436 



HYDRODYNAMICS IN SHIP DESIGN 



Sec. 62.6 



B and the section draft, and (2) the ratio h/H, 

 relating the depth of water to the ship draft. 



Koch's shallow-water shape factors <^ apply to 

 rectangular sections only, so to use them for a 

 ship one would have to assume a constant section 

 for the entire length. Prohaska's shallow-water 

 shape factors apply to sections of varying fullness, 

 from 0.99 to 0.37, so that a calculation could be 

 made for a ship with somewhat tapering ends by 

 using segments of diminishing section coefficient 

 toward the ends. 



Applying the data of Koch and Prohaska to the 

 vertical-vibration problem of the ABC ship of 

 Part 4, transiting the 30-ft river between Port 

 Correo and the sea, the basic data are: 



Draft, i? = 26 ft 

 Depth of water, /i = 30 ft 

 Bed clearance, h — H = A it 

 Ratio of draft to half-beam, 2HfBx = 52/73 = 

 0.71 



Maximum-section coefficient, Cx = 0.956 

 Ratio of depth of water to draft, h/H = 30/26 



= 1.15 

 Ratio of half-beam to bed clearance, 



Bx/[2{h - H)\ = 73/[2(4)] = 9.1 



Applying these data to Koch's graph with 

 ITTC notation. Fig. 62.J, they fall far beyond the 

 limits of the graph. Applying the equivalent 

 ratios to Koch's original graph. Fig. 8 of reference 

 (1) at the beginning of this section, and extra- 

 polating roughly, ^ appears to have a value of 

 about 5.0 for a ship having a constant rectangular 

 section throughout. Since Koch's^ = mi/(p5.Y/2), 

 the added-liquid mass is of the order of 



rriL = (0.5)^p5x = (0.5)(5.0)pBl- 

 = 2.5pBx per tmit length. 



From Prohaska's original data, reproduced in 

 diagram 1 of Fig. 62. L, his "coefficient" C is about 

 2.9 for a ship having a constant section coefficient 

 of 0.956. The added-liquid mass is therefore of 

 the order of 



= C 



Bl- 



= (2.9)(0.125)7rpB^ 



= 1.14pB.v per unit length. 



The weight of the added-liquid mass is g{mL) in 

 each case. 



This is a rather large discrepancy but the 

 comparison is hardly fair to the data of Koch 



because, as indicated in Fig. 62.. I of the present 

 section, as well as in Fig. 31 on page 203 of 

 Prohaska's ATMA, 1947 paper, Koch's experi- 

 ments do not cover such a small depth-of-water 

 to draft ratio. 



62.6 Estimating the Added-Mass Coefficients 

 for Vibrating Propulsion Devices. Data on the 

 added mass of entrained water surrounding the 

 thrust-producing blades of any type of mechan- 

 ically driven propulsion device are required for 

 predicting the vibration characteristics of the 

 component parts or of the entire mechanical 

 propelling system. 



For a propulsion device like a paddlewheel, 

 with blades generally normal to their direction of 

 motion relative to the surrounding water, the 

 added mass of entrained liquid may be approxi- 

 mated by the known tkl for a submerged fiat 

 plate of rectangular outline, in unsteady motion 

 in a direction normal to its surface. For this case, 

 the added mass is given in Fig. 62.A for the 

 rectangular flat plate having dimensions of 2a 

 and 2b. However, the paddlewheel case is by no 

 means simple because all the submerged blades 

 create surface waves, and one or two or more 

 blades are always partly in and partly out of the 

 water. So far as known, no engineering rule has 

 been developed for estimating the added mass 

 niL of paddlewheels or sternwheels. 



The rotating-blade propeller presents a much 

 different case because the several blades usually 

 (except in maneuvering) lie at only a small angle 

 (the attack angle) relative to their direction of 

 motion through the surrounding water. However, 

 what is wanted in this case is the added mass for 

 a mode of motion which is tangential to the 

 spindle circle at each spindle position. This 

 quantity depends upon the pitch ratio and the 

 exact type of blade motion. Moreover, it changes 

 with the position of a blade on the blade orbit 

 or spindle circle. It can be assumed roughly as 

 one-half the added liquid mass for the blade, 

 reckoned for a mode of motion normal to the 

 projected area of the blade [Mueller, H. F., un- 

 publ. Itr. to HES, 6 Jul 1956]. This Hquid mass, 

 added to the mass of the blade and summed up 

 for all the blades, plus the mass of the supporting 

 and actuating machinery, gives the polar moment 

 of inertia of the whole assembly about the axis 

 of propeller (not blade) rotation. 



For the screw propeller the marine architect 

 is interested in the added-liquid masses and the 

 corresponding added mass moments of inertia 



