3.1. Deformations Influence of Elliptical Leading Edge
Figure 4 shows the distribution of
Cp in the front of the surfaces of ORI; 10% BLUNT, the BLUNT at
ε = 10% and 10% WEDGE, respectively, at
Tu = 5%. The relative arc length
S is used for the X-axis, which demonstrates the relationship between pressure and the distance of development in the early stage. The
Cp distribution of the ORI is relatively smooth. By contrast, the 10% BLUNT and 10% WEDGE appear as spikes because of the metal angles. After the spikes, the
Cp of different leading edges were almost the same, indicating that the deformations at
ε < 10% can only affect the early flow environment. It can be seen that the curve of the 10% BLUNT is nearly horizontal at the beginning and nearly vertical before the spike, showing that the BLUNT can represent the worst condition at certain
ε values and the WEDGE can represent an intermediate level.
Shape factor
H (defined in Equation (3)) and the normal average turbulence intermittency
γ’ reflect the development process of the suction surface boundary layer in a more detailed manner. The definition of
γ’ is
where
γ is the turbulent intermittency.
H can reflect the velocity profile in the boundary layer. The increase of H indicates that the velocity profile has deteriorated due to the wall curvature or adverse pressure gradient. The laminar boundary layer separates when H reaches 2.5 to 3 and reattaches when H comes back to around 3. The turbulent boundary layer stays when H is around 2. The maximum of H in the boundary layer separation region Hmax indicates the strength of the separation bubble. γ’ can reflect the process of boundary layer transition. It is less than 0.2 for the laminar boundary layer. The transition appears when the value of γ’ begins to increase and competes when the value of γ’ approximates to about 0.9.
At
Tu = 0, The ORI boundary layer can be regarded as the reference, as shown in
Figure 5. A few spikes of
H and
θ formed before
S = 20.78 (the reattachment point) because there are some strong laminar separation bubbles. In the separation region, the
Hmax amounts to 7.17,
θ reaches 0.516 mm, and
γ’ achieves 0.087. After reattachment,
γ’ stays at a low level, showing that the boundary layer stays in the laminar region. It can be seen from
Figure 5 and
Figure 6 that different boundary layers all have steady states after the point of about
S = 20. Therefore, this point can be taken as the investigation point
S0 (it also applies to the following research). For ORI, at
S0,
θ starts to rise from 0.452 mm in a nearly linear way.
γ’ achieves 0.041 and it grows slightly.
The BLUNT series promotes the transition process and suppresses separation. For 2% BLUNT, the reattachment point is put forward to
S = 12.3. At
S0,
θ reaches 0.455 mm and
γ’ achieves 0.106 (+0.065). Unexpectedly, the WEDGE series leads to stronger separation, as shown in
Figure 6. For 2% WEDGE, the reattachment point is at
S = 13.99, at
S0,
θ achieves 0.559 mm (+0.107 mm), which is obviously higher than the others. It should be noted that for all leading edges, after reattachment,
θ with a higher
γ’ increases faster (this rule applies to other computation conditions).
Figure 7 shows the influence of BLUNT at
Tu = 5% (
Tu = 3% is similar). For ORI, a relatively weak leading edge separation bubble forms. The reattachment point is at
S = 7.58. In the separation region, the
Hmax amounts to 5.56 and
γ’ achieves 0.43. After reattachment,
γ’ drops to 0.12 (relaminarization). However, it grows again. At
S0,
θ reaches 0.445 mm and
γ’ achieves 0.232.
Deformations bring two negative influences: intensifying the strength of the leading edge separation bubble and the promoting transition process. However, the impact degree of either the BLUNT or WEDGE is small unless the transition is completed in a separation bubble. For 6% BLUNT, the
Hmax amounts to 5.59. At
S0,
θ reaches 0.455 mm (+0.01 mm) and
γ’ achieves 0.25. This phenomenon is similar for the WEDGE. However, it is weaker than the BLUNT’s phenomenon. For 6% WEDGE, as shown in
Figure 8, the
Hmax amounts to 5.57. At
S0,
θ reaches 0.45 mm and
γ’ achieves 0.243. It should be noted that 10% BLUNT can be much more influential. The
Hmax reaches 5.7 (+0.14) and the separation directly leads to the transition completion, which leads to a dramatical loss increase. At
S0,
γ’ stays at 0.87 (the total turbulence boundary layer) and
θ reaches 0.56 mm (+0.105 mm) with much more growth than others.
