The results reported in this section are for the þstage compressor described above. These results were all computed at an inlet Mach number of 0.07, an inlet Reynolds number of 39,370 per cm, and a pressure rise coefficient of . Several approximations should be considered when interpreting the predicted results. First, the flow in the compressor is three-dimensional with end-wall boundary layer growth, hub corner stall and tip leakage effects. Because STAGE-2 is a two-dimensional code, it is unable to compute these three-dimensional effects. Stream-tube contraction terms have not been used for these computations, so the effect of the end-wall boundary layer growth is not modeled. Second, the effects of rotor clocking are not being studied in this investigation, and may have as much influence on the performance of the compressor as stator clocking.
For these computations, 2 Newton-Raphson sub-iterations per timestep and 1000 time-steps per cycle have proven sufficient to provide both time-accuracy and stability. Here, a cycle is defined as the time it takes a rotor to move through a distance equal to , where r is the midspan radius of the compressor, n is the number of stators in the simulation and N is the number of stators around the full annulus. Each simulation was run in excess of 100 cycles to ensure time-periodicity. The computations were performed on a Cray J90 with between 4 and 12 processors per node. Four cases were run at a time in parallel in order to make optimal usage of the machine configuration. Each individual computation required -secs per iteration per grid point and ran at 73 Mflops. Because of the parallelization, the combined throughput is 4 times these figures.
Time-averaged pressures will be presented for stator-2 only in this paper. Other detailed comparisons with experimental data can be found in Gundy-Burlet, et al. . Time-averaged surface pressures have been compared with experimental data in Fig. 3. for stator-2. The time-averaged pressures are obtained by averaging the instantaneous static pressure over 10 cycles. The pressures are then non-dimensionalized and plotted with respect to axial distance. Time-averaged pressures for each of the 8 different stator positions and 3 axial gaps are plotted here. No attempt has been made to distinguish between the different cases because the time-averaged pressures are quite similar to each other. Minor differences do not affect the overall good comparison with the experimental data. Simple time-averaged pressure plots, however, do not adequately elucidate the effects of airfoil clocking. An unsteady analysis is required to investigate performance variations due to airfoil clocking. The predicted results for the configuration with the largest axial gap (50% of the rotor chord) will be presented first, followed by the results for the smaller axial gaps.
The greatest potential benefit of airfoil clocking is a gain in compressor efficiency. Figure 4. Figure 4 illustrates the variation in the compressor efficiency as a function of clocking position, where the efficiency is given by 
Note that the average efficiency is calculated using the time-average of the area-averaged total pressure and total temperature. The total temperature and total pressure used to calculate the efficiency are temporally and spatially averaged over one rotor blade-passing cycle at a position approximately 17% of the rotor axial chord aft of the second-stator trailing edge. Ensemble averaging of the efficiency over multiple cycles is used to average out small variations in efficiency due to the reflective boundary conditions used in these computations, as well as non-blade-passing frequency trailing edge vortex shedding. The results in Fig. 4 were averaged over 20 cycles. The efficiencies for the 8 separate displacements are also shown in Table 1, along with the deviation from the average value. The efficiencies for the 50% axial gap remain nearly constant as stator-2 is clocked over the first half of the the pitch. In this clocking regime, the wake of the first stator (in a time-averaged sense) is on the suction-side of the mid-passage region. The relative impact point of the first-stator wake on the second stator passage has a significant effect on the unsteady potential field of the second stator. The wake also influences the angle of attack which the second stator experiences. As stator-2 is clocked between 50% and 87.5% of the pitch the efficiency increases rapidly. In these clocking positions, the first-stator wake moves from mid-passage onto the pressure side of stator-2. The difference of approximately 0.8% between the minimum and maximum efficiency is consistent with estimates of efficiency gains between 0.5% and 1.0% when turbine airfoils are clocked [1, 2].
