E L S E V I E R
P I I : 8 0 1 4 1 - 1 1 8 7 ( 9 7 ) 0 0 0 0 9 - 6
Applied Ocean Research 19 (1997) 49-60 © 1997 Elsevier Science Limited All rights reserved. Printed in Great Britain 0141-1187/97/517.00
On the estimation method of hydrodynamic
forces acting on a very large floating structure
Hiroshi Kagemoto, Masataka Fujino & Tingyao Zhu
Department of Naval Architecture and Ocean Engineering, Graduate Sciwol of Engineering, Tiie Univeisity ofToio>'o. 7-3-1 Hongo, Bimicyo-l<u, Tokyo 113, Japan
(Received for publication 22 March 1997)
The floating structures that may be used for such purposes as an international airport or an offshore city are expected to be as large as several kilometers long and wide. For the estimation of hydrodynamic forces due to waves or motions that will act on such huge structures, a direct application of conventional numerical methods is practically prohibitive, because the required computational burden is enormous. In order to avoid this difficulty, an approximate method is developed in which computational time is drastically reduced without appreciable loss of accuracy. Although a direct application of conventional numerical methods is difficult for the reason that the corresponding structure is so large, the method proposed in this paper exploits the very fact that a structure is veiy iarge to simplify the calculation. The effectiveness of the new method is demonstrated in comparisons with results obtained by the direct application of a conventional numerical method. © 1997 Elsevier Science Limited.
Key words: very large floating structure, hydrodynamic force, floating airport.
1 I N T R O D U C T I O N
A l t h o u g h Japan does not have large land areas, i t is sur-rounded by the oceans and the territorial sea area is quite large. E x p l o i t i n g this fact, reclamations o f the sunrounding sea areas have been extensively carried out since an ancient period i n Japan i n order to secure the land spaces that can be used f o r various human activities. I n recent years, other than the seaward extension of existing lands, several artificial islands are being constructed o f f shorelines. A typical exam-ple is the new international airport constructed o f f Osaka, the largest city i n western Japan, i n 1994. As a construction method of artificial islands, a floating island is now being considered to be feasible. A n offshore floating o i l storage system, w h i c h is as large as 400 m l o n g and 100 m wide, was actually constructed and installed o f f Japan's coast and has been w o r k i n g so far without any serious accidents. It is understood that a floating type island is more ecoriomical than a reclaimed artificial island when the water depth is larger than 2 0 - 3 0 m . Moreover, a fioating one is preferable f r o m the viewpoint o f the preservation o f the surrounding environment.
I n exploiting such advantages o f floating structures, there now exist several proposals i n Japan to construct a floating airport o f f some metropolitan areas. Even i n the countries
that possess vast land areas such as the U n i t e d States, a floating airport is now being considered seriously as a pos-sible alternative to a land-based airport because o f t h e envir-onmental issues [ 1 ] . I n order to f u l f i l the role as an international airport, the length of the structure should be at least 5000 m and the w i d t h w i U also be needed to be as wide as 1000 m . A s menfioned before, the largest floating structure that ever has been built is 400 m i n length and
100 m i n w i d t h . Therefore the projected floating interna-tional airport is more than 10 times the length and 100 times the area o f the existing largest floating structure. I n order to examine the feasibility of such a huge floating structure, accurate estimation of wave induced forces is very important. A s f o r the estimation o f w a v e forces o n floating structures, several numerical methods have already been established. O f those, one that is used routinely is a singularity distribution method, i n w h i c h the wetted surface of a corresponding structure is divided into small panels and hydrodynamic singularities, such as a source or a doublet are distributed on each panel so that the boundary conditions on the surface is satisfied. A l t h o u g h , i n principle, larger number o f panels result i n a more accurate answer, i t is understood that solutions o f practically enough accuracy can be obtained i f one wavelength is discretized into, say, 10 panels. (Here we assume that the singularity strength o n
each panel is constant.) A b i d i n g by this m l e , f o r the analysis o f a structure o f 5000 m long and 1000 m wide i n a regular wave train whose period is 5 s, which is commonly observed near-shore areas, about 1250 panels are needed i n length-wise direcdon and 250 panels are needed i n widthlength-wise direcdon, w h i c h amounts to as many as 312 500 panels i n total. (Since the vertical dimension is quite small (may be less than 10 m ) compared to the horizontal dimension, the discretization i n vertical direction is not accounted f o r i n the present analysis). This, i n turn, requires us to solve a set o f 312 500 simultaneous equations o f complex coefficients, w h i c h is extraordinarily time consuming even i f we have an access to the fastest computer machine i n the w o r l d .
