Numerical Studies on the Effect of Leading Edge Tubercles on a Low-Pressure Turbine Cascade

Abstract

The influence of the bionic leading edge tubercles on the aerodynamic performance of a high-speed low-pressure turbine has been numerically studied. Nine different tubercle configurations were designed to reduce the loss of the cascade at high Mach and low Reynolds numbers. Firstly, the effect of the geometric parameters of the bionic leading edge tubercles on the total pressure loss of the turbine cascade is discussed. Then, the flow control mechanism of the bionic leading edge tubercles is discussed. The results indicate that the total pressure loss of the turbine cascade could be reduced effectively by the leading edge tubercles. A larger tubercle size and larger amplitude-to-wavelength ratio can achieve a better loss reduction effect. The streamwise vortex generated by the leading edge tubercles continuously promotes the momentum exchange of the boundary layer in the process of downstream development, which makes the boundary layer shape plumper, and the ability to resist separation is enhanced. At the same time, the disturbance introduced by the streamwise vortex enables the separated boundary layer to transition quickly.

1. Introduction

A turbofan engine with a large bypass ratio is the most important power unit of modern civil aircraft. In order to continuously pursue the economic benefits of flight, the bypass ratio of the engine keeps increasing [1]. With the increasing bypass ratio, the matching problem between fan speed and low-pressure turbine (LPT) speed is becoming increasingly serious. To solve this problem, the geared turbofan (GTF) engine was developed and has been successfully deployed in commercial aviation. The high-speed LPT is one of the most important components of the GTF engine. The working environment with both a high Mach number and a low Reynolds number is its most important aerodynamic characteristic [2]. Under the low Reynolds number environment, a closed separation bubble or even an open separation may appear at the trailing edge of the blade suction surface, which will greatly reduce the efficiency of the turbine. The aerodynamic performance of LPT has a significant impact on aircraft flight costs. The research results of traditional turbofan engines show that a 1% increase in LPT efficiency could obtain a reduction of fuel consumption by 0.5–1.0% [3]. The application of appropriate flow control methods to reduce blade surface separation and LPT blade losses is a hot research area.

For the boundary layer control of the blade surface at a low Reynolds number, there are two main methods: active control and passive control. For active flow control methods, the jets [4,5,6,7] and plasma actuators [8,9], etc. are introduced as vortex generators, which could enhance the energy mixing between the boundary layer and the main flow and promote the transition of the separated laminar boundary layer. The size of the separation bubble is suppressed and the loss of the blade is reduced. Passive flow control methods mainly include the introduction of rectangular bars [10], two-dimensional spanwise grooves [11,12], three-dimensional dimples [13], and surface roughness [14], which could induce the boundary layer to become turbulent and then achieve the purpose of separation suppression and loss reduction. Although the active control methods have the advantages of good adaptability and high controllability compared with the passive control methods, the cost is too high and the system is too complex to be realized, so it is not suitable for engineering practice for the time being. Passive control methods are more feasible in engineering practice because of their advantages of low cost and ease of application.

