Numerical Analysis of Transom stern high speed ship
140
SEAKEEPING OF HIGH SPEED SHIPS WITH TRANSOM STERN AND THE VALIDATION METHOD WITH UNSTEADY WAVES AROUND SHIPS A Dissertation by Muniyandy ELANGOVAN Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor of Engineering Graduate School of Engineering December 2011 JAPAN
Numerical Analysis of Transom stern high speed ship
1. SEAKEEPING OF HIGH SPEED SHIPS WITH TRANSOM STERN AND THE
VALIDATION METHOD WITH UNSTEADY WAVES AROUND SHIPS A Dissertation
by Muniyandy ELANGOVAN Submitted in Partial Fullment of the
Requirements for the Degree of Doctor of Engineering Graduate
School of Engineering December 2011 JAPAN
2. Abstract Improvement in seakeeping qualities of the ship
hull through numerical computation isalways a demand from shipping
industry to have the better hull to reduce the resistance whichis
related with the engine power requirement. So far, plenty of eort
has been made to improvethe estimation accuracy of the seakeeping
qualities that are the hydrodynamic forces, motions,wave eld around
a ship and added wave resistance. Nowadays, Rankine Panel Method is
popularto carry out this numerical analysis. In this thesis, the
Rankine panel method in the frequencydomain is developed and
applied to several kinds of ships including the high speed
transomstern for the purpose of conrming its eciency. The
Neumann-Kelvin and Double-body owformulations are examined as a
basis ow. As numerical estimation methods become higher grade, more
accurate and detailed exper-imental data for their validation is
required. An unsteady wave pattern is being used as one ofthe
methods for that purpose. To capture the wave pattern, Ohkusu,
RIAM, Kyushu University,Japan has developed a method to measure the
waves and analysis the added wave resistanceby means of unsteady
waves. In this thesis, the analysis method is improved by including
theinteraction eect of steady wave and the incident wave in the
original method. A ModiedWigley hull is analyzed numerically for
the unsteady waves and compared with experimentaldata obtained by
the present method. This interaction eect has been observed
remarkably inthe comparison. In addition to the unsteady wave eld,
hydrodynamics forces, motions andadded wave resistance are also
compared with experimental data. To treat the high speed vessel
which has a transom, a new boundary condition has beenintroduced.
This condition has been derived from the experimental observation
which conrmsthat the transom stern part is completely dry at the
high forward speed. From this point ofview, the boundary condition
is formulated at the transom stern just behind stern to implementin
the potential theory panel code. This condition corresponding to
the Kutta condition inthe lifting body theory. High speed monohull
and trimaran are taken for the analysis, and thecomputed numerical
results are compared with experimental data. Inuence of a transom
sternis observed in hydrodynamic forces and moments, ship motions,
unsteady waves and addedwave resistance. It is concluded that the
new transom boundary condition can capture thehydrodynamics
phenomena around the transom and this can improve the estimation
accuracyof the seakeeping qualities in numerical computation for
this kind of vessel. ii
3. Acknowledgements The completion of this thesis has been
facilitated by several persons. I would like to thankall of them
for their help and cooperation during this research. First, I would
like to express my deepest gratitude to my academic supervisor
Prof. Hidet-sugu Iwashita, who has given me an opportunity to do
research under his supervision. Histechnical advice and guidance
have signicantly contributed to the success of this research
andalso thank him for his valuable time to explain the critical
research point which was raised duringthe research period. He has
helped me and my family in terms of advice and nancial
supportwithout which will be dicult to complete this research.
Myself and my family members willremember in our lifetime and
thankful to him. It is my great pleasure to thank Prof Yasuaki Doi,
Prof Hironori Yasukawa and Prof. HidemiMutsuda for evaluating this
thesis and for their valuable suggestions. I take this opportunity
tothank Prof. Mikio Takaki, and we had a departmental party in the
rst year which is memorablein my life. I would like to thank
students from Airworthiness and seakeeping for vehicles
laboratory,for their direct and indirect support. I also thanks to
Mr. Tanabe and Mr. Ito for their supportin the nal stage of thesis
preparation. I very much acknowledge the help of the sta, Facultyof
Engineering Department and Graduate School of Engineering. My
deepest thanks go to my father, mother, wife, two daughters and
father in-law whoselove, aection and encouragement during the
period of course had been main back born to keepme with more
condent and motivation towards the completion of this research. My
stay and studies in Japan have been supported in-terms of
scholarship of the JapaneseGovernment, Ministry of Education,
Science, Sports and Culture, for which I am very thankful. I would
like to thank www.google.com making the search part and translation
of Japanesedocument very easy and fast. My last thanks but not
least, will be for the citizens of Japan and especially Saijo city
oceand people for their great cooperation for my family stay and me
during our stay. I had a fewopportunities to participate in some of
the Japanese cultural program in Japan, which madeunforgettable in
my life and love their traditional culture. Therefore, once again,
I am thankfulto the people in Japan. Muniyandy Elangovan December,
2011 Higashi-Hiroshima Shi iii
4. DedicationTo All Citizens of Japan and My Family Members
iv
5. NomenclatureAcronyms j Radiation PotentialBEM Boundary
Element Method Total Velocity PotentialBVP Boundary Value Problem
Fluid DensityCFD Computational Fluid Dynamics Source StrengthEUT
Enhanced Unied Theory Reduced FrequencyGFM Green Function Method j
Motion in j-th DirectionHSST High Speed Strip Theory Free Surface
ElevationLES Large-Eddy Simulation 7 Diraction waveRANS Reynolds
Averaged Navier-Stokes j Radiation wave in jth DirectionRPM Rankine
Panel Method s Steady Wave ElevationTSC Transom Stern Condition
Mathematical SymbolsGreek Symbols Two Dimmensional Laplacian with
re- spect to x and y Displacement Vector GML Longitudinal
Metacentric Height Encounter Angle of Incident Waves GMT Transverse
Metacentric Height Wave Length mj jth Component of m Vector0 Wave
Circular Frequency nj jth Component of n Vectore Encounter Circular
Frequency n Unit Normal Vector Double body ow potential A Wave
Amplitude Unsteady velocity Potential Aij Added Mass acting in i-th
direction due0 Incident Wave Potential to j-th Motion7 Scattering
Potential v
6. viBij Damping Coecient acting in i-th di- Miscellaneous
rection due to j-th Motion V Steady Velocity VectorCb Block
Coecient B BreadthCij Matrix of Restoring Coecients d DraughtD/Dt
Substantial Derivatives g Gravitational AccelerationEj Exciting
Force in j-th Direction H Kochin FunctionFj Steady Force in j-th
Direction k Moments of Inertia and Centrifugal Mo-Fn Froude Number
mentG Green Function k1 , k2 Elementary WaveH Wave Height L Length
Between PerpendicularsK Wave Number NF Number of Elements on Free
SurfaceK0 Steady Wave Number NH Number of Elements on Hull
SurfaceKe Encounter Wave Number NT Total Number of ElementsMij Mass
Matrix Associated with Body NF A No. of Elements on Transom
Surfacep Unsteady Pressure P (x, y, z) Field Pointps Steady
Pressure Q(x, y, z) Source PointU Forward Speed of the Ship RAW
Added Wave Resistancexg , zg Coordinates of the Center of Gravity
SC Transom Surface SF Free Surface SH Hull Surface Sw Waterline
Area xw Center of Water Line Area
10. List of Figures2.1 Body boundary bondition - coordinate
system . . . . . . . . . . . . . . . . . . . 132.2 Transom stern
with steady wave . . . . . . . . . . . . . . . . . . . . . . . . .
