19 th WEGEMT School
Numerical Simulation of Hydrodynamics:
Ships and Offshore Structures
Offshore Structures
Hydrodynamic problems of offshore structures
Mr. BERHAULT Christian
PRINCIPIA R.D.
ECOLE
Centrale
Nantes
NANTES 20-24 September 1993
P1993-14
Part 2
'KC:MISCUE UNIVERSITET Laboratorium voor Schespohydromecharka Archlef Mekelweg 2, 2628 CD Doff t Tel.: 015 786373 Fa= 015 7818311HYDRODYNAMIC PROBLEMS
OF OFFSHORE STRUCTURES
C. BERHAULT
19th WEGEMT School
1
Introduction
The design of the offshore production systems for the oil industry requiresan
important knowledge on the forces and responses induced by waves and cur-rent actions. The prediction of theses forces is a complex problem because of:
the environmental field conditions (water depth, waves, wind and current characteristics) are quite different according to the geographic areas (North Sea, Gulf of Mexico, West Africa Coast, ...)
a large variety of systems are designed : fixed structures (ranging from
steel jacket to concrete large volume structure), floating structures (ranging from submersible platform and TLP to moored tankers or barges), rigid or flexible risers and flowlines, mooring systems (Fig.1).
the action of the environment
on the structures has to be taken into
ac-count at any stage of their life (construction, transportation and/or towing, installation, operational and survival conditions Fig.2).the induced hydrodynamic effects are of different natures : wave diffraction around the structures, viscous effects (friction, vortex-shedding).
each system has its own natural frequencies; the wave and current actions
induce dynamic effects, resonant behaviour may occur : roll motion of ships,
low frequency response of moored structures, springing / ringing effects on
TLP or fixed "monotower"
In fact hydrodynamic analyses are necessary at any stage of the projects. In 1
that purpose, the numerical models are used as basic tools by the offshore industry. Now one may calculate in a standard way
waves and current forces on the different parts of system. The hydrody-namic tools are generally interfaced with other numerical models to perform the structural analysis (Fig.3).
dynamic responses to environmental forces (wave frequencies, natural
fre-quencies) : motions, mooring lines tensions, flowlines behaviour, acceleration on deck, air-gap prediction (Fig.4).
Recently interest in the oil and gaz production in deep water depth (gulf
of Mexique, west Africa, Brazil, north) have reinforced the hydrodynamic research activities, in Europe particularly.
The present lecture notes are composed of two parts. A description of the typical offshore structures from a hydrodynamic aspects is done in a fist part. Then the hydrodynamic problems to be solved and the associated numerical tools generally used are rewiewed in the second part. In the conclusion the main research subjects in progress are briefly presented.
FE
MEIN
X aminmi iskokei
Figure 1-b (from Faltinsen
(ThlThrTh
vsmovemooripm...rlomm....
Figure 3 (from Kvaerner)
Figure 4 (from Faltinsen)
5
'Nater on deck
2
Offshore structures characteristics
From a hydrodynamic point of the view, the offshore systems can be classified in five categories. For each of them, hydrodynamic problems are described. The following table gives the natural periods for different structures and as-sociated degrees of freedom (Fig.5).
2.1
Fixed structures with small members
One includes in this type of structures : jacket, jack-up and compliant towers for deep water depth (Fig.6). As they are composed of slender cylinders of small diameters, the incident waves field is not modified in its vicinity. The main forces are induced by inertia and drag effects. The difficulty lies in the
6 Structure Surge/sway bending Heave roll/pitch torsion Jacket small water < lsec. < lsec. Compliant tower deep water > 20 sec.
-Gravity base depth 150 m < lsec. < lsec. Gravity base in towage-
'.-:_-40 sec. 60 sec.Monotower gravity base depth 350 m
2 to 5 sec.
-Semisubmersible
drilling
:-_-150 sec. 18 to 25 sec. - -'30 sec.
Semisubmersible production / concrete
- -150 sec. L-_40 sec. '- -.60 sec.
T.L.P. :-_-150 sec. 3 to 5 sec. 3 to 5 sec.
Tanker SPM > 200 sec. _-L-'10 sec. --'.10 sec.
Tanker on turret 100 to 150 sec. :-._10 sec. 10 sec.
Mooring barge multi-lines
estimation of the drag coefficient. The reasons are the following
forces are induced by complex oscillatory flows (irregular waves, superim-position of currents) generally at high K, number. An additional problem is the number of elements and their proximity inducing high interaction effects (Fig.7).
inertia and drag coefficients are generally obtained from model tests and the extrapolation from model scale to real scale is quite difficult when flow is governed by viscous effects. A recent alternative is to use numerical meth-ods solving the Navier-Stokes equation (or its different approximations) for unsteady flows (Fig.8).
the waves field is modelled from the Stoke's wave solutions (1st order to 5th order). Hence the velocity is not defined in the wave crest. Its estimation requires extrapolation or stretching methods (Fig.9).
vibration from vortex-shedding may occur. This depends on the Strouhal number in connection to the cross sectional shape of the cylinders and to the Reynolds number (Fig.10).
A particularity of the compliant towers is their low natural frequency in
flexion compared to classical jackets. So the waves forces may induce large
horizontal motions at the deck level. An additional large box is generally
fixed far from the free surface level In order to reduce these motions (Fig.6). The resulting increase of added mass requires the first natural frequency to
be smaller than wave frequencies. As one obtains low frequency motion, the complexity of velocity field around the bracings is increased and the estima-tion of drag coefficient becomes more difficult. The box can be composed of porous elements to induce damping from viscous energy dissipation.
Critical phases of a jacket life is its transportation and installation (Fig.11). These operations require the use of barges or other floating systems. This implies new hydrodynamic problems
the transportation may lost few days or even few weeks in case of transocea-nis trip.
Severe weather conditions can be encountered and the dynamic
behaviour of the barge may influence strongly the structural behaviour of
the transported structure : roll and pitch motions, longitudinal deformation, "fish-tailing" effects, slamming, deck-wetting.
the installation procedures include critical configurations with permanent waves and current.
In the design stage, these configurations have to be
modelled by taking into account the coupling behaviour of both the floating system and the transported structure.
MAJOR MOTIONS EXPERIENCLO BY FLOATING I'l A'l FORMS
Figure 6 (from Aker and TPG)
Ft- 1 ./ 9 W A TER DEF'l H 1 Crn. 1)4/A\\P-SSC
3
P-558
CL-7)1Coupled
svtorreteix
Figure 7 (from Blevins)
7/0 W-5G
II
w-T2 30---)--
-. Biased gap (e ) flow (bistobte) C=C2---Single w-T0(124-2)(d) vortex street2)I
\
(0) 3 Bistoble 4 5 6 L/0 ( b) ( c )I-awill amollimr. ammillImme...0
2
( d 1
( f 1
I el ( f
Two vortex streets One vortex street
Interference flow regimes for side-by-side, tandem and staggered arrangements of two pipes For
Indem regimes: (a) single slender body; (b) alternate reattachment: (c) quasi-steady reattachment: (d)
inter-mittent shedding: (c) discontinuous jump (d) (f). (1) hinary vortex street
(
W - SD
e)
Figure 8 (from Scolan)
21.0
10 w- TI
C CM)
0.47
0,4
03
02
0.1
Figure 9
Extrapolation in the wave crest
1 1 I 103 11 LINEAR EX TR A POL AT ION DELTASTRETCH W/ DELTAOEPTH
Figure 10 (from Blevins)
1 SMOOTH SURFACE / 1 ROUGH SURFACE 1 1 II
itl
It]
REYNOLDS NUMBER (110/0-Strouhal numberReynolds number relationship for circular cylinders
(Lienhard, 1966; Achenbach and Heinecke, 1981). S 0.21 (I-21/Re) for 40 <Re <200
(Roshko, 1955).