I n order to avoid this d i f f i c u l t y , we propose a new method, i n w h i c h solutions can be obtained w i t h reasonable computational time without appreciable loss o f accuracy. A m o n g various types o f structures that are n o w being con-sidered as possible candidates o f a floating airport, we deal w i t h a simple thin box-shaped structure i n the present study. W e also assume that a singularity distribution method is used as a numerical tool o f the analyses. However, as w i l l be described later i n this paper, similar ideas to those that w i l l be presented i n the f o l l o w i n g sections can be applied to other types o f structures or to other existing numerical meth-ods. For the analyses we assume that the usual small ampli-tude linear potential theory can be applied.
2 W A V E F O R C E S ON A V E R Y L O N G AND S L E N D E R S T R U C T U R E
2.1 In beam waves
First let us consider an analysis o f wave forces on a very long structure fixed i n regular beam waves. W e assume that the length o f the structure is very long compared to the wavelength w h i l e the width is comparable to or smaller than the wavelength. I n order to can-y out the analysis by a singularity distribution method, we discretize the bottom surface o f the structure into small panels. ( W e assume the draft o f the structure is much smaller than its horizontal dimension and thus we neglect the discretization i n draft-wise direction.) W h e n the length o f a structure is much
X=100in (a)
(b)
Fig. 1. (a) Discretization of a very long structure into 500 panels,
(b) Discretization of a very long structure into 105 panels.
larger than the wavelength, it can be easily understood w i t h -out detailed calculations that the lengthwise variation o f the singularity strength should be very smafl, except at the ends o f the stmcture. I f this is the case, i t is not necessary to divide the surface o f the structure into small panels o f equal size but, instead, a certain inner part o f the structure can be represented by a small number o f l o n g panels on w h i c h the singularity strength is assumed to be constant. I n order to c o n f i r m this assumption, we earned out an exam-ple numerical analysis o f wave forces on a stmcture w h i c h is 1000 m long and 50 m wide fixed i n beam waves. The wavelength was assumed to be 100 m . A m o n g the two c o m -putations conducted f o r this analysis, one was earned out, as shown i n F i g . 1(a), by discretizing the stmcture w i t h 10 m X 10 m square panels (10 m is 1/10 o f the wave-length). The total number o f panels is 500. The other compu-tation was caixied out, as shown in Fig. 1(b) by discretizing the stmcture into 5 strips i n beamwise dii'ection while each o f the strips was divided into a single long panel, plus 10 small panels at both ends that account f o r the end effects. The total number o f panels is 105. As already described, i n the analysis by 105 panels, we assume the singularity strength on each long panel is constant. I n Fig. 2(a) and (b), the lengthwise
2.0 1.5 1.0 0. 5 I " ^ - 0 . 5 -1.0 - 1 . 5 -2.0 2.0 1. 5 1.0 . 0. 5 3 -0.6 -1.0 - 1 . 5 1 — ' — I — r - r i — ; real ; _ l _ 1 10 20 30 40 50 60 70 80 90 100 (a) p a n e l n u m b e r ' ! ! ' ! ' ! , 1 > 1 . 1 . 1 . 1 1 r real Inag \ y \ \ \ \
" ~ i — \ — \ — \ —
\ \ \ \ \ \ \ \ \ -\ \ \ \ \ \ \ \ \ -i -i -i -i , i , i . i , i . i . i , i , i . i . 30 50 60 70 (b) 80 90 too p a n e l n u m b e rFig. 2. (a) The lengthwise distribution of the singularity strength obtained by 500 panels in beam waves (reahreal part, imag:ima-ginary part), (b) The lengthwise distribution of the singularity
Estimation metiiod of liydrodynamic forces 51
- i ; > 1000 m
Fig. 3. A very long structure in oblique waves.
distribution (along the most weatherside column of panels) o f the singularity strength obtained by 500 panels and those obtained by 105 panels are shown respectively. The singular-ity strength on the inner 80 panels is the same i n the solution obtained by 105 panels i n consequence o f the assumption, whereas i n those obtained by 500 panels, although no such assumption is made, the singularity strength is also o f almost the same value on the corresponding inner panels. Moreover,
the values o f the singularity strength on the panels at the ends also agree quite w e l l between the two results. Since the time needed f o r solving the final simultaneous equations, which accounts f o r a large pait o f the entire calculation, is approxi-mately proportional to the square o f the number of panels, the computation time for that part is reduced to (105/500)^ — 0.0441 by employ mg the present approximation.
' I ' ! ' ! ' ! ' ! ' 1 ' 1 ' 1 ' ! ' i : -1— r e a l \ ! imag V \ : :, : / : \ : \\ \ ^ •
/
\ !
i \ i \ i / . i i , i M . / ' i . i , i , 10 20 30 40 50 (a) 60 70 80 90 100 p a n e l n u m b e r 90 too p a n e l n u m b e r Fig. 4. (a) The lengthwise distribution of the singularity strengthobtained by 500 panels in oblique waves (x = 80°) (realireal part, imag:imaginary part), (b) The lengthwise distribution of the
singularity strength obtained by 105 panels in oblique waves.