When the inlet turbulence achieves 7%, as shown in
Figure 9 and
Figure 10, the boundary layer separation can directly lead to the complete transition. For ORI, the
Hmax amounts to 4.78. The reattachment point is at
S = 5.7. At
S0,
θ reaches 0.534 mm and
γ’ stays at around 0.87. The main influence of the deformations is putting the transition forward and, thus, increasing the loss. For 10% BLUNT, as shown in
Figure 9, the point where
γ’ achieves 0.6 is put forward from
S = 7.32 to
S = 5.96. At
S0,
θ reaches 0.586 mm (+0.052 mm). Similarly, for 10% WEDGE, as shown in
Figure 10, at
S0,
θ reaches 0.55 mm (+0.016 mm).
Loss penalty ∆
θ is the loss caused by deformations; the definition is
It should be noted that after S0, there is no difference among γ’ in all the series, but deformations also accelerate the increase of θ. This means that ∆θ keeps growing as the boundary layer develops downstream. For instance, the γ’ of ORI and 10% BLUNT both stay at 0.87 after S0, while at S0, ∆θ is 0.052 mm and at S = 40, ∆θ increases to 0.063 mm.
Through the above analyses of the boundary layer parameters at turbulence ranges from 0 to 7%, the influence mechanisms of leading edge deformations can be summarized. This should be divided into three parts. First, at low free-stream turbulence, the boundary layer keeps laminar and has a large leading edge separation bubble, as shown in
Figure 11a. In this condition, relatively serious deformations (such as BLUNT) will make the separation bubble weaker and
θ drops due to the large pressure disturbance. On the contrary, small deformations (such as WEDGE) will make the separation bubble stronger and
θ increases. Second, at the middle level of free-stream turbulence, the boundary layer transition process is slow and the separation bubble is medium, as shown in
Figure 11b. Deformations make a small influence at low
ε levels. With the expansion of
ε, deformations can promote the transition process obviously and bring about higher loss. Third, when the inlet turbulence level is up to 7%, the leading edge separation bubble is very weak because the transition completes just after the separation begins, as shown in
Figure 11c. The main influence of the deformations is in promoting the transition process. In this condition, the loss penalty ∆
θ has a positive correlation with
ε.
It should be noted that instead of free-stream turbulence, the separation bubble and transition process both decide the influence mechanism. These two questions have captured researchers’ attention for decades [
34,
35,
36,
37,
38]. Hatman and Wang [
35] distinguished the separation bubble as three modes, shown in
Figure 12. The scale of the separation bubble and transition speed has a negative correlation. It is very meaningful that there is a one-to-one correspondence between
Figure 11 and
Figure 12. In other words, different influence mechanisms are only suited to different separation modes.
The influence of leading edge deformations can be extended to a bigger incidence range. Walraevens and Cumpsty’s study [
5] showed that increasing incidence can generate transition and suppress the separation bubble, which makes the boundary layer similar to a higher turbulence. Therefore, it can be speculated that the influence of the leading edge deformations increases with the incidence.
Figure 13 shows the result of ORI and 6% BLUNT at
i = 2 deg. For ORI, at
Tu = 0, the leading edge separation is extra strong. The
Hmax amounts to 11.2 and the reattachment point is at
S = 10.14. At
S0,
θ reaches 1.43 mm. For 6% BLUNT, the
Hmax amounts to 10.41 and
θ reaches 1.52 mm (+0.09 mm) at
S0. With the turbulent inlet, the
Hmax and
θ decrease obviously, indicating that a short transitional separation bubble occurs, which is similar to the condition at
i = 0 deg and
Tu = 7%. As expected, the deformation influence is also similar to that condition. Taking the result at
Tu = 3% as an example, for ORI, the
Hmax amounts to 7.88, and the reattachment point is at
S = 5.47. At
S0,
θ reaches 1.427 mm. For the 6% BLUNT, the
Hmax amounts to 7.46 and the reattachment point is at
S = 5.51. At
S0,
θ reaches 1.525 mm (+0.1 mm).
