HELSINKI UNIVERSITY OF TECHNOLOGY SHIP HYDRODYNAMICS LABORATORY
OTANIEMI FINLAND REPORT NO. 9
ON THE EVALUATION OF UNDERKEEL CLEARANCES IN FINNISH WATERWAYS
OLAVI HUUSKA
This study was carried out for The Finnish Board of Navigation in the Ship Hydrodynamics Laboratory at The Helsinki University of Technology, Otaniemi.
A literature study started the work in 1972. Experi-mental stages were carried out in
1973
and1974,
analysis and final calculations in1975.
The author is indebted to Professor V. Kostilainen, principal of the Ship Hydrodynamics Laboratory at The Helsinki University of Technology, for his help and support during every phase of the work.
I also thank the staff of the Laboratory for their help. Especially the assistance of Messrs P. Hervala, J. Sukse-lamen and H. Lindroos is appreciated. The help of Messrs T. Ovaska, E. Laustela and J. Lund during the experimental part of the study is also acknowledged.
The advices of Miss S. Kuosma for the English text is appreciated.
CONTENTS PREFACE 3 LIST OF SYMBOLS 8 INTRODUCTION 13 BASIC CONCEPTS 15
2.1
Depth 152.2
Underkeel clearance 172.3
Factors affecting underkeel clearance 172.1
Necessity of probabilistic approach 182.5
Suitable accuracy 19METHODS FOR THE EVALUATION OF TJNDERKEEL
CLEARANCES 20
3.1
General 203.2
The design-ship method21
3.3
The traffic stream simulation method23
3.l
Present methods 263.5
The method used in this report 26SQUAT 27
Basic squat 27
Corrections to the basic squat 28
¿1.2.1
The area ratio 284.2.2
Ship location on the fairway 31I.2.3
Other ships 3¿1.2.i
Bulbous bow 31414.2.5
Static trim 3514.2.6 Bottom topography 35
EFFECT OF THE WAVES 36
5.1
General 365.2
The imput: waves 365.3
The response operator 385.3.1
Theoretical results 385.3.2
Experimental results 425.3.3
Response operators used in evaluation ofumderkeel clearances 51
5.4
The output: vertical motion of bow 545.5
Total increase of draught in waves 55ROLLING 56
EXAMPLE OF UNDERKEEL CLEARANCE DETERMINATION 57
7.1 General 57
7.2 Procedure of calculation 57
7.2.1 Initial data 57
7.2.1.1 The fairway 57.
7.2.1.2 The sea spectrum 58
7.2.1.3 The ship 61
7.2.2 Squat 6k
7.2.3 Vertical motions of the bow due to waves 68
7.3 Evaluation of results 70
8. CONCLUDING REMARKS 7'
8.1 The applicability of the presented method 7
8.2 Probability aspects 75
8.3 Recommendations for fairway dimensioning 76
8.11 Recommendations for future research 76
8.5 Final remarks 77
BIBLIOGRAPHY 78
APPENDIX I SOME RESULTS OF SQUAT TESTS 95
APPENDIX II ARRANGEMENT OF TRE EXPERIMENTS 98
1. Introduction 98
2 General arrangement 102
2.1 General 102
2.2 Tests at zero speed 102
2.3 Tests with the self-propelled model 102
2.11 Calibration of instruments 103
2.5 Procedure of a single run 101$
2.5.1 General 1011
2.5.2 The tests at zero speed 10k
2.5.3 The tests with the self-propelled models 10k 3. The outfitting of the models
105
The waves 110
11.1 Generation of waves 110
11.2 Measurement of waves
112
APPENDIX III DATA PROCESSING 113
1. General 113
2. Correction of the signals
115
3. Partition of the signals
7
Determination of the response operator 119
¿1.1 Theoretical background 119
Digitizing the signal 120
The response operator l2
F fetch
A coefficient in wave spectrum formula
A 27/Lpp
-A amplitude of j: th harmonic of Fourier spectrum Ax area of maximum transverse section of h-ip ACH area of transverse section of channel
area of transverse section of a channel 5Áagined in an open type fairway
a coefficient
a T2 corresponding with given probability a signal ôf ultra-sonic detector at sterñ B breadth of vessel
B coefficient in wave spectrum formula
b coefficient
b siia1 of ulra-soñic detector at bow
b breadth of channel in general breadth of chànnel at bottom CB block coefficient
CSQ squat coefficiént Cz,C0 shape factors of ship
e signal of wave probe iñ model
c coefficient : CL -coefficient
C ,C1,C,
O coefficients C C C7 wave velocity c coefficient a e 2.71828...9
FFAS fetch for fully arisen sea Fn Froude number
Fflh Froude depth number
f frequency =
Nyqvist (or cut-off, folding) frequency sampling frequency
f(t) time dependent signal
time dependent descrete signal
as above, but corrected to the same time instant in all measuring channels
f(T2)
probability density function ofG center of gravity of ship
g 9.80665 mis2
H(jii) response operator
h depth in general
hdr dredging depth from mean water level hue underkeel clearance
hw water depth
h' height of walls of open type channel
th permanency correction of water level i index of descrete time instant
i
j index of measuring channel
K(u,),K(x) correction factor of sea spectrum due to shallow water
K1 correction factor of area ratio in open type channel
correction factor of squat due to area ratio
KL as above, but due to ship's location
k wave number = 21T/Lw. k factor in Fourier erieS
longitudinal radius of gyration length between perpendiculars resonance length
L wave leñgth
i distance of ship's centerliné from the side of the fairway at the bottom
M number of seuence of the measuring channel
In0 first ioment of spectrum
N. number of pòints in a. digital signal
Ò( )
oforder( )
P{ } probability of the conditionin {
p,p' probability .
p(t) functioñof t . ..
q(x) function; dw = q(x)dx/2w . . S,S1,S2 stroke of wave generator
sB(w),SB(x) spectrüm of vertical motions of bow
Sz(w) spectrum of ship motion, in gênerai
S(w),S'x) wave spectrum . .
Shi(x)Sh wave spectrum in shallow water.
S area ratio
s' area ratio
draught
T period
T apparent wave period
T0 fundamental period
TE period of encounter
11
T wave period at which the sea spectrum has
m a maximum
AT increase of draught
AT1 squat
AT1 squat in very wide channel, basic squat AT2 increase of draught due to waves
safety margin of underkeel clearance
t time
tj discrete time
At sampling interval 1/rs
time lag for the A/D conversion of two adjacent measuring channels
U wind velocity at a height of 19.5 n
V ship speed
x Lpp/Lw
XG,YG,ZG coordinates of the center of gravity
XF distance of the U-sonic detector from the
perpendicular at bow
XA same, but from the perpendicular at stern
distance of the wave probe in model from the perpendicular at bow
Z vertical motion at bow perpendicular during
measurement s
ZA same, but at stern
Z1 sinkage
Z2 heave amplitude
ZB amplitude of vertical notion at bow
a,ß coefficients in the wave spectrum equation
A deplacement
c. phase angle of j:th harmonic of Fourier spectrum
A scalé factor heading angle
w circular f±'euency
w.. fuñdamenta circular frequency of the digital signal 2,r/T
WE circula frequency QQ encountér
w ciràulì frequency at which sea s.pecrm
m has a maximum
O pitòh angle
e1 change of trim dùe. tò sqüat
e - factor in Fourier eriès = O or 1
CA wave arnplitùdê . .
-wave height
Ç
mean wave heightW1/3 significânt wave height wl/lO mean height of 1/b heightest
Ves a standard deviation.
y volume of deplacemet
I,II,IIIIV indexes indicating the measuring sage
X mean value of the variable. X X
-.
x is random variableNOTE.
Metiic system of measures is applied if not state4 otherwie
13
1. INTRdDUCTION,
Dur.ng the last few years the depth of many fairways and sea areas has decreased for the increased ship sizes.. This development has made it necessary to deepen the fair-ways. Thé dredging costs form a considerable part of the
total investment for building a fairway. Therefore, a method for the determination of a required and safe but not too great fairway depth for anticipated traffic should be available.
