt [33].
The second calculation case is an axisymmetric body, the Afterbody 1, with and without an operating propeller tested by Huang et al [34], [35]. Two near-wall
treatments are tested for the case without a propeller. One is the two-point wall-function approach of Chen and Patel [5], in which the boundary values of the velocity
components at the first two grid points off the body are determined from the wall-function and the geometry of the wall surface by an interactive procedure, and the boundary values of the k and E are obtained from the equilibrium relations. The other is the two-layer model [27], in which the flow domain is divided into two regions : the near-wall region and the outer region. The near-wall region contains the sublayer, the buffer layer and a part of the fully turbulent layer. Only the transport equation for
turbulent kinetic energy needs to be solved in this region. The rate of dissipation eis calculated by the one-equation model of Wolfshtein [36]. The standard k-cmodel is
used in the outer region. It is concluded that a more accurate resolution ofthe flow in the near-wall region is obtained by the two-layer model than by the two-point wall-function approach.
The calculation results from the with propeller case indicate that the present method can accurately predict the interactive flow. The calculated velocity profiles are in good agreement with the experimental data, especially at the station just ahead of the propeller where the inflow is an important input data for the design of a propeller. The calculation results show that the flow field is influenced by the operating propeller upstream to about twice the propeller diameter and the propeller performance is also very well predicted by the present method.
Paper D: Numerical Computation of Propeller/Hull Interaction in Viscous Flow. Part II: Three-dimensional Flow
The method for calculating propeller/hull interaction in viscous flow developed in Paper C was extended to calculate three-dimensional flow, i.e. propeller/ship stern
interaction, in this paper. Since the rotation of the propeller introduces asymmetric flow the vertical centreplane of the ship is no longer a symmetry plane for the stern flow, even though it is a plane of symmetry for the geometry. The so called full-plane calculation including both port and starboard sides has to be considered, in which case the vertical centreplane is not a boundary plane but in the calculation domain. There will then emerge some difficulties around the grid singularity at the keel level, where the transverse curvature of the grid becomes very large. An interactive procedure between the port side and starboard side is introduced to overcome the difficulties. In this procedure the vertical center plane becomes a boundary plane and boundary conditions
are needed to connect the flows between the two sides. The boundary values of ui, p, k and c at one side are obtained from the corresponding grid points at the other side. Since there is no momentum equations being set up at the vertical centreplane the velocity components at this plane, i.e. the u2 component above the grid singularity are obtained from the local continuity and u3 component just below that are obtained by interpolation between the the sides. For each time step two longitudinal marching sweeps are required, one for the port side and another for the starboard side.
For a three-dimensional nonuniform inflow, the thrust and torque of a propeller blade will vary with the angular and radial position. It is assumed that at a certain grid point on the propeller disk with the specified inflow there is one propeller with an infinite
number of blades. The local thrust and torque of this propeller at that grid point is calculated by a simple lifting-line propeller analysis method based upon Goldsteins
Kappa theory. The total thrust and torque of the propeller can be obtained by integrating the local thrust and torque over the propeller disk. The body forces calculated fromthe local thrust and torque will be put into the momentum equations just at the grid points of the propeller disk to take the effect of the propeller into account in the stern flow.
The Series 60 CB = 0.6 hull form was chosen as a test case for which detailed experimental data with and without propeller by Toda et al [26] are available. The calculated results indicate that the computational method can capture almost all the characteristics of the three-dimensional interactive flow between a ship stem and a propeller, such as the acceleration of the flow ahead of the propeller, the lower pressure
at the stern region and the rotation of the slipstream. The socalled Hovgaard Effect is also predicted by the calculation. This means that a single right hand screwpropeller operating behind a ship develops a larger portion of its thrust on the starboardside, where the blades are moving downward. The center of the rotating slipstreamis shifted to the port side. The calculation results are in good agreement with themeasurements.
