Results for the - stage turbine configuration are presented in this section. Nineteen-hundred time-steps are used for each blade-passage period and three subiterations are used per time-step to reduce the maximum residual in each grid by approximately two orders of magnitude. For the geometry and fine grid density used in this calculation approximately 48 single-processor Cray-C90 CPU hours are required per one blade-passage period. The code operates at -secs/it./grid point and 356 Mflops on the Cray-C90. The solution startup costs for the turbine were minimized by interpolating a two-dimensional midspan solution onto a coarse three-dimensional grid system. The coarse-grid solution was then interpolated onto a fine grid.
In Joslyn and Dring, the experiment was reported to have been performed at a Mach number of 0.07 based on inlet conditions. The flow coefficient, , which is defined as the ratio of inlet axial velocity to the midspan rotor speed , was for the on-design case. The midspan pressure coefficient 14% of chord aft of the second-stator trailing edge was where is defined as
Here, P is the local static pressure, is the total pressure at the inlet and is the density at the inlet. Dring had earlier estimated the midspan exit pressure at -8.924 from a free-vortex distribution. In order to match the turbine operating conditions in this computation, an exit pressure coefficient of Cp = -9.35 was imposed at the computational exit plane (approximately 1.5 chords downstream of the second-stage stator trailing edge). Other conditions in the computation are equivalent to those of the experiment.
Time-averaged pressure and the pressure amplitude envelope for the first stage stator at the 12.5%, 50% and 87.5% span stations are shown in figure 2, figure 3, figure 4, The time-averaged results from the STAGE-3 code are represented as a solid line, while the experimental suction side results are shown as squares and the pressure side results are indicated by circles. The time averaged data in this paper are obtained by averaging the variable of interest over a cycle, where a cycle is defined as the time it takes for a rotor to move from its position relative to one stator to the corresponding position relative to the next stator. The comparison between the time-averaged computational results and the experimental data is good in general on both suction and pressure surfaces. The "footprint" of the hub and tip passage vortices on the suction surface agrees well with that of the experimental data. In computations with fewer points in the radial direction, these vortices are larger in extent and impact the pressure field upstream of the position indicated by the experimental data and current results.
In Figure 2, figure 3, and figure 4, , pressure amplitude is represented by the dashed lines. The range in pressure is determined by finding the maximum and minimum pressures at each point on the surface over a cycle. Experimental pressure amplitudes are not available for this turbine configuration and therefore a quantitative analysis of these effects on the pressure amplitude is not possible. However, the results can be evaluated in a qualitative sense. As expected, the first-stage stator shows less unsteadiness than the downstream airfoils. The unsteady regions are generally concentrated on the suction surface near the trailing edge where the influence of the downstream rotor potential field would be greatest.
First stage rotor time-averaged pressures and pressure amplitudes at the 12.5%, 50% and 87.5% span stations are shown in figure 5, figure 6, and figure 7 . The time-averaged pressures are in good agreement with the experimental data on each of the pressure surfaces. The results on the suction surface are typical of earlier comparisons with a single-stage version of this experimental configuration. In particular, the computed suction-surface pressures follow the general trends shown in the experimental pressures, but are somewhat over predicted. The pressure amplitudes are in general larger than those seen on the first-stage stator and are fairly evenly distributed from leading edge to trailing edge. These amplitudes are due to the rotor passing through the potential fields of both the first- and second-stage stators as well as from the convection of the first-stage stator wakes across the surface of the rotor. Sharma et al. indicates that the periodic passage of the first-stage stator secondary and wake flows interact with the rotor secondary flows to further increase unsteadiness, in particular on the suction surface of the rotor.
