420 



HYDRODYNAMICS IN SHIP DESIGN 



Sec. 62.2 



Form of 

 Two-Dimensionol Bod\( 



Added Moss of 

 Entrained Li({uid 



Rod of 

 Circular Section 



Major A]<is 



— *- Mode of 

 Motion 



/oir(b^cos^a*- 

 a^ein^a,) 



Lonq, 

 Flat Plate 



K-2a >\ 



2b 



Rod of Square r~2a~ 

 or Rectonqulor 

 Section 



Square-Sectionr 

 Rod with 2L 

 Fins on 



the Corners 



i_ 



Vza^^ 



Floating 

 Rectonqulor \zz 



V///////////////A 



h-2a^-1 



Mode of Motion 

 mL"/Otra 



Added Moment 

 of Inertia of 

 Entrained Liquid 



HL'/Jira 



m|_"k|/)ira 



a/b 



Mode of 

 Motion 



Mode of 

 Motion 





•^l'qp^'^ 



2.23 

 1.98 

 1.7 



_JjjJ<2/oira 



94 

 24 



0.234 

 015 

 0.15 



0.147 



ni_-k3/)1Ta 



% 



005 

 Dl 

 0.25 



JL=k4/)ira^ 



tTH_"076/)ira 



mi_-Q25/)Tra 



mL"kg/jTra 



% 



n|_= a76/)Tra^ 



m|_- kg/jtra 



■' ^ Mode of 

 Motion 



JL=QII7/3ira'* 



JL-0.059/)Tra* 



0.61 

 0.67 

 085 



JL=0,055jO-rTa^ 



Fig. 62. a Added-Liquid-Mass Values for Some Two- 

 dlmensional geometric shapes in unsteady motion 

 All values given are for unit lengths normal to the page. 

 The respective modes of motion are indicated by the 

 double-headed arrows. 



and stream functions can be set up, applying to 

 simple modes of body motion. Of these bodies 

 the 2-diml elliptic-section cylinder, depicted near 

 the top of Fig. 62. A, is a well-known example. 

 The velocity-potential expressions make it pos- 

 sible to calculate, for an ideal liquid, the total 

 amount of kinetic energy involved in the intricate 

 particle motion around such a body, out to 

 infinity distance, when it moves in one of the 

 given modes of unsteady motion. The result is a 

 function involving the square of the body velocity 

 Ub and the first power of the mass density p of 

 the Uquid, no matter what the shape of the body 

 or the direction in which it is moving with respect 

 to its own axis. From the kinetic-energy function 

 the added mass of the entrained liquid for the 

 corresponding mode of motion is readily deter- 

 mined, as indicated in Sec. 3.4. In general, the 

 added mass of the entrained liquid can be calcu- 

 lated for the motion of any body for which a 

 velocity potential and a stream function can be 

 set up and for which the kinetic energy in the 

 flow can be derived. 



Fig. 62.A contains diagrams of a number of 

 2-diml geometric shapes, it indicates one or 

 more modes of motion for each, and it gives the 

 added-mass values in terms of the mass density 

 p of the surrounding liquid and the physical 

 dimensions of the bodies. Most of the data in this 

 figure were derived from those given by K. Wendel 

 [STG, 1950, Vol. 44, pp. 207-255; English version 

 in TMB Transl. 260, Jul 1956]; those for the 

 general case of the 2-diml elliptic-section cylinder 

 are from L. M. Milne-Thomson [TH, 1950, p. 239]. 

 Except as indicated in the diagrams, all the values 

 listed are for bodies submerged at a considerable 

 depth in an infinite expanse of liquid, so that at 

 infinite distances from the moving bodies the 

 particle motions are all zero. In a practical sense, 

 therefore, the diagrams apply only to certain 

 appendages on a surface ship having a roughly 

 geometric shape, lying well below the surface, 

 and to fully submerged submarine vessels. 



Fig. 62. B gives corresponding added-Iiquid-mass 

 data for a series of 3-dinil geometric bodies, and 

 for circular and elliptic discs, derived from data 

 on standard reference works on hydrodynamics 

 by Sir Horace Lamb and L. M. Milne-Thomson. 



A considerable number of references dealing 

 with the added mass of entrained liquid around 

 bodies of various types is listed on pages 100 and 

 101 of the book "Hydrodynamics," prepared and 

 published by the National Research Council, 



