by
J Moes
COASTAL ENGINEERING AND HYDRAULICS DIVISION NATIONAL RESEARCH INSTITUTE FOR OCEANOLOGY COUNCIL FOR SCIENTIFIC AND INDUSTRIAL RESEARCH
NRIO TECHNICAL REPORT T/SEA 8138 Stellenbosch June 1982
CONTENTS
Page
LIST OF TABLES iii
LIST OF FIGURES iv
LIST OF APPENDICES V
LIST OF SYMBOLS vi
SUMMARY ix
INTRODUCTION
2 REQUIREMENTS OF MATHEMATICAL MODELS 4
3 BASIC THEORIES 10
4 THREE-DIMENSIONAL SOURCE TECHNIQUE MODELS 13
4.1 General Theory 13 4.2 ENSM Models 19 4.3 MARIN Model 22 4.4 OEC Model 26 4.5 CJG Model 28 4.6 NMI Model 30 4.7 DnV Model 32 4.8 Conclusions 33
5 STRIP THEORY MODELS 34
5. 1 General 34
5.2 IFS Model 36
5.3 NSRDC Models 38
5.4 Conclusions 41
6 FINITE ELEMENT MODELS 42
6. 1 General 42 6.2 UCL Model 44 6.3 IFP Model 45 6.4 MIT Model 46 6.5 ISH Model 47 6.6 Conclusions 47 7 NON-LINEAR MODELS 49 7. 1 General 49 7.2 LNH Model 51 7.3 Conclusions 54
CONTENTS (continued) Page
8 COMPARISONS OF MODEL RESULTS 55
9 GENERAL DISCUSSION 58
10 GENERAL CONCLUSIONS AND RECOMMENDATIONS 61
REFERENCES 66
TABLES FIGURE S
APPEND I CES
LIST OF TABLES
Table I Comparison of mathematical models for ship motions II Comparison of available 3D source technique models
for computation of ship motions
III Principal ship motions for test run ROO2A computed by OEC model VESDYN
IV Principal ship motions for test run ROO2A computed by CJG model DYNRES6
V Principal ship motions for test run ROO2A computed by NMI model NNIWAVE
VI Principal ship motions for test run ROO2A computed by NSRDC model HANSEL
LIST OF FIGURES
Figure 1 Layout of Richards Bay Harbour
2 Layout of Saldanha Bay Harbour
3 General picture of factors involved in design of channel depth
4 Organisation diagram of the MARIN computer programs 5 Determination of viscous damping coefficient
6.1 Main particulars of the 200 000 dwt tanker used for comparison of models
6.2 Comparison of surge response for wave direction 1800 6.3 Comparison of heave response for wave direction 1800 6.4 Comparison of pitch response for wave direction 1800 6.5 Comparison of surge response for wave direction 225° 6.6 Comparison of sway response for wave direction 225° 6.7 Comparison of heave response for wave direction 225° 6.8 Comparison of roll response for wave direction 225° 6.9 Comparison of pitch response for wave direction 225°
6. 10 Comparison of yaw response for wave direction 225°
6.11 Comparison of surge response for wave direction 250° 6.12 Comparison of sway response for wave direction 250°
6.13 Comparison of heave response for wave direction 250°
6. 14 Comparison of roll response for wave direction 250°
6.15 Comparison of pitch response for wave direction 250° 6.16 Comparison of yaw response for wave direction 250° 6.17 Comparison of sway response for wave direction 270° 6.18 Comparison of heave response for wave direction 2700
6. 19 Comparison of roll response for wave direction 270°
7 Comparison between prototype roll spectrum of ROO2A and that computed by NSRDC's model HANSEL
8 Scheniatization of fluid around a ship using finite elements 9.1 Surge motions computed by various mathematical models
9.2 Sway motions computed by various mathematical models 9.3 Heave motions computed by various mathematical models 9.4 Roll motions computed by various mathematical models 9.5 Pitch motions computed by various mathematical models 9.6 Yaw motions computed by various mathematical models
LIST OF FIGURES (continued)
APPENDIX B
Figure Bl General ship layout and co-ordinate and sign conventions B2 Ship's lines drawings
B3 Time history of wave conditions during RUN ROO2A
B4 Instantaneous levels of target points on ship-F(I), RUN ROO2A B5 Wave-induced principal motions of ship-F(I), RUN ROO2A
B6 Vertical motions at critical points on ship-F(I), RUN ROO2A B7 Route, heading and speed for ship-F(I), RUN ROO2A.
APPENDIX C
Figure Cl Surge and sway response by VESDYN and HANSEL C2 Heave and roll response by VESDYN and HANSEL C3 Pitch and yaw response by VESDYN and HANSEL
LIST OF APPENDICES
Appendix A: Itinerary of overseas visit by J Moes, 10 April to 9 May 1981 Data for test run with mathematical model for a free-moving ship
Computed response functions by O.E.C. using VESDYN and NRIO using modified HANSEL program (Te 10, I s; d = 24,L ni)
LIST OF SYMBOLS
wave amplitude (m)
added mass or inertia coefficients
frequency-independent added mass coefficients damping coefficients
critical damping coefficient viscous damping coefficient
boundary between ship's hull and RI boundary between Rl and R2
wave celerity (m/s)
intersection of the ship's hull and the free surface restoration coefficients
sectional area coefficient water depth (m)
ship's draught (m)
wave encounter frequency (Hz) external forces or moments Froude number
free surface
acceleration due to gravity (mis2)
= the Green's function or source potentials functional of velocity potential
wave number = 2ïr/X (rn1) viscous roll damping factor retardation function
= ship's length between perpendiculars (m) = mass or inertia of the ship
n = outward normal to the hull surface
Cw = C = Cj = Cs = d = D = = F F = = g = = k = K = K1 = a =
Aj
= =Bj
= Bcrit = By = BI = B2 =p = pressure in the fluid (N/rn2)
Q. = source density function (complex source strength) Rl = inner finite element region
R2 = outer finite element region S = wetted hull section (m2) S = water plane area of ship (m2)
wp
t = time (s)
T = wave period (s)
u = x-cornponent of U (m/s)
Um = maximum velocity amplitude at B] (m/s) U = fluid velocity (m/s)
y = y-component of U (m/s)
V = average forward speed of ship (m/s) = displaced volume of water by ship (m3) w = z-component of U (mis)
x = n = horizontal bow-ward coordinate axis of ship, surge motion (m) y = fl horizontal port side-ward coordinate axis of ship, sway motion (m)
z = = vertical upward coordinate axis of ship, heave motion (m) ZB = z-coordinate of centre of buoyancy (m)
= angle of wave incidence
parameter for an "almost perfect" fluid
= displacement or rotation, namely, x, y, z, , O or 4
T.
