HYDRODYNAMIC COEFFICIENTS FOR AXISYMMETRIC BODIES AT FINITE DEPTH by JOHNSON L.K. CHAN Bachelor of Applied Science, U n i v e r s i t y of Toronto, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE l n THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1984 © Johnson L.K. Chan, 1984 In presenting t h i s t h e s i s in p a r t i a l f u l f i l l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by his or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Johnson L.K. Chan Department of Mechanical Engineering The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, B r i t i s h Columbia Canada V6T 1W5 Date: January 31, 1984 ABSTRACT A numerical procedure tor the c a l c u l a t i o n of v e l o c i t y p o t e n t i a l , f o r axisymmetric shapes i s st u d i e d . The procedure incorporates l i n e a r i z e d f r e e surface and r a d i a t i o n c o n d i t i o n s . Forced heave, surge, p i t c h induced v e l o c i t y p o t e n t i a l s and wave d i f f r a c t i o n p o t e n t i a l s are considered as s p e c i f i c a p p l i c a t i o n s . D e t a i l s of the a n a l y t i c a l steps necessary to reduce the three dimensional i n t e g r a l equations to two dimensional e q u i v a l e n t equations are given in the appendix. The hydrodynamic c o e f f i c i e n t s and e x i s t i n g forces f o r four well studied examples are presented and compared to published r e s u l t s by other numerical methods. The method studied here o f f e r s c o n s i d e r a b l e savings i n computer time, as the input requirements and the computational e f f o r t s are reduced by intermediate a n a l y t i c a l steps. iii TABLE OF CONTENTS Page ABSTRACT i i CHAPTER I -- INTRODUCTION 1.1 ' CHAPTER II — MATHEMATICAL FORMULATION 2.1 4 1. D e f i n i t i o n of the Coordinate System and Major V a r i a b l e s 2.1 * 2. General Formulation 2.2 ^ 3. Method of S o l u t i o n 2.4 7 CHAPTER III -- HYDRODYNAMIC COEFFICIENTS 3.1 /4 1. General Formulation of Hydrodynamic C o e f f i c i e n t s 3.1 /4 2. Computation of Heave Added Mass and Damping C o e f f i c i e n t 3.5 »8 3. Computation of Surge Added Mass and Damping C o e f f i c i e n t 3.6 l9 4. Computation of P i t c h Added Mass and Damping . C o e f f i c i e n t 3.8 Z l CHAPTER IV — EXCITING FORCE AND MOMENT 4.1 *S 1. General Formulation of E x c i t i n g Force and Moment due to Incoming Waves 4.1 25" 2. Heave E x c i t i n g Force 4.5 29 3. Surge E x c i t i n g Force 4.7 52 4. P i t c h E x c i t i n g Moment 4.9 33 5. An A l t e r n a t i v e Method f o r Computing E x c i t i n g Forces 4.10 34 CHAPTER V — RESULTS 5.1 36 1. S i n g l e C y l i n d e r at F i n i t e Water Depth 5.1 36 2. Composite C y l i n d e r 5.3 58 3. Hemisphere 5.5 4o iv Table of Contents (continued) Page 4. Axisymmetric F l o a t i n g Rig 5.6 4' 4 2 5. Discussions and Conclusion 5.7 LIST OF FIGURES F . l 44-NOMENCLATURE N.l 7 2 APPENDIX A A.l 74 APPENDIX B A.2 7S APPENDIX C A.4 7 g APPENDIX D A.6 8o APPENDIX E A.13 87 APPENDIX F A.22 9£ APPENDIX G A.24 98 APPENDIX H A.26 /oo REFERENCES R.l /o3 - 1.1 -! I. INTRODUCTION The motions of a platform at sea i s one of the most important t o p i c s t h a t Naval A r c h i t e c t s are concerned about. In the l i n e a r frequency domain a n a l y s i s of a f l o a t i n g o b j e c t , the computation of the added masses, the damping c o e f f i c i e n t s , and the e x c i t i n g forces i s necessary. Two well developed t h e o r i e s can be applied to solve the problem. One of these t h e o r i e s i s the c l a s s i c a l S i n g u l a r i t y Method. Using conformal mapping and a d i s t r i b u t i o n of sources and sinks of various strength i n s i d e or on the f l o a t i n g o b j e c t , an a n a l y t i c f u n c t i o n d e s c r i b i n g the f l u i d flow can be obtained. The a l t e r n a t e method i s to solve the Integral Equation of the system coupled with the Green's Function. The v e l o c i t y p o t e n t i a l value on the boundary surface and in the f l u i d f i e l d can then be found. In 1949, U r s e l l s u c c e s s f u l l y solved the forced o s c i l l a t i o n problem of a semi-submerged c i r c u l a r c y l i n d e r f o r the deep water case. The a n a l y s i s was done using the S i n g u l a r i t y Method and the waves were assumed to be long. However, only the heave motion was considered. His work was followed by T a i s a i [1959] and Porter [1960] who extended h i s method with the assumption of John [1950] that the body's surface must be perpendicular to and at the f r e e s u r f a c e . In 1969, Kim obtained the added masses and damping c o e f f i c i e n t s by using the Lewis Transform f o r f i n i t e water depth. Three modes of motion were considered. The p o t e n t i a l of Integral Equation Method was not obvious u n t i l the development of the computers. In 1953, U r s e l l developed a theory f o r an o s c i l l a t i n g c y l i n d e r at high frequency with heave motion and shallow draught approximation. His theory was f i r s t a p p l i e d s u c c e s s f u l l y by - 1.2 - 2 MacCamy in 1961. However, the Integral Equation Method had obtained i t s success three years p r i o r to that, when an acoustic problem was solved by Smith and Hess. In 1967, Frank's removed the geometric body r e s t r i c t i o n s from the problem which John [1950] had o r i g i n a l l y imposed. The disadvantage of the c l a s s i c a l approach i s that a long s e r i e s of terms i s required f o r convergence and the a n a l y s i s i s r e s t r i c t e d to two dimensions o n l y . On the other hand, the Integral Equation Method also has i t s r e s t r i c t i o n s . The bottom of the control volume must be f l a t and the Green's Functions are hard to compute. Moreover, the body geometry r e s t r i c t i o n was not removed u n t i l Frank's work in 1967. In 1955, Havelock had solved the problems of a semi-submerged sphere by the Integral Equation Method. In 1966, S. Wang extended h i s work from deep water to f i n i t e water depth. The f i r s t complete three-dimensional a n a l y s i s was done by Kim [1965], who used an e l l i p s o i d a l - p o l a r coordinate system. S o l u t i o n s f o r several spheroids and e l l i p s o i d s with surge, heave, and p i t c h motions were obtained. In 1972, Bai obtained successful r e s u l t s by using the F i n i t e Element Method with an approximate r a d i a t i o n boundary c o n d i t i o n . One year l a t e r , Yeung solved s i m i l a r problems in two and three dimensions with no r e s t r i c -t i o n on the body's geometry. A general form of the Integral Equation f o r f o r c e d - o s c i l l a t i o n problems was then set up. In 1979, K r i t i s a p p l i e d Yeung's equation to obtain a s a t i s f a c t o r y r e s u l t f o r heave motion of some axisymmetric o b j e c t s . - 1.3 - s In 1981, S.H. C a l i s a l and Sabuncu developed a matching technique f o r composite c i r c u l a r c y l i n d e r problems. In the f o l l o w i n g chapters, the author presents more s p e c i a l i z e d Inte-gral Equations f o r s o l v i n g axisymmetric body f l u i d flows in f i n i t e water depth. The i n t e g r a l equations given in the fo l l o w i n g chapters are a modi-f i c a t i o n of Yeung's equations. These enable one to solve a three-dimen-sion a l problem in two dimensions. The only r e s t r i c t i o n i s that the body must be axisymmetric about the v e r t i c a l a x i s . Equations f o r determining hydrodynamic c o e f f i c i e n t s as well as e x c i t i n g forces are developed. - 2.1 - 4 I I . MATHEMATICAL FORMULATION I I . l D e f i n i t i o n o f the Co-ordinate System and Major V a r i a b l e s The f i r s t step in s o l v i n g the problem i s to set up a co - o r d i n a t e system, so that the geometry of the whole system can be c l e a r l y d e s c r i b e d . The o r i g i n of the co-ordinate axes i s l o c a t e d i n s i d e the f l o a t i n g body. The x and z axes are at the undisturbed free surface while the y axis i s p o i n t i n g upward. Angle 6 i s measured from the x axis as shown in F i g . A. h i s defined as l o c a l water depth. A c y l i n d r i c a l outer boundary i s chosen to define the c o n t r o l volume which can be described by a r a d i a l d i s t a n c e R D. Any a r b i t r a r y point can be descri b e d by (x,y,z) or by (R,y ,6) c o o r d i n a t e s . They are r e l a t e d by Let P be an "observation point" l o c a t e d e i t h e r i n s i d e or outside the con t r o l volume. The p o t e n t i a l value at P i s to be found. A l s o , l e t point Q be a "control p o i n t " l o c a t e d on the c o n t r o l s u r f a c e . Q i s a running parameter when taking area i n t e g r a t i o n f o r computing p o t e n t i a l value at P . Let the observ a t i o n point P be descr i b e d by (R^, y^, 0) and the c o n t r o l point Q be (RcosQ, y, -Rsin6). The dis t a n c e r between P and Q i s given by x = RcosO z = Rsine . (1) (2) the gradient of r , Vr i s (Rcos6- Rp, y-y , -Rsine) Vr = (3) - 2.2 -5 where Vr is given in x-y-z co-ordinates. F i n a l l y , le t n be the unit normal vector on the control surface pointing out of the control volume. 11.2 General Formulation If an i n v i s c i d and incompressible f l u i d flow ex i s t s , a complex potential function <& can be found. In order to predict the response of an arb i t rary f loat ing body in waves, the computation of the added masses, damping coe f f i c i en t s , and exci t ing forces is necessary. (See Wehausen, 1971.) These hydrodynamic coef f ic ients which are related to the hydrodynamic pressure force can be calculated i f the motion induced ve loc i ty po ten t i a l , $ is known. $ in general is a function of posit ion as well as time and must sat i s fy the Laplace Equation because of continuity of incompressible f l u i d flow, i . e . V 2 * = 0. (4) This is an e l l i p t i c par t ia l d i f f e r e n t i a l equation to be solved with the appropriate boundary condit ions . For this type of problem with a f i n i t e depth and free surface, there are four di f ferent kinds of boundary condit ions . The f i r s t one is the r i g i d impermeable boundary condit ion. That i s , the component of f l u i d ve loc i ty in the d irect ion normal to the boundary surface is zero. * , = V" • n S o = 0 , (5) where V is the ve loc i ty of f l u i d at the boundary, and SQ is the impermeable boundary surface. - 2.3 - 6 The second boundary c o n d i t i o n i s of the f l u i d moving in contact with the body. In t h i s case, the component of f l u i d v e l o c i t y in the d i r e c t i o n normal to the body's surface must be equal to the component of the body's v e l o c i t y along the d i r e c t i o n n. This i s given by ••n = V „ , (6) where V n i s the v e l o c i t y of the boundary in the n d i r e c t i o n and i s the surface area of the body. On the fr e e surface, two boundary c o n d i t i o n s have to be s a t i s f i e d at the same time. The kinematic boundary c o n d i t i o n which says that the f l u i d p a r t i c l e s at the surface w i l l remain at the surface or Dt <*-y) S f = n , t + U T 1,x " v + W V ( 7' a ) = 0 on y = n as given in Mewman (1977). The second c o n d i t i o n on the fr e e surface i s the dynamic boundary c o n d i t i o n which i s a B e r n o u l l i Equation and which gives the pressure at the surfa c e . This c o n d i t i o n can be wr i t t e n as $ . + £- + i ( u 2 + v 2 + w 2) + gy = 0, on y = n (7.b) , t p c. where n i s the wave e l e v a t i o n , p i s the pressure, p i s the f l u i d d e n sity, and y = n on the free surface. The l a s t boundary c o n d i t i o n i s the r a d i a t i o n c o n d i t i o n . This boundary c o n d i t i o n concerns about the propagation of energy in wave form away from the body. The l i n e a r i z e d form of t h i s boundary c o n d i t i o n i s given by Wehausen (1950) as: l i m R > o o (*,R + 1k*) = 0 , (8) where k, the wave number, f o r p e r i o d i c waves, can be found by s o l v i n g the surface wave equation k tanh(kh) = |^ . (9) u> i s the angular frequency of the waves. 