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SSPA Report No. 90, Gothenburg, Sweden, 1981.
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Proc. 14th ONR Sym. Naval Hydrodynamics, Ann Arbor, Mi, 1982. Broberg, L.: "Numerical Calculation of Ship Stern Flow", PhD Thesis,
Division of Mechanics, Chalmers University of Technology, Gothenburg, 1988. Chen, H.C. and Patel, V.C.: "Calculation of Stern Flows by a Time-Marching
Solution of the partially-Parabolic Equations", Proc., 15th ONR Sym. on
Naval Hydrodynamics, Hamburg, 1984.
Chen, H.C. and Patel, V.C.: "Calculation of Trailing-Edge, Stern and Wake Flows by a Time-Marching Solution of the Partially-Parabolic Equations", IIHR
Report No. 285, Iowa Institute of Hydraulic Research, University of Iowa, Iowa City, Ia., 1985.
Chen, H.C. and Patel, V.C.: "Numerical Solutions of the Flow over the Stern and in the Wake of Ship Hulls", Proc. 4th Int. Conf. Numeric. Ship Hydrodyn. Washington D.C., 1985.
Hoekstra, M. and Raven, H.C.: "Ship Boundary Layer and Wake Calculation with a Parabolized Navier-Stokes Solution System", Proc. 4th Int. Conf.
Numeric. Ship Hydrodyn., Washington D.C., 1985.
Hoekstra, M. and Raven, H.C.: "Application of a Parabolized Navier-Stokes Solution System to Ship Stern Flow Computation", Proc. Osaka Int. Colloq.
Ship Visc. Flow, Osaka, Japan, 1985.
Huang, S. and Zhou, L.D.: "A Streamline Iteration Method for Computing the Three-Dimensional Turbulent Flow around the Stem and in the Wake of Ship;
First Report: Wigley Ship Model", Proc. 2nd Int. Sym. Ship Viscous Res. Gothenburg, Sweden, 1985.
Janson, C.E. and Larsson, L.: "Ship Flow Calculations using the Phoenics Computer Code", Proc. 2nd int. Symp. Ship Viscous Res., Gothenburg, Sweden, 1985.
Kodama, Y.: "Computation of 3-D Incompressible Navier-Stokes Equations for Flow Around a Ship Hull Using an Implicit Factored Scheme", Proc. Osaka Int.
Colloq. Ship Viscous Flow, Osaka, Japan, 1985.
Kodama, Y.: "Computation of High Reynolds Number Flows past a Ship Hull
Using the IAF Method", J. Soc. Nay. Arch. Japan, Vol. 161, 1987.
Raven, H.C. and Hoekstra, M.: "A Parabolized Navier-Stokes Method for Ship
Stern Flow Calculations", Proc. 2nd Int. Sym. Ship Viscous Res., Gothenburg, Sweden, 1985.
Tzabiras, G.D.: "On the Calculation of the 3-D Reynolds Stress Tensor by Two
Algorithms", Proc. 2nd Int. Sym. Ship Viscous Res. Gothenburg, Sweden,
1985.
[151 Piquet, J. and Visonneau, M.: "Computation of the Flow past Shiplike Hulls",
Proc. 5th Int. Conf. Numeric. Ship Hydrodyn., Hiroshima, Japan, 1989.
Patel, V.C.: "Ship Stern and Wake Flows: Status of Experiment and Theory",
Proc. 17th ONR Sym. Naval Hydrodynamics, The Hague, 1988.
Broberg, L. and Larsson, L.: "A Calculation Methodfor Ship Stern Flows using an Analytic Body-Fitted Coordinate System", Proc 15th ONR Sym. Naval
Hydrodynamics, Hamburg, 1984.
Broberg, L.: "Calculation of Partially-Parabolic Stern Flows", SSPA Report No. 2801-2, 1987, or PhD Thesis, Div. of Mechanics, Chalmers University of
Technology, Gothenburg, 1988.
