Xiutao BIAN, Qingsong WANG, Xinrong SU, Xin YUAN

Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Tsinghua University, Beijing 100084, China

KEYWORDS

Abstract Accurate predictions of Shock Waves and Boundary Layer Interaction (SWBLI) and strong Shock Waves and Wake Vortices Interaction (SWWVI) in a highly-loaded turbine propose challenges to the currently widely used Reynolds-Averaged Navier-Stokes (RANS) model. In this work,the SWBLI and the SWWVI in a highly-loaded Nozzle Guide Vane(NGV)are studied using a hybrid RANS/LES strategy. The Turbulence Kinetic Energy (TKE) budget and the Proper Orthogonal Decomposition(POD)method are used to analyze flow mechanisms.Results show that this hybrid RANS/LES method can obtain detailed flow structures for flow mechanisms analysis.Strong shock waves induce boundary layer separation, while the presence of a separation bubble can in turn lead to a Mach reflection phenomenon. The shock waves cause trailing-edge vortices to break clearly, and the wakes, in turn, can change the shocks intensity and direction. Furthermore,the Entropy Generation Rate(EGR)is used to analyze the irreversible loss.It turns out that the SWWVI can reduce the flow field loss. There are several weak shock waves in the NGV flow field, which can increase the irreversible loss. This work offers flow mechanisms analysis and presents the EGR distribution in SWBLI and SWWVI areas in a transonic turbine blade.

1. Introduction

Gas turbines play an important role in aviation and power generation fields. The flow inside a turbine is characterized by flow structures of different length scales, and in a highpressure turbine,the shock wave may interact with the boundary layer and the wake vortex. The flow field in the turbine is also affected by the swirling flow from the combustor.1,2With ever-increasing turbine inlet temperature and pressure ratio,the flow inside the turbine is further complicated, which proposes challenges to accurate prediction and design optimization.3For a better understanding of flow structures and loss mechanisms, as well as design of high-performance turbines,an accurate prediction of the complicated turbine flow is necessary.

A series of methods, ranging from the simple through-flow model to the DNS,4–9are available, among which the current workhorse in predicting turbomachinery flow is RANS modeling, and with an increase of computation power, LES and DNS have been increasingly used. RANS is the least costly;however, when studying complex flow phenomena such as SWBLI and SWWVI,the RANS method cannot offer enough flow details which are extremely important to study the interaction mechanisms between the shock waves, the boundary layer,and the wakes.LES and DNS can provide high accuracy and detailed knowledge about flow structures, but for high-Reynolds number flow, both of them will consume huge computing resources.8,9A hybrid of RANS and LES, which models the boundary layer with RANS and the rest of the domain with LES, offers an opportunity for LES resolution of important flow structures at a much reduced cost.In 1997,Spalart10proposed a hybrid approach named DES, using the RANS method near the wall area while using the LES method away from the wall area. However, the DES method suffers from modeled stress depletion problems. In 2006, Spalart et al.11improved the DES method and proposed a DDES method.Compared with the LES method and the DNS method, the DDES method can significantly reduce the demand for computing resources. Meanwhile, The DDES method can obtain more detailed flow field structures than those of the RANS method.

Wheeler et al.12applied the DNS method to investigate the flow phenomenon in a transonic NGV and studied turbulence influences on flow predictions and heat transfer. Segui et al.13applied LES to the same case to study the effects of inlet turbulence on the heat transfer coefficient. Le′onard et al.14compared RANS and URANS with LES in high-pressure turbine NGV and found that, at a high Reynolds number, compared with RANS and URANS methods, only the LES method could provide a complete view of the complex flow.Lin et al.15applied the DDES method in a high-pressure turbine case to study the main flow entropy loss of a transonic flow field.Results showed that DDES was able to predict complex flow structures and also the interactions between different structures.

Summarizing existing high-fidelity simulations about the transonic flow inside a turbine, in most cases, the shock wave is of moderate amplitude,and in this work,we are interested in the prediction of transonic flow at a higher outlet Mach number where the SWBLI induces flow separation. Shock waves and wakes dominate unsteadiness characteristics. In this paper, compared with the RANS method, the DDES method can capture a detailed flow field. Different terms of the TKE equation in a separation area are analyzed. The POD method is applied to obtain flow field unsteadiness dominant factors.Based on the second law of thermodynamics,the total aerodynamic loss is separated into two parts to evaluate the losses in the flow field.

