343 



2io 



0) ' = W9 

 S2 



WR' 



ds' + W, 



2Q^ ' u ' 

 b n 



ds' 



- W2 



2fi V ' 



ds' + M 



(4) 



where the primes refer to a rotating frame of 

 reference and the subscripts, 1, 2, refer to com- 

 puting stations along a streamline within the rotor. 

 As shown in Figure 6, s', n', b' represent the 

 natural coordinates for the relative flow, W is the 

 relative velocity, 033' and ui^' are absolute vorticity 

 resolved along the relative streamline, s', and the 

 principal normal direction, n' , f! is the rotor 

 rotation vector, and R' is the radius of curvature 

 of the relative streamline. 



The means by which the streamwise component of 

 vorticity is produced in this relative flow are 

 similar to those discussed by many investigators 

 for a stationary system. However, it is important 

 to note that additional secondary vorticity is 

 generated when fl x W has a component in the relative 

 streamwise direction. Rotation has no effect when 

 the absolute vorticity vector lies in the s'-n' 

 plane and the rotation, Q, has no component in the 

 binormal direction, £' . 



These equations were employed to calculate the 

 secondary vorticity along a relative streamline 

 through the rotor. All of the quantities in the 

 equations were calculated by an iterative procedure 

 using the primary flow calculations. The initial 

 normal component of absolute vorticity, Uj^, , for a 

 streamline was calculated from the incoming axial 

 velocity profile to the rotor. In all, the vorticity 

 along twenty-eight streamlines was calculated. 



As an example. Figure 7 shows the importance of 

 each term in Eq. 4 in the rotor exit plane for Basic 

 Flow No. 1. The sum of these terms is given in 

 Figure 8. The secondary passage vorticity is the 



difference between the exit vorticity, co 

 inlet vorticity, 0)3 J, along a streamline. 



S2 ' 



and the 



CALCULATION OF FLOW FIELD THROUGH ROTOR IN RELATIVE COORDINATE SYSTEM 



VELOCITY COMPONENTS VORTICITY COMPONENTS ROTATION COMPONENTS 



WITHOUT UPSTREAM STRUTS 

 WITHOUT SCREEN 

 DESIGN FLOW COEFFICIENT 

 (BASIC FLOW NO. II 



2 3 



u.. R„ 

 RELATIVE ABSOLUTI STT^EAMWISE VORTICITY, u =- 



S' R 



FIGURE 

 no. 1. 



7. Streamwise passage vorticity for basic flow 



The effect of this additional vorticity. 



"S2 



cogj, is to induce secondary velocities which are 

 assumed to occur at the exit plane of the rotor. It 

 is important to note that the normal component of 

 vorticity, ai^^^, is accounted for in the axisymmetric 

 flow analysis. Thus, only streamwise secondary 

 vorticity calculated as a function of radius influ- 

 ences the flow field. 



The effect of the streamwise component of vorti- 

 city within the blade passage is similar to that 

 obtained in the flow through a curved duct [Hawthorne , 

 (1961), Eichenberger, (1953)]; however, there is 

 the difficulty of devising a reasonable approximate 

 method of satisfying the Kutta-Joukowski condition 

 at the exit of the rotor. The method used in this 

 investigation assumes that the flow is contained in 

 a duct defined by the blades and streamlines of the 

 primary flow leaving the exit of each blade. In 

 this exit plane, a flow solution devised by Hawthorne 

 and Novak (1969) was applied. The secondary stream- 



FIGURE 6. Description of relative coordinate system. 



-3 -2-10 1 2., 



RELATIVE STREAMWISE VORTICITY, u , 



FIGURE 8. Relative passage streaniwlse vorticity at 

 rotor exit plane for basic flow no. 1. 



