The Coupling Problem for a Pump 



We sketch the blades of an axial inducer together with the types of 

 cavitation in Figure 36. We imagine that the upstream pressure and inlet 

 velocity are fluctuating. We anticipate that the outlet quantities of the 

 pump will similarly fluctuate. There is upstream structure and downstream 

 structure to input and receive the unsteady flow through the pump. Our 

 coupling problem here is to provide the effect due to the insertion of the 

 pump or other cavitating device on these up and downstream flows. It is 

 natural, then, to think of the pump as having a "transfer function" whereby 

 periodic input disturbances are connected to output ones. An extended 

 system can then be treated by connecting its various transfer functions 

 appropriately. In what follows, some components of such a transfer 

 function are estimated from the standard reference problem in turbo- 

 machinery analysis, the flow through a cavitating cascade. The flow model 

 is that of an attached blade cavity with a fluctuating axial inlet flow 

 velocity. The cavity is assumed to be at constant pressure; its boundaries 

 and volume change in some relationship to the inlet flow perturbations. 

 To focus on the effects of cavitation, the wetted part of the cascade is 

 taken to be infinitely long. This representation, at first sight, highly 

 artificial, is actually a good one for the partial cavitation within the 

 inducer portions of cavitating pumps. A sketch of the cascade is given in 

 Figure 37a and linearized boundary conditions in Figure 37b. This 

 linearized boundary value problem implicit in Figure 37b is capable of 

 formal solution in an auxiliary plane where the entire blade and cavity 

 geometry appears on the real axis by the methods outlined in Wu's review. 

 We need not concern ourselves herein with these details except for several 

 basic features of the solution. There must be a fluctuating pressure gra- 

 dient approaching the cascade because of the fluctuating inflow velocity. 

 There will be a similar fluctuating downstream pressure gradient, possibly 

 with a different amplitude and phase, because of the changing cavity 

 volume. We can anticipate that the actual pressure will be of the form 



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