334 



log (cS 



"*V^ - ■"• 



In words, we again require a pressure fluctuation 

 to persist for a time which is long in comparison 

 to the response time of a typical nucleus. 



Since ^'/u^ is the shortest time scale in a 

 smooth wall boundary layer, all of the pressure 

 spectrum is sampled by the nuclei when 



T /V < 1 



B 



This criterion is especially important in view of 

 the highly intermittent process near the wall. 

 For rough walls, the last criterion can be 

 expressed in terms of the roughness height h by 



log Ki/Zu. 



FIGURE 7. Wall pressure spectra: (a) outer scaling, 

 (b) inner scaling. 



be u^^/v or u*/h, depending on whether the wall is 

 hydraulically smooth or rough, and there V7ill be 

 increasing intermittency with increasing Reynolds 

 number. The latter effect is most interesting and 

 is quite evident in the many observations of dye 

 streaks in the wall layer [cf Kim, Kline, and 

 Reynolds (1971)]. 



Effect of the Pressure Field on Cavitation 



Whether or not the pressure fluctuations play a role 

 in the cavitation inception process, depends on 

 the previously cited criteria: 



1) The minimum pressure must fall below a 

 critical level. 



2) The minimum pressure must persist below the 

 critical level for a finite length of time. 



The first criterion depends greatly on the yet 

 unresolved question of intermittency and its effect 

 on the probability density of the pressure fluctua- 

 tions. At this point in time we can say that the 

 critical cavitation index will increase with 

 Reynolds number because larger excursions from the 

 mean pressure are more likely. Without justification, 

 it is hypothesized that the effect on the pressure 

 variance will be approximated by a log-normal 

 dependence on thie Reynolds number. Detailed study 

 of the wall pressure such as that proposed by 

 George (1975) should aid considerably in resolving 

 this question. 



The question of time scale is more easily con- 

 fronted. Since most of the energy in the pressure 

 spectrum scales with u^ and 6 it is clear that the 

 criteria for bubble growth without appreciable 

 tension reduces to 



,T^/h < 



Since in fully rough flow u h/V > 1, it is clear 

 that the small scale criterion is more easily 

 satisfied with rough wall experiments. 



In summary, the information we have on pressure 

 fields in turbulent boundary layers and its 

 relationship to cavitation inception can be 

 summarized as follows: 



Significant scale effects can be expected when 



u'T /6 > 1. As the ratio of T to the smallest 



time scale in the flow decreases, the scale effect 



2 

 would be expected to level off i.e. when u*Tg/v or 



u^T /h < 1. Further increase in the cavitation 



number with Reynolds number will be due to the 



Reynolds number dependent effects on the probability 



density of the pressure fluctuations as a result of 



increased intermittancy of the small scale structure. 



The latter effect should produce a more gradual 



dependence of the cavitation index on Reynolds 



number than the former effect. 



The picture, as displayed above, is plausible 



and perhaps even appealing, but it must be viewed 



simply as conjecture until definitive experimental 



information is made available. An important hint 



of the relevance of these results can be found in 



the work of Arndt and Ippen (1967) where it was 



found that the region of maximum cavitation in a 



rough boundary layer shifted inward with a decrease 



in u^T /h. However, the change in this parameter 



varied only by a factor of 15 in their experiments. 



This will be discussed in more detail in subsequent 



sections . 



4. CAVITATION INCEPTION DATA 



A rather limited amount of experimental data have 

 been collected under controlled conditions. The 

 types of flows considered to date include the wake 

 behind a sharp edged disk, submerged jets from 

 nozzles and orifices, and smooth and rough boundary 

 layers. There is a dearth of information relating 

 the observed cavitation inception with the turbulence 

 parameters. Some of the earlier efforts in this 

 direction are summarized in a paper by Arndt and 

 Daily (1969) and by Arndt (1974b). A collation of 

 available data is presented in Figure 8. Here the 

 data are presented in the form of Eq. (1) : 



+ C = f (C^ 