3.2. Different Leading Edge Sensitivities
In order to investigate the sensitivity of different leading edge shapes, apart from the involved elliptical leading edge, named E1.89 in this subsection, another two series are added to the research, as shown in
Figure 14. One is an elliptical leading edge with a ratio of 1.2, named E1.2. The other one is a continuous curvature leading edge with a ratio of 1.89, named CL. The continuous curvature design uses the three-order Bézier curve [
39]. The design principle is to ensure that the wall curvature is continuous at the tangent point and to keep the shape as similar to E1.89 as possible. The results show that the continuous curvature design could not perform better than E1.89, which is opposite to the previous research articles [
8] (a real blade always has a positive wedge angle and larger leading edge length). However, it is still significant to compare and help summarize the sensitivity rules of different leading edge shapes.
The research compares the θ of different leading edges at the point of S = 50 (approximating to the chord length). Based on the above analysis, the influence mechanisms of the leading edge deformations are different in low free-stream turbulence, middle free-stream turbulence, and high free-stream turbulence, respectively. Therefore, the three states should be distinguished.
The low turbulence condition is shown in
Figure 15. It should be especially explained that in the following figures, there are two vertical axes both in the left and right because the
θ of E1.2 is much higher than others and, thus, it is based on the right-side vertical axis. At the same time, the increments of the two axes are the same in order to reflect the deformation influence. For the E1.89 series, the BLUNT makes the profile loss increase and then decline. The greatest change occurs at
ε = 2% and
θ grows by 10.19% (+0.063 mm). The smallest influence occurs at
ε = 10% and
θ decreases by 1.7% (–0.011 mm).
The E1.2 series has a high θ level as a result of more serious separation. At ε = 2%, θ increases by 2.59% (+0.032 mm) and at ε = 10%, θ decreases by 2.79% (–0.035 mm). The WEDGE’s influence is larger than the BLUNT’s influence. At ε = 2%, the loss of E1.89 increases by 10% (+0.061 mm), and the loss of E 1.2 increases by 11.48% (+0.142 mm). For the CL series, the deformations influence is much more obvious. The greatest decline amounts to 37.25% (–0.372 mm) at 6% BLUNT. This is because the CL leading edge causes a larger separation bubble than E1.89 and the deformations ability of the suppressing separation is strengthened.
The trend of E1.89 and E1.2 is similar. When ε is small, profile loss increases. However, with the expansion of ε, the loss can decrease because of suppressed separation bubble. This conflict effect indicates a maximum loss penalty ∆θmax. The result shows that ∆θmax of E1.89 is larger than 19.58% (+0.121 mm), and larger than 15.6% (+0.193 mm) of E1.2 series.
Figure 16 shows the condition of middle free-stream turbulence. E1.2 only has this condition at
Tu = 3%. At
Tu = 5%, a short transitional separation bubble is formed, and the condition should be classified to high turbulence. Similarly, CL does not have this condition at all. The state at
Tu = 3% is taken as an example. For E1.89, 2% BLUNT reduces
θ by 0.78% (–0.05 mm), and 6% BLUNT rises
θ by 1.67% (+0.011 mm). Indeed, 10% BLUNT leads to transition near the leading edge and raises
θ by 68.6% (+0.45 mm). For E1.2, the trend is similar and gentler. ∆
θ ranges from –0.7% to 0.17%. The influence of WEDGE is similar to BLUNT’s, but the impact degree is much lower.
Middle free-stream turbulence is the most common working condition of turbomachine blades at present. The separation bubble is reduced to a great extent and the transition process is mainly related to free-stream turbulence. This makes the deformations influence insignificant. If the deformations are large enough to affect the transition process, the loss changes obviously. The results of the E1.89 approximate to a second-order function and the results of E1.2 approximate to a linear function. This indicates that E1.89 is more sensitive to deformation even though the θ of E1.89 is still lower than E1.2 at ε = 10%. Besides, it should be noted that for E1.89, the θ at Tu = 5% is almost the upper shift of Tu = 3%, regardless of BLUNT or WEDGE. The separation bubble at Tu = 5% is very similar to that at Tu = 3%, with an incomplete transition and relaminarization. Therefore, a reasonable hypothesis is proposed. The influence of the leading edge deformations cannot be changed by turbulence at a certain mode of the separation bubble.