This paper contains five parts.Firstly,the simulation object and the numerical method are presented.Next,the interaction between the shock waves and the boundary layer is analyzed.Then, an analysis of the interaction between the shock waves and the boundary layer is conducted. Afterwards, the entropy generation rate is introduced to investigate the flow field.Finally, conclusions are given.

2. Simulation object and methodology

2.1. Numerical methods

The turbine cascade model is a highly-loaded turbine NGV.16Table 1 shows the primary geometric parameters of the turbine cascade.The computation domain is shown in Fig.1.The inlet surface is 0.72 chord upstream from the blade leading edge,while the outlet surface is 1.5 chord downstream from the blade trailing edge.Given the three-dimensional characteristics of vortex structures in an actual flow field,the spanwise size of the computational domain is selected to be 10 mm based on experiences gained in similar cases.17.

The topology contains 95 blocks,and the multi-block structured mesh consists of 14 million elements. Besides, the number of grids in each block can meet the requirement of a multi-grid method to accelerate the convergence. The grids near the leading edge and the trailing edge are shown in Fig.2, and the mesh is coarsened two-times for clarity. The outlet Mach number of this case is 1.16, and the Reynolds number based on the chord and outlet flow velocity is about 3×106. The boundary conditions used in this case are given in Table 2.

In this work,a well-proven in-house multi-block solver17–22is used to carry out numerical simulation. Turbulence simulation in the RANS is based on the Spalart-Allmaras model,23while in the hybrid RANS/LES simulation,the DDES model11is used. For high resolution of turbulence fluctuations, the fifth-order high-accuracy upwind method24is used, and an adaptive methodology25is also used to further reduce the numerical dissipation, which helps to improve the resolutionof small-scale structures. Boundary conditions are implemented in a non-reflecting fashion to avoid contamination of the numerical results by reflections at the boundaries. For unsteady computation,the dual time-step method is employed,and according to the wake vortex shedding period of the case,the physical time step is chosen to be 1×10-6s, which corresponds to about 13000 physical time steps during a flowthrough period.During each physical time step,the governing equation is solved using an efficient implicit method assisted with multi-grid for rapid convergence. In general, with about 30 iterations, the residual can be dropped by 4 orders of magnitude.

Table 1 Parameters of blade.

Fig.1 Computation domain.

Fig.2 Two-times coarsened mesh details.

Table 2 Inlet and outlet parameters.

With a careful design of the mesh resolution and efficient numerical methods, the current numerical study satisfies the accuracy requirements of hybrid RANS/LES simulations.Besides, checking that the wall normal directiony+≈1, the streamwise direction Δx+≈70, and the spanwise direction Δz+≈30. The ratio of the resolved turbulent kinetic energy in Eq. (1) is also checked, wherektresolvedis the resolved turbulent kinetic energy, andktmodeledis the modeled turbulent kinetic energy.

As the LES model is activated in the off-wall region, the numerical settings should satisfy the LES requirements, and one criterion is that at least 80% of the turbulent kinetic energy should be resolved, other than modeled by sub-grid scale modeling.26Fig.3 gives the distribution ofr. It is clear that in the near-wall region,r?1 because RANS modeling is used in this region,while in the rest of the domain,ris close to 1, and the mesh resolution meets the LES requirement.

2.2. Comparison between DDES, RANS, and experimental results

Fig.3 Ratio of resolved turbulent kinetic energy.

Fig.4 Overview of instantaneous flow field (Q=500).

After 4.5 flow-through periods, the unsteady calculation reaches the statistical steady state, and the instantaneous flow field and vortex structures are shown in Fig.4.URANS results are also given in Fig.4(b)for comparison. Mach number distributions are given in Fig.4,and the vortex structures are represented by theQ-criterion. Fig.4 shows that compared with the URANS result, the DDES method can obtain rich threedimensional vortex structures. In Fig.4(a), there are two oblique shock waves originated from the trailing edge of each guide vane. Taking blade No. 3 as an example, the right branch of the shock wave directly enters the downstream flow field and interacts with the wake vortex of the neighboring blade. The left branch of the shock wave interacts with the suction-side boundary layer of blade No.2,then it is reflected,and the reflected shock wave interacts with the wake vortex again. It is clear that several complicated flow phenomena,including the shock wave, SWBLI, and shock-wake vortex interaction,co-exist in this case,and this proposes severe challenge to turbulence modeling.