^ 1.0 h 3 2. 5 2.0 1.5 >< 1.0 s 1 0. 5 0 a -0.5 b -1.0 1 1 1 1 1 ' ! ' ! ' ! ' ! ' ! ' ! ' i l l ; — r reali i i i i i i / 1 , i , i , , i , i , i . i , i . i , 1 10 20 30 40 50 60 70 80 90 100 (a) p a n e l n u m b e r 1 — r T " ' — r ~ ' ~ ~ ! — ' ~ T — • real L..::.:rL.ini3g.;.. J -10 20 40 50 (b) 60 80 90 too p a n e l n u m b e r Fig. 5. (a) The lengthwise distribution of the singularity strength
(divided by e''-*/"-'*"^") obtained by 500 panels in oblique waves (x = 80°) (reahreal part, imag:imaginary parO. (b) The lengthwise distribution of the singularity strength (divided by e''-('-i^ ''''"°«)
X=100lD
Fig. 6. Discretization of a very long structure in head waves (205 panels). 2.2 In oblique waves
The idea described i n Section 2.1 can be easily extended to the case i n oblique waves. W h e n we consider a very l o n g and slender structure fixed i n oblique waves as shown i n Fig. 3, instead of assuming that the singularity strength on inner panels is o f constant value as i n beam waves, i t may be able to assume that the singularity strength on each o f the inner panels varies according to the f o l l o w i n g law:
a; = ff,-e'**'^ (1) where Oj, d,- denote the singularity strength on y-th and (-th
panel respectively and Xj, Xi stand f o r the x coordinates o f (the center o f ) the conesponding panels, k represents the wave number o f the incident wave. I f eqn (1) does hold, i t is not necessary to discretize a structure into a large number o f small panels of equal size but a certain inner part o f the
1.0 0.5 3 -0. 5 -1.0 1 ' ! ' ! ! ! ' ! ! ' ! ! • = - real Imag
'. / A • /
A V / .
V \
A " - . A ' \ /k\ Ac. A<\
r
, i , i . i , i , i , i , i , i , i ,
1 10 20 30 40 50 60 70 80 90 100 (a) p a n e l n u m b e r 3 - 0 . 5 90 100 p a n e l n u m b e r Fig. 7. (a) The lengthwise distribution of the singularity strength obtained by 500 panels in head waves (reahreal part, imag: imaginary part), (b) The lengthwise distribudon of the singularitystrength obtained by 205 panels in head waves.
Structure can be represented by a small number o f l o n g panels, as was the case i n beam waves, on w h i c h the sin-gularity strength is assumed to vary according to eqn (1). Example calculations were carried out f o r a 1000 m X 50 m structure fixed i n obtique waves (wavelength: 100 m , X = 80°).
Figure 4(a) and (b) show the lengthwise singularity strength distribution obtained by 500 panels and that obtained by 105 panels respectively. (The discretizations are the same as that shown i n F i g . 1(a) and (b).) The results obtained by 105 panels agree quite w e l l w i t h those obtained by 500 panels. This agreement is further manifested i f we divide the obtained singularity strength by e'^'^^'~^''^'^°'^^ as shown i n Fig. 5(a) and (b). I n F i g . 5, the straight fine parallel to the horizontal axis is where the assumption given by eqn (1) holds. I n the results obtained by 105 panels, aU but those on the 10 panels at both ends are parallel to the horizontal axis because o f the assumption we made, whereas those obtained by 500 panels, which does not use such an assump-tion, are also almost parallel to the horizontal axis on the corresponding panels. The end effects are w e l l accounted f o r by 105 panels.