The bionic leading edge tubercle is used as a passive control method, imitating the protrusions on the humpback whales flipper. Fish et al. [15] analyzed the anatomical characteristics of humpback whale flippers for the first time and proposed that the leading-edge tubercle structure could improve the hydrodynamic performance of humpback whale flippers. Since then, a large number of researchers have applied the leading edge tubercle structure to the study of the aerodynamic performance of symmetric airfoils [16,17,18]. The results show that the leading edge tubercle structure can promote the boundary layer transition to turbulence, thus inhibiting the separation of the boundary layer, improving the lift of the airfoil, and delaying the stall of the airfoil. Seyhan [19] et al. applied a new kind of leading-edge tubercle with amplitude modulation to the airfoil. The leading edge tubercles greatly improved the aerodynamic performance. The airfoil, having a leading edge tubercle with amplitude modulation, shows better lift characteristics than the airfoils with constant amplitude. Simanto [20] et al. experimentally investigated hydrofoils with leading-edge tubercles. The influence of leading-edge tubercles on cavitation and noise is analyzed. The results showed that the cavitation could be reduced by 25–60%. The leading edge tubercles could also decrease the noise at high flow velocity. Ke [21] numerically studied the influence of leading-edge tubercles on a wind turbine. The results indicated that the tubercle geometry which can improve turbine performance varies depending on the wind speed. The applications of bionic leading-edge tubercles in turbomachinery are relatively few. Keerthi [22] et al. applied the leading edge tubercles to a linear compressor cascade, successfully suppressed the ground stall of the blade, and extended the working envelope. The stall angle of the cascade is delayed by the tubercles. A compressor cascade with leading-edge tubercles was evaluated at a high Mach number by Sidhu [23] et al. They reported that all tubercle geometries could delay the stall but the best geometry depended on the incidence angle. Bouchard [24] et al. applied different leading edge tubercle configurations to a transonic turbine vane and achieved loss reduction under the condition of a positive incidence angle. They proposed that the tubercles should have a better application prospect in low-pressure turbines. Asghar et al. [25] applied a leading edge tubercle configuration to an LPT cascade and conducted experiments under very low Reynolds number conditions. The experimental results show that the tubercle configuration greatly delays the separation position of the boundary layer, but does not significantly reduce the total pressure loss of the cascade. In the few studies on the application of tubercles in turbine blades, there is no effective rule for the design parameters of the tubercle configuration. The influence mechanism of this configuration on the blade boundary layer flows has not been clearly understood.

In the present study, the aerodynamic effects of the bionic leading edge tubercles applied to a high-speed low-pressure turbine cascade designed for the GTF engine working environment are numerically investigated. Nine variations of the reference high-speed low-pressure turbine blade were obtained by varying the characteristic parameters of the tubercles. Under the condition of high-speed and low Reynolds numbers, the influence of different characteristic parameters of the tubercles on the aerodynamic performance of the cascade is studied by numerical simulation. The influence law of each design parameter of the tubercles on the aerodynamic performance of the turbine cascade under this high-speed low Reynolds number condition is preliminarily obtained. The mechanism of how the tubercles affect the boundary layer flow on the turbine blade surface is discussed.

2. Model and Method

2.1. Reference Turbine Cascade

A high-speed LPT blade section is investigated as the reference cascade. The characteristics of the cascade are summarized in

Table 1. Characteristics of the cascade.

Parameter	Value
Chord C/mm	81.24
Axial chord Cx/mm	68.68
Pitch s/mm	65.2
Pitch chord ratio s/C	0.80
Inlet angle β1/(°)	39.9
Out let angle β2/(°)	65.2
Stagger angle βs/(°)	32.3

. The after-loaded method is adopted for the profile design. The isentropic Mach number at the outlet of the cascade could reach 0.87 at the design point. The minimum Reynolds number based on the axial chord length at the cascade exit can reach 1 × 10⁵, and the incidence angle at the design point is −9.5°. In the study of this paper, the Reynolds number at the outlet of the cascade is about 1.6 × 10⁵, and the isentropic Mach number at the outlet and the incidence angle are consistent with the design point. The working point of the cascade is consistent with the experiment.

2.2. Tubercle Configurations

The leading edge tubercle designed in this paper is shown in Figure 1. The amplitude A and the wavelength W are the main design parameters for the geometry of the leading edge tubercles. In order to explore the influence of geometric design parameters on the flow control effect, a total of 9 different amplitude and wavelength combination schemes were designed in this paper. The amplitude A ranges from 5% to 20% of the chord length, and the wavelength W ranges from 2.5% to 40% of the chord length. The specific geometric design parameters are given in

Table 2. Characteristics of the tubercles.

Case	A	W	A/W
A5W2.5	0.05C	0.025C	2
A5W5	0.05C	0.005C	1
A5W10	0.05C	0.1C	0.5
A10W5	0.1C	0.005C	2
A10W10	0.1C	0.1C	1
A10W20	0.1C	0.2C	0.5
A20W10	0.2C	0.1C	2
A20W20	0.2C	0.2C	1
A20W40	0.2C	0.4C	0.5

.