11.3 Method of S o l u t i o n In order to solve (4) with the appropriate boundary c o n d i t i o n s , a s u i t a b l e s o l u t i o n method must be chosen. Numerous methods have been developed f o r the s o l u t i o n of p o t e n t i a l problems, and most of them can be used s a t i s f a c t o r i l y . A n a l y t i c a l s o l u t i o n methods are p r e f e r a b l e in general f o r simple geometries but they become highly complicated f o r problems which w i l l be considered here. The F i n i t e Element Method i s the most popular to use but the r e s u l t s are expensive to o b t a i n . This i s due to two main reasons. The f i r s t reason i s that the control volume must be d i v i d e d into a l a r g e number of f i n i t e elements; the input preparation takes much computer time. The second reason i s that a l i n e a r equation with a large matrix has to be solved during computation. As the cost of computing time v a r i e s with the t h i r d power of the order of the matrix, the computing charge w i l l be rather high. The Surface S i n g u l a r i t y Method and the Boundary Element Method each o f f e r an a l t e r n a t i v e s o l u t i o n , though they are not so commonly used. However, the Boundary Element Method fo r p o t e n t i a l problems has c e r t a i n advantages over the F i n i t e Element Method in terms of the computer input time and computational time. Moreover, the reduction of number of the elements can decrease the complexity of the problem. - 2.5 - 6 Before going any further into the computing technique, some assumptions have to be made to s implify the problem defined above. It is obviously non-l inear because of the free surface boundary condit ions . One major assumption can be made about the complex potential function. It is assumed that $ can be represented by a series of terms of dif ferent orders of magnitude. That i s , • = * ( 0 > + ^ + e 2 * ^ + e%<n> , (10) where e is a small number and the superscript inside the bracket indicates the order of that term. For e, one can take i t equal to A/A, the non-dimensional wave amplitude which is the wave amplitude divided by the wave-length. It can be proved that each of these terms should sat i s fy (4) and the following statement can be shown to be true 3 x m 8 x m That i s , the order of the derivat ive of a term is the same as the order of that term. Since the veloci ty component, U. , is given by $, , i t is also true to assume that U- = U - ( 0 ) + s U . ( i ) + £ 2 U . ( 2 ) + •1 -1 - 1 - "1 - * <12> Moreover, the free surface and pressure functions are assumed to exist respectively in the same form. (0) A (1) A 2 (2) A n = n + en + e n + • • • • , (13) p = p ^ + e p ^ + e 2 p ^ + • • • • . (14) - 2.6 - 9 One of the major r e s u l t s of l i n e a r i z a t i o n i s that the boundary c o n d i t i o n s are s a t i s f i e d at the mean p o s i t i o n of the body's surface, and at the free surface (see Appendix A). The mean p o s i t i o n of the free surface i s taken as the undisturbed f r e e s u r f a c e . One can show that « ^ i n (10) has to be equal to a constant to s a t i s f y the boundary c o n d i t i o n s and i s taken to be zero since the reference atmospheric pressure i s also taken as zero. Therefore, the zeroth order terms of p, IK n can be equalled to zero. Thus, $ = e 2J2"> + £ $ + • • • • p = e e p + • • • • U. = e eU.j + •••• n = e 2 (2) . e n + (15) I t must be noted that p i s the change in pressure due to o s c i l l a t i o n of the body and does not include the h y d r o s t a t i c pressure. By b r i n g i n g (15) i n t o (4), (5), (6), (7.a), (7.b) and keeping only the terms of order e, i . e . the f i r s t order terms, the l i n e a r i z e d problem can be wr i t t e n as 'n = 0, = o , (16.a) (16.b) 'n = V (1) .(1) Al) S f (1) ' t + gy 'y = o , (1) - n fl) at y = 0, at y = 0, (16.c) (16.d) (16.e) but in (16.e), i s equal to zero at y = n. t h e r e f o r e , + g n ( 1 ) = 0 at y = 0. (17) s And since (16.d) and (17) are boundary c o n d i t i o n s f o r the free surface, they can be combined as a s i n g l e equation as $ U ) ' t t | + 9 $ U ) ' y = 0 a t y = 0 * { 1 8 ) |S f Now, with the understanding that only the f i r s t order terms are to be solved, a l l the s u p e r s c r i p t s w i l l be dropped. For the study of p e r i o d i c motions, the body i s assumed to be o s c i l l a t i n g at a c e r t i n frequency, u>, and the induced waves are generated at the same frequency. Therefore, the form of the expected $ can be imposed as: $ = <|>(x,y,z)e"iu)t , (19) where <)> i s the amplitude of $ and i s a f u n c t i o n of p o s i t i o n only. S i m i l a r l y , n and can be expressed as re" 1 ^ and e""1^. Thus, the f i n a l s i m p l i f i e d form of the equations i s as f o l l o w s : From (16.a), the p a r t i a l d i f f e r e n t i a l equation can be r e w r i t t e n as: v2^ = 0 , (20) the impermeable boundary condition. (16.b) becomes * , J = 0 , (21) the boundary c o n d i t i o n on the body's surface (16.c) becomes ••nL = V n • ( 2 2 ) b and f o r the f r e e surface (18) -o>2<t> + g<hy|sf = 0 • ( 2 3 ) - 2.8 -In general, (20), (21), (22) and (23) are f o r any a r b i t r a r y f l o a t i n g body i n p e r i o d i c waves. The r a d i a t i o n boundary c o n d i t i o n i s dependent on the type of fi^w. For d i f f e r e n t problems, there i s a numerically d i f f e r e n t form f o r the r a d i a t i o n c o n d i t i o n . In some a p p l i c a t i o n s , some of the f l o a t i n g objects are axisymmetric i n shape. Included are composite c y l i n d e r s and objects that are c y l i n d r i c a l , s p h e r i c a l , and coni c a l . For any axisymmetrical body, the r a d i a t i o n c o n d i t i o n i s given by Bai(1972) as 4>,R = (- + 1k) 4> , (24) S „ R where RD i s defined as in F i g . A, S„ i s the outer boundary surfac e , and k i s defined by (9). The equations (20), (21), (22), (23) and (24) form a complete set of equations f o r the f l u i d flow system under the assumption of simple harmonic o s c i l l a t i o n of an axisymmetric body. The s o l u t i o n of (20) can be obtained by using an i n t e g r a l equation suggested by Brebbia (1978). The i n t e g r a l equation i s given as CQ*(P) + / s 4>(Q)^ ( P' Q ) dS = J s G(P,Q)dS , (25) where S t i s the t o t a l surface area bounding the control volume. P and Q are points defined in se c t i o n II. 1 . Q i s equal to 4u i f P i s i n s i d e the control volume and equal to zero when P i s outside the control volume. When P i s at the boundary surfac e , Q i s equal to 2 it. C i s a constant equal to 1 in three-dimensional problems and equal to 1/2 f o r two-dimensional ones. G i s the well-known Green's Function which i s equal to 1/r in three-dimensional formulations and equal to l n ( l / r ) i n two-dimensional ones, r i s as defined in se c t i o n II.1 and i s a f u n c t i o n of P and Q. Therefore, G i s a fun c t i o n of P and Q only. For area i n t e g r a t i o n i n (25), a numerical computation i s req u i r e d . In gener a l , the control surface i s d i v i d e d i n t o elements of f i n i t e area. C i r c u l a r r i n g elements are chosen f o r t h i s study as the v a r i a t i o n of po t e n t i a l value on each of these elements i s a known fu n c t i o n of e. The mid point of each element i s defined as the node point of that element.(Fig. B) For three-dimensional problems with P loc a t e d on the boundary, equation (25) can be rewritten as 2n*(P) + Js • ( Q ^ W e d * = Js M i l I Rdedi . (26) This equation can be rewritten as a matrix equation [ ^ [ • j ] = [B,] (27) as shown in Appendix B. By i n c l u d i n g the boundary c o n d i t i o n s (21), (22), (23) and (24) i n t o (26) , the f o l l o w i n g equation i s obtained. (See Appendix C) 2*<fr(P) + / s | ? r { ^ ) < D ( Q ) R d e d x + J s !^)<|,(Q)RdFdx • K + / S f & (7> " F ^ » ( Q ) R d * d R = J $ b 7 V n RdedJl . (28) Although (28) i s d i f f e r e n t i n form from (26), i t can als o be wr i t t e n as a matrix equation s i m i l a r to (27). A f t e r the matrix equation i s set up, the 4> value at the node of each element can be solved by the well-known methods - 2.10 »3 of l i n e a r algebra. In general, $ w i l l be a complex number since some of the matrix e n t r i e s are complex in value because of the r a d i a t i o n boundary c o n d i t i o n s . Equation (28) seems to have surface i n t e g r a l s , but f o r axisymmetric cases, i t can be s i m p l i f i e d i n t o a l i n e i n t e g r a l equation. The d e t a i l s of the s i m p l i f i c a t i o n are explained i n the next chapter. - 3.1 - 14 I I I . HYDRODYNAMIC COEFFICIENTS I I I . l General Formulation of Hydrodynamic C o e f f i c i e n t s The motion of a body can be described with the supe r p o s i t i o n of 6 independent motions. Three of these are t r a n s l a t i o n a l motions i n the x, y, and z d i r e c t i o n s and the three others are r o t a t i o n a l motions about the axes. The three t r a n s l a t i o n a l motions represented by x x , x 2 , and x 3 are known as surge, heave and sway. The r o t a t i o n a l motions x^, x 5 , and x 6 are named r o l l , yaw, and p i t c h . With the above n o t a t i o n , the most general dynamic equation of the body i s 6 j = l K-j + a i j ) x i + b i j x i + c i j x i (29) where m.