330 



The first theoretical arguments on the pressure 

 fluctuations associated with turbulent flow appear 

 to be due to Obukov and Heisenberg [Batchelor 

 (1953)]. Heisenberg argued that Kolmogorov scaling 

 should be possible for small scale pressure fluc- 

 tuations. Batchelor (1951) was able to calculate 

 the mean square intensity of the pressure 

 fluctuations as well as the mean square fluctuating 

 pressure gradient in a homogeneous, isotropic 

 turbulent flow. This work was extended by Kraichnan 

 (1955) to the physically impossible but conceptually 

 useful case of a shear flow having a constant mean 

 velocity gradient and homogeneous and isotropic 

 turbulence . 



Apparently there were no attempts made to extend 

 this theoretical work until the 1970 's when George 

 (1974a), Beuther, George, and Arndt {1977a, b, c) 

 and George and Beuther (1977) applied the concepts 

 developed by Batchelor and Kraichnan to the calcu- 

 lation of the turbulent pressure spectrum in 

 honogeneous, isotropic turbulent flows with and 

 without shear. When compared with experimental 

 evidence gathered in turbulent mixing layers, the 

 theory is found to be remarkably accurate. The 

 predicted spectrum (with no adjustable constants) 

 agrees with pressure measurements in turbulent jet 

 mixing layers from several sources, including 

 those of Fuchs (1972a) , Jones and his co-workers 

 (1977), and the authors themselves. As shown in 

 Figure 1, the experimental data and the theory are 

 remarkably consistent, especially in light of the 

 fact that several different experimental techniques 

 and different flow facilities are involved. 



The current state of knowledge of turbulent 

 pressure fluctuations can be summarized as follows: 



1) Pressure fluctuations in a shear flow can 

 arise from three sources. The first two involve 

 interaction of the turbulence with the mean shear. 

 These are second order and third order interactions , 

 of which only the second order interactions are 

 important at small scales. The last involves only 

 interactions of the turbulence with itself. 



2) Kolmogorov similarity arguments can be 

 applied to each of the spectra arising from these 





1.00 



aoo 



log K|X 



3.00 



terms. These arguments are valid for the small 

 scale fluctuations. 



3) If the turbulent Reynolds number is high 

 enough, there exists an inertial subrange in each 

 of the three spectra in which 



MSI si 



\S2 "'' = "S2P' ^ ^1^"'^' 



„ ^ 2 4/3, -7/3 

 Tr^(k) = a^p' £ ' k ' 



wherein a. 



= 2, a 



., ^,„ 0,a=1.3,eis the rate of 

 si s2 T I ^ 

 dissipation of turbulent energy per unit volume, 



K is the mean shear, and k is the disturbance wave 



number . 



4) There is considerable evidence that coherent 



structures play an important role in determination 



of at least the large scale pressure fluctuations 



[Fuchs and Michaike (1975) , Fuchs {1972a, b) , Chan 



{1974a, b) , and Chan (1976)]. 



Relation to Cavitation 



Since the above spectral results are expressed in 

 Eulerian frames , they cannot be directly applied 

 to the problem of cavitation inception which is a 

 Lagrangian problem. Nonetheless , Kolomogorov scaling 

 has been successful in an Eulerian frame of reference 

 and therefore we can, with some confidence, infer 

 that similar scaling will be valid for Lagrangian 

 time spectra (i.e. the frequency spectra that would 

 be seen by a moving material point) . The results 

 of such an exercise are as follows: 



1) The Lagrangian turbulent spectrum can be 

 separated into interaction of the turbulence with 

 the mean shear and the interaction of the turbulence 

 with itself. 



2) The high frequency (analogous to small scale) 

 will be well described by Kolmogorov scaling such 

 that 



^ Apps (0.) = K= v^/2 e-"" f (^ 



1 . „ , ^ 3/2 % ^ fbi 



-~2 AppT (W) = V £ f^ 



P 

 where 



T V W 



3) In the inertial subrange these reduce to 



-5 



^ Apps (00) = K^ v^/2 ^-h /o^ 



FIGURE 1. Experimental confirmation of the theoretical 

 pressure spectrum for a turbulent jet. 



3/2 i' 

 AppT (lo) = V e ■ 



