332 



FIGURE 4. Filtered hot-wire signals in grid- 

 generated turbulence (adapted from Kuo and Corrsin 

 (1971)]. (i) f = 200 Hz, f/f = 0.52, 20 ms/division 

 (horizontal scale); (ii) 1 kHz, 0.52, 4; (iii) 6, 



0.52, 1; (iv) high-pass signal, f = kHz, 1 ms/ 



.... c 



division. 



will contain more intense deviations from the mean. 

 In particular, with all other factors held equal 

 it is more likely that the local pressure will fall 

 below the critical pressure when the Reynolds number 

 is high, even though the spectra are identical. 

 This is shown in Figure 5. If the log normal 

 arguments were applicable, then it can be expected 

 that this will depend on the Reynolds number. 



The effect of intermittancy coupled with effects 

 cited earlier could be of considerable importance 

 to the problem of predicting cavitation inception 

 in the prototype from small scale experiments in 

 the laboratory. The Reynolds number in model and 

 prototype can vary by many orders of magnitude. 

 For example, experimental observations of boundary 

 layer cavitation by Arndt and Ippen (1968) were 

 carried out at Reynolds numbers, u'6/v, of the 

 order 5000. On large ships, Reynolds numbers of 

 10 and greater are not uncommon. 



Coherency of the Pressure Field 



Kolmogorov (1962) to reformulate his original 

 similarity hypothesis in terms of the average rate 

 of dissipation of turbulent energy <£> , and to 

 assume that the logarithm of e was governed by a 

 normal distribution. Later work by Gurvich and 

 Yaglom (1967) showed that any non-negative quantity 

 governed by fine scale components has ajlog normal 

 distribution with a variance given by 2 = A + B 

 In R„ where A is a constant depending on the 

 structure of the flow, B is a universal constant and 

 R„ is the turbulence Reynolds number. 



These results have implications for the cavita- 

 tion problem at hand. Beuther, George, and Arndt 

 (1977a, b) have shown that Kolmogorov similarity 

 scaling is applicable to the high wave number 

 turbulent pressure spectrum. As a consequence of 

 this and the observed intermittancy and spatial 

 localization of small scale velocity fluctuations, 

 it is reasonable to expect the same trend in the 

 small scale pressure fluctuations. This could 

 result in an important cavitation scale effect. 



To make this point clear, a set of hypothetical 

 band passed pressure signals at high and low 

 Reynolds number are presented in Figure 5. For the 

 sake of argument, assume that the filter is set 

 around a range of frequencies which will result in 

 bubble growth (aiTg 1 1) . Since the spectra of these 

 two signals will be identified in terms of Kolmo- 

 gorov variables and since the low Reynolds number 

 signal is less intermittant, there is a greater 

 probability that the high Reynolds number signal 



An important factor related to cavitation in- 

 ception in jets is the existence of coherent 

 structure in the flow. Cavitation in highly turbu- 

 lent jets is observed to occur in ring like bursts, 

 smoke rings if you will. These bursts appear to 

 have a Strouhal frequency fd/U of approximately 

 0.5. This point is underscored by some recent work 

 of Fuchs (1974) . Fuchs made 2 and 3 probe pressure 

 correlations as shown in Figure 6. His results are 

 summarized in Table 1. Signals filtered at a 

 Strouhal number of 0.45 were highly coherent. For 

 comparison, velocity correlations are shown in 

 parentheses indicating that the velocity field is 

 much less coherent than the pressure field. 



The Turbulent Boundary Layer 



Because of the relative ease of measurement, there 

 exists a considerable body of experimental data 



Jet Nozzle 



Probes (ia2) 



' \A/|/|/Vl/»AAAAMjl)fl/^^ 



(t) 



Aw 



FIGURE 5. Hypothetical band-passed pressure signals: 

 (i) low turbulent Reynolds numbers, (ii) high turbu- 

 lent Reynolds number. 



General Arrangement 



PqP, PotPrPa' 





(b) 



(c) 



P. P, 



FIGURE 6. Measurement of pressure coherency in a 

 turbulent jet [adapted from Fuchs (1974)]. 