^ i s e i t h e r the mass or the mass moment of i n e r t i a of the body, a., and b.. are known as the hydrodynamic c o e f f i c i e n t s . F' i s the e x c i t i n g f o r c e or moment in the i d i r e c t i o n . [See Wehausen 1971]. m.. i s the actual mass of the body when i i s equal to j and l e s s than 4. I t i s equal to zero when e i t h e r i or j i s l e s s than 4 and i i s not equal to j . When neit h e r i nor j i s l e s s than 4, m.. i s the mass moment of i n e r t i a . For s i m p l i c i t y , m.. i s u s u a l l y expressed in matrix fo * J rm. m.. = U 0 0 0 0 0 0 M 0 0 0 0 0 0 M 0 0 0 0 0 0 0 0 0 0 0 0 Ixx Ixy Ixz Iyx Iyy Iyz Izx Izy Izz (30) - 3.2 - 15 a., i s known as the "added mass" which when m u l t i p l i e d by the corresponding a c c e l e r a t i o n gives the hydrodynamic pressure force component in phase with the a c c e l e r a t i o n , b ^ i s the "damping c o e f f i c i e n t " which when m u l t i p l i e d by the corresponding v e l o c i t y gives the hydrodynamic pressure force i n phase with the v e l o c i t y , c . i s the " r e s t o r i n g force c o e f f i c i e n t " and when m u l t i p l i e d by the corresponding body displacement (from the e q u i l i b r i u m p o s i t i o n ) gives the h y d r o s t a t i c force component opposite to the displacement. A general r e l a t i o n s h i p between the hydrodynamic forces and the added masses and damping c o e f f i c i e n t s i s given by Newman [1977] as where U. and II. are the v e l o c i t y and a c c e l e r a t i o n of the body in the j J J > t I I I I I d i r e c t i o n . ( E ^ E 2 , E 3) are the hydrodynamic pressure forces and (E^, E 5, i i E 6 ) are the hydrodynamic pressure moments. E^ i s induced by the motion of the body and the h y d r o s t a t i c pressure force i s not included because of l i n e a r i z a t i o n . For p e r i o d i c motion, l e t u. be the amplitude of the body's v e l o c i t y and 3 E.. be the force amplitude. That i s , U. = u e ~ i a ) t , (32) J J U. . = -iuAJ. , (33) and E^ = E . e ~ i a ) t . (34) Therefore, (31) can be rewritten as - 3.3 -6 b.. E. s (a.. + i - L i ) u. = • Another formula f o r computing added masses and damping c o e f f i c i e n t s i s given by Wehausen [1971] i j + f T T - IK ' • j " t d S • , 3 6 > where p i s the f l u i d density, <)>'. i s the p o t e n t i a l f u n c t i o n induced by u n i t 3 v e l o c i t y of the body in the j d i r e c t i o n and n i i s the i component of the u n i t normal vector. From (16.e), only the hydrodynamic pressure, p, can be a f u n c t i o n of time and i t can be equated to the negative of the time d e r i v a t i v e of the p o t e n t i a l f u n c t i o n , i . e . P = " P * , t = iu><|>e~1a)t . (37) Hence, the hydrodynamic pressure force or moment can be computed as: E. = L p n. dS i = 1 to 6 , (38) 1 5 b 1 where ( n 1 , n 2 , n 3) = ( n x > n y » n z ) (39.a) and ( n 4 , n & , n g) = (r x n) . (39.b) r Q i s the p o s i t i o n vector measured from the o r i g i n . If (37) i s s u b s t i t u t e d into (38), the reader can prove (36) e a s i l y . Moreover, i f i s used to represent ^, (36) can be rewritten as 3 - 3.4 - 17 a . j + i ^ - = n.dS . (40.a) The non-dimensional form of the above equation can be obtained by d i v i d i n g i t by the f l u i d density and the body's d i s p l a c e d volume, v. w + 4p4 - 4 v ^ - d s • (4o-b> Equation (36) and (40.b) are the general'.forms f o r computing added masses and damping c o e f f i c i e n t s . The area i n t e g r a t i o n i s u s u a l l y done by a numerical method. However, the value of <j>. on each element must be known J before the area i n t e g r a t i o n i s done. Thus, (28) must be solved f o r the c a l c u l a t i o n of hydrodynamic c o e f f i c i e n t s . Before proceeding on to the next s e c t i o n , some assumptions about $ must be made. I t was mentioned by Wehausen [1971] t h a t $ can be expressed as the sum of three d i f f e r e n t terms, i . e . , $ = * + $ D + $ F (41) $j i s the p o t e n t i a l value of incoming waves, ^ i s the p o t e n t i a l value r e s u l t e d from r e f l e c t i o n of $>j on the body's surface , and ^ i s the po t e n t i a l value induced by the motion of the body. $j and $ are important i n e x c i t i n g f o rce and moment computation. In l i n e a r i z e d c a l c u l a t i o n s , the body i s assumed to be f i x e d in space and experiences a force due to $j and For the computation of the added masses and damping c o e f f i c i e n t s , ^ i s induced by the motion of the body. The d i s t r i b u t i o n of <s>p on the submerged surface of the body w i l l determine the value of a., and b^.. - 3.5 - /8 The p o t e n t i a l f u n c t i o n , $, which appears in t h i s chapter i s a c t u a l l y the $p; the i n t e r a c t i o n between $ , $ D and ^ i s assumed to be n e g l i g i b l e . III.2 Computation of Heave Added Mass and Damping C o e f f i c i e n t For an axisymmetric f l o a t i n g body o s c i l l a t i n g i n the y d i r e c t i o n , a ^ and b ^ can be computed as: a „ b y=0 2 it <u where y t i s the keel of the body. Since the body i s axisymmetrical about the y a x i s , <}>2 can be assumed independent of 0; that i s , ^ i s a constant value on a c i r c u l a r r i n g . Then, (28) can be s i m p l i f i e d as y=0 -4 R 2n + (y-y_)n ) 2n 2 * U P ) + / [ 5 E L J L E + —B- ( - i - E - F ) ] c y=y + (a-b)/a+F /(a+F) (a-b) 6 R D 4(R 2n + (y-y ) n j 2n„ + J R [- _ P _ J L E + (— r _ E - F ) ] M A 0 (a-b) / a+F /a+F (a-b) L y=-h /a+b (a-b)/a+b RR ,„#.. ..^ » 2 r + j r . 5 l [ y i y p J _ E . i £ L i _ F ] M R RC /a+TTla-b) g/a+F - 3.6 - /9 ry=° 4R = / ™ _ F u n dA , (42) y=yt /a+b" c y where F and E are Complete E l l i p t i c Functions of the f i r s t and second kind, a and b are constant values. D e t a i l s of the s i m p l i f i c a t i o n are shown i n Appendix D. (41) can e a s i l y be rewritten as a 22 boo i y=o <t>o y " t As <t>2 on each element can be found by s o l v i n g (42), and can be evaluated from (43). A computer program, named HEAVE.FTN, was w r i t t e n to solve f o r <^ and a22 ^22 then compute and . This computer program sets up the matrix a 0 0 equation (42), and then solves f o r <|>0 on each c i r c u l a r r i n g element. -=^- and c pV b 2 2 are then computed by (43). I t must be noted that (42) and (43) are now l i n e i n t e g r a l equations instead of area i n t e g r a t i o n as in (28). III.3 Computation of Surge added Mass and Damping C o e f f i c i e n t I f the body i s o s c i l l a t i n g i n the x.jdirection, the normal v e l o c i t y component on the moving boundary i s in the form $ I = u, n ncose . (44.a) ,n c 1 R That i s , the motion of the body has a maximum e f f e c t when e i s equal to 0 and % and has a minimum e f f e c t when 9 i s equal to ± This suggests ^ w i l l be in the f o l l o w i n g form - 3.7 - 20 <t>1 = <J>^ cose . (44.b) By taking (44.b) into (28), (28) becomes 2TUD1'(P) + J | n ( l / r ) cos9 RdedA S o + / - | r ( l / r ) (Dj cose RdedA S b . 2 + J [ -^f (1/r) - |- l/r]<^cose RdedR S f + ' f | n " ( 1 / r ) " ^ + J o c o s e Rdedy R = / t^cose n R 1/r RdedA . (45) S b The f i r s t term i n (45) i s 2n<^ instead of 2it<|>Jcose because point P i s on the x-y reference plane. (See Section I I . l ) . Equation (45) can be s i m p l i f i e d i n t o a l i n e i n t e g r a l equation by i n t e g r a t i n g with respect to 9. The d e t a i l s are in Appendix E and the r e s u l t i s 2n*J ( P ) + [-4= ("RR2 + n (y-y )R)(F - * E) y=y t b/a+b K y p u DJ 2 + j n nf/a+¥ E - ^ — F + a E ) ] ^ d A D K /a+b /a+b(a-b) 1 RR 0 b/a+F K y P 1 3 0 1 + £ n D (/a+b" E - -2-A- F + — E ) ] M A D K /a+b /a+b (a-b) 1 - 3.8 - 21 RR - 2 2 c ( y - y J a R c 9 p 9 /a+b /i+bla-b) ^ 0 2 r r 2(2R -a) r c a r >, , 4ikR , ,—r- r a c^ n ' ., J L — — — (F - r r ^ y - E) + — g - (/a+b E - F) fcdy y=-h b/a+F u D' D /a+b 1 y=0 4u, n DR / 1 , R [ - A — F - E/a+F ] d i . (46) y=y t /a+F A f t e r on each element i s known by s o l v i n g (46), the p o t e n t i a l value at any point on the boundary elements i s given by <t>'^cose. By b r i n g i n g (44.b) into (40.b), the f o l l o w i n g equation can be obtained. a n b n 1 y=0 2u <t>-!cose -4r + -4r~ = i / / -r, n D c o s e Rdedi pV pVu v y = y^ o u l R 1 ,y=0 *i = ^ / ™ R TT R d* • (47) v y=yt u i A computer program named SWAY.FTN was w r i t t e n to solve equation (46) a l l b l l and then evaluate -^=- and •—^ . The r e s u l t s computed by SWAY.FTN are shown in Chapter V. III.4 Computation of P i t c h Added Mass and Damping C o e f f i c i e n t In computing agg and bgg, the body i s assumed to p i t c h about a f i x e d p o i n t and induces the p i t c h p o t e n t i a l value $g. The choice of t h i s centre of r o t a t i o n i s a r b i t r a r y . ( O r i g i n of coordinate axis i s chosen here). In t h i s case the z-axis i s taken as the axis of r o t a t i o n . I f the angular v e l o c i t y of the body i s assumed to be y e _ U ) t , the boundary c o n d i t i o n (22) on the body becomes 6 ' n (y x r Q ) • n S b = [(0,0,y) x (Rcose,y,-Rsine) ] • ( n R c o s e , n y , - n R s i n e ) = (-rij^y + n y R ) Y cose , (48) where r o i s the p o s i t i o n vector from the o r i g i n . Equation (48) suggests the term 'cose' w i l l be present in <t>g, t h e r e f o r e , ^ i s assumed to have the form 4>6 = <t>g cose (49) where <|>g w i l l be at i t s maximum value at e equal to zero. Since both the p i t c h p o t e n t i a l f u n c t i o n and the surge p o t e n t i a l f u n c t i o n have the 'cose' term, i t i s reasonable to say that the p i t c h motion of the body has induced a surge moment on the body and the coupled hydrodynamic c o e f f i c i e n t s , a g ^ and b ^ , should e x i s t . By taking (48) and (49) i n t o (28), the i n t e g r a l equation f o r p i t c h i n g motion i s 2Ti<Dg (P) + / f n - d / r H g - cose RdedA + / - | r ( l / r ) <DgCOS e RdedA S o S b a 2 + / [ Ipj-d/r) - |-(1/D ] <D6cose RdedR S f + / [|„ d / r ) - ( i + i k ) l / r J o c o s e Rdedy S r R J (-n^y + n R ) Y c o s e ( l / r ) R d e d i . (50) r iv y b b Equation (50) i s s i m i l a r to (45) except that the term on the right-hand side i s d i f f e r e n t . The evaluation of the l e f t - h a n d side i s e x a c t l y s i m i l a r - 3.10 -to (E.2), (E.3), (E.4) and (E.5) mentioned i n Appendix E. The i n t e g r a t i o n of the right-hand side of (50) i s shown i n Appendix F. (50) can be re w r i t t e n in the fo l l o w i n g form. 2 7 t < f , 6 ( P ) + L ° ( n R R 2 + n y { y " V R ) ( F " i ? F E ) y=y t b/a+b 2 + I n J / i + F E - ^ - F + — E)]o>' d l 0 K /a+b /a+b(a-b) 0 2 + ?- n D ( / a + F E — F + E)]<t>'di D K / a + b / a + b ( a - b ) ° RR A 2 2 p ( y - y n ) a R„ 9 P 9 /5+b" /aTF(a-b) ^ + j° [ 2 ( 2 R 2 - a ) ( r a r ) + 4ikR ( / i T F £ _ _j_ p ) ] , y=y h b/a+ F ^ / i + F J 6 = / -4(-n Dy + n R ) J ^ [/a+F E — - — F ] d i . (51) y=y t K y /a+ F A f t e r <))g i s obtained by s o l v i n g (51), the added masses and damping c o e f f i c i e n t s can be computed by using (40). For the eva l u a t i o n of a „ and 00 bgg, Uj i s equal to y, i s equal to ^ c o s e , and n.. i s equal to (r x n)•(0,0,1). Therefore, (40) becomes - 3.11 -a66 b66 1 y = 0 2 % *6 W+ 1 = v / J — cos e (Rn - n y) cos e Rd 9d H M y=y t o y * + 1 / -4 (Rn - r y ) RTI dx . (52) y=yt Y The non-dimensional form of the above equation i s obtained by d i v i d i n g 2 2 both sides by R^ • R M i s a length parameter of the body. F i n a l l y , the non-dimensional form of a „ and b „ becomes 66 66 _ 2 | + A ' / _ | ( R N . N „) D J L . (53) PVRM PVwRM VRM y=y For a computation of ag.| and bg.|, the u n i t normal vector component, n. i s equal to n Rcose, and (40) becomes a t l b,, , y=0 it i>l W+i-M, - 1 I * I % ^ n R c o s e R d e d * H y=yt o Y l r° *e cose . p H o - J n R Rn dA . y=yH t The non-dimensional form i s given as a.