Figure 11
4n-;
Rgure 6 Installation of Steel CompUant Tower
2.2
Gravity base structures
Typical fixed concrete structures are composed of a large volume base which
is designed in order to counterbalance the hydrostatic pressure. The base
supports columns of large diameters (10 to 20 meters). The deck is placed above the columns (Fig.12).
The dimensions of the structure can induce large deformation of the wave fiels around the columns. This is of importance in the estimation of the deck level (air-gap prediction). The wave surelevation is increased for long wave period if the base is not to far from the free surface. Non-linear effects along the columns are also expected such as breaking waves on the columns, inter-action effects between columns, run-up.
The wave forces estimation must be taken into account the diffraction effects. For specific structures, such as "monotower", the first bending mode can be excited by second order or non-linear effects (springing / ringing). Slamming and impact forces may also occur in extreme wave conditions. However the columns diameter are not large enough to neglect drag effects.
A particular situation is the towing condition of the structure as it is still
in flotation configuration.
Due to the small waterplane area, the natural
frequencies are lower than wave frequencies. The resulting motions in heave, pitch and roll is of low frequency nature. It should be noted that the forward speed has to be taken into account in order to estimate the wave diffraction loads and the drift forces (added resistance in waves). Another problem is the vortex-shedding which can induce lateral motions during the towing if the forward speed (and the current) corresponds to the Strouhal number as-sociated to the cross sectional shape of the tower.
During the installation, the ballast operation modifies the natural frequen-cies. A critical point is when the structure base is at the immediate vicinity of the sea bed.
2.3
Semisubmersibles and TLPs
Semisubmersible platforms and Tension Leg Platforms are composed of a group of columns of large diameter (typically 20 to 30 meters). These columns are mounted on horizontal pontoons. The cross section of the pontoons are
generally cicular or retangular with sharp or rounded edges. The draft of
those structures varies form 25 m (drilling semisubmersibles) to 150 m (deep draft floater) (Fig.13). Generally concrete concepts have greater displace-ment than steel concepts. Semisubmersible concepts and TLP differ by their mooring system
catenary lines which restrain the horizontal offset (semisubmersible) rigid tethers which counterbalance the positive hydrostatic forces (TLP)
Figure 13
Deep draft floater and TLP
15 Pontoon Well Template Column Tubular Steel Tethers FoundatIons
The design of such structures depends on the following hydrodynamic crite-ria : small heave motion (Fig.14), limited horizontal offset (if rigid risers are used), minimum relative wave elevation under the deck, minimum tension in the mooring lines, safety insured in survival and extreme conditions.
The following table gives the typical response of their structures. Towing and installation configurations are included, due to the fact that the behaviour can be significantly modified. The contribution ratios of the different effects given in the table have to be taken with care. They depend of the mooring system, the weather conditions and so on ...
The main conclusions are the following
all these structures are concerned with low frequency motions for horizontal and vertical modes. The main difficulties ly in an accurate prediction of the vertical low frequency wave forces and of the associated damping : wave drift
16
Structure Surge / sway / yaw Heave Roll / pitch
Small semi.
15.000 t
wave freq. : 80%
+ mean offset : 10%
+ low freq. : 10%
wave freq. : 100% wave freq. : 90%
+ mean offset : 5% + low freq. : 5% Large semi. 200.000 t wave freq. : 30% + mean offset : 40% + low freq. : 30% wave freq. : 45% + mean offset : 5% + low freq. : 50% wave freq. : 30% + mean offset : 10% + low freq. : 60%
Deep draft floater 200.000 t wave freq. : 20% + mean offset : 40% + low freq. : 40% wave freq. : 20% + mean offset : 5% + low freq. : 70% wave frequency : 10 + mean offset : 10% + low freq. : 80% T.L.P. 150.000 t wave freq. : 30% + mean offset : 40% + low freq. : 30% wave freq. : 70% + high freq. : 30% wave freq. : 70% + mean offset : 10% + high freq. : 30% T.L.P. towage wave freq. : 30% + mean offset : 40% + low freq. : 30% wave freq. : 45% + mean offset : 5% + low freq. : 50% wave frequency : 30 + mean offset : 10% + low freq. : 60%
damping, viscous and drag damping from columns and pontoons, mooring lines and tether damping, drag on risers.
for TLP the high frequency resonance effect has to be analyzed since it may increase the tethers tension up to 40% of the total tension in extreme wave conditions. Its contribution to fatigue analysis of tethers is also significant,
(Fig.15). The difficulty lies in the estimation of the non-linear wave forces inducing the ringing effects. The corresponding damping, coming from hy-drodynamic, soil, structural behaviour of tethers and hull, is alsodifficult to predict.
Figure 14
Typical response of semisubmersible
10 12 14 16 18 20 22 24 26 I I Perlode (s) T. To 17 0.01 -5 o a, a, 0.005 02 0 0.6 0,8 1.0 Pulsahon w (rd sl wo pulsation propre wu plc du spectre de houle GM 2.1m H, 11.6m T 16.1s 2,7 T calcul ezpenence Sernisubrnersible I .1 Tr T. penode Propre penode d eclud'brage 1 I I
/
/ \ , / I //
---_....,.... I // . ,..)if'
P./LT .--,,,42./.._ 1,2 1.0 08 0,6 0,4 0.2 O20000 10000 o i0000 20000 30000 8500 86000 84000 82000 80000 78000 76000 8500
Figure 15
Tether response of a TLP (from N.C.)
i40000 120000 100000 80000 60000 40000 20000 8500
A-\A
60000 40000 20000 O .s4. 20000 40000 60000 8500 8600TOTAL RESPONSE
8600 8700 8800 8900 9000HIGH FREQUENCY
RESPONSE
,isy ,,ss\ms,
8700
18
8800 Time (Sec.)
WAVE FREQUENCY
RESPONSE
8600 8700 8800
Time ("`.)
LOW FREQUENCY
RESPONSE
8600 8700 8800 rime (sec.) 8900 9000 IIMALMEWEIMENif
Krirarfisummpli 1I
01,--winowneram
raawayifflimur 11111,am=
\I
Nsy 8900 9000 TOT HF LF 8900 90002.4
Tankers and barges
Moored tankers are widely used in the offshore industry as : early production
system, storage unit, shuttle for oil transfer. They are generally moored at a single point system (calm buoy, yoke, turret) but they can be also moored with a multi catenary lines system (Fig.16). Production is insured by using flexible flowlines. Recently specific hull ships have been designed according to large barge concepts. (200 m x 40 m). The draft varies from 5 m to 15 m. A particular configuration is the oil transfer phase from the storage tanker to the shuttle; the latter moored in tandem or side by side.
The hydrodynamic behaviour of moored ships is gouverned by two non-linear components
a wave frequency resonance for the roll, pitch and heave motions. The roll motion is the more critical (light damping).
a mean offset and a low frequency resonance for the horizontal motions induced combined actions of waves and wind.
Most of the tension in the mooring lines is given by the low frequency compo-nent. A correct prediction of extreme values imposes an accurated estimation of damping from wave drift, mooring lines, friction and drag on the hull.
Figure 16 - Moored tankers
SBM Converted Tanker Floating Production Storage
artd Offloading (FPSO)
19
f
et,a:A !alooLW.'"'" -F- loating Storage & Offtake Vessel2.5
Flowlines and mooring lines
The main forces on flowlines and mooring lines come from viscous and drag effects. The combination of waves and current induced dynamic response. The hydrodynamic problems are in
the complexity of the flow : irregular waves, current, wave frequency and
low frequency motions of the floating support.
the large motion amplitude of the flexible risers or the mooring lines
(Fig.16). Resonance can be induced if excitation frequencies corresponds
to natural frequencies of the structure.
the interaction effects in a multi slender bodies configuration.
