compliance times the inertance of the inlet line. It is the compliance 

 that becomes singular for the fixed-terminus cavity model, thereby showing 

 this model to be unsuitable for an unsteady internal flow. 



Thus far, in discussing unsteady flow interactions, we have focused on 

 the cascade idea. This was, primarily, because the natural form of the 

 solution led immediately into the transfer function description, more 

 common in system analysis. As mentioned in Table 5, not all flows of 

 interest are attached cavity flows; in fact, these may not be, dynamically, 

 the most important forms. Observations of some pump inflows reveal a 

 significant number of microbubbles. These undergo growth and collapse 

 histories within the impeller which are modified by the fluctuating inlet 

 pressure. The "compliance" of a stream of microbubbles undergoing such a 

 growth and collapse process can be determined from bubble mechanics. This 

 is, in fact, what Brennen, in a most influential paper, has carried out in 

 great detail (Brennen 1973), using, for want of a better choice, microbubble 

 distributions derived from water tunnel measurements. The results are most 

 interesting in that the response of individual bubbles shows that the 

 compliance slowly decreases from its quasi-static value as frequency in- 

 creases with a gradually increasing phase lag. Thus, in the present 

 notation, z becomes complex with increase in frequency. We defer these 

 interesting results for comparison later with cavitating pump applications. 



Application to Pumps 



Pump engineers, historically, have carried out two types of steady 

 tests; namely, a performance test in which the (total) pressure difference 

 at fixed rotative speed is measured as a function of flow rate, and then, 

 a cavitation test in which the pressure rise at fixed speed and flow rate 

 is measured as a function of inlet pressure. It is customary to normalize 

 the total pressure rise by the quantity (q) , and the inlet pressure by (q) , 

 where q is the dynamic pressure based on tip speed. The axial inlet 

 velocity is normalized by the tip speed (different, thereby, by it from the 

 advance ratio of propellers). Curves, representative of the performance of 

 real pumps, are sketched in Figure 38. From our previous discussion of 



80 