Figure 17 shows the condition of high turbulence. This condition applies to E1.89 at
Tu = 7%, E1.2 at
Tu = 5%, 7%, and CL at
Tu = 3%, 5%, 7%. In this condition, the size of the leading edge separation bubble is very small and the transition is completed just after the separation begins. The profile loss has a close positive relationship with
ε. At
Tu = 7%, for BLUNT, the
θ of E1.89 ranges from –0.26% to 5.24% (+0.057 mm), and the
θ of E1.2 ranges from 0.09% to 0.8% (+0.01 mm). Their trends both approximate to a second-order function. The
θ of CL ranges from –0.27% to 1.8% (+0.021 mm) in its own trend. Similar to the middle free-stream turbulence condition, WEDGE is consistent with BLUNT, while the impact degree is smaller.
For both the E1.2 series and CL series, the results at different turbulence values are almost parallel, which are similar to the E1.89 series at middle turbulence. This supports the hypothesis that the influence of the leading edge deformations remain the same when the basic boundary layer separation mode is maintained unchanged. This phenomenon can also be applied to the Reynolds number, as
Section 3.3 discusses. Therefore, when the leading edge sensitivity is investigated, rather than other working conditions, only the separation mode is required.
By comparing the sensitivity of different leading edge shapes, it can be concluded that the sensitivity is closely related to the original boundary layer development in the early state. For instance, at low free-stream turbulence, the separation bubble brings a large loss and it is very sensitive to the disturbance brought by deformations. At this time, the excellent leading edge design is endowed with not only good aerodynamic performance, but also an optimized sensitivity. When the free-stream turbulence increases, the loss caused by the turbulent boundary layer gradually occupies a larger percentage. The excellent design can usually effectively restrain the transition process, but the weakness is that it easily becomes very unsteady, making it very sensitive to deformations. Thus, for the leading edge optimization, it is important to consider the influence of deformations on the blade performance. Even if the problem is ignored, the leading edge optimizing still has a practical meaning. As shown in
Figure 15 to
Figure 17, the aerodynamic performance of E1.89 is better than E1.2 in all cases, even though a 63.4% loss penalty occurs.
3.3. Compressor Cascade Investigation
In practice, there are two questions that flat plate research has neglected that should be further discussed. One is the influence of deformations on the velocity profile in the downstream boundary layer. The other is the influence under 3D condition (the region near the hub). Therefore, an investigation on real compressor cascades is conducted.
In the flat plate, after the leading edge separation, the boundary layer develops with zero pressure gradient and zero wall curvature. Thus, the shape factor
H remains stable. However, in the real cascade, the boundary layer inevitably faces a negative pressure gradient and convex wall curvature. Once the early velocity profile has changed, it is difficult to recover and the whole boundary layer is affected. The loss cannot be ignored, especially in the high-lift blade design [
40,
41], where the mid-blade separation must be paid attention to.
Table 1 depicts the cascade physical parameters. The blade is designed by the National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, Beihang University. The leading edge includes two original shapes. One is an ellipse with a ratio of 1.89; the other is an integrated design with a continuous curvature, as shown in
Figure 18. Owing to the larger wedge angle and leading edge length, the integrated design performs better than an ellipse for the disappearance of the leading edge separation. The deformations include the BLUNT and WEDGE of
ε = 2%, as
Table 2 describes. It ensures there is no influence on blade load.
The experiment is carried out in a low-speed tunnel of the laboratory. The conditions are as follows. The environment pressure Pout = 101,000 Pa, the Temperature T = 15 °C, the inlet free-stream turbulence Tu = 2.5%. the inlet relative total pressure Ptin = 80.5 Pa and 160.5 Pa, which makes the Reynolds number based on the leading edge thickness Ret = 2500 and 3500. At Ptin = 80.5 Pa, the middle area separation appears in the suction surface, however, at Ptin = 160.5 Pa, the middle area separation does not appear.
CFX-12.0 is used for numerical simulations. The SST turbulence model and gamma-theta transition model are used similarly for flat plate computation. The type of the mesh is also H-O-H, and there are two layers of the O-type mesh surrounding the blade with local refinement. The convergence of the mesh was investigated similar to the flat plate. The total number of nodes amounts to around five million, as shown in
Figure 19. The simulation used the total pressure inlet and static pressure outlet as the boundary condition. The inlet total pressure and free-stream turbulence were the same as that of the experiments. The outlet static pressure was atmospheric pressure.