2.3 In head waves
I f we simply apply eqn (1) to the analysis of 1000 m x 50 m structure fixed i n head waves (wavelength: 100 m ) , the results shown i n F i g . 6(a) and (b) are obtained. Compared i n the figures are the lengthwise singularity strength distri-bution obtained by 500 panels and that obtained by 205 panels, i n w h i c h , as shown i n F i g . 6, the upstream and the downstream o f each o f 5 strips are divided into 20 small panels while the remaining inner part is represented b y a single long panel. I n the calculation by 205 panels, i t is assumed the singularity strength varies according to eqn (1) along the inner long panels. A l t h o u g h the singularity strength at the downstream is not property accounted f o r by the 205 panels, i t seems that i t is w e l l predicted on the other parts. However, i f we observe carefully, the singular-ity strength decays gradually f r o m upstream to downstream i n the results obtained by 500 panels (Fig. 7), whereas i n those obtained by 205 panels it does not decay at all due to the assumption (eqn (1)) we made. Since waves are reflected more or less at each panel as they proceed to downstream, the wave energy should eventually be diminished at far downstream, w h i c h i n turn means that the singularity strength at the con-esponding downstream area should be zero. However, as far as we stick to the present approxima-tion (eqn (1)), the singularity strength cannot be zero how-ever f a r downstream we go. I n other words, eqn (1) holds at sufficientiy downstream, where the singularity strength is
Estimation metiiod of liydrodynamic forces 53
?v=100n
Fig. 8. Discretization of a very long structure in head waves
(105 panels). 1.0 0.5 3 -0.5 -1.0 1.0 0. 5 -0.5 -1.0 1.0 0. 5 -0.5 -1.0 1 ' ! ' ' 1 ' 1 ' 1 ' 1 ' : = — ireal { 500 f a n e l s ) ' : iimag i -o real 1 105 )anels )
• / M M $'. fit'' A f •* Jit'' jitK A w w . i , , i , . i . i , i , . 1 10 20 30 40 50 60 70 80 90 100 (a) p a n e l n u m b e r t ! ' ! ' ! ' ireal ( 500 ; iimag i i r e a l i; 55 ' imag lanels panels ) t \ (1 , i , , i , i , i , i , • V 10 20 30 40 50 60 70 80 90 100 (b) p a n e l n u m b e r ' ! ' ; — 'real i 500 i 1° | r e a l ( 30 i " iimag i anels janels ) ' ) • iK\ [ .^^ r i \ t ^ a aT V .1 f t : : V v / i \>3t/: V K j i ' • , i , , i , i ' , i , i , . 10 20 30 60 70 80 90 100 p a n e l n u m b e r 40 50 (c)
Fig. 9. (a) Comparison of the lengthwise singularity strength
dis-tribution in head waves obtained by 500 panels and that obtained by 105 panels (reahreal part, imag:imaginary part), (b) Compar-ison of the lengthwise singularity strength of distribution in head waves obtained by 500 panels and that obtained by 55 panels, (c) Comparison of the lengthwise singularity strength distribution in head waves obtained by 500 panels and that obtained by 30
panels.
zero, and thus the approximation is meaningless f r o m the pracdcal point o f view. However, although eqn (1) cannot be used f o r this reason, it has been shown that the waves propagating along a slender body decay i n proportion to 5^"', where s is the distance measured f r o m the upstream end o f the body, while oscillating i n phase w i t h the ambient incident waves [ 2 ] . Then f o r the analysis o f a very long structure i n head waves, we w i l l be able to assume that the singularity strength varies on the downstream long panels as:
. | - l / 2 . ^ / . ( ^ - . . )
aj = ar\xj (2)
Replacing eqn (1) w i t h the above equation, we still may be able to reduce the number o f panels even i n head waves. I n order to c o n f i r m this speculation, calculations were canied out f o r a 1000 m X 50 m structure fixed i n head waves (wavelength: 100 m ) . Figure 9(a) compares the results on the lengthwise singularity strength distribution obtained by 500 panels o f equal size and those obtained by 105 panels. (The discretization w i t h 105 panels is shown i n F i g . 8.) O n the inner long panels, it was assumed eqn (2) holds. W e can observe the results obtained by 500 panels and those obtained by 105 panels agree quite w e l l w i t h each other. Although, i n order to account f o r the end effects, it may be better to discretize the downstream end into small panels, the numerical fact shows such care is not necessary, w h i c h means that the effect o f the downstream on its upstream is small. Results obtained by 55 panels, i n w h i c h the number of small panels at the upstream is reduced to 10, and those obtained by as f e w as 30 panels, i n w h i c h the number o f small panels at the upstream is further reduced to 5, are shown i n F i g . 9(b) and (c), respectively. I n Table 1, the heave force and the pitch moment (about y axis) that w i l l act on the structure calculated using the obtained singular-ity strength are compared. W e can see that the forces/ moments obtained by 55 panels d i f f e r by only less than 2% f r o m those obtained by 500 panels. Even w i t h 30 panels, the error is 10% at most.
3 W A V E F O R C E S ON A V E R Y L O N G AND V E R Y W I D E S T R U C T U R E
3.1 In head waves and in beam waves
I n the previous section, we dealt w i t h a structure w h i c h is very l o n g but not quite w i d e compared to wavelength.
Table 1. Comparisons of the heave force and the pitch moment obtained by 500, 105, 55 and 30 panels
Number heave f. pitch m. heave f. pitch m. of panels Pg^a BL {b),{c),(d)iia) (b)Xc),{d)/ia)
500 (fl) 1.13E-02 1.18E-02
105 ib) 1.12E-02 1.22E-02 0.988 1.036 55 (c) 1.15E-02 1.19E-02 1.018 1.007 30 (d) 1.24E-02 1.06E-02 1.097 0.898
6 t h c o l u m n of p a n e l s
.1 5 0 0 m
Fig. 10. Discretization of a very long and very wide structure (1800 panels).