Figure 17
Risers and mooring lines configurations
Single Point Loactzng Buoy Rowline Lary Wave Direct:1cm of Surface Waves 20 Mooring
'/
Lines Floating Prod urn on tein QC/DC Vortes Shedding Production Ruer Fr- Hanging Catenary 'Lser LTV Protection\
W.:mg \ Lines Current Density Wind Current R.Sser Banait
..
Flow Uricararagrane-m-.. ...La.ry
3
Hydrodynamic problems and numerical tools
For the last decades, numerical models have been developed to predict the dynamic behaviour of the offshore systems. These models are mainly con-cerned with the linear and non-linear wave diffraction loads, the viscous / drag effects and the linear and non-linear dynamic response of the structures.
3.1
Wave diffraction effects
Wave diffraction (and radiation for floating structures) are significant for
structures with large dimensions. The potential theDry is generaly used,
as-suming a perfect fluid and an nonrotationnal flow. The Laplace equation
associated with boundary conditions on the hull, on the bottom, on the free surface and at infinity is solved in analytical or numerical ways (Ref.1). Free surface condition and hull condition are non-linear and the free surface elevation is unknown. The problem may- be solved directly using time domain
approaches. However numerical difficulties exist and the time consuming may
be very large. Based on perturbation series and Taylor's expansions of the main variables (velocity potential, free surface elevation, ...), different level of approximations have been developed. The wave steepness E is used as the
perturbation parameter. From these approximations the velocity potential is calculated in the frequency domain by using Boundary Integral Equation Methods (Rankine or Kelvin sources distribution) or Finite Element Method.
The first order approximation is called the "linear theory". The incident
potential is given by the Airy wave formulation. The diffraction velocities and pressures vary linearly with the incident wave amplitude and take place at the wave frequency. Hence only the wave frequency response can be
pre-dicted by using this approximation. Forward speed or current effects may be included with different approximations. This depends on the terrns retained
in the linear free surface condition (Ref.6). The slender body assumption
is generally made. The steady velocity is taken equal to its value
at infin-ity. For offshore structures, as large semi., TLP or barge, the influence of the modified steady velocity on the diffraction effects is cjiite important (Fig.18). Since the free surface condition is more complicatel, new numerical meth-ods have been developed (Ref.8,9). The following table gives the theoreticallimits for the different approximations (7 = wC1g, .4.) is the wave frequency
and C is the forward speed) (Ref.4).
Figure 18
Steady flow around a circular cylinder
o
aD
-78.00 -52.00 -26 00 0.00 26.00 52.00 78 00
Formulation Free surface condition validity
to compute drift forces
up to
a24,
at2 , a+az
2C
ata. 02...--
a +a. 2Complete formulation no limit
Dev in Strouhal 2- < 1/4
Grekas, Zhao, Faltinsen
.
7 < 1/2frequency of encounter no limit
o o o o N o o.
The extension of the diffraction theory to the second order approximation is necessary to predict
the mean loads from waves : drift forces, added resi;:tance in waves and wave drift damping. These loads are computed directly foiln the first order poten-tial. They vary with the square of the wave amplitude. Wave drift damping may have an important contribution in the horizontal low frequency- motion of moored ships and semisubmersible. Specific apprcximations have been de-velopped to save time consuming associated with forward speed calculations, particularly if interaction with the steady flow is t aken into account. The drift forces and wave drift damping are generally computed by using pres-sure integration on the hull. For horizontal components the convergence is
poor with the number of panels used for the discretization. Special attention must be paid for the mesh generation at the vicinity of the free surface and at any change of shape along the hull surface (Fig.19, Ref.15). An alternative is to compute drift forces using a momentum conservation formulation (Fig.20).
Figure 19
Typical mesh gene ration
Figure 20 - Wave drift damping from different formulations
cR3LLAß. C.."e c_ is3 z (2,
o o
24
Nossen Grue Palm Zhao & Faltinsen Integration des pressions + potentiel stationaire Lagally Integration des pressions
o
820 0 BO 1.00 1.20 1.40
the 2nd order wave loads in regular waves. If the Tiaskind method is used, the 2nd order diffraction potential is not computed (Ref.14). However 2nd order pressure on the hull can not be obtained. The extension of the method to bichromatic waves provides respectively the low frequency and the high frequency quadratic transfer functions (QTF-, QTF+)(Fig.21). The low fre-quency and high frefre-quency loads in irregular waves are deduced from the QTF or the drift forces. Newman's approximation can be used for very low frequency loads.
the 2nd order wave elevation. In that case the 2d order potential must
be computed (Fig.22, Ref.11). The extension to bichromatic waves is more complicated.
The numerical methods used to predict diffraction loads are now well val-idated. Comparison with model tests or with analytical formulations have been performed extensively for the 5 last years (FPS2000, Fig.23). The
cor-responding codes are now widely used for the offshore projects.
However the perturbation series are valid only for infinitely small wave steep-ness. Effects of steep waves or breaking waves car be analysed only from
non linear theory. In that way numerical methods .aave been developed for 2-dimensional flows. Most of them are based on the perfect fluid and irro-tational flows assumptions. They generally use the "numerical wave basin" concept (Ref.12). The problem is solved in a confined domain. Wavemaker and attenuation zone are modelled. Extension for 3D dimensional problem and for infinite domain are in progress (Fig.24). In the same way numerical methods are developed to solve Navier-Stokes equations with a non-linear surface wave condition, such as mixed Euler-Lagrange method (Ref.3) or VOF method (Ref.10). The main objective is to predict interaction between vortex shedding and free surface waves (Fig.25).
Figure 21
Low frequertcy QTF
Figure 22
Second-order diffraction wave elevation
I ni(77g)) Re(i7) um ABOVE 0 120 Mil 0 090 0 120 0 060 0 090 0 030 0 060 0 000 0 030
fl
0 030 0 000 1.= 0 060 0 030 0 090 0 060 I I 0 120 0 090 I 1 0 150 0 120 I 1 BELOW 0 150Figure 23
FPS2000
Comparison between numerical codes
0.4 0.5 0.6 0.7 0.8 0.9 11.0 11 1.2 113
Frequency, (..) (rad/sec)
2nd order excitation force. Mean drift force Surge Wave directon O deg
Figure 24
Non-linear free surface wave from potential flow theory
FIGURE 8. Champ de vagues d'un mat Wigley á Fr=0.25
14 Inst 02 Inst 03 Inst 04 Dzodore Inst 05 Inst 06 Mean Mean- 1.st.dev Mean+ 1.st.dev
Figure 25
Non-linear free surface wave in rotational flow
Q o aD 28 0.20 0.40 060 080 1.00 a) Champ de vitesse, 26 DT3.2
Viscous and drag effects
The viscous forces are dominant on slender bodies of small cross section
(bracings of jackets, risers, tethers, mooring lines). At resonant behaviour of structures (roll motions, low frequency and high frequency motions), it is also crucial to take into account the viscous effects for the damping estimation.
As the flows induced by the waves and the motions of the structures
arecomplex, one uses a simplified formulation ! That is the well-known Morison
equation adapted to unsteady flows, 3D flows, ... Generaly, in spectral
anal-ysis the Morison equation is linearized. Two problems have to be solved if Morison equation is used to compute drag forces
the formulation should model the interaction between the different velocity components : wave velocity, wave frequency motion, low frequency motion, current. Depending on the flow, relative formulation or independant flow
/ / Q ao ; ; ; ////// / / / I / / / o
s,
Ss\ \ ''' / / // / \\Vs,"s C,1formulation are used.
the drag coefficient Cd must correspond to the "real "flow. In that way Cd depends on the Reynolds number, the Keulegan and Carpenter number, the Strouhal number, the reduced velocity (ratio of the wave velocity over the current velocty). An associated problem is the definition of these coefficients : what velocity should be used ? The drag coefficient comes generally from model tests. However extrapolation from model scale to real scale may be
difficult.