Force Polar plots are used to investigate both the frequencies and amplitudes associated with the unsteadiness. These plots are generated by integrating the instantaneous surface pressure field and resolving the resultant force into its axial and tangential components. Because of reflective boundary conditions and off-frequency shedding by the IGV, the forces and efficiencies are ensemble averaged over 20 cycles. The tangential force is then plotted against the axial force. The symbol "X" on each of the plots indicates the time average of the force over 20 cycles. Figures 5 through 12. show the force polars for the 0.0% through 87.5% displacements, respectively, for a 50% axial gap. Each of the force polar plots has some characteristics in common. The average force is approximately the same for each displacement. There is a large excursion corresponding to the interaction of stator-2 with the rotor-2 wake. The interaction of stator-2 with wakes from upstream airfoils results in several small force variations. There are, however, large differences in the total amplitude of the forces about the time-averaged values. The smallest amplitude is for the 12.5% displacement case followed by the 62.5% displacement case. The 25.0%, 37.5% and 75% displacements are similar to each other and have slightly larger amplitudes than the 12.5% and 62.5% cases. The 50.0% and 87.5% cases have similar but slightly higher amplitudes yet again, and the largest amplitude case (0.0% displacement) has an amplitude of a little more than double that of the 12.5% displacement case. This variation occurs because of the differing interactions between stator-2 and the convected wakes from upstream airfoils. One interesting point is that, in general, larger amplitudes of unsteadiness on stator-2 correspond to higher efficiencies. This observation agrees with the findings of Dorney and Sharma for turbines .
Figure 4 illustrates the efficiency as a function of clocking position for the 35% axial gap, while Figures 13 through 20 illustrate the unsteady force polars. The efficiencies shown in Fig. 4 remain nearly constant over a larger span of clocking positions than at the 50% axial gap. There is still a significant rise in the efficiency between the 75% and 87.5% clocking positions, but the region of high efficiency is smaller than at the larger axial gap. The higher efficiencies are again associated with increased levels of unsteadiness. Overall the amplitude of the unsteady forces are slightly smaller than at the 50% axial gap (see Figs. 5-12 and Figs. 13-20), and are more complex. The efficiencies for the 35% axial gap are generally lower that at the 50% axial gap. However, if one were to perform an experiment where stator-2 was arbitrarily clocked to the 0% position, and data was taken for both 35% and 50% axial gaps, it would appear that the efficiency was higher for a smaller axial gap (see Table 2 or Fig. 4).
Figure 4 illustrates the efficiency as a function of clocking position for the 20% axial gap, while Figures 21 through 28 illustrate the unsteady force polars. At this axial gap the efficiencies remain relatively constant over the majority of clocking positions, except for lower efficiencies between the 50% and 75% clocking locations. The unsteady forces are much more complex than at the 35% and 50% axial gaps, indicating that the vortical structures in the wakes impacting stator-2 have not been mixed out. The force amplitudes are on par with those of the 35% case and generally less than those of the 50% case. This is somewhat counterintuitive but the non-linear nature of the wake-wake interactions produces these effects. The average efficiency for the 20% axial gap is lower than for the 35% and 50% axial gaps (see Tables 3 and 4).
Figure 29 shows time-averaged entropy on the second-stage stator for each of the axial gaps at the the lowest efficiency (20% gap-62.5% displacement, 35% gap-25.0% displacement and 50% gap-12.5% displacement) and at the highest efficiency (87.5% displacement for all 3 gaps). In the figure, red indicates the highest levels of entropy and blue the lowest. Different color maps are used for each of the cases in order to bring out the salient features. One thing to note is that in each of the higher efficiency cases, there is a layer of lower entropy fluid sitting just off the suction surface of the airfoil and higher entropy fluid from the first-stage stator wake in the pressure side of the passage. For the low efficiency cases, the airfoil is generally entrenched within the wake of stator-1 (35% gap and 50% gap). The 20% gap case doesn't exihibit this behavior as clearly, but the higher-entropy wake fluid still partly impacts the leading edge. During the course of these computations, this particular case (20% axial gap, 62.5% displacement) has exhibited the most variability in efficiency. From an examination of this figure, it is likely that small differences in the time-averaged stator-1 wake position relative to stator-2 have caused the large changes in efficiency.