= first derivative of r with respect to time (velocity)fj. second derivative of fl with respect to time (acceleration)
O = rotation around y-axis, pitch motion A = wave length (m)
p = mass density of the fluid (kg/rn3) = = rotation around x-axis, roll motion
= velocity potential (m2Is) = impulsive displacement
= n6 = rotation around z-axis, yaw motion = normalized velocity potential w = radial wave frequency = 2Tr/T (s')
Subscripts: i and j = modes of motion, namely:
e.g. A35 added mass in heave due to pitch motion O = incident wave
7 = diffracted wave w = wave
Mathematical model names:
AQUADYN: model of ENSM - section 4.2 DIFFRAC: model of MARIN - section 4,3 DYNRES6: model of Garrison - section 4.5 HANSEL: model of NSRDC - section 5.3 MODEL 2.4: model of ENSM - section 4.2
NMIWAVE: model of NMI - section 4.6 NV459: model of DnV - section 4.7 TRITON: model of IFP - section 6.3
1 2 3 4 5 6
i = I surge-surge surge-sway surge-heave surge-roll surge-pitch surge-yaw
2 sway-surge sway-sway sway-heave sway-roll sway-pitch sway-yaw 3 heave-surge heave-sway heave-heave heave-roll heave-pitch heave-yaw 4 roll-surge roll-sway roll-heave roll-roll roll-pitch roll-yaw 5 pitch-surge pitch-sway pitch-heave pitch-roll pitch-pitch pitch-yaw 6 yaw-surge yaw-sway yaw-heave yaw-roll yaw-pitch yaw-yaw
SUMMARY
An account is given of discussions held at a number of overseas univer-sities and institutions and with various consultants to determine the suitability and availability of mathematical models for the computation of wave-induced motions of large ships in shallow water and having forward speed.
Section 1 gives an introduction to the research on ship motions at NRIO
and specifically the efforts and achievements so far. In Section 2 are listed the specific requirements which should be satisfied by a mathe-matical model to be obtained by NRIO. Furthermore, a list of preferred properties of such a model is given. Section 3 gives a brief review of the basic theory used in the majority of the models. Four different theories have been used in the development of mathematical models, namely, the three-dimensional source technique (3DST), the strip theory, the finite element technique and the long-wave approach.
In Section 4 síx 3DST mode/_s are reviewed, after an introduction to the underlying theory. It appears that models based on the 3DST have been widely and successfully developed. Two of these models, namely, those
of Det norske Ventas (DuV) and the National Maritime Institute (NNI), are not available to NRIO. Of the remaining four models those of the Ecole Nationale Supérieure de Mecanique (ENSM) and C J Garrison and Associates (CJG) are basic diffraction models for the computation of
forces and motions. The ENSM model is further developed but has only recently been completed and has not yet been calibrated and documented. The other two models, namely, those of the Maritime Research Institute Netherlands (MARIN) and the Ocean Engineering Consultants (OEC) have been developed as part of a suite of programs for the computation of moored ship dynamics.
Many strip theory models have been developed (see Section 5), but almost all of them are applicable only to deep-water conditions. One model at the Institut fir Schiffbau (IFS), however, incorporates both shallow-water and forward-speed conditions. Model output compares rather well
with physical model test results. Shallow-water corrections are made by using empirical coefficients. The theory as such is less applicable to shallow-water conditions than the 3DST and may be applied only in the case of slender bodies, such as ships.
Four hybrid finite element models have been found (see Section 6), of which three are not (yet) sufficiently well-developed to suit NRIO's requirements. The fourth model, that of the Institut Français du Pétrole
(IFP), has been abandoned by its developers in favour of a 3DST model and is not in active use any more.
The only non-linear model that has been traced is the long-wave theory model of the Laboratoire National d'Hydraulique (LNH) (see Section 7). Although this model is not subject to the restriction of linearity as the
other models áre, a number of other simplifications have been introduced in the theory which causes its domain of application to deviate from the domain of interest of NRIO. Furthermore, the model is not documented, is expensive and is not in active use any more.
In Section 8 the results of a modelling workshop organised by the IFP are discussed. At this workshop a number of the models mentioned above were
compared, but no definite conclusions were drawn as to which model is most efficient or accurate.
Test runs have been performed with a number of models using data pertaining to a prototype run monitored at Richards Bay. The results of these test runs and some additional information on later modifications of the MARIN and OEC models are discussed.
Apart from discussions on specific models, discussions were held at the Massachusetts Institute of Technology (MIT) on the theory of ship motions and to obtain their advice as to which theory or model would be most Suit-able for NRIO's needs (see Section 9).
In a final section the results of the discussions are summarized and recommendations are made. It appears that the 3DST will be the most suitable and flexible theory to be used as a basis of a mathematical
model. In this respect the package of models of MARIN appears to Suit almost all the requirements while the model of OEC appears to be the next best. The conditions discussed with MARIN, namely, an exchange
agreement for prototype data, appear attractive.
Besides the 3DST it will be useful to install the (gratis) strip theory program of the IFS on the CSIR computer for the purposes of comparison and as a less CPU time-consuming "pre-run" model.
INTRODUCTION
It is expected that the bulk export of two major raw materials, namely, iron ore and coal, from South Africa will increase during the coming years. It has already, therefore, been decided to increase the export of coal from the present 28 million tons to 44 million tons per year by 1985. Most of the raw materials will be exported via Richards Bay (coal) and
Saldanha Bay (iron ore). The present layout of these harbours is shown in Figures 1 and 2, respectively. This expected increase in the volume
of exports has raised two questions with regard to the economy of the shipping
Can the allowance criteria regarding the maximum draught of ships using the two harbours be relaxed and, if they can, to what extent?
What will be the required channel depth if ships with draughts greater than allowed at present are permitted to use the harbours?
The South African Transport Services requested NRIO to provide answers to these questions.
Figure 3 shows the factors which have to be quantified in order to deter-mine the depth of a navigation channel. A major contributing factor in determining the depth, besides the ship's draught, is the maximum vertical downward motion of the ship's bottom due to wave action. This maximum motion can be assessed from prototype measurements, from physical model
tests, or by using theoretical computation techniques, which are often called mathematical models, when they are assembled as suites of computer programs.
All the techniques incorporate assumptions and simplifications. lt is accepted, therefore, that a more accurate estimation of the vertical ship motions can be obtained from prototype measurements in conjunction with physical and mathematical model studies. Prototype monitoring of large ships at Richards Bay has been performed since August 1978 and at Saldanha Bay since June 1980. Descriptions of the techniques used and of the
accuracies obtained are given in an earlier report (CSIR, 1981). Physical model tests are planned to start during 1982.
Two surveys were made (Zwamborn, 1979, and Moes, 1980) to investigate
whether suitable mathematical models had been developed elsewhere and if so, whether they could be obtained by NRIO and applied to South African condi-tions. Seven models were found, while the source listing of one of the computer programs appeared in a report (NSRDC's program HANSEL in Meyers et al, 1975). This program has been installed on the CSIR's Control Data Cyber 174. However, this program is based on strip theory and is applicable only to deep-water conditions. The search for a more suitable model has
therefore been continued by further literature surveys and correspondence. As a result, eleven apparently suitable models have been found. Subsequently,
a trip was made by Messrs. J A Zwamborn and J Moes to investigate in more detail the applicability of these models to conditions at Richards Bay and Saldanha Bay, and to discuss the possibility of NRIO's acquiring the computer programs. On this trip, which took place during April and May
1981, two more models were found and a visit to one of the relevant in-stitutions was included in the itinerary. The itinerary is included as Appendix A.
In more detail the aims of the discussions with the developers of the various models were to
explain NRIO's specific requirements;
determine the quality and availability of the models;
discuss the possibility of extensions to, or changes in, the programs;
find out about possible cooperation in changing the model when necessary;
investigate the available documentation of the program and the related theory;
learn about specific computer and program-related aspects of the models;
exchange prototype data in return for the mathematical models, and other commercial aspects;
learn in general about the status of research on ship motions and associated investigation techniques; and to
discuss the results of a test run with conditions at Richards Bay (monitoring run ROO2A). These data (Appendix B) were sent to most of the organizations to be visited with the request that a model run be performed using these data. The results could then be com-pared with the prototype data during the visit.
