continue smoothly far downstream, but roll up to form vortices which in turn diffuse 

 and eventually are dissipated in the wake. Consequently, when the hydrodynamic 

 force exerted on the body is calculated by integrating the local pressure over the body, 

 the flow may be assumed to be irrotational provided the dissipative wake can be 

 represented by an equivalent, properly chosen, potential flow model. This argument, I 

 believe, should provide the physical justification of any workable mathematical model 

 for cavity flows. A model is satisfactory as long as it gives a good description of the 

 flow near the body; and it is artificial in the sense that it fails to describe the wake far 

 downstream. 



Consider a two-dimensional flat plate set normal to a uniform stream of 

 velocity U. The image model introduces an image plate, of the same size as the real 

 plate, to be put at a certain distance downstream of the real plate, the pair of plates 

 being joined by free streamlines. From the potential theory, the force exerted on the 

 image is then equal and opposite to the drag D on the real plate. In a coordinate 

 system where the fluid at infinity is at rest, the force holding the moving image plate 

 does a negative work equal to ( — DU), which should be removed from the flow field 

 if the resulting flow is assumed to be potential. An estimate of this amount of work 

 corresponds approximately to the energy dissipated in the wake. Along the same line 

 of reasoning, the jet momentum of the reentrant jet model carried away from the 

 first mathematical sheet should be associated physically with the wake dissipation. For 

 even though the jet can often be observed, it is quite weakened by the turbulent mixing; 

 certainly its observed width is much smaller than its theoretical value (0.22 (1 + a) 

 X plate width). In the dissipation model the viscous dissipation is assumed to take place 

 within a strip in the wake, aligned parallel to the free stream; the detailed mechanism 

 as to how the flow is dissipated is rather immaterial. All these models have in their 

 mathematical formulations one free parameter (such as the location of the image, the 

 jet width, or the starting point of the dissipation zone), which can be expressed in 

 terms of the cavitation number a. Perhaps for this reason they yield about the same 

 value of C D . For small values of a, the first two models give 



2tt ( a 2 ) 



C D (a) « {1 + a + \ (21) 



7T + 4 f 8(tt + 4)) 



whereas the coefficient of o- 2 is [6(tt + 4) J -1 for the dissipation model. Other than 

 this one parameter, no freedom is left elsewhere to adjust the models. For example, 

 there is no solution of the potential flow when the image plate is different in size from 

 the real one. 



Furthermore, the application of these models to more general cases, such as 

 a cavitated lifting surface, must be carried out with care. For instance, the image of a 

 lifting surface could be made symmetric with respect to either a central plane or only 

 a certain point. To make sure that the formulation of any model is correct and com- 

 patible with the potential theory, I would believe that nothing else could serve as a 

 better guide than the physical significance underlying the model itself. Compared 

 with this point, the relative simplicity of the mathematical details involved in the 

 analysis of each model is rather a minor matter. To name one example, the cavitated 

 lifting surface with the cavity length equal approximately to the body length presents 

 a difficult mathematical problem; apparently the difficulties could not be removed 

 without a thorough understanding of the physical features of the flow near the rear 

 of the cavity. Therefore it seems fruitful to devote some effort to the investigation of 

 the physical basis of the cavity models; such an investigation would undoubtedly benefit 

 future research in this field. 



M. P. Tulin 



I know that time prevented Dr. Gilbarg from discussing aspects of this rich 

 subject which now have important implications for the design engineer, so permit 



294 