2.3. Numerical Method

Due to the high cost of experiments at a high Mach number and low Reynolds number, the method of numerical simulation is mainly used to study the flow control effect of leading-edge tubercles. The commercial software package Ansys Fluent (version 19.4) is used here to solve the three-dimensional steady Reynolds-averaged Navier–Stokes (RANS) equations. The RANS computations are performed with the k-ω shear-stress transport (SST) model of Menter [26]. The γ (intermittency) transition model proposed by Menter [27] is used. The γ transition model is a one-equation transition model evolved from the γ-θ transition model. The main advantage of this model is that the turbulence intermittency γ is solved by the transport equation and the critical Reynolds number Re_θ is solved algebraically. The usage of computing resources is reduced compared to the γ-θ transition model, which needs to solve two transport equations. The γ transition model is Galilean invariant due to the movement of Re_θ equation. It is easier for users to fine-tune the parameters as the model has a simple formulation.

In order to reduce the consumption of computing resources, the computational domain consists of only one blade passage. Periodic boundary conditions are applied in the pitchwise direction. As the influence of the endwall is not considered in this study, periodic boundary conditions are also applied in the spanwise direction. The size of the computational domain in the spanwise direction is approximately 20% of the chord length (for the A20W40 case it extends 40% of the chord length). The results of direct numerical simulation and large eddy simulation show that this spanwise height is sufficient for the development of a three-dimensional vortex structure [28,29]. For schemes with different leading edge tubercles, the spanwise height can cover 1 to 8 complete wavelengths. The inlet of the computational domain is set at 1C_x upstream of the blade leading edge. The outlet of the computational domain is set at 2.5C_x downstream of the blade trailing edge. At the inlet of the domain, total pressure, total temperature, and flow direction are enforced. An adiabatic no-slip boundary is applied at the blade surface. The free stream turbulence intensity is set to zero at the inlet.

Structured meshes are employed in this study. The mesh tool used to generate the grids is ANSYS ICEM. The mesh is characterized by an H-O-H topology. The O-type block is present around the blade and the H-type block is present in other areas. The maximum expansion ratio is enforced to be 1.1 in the wall-normal direction to ensure the resolution of the boundary layer. The non-dimensional distance from the blade surface to the first layer of the grid (y+) is less than 1. There are 556 nodes positioned around the blade. For the reference cascade, there are 41 nodes positioned along the spanwise direction of the computational domain. For the cascade with tubercles, the grid in the spanwise direction is densified and 65 nodes are given. Finally, the total number of nodes of the meshes is 3.66 million and 5.81 million for the reference cascade and bionic cascade, respectively. The results of the mesh dependency study show that mesh convergence has been achieved. The mesh of the bionic leading edge tubercles is shown in Figure 2.

2.4. Verification of Numerical Method

In order to verify the reliability of the numerical simulation method adopted in this study, the numerical results of the reference cascade are compared with the experimental data. The experiments were conducted in the variable-density wind tunnel of the China Aerodynamics Research and Development Center. The wind tunnel can realize the continuous adjustment of the test chamber pressure through suction, so as to achieve the purpose of changing the Reynolds number while guaranteeing the outlet Mach number. The test turbine cascade is shown in Figure 3. The isentropic exit Mach number is 0.87 in the experiment. The Reynolds number at the cascade exit is about 1.6 × 10⁵ in the experiment. The incidence angle in the experiment is set to −9.5°. No free stream turbulence is introduced in the experiment. Figure 4 shows the comparison between the numerical results and the experimental results of the static pressure coefficient distribution on the surface of the turbine blade under this working condition. The static pressure coefficient C_p around the blade is defined as follows:

$$C_p=\frac{p_s-p_{s,in}}{p_{t,in}-p_{s,out}}$$

(1)

where $p_s$, $p_{s,in}$, $p_{t,in}$, and $p_{s,out}$ are, respectively, the local static pressure, the static pressure at the inlet, the total pressure at the inlet, and the static pressure at the outlet of the blade cascade. It can be seen from the figure that the numerical results are in good agreement with the experimental results, and the separation at the trailing edge of the blade is successfully predicted. It is believed that the numerical simulation method adopted in this study is relatively reliable under the condition of high Mach number and low Reynolds number.

