■126 



HYDRODYNAMICS IN SHIP DESIGN 



Sec. 623 



ends; almost invariabl}' it tapers toward those 

 ends, so that 3-dinil flow is involved. 



At the time of writing (1956), the general pro- 

 cedure whereby the underwater hull of a surface 

 ship is compared to one or more geometric forms 

 to determine the added mass of entrained liquid 

 has imdergone about a quarter-centurj^ of develop- 

 ment, applying to one particular degree of 

 freedom. This mode of motion, as mentioned 

 previously, is in addition to the usual six degrees 

 of freedom in that it embodies periodic bending 

 or flexure in a vertical plane, usually with two 

 nodes and three loops or antinodes, such as that 

 encountered in the fundamental resonant vibra- 

 tion of a ship hull in the vertical plane. The 

 processes in this development, which should be 

 understood clearly by the marine architect who 

 undertakes to calculate and to use intelligently 

 the added-mass values for his ship design, are 

 outlined here in a combination of diagrams and 

 words. 



Diagram 1 in Fig. G2.E illustrates a circular- 

 section bar of radius a, with its axis horizontal. 

 The bar is oscillating vertically (actually in the 

 plane of the page) in a pure translational mode, 

 embodying rising and dropping for equal dis- 

 tances above and below its normal position. It is 

 assimied that the rod is buoyant, having the 

 same mass density as the surrounding liquid, 

 and that at this stage it is deeply submerged in 

 an ideal, non-viscous liquid. For such a 2-diml 

 body, a section of which is pictured at the top of 

 Fig. 62.A and in diagram 1 of Fig. 62.E, the 

 added-liquid mass -per unit length is irpa^, where 

 p is the mass density of the buoyant rod and of 

 the surroimding hquid. All the unsteady flow 

 in this liquid takes place in vertical planes 

 bounding unit lengths, normal to the horizontal 

 midposition axis of the rod. 



The rod is next cut off to a given length L, 

 with free or exposed ends. It is then made to 

 vibrate in the vertical plane (that of the page) 

 with two nodes, at its fundamental frequency 

 in the surrounding hquid, taking both the rod 

 mass and the added-hquid mass into account. 

 The problem of predicting the fundamental 

 frequency in advance then resolves itself into one 

 of finding the numerical value of the added- 

 liquid mass for this mode of motion. This may be 

 tackled in two ways; they are described separately 

 in the paragraphs which follow. 



For the first method it is assumed that the rod, 

 although a single solid entity so far as its elastic 



characteristics are concerned, is made up of a 

 number of separate but adjacent length seg- 

 ments in the form of thick circular discs, indi- 

 cated in diagram 2 of Fig. 62.E. The unsteady 

 flow around each of these circular discs is assumed 

 to take place in vertical planes normal to the 

 straight or midposition axis of the rod, so that 

 for each segment the added-liquid mass is irpa' 

 times the length s of the segment. Although the 

 discs near the ends move up and down with a 

 greater amplitude, and a greater velocity, than 

 the discs at the center, the added-liquid mass for 

 each segment is not changed because of this situa- 

 tion, at least not for the low frequencies involved 

 here. However, since there are no imaginary 

 planes beyond the end segments to insure pure 

 2-diml flow there, it is obviously not acceptable 

 to multiply the added-liquid mass for each disc 

 by the number of discs and to say that this is the 

 added-liquid mass for the whole flexing or bending 

 bar. One reason is that the bar in diagram 2 is 

 deformed in pure shear rather than in pure 

 bending, or in a combination of the two. Another 

 reason is that the flow near the exposed ends is 

 certainly 3-diml in character. 



F. M. Lewis, who developed this method in 

 1929, utihzed as a solution for the second portion 

 of this problem a longitudinal reduction factor. 

 This repi-esented the ratio between (1) the added- 

 hquid mass around an ellipsoid of revolution in 

 vertical vibration and (2) the added-liquid mass 

 around a circular bar undergoing pure shear 

 deflection, having the same length L as the bar 

 in diagram 2 of the figure but in effect forming 

 part of an infinitely long circular rod. For his 

 solution, embodied in SNAME, 1929, pages 6-11, 

 Lewis assumed that the 3-diml circular-section 

 ellipsoid also deformed in pure shear. This meant 

 that the flow around any section took place 

 between vertical planes normal to the horizontal, 

 straight midposition of the ellipsoid, represented 

 in diagram 3 of Fig. 62. E. He thus obtained a 

 reduction factor J2-Nodc which he applied to the 

 added-hquid mass around each of the circular 

 segments, from one end of the bar to the other. 

 The reason for doing this, instead of applying 

 </2-Nod6 to the whole added Uquid mass, appears 

 presently. The reason for not summing up the 

 varied added-hquid masses for the segments of 

 radius a, a, , Oa , fla • • • of the ellipsoid of diagram 

 3 is also explained presently. 



To make this elongated and pointed ellipsoid 

 resemble the imderwater hull of a surface ship 