Time-averaged pressures and pressure amplitudes for the second-stage stator for the 12.5%, 50% and 87.5% span stations are included in figure 8, figure 9, and figure 10 . The comparison with experiment is again good for both the suction and pressure surfaces at the three span stations. Pressure amplitudes are generally slightly larger near the leading edge as compared to the trailing edge of the second stage stator. This is partly due to the influence of the upstream rotor potential field. The first-stage rotor wake also diffuses and dissipates as it convects along the second-stage stator surface, so the wake/vane interaction is strong at the leading edge. In addition, the pressure amplitude is affected by the convected wake from the first-stage stator and the wake/wake interactions that occur between the upstream stator and rotor wakes. For this large axial gap turbine, these interactions have only a minimal influence on the time-averaged pressure field within the turbine. However, Rangwalla et. al have shown that the time-averaged flow field can be affected by these interactions in a turbine with realistic axial gaps at flight operating conditions.
Several factors are present in the computation which could affect the amplitude and phase of the unsteady pressures within the turbine. Madavan et al. have shown that a modification of the stator to rotor airfoil count to one-to-one in a single-stage turbine may have a significant affect on both the amplitude and phase of unsteadiness. The current computation has reflective-type boundary conditions at the inlet and exit, which could change the unsteadiness within the turbine. This effect would be most evident on the first-stage stator pressure surface because there is no natural unsteadiness induced by upstream turbomachinery components. In addition, numerical dissipation may weaken the strength of blade and vane wakes as they convect downstream in the turbine. This reduces the pressure amplitude predicted by the computation in downstream blade and vane rows of multistage turbomachines.
Time-averaged pressure contours for an axial plane 17% aft of the first stage stator are presented in figure11 and figure 12 . Experimental data are presented in figure 13 while computational results are presented in figure 14
. The computational results were obtained by averaging the static pressure over a cycle at each point across the radius and circumference of a single passage. The passage has been duplicated for the purposes of a clearer comparison with experiment. The increment between contours levels is for both the computation and the experiment. However, the contours themselves may be affected by the difference in the distribution and number of points in the experimental and computational datasets.
The qualitative agreement between the results and the experimental data is good. Contours for both datasets show steeper gradients for the pressure-side of the wake, in particular near the tip. Total pressure contours for the same axial plane behind the first stage stator are shown in figure 12 and figure 14 for the computed results and experimental data respectively. The computed results show the same qualitative trends as the experimental results, including the movement of the tip and hub passage vortices toward the hub.
Figure 15 and figure 16 show computed and experimental time-averaged pressure contours at an axial plane 36% aft of the rotor. The contours are time-averaged relative to the rotor motion in order to show structures in the rotating reference frame. Both the experiment and computation show a much smaller range in static pressure behind the rotor than the first-stage stator. However, the computation appears to have more structure in the flow behind the rotor than does the experimental data. This may be in part because of passage-to-passage periodicity enforced on the computation. Because the ratio of first-stage stator and rotor airfoils are different, the experimental rotor flowfield is not periodic from one passage to the next because of the influence of the upstream stators. Figure 17 and Figure 18 show rotary total pressure contours at the same axial location behind the rotor. Here the qualitative agreement is good because the rotary total pressure gradients produced by the rotor dominate the aperiodic total pressure increase induced by the upstream stators. The low total pressure region formed by the merger of the hub and tip vortical flows is seen in both the computational results and the experimental data. The computed tip leakage flow is less evident than that of the experiment.
Computed and experimental static pressure contours 14% aft of the second-stage stator are presented in figure 19 and figure 20 respectively. Absolute total pressure contours for the computed results and the experimental data are presented in figure 21 and figure 22
respectively. Both computed total and static pressures are again in good qualitative agreement with the experimental data with both hub and tip low total pressure regions evident. A slight aperiodicity absent in the computed results but seen in the experimental results is because of the different numbers of stator airfoils in the upstream stator rows. This probably also accounts for the additional structure in the passages between the wakes in the computation.
Computed and experimental time-averaged surface streamlines on the hub and airfoil surfaces are presented in figure 23, figure 24, figure 25, and figure 26 The experimental streamlines were visualized by injecting ammonia through static pressure taps and allowing it to interact with Ozalid paper attached to the surface. The computed streamlines were determined by averaging the components of velocity over a cycle and then integrating the streamlines through the averaged velocity field. The computed hub passage surfaces are duplicated for the purposes of allowing a clearer comparison with experiment. In some figures, streamlines from the rearward-facing surface are visible through the open tips because of graphic-software difficulties with hidden-line removal.