2 REQUIREMENTS OF MATHEMATICAL MODELS
A mathematical model must meet a number of requirements if it is to be applicable to local conditions and if it is to yield reliable results. These requirements are
The model should operate in six degrees of freedom. Although the principal motions of heave, roll and pitch determine the vertical motions of the ship, these motions are coupled with the other three principal (horizontal) motions. For a ship with lateral symmetry, heave and pitch are coupled only to surge, while roll is coupled only to sway and yaw. Therefore, all six degrees of freedom should be computed, including appropriate coupling coefficients.
The model should be applicable to finite water depth with a ratio of water depth over ship's draught down to about 1,1 (10 per cent of the water depth as total underkeel clearance). At present the minimum required total underkeel clearance at Richards Bay is about 40 per cent and at Saldanha Bay about 15 per cent. It is accepted that calculations for underkeel clearance down to about IO per cent will cover the full range of practical conditions. Van der Doel
(1971) reported that for the Europort Harbour entrance a minimum underkeel clearance of 20 per cent is required for the outer channel,
15 per cent for the outer harbour area and 10 per cent for the inner harbour area.
It has been shown by Kim (1968) and van Oortmerssen (1976) that for a depth-to-draught ratio of less than 1,5 the underkeel clearance has a considerable influence on the vertical motions of the ship. This influence should be allowed for in the model.
The effect of forward speed of the ship should be allowed for in the computation routines. The expected maximum speed of the ships is about 6 rn/s (12 knots). This results in a maximum Froude number of about 0,12 calculated from the following equation
vs
F=
where F = Froude number n
V = ship's speed (m/s)
g = acceleration due to gravity (taken as 9,80 mIs2)
= length of ship between perpendiculars (a minimum of 250 m has been used).
the model should have been sufficiently calibrated and verified either with prototype measurements or with physical model test results. It
is realised that accurate prototype data on (vertical) ship motions are very scarce, so that calibration or satisfactory comparison with phy-sical model test results would be acceptable. The results from physical and mathematical models for similar conditions should show good agreement, particularly for extreme motions, that is, at the resonance frequencies for heave, roll and pitch. Calibration or satisfactory comparison should have been obtained for a number of wave directions, especially for waves from the beam, at which wave direction roll motions will be greatest. (In order to facilitate the evaluation of the models, a set of proto-type data was sent ahead to the various persons to be visited with the request that a trial run be performed with their model at the specified conditions. These conditions were those of monitoring run ROO2A, which showed large roll motions. The data for this run are included in
Appendix B).
The computer program should be available as a symbolic source listing in order to be able to read the computation statements. It is then possible to (re-)calibrate the mathematical model at NRIO with
proto-type data obtained from the monitoring runs at Richards Bay and Saldanha Bay. This means that changes would possibly have to be made to the program. Furthermore, the program should not be used as a "black box", but should rather be used to investigate the effects of certain changes, and the program should be updated when better techniques or more
accurate coefficients appear in the literature or are derived from basic physical model tests at NRIO.
In order that the user will be able to learn the working of the model the program should be reasonably well documented. This means that the theory which forms the basis of the model should be described in de-tail. Furthermore, a user's manual should be available so that the operation of the model can be learnt easily. The program should also contain internal comments explaining the functions and purposes of the main parts of the program.
The computer memory storage requirements should not exceed about 250k (decimal) words. This is the maximum central memory space avail-able at present (using 60-bit words) at the CSIR's Cyber 174 computer to which NRIO is linked. Expansion of the memory in the near future is not expected.
The possible cost of the model should be within the limits of the amount allocated for this purpose in the budget.
There are a number of aspects which are not requirements, but which would be advantageous to take into account and which should preferably be included
in the evaluation of the various models. A number of these aspects are related to the use of the model for computing the motions of and forces on moored ships. At Saldanha Bay the problem has been experienced of moored ships breaking their mooring lines. The South African Transport Services have requested NRIO to investigate this problem. One of the tools in such an investigation may be a mathematical model. Since the basic theory for the computation of the motions of free-moving and moores ships is the same, it would be advantageous to obtain a mathematical model that could also compute moored-ship conditions realistically.
Specific aspects that would allow the computation of moored ship motions and related mooring forces are
The model should allow for (preferably non-linear) mooring forces from specified force-elongation characteristics of the mooring lines and force-deflection characteristics of the fenders for at least twenty mooring lines and eight fenders.
The model should be capable of computing low-frequency drift forces (second-order horizontal wave forces) which generate slow oscillations of a moored ship when these oscillations coincide with, or are close to, the natural frequency of
(horizontal) oscillation of a moored ship.
The effects of wind and current should be allowed for in the model.
The effect of the presence of a quay or jetty should be allowed for in the model.
Furthermore, the following aspects would in general be of advantage if allowed for in the model
(y) Additional damping of the ship's motions due to viscous effects, vortex shedding and the presence of bilge keels and a bulbous bow should be computed by the model and not have to be specified by the user as additional damping coefficients.
For models in which the Fredholm integral equation is used for the computation of source densities, so-called irregular frequencies exist for which the solution is undefined (see John, 1950). The model should incorporate a routine for overcoming the effect of
these irregular frequencies.
The general structure of the model should be logical and the model should be efficient in using the computer to reduce running
(CPU) time. Since FORTRAN is the most widely used programming language at NRIO the program should preferably be written in
FORTRAN. If any plotting routines are incorporated in the program it would be advantageous if CALCOMP routines were to be used for them since they are also used for the control of NRIO's Houston plotter.
Since it is expected that in the future the model will be changed or extended, it would be very helpful if the developer of the program were to give further assistance in this respect, as he will be familiar with the theory and the program.
Since the first application of the model will be to ships (slender elongated bodies) it would be economic if the ship's hull could be schematized to facets of similar elongated shape. If these facets are described by more than three points, pro-vision should be made to obtain the most realistic position and orientation of the facets.
Because of the nature of the prototype data it would be useful if input to and output from the model were in the time domain.
It would be advantageous if the SI system of units was used in the program since most available data are expressed in these units.
The above requirements and preferred properties of a mathematical model are summarized as follows
Requirements
Six degrees of freedom.
Finite water depth (d/D down to 1,1). vs
Ship's speed (Fn
= up to 0,12). pp
Model calibrated/verified for various wave directions including resonance frequencies.
Source code listing of programs. Theory and programs documented.
Required computer memory < 250 k words. Costs less than budgeted.
Preferred Properties
Non-linear mooring forces
low-frequency drift forces for the simulation of wind and current forces mooring conditions presence of quay, jetty
(y) computation of additional (viscous) damping no, or overcome "irregular frequencies"
structured and CPU efficient programs written in FORTRAN cooperation in further developments
elongated projected facets time domain input and output SI units used.
3 BASIC THEORIES
The motions of a floating object subject to wave action can be described by the following matrix equation of motion
(M.. + A..) . + B. .fì. + C. .fl. = F.