. . 252.3 Denition of problem . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 282.4 Basis ow approximation . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 283.1 SPM -
Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 5.0, =
0.447 . 413.2 SPM - Comp. of wave pattern and pers. view for Fn =
0.2, Ke = 10, = 0.633 . 413.3 SPM - Comp. of wave pattern and pers.
view for Fn = 0.2, Ke = 20, = 0.894 . 423.4 SPM - Comp. of wave
pattern and pers. view for Fn = 0.2, Ke = 30, = 1.095 . 423.5 SIM -
Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 5.0, =
0.447 . 433.6 SIM - Comp. of wave pattern and pers. view for Fn =
0.2, Ke = 10, = 0.633 . . 433.7 SIM - Comp. of wave pattern and
pers. view for Fn = 0.2, Ke = 20, = 0.894 . . 443.8 SIM - Comp. of
wave pattern and pers. view for Fn = 0.2, Ke = 30, = 1.095 . .
443.9 Numerical treatment of transom stern . . . . . . . . . . . .
. . . . . . . . . . . . 453.10 Panel shift method for the transom
stern problem . . . . . . . . . . . . . . . . . 454.1 Motion free
experimental setup diagram . . . . . . . . . . . . . . . . . . . .
. . . 484.2 Forced motion experimental setup diagram . . . . . . .
. . . . . . . . . . . . . . 494.3 Schematic diagram for the wave
measurement by multifold method . . . . . . . . 554.4 Wave
propagation with respect to time . . . . . . . . . . . . . . . . .
. . . . . . . 565.1 Plans of the modied Wigley hull . . . . . . . .
. . . . . . . . . . . . . . . . . . 625.2 Computation grids for
Rankine panel method . . . . . . . . . . . . . . . . . . . 635.3
Eect of number of wave-probe in accuracy of wave pattern analysis
(Diraction wave at y/(B/2) = 1.4 for Fn = 0.2, /L = 0.5, = ) . . .
. . . . . . . . . . . . 65 x
11. List of Figures xi 5.4 2nd-order term of wave pattern
analysis (Diraction wave at y/(B/2) = 1.4 for Fn = 0.2, /L = 0.5, =
) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.5 Interaction eect between double-body ow and incident wave in
wave pattern analysis (Diraction wave at y/(B/2) = 1.4 for Fn =
0.2, /L = 0.5, = ) . . . 66 5.6 Kelvin wave at Fn = 0.2, y/(B/2) =
1.4 . . . . . . . . . . . . . . . . . . . . . . . 66 5.7
Interaction eect between Kelvin wave and incident wave in wave
pattern analysis (Diraction wave at y/(B/2) = 1.4 for Fn = 0.2, = )
. . . . . . . . . . . . . . 67 5.8 Kochin functions computed with
diraction waves measured at Fn = 0.2, /L = 0.5, = , y/(B/2) = 1.4 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.9
Steady Kelvin wave pattern of modied Wigley model (blunt) at Fn =
0.2 . . . . 70 5.10 Added mass and damping coecients due to forced
heave motion at Fn = 0.2 . . 71 5.11 Added mass and damping
coecients due to forced pitch motion at Fn = 0.2 . . 72 5.12 Wave
exciting forces and moment at Fn = 0.2, = . . . . . . . . . . . . .
. . . 73 5.13 Ship motions at Fn = 0.2, = . . . . . . . . . . . . .
. . . . . . . . . . . . . . 73 5.14 Steady pressure distribution of
modied Wigley model (blunt) at Fn = 0.2 . . . . 74 5.15 Wave
pressure (cos component) at Fn = 0.2, /L = 0.5, = . . . . . . . . .
. 75 5.16 Wave pressure (cos component) at Fn = 0.2, /L = 1.3, = .
. . . . . . . . . 75 5.17 Unsteady pressure (cos component) at Fn =
0.2, /L = 0.5, = . . . . . . . . 76 5.18 Unsteady pressure (cos
component) at Fn = 0.2, /L = 1.3, = . . . . . . . . 76 5.19 Contour
plots of heave radiation wave at Fn = 0.2, KL = 30 . . . . . . . .
. . . . 78 5.20 Contour plots of diraction wave at Fn = 0.2, /L =
0.5, = . . . . . . . . . . 78 5.21 Contour plots of total wave at
Fn = 0.2, /L = 0.5, = . . . . . . . . . . . . . 79 5.22 Contour
plots of total wave at Fn = 0.2, /L = 1.3, = . . . . . . . . . . .
. . 79 5.23 Heave radiation waves at y/(B/2) = 1.4 for Fn = 0.2, KL
= 30, 35 . . . . . . . . 80 5.24 Diraction waves at y/(B/2) = 1.4
for Fn = 0.2, /L = 0.5, 0.7, = . . . . . . 80 5.25 Wave prole for
Wigley model at y/(B/2) = 1.4 for Fn = 0.2, /L = 0.7, = . 82 5.26
Wave prole for Wigley model at y/(B/2) = 1.4 for Fn = 0.2, /L =
0.9, = . 83 5.27 Wave prole for Wigley model at y/(B/2) = 1.4 for
Fn = 0.2, /L = 1.1, = . 83 5.28 Wave prole for Wigley model at
y/(B/2) = 1.4 for Fn = 0.2, /L = 1.4, = . 84 5.29 Added wave
resistance at Fn = 0.2, = . . . . . . . . . . . . . . . . . . . . .
. 84
12. List of Figures xii 6.1 Plans of the monohull . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.2
Perspective view of the monohull with computation grids . . . . . .
. . . . . . . 88 6.3 Computation grids (NH = 1480(74 20), NF =
3888(162 24), NF A = 297(99 3)) . 89 6.4 A snapshot of the transom
stern in the motion measurement test . . . . . . . . . 90 6.5
Steady Kelvin wave pattern at Fn = 0.5 . . . . . . . . . . . . . .
. . . . . . . . . 91 6.6 Measured steady resistance (total),
sinkage and trim . . . . . . . . . . . . . . . . 91 6.7 Added mass
and damping coecients due to forced heave motion at Fn = 0.5 . . 92
6.8 Added mass and damping coecients due to forced pitch motion at
Fn = 0.5 . . 93 6.9 Wave exciting forces and moment at Fn = 0.5, =
180degs. . . . . . . . . . . . . 94 6.10 Ship motions at Fn = 0.5,
= 180degs. . . . . . . . . . . . . . . . . . . . . . . . 94 6.11
Wave pressure on the hull at Fn = 0.5, /L = 1.1, = 180degs. . . . .