3. Results and Discussions

In this paper, the numerical simulation of 9 leading edge tubercles design schemes is carried out under the above experimental condition.

3.1. Influence of Tubercle Geometric Parameters on Cascade Loss

Figure 5 depicts the influence of wavelength (W) on the total pressure loss coefficient ($\xi$) of the cascade. The total pressure loss coefficient $\xi$ is defined as follows:

$$\xi =\frac{p_{t,in}-p_{t,out}}{p_{t,in}-p_{s,out}}$$

(2)

where $p_{t,out}$ is the average total pressure at the cascade inlet. The vertical ordinate in the figure is the dimensionless result after dividing the total pressure loss coefficient of each tubercle scheme by the reference total pressure loss coefficient (ξ_ref). The reference total pressure loss coefficient ξ_ref is the total pressure loss coefficient of the reference turbine cascade. Therefore, a value less than 1 in the figure represents a decrease in loss compared with the reference cascade, and a value greater than 1 represents an increase in loss. It can be seen from Figure 5 that for different design amplitudes A, the variation trend of the total pressure loss coefficient with wavelength W is quite different. Under the condition of a small amplitude value (A = 5), the loss of the three schemes increases relative to the reference cascade, and the loss tends to decrease with the increase of wavelength W. Under the condition of medium amplitude value (A = 10), the loss increases first and then decreases with the increase of wavelength W. Both the large wavelength (A10W20) and the small wavelength (A10W5) schemes have achieved the effect of reducing the total pressure loss. The A10W5 scheme has the best total pressure loss reduction effect among the nine design schemes, with a total pressure loss reduction of 21.2%. For the cases of large amplitude value (A = 20), all the schemes with tubercle wavelength in the range of 10% to 40% chord length show the effect of reducing loss. The loss shows a tendency to increase with increasing wavelength. The low loss area in the figure is concentrated in the lower left corner.

It can be seen from Figure 5 that, overall, the total pressure loss of the turbine cascade tends to decrease as the tubercle amplitude A increases. In order to further clarify the influence of tubercle amplitude on the cascade total pressure loss coefficient, Figure 6 shows the influence of amplitude on the cascade total pressure loss coefficient under the same wavelength condition. Although the number of examples is limited, it can still be found that under the same wavelength condition, the cascade loss decreases as the amplitude increases. The range of parameters to achieve a large loss reduction effect is in the lower right-hand corner of the figure.

While the effects of tubercle wavelength or amplitude have been analyzed in isolation, the natural idea is to combine these two parameters and see their effects on the cascade loss. The simplest method is to take the ratio of amplitude to wavelength, A/W, as a design parameter to investigate its influence on cascade flow loss. The trend of the cascade loss as a function of the amplitude-wavelength ratio is presented in Figure 7. It can be seen from the figure that, under different amplitude conditions, the influence of the amplitude-wavelength ratio on the cascade loss leads to completely different results. In the case of small amplitude (A = 5), the cascade loss increases with the increase of the amplitude-wavelength ratio. In the case of moderate amplitude (A = 10), the cascade loss first increases and then decreases with the increase of amplitude-wavelength ratio. Under the condition of large amplitude (A = 20), cascade loss decreases with the increase of amplitude-wavelength ratio. The range of parameters leading to a low cascade loss is located in the bottom-right region of the figure.