Huang, T.T. and Groves, N.C.: "Effective Wake: Theory and Experiment", Proc., 13th ONR Sym. Naval Hydrodynamics, Tokyo, 1980.
Dyne, G.: "On Optimal Wake Vorticity Adapted Propellers", Proc. 2nd Int. Sym. on Practical Design in Shipbuilding, Tokyo and Seoul, 1983.
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15th ONR Sym. Naval Hydrodynamics, Hamburg, 1984.
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Hydrodynamics, Berkeley, Caliv., 1986.
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Practical Hull Form", Proc. 5th Int. Conf. Numerical Ship Hydrodyn. Hiroshima, Tokyo, Japan, 1989.
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the Boundary Layer and Wake of a Series 60 CB=0.6 Model Ship with and without Propeller", Ill-IR Report No. 326, Iowa Institute of Hydraulic Research,
University of Iowa, Iowa City, 1988.
Chen, H.C. and Patel, V.C.: "Practical Near-Wall Turbulence Models for Complex Flows Including Separation", 19th AIAA Fluid Dynamics, Plasma Dynamics and Lasers Conference, Honolulu, Hawaii, 1987.
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DEPARTMENT OF MARINE HYDRODYNAMICS
Gothenburg - Sweden
Body Fitted Coordinate Systems for Ship Stern Flow Calculations
by
Dao Hua Zhang
Report NO 73 ISSN 0284-7760
Submitted to Chalmers University of Technology in partial fulfillment of the requirements for the degree of licentiate of engineering
ABSTRACT LIST OF SYMBOLS INTRODUCTION 1 GENERATION EQUATIONS 4 11.1 Basic Principles 4 11.2 Generation Equations 5 11.3 Transformation Relations 6
NUMERICAL SOLUTION PROCEDURE 12
111.1 The Choice of Coordinates in the
Physical Domain . 12
111.2 Numerical Solution 15
111.3 Evaluation of the Control Functions 18
111.4 Boundary Conditions 20
THE OVERALL SOLUTION ALGORITHM
23
APPLICATIONS 26
EXAMPLES OF FLOW CALCULATIONS 28
ACKNOWLEDGEMENTS 29
REFERENCES 30
ABSTRACT
In the present paper body-fitted coordinate systems suitable for ship stern and wake flow calculations are generated by numerically solving a set of Poisson equations. The stretching in each coordi-nate direction. is implemented by the control functions on the right hand side of the Poisson equations, and the method for evaluating the control functions numerically from the prescribed distributions of the boundary nodes is explained in the paper. Neumann boundary conditions are specified on all the boundaries of the cross section
in order to get as closely as possible orthogonal coordinates in each axial cross section. The techniques for treating boundary con-ditions in different complex geometries are discussed in more
detail in the paper.
The method has been applied to several different bodies including four typical ship hull forms, i.e. the SSPA model 720, the HSVA tanker, the Series 60, and the Wigley hull. It is concluded that the method is very flexible and capable of generating body-fitted coordinate systems for different ship hulls. Of the grids generated by the method, the body-fitted coordinates for the SSPA 720 hull and the Wigley hull have already been used for partially-parabolic and fully-elliptic flow calculations. Good results were obtained.
Alphabetical Symbols
A a scalar-valued function
A a vector
a,b,c convection coefficients in linearized grid generation
equations
a.,a- covariant and contravariant base vectors in the
-general curvilinear coordinates
e. covariant base vectors in orthogonal coordinates x.
fi
F2
ij
grid control functions
control function in Ti -direction for r-equation
square of Jacobian
covariant metric tensor in the general curvilinear
coordinates 1
contravariant metric tensor in the general curvili-near coordinates
metric coefficients or scale factors in the orthogo-nal coordinates x.
covariant metric tensor in the orthogonal coordina-tes x.
