The results reported in this section are for the þ stage compressor described above. These results were all computed at an inlet Mach number of 0.07, an inlet Reynolds number (based on the first-stage rotor chord) of 39,370 per cm, and a pressure rise of . Several approximations should be considered when interpreting the following results. 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 implemented in the code, so the effect of the end-wall boundary layer growth is not modeled.
For these computations, 2 sub-iterations per time-step 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 travel a distance equal to , where r is the radius at midspan, n is the number of rotor blades being modelled and N is the number of rotor blades in the actual machine. Each simulation was run in excess of 100 cycles to ensure time-periodicity. The computations were performed on a Silicon Graphics PowerChallenge Array with 8 compute nodes each comprised of between 2 and 8 R8000 cpus. Shell scripts were used to automate the process. Eight separate clocking positions were chosen in this study in order to make optimal parallel use of an 8 cpu compute node. Each individual computation required -secs per iteration per grid point and ran at 46 Mflops. Because of the parallelization, the combined throughput is 8 times these figures.
Results will be presented for stator-2 only in this paper. For other detailed comparisons with experiment of the flow through the compressor, please see Gundy-Burlet, et al. (1991). 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 one cycle. The pressures are then non-dimensionalized and plotted with respect to axial distance. Time-averaged pressure for each of the 8 different stator positions is 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 do not adequately elucidate the effects of clocking. An unsteady analysis is required to investigate airfoil clocking in turbomachines.
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. The two force components are plotted as a function of time. Because of the reflective boundary conditions and non-blade-passing-frequency shedding by the IGV, cycle-to-cycle periodicity cannot be obtained for the more sensitive variables, like forces or efficiencies. The forces are therefore ensemble averaged over 10 cycles. The reflective boundary conditions, which are applied 25 chords upstream of the IGV and downstream of the second stator, are used because non-reflecting boundary conditions do not maintain mass-flow rate. 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 10 cycles. Figure 4 figure5, figure 6, figure7, figure 8, figure 9, and figure 10 and figure 11 show the force polars for the 0.0% through 87.5% displacements respectively. Each of the force polar plots has some characteristics in common. The average force appears to be the same for each displacement. There is a large excursion from the time-averaged force 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 62.5% displacement case. The 25.0% and 37.5% displacements are similar to each other and have slightly larger amplitudes than the 12.5% and 62.5% cases. The 50.0%, 75.0% and 87.5% cases all 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.
Stator clocking can also be seen to have an effect on the compressor efficiency. Using the definition from Oates (1984), compressor efficiency is given by
Note, the average efficiency is calculated using the time-average of the area-averaged total pressure and total temperature. The efficiency is temporally and spatially averaged over ten rotor blade-passing cycles at a position approximately 1.5 cm (17% of the rotor axial chord) aft of the second-stator trailing edge. Ensemble averaging over multiple cycles is used to average out small variations in efficiency due to the reflective boundary conditions used for this computation, as well as off-frequency shedding by the IGV. The efficiencies for the 8 separate displacements are shown in Table 1, along with the deviation from the average value.. With the exception of the low efficiency at 37.5% displacement, the efficiency roughly forms a sign wave with a peak value of 97.81% at 25.0% displacement and a minimum value of 97.20% at 87.5% displacement. The lower efficiency at the 37.5% displacement occurs because the wake of the first stator (in a time-averaged sense) begins to impact the suction surface of the second-stage stator, instead of the pressure surface. The relative impact point of the first-stator wake on the second stator 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. In general, it was observed that higher efficiencies are observed when the first-stator wake is located on the pressure side of the second-stator passage.
A difference of approximately 0.6% in efficiency is consistent with estimates of efficiency gains of between 0.5% and 1.0% when turbine airfoils are clocked (Huber et al. 1995, Griffin et al. 1995). One interesting point is that the minimum amplitude on stator-2 (12.5% displacement) has an efficiency that is close to the peak value. Appropriate tuning of this compressor could provide both higher efficiency as well as lower time-varying forces on stator-2.
Table. 1. Compressor efficiency.
The interactions between the convected wakes and the airfoils are visualized through the use of entropy contours. Blue indicates a lower level of entropy and red signifies a higher entropy region. In the interests of brevity, contours for a few illustrative cases are shown. Contours for the 12.5%, 37.5%, 62.5% and 87.5% displacements are shown in Figs. 12 through 15, respectively. The viscous flow field upstream of the second-stage stator is not significantly affected by the position of the stator. There are several important flow features whose convection paths need to be followed in order to understand the changes in amplitude and efficiency. The rotor-1 wake convects downstream and is cut by stator-1. The faster convection rate along the suction surface (relative to the pressure surface) of stator-1 causes the cut ends of the rotor-1 wake to be displaced from each other about 40% of average chord. Rotor-2 then cuts the stator-2 wake, and the relative positions of the rotors causes one end of the rotor-1 wake to convect along the pressure surface of rotor-2. Thus, ahead of stator-2 for each case, there is a strong rotor-2 wake which has a complex grouping of interacting wakes on its pressure side and a single stator-1 wake intersecting its suction side. This group of wakes is circled near the trailing edge of rotor-2 in each of Figs. 12 through 15. It should be noted that clocking rotor-2 with rotor-1 and stator-1 with the IGV would change the geometry and characteristics of these convected wake forms.The 12.5% displacement case (Fig. 12) is characterized by both high efficiency and low force amplitudes. It can be seen that stator-2 cuts the rotor-2 wake where it intersects with the "isolated" leg of the stator-1 wake. The "isolated" leg of the stator-1 wake convects relatively quickly just off the suction surface of stator-2. Most of the circled wake grouping convects relatively slowly down the pressure surface of stator-2. The buildup of the wakes on the pressure surface of stator-2 is quite evident in Fig. 12. This stable arrangement interacts with the stator-2 flowfield to cause a slightly larger pressure difference between the pressure and suction surfaces which results in lower losses and force amplitudes on stator-2.
For the 37.5% displacement case (Fig. 13), the circled grouping of wakes convects almost entirely along the suction surface of stator-2. The low-energy fluid sitting near the suction surface reduces the pressure difference from the pressure to suction surfaces of stator-2 thereby reducing the efficiency of the stator. A slight displacement of the stator-1 wake off the surface of stator-2 results in an efficiency improvement.
The 62.5% case (Fig. 14) had a moderate efficiency and the second-lowest force amplitude. The circled grouping of wakes convects through the stator-2 passage well off the suction surface of stator-2. The 87.5% case (Fig. 15) shows the lowest efficiency of any stator-2 displacement. In this case, stator-2 cuts through the center of the circled wake/wake interaction region. The stator-1 wake convects down the pressure-side of the stator-2 passage without coming in contact with stator-2 while a large portion of the circled wakes convect down the suction surface. Small changes in the relative position of the second-stage stator with the upstream stator wake can have large non-linear effects on efficiency and force amplitudes.