Results for the þstage compressor configuration shown in Fig. 1 are presented in this section. For the 53,000-point coarse grid used in this study, 500 time steps are used for each blade-passing time. Each time step requires 3.9 CPU seconds on a Cray YMP. Time-averaged quantities converge within 10 blade passings, but the presence of multiple stages increases the time required to achieve a time-periodic solution, as compared to a single-stage system. Acoustic signals must travel a larger distance to exit from a multistage system. In addition, these signals reflect off airfoils and the reflections take time to exit the system. Depending on the level of convergence required of the time-periodic solution, between 5 and 40 hours of CPU time are needed for the calculations.
Several items must be taken into consideration when comparing these results with the experimental data. The experiment is inherently three-dimensional, while the computations are two-dimensional. Hub-corner separation, tip-leakage effects, and end-wall boundary-layer growth are not modeled. In the experiment, the end-wall boundary-layer growth generates a stream-tube contraction that increases the axial velocity at midspan along the axis of the compressor; the midspan exit axial velocity is increased to 110% of the midspan inlet velocity. This increase is not reflected in the calculation because the stream-tube contraction is currently not modeled.
The calculation was performed at an inlet Mach number of 0.07 and a Reynolds number of 100,000 per inch based on inlet conditions. These values were set to match the experimental operating conditions. The experimental flow coefficient is . The exit static-pressure coefficient for the experiment is . It is not possible to match the computational values with both the experimental pressure rise and the experimental flow coefficient because the stream-tube contraction is not modeled.
The first comparison typically made between calculation and experiment is of the time-averaged pressures on the rotors and stators. As data is available only for the second stage of the compressor, setting the exit pressure equal to the experimental value and allowing the flow coefficient to float provide a reasonable comparison between computation and experiment. The mass flow rate in the calculation then adjusts itself to the pressure rise specified. Only the Riemann invariants (functions of velocity and speed of sound), total pressure, and flow angle are specified at the inlet, so the final time-averaged values of velocity and static pressure may vary from the initial condition. The flow coefficient in the calculation increased to as the periodic steady state was reached. The flow coefficient for the experiment was . In order to demonstrate the complexity of the flow field, instantaneous entropy contours are shown in Fig. 2. Entropy was chosen to display the viscous nature of the flow because, unlike total pressure, entropy is invariant with reference frame. These results were computed using refined grids in the second stage of the compressor. The entropy contours illustrate the depth and width of the wake for the inlet guide vane as well as for the rotors and stators. The wakes generated in the second stage of the compressor are better resolved than the wakes generated in the first stage of the compressor. Although it is difficult to see this in Fig. 2, the difference in resolution of the wakes becomes clear in subsequent figures (in particular, Fig. 5). The numerical dissipation associated with the coarse-grid resolution of the first stage of the compressor causes the wakes to diffuse more rapidly than they would have with a fine grid. Because the wakes from the first stage of the compressor are more diffuse, the general level of unsteadiness seen by the second stage of the compressor (especially the second-stage rotor) is expected to be underpredicted. All grids would have to be refined to get a more accurate estimate of the amplitude of the unsteady behavior in the compressor.
The flow field in the second stage of the compressor is complicated by the accumulation of the convected wakes. Wakes from the first-stage rotor and stator penetrate into the second stage of the compressor, despite the lack of grid resolution in the first stage. Both wake/wake interactions and wake/airfoil interactions occur in the flow field. The impact of the difference in convection rate from the suction side of an airfoil to the pressure side of an airfoil is also evident, particularly on the second-stage rotor. The circulation around the airfoils produces a streamwise displacement of the wakes as they are convected along the suction and pressure surfaces. The circulation also causes a rotation of the chopped-wake segments as they are convected through the airfoil-to-airfoil passages. It is clear from Fig. 2 that wakes can be convected many chords downstream to interact with other airfoils. This causes unsteady variations in flow variables in the later stages of the compressor to be more complex than in the early stages.