-, b ^ 1 y=0 <D6' fii fii l J r u ita - § 5 - + 1 -?T5- = ITS" / — n D Rit d l . (54) pVR M uVa,RM VR M ^ . R Since (51) i s s i m i l a r to (46), the computer program SWAY.FTN was converted to solve (51) and give <|>g. a g ^ , a g g , b g^ and b g g can be computed by (53) and (54). A computer program PITCH.FTN was w r i t t e n to do the j o b . Results and output of a l l the hydrodynamic c o e f f i c i e n t s mentioned in Sections III.2, III.3, and III.4 w i l l be discussed in the l a s t chapter. - 4.1 -25" IV. EXCITING FORCE AND MOMENT IV.1 General Formulation o f E x c i t i n g Force and Moment Due to Incoming Waves For an axisymmetric f l o a t i n g body, the external f o r c e s and moments a c t i n g on i t are mainly due to surface waves. The surface waves can be considered as u n i d i r e c t i o n a l . They i n t e r a c t with the f l o a t i n g body, and are r e f l e c t e d and r e f r a c t e d from the body's s u r f a c e . These i n c i d e n t waves and r e f l e c t e d waves i n d i c a t e a t r a n s f e r of wave momentum between a s o l i d o b j e c t and the f l u i d flow. I t i s the t r a n s f e r of momentum th a t sets the body i n t o motion; and the f o r c e s experienced by the body at t h i s i n s t a n t are known as the e x c i t i n g f o r c e s or moments. And they are represented by F.J on the r i g h t -hand side of (29). The p o t e n t i a l f u n c t i o n of incoming waves, $ j , i s assumed to be a two-dimensional p e r i o d i c f u n c t i o n . I f the waves are approaching from the p o s i t i v e x - d i r e c t i o n with an angular frequency w, $j i s u s u a l l y given as in Newman [1977] as, * . Ag cosh k(y+h) J(kx+cot) ®T ~ ~ 1 L,U 6 > (00) I to cosh kh ' where A i s the wave amplitude and g i s the g r a v i t a t i o n a l a c c e l e r a t i o n . $ D i s the r e f l e c t e d wave p o t e n t i a l f u n c t i o n due to the presence of the body. T h i s r e f l e c t e d wave p o t e n t i a l f u n c t i o n i s a l s o assumed to be p e r i o d i c because <t>j i s a p e r i o d i c f u n c t i o n . In most cases, the amplitude of $ D i s not known and i s obtained as par t of the s o l u t i o n f o r the given geometry o f the f l o a t i n g body. When the amplitude of e x c i t i n g f o r c e s i s computed by a l i n e a r theory, the body i s assumed f i x e d i n space. The o s c i l l a t i n g frequency of the e x c i t i n g f o r c e w i l l be the same as the waves' frequency though there may be a phase d i f f e r e n c e between them. -4.2 - 26 When s o l v i n g $ D from (26), i t must be noted that on the surface of the body the f l u i d v e l o c i t y i s equal to zero. This boundary c o n d i t i o n implies that on the body's surface the normal d e r i v a t i v e of the t o t a l p o t e n t i a l i s zer o , t h a t i s , '^b and i t can be rewri t t e n as D,n = _ a> c I.n b b (56) S b For an axisymmetric body, 4^ must s a t i s f y the i n t e g r a l equation (26) as well as the boundary c o n d i t i o n s (21), (23), (24) and (56). These f i v e equations w i l l be combined to give a s i n g l e i n t e g r a l equation s i m i l a r to (28). By assuming ^ as the time independent part of $ D, the i n t e g r a l equation (26) becomes * l o<t>n(Q) -, 2* <KD(P) + J 4>D -§jj (-p) RdedA = j—iL—1 RdedA . (57) S t S t The impermeable boundary c o n d i t i o n becomes *D,n L " 0 • ( 5 8 ) >bo The free surface boundary c o n d i t i o n (23) i s now writte n as - w 2 < t > D + 9< t > D j V j = 0 . (59) ' S f The r a d i a t i o n c o n d i t i o n i s • D . R s " (- 2TR + ik) V ( 6 0 ) r - 4.3 - 27 For the boundary c o n d i t i o n on the body, the right-hand side of (56) has to be evaluated. Since $j i s given i n (55) and i f ^ i s assumed to be the amplitude of $ j , <J>J i s given i n the form o f < Ag cosh k(y+h) ikx *I " " 1 ui COsh kh e • ( 6 1 ) ikx A s e r i e s representation of the term e i s given by Abramowitch and Stegun [1964] as . Ag cosh k(y+h) ikRcose $ I " 1 a) cosh kh e m=U where J i s the Bessel Function of order m and B i s equal to one when m i s m "m ^ equal to zero, otherwise, 6m i s given by 2 i m . Therefore, (56) can be rewr i t t e n as pD,n where and = 1 ^ c ^ s T T h Jn ^ c o s m 9 t J m ( k R ) nR c o s h k (* + h ) m=0 + J n ( k R ) n si n hk(y+h) ] , (63) 1' ^m-l ^ m+1 m = 2 { 6 4 ) J _ = (-1) J m . (65) - m m D e t a i l s f o r the d e r i v a t i o n of (63) are shown i n Appendix G. With (58), (59), (60) and (63), the reader can e a s i l y rewrite (57) in t o the f o l l o w i n g form 2 ^ D ( P ) + / ^ ( 1 ) <t>p(Q) RdedA + / ^ ( 1 ) <t>D(Q) RdedA S o S b + / [|n <^ " ( 1 / r ) + i k ) l % R d 6 d y b r K + / $ - i«b «QJ RDEDR = / 1 ? O T i v o s m e t j ; ( k R ) v o s h k(y+h> m=0 + J m ( k R ) n y s i n h k(y+h) ]-p R dedA . (66) From (66), one can observe that i n order to solve f o r 4 ,^ a large number of terms must be used on the right-hand side of the equation. The higher the value of m, the more p r e c i s e the value of 4^ w i l l be. Moreover, since 4>j appears as a functi o n of cos me, (66) can no longer be s i m p l i f i e d unless an assumption i s made on 4^. The most straightforw 3'"* way of doing t h i s i s to assume that 4>D i s also e x p r e s s i b l e as a s e r i e s of terms i n v o l v i n g cosme. That i s , I 4>n cos me . (67) m=0 m Then, (66) can be written as 2u4>D (P) + / | r 7 ( 1 / r ) , t b (Q)cosme RdedA + / ' - ^ l / r ) (Q)cosme RdedA m S„ m S. m 0 b + / fel/r) - d / r H ik)]4h (Q)cosme Rdedy S a n ^ KR Um r 2 + / [|n" ( 1 / r ) " f~ ( 1 / r ) ] " b (Q)cosmeRdedR 9 m - 4.5 -2 9 = / 1 £ c F i O h P mcosme[j' m(kR)n Rcosh k(y +h) + J m ( k R ) n y s i n h k(y+h)jl/rRdedA . (68) Equation (68) has to be solved f o r d i f f e r e n t m values, and 4^ can be computed by (67). For the c a l c u l a t i o n of e x c i t i n g forces and the p i t c h i n g moment, evaluation of the pressure d i s t r i b u t i o n on the body's surface i s necessary. From (37), the expression of pressure i s w r i t t e n as P = I - P U T + <t>D cosme) iwe l a ) m=0 m m • t * - « « "Hit" - **o rfc^ee11* • (69) m=0 m The e x c i t i n g forces are given by s u b s t i t u t i o n of (69) into (38) as - U l P["AS V- lkR> m=0 - i4>D oj ] cos me n.R dedjt. } e i ( i ) t , (70) m where ( n ^ n 2 , n 3 ) = (n R c o s e , n y , -n R s i n e ) (71a) and ( n 4 , n g , n g) = (Rcose, y,-Rsine) x ( n R c o s e , n y , - n R s i n e ) . (71b) IV.2 Heave E x c i t i n g Force For the heave e x c i t i n g force computation, (70) can be used by l e t t i n g n.. be equal to n y . From Appendix H, F ^ i s given as 30 4.6 -F 2 = { f °-2n PRn [Ag " g ^ f f 0 ,J 0<kR) + 1 » D »] d i j e 1 ^ . 72) y=yt ^ o That i s , F 2 depends only on the terms with m equal to zero. Therefore, only the value of <t>D w i l l be necessary f o r computation of F' . Then o Equation (68) can be w r i t t e n as 2 % % ( P ) + 'In" ( 1 / r ) < t b ( Q ) R d e d A + / "IT (7) <trj (Q)RdedA 0 S Q 0 S b 0 + / " ^ + i k ) ( ? n v ( Q ) R d e d 5^ K 0 = | A 1 "^w cosfi~~kh~ P 0 [ j ; ( k R ) n R c o s h k ( y + h ) b b + J Q ( k R ) n y s i n h k(y+h) ](l/r)Rdedjt . (73) Equation (73) i s s i m i l a r to (28) except that a d i f f e r e n t f u n c t i o n independent of 9 i s on the right-hand s i d e . Therefore, by f o l l o w i n g the procedure described in Appendix D, the reader can e a s i l y obtain y=0 -4(R 2n + (y-y_)n 2 7 % (P) + / [ = E u o y=yt (a-b)/a+F \ -4(R 2n + ( y - y J n R ) + J [ P—K- E 0 (a-b)/a+b~ + 3i (T^T E- F»1 V Q , < U • r ^p . m - R p-^ -v 2 E H d ( Q l d y y=-h /i+F (a-b)/a+F u o RR 4R(y-y n) - 2 D + / [ E - — F ] «u dR R c /a+F~(a-b) g/a+F u o y-y t + J ( k R ) n s i n h k(y+h)] F d* . (74) 0 y /a+F Now, <{>n can be obtained by s o l v i n g (74) and F' can be computed by o using (72). A computer program, F2.FTN has been w r i t t e n to evaluate the heave e x c i t i n g f o r c e experienced by an axisymmetric body due to incoming waves. IV.3 Surge E x c i t i n g Force To compute the surge e x c i t i n g f o r c e , equation (70) should be used by l e t t i n g n. equal to n D c o s e . T h e r e f o r e , (70) i s r e w r i t t e n as 1 K - i<t»D to] cosmecose n Rdedi} e 1 u ) t (75) m By i n t e g r a t i n g with r e s p e c t to e, (75) can be expressed as F J = { f ° - upn R R[Ag ^ o s ^ k h ^ ¥ l ( k p ) + HD ^ d * } e 1 U j t ( 7 6 ) y=yt l - 4.8 - 32 D e t a i l s of the proof are given i n Appendix H. From Appendix I, i t i s known that F' i s dependent only on ^ a n d <t>T • For <|> and <{>.. with m not equal to one, there w i l l be no c o n t r i b u t i o n to m m the surge e x c i t i n g f o r c e . Therefore, by keeping only the terms of m = 1 (68) becomes 2lt*D ( P ) + J f n ( 1 / r ) < , b (Q)cos9Rded* + /|n" ( 1 / r H (Q)coseRd9d*-1 S o 1 Sb 1 + / [fn" (1/r) ~ (" + 1 k ) ( l / r ) ] ^ (Q)cosGRdedy S^ , R 1 - 2 + / [|jr - | - ( 1 / r ) ] ^ (Q)coseRdedR S r 1 = / 1 ^ H s T 1 ^ P l t j ; ( k R ) V o s h k ( y + h ) b b + J 1 ( k R ) n y s i n h k ( y + h ) ] ( l / r ) RcosededA . (77) (77) i s s i m i l a r to (45) except that a d i f f e r e n t f u n c t i o n independent of e i s on the right-hand side of (77). Therefore, (77) can be in t e g r a t e d with respect to e and can be s i m p l i f i e d i n exa c t l y the same way as explained i n Appendix E to give 2*VP1 * Xt feiS ( n« R 2 + V ' - V W " T^ T E> 2 - 4.9 2 + | n (/i+¥ E - 4 = F + - = E ) ] * j d x D K /a+b /a+b(a-b) u l RR A 2 . 2 . _ ( y - y n ) a c 0 2 , t r2(2R - a) (c aE > . 4ikR , r- c a r n A , + y i - h [-T^r Ta^ rST^ 5 W - — F ) ] ^ d y 33 j ^ 1 c^osTTTrT hWk*)nRC05h k ( y + h ) + J.(kR)n slnh k(y+h)] £*- [ — — F - /a+E" E] djj. . (78) 1 y D /a+b" Now, <t>_ can be obtained by so l v i n g (78) and F' can be computed by u l 1 using (76). IV.4 P i t c h E x c i t i n g Moment In the computation of the p i t c h e x c i t i n g moment, n. i n (70) i s equal to n c which i s given by (71 .b) as b "S = ( n y R " C 0 S Q » ( 7 9^ and (70) becomes • n S b J P [ . » , f i ! | h ^ h i ^ 0 ( k R ) b m=0 - i 4>D oo] (n R - npy) cosmeRdedx } e 1 a ) t . (80) m * 4.10 As in Appendix I, i t can be proved that the only c o n t r i b u t i o n to Fg i s from the terms with m equal to one. Therefore, Fg i s w r i t t e n in the f o l l o w i n g form: F6 " < yCt - *PR <V " V H * C°cSsh(rhh) ¥l™ + i <un co ] d* } e i t j t . (81) 1 i A f t e r <)>D i s solved from (78), F g can be computed by (81). As both F-| and Fg are dependent only on 4^ , they are computed at the same time in computer program Fl.FTN. IV.5 An A l t e r n a t i v e Method f o r Computing E x c i t i n g Forces Wehausen [1971] gives a r e l a t i o n s h i p between the damping c o e f f i c i e n t and the e x c i t i n g force due to incoming waves f o r three-dimensional f l ow. k l F J / A 12 b.. = 1 ' 1— (82) D i i 2 Pg vg ' ( * a where V i s the group v e l o c i t y of waves, b.. i s the damping c o e f f i c i e n t , |F. | i s the amplitude of e x c i t i n g f o rce or moment and A i s the wave amplitude. Vg can be found in Newman [1977] as % • - . k where V p i s the phase v e l o c i t y , k i s the wave number and h i s the l o c a l water depth. - 4.11 -Equation (82) i s a good way of checking the amplitude of the e x c i t i n g f o r c e computed using the formulations of t h i s chapter. Chapter V w i l l e x p l a i n the r e s u l t s of the computation by (72), (76), (80), and (82). - 5.1 -36 V. RESULTS In t h i s chapter, the hydrodynamic c o e f f i c i e n t s and e x c i t i n g forces of some axisymmetric bodies are computed using the equations presented in the previous chapters. The r e s u l t s are then compared to those obtained by some other s o l u t i o n methods. The d i f f e r e n c e s between a l t e r n a t e methods are also discussed. The r e s u l t s presented by the author in t h i s t h e s i s are under the assumption that the p o t e n t i a l value i s a constant on each r i n g element. Therefore, when the hydrodynamic c o e f f i c i e n t s and e x c i t i n g f o r c e s are computed by taking the i n t e g r a l of the p o t e n t i a l values on the body's surface, the p o t e n t i a l value at the node of each r i n g element i s used. Although, a more pre c i s e r e s u l t could be achieved by c o n s i d e r i n g the p o t e n t i a l value varying across an element, the constant p o t e n t i a l value assumption i s the simplest assumption that one can make. The examples studied and the comparisons done f o r each p a r t i c u l a r geometry are explained below. V.l S i n g l e C y l i n d e r a t F i n i t e Water Depth The f i r s t example to study i s a s i n g l e c y l i n d e r at f i n i t e water depth. The radius of the c y l i n d e r i s designated by R, the draught by T, and the water depth by D. ( F i g . C) The hydrodynamic c o e f f i c i e n t s f o r heave, surge and p i t c h cases are computed and given in non-dimensional form (41). Results are p l o t t e d against the non-dimensional frequency, 2 to R/g, and are shown in F i g s . 