The rules of thumb applied so far may have resulted in overdimensioning due to safety reasons. 1n order to arrive
at a reliable method theoretical and experimental wOrk is required in the field of ship motions in shallow water,
especially in shallow water waves. Indeed, the activity in this research area has grown recently. Some theoretical results have already been issued, but their accuracy has not been known due to the lack of experimental work. It is to be emphasized, that the methods used succesfully for deep water conditions cannot be applied in shallow water
conditions without an adjustment.
The aim of this report is to present a method for the determination of a reasonable fairway depth. Main purpose will be to find a suitable method for the conditions pre-vailing in Finnish waterways. Moreover, main attention is paid to the evaluation of underkeel clearance. This is the distance between the bottoms of the ship and the fairway -meásured when the ship is at rest which is sufficient in order to avoid bottom contact in spite of ship's vertical motions caused by speed and waves.
In this report the theoretical background is based on a literature study. The effect of shallow water waves on the underkeel clearance has been studied experimentally. The theoretical arid experimental results are compared and the method is presented in the light of an actual example. Conclusions and recommendations for the future research close the first part of the report. The second part
con-sists of the underkeel clearaiice bibliography. The details of experïnients and data processing are presented in the last, part.
15
2. BASIC CONCEPTS
2.1 Depth
By the nominal deQth of the fairway one means - in Finland - the draught which cannot normally be exceeded by any ship entering the fairway. Hence, nominal depth max. allowable draught.
By the water de2th one means the minimum distance from the fairway bottom to the water level having the planned permanency (probability). The permanency of the mean water level is 0.5. Normally a higher figure is used, e.g. 0.90. The third depth used is the dredging death. By this one means the minimum distance from the fairway bottom to the mean water level. Therefore, the water depth and the dred-ging depth are equal if the permanency of the water level is taken equal to 0.5.
From these definat ions follows
hdr = hw + th (2.1)
where
h dredging depth hw water depth
1h permanency correction
The permanency correction is based on the statistics of the water level fluctuations. One example is shown in Fig. 1 /188/.
Three depth ranges can be distinguished according to the value of h/T (h is the actual depth, T is the draught of the vessel)
/73/:
deep water, h/T >
finite depth, 2 < h/T < k shallow water, h/T 2
Moreover, for the ships of maximum sizes allowed to enter a fairway, h/T is 1.1.. .1.25.
gag gas 9g-o 9go gas 900 alo '700 we 500 400 300 200 '00 he 20 05 al -26 Io h [mJ
FIGURE 1 WATER LEVEL AT MXNTYLUOTO IN JANUARY AND JUNE.
PLOTTED ON NORMAL PROBABILITY PAPER
T - draugM
- increase of drought due to waves and sauat
bT - safety margin
Bac - darot'tce
water tIf
bd,. - dredging gth
I mean water ei. permanency - 0.5
2 - water imwI w,th a greater permanency
A bow at rest
B bow when immersed to the rnoairnum pth dr,ng rttion
FIGURE 2 FAIRWAY DEPTH AND ITS COMPOSITION
JA UAaY Probabhty
that ti
17
2.2 TJnderkeel clearance
The water depth consists of two parts: the static
draught of the ship and the underkeel clearance. The under-keel clearance is considered here to depend only on the vertical movements of the propelling vessel. Other aspects could be strength arid manoeuvrability /75/.
The draught must be corrected to correspond with the prevailing water density. The nominal draught of a ship is usually given with the density of 1.025 kg/rn3.
Figure 2 shows the parts of the dredging depth and the underkeel clearance. This figure corresponds to the situa-tion where the mositua-tions of the bow are crucial. Other points which should be considered are at stern and at bilge.
The part of the underkeel clearance depending on the vertical movements of the vessel is indicated by ¿T. This can be divided into two parts. Firstly, the increase of draught in shallow, calm water due to the speed, called squat and indicated by AT1. Secondly, the increase of draught due to vertical movements caused by waves and indi-cated by ¿T2. When a safety margit 6T is used, then:
huc AT1 + AT2 + T (2.2)
hw T + (2.3)
where
huc underkeel clearance
Squat in turn is composed of the vertical sinkage Z1 and of the change of trim, Ol
Vertical movements due to waves are caused b7 the summa-tion of pitch, heave and roll. In the summasumma-tion the phase angles of these motions and the geometry of the ship should be considered. However, in this report only the effect of pitch and heave will be dealt with in details.
2.3 Factors affecting underkeel clearance
Underkeel clearance depends on many factors as shown in Pig. 3. Prom these the following have been selected for a more detailed analysis:
WS PV ON
FOGY
- TER PTH
WI4D SPUD. DECT TGN
- GPW fl
4. 6S E)
B WATER-pTh
- OENSØV ES-
CURRENTS-
TIDES ICE NTIORS C. FAèIWAY - Speed Size Geometry, mainly C8 WATER - Depth - Waves FAIRWAY - GeometryThe reasoning for this selection
A. S&41P -
PRflES
-
METRY. -OThER SHIPS - SW1C AMT - sT*T R4- TmM PRTES
-
STALJY PRTS
- STAUZIG DEVISES - MESSDTR18UT-
SHIP SIZE - SHIP SPEEDr
J-FIGURE 3 FACTORS AFFECTING THE UNDERKEEL CLEARANCE
2 k Necessity of probabilistic approach
The waves are handled with he aid of statistics as well as the fluctuations of the water level.
Therefore one can èalculate only the probability that a certain underkee. cleararce is adequate.. Hence, an under-kéel clearance given without the corresponding prôbabilitY
19
is not the Cull story.
Determination of a safe probability level has no relevancy in this report.
25 Suitable accuracy
The complexity of the calculation method for the under-keel blearances depends strongly on the accuracy intended. On the other hand, many factors affecting the unde±'keel clearance and the dredging depth are not known with great accuracy.
Draughts at stern and bow measured by the instruments on board are affected by the waves. The salinity of the sea water may alter and the load condition (ballast and stores) of the ship changes during the passage /9I. Therefore, the exact mean draught and trim of the ship are not known when entering the fairway.
The inaccuracy in the evaluation of the water lével and thé distance from the mean water level to the dredged bottom results im an error in the wate± dépth. The errbr in the dredging depth increases as the fairway gets older. The prevailing water level is known quite exactly, but only at the points of observation, which may be situated a long way from the fairway.
All these errors - in draught, in water level and in dredging depth - are supposed to fóllow normal probability laws. Moreover, they are considered to be uncorrelated arid independent. The error is here defined equal to ? x o 2 x standard deviation), which means that the measured values are within the boundaries ±2o from the mean value with a probability of 0.955.
We assume here roughly as follows draught 2 x a 10 cm /92/ water level 2 x a : cm
dredging depth 2 x a 50 cm /7/
The result from Eq. (2.1) and (2.3) is that the error of underkeel clearance could be about 0.5 m. However, an accuracy of 0.3 rn is aimed at because the accuracy in dredging depth is supposed to.increase in years to come.
3. METHODS FOR THE EVALUATION OF UNDERIEL CLEARANCES 3.1 General
This section will describe the principal ways for the determination of a reasonable uriderkeel clearance and dred-ging depth.
One may use two methods for the calculation of the underkeel clearance, namely
- the design ship method
- the traffic stream simulation method
In order to use the second method the first one must be available.
'/7/
///7//,// 1/
7//, ///////j// 'j i //, /J
///J
i. SIDEWAYS UNRESTRICTED FAIRWAY
2. OPEN
7/
'7/V/ Z////
/ ,///y
// '/ //
1'1
TYPE CHANNELi
3. CHANNELFIGURE L4 CROSS SECTION TYPES 0F FAIRWAYS
21
of the fairway and the initiál estimate for the water depth (the breadth b is held constant during the calcula-tions). The types of the fairways are illustrated in Fig..
