XI 

 FREE-STREAMLINE THEORY AND STEADY-STATE CAVITATION 



David Gilbarg 

 Indiana University 



The following is a brief review of several aspects of free streamline theory, 

 with particular reference to steady state cavitation [1J. 



Cavity models 



The Helmholtz concept of free streamline long had its principal applications 

 in the theory of wakes and jets. The well-known Kirchhoff model of the infinite wake, 

 in which free streamlines detach from a body and enclose a constant pressure stagna- 

 tion region, provided a theory of fluid resistance within the framework of classical 

 hydrodynamics, but was recognized, practically from inception, to be unrealistic in its 

 essential features [2]. It has recently become apparent that the natural application 

 of free streamline theory is to the phenomena of cavitation rather than wakes. 



In steady state cavitation, as observed in water tunnel studies, a vapor-filled 

 cavity forms behind a body past which the fluid is moving at sufficiently high speed, 

 the cavity being essentially at the vapor pressure of the liquid and its boundary a 

 constant pressure free surface. The conditions here differ from those of the classical 

 wake theory in that the cavity pressure p c is below rather than equal to the static 

 value p co . The pressure difference is usually measured in non-dimensional units by 

 the cavitation parameter 



P=o - Pc 



« = , (1) 



%pu* 



where it is the free stream velocity and p is the fluid density, a is the basic similarity 

 parameter of the theory of cavitation, and in the absence of gravity, viscosity, etc., 

 all non-dimensional quantities are functions of it alone. 



Observed cavities have positive cavitation number and of course finite dimen- 

 sions. It is the function of the theory to describe these flows and the associated physical 

 quantities. The limiting case a = — which yields an infinite cavity corresponding to 

 the classical wake — often proves useful in studying the finite cavity. Mathematically, 

 the essential difference between the problems of cavitation and the traditional problems 

 of hydrodynamics is that in the former the shape of the flow region is unspecified and 

 has to be determined from the condition that the pressure, and hence flow speed, 

 is constant on the free surface. 



In attempting to describe a finite cavity within classical hydrodynamics — that 

 is, under the assumptions of steady, gravity free, irrotational flow — we run at once into 

 the paradox that such a flow cannot be realized physically. To be specific, let us 

 consider the behavior of the free streamlines that detach from a body in cavitational 

 flow in an unbounded stream. Because the minimum pressure, and therefore the maxi- 

 mum speed, must occur on the free streamlines (the cavitation hypothesis), it follows 

 that these curves are convex to the flow [3]. This allows only the following possibilities: 

 i. The free streamlines may extend to infinity downstream without intersecting one 

 another; this can be shown impossible except when a = 0. ii. They may intersect, 



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