The low-energy flow inside the endwall boundary layer accumulates under the transverse pressure gradient from the blade pressure side to the suction side, and it separates under the streamwise unfavorable pressure gradient, as is well known, giving rise to the corner separation. When the stator hub gap is generated, a portion of the flow near the pressure side of the gap will pass through the gap and inject into the suction side, creating the leakage flow. This happens because of the pressure differential between the blade pressure side and the suction side. In other words, the leakage flow has the opposite effect on the endwall boundary layer compared to corner separation. On the one hand, the transverse component of the leakage flow momentum can prevent the transverse migration of the low-energy flow inside the endwall boundary layer. On the other hand, the streamwise component of the leakage flow momentum can help the flow to resist the streamwise adverse pressure gradient. Therefore, the interaction between the hub corner separation and the leakage flow is in fact the flow mechanism of the hub clearance’s influence on the stator’s three-dimensional flows.
5.1. Interaction between the Hub Corner Separation and the Leakage Flow
In this section, to clarify the interaction between hub corner separation and leakage flow, and the limiting streamline on the blade suction surface and the hub, the 3D streamline originates from the stator hub gap and the stator inlet boundary layer is depicted along with the stator outlet flow field (the axial velocity).
Figure 9 shows the 3D flow structures inside the stator passage at various stator hub gap configurations in the DE condition. At 0.00τ, only very weak corner separation occurs at the hub corner, so the 3D flow structure is relatively simple. The accumulation of the low-energy flow inside the endwall boundary layer towards the hub-SS corner is significant from the limiting streamline on the hub, and the slight separation of the accumulated low-energy flow is easy to see from the limiting streamline on the blade suction side.
At 0.25 τ, it is still clear that low-energy flow is accumulating within the endwall boundary layer at the hub–SS corner, and it also seems that the backflow on the blade suction side is stronger near the trailing edge. Backflow also occurs on the endwall, as can be observed from the limiting streamline on the hub, and no obvious leaking vortex can be detected from the 3D streamlines coming from the hub gap. This means that in this instance, the presence of the stator hub gap does not prevent the hub corner separation; rather, the leakage flow separates and causes the corner stall to occur. Therefore, in this case, the induced low-velocity region of the stator outlet is much larger than that at 0.00 τ.
At 0.50 τ, backflow on the blade suction side close to the trailing edge vanishes when the accumulation of low-energy flow inside the endwall boundary layer weakens significantly and approaches the hub–SS corner. Additionally, a clear separating line shows up on the hub where the boundary layer and leakage flow connect, and a line of attachment shows up on the blade suction surface, indicating the formation of the leakage vortex, which is also visible from the 3D streamlines coming from the hub gap. Therefore, the hub corner separation is removed in this instance and the leakage flow dominates the hub corner, resulting in a significantly smaller induced low-velocity area at the stator outlet than at 0.25 τ, but still bigger than at 0.00 τ.
At 1.00 τ, the concentration of low-energy flow within the endwall boundary layer towards the hub–SS corner weakens even further. The separating line on the hub moves further away from the blade suction side, the attachment line on the blade suction side indicated above vanishes, and the leaking vortex completely separates from the blade suction surface. Additionally, the leaking vortex grows stronger as seen by the 3D streamlines coming from the hub gap. As a result, in this situation, the induced low-velocity area changes from being smaller in one direction (spanwise) to being bigger in another (pitchwise).
At 1.50 τ and 2.00 τ, the feature of the 3D flow structures is similar to that at 1.00 τ, but the leakage vortex leaves further away from the blade suction side, so the accumulation of the low-energy flow inside the endwall boundary layer towards to the hub–SS corner becomes even weaker.
Figure 10 shows the 3D flow structures inside the stator passage at various stator hub gap configurations in the NS condition. At 0.00 τ, a focus point and backflows appear on the hub, which is the typical feature of the corner stall. From the 3D streamlines originating from the boundary layer, it also can be seen that the low-energy flow begins to roll up near the blade leading edge, and around the blade trailing edge it goes back to upstream. Hence, in this case, the hub corner stall happens; as a result, the induced low-velocity region at the stator outlet increases enormously.