3.2 The design-ship method .
-In this method one continues by selectiñg the permanency of the water level (or levels) at which the calculations 'are carriéd out. This must be done because at a constant
water, depth a changing water level affects the geometrical properties of the fairway cross-section. For a block diagram, see Fig. 5a.
Methods for the selection f water level are not dealt with in this report. The selection of te design 'ship is bäsed on the forecasts of the dimensions, of vessels expec-ted to use the fairway during the nearest 10. .20 years. Generally the design ship is thé largest vesse-1 allowed to enter the fairway. One must seléct the length, breadth, draught, block coefficient and speed, see 2.3.
Ship motioñs'in waves - in deep water,- are càlculated with methods based on the linearity and superposition prin-.ciple. In this report it is assumed that these principles
are val-id in shallow water conditions. Therefore., one needs the response operator for pitch, heave and roll (or for the motìon of stern, bow and bilge) and the spectral density function of the sea. Response operators can be determined from experimeñts or from theory. The spectral density func-tion is known from the measurements or it can be calculated approximately w.ith the help Of wind speèd or atmospheric pressure, see 5.2.
Squat is calculated. first. Then the sea level is decrea-sed by the amount of the sinkage and a new ratio h/T is udecrea-sed when computing the vertical motions. The necessity of this adjustment depends on the way the response operators are determined. The result will be 'the probability distribu-tion funcdistribu-tion of the vertical 'movements Of the ship's criti-cal points. By selecting .a suitable' probability one gets the vertical movement. Adding this to the sum of the squat and the, safety margin gives the underkeel clearance.
Calculate:
Response
operator
not
BEGIN
FIGURE
5a.
BLOCK DIAGRAM OF DESIGN SHIP METHODL ¿
Select:
Water dept h
Select r
Select:
Select:
Select'
Ship Water level Sea state Safety margin
Calculate: Squat sin kaq
Adjust:
h/T (if necessaryl Calculate: Probability distrijt ion of vertical motionSelect:
ProbabilityCalculate:
Vertigal matiors due to waves Calculate: The necessary water depth? 3
It is recommended to g-through'this calculation proce-dure with some 3.. .5 ships and at various speeds. It is advisable to vary the sea tate, too.
If the estimated water depth turned out to be suitable the calculations can be stopped, otherways one must select a new depth.:
The-xùain difficulty of this methodare the number of selections one must do: thé water level, the ship, thé sea. If these all are selected according to the worst possible conditions (dúring the 10.. .20 years) the result nay be a water depth which is insufficient only once in 1000...
10000 years. : .
There are two ways to cure this Situatión. First, the selections are not óarried out on the "worst possible"-prin-ciple, but a probability of 0.05.. .0.15 is allowed instead
of 0.0005.. .0.01.
The second. way is: to take as the basis the whole yearly traffiç and the fluctuation of the environmental conditions. Only then one can clear up the true probability for the hitting of the fairway's bottom by any ship in one year. Also the probability that a ship cannot enter safely the fairway (it must wait) can be determined. For these compu-tations one can use e.g. the Monte-Carlo simulation method together with the Extreme-Value Statistics, see 3.3.
3.3 The traffic stream simulation method
In this method the selections are mäde with the help of random number generation. For this one must know the proba-bility distribution functions of all the parameters needed. It isemphasized, that the selection of the value of the pa-rameters- is not arbitrary, it is random.
One simulation loop (the ship steaming through the fair-way) consists of the selection of a combination of the ship and its environnent. (For the block diagram, see Fig. 5 b). The probability distribution function of the vertical
mo-tions corresponding to this combination is calculated. The number of motions during the passage is computed. A corres-ponding quantity of vertical motions -is generated at random
Calculate Response operator BEGI N Select.Water depth ...jGerterote. Month Generate Generate
Ship Water level
yes yes Calculate Squat sun kae I Adjust
I h/T
(ut necessafyl CalculatePro bob u lit y dustru buhan ot vertucau rrtuor Generate Motion repeat number of swings times Select The greatest Calculate The vertical motion Select The greatest for year New ear riot Ext reme value stat Stucs not yes END
)
Generate SelectSea state Safety margin
25
from the distribution function. The greatest motion is selec-ted. Adding this to the squat gives the increase of the draught for this ship.
The simulation loop is repeated until the number corres-ponding with the quantity of one year's traffic is reached. This in turn is repeated n times, Which corresponds with the traffic during n years. The highest values in each year are collected and analyzed according to the methods of Extreme Value Statistics. This procédure is carried out at various water depths.
One possibility for the basic simulation loop is as follows. The traffic of a year is indicated by a histogram fOr each month. The month is selected at rándom with the help of this probability distribution function. As a result the parameters of the distribution functions of the wind direction, wind speed and water level will be known. Also the parameters of the distribution function of the ship's size and speed may depend on the month. The draught deser-ves to be specially considered, because it can differ from the construction draught according to the load condition. It may be necessary to distinquish the arrival and departure situations, too. With the aid of thése distribution func-tions it is possible to generate all the necessary initial
data.
Some combinàtion of the ship and her environment may be definitely impossible e.g. a ship has the maximum draught, the sea level is low and the wind speed is high. This case corresponds with the situation where the ship must wait for better conditions.
The drawback of this method is the great quantity of th initial data. For the moment thère are no direct sea state statistics in Finland, but the collection work is going on. However, the wind and water level statistics are at hand. Also the statistics of the quantity of ships during a month and year exists for all main ports. On the other hand the statistics of the geometricàl properties and speed is lacking.
3.l Present methods
So rar the design ship method has been the only one used. The way of application varies according to the local conditions. For details, see /2,7,l6,5l,l0l,l73/.
In Finland the determination of the depth of the fair-way has been based on the norms of the Finnish Board of Navigation, see Table i /1I7,9,92,l08/.
Minimum 0.6 n
Fairway in protected archipelago 0.1 X T
Fairway in unprotected archipelago up to 2 n
Open sea 0.25 x T
Open sea, minimum
2m
At port 0.6 m
TABLE i NORMS FOR h IN FINLAND
3.5 The method used in this report
Due to the inadequateness of the initial data the traffic stream simulation method is not applied here. The design ship method is used, instead.
It is assumed, that the vertical motion of the bow is crucial in the evaluation of the underkeel clearance. Some aspects of rolling are dealt with in chapter 6.
14 SQUAT
4.1 Basic squat
The simplest way to handle squat theoretically is to use Bernoulli equation. The results are accurate enough for chane1s. A survey of these theories can be found e.g. in Re-ferences /7,34,75,171/. Unfortunately, the most common cross-section of a fairway is a channel totally under the water level, see Fig. 4.
One way to cope with this situation seems to be the fol-fowing: First the basic squat is calculated for sideways un-restricted waters. After that all the necessary corrections for the actual fairway are carried out.
According to the 2-dimensional theory of Tuck and Taylor /171/ the basic squat can be calculated in the following way as noted by Hooft /73/:
F2 C
-_-fl--(C+2-)--Y--
T1 (4.1)where
r squat in very wide channel
Fh
Froude depth numberV(gh)05
h water depth, actualV ship speed
V
r
volume of displacementr
Length of ship between perpendicularsC,Cr
shape factors of the ship's hullAs noted by Hooft CZ and C0 do not vary much for normal ships and we can take
c % 1.5 C0 1
Formula (4.1) can be written /73/:
T B CB S
where
CSQ squat coefficient, and 27
C3 = 1FnhZ +
0.5 c0)(l - FhY°5
(4.4) In Fig. 6 values. of C fröm tests carried out by seve-ral laboratories have been plotted together with the re-sults from ('4») when C+ 0.5 C9: 2.0 and 2.4. As seen, the curve with the coefficient 2. gives better correlation 'with expriments. Therefore, the value 2.I is used iñ this report.The scatter of the results is about ±20. . .25 %. One rea-son for this is the nonuniformity of test conditions in various laboratories: models are different, of different sizes (scale-effect possibility), measuring methods are dissimilar, shallow-watér conditions vary 'and so on. One possibIlity of errors is also the data collection and transform work done by the author. The points in Fig. 6 are based onthe literature study carried out in /75/, seé Appendix I.
