Some Hydrodynamic Aspects of Ship Maneuverability 



An altogether different objection to the Taylor series approach arises from 

 the existence of nonlinear effects associated with separation drag. This can be 

 illustrated in the simplest case of steady motion at a drift angle /?, where there 

 is both theoretical and experimental evidence (Thwaites, 1960) that the side 

 force is of the form* 



Y = A sin 2/3 + B sin /3 |sin/3| 



= 2 A/3 + B/5 |/3| + Oi/3^) 



for a slender body with transverse symmetry. (Here A and B are constants 

 which depend on the body shape and Reynolds number.) In the Taylor series 

 approach, on the other hand, second-order effects would of necessity be even 

 functions of the drift angle and are ruled out by the transverse symmetry con- 

 dition, so that the nonlinear effects are by assumption of third order in the drift 

 angle. This particular point has led to some controversy regarding the correct 

 mathematical model for curve -fitting of experimental data. Martin (1961) and 

 Norrbin (1965b) lump all of the nonlinearities in a second-order term, whereas 

 most other workers assume that the third-order correction is valid, unless 

 second-order (even) terms are present as a result of the asymmetry introduced 

 by the ship's propeller (Suarez, 1963). 



It should be emphasized that those components of the hydrodynamic force 

 and moment which can be regarded as inviscid in their origin can in fact be 

 represented by finite Taylor series in powers of the drift angle, so that the non- 

 linearities of these forces will be more amenable to the Taylor series approach. 

 These will include the higher order effects associated with both circulation (in 

 the idealized lifting-surface sense) and with the free surface. In summary, both 

 second- and third-order terms should be included in a nonlinear model, and it 

 may be expected that these will not always scale simultaneously since the sepa- 

 ration drag will generally depend on the Reynolds number. 



SOME SPECIFIC THEORETICAL MODELS 



As stated in the Introduction, progress with theoretical predictions of ship 

 maneuvers requires that the various hydrodynamic processes be treated sepa- 

 rately, or at most in pairs, since the complete problem is intractable. Thus it 

 is necessary to restrict ourselves successively to individual details of the hy- 

 drodynamics in order to discuss specific theoretical techniques for prediction. 

 The present section will be devoted to the discussion of some of these, with em- 

 phasis on those aspects wherein theoretical predictions appear to be most 

 promising. 



Classical Added Mass 



The oldest and simplest mathematical model is formulated by assuming 

 that the fluid is ideal and ignoring the wave effects of the free surface. The 



*The subsequent notation is as given in the Appendix. 



213 



