329 



or explosive growth of small bubbles or nuclei 

 which become unstable due to a change in ambient 

 pressure. These nuclei can be either imbedded in 

 the flow or find their origins in small cracks 

 and crevices at the surfaces bounding a given flow. 

 The details of how these nuclei can exist have been 

 considered by many investigators. A summary of 

 this work is offered by Holl (1969, 1970). 



Theoretically, liquids are capable of sustaining 

 large values of tension. However, the nuclei in 

 the flow act as sites for cavitation inception 

 and prevent the existence of significant tensions. 

 The mechanics of the inception process are adequately 

 described by the Rayleiqh-Plesset equation, which 

 considers the dynamic equilibrium of a spherical 



bubble containing vapor and non-condensable gas 



and subject to an external pressure p , (t) : 



mi 



3 A2 1 



RR + - R = — 



2 P 



+ P. 



, , 2S 

 P , (t) - — 

 ml R 



4 u 



(2) 



wherein R is the bubble radius and dots denote 

 differentiation with respect to time. It should 

 be emphasized here that even for the case of steady 

 flow over a streamlined body, p (t) is a function 

 of time since we are concerned with the pressure 

 history sensed by a moving bubble. If the problem 

 is simplified to consider the static equilibrium 

 of a bubble, we find that there is a critical 

 value of p ~ P 1 below which static equilibrium 

 is not possible. This is found to be 



(p., 



' ml c 



4S/3R* 



(3) 



wherein R* is defined as the critical bubble radius. 

 Substitution of Eq. (3) into Eq. (2) with dynamical 

 terms identically zero will indicate that R* is a 

 function of the partial pressure of noncondensable 

 gas within the bubble. If p , (t) varies rapidly 

 in comparison to the response time of the nuclei, 

 then even greater values of tension are possible. 

 Thus in general we can write 



Pg ~ -3- 

 ^ R 



and the growth rate stabilizes at a value given by 



■> P - P 1 

 2 V ml 



(4) 



Assuming a characteristic bubble response time 

 given by R*/R, with p^ - p^^ = 4S/3R*, we obtain 



R* 



T i -^ = 0.87 

 B R 



PR 



(5) 



A typical variation of if based on the theoretical 

 computations of Keller (1974) is given in Arndt 

 (1974). 



The Influence of Dissolved and Free Gas 



The discussion in the previous section is based on 



the assumption of a healthy supply of free nuclei 



which is generally the case in recirculating water 



tunnels and in the field. Generally speaking, a 



reduction in a due to bubble dynamic effects 



usually only occurs on model scale. To some extent 



the level of dissolved gas and the number and size 



of free nuclei are interrelated. Some recent 



experimental results are documented in Arndt and 



Keller (1976) . The level of dissolved gas can 



play an important direct role when the time of 



exposure to reduced pressure is relatively long. 



Under these circumstances Holl (1960) has shown 



that gaseous cavitation can occur at values of a 



much greater than those for vaporous cavitation. 



Using an equilibrium theory, Holl (1960) deduced 



an upper limit on a given by 

 c 



C + 

 pm 



at 



Jj p U 



(6) 



P. - P 



ml 



4S/3R* 



PR* 



where 



(|>(o) 



(t)(oo) 



The function (}) depends on the flow field. The 

 argument of (j) contains a characteristic time scale 

 of the pressure field (t ) and a characteristic 

 response time of the nuclei, (PR^^/S)^ . In the 

 case of a streamlined body in the absence of viscous 

 effects, t would be proportional to the quotient 

 of body diameter and velocity. In the case of 

 cavitation induced by turbulence, the characteristic 

 time scale could be any of the turbulence time 

 scales. For example. 



is 



(2) 



Under 



11 = V 



is often appropriate. The factor (PR^ /S) ^ 



derived from the asymptotic solution to Eq. 

 for the case of negligible gas diffusion, 

 these conditions 



wherein a is the concentration of dissolved gas 



and 3 is Henry's constant. 



In summary, an overview of the effects of bubble 



dynamics and free and dissolved gas indicates that 



short exposure times such as are the case in a 



model implies that cavitation will occur at pressures 



lower than vapor pressure and a is less than 



expected. Long exposure time, such as can occur 



in vortical motion of all types , including large 



scale turbulence, implies the possibility of gaseous 



cavitation with O being greater than expected, 

 c 



3. PRESSURE FLUCTUATIONS IN TURBULENT SHEAR FLOWS 



Background 



Considerable progress has been made over the last 

 five years in the understanding turbulent pressure 

 fluctuations in free shear flows in an Eulerian 

 frame of reference. Of particular importance is 

 the development of pressure sensing techniques 

 which under certain circumstances can lead to 

 reliable measurements of pressure fluctuations. 



