406 



HYDRODYNAMICS IN SHIP DESIGN 



Sec. 61.11 



speed T',, is practically equal to the intermediate 

 speed V I . In the ranges of critical-speed ratio 

 and square-draft to water-depth ratio between 

 those mentioned the sum of the speed reductions 

 due to both causes can not exceed 0.01 for the 

 depth to be determined. This is on the basis that 

 for small values of these differences, not exceeding 

 0.01, they may be determined accurately either 

 by addition of the differences or by multiphcation 

 of the corresponding speed ratios. For example, 

 0.99(0.99) = 0.9801 while {1.0 - [(1.00 - 0.99) + 

 (1.00 - 0.99)11 = 0.9800. 



Fig. 61. K is a diagram for use in calculating the 

 required limiting depth. Unfortunately, until 

 further developed, it involves a trial-and-error 

 procedure. At and beyond the left end of the 

 diagram the speed reduction of 0.01 is assumed 

 to be due entirely to augmented potential flow 

 while at and beyond the right end it is due entirely 

 to reduced wave speed in the shallow water. 

 Between the ends both effects occur, and they 

 are additive, as for EA + AH = EH. Here EA 

 is the speed reduction due to wave speed, cor- 

 responding to the right-hand scale, while AH is 

 that due to potential flow, corresponding to the 

 left-hand scale. The values of each are taken 

 from the theoretical and experimental curves of 

 Figs. 61.E and 61. G for a critical-speed ratio of 

 0.602 (top scale) and a square-draft to water- 

 depth ratio of 0.323 (bottom scale). 



The method of using the diagram is explained 

 in the examples which follow. The nomograms 

 of Figs. 61. D and 61. F may be entered for ready 

 determination of the values y/Ax/h and V^/ 'vgh, 

 or these values may be calculated, as in Examples 

 61. IV, 61.V, and 61.VI. 



To take care of the situation on low-speed and 

 high-speed ships, where the total resistance may 

 vary at less or more than the square of the speed, 

 additional diagrams of this type may be con- 

 structed from the data on Figs. 61. E and 61. G, 

 for overall speed ratios correspondingly greater 

 or less than 0.99. 



Exainple 61. IV. For the sake of simplicity, since only 

 the deep-water speed and the square-draft enter into the 

 problem, it is assumed that the vessel selected for this 

 example has a maximum-section area of 1,600 ft', with a 

 square draft of 40.00 ft. It is desired to find the limiting 

 depth of water, at sea level, in which the resistance does 

 not exceed 1.02 times the deep-water resistance at speeds 

 of 10, 20, and 30 let. These speeds are equivalent to 16.89, 

 33.78, and 50.67 ft per sec, respectively; g = 32.174 ft 

 per sec2, ^/g = 5.672. 



For the lowest speed of 10 kt assume first that the 



potential-flow effect limits the depth of water. At the 

 extreme left of Fig. 61.K, where Vh/Vt = 0.990, the 

 square-draft to water-depth ratio s/Ax/h is 0.393, 

 whence the Hmiting depth h is 40.00/0.393 = 101.8 ft. 

 Assuming on the other hand that all the speed reduction is 

 due to wave effect, for an intermediate-speed ratio Vi/Va, 

 of 0.990 the critical-speed ratio Vo,/Vgh is 0.658. Then 



F„ , . /r T„ 



= 0.658, whence \//i = 



0.658 Vfif 



h = 



(16.889)' 



(O.QdSyg (0.433)(32.174) 

 = 20.5 ft. 



For any square-draft to water-depth ratio less than the 

 lowest value 0.393 at the extreme left of the diagram of 

 Fig. 61. K, the limiting depth is greater than 101.8 ft, 

 whereas the common value for h is somewhere between 

 that value and 20.5 ft. The critical-speed ratio Va:,/\/gh 

 is therefore smaller than 0.658. In fact, it may be smaller 

 than 0.400, at the left end of the diagram. For a value of 

 Va,/Vgh = 0.4, 



h = 



VI 



285.27 



(O.-ifg (0.16)(32.174) 



= 55.4 ft. 



The fact that it is not possible, within the limits of the 

 diagram, to achieve a common value for h indicates that 

 the potential-flow effect is the determining one while the 

 wave-speed effect is zero. The required depth is therefore 

 101.8 ft. 



For the 20-kt speed assume, as a starter, that the point 

 A represents the operating condition. Here the square- 

 draft to water-depth ratio is 0.323, and the depth deter- 

 mined from that ratio is 40/0.323 = 123.8 ft. The cor- 

 responding critical-speed ratio is 0.602 and the depth 

 derived from it is 



h = 



F = 



(33.78)' 



{0.&02yg (0.36)(32.174) 



= 98.4 ft. 



It is obvious that the first depth is slightly too large and 

 that the square-draft to water-depth ratio should therefore 

 be larger than 0.323. Assume a value of 0.362, at the point 

 B. This gives a depth h of 40/0.362 = 110.5 ft. The corre- 

 sponding critical-speed ratio of 0.567 gives a depth of 



(33.78)' 

 (0.567)V 



110.3 ft. 



The estimate of the position of B was excellent in this case, 

 so that no further computation is necessary. 



For the 30-kt speed, assume the point C where the 

 square-draft to water-depth ratio is 0.199 and the critical- 

 speed ratio is 0.647. Then h = 40/0.199 = 201.0 ft, and 



^^ = ?7r?^ = 190-6 ft. 



(0.647) g 



Making another calculation for the point D, where the 

 square-draft to water-depth ratio is about 0.209, gives a 

 limiting depth h of 40/0.209 = 191.4 ft. Using the critical- 

 speed ratio of 0.645, the depth is 