For normal ships at speeds used in fairways the squat is aboút 0.3. .0.8 m. The possible 2 x o error, due to the inaccuracy of CSQ is then about 15 cm. This is close to-the accuracy given in 2.5. Therefore, using Fig. 6 or Equation (4.3) for the determination of the basic squat is accurate enough fôr engineering purposes.
Note that the maximum value of C depends on the ratio
h/T, seeFig 7. At this maximum value the ship touches the bottom.
k.2 Corrections to the basic squat
¿4.2.1 The area ratio
The area ratio s is defined as the ratio of the ship's main frame area to the' channel crossection area (ses Fig.I).
A very recent report /15/ gives theoretical means to deal with the effect of s in case of an open type channel, but it has not been possible to take it into consideration in this report.
0.5 0.4 0.3 0.2 0.1 1.4 1.2
to
0.8 0.6 0.4 0.2 O o £ P FIGURE 7. TMAXIMUM NONDIMENSIONAL SQUAT
FIGURE 6. SQUAT COEFFICIENT CSQ
'SQ
-JScatter
of resuIt/
/
/-/
/
Cz+2.a
/
-
/
29/
/
-7
/
/
/
-Fflh 1.2 1.6 1.8 2e 29 maxvariable have been carried out by Guliev /6o,61,64/. The results of his tests can be seen in Appendix I. Figure 8 is drawn according to this data. For the normal fairways s < 0.15, and one can take:
2.5
2.0
1.5
to
SQUAT
FIGURE 8 CORRECTION COEFFICIENT K DUE TO AREA RATIO S
3i
K5
r
7»45.s + 0.76S ?
0.03K.: 1-
s <0.03
where
K5 = àorrection coefficient due to s.
aréa ratio, calculated with the aid of Fig. 9 as follows:
.i (46)
-1 CH
where
-K1 correction factor, Fig. 9 (or (7.9))
r
cross section arèa of' a channel formed by thelenghtening of the slopes of the open type channel
area of main frame of the vessel 0.98 x B x T
4.2.2 Ship location on the fairway
Acôording to the results in Referencé /55/ there is a considerable increasé in squat when the ship moves away from the centerline of the fairway Based on this refe-rence the correction factor KL is plotted in Fig. 10 when thé ship ié on the other sIde of. the fairway (the cross section of thé fairway is supposed to be symmetrical). The curve is based on thé iesults at 5 0.11, 0.114 and 0.24
(hIT 1.4).
This is the only existing report found about this fact6r Therêfore, one can òrilyguéss the accuracy of' the cùrve in
Fig. 10. P1urthermore, it holds good onlyfora channel-type fairway. Linear interpolàtion is proposed to be used for
fairways f other tes. Correction factor KL s 1 for an-unrestricted shallow fairway when h'/h -: 0, see Fig. 9.
When s gets smaller the width df the fairway increases if the depth remainscoflstaflt. According to the model tests at Sogreah /1/, one can consider the fairway width to equal infinity, if that is greater than 10 X B. Therefore, in a
wide channel the locatiOn of the ship does not affect squat when the distance from the ship to the side of the fairway
FIGURE 9.
/
/
/
/
/ / I If / I
K SQUAT CORRECTION COEFFICIENT K1 ACC. TO GULIEV h 2 6 .8-h'/h
4
A 4
dll
o._
CAx! ¼H
05
alo o.o2.50 2.00 KL h/T r 33 g 1.5 1.4 1.3 1.2 1.1 1.3 o X Fflh O = 0.5
+ = 0.4
O = 0.3 X = 0.2h/I = 1.4
0 0.05 0.10 0.15 0.20 0.25 S=A)( ACH B/1/ / f/f/f/I
b0FIGURE 10. INCREASE OF SQUAT WHEN THE SHIP
is more than 5 x B. Hence KL in Fig. lo has an upper limit
depending on the ±atio h/T as shown.
Liñear interpolation is also proposed foí' te situation-where the ship is somesituation-where between the centerline and the side of the fairway. This situation cover two pössibilities. Firstly, on a wide fairway KL: 11f the distance to the side is more than 5 x B. Secondly, when the width of the fairway is smaller than 10 X B, KL i at the centerline.
Moreover,' linear interpolatipfl is also applied if the slope differs from Z45. The interpolating variable is in this casethe ratiO of the crossection areas of the channel having the actual slope and the channel having a slope of 145°. If the slope is 0° then KL 1. For equations, see chapter 7.
Combining the effects of the area ratio and the location of' the ship gives the squat AT1 on the actual fairway.
AT1 = K KL (14.7)
14.2.3 Other ships
If' there are two ships in the same cross section of the fairway, the ratiO s will grow from the one-ship situation. The effect of the meeting, overtaking and passing ships on squat is emphasized because now they must move away from the centerline of the fairway.
-TO coe with this case, it is proposed
/69/
to use s cal-. culãtèd with the aid of the sum of the cross section areas of the ships concerned. Then the methods mentioned in the previous paragraphs can be applied,Note that the speed of the two ship situation may be de-creased from the one-ship situations The basic squat must then be calculated corresponding to the actual speed.
14.2.14 BulbouS bow
-In Reference /514/ some results aboit the effect of the buibbus bow òn squat can be-found. Also Dand has carried out
some experiments with buiboi5 bow
/33/.
Oi the basis of these results it is not possible to say how bulbous bow in35
general affects the squat. Anyway the effect is small and may be ignored until more knowledge is available.
L2.5 Static trim
According to /33/ the initial trim (if not unnormal) can be taken into consideration with sufficient accuracy using summation only.
L2.6 Bottom topography
An uneven bottom may cause an alternating squat. When steaming from deep to shallow water at full speed, the squat may increase considerably till the speed is decreased.
Only one report concerning this case has been found /23-3/, but the results hold good only for a single case. The bottom topography will not be dealt with here in more details, because there is a special research programme on it at the ship Hydrodynamics Laboratory in Otaniemi. (See Ref. /205-210/).
5. EFFECT OF THE WAVES 5.1 General
The calculation of the vertical motions of a ship operating in shallow water waves is principially carried
out with the sane method, which is widely accepted for deep water conditions. This means that the ship is consi-dered as a linear system, see Fig. 11 /75,190/. Then, from the spectral density function of the sea, S(w), one can
calculate the spectral density function of the ship's motion, Sz(w), with the aid of the response operator H(jui), as Fig. il shows.
H(jw) sea
x IH(iw)t
motion
FIGURE 11 SHIP AS A LINEAR SYSTEM
So far the suitability of this method for shallow water conditions is not known. Its application is based on the analogy and on the lack of any better method.
5.2 The input: waves
Measurements are going on in Finland to find out the actual spectras along the Finnish coast. Unfortunately, only tentative results are available for the present. Therefore, a semiempirical spectral density function is applied in this report. Moreover, only one-dimensional spectrum is considered. The application of a two-dimensio-nal spectral density function would be possible only if the amplitude response operator would be known for all
37
heading angles, from 00 to 1800. However, the response operator is known only at angles of 180° ±
The spectral density function is presented as follows: AB
- - exp (-BIW ) (5.1
This is the well known standard spectrum, recommended by ISSC. The coefficients A and B are
A = 0.25
(5.2) B =
where
height of the highest 1/3 of the waves, significant wave height
T = apparent wave period according to the zero crossings
If these parameters, Cw113 and T are not known, one can apply spectral density function given in Reference /189/. This function is applicable for restricted water areas. Then
A = aU4/ßg2
B = ßg/u
where
a = 0.0081
ß 0.1 exp [(in 7.4)(FIFpAs)_0281
FFAS 16030 U"5
F fetch
U wind velocity at a height of 19.5 m [in/si FFAS fetch for fully arisen sea
These spectral density functions correspond to the deep water conditions. These are transformed for shallow water conditions according to the method presented in /7/.