After analyzing the influence of the above three design parameters, the overall size of the tubercle becomes the next factor to consider, i.e., if the shape of the tubercle is fixed, the influence of the size of the tubercle on the cascade loss. Since the shape of a tubercle is determined by the amplitude-wavelength ratio, when the amplitude-wavelength ratio is constant, either the amplitude or the wavelength can be used to represent the size of the tubercle. Figure 8 shows the changing trend of the cascade loss with tubercle size when the tubercle shape is fixed. It can be found that in the parameter range studied, the effect of loss reduction cannot be obtained regardless of the tubercle shape when the tubercle size is too small. When the tubercle size is large enough, all three tubercle shapes can achieve the effect of loss reduction. In general, the cascade loss decreases first and then increases with increasing tubercle size.

3.2. Analysis of Flow Control Mechanism

In this section, the A10W5 scheme, which has the best loss reduction among the 9 schemes, is taken as the research object, and its flow control mechanism is discussed.

Figure 9 shows the comparison of the static pressure coefficient distribution around the blade surface at four different spanwise positions of the A10W5 blade cascade. The static pressure coefficient distribution at the mid-span of the reference turbine cascade is taken as the baseline. The four spanwise locations are typical sections in a wavelength structure respectively, and the specific positions are shown in Figure 2. As shown in Figure 2, span1 is located at the trough, span4 is located at the peak, and span2 and span3 are located in the middle of the peak and trough. It could be found that the structure of the tubercle changes the load distribution on the blade surface, causing the peak velocity point to move backwards, the diffuser area at the rear of the blade to decrease, and the adverse pressure gradient to increase. Although the tubercles increase the adverse pressure gradient at the rear of the blade, the separation of the A10W5 blade is weaker than that of the reference blade, which has a large platform area in Figure 9. In addition, it can be found that the static pressure coefficient distributions around the blade surface at the four spanwise locations are almost identical, except for the differences at the leading edge of the blade due to the large geometric differences.

To further clarify the influence of the tubercles on the separation zone at the trailing edge of the blade, the distribution of the friction coefficient on the blade surface is shown in Figure 10. The friction coefficient C_f around the blade is defined as follows:

$$C_f=\frac{{\tau }_w}{p_{t,in}-p_{s,out}}$$

(3)

where τ_w is the wall shear stress around the blade. The position where the friction coefficient is less than 0 indicates the region where the flow is separated. Comparing the friction coefficient distribution curves, it can be found that, compared with the reference cascade, the separation starting position of the tubercle cascade at different spanwise positions is moved backwards, and the length of the separation zone is reduced. The separation at the trough position (span1) is almost completely eliminated, and the separation positions at other spanwise positions are different.

A comparison of the velocity profiles of the boundary layer on the suction side surface of the blade is presented for six axial locations in Figure 11. The velocity profiles are normalized to the velocity at the edge of the boundary layer. It can be found that the boundary layer of the blade suction surface shows different morphological characteristics in the acceleration region due to the influence of the leading edge tubercles.

At 51% of the axial chord length position, the thickness and shape of the boundary layer of the tubercle cascade show different characteristics along the spanwise direction. The thickness of the boundary layer at the peak position (span3) is significantly smaller than that at other spanwise positions, and also smaller than that of the reference cascade. The thickness of the boundary layer at the other three spanwise positions is significantly greater than that of the reference cascade. The boundary layer of the reference cascade shows obvious laminar boundary layer characteristics. As for the tubercle cascade, the boundary layer at the peak position (span3) exhibits obvious laminar boundary layer characteristics, while those at the other three spanwise positions all exhibit some turbulent boundary layer characteristics.

At 72%C_x, the shape of the boundary layer at the peak (span3) is almost identical to that of the reference cascade. The boundary layers at other locations show the characteristics of turbulent boundary layers and are thicker than the reference cascade.

As the boundary layer develops to 88% of the axial chordal length, a significant separation has occurred for the reference cascade boundary layer. For the tubercle cascade, separation also occurs at the peak position, but the thickness of the reverse flow region (the distance from the zero-velocity point to the wall) is significantly less. The boundary layer at the middle position of the tubercle (span2 and span4) is at the critical point of separation.

When the boundary layer continues to develop downstream to the 90% axial chord length position, the thickness of each separated boundary layer further increases, and the boundary layer at the peak position still maintains the state of attached flow.