13 13 1 13 1 13 1 1
where M. . = mass or inertia of the object in the i-th mode of 13
motion due to motion in the j-th mode
= added mass coefficient matrix
13
= damping coefficient matrix
C. . = restoration coefficient matrix, due to hydrostatic
13
or (linear) mooring forces
= external forces or moments in the i-th mode
= displacement or rotation in the i-th mode of motion
6rì
(velocity)
-2n
-
(acceleration)In general, the values of M.. and C.. are known. In order to solve
1]
13Equation (3.1) and to compute the displacements and rotations the values of A. ., B. . and F. must first be determined.
13 13 1
For a floating ship Equation (3.1) is formulated for all six degrees of freedom, where each coefficient constitutes a six-by-six matrix, namely, coefficients for the principal motions and for the coupling between the various principal motions. The restoring coefficients for the vertical motions follow from the hydrostatic restoring, while for a moored ship
they may also include the restoring action due to the mooring lines and fenders. The external force Fj includes the oscillatory wave exciting force and moment, and possibly the wind and current force and moment. The main problem in solving Equation (3.1) is to find the values of the added mass and damping coefficients for a given ship, ship speed, wave condition and water depth.
Usually the procedure used to compute these coefficients is to use a velocity potential technique. The added mass and damping coefficients are related to the real and imaginary parts of the integral of the velocity potentials for the ship's hull due to the ship's motions (see next section). The external wave force can be computed from the incident and diffracted wave velocity potential. The diffracted wave and ship motion velocity potentials can be expressed as the integral of the source density function and the Green's function. The Green's function can be solved by the procedures described by Wehausen and Laitone (1969). The source densíties can be found by solving the Fredhoim integral equations of the second kind.
The integration is usually performed over small hull surface parts (facets or panels) into which the total hull surface is divided. The wave-exciting forces and moments can be computed from one of the available wave theories, for example, the linear or the vocoidal wave theory.
The above method of solving the set of equations of motion is applied by the so-called three-dimensional source technique (3DST) models, of which a number appear to be available and which are rather well developed and applicable to shallow-water ship motions. For slender bodies in deep water strip theory models, which are basically two-dimensional, have been developed, while for applying these models to shallow-water conditions empirical correction coefficients have been used. Another series of models based on potential wave theory has been developed by the use of finite elements to represent the fluid region around the ship's hull. Although this latter type of model can also be applied effectively to shallow-water conditions they are in general not as far developed as the 3DST and strip theory models.
In all the above models it is assumed that there is a
linear relationship
between wave amplitude and amplitude of principal ship motion, which is valid only for small motions. Non-linear effects may be due to viscous damping and high flow velocities of water under the keel at very smallunderkeel clearances. One model has been found which copes with non-linearity by using long-wave equations to compute the pressures between the ship and the bottom and the resulting ship motions.
In the following sections six 3DST models which have been found will be described and discussed. Thereafter, two strip theory models will be re-viewed. Four finite element models will then be discussed to illustrate their capabilities. Finally, a non-linear model will be reviewed. For each category of model a brief review of the underlying theory wíll precede the description of the model.
4.1 General Theory
The general aspects of the three-dimensional source technique (3DST) to solve the equations of motion and to compute the principal ship motions were presented in Section 3. The name of this technique is derived from the method of describing a flow field by using flow potential sources in three dimensions. (This technique may also be used in strip theory in which it is applied to two-dimensional sections). The hull of the ship is divided into facets (or panels) and a flow potential source is located at the centre of each facet. The characteristics of these sources are deter-mined by the surrounding flow conditions.
The basic formulae used in the 3DST will be summarized briefly. A more extensive description of the theory can be found, inter alia, in Newman
(1978).
If the fluid around the ship is irrotational, a velocity potential exists and the motions of the ship in a wave field can be solved by means of the velocity potential theory. In this case the ship is subject to the incident wave potential Because of the presence of the ship there is also a wave diffraction potential When the ship is allowed to move in six degrees of freedom each principal motion generates a radiation potential
with j = 1, 2, 3, 4, 5 and 6 for the motions in surge, sway, heave, roll, pitch and yaw, respectively. When linearity is assumed, that is, when only small motions occur, the total velocity potential is the sum of all
potentials.
4 THREE-DIMENSIONAL SOURCE TECHNIQUE MODELS
= +
+.
.o :1=1 3
(4.1)
The space derivative of the velocity potential yields the velocity components of the fluid, namely,
Sct
u
-,
y -a----andw
The real part of the incident wave potential o is
ag
w cosh{k(d+z)} cos{k(xcoscx + y sin)} w cosh{kd} wave amplitude (m)acceleration due to gravity (mis2)
radial wave frequency = 2rr/T (s')
wave period (s)
wave number = 2'iT/A
(1n1)
wave length (m)
water depth (m)
= angle of wave incidence
The diffraction and radiation potentials are solved by applying the equation of continuity and relevant boundary conditions. The three-dimensional
equation of continuity for an incompressible fluid of constant density is the Laplace equation
52
V2.
= J + J + J - 0 (4.3)x2 Sy2 Sz2
which is valid for j = 0,.. .7
The relevant boundary conditions for a coordínate system linked to a moving ship with average forward velocity V are
(i) at the bottom (z = -d) there is no vertical velocity
5z - O for j = 0,.. .7 (4.4) (4.2) where a = g = w = T = k = X = d =
(ii) at the ship's hull the normal velocity is equal to the local hull velocity
- V n. for j = 1,.. .6
sj
where n is the outward normal vector to the hull surface.
(iii) at the ship's hull the normal incident and diffracted wave velocities are opposite
(iv) at infinity the radiation condition yields
I 2
lun
r (__J - i u- (i).) = O for j = 1,.. .7
r±
(Sr g j(y) at the free surface (z = O)
(S2, (S2 (S2c
2V
J+V2
+ g(StZ s (Sx6t s cSx
The added mass coefficients are defined as
A. . = -p Re{f J.n.ds}
ij s
ji
and the damping coefficients are defined as
B.. = -pwIm
U
.f.n.ds} ijSji
(4.5) (4.6) (4.7) O for j = 0,.. .7 (4.8) (49) (4. 10)where Re and Im are the real and imaginary parts, respectively. The values of the indices i and j range from I to 6.
The wave-exciting forces and moments are obtained from the incident and diffracted potentials using the Bernoulli equation and by integrating
The Bernoulli equation is
+ + gz + = O for j = O,...7 p
where P = pressure in the fluid (N/rn2) p = mass density of the fluid (kg/rn3)
The pressure force F., as required in Equation (3.1), is
F. = -f
f
P.n. .ds for i = 1,...6 (4.12)i
si
In order to determine the velocity potentials ., j = 1,.. .7 from Equation (4.3), (4.4), (4.5), (4.6), (4.7) and (4.8) use ís made of a source distribution integral technique (see Chang, 1977)
j
4rry2
f
f.G.ds +QGnldY
S
jj
gc
where Q. = the source density function, defining the complex source strengths;
G. = the Green's function or source potentials of a source distribution over the ship's hull;
S = wetted hull surface;
C = intersection of the hull and the free surface;
n1 = the x-cornponent of the outward normal to the hull surface, i.e. in the direction of the forward speed of the ship.
The second part of that between brackets in Equation (4.13) is tue con-tribution of the forward speed. If there is no forward speed this term becomes zero.
The Green's function or source potential is the flow potential at a
particular facet due to a unit displacement of another facet of the ship's hull.