1, 2 and 3. In F i g . 1, the heave hydrodynamic c o e f f i c i e n t s f o r T/R = 0.5 and D/R = 1.5 are c a l c u l a t e d and p l o t t e d against the published r e s u l t s of K r i t i s [1979]. The r e s u l t s by K r i t i s and the author are both obtained - 5.2 - 37 through the Integral Equation Method discussed in previous chapters. Good agreement between the r e s u l t s i s observed. In F i g . 2, the surge hydrodynamic c o e f f i c i e n t s f o r T/R = 0.5 and D/R =2.0 are p l o t t e d and compared to the values obtained by the Matching Technique (M.T.) of Sabuncu and C a l i s a l [1981]. The r e s u l t s obtained usingthe Boundary Element Method (B.E.M.) are lower than those of the M.T. There i s a d i f f e r e n c e of about 6% and 10% at the peak values of the added mass and damping c o e f f i c i e n t r e s p e c t i v e l y . The p i t c h added mass and damping c o e f f i c i e n t f o r a simple c y l i n d e r are p l o t t e d in F i g . 3 and compared to the values given by M.T. The added mass by B.E.M. has a lower value over the low and high frequency range. The damping c o e f f i c i e n t by B.E.M. i s lower than that of M.T. at low frequency but i s higher at the high frequency end. The maximum d i f f e r e n c e between the two curves i s at the peak value and i s l e s s than 5% f o r both the added mass and the damping c o e f f i c i e n t . The heave e x c i t i n g force experienced by the s i n g l e c y l i n d e r due to incoming waves i s shown in F i g . 9. The heave e x c i t i n g force per u n i t wave amplitude, f ^ , i s non-dimensionalized by the weight of the f l u i d d i s p l a c e d by the c y l i n d e r . Another way to p r e d i c t f^ i s by using the r e l a t i o n s h i p in (82) and the pr e v i o u s l y computed b^ values shown in F i g . 1. F i g . 9, shows that the B.E.M.1s r e s u l t i s lower than the r e s u l t obtained by using equation (82). T h e i r maximum d i f f e r e n c e i s l e s s than 5%. The surge e x c i t i n g f o r c e , F n , i s non-dimensionalized and p l o t t e d i n F i g . 10. Result p r e d i c t e d by equation (82)(with b ^ on F i g . 2) are als o shown on the same f i g u r e . Both r e s u l t s e x h i b i t the same peak value but 2 there i s a d i f f e r e n c e of about 10% at u R/g = 3.0. The p i t c h e x c i t i n g moment i s non-dimensionalized by the weight of d i s p l a c e d f l u i d and the radius of the c y l i n d e r . The moment i s c a l c u l a t e d about the z - a x i s . Both the r e s u l t s of B.E.M. and of equation (82) (with b g 6 on F i g . 3) are p l o t t e d in F i g . 11. Peak values are comparable but o there i s a maximum d i f f e r e n c e of about 15% at to R/g = 3.0. From F i g s . 9, 10, and 11, one can see that the e x c i t i n g forces p r e d i c t e d by B.E.M. have a b e t t e r agreement with those from (82) in the low frequency range. At the high-frequency end, the r e s u l t s of the two methods, though they e x h i b i t some d i f f e r e n c e , show the same trend and are comparable in magnitude. V.2 Composite C y l i n d e r As a second example, the hydrodynamic c o e f f i c i e n t s of a composite c y l i n d e r are computed. The dimensions of the composite c y l i n d e r are shown in F i g . D. The s o l u t i o n method the author w i l l compare with i s the Matching Technique of Sabuncu and C a l i s a l [1983]. The non-dimensional hydrodynamic c o e f f i c i e n t s are p l o t t e d against the 2 2 parameter to /g. to /g i s used instead of the non-dimensional frequency because in t h i s case i t i s hard to f i n d a s i g n i f i c a n t length parameter as the c r i t i c a l dimension of the composite c y l i n d e r . However, to non-dimensional i z e the p i t c h hydrodynamic c o e f f i c i e n t s (53), the l a r g e r radius of the composite c y l i n d e r i s used. F i g . 4 shows the heave added mass and damping c o e f f i c i e n t . The r e s u l t by B.E.M. e x h i b i t s an o s c i l l a t i n g behavior both in the added mass and damping c o e f f i c i e n t . This o s c i l l a t i o n i s b e l i e v e d to be caused by numerical problems in the computer program and w i l l be discussed l a t e r . I r r e s p e c t i v e of the o s c i l l a t i n g behavior, the r e s u l t s by B.E.M. e x h i b i t a s l i g h t l y higher value in the added mass computation. The damping c o e f f i c i e n t by B.E.M. agrees quite well with that computed by M.T. The maximum d i f f e r e n c e between the added mass value computed by the two 2 methods i s l e s s than 10% and occurs at to /g = 1.5. The surge hydrodynamic c o e f f i c i e n t s are shown in F i g . 5. The maximum d i f f e r e n c e in the added masses computed by the B.E.M. and M.T. i s at the peak value and i s l e s s than 5%. The damping c o e f f i c i e n t s obtained by the two methods e x h i b i t a d i s p a r i t y of about 10% at the peak value. The non-dimensional added mass approaches a constant value of about 3.75 at high frequency and the damping c o e f f i c i e n t goes to zero. F i g . 6 i s a p l o t of the p i t c h hydrodynamic c o e f f i c i e n t s ( a gg and bgg). The values by B.E.M. are lower in magnitude in an average sense. The maximum d i f f e r e n c e of damping c o e f f i c i e n t s i s observed at the peak value and i s about 5%. However, the added mass has a large d i f f e r e n c e of 2 about 35% at to /g = 1.5. The non-dimensional added mass value has a l i m i t equal to 0.4 at the high frequency while the damping c o e f f i c i e n t goes to zero. R e f e r r i n g to the f i g u r e s mentioned above, one can see that the o s c i l l a t i n g behavior of r e s u l t s by B.E.M. i s always about some mean value. I t i s bel i e v e d that t h i s numerical o s c i l l a t i o n i s caused by the constant p o t e n t i a l value assumption discussed in the e a r l y s e c t i o n s , and i s r e l a t e d to the s i z e of the r i n g elements chosen. In F i g . 12, the heave hydrodynamic c o e f f i c i e n t s of a s i n g l e c y l i n d e r are shown f o r two d i f f e r e n t element s i z e s . The r e s u l t with a l a r g e r - 5.5 -40 element s i z e e x h i b i t s o s c i l l a t i o n s , while the r e s u l t s with a smaller element s i z e does not. A p o s s i b l e explanation f o r t h i s i s that as the element s i z e becomes l a r g e r and l a r g e r , the constant p o t e n t i a l value assumption can no longer represent the actual p o t e n t i a l value on the surface of the body. This i s e s p e c i a l l y true f o r an object with complicated geometry. Therefore, in order to suppress t h i s o s c i l l a t i n g behavior, a f i n e r element s i z e must be used. V.3 Hemisphere In the t h i r d example, the heave and surge hydrodynamic c o e f f i c i e n t s of a hemisphere are computed. As in our f i r s t example, general notations such as R and D are used to represent the radius of the hemisphere and the water depth r e s p e c t i v e l y . In t h i s example, the published data of Garrison [1974] by Surface S i n g u l a r i t y Method (S.S.M.) are used f o r comparison. The heave hydrodynamic c o e f f i c i e n t s of a hemisphere with D/R equal to 2 2.0 are p l o t t e d against the non-dimensional frequency (co R/g) in F i g . 7. I t can be seen that the added mass computed by B.E.M., though o s c i l l a t i n g about some value, has in general a magnitude c l o s e to the r e s u l t by S.S.M.'s value in general. For the surge case ( F i g . 8 ) , the hydrodynamic c o e f f i c i e n t s obtained by the two methods e x h i b i t e d a d i f f e r e n c e of about 10%. The r e s u l t s by B.E.M. are lower than the r e s u l t s by S.S.M. V.4 Axisymmetric F l o a t i n g Rig The l a s t example studied i s an axisymmetric f l o a t i n g platform with a c i r c u l a r d r i l l i n g well at i t s centre ( F i g . F ) . The hydrodynamic c o e f f i c i e n t s as well as the e x c i t i n g forces are computed and compared to the unpublished r e s u l t s of Bai who used a F i n i t e Element Method (F.E.M.) in 1981. The r e s u l t s are presented in real value instead of non-dimensional 3 value. The hydrodynamic c o e f f i c i e n t s are given in un i t s of volume ( f t ) and the e x c i t i n g forces i n pounds per u n i t wave amplitude. The heave hydrodynamic c o e f f i c i e n t s are p l o t t e d against OJ in F i g . 13a. The r e s u l t s by B.E.M. e x h i b i t an o s c i l l a t i n g behavior while the r e s u l t s by F.E.M. seem to e x h i b i t an " i r r e g u l a r frequency" at co = 1.0. The B.E.M. r e s u l t s are lower than those by F.E.M. i n the low frequency range but higher in the high frequency range. Since both r e s u l t s show some numerical e r r o r , a general remark about the heave hydrodynamic c o e f f i c i e n t s i s hard to make. The surge hydrodynamic c o e f f i c i e n t s are shown in F i g . 13b. The r e s u l t s by B.E.M. are lower in value throughout the frequency range. The d i f f e r e n c e between the two methods i s about 10% f o r the added mass and l e s s than 8% f o r the damping c o e f f i c i e n t . The p i t c h hydrodynamic c o e f f i c i e n t s shown in F i g . 13c are in much bette r agreement. The r e s u l t s by B.E.M. are lower than those by F.E.M. However, the maximum d i f f e r e n c e i s l e s s than 3%. The heave e x c i t i n g force presented in F i g . 14a shows some agreement between the two methods. The r e s u l t by F.E.M. seems to show an " i r r e g u l a r frequency" at w c l o s e to one. The r e s u l t by B.E.M. i s lower - 5.7 - 42 than that of F.E.M. by 4% in the low frequency range but the two approach each other at high frequency. The p r e d i c t i o n of the phase angle between the heave e x c i t i n g force and the incoming waves using two methods i s shown in F i g . 14b. One can see that the r e s u l t s are extremely c l o s e to each other. The surge e x c i t i n g force and phase angle between e x c i t i n g force and waves are shown in F i g . 15a and F i g . 15b. F i g . 15a shows that the surge e x c i t i n g force by B.E.M. i s lower by 10% at the peak value but the phase angles p r e d i c t e d by the two methods are almost i d e n t i c a l . The p i t c h e x c i t i n g moment and i t s phase angle are c a l c u l a t e d and shown in F i g . 16a and F i g . 16b. In F i g . 16a, the p i t c h i n g moment by B.E.M. i s observed to be 4% lower than the values c a l c u l a t e d by F.E.M. The phase angles p r e d i c t e d by the two methods show an extremely good agreement. V.5 Discussions and Conclusion Following the r e s u l t s of the examples presented in the previous s e c t i o n s , a few remarks can be made. 1. The magnitude of the hydrodynamic c o e f f i c i e n t s , a., and b.. i s 1 J 1 J frequency dependent. 2. The r e s u l t s computed by B.E.M. are us u a l l y lower in magnitude value than the r e s u l t s by other methods. 3. With a constant p o t e n t i a l assumption, the r e s u l t s computed by B.E.M. e x h i b i t an o s c i l l a t i n g behavior which i s be l i e v e d to be as s o c i a t e d with the s i z e of surface elements. T h i s o s c i l l a t i n g behavior i s l e s s s i g n i f i c a n t as the element s i z e i s decreased. ( F i g . 12) - 5.8 - 43 4. The o s c i l l a t i n g behavior of B.E.M.'s r e s u l t s can be suppressed by decreasing the s i z e of the surface elements. ( F i g . 