Sec. 62.2 



ESTIMATE OF ADDED LIQUID MASS 



419 



and entrained liquid, for unit length and for 

 up-and-down unsteady motion, is thb + vii, = 

 2pa' + 0.76pTa = (2 + 0.767r)pal The virtual- 

 mass coefficient, from Sec. 3.4, is (wib + mi)lmB or 



(^ VM — 



(2 -V 0.767r)pa" ^ 2 -\- 2.388 

 2pa' ~ 2 



= 2.19 



The corresponding added-mass coefficient is 

 simply mL/niB or 



i^AM — 



.76p7ra ' _ 2.388 

 2pa^ 



= 1.19 



This coefficient is always, by the definitions of 

 this book, equal to {Cvm — 1-0). 



62.2 Added-Liquid Masses for Some Geo- 

 metric Shapes and for Selected Modes of Motion. 



It is stated in Sees. 3.4 and 3.5 of Volume I that 

 the added mass of the entrained liquid around a 

 body in unsteady motion, symbolized by m^ , 

 is determined by the combination of size, volume, 

 shape, and mode of motion of the body and the 

 mass density p of the surrounding liquid. The 

 mass density of the body, symbolized by mg , 

 is the ratio of its own mass to the mass of the 

 volume of liquid that it displaces. In the, general 

 case the added liquid mass m^ has no relation to 

 the body mass me . 



If the body is a 1-ft cube of cork its mass is 

 small; if it is a 1-ft cube of lead, its mass is large. 

 However, for a given mode of motion in each 

 case, in a given liquid, the added mass of en- 

 trained liquid for each cube would be exactly 

 the same. There is not much point, therefore, in 

 relating these cork and lead body masses to a 

 given added mass of some liquid surrounding 

 them, for example water, while they execute this 

 unsteady motion. 



If, however, the submerged 1-ft cube is of 

 heavy wood, so that its weight is exactly equal 

 to that of a 1-ft cube of the adjacent water — in 

 other words, if the cube is buoyant — then the 

 ratio mi^lniB becomes most useful in ship design. 

 It is called the added-mass coefficient, symbolized 

 by Cam ■ The ratio (mt -\- mBiImB for a buoyant 

 body is called in this book the virtual-mass 

 coefficient, symbolized by Cvm ■ In some quarters 

 the latter name is applied to the former ratio, 

 and added-liquid mass is called virtual mass. In 

 other quarters the added mass is called the 

 hydrodynamic mass. It is most important, 

 therefore, in any discussion of this kind, that the 

 marine architect know exactly what is meant in 

 every case. 



It is equally important that he know what is 

 meant by inertia coefficients, mentioned in Sec. 

 62.1. In most technical books and papers the 

 inertia coefficients, linear and angular, correspond 

 to the added-mass coefficients described in the 

 foregoing, for translational and rotational motion, 

 respectively. Sometimes other names, or addi- 

 tional names, are applied to them to indicate the 

 exact mode of motion. 



It is customary, although writers are by no 

 means always specific in this matter, to base the 

 inertia coefficients on the mass (or mass moment 

 of inertia) of a buoyant body which has the same 

 mass as the identical volume of liquid would have. 

 This is always the case for the added-mass 

 coefficient defined here. A. F. Zahm, for one, 

 puts the matter this way: 



"Each inertia coefficient therefore is a ratio of the 

 body's apparent inertia, due to the field fluid, to the like 

 inertia of the displaced fluid moving as a solid" [NACA 

 Rep. 323, 1929, Part V, p. 437]. 



In this case the last four words are the important 

 ones, because the mass density for the sohd body 

 is then the same as for the liquid displaced by it. 

 This method breaks down for the infinitely thin 

 flat plate which has finite added liquid mass for 

 unsteady motion normal to its plane but zero 

 buoyant or displaced volume. However, it serves 

 very well for all practical purposes. 



The foregoing is a necessary preliminary to a 

 discussion of the added masses of a variety of 

 geometric shapes and of ship hulls because of the 

 presence, in the technical literature on this 

 subject, of certain form, shape, and proportion 

 coefficients involving added mass. These can 

 easily be confused with the added-mass coefficients 

 and the inertia coefficients for buoyant bodies, 

 in which the displaced-Uquid mass equals the 

 body mass. The text endeavors to make the 

 distinction clear as each of these form coefficients 

 is encountered. 



As a means toward this end, the lead of K. 

 Wendel is followed in stressing the added-hquid 

 masses themselves instead of the added-mass or 

 inertia coefficients. These added-liquid masses 

 and added-liquid weights are the numerical values 

 required by the marine architect; the coefficients 

 are convenient tools with which to make early 

 estimates, and the necessary tools with which to 

 conduct analytic investigations. 



There are a considerable number of geometric 

 shapes or bodies for which velocity potentials 