. . . . . . 96 6.12 Total unsteady pressure on the hull at Fn =
0.5, /L = 1.1, = 180degs. . . . . 96 6.13 Wave pressure on the hull
at Fn = 0.5, /L = 1.1, = 180degs. . . . . . . . . . . 97 6.14 Total
unsteady pressure on the hull at Fn = 0.5, /L = 1.1, = 180degs. . .
. . 97 6.15 Comparisons of measured and computed wave patterns . .
. . . . . . . . . . . . . 98 6.16 Comparisons of measured and
computed wave proles along y/(B/2) = 1.52 . . . 98 6.17 Added wave
resistance computed by the wave pattern analysis . . . . . . . . .
. 101 6.18 Plans of the Trimaran . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 103 6.19 Perspective view of the
model with computation grids . . . . . . . . . . . . . . . 104 6.20
Computational grids used for Trimaran . . . . . . . . . . . . . . .
. . . . . . . . 104 6.21 Steady Kelvin wave pattern at Fn = 0.5 . .
. . . . . . . . . . . . . . . . . . . . . 105 6.22 Steady wave view
at Fn = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 105 6.23 Steady pressure on hull at Fn = 0.5 . . . . . . . . . .
. . . . . . . . . . . . . . . . 106 6.24 Added mass and damping
coecients due to forced heave motion at Fn = 0.5 . . 107 6.25 Added
mass and damping coecients due to forced pitch motion at Fn = 0.5 .
. 107 6.26 Wave exciting forces and moment at Fn = 0.5, = 180degs.
. . . . . . . . . . . . 108 6.27 Diraction and Radiation Wave
Pattern . . . . . . . . . . . . . . . . . . . . . . . 109 6.28
Perspective view of the model with sinkage and trim . . . . . . . .
. . . . . . . . 109
13. List of Figures xiii 6.29 Computation grids which include
sinkage and trim . . . . . . . . . . . . . . . . . 110 6.30 Ship
motions at Fn = 0.5, = 180degs. . . . . . . . . . . . . . . . . . .
. . . . . 111 6.31 Pressure p/ g A at Fn = 0.5, /L = 1.1 = 180degs.
(with TSC, sinkage & trim eect) . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.32
Comparison of wave pressure on monohull Vs trimaran . . . . . . . .
. . . . . . . 112 6.33 Comparison of total unsteady pressure on
monohull Vs trimaran . . . . . . . . . 112
14. List of Tables5.1 Principal dimensions of the model (Modied
Wigley Hull) . . . . . . . . . . . . . 626.1 Main particulars of
monohull . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
896.2 Main particulars of trimaran . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 103 xiv
15. Chapter 1 Introduction1.1 Background of Theoretical
Estimations of SeakeepingA ship operated at sea is exposed to
forces due to waves, current and winds. These forcesnot only cause
the motions of the ship, which can be very annoying for its
passengers but alsoaccount for the resistance of the ship or drift
the ship away from its course. Resistance due towind is very low
when compared to water resistance. Estimations of resistance are
important fordeciding the power requirements of any ship. The
resistance is decomposed into the resistanceoriginated in the
viscosity and the resistance due to the wave making. Before a ship
is built, knowing the maximum data about the ships performance in
calmwater, and in waves can lead to a better ship construction.
Therefore, a scale model of theship is built and tested in a basin
with and without incoming waves. These model tests arevery
expensive, and it is time consuming. Making new computer simulation
software, which canpredict well about the ship behavior can replace
a model test. With the increase of computertechnology, it has
become possible to simulate a ships behavior in waves numerically.
Thesesimulations are based on mathematical descriptions of the
physics of the ship and the sea whichare extremely complex. To
understand the gross uid motion and corresponding interaction
withthe ship, one must understand and predict turbulence, wave
breaking, water spray, non-linearmotion, slamming, green water on
deck, sloshing, acoustics, etc. Viscous uid motion is governed by
the continuity equation and the motion equations of therigid body
must be coupled when we treat the xed-body interaction problem.
When we focuson the ocean waves and the interaction between the
ship and ocean waves, the gravity eect isdominant and the viscous
eect is negligible. Therefore, the uid can be treated as an ideal
uidand the potential theory can be applied. Potential ow solvers
are usually based on boundary 1
16. 1.1. Background of Theoretical Estimations of Seakeeping
2element methods and need only to be discretized the boundaries of
the domain, not the wholedomain. This reduces the load in grid
generation and less time computation. Potential owsolver needs
suitable boundary conditions that consist of the body boundary
condition, the freesurface boundary condition and the radiation
condition to satisfy the physical condition on thefree surface. The
well-known strip theory was the rst numerical method used as a
practical designtool for predicting ship hydrodynamic forces and
ship motions. This method solves the 2-Dow problem for each strip
of the ship and integrates the results over the ship to nd out
thehydrodynamic forces and motions. This problem was solved by U
rsell[1]for heaving motion ofa half-immersed circular cylinder.
Other extensive works were done afterward by Korvin Kroukovski[2],
T asai[3] and Chapman[4]. The rational foundations for the strip
theory wereprovided by W atanabe[5], T asai & T akagi[6] ,
Salvasan et al. [7] and Gerritsma[8]. Now alsoit is popular in this
eld because of satisfactory performance and computational
simplicity. When the forward speed increases, the eciency of the
strip theory based on the 2D theorygot reduced, due to a strong 3D
eect near the bow part and forward speed inuence from thesteady ow
to the unsteady wave eld. It has been reported by T akaki &
Iwashita[9]that theapplicable limitation of the strip theory is
around Fn = 0.4 for the typical high speed vessel.The high speed
strip theory (HSST), so called 2.5D theory, can be applied
eectively for thehigh speed vessels. The theory originated in
Chapman[10] and developed subsequently by manyresearchers, Saito
& T akagi[11], Y ueng & Kim[12][13], F altinsen[14] ,
Ohkusu & F altinsen[15]can capture the forward speed eect
within the framework of uniform ow approximation in thefree-surface
condition. The rational justication of strip theory, as a method
valid for high frequencies and moderateFroude numbers, was derived
from systematic analysis based on the slender-body theory byOgilvie
and T uck[16]. This theory was extended to the diraction problem by
F altinsen[17]and was further rened by M aruo & Sasaki[18]. The
high-frequency restriction in slender shiptheories was removed by
the unied theory framework presented by N ewman [19]. Its
extensionto the diraction problem was derived by Sclavounos[20] and
applied to the seakeeping of shipsby N ewman and Sclavounos[21] and
Sclavounos[22]. The increasing accessibility of computers of high
capacity led to the development of three-dimensional theories that
removed some of the deciencies of strip theory. The choice of
the
17. 1.1. Background of Theoretical Estimations of Seakeeping
3elementary singularities leads to the classication of these
methods into the Green functionmethod (GFM) and the Rankine panel
method (RPM). In GFM, the wave Green function isapplied only at the
ship surface and in the RPM, simple source will be applied to ship
surfaceand free surface. The 3-D Green function method has been
applied and succeded for the oating bodieswithout forward speed and
extended for the forward speed. In the Green Function method,the
unsteady wave source, which satises the radiation condition and the
linearized free surfacecondition based on the uniform ow, is chosen
as the elementary singularity. Important devel-opments for its fast
and accurate evaluation were made by Iwashita & Ohkusu[23]
based onthe single integral formulation derived by Bessho[24].