As the downstream airfoils move through the wakes of the upstream airfoil rows, the wake/airfoil interaction causes the surface pressures to fluctuate. The airfoil surface pressures also fluctuate because of interactions with the potential flow fields of the upstream and downstream airfoil rows. The pressure distributions for the second-stage rotor and stator are presented in Fig. 3 in terms of the time-averaged surface pressures. The shaded band indicates the temporal maximum and miminum over a cycle caused by the fluctuations mentioned above. The time-averaged static pressure is arrived at by averaging the instantaneous static pressures over one cycle. A cycle is defined as the time it takes for the rotor to move from its position relative to one stator to the corresponding position with respect to the next stator. Pressure is plotted against axial position in the compressor. The axial distance between the rotor and stator results is the actual axial gap.
The symbols in Fig. 3 represent measured data for this compressor that had been reported previously by Joslyn and Dring (1989a). The data was acquired using long lengths of instrumentation tubing between the rotor and stator surface pressure taps and the rotating and stationary frame transducers, respectively. Thus, the measurements are presumed to be indicative of the time-averaged surface static pressure.
The agreement between the calculation and the experimental data is good. The small differences between them at the leading and trailing edges are due primarily to the mismatch in axial velocity. Although the flow coefficient in the experiment was 0.51 when averaged from hub to tip, a stream-tube contraction caused the flow coefficient at the midspan to be 0.54 at the inlet to the second stage.
The band in Fig. 3 indicating the temporal maximum and minimum of the local surface static pressure on the rotor and stator is nearly uniform in width around both sides of both airfoils. The increase in this unsteady amplitude near the leading edges may be related to the wakes of the upstream airfoil. However, it may also be related to the large gradient in the time-averaged pressure at the leading edge. The amplitude of the unsteady static pressure is approximately 10% of the airfoil loading. The pressure amplitude is also evident in the pressure-force results presented in Fig. 4 and discussed below. Both the rotor and stator airfoils show a pressure spike at the trailing edge. The trailing-edge spikes in this fine-grid calculation are one-third the size of the trailing-edge spikes in the coarse-grid calculation; this shows that the spikes are strongly dependent on grid resolution.
Close inspection of computed flow velocities at one instant in time near the second-stage stator shows the presence of a thin separation bubble from about 80% chord to the stator trailing edge. The separation bubble became apparent only under very high magnification of the results. It is undetectable at the scale of the results presented in Fig. 2 and does not appear to strongly affect the comparison with experment in Fig. 3. The temporal behavior of the separation bubble has not yet been examined. The existence of the separation bubble contrasts with previous flow visualization of the airfoils, which had indicated that at midspan both the rotor and stator suction-surface boundary layers are attached all the way to the trailing edge (Joslyn and Dring, 1989b). These two results are reconciled as follows. The turbulence model used in the present calculation is the fully turbulent Baldwin and Lomax (1978) model; Joslyn and Dring (1989b) showed that if the boundary layers were fully turbulent, both the rotor and stator would experience boundary layer separation at about the 80% chord location. They also demonstrated that the boundary layers had to be transitional to remain attached to the trailing edge. Transitional modeling was beyond the scope of the present calculation.
In spite of the fact that the separation bubble on the stator did not seem to have any major impact on the pressure distribution, its presence should be kept in mind throughout the remainder of this assessment.
The instantaneous surface static pressures were integrated around the perimeter of the rotor and stator to produce the periodic variation of the pressure-force vector on each airfoil for both a coarse-grid and a fine-grid calculation. A similar integration for the skin-friction force has not been carried out yet. The instantaneous axial and circumferential components of the force are plotted as a polar curve for the second-stage rotor in Fig. 4a and for the second-stage stator in Fig. 4b. The squares indicate the time-averaged values of the forces, computed with the fine grid. The direction of the time-averaged force and the normal to the stagger angle of each airfoil are included in the plots as reference.