However, structures such as a floating airport should be very w i d e as w e l l as being very long compared to wavelength. As an example, we carried out an analysis o f a structure w h i c h is 1500 m long and 300 m wide flxed i n head waves. The wave length was assumed to be 100 m . For this analysis, the stmcture was discretized into 1800 (150 X 12) panels o f equal size as shown i n F i g . 10. Figure 11(a) and (b) show the obtained lengthwise singularity strength distribudon along the 1st column o f panels and that along the 6th c o l u m n o f panels respectively. F r o m these results, i t can be observed that there is a distinct difference between
120 135 150 p a n e l n u m b e r 2.0 1.5 1.0 0.5 0 - 0 . 5 - 1 . 0 - 1 . 5 - 2 . 0 ! r r e a l • ; - - ' " i m a g : " 751 766 781 796 811 826 841 856 871 886 900 (b) p a n e l n u m b e r Fig. 11. (a) The lengthwise distribution of the singularity strength obtained by 1800 panels in head waves (1st column of panels), (b) The lengthwise distribution of the singularity strength obtained by
1800 panels in head waves (6th column of panels).
the singularity strength distribudon along the 1st column and that along the 6th column. A l o n g the 1st column, w h i c h is exposed to the outer sea area, the singularity strength decays gradually f r o m upstream to downstream w h i l e varying i n phase o f the ambient incident waves. On the other hand, along the 6th c o l u m n o f panels, w h i c h is located about 1.5 dmes the wavelength f r o m the side fringe, the singularity strength does not oscillate but simply decays quite fast.
Under the panels located far f r o m the fringes of a struc-ture, the flow field is almost 2-dimensional and is covered w i t h the bottom o f the stmcture w h ü e it is bounded by a sea bottom as schematically shown i n F i g . 12. Therefore, the velocity potendal < / ) ( A ' , z ) e m u s t satisfy the f o l l o w i n g equations. A<^ = 0 at - ft < z < - <i (3) d4> = 0 a t 2 = : 0 a t z : h (4) (5)
F r o m eqns (4) and (5), i t can be k n o w n that 0 is written i n the f o l l o w i n g f o r m .
nitiz + h)
4> = / W ' C O S - (6)
(/z = 0 , l , 2 , . . . )
Then f r o m eqn (3), it can be deduced that/(a-) i n eqn (6) should satisfy the f o l l o w i n g equation.
dA-2 h-d iJ)
bottom of the structure ( z = - c i )
bottom of the sea ( z - - h )
Fig. 12. The flow field under a very long and very wide floating structure.
Estimation method of hydrodynamic forces 55
j i i i p | j S i
O / O CO 1 b u u m / O COFig. 13. Discretization of a very long and very wide structure (372 panels).
w h i c h gives:
f = e^^"^ (8)
Altogether 4> is written i n the f o l l o w i n g f o r m .
0 = (A + e""' + A,7 e - ''•"•^')-cos/c„ (z + /z) (9) F r o m this, we now k n o w that waves o f progressing mode
cannot exist, w h i c h explains the fact that the singularity strength does not oscillate but simply decays i n x direction along the 6th column o f panels as was shown i n F i g . 11(b). Moreover, the decay rate is now k n o w n to be e " ""^ where
Kn is given i n eqn (7).
Consequently, i t can be said that: W h e n head waves are
Fig. 14. Comparisons of the overall singularity strength
distribu-tion over the entire bottom surface of the 1500 m X 300 m struc-ture in head waves (X = 100 m) obtained by 1800 panels and that
obtained by 372 panels (reahreal part, imag:imaginary part).
incident to a stmcture which is very long and veiy wide com-pai'cd to the wavelength, the lengthwise singularity strength on the panels located near the outer sea area varies as A-""^e*', where /: represents the wave number o f the incident wave, while that on the panels located far f r o m the outer sea area does not oscillate but simply decays as e'""''.
