253 



Go(C,n,C;x,y,z) 



v^(C-x)2 + (n-Y)2 + (t-z)^ 



/{5-x)2 + (n+y)2 + (C-z)2 



Numerical Procedure 



The purpose of this section is to describe the 

 numerical procedure for the method explained in 

 the previous section. Here, for convenience' sake, 

 let us denote ipi^' in (62) by ijj* . 



dk 



k exp (k ri+y + ikp' ; 

 k - Kosec 9 



(63) 



where p' 



= (?-x) cos e + (C-z) 



(64) 



We can get Wo by using Eq. (46) as follows. The 

 first step is to rewrite the integrated term in the 

 right side of Eq. (46) by the transformation 



3 



an 



'i' 



3 

 3n 



Go 



(65) 



Then, using the rewritten expression, we can obtain 

 Wo as follows : 



Wo (x,y,z) 



W*(x,y,z) 



jvf2t| 



v=0 



(66) 



It should be understood from the above expla- 

 nation that Lx ' must be given to calculate Wq . In 

 order to obtain L)^ ' precisely, we must consider the 

 boundary conditions on the propeller surface which 

 have been disregarded in the discussion up to this 

 step. To do so, however, requires complicated 

 calculations as seen in the conventional methods 

 for the problems of the hull-propeller interaction. 

 The complexity of the calculations have caused the 

 conventional methods to be impractical as described 

 in the Section 1 . Hence , the author introduces the 

 following approximation. The steady change of 

 pressure on a hull surface which we are now examining 

 corresponds to a pressure component of thrust de- 

 duction. We can consider that obtaining the thrust 

 deduction is the same as obtaining pressure on the 

 hull surface as a percentage of the mean propeller 

 thrust, Tq . Hence, the relationship between un- 

 known Lq ' and known Tp can be given as follows: 



-g^i 



ds[L;(5' ,j,eo)] 



(67) 



where [ ] denotes the component in x direction. 

 Now, the L, ' can be considered as the jump of the 

 pressure change across the propeller surface due 

 to the interaction of the hull and propeller, and 

 consequently, Eq. (67) may be considered as the 

 approximate boundary condition on the propeller 

 surface for ijjCe) (Q ) . By giving an arbitrary 

 function, L^ ' , which satisfies the auxiliary Eq. (67) 

 and calculating Wq by (46) , (65) , and (66) , we can 

 solve the integral equation, (62). This is the 

 approximate calculation method proposed in this 

 paper . 



Numerical Calculation 



The integral equation, (62) is an integral equa- 

 tion of Fredholm type of the 2nd kind. Generally, 

 it is impossible to obtain analytic solutions of 

 the integral equation for S in an arbitrary form. 

 Thus, various approximation methods have been 

 suggested. In this paper, a definite integral is 

 approximated by a finite sum, the equation is con- 

 verted to a linear equation, and this equation is 

 solved numerically. 



At first, the following approximations are used: 

 (i) A hull in an arbitrary form is replaced by a 

 polyhedron. The form of each surface named 

 "element" is a plane quadrilateral, 

 (ii) On each element, the unknown function i>* (.Q ) 



is assumed to be constant. 

 Using this approximation, the continuous function 

 ij^* (Q ) is replaced by the discrete quantities, \p* 



(i=l,2, , M) , for the total number, M, of the 



elements. A control point, Q , where Wo (Q ) must 

 be calculated, is selected for each element. Thus, 

 we have the following transformation: 



r a . " * r ^^0 



dS lp*(Q)3^- G (Q;0 )^ I i>. dS' ^— r(Q';Q ), (6£ 

 J ^ an^ ^ -o '. ij Q an^ 



■th , 

 S 1 element 



where dsA, nA, and Q' denote values on the elements. 

 The definite integral in the right side of this 

 equation is an influence function from point Q to 

 point Qo and we denote this function by Aq^,Q. 

 On calculating Plq^,Q, the existence of a singular 

 point, a so called doublet, becomes a problem. How- 

 ever, there are many numerical calculation methods 

 for this case. In this paper, the Hess-Smith method 

 is used [Hess and Smith (1967)]. Further, selection 

 of a control point is also a problem. However, for 

 this problem various methods have also been suggested 

 in the analysis of potential flow field. In this 

 paper, each element is selected to be similar to a 

 rectangle, and the point of intersection of its 

 diagonal lines is employed as the control point. 

 Finally, the hull surface after St. 1^/2 is taken 

 into consideration, and it is divided more nar- 

 rowly near stern in the longitudinal direction and 

 approximately equally in the depth direction. 



Thus, each element, Ai , i'(i,i'=l,2, ,M) , 



which corresponds to Aqq,Q can be calculated and 

 Wo(Qo)' "^^n be calculated for each control point. 

 Then, the integral equation of unknown function, 

 ip* (Q ) , is converted to a linear equation of un- 

 known, iIJ-l*- 



Now, in the calculation of Wo (Q ), the author 

 uses the approximation that the number of propeller 

 blades is infinite. Then, in correspondence with 



(46), (47), (65), and (t6), we can get the following 



relations : 