12) 5. I r r e s p e c t i v e of the o s c i l l a t i n g behavior of the r e s u l t s by B.E.M., the method gives good p r e d i c t i o n s f o r hydrodynamic c o e f f i c i e n t s as well as e x c i t i n g f o r c e s . 6. The pr e d i c t e d phase angles between the e x c i t i n g forces and waves are extremely close to those p r e d i c t e d by F.E.M. In general, i t i s hard to state that any s o l u t i o n method i s superior to the others, f o r every s o l u t i o n method may have i t s own advantages and disadvantages. However, the B.E.M. i s a ge n e r a l l y accepted method f o r so l v i n g forced o s c i l l a t i o n problems and p r e d i c t i n g e x c i t i n g f o r c e s . In t h i s t h e s i s , though the d i s c u s s i o n i s r e s t r i c t e d to axisymmetric bodies, the general i n t e g r a l equation, (28), can be ap p l i e d to f l o a t i n g objects of a r b i t r a r y shape. The cost of the computation i s u s u a l l y lower than f o r other s o l u t i o n methods. A problem of B.E.M. presented in t h i s t h e s i s i s the i r r e g u l a r o s c i l l a t i n g behavior of hydrodynamic c o e f f i c i e n t s e x h i b i t e d in the r e s u l t s under the constant p o t e n t i a l value assumption. In f a c t , as i n d i c a t e d before, t h i s o s c i l l a t i n g behavior can be decreased by decreasing the s i z e of the surface elements bounding the control volume. However, f o r some f l o a t i n g objects with complicated geometries are complicated such as the f l o a t i n g r i g studied ( F i g . F ) , i t i s extremely c o s t l y to decrease the element s i z e even by 50%. Therefore, i n order to avoid the o s c i l l a t i n g behavior of the r e s u l t s , as well as to achieve a more p r e c i s e r e s u l t , the po t e n t i a l value should be assumed varying l i n e a r l y across an element i n s t e a d of taking i t as a constant. In other words, a higher l e v e l s o l u t i o n technique ( l i n e a r elements) seems to be p r e f e r a b l e . - F . l - 44 LIST OF FIGURES Figure Page A. Coordinate System F.2 4-5 B. Axisymmetry of Geometry F.3 4 6 C. A S i n g l e C y l i n d e r F.4 47 D. A Composite C y l i n d e r F.5 48 E. A Hemisphere F.6 49 F. A D r i l l i n g Platform F.7 So 1. A 2 2 and B 2 2 of S i n g l e C y l i n d e r F.8 s i 2. A-JI and B ^ of Sin g l e C y l i n d e r F.9 52 3. A g 6 and B g 6 of Single C y l i n d e r F.10 S3 4. A 2 2 and B 2 2 of Composite C y l i n d e r F . l l 54 5. A ^ and B ^ of Composite C y l i n d e r F.l2 55 6. Ag 6 and Bg g of Composite C y l i n d e r F.13 56 7. A 2 2 and B 2 2 of Hemisphere F.l4 57 8. A ^ and B^ of Hemisphere F.l5 SB 9. f 2 of Single C y l i n d e r F.16 SB 10. f| of S i n g l e C y l i n d e r F.17 Bo 11. fg of S i n g l e C y l i n d e r F.18 61 12. A 2 2 and B 2 2 of Single C y l i n d e r F.l9 62. 13a. A 2 2 and B 2 2 of D r i l l i n g Platform F.20 €3 13b. A-JI and B ^ of D r i l l i n g Platform F.21 6* 13c. Agg and B gg of D r i l l i n g Platform F.22 65" 14a. F 2 of D r i l l i n g Platform F.23 66 14b. Phase D i f f e r e n c e between F 2 and Wave F.24 67 15a. F 1 of D r i l l i n g Platform F.25 6g 15b. Phase D i f f e r e n c e between F-j and Wave F.26 69 16a. F 6 of D r i l l i n g Platform F.27 7o 16b. Phase D i f f e r e n c e between F g and Wave F.28 11 - F.2 -- F.3 -I - F.4 - 47 FIG D A . COMPOSITE CYLINDER 00 - F . 6 -49 FIG F A DRILLING PLATFORM o 1 . A 2 2 & of 7 S i n g l e C y l i n d e r ^ 1. 60^_ © 0 1.40J. 0 0 1.201 TJ ^ 1.00L "TJ C 0- 80}. 0 0 0 ^ 0. 40|. a. E U 0. 201 0. 00L 0. 00 _L_J I L I J I L J I I L 0. 50 1. 00 1. 50 w * w * R / g L E G E N D o A22 b y B . E . M-*• A22 b y K n i t i e + B22 b y B . E . M . x B22 b y K n i t i e J I L J I I L J I I L 2. 00 2. 50 3. 00 F i g 2 . A l l . S B l l o f S i n g l e C y l i n d e r - C T / R = 0 , 5 , D / R = 2 , 0 ) 0. 80 L E G E N D © A l l b y B. E . M . * A l l b y M . T . + B l l b y B. E . M . x B l l b y M . T . 0. 00 0. 00 1.00 2. 00 4. 00 5. 00 6. 00 w*w*R/g i I F i g 3 , A 6 6 8, B66 o f S i n g l e C y l i n d e r - C T / R = l . 0 , D / R = 2 , 0 ) 0. 20^_ • 0 0 0. 17 0 j? 0. 15| 0 TJ 0 0. 12L_ "0 C 0 . 101 0 ^ 0.08 (D 0 0 P 0.051 •r>( £ 0 ~0 0.03! 0. 00! 0. 00 L E G E N D A66 b y B . E . M . * A66 b y M . T . + B66 b y B . E . M. x B66 b y M . T . 1.00 2. 00 3. 00 w*w*R/cj 4. 00 -©-5. 00 -€5 6700 F i g 4 , A 2 2 8c B 2 2 o f C o m p o s i t e C y l i n d e r - 1 0. 80r ® 0 0- 70i 0 CO 0 0.60J TJ 0 TJ TJ 0 0. 50 ~C 0. 40, 0 0. 00 1. 00 2. 00 w*w/g 3. 00 Cl. / f t . ) L E G E N D © A22 b y B. E . M . * A22 b y M . T . + B22 b y B . E . M . x B22 b y M . T . F i g 5 . A l l & B l l o f C o m p o s i t e C y l i n d e n 1 1.20, 0. 00L 0. 00 L E G E N D © A l l b y B . E . M . * A l l b y M . T . + B l l b y B . E . M . x B l l b y M . T . 1.00 2. 00 w * w / C J r * H - * i x1 * i i i >k i i i i ik 3. 00 4. 00 5. 00 6. 00 < 1 . . / f t . ) F i g 6 . A 6 6 & B 6 6 o f C o m p o s i t e C y l i n d e r - 1 . L E G E N D w*w/g CI. / f t . ) F 7. A22 & o f h e m i s p h e r e ( D /R = 2 . 0 ) 0. 80 0 0. 70^_ 0 r-(0 0 0. 60. 0 0.50 C 0. 40 0.301— 0 0 (P 0. 20L CL o U 13 0. 1EL-0. 001 I I. 0. 00 J — I J 0. 50 L „ i i i J L 1.00 1.50 J I L E G E N D A 2 2 b y B . E . M . * A 2 2 b y G A R R I S O N + B 2 2 b y B . E . M . x B 2 2 b y G A R R I S O t U H 2. 00 2. 50 . J L 3. 00 w * w * R / g F i 3 . _ a . A l l 8, _B1 1_ o_f_ h e m i s p h e r e C _D/R=2. 0 ) 0. 80^ _ L E G E N D A l l A l l B l l B l l b y B . E . M . b y G A R R I S O N b y B . E . M . b y G A R R I S O N 0. 00 0. 00 0. 50 —I- t—1—I I X. 1. 50 w * w * R / g j L—i~_ i 2. 00 I L 2. 50 A I I L 3. 00 I 1 f 2 o f S i n q l e C y l i n d e r - C T / R = 0 . 5„ D / R = 1 . 5 ) 2. 00 1. 75! 1. 50 -P 1. 25 <+-\ 1. 00! < OJ 0. 75] 0. 50 0. 25 L E G E N D © f 2 b y B. E . M. * f 2 f r o m B 2 2 0. 001 J i I I I 0. 00 0. 50 J I L L _ 1 J I 1- 1. J L__ 1 — 1 1. 00 1. 50 2. 00 — J I L„..i I I 2. 50 3. 00 ! i F i q 10. f l o f S i n q l e C y l i n d e r - C T / R = 0 . 5 . D / R = 2 . 0 ) 1. 20, 1. 05j 0. 90 •P 0.75| E* < 0. 601 0. 45L_ 0. 30L_ 0. 15 0. 001 I 1 I I J L L L 0.00 1.00 _ 1 _ L 2. 00 - I I L 1. L E G E N D co f 1 b y B . E . M. * f l f r o m B l l J I l J J i L 3. 00 W * W * R /C | 4 . 00 5. 00 6. 00 F i g 1 1 . f 6 o f S i n g l e C / y l i n d » 0. 30r 0. 26L 0. 221 •P 0. 19U *+• \ C 0. lSf < (0 0 . u } _ 0. 08L. 0. 04; 0. 00L 0. 00 - i J I I .1 L L 1. 00 J J I I L 1 X 2. 00 3. 00 w * w * " R / g C T / R = l . 0 , D / R = 2 . 0 ) LEGEND © f 6 by B. E . M. * f 6 f r o m B 6 6 ! 1 J I L 1 I X 4. 00 5. 00 6. 00 i F i g 12 , A 2 2 & B22 o f S i n g l e C y l i n d e r - C T / R = 0 . 5 . D / R = l . 5 ) 1. 60 <D 0 1. 40 0 0 1.20 "0 <D ^ 1. 00 0 £ 0.80 0 0.60 <D 0 0 ? 0. 40 •H CL E 0 "D 0.20 0. 00 0. 00 co L E G E N D A 2 2 * A 2 2 + B 2 2 x B 2 2 d L = 0 . 2 d L = 0 . 4 d L = 0 . 2 d L = 0 . 4 •—e-J I L 0. 50 I I I L J L J I L J I L 1.00 1.50 w*"w*-R/g 2. 00 2. 50 3. 00 0. 24 r 0. 2 l L_ 0. 18L 0. 15L_ 0. 12f__ (\l CM OQ 0. 0 9 C <\1 ^ 0 . 0 6 L 0. 03L_ + III 0. 00L F i a 13 . a A 2 2 & B 2 2 o f D r i l l i n q P l a t f o r m 0. 00 L E G E N D © A 2 2 b y B . E . M . ^ A 2 2 b y F . E . M . + B 2 2 b y B . E . M . x B 2 2 b y F . E . M . J I L 1 0.25 J I I L J I L 0. 50 0. 75 w CI. / e e c ) J — I 1 — L 1 1.00 J I I L 1 1. 25 J I I L 1.50 i O 0 . 2 4 _ 0. 2lL_ ro * 0 . 18 » -p (D + LLJ 0 . 15 ^ 0 . 12 i — i «—i CD 0 . 09 cx3 «—• < 0 . 06 0 . 03j F i q 13 . b A l l & B l l o f D r i l l i n g P l a t f o r m L E G E N D © A l l b y B . E . M . ^ A l l b y F . E . M . + B l l b y B . E . M . x B l l b y F . E . M . 0 . 00L_J__i__l I I I I — I — I — I — I — l — ' — L I I 1 1 1 1 I I I 1 1 1 1 1 L 0. 00 0. 2 5 0. 50 0 . 75 w ( 1 - / s e c ) 1. 00 1. 25 1. 50 F i g 1 3 . c A 6 6 & B 6 6 o f D r i l l i n g P l a t f o r m 0. 56 r 0. 49. in * 0 . 4 4 • +> tr-IS 0. 35U r - l + LL) 0. 28 _ (0 (D m 0-21 (0 (0 0. 14. 0. 07. 0. 00. J L 0. 00 L E G E N D © A 6 6 b y B . E . M . ^ A 6 6 b y F . E . M . + B 6 6 b y B . E . M . x B 6 6 b y F . E . M . 1 J L J I I L J I L J I L J I I L 0. 25 0. 50 0. 75 w Cl. /sec) 1. 00 1. 25 1. 50 1 F i g 14 , g F2 o f D r i l l i n g P l a t f o r m 0. 24r L E G E N D F2 b y B. E . M. F2 b y F . E . M. 0. 21 0. 18L_ 0. 15 + LU . 0. 12L ^ 0. 09i \ C\J LL 0. 06L 0. 03L 0. 00L J I L J I L J I L J I L J I 1 L J I I L 0. 00 0. 25 0. 50 w 0. 75 C1. /sec) 1. 00 1. 25 1. 50 1 4 . b P h a s e D i f f © n o n e © b e t w e e n F 2 a n d Wave L E G E N D © b y B . E . M . ^ b y F . E . M . 0. 180, 0. 135 Q CO + Ul 0. 045 0. 000[_ •H D-0. 045| <D (0 0 Q_-0- 0901 -0. 135L_ -0. 1801 J I L 0. 000 J I L 0. 250 J I I L 1 J I L 0. 500 w 0. 750 CI. /sec) 1. 000 1 I L J I I L 1. 250 1. 500 F i g 15 . a F l o f D r i 1 1 i n g P l a t f o r m L E G E N D © F l b y B . E . M . * F l b y F . E . M . w C1. / s o c ) OS 0 0 15. b P h a s e D i f f e r e n c e b e t w e e n F l a n d Wave 0. 180_ 0. 135 0*0. 090 Q CO LU 0. 045 0. 000| ^0. 045U <D (0 0 J : Qr-0. 090! -0. 135 -0. 1801 0. 000 L E G E N D © b y B . E . M. b y F . E . M. J I I L J L J I I L J I L J I 1 L J I I L 0. 250 0. 500 w 0. 750 (1. /sec) 1. 000 1. 250 1. 500 F i g 16 , q F 6 o f D r i l l i n g P l a t f o r m , - I L E G E N D w C I . / e e c ) F i q 16 . b P h a s e D i f f e r e n c e b e t w e e n F6 a n d Wave 0. 180_ 0. 135 ^0. 090| a co LU 0. 0451 0. 000 t <+• •ri C5-0. 045|_ (D 0 Ct-0. 0901 •0. 135 -0. 1801 I 1—L 0. 000 L E G E N D co b y B . E . M. b y F . E . M. i i 00 J I L J I L L J I I L . 1 J I L J L 0. 250 0. 500 w 0. 750 CI. /sec) 1. 000 1. 250 1. 500 LIST OF SYMBOLS wave amp!i tude constant water depth complete E l l i p t i c Function of f i r s t and second kind hydrodynamic pressure force e x c i t i n g force time independent part of F'. J Green's Function Bessel Function observation point co n t r o l point length parameter f o r non-dimensioning radius of control volume r a d i a l measurement from z axis surface area area of r a d i a t i o n boundary area of free surface area of f l o a t i n g body area of r i g i d wall t o t a l surface area draught of f l o a t i n g body v e l o c i t y of f l u i d p a r t i c l e added mass in i d i r e c t i o n induced by motion in j d i r e c t i o n damping c o e f f i c i e n t r e s t o r i n g force c o e f f i c i e n t g r a v i t a t i o n a l constant wave-number integer actual mass of f l o a t i n g body normal un i t vector on the control surface pressure or hydrodynamic pressure d i s t a n c e between P and Q p o s i t i o n measured from o r i g i n 73 - N.2 -t time u.j ve loc i ty component of f loa t ing body Vg group veloci ty v phase ve loc i ty p R ,y radial and ver t i ca l measurement of point p n D ,n radial and y component of unit normal vector K y a,b constant u,v,w ve loc i ty component of f l u i d par t i c le s X,Y,Z coordinate axes $ tota l ve loc i ty potential <|>, <t>', ^ time independent part of $ 4j incoming wave potential $D refracted and ref lected potential $P potential value induced by motion of f loat ing body e , n constant m a) angul ar frequency p f l u i d density e small parameter for l i n e a r i z a t i o n e angular measurement n wave elevation v f l u i d volume displaced by f loa t ing object Y angular ve loc i ty of f loat ing object. X wave-length APPENDIX A L i n e a r i z a t i o n of the Kinematic Boundary Cond i t i o n From (16.d), which i s the free surface boundary c o n d i t i o n , rj - 4 = 0 . ... .(16.d) »t »y P h y s i c a l l y , i t means t h i s equation i s to be s a t i s f i e d at the p o s i t i o n of y = TI. However, by (15) T) = e n ( 1 ) + e 2 n ( 2 ) + . . . . . . . .