328 



topic is of practical significance. Turbulent 

 shear flows are very common in practice and what 

 cavitation data are available for these flows 

 indicate that there can be significant scale effects. 

 For example, Lienhard and Goss (1971) present a 

 collection of cavitation data for submerged jets. 

 It is observed that the critical value of the 

 cavitation index increases with an increase in 

 jet diameter, with no upper bound on the cavitation 

 index being defined by the available data. The 

 cavitation index is observed to vary from . 15 to 

 3.0 over a size range of 0.1 cm to 13 cm. Arndt 

 (1978) reviews the available data for cavitation 

 in the wake of a sharp edged disk. These data 

 increase monotonically with Reynolds ntimber and 

 again no upper limit on the critical cavitation 

 index can be determined from the available data. 

 At present, it can be said that laboratory experi- 

 ments do not provide a reasonable estimate of the 

 conditions that can be encountered under prototype 

 conditions . From a practical point of view the 

 situation is much more critical than the scaling 

 problems associated with streamlined bodies since 

 at present there is no definable upper limit on 

 the cavitation index for these free shear flows. 



There are a myriad of factors that enter into 

 the inception process in turbulent shear flows. 

 As a minimum, we need information on the turbulent 

 pressure field, such as spectra and probability 

 density. We require an understanding of the diffu- 

 sion of nuclei within the flow, and we need to 

 know how these nuclei respond to temporal fluctu- 

 ations in pressure. In taking into account the 

 bubble dynamics inherent in the problem, consider- 

 ation must also be given to gas in solution which 

 can have an influence on both bubble growth and 

 collapse. 



The theory of bubble dynamics is well founded 

 and reasonable estimates of critical pressure can 

 be determined under flow conditions that are well 

 defined. Needless to say, the flow conditions in 

 a turbulent shear flow cannot be defined in 

 sufficient detail. However, the problem of flow 

 noise has led to a more comprehensive understanding 

 of turbulence; in particular, recent aeroacoustic 

 research has provided a wealth of data on turbulent 

 pressure fluctuations. These data are a by-product 

 of the need for understanding turbulence as a source 

 of sound. At this point in time, it seems only 

 logical to review the inception problem in terms 

 of both classical bubble dynamics and the more 

 recent results of the field of aeroacoustics. 



ated with this flow condition, which for convenience 

 will be defined as the critical index: 



p - p 

 oc V 



hpv 



If it is necessary to have completely cavitation 

 free conditions, one design objective for various 

 hydronautical vehicles is the minimization of a . 



Cavity flows are assumed identical in model 

 and prototype for geometrically similar bodies 

 when O is constant, irrespective of variations 

 in physical size, velocity, temperature, type of 

 fluid etc. In practice is found to vary over 

 wide limits. Simply stated, these so-called scale 

 effects are due to deviations in two basic assump- 

 tions inherent in the cavitation scaling law; namely 

 that the pressure scales with velocity squared and 

 the critical pressure for inception is the vapor 

 pressure, p . As will be shown, the two factors 

 can be interrelated, since in principle the critical 

 pressure is a function of the time scale of the 

 pressure field. 



In order to provide a foundation for the ensuing 

 discussion, consider a steady uniform flow over a 

 streamlined body devoid of any viscous effects. 

 The following identity can be written: 



p - p 



S^ P U 



^-^ 



wherein C is a pressure coefficent defined in the 

 usual manher. Generally speaking, C is defined 

 by the pressure on the surface of a given body. It 

 is generally assumed that cavitation first occurs 

 when the minimum pressure, p , is equal to the 



vapor pressure, 

 scaling law 



This results in the well-known 



Consider next the case where the pressure in the 

 cavitation zone is less than the minimum pressure 

 measured on the surface of the body, then 



^ml 



p - p 



ml 



P u„ 



P u_ 



2. THEORETICAL CONSIDERATIONS FOR CAVITATION 

 Cavitation Index 



Here we have to distinguish between the pressure 

 at the surface of the body p, and the pressure 

 sensed by cavitating nuclei, p . Assioming 



cavitation occurs when p , = p we have 



ml V 



The most fundamental parameter for cavitating flows 

 is the cavitation index 



P - P, 



a = 

 c 



C + 

 P 



ml 



P U 



(1) 



PU. 



wherein p is a reference pressure, p the vapor 

 pressure, U a reference velocity, anS p the 

 density of the liquid. The flow state of primary 

 interest in this paper is characterized by a 

 limited amount of cavitation in an otherwise single 

 phase flow. There is a specific value of O associ- 



Equation (1) is one version of the superposition 

 equation that is commonly referred to in the 

 literature. 



Bubble Dynamics 



It is generally accepted that the process of 

 cavitation inception is a consequence of the rapid 