Sc(w)sh = S(u) K2 (w) (5.5)
(5.3)
If the ratio LPp/Lw is used instead of w, then ¿1Th
r-l/K2(x) tanh( x)[1 + sicth( x)1 where X Lpp/Lw= length between perpendiculars Lw wave length
The relation between w and x is determined by
w (kg tanh kh)05 (5.7) where k = wave number 21T/L Moreover dw q(x)dx/2w (5.8) q(x): Ag tanh Ahx
AX
(5.9) cosh Ahx A 21T/L5.3 The response operator
5.3.1 Theoretical results
There are some theoretical methods for the calculation of the response operators of pitch and heave in shallow
waters.
Beck and Tuck have developed theories for an arbitrary value of h/T but so far only for zero speed /i2,l3,l4, 170/. On the other hand, Kim has a theory for all speeds, but h/T must be greater than 1.5 /81.. .86/. Guliev has in turn developed an approximate theory, which gives nearly the same results as Kin's i9,6O,62,63/. Therefore, there is no general theory applicable for underkeel clearance calculations, when h/T 1.1... 1.3 and the speed varies
between 0.. .8 rn/s.
Por the time being one possibility is to use these theories together. According to the theory of Kim the
39
wavelength is of order ship's breadth. Tuck in turn assumes that the wavelength is of order ship's length. Therefore Tuck's theory should give better results when c is small
(Lpp/Lw is of O (1)), and Kim's theory is more accurate when w is great (Lpp/Lw » 1) /12,13/. But these theories can be applied also so, that Tuck's theory is used when Lpp/Lw < 1.5 and Kim's theory otherwise. For example, according to /13/ the validity of Tuck's theory becomes questionable if Lw smaller than 10 X h. For the critical value of 1.2 of the ratio h/T and when LIT : 16 the results of Tuck may be of no use when Lpp/L > 1.33.
The effect of the speed may be considered according to the trend given by Kim.
For the heading angles not equal to 1800 the Lpp/Lw-scale can be streched out as the first approximation /12, 13/. This is done by using Lw/cos (i18Oo) instead of Lw.
h/T F
ca
FIGURE 12 EVALUATION OF BOW MOTION AMPLITUDE RESPONSE OPERATOR AT VARIOUS FIEADING ANGLES
2.50
0.00 o.,o
Theory. BECK and TUCK
Calculated from the theorettca! value when ,., -BO
by Strechirig L f L -axis
In Fig. 12 the theoretical results are shown at diffe-rent heading angles according to /l4/ compared with the curves based on the curve at i 180°modified as stated. These curves are not complete, because they do not include the effect of the wave diffraction forces as noted by Beck /12/. Furthermore, these curves apply only to the condition when h/T : 2.5. In spite of these defects it is assumed that they show the general trend. So it seems to be possible to deal with heading angles of 1800 ±Z5° in this way.
Consequently, the changing of ship's heading is equiva-lent to the changing of ship's length if the sea spectrum is kept constant. For example, calculations of the motions at lengths of 120, 160 and 200 n in head waves give the
same results as at angles of 530 370 and O for a ship's length of 200 m. However, on a part of the fairway the course of the vessel is constant. The sea spectrum at a constant wind speed can vary at different angles because the fetch can depend ori the direction of the wind.
There-fore, this rule of equivalence may be violated on an actual seaway.
According to all these theories the amplitude response operator decreases with the increasing block coefficient. Fig. 13.1 shows the results of Tuck and Beck for a SERIES 60 ship with CB 0.70 /13/ and with CB 0.80 /12/. The effect of CB is clear. Unfortunately, these results apply only at zero speed.
According to Tuck & Beck /12/ these amplitude response operators have limits when h/T approaches to unity. These limits are also shown in Fig. 13.1 at CB 0.80.
lp'. PCH
02 L/5A
HEAVE Z2 / SA f4 J' I 1 CH-OJOHEAD WAVES TI/T
-C8 - 0.80
SHIP. SERIES 60 ACC. TO THE TI-IEOTTY OF TUCA & BECII - --LIMIT AT C6-Q80 WHEN hIT-10
FIGURE 13.1
PITCH AND HEAVE AMPLITUDE RESPONSE OPERATOR FOR ZERO SPEED AT DIFFERENT CB
FIGURE 13.2
Tie' experimental programme consisted of the determina-tion of the amplitude response operator for pitch, heave and vertical motions of stern and bow. In the main, only the bow motion is considered in this context. For the details of the experiments and data processing, see Appendices II and III.
Fig. 13.2 shows the results at zero speed for a SERIES 60, Cß = 0.70 model. In the tests in transient waves the ratio h/T was equal to 1.50 and 1.25. Some tests were carried out alsO in tregular waves when h/T 1.05. (In fact, the waves were not regular because of the nearness of the bottom). Thé results of these "regular" wave tests support the view that the amplitude response operators have a limit when h/T approaches to ûnity, see 5.3.1.
In Fig. l4 the results of various theories are compared with thôse of the tests. This combination of the parame-ters (h/I : 1.5, c8 0.70, V : 0, u :180°) is the only one enabling Such a comparison so far. The experiments give greater résponse operators, but thé discrepancies are not serious.
Fig. 15.1.. .15.11 shows the results of the tests with the self-propelled models. The models used were SERIES 60,
CB : 0.-70 and "ENSKERI", a bulbous bow tanker with CB:0.814.
As it can be seen, there is a contradiction between the results based on the wave rneasuremeñtS from the static and from the moving wave probes. One possible explanation is that the wave system of the ship and the sinking water level have disturbed the measurements. Therefore, the resul-ts based on the static wave probe are believed to be more- accurate, and only these curves will be commented upon in the following. (Also the nonlinea.rities of shallow water waves maybe the cause /213/).
The curves showà general trend: as the speed
in-creases, the amplitude operator decreases. This effect is most obvioüs at small válues of h/T. This happens because the actual válue of h/T decreases with an increasing speed
due to the effect of sinkage. (h/T values given in this report are measured at zero speed).
The three curves in Fig. 15.3 for run 12 are calculated at different record lengths. As seen, this has only a negligible influence on the results.
tH(jw)I PITCH
02 L/25A
Ie/E
z2 / A B ISA EXPERIMENT -IN OtANIEMITY
¡ruK I. BK KIM
EL SERIES 00. C8 - 70 AD S hIT - 1.5 Lpp/LFIGURE IL+ PITCH, HEAVE AND BOW MOTION AMPLITUDE RESPONSE OPERATOR FOR ZERO SPEED
5A A.l.[I ,)I, '..-. M A 'IL -R 'I'll Fj,8E I. BASIl' l/T - I2 + 0052 0 0.087 D 0.115 Ipp/ LW FIGURES 15.1-2
BOW MOTION AMPLITUDE RESPONSE OPERATOR, 11ENSKERI", C8
0.84 LL1
'\
1 'I' A-'.' 1'Fl.lE I fl, 11.1W 18 + 0052 9 0 0087 20 -0 -0115SitI
'N .W.t f.'-1«,.. -.S.f&f5
n/ - 1.56 F,. 1 094 II 0 046 12 0--- 0046 2/I 0-- 0046 121,1 O 0.090 7 ¿iir
PR,6 .S SSS. 1 Lpp/IW S FIGURES 5.3-kBOW MOTION AMPLITUDE RESPONSE OPERATOR, "ENSKERI", CB
0.8k ,L/I 156 1 L 0044 5,11, 11 0 0046 12 O 0.090 17
h/I- 2.00
RUN + 0.062 7061
9 0.091 lo Lpp/ LwFIGURE 15.5 BOW MOTION AMPLITUDE RESPONSE OPERATOR "ENSKERI", CB 0.8k
Fig. 15.6 and 15.8 show also the results at zero speed for comparison (SERIES 60). These curves support the theo-retical results of Kim /86/, indicating that the amplitude response operator decreases greatly with an increasing speed at higher frequencies (Lpp/Lw > 1.5).
bASe ZERO SPEED 5/1. 1.25, hg 32 B A i . l-p / L., FIGURES 15.6-7
BOW MOTION AMPLITUDE RESPONSE OPERATOR, SERIES 60, CB
0.70 14SEC)
.