(15) (1) 2 (2) $ can be writt e n as $(x , e T ] •+ E n + z , t ) . I f $ i s expanded about y = 0 as: (1) 2 2 $ = <s>(x,o,z,t) + st) $ (x,o,z,t) + e n $ u v / ( x , o , z , t ) + •< »y >yy Then, to the lowest power i n e, $ = $ (x,o,z,t) + e n ( 1 ) $ v v ( x , o , z , t ) + ... Therefore, $ i s evaluated at the mean p o s i t i o n , y = 0, »y APPENDIX B Matrix Representation of the Integral Equation From (26), 2ir<|.(P) + /s *(Q) - | r (7) M e d * = J s |* (Q)-Jr RdedA • ( 2 6 ) Now, suppose the control surface i s d i v i d e d into n c i r c u l a r r i n g elements, n equations can be written f o r each point P located on a d i f f e r e n t element. For each of these equations, there are n terms on the r i g h t hand side of the equation as well as n terms on the l e f t hand s i d e . When P and Q are on d i f f e r e n t elements, the term on the r i g h t hand side i s ^ ¥ Q ) I n (F> R d e d * = A i j * j ' When P and Q are on the same element, 2wfj + /* V,(Q) ^ (1) Rde d* = A j j*. . On the l e f t hand s i d e , n 2 TT 9<|>. , ^Q Hn~ 7 R d 9 d * = B i • Therefore, the equations can be represented by (27) [A..] [•.] = [B.] i = 1, ...n, 1 J J 1 j = 1, ..-n. (27) In some cases, not a l l <|>. are known on each element, then the term J »n j ,n i s expressed as a fun c t i o n of 4>. and moved to the r i g h t hand side (as shown in Appendix C). I f <j>. cannot be expressed as a fun c t i o n of A., J ,n j some reasonable constant are assumed to represent , and exchange J p o s i t i o n with <{>. on the l e f t hand s i d e . I t must be noted that, the j , n - A.2 -APPENDIX B Matrix Representation of the Integral Equation 7 6 From (26), 2*<t>(P) + JS *(Q) IJJ- (1) RdecU = JS |4 (Q)l Rdedi . (26) t t Now, suppose the control surface i s d i v i d e d i n t o n c i r c u l a r r i n g elements, n equations can be written f o r each point P located on a d i f f e r e n t element. For each of these equations, there are n terms on the r i g h t hand side of the equation as well as n terms on the l e f t hand s i d e . When P and Q are on d i f f e r e n t elements, the term on the r i g h t hand side i s C • j W In ( 7 } R d e d * " A i j * j ' When P and Q are on the same element, 2«*S + C^ ( Q ) In (?) R d 9 d * = A j j * j ' On the l e f t hand s i d e , n 2n a*. , s / - r - i 1 Rdedi = B. . j=l 0 5 n r 1 Therefore, the equations can be represented by (27) CA..] [*.] = [B.] i = l,...n, 1 J J 1 j = l , . - n . (27) In some cases, not a l l (t>. are known on each element, then the term *. i s expressed as a fun c t i o n of A. and moved to the r i g h t hand side (as J shown i n Appendix C ) . I f <)>. cannot be expressed as a functi o n of A., J , n j some reasonable constant are assumed to represent A.., and exchange p o s i t i o n with *. on the l e f t hand s i d e . I t must be noted that, the J»n - A.3 - 77 matrix equation must have a column vector, which e n t r i e s are known, on the r i g h t hand s i d e . The only unknown i s the column vector on the l e f t hand s i d e , of which i t s e n t r i e s may be A. or <(>. . - A.4 -APPENDIX C Combining the Boundary Conditions with the Integ r a l Equation 78 The i n t e g r a l equation i s 2Tc<t»(P) + 4>(Q) IJJ- (1) RdedA = J, ^ 1 On the f i x e d boundary (at the bottom) S , S T an ( ( » r R d 9 ( U * (26) (21) On the moving boundary, S^, = V. (22 ) On the free surface, S f ! OJ 9 (23) and on the r a d i a t i o n control surface, S^, '.R + Ik) (24) Now, with S o + S b + S r + S f equation (26) can be decomposed i n t o b ,1 9 ,1 2rc<t> + J s 4> (7) RdedA + J s <|) (^) RdedA 0 b + J c • h <i> RdedR + L 4) £ r (±) Rdedy a , 1 . 'S f T an l r an r 's Sf 7 R d e d * + /• ^ 1 RdedA S. an r W S f ^ RdedR • ^ | i I Rdedy . - A.5 - 79 By s u b s t i t u t i n g Equations (21), (22), (23), and (24) i n t o the r i g h t hand si d e of the above equation and moving the unknown expressions to the l e f t hand s i d e , the f o l l o w i n g equation i s obtained. 2*» + 's0 Ir (?> *R d 9 d* + kbh <?> •Rded* + V i ' ? ' " (4+ 1k)^] *Rd9dy = Jo i (v • n) R dedA . (28) b b r Bo - A.6 -APPENDIX D S i m p l i f i c a t i o n of Equation (28) i n t o (42) Equation (28) i s given i n Appendix C as: 2*+2 * >s0 h <?> *2 R D E D * * kb h <?> ^Rd9d^ 2 + /Sf[lr7 (F} " F f" ] «feRdedR = 'sb F {* ' " ) R d e c U ' ( 2 8 ) Let R be the radius of the body at the free surface, y. be the point on the keel of the body, and the normal u n i t vector, n, be in the f o l l o w i n g form: n = ( n R cos e, n , - n R s i n e ) , (D.l) where n D and n be the R and y component of n. Then, with r and vr as K y s i 1 given in (2) and (3), -^ r(^ r) i s equal to — ^ V r * " which can be represented by the f o l l o w i n g expression: a 1 (Rcose - R p ) n R c o s e + ( y - y p ) n y + Rsine n R s i n e ^ ~ " { [(R cose - R p ) 2 + ( y - y p ) Z + (-R s i n e ) 2 ] 3 / 2 1 Rn R + (y-y )n - R n R c o s e = - { , o o o o / o } • (D.2) ( R 2 + R 2 + ( y - y D ) 2 - 2RR c o s e ) 3 / 2 r r r By l e t t i n g a = R 2 + R 2 + (y - y ) 2 (D.3) P P and - A . 7 - Bl b = 2 RRp , ( D . 4 ) (D.2) can be rewri t t e n as: a i R nR + ( y _ y n ) n v " R n n R C O s e IJJ (1) = - { — P Y } • ( D . 5 ) 9 n r ( a - b cos Q ) 6 / d ( i ) By s u b s t i t u t i n g ( D . 5 ) into the second term of (28), /s0 In (7} h R d6 d* RR 2TT R 2 n D + ( y - y n ) R n w - R R n B c o s e = / / " { — P y 3/2 P J *2 d 9 d* 0 0 (a - b cos %V,C L R r ? 217 rift = J - (R n R + (y-y )Rn \ J ^ ™ d£ 0 K P Y ^ 0 ( a - b c o s e r ' * D R 2u , ( a - b cose - a ) n R + / / - I j T P — A, de dA 0 0 * ( a - b cose) 1 R R = / - 2(R 2n R + (y-y )Rn ) <fr2 f — — - J J ^ de dz 0 K P Y ^ 0 (a - b c o s e r ' * R D 1 a 0 K L 0 (a - b c o s e ) 1 ^ (a - b c o s e r ' * From page 154 and 156 of Gradshteyn and Ryzhik [ 1 9 6 5 ] , one has I d x = 2 F (J.6) ( 0 . 6 ) 0 / a - b cosx / a + b and / D X = 2 E 6 ) , ( D . 7 ) 0 ' (a - b c o s x ) 3 ( a - b ) / a + b 82 - A.8 -where 6 = / 2b • a+b Therefore, the integration can be further s impl i f ied into RR 9 2E <|u RR 2n p F $ / - 2 ( R S R + ( y - y j R n ) d£ + / - ^ dA 0 (a-b) / a + b 0 / a + b RR 2anD A„ E + / R 2 dA , 0 / a + b (a - b) that i s , a i RR - 4 ( R 2 n D + ( y - y j R n ) E / s ^ ( l ) , 2 R d e d A = J [ o ° " ' ^ o ( a - b ) r m 2nD c (D.8) ( i i ) The integration of the th i rd term of (28) is s imi lar to that of the second term, therefore, from (D.8) , 3 1, y = 0 - 4 ( R 2 n R + ( y - y j R n l E ' s k I n ^ *2 R d 9 d* " / t " y=y ( a - b ) / a + b 2nr / a + b i a D ; * ( i i i ) For the fourth term, - A.9 - 83 /s t f e ^ - F ^ + 1 ' k ) ] ^ R d e d y r K 0 2it R 2 n R + R ( y - y p ) n y - RR pn Rcos 9 -h 0 (R^ + R^ + ( y - y D ) - 2RR cos e y ' c r r r -R/2RR + ikR (R + R 2 + ( y - y p ) 2 - 2RR pcos 9 ) 1 / Z 2 but on S . n = 0, n D = 1 and R = R D. Then, the above expression v y K K becomes 2 0 2TC R - RR p cos 9 / / [ - —* * * ~TT7 -h 0 (IT + IT + ( y - y n r - 2RRcos ^ p p p ' j - i k R + — 5 Q 5 "T79 ] <t>o d9 dy . (R + R 2 ( y - y p ) 2 - 2RR pcos e ) 1 / z 2 Using (D.3) and (D.4), the above expression becomes Zn ^ 1 R 2 + R 2 + ( y - y p ) 2 - 2RR pcos 9 - R 2 - ( y - y p ) 2 + R 2 -h 0 ? ( a - b cos 9 ) 3 / 2 i - i k R •jjZ ] *2 d e dy ( a - b cos 9) -h 0 * ( a - b cos 9 ) ' ^ ( a - b cps e r ' * i - 1 k R + ] * d e dy (a - b cos 9 ) ' ^ L - A.10 -o i t R 2 - R 2 - (y-y J 2 2 i k R = / / [ E _ __ ] * de dy -h 0 ( a - b cos er u ( a - b cos e ) 1 ^ c 0 (R 2 - R 2 - ( y - y J 2 ) E 4 i k R F • / t " 2 P , P ' ^ ] *2 dy . -h (a - b) / a + b / a + b c Therefore, r K 0 2 ( R 2 - R 2 - ( y - y J 2 ) E 4 1 k R F = / [ E B- _ _ _ _ _ ] . d y > ( D > 1 0 ) -h (a - b) / a + b / a + b * ( i v ) I n t e g r a t i n g the f i f t h term, RR 2* Rn R + ( y - y p ) n y - R p n R cose J- J |_ - - o 5 5 Q/o K c o ( R * + R „ + (y-yJ - 2RR cos e ) J / i 2 1 - — 5 * 5 T-75- ] <tv>R de dR. 9 (R + Rp + (y-y r - 2RR p cos e ) l / 2 1 On S^, n R i s equal to zero and n^ i s equal to one. A l s o , by using (D.3) and (D.4), the above expression can be r e w r i t t e n as: RR ii (y-y.) R 2 R / 2/ [ E " R ] , de dR 0 0 (a - b cos %r ,c g(a - b cos e) l c c - A.11 -R (y-yJ R E 2 R F / 4 ( . P M R F ) 4, dR . 0 / a + b ( a - b ) g / a + b That i s , / S f t Sir & " F H R d e d R RR (y-yJ R E 2 R F = Jo 4 ( - P M R F ) dR . ( D . l l ) K c / a + b ( a - b ) g / a + b (v) The f i n a l term to be considered i s the only term on the r i g h t hand si d e . L -J- (v • n) R de dA 5 b r y=0 2TC R U „ n = 1 1 _ de dA y=y t o (R + Rp + (y-y p) - 2RR pcos e ) 1 ^ y=0 TC . = J 2 R u 2n / ^ m dA y=yt * y 0 (a - b cos e) x'c y=0 4 R u ?n F = / L ? dA . (D.12) y=yt / a + b Therefore, (28) can be rewri t t e n as: 2n4> + (D.8) + (D.9) + (D.10) + ( D . l l ) = (D.12) , and becomes - A.12 -86 2 n* 2 + y=0 4(R n + (y-y )Rn ) E J [ H 3 y=y + ( a + b ) / a + b 2n / a + b R C T a ^ b T " M ] *2 dx ;R R 4 ( R 2 n R + ( y - y p ) R n y ) E 0 ( a - b ) / a + b 2n R ( - n r r r r - F ) ] * 9 d i y r ° r 4 k R F 2 ( r 2 - Rp - t y - y 2 E . a . J L- , - v v — J *? dy y=-h / a + b (a - b) / a + b + ;R R r. 4 R ( ^ P ) E 4a)2RF lc / a + b ( a - b ) g / a + b) ] *2 dR 4 R F 7 y=y t / a + b u 2 n y d i . (42) 87 - A.13 -APPENDIX E S i m p l i f i c a t i o n of Equation (45) i n t o (46) Equation (45) i s given in Chapter 111.3 as: 2 7 l < t >l + ^S 0 fn & *} c o s 9 R d e d i + kh In & < h ' c o s 0 R d 0 ( U + / S f [ In & " F ] *i c o s 9 R d e d R + 's t In " C Ik) 1 3 *J cos 9 R d e d y r R J s Ujcos 9 nR 1 R d9 dA . (45) ( i ) The second term in (45) w i l l be considered f i r s t . RR 2TI ' ' * IrT & *i c o s e R d9 d i 0 o RR 2TC (n DR + n ( y - y ) - n D R „ cose)*; R cos 9 * = / / - R 2 y 2 P - P ] 3 / 2 - d9 dA 0 0 (R + R„ + (y-Yn' - 2 R R n C 0 S G ) By using (D . 3 ) and ( D . 4 ) , the expression i s r e w r i t t e n as: R 2 R 2TC (n DR + n , ( y - y J R - nDRR cos e ) * i cos 9 / / - - i y _ _ £ P 3 / 2 — J d e d i 0 0 (a - b cos 9)°'* RR 2u ( n D R 2 + n ( y - y n)R) cos 9 • ' I I ' — - y P M 0 0 (a - b cos 9)' 88 - A.14 -2 bn R cos 6 + ^ ] de d i 2(a-b cos 6) 6/d 1 RR TC ( n D R 2 + n (y-y )R) ( a - b cose - a) = J 2 J [ - —5 * 2 0 0 b (a - b cos e r ' * (a - b cos e - a) cos e -> , . . - n„ — T ^ — J A. de d i R 2(a - b cos e ) 3 / 2 1 . j \ [ ^ - ') ( 0 0 D (a - b cos e ) 1 7 ^ a ) nR_ p i (a - b cos e - a) ( a - b cos e ) 3 / Z " 2 l F " ( a - b cos e ) 1 / Z , a (a - b cos e - a) -> -, .... + j A, de d i D (a - b cos e ) 3 / ^ 1 . ,\ [ - v y-vR ( — J — M 0 0 D (a - b cos e) W £ - ^ ) + R ( ( A . B C O S Q)V2 ( a - b cos er u 60 ) - X (- 1 (a - b cos e ) 1 / 2 " (a - b cos e ) 1 / 2 - - 0 7 0 ) ] <th de dA . ( a - b cos er /d 1 From p. 156 of Gradshteyn and Ryzhik [1965], one has / / a - b cos x dx = a / a + b E {j, 6 ) , ( E . l ) where - A.15 - 89 6 = / _2b_ • a+b With the help of (D.6), (D.7) and ( E . l ) , the above integral expression can be rewritten as: ;R R 2 { (n RR 2 + n y ( y - y p ) R) ^ 2 F _ 2 a E 0 b / a + b / a + b (a - b) + ^ ( 2 / T T T E - 2 a F ) / a + b n„a f y ; / a + b / a + b (a - b) R r __1L_ 2 a E i i i J o RR / I 4 ( " / + n ( y - y j R ) (F - a E 0 b / a + b R y p ( a - b ) 2 + | n D ( / a + b E - 2 a F + a E )} d A. (E.2) D K / a + b / a + b (a - b) 1 ( i i ) The th i rd term in (45) is exactly the same as the second term except the l i m i t is d i f f e rent , therefore, J s (7) cos e R de d£ b I 0 [ — 4 = ( » / - n ( y - j , )R) (F - ) y=y t b / a + b ^ K v ' - A.16 -2 9 L ) ] 0>; d i . ( E / a + b ( a - b ) 1 ( i i i ) The fourth term i s s i m p l i f i e d as the f o l l o w i n g way: 5 A ^ u ) 2 >Sf I ?» ^ " ?F 1 *i c° s 9R d e d R R R TC ( n R R + n y ( y - y p ) - n R R P cose) R cose = / 2 / [ - - — 7 j 5 : • l / o o o (R + + (y-y J - 2RR cos s y ' c r r r 2 QJ R cos e -j ^ <j0 dR 9 ( R 2 + R 2 ( y - y n ) 2 - 2RR cos e ) 1 / 2 ^ but f o r Sf, n R = 0, n = 1. Thus, the expression becomes R R TC (y-^p^ R c o s 9 J 2 / [ - ——.5 x o T77 o o ( R + R P + ( y - y p ) - 2RR p cos e ) l / < : 2 a) R cos e • K i u b o -j i de dR 9 (R 2 + R 2 + ( y - y n ) 2 - 2RR cos e ) 1 / 2 ^ p -p- p By (D . 