Iwashita et al.[25][26] have rigorouslyexamined the wave pressure
distribution on a blunt VLCC advancing in oblique waves by
ap-plying the Green function method and demonstrated that the strip
method practically used forthe estimation of ship motions is
insucient for this purpose. They have also showed that a signicant
discrepancy of the wave pressure between numericalresults and
experiments still remains at blunt bow part even if the
three-dimensional methodis applied. It was decided to include the
inuence of the steady eld in an unsteady wave eldin dierent
boundary condition. Then some improvements have been reported by
Iwashita &Bertram[27], where the inuence of the steady ow on
the wave pressure is taken into accountthrough the body boundary
condition. The problem is often formulated in the frequency
domain,which assumes that the body motions are strictly sinusoidal
in time. First achievements werereported by Chang[28],
Kabayashi[29], Inglis & price[30], Guevel & Bougis[31] and
they foundgood agreement with experimental data. The Ranking panel
method was proposed by Gadd[32] and Dawson[33] for the
steadyproblem and extended by N akos & sclavounos[34] to the
unsteady problem. Y asukawa [42]and Iwashita et al.[35] suspected
that alternative inuence of the steady ow through thefree surface
condition might aect more strongly the local wave pressure,
especially at the bowpart. They used for the computation a Rankine
panel method based on the double body owlinearization for steady
wave eld, and its inuence is taken into account in the unsteady
problemthrough the free surface condition and the body boundary
condition. This formulation will becalled as double body ow
formulation. It was considered that the inuence of the steady
doublebody ow through the free surface condition is important for
estimating hydrodynamic forces
18. 1.1. Background of Theoretical Estimations of Seakeeping
4and local pressures. In RPM, the radiation condition must be
satised numerically, and this numerical radiationcan be solved in
dierent method. The typical one is the nite dierence method
originated inDawson[33]. The upward dierential operator is used to
evaluate the partial derivative of thevelocity potential on the
free surface and the radiation condition which proves
non-radiatingwaves in front of a ship is satised. Sclavounos &
N akos[36] introduced the B-Spline functionto express the potential
distribution on the free surface and satised the radiation
conditionby adding a non wave condition at the leading edge of the
computation domain of the freesurface. They also applied their
method to seakeeping problem provided the high reducedfrequency (=
U we /g) > 0.25 where any disturbed wave due to a ship does not
propagateupward. The alternative method is Jensen s[37] method so
called collocation method. Thecollocation points on the free
surface are shifted just one panel upward to satisfy the
radiationcondition numerically. This method has been applied to the
seakeeping problem by Bertram[38],and the analytical proof on the
numerical radiation condition has been done by Seto[39] for the2D
problem. For the investigation of fully non-linear steady kelvin
wave eld and its inuence on theunsteady wave eld, Bertram[40]
employed a desingularized Rankine panel method proposedinitially
for steady problem by Jensen et al.[37]. The steady problem is
solved so that thefully nonlinear free surface condition is satised
and the inuence terms of the steady wave eldon the unsteady ow are
evaluated by assuming the small amplitude of the incident wave
andsmall ship motions. The boundary conditions are satised on the
steady free surface and wettedsurface of the body. Rankine source
method is a highly ecient method for seakeeping analysis. There
aremany researchers working with RPM to satisfy the radiation
condition in the frequency domainin dierent ways. T akagi[41] and Y
asukawa [42] have introduced Rayleighs viscosity numeri-cally to
satisfy the radiation condition numerically and getting the
adequate value of Rayleighsviscosity is not strait forward. Bertram
[43] and Sclavounos et al. [44] has introduced desingu-larized
panel method to satisfy the radiation condition, and this is
applicable when the reducedfrequency is > 0.25. Iwashita et
al.[45], Lin et al.[46], T akagi et al.[47] proposed a hybridmethod
named combined boundary integral equation method (CBIEM) which is
the combina-tion of Rankine Panel Method and Green Function Method.
The method makes possible to
19. 1.1. Background of Theoretical Estimations of Seakeeping
5satisfy the radiation condition accurately by introducing the
Green function method in the fareld. Three dimensional computation
method was extended to the time domain byLin et al.[50],[51] had
developed in Green function based 3D panel method satisfying the
linearfree surface condition, and the ship surface is treated
nonlinear boundary condition. The freesurface is to be re-meshed
according to ship position. This program is called as LAMP
(LargeAmplitude Motion Program), and it was made commercially
available. Later, free surface hasbeen improved with partially
nonlinear and further; there are many series as Lamp-1 to
Lamp-4.This software has been extended as a Non-linear Large
Amplitude Motions and Loads methodby Shin et al. [52] used for
oshore structure application. Using this program, many
calculationshave been carried out by W eems et al. [53], [54] for
the trimaran and wave piercer. M askew [55] has developed Rankine
source based 3D panel method where the three surfacetreated as a
fully nonlinear and the ship surface is treated as partially
nonlinear. It has beenapplied for Frigate ship with waves. This
program also considered lift eect and made availablecommercially.
Though it is the rst program based Rankine source in time domian
but there aresome disadvantages in accuracy of the calculation.
This has been applied for the high speed anddemonstrated few
results for S-60, SAWTH, S175 and Frigate with waves by M askew
[56], [57].Beck.R.F [58], [59]. has developed Rankine source based
fully nonlinear boundary condition forfree surface as well as ship
surface and this method used the desingularized panel source.
Thiscode has been applied for Wigley hull for ship motion problems.
This has been extended byScorpio et al.[60] for the container ship.