These polar plots provide a good measure of the periodic convergence of the solution. At the periodic steady state, the force polar plot for one cycle should lay on top of the curve for the previous cycle. The coarse-grid results shown here are converged to this criterion. The fine-grid results show small variations from cycle to cycle, but are considered close to convergence. A time-accuracy study has been carried out for the coarse grid. Solutions were carried out at time steps of 500, 1000 and 2000 steps per cycle. Although not shown here, the force polar plots for each of these time steps compared well with those for the other time steps. This indicates that the periodic solution is independent of the time steps used in this study.
The coarse- and fine-grid solutions are qualitatively similar for the second-stage rotor shown in Fig. 4a. Because only the grids in the second stage of the compressor were refined for the fine-grid calculation, the wakes for the airfoils upstream of the second-stage rotor are computed on identical grids for both the fine and coarse grids. There are larger differences between the coarse- and fine-grid solutions for the second-stage stator shown in Fig. 4b, because of the difference in resolution of the second-stage rotor wake. The amplitude of the fine-grid force on the second-stage stator is approximately 40% larger than the amplitude of the coarse-grid force.
The unsteady forces are dominated by the variation in the lift component of the force. For brevity, "lift" is defined as the component of force normal to the stagger angle and "drag" as the component of force parallel to the stagger angle. The magnitude of the unsteady force is typically 10% of the time average for the relatively large rotor/stator axial gaps found in this compressor. These axial gaps are twice as large as those found in a typical modern engine.The unsteadiness is largely contained in the lift component of the force. Fig. 4c and Fig. 4d. show the lift and drag components of the force plotted against time for the second-stage rotor and stator, respectively. The curves vary smoothly, indicating that the unsteadiness in the force is not due to a narrow wake, but is distributed more uniformly, as one might expect for a potential interaction. Had the axial gaps between airfoil rows been smaller, the wake contribution might have been more pronounced.
The effect of grid refinement on wake profiles can be seen in Fig. 5. Time-averaged relative velocities for the coarse grid, fine grid, and experiment (Stauter, et al. 1990) are plotted against tangential displacement from the rotor trailing edge. This comparison is at an axial location 26% of the rotor axial chord downstream of the second-stage rotortrailing edge. The fine-grid calculation is in good agreement with the experimental data in terms of wake depth, width, and location. The wake depths for the coarse-grid results are shallower than for the experimental data and are visibly displaced. The fine-grid and experimental results have a displacement approximately 50% greater than those of the coarse grid. This is caused by the different computed rotor exit-flow angles which resulted from the difference in airfoil loading between the coarse- and fine-grid solutions seen in Fig. 4a; these results also suggest that solutions should be computed with several different grid densities in order to determine the asymptotic behavior of the solution with grid refinement. The remainder of the wake profile comparisons will be between the measured data (Stauter, et al. 1990) and the fine-grid results.
A comparison of the fine-grid and the experimental data is made at axial locations nominally 2%, 8%, 15%, and 26% aft of the rotor trailing edge (Figs. 6a-6d). Both the computed and the measured results are presented in terms of their temporal maximum, temporal minimum, and time average. The temporal fluctuation is caused by the interaction with the upstream stator's wake (see Fig. 2), as well as by the potential field of the downstream stator.
Generally, at each axial location, the agreement is good between the measured and computed time-averaged results for wake depth, width, and tangential location. The greatest differences between the measured and computed results are in the regions between wakes, where there are systematic (albeit small) differences. This can be seen most clearly on the suction-surface side of the wake, where the wake merges with the "free stream". At all axial locations, the prediction at this tangential location is typically 5% greater than the measured results. These differences are probably related to the nature of the mixing of the wakes, the lack of resolution of the upstream wakes, and the inability to model three-dimensional effects with a two-dimensional calculation.
Flow-speed fluctuations are given, in Figs. 6a, 6b, 6c, and6d by the width of the band between the instantaneous maximum and minimum relative velocities. The agreement between the measured and computed fluctuations is reasonably good. While the maximum-to-minimum band width of the computation is generally smaller than that of the experiment, many of the features have been captured in the computation, most notably the increase in band width with distance aft of the rotor. As pointed out by Stauter et al. (1990), this increase is due to the potential field of the downstream stator.