356 



HYDRODYNAMICS IN SHIP DESIGN 



Sec. 60.3 



changes without recourse to additional lengthy 

 calculations or model tests, data have been 

 analyzed from the tests of some two dozen models 

 of ships of various types. The aim of this analysis 

 is to determine the exponent n in the relationship 



Pe for (A ± gA) 



Pe for A 

 or Pe for (A ± 



SA 



F± b¥ 



¥ 



sA) 



f„..A,(^)" 



(60. i) 



It may be expected that these relationships will 

 vary somewhat with speed-length quotient; pos- 

 sibly also with hydrodynamic ship proportions 

 such as prismatic coefficient and fatness ratio. 



Fig. 60. A gives a tentative mean line for a 

 variation of n in Eq. (60.i) with T„ or F„ , as well 

 as a lane in which the majority of n values may 

 be expected to lie. They conform reasonably 

 well for SA, bW, or 5¥ values ranging from +0.18 

 to —0.40, and for large-trim as well as zero-trim 

 changes accompanying the changes in displace- 

 ment. The lane appears to be as valid for large 

 gA or bW percentages as for small ones. 



In general the exponent n is greater for a 

 -t-5A or -\-bW than for a - SA or - bW, especially 

 when r, > 1.0, F„ > 0.30. 



The plotting of Fig. 60. A is based only indirectly 

 on physical reasoning. The hulls which are less 

 deeply immersed, more deeply immersed, or 

 inclined with trim by the bow or stern, can only 

 in exceptional cases be geosims of the designed 

 underwater hull. Further, because of the different 

 proportions in each case, the surface waterline 

 changes, the wetted surface varies, and the flow 

 pattern is different. 



It is to be noted that, because of the arithmetic 

 of the situation, the exponent n is extremely 

 sensitive to changes in the power ratio. For 

 example, for a displacement ratio (A — 5A)/A 

 of 0.9, and a power ratio [{Pe for 0.9A)/(P£ 

 for A)] of 0.9, the exponent n = 1.0. For the same 

 displacement ratio and a power ratio of 1.0, the 

 exponent n = 0, Avhereas for a power ratio of 

 0.81 the value of n = 2, since 1.0 = (0.9)° and 

 0.81 = (0.9)". Thus, while the possible selections 

 of n for a given T, from Fig. 60. A vary rather 

 widely, the range of estimated Pe derived from 

 them is rather small. 



To avoid multiplying large numbers by factors 

 very close to unity, which is the case when sA is 

 small, and to avoid taking powers, the following 

 formula can be used: 



Pe = A-A" 



cIPe = /,:nA""' f/A or bPe = hiA"~' sA 



whence 

 bP, 



Pe 



n(5A) 



and bPs = Pf 



"n(5A)"| 



(60. ia) 



For small percentage changes in the displacement 

 A, Eq. (60. ia) is more accurate than Eq. (60. i) 

 and easier to evaluate. 



Take for example the ABC ship of Part 4, 

 at a T^ of 0.9. Assume that the effective power 

 Pe for the designed displacement and trim is 

 11,902 horses, equal to the estimated 10,820 

 horses of Sec. 66.9 plus 10 per cent for appendages 

 and other factors. Assume also that n is selected 

 from the mean line of Fig. 60.A as 0.72. Then for 

 a partial-load displacement of 16,400 — 2,425 = 

 13,975 t, as listed in Table 66.f of Sec. 66.16, 



■5 0.8 



0.4 0.6 0.8 1.0 1.2 \A 1.6 1.6 2.0 



Fig. 60.A Graphs for Predicting Change in Effective Power Due to a 10 Per Cent Change in Displacement 