N akos et al.[61] have developed the Rankine source based time
domain program wherethe double ow is introduced, and it has been
applied for Wigley hull with linear free surfacecondition. Later,
it has been improved by Kring et al. [62] for the application of
contained shipmotion and load estimation where the free surface and
body surface have taken as nonlinear.This program gives much higher
accuracy without many panels. However, this software hasbeen made
commercially for the ship motion and load estimation with the name
called SWAN.Adegeest et al[63] has improved in the Rankine based
time domain program for the containership in unsteady waves. Bunnik
and Hermans [64] & Buunik [65]have developed code based Rankine
sourcemethod for the estimation added wave resistance and ship
motion. It used the numerical beach
20. 1.1. Background of Theoretical Estimations of Seakeeping
6to satisfy the radiation condition. However, it cannot be used for
blunt ship and high speedvessel. Rankine source based code is
developed by Colagrossi et al.[66], [67] with non linearfree
surface condition and applied for slender ship body. This method
used the desingularizedmethod for free surface condition and
numerical beach for radiation condition. It has beenapplied for
catamaran and trimaran. This method has been developed and applied
for the S-60(Cb=0.8) by Iwashita et al.[68]. T anizawa[69] and
Shirokura[70] has developed the fullynonlinear free surface and
body boundary condition seakeeping code based on Rankine
sourcemethod. This code is applied for the Wigley hull and
demonstrated high-accuracy results. Y asukawa[71], [72] has
developed nonlinear code for seakeeping problem based on
Rankinesource. For the high speed calculation SOR method has been
introduced where after the transomis modelled with a dummy grid
surface to get the smooth ow. It has been applied for the
Wigleyhull and S175 by Y asukawa[73], [74] and validated with
experimental results. Green function based time domain code is
developed by Kataoka and Iwashita [75], [76]in which the uid domain
is decomposed into near eld and far eld, and the space xed
articialsurface like side-walls of a towing tank is employed to
separate them. The Rankine method andthe Green function method are
applied to two domains respectively, and the solutions in
bothdomains are combined on the articial surface. This method is
called hybrid method. It hasbeen applied for the modied Wigley hull
and S-60 to estimate the forces and motions. It hasbeen extended
for the dierent kind of vessel by Kataoka & Iwashita [77] [81]
for the S175,S-60 (Cb=0.7, Cb=0.8) and modied Wigley hull. The
added wave resistance and pressure arecalculated and compared with
an experimental result for the validation. In practical uid motion
problem, due to the presence of turbulence suggests using
theunsteady Reynolds Averaged Navier-Stokes (RANS) equations or
Large-Eddy Simulations (LES)to model the uid motion or seakeeping
problem. In recent years, the CFD simulation toolsare applied to
seakeeping. In grid-based method, three-dimensional CFD simulations
of verticalplane motion in waves are reported by Sato et al.[82]
and W eymouth et al.[83]. A non-linear motion for a high speed
catamaran vessel, and the ship resistance in 3D was presentedby P
anahi et al.[84]. On the other hand, in particle based method,
Shibata et al.[85]&[86]computed a shipping water and pressure
onto moving ship. Oger et al.[87] SPH method appliedto a ship
motion at high Froude number. M utsuda.H. et al.[88] [90], has
validated for freesurface and impact pressure problems.
21. 1.2. Validation Methods of Theoretical Estimations 7 M
utsuda.H.etal.[91], the seakeeping performance of a blunt ship in
nonlinear waves studiedand validated. As an improvement, CFD has
been applied by Suandar.B. et al.[92]. for shingboat and high speed
ferry. Ship motions, pressure distribution on the hull and velocity
eldhave been validated with panel method and experimental results.
A practical computational method to analysis the transom stern ow
is important in thearea of marine hydrodynamics. Research on the
transom stern steady ow was conducted bySaunders[93]. He provided
advice that the speed of ventilation occurs at a FT 4.0 to
5.0(where FT is the Froude number based on the transom draft). More
recent observation onnaval combatants put the ventilation in the
range of FT 3.0 to 3.5. V anden Broeck andT uck[94] describes a
potential ow solution with series expansion in FT that is valid for
lowspeeds where the transom stagnation point travels vertically
upward on the transom as theFT increases. V anden Broeck[95] oers a
second solution applicable to the post ventilationspeeds. For a 3D
full ship problem, Subramani[96] extended the fully nonlinear
desingularizedpotential code (DELTA) to analyze a body with a
transom stern. Additionally, Doctors[97] hasstrived to suitably
model the transom hollow for use in linearized potential ow
program. ByDoctors et al.[98] non linear eect has been introduced
to analyze the transom stern vessel.Because of its advantage, most
of the high speed vessel and naval ships are built with transom.It
is also important to study hydrodynamic ow behavior at transom and
behind the ships.When the ship reaches to a sucient speed, ow
leaves at transom and the transom area isexposed to air. The
transom ow detaches smoothly from the underside of the transom,
anda depression is created on the free surface behind the transom.
This creates inuence in thepressure reduction and resistance on the
hull. For unsteady ow, many researchers are working on this to get
transom ow with dierentapproaches but there is no exact solution is
arrived yet. Considering the need and a newboundary condition has
been proposed to treat the transom part and details will be
coveredinside the thesis.1.2 Validation Methods of Theoretical
EstimationsA ship advancing in waves generate waves. The generated
waves are classied into the time-independent and time-dependent
waves. The former is the steady wave called Kelvins wave and
22. 1.3. Scope of Present Research 8corresponds to the wave
that is generated by the ship advancing in calm water surface. The
waveis independent of time when we observe it from the body-xed
coordinate system. The secondone is the unsteady wave that consists
of the radiation waves and the diraction wave. Theradiation waves
are the wave generated by the ship motions, and the diraction wave
is scattedwave of the incident wave by the hull. It is understood
that both the steady and unsteady wavesdissipate more energy, which
leads to a power loss. The resistance of a ship in wave minus
thecalm water resistance is called added wave resistance. Besides
this power loss, a transverse forcewill drift the ship from its
course, and a rotating moment about the ships vertical axis leads
toa change in its course. Therefore, it is necessary to consider
the above when designing the ship,especially its hull. The
asymptotic unsteady wave patterns which are for away from the ship
were studiedinitially by Eggers[99]. Ohkusus.M [100] proposed a
method for measuring ship generatedunsteady waves and then
evaluating the wave amplitude function and the added wave
resistance.Iwashita[101] has made systematic work, including the
analysis near the cusp (the caustics)wave pattern. That was
successful in treating with the surface elevation near the cusp for
awave system of shorter wave components with introducing Hogners
approach [102]. A 2nd orderunsteady wave pattern has been
investigated by Ohkusu.M [103]. These unsteady wave
patternsphysically show the pressure distributions over the free
surface. Therefore, the comparison ofunsteady wave distribution
between computed and measured corresponds to the comparison asfor
the pressure distributions that can be considered as the local
physical value. From thesereasons, it is very valuable to utilize
the unsteady waves in order to validate the numericalcomputation
methods more precisely. In this paper, the interaction eect between
the incident wave and steady wave, which hasnot been considered in
the measurement analysis up to now, is investigated in order to
makethe experimental unsteady waves more accurate.1.3 Scope of
Present ResearchTo the analysis of seakeeping problems, Rankine
panel method based mathematical formulationis derived which
includes the body boundary condition, the free surface boundary
condition andthe radiation condition. Computational code is
developed to handle the conventional ship hull.