I n order to c o n f i r m this, comparisons were made on the singularity strength f o r a 1500 m X 300 m stmcture i n head waves (wavelength: 100 m ) . One calculafion was canied out w h i l e discretizing the structure into 1800 panels o f equal size as was shown i n F i g . 10. The other calculation was conducted, as shown i n F i g . 13, by d i v i d i n g the struc-ture into 12 strips i n beamwise direction w h i l e each o f the strips was divided into 30 small panels at the upstream area plus a single long panel. The total number o f panels used f o r the latter case is 372. I n the latter calculation, up to 2 columns o f panels f r o m the side fringes, it is assumed that the singularity strength varies according to A-~"^e''''' along the l o n g panels, while, on the other inner columns, it is assumed that the singularity strength varies according to e " " ' ' along the panels. I n F i g . 14, the overall singularity strength distribution over the entire bottom surface o f the structure obtained by 1800 panels and that obtained b y 372 panels are compared. W i t h only 372 panels, almost c o m -plete agreements w i t h the results obtained b y 1800 panels are attained. I t may be able to further s i m p l i f y the analysis by assuming the singularity strength is zero on the inner long panels, instead o f assuming that i t varies as e I n Table 2, resultant wave forces obtained by 1800 panels and those obtained by 372 panels are compared. The difference between the t w o results is less than 2 % .
I f the w i d t h o f a structure is very large compared to wavelength, since it can be assumed that the w i d t h w i s e singularity strength is o f almost the same value i n head waves, the number o f panels can be further reduced b y dis-credzing a structure as shown i n Fig. 15. O n the single large panel at the inner part o f the stmcture, it is assumed that the singularity strength varies as e^"''' i n x direcdon but does not vary i n y direction. The number o f panels shown i n
Table 2. Comparisons of the heave force and the pitch moment obtained by 1800 panels and those obtained by 372 panels
1800 panels (n) 372 panels (b) bla
Heave F 5.93E-03 5.87E-03 0.989
fygLBL
IIII IIII 1II III nil 1 I I 1 n i| II 1 II III I =! III 1 II III III II II III 1 II 11 IJ II 1 II II 1 II 11II 1 II 11" - I L . i s o o m
Fig. 15. Discretization of a very long and very wide structure (155 panels).
Fig. 15 is 155. The lengthwise singularity strength distri-butions along the 1st and the 6th c o l u m n o f panels obtained by the 155 panels are compared w i t h those obtained by 1800 panels i n F i g . 16(a) and (b) respectively. I t can be seen that w i t h only 155 panels, quite accurate results are already obtained.
W i t h this approximation, we are now ready to go on to the analysis o f wave forces on such a very large floating struc-ture as long as 5000 m and as wide as 1000 m , w h i c h is, as described i n Section 1, the typical dimension o f a floating international airport. -0.50 ' ! ' ! ' ! • ! • ^ ^ ^ ' I ' l ' real '{ 1800 — " iBiag o r e a l :{ 155 .dimag.-i ' panel panei E ) S ) T v' i J i X ? i • a g™ ; ! L : f i i i i • . i . i , i , i , i . i , i , 1 15 30 45 60 75 90 105 120 135 150 (a) p a n e l n u m b e r 2.0 I I I ' I ' I ' I ' I ' I ' I ' I ' I ' — real ( 1800 panels ) ' o real i( 155: panel:s ) 1, 0 i i \ \ i " itmag.| f • --1.5 H 751 766 781 796 811 826 841 856 871 886 900 (b) p a n e l n u m b e r Fig. 16. (a) Comparisons of the lengthwise singularity strength
distribution in head waves obtained by 1800 panels and that obtained by 155 panels (1st column of panels), (b) Comparisons of the lengthwise singularity strength distribution in head waves obtained by 1800 panels and that obtained by 155 panels (6th
column of panels).
Assuming that the wavelength is 40 m , w h i c h corresponds to the length o f the waves that often appear at coastal zone areas, i f we simply apply a singularity distri-bution method and discretize the stmcture into 4 m X 4 m (1/10 o f the wavelength) panels, the number o f panels amounts to as large as 312 500. O n the other hand, i f we use the present approximate technique, the structure can be represented w i t h only, say, 341 panels as shown i n Fig. 17. The distribution o f the singularity strength ( ^ ( r e a l part)^ -|- (imaginary part)^) over the entire bottom surface o f the stmcture obtained by the 341 panels is shown i n F i g . 18. W i t h this approximation, the time needed f o r solving the final simultaneous equations is reduced to (341/312 5 0 0 ) ^ - 1 0 " ^ Since i t is assumed that the w i d t h as w e l l as the length o f a structure is very large compared to wavelength, the analysis i n beam waves can be done i n the same way.