(4.11)
M.. =
1]
The Green's function can be computed using one of the formulations of Wehausen and Laitone (1969). The source densities are found by means of Equation (4.13) together with the boundary conditions at the ship's hull, namely, Equations (4.5) and (4.6). When the Green's function and
the source density function are computed the velocity potential can be determined from Equation (4.13). Thereafter, the added mass and damping
coefficients can be computed, as well as the wave-exciting forces, by the use of Equations (4.9), (4.10), (4.11) and (4.12).
It can be seen from the above equations that the only influence of the ship's speed on the boundary conditions is the free-surface boundary condition, Equation (4.8). If the ship's speed is zero the free-surface boundary condition reduces to
52
- O for j = 0,.. .7 (4.14)
Where the ship has lateral symmetry the principal motions of surge, heave and pitch are independent of the motions of sway, roll and yaw. This can be seen from the resulting matrices of the ship's mass, added mass and damping coefficients in the case of lateral symmetry
M O O O M O O O M O _MZG O MzG O O O O O (4.15)
where M = mass of the ship (kg) Ii = moment of inertia (kg m2)
= product of inertia (kg m2)
ZG = vertical coordinate of centre of gravity (usually the origin of the coordinate system is at the intersection of the ship's vertical centre line and the still-water level and is fixed
to the ship) (m). O Mzc O _MZG O O O O O O 0 15 0 0 16
=
is the damping coefficient matrix analogue to the added mass matrix. The restoring matrix C.. is due to hydrostatic forces (only vertical)
rilO
A13 O A15 O
O A22 O A O A26
A31 O A33 O A35 O
O
A2
O A4 O A46A51 O A53 O A55 O
O A62 O A6
o A66
where S = water plane area of ship (m2)
= displaced volume of water (m3)
ZB vertical coordinate of centre of buoyancy (m)
The above integral equations are transformed into finite integral equations and solved for each facet on the ship's hull. The coefficients and forces for the entire ship are determined by integrating over the hull surface. Thereafter the equations of motion (3.1) can be solved to yield the dis-placements and rotations of the ship.
Of six 3DST models which have been found five are rather similar. These models do not incorporate the forward speed of the ship in the equations, that is, they use Equation (4.14) instead of (4.8).
(4. 16) 13 values of C.. are 13 C33 pgS wp
or (linear) mooring forces. For a free-moving ship the only non-zero
(4. 17) CRLf = pg (zB - zG) + pgfy2ds wp (4. 18) C55 = pg (zB - zG) + pgfx2ds w? (4.19) C35 = C53 = -pgfxds wp (4.20)
These models are those of
the Maritime Research Institute Netherlands (MARIN)
the Ocean Engineering Consultants (OEC)
Dr C J Garrison (CJG)
the National Maritime Institute (NMI) and
(y) Det norske Ventas (DnV).
The sixth model, which includes the effect of forward speed is that of
(vi) the Ecole Nationale Supérieure de Mécanique (ENSM)
These models will be discussed in some detail in the following sections.
4.2 ENSM Models
During the past five years a number of postgraduate students of the Ecole Nationale Supérieure de Mécanique of the University of Nantes in France have written doctoral theses on the subject of wave-induced ship motions. Most of the work is rather theoretical and is done under the supervision of Professor P. Guevel. The latest thesis of this series was written by Dr J. Bougis and as part of this work he compiled a mathematical model
for the computation of wave-induced ship motions in shallow water, in-cluding the effect of the forward speed of the ship (Bougis, 1980).
In his model, which in correspondence is identified as Model 2.4, Bougis applies Equations (4.2) and (4.4) for shallow water and Equation (4.8) for
the influence of forward speed. Furthermore, in Equation (4.8) two addi-tional terms are included which account for the fact that the fluid around the ship is not "perfect" but "almost perfect".
Equation (4.8) then becomes
2V + V2 + g + 2E - 2uE = 0 for j = 0,.. .7 (4.21)
S St óx
where the value of E is very small and represents the influence of the "almost perfect" fluid in Euler's equation.
Although the model is theoretically well-developed, it has not yet been calibrated, nor has the influence of the various forward speed-dependent terms been quantified. Physical model tests are under way at the Basin d'Essais des Carènes, a naval ship-testing basin in Paris, to provide cali-bration data for the mathematical model. It was intended to have the program documented and available to clients by September 1981, but by January 1982 this had not yet been confirmed.
As with the other 3DST models the model of Bougis uses a single source distribution technique. When a single source distribution is applied, irregular frequencies may occur (see, for example, Faltinsen and Michelsen,
1974, and John, 1950). One possible technique which may overcome irregular frequencies is the application of a source distribution not only along the wet part of the hull, but also at the inside of the ship at a plane just below still-water level. Bougis, however, applies a technique developed by Kobus (1976) namely, an interpolation technique which is much cheaper
than the technique mentioned above. (The running time of a 3DST model is a function of about the square of the number of facets). If a method can be developed to overcome the irregular frequencies in a mixed-distribution technique in which sources and doublets are used, then a mixed-source distri-bution version of Program 2.4 will also be compiled. Model 2.4 does not (yet)
include moored-ship conditions, viscous effects or drift forces. Because of the absence of viscous effects the ship motions at roll resonance will be strongly overpredicted. Because of the inclusion of the speed-dependent terms the running time of the model is relatively large.
Besides Model 2.4 another model has been developed at the ENSM, namely, Model 2.2: AQUADYN, which is rather similar to Model 2.4 but which does
not include the speed terms. This model is fully operational and documented and has been sold to a number of French clients such as the Bureau Ventas (the French Ship Registration and Classification Bureau), the Institut Français du Pétrole (see Section 6.3) and the Chantier de Bretagne (a shipyard).
AQUADYN is available in two versions, one of which uses a single-source distribution technique while the other uses a mixed-distribution technique of sources and doublets. AQUADYN includes the computation of drift forces, but viscous effects are not included. Model 2.4 requires about four times
as much CPU time as AQUADYN for a similar run. Even for zero speed Model 2.4 requires a relatively large amount of CPU time. If, therefore, a model is required to be used for the computation of moored-ship conditions (zero speed) AQUADYN would be the most economical one to use for the computation of the
added mass and damping coefficients.
Model 2.4 can be bought for FF 300 000 (about R50 000) and AQUADYN for FF 200 000 (about R35 000). ENSM is not interested in obtaining NRIO's prototype data in (partial) exchange for their model(s). Professor Guevel
is willing to make any additions to the program on a services-rendered basis. The required central computer memory for Model 2.4 is about 100k decimal words.
It appears that as regards the requirements for a mathematical model for the computation of free-moving ships as formulated in Section 2, the Model 2.4 satisfied conditions (a), (b), (c), (e), (g) and (h), while conditions (d) and (f) are worked at. With regard to the preferred properties, ít appears that Model 2.4 is not very suitable as a basis for extensions toward meeting the preferred properties. For zero-speed conditions (e.g. moored-ship condi-tions) model AQUADYN appears to be more suitable, because it requires conside-rably less CPU time. Although all models are written in FORTRAN4 the
relevant documentation is in French.
No test run
with NRIO's prototype data was performed, nor have any test results from Model 2.4 been received.The most important aspects of the various models are listed in Table I, while a number of more detailed aspects of 3DST models are listed in Table II. These tables ease evaluation of each model and will facilitate
a final choice as to which model to acquire. Mr Berhault has indicated (see Section 6.3) that a (probably binary) version of model AQUADYN may be available from Control Data Cybernet Services (CDCS), but upon enquiry no response from CDC, South Africa, has been received yet.
