In this section, the main findings of the investigation are analyzed. At first, the predictions of the blade loading are compared with experimental measurements, with a particular emphasis on the effect of the Reynolds number and impingement of shocks on the blade suction side. The last sections are dedicated to the discussion of the wake development and the prediction of the boundary layer from both models.
3.1. Blade Loading
The predictions of the blade loading, both for the
-
and the
k-
-
models, are shown in
Figure 5a–c, respectively, for the cases at Re
= 70,000, Re
= 100,000, and Re
= 120,000. Crosses represent experimental points, while numerical predictions from the
-
and the
k-
-
models are reported with continuous and dashed lines, respectively. The two models present a quite different behavior at low and moderate exit Mach numbers (0.7–0.8), while the predictions are very similar at the highest Mach number (0.95).
Regarding the
-
, the prediction of the blade loading follows the experimental trend for the whole range of numerically tested conditions. At the lowest Mach number, the experimental blade loading evidences the occurrence of a separation bubble with a consequent reattachment on the blade suction side. This is clearly seen, especially at the lowest Reynolds number (
Figure 5a). For this case, the model fairly predicts the occurrence of the bubble, even though it can be inferred that the size of the separation bubble is too small, which leads to a slight misprediction of the suction side pressure distribution in the range 0.4 < S/S
< 0.9. On the other hand, the reattachment region is adequately predicted and the model recovers the experimental diffusion rate in the last part of the suction side closer to the TE. When the Mach number is increased at the same Reynolds number, the model is still able to correctly capture the loading, especially at Ma
= 0.95, where both the suction side peak isentropic velocity and the later diffusion are captured properly. At Ma
= 0.8 instead, the diffusion is slightly faster than the experimental one. This misprediction was already noticed in
Section 2.4. For the medium and the high-Reynolds cases, the performances of the model are approximately the same. At the highest Mach, the load is correctly captured, while at the lowest Mach, the prediction of the rear diffusion after the formation of the separation bubble is correct but the isentropic velocity is slightly underestimated in the central region of the blade. On the contrary, at Ma
= 0.8, predictions improve at a higher Re
. As far as the
k-
-
is concerned, the model predicts an open separation bubble with no evident reattachment at the lowest Mach numbers. For Ma
= 0.95, the results are comparable to the ones obtained with the
-
, with a fair prediction of the peak velocity point and the later diffusion down to the TE. The
k-
-
also exhibits a worse prediction of the pressure side separation bubble occurring in the region 0.15 < S/S
< 0.3 for all tested conditions.
A more detailed analysis of the effect of the Reynolds and Mach numbers on the loading distribution obtained using the
-
model is presented in
Figure 6 and
Figure 7, respectively, where the acceleration parameter K
and its derivative along the curvilinear coordinate are shown. The figures also report the experimental values of the acceleration parameter only, which have been obtained via a spline interpolation of the experimental insentropic Mach number distribution on the blade suction side.
As can be observed in
Figure 6a, the Reynolds number majorly affects K
values in the central region of the blade, but, in general, does not change the loading distribution. The zero of K
, which indicates the peak velocity point over the suction side does not change with Reynolds number, while the relative minimum position located in the rear part of the suction side moves downstream along the curvilinear coordinate. This is caused by a change in the separation bubble formation and growth at a different Reynolds number. From a loading perspective, the size and the location of the separation bubble recovery point and reattachment point can be inferred from the distribution of the acceleration parameter [
26], even though the exact location of the separation point and the reattachment point are analyzed better by resorting to the wall shear stress. The relative minimum of K
approximately indicates the initiation of a separation bubble (at S/S
0.52 for Re
= 70,000), while the peak (S/S
0.76 for Re
= 70,000) indicates the recovery point, where the separation bubble reaches its maximum size. By increasing the Reynolds number, the peak moves upstream, and the acceleration curve is flatter, indicating a lower size of the bubble. Moreover, the diffusion in the rear part at a lower Reynolds is faster, due to a later reattachment of the bubble itself. This point can be approximately located at the peak of the acceleration derivative (
Figure 6b), which moves from S/S
= 0.85 at Re
= 120,000 to S/S
= 0.95 at Re
= 70,000.