A'-. ROM WA,L 1,1, -N 8AlN nI -1.185 RUN + 0078 24 0 0.116 25 o 0.116 27 X 0115 28 O 0108 42 +-- 0.073 41.'
--/1 -1185 O 0116 25 O 0116 27 X 0115 25 O 0O9 42 + 0.073 41ZE SED fig 14 Lp./1w Lpp/&w FIGURES 15.8-9
BOW MOTION AMPLITUDE RESPONSE OPERATOR, SERIES 60, CB
0.70 4 081 38 + 0.081 38 0 0.112 39 0 0.112 39 .SF) . B4SE) )1, WAVr r
.-;
FFfCf. T..r WA -. r. MooEL ¿u r. .50 -1.50t
Lpp /
..,
Lp / Lw
FIGURES15.10-11
BOW MOTION AMPLITUDE RESPONSE OPERATOR, SERIES 60, CB
0.70 4. 0066 44 + 0.066 44 0 0.107 45 o 0.107 45 BASED N FRONI
.AVl-.
\AI l'11. I . 0/1 F0 bA,II'. - 2.02 RUN £ z6 BASED ON AAVL NQA.I,Mf I FRON T14L WRVE PROUL N MODEL 0/1 - 202 F,, RUN£ Z8 Oi-o HF SER6O, C5-Q8 h/I.. L875
FO.2
- curve theoretical
points experimental KIM SER.60,C80.7O-h/I -20 , FnO.
ÖTANIEMI SER. 6O,C-O.7O, F-O.O7
h/T-2re
ENSIERI, B0. FnO-09
h/T2.00
Lp / LW
FIGURE 16 80W MOtION AMPLITUDE RESPONSE OPERATOR
COMPARISON ÖF VARIOUS RESULTS
In Fig. 16 the results of these.teStS are compared with the experimental and theoretica]. results given by Hooft /73/ and theoretical results given by Kim /83/. The parametérS are not exactly the same in various data. The
51
curve giving the results of the tests with the ENSKERI is based on the measurements with the moving wave probe and therefore may not be reliable. The theoretical curve by Hooft gives possibly too high values, but otherwise quite a good correlation is achieved.
5.3.3
Response operators used in evaluation of underkeel clearancesPig. 17.1.. .17.5
show the amplitude response operators of bow motion used in this report. When only the vertical notion of the bow is calculated one does not need any information about the phase angles.LÌ
SA
i,/ LW
FIGURES 17.2-3
BOW MOTION AMPLITUDE RESPONSE OPERATOR
nfl -2Y) CB -OE7O -18O
70
-8OL/ Lw
Lpp/ Lw FIGURES 17.1.-5The curves are composed mainly by help of the results from model tests carried out in Otaniemi (static wave probe). The results of Kirn a.re considered when extrapola-ting to F= 0.2. (For the effect of the speed.) The
re-sults f tuck and Beck are considered when F: 0. At h/T 2 and more the results of Hooft and Vugts (at hIT ) are consulted. /73,193/.
A smaller lower limit than could be concluded in view of the' results by Tuck & Beck (Fig. 13.1) is accepted based on the experiments.
Theê curves hold good srictly speaking only in the case when CB: 0.70. However, th effèct of the block coefficient can be ignored when dimensioning fairways according to the design ship method This is due to the fact that squat increases but bow motion in waves decrea-ses with increasing CB. It is asSumed here that these opposite effects are liable to cancel each other.
Inthe calculations linear interpolation is used for other values of and hIT than shówn in these figures. The c'ves at h/T' 2 and more and at 0.2 are given only for complèteness, because on the fairways hIT is normally below 1.5 and the speed is iowa
5iI The output: vertical motion of bow
If it is asSuifted that the sea spectrum is limited toa narrow band añd the wave height follows the Rayleigh-den-sity function, by help of linearity assumption we come to the following conclusions (see e.g. /190/:
SB(w) :!H(jw)!2'S(W) m
: f3B)d
(5.12) ATAT2
exp [---]
where H(jU)55
Then /190/:
a
-P{AT2 > a}. 1 - ff(AT2)dAT2 (5.13) o
which is the probability that T2 is greater thàn a. From (5.8) and (5.10) follows
m0
= OCA
a4l
Sc(x)shdxand from (5.13)
a = (-2m01n[P.{AT2 >
a}]°5
()
where, fiñally, a is the value, of AT2 which must be
consi-dered in the evaluation of underkeel. c'lêararices.
5.5Totalincrease of draught in waves
In this report it is assumed that squat is greatestat bow. This assumption is supported by the majority of the experimental results referred to in /75/.
Mor.eover, according to Guliev /60,61/ and the tests carried out in Otaniemi in transient shallow water waves squat is nearly the same in calm waters as in waves. There-fore one can calculate squat separate1y This is-
advanta-geous because at a given water depth only one squat cttrve (as a function of the speed) is needed to which one can add the effect df diffeient sea states.
The ratio h/T needs a special consideration. As noted in 5.3..2 the actual value of h/T decreases wt'h increasing speed. Consequently, one has two possibilities whén déter-mining the amplitude response operators. Firstly, hIT is calculated for a ship at rest., in which case the response operators at different speeds include the effect of the changing water depth. Secondly, h/T is the actual value. In this case the water level must be lowered by the amount of sinkage befbre calculating the new ratio h/T.
6. ROLLING
Rolling is not considered in this report. This does not exclude its importance in some cases, e.g. in beam and oblique seas, when calculating uriderkeel clearances.
Indeed, according to some model tests /66,212/ vertical motions at bilge were greater than the motions of bow.
The severity of rolling can be evaluated with the help of the ratio T3/Tm where T3 is the natural rolling period of the ship and Tm is the period at which the wave spectrum is at its maximum. Figure 21 shows the shallow water sea spectrum at various wind speeds in the Rauna fairway. At the wind speeds of 7 beauf. and below Tm is smaller than 9 s. On the other hand the natural rolling period for a normal ship is greater than 10. . .12 s de-pending on the metacentric height. In addition, when the sea is not fully arisen (the wind has not been blowing sufficiently long), the wave periods are smaller. One can conclude, that the ships entering the Rauma fairway will not roll seriously in beam seas.
57
7.
EXAMPLE OF UNDERKEEL CLEARANCE DETERNINATION7.1
GeneralThe procedure for the determination of underkeel clea-rance is shown here illustrated by an actual problem. The example will be the new, deep fairway to the city of Rau-ma, ori the Finnish west coast /191/. The calculations Were carried out by the computer programme UKCL which exists in two versions: FORTRAN IV language (UNIVAC 1108) and BASIC (HP 9830). The block diagram is essentially the same as shown in Fig. 5a. This programme is quite simple, because the response operators are given as initial data.
7.2
Procedure of calculation7.2.1
Initial data7.2.1.1
The fairway CENTERLINE CH -looh + 3 h2 Mean length-
4 20 m Breadth. b 100 m Slope -h',meanSm
MAIN DIMENSIONS ARE FROM 119V. PLACE 3a.
Fig. 18 shows the cross-séction of he fairway with all the necessary parameters. It is assumed that only one ship is in this part of the fairway at a time. A devia-tion of 20 m from the centerline of the fairway is allo-wed fOr the ship propelling at full speed.
The first estimate for the. water depth is calculated from (7.1)(see Table 1).
h 1.15. ...l.20 x T .. (7.1)
where T = design draught, here T 9.0 n
The water depth is selected equal to 10.5 n as in
/191/, which givesh 1.16 T.