4.3 MARIN Model
At the Maritime Research Institute Netherlands (MARIN), the former Netherlands Ship Model Basin (NSMB), Dr G van Oortmerssen has developed a suite of computer programs which constitute a mathematical model for the computation of
moored-ship conditions (van Oortmerssen, 1976). The effect of forward speed of the ship has therefore not been included. This model consists of a number of computation steps. Program DIFFRAC is a 3DST program that computes the added mass and damping matrix due to first-order wave-force components. Subsequently, the motions of the ship in six degrees of freedom are computed for each spe-cified wave frequency. Such a solution in the frequency domain allows the inclusion of only linear mooring forces in the restoration coefficient matrix
(see Equation 3.1). However, large horizontal motions of moored ships can result from the non-linearity of the mooring forces. Therefore, a time-domain convolution is applied, using the theory of Cumins, based on an impulse
response function technique (Cummins, 1962). The equation of motion (Equation 3.1) is then written as
(M.. + + K..(t - T).(T)dT + C... = F. for i and j = 1,...6 (4.22)
Compared with Equation (3. 1) the damping coefficient matrix B.. is replaced by an integral of the retardation function K.. while the frequency-dependent added mass coefficients A.. are replaced by frequency-independent added mass coefficients At.. The parameters are defined as follows
13
6.(t)
with
x.
&p. 3-
g J K. .(t) 2 B..(w) cos wt dw 13TÍo
(4.25)The relationship between the retardation function and the frequency-dependent damping coefficients is
(4.26)
The relationship between the frequency-independent added mass coefficient and the frequency-dependent added mass coefficient is
A.
= A. .(w1)17 K..(t)
sin w1t dt13 13 +
wo
13where in Equation (4.27) w1 is an arbitrarily chosen value of w.
(4.27)
This time-domain convolution and subsequent mooring condition computations are performed by program MOORSIM, using the output from DIFFRAC. The mean and the low-frequency second-order wave-exciting forces and moments are computed by the programs DB DRIFT and DRIFT P, which are based on the re-search of Pinkster (1980). Dr R G Standing of NMI (see Section 4.6) has some reservations about the accuracy of Pinkster's method. This could, however, be investigated in more detail at a later stage. The retardation functions are computed by program IRFUN and the first- and second-order wave-force impulse response functions by IMPRESI and INPRES2, respectively.
where x3(t) = normalized velocity potential of the ship with respect to an impulsive displacement Ax.
= the normalized velocity potential of the ship with respect to an impulsive velocity of the ship V..
A.
= pf..ds
(4.24)The model results are analysed statistically using the program STATAN. Figure 4 shows the organization diagram of the MARIN model. This package of programs has been developed mainly as an in-house system, but a limited part is documented and available to external users through a computer bureau in Petten, the Netherlands. These MARIN programs have never been sold as a source code listing.
Although this program package appears to be well developed and to have been extensively tested, the major drawback is that the effect of
forward speed
ofthe ship is not included. In discussing this shortcoming, there appear to be three possible approaches
1. Include the effect of forward speed in the frequency of wave encounter and neglect further speed influences. The encounter frequency e is
(Moes, 1981)
(c - V cosa)/X
wave celerity (m/s) speed of the ship (m/s)
angle between ship's velocity vector and wave celerity vector
X = local wave length (m)
Include the effect of the ship's speed using the formulation of Salvesen et al (1970). In this approach Equation (4.14) is used instead of Equation (4.8), but speed-dependent terms are included in a number of the added mass and damping coefficients and thus in the equations of motion. This method has been incorporated by Meyers et al (1975) in the NSRDC strip theory model (see Section
5.3) and by Inglis and Price (1981).
Use the correct formulation of the speed effect in the free-surface boundary condition (see Equation (4.8)) as applied by Chang (1977) and Inglis and Price (1981).
(4.28) =
where c =
V =
Inglis and Price (1981) have compared the last two alternatives for deep-water conditions. The results are also compared with the results of a strip theory model. Inglis and Price conclude that alternative No. 3 gives the most accurate results, but that it requires considerable CPU
time. Alternative No. 2 appears to be a good compromise in required CPU time and accuracy. Van Oortrnerssen therefore recommends alternative No. 2 for adjusting program DIFFRAC to include the effect of the forward speed of the ship. It is assumed that the conclusions of Inglis and Price also hold for shallow-water conditions.
In correspondence since the visit MARIN indicated that under certain conditions
they are willing to make the source code of the programs mentioned above available to NRIO;
they are interested in obtaining prototype data as part of an exchange agreement; and
they will incorporate the effect of the forward speed of the ship in DIFFRAC as indicated above under alternative No. 2, to be performed by the end of 1981.
For realistic predictions of the motions of the ship at near-to-roll resonance frequencies additional viscous damping has to be allowed for. For low-frequency motions potential flow damping tends to zero and the damping can be only viscous. Provision has been made in DIFFRAC to specify viscous damping coefficients.
Furthermore, for economic and realistic simulation the facet dimensions should be between 0,1 and 0,2 of the wave length. The aspect ratio of the facets in DIFFRAC should not be larger than about 2. In general, this will require between 150 and 200 facets for a large ship.
After the changes the MARIN suite of computer programs will be very close to the requirements and preferred properties as formulated in Section 2. MARIN also appears to be prepared to cooperate further with NRIO with regard
to mathematical modelling and research of free-moving and moored-ship motions.
No test run
with NRIO's prototype data was performed although an assurancewas made that this would be done after the forward-speed adjustments had been made at the end of 1981.
4.4. OEC Model
Professor L H Seídl of the Department of Ocean Engineering at the University of Hawaii is also a partner in Ocean Engineering Consultants, Inc., in
Honolulu. Initially he developed a strip theory model for the computation of moored-ship conditions called MOSA2 (OEC, 1978). MOSA2 was later in-corporated into a modularized program package VESDYN. Recently, program HC3AHI was compiled and included in VESDYN. This program utilizes a three-dimensional source distribution technique as developed by Faltinsen and Michelsen (1974) and can be applied to arbitrarily shaped hulls. As with
other 3DST models this program uses quadrilateral facets, defined by the coordinates of four corner points. These corner points at curved sections of the hull will usually not lie in a plane. The program includes a routine
for determining the best plane for each facet and also for finding the projections of the corner points onto the facet plane.
VESDYN is very similar to the suite of programs of MARIN described in the previous section and also does not include the effects of forward speed of the ship and viscous damping. Together with van Oortmerssen of MARIN, Seidi believes that the approaches of Bougis (see Section 4.2) and of M.-S.Chang
(1977), who apply the correct formulation of the free-surface boundary con-dition according to Equations (4.21) and (4.8), respectively, are rigorous, but may not be worth the effort in programming and CPU time having regard to the accuracy of the results obtained. Seidi also favours the approach of Salvesen et al (Alternative No. 2 in Section 4.3). In correspondence after
the visit Seidl has indicated that he has included the speed effect in this way in VESDYN. He believes this method to give good results for Froude numbers of up to 0,3, while NRIO requires values of up to only 0,12. Apart
from the possibility of being able to simulate free-moving and moored-ship conditions, VESDYN can also be applied to simulate berthing impacts, which may later be of advantage to NRIO. For the inclusion of viscous damping it
seems best to obtain coefficients from physical model tests. Reliable theo-retical methods are still lacking and some concentrated research is needed. Moreover, viscous effects are non-linear and it will be difficult to incor-porate this non-linear effect into a linear model. However, this may be
realised in the time domain, similar to the non-linear mooring forces approach.
