Secondary Flow Generated 

 Vortex Cavitation 



Michael L. Billet 



The Pennsylvania State University 



State College, Pennsylvania 



ABSTRACT 



Secondary flow theories are employed to calculate 

 the secondary vorticity near the inner wall of a rotor 

 for several flow conditions. This calculated vortic- 

 ity is used in a simple vortex model to calculate the 

 minimijm pressure coefficient of the resulting vortex 

 behind the rotor. The influence of inflow velocity 

 distributions on the generation of secondary vortic- 

 ity is discussed. Comparisons are given between the 

 calculated pressure coefficients and the measured 

 cavitation indices of the vortex. 



1 . INTRODUCTION 



Secondary flows generate additional streamwise vor- 

 ticity when a boundary layer flow is turned by a 

 rotor. The apparent effect of this additional vor- 

 ticity is evidenced by the high cavitation numbers 

 of the vortex formed downstream of the rotor plane . 

 One example of the cavitation associated with a 

 vortex can be found in the draft tube of a Francis 

 turbine operating in the part load range . The 

 cavitation depends directly on the square of the 

 streamwise vorticity associated with the vortex. In 

 most cases, the critical cavitation numbers typical 

 of this vortex are often higher than those associ- 

 ated with any other type of rotor cavitation. 



Previous experimental results have shown that a 

 cavitation inception prediction of this vortex is a 

 very difficult problem. All rotors operating with 

 a wall boundary layer have a vortex along the inner 

 wall. The appearance of this cavitating vortex varies 

 from rotor to rotor. The critical cavitation number 

 can vary as much as an order of magnitude. Small 

 variations in the wall boundary layer can cause a 

 significant change in the critical cavitation number. 



Some confusion in cavitation inception data asso- 

 ciated with this vortex is due to a confusion of 

 types of cavitation, i.e., vaporous versus nonva- 

 porous cavitation. Vortex flows tend to be good 



340 



collectors of gas bubbles which can cause non- 

 vaporous cavitation. This often leads to confusing 

 nonvaporous for vaporous cavitation giving high 

 cavitation numbers. In general, results indicate 

 for vaporous limited cavitation that 



o, 



1 -C 



Pmi 



(1) 



Thus, the minimum pressure coefficient is of partic- 

 ular importance in a study of vortex cavitation in- 

 ception. 



It is appropriate then to find a simple descrip- 

 tion of the vortex in order to calculate its minimum 

 pressure coefficient. Unfortunately, the vortex is 

 composed of a finite number of vortex filaments 

 and a difficulty arises in specifying this number. 

 This is particularly difficult when the vortex exists 

 in the low pressure region near the inner wall of the 

 complicated flow behind a rotor. In this region, 

 there are vortex filaments in the primary flow in 

 addition to the secondary vortex filaments which can 

 influence this vortex. The combined effect of these 

 filaments is to induce a swirl velocity distribution, 

 Vq , which can be easily measured. 



Some preliminary tests show that in many cases 

 small changes in the incoming velocity profile near 

 the inner wall cause large differences in the crit- 

 ical cavitation number of the vortex. Measurements 

 of the primary flow field show only a change in down- 

 stream velocity profile near the inner wall. This 

 is especially true if the rotor was designed to be 

 unloaded near the inner wall. For these cases, 

 changes in the critical cavitation number can be 

 directly related to changes in the secondary vortic- 

 ity near the rotor inner wall. 



The secondary vorticity can roll-up into a vortex 

 like flow in the blade passage or it can simply com- 

 bine with other vortex filaments aft of the rotor to 

 form a larger vortex flow. In either case, there 

 will be a circulation and a characteristic dimension 

 of the passage vorticity which will determine the 

 critical cavitation number of the resulting vortex. 



