Sec. 62.3 



ESTIMA IE OF ADDED LIQUID MASS 



429 



Lewis, and having approximately the correct 

 ratio [(beam) /(section draft)] of the ship section 

 under consideration. Lrtroducing the shape factor 



KSoot I 



Added mass of ship-shaped section of unit 

 length 



and / 



Added weight of added mass of liquid surround- 

 ing ship-shaped section of unit length 



= h..m->^p{g)B' 



(62. iv) 



where the beam B is that at midlength of the 

 constant-section segment in question; in other 

 words, the local beam. 



F. H. Todd made it unnecessary to compare 

 the shape of the given ship sections with the 

 transformations of F. M. Lewis by publishing, 

 in reference (24) of Sec. 62.8, a graph which 

 gave the shape factor direct from known values 

 of the ratio [(beam)/(section draft)] and the 

 section coefficient estimated from the body plan. 



C. W. Prohaska modified the diagram somewhat 

 so that the abscissas were values of the ratio 

 [(beam) /(section draft)] for the section in question 

 and the ordinates were values of the section 

 coefficient [ATMA, 1947, Vol. 46, Fig. 24 on p. 

 196; TMB Rep. 739, Oct 1953, Fig. 1 on p. 14; 

 SNAME, 1955, Fig. 34 on p. 471]. In all three 

 references cited the ordinates were labeled 

 iS(beta), which is the alternative ITTC symbol for 

 midship-section coefficient. This is misleading 

 because the section coefficient has a value of /3 

 only at the midship or maximum section. The 

 present author has further modified the Prohaska 

 graphs by substituting the shape factor fcsect for 

 the "coefficient" C. In their new form the graphs 

 are reproduced here as Fig. 62. H; the method of 

 defining the section coefficient is clearly illustrated 

 in diagram 4 of Fig. 62. G. 



It is to be noted that for a section coefficient 

 of 0.7854, corresponding to that of a semicircle, 

 the value of fcgect is constant for all values of the 

 ratio [(beam) /(section draft)]. When the latter 

 ratio is 1.0, the ship section is a semicircle and the 

 shape factor fcseot is 1.00. When the [(beam)/ 

 (section draft)] ratio has values other than 1.0, 

 the ship sections having shape factors fcgect of 

 1.00 are all ellipses, because their added-liquid 

 masses are, from Fig. 62. F, the same as for a 

 semicircular segment of unit length having the 

 same beam. 



i. 5 



Beam- Draft Ratio B/H 



Fig. 62. H C. W. Prohaska's Graphs for Deter- 

 mining Section-Shape Factors by Inspection 



Prohaska has supplemented the fcsj„t-values for 

 the ship-shaped sections of F. M. Lewis by values 

 for other more intricate sections, resembling those 

 of ships with bossings and other projecting 

 appendages [ATMA, 1947, Figs. 16, 17, 18, pp. 

 191-192]. These section shapes and shape factors 

 are also derived by conformal transformation. 



The added mass of the entrained liquid around 

 the transverse sections of a ship vibrating ver- 

 tically, calculated by the methods just described, 

 is still valid only for the flow around each 2-diml 

 segment, where the segment motion and the sur- 

 rounding flow are confined between two vertical 

 planes at the ends of the segment. This situation 

 is represented by diagrams 1 and 2 in Fig. 62. E, 

 drawn for a cylindrical bar and for an ellipsoid 

 of revolution, respectively. It is now necessary to 

 apply to the added mass around this segment a 

 longitudinal reduction factor to compensate for 

 the 3-diml nature of the actual flow, equal or cor- 

 responding to Lewis' factor Ja-wode mentioned 

 earher in this section. Because of the use of the 

 standard sysbol J for mass or polar moment of 

 inertia, the 3-diml reduction factor is symbolized 

 by R. However, concerning a factor R derived 

 for an ellipsoid of revolution and applied to a 