23. 1.4. Organization of the Thesis 9As a basis ow, uniform ow
approximation and the Neumann-Kelvin ow approximation areconsidered
in the formulation and applied for the modied Wigley hull to study
the seakeepingqualities. Improve the measured unsteady wave taking
into account of the interaction between theincident wave and the
steady wave. A new analysis method of the measured unsteady wave
isproposed and the interaction eect related above is conrmed by
applying the method for twomodied Wigley hull models. Additionally,
the RPM code is developed in the thesis is validatedthrough the
comparisons with experimental data, that is, hydrodynamic forces,
motions, addedwave resistance and wave elds. To treat the transom
stern condition, until now there is no proper boundary condition
totreat the transom stern by panel method. A new boundary condition
is derived to treat thetransom eect mathematically in numerical
method. Therefore, the developed code is extendedfor the
application of high speed monhull to predict the transom stern eect
at the transom.The RPM is applied to the high speed ships with the
newly introduced transom stern conditionand validated through the
comparison with experiments. Transom eect is compared betweenby
numerical method taking into account of transom condition and
without transom condition.This method is also applied to a trimaran
and the seakeeping data is compared with experimentaldata.1.4
Organization of the ThesisThe general scope of the thesis is
divided into ve chapters excluding the rst chapter is
theintroduction which covers the previous research and the
conclusion as a last chapter. The thesis is outlined as
follows:Chapter 2: This chapter covers the mathematical formulation
of a boundary value problem (BVP) for seakeeping analysis. As a rst
step, to formulate BVP, appropriate bound- ary conditions need to
be derived for frequency-domain formulation. Derivation of body
boundary condition, free surface boundary condition and radiation
condition derivation are given. A new boundary condition to deal
with transom stern is developed and the details also covered. The
nal section is the formulation of the boundary value problem
to
24. 1.4. Organization of the Thesis 10 solve the potential
equation in the panel method. And also the formulation of
estimating the hydrodynamic forces, exciting forces and moments,
motions and waves are derived.Chapter 3: In panel method, it is
very critical to select the suitable formulation to solve the
problem numerically. To calculate the potential in RPM, there are
two methods available i.e., (i) direct method and (ii) indirect
method. Each method has been discussed. As a next step, it is very
important to select the appropriate method to satisfy the radiation
condition either panel shift method or spline interpolation method.
Therefore, attempt has been made to plot wave pattern by both the
method and compared with an analytical wave pattern. This chapter
also covers explanation about the treatment of the transom boundary
conditions in panel method.Chapter 4: In Engineering eld, any
numerical solution must be veried and compared with experimental
results. Therefore, required tests are identied based on
requirements. Mea- suring method for forces, motions and unsteady
wave method is discussed towards im- provement in capturing the
waves around the hull. Formulation is derived to capture
interaction eect of incident wave and steady wave in an unsteady
waves.Chapter 5: To investigate the wave interaction eect, modied
Wigley hulls have been numer- ically analyzed and the results are
compared with experimental results. Hydrodynamics forces, exciting
forces and moments are compared with experimental results. Ship
motions and added wave resistance are also compared with
experimental results. This chapter is to deal with the validation
of incident wave and steady wave interaction eect with the unsteady
waves.Chapter 6: High speed monohull has been taken for the
analysis. Radiation forces and ex- citing forces are calculated
with transom and without transom stern condition to see the eect of
a newly introduced condition. All the results are compared with
experimental results. Unsteady waves and pressure plots are also
validated with an experimental data. Ship motion and added wave
resistance are also compared with experimental results. It has been
extended for the trimaran application and numerical results are
compared with experimental data. This chapter is to deal with
transom stern boundary condition with an experimental
validation.Chapter 7: In this chapter, the thesis is concluded
making clear the obtained results.
25. Chapter 2 Mathematical Formulation of Seakeeping2.1
IntroductionDue to the development in computer technology and
advanced applied mathematics, engineeringproblems are solved by a
numerical method, and being validated by experimental results.
Inmarine hydrodynamics, ship/oshore structures problems are
formulated as a boundary valueproblem where the physical parameters
are to be dened well in the form of mathematicalexpression to solve
the problem numerically. It is very important that boundary
conditions ofreal physics are to be expressed mathematically, which
contribute to the prediction accuracy ofthe expected results. In
this research, boundary value problems are solved by boundary
element method (BEM)also referred as a panel method or boundary
integral equation methods. BEM is more suitablefor the marine
structure hydrodynamics problem which takes the surface of the uid
domain, andit is the best numerical method when compare to other
methods like nite dierence method,nite-element method, nite volume
method, etc. in terms of computational time because itsolves for
entire domain, which is not required for practical needs. Consider
a ship which is oating on water and the water surface which is in
turn touchwith the air is called a free surface. The most exact
description of the ow of water is given bythe Navier-Stokes
equations, which take into account of the water viscosity.
Viscosity in shiphydrodynamics can be important in turbulent areas
like, for example, near a rudder, propulsionor sharp edges of hull,
but none of these is considered in this research. Near the hull, a
smallboundary layer exists in which viscous eects dominate, but
this layer does not really aect thelarge-scale interactions of
ocean waves and ship motions. Assuming that the ow is irrotational
and incompressible, the ow can be described using 11
26. 2.2. Body Boundary Condition 12potential theory. By dening
the uid velocity by a scalar potential, the velocity eld of theow
can be expressed as the gradient of a scalar function, namely the
velocity potential, u(x, t) = (x, t) (2.1). The continuity equation
or conservation of mass reduces to Laplaces equation because theuid
is incompressible, which states that the divergence of the velocity
eld is equal to zero. V =0 (2.2)and the velocity potential must be
a harmonic function which satises the Laplace equation. 2 2 2 2 = +
+ =0 (2.3) x2 y 2 z 2The conservation of momentum equation can be
reduced to Bernoulis equation. 1 1 p pa 1 2 z= + + U (2.4) g t 2
2where is the uid density, g is the gravitational constant and pa
is the atmospheric pressure.The total velocity potential can be
decomposed into steady velocity potential and unsteadyvelocity
potential. (x, y, z; t) = s (x, y, z) + t (x, y, z; t) (2.5)For a
real ship in a seaway, the uid domain is eectively unbounded
relative to the scale of theship. For the computational purpose,
the uid domain must be truncated. On the boundariesof the truncated
uid domain, the hull of the ship SH , the free surface SF , the
bottom of thewater SB and the control surface (truncation surface)
SC are covered. The total surface canbe written as S = SH + SF + SB
+ SC . In addition to the above boundary condition, it isnecessary
to satisfy the radiation mathematically to get unique results. The
control surface andthe bottom surface need not be considered for
analysis because of the simple source in RPMmethod. We discuss on
each boundary in details on further section.2.2 Body Boundary
ConditionThe boundary condition on the hull should take into
account the interaction between themotion of the hull and the
motion of the water at the hull. Just like the water, thehull of
the ship cannot be crossed by a uid particle. The water should
therefore havethe same normal velocity as the ships hull and the
water does not penetrate the hull.
