254 



W(,(Q_) = 



Go (r,e;x,y,z) = Go(Xp, r cos6 -f, sinB; x,y,z), (70) 



h 3 



3x 



H 



2tt 



(71) 



T ship 

 FIGURE 2. Body plan. 



L ship 



where H represents a mean pitch of the propeller 

 blade, and r and r^ represent respectively radius 

 of the boss and radius of the propeller. Moreover, 

 r(r,9) represents the thrust per unit length in the 

 radial direction of the propeller blade elements 

 and can be developed as follows : 



approximation used in the calculation of Wo, the 

 procedure as follows is performed. First, perfor- 

 mance of the propeller in the nominal wake is calcu- 

 lated to obtain T-^ and L)^'. Next, by using the 



five combinations of distribution forms of Ty 



H 



and the number of the propeller blades as follows: 



^'^'9' = xio ^'^>^ 



-jxe 



(72) 



We can also get the following equation in correspon- 

 dence with the Eq. (57) : 



dr Nrn(r) 



(73) 



(a) N; finite, using Lo',Li', , L7 ' in (66) 



(b) N; infinite, using ro,ri, , r7 in (59) 



(c) N; finite, using Lq ' only in (66) 



(d) N; infinite, using Fq only in (69) 



(e) N; infinite, using To : elliptic in (69) , 



The Wo are calculated. Then, by substituting these 

 Wo in (62), the pressure changes, ijj*, are calculated 

 and indicated in a non-dimensional form in Figure 3. 

 As shown in Figure 3, the ijj* barely differ due to 

 the distribution form of r,L' , and the number of 

 propeller blades. Hence, the approximation of the 

 elliptic distribution is reasonable. 



Further, for the calculation of Wo in Eq. (69) , it 

 is approximated that To is an elliptic distribution 

 against r, and Fi, F2 ....are disregarded. 



Examples 



The numerical calculations are performed in the 

 case of two combinations of the hull and propeller 

 shown in Table 1. Figure 2 shows the body plan of 

 hulls. In order to examine the correctness of the 



TABLE 1 Particulars and Operating Condition 



SHIP L L /B B/T C„ D 

 PP pp B 



Z U F n., 

 n M 



Experiment 



The experiment was performed at the towing tank of 

 IHI by applying a standard hul,l surface pressure 

 measurement [Namimatsu, (1976)]. For the ships 

 indicated in Table 1, pressures on the hull surface 

 are measured under both the towed and the self- 

 propulsion condition. Differences of the measured 

 pressure between the towed and the self-propulsion 

 condition are used for the experimental values of 

 the pressure change caused by the propeller. 



Figure 4 shows the comparison of the experimental 

 values to the calculated values, which are obtained 

 by approximating Fo as the elliptic distribution. 

 In addition. Table 2 shows the pressure component, 

 tp, of the thrust deduction fraction, t, which is 

 the sum of the pressure change. The comparison in- 

 dicates better agreement for the L ship (a thinner 

 ship) . 



L 6.00 6.50 2.86 .572 .215 5 2.05 .267 9.55 

 T 7.00 6.00 2.63 .829 .210 5 1.27 .153 8.52 



Lpp = Length between perpendiculars (meter) , B = 



Breadth, 

 T = Draft at mid-section, Cg = Block coefficient, 

 D = Propeller diameter (meter) , Z = Number of 



propeller blades, 

 U = Ship speed (meter/second) , Fj, = Froude number. 



"M 



Propeller's number of revolution per second. 



Discussion 



The calculation method in this paper is derived by 

 expressing the equations and boundary conditions 

 (which determine the change of the flow field due 

 to the interaction of the hull and propeller) in 

 the form of an acceleration potential. For this 

 reason, this method nominally requires calculations 

 of pressures induced by the hull and propeller, 

 while the conventional methods, which express flow 

 fields in the form of a velocity potential, require 