3 ) , (D.4) the above expression i s given as: RR TC ( y - y p ) R ( a - b cos e - a) J 2 / [ 0 7 5 0 0 b (a - b cos e ) ' 2 R(a - b cos e - a) + £ L ] d e d i 9 b(a - b cos e) ]/d 1 RR TC (y-y )R 1 = / 2 / [ rf-— ( fpr - o / o 0 0 D (a - b cos e) (a - b cos e r ^ a - A.17 -2 + ((a - b cos 9 ) 1 / 2 ^ )] <jH de dA . 9 b (a - b cos e ) 1 / 2 ^ With the help of (D.6), (D.7) and ( E . l ) , the expression can be re w r i t t e n as: RR ( y - y j R or ? a F / 2 [ P — ( 2 F 2 3 E 0 b / a + b / a + b (a - b) 2 + |_R ( 2 / T + T E - ) ] ^ dA 0 D 9 P 9 / a + b (y-y ) a E P ] *; dA . (E.4) / a + b ( a - b ) 1 ( i v ) For the f i f t h term of (45), 's C In ^ ' [ 2 T " + 1 k > 7 ] *i cos 9 R ^ dy x* R 0 2TC ( n R R + n y ( y - y p ) - n R R P cos e)R cos e -h o (R 2 + R p + (y-y p ) 2 - 2RR p cos e ) 3 / 2 2 R R ( R * + R „ + (y-yJ - 2RR cos e ) 1 ^ 1 With (D.3) and (D.4), n = 0 and n = 1 on S . the above expression y K r can be writte n as: - A.18 - 92 o TC ( R - R R cos e) cos e / 2 / [ - - — * -h 0 ( a - b cos e) 172 I "I -I 1 R COS e "1 | A rt A ' 2R7 1 k ) ~ — 1 / 2 J <h d e d y ( a - b cos e) 0 *r ( r 2 + R p + ( y - y p ) 2 - 2 R R P c o s e + R 2 - Rp - ( y - y p ) 2 ) c o s e / 2/ [ -h 0 2(a - b cos 0) 17? r -1 , . . •> R ( a - b cos e - a) n , . „ . ^ K R D (a - b cos e ) 1 7 ^ 1 0 it 2 r o (% r ( a - b cos 9 + 2R - a) cos e J 2 J L - ; 7372 -h 0 2(a - b cos B y " + ( ijT + i k ) £ ( ( a " b c o s e ) 1/2 ( a - b cos e ) y/2 ) ] ^ de dy r 9 f 1 1 r (a - b cos e - a) . (2R - a ) f a - b cos 9 - a) -h 0 2b(a - b cos e ) l / d 2b(a - b cos e ) J / + ( ^ - + 1k) £ ((a - b cos e) 1/2 ( a - b cos e ) T 7 2 ) ] * i d e d^ On S_, R = R R, and the above expression can be decomposed i n t o - A.19 J°2 j% [ + L (a - b cos e ) 1 / 2 5 -h 0 ^D 2b(a - b cos e) ' + (2R 2 - a) _ (2R 2 - a) a 2b(a - b cos e ) 1 / 2 2b(a - b cos e ) 3 / 2 + (- Y + 1kR) ((a - b cos e ) 1 / 2 95 ( a - b cos 9) 172 ) ] 4>J d9 dy - / ° 2 /" [ J ^ ^ - l ( 3 _ _ ^ ) -h 0 ^D ( a - b cos QVU (a - b cos % ) i U + i-M ((a - b cos e ) 1 / 2 a ^ )] *; de dy D ( a - b cos e ) s u 1 Using (D.6), (D.7) and ( E . l ) , the above expression becomes j° [ (2R 2 - a) 2F 2aE -h u / a + b /a + b (a - b) + ^ (2 / a + b E - 2 a F )] * i de dy D / a + b 1 j° r_ 2(2R 2 - a) ( p _ _ _ J E _ ] + _4ikR_ ( / - F T T . -h b / a + b / a - b aF / a + b ) ] <t>} dy . (E.5) A.20 (v) The only term on the r i g h t hand side of (45) i s y=0 2rt R cos 0 / / u,n R — * j-{ de dA . y=y t 0 (R + Rp + ( y - y p r - 2RR p cos e ) " * By using (D.3) and (D.4), the above expression becomes y=0 TX - u,n DR ( a - b cos e - a) / 2 / [ — L B ] de d A y=y t 0 b(a - b cos e) ' y=0 TI fTf ipR = / 2 / - — L [ / a - b cos e 3 ] de dA y=y t 0 0 (a - b cos e ) 1 ^ With (D.6) and ( E . l ) , the above equation can be rewr i t t e n as: y=0 2u.n_R , _ / . _ I L . [ 2 / T H b E - 2 a F ] dA y=y + / a + b y=0 4u.n R / ^ L [ / a + b E - a h 1 dA . (E.6) y=y t / a + b Therefore, equation (45) i s eq u i v a l e n t to the f o l l o w i n g equation: 2 ™ * i < P > + 7 t (" r R 2 + " y ( y - y P m » F Ta?ET E ' + -I n D (/ a + b E - 2 9 F + 3 E )] Ai dA D K / a + b / a + b ) ( a - b ) 1 9 4 - A.21 - 9S + / [ 4 ( n D R 2 + n ( y - y n ) R ) ( F a 0 b / a + b 'IT ' ' V J V n a - b E ) + | n R (/ a + b E ^ — F + a / a + b / a + b ( a - b ) + / F R t f - y a + b E + ( y- y F a 7 r ) - = = 9 / T T T (y - y n ) aE e ] d R / a + b (a - b) + /°[212BL^|1(F..^E) y=-h b / a + b u D ) + ( y-g- r-b E . — J — F ) ] <t>l dy D / a + b 1 y=0 4u.n RR J b y =y+ [ a F - E / a + b ] d i . / a + b (46) - A.22 -APPENDIX F S i m p l i f i c a t i o n of the Right Hand Side of (50) The s i m p l i f i c a t i o n of the l e f t hand side of (50) i s e x a c t l y s i m i l a r to (E.2), (E.3), (E.4) and (E.5) in Appendix E. The r i g h t hand si d e of (50) i s given as: J s (- V + nyR) Y cos e R 1 de di y=0 2TC m cos e y=y t 0 y (R + Rp + (y-y p) - 2RR pcos e)1^ y=0 / 2(-n Ry + n R) YR y=y t TX r cos e .„ . x j — 5 9 5 j-pr de di . o (R + R^ + (y - y r - 2RR COS e)1^ r r r By (D.3) and (D.4), the above expression becomes y = 0 . 7 1 r n c ft / 2(- n Ry + n R) YR / iy» de d i ' R -*- " V K J Y K J ; 7T72y=y t K y 0 (a - b cos e) 1 y7° o r ^ „i yR r 7 1 (a - b cos e - a) , J " 2(- npy + n R) \ j - de di y=y. ^ y . 0 (a - b cos e) 1 y=0 « R TT / - 2(- n R y + n i ) \ j [/ a - b cos e y=y t y 0 5 m ] de di ( a - b cos e)1^ A.23 - ^ F i n a l l y , w i th ( E . l ) and (D.6), the i n t e g r a l has the form y=0 / - 4(- n y + n R ) l£ [ /~a~+~¥ E - a F ] d i . ( F . l ) y=y* R y b / T + T - A.24 -APPENDIX G The S e r i e s Representation of 4^ n From (62), 4>j i s given as: i M cosh k (y + h) ( k R ) C Q s mQ { 6 2 ) co cosh kh r. rm m ' m=Q and with >I,n = V * i ' " = ( W' ay» R 5 0 H ' ( nR» ny» n 9 ) ' (--D However, f o r axisymmetric body, n i s always equal to zero. Therefore, • i . n = •i.R n R + • i . y n y ' ( G' 2 ) By taking the d e r i v a t i v e of 4>j with respect to R and y, one can get = _ i M cos;1 k(y+h) * ,( , I,R 00 cosh kh m_Q pm m and _ . i _ k sinh k(y+h) " J ( k R ) c o s m 0 . I,y OJ cosh kh __Q Km m - A.25 - 99 Therefore, and • = _ i M k z B cos me [J'(kR)n Dcosh k(y+h) I,n (j cosh kh n rm L m R m=0 + J m ( k R ) n y sinh k(y + h) ] , (G.3) D,n to cosh kh m_Q L m R + J m ( k R ) n y s i n h k(y + h) ] . (63) - A.26 -APPENDIX H S i m p l i f i c a t i o n of E x c i t i n g Force and Moment /oo ( i ) From (71) one has y=y t 0 M=0 i <t>D_ w ] cos me n yR de dA } e 1 w t . (71) However, only the term 'cos me' in (71) i s a function of e. A l l other v a r i a b l e s i n s i d e the i n t e g r a l are functions of R and y only. Therefore, by taking i n t e g r a t i o n with respect to e, one can get F2 = t P [" Ag C ° S h C 0 ^ y k ; h ) P0 J 0 ( k R ) - i *QQW]2nn R dA y=y+ J + y / ° s P [- Ag c o s h k l \ t h ) p J (kR) v_ v m = i c o s n k n m mv •'t Hn-W ] s i n me Dm" J m 2 U n R dA } e 1 u ) t . (H.l) 0 y I t i s obvious from (H.l) that f o r m greater than zero, the i n t e g r a l i s equal to zero. Therefore, (71) can be s i m p l i f i e d as: F' = {V - 2* PRn y[Ag ^ ^ l ^ h ) P 0 J 0 ( k R ) + ^DcH { 7 2 ) y=y t - A.27 ( i i ) From (75), the surge e x c i t i n g f o r c e i s given as: j y = f ° r 2 % " r fln cosh k(y + h) a , 0, F 1 = * J J n s P L- Ag c o s h kh pm Jm { k R ) y=y t 0 m=0 2u . t - i * D m to ] n RR / cos me cos e de d i } e w y=y t m=0 2n . t i * D m u ] n RR / cos me cos e de d i } e w However, 2n / cos me cos e de = n m = 1 0 = 0 m * 1 Therefore, Fj - ( T - * R p [ Ag *> M l ( k R ) y=y H t + i * D 1 to ] d i } e 1 u t ( i i i ) From (80), the e x c i t i n g moment Fg i s A.28 102 i<t>D_w ](n R - n ^ j c o s me cos e Rde dA } e 1 w t . (80) f 2 Jo - R(v - v> ^ c o v i \ : h ) iw.<«) y y^. 2TC i<l>D_ w ] / cos me cos e de dA } e By using (H.2), the above equation becomes y y t + i< t . D 1 o) ] dA } e 1 w t . (81) - R.l -REFERENCES BAI, K.J., A V a r i a t i o n a l Method i n P o t e n t i a l Flows with a Free Surface, U n i v e r s i t y of C a l i f o r n i a , College of Engineering, Rep. NA 72-2 (Sept. 1972), vi + 137 pp. BREBBIA, C.A., The Boundary Element Method f o r Engineering, Pentech Press, 1978, pp. 46-72. FRANK, W. " O s c i l l a t i o n of C y l i n d e r s i n or Below the Free Surface of Deep F l u i d s , " Naval Ship Research and Development Center, Rep. 2375 (1967), vi + 40 pp. FRANK, W., "The Heave Damping C o e f f i c i e n t s of Bulbous C y l i n d e r s , P a r t i a l l y Immersed i n Deep Water," Journal of Ship Research, V o l . 11, (1967), pp. 151-153. GARRISON, C.J., Hydrodynamic Loading o f Large Offshore S t r u c t u r e : Three Dimensional Source D i s t r i b u t i o n Methods, John Wiley & Sons, 1974, pp. 130-131. GRADSHTEYN, R., Table of I n t e g r a l s , S e r i e s , and Products, Academic Press, 1983, pp. 153-156. JOHN, F., "On the Motion of F l o a t i n g Bodies. I I . Simple Harmonic Motions," Comm. Pure Appl. Math., V o l . 3, (1950), pp. 45-101. KEULEGAN, G.H. and CARPENTER, L.H., "Forces on C y l i n d e r s and Plates i n an O s c i l l a t i n g F l u i d , " Journal of Research of the National Bureau o f Standard, 1958. KIM, W.D., " O s c i l l a t i o n s of a R i g i d Body in a Free Surface," Journal o f F l u i d Mechanics, V o l . 21 , (1965), pp. 427-451. KRITIS, I r . B., "Heaving Motions of Axisymmetric Bodies," Journal of Ship Research, 1979, pp. 26-27. MacCAMY, R.C., "On the Heaving Motion of Cy l i n d e r s of Shallow D r a f t , " Journal of Ship Research, V o l . 5, No. 3, (1961), pp. 34-43. NEWMAN, J.N., Marine Hydrodynamics, The MIT Press, 1980. References (continued) PORTER, W.R., "Pressure D i s t r i b u t i o n s , Added Mass, and Damping C o e f f i c i e n t s f o r C y l i n d e r s O s c i l l a t i n g in a Free Surface," U n i v e r s i t y of Cali f o r m ' a , Berkeley, I n s t i t u t e of Engineering Research Rep. 8216 ( J u l y 1960), x + 181 pp. SABUNCU, T. and CALISAL, S.M., "Hydrodynamic C o e f f i c i e n t s f o r V e r t i c a l C i r c u l a r C y l i n d e r s at F i n i t e Depth," Ocean Engineering, V o l . 8, 1981, pp. 25-63. TASAI, F., "Damping Force and Added Mass of Ships Heaving and P i t c h i n g , " Rep. Research I n s t i t u t e of Applied Mechanics, Kyushu U n i v e r s i t y , V o l . 7 (1959), pp. 131-152. URSELL, F., "On the Heaving Motion of a C i r c u l a r C y l i n d e r in the Free Surface of a F l u i d , " Quart. J . Mech. Appl. Math., V o l . 2 (1949), pp. 218-231. URSELL, F., "On the R o l l i n g Motion of C y l i n d e r s i n the Surface of a F l u i d , " Quart. J . Mech. Appl. Math., V o l . 2, (1949), pp. 335-353. WEHAUSAN, J.V., "The Motion of F l o a t i n g Bodies," Annual Review of F l u i d Mechanics, V o l . 3, 1971, pp. 237-267. YEUNG, R.W., A S i n g u l a r i t y -- D i s t r i b u t i o n Method f o r Free Surface Flow Problems with an O s c i l l a t i n g Body, U n i v e r s i t y of C a l i f o r n i a , Berkeley, 1973, pp. 9-18.
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Hydrodynamic coefficients for axisymmetric bodies at finite depth Chan, Johnson L. K. 1984
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Title | Hydrodynamic coefficients for axisymmetric bodies at finite depth |
Creator |
Chan, Johnson L. K. |
Publisher | University of British Columbia |
Date Issued | 1984 |
Description | A numerical procedure for the calculation of velocity potential, for axisymmetric shapes is studied. The procedure incorporates linearized free surface and radiation conditions. Forced heave, surge, pitch induced velocity potentials and wave diffraction potentials are considered as specific applications. Details of the analytical steps necessary to reduce the three dimensional integral equations to two dimensional equivalent equations are given in the appendix. The hydrodynamic coefficients and existing forces for four well studied examples are presented and compared to published results by other numerical methods. The method studied here offers considerable savings in computer time, as the input requirements and the computational efforts are reduced by intermediate analytical steps. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-05-17 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0228489 |
URI | http://hdl.handle.net/2429/24808 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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