337 



wherein t^^ is a characteristic bubble collapse 

 time, Rjj, is the maximum bubble radius, and R is 

 the distance to the observer. In addition, it 

 appears reasonable to assume that N is related to 

 the number of nuclei per unit volume, n, the 

 velocity, the size of a given flow field, and the 

 relative level of cavitation. Therefore we write 



N/nU d2 = f (a/a ) 

 c 



Thus a normalized version of Eq. (8) would be 



S(f) f 

 d 



pU p nR 



m 



f(a/a ) G(fT ) 

 c 



(9) 



It is difficult to obtain appropriate scaling 

 factors for R and t in a turbulent shear flow. 

 The problem is discussed briefly by Arndt and Keller 

 (1976). Lacking more detailed information, the 

 following assumptions can be used 



R , d 



T ~ d/Ua' 



o 



If we interpret S(f) as the mean square acoustic 

 pressure in a frequency band Af, Eq. (9) can be 

 written in the form 



p 2/p2u 



a C 



(Afd/U ) 



(r/d) 



,3 

 and 



f {a/a ) G(fd/ua^) 

 c 



(10) 



(Afd/U ) 



(r/d) ^ 

 a3/2 



N G' (fd/Ua' 



(12) 



Blake et al. were able to determine S(Tof) for the 

 case of noise due to cavitation on a hydrofoil 

 using measured values of R^. They assumed N equal 

 to unity and found that Eq. (11) resulted in 

 excellent collapse of the data. 



Arndt (1978) used Eq. (12) to normalize cavitation 

 data previously reported by Arndt and Keller (1976) . 

 These data correspond to noise from cavitation in 

 the wake of a disk and were collected under a 

 variety of conditions in both a water tunnel and in 

 a depressurized towing tank. Both the level of 

 dissolved gas and the number of free nuclei were 

 monitored. As shown in Figure 13, the normalization 

 is not very successful. It would appear that Eq. (10) 

 would be more effective in taking all of the 

 variables into account. However, n could only be 

 measured in unison with acoustic observations in the 

 water tunnel. Because of the nature of the laser 

 scattering measurements used to determine n in the 

 depressurized towing tank, these measurements had 

 to be made separately from the acoustic measure- 

 ments. The assumed form for S(fT_) in Eqs. (10) and 

 (11) varies by a factor nd^/a'2. As an example, n in 

 the depressurized towing tank appeared to be rela- 

 tively constant and equal to about IS/cm^. Therefore 

 the factor nd /a 2 was found to have a maximum varia- 

 tion of 23 dB. This does not account for the scatter 

 shown and one can only assume that there are other 

 complicating factors. It should be emphasized that 

 these data were collected under carefully controlled 

 conditions. This underscores the fact that the 

 current state of knowledge in this area is poor. 



Blake et al . (1977) circumvented the requirement 

 of measuring n. They reasoned that 



6. CONCLUSIONS 



G(f)df 



Pb "^^ 



YT P, 

 o -^b 



wherein pj-, is the time mean square of pb and Yxq 

 is the total lifetime of the bi±>ble (including 

 growth, initial collapse times and rebounding times) 

 Further, they simply reasoned that 



2 = 



Pb 



or that 



S(t f) = NT YG(fT ) 

 00 



This results in the normalized spectrum 



S(T f) 







p ^(f ,Af) YT r^ 

 a o 



AfN 



R '*pp 

 m o 



(11) 



Making the same assumptions as before, we would 

 expect that 



Cavitation inception in turbulent shear flows is 

 the result of a complex interaction between an 

 unsteady pressure field and a distribution of free 

 stream nuclei. There is a dearth of data relating 

 cavitation inception and the turbulent pressure 

 field. What little information that is available 

 indicates that negative peaks in pressure having a 

 magnitude as high as ten times the root mean square 



OdB 

 20 

 40 

 60 

 ' 80 

 100 

 120 

 140 



'^°| 10 100 1000 10000 100000 



fd _l_ 

 U '^"^ 



FIGURE 13- Normalized cavitation noise spectra. 