The Mach number mainly affects the acceleration in the front part, bringing the peak velocity point from S/S
0.37 for Ma
= 0.7 to S/S
0.61 for Ma
= 0.95, as can be seen in
Figure 7a by tracking the K
= 0 abscissa for the three Mach numbers. Also, with an increase of the Mach number, the acceleration derivative tends to be 0, downstream of the velocity peak at S/S
0.7 (see
Figure 7b), and the K
curve tends to be flatter in the high-Mach case if compared to the low-Mach number case. This is caused by the fact that the separation bubble tends to be suppressed and does not retain a major effect on the acceleration over the profile. Another major feature of the high-Mach case is the presence of a small bump in
close to S/S
0.7. This is not present in the other cases and is due to the presence of a shock in the throat section of the cascade, as shown in
Figure 8 using the numerical Schlieren.
The simulations underestimate the experimental acceleration parameter at a low-Mach (
Figure 6) for all Reynolds numbers in the front region up to S/S
= 0.7. At Re
= 70,000, numerical predictions underestimate the acceleration parameter until S/S
= 0.91, thus including the minimum value. At the highest outlet Reynolds number, the underestimation stops at S/S
= 0.7 and the minimum value is overestimated. The best prediction in the aft region is obtained at the medium outlet Reynolds number, where the minimum value of the acceleration parameter is retrieved correctly.
A better agreement between simulations and experiments in the front part of the blade is found at Re
= 120,000 (
Figure 7). For the medium Mach case, the simulations correctly capture the experimental values up to S/S
= 0.8. Experiments show an almost constant diffusion rate in the region 0.85 < S/S
< 0.95, while the simulations show a clear minimum at S/S
= 0.91 with a less smoother trend. At the highest Mach number, simulations overestimate the acceleration in the region 0.51 < S/S
< 0.61. Moreover, the diffusion rate is faster in the region 0.68 < S/S
< 0.88, where the acceleration parameter is underestimated. Finally, an overestimation is found in high-Mach conditions for S/S
> 0.88.
3.2. Wake Prediction
The flow field measured at Plane 06 was compared to the numerical results from both models in terms of the mass-flow-averaged total pressure loss and deviation angle (Equation (
10)) in
Table 4 and
Table 5.
Regarding the total pressure losses, both models manage to satisfactorily predict the experimental values. The maximum discrepancy occurs for the Ma
= 0.95 Re
= 70,000 case when using the
-
model. The two models tend to overestimate the losses, but in both cases, they manage to correctly retrieve the experimental trend which sees an increase of the losses with Ma
. Moreover, both models predict a reduction of losses when Re
is increased from 70,000 to 120,000 for Ma
= 0.9. The same trend has been found for the deviation angle.
Table 5 also reports the experimental uncertainty over the measured angle. It can be observed that the predictions obtained with the
-
model are generally in better agreement than the ones produced by the
k-
-
in terms of the deviation angle.
Figure 9 shows the predicted shape of the wake, compared with the experimental measurements. The effect of the Reynolds number is shown in
Figure 9a, where the wakes are compared at Ma
= 0.9. Both models overpredict the peak of losses at the center of the wake and underestimate the wake spreading, as already mentioned in
Section 2.4. The
-
model performs better than the
k-
-
model, especially at Re
= 120,000, with a maximum deviation from the experimental measurement equal to ≈0.045. However, the
-
exhibits very little sensitivity to the Reynolds number at Ma
= 0.9. Indeed, the peak of the losses only reduces by ≈0.01 when the outlet Reynolds number is increased from 70,000 to 120,000. The experiments show a reduction of ≈0.02. A similar reduction in the peak of losses is found in the predictions from the
k-
-
model (≈0.023).
The effect of the Mach number is instead shown in
Figure 9b, for Re
= 70,000. The two models perform similarly for the lowest isentropic Mach (0.7), but the
k-
-
performs sensibly worse at the highest Mach number, predicting a very thin wake, with a very high peak in the losses which exceeds the experimental one by more than 50%. The
-
also retrieves the experimental increase in the peak of losses (≈0.05) passing from Ma
= 0.7 to Ma
= 0.95. It is worth mentioning that the
k-
-
wakes are slightly misaligned with respect to the experimental one, being a little closer to the blade SS. On the other hand, the
-
correctly predicts the position of the wake for all the combinations of outlet isentropic Mach and Reynolds numbers. It must be underlined that the two models have a similar treatment of the wake region, both implementing a shear stress transport term, so the different prediction of the wake region is to be addressed to the different state of the boundary layer in the rear part of the SS, which affects the wake growth and position, especially in the case of open and long separation bubbles. This topic will be discussed in
Section 3.3, where the boundary layers predicted by the two models will be compared.