259 



E(Ho - H ) 

 sp 



P^g 



^^ <«0 - %'' ■ (89) 



Using (83) , the following equation is obtained: 



P^g 



dS(Ho 



H ) 

 sp 



AR 



(90) 



where R denotes a wave making resistance under a 

 self -propulsion condition. This R might be 

 approximately equal to a wave making resistance 

 under the towed condition. Furthermore, AR can 

 also be given by the total resistance under the 

 towed condition. Hence, it can be considered that, 

 under the self-propulsion condition, the following 

 equation is given: 



(92) is the condition for minimizing the kinetic 

 energy left in the wake by recovering the kinetic 

 energy of the viscous wake with the propeller. 



The optimum condition for this energy recovery 

 is obtained under the assumption that the constant 

 C of Eq. (91) is given as the constant decided by 

 the towed condition. In other words, it is con- 

 sidered that condition (92) gives only the condition 

 for the propeller to accelerate flow effectively 

 under the assumption. If, however, the wave making 

 resistance is zero under a purely self-propulsion 

 condition, then (ArHO) C can be expressed as CHO 

 regardless of the towed condition. Therefore, it 

 can be considered that this fact indicates condition 

 (92) applies not only to the optimization of the 

 flow acceleration by the propeller but also to the 

 optimization of the hull-propeller combination for 

 effective recovery of the wake energy. 



The author proceeds to examine the correctness 

 of this condition in the following sections by 

 using results of the self-propulsion tests and 

 wake survey measurements . 



P^g 



dS(Ho 



H ) = 

 sp 



C, 



(91) 



Experiment 



where C is constant and can be decided by the towed 

 condition. Thus, the problem of minimization of E 

 is converted to the problem of variations for 

 minimization of E given by (89) under the constraint 

 condition (91) . It is obvious that the following 

 solution exists for the problem of variations: 



Ho 



H 



sp 



constant. 



(92) 



Furthermore, although it is omitted here, at least 

 the conditions that the ship speed and displacement 

 are constant are implicitly required in addition 

 to this constraint condition. 



Let us consider the meaning of Eq. (92) . Since 

 Hg - Hg and Kg - Hgn are proportional to the viscous 

 wake in a position far from the hull as indicated in 



and (Hq - Hgp)'^ are propor- 



the Appendix, (Hq - Hg 



tional to the kinetic energy of the viscous wake . 

 Hence, the minimization of E corresponds to the 

 minimization of the kinetic energy of the viscous 

 wake. And, it can be considered that the condition 



Total head at a wake far from the hull was measured 

 at the towing tank of IHI . The measurements were 

 performed for the ships and operating conditions 

 indicated in the Table 1 under both the towed and 

 the self-propulsion conditions. The measurement 

 cross-sections which correspond to plane S;^ were 

 three vertical cross-sections of 0.3Lpp, O.SLpp, 

 and O.VLpp behind A. P. Figure 6 shows the total 

 head loss distribution of the towed condition in 

 the non-dimensional forms and also shows H * which 

 is the change of total head loss by the propeller 

 action. Here, H * is obtained as follows: 



H* 

 P 



= (Ho - H 



sp 



(Ho 



H ) 

 s 



(93) 



We observe that in the towed condition the wake of 

 the T ship spreads to the relatively lower region 

 of the flow field. Further, we can see that the 

 peak of the total head distribution in the towed 

 condition agrees well with the peak of the change 

 distribution for the T ship, but not for the L 

 ship. In addition. Table 3 shows results of the 



TABLE 3 Self-propulsion and Towed Test Data and 

 Wake Survey 



SHIP F 



n 



w 



H 



R 



R A„ F 

 w R E 



sp 



L .267 .166 .271 1.14 5.57 .408 2.11 5.42 1.50 

 T .153 .201 .495 1.58 3.92 .130 1.60 3.87 1.39 



Rw 



AR 

 fs 



fsp 



= Effective wake, R,- = Total resistance from towing 

 test (kg.) , 



= Wave resistance from wave analysis at towed condi- 

 tion (kg. ) , 



= Skin friction correction (kg.), 



= pfg / dS(Ho-Hs) at 0.7 Lpp behind ship in towed 

 condition (kg.), 



= Pfg / ds(Ho-H3p) at 0.7 Lpp behind ship in self- 

 propulsion condition (kg.). 