At 0.25 τ, though a small hub gap is introduced, the backflows still exist, as can be seen from the limiting streamline on the hub and the 3D streamlines originating from the boundary layer. However, compared with the situation at 0.00 τ, the focus point on the hub becomes hard to see, and the place where the low-energy flow begins to roll up moves far away from the blade leading edge. That is to say, in this case, the hub corner stall still exists, but becomes weaker in contrast to that at 0.00 τ. Hence, the induced low-velocity region at the stator outlet becomes smaller.
At 0.50 τ, no limiting streamlining on the hub points upstream, and the concentration point on the hub totally disappears. The hub corner stall, therefore, vanishes. The hub corner separation still persists, though, and no concentrated leakage vortex arises, as can be seen from the limiting streamline on the blade suction side and the 3D streamlines coming from the hub gap and the boundary layer. Because of this, the hub corner is still primarily affected by the corner separation in this example, but when the corner stall is removed, the induced low-velocity zone at the stator outlet shrinks significantly.
At 1.00 τ, the separating line on the blade suction side (near the hub) is replaced by an attachment line, and the concentrated streamwise leakage vortex can be seen from the 3D streamlines originating from the hub gap. Hence, in this case, the hub corner separation disappears thoroughly and the leakage flow dominates the hub corner, so the induced low-velocity region at the stator outlet becomes even smaller. However, as the leakage flow is so close to the blade suction side, some interaction still exists between the leakage flow and the blade surface boundary layer. This is similar to that at the DE condition of 0.50 τ.
At 1.50 τ, the attachment line that was previously present on the blade suction side (near the hub) vanishes and the separation line on the hub moves further away from the blade suction side, i.e., the leakage vortex breaks away from the blade suction surface thoroughly. In addition, from the 3D streamlines originating from the hub gap, it can be seen that the leakage vortex becomes even stronger. Hence, in this case, the induced low-velocity region becomes even smaller. This is similar to the DE condition at 1.00 τ.
At 2.00 τ, the features of the 3D flow structures are similar to that at 1.50 τ, but the leakage vortex moves further away from the blade suction side, so the accumulation of the low-energy flow inside the endwall boundary layer towards to the hub–SS corner becomes even weaker. This is similar to the DE condition at 1.50 τ.
From above, the interaction between the leakage flow and the hub corner separation varies with the stator hub gap, and at different mass flow conditions the flow mechanism is also different. Generally speaking, when the stator is shrouded or a small hub gap is introduced, the hub corner will be dominated by the corner separation/stall, and when a big hub gap is introduced the hub corner will be dominated by the leakage flow. The specific situation depends on the momentum of the leakage flow.
5.2. Effect of the Hub Clearance on the Leakage Flow
As mentioned above, the transverse component of the leakage flow momentum can prevent the transverse migration of the low-energy flow inside the endwall boundary layer and the streamwise component of the leakage flow momentum can help the flow to resist the streamwise adverse pressure gradient. When the stator hub gap varies, the leakage flow velocity changes too.
Figure 11 and
Figure 12 show the distribution of the normalized streamwise component (tangent to the blade suction side) and the normalized normal component (perpendicular to the blade suction side) of leakage flow velocity at the suction side of the stator hub gap along the blade chord, respectively. Additionally, to make a comparison with the situation with no stator hub gap, in
Figure 11, the normalized streamwise component of the velocity of the separating flow in the hub corner at 0.00 τ is also depicted.
It can be seen that in the DE condition, at most places the streamwise velocity of 0.00 τ is higher than that with a stator hub gap. The smaller the hub gap is, the lower the streamwise velocity will be, and so is the normal velocity. At 0.25 τ, the normal velocity of the leakage flow is too small to prevent the transverse migration of the low-energy flow inside the endwall boundary layer; instead, due to the very low streamwise velocity, when the flow in the boundary layer mixes with the leakage flow, the streamwise velocity of the flow in the boundary layer decreases sharply. Hence, under the effect of the streamwise adverse pressure gradient, the boundary layer separates. At 0.50 τ, both the streamwise velocity and the normal velocity of the leakage flow increases significantly. The momentum is the product of the mass flow and the velocity, and the mass flow is the product of the stator hub gap and the velocity; hence, the momentum of the leakage increases enormously. For example, the averaged normal component of the leakage flow velocity at 0.50 τ is 1.4 times of that at 0.25 τ; as a result, the averaged normal component of the leakage flow momentum at 0.50 τ is 4.0 times that at 0.25 τ. Hence, in this case, the transverse migration of the low-energy flow inside the endwall boundary layer is prevented effectively. Moreover, the averaged streamwise component of the leakage flow momentum at 0.50 τ is also about 4.0 times that at 0.25 τ; hence, after the mixing with the leakage flow, the streamwise momentum of the flow in the boundary layer is still able to resist the streamwise adverse pressure gradient. Therefore, in this case, no obvious backflow generates, and the hub corner is dominated by the leakage flow. When the stator hub gap increases continuously, both the streamwise velocity and the normal velocity of the leakage flow increase further, so the intensity of the leakage flow increases. However, as the stator hub gap exceeds 1.00 τ, the growing rates of both the streamwise velocity and the normal velocity of the leakage flow become very small.