27. 2.2. Body Boundary Condition 13 it Z r = r + e Z = V SH n
on SH (2.6) n YWhere the normal vector n is from hull to- Ywards
the uid domain. This condition must rbe simplied to the mean wetted
surface from ran instantaneous surface. Let us consider a it SH
ethree-dimensional object in a uid with a free SHsurface. The
object, for instance, a ship, sail Instantaneous Wetted Surface
SHthrough an incident wave eld with a velocity SH Mean Wetted
SurfaceU (t) in the negative x direction; this is equiv- Figure
2.1: Body boundary bondition - coordinatealent to an object with
zero speed in current systemU (t) in positive x direction. The
object is free to translate in threedirections and to rotate around
the three axes. Therefore, six motions are taken to representthe
body boundary condition. To derive the mathematical equation of the
moving body, thereare two coordinate systems are followed. The rst
coordinate is the space coordinate systemwhich can be represented
as, r = (x, y, z) (2.7)The second coordinate system is body xed
coordinate system is represented as r = (x, y, z) (2.8)Bring the
relation between teh Space and body xed coordinate system, the
equation shall bewritten as r = r eit (2.9) Where is a displacement
vector of a point on the body and is a circular frequency
ofoscillations. = j ; (j=1,2..6 surge, sway, heave, roll, pitch,
yaw). = i1 + j2 + k3 + (i4 +j5 + k6 ) r. The r coordinates are xed
with respect to a body which is dened by theequation F (x, y, z) =
0, then the potential ow kinematic boundary condition to be satised
on DF (r)the surface is 0 = . Dt
28. 2.2. Body Boundary Condition 14 The velocity potential is
decomposed into a steady mean potential and unsteady perturba-tion
potential. The Velocity of the uid is represented by the vector = V
(r) + eit (r)and V is the steady ow eld due to forward motion of
the body(negative x direction ) i.e.,V (r) = U (x + ) and (r) is
the potential of the oscillating velocity vector. The
coordinatesystem chosen such that undisturbed free surface
coincides with the plane z=0 and the centreof gravity of the object
is on the z axis, with z pointing upwards. In this formulation, SH
is themean wetted surface and S H is the instantaneous wetted
surface, Fig. 2.1The boundary condition on the body is DF (r) 0= =
+ . F (r) (2.10) Dt t DF (r) F (r) 0 = = + V (r) + eit (r) . F (r)
on S H (2.11) Dt tThe above equation can be written as F x F y F z
0 = + + + V (r) + eit (r) . x t y t z t F x F y F z F x F y F z i +
+ +j + + x x y x z x x y y y z y F x F y F z +k + + on S H (2.12) x
z y z z zIn the above equation, equation can be simplied using the
below relation as it . F (r) jeit . F (r) F (r) = F (r) ie x y ke
it . F (r) z (r ) = i x + j y + k z Now the equation can be
simplied as F x F y F z 0 = + + + (V + eit ). x t y t z t F (r)
ieit . F (r) jeit . F (r) x y keit . F (r) on S H (2.13) z
29. 2.2. Body Boundary Condition 15Now the equatin must be
simplied. Let us see each term individually F x F y F z First Term
: + + = ieit . F (r) (2.14) x t y t z tNow the equation can be
rewritten as 0 = ieit . F (r) + (V + eit ). F (r) ieit . F (r) jeit
. F (r) x y keit . F (r) on S H (2.15) z The above formulation is
satised on the instantaneous wetted surface (Exact body surfaceF (x
, y, z , t). This can be linearized under the assumption that
unsteady displacement amplitudesmall. So Taylors expansion can be
applied for both steady and unsteady velocity eld on themean wetted
surface. Taylors expansion for V and V (r) = V (r)mean + (. )V (r)
eit + 0(2 ) (2.16) mean = (r)mean + (. ) (r) eit + 0(2 ) (2.17)
meanSubstituting the above formation in boundary condition ( 2.15
), we can obtain the equationon mean wetted surface 0 = ieit . F
(r) + V (r) + eit + eit (. )V (r) . F (r) i eit . F (r) j eit . F
(r) x y k eit . F (r) on SH (2.18) z Last Term : F (r) i eit . F
(r) + j eit . F (r) x y +k eit . F (r) (2.19) zBoundary Condition
by Steady State Term for: V V (r). F (r) = 0; or V . F = 0
30. 2.2. Body Boundary Condition 16Boundary Condition by
Oscillatory Function for: eit . F (r) = ieit . F (r) eit (. )V . F
r + (V )mean . mean i eit . F (r) + j eit . F (r) x y + k eit . F
(r) on SH (2.20) zAfter removing the time part out of this
equation, the equation becomes . F (r) = i (. F (r)) (. )V . F (r)
+ (V )mean . mean i . F (r) + j . F (r) x y +k . F (r) on SH (2.21)
z or . F (r) = i . F (r) (V . ) (. )V . F (r) on SH (2.22) meanAll
the terms are small, of the same order as or . Thus to this order
of approximation itis no longer necessary to distinguish between
the actual position and of the body and its meanposition, or
between the co-ordinates r and r F = {i F ( )V F + (V ) } F
(2.23)Vector Identity: (V . )A (A. )V = (A V ) A .V + V .A (2.24)
.V = 0 for Incompressibility;V. F =0 from the Steady Potential
Condition;so the unsteady condition shall be written from the
equation ( 2.23 ) as . F = i. F + ( V ) . F on SH (2.25) since: F
vector normal to the body surface and n. = nBoundary Condition for
on the Body is : = i + ( V ) .n on SH (2.26) n
31. 2.2. Body Boundary Condition 17(i) Zero Speed CaseThe
derived body boundary condition shall be written as = i + ( V ) .n
on SH (2.27) nThe forward velocity is considered zero. So the
second term will be neglected from equation (2.27 ). So the
boundary condition shall be, = i.n on SH (2.28) nSubstituting the
in the above equation = i i1 + j2 + k3 + (i4 + j5 + k6 ) r .n on SH
(2.29) nVector relations and the normal vectors are written as . .
