346 



TABLE 2 - Vortex Circulation and Core Size Calculated Flow Vorticity Data 



Basic Flows 



Circulation 



r 



(in /sec) 



Basic Flow No. 1 



without upstream struts - 11.64 



without screen 



design flow coefficient 



Basic Floxi; No. 2 



without upstream struts - 8.23 



without screen 



0.9 design flow coefficient 



Basic Flow No. 3 



without upstream struts - 10.99 



with screen 



design flow coefficient 



Basic Flow No. 4 



with upstream struts - 8.29 



without screen 



design flow coefficient 



Characteristic 

 Dimension 

 R (inch) 



0.81 



0.57 



0.20 



0.45 



Nondimensional 

 Ratio 



-0.080 



-0.091 



-0.076 



-0.102 



Planar Momentum 

 Thickness 

 e (inch) 



0.85 



0.71 



0.94 



1.01 



the inner wall has a characteristic dimension 

 associated with it. A measure of the circulation 

 associated with this vorticity can be found by 

 integrating the vorticity from the inner wall to 

 the radius where the vorticity changes sign. In 

 addition, the characteristic dimension of the 

 passage streamwise vorticity must be related to the 

 difference between the radius where the vorticity 

 changes sign and the inner wall radius. The results 

 for several basic flow configurations are shown in 

 Table 2. Also, the nondimensional ratio, r/r^V„, 

 which is a measure of the minimum pressure coef- 

 ficient of the vortex is given in addition to the 

 planar momentum thickness of the mean boundary 

 layer profile entering the rotor for each flow 

 configuration . 



In order to make absolute comparisons between 

 calculated minimum pressure coefficients and 

 cavitation data, a reference point is necessary 

 and the effect of Reynolds number must be calculated. 

 A reference point for Basic Flow No. 1 of a = 2.8 at 

 a velocity of 15 ft/sec was chosen. The influence 

 of Reynolds number was determined by solving for the 

 relative streamwise vorticity at two different free 

 stream velocities. For these calculations, a bound- 

 ary layer profile at the reference Reynolds number 

 was used in one calculation and the boundary layer 

 profile at three times the reference number was 

 used in the other calculation. 



Now using Eq. (8) with Basic Flow No. 1 as the 

 reference point, comparisons between cavitation 

 data and Cpj^i^ calculated using the passage stream- 

 wise vorticity can be made. Some of the results 

 are shown in Figures 9, 10, and 11. As can be noted, 



the changes in Cp^ 



or a for the vortex as calcu- 



lated, using secondary flow theory, correlate well 

 with the cavitation results. Only the correlation 

 with the rotor operating off-design (Basic Flow No. 

 2) is poor at the higher velocities. It is felt 

 that this is due to primary flow problems. 



5 . SUMMARY 



A secondary flow analysis has been developed which 

 can be employed to assess the effect of inflow 

 velocity distribution on the strength and core 

 size of a vortex. This analysis has been success- 

 fully applied to a rotor where the secondary flows 

 dominate the flow field near the inner wall. 



NOMENCLATURE 



aj^' - streamline spacing in bi-normal direction 



Rj^ - radius of rotor 



W - relative velocity 



&2 ~ relative outlet metal angle 



gf - relative outlet air angle 



A6' - deviation angle due to axial velocity accel- 

 eration 



ASg - deviation angle due to secondary flows 



Oq - deviation angle due to blade camber 



a - cavitation number = (Pod - Pv)/(l/2pVoo ) 



Oji - limited cavitation number 



0(j - desinent cavitation number 



Us' - component of absolute vorticity vector in 

 relative streamwise direction 



LOn' - component of absolute vorticity vector in 

 relative normal direction 



Ub' - component of absolute vorticity vector in 

 relative bi-normal direction 



fln' - component of rotation vector in relative 

 normal direction 



flj-,' - component of rotation vector in relative bi- 

 normal direction 



ACKNOWLEDGMENT 



This research was carried out under the Naval Sea 

 Systems Command General Hydromechanics Research 