An alternative to Morison equation is to solve the Navier-Stokes equation for unsteady flows. The hardware performance and the development of the numerical methods are in a rapid progress. Methods are validated and can
be presently used for simple geometry configurations, as slender bodies
the vortex methods family : Discrete Vortex, Vortex-In-Cell (Fig.26, Ref.13)). the direct integration of the Navier-Stokes equation (Reynolds Average Navier-Stokes) : Finite Difference scheme, Finite Volume scheme (Fig.27). These methods have been recently compared for a 2D unsteady flow configu-ration on a semi pontoon. The following table gives the main results obtained on a square section for an oscillating flow at wave frequency (Ref.7).
Euler equations are the Navier-Stokes equations where the viscosity terms are neglected. This method gives accurate results for
a body shape with
sharp edges which impose the vortex generation. However it cannot be used for cross section with rounded edge.
29
Method
Damping coefficieent Inertia coefficientModel test 0.57 1.38
Vortex-In-Cell 0.51 1.19
Navier-Stokes 0.62 1.35
Figure 26
Vortex-shedding from the V.I.C. Method
CD
rD
Figure 27
Vortex-shedding from Euler approximation method
/1Sf°
CD 6 -..._ -__ ..to '.) / a....-,,, .:''.'r ; --t,,....--,,, _-_--- .--- _ ---__ --- ----_----=______________________-______---. ___,,_.--- - ,---N ____. -'-- -r----.-<' ----.--- --__ --__.---..----.- ___.--7- ---... -"--...,\ ,..--...---Z---_1 ---: ---___.---__, --- ___--
__-,
c,:, ----I--12-14 128 I t 0.4-3 0.43 126 2.143.3
Dynamic response of structures
Wave fequency responseThe wave frequency response may be easily predicted from linear theory in the limit of small amplitude of motion. The equations of motion are solved in the frequency domain after linearization of the restoring forces (hydrostatic, mooring, structural stiffness). Transfer functions of loads and motions are obtained. In irregular waves the response spectra are directly deduced from the wave spectrum.
However when a resonant effect appears in the wave frequency domain (roll ship motion for example), non-linear effects have to be taken into acount, including
restoring forces : hydrostatic, mooring, ...
drag forces on the hull and appendages (bilge keels) particular damping systems
modification of the wave diffraction and radiation loads
non-linearities coming from the motions equations formulation In that way time domain analysis is generally used (Fig.28).
Figure 28
Irregular wave and non-linear roll motion
ci) a)
o
JI 1 t 1500. 3550. 3600. 3650. 3700. 3750. 3800. 3850. 3900. 3950. TEMPS (s)Hs=5m, Tp=5s
4000 I -11 11111 III 111_111' 1-500. 355 0. 3600 3650. 3700. 3750. 3800. 3850, 3900. 3950. 4000. TEMPS (s)Low frequency responses
The prediction of the low frequency response may be performed in two ways the frequency domain approach. The low frequency response is assumed to take place at the natural frequency of the motions. The response spectrum is of strait band. The natural frequency of the motion is far from the wave fre-quencies. Linear damping and linear restoring forces are assumed. Then the low frequency loads spectrum can be computed from QTF and the response spectrum is directly deduced (linear transfer function). The mean offset and the r.m.s. may be correctly predicted for horizontal motions. That is of in-terest for fatigue life analysis. However the main limitation of the frequency domain approach is the accuracy in the extreme value estimation.
the time domain approach. The external loads on the structure
can be estimated from non-linear formulations. That allows one to predict the low frequency loads, the damping and the restoring forces as well.In this
ap-proach one may include current effects, wind loads and interaction between wave frequency motions and low frequency motions This is important for the prediction of the roll / pitch / heave motions of semisubmersible struc-tures (Fig.29). The time domain method is particularly suitable to estimate the maximum loads and motions in the extreme environmental conditions
(Ref.2).
The low frequency prediction methods are generally included in the hydro-dynamic software. Statistical and spectral analyses are performed in post-processing. Validation of these methods are performed by comparison of the statistics with model tests results.
Figure 29
Low frequency response from time simulation model
7 Oc o 33 100 150 200 250 300 350 400 450 500 550 600 Time (s) (E+01) 100 150 200 250 300 350 400 450 500 550 600 Time (s) (E+01) 100 200 250 300 350 400 450 500 550 600 Time (s) (E+01) 15 30 7560105
I20 135 Time (s) (E+01) Oc 75 7High frequency responses
The prediction of the high frequency response, said "springing / ringing", is much more difficult. Our philosophy is to separate the following two phe-nomena
the springing effect induced by high frequency 2nd order loads. Assuming a linear transfer function, the response in irregular waves can be easliy obtained from the QTF+ (frequency or time domain approach) (Fig.30). However only the loads induced by waves with frequency half of the natural frequency of the motion can be predicted. In reality springing has an effect mainly on the fatigue behaviour of the structure (Ref.5).
the ringing effect taking place for extreme waves. The origin of ringing is
not well-kown. It depends probably on the type of structure (mainly TLP
and monotower gravity structure). Analysis of model tests results seems to show that the ringing is correlated with the wave steepness. In thatway,
ap-proximations of the non-linear loads in the wave crest have been developed. The first results confirm the methodology (Ref.5,16, Fig.31). However more research works have to be done on this subject in different way. Third order diffraction theory, non-linear diffraction theory, modified Morison formula-tion are the present ones. Associated problems are the damping predicformula-tion and the statistic distribution analysis of ringing apparition.
Figure 30
Springing time domain simulation
f
WAVE ELEVATION
from jons,rap spectrum
Tp-7s ; gsmanas..4.3
100.00 20CLDO 300_00 4.03.00 60CL CO (SOCLOO
Time (s)
100.00 203 00 300.00 400.00
A
figure 4.a : effet de ringing calculé - tension dans 1 tendon
Figure 31 - Ringing simulation
36 100 200 300 400 500 600 time (s.) i 1 1 100, 200. 300. 400 500 600 time (s.)
Slender bodies responses
For structures composed of slender members of small cross section (risers, jacket) the dynamic response is obtained from the following way
the hydrodynamic loads are used in a classical structural model, generally based on Finite Element Method.
then the hydrodynamic loads are computed from Morison formulations on each element of the structural model. Generally a local 2-dimensional flow is assumed. the latter takes into account current, wave velocity and motions of the structure at the element position.
3.4
Conclusion
Numerical models are largely used for offshore projects. Their validation have been performed and a good level of confidence can be generally given to the user. However they cannot give answer for some important problems. In that way the main future research activity has probably to be concerned
with
the viscous and drag prediction in unsteady complex flows from numerical methods.
the non-linear wave diffraction loads induced by steep waves and breaking
waves.
the interaction effects between vortex-shedding and non-linear free surface
waves.
Acknowledgements
Some results and comments presented in this paper come from French projects on "damping of low frequency motions" and on "high frequency resonance". Partners in these CLAROM projects are Bureau Ventas, Doris Engineering, Elf, IFP, Ifremer, Principia and Sirehna. The main illustrations come from research works done by my colleague at Principia. Thanks to Y.M. Scolan for trying to correct my english.