( r) n = (n r) n = n1 , n 2 , n 3 (2.30) nr =n , n , n 4 5
6Applying the above relation in equation ( 2.29 ), the equation
shall be written as 3 6 n = i j nj + i . j (n r)j on SH (2.31) j=1
j=4To consider the six degree of motion, boundary condition shall
be written as 6 = i j n j on SH (2.32) n j=1(ii) With Forward
Speeda) Translatory Motion (j = 1 3): = 1 i + 2 j + 3 k ANow the
derived boundary condition shall be applied for the translatory
motion as = i + ( V ) .n on SH (2.33) nLet us rewrite the vector
identity and the motions are considered only three direction. (A V
) = (V )A + V ( V )A V ( A) (A )V (2.34)
32. 2.2. Body Boundary Condition 18Considering the translatory
motion, the below mentioned relation can be used A=0 V =0 Now in
the vector identity, applying the above relations, the last term n
(A ) V can bemodied using given equation as vi vj = = (1 i, j 3)
(2.35) xj xj xi xi n (A ) V = A (n )V (2.36)Now applying above
relation in the body boundary condition, the nal equation for
translatorymotion with forward speed shall be written as = i F+ ( V
) n n = i n (n )V A = in (n )V A (2.37)b) Rotational Motion (j = 4
6): = 4 i + 5 j + 6 k r B rBody boundary condition shall be written
as (A V ) = (V )A + V ( V )A V ( A) = (A )V (2.38)Let us apply only
the rotation motion in the condition as in (B r) = iB (n r) = iB (r
n) (2.39)The above equation can be applied to the vector identity
asn {(B r) V } = n {(V ) (B r) + (B r) ( V)V [ (B r)] [(B r) } ]
V(2.40)Using the below mentioned mathematical relation u v w (B r)
+ + =0 x y z (2.41) V + + (B r) = 0 x y zThe vector identity term
can be simplied as n (V ) (B r) = n (B V ) = B (n V ) (2.42) n (B
r) V = (B r) (n ) V = (n ) r B = r (n ) B
33. 2.3. Free Surface Boundary Condition 19Applying to the
boundary condition equation ( 2.26 ) as = i + ( V ) n n = i (r n)
(n ) r (n )V B (2.43)To simplify the above equation, tensor
relation can be used vj xi + vj = (xi vj ) xi xi vk (xj vk ) = xj
(1 i, j, k 3) (2.44) xi xiVector idetity which can be used (n ) + r
(n ) V = (n ) (r V ) (2.45)Applying the above mentioned relations,
the nal equation for body boundary condition shallbe written as j =
(inj + U mj ) j (j = 1 6) (2.46) nThe full form of m and n vectors
are written as n = n1 i + n2 j + n3 k r n = n4 i + n5 j + n6 k (n
)V = m1 i + m2 j + m3 k (n ) (r V ) = m4 i + m5 j + m6 k (2.47)2.3
Free Surface Boundary ConditionIn the free surface, two boundary
conditions are to be satised. The rst one is (a)
kinematic-freesurface boundary condition, and the second one is (b)
dynamic free surface boundary condition.The kinematic-free surface
boundary condition states that the normal velocity of the uid
surface& air surface must be equal, and the water surface does
not penetrate the air surface. The uidparticle on the uid surface
remains at the free surface. The exact free surface shall be
writtenas z = (x, y; t)
34. 2.3. Free Surface Boundary Condition 20 Equation (2.5) is
the total velocity potential. Now this can decomposed into steady
waveand unsteady wave eld as (x, y, z, t) = U S (x, y, z) + (x, y,
z) eie tIn the abve equation, steady ow can be further expanded as
(x, y, z, t) = U [ (x, y, z) + (x, y, z)] + (x, y, z) eie t
(2.48)Where = x + D The unsteady velocity potential equation shall
be written as, 6 gA = (0 + 7 ) + ie j j (2.49) 0 j=1where 0 =
ieKziK(x cos +y sin ) means the double body ow, the steady wave eld
and the unsteady wave eld whichconsists of the incident wave
velocity potential, the radiation potentials j (j = 1 6) andthe
scattering potential 7 that represents the disturbance of the
incident waves by the xedship. The radiation potentials represent
the velocity potentials of a rigid body motion with unitamplitude,
in the absence of the incident waves. The total velocity potential
must be appliedover the uid domian and potential equation has been
derived based Greens theorem as follows, j (Q) j (P ) = j (Q) G(P,
Q)dS (2.50) SH n n 1where G(P, Q) = and r = (x x)2 + (y y)2 + (z
z)2 , P (x, y, z) is the eld point 4rand Q(x, y, z) is the source
point. Now let us derive the free surface boundary condition.
Kinematic Free surface Boundary Condition: The substantial
derivative D/Dt of a functionexpress that the rate of change with
time of the function, if we follow a uid particle in the
freesurface. This can be applied in our exact free surface as, D (z
) = + z (x, y; t) = 0 on z = (x, y; t) (2.51) Dt tD/Dt means the
substantial derivative and is dened as a two-dimensional Laplacian
withrespect to x and y on the free surface. z = (x, y; t) is the
wave elevation around a ship and
35. 2.3. Free Surface Boundary Condition 21considered to be
expressed by the summation of the steady wave s and the unsteady
wave tas follows (x, y; t) = s (x, y) + t (x, y; t) (2.52)Dynamic
Free surface Boundary Condition: The dynamics-free surface boundary
condition isthat the pressure on the water surface is equal to the
constant atmospheric pressure pa and thiscan be obtained from
Bernoullis equation. 1 p pa 1 2 t + + gz + = + U (2.53) 2 2where U
is the forward speed of the vessel. Now, shifting the pressure to
one side, the equationshall be written as 1 1 p pa = t + U 2 + gz
(2.54) 2 2This pressure equation is applied on the exact free
surface z = (x, y; t). Equation shall berewritten considering the
uid pressure is equal to atmospheric pressure as, 1 1 pa pa = + U 2
+ g (x, y; t) (2.55) t 2 2Keeping the free surface elevation in one
side and the free surface is written as 1 1 1 (x, y; t) = + U2 on z
= (x, y; t) (2.56) g t 2 2 Now substituting the dynamic free
surface boundary equation ( 2.56 )in the kinematic-freesurface
boundary equation eq.(2.51), the free surface boundary condition is
written as, 1 1 1 0= + z+ t + U2 t g 2 2 1 1 = z+ tt + t + t + ( )
g 2 1 1 = z + tt + 2 t + ( ) on z = (x, y; t) (2.57) g 2Now the
exact free surface boundary condition shall be written as 1 tt + 2
t + ( ) + gz = 0 on z = (x, y; t) (2.58) 2 Total velocity potential
equation (2.48) shall be substituted in the free surface
bounndarycondition eq.(2.58) as tt + 2 ( + + ) t 1 + ( + + ) ( + +
) ( + + ) 2 + g (z + z + z ) = 0 (2.59)
36. 2.3. Free Surface Boundary Condition 22 tt + 2 t + ( + ) 1
+ ( ) ( + ) + g (z + z ) 2 1 = gz ( ) on z = (x, y; t) (2.60)
2Similarly, total velocity potential shall be applied to the
dynamic free surface as 1 1 1 (x, y; t) = t + U2 on z = (x, y; t)
(2.61) g 2 2 1 1 1 (x, y; t) = t + ( + + ) ( + + ) U 2 on z = (x,
y; t) (2.62) g 2 2 1 1 1 (x, y; t) = U2 ( ) (t + ) on z = (x, y; t)
(2.63) 2g g gIn the above wave equation, the rst part of equation
is the wave which is the function of onlyforward speed. The double
body potential satises the Miller condition on z = 0, which do
notgenerate waves and the double body ow is satised on the free
surface i.e, /z = 0 on z = 0.Considering the above condition, the
forward speed wave shall be written as, 1 (x, y) = U 2 on z = (x,
y) (2.64) 2gNow the free surface equation eq.(2.63) shall be
applied to the forward speed wave z = byapplying Tailor series 1 1
(x, y; t) = ( ) (t + ) g g 1 + on z = (x, y) (2.65) z 2g ( ) + (t +
) = on z = (x, y) (2.66) g + zSimilarly the free surface equation
equation (2.60) shall be applied on the dou
![Untitled-1 [] · Fast patrol Vessel designed with hull, raked Stem, Chine bilge, transom Stern and built as a medium range weapon fitted and with water jet propulsion System, driven](https://img.dokumen.tips/doc/110x75/5e84adf8c83e331fa6356abd/untitled-1-fast-patrol-vessel-designed-with-hull-raked-stem-chine-bilge-transom.jpg)