A special routine has been compiled to simulate the effect of low-frequency wave forces, by which hardly any vertical motions will be induced. There-fore only planar motions are computed. The driving force for these slow motions is the slowly varying wave drift force, which is computed by using two trains of regular waves and slightly different periods (so-called bi-chromatic waves). Furthermore, a directional spectrum consisting of seven wave directions can be applied, each direction being governed by a spreading
function. The model operates in imperial units.
The test run using NRIO's monitoring data (see Appendix 3) could not be performed at the time of the visit, but Seidi promised to perform the test run as soon as the effect of forward speed had been included in the model. The effect of forward speed was included during October 1981 and the
results of the subsequent test run are presented in Table III and Appendix C. Results of runs with the model HANSEL (see Section 5.3) are also plotted
in Appendix C. Seidl is also prepared to act as a consultant to
NR1O for
any future work that may be required for the ship motion studies. No publications were available to show the quality of the computations for conditions of a large ship in a relatively shallow channel.The cost of VESDYN for the computatíon of free-moving and moored-ship conditions is US$ 68 500 (about R65 000), but the immediate requirement
(only free-moving ship conditions under regular waves) of NRIO could be met for US$ 31 500.
In exchange for prototype data Seidi is prepared to devote some of his time to NRIO's specific requirements. It will be possible to buy the rest of VESDYN later for about US$ 30 000. Updating of the program can be done an-nually for about 2,5 per cent of purchase costs. Notwithstanding the lack of proof of good correlations of model results with prototype or physical model data, the VESDYN package appears to be suitable for NRIO's present
needs as well as for possible future applications to the computation of moored-ship conditions.
4.5- CJG Model
Dr C J Garrison developed his mathematical model DYNRES6 while he was a professor at the Naval Postgraduate School in Monterey, California (Garrison
1977). Garrison was one of the first to compile a 3DST model. The theory is similar to that applied in van Oortmerssen's program DIFFRAC (see Section 4.3) and in Seidi's program HC3AHI (see Section 4.4), and as described in some detail in Section 4.1.
No forward-speed effects and additional viscous damping are incorporated. However, the computation of mean drift forces is included. This can be done
either by using the momentum approach of Faltinsen and Michelsen (1974) or by using the pressure integral approach of Pinkster (1980). The Green's function as formulated by Wehausen and Laitone (1969) is also used but the Primary Value integral is solved in two ways. For facets which are not too close together the series solution of John (1950) is followed, using a hy-perbolic function which can be pre-computed and stored. For those facets which are close together an integral solution is applied which is more time-consuming. Because Garrison uses a linear variation of the hydraulic conditions over the length of each facet, his model may use rather elongated facets. This could restrict the number of facets and consequently result in saving of CPU time. Garrison has also tried to schematize the ship's hull by using triangular facets, but so far without success. A method of checking the numerical model results comprises comparing the wave energy that is transmitted through a remote control plane formed around the ship and the energy contained in the motions of the ship (the so-called Haskind relationships). A table which gives the various energies and the percentage
difference is included in the computer printout of the model. The theory as well as the use of the program are documented in detail, and the model is being used by a number of clients. Computations of the motions of barges and pontoons show good agreement with physical model test results.
With regard to the possible inclusion of forward speed in the model, Garrison, together with van Oortmerssen and Seidl, is of the opinion that the inclusion of speed-dependent coefficients will be a very practical solution (see
Section 4.3, Alternative 2). Although Garrison is interested in including forward-speed effects he has no intention of doing so at present on his own initiative. However, he is prepared to do this for NRIO on a services-rendered basis. The inclusion of viscous damping can be performed easily, but the
damping coefficients have to be supplied since at present there is no accurate theory according to which these coefficients can be computed.
Garrison has performed the
test run
with the Richards Bay prototype data supplied by NRIO (see Appendix B). Because of the relatively high expense of running this 3DST model, only one frequency, the encounter frequency, is specified. It appears that the computed vertical motions are much smaller than the prototype motions. However, this is because response functions of the ship are narrow and have their peak values well above the encounter period of 10,1 s. The computed motions are presented in Table IVa, while an example of the output tables is presented in Appendix D.In order to obtain further information on the model results for frequencies close to the resonance frequencies of the vertical motions it was decided during the visit to perform another test run for an encounter period of
12,7 s (the measured periods of oscillation from the prototype recording -see Appendix B, Figures B5 - are T = 12,7 s, T = 12,9 s and T
heave roll pitch
= 12,5 s). The cost of this additional run, namely, US$ 500, was paid by NRIO. The results of this second run are summarized in Table P/b, from which it can be seen that the results are of the same order of magnitude
as that of the prototype measurements. Furthermore, it can be seen that the vertical motions are much larger than those for the encounter period of 10,1 s. However, it should be borne in mind that because the response
functions have relatively narrow peaks, the real wave energy for each frequency should be supplied to the model in order to check the quality of the model against prototype data. A comparison using the full frequency range will be required ultimately.
With regard to mooring conditions, model DYNRES6 does include the effect of linear mooring forces. Garrison has a pre-processor program to transform a mooring layout into a (linear) six-by-six mooring force matrix. This matrix
is added to the hydrostatic restoration coefficient matrix (see Equation 3.1). Garrison is of the opinion that the non-linear characteristics of the mooring
forces are not very important and therefore he does not plan to add a time domain convolution to the frequency domain routine.
A spectral post-processor program is available, which performs statistical computations for the horizontal motions resulting from a wave spectrum. This program, SPECMOR6, will later be extended to include all six degrees
of freedom. The cost of DYNRES6 is US$ 14 000 (about R 13 000), which includes the cost of the mooring pre-processor, a tape with the programs in source code and example runs, the manual and relevant literature on the theory. A possible future option in which a triangular facet is used instead of
quadrilateral facets must be bought separately (estimated cost about US$ 7 000). Garrison is prepared to assist in making changes to the program. It is es-timated that the inclusion of the ship's speed and viscous damping will cost between US$ 5 000 and US$ 10 000. Garrison is not interested in an exchange agreement including NRIO's prototype data.
4.6 NMI Model
Dr R G Standing, together with Dr H Hogben, of the National Maritime In-stitute, have developed a 3DST model called NNIWAVE. This program is very similar to van Oortmerssen's DIFFRAC, Seidi's HC3AI{1 and Garrison's DYNRES6.
In addition to being able to compute forces and motions on structures which have dimensions which are large compared with the wave length, the model
also includes routines to perform calculations of wave forces on structures or structural parts that have dimensions which are small compared with the wave length. This is performed by applying the Morison equation. This com-bination of computation principles is particularly useful for the computation of motions of, and loads on, the large offshore structures used in the North Sea by the oil industry. In the computation of ship motions the Morison
subprogram can be bypassed. The model can compute linear mooring conditions.
The theory of the model is well documented and a user's manual is available. Agreement with physical model results is good and the model can include viscous damping coefficients, although the values must be supplied by the
user. The fixed, and thus linear, viscous damping coefficients can be esti-mated from a damping test with a scale model of the ship in a basin. When the model is given a certain displacement (heave) or rotation (roll or pitch) and is then left to oscillate back to its rest position, the decay of the motion is a measure of the viscous damping. Usually the viscous damping is about 10 per cent of the critical damping. This is illustrated in Figure 5.