3.5
References
Réf.1 O.M. Faltinsen, "sea loads on ship and offshore
structures", Cam-bridge University Press, 1990Réf.2 C. Berhault, Ph. Le Buhan, B. Molin, J. Bougis, "Diodore : A numer-ical tool for frequency and time domain analysis of the behaviour ofmoored or towed floating structures", CADM0'92, Madrid, October 1992,
Réf.3 C. de Jouette, "A numerical method for unsteady wave flows around submerged obstacles", 7th Workshop on Water Waves and Floating Bodies, Val de Reuil, May 1992
Réf.4 C. Berhault, T. Coudray, F. Villeger, "A scope of different methods to compute wave drift damping in regular waves", 7th Workshop on Water Waves and Floating Bodies, Val de Reuil, May 1992
Réf.5 C. Berhault, T. Coudray, E. Magne, "Springing and ringing effects on offshore structures", Conf. on Floating Production Systems, Blueprints for the '90s, Oslo, October 1992
Réf.6 T. Coudray, J.F. Le Guen, "Validation of a 3D-seakeeping software", CADM0'92, Madrid, October 1992
Réf.7 C. Berhault, C. de Jouette, J.M. Le Gouez, "The prediction of low
frequency motion damping for semi-submersible and TLP structures", Conf. on Floating Production Systems, London, December 1992
Réf.8 J. Nossen, J. Grue, E. Palm, "Wave force on 3-dimensional floating bodies with small forward speed", Jour. of Fluis Mechanics, Vol.227, 1991
Réf.9 R. Zhao, O.M. Faltinsen, "Interaction between current, waves and
marine structures, 5th Int. Conf. on Numerical Ship Hydrodynamics, 1989 Réf.10 C.W. Hirt, B.D. Nichols, "Volume of fluid method for the dynamics of free boundaries", Jour. of Comp. Physics, Vol.39, 1981
Réf.11 L. Boudet, Y.M. Scolan, "Diffraction du second ordre sur un cylindre vertical", 2èmes Jour. de l'Hydrodynamique, 1989
Réf.12 R. Cointe, "Quelques aspects de la simulation numérique d'un canal houle", Phd from ENPC, Paris, 1989
Y.M. Scolan, O.M. Fa1tinsen, "Numerical prediction of viscous flows around multi-bodies by a vortex method", 6th Int. Conf. on Numerical Ship Hydrodynamics, 1993
Réf.14 X.B. Chen, B. Molin, "Calcul du 2ème ordre à très haute fréquence sur des plateformes à lignes tendues", 3èmes Jour. de l'Hydrodynamique, 1991
Réf.15 J.N. Newman, C.H. Lee, "Sensitivity of wave loads to the discretiza-tion of bodies", BOSS, 1992
Réf.16 R. Jefferys, "A slender body model of ringing", 8th Workshop on
Water Waves and Floating Bodies, 1993
19 th WEGEMT School
Numerical Simulation of Hydrodynamics:
Ships and Offshore Structures
Offshore Structures
Seakeeping codes AQUADYN and AQUAPLUS
Doctor DELHOMMEAU Gérard
Ecole Centrale de Nantes
ECOL E
Centrale
Nan tes
THE SEAKEEPING CODES AQUADYN AND AQUAPLUS
G. DELHOMMEAU L.M.F.-D.H.N. U.R.A. 1217 du C.N.R.S.
Ecole Centrale de Nantes
I - INTRODUCTION :
This lecture describes briefly the theoretical basis of the two computer codes AQUADYN for seakeeping without forward speed and AQUAPLUS for seakeeping with the approximation of encounter frequency. The code CUVE solving the inner problem will also be described.
The seakeeping problem is the problem of bodies, floating or immersed, in a fluid of infinite or constant finite depth, with or without forward speed and submitted to sinusoidal waves. To describe the problem in mathematical terms, we use the theory of perfect fluid, without viscosity. The theory of nearly perfect fluid allows us to take into account the radiation condition at infinity. First, we set the exact non linear problem, with the exact free surfaces conditions, then this problem
is linearized and all quantities are developed up to the second order. We have to solve the
hydrodynamic problem at the first order and to apply the results in the mechanical equations at the same order. We will first solve the problem of bodies in waves without forward speed, then the problem of oscillations of fluid in tanks, which is a problem of the same lcind as the previous one, and we will conclude by the problem with forward speed and its approximations.
We will describe the computer codes AQUADYN 2.1, CUVE and AQUAPLUS. Several practical examples will be given, with discussion of the main particularities of these codes:
recommended number of panels, location and removal of irregular frequencies, need of
experimental damping at the resonance of motions, comparison of calculations methods for drift and second order forces.
II- THEORETICAL BASIS :
2.1 General equations for a perfect fluid
2.1.1 Hypothesis :
The fundamental hypothesis allowing us to describe mathematically the behaviour of a fluid is the continuity hypothesis. Other hypotheses are :
H1: Strains are proportional to deformation velocities ( Newtonian fluid ).
-112: The fluid is homogenous and isotropic. Lame's viscosity coefficients are slowly varying functions of density and temperature.
1-13: The viscosity coefficients are zero. The density is constant.
H4: The only external forces are gravity forces. The fluid is initiallyat rest.
-115: When there is a free surface, we neglect the surface tension effect and suppose that pressure is constant above free surface.
Hypothesis H1 and H2 lead to the Navier-Stokes equations. With H3, we obtain the perfect fluid equations for an irrotational flow.
2.1.2 General equations :
In the Euler's representation mode, the mass conservation principle gives:
-> ap
In perfect fluid, we have :
divT. = 0
With H4 hypothesis, we have a potential 0 such as :
-> >
V = grad (1)
Continuity equation is then :
A(13 = 0
The fundamental principle of dynamics is expressed by the Navier-Stokes equations :
1 > _, -> X1 + X.2 > -> X2 ->
grad p = F - y +
grad (div V) + A VP P P
where, in the Euler's representation mode, the acceleration is given by Helmholtz's formula: ->
->
> V2 ----> -> -> DV y = grad (T) + (Rot V) A V + -D-F With H3, we have Euler's equation :
1 > ->
-'
grad p = F - y
P H4 gives : -> >F = grad U
with U=-gz, the z axis being directed upward.
Integration of this equation gives Lagrange's equation, giving pressures at any point of the
fluid. P
+ gz +
y2ao
2+ -,-- = F(t)
dt P 2.1.3 Boundary conditions : - Body condition :On a fluid proof body C, the normal relative velocity is zero. The normal velocity of the fluid is then equal to the normal velocity of a point taken on the body:
V.nlc = VE.flic
and with the potential :
ao
_>...>1
= y
n1Fi C
E CFree surface conditions :
Equation of free surface is
: .z = (x,y,t)
The dynamic condition of free surface is given from Lagrange's equation : Do
Pa V2
gz . F(t) -
- (T + w) I
74(x,y,t)P
For all following problems, we take pa=0.
The kinematic condition is obtained from continuity hypothesis :
dpap
_>. >= o =
+ V. grad pdt
a
Which gives, combined with Lagrange's formula :
a24)
ao
av21 -» 2
+ g -, + -, + -y V. grad V I
(x,y,t) = F' 0)
- Radiation condition at infinity :
To provide for the unicity of the mathematical solution, it is often necessary to have a supplementary condition expressing the behaviour of the fluid at infinity. One method to obtain this condition is due to Rayleigh and was used by H. Lamb. This is the nearly perfect fluid hypothesis, where we add to the second member of Euler's equation a dissipative term, expressing that the behaviour of the fluid at infinity follows the physical reality, which will be cancelled at the end of the calculations. The latter are transformed as indicated below
Euler's equation 1 -> -> grad p = lim (F - y - 2 e' V) e-->o Lagrange's equation : V2 ae. lim ( + gz + + + 2 e'
) = F(t)
2 dtkinematic and dynamic free surface conditions :
a20
acD av2 ---> V2 ati)Hm
(+g
++ 7: V. grad T + 2E'T+E'
V2 )I = F'(t)e--->0
at'
ao
v2= [- - - 2 e' cD + F' (t)
e__>0 g dt 2
- Absolute values expressed in moving axis :
Previous boundary conditions can be written in a moving frame parallel to the absolute frame and moving with the velocity Vo. In this case, we have :
a a .> >
1
-V
a
Ifixed axis = a moving axis 0' grad Imoving axis
Other derivatives in respect with space variables are the same in the two axes. 2.1.4 Similitude laws :
The similitude of perfect fluid submitted to gravity forces with a free surface makes it necessary to respect the Reech-Froude number :
Fn =
1./gL
If we consider two fluids of density Pl and p2, with lengths Li and L2, with gravity fields gland g2, similitude of velocities give :
V1 V2
speed :
ig2 L2
and for mass and time :M1 M2 mass time P1 14 P2 14 1 Li 1 gl T2 g2
and for other quantities : X/ X2 L11
17
2 71 72 displacement acceleiation forceAll calculations made for a given fluid will be available for a fluid in similitude. If density and gravity are the same, these laws give the variation of quantities with geometric scale.