In the NS condition, when the stator is cantilevered, both the streamwise velocity and the normal velocity of the leakage flow are almost the same as that of the DE condition. However, for the shrouded stator, as the hub corner separation turns into the corner stall, many backflows are generated in this condition. Hence, in most places, the streamwise velocity of 0.00 τ is negative. At 0.25 τ, though, the normal velocity of the leakage flow is still too small to prevent the transverse migration of the low-energy flow inside the endwall boundary layer; the mixing between the leakage flow and the separating flow can also increase the streamwise velocity of the flow in the boundary layer. Therefore, even though the hub corner stall still exists, the intensity of it becomes weaker. At 0.50 τ, as mentioned above, both the normal component and the streamwise component of the leakage flow momentum at 0.50 τ are about 4.0 times that at 0.25 τ. Even so, the normal velocity of the leakage flow is still not big enough to prevent the transverse migration of the low-energy flow inside the endwall boundary layer. However, after mixing with the leakage flow, the streamwise momentum of the flow in the boundary layer is big enough to resist the streamwise adverse pressure gradient. Hence, in this case, the hub corner stall turns back into corner separation. At 1.00 τ, both the streamwise velocity and the normal velocity of the leakage flow increase continuously, and so does the momentum of the leakage flow. This time, the hub corner separation completely vanishes, and the leakage flow predominates the hub corner because the normal velocity of the leakage flow is high enough to prohibit the transverse migration of the low-energy flow inside the endwall boundary layer. There is no need for further explanation, because the scenario is identical to the DE circumstance when the stator hub gap is continually increasing.
5.3. Discussion of the Optimum Hub Clearance
As mentioned above, the hub corner will separate or stall when the stator is shrouded or when a small hub gap is introduced; when the stator hub gap is sufficiently wide, the leakage flow will predominate the hub corner. Large flow blockage and flow loss are caused by hub corner stalls, and when leakage flow is too strong, the interaction of the leakage flow and the mainflow also causes significant flow blockage and loss. As a result, there typically is an ideal stator hub gap that results in the best compressor performance.
The circumferentially averaged normalized axial velocity at the stator outlet and the circumferentially averaged total pressure loss coefficient of the stator are shown in
Figure 13 and
Figure 14, respectively. In the DE condition, the flow loss and the flow blockage of the shrouded stator are the lowest. At 0.25 τ, the hub corner separation turns into a corner stall, so the spanwise range and the value of the flow blockage or flow loss increase abruptly; at 0.50 τ, the hub corner stall turns back to corner separation, so the spanwise range of the flow blockage or flow loss decreases significantly, but the value does not decrease; at 1.00 τ, the hub corner separation is replaced by the leakage flow, so the spanwise range of the flow blockage or flow loss decreases further, but the value becomes larger. As the stator hub gap increases continuously, both the spanwise range and the value of the flow loss begin to increase again, which is due to the strength of the leakage flow. In the NS condition, the situation is a little different from that in the DE condition. In this condition, the flow loss and the flow blockage of the shrouded stator are the highest. As the stator hub gap increases from 0.00 τ to 1.00 τ, both the spanwise range and the value of the flow blockage and flow loss decrease continuously. However, when the stator hub gap increases from 1.00 τ to 1.50 τ, the spanwise range and the value of the flow blockage decrease and the spanwise range of flow loss decreases, but the value increases, and when the stator hub gap increases from 1.50 τ to 2.00 τ, the spanwise range of the flow blockage and flow loss almost do not change and the value increases continuously.
