418 



HYDRODYNAMICS IN SHIP DESIGN 



Sec. 62.1 



entrained liquid are constant, independent of the 

 frequency or tlie amplitude of unsteady motion 



(3) There are no discontinuities in the liquid 

 surrounding the body or ship, which means that 

 no separation or cavitation exists 



(4) There are no damping forces or moments 

 acting on the body or ship, because of the lack 

 of viscosity in the liquid 



(5) For those modes of unsteady motion which 

 do not involve directly the speed of the body or 

 ship along its major axis, the added mass of the 

 entrained Uquid is independent of this speed 



(6) For a body floating on water, in a state of 

 equilibrium, the kinetic energy and the added- 

 liquid mass are assumed to be half of the respec- 

 tive values for a fully and deeply submerged 

 "double body" composed of 'the underwater form 

 plus its mirror image above the free surface of 

 the liquid 



(7) For some of the analytic procedures developed 

 to determine the kinetic energy in the Uquid 

 surrounding the underwater hull of a surface 

 ship, such as the 1929 method of F. M. Lewis, 

 described in Sec. 62.3, it is assumed that the 

 ship has vertical or wall sides all around at the 

 surface waterline. This means that there is no 

 discontinuity in the "double body" at the surface- 

 water line level. 



(8) So far as the 3-diml effects of finite length 

 and tapering ends on the added mass of entrained 

 liquid are concerned, the effects on the underwater 

 hull of a surface ship are assumed to be half of 

 those on an elliptic ellipsoid having the same 

 proportions of length, beam, and draft. 



No great study is required to reahze that in 

 practice, with ships and their parts, practically 

 none of these assumptions are truly valid. When 

 the ships and appendages are moving through a 

 real liquid Uke water, they are surrounded by 

 boundary layers, but the viscous effects appear 

 to be minor except for very small bodies. There 

 is increasing evidence that the added-liquid 

 masses around a vibrating or oscillating body 

 change with frequency and amphtude of vibra- 

 tion, especially at the higher frequencies. This 

 means that the motion is not that of a body in an 

 ideal liquid, surrounded only by potential flow. 

 E. Schadlofsky, in reference (14) of Sec. 62.8, 

 went so far as to say that for these reasons it was 

 hopeless to attempt an added-mass determination 

 by analytic methods. At high frequencies and 

 large amplitudes there may easily be cavitation. 



especially in regions where — Ap's exist because 

 of normal ship motions. Further, as R. Brahmig 

 points out [TMB Transl. 118, Nov 1943, pp. 2-3]: 



"Whereas the calculated hydrodynamic (added) mass 

 depends only on shape (of the body), its value may vary 

 with flow conditions in a real, eddying medium. A satis- 

 factory agreement of the calculated result with the mass 

 increase in the actual flow is therefore possible only when 

 the flow patterns of the two differing phenomena are 

 identical." 



There is damping of some sort in practically 

 all unsteady motion; certainly in all ship vibra- 

 tion. Assumption (6) requires that the flow 

 pattern around the actual underwater ship form 

 be half of that around the "double body." It 

 neglects the free-surface and gravity effects, 

 whatever they may be, and the dissipation of 

 energy by waves generated around the sides of 

 the ship and moving away from it. 



Despite all these drawbacks and disadvantages 

 the data derived from potential theory have been 

 most useful. In many cases the simplifying 

 assumptions have only minor influences, and in 

 most cases one can be reasonably certain that the 

 effect of factors not allowed for are definitely 

 additive or subtractive. 



It is again emphasized here, as' is pointed out 

 in Sec. 3.4 and illustrated in Fig. 3.F, that the 

 added mass of the liquid set in motion during 

 acceleration or deceleration is primarily a func- 

 tion of the mode of motion of the body or ship. 

 That mode must be known or assumed before 

 one sets out to estimate or to calculate the added- 

 mass effect. For example, in the case of an ellip- 

 soid of revolution, the field kinetic enei'gies and 

 added-liquid masses are by no means the same 

 for (1) translational motion in a given plane 

 parallel to the major axis and (2) bending or 

 flexural vibration, with two nodes and three 

 loops, in the same plane. For a 2-diml body of 

 rectangular section they are not the same for 

 translational motion in a plane parallel to the 

 long sides as for that type of motion in a plane 

 parallel to the short sides. 



The latter difference is illustrated quantita- 

 tively for the floating box of unit length and 

 rectangular section of Fig. 62. A of Sec. 62.2, 

 having a beam 2a and a draft a. The added- 

 liquid mass for up-and-down unsteady motion 

 is 0.76pTra^, while for right-and-left sidling motion 

 the added-liquid mass is 0.25p7ra". 



The mass of the floating box, for unit length, 

 is 2pa'. Therefore the virtual mass of both box 