3.2 In oblique waves
Next we consider an analysis o f a very long and very w i d e structure i n oblique waves. As an example, let us carry out an analysis o f a 1500 m x 300 m structure i n oblique waves (wavelength: 100 m , x = 80°) as shown i n F i g . 19. F i g . 20(a) and (b) show the lengthwise singularity strength dis-tributions along the 1st c o l u m n o f panels (Fig. 20(a)) and along the 6th column o f panels ( F i g . 20(b)) obtained by 1800 panels o f equal size. As can be observed i n F i g . 20, the singularity strength does not decay neither as A-~"^e''" nor as e"""'. Instead, it varies according to eqn (1). I n obtaining the results by 612 panels shown i n F i g . 20(a) and (b), it is assumed that the singularity strength varies according to eqn (1). I n head waves progressing f r o m -x to -\-x direction, the waves are transmitted toward inner panels only f r o m —x direction, whereas i n oblique waves, on top o f the waves coming f r o m —x direction, waves are also incident f r o m the side edge. Therefore the singularity strength decays little i n lengthwise direction but varies i n phase o f the ambient incident waves w i t h almost constant amplitude as described by eqn (1). O n the other hand, the singularity strength decays exponentially i n beamwise direction at the places far f r o m the fringes o f the stmcture. I n F i g . 2 1 , the overafl singularity strength distribution over the entire bottom surface o f the structure obtained by 1800 panels and that obtained by 612 panels are compared.
Estimation metlwd of hydrodynamic forces 57
Fig. 17. Discretization of a 5000 m X 1000 m structure in head waves (341 panels).
-2500 m -1250 1250 2500,
Fig. 18. The singularity strength distribution (A/(real part)^ + (imaginary part)-) of a 5000 m X 1000 m structure in head waves
(X = 40 m). wave forces on a very long and very wide structure i n oblique waves, the panel discretization such as that shown in F i g . 22 is appropriate. The example shown i n F i g . 22 is a discretization o f a 5000 m X 1000 m structure w i t h 1681 panels. The distribution o f the singularity strength ( \ / ( r e a l part)^ -F (imaginary part)^) over the entire bottom surface o f the structure i n oblique waves (wave-length: 100 m , X = 80°) obtained by the 1681 panels is shown i n F i g . 23.
4 R A D I A T I O N F O R C E S
4.1 Hydrodynamic forces due to motion as a rigid structure
The analysis o f hydrodynamic forces due to motions o f a very large structure can also be conducted i n a similar way to that applied for the analysis o f wave forces. As an exam-ple, we consider an analysis o f an added mass and a damp-ing coefficient due to heave motions as a r i g i d body. I f the structure is very large both i n length and i n w i d t h , it can be easily understood that the singularity strength on all the panels except f o r those located along the fringes o f the structure is o f almost the same value. Then, as shown i n Fig. 24, a certain inner part o f the structure can be repre-sented by a single very large panel, on w h i c h the singularity strength is assumed to be constant. The overall singularity
X = 8 0
Fig. 19. Discretization of a 1500 m X 300 m structure in obhque
waves (612 panels). 1 — ' — I — ' — 1 — ' — ' ireal { 1800 iimag i real ( 612 imag panels )' panels ) 1 15 30 45 60 75 90 105 120 135 150 (a) p a n e l n u m b e r 0.010 0.005 -0.005 -0.010 1 ! _ — real •• imag » reail ' imaig ' 1 ' 1 ' 1 ' ; ( 1800 panels ) i ; ( 6il2 pariels ) i j ; : : A . • • 1 1 • , i , , i , i , i , i , • 751 766 781 796 811 826 841 (b) 856 871 886 900 p a n e l n u m b e r Fig. 20. (a) Comparisons of the lengthwise singularity strength
distribution in oblique waves (x = 80°) obtained by 1800 panels and that obtained by 612 panels (1st column of panels), (b) Com-parisons of the lengthwise singularity strength distribution in obli-que waves (x = 80°) obtained by 1800 panels and that obtained by
real (obtained by 1800 panels) real (oblained by 612 panels) o
imag (obtained by 1800 panels) imag (obtained by 612 panels) o
Fig. 21. Comparisons of the overall singularity strength
distri-bution over the entire bottom surface of the 1500 m X 300 m structure in oblique waves (X = 100 m, x = 80°) obtained by 1800 panels and that obtained by 612 panels (reahreal part,
imag:imaginary part).
strength distribution over the entire bottom surface o f the structure (1500 m X 300 m) due to forced heave motions (period: 8.68 s) obtained by 1800 panels and those obtained by 369 panels, w h i c h is shown i n F i g . 24, are compared i n
Table 3. Comparisons of added mass and damping coefficients due to forced heaving motion obtained by 1800 panels and those obtained by 369 panels (period:8.68 s)
1800 panels {a) 369 panels {b) bla
2.256E-01 2.236E-01 0.991 1.158E-02 1.177E-02 1.016
Fig. 25. I n Table 3, nondimensionalized added mass coef-ficients and damping coefcoef-ficients calculated using the obtained singularity strength are compared. F r o m F i g . 25 and Table 3, we can see that the results obtained by only 369 panels have quite good accuracy i n comparison w i t h those obtained by 1800 panels.