Jacobian
pcsition vector
radius of the cylindrical external boundary surface dimensionless cylindrical-polar coordinates
dimensionless orthogonal coordinates (j=1,2,3)
Greek Symbols
6). Kroneoker del ca
tranformed coordinates (Tr the indices in corresponding
coordinate directions
general curvilinear coordinates (i=1,2,3)
0 transport quantiti-T's V gradient V2 Laplacian h. 1 h. . x, r, x.
I. INTRODUCTION
At the international SSPA-ITIC workshop on ship boundary layers 1980 [1] Larsson gave a summary of the different coordinate sys-tems for ship boundary layer calcultion. Generally speaking they can be classfied into four types: the coordinate system based on the streamlines and eqpipotential lines of the potential flow on the surface, the system based on a conformal mapping of the hull sections, the system based on equal division of the girth length of each section and the distorted cylindrical system. All of them except the last one are coordinates only on the body surface. They are suitable for the boundary layer calculations but not for the ship stern flow calculafions.
It has been pointed out in the report of 18th ITTC Resistance and Flu,: Committee that it is now well known that thin boundary layer theory fails to predict the thickening viscous flow over the stern and the near wake of a ship hull [2]. The general features of the flow in the thick boundary layer over the stern and in the near wake of a ship have been reviewed by Patel and Chen [3],
[4]. The flow over the stern and in the near wake of a ship is characterized by rapid thickening of the viscous layer, strong viscous7inviscid interaction, longitudinal vortex formation, and a general reduction in the level of turbulence. There is a predomi-nant flow direction in the ship stern flow. The diffusion in the local mean streamline direction therefore can be neglected. This type of flow, sometimes called partially-parabolic flow, can be precisely discribed by partially-parabolic equations, which are obtained by neglecting the molecular and turbulent transport terms in the longitudinal direction in the Reynolds-averaged Navier-Stokes equations.
Methods for calculating ship stern flows involving numerical solu-tions of the complete Reynolds-averaged Navier-Stokes equasolu-tions or the parabolized approximation thereof call for an accurate numeri-cal representation of the boundary conditions. Such a representa-tin is best fullfilled when the boundary coincide with a coordina-te surface. This means that the co.prdinacoordina-tes should be "body-fitcoordina-ted"
or "boundary-fitted". It is also vitally important that the grid is smooth, particulary if the flow equations are fully transformed, i.e. the velocities are directed along the coordinate lines (cont-ravariant or physical components), since in this case, third order grid derivatives will appear in the flow equations. The generation of a body-fitted coordinate system is thus an essential part of a
general numerical method for computing the ship stern and wake flow.
A body-fitted coordinate system suitable for ship stern flow cal-culations was generated by L. Broberg [5]. The sections of the hull are represented by a n-parameter conformal mapping. The coef-ficients of the mapping function are obtained from a set of coordi-nates for each section, and written as polynomials in x, which is the coordinate along the hull. The entire hull may then be repre-sented analytically, as well as the coordinate system outside the hull. This system was, however, originally generated for boundary
layer method calculations so no consideration was given its
exten-sion into the wake. Therefore a more flexible method for generating a body-fitted coordinate system is desired.
Tzabiras and Loukakis [6] have also used conformal mapping
tech-niques to generate locally orthogonal coordinates at each section for predicting the flow around the stern of ship hulls. Abdelmeguid
et al. [7] on the other hand used a non-orthogonal distorted
polar coordinate system for a partially-parabolic flow solution
procedure. It seems that the results are greatly dependent upon the
choice of the coordinate system. Raven and Hcekstra [8] used a
conformal mapping procedure based on a generalized
Schwarz-Christoffel transformation to generate the coordinate system.
In the past few years, there has been a great deal of interest in the developrent of numerical grid-generation techniques for complex geometries. Computer-generated coordinates for numerical calcula-tions of flows involving complex boundary surfaces have received
much attention, particularly for aeronautical and internal-flow