The change in the cross section-averaged total pressure loss along with the stator hub gap is displayed in
Figure 15 to provide a clear understanding of the impact of the stator hub gap on the stator’s flow loss. As can be observed, the ideal stator hub gap should be in the range of 0.50 τ to 1.00 τ, which is consistent with the study of compressor performance covered in
Section 4.1. From the analysis above, this optimal stator hub gap only occurs when the induced leakage flow is similar to the corner separation, meaning that under situations of high mass flow rates, the induced leakage flow could reduce the corner separation, and under conditions of low mass flow rates, it could remove the corner stall, but does not result in additional mixing loss. The compressor performance will increase greatly under conditions of low mass flow rate and remain unchanged under situations of high mass flow rate.
As is known, both leakage flow and corner separation are promoted by the transverse pressure difference between the blade pressure side and the suction side. Hence, if without loss, the leakage flow is supposed to be comparable to the corner separation. However, in practice, due to the existence of the flow viscosity, loss will generate when leakage flow passes through the gap between the blade tip and the endwall. Therefore, for incompressible flow, the normal velocity can be calculated by Storer and Cumpsty [
20]:
Here,
CD is the discharge coefficient, and is dependent on the viscosity in the gap.
Figure 16 shows the variation in
CD along with the stator hub gap. It can be seen that in the DE condition and in the NS condition the value of
CD is almost the same. When the stator hub gap is small, the viscosity in the gap is relatively high, so
CD is small. As the stator hub gap increases, the viscosity in the gap becomes relatively lower, so
CD increases. Until the stator hub gap is bigger than a certain value, the viscosity in the gap can be ignored and
CD almost does not change. Similarly, the streamwise velocity of the leakage flow is also dependent on the viscosity in the gap. Define the coefficient
CL as the ratio of the streamwise velocity at the suction side of the gap to the streamwise velocity at the pressure side of the gap. In
Figure 16, the variation in
CL along with the stator hub gap is also displayed. It can be seen that the distribution of
CL is similar to that of
CD, i.e., it increases with the increment of the gap but almost does not change when the gap is bigger than a certain value. Hence, the optimum stator hub gap should be this critical value where the viscosity in the gap can be ignored.
Actually, through abundant data, Rains [
21] has already proposed that the flow viscosity in the gap can only be disregarded when
, where
is the Reynolds number based on the blade chord and the mainflow velocity near the hub,
is the ratio of the blade maximum thickness to the blade chord, and
is the ratio of the clearance size to the blade maximum thickness. The Reynolds number for the compressor under investigation is around
, and the blade maximum thickness close to the hub is 10%. Hence, when the stator hub gap is 0.25 τ, the corresponding
R equals 25; as a result, a lot of loss is generated as the flow passes through the gap and the leakage flow is much weaker than the corner separation; when the stator hub gap is 0.50τ, the corresponding
R equals to 100, which is very close to the critical value, 125, so in this case the leakage flow is nearly comparable to the corner separation. By this analogy, the optimum stator hub gap should be 0.56τ, which coincides well with the results shown in
Figure 15 and
Figure 16. Hence,
could be a simple metric to choose the optimum stator hub gap.
To verify the metric above, some typical results from published papers are discussed here, as shown in
Figure 17. Detailed information about the quoted results is summarized in
Table 1. It can be seen that, for Wang’s results [
22], in the P1 condition (where corner stall does not occur) when
, the loss reaches the local minimum and in the P2 condition (where corner stall happens) when
, the loss reaches the minimum, which coincides well with the results in this paper. In Chen’s [
23] and George’s [
7] results, when
, the loss is also near the minimum. In Gbadebo’s results [
9], as the stator hub gap increases, the loss increases firstly, then decreases, and at last increases again; the evolution tendency of the loss with the variation in the gap is similar to the results in this paper, and when
the loss does not reach but is very close to the local minimum. In Sakulkaew’s results [
11], when
the compressor efficiency is about the maximum. In Yang’s results [
24], it is hard to judge if this metric is right as there are no investigated points between zero clearance and
, but for the results where
, the evolution tendency of the loss with the variation in the gap coincides well with the results in this paper. In McDoungall’s results [
13], in both the DE and NS conditions the pressure-rise coefficient increases at first, then decreases as
R increases, and the pressure-rise coefficient reaches the maximum around
. Hence, generally speaking, the metric proposed in this paper is feasible.