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 100,000 per inch, 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 the coarse grid computation, 2 subiterations per timestep and 500 time-steps per cycle are sufficient to provide stability while eliminating transients from the solution. Here, a cycle is defined as the time it takes a rotor to move from its position relative to one stator to the corresponding position relative to the next stator. The code was benchmarked on several different workstations for a 50,367-point grid and for 2 subiterations per timestep. Five hundred time-steps per cycle and 2 subiterations per time-step have been found to be sufficient for converging this coarse grid þstage compressor calculation. Table 1 gives the code's performance on several workstations, ranging in price from $10,000 to $100,000. It is not the purpose of this study to present price/performance comparisons between different workstations. Instead, it is meant to show that STAGE-2 will operate on a wide variety of workstations, and to give a general idea of its performance on these workstations. The performance is measured both by cpu time per iteration per grid point and by the MFLOP rate. The MFLOP rate for the workstations was computed by determining the number of floating point operations in a run using a profiler on a CRAY-YMP. The number of floating point operations is assumed to be the same for the workstations, and is divided by the cpu time to get an overall MFLOP rate.
Table. 1. STAGE-2 performance statistics. STAGE-2 scalar version, STAGE-2 vector version.
Timings for the CRAY-YMP have been included here for a comparison between supercomputer rates and workstation rates. All the timings reported here are for single-processor hours. The timings on the CRAY-YMP illustrate the benefits of using additional memory to enhance vectorization. The overall cpu time of the code is decreased by a factor of 2.3 if additional memory is used to perform several inversions at once in the block tridiagonal solver. The vectorization in the block tridiagonal solver is the only difference between the scalar and vector versions of the code in this study. The scalar version of STAGE-2 runs at 2.0 MFLOPS on even the least expensive workstation used. With 2 sub-iterations per timestep, 500 timesteps per cycle and a 50,367 point grid, this translates to a turn-around time of 16 clock hours for a cycle on a dedicated low-end machine. For the fastest workstation used here, a cycle can be obtained in less than 2.6 clock hours. With the continuing rapid improvement in workstation technology, these timings will improve dramatically in the near future.
One concern when implementing a CFD code on a workstation is the effect of word length on the accuracy of the solution. CRAY-class supercomputers have a 64 bit word length while the workstations used in this study have a 32 bit word length. To address this issue, a coarse grid calculation using the experimental axial gap spacing has been performed. Workstation generated time-averaged surface pressures have been compared with experimental data in Fig. 2. The time-averaged pressures are obtained by averaging the instantaneous static pressure over one cycle. The pressures are then nondimensionalized and plotted with respect to axial distance. The workstation results compare well with the experimental data and are nearly identical to supercomputer results. This indicates that the 32 bit word length on the workstation is sufficient to generate accurate solutions.
Time-averaged pressure contours ( Fig. 3a, Fig. 3b, Fig. 3c) and standard deviation of pressure contours ( Fig. 3d, Fig. 3e, Fig. 3f) are presented for the field around the second-stage rotor for three different axial gaps. The pressure is averaged in the rotor frame of reference in Fig. 3. The standard deviation is also computed in the rotating frame of reference for the second-stage rotor. The standard deviation of the pressure field at each point is computed as:
where n is the number of time-steps in a cycle. Darker colors indicate higher pressures or higher levels of unsteadiness. The locus of points described by the the trailing edge of the first-stage stator and the leading edge of the second-stage stator are plotted as dashed lines. The time-averaged flow fields are qualitatively similar for each of the different axial gap cases. Contours of show that the greatest unsteadiness is near the leading edge of the stator. This is most pronounced for the 20% axial gap case (Fig. 3d).
Time-averaged pressure contours ( Fig. 4a, Fig. 4b, Fig. 4c) and standard deviation of pressure contours ( Fig. 4d, Fig. 4e, Fig. 4f) for the second-stage-stator are now presented. Time-averaged pressure and standard-deviation are computed in the stator frame of reference. The locus of points described by the trailing edge of the second-stage rotor is plotted as a dashed line in this figure. The time-averaged field pressures for the 20%, 35% and 50% cases are similar to one another. The 20% gap case (Fig. 4d.) shows a higher level of unsteadiness near the second-stage rotor trailing edge locus than in the rest of the field. The area immediately surrounding the leading edges of the second-stage stator are also more unsteady than the rest of the field for each of the axial gaps.
Figures 3a-f and 4a-f give a qualitative view of the steady and unsteady flow features in the second stage of the compressor. As can be surmised from these figures, time-averaged surface pressures for all three axial gaps are similar to each other. The time-averaged surface pressures closely resemble those for the experimental gap configuration in Fig. 2, and, hence, they are not reported here. The surface pressure amplitudes do vary with axial gap, and are shown in Figs. 5abc for the second stage rotor. The pressure amplitudes are computed by determining the maximum and minimum pressure at each point on the surface over a cycle. The minimum pressure is then subtracted from the maximum pressure to determine the amplitude. As expected, the 20% gap case ( Fig. 5a) shows the greatest level of unsteadiness and the 35% gap case ( Fig. 5b) generally shows more unsteadiness than the 50% gap case ( Fig. 5c). Because the airfoils are further apart in the 35% and 50% gap cases, the effect of the potential fields are reduced. This reduces the overall level of unsteadiness of pressure in the compressor.
Pressure amplitude plots yield information regarding the level of unsteadiness in the compressor, but do not contain phase information. Force Polar plots are used to investigate both the frequencies and amplitudes associated with the unsteadiness. Force polar plots are presented for the second stage of the compressor for all 3 axial gap cases in Fig. 6a, Fig. 6b, Fig. 6c, Fig. 6d, Fig. 6e and Fig. 6f). These plots are generated by integrating the instantaneous surface pressure field and resolving the resultant force into its axial and tangential components. The tangential force is then plotted against the axial force. For a periodic solution, this curve should close on itself at the end of a cycle, and is a good measure of the convergence of a solution to a periodic state. From Fig. 5, it is seen that the overall unsteadiness in the compressor increases as axial gap decreases. However, the integrated force field does not necessarily become more unsteady as the axial gap decreases. The force polar for the second-stage rotor at an axial gap of 20% (Fig. 6a) shows more unsteadiness than either the 35% axial gap case (Fig. 6b) or the 50% axial gap case. However, the integrated forces are more unsteady for the 50% axial gap case as compared to the 35% axial gap case. Animations of these flows indicate that the second-stage rotor interacts with wakes that interacted with each other for the 50% gap case. This reduces the frequency with which the rotor passes through upstream wakes, but increases the amplitude of the force polar. For the 35% gap case, the wakes from upstream airfoils are encountered at different times, so the frequency of the force variation is higher, but the amplitude is reduced.
A similar effect is seen for the forces on the second-stage stator in Figs. 6def. The amplitude of the forces is the least for the 20% gap case (Fig. 6d) and is the largest for the 50% gap case (Fig. 6f). It is interesting to note that the passage of each individual wake can be seen in the force polar for the 35% gap case. The IGV wake is seen as the smallest amplitude loop (on the left). The further downstream the wake was generated, the larger the amplitude of the loop. Despite the fact that the unsteadiness of the pressure field increases as the axial gap decreases, the actual force amplitude on the airfoil may decrease.