At 97%C_x, the thickness of the reverse flow region of the boundary layer of the reference cascade further increases. The thickness of the reverse flow region at each spanwise position of the tubercle cascade has begun to decrease and the separated boundary layer shows a tendency to reattach. In addition, very small separation also occurs in the boundary layer at the peak position.

At the position of 99.6% axial chord length, the thickness of the boundary layer reverse flow region of the reference cascade is still further enlarged, while the thickness of the boundary layer reverse flow region of the tubercle cascade is further reduced, which is only one step away from the complete reattachment.

It can be seen that the length of the separation region and the thickness of the reverse flow region at the trailing edge of the blade is significantly reduced by the tubercles, and thus the cascade total pressure loss is significantly reduced.

A comparison of the evolution of the boundary layer shape factor H on the suction surface of the blade is given in Figure 12. It can be found that for the reference cascade, the boundary layer shape factor is about 2.6 at the beginning, which is a typical value for the laminar boundary layer. The shape factor starts to increase from the vicinity of the diffuser area, and the increase is more rapid after separation. For the tubercle cascade, the shape factor is smaller than that of the reference cascade at every spanwise position except the peak position. This is consistent with the fuller boundary layer shape with more turbulent boundary layer characteristics shown in Figure 11, while the fuller boundary layer is more resistant to separation. In general, the decrease in the shape factor indicates the beginning of the transition of the boundary layer. It can be seen from Figure 12 that, relative to the reference cascade, the position where the shape factor starts to decrease at each spanwise position of the tubercle cascade moves forward relative to the reference cascade, and the boundary layer transition position moves forward. The decrease in the peak value of the shape factor also indicates the weakening of the boundary layer separation, which is consistent with the development of the boundary layer in Figure 11. Since the boundary layers at each location fail to reattach and develop into fully developed turbulent boundary layers, the shape factor values are not reduced to the range of the turbulent boundary layer.

Figure 13 shows the distribution of surface streamlines near the trailing edge of the suction surface of the tubercle cascade. The separation lines identified by the wall shear stress are marked in red. It can be found that the initial position of the separation presents a wavy distribution along the spanwise direction, corresponding to the structure of the leading edge tubercles. The spanwise position of the saddle points of the surface streamline corresponds to the position of the leading edge tubercle peak position. Downstream of the saddle points are pairs of foci, indicating that pairs of streamwise eddies are present in the boundary layer and that the streamwise eddies are lifting upward away from the wall. The foci mark the locations at which concentrated vortices separate from the surface [30]. This flow topology is consistent with the separation flow structure described by Perry [31] et al. The three-dimensional streamlines (which is colored by velocity) in Figure 14 show the flow structure well around the saddle point and the foci.

Due to the wavy distribution structure of the separation zone at the trailing edge along the spanwise direction, the cascade outlet loss also presents a wavy distribution along the spanwise direction. Figure 15 shows the distribution of the total pressure loss coefficient along the blade spanwise direction. It can be found that the position of each loss peak corresponds to the position of the peak of the leading edge tubercle, that is, the position of the short separation zone at the trailing edge.

So how do the streamwise vortices in the boundary layer form and evolve? Figure 16 shows the contour of the static pressure coefficient of the blade surface at the leading edge. It can be seen from the figure that the trough position of the tubercle is a low-pressure area, and the spanwise distribution of the blade leading edge pressure has a pressure gradient pointing towards the trough. This pressure gradient will cause the fluid near the blade surface to flow towards the location of the trough section, producing a spanwise velocity component. The fluid gathered towards the trough position will form streamwise vortex pairs with opposite rotation directions at the trough position.