7.2.1.2 The sea spectrum
Due to the lack of sufficient data, the parameters of the sea spectrum function are calculated from Eq. (5.3) arid (5.11). This function makes allowance to 'the limited fetch. But, the duration of the wind is not considered in these equations. Now, most often the duration of strong winds is mot long enough to arise the sea fully im Finnish water areas. Therefore, the wind speed used in the dimensioning of fairways must be selected to be smaller than the wind speed based purely on the wind sta-tistics.
A rough estimate, of the amount by which the wind speed must be decreased is made as follows. Àccordingto Eq. (5.2) and (5.3) the significant wavé height is pro-portional to U24 if a arid ß remain constant. This last condition is approximately trué due to the form of Eq. (5.24). If we say roughly, that the waves calculated by help of Eq (5.3), (5.14) and (5.1) - see Table 3 - aré two times too high then we must, decrease the wind speed by a factor of about 1.2.
Consequently, the combination of wind velocity, wind direction and fetch is determined first. Data for' this case are shOwn in Fig. 19. Encircled numbers indicate
59
fetch in kilometers. Also the direction of the fairway and the directions of the wind considered in
0+
0 are shown. Only heading angles of 180 - I5dealt with.Table 3 shows heading angles and max. corresponding to each other.
70
this report will be
fetches
TABLE 2 WIND DIRECTIONS / FETCH COMBINATION IN RAUMA FAIRWAY o. SW SE
0
225 135E Q90
°U'ect,on of the fairway
( \
Directions of the wind (directions of the wove front)considered when calculating the motions of the bow. The numbers in circles indicate the fetch in kilometers. FIGURE 19 WIND DIRECTION / FETCH COMBINATIONS
AT THE RAUMA FAIRWAY
Heading angle [degi fetch 1m] resonance length 1m]
180 230 58 150 2IO 69 135 2I5 85 120 260 121 315
NP
loo go 80 5 P
tjjuI
[ei.]The wind velocity distribution function, shown in Fig. 20 is based on the statistics of reference /192/. In this case the wind speed is considered to be independent on the direction.
In add-ition, a fetch equal to 20 km is used through-oùt the calculations, because it does not vary much at different heading angles, see Table, 2.
i
(.su)
is the probability that therandom wind speed is
less or equal than
U
12 15 19 23
il [BeautJ
Wind speed
[m/s].
FIGURE 20 CUMULATIVE WIND VELOCITY DISTRIBUTION FUNCTION AT TRE RAUMA FAIRWAY
61
Fig. 21 illustrates the shallow water sea spectrum calculated at different wind speeds. In the same figure the scale of the ratio Lpp/Lw is shown as a function of
The Tmvalue gives the wave period at which the spectrum is at its maximum.
Fig. 22 shows the square root of the first moment in0 of the sea spectrum in deep and shallow (h = 10.5 m) water. By the help of m0it is possible to calculate some pro-perties of the waves /190/, e.g:
cw
2.5/s
mean wave height(7.2)
s mean height of the 1/3 highest waves, called the significant height
Cw1/]0 5.lv'i0 as above but for 1/10 highest waves
Table 3 shows some wave properties at different wind speeds. The wave heigbt-values seems to be too great by a factor of 1.5.. .2. (In this table the duration of the wind is not limited).
TABLE 3 CALCULATED WAVE HEIGHTS IN THE RAUMA FAIRWAY, IN METERS
7.2.1.3 The ship
The motions of the vessel are greatest at the reso-nance. This means that the maxima of the sea spectrum and of the amplitude response operator of the bow occur at
the sane Lpp/L-value. In the case of amplitude response operator this value depends only slightly on the speed of
Sea state 110 ÇW1/3 5 0.38 0.95 1.52 l.9 6 0.60 1.50 2.kO 3.06 7 0.83 2.08 3.32 .23 8 1.17 2.92 4.68 5.98
THE RAUMA FAIRWAY. PIERSON MOSKOWITZ
EQUATION ADJUSTED FOR SHALLOW
V [rn] 2.0 1.5 1.0 0.5 0 Wind Speed
rrn
L 0 1 2 3 4 5 6 7 8 9 10 [Beau1.FIGURE 22. THE SOUARE ROOT OF THE FIRST MOMENT OF THE SEA SPECTRUM AT THE RAUMA FAIRWAY
3w
2.5 su34.0 V
Si/IO/1
1'
/1
/
Fetch hPieron/
- 240
krn 10.5 Moskowitz Deep water Shallow/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
n qn nr, nthe ship, see Figures 17. If this value is taken equal to 0.9 then:
LR = 1.8w/k cos(ii - 1800) (7.3)
where LR = resonance lenght of the ship heading angle
k wave number
Wave number k must be solved by some iterative method
from
2gI/J U2
k tanhkh (7)4)where ii wind speed B, see (5)4)
Equation (7)1) is based on the fact that the sea spectrum calculated according to (5.l)(5.3) arid (5)4) has a maximum at circular frequency w as follows /189/
0.25
(7.5)
In addition, Table 2 shows the resonance lengths for different heading angles. A usual ship having a draught of 9rnhas a length of 11I4 m (L/T = 16). As one can see, the resonance length is smaller than 11444 n. Therefore the situation will be worse when the ship's size is decreased while draught remains constant. This means, that the design
ship must have as small value of the ratio LIT as would
be expected to occur with a reasonable probability. More-over, in oblique seas the effective length of the vessel
is still decreased by the factor cos(u-180°).
Block coefficient was selected equal to 0.70 due to the reasons stated in 5.3.3.
7.2.2 Squat
Squat was calculated in accordance with the section 44. For convenience, all the necessary equations are collected
where
65
here. For the computer the curves in Figures 9 and 10 are presented in a form of formulas, see below.
Squat is calculated from
AT1 = 2.ZI KSKLFh -
Fh)°5 T B CB/LPP
(7.6)i Ax
Sç
Factors K1, K, KL are calculated as follows
K3 7.145s + 0.76 s > 0.03 K5 1 S < 0.03 h' K1
2asbecl
- + 1 where e 2.718... a 3282.205 b = 3.027662 -10.76032The uncorrected KL = K is (Fig. 10)
= I40( - 0.25) + 1 (7.10)
When calculating the area ratio s, the breadth of the channel is selected as follows
channel breadth = b0 if b0 < lOB channel breadth lOB if b0 lOB
Then the crossection area of the actual channel is
divided by that of a channel having a slope of k5°. If the
ratio is indicated by c, then
KL = cL(KL -
l)L L.
+ 1 (7.11)(7.7)
(7.8)
EFFECTIVE LENGTH [n,]
FIGURE 23.1 VERTICAL 80W MOTION DUE TO WAVES, U=1O rn/S £
L 12 Cm]
200
EFFECTIVE LENGTh [m]
67
u FIGURE 24 VERTICAL BOW MOTIONS DUE TO WAVES
(FFECTIVE LENGTH (mJ
DUE TO WAVES U 15 m/
VERTICAL BOW MOTION FIGURE 23.3
CL O b0 - 2 1 CL - b0 - B lOB - 2 1
CL_
9B 1 see Pig. loThe squat of some ship sizes Calculated by this proce-dure is shown in Figures 25.
7.2.3 Vertical motions of the bow due to waves
The calculations were carried out with the following values of the parameters
h: 10.2, 10.5, 10.8 m
0, 10, 12.5, 15 rn/s 0, 2, ,
6 rn/s
Lpp: 60(20)200 or 120, 160, 200 m Fetch: 2IO kn
The computations were carried out in accordance with
5Z First moment m0 was calculated from Eq. (5.14) and
the bow motion from (5.15). Probability was selected equal to 0.10. The reasons for this rather high value are given
in 3.1.
Figures 23.1.. .23.3 show the effect of the ship's length at various ship and wind speeds. Fig. 2k shows same results, but in another way.
Figures 25 show the total motion of the bow at various wind speeds and water depths as a function of ship speed.