2.2 Eauations of mechanics applied to the problem without forward speed 2.2.1 System of coordinates :
. -.
gl g2 F1 F2 = T 3 T 3 P1 gl '-' 1 P2 g2 '-'20
: Origin of the fixed axis.O' : Origin of axis moving with the body and parallel to the fixed axis.
The height of free surface at rest in these axis is zF. Vanishing viscosity of nearly perfect fluid.
e : Little parameter of the development in series.
2,2.2 Equations of the problem :
The equations of the whole problem in perfect fluidare : AO = 0 C= I.E.IIIC aCD
--,
-o
dz z=-h 0:1) > cl) 1 at inf initya2o
ao
av2
1 ,.. 2ao
2 ii 1 m+ g+ + v.gradV + 2e
t + e e -)0+ at az at 2 z=C(x,y,t) C =If we develop the problem in perturbation series of the little parameter c, we have : () = EOM +c20(2) + 0(62) = (1)i + 432 + o(c2)
V. = gra-d(1) = ci./(1) + 2V.(2) + 0(62) = -\-T1 + V-2 + 0(62)
where 0(62) are terms of order greater than £2
2.2.3 Hydrodynamic problems at first and second order ;
Developing the incident wave, the body condition and the free surface condition at second order, we have the following equations :
For the incident wave :
OD = ag ch mo(z+ h)cos moRxcos f3 + ysin 13) ox]
i co ch mph
with : co2 = gmoth mph ,and 3m0a2g ch 2m0(z+ h)
sin 2m0 [(x cos f3 + y sin (3) cot]
(1)12 8c0 sh3 mph ch mph
F(t) = F2 = m0a2g 1
For the hydrodynamic problem at first and second order :
A(1)1 =0 a(Di = T'Ei . '10 IC0 an0 co
aoi
=0
az z=-h
ot -> oil at inf inity
lim + a2(1'1 +2e. a(1)1 +gaCbi
E. ,0 at2 at az
1 ao,
=g at z=0
A02 = 0 as:1)2= .V.
E2011C0
ii IC0 + -VE1-1-5
ano co
acD2
=0
az z=-h
02 --) CDI2 at infinity
4 sh mph. ch mph
E021z=0 = 2 gfa-clozbi.grada(1), +
at z=0 1 rao2 ,,
a ao,
2 =g ataz at
+1-
LJ- + - r21
2=Eoilz=0 =0
z=0 acI31 aniaoi acEoi)
at az z=0-
PoPi .grddi.Ei -5.0 Ice,Co
+
lime--)0+ ei (grdd4)1)21
=0
The total potential at second order being :
cl) = 0, + (1)2 + o(E2)
2.2.4 Description of the motion of a body :
To describe the motion of a body, we divide the motion into two parts, translations and
rotations.
Vector translation : i = 00
Vector rotation : 6= 04 around -4
We can represent the rotation of any vector by:
_ sin 0 ,- _ 1- cos 0 -
-R(5) =u+uA 11+
2
e A (0 A u)
0 0
The displacements of any point of the body in respect to its location at rest are
PAM =6
PA = tii +R1(0P0)= 'El +01 A0130
Poii2 = ti2 + R2 (0P0 )= ;i2 +82 Aofio +-01 A(81 A ofk)
2.2.5 Pressures, forces:
Pressures at different orders are obtained from Lagrange's equation : pp = -pg(z - zF)
-- lad)
131 = -PgPOPt- iz P at
-
-
ao2
P2 = -PgPo2.P- iz - P
at
PPoPi-grddT-t
Doi 12 P(grddc1)1)2+ F2
F2 = m0a2g 1
4 sh mph. ch mph
The position taken by the body during its motion are given on the figure below :
5 L
,
C,
co u S.,With the following hypothesis on the order of magnitude of the integrals, the forces exerted by the fluid on the body can be written :
ff = o(e) , SS =o1
ow
si Co
fig = iio for the forces
fig = oPo A fio for the moments around O' moving with the body
'g0
-
ff poi-1g dSCo
Po
= 61 A Po-
ff piiig dS-
SS pojig dSCo Si
Pg2 =02 A fgo +12 61 A(81 A fig° ) + 61 A (Po - 61 A fg0)- If p2.fig dS - if pi.fig dS
Co s1
2.2.6 External forces :
External forces can be divided into hydrostatic forces and hydrodynamic forces. The hydrostatic forces are the resultant of gravity forces and buoyancy forces. For a floating body at equilibrium we have :
Mg = pgVo ,Xco = XG0 ,Yco = YGO
Go being the gravity centre and Co the hull centre, defined by :
OP0 = XTx + Yi.), + Z-i; ,(:) .0 = --1- f oPo dV,
060
=-1- SON dmVo vo
M Vo
The hydrostatic restoring matrix is defined by :
SF. S34 =pgif Y dS=S43 SF0 S35 =-pgff X dS=S53 SFo
s
pgif Y2 dS+ pgV,(ZG0 - ZGO) SFo S45 =-pgif XY dS =S,
SF S = pgif X2 dS + pgV0(Zco - ZG0 ) SF. -0 0 0 0000
0 000
0-0 0 533 S-4 3 S35 0 §= 0 0 S43 S44 S45 0 0 0 S53 S54 S55 0 0 0 0 0 0 0 with-: _S = pgif dS algSF,
The hydrostatic forces at a point O' moving with the body, are given by : ftiso = = [S33tz1 + S340,1 + S35ey1 1. Tz + 02
1Y1
zF) s34 (Tx2 eY1:
zl Ns2 = A È-t1S1 [S33(TZ2 2mhso (a)=
hsi ) = [S43tz1 S448x1 S450y1 1. 1x [ S53tzi + S540,1 + S55vyi ]. 02 2Mhs2 (CY )
= [s43(tz2
.1 0Y1 zF ) s44(tx2 13Y1ez12 2
02x +82 0 ez,
[S53 ('r2 1 Y1 zF ) + S54 (Tx2 + " ) + S55 (Ty2 The transportation at a point 0 fixed being done by :
Mhs2(0)=Mhs2(O')+tl A PhS1
And the hydrodynamic forces at a point O' moving with the body, by :
aoi
CI =g at
)+ S35(ty2 0x1 'ezi)171Z 2°xiez1
)+ S45(TY2 2 )].1. 2 Y+61 A IkS1 (CY ) z=0: wave height ; Z1 P0P1. iJ0 vertical displacementat free surface z=
fig = fio for the forces
fig = OP0 A fio for the moments around G moving with the body
Phdo =
fridi = if piiig as = p ffs71-a"D i-ig as
co co rfr
ao
ad) Éhd2 = A fhdi at2 p J, +p 0.p-1.gr,.