The contour of streamwise vorticity near the suction surface of the blade along different axial positions is presented in Figure 17. It can be seen from the figure that, affected by the tubercle structure of the leading edge, a complex streamwise vortex pair structure is formed at the trough position. At the position of 40% of the axial chord length, where the geometric structure of the tubercle can be ignored, a well-defined structure of streamwise vortex pairs is formed. The streamwise vortex pairs A and B are located above the boundary layer, and the streamwise vortex pair below them are located in the boundary layer. It can be found that as the streamwise vortex pair develops downstream, the vortex pair A and B gradually move away from the blade surface, and the intensity is also weakened, but its shape can still be maintained. The fluid in the boundary layer is disturbed by the vortex pair B, which continuously induces new streamwise vortex structures, and the new streamwise vortex structures are dissipated as they develop downstream. The generation and dissipation process of the streamwise vortex in the boundary layer can be observed through the change of the vorticity contour at 40% to 80% of the axial chord length. At the separation location, the streamwise vortices within the boundary layer are lifted upwards, leaving the topology of Figure 13. In the downstream development process, the streamwise vortex outside the boundary layer interacts with the fluid in the boundary layer, constantly exchanging momentum, so that the energy of the low-energy fluid in the boundary layer is continuously replenished. The velocity profile of the boundary layer is fuller and the shape factor is smaller, which is consistent with the results in Figure 11 and Figure 12.

The momentum exchange between the boundary layer and the main flow will inevitably lead to a greater mixing loss than in the undisturbed laminar boundary layer. The distribution of the total pressure loss along the flow direction of the reference cascade and the tubercle cascade is given in Figure 18. It can be found that; starting from the leading edge of the blade, the total pressure loss of the tubercle leading edge blade is slightly greater than that of the reference cascade, and the gap between the two total pressure loss coefficients gradually increases along the flow direction. This is because the streamwise vortex formed by the tubercles strengthens the mixing effect between the boundary layer and the main flow, resulting in a greater mixing loss. Downstream of the trailing edge of the blade, the loss of the reference cascade increases sharply. This is due to the fact that, in the reference cascade, the wake width is much larger than the trailing edge thickness due to the severe opening separation structure at the trailing edge, and the mixing effect of the separated suction surface shear layer, pressure surface boundary layer and mainstream is more intense, resulting in greater losses. For the tubercle cascade, the separation of the trailing edge is greatly suppressed. On the one hand, the thickness of the separation zone has been greatly reduced. On the other hand, the airflow at the trailing edge is already close to the attached flow. The wake width is greatly reduced compared to the reference cascade, which significantly reduces the mixing loss behind the trailing edge of the blade. The benefit of reducing losses in the wake is greater than the mixing losses in the blade surface boundary layer, resulting in a reduction in overall cascade losses.

4. Conclusions

In this study, nine different leading edge tubercle configurations are designed for a high-speed and low-pressure turbine blade, and the aerodynamic performance of various configurations under low Reynolds number and high Mach number state is numerically simulated. The following conclusions are obtained:

• The numerical simulation results are in good agreement with the experimental results. The numerical simulation method adopted in this study can obtain accurate prediction results in the state of high Mach number and low Reynolds number.
• Under the high Mach number and low Reynolds number condition studied here, by selecting appropriate geometric design parameters, the tubercle configuration can effectively reduce the cascade loss, and the total pressure loss can be reduced by up to 21.2%.
• The amplitude of the tubercles has the most obvious influence on the loss reduction effect. If the amplitude is too small, no matter how other geometric parameters are designed, the loss reduction effect cannot be obtained.
• Under the research condition, the optimal design parameter range to obtain the best loss reduction effect is approximately the amplitude is 0.1C~0.2C; the amplitude-wavelength ratio is 1~2.
• As the streamwise vortex induced by the leading edge tubercles develops downstream, the low-energy fluid in the boundary layer is continuously disturbed during this period; so that the boundary layer and the main flow continuously realize momentum exchange. The resistance of the boundary layer to separation is enhanced after the energy supplement, and the separation position moves backwards. The separated boundary layer quickly reattaches to the wall, which greatly reduces the thickness and length of the separation region and reduces the mixing loss after the trailing edge of the blade.
• The leading edge tubercles increase the mixing loss between the boundary layer and the mainstream, and when the mixing loss in the boundary layer takes a secondary position compared with the loss reduction benefit in the wake, the overall loss of the cascade is reduced.