The program NMIWAVE includes the computation of wave drift forces, using either the momentum method of Faltinsen (1979) or the pressure method of Pinkster (1980). Standing, however, believes that Pinkster's method contains
errors, and that the method of Faltinsen is correct. No special measures are taken with regard to the possible occurrence of irregular frequencies.
NMI has performed the
test run
for the prototype data supplied by NRIO. The computed motions for an encounter period of 10,1 s are very small, probably because of the narrow response functions for vertical ship motions. The results of this test run are summarized in Table V. Comparison with the data in Table IVa shows very close agreement between the results of NNIWAVE and DYNRES6. If NRIO requires this, tests for additional frequencies can be run for £300 to £400 (R550 - R700) per frequency. The model accepts both quadrilateral and triangular facets. Spectral, statistical and plotting post-programs are also available at NMI. The cost of obtaining NMIWAVE is£30 000 (about R55 000), but only the binary code will be supplied. There-fore one of the basic requirements of NRIO cannot be met, although NMI is
prepared to enter into research cooperation with NRIO. In that case the model would stay with NMI.
4.7 DnV Model
Det norske Ventas (DnV) is the Norwegian shipping registration and classi-fication organization. Dr O M Faltinsen, presently professor in Trondheim, together with Mr O A Olsen at DnV, has developed the model NV459, which is a 3DST model similar to those of van Oortmerssen, Seidi, Garrison and NMI for ship motions in the frequency domain. Besides NV459 DnV also has a strip theory model NV417 (similar to the NSRDC model - see Section 6.3), but this model is seldom used.
For shallow-water ship motion studies DnV would use the 3DST model. Model NV459 can accept viscous damping coefficients, but does not include the effect of forward speed. For their own applications, forward speed is not very important and they do not plan to allow for this effect in their model. NV459 has been tested for underkeel clearance down to about IO per cent of the water depth. Care must be taken to ensure that the facet size is not larger than the underkeel clearance. If this is not so the results will become inaccurate. Furthermore, resonance can occur for small underkeel clearances, because of oscillation of the water under the keel. Research at DnV is aimed at drift forces and the development of non-linear models, the latter work being a long-term project. At present NV459 can work with only a linear mooring condition and no time domain convolution is available. Also, no special measures are taken to suppress irregular frequencies.
Program NV459 has been sold to clients. The present price is N.Kr. 300 000 (about R50 000), which includes documentation, a training course and one year of program maintenance. The program is supplied in source code so that the client can make his own modifications. A version of this model is also at the Norwegian Hydrodynamic Laboratory in Trondheim. DnV regrets that they are not able to obtain an export licence from the Norwegian government to export this piece of technology to the RSA. Most of DnV's application of
the model is to offshore structures in the deep-water conditions of the North Sea. They are therefore not very interested in obtaining NRIO's prototype data for shallow-water ship motions.
4.8 Conclusions
The discussion in the previous sections of various 3DST models has indicated that this technique is now relatively well developed and that it has been applied by a number of well-known institutes and laboratories. For shallow-water conditions, particularly, it appears that the theory of the 3DST models
is superior to the strip theory.
In addition to the six models discussed in the previous sections a model has been developed by Inglis and Price (1981), but few details are known about
the availability and stage of development of their model, which is in use at University College, London, (see also Section 6.2). Of the other six models the DnV model is not available at all, while the NMI model is not available in source code. Of the remaining four models the ENSM model is expensive to run, is not yet calibrated and because of the required CPU time not very suitable for mooring conditions. The models of MARIN and OEC are the most highly developed, especially since they are being adapted
to include the effect of forward speed. Garrison's model has the same potential, but will require a considerable amount of development before it reaches the same stage as the MARIN and OEC models.
In choosing a 3DST model, secondary factors must be taken into account. The MARIN model appears to be the one that has been longest in use. Furthermore, it may be possible to reach an agreement whereby this model could be ex-changed for prototype data.
Many researchers agree that allowance for the forward-speed effects in the added mass and damping coefficients according to Salvesen et al (1970), will be the best solution until the computations of the more exact model routines are made more efficient.
Of the models run for prototype test condition ROO2A the results of VESDYN are rather close to the prototype values for roll and pitch, while heave is too small as is also the case for the other models (see Table III).
5 STRIP THEORY MODELS
5.1 General
The strip theory is the predecessor of the 3DST. The basic strip theory was originally developed by Korvin-Kroukovski and Jacobs (1957). As the name of the theory indicates, the ship is divided into a number of strips, usually about 20. The hydrodynamic coefficients and forces are computed for each strip by assuming it to be part of an infinitely long cylinder with a constant cross-section equal to the average cross-section of the strip. When the hydrodynamic coefficients and forces have been determined for each strip the coefficients and forces for the complete ship are determined by integration over the ship's length. Subsequently, the equations of motion -Equation (3. 1.) - are solved and the principal motions and rotations deter-mined. Because of the division of the ship into two-dimensional sections
the surge motion cannot be included directly.
The simplifications in the strip theory are acceptable for slender bodies, such as ships, in deep water. As soon as the value of the length-to-width ratio becomes smaller than about four, the fore and aft ends will influence the water movements at the middle sections. In this case the assumption that the strips can be regarded as part of a long cylinder which can be approached two-dimensionally is no longer valid. This also holds for the case where the underkeel clearance becomes so small that the flow pattern beneath the ship is influenced by the ends although the ship is a slender body. This influence becomes noticeable if the depth-to-draught ratio becomes smaller than about 2.
However, a number of the mathematical models based on strip theory which have been developed since the work of Korvin-Kroukovski and Jacobs, have introduced methods to include the surge motions in the computation, as well as correction coefficients to compensate for the effects of small underkeel clearances.
A number of techniques exist for the computation of the two-dimensional added mass and damping coefficients, for each strip. The three techniques found so far are the close-fit source distribution technique developed by Frank, the Lewis form method (see for both: Frank and Salvesen, 1970) and the two-dimensional finite element technique as applied by Andersen (1978). This latter finite element strip theory model will be discussed in Section 6.5 after other finite element models have been discussed.
The
close-fit method is
a two-dimensional source technique. Source irregularitiesare distributed over the submerged part of the sections and the Laplace
equation and the various boundary conditions are applied. After the velocity potentials have been determined, the pressures for each segment between offset points of the cross-sections are determined by applying the linearized Bernoulli equation. Integration of the pressures over the immersed part of the sections yields the hydrodynamic force. The close-fit technique is not valid for
high frequencies, that is, for w>
IJa
(see Frank and Salvesen, 1970).The
Lewis for'in method
applies a conformal mapping routine whereby the cross-sections are mathematically transformed into Lewis forms, which have the same beam, draught and area as the actual cross-section. The restriction on the application of the Lewis form method is that the sectional area coefficient Cs is between about 0,5 and 1,1, where- beam x draught
This holds for nxst merchant ship forms with a monohull. Only for a bulbous bow section may the above requirement not be met.
The velocity potential is represented by a series of multipoles each of which satisfies the free surface and other boundary conditions. The density of the multipoles is also transformed and the velocity potentials are solved in the Lewis domain. Again, the pressures and hydrodynamic forces can be computed by means of the linearized Bernoulli equation; they can then be transformed back into real values. Thereafter, the computations are similar to those of the close-fit approach.
