341 



In this paper, a brief summary is given of the 

 method for calculating the secondary vorticity in 

 the blade passage with comparisons to flow field 

 measurements. Initially, the primary flow field 

 through the rotor had to be determined in order to 

 calculate the passage secondary vorticity. This was 

 accomplished by using a streamline curvature method. 

 Flow field results are given in detail for one basic 

 flow configuration so named Basic Flow No. 1. Com- 

 parisons between the calculated minimum pressure co- 

 efficients and measured critical cavitation indices 

 are given for several basic flow configurations or 

 inflow velocity distributions. 



CALCULATION OF FLOW FIELD 



VELOCITY PROFILE 

 (B.C. t}l 



BOUNDING SIREAMLINES (B.C. »4I 



ROTOR RPM (B.C. 121 

 FIGURE 2. Schematic of boundary conditions. 



Primary Flow Field 



A schematic of the calculation procedure for the 

 flow through a rotor is given in Figure 1. This 

 outlines the iterative procedure for the calcula- 

 tions and indicates the point at which refinements 

 to the deviation angle are necessary and where 

 secondary flow calculations are employed. 



It is important to realize that in this discus- 

 sion the flow field is being solved for a given 

 rotor configuration. For this case, the boundary 

 conditions are (1) the geometric or metal angles 

 of the blades, (2) the rpm of the rotor, (3) the 

 velocity profile far upstream of the rotor plane, 

 and (4) the bounding streamlines of the flow. 



After solving for the bounding streamlines, the 

 iterative calculation procedure is started by 

 establishing the velocity profile far upstream of 

 the rotor. The initial conditions (Step 1) to the 

 solution for this boundary condition are (1) bounding 

 streamtube and (2) velocity profile in rotor plane 

 without rotor. With this information, the initial 

 streamlines without rotor can be calculated using 

 the streamline curvature equations (Step 2) . The 

 result of this calculation is the boundary condition 

 of an initial velocity or energy profile at a station 

 far upstream of the rotor plane. 



CALCULATION OF PRIMARY FLOW FIELD 



STEP 1 - 



INITIAL CONDITIONS 



CALCULATION OF FLOW WITHOLn' ROTOR 



STEP 3- 



STEP 4- 



STEP 5- 



STEP 6- 



STEP 7- 



FIRST ESTIMATE OF ROTOR OUTLET ANGLE 



CALCULATION OF FLOW FIELD WITH ROTOR 



SECOND ESTIMATE OF ROTOR OUTLET ANGLE 



CALCULATION OF FLOW FIELD WITH ROTOR 



SECONDARY FLOW CALCULATION 



THIRD ESTIMATE OF ROTOR OUTLET ANGLE 



STEP 9- 



FINAL CALCULATION OF FLOW FIELD WITH ROTOR 



FIGURE 1. Schematic of calculation procedure for 

 primary flow field. 



Knowing the blade metal angles, the first estimate 

 of the flow outlet angles (Step 3) can be calculated. 

 These flow outlet angles depend on the blade metal 

 angles and on a deviation angle. The deviation 

 angle correlation developed by Howell as discussed 

 in Horlock (1973) is initially applied. This 

 relationship considers only thin blade sections and 

 assumes that each blade secticn operates near design 

 incidence. As shown in Figure 2, all of the boundary 

 conditions are now known and the flow field can be 

 solved with the rotor included (Step 4) by using 

 the streamline curvature equations [McBride (1977)]. 



Once a converged solution is obtained for the 

 flow field using Howell's deviation angles (Step 4), 

 the axial velocity distribution is known whereby the 

 inlet angles can be estimated in addition to the 

 acceleration through the rotor. Now a second 

 estimate of the rotor outlet angles (Step 5) can be 

 made. For this deviation angle, the effects of 

 acceleration, A6', blade camber, Sq' ^nd blade 

 thickness, A6*, are calculated separately. For the 

 calculation of the deviation term due to axial 

 acceleration through the rotor, an equation developed 

 by Lakshminarayana (1974) is applied. For the 

 calculation of deviation terms due to camber and 

 thickness effects, the data obtained by the National 

 Aeronautics and Space Administration [Lieblein 

 (1955)] are used. The result is an improved outlet 

 flow angle profile which can be used to again calcu- 

 late the flow field (Step 6). 



The converged solution of the flow field (Step 6) 

 is then used to solve the secondary vorticity 

 equations (Step 7) and to determine a deviation term, 

 A63, which is due to nonsymmetric flow effects. The 

 details of the secondary flow calculations will be 

 discussed later in this paper. An improved outlet 

 flow angle profile (Step 8) is obtained by adding 

 this secondary flow term to the deviation terms 

 thus far calculated to obtain 



A6' + A6* + 60 + A6 



(2) 



where 62* i^ the outlet flow angle and Bz is the 

 blade metal outlet angle. This outlet flow angle 

 distribution is then used as a boundary condition 

 in the calculation of the flow field (Step 9) . 



Finally, all of the deviation angle calculations 

 are checked based on the flow field calculated in 

 Step 9. If the angles did not change significantly 

 then the result obtained in Step 9 is used as the 

 final flow field. 



In all, twenty-eight streamlines were calculated 



