Newman 



SO that the problem reduces to that of an elongated yawed body in an infinite real 

 fluid. This is of course one of the fundamental unsolved problems of subsonic 

 aerodynamics; if the body is basically flat a lifting-surface theory is appropri- 

 ate and the circulation is prescribed by the Kutta condition at the trailing edge, 

 but for a bluff body such as a body of revolution or ship hull there is no conven- 

 ient way of prescribing either the location or strength of the shed vorticity. It 

 is known that the vorticity and lift force arise from separation of the cross flow, 

 with regions of large vorticity at the boundaries of the separated region which 

 can be idealized in terms of vortex sheets. A detailed mathematical model 

 based upon this idealization has been constructed by Brard (1964) for application 

 to ship maneuvering problems, and in that work unsteady effects are included, 

 but the final analytical results are limited to rather elaborate convolution inte- 

 gral representations for the forces, whose principal utility lies in indicating the 

 proper form for the mathematical modeling and interpretation of experimental 

 data. (Brard's experimental results will be discussed in a subsequent section.) 



The lift on an oscillating body of revolution has been analyzed in consider- 

 able detail by Sevik (1965a, b). General momentum relations are derived for the 

 lift in terms of the far-field circulation, which is determined by means of an 

 unsteady laminar boundary -layer theory. The results are compared with ex- 

 periments in the case of a slender spheroid and show substantial qualitative 

 agreement, although the magnitude of the lift force is overpredicted in the theory 

 by a factor of two. The moment is dominated by the inviscid potential flow re- 

 sult, and is reduced by only 16% due to viscous effects. A comparison is made 

 between the pseudo- steady flow and that for high frequency oscillations, with 

 substantial differences noted both in magnitude and phase. Comparison of the 

 experimental pressure distribution with potential theory shows good agreement 

 except over the after 20% of the spheroid. The unsteady results are limited to 

 one value of the reduced frequency, L v = 9.68. 



The steady lift and associated boundary -layer flow on a body of revolution 

 at constant drift angle have been studied by Nonweiler (see Thwaites, 1960). 

 Rather complicated flow patterns are described particularly in the case of large 

 angles of incidence, but the form of the side-force coefficient is given very sim- 

 ply by the equation 



Y^ = (a^ cos /3 + Cj i sin /3| ) sin /3 , 



where a^ is the lift-curve slope appropriate to small angles of incidence andc^ 

 is the cross-flow drag coefficient. In practice these two coefficients must be 

 determined experimentally, but the results appear to be valid with engineering 

 accuracy throughout the range of angles up to and including normal incidence. 

 It is pertinent to the correlation of model and full-scale data to note that there 

 is a substantial Reynolds number dependence for the drag coefficient but not for 

 the coefficient a^, so that strictly the results from small-scale models should 

 be used only within the linear regime. 



Propeller Influence 



It has long been known that the action of a screw propeller or propellers 

 during a maneuver was stabilizing or, in effect, that of a skeg (Davidson and 



222 