Computed and experimental results for the first-stage stator suction surface are shown in figure 23 and figure 24 , respectively. The computed results follow the trends indicated by the experiment including the relative two-dimensionality of the flow in the midspan region, the hub-direction migration of the tip vortical flow and the greater strength of the tip vortex relative to the hub vortex. In addition, the footprint of the hub passage vortex and the general migration of the flow from the pressure surface to the suction surface are also in good agreement with the experimental data. Results for the first-stage stator pressure surface streamlines for the computed and experimental are shown in figure 25 and figure 26
respectively. Both show a mild outward radial flow near the tip which is induced by the tip passage vortex. The pressure gradient driven radial flow toward the hub in both figures aft of approximately 1/3 chord from near the hub to approximately 3/4 span.
Computed and experimental time-averaged surface streamlines for the suction surface of the rotor are shown in figure 27 and figure 28 , respectively. The footprint of the hub and tip passage vortices are clearly evident in both figures. The hub and tip passage vortices in both figures converge at the trailing edge near 60% of midspan. The hub surfaces also both show the general migration of the flow from the pressure surface to the suction surface. The results for the pressure-side streamlines are shown in figure 29 and figure 30 for the computed results and experimental data respectively. The radial flows toward the tip near the leading edge in the computed results compare well with those of the experimental data.
Suction surface visualizations for the second-stage stator computed results ( figure 31 ) and experimental data ( figure 32 ) also compare well. The extent of the hub and tip vortical flow regions appears to be well predicted as is the footprint of the leading edge passage vortex and the general flow from the pressure to suction surfaces at the hub. The pressure surface results are shown in figure 33 and figure 34 for the computed and experimental streamlines respectively. As pointed out by Adamczyk, the flow field shows more three-dimensionality than would be expected for a steady cascade flow. The computed results have a region of low-momentum fluid with some radial components of velocity which exists just aft of the leading edge on the pressure surface of the second-stage stator. This region of low-momentum fluid is probably induced by the presence of the mid-passage rotor vortex. This separation is not evident in the experimental results, however the experimental data is sparse in the region where the computation shows the region of separation. In particular, the third streamline from the bottom is foreshortened. That may indicate that the ammonia flow through that tap was less than of the surrounding taps, or that a separation region may have inhibited the axial progression of the flow. An adverse pressure gradient just aft of the tip does exist for both the computed and experimental data in figure 9
. The width of the surrounding experimental streamlines suggests that the flow on the pressure surface of the stator is quite unsteady and does have fluctuating velocity components over a cycle. The coalescence of streamlines near the tip region is similar between the two visualizations as is the inward turning of the flow very near the hub.
A third-order-accurate, upwind-biased, Navier-Stokes zonal code (STAGE-3) has been developed to compute three-dimensional flows in multistage turbomachines. Overlaid grids are used to resolve viscous flows near the hub, casing and airfoil surfaces. Rotor grids move relative to stator grids, allowing the rotor/stator interaction problem to be investigated. Flexible database and bookkeeping systems are used to allow flows to be computed in turbomachines with any number of stages. The flow within a - stage turbine has been computed and time-averaged pressures over a range of spanwise stations compared well with experimental data for the configuration. Good agreement was obtained for surface streamlines on each airfoil and static and total pressure contours in axial planes aft of each airfoil. Physical and computational mechanisms affecting the prediction of pressure amplitudes on blades and vanes were discussed. Physical mechanisms such as wake/airfoil and wake/wake interactions generate a complex unsteady flowfield in the later stages of a multistage turbomachine. A time-accurate, viscous multistage method can provide insight into the fluid physics found in complex flowfields.
A capability for three-dimensional computations of unsteady flows through multistage turbomachines has been demonstrated. These results represent an initial validation of the STAGE-3 code for unsteady computations through multistage turbomachines. This multistage turbine will be used to validate the multi-airfoil capability of the STAGE-3 code. Further validation is also currently underway for a þstage compressor configuration.