Analyses o f hydrodynamic forces due to antisymmetric mode of motions such as pitch or roll can be caiiied out i n a sinular way. For example, i f we consider the radiation analysis due to pitch motions about y axis, the inner part o f a structure can also be represented by a single large panel, on which the singularity strength is assumed to vary i n propordon to x.
4.2 Hydrodynamic forces due to motions as an elastic
structure
Since the structures that are dealt w i t h i n the present study are several kilometers i n length (L) and i n w i d t h (B) w h i l e the thickness is 10 m at most, the relative bending rigidity
{EliÜ, EIIB^), where represents the bending rigidity o f a
vertical cross section, is expected to be very small compared to that o f the conventional floating structures. Therefore, i n wave excitations, elastic deformations may be more rele-vant than r i g i d body motions. Hydrodynamic forces due to such elastic deformations can be calculated i n a similar way to those used f o r rigid-body motions. For example, hydro-dynamic forces due to such elastic deformations as cos a,fX--e"'"' can be analyzed i n the f o l l o w i n g way.
Since the deformation can be decomposed into two defor-mations progressing i n -|- x and - x directions respectively as:
cosci„x-e - = ^(e'""" + e - '""'ye " ™' (10)
Estimation metiiod of liydrodynamic forces 59
0 M « u M aa a< a? aa
Fig. 23. Singularity strength! distribution (A/(real part)^ + (imaginary part)^) of a 5000 m X 1000 m structure in oblique waves ( X = 1 0 0 < ? / ; / i > m , x = 80°).
Fig. 24. Discretization for the analysis of a radiation problem due to heave motions.
Fig. 25. Comparisons of the overall singularity strength distri-bution over the endre bottom surface due to forced heave motions of the 1500 m X 300 m structure obtained by 1800 panels and that obtained by 369 panels (period: 8.68 s) (reahreal part,
imagiimaginary part).
then, f r o m the same argument used f o r the analysis o f wave forces i n oblique waves, i t can be said that the singularity strength due to the deformation expressed by cos a,pc-e^"^' varies lengthwise as 5(e'""'*'-|-e^'""''')( = cosa„A-) except at the places near the fringes o f the structure.
5 A P P L I C A T I O N S T O O T H E R N U M E R I C A L T E C H N I Q U E S
So far, i t has been assumed that a singularity distribution method is used f o r the analyses. However, the basic idea o f the approximations described i n the preceding sections can be applied to other existing numerical techniques such as a finite element method or an eigenfunction expansion method. For example, i n an eigenfunction expansion method, the velocity potential that represents the flow field under a structure is expressed w i t h a liner superposition o f a complete set o f certain functions that satisfy the Laplace equation and the boundary conditions at the b o t t o m o f a structure and the sea bottom. I f we consider a wave force analysis o f a very long structure i n beam waves, since we can assume that the flow field under a certain inner part o f the structure does not vary i n x direction, we may be able to write the velocity potential as f o l l o w s .
0 = flo + a i 3 ' + Z!)'= 1 (^«« ~ + ^ne""^) cosK„{z + h) (11)
I n this way, the number o f unknowns can be reduced.
6 CONCLUSIONS
New approximate but quite accurate numerical methods f o r the analyses o f hydrodynamic forces on a very large floating
structure were presented. I t has been shown that there is a distinct difference between a very long and slender structure and a very long and very wide structure. W h e n a structure is very large both i n length and i n width, wave disturbances cannot reach into the flow field under a certain inner part o f the structure.
Through comparisons w i t h results obtained without the present approximations, the validity o f the present methods was demonstrated. Although a simple box-shaped thin structure was dealt w i t h i n the present study, i t has already shown by one o f the authors and Yue [3] that similar ideas can be used f o r the analysis o f a structure supported on a very large number o f legs, w h i c h is another type o f structure now being considered as a candidate o f a floating airport. (Or it should be said that the present study is the application of the idea o f Kagemoto and Yue to a box-shaped structure.) A l t h o u g h the attention has been l i m i t e d to the analyses o f
hydrodynamic forces, a s i m l a r approximation w i l l be able to be extended to the analysis o f motions both as a r i g i d body and as an elastic body, provided the distnbutions o f such properties as mass or bending rigidity is u n i f o r m .
R E F E R E N C E S
1. Lee, Seok-Won and Webster, W. C , A preliminary to the design of a hydroelastic model of a floating airport. In
Hydro-elasticity in Marine Teclmology, eds O. Faltinsen et al., 1994,
pp. 351-362.
2. Faltinsen, Odd, Wave forces on a restrained ship in head-sea waves. 9tii symp. on Naval Hydrodynamics, 1972, pp. 1763¬
1843.
3. Kagemoto, H . and Yue, D . K. P., Wave forces on a platform supported on a large number of floating legs. Proc. 5tii Int.
OJfsiwre and Arctic Engineering Symp., V o l . 1, 1986, pp.