1> 5xB
b0 < 10 B
b0 > 10 B
FIGURE 25.1 BOW MOTIONS AT THE RAUMA FAIRWAY,
120 m
FIGURE 25.2
BOW MOTIONS AT THE RAUMA FAIRWAYS, L1,
7.3 Evaluation of results
- This subchapter will cômmént upon Figures 23.. .25. One
must béar in mind, that these conclusions only apply to waterwáys typical of Finland òr similar circumstances.
The effect of the ship's length on the bow motion due to waves when the draught remains unchanged is shown in Fig. 23.1.. .23.3 and 2. On the basis ôf these figures the length -of the design ship must be selected as small as seems reasonably probable. This minimum length may be estimated as follows. A ship with a draught of .9 n may have â length of 135 (then Lp?/T 15). When the heading angle is 135 the effective length is 135 x cos ¿5 95 in. Then
the bow motion due to waves is about 0.5, l.Ò and 2.0 m atwind speeds of 10, 12.5, 15 mIs, respectively. To these values one must add squat. (The effect of ship speed (on motions due to waves) is ignored so far). Due to the limi-ted duration of the wind - the sea will be fully arisen very seldom - one can decreasè the wind speed and inérease the length of the ship applied to in the computations.
The wind speed is dealt with in 7.2.1.2. The waves for a developing sea are shorter than fdr a fully ariseñ sea. Therefore, the resonance length will be shortér, or equi-- valently, the length of the ship may be increased. As a
rough estimation it may bè said that instead of n the length of 110 n can be applied. Putting these effects together, we get new valües -for the bow motion: 0.1; and 0.9 m t wind speeds öf 12.5 and 15.0 mIs-, respectively.
-The selection of the wind speed is important. As seen in F-ig. 21;, the bow motion due to waves will increase from 0.5to 1.5 m - or 200 % - when wind speed is increased from. 10 to 15 m/s - ôr 50 % - at hip lengths of 100... 120 n.
The effect of the ship's speed is not so important when considering the bow notioné due to waves. In fact, increa-sing thé ship's speed decreases the bow motions. This .s
a result from the fact, that the effective ship length increases when the speed is increased. In addition, the
71
ratio h/T decreases due to sinkage.
On the contrary, the ship speed is of decisive impor-tance when we consider the total bowmotion tT. This is
caused by the strong effect the speed has on squat, see
Fig. 2.5. Hence, the maximum speed of the design ship per-mitted in the fairway needs a special consideration, for
instancé,, a speed limit might be thinkable.
One remarkable fact - see Figure. 25 - is that small variations in water depth affect only slightly the under-keel clearance. In Fig. 25 the results are nearly the same at water depths of 10.2, 10.5, 10.8 n when ship speed
varies between 4. ..6 rn/S. (Thi speed rangê is décisivé in the dimensioning of fairways). Consequently, a new
calcu-latioñ loop is not needed if the final and irdtial water depths do not differ more than about 5 per cent. For
example, in this case we can select añy depth between 10 and 11 m based on the results at 10.5 m. This fact is worth commenting upon. Firstly, sqúat incrèases but bow motion decreases as the speed of ship is increased. These opposite effects cancel each other at the speed range of
4. . .6 rn/s. Secondly, the possible max. error in underkeel clearance due to this simplification is .about 5. . .10 cm.
Another interesting fact which in itself does not
affect. the underkeel clearance .calcu_lat ions - is that at
higher wind speèds the total bow motion is smallest, when the speed of the vessel is 1.. .2 m/s. Consequently, a ship propelling in shallow waters in a severe sea state must select her speed carefully in order to avoid bottom cön-tact - she can neither steam at full speed nor slow down too much.
Finally, we calculate the water depth of the pauma fair-way. The probability of' thè wind speed is selected equal
tó 0.9.5. Froín Fig. 20 we get U 12.5 rn/s.. This.is reduced
to 11 m/s(the effect of limited duration). Efective length is selected equal to 100 m which corresponds to 142
m. (A heading angle of 45° and the effect of limited wind duration). The probability of bow motion due to waves
is chosen to be 0.10 as above. One can use anôther probabi-lity if âT2 is multiplied by the factor
(lnp'/1np01)°5
wheré p' thé new probability 0.10
Theñ we get the following results, Table .
TABLE . Sorné results of dredging depth calculatioñs in thé Rauma fairway.
For comparison in /191/ water depth - which now equals the dredging-depth - is selected equal to 10.50 m. The norms (Table 1) give
i.: 9.9
m (protected archipelago) <11.0 m (unprotected archipelago)11.:2.5 ni (open sea)
Place a can be placed in the second group. This ealculatioñ is p±esented only as an example; during the actual planning the selections may be carried out otherwise However, the author believes that the Figures 23-25 are generally applicable to the Finnish
seaways, when thê depth iS about 10.5 m and wien the sea-way is not protected by an archipelago.
-Speed of vessel rn/si Em]
. 3
11 6
Based on Squat 0.25 0.47 0.80 1.20 25.1,2.5.?
Bow motion due to
waves 0.57 0.511 0.50 0.116 Fig.2h4
Safety. margin . . 0.30 0.30 0.30 Chapter 2.5.
Totally 1.12 1.31
1.60
1.96
Draught 9.00 9.00 9.00 9.00
Permanency correction 0 15 0 15 0 15 0 15 Fig 1, ixa
73
Which speed should be selected for the basis of dimen-' sioning? An interesting fact revealed from Figure 25 is, that the depth of the fairway has an optimum. This means, that the required depth can be decreased with the decrea-Sing ship speed - but only to a certain limit which seems to be here at 3 or rn/s. If the ship speed is furthér decreased, the total bow motion remains nearly constant. Consequently, the dredging depth of 9+b.8+0.3+0.l5l0.25 seems to be sufficient instead of 10.5.. .10.7 m. (If a speed limit of 3 rn/s is applied).
One can get a rough idea about the order o the total bottom contact probability, if some simpliications are carried out. Ship size, wind speed, wind direction, water level etc. are assumed to be independent. Then
P{AT > l.3} Z 0.1 x 0.5-x 0.05 x 0.15 x OfOl
-
3.75 x iowhere 0.1 prob. of bow motión
0.5 = prob. of wind diection
0.05 r prob. of wind speed 0.15 r prob. ôf water, level 0.01 r prob. of ship size
This value is not the right one because iT l3 could
be reached for exàrnple at a lower wind spéed but with a lower probability of bow motion. Therefore we take here
P{T > 1.3) r
This means, that when the yearly traffic equals 1000 ships, a bottom contact occurs once in 100 years.
8. CONCLUDING REMARKS
8.1 Thé, applicability of the presented method
This report gives no general method for the determina-. tion of underkeel clearances. Actually, only the vertical
-motion of the bow in shallow water waves is computed. More-over, only heading angles of 1800 ±145° are dealt with. The cases of beam seas (rolling) and following or overtäking seas are môt cnsidered due to the lack of theore.tiàal
and experimental knowledge. It is believed, that the astern sea case is not so serious as -the head sea case. Therefore, the former is not dealt with because one can always imagine the sane sea state for a head sea case. Instead, a possible fairway direction can .be perpendicular -to the most severe wave directiOn. Then the beam sea case might bé decisive.. (This report give8 no way, to cope with
this situation). . . .
According to the presented method one- can calculate squat in cómpletely different fairways and for a wide variety Of ships. The bow motions, on the 'contrary, are
calculated for a special ship, SERIES 60 with block coeffi-cient of 0.70. All the response operators are given for this ship type But, in principle, one can take any other type of ship añd calculate or determine by experiments the necessary response
operators.-For the present, however, the determination of under-keel clarances based on block coeffiòient equal to 0.70 'is recommended because the majority of theoretical and
experimèntal results apply to this value. In addition, -squat and-bow Ñotions are affected in an opposite way if. the block coefficient is changed.
One unclear question -is the influence of the scale-effect when applying the results of model tests to the ships of full size. In fact all the results of this. report
apply to the model scale. The seriousness of this effect is not known and it is possible that it has only a minor influence. The effects of non)inearities in shallow water wàves remain unclear, too. .