The verification of the simulation includes two aspects: the wall pressure confident distribution at the middle height of the blade (
Figure 20a) and exit flow field (
Figure 20b). The difference between the experiment and CFX computation was located mainly in the region near the hub since the hub boundary layer scale is larger in the experiment than the CFX computation. However, the accuracy of the suction surface boundary layer can be ensured before the hub boundary layer influence.
The influence on the boundary layer velocity profile can be studied by analyzing the flow field at a middle height (
z/
l = 0.5) where no 3D flow appears.
Figure 21 shows the
θ,
H and
γ’ among the suction surface at
Tu = 2.5%,
Ret = 2500. For ORI, the same as the flat plate, a laminar separation/short bubble occurs in the leading region.
γ’ increases at the leading edge separation but then decreases after reattachment (relaminarization). While downstream,
H rises again, and a middle area separation occurs. The maximum level of
H in this region is named
Hmax’. In the middle separation region,
γ’ reaches 0.9, indicating that the transition is completed. The condition is similar at
Ret = 3500, thus, it is not further discussed. However,
Hmax’ under this condition does not exceed 3.
For ORI, Hmax amounts to 4.22 and Hmax’ amounts to 3.38, showing that the mid-bubble separation is much weaker than the leading edge’s. For BLUNT, Hmax amounts to 4.65 while Hmax’ amounts to 3.19, meaning that the leading edge separation is promoted (similar to a flat plate) and the middle area separation is suppressed. Remarkably, at S = 8 (after leading edge separation), ∆θ is 0.005 mm at S = 35 (after leading edge separation) and ∆θ is 0.011 mm, The result is similar to the plate form aerofoil research. Therefore, it can be asserted that the reason for the major loss penalty caused by BLUNT is the larger leading edge separation bubble. For CL-ORI, there is only a middle area separation bubble and Hmax’ amounts to 3.6. For CL-WEDGE, the H distribution changes slightly. However, the γ’ decreases significantly, which makes its loss lower than other leading edges.
Although the horseshoe vortex is directly related to the leading edge geometry, in some research, this vortex is not obvious (as in this computation) or it dissipates quickly in the compressor [
42]. Besides, in Goodhand’s research [
27], it is the transition process that leads to the change of corner separation. Based on this fact, the suction surface boundary layer development process at a 0.05 blade height is investigated before it is influenced by the hub boundary layer. In this section, corner separation can be regarded as trailing edge separation [
22].
Figure 22 shows the distribution of
H,
γ’, and
θ for both series. The leading edge separation bubble is slightly put forward and the middle area separation bubble is put forward normally. After this (at the point of
S > 0.4), H increases sharply, indicating a large trailing edge separation (corner separation). For ORI,
Hmax amounts to 4.16 and
Hmax’ amounts to 3.33. For BLUNT,
Hmax amounts to 4.57 and
Hmax’ amounts to 3.28. For CL-ORI,
Hmax’ amounts to 3.82. For CL-WEDGE,
Hmax’ amounts to 3.78. It should be noted that in the 0.1 blade height region,
H is more critical than
θ because the trailing edge separation has a decisive role in the flow. At the point of S = 30, the
H of the ellipse series is about 2.6 and the
H of the continuous curvature series is about 2.3. This means at this point the velocity profile of continuous curvature series is better than the ellipse series.
Figure 23 shows the
Cp (defined in Equation (1)) contours at the 1.1 chord length exit of the BLUNT and CL-WEDGE, as the biggest difference in the four leading edge shapes. The contour of ORI is very close to BLUNT and the contour of CL-ORI is very close to the CL-WEDGE’s. Contours reflect the scale of corner separations. It can be seen that BLUNT is larger than CL-WEDGE, which means the trailing edge separation of BLUNT is larger than CL-WEDGE’s at a 0.05 blade length. Therefore, it can be inferred that the velocity profile penalty caused by leading edge deformation is retained afterwards in the downstream in the suction surface boundary layer. Besides, it reflects that the Reynolds number has few influences.
Figure 21 to
Figure 23 show that CL-WEDGE has an excellent aerodynamic performance. Similar research has been investigated by Lu Hongzhi [
43,
44]. She improved a circular leading edge design with a tiny flat, as shown in
Figure 24. The double suction spikes on the flat edges were much weaker than the single spike that appeared on a circular leading edge. Therefore, the loss of the boundary layer was reduced.