+ 1p(gra'd 01)2
F2]. g dSJ(i
Z )2 dS at 2 Co roThe transportation at a point 0 fixed being done by :
2.2.7 Inertial forces :
The inertial forces, developed at first order give :
k1
=M'il +M61 A °Ò. 0KIM' =M060 A
+ i
[61 (OPO )2-
0130 (0P0.61)] dmM
144
f(y2 +z2) dm
; 155f(z2 +)(2)dm;I66
f(x2 + y2) dm145=154 = PCY dm ; 146 =164 = PCZ dm ; 156 = 165 = f YZ dm M N4 NI
M 0
01
[
P= o
NIo ;I=
00 M
rkii FP
TiFid__,,,-,--dii
Lm-,,, ]=L-T, Ti[ii, ]-[6:, i
2.3 Solution of the hydrodynamic problem at first order : 2,3,1 Decomposition of the problem :
For the problem of N bodies oscillating independently in waves, we have at first order :
M=0
0ad)
an 11 = ..\-/Ei An ; i= 1,2,...,N pa)
az z=-h
(I) ) 0 1 at inf inity
a2o, acl)
acI)lim
E'>0+
at2 +LEeat +gaZ
= Ecblz=0 =0 z=0lace
C--g at z=0
ag ch mo (z + h)
cos mo [(x cos fi + y sin po cot] (DI = co ch moll and : 0.)2 = gmoth mph
=0
II44 145 _ 146 _ o MzGo-MYG0
154 155156 ;T=
-MzGO O mxGo 164 165 166_MYGO MxGo
o _with the unknown velocities obtained from unknown amplitudes of motions :
VEi
6 =V'Ei* cos COt+V.Ei** sin COt =
q=1 and :
3 * 6 *
.T.Ei = CO sin cot[ E All Eq + 1 All (Eq_3 A OP0)]
q=1 q=4
3 ** 6 **
+COCOS COt[ I Aiil Eq + E A ? (Eq_3 A 0130 )]
q=1 q=4
with :
**
A? = A? cos cot + A? sin cot
eq, q=1,2,3 are the unit vector of the axis x,y and z.
Considering a perturbation potential, this problem can be divided into N+1 problems :
ozto = (1)* cos cfm + (I)** sin cot (I) = (1) +
N
(bp =D +
CD Rii=1
One diffraction problem, for which the body condition is :
aCren = a(7>
= 2
Nan xi
an xi
and 2N radiation problems, for which the body condition are :
ac-DRi = (VE:cos cot + V.Ei** sin(ot)! ; i = 1,2...,N
an zi Ei
acDRi
=0;j#i
an zi
which are known after having solved only N elementary radiation problems :
= clEi*COS Onlyi ; = 1
= 0 ;
j
i
Ej
2.3.2 Complex notation :
For quantities varying sinusoidally with time, we canuse the complex notation : A = A* cos cot + A** sin cot = Re A e-iwt ; A- = A* + iA**
1
< A.B >=
Re( AB)=
1Re( BA)= Re( AB+ BA)
12 2 4
2.3.3 Solution of the problem by boundary element method :
To solve these problems, we use the boundary element method, with Green's function allowing to take into account the boundary conditions on body, bottom and free surface.
There are two main differences between all the diffraction-radiation programs :
The first difference is in the type of singularity distribution used. In the program
AQUADYN, we use a mixed distribution of sources and normal dipoles, where the sources are known from the body condition and the normal dipoles are unlcnown. The solution of the problem is the potential. The velocities outside of the bodies, which are not known by the body condition need computation of influence coefficients of velocity for normal dipoles, a process known as being of poor accuracy ( derivative operator ) . In the AQUAPLUS program, we use only a distribution of sources. The solution of the problem is the velocity and the potential is calculated from the sources by influence coefficients with a good accuracy ( integration operator ).
The second difference is in the way to compute Green's function. and the discretized influence coefficients integrated on a panel. In the general case, we can write:
C = Ci + C2
=fff(1
) dS(M'i ); Mils MM 1
C2 = [ fk() de] dS(M' )
S n
= A[z + CZ' +RD] ; 153 = (x x' )cos 0 + (y y' )sin O
Due to the cylindrical symmetry of the Green's function, we can also write :
TC TC
2 2
k() de =
g(C) de7C 7C
with: :
= Z + iR cos ; Z = A(z + ez' ) ; R = AV(x x' )2 + (y y' )2
The terms Ci are calculated analytically by classical Hess and Smith formulas (1966) or similar approximations (P. Guével: 1975) [14].
For the terms C2, there is a choice:
- We can first compute numerically the simple integral, and then analytically the double
integral on S. This choice was made for the first version of AQUADYN (1976)[19].
- We can first compute the simple integral in (3 and then compute numerically the double integral on S. In practice, due to the slow variation of the integration term on the panel, the double integral on S can be approximated by only one point formula, which makes this method very efficient. There is also two ways of computing the simple integral inO:
we can approximate the integral by several analytical formulas, each available in a domain
of variation of R and Z, as proposed by F. Noblesse (1982) andJ.N. Newman (1985)[17].
we can obtain the same results by using a interpolation into a file of four elementary functions of the two variables R and Z, created only one time and loaded at each execution of the program [21]. This file is available both for infinite and finite water depth. The size of the file is
less than 512 KBytes and this method is used in the codes AQUADYN 2.1 and AQUAPLUS.
(1987) with a quadratic interpolation. The resulting computation time are of the same order as with analytical formulas, but the development effort is less for the second method.
The asymptotic values of the Green functions are obtained by the anti mirror image (1/R-1/Ri
2.3.4 Discretization of integral equations :
When written on the control point of each panel of a body, the integral equations of the boundary element method are transformed in linear systems, we have :
For the mixed distribution , N being the total number of panels on the bodies:
N - -
N ao
+ Dij = e
And for the source distribution :
a- N
+ Cr, K, =
2
j.i
anThe potential of sources are obtained by :
N
Mj j=1
and the influence coefficients are obtained by:
-1 s-(
M' ) ds(M' )4n
s 1 a Dii = 4nSIan a = 47c anS being the Green's function of the problem.
For the elementary radiation problem (1:0-ai:
q=1,2,3
= G9 with G9 =
(Eq_3 A 0130).ñ q= 4,5,6
For the diffraction problem (I)D:
acD-e
aoi
an an
All coefficients of the linear systems are complex. They can be written in real form by separating the contribution of terms in sinus and terms in cosines, but the real system will have twice as many unknowns as the complex one. The computation times of a real system of order 2M
is in 8M3 and those of a complex system of order M is 4M3. The complex notation is not only a
writing facility, it also saves computation time.
Mi S Mi Ski Mj S-(Mi,MI ) ds(M' ) M' if -S(Mi, ) ds(M' ) , Dii = 0.5
,K j= 0.5
ae
an2.3.5 Irregular frequencies :
The previous integral equations are Fredholm's integral equations of the second lcind. These equations have generally an unique solution, except for certain discrete frequencies when the solution of the inner associate problem is not identical to zero.
{AO= 0
E=0
(131z = O= 0
This can occur only when the body is not fully immersed. The general principle to suppress these irregular frequencies is to write on the free surface a supplementary condition which makes the potential be identical to zero inside the body. This can be made by writing that the potential ( or its normal derivative in z ) is zero at certain points of the free surface. In practice, these frequencies are generally rather high and are not always in the domain of calculation for wave problems. We can approximate the period of the lowest irregular frequency by the formula
T = 27c.. 7c1-1 .1L2
th(-- + 1)
\ L 132 itgli L2 L B2where L,B are the length and breadth at waterline and H is the draft.
For L=20 m and B=6 m, we have T=2.6 s and for Land B=90 m and H=40 m, T=8.857 s.
In practice, the influence of irregular frequencies is identified in a narrow strip of
approximately 0.2 s in period, depending of the mesh ( decreasing with refined mesh ). This region is slightly greater for source distribution than for mixed distribution for three dimensional cases.
2.3.6 Radiation problems :
The elementary radiation problem is the problem of a body with a forced sinusoidal motion,
in initially calm water. The forces on the body and on the other bodies are obtained from
Lagrange's equation:
For the motion q of body i, we have :
6 q
2 A
p = pco selti q=1
Force p on the body j is given by :
6
= fmPq [--0)2 (Aru cos OA +
A**
cot)] q=1i-BP(1[co(Ar sincot -
A**
cot)])acDP*.
