36 



ANALYTICAL DESCRIPTION OP STEADY-STATE CAVITIES AND DRAG OP CAVITATING MODELS 

 IN REAL AND IDEAL LIQUIDS 



The conditions in steady-state cavities are such that the velocity 

 magnitude (but not necessarily the direction) along the boundary streamline 

 may be taken as constant. Thus, the flow about such cavities falls in the 

 class of problems which compose the free-streamline theory in theoretical hy- 

 drodynamics. The earliest interest in flows with free streamlines was in con- 

 nection with the drag of bodies in a real fluid. The treatment of the wake 

 of bodies as free -streamline flows resulted in the now classical conformal 

 mapping methods of Kirchhoff and Helmholtz. 60 The Kirchhoff flow, in which it 

 is assumed that the pressure in the wake is the same as that at infinity cor- 

 responds to a cavity flow with cavitation number zero. Work in the mathemat- 

 ical theory of wakes, jets, and cavities has resulted in the accumulation of 

 a voluminous literature on free streamline theory. A bibliography of work on 

 two-dimensional wakes will be found in Reference 6l , and a critical examina- 

 tion of the progress made during the last 25 years has been carried out by 

 Weinstein. 62 The problem of two-dimensional free -streamline theory is to find 

 the cdmplex potential w(z) = <f> + i\p where <t> is the velocity potential, \p is 

 the stream function, and z = x + iy, with the condition 



p ♦ %*. - P ♦ urn ♦ m 



(d<t>\2 /d<t>\2 



where [T^j + \~q^~) = constant on the bounding streamline of the cavity. The 

 nonlinearity of the boundary conditions has restricted successful solutions 

 to the two-dimensional case. Much work remains to be done for three dimen- 

 sions, and, so far, useful results for three-dimensional flow are still lack- 

 ing. However, the results of two-dimensional treatments and approximations to 

 axisymmetric three-dimensional problems are of immediate usefulness in design. 

 The results that are of especial interest in engineering applications are the 

 drag associated with cavitating flows and the extent of the cavitating regions, 

 the latter data being of interest in the prediction of regions where collapse 

 may be expected and, thus, where damage may occur. 



The extensive mathematical analysis associated with obtaining drag 

 and cavity shape will not be reproduced here; the details will be found in the 

 references cited. However, the results of these analyses will be discussed. 

 There are two mathematical models of two-dimensional cavities that are of spec- 

 ial interest to the present discussion. The first, which is a flow suggested 

 by Riabouchinsky, 63 artificially introduces an after plate in tandem with the 

 forward one — with the cavity formed between them, Figure 25. It should be 