Mrici = pff 44: RI as
an
Ej
a** aeopP:
BFiq =
pcoi
Crkian dS
Ej
Pr operties :
= MT) BP(1 = B9P
7
BPP >= 0 ;M=:?
9
Ji 1J Ji n 11
The diagonal of the matrix of damping coefficient must be positive, and the matrices
synunetrical by blocks. This gives a good estimation of the numerical error due to discretization. In practice, we also find that the diagonal of the added mass matrix is positive, except at certain frequencies for special geometries ( catamaran or moon pool )..
The terms of added mass and damping coefficients are the resultant of dynamic pressures.and are not invariant. The added mass and damping coefficients are function of the period of the motion
and of water depth. At high frequencies (T=infinite, acoustic in incompressible water) and at low frequencies (T=0) the damping coefficients are zero. The asymptotic values of the added mass coefficients are obtained by the anti mirror image ( 1/R-1/R1 ) and the mirror image ( 1/R+1/Ri: double model ) in Green's functions. For a group of N bodies, the added mass and damping coefficient matrices are two real matrices (6Nx6N) or one complex matrix of the same dimension.
2.3.7 Diffraction problem :
Excitation forces come from diffraction potential and incident potential:
eDex = OD+ (DI
Force q on the body i is given by :
acl)R
Fexi = ipcon (OD+ )dS
anOr by Haslcind' s formula : q
-q
- a cDRiFexi = -ipcoff ((DI
wRi )dS
an an
u
u--uEN
The part of excitation forces due to the incident wave are often called Froude-Krylov forces. The excitation forces can be calculated by direct pressure integration of the diffraction and
incident potentials or after having solved the radiation problem, without having solved a
supplementary problem, by the Haslcind's formula. When we use this formula, we don't know the local pressures on the bodies , which forbid to reach second order forces. This formula is only used to verify the quality of the numerical solution by pressure integration. Unfortunately, the agreement with pressure integration is too good for this comparison to be useful.
2.18 Equations of motions
The unknowns are the motions of the bodies. If we make the sum of all forces exerted on the bodies, we have :
FM = Fhs FR + Fex + FL
where FM are inertial forces, Fhs the hydrostatic restoring force, FR the radiation forces, Fex the excitation forces (the sum of Froude-Krylov forces and diffraction forces) and FL external linear forces which may be due to the stiffness and the damping of a mooringsystem.
The line corresponding to the motion q of body i is written :
2
6Pq -(1
6 -Pq -qI
N 6 -Pq -P
6 - Pq -q
_2
Ai + Ai _CO21M A j + EFL
Ai = Fexip=1 p=1 j=1p=1 P=1 i
The linear system for N bodies is :
M'
The case of internal forces can be solved by separating the body into several bodies tied by internal forces, with connection equations giving the unlcnown forces (for example, the beam of a catamaran).
2,3,9 Drift forces :
For a distribution of singularities kinematically equivalent to bodies oscillating in waves, we define the Kochin function by :
1
H(0) =if (
a
+ Pt )ch m0 (z' +h)e
(x' cos 8+y'sin e)ds(m,)47c sh mph an with for a mixed distribution :
-
v6 -q ; izy= L,; =,
an q=1-
N 6 -9
-= 11D+ E Ri i=lq=1and for a source distribution :
N 6
-9
-a = crD+ E E
i=101=1
By application of the momentum equation, we obtain the Maruo-Newman formulas, which give two horizontal resultant drift forces and the vertical resultant drift moment exerted on all the
bodies:
27c _
cos f3
(kh)2
< Fx -27tapo)(
)Im H(P)- 2n
pmoo
f [H(0)] (
2 COS 0) desin P h[(moh)2 -(411)2 +kph] o
Y sin O
and for the moment at a fixed point 0 (mooring) :
(411)2
2n7
< Mz (0) >= -2napw Re H(P) - 27cp f H(0).111(0) dO
h[(moh)2 -(1(0)2+ kph] 0
For a point O' moving with the bodies (dynamic positioning),we have :
<M (0' ) >=< Mz (0) > - < t,.Fmy - Ty >
2
Sl+FL11-W (M1 + M11) FIAN-
2 -
MN1 Al Fexl-
-
-
-
-2
It can be demonstrated that, without external forces, the drift forces in the wave direction are positive : 2n MO(k0h)2 <F >= 27cp j. [H(0)]2[1 cos(0 [3)] de >= h[(moh)2
-
(411)2 + kohl oThe second method is the use of direct pressure integration at second order. If must be noticed that at second order, the average value of second order potential is zero, and the constant issue of Lagrange's equation F2 is zero in infinite depth and also in finite depth for a freely floating body ( not for a body artificially maintained at a constant height ). The expression of average forces and moment at second order on the body number i is then given by :
[
_0,32[
'
T"]ri]
-1
I 6i2
°xi +O2w ovi.ez. oxi.oz
-Pder =< Afivfj [S33 ( ZF)± S34 (± 2 )+ S35( > 2 2
ao
1 pg +pff [PoP.grdd-Tt +2p(grdd
4:13.)2 F21. fig dS Fi(C Z)2 fig dS > +8?,i Ow; 07; e-&Ideri (O' )=< Mmi(0' ) [S43 ( 2 zF ) +S44(+ 2 ) 345( 2 )i. lx 02 n2
-[S53(
vxiuyi
zF)+
S54 (4- 0)11..0zi)-/- S5" ( exi.ezij
2 2
"
2".
Y+pjj[POP.grdd +-1-p(gr'd 0)2 F2].i dS z>2. fig dS>
at 2 2 Fi
1 a(1)
= : wave height ; Z = P0-15.izlz=0 : vertical displacement at free surface
g at z=0
fig = fio for the forces
fig = OP0 A lip for the moments around O' moving with the body
and the transportation to a fixed point 0 is given by :
<Mder,(C) >=< Mder, (C1' ) > <1-i A FM-1 >
For a mixed distribution, the difficulty lays in the poor accuracy of velocity calculations for normal dipoles. To avoid this problem, we can solve one supplementary integral equation with a source distribution, once the motions are known, with one second member equal to the normal perturbation velocity of the bodies. The only benefit would be in the case of the calculation of motions by sources being wrong, which is not the case. The solution is then the same as with source distribution. In any cases, the calculation of second order forces with pressure integration requires the solution of at least one integral equation withsources.
2.5 $olution of the inner problem of oscillations in a tank: 2.5.1 The inner problem :
We consider a tank with forced motions, as shown on the figure below :
The governing equations of the inner and outer problems are [23]
O
Inner problem : uter problem :
464:0=0 AO = 0
VE = -N.TE* cos Cot
and :
3 * 6 *
= sin cot[Igil Eq. 11 + (ig_3 A oP0).fii
q=1 q=4
In complex notation :
3 6
= [ Igil(è'q_3 A Of30).ii]
q=1 q=4
For this problem, the condition at infinity does not exist, so the response has no phase shift with excitation. For an excitation in sin cot, the response will be in sin cot. and the elementary inner problems of radiation are :
Inner problem :
=0
acIg -6 q = 1,2,3 = an Eq _3 AoPo q = 4,5,6 Co 0)2 -kod)q + ga,cbc1 = 0; ko =°
z=0 g 6 cb = -ico I cDc1Ac1 q=12.5.2 Solution by Rankine singularities method :
The elementary problem can be solved by a Rankine singularity method, with a mixed distribution of sources and normal dipoles, with M panels on the free surface and a total of N panels on the free surface and on the body. We have :
ad)
= ---; p, = <I>
an
alc =
o an co ;alsL = 1(04) =And the discretized integral equations are :
N N
Ili unknowns
j=1 j=1
= o urn a2(D+2E, all)+gacb
E'->O+ at2 at
- az
z=0=0
a24:13 aao ,.., act,
+ g az