After reaching the statistical steady state, the unsteady computation has been continued for long enough time steps to obtain a time-averaged flow field.The time-averaged dimensionless pressure (p/p0) distribution on the blade surface is given in Fig.5 and compared with experimental results(EXP). On the whole, Fig.5 shows that the calculated values are in good agreement with the experimental ones16; however,from 51%to 58%axial position,there is a pressure fluctuation area predicted by the simulation method.

Fig.6 shows the PSD result obtained by Fourier transformation using the velocity data at a monitor point near the trailing edge of the blade. The DDES results obey the -5/3 law well in the inertial sub-region, and the calculated vortex shedding frequency of 23.4 kHz is basically the same as the experimental measurement from 24 to 26.5 kHz.27

Fig.5 Pressure distribution on blade surface.

Fig.6 PSD result.

3. Interaction between shock wave and boundary layer

Fig.7 shows the time-averaged DGM results which are defined by Eq. (2). The DGM is normalized by the inlet density and the axial chord.

A series of important flow features can be found in this case.Besides the shock wave near the trailing edge,two normal shock wave structures denoted by a1 and a2 in Fig.7(a) are observed, which are consistent with the pressure fluctuations between 51% and 58% axial chord given in Fig.5. These two weak shock waves are also observed in the experiment,28as given in Fig.7(b). Fortunately, these two normal shock waves are of medium amplitude, and shock-induced flow separation is not observed. In the following, the shock wave near the trailing edge and the SWBLI will be analyzed.

3.1. Boundary layer separation and Mach reflection

Fig.7 Comparison between numerical and experimental Schlieren.

Fig.8 Time-averaged Mach number contour.

Fig.8 shows the time-averaged Mach number contour,and the SWBLI between the oblique shock wave and the suction-side boundary layer mainly occurs in the region denoted by a black dashed rectangle. Fig.9 gives an enlarged view of the SWBLI region.The streamlines near the surface indicate the separation bubble. The distribution of the shape factorH12along the streamwise direction is also given in Fig.9 by a red dashed line,and the distance between the dashed line and the solid surface represents the amplitude. After the SWBLI,H12rises sharply and reaches the maximum value, which clearly indicates flow separation. Fig.10 shows the time-averaged skin friction lines on the suction-side blade surface,and it can be seen that a flow reversal occurs in the range of 71%-73%axial chord,which is induced by the SWBLI.

Fig.9 Boundary layer separation.

Fig.10 Skin friction lines on suction surface.

Fig.11 Mach reflection at suction surface (DDES result).

With the boundary layer separation caused by the SWBLI,the emergence of backflow and the separation bubble increase the incident angle between the oblique shock wave and the suction-side blade surface, which results in Mach reflection29,30, as shown in Fig.11. The black solid line in Fig.11 denotes the isoline of unit Mach number,and the dashed lines represent the major flow structures, including the ISW, the RSW, and the MSW which is nearly perpendicular to the streamlines. The flow before the Mach stem is supersonic,but after the Mach stem, the fluid speed quickly reduces to subsonic, and then expands to supersonic again. Results yielded by RANS are also given in Fig.12 for comparison.The same magnification factor and flow field position are taken in Fig.9, Figs. 11–15. Comparing Fig.12 with Fig.11,it can be seen that the two methods successfully predict the same flow structures; however, the shock wave amplitude and the size of the separation bubble are much smaller than those of DDES. As a result, there is a difference of axial position for the interaction area between DDES and RANS results. Meanwhile, the SWBLI-related loss will be underestimated by RANS.

3.2. Turbulent kinetic energy in SWBLI area

Fig.12 Mach reflection at suction surface (RANS result).

Fig.13 Turbulent kinetic energy in separation area.

Fig.14 Reynolds stress in flow direction in separation area.

In order to further analyze the mechanism of TKE in the interaction area, the turbulence kinetic energy budget32,33is used in this paper, and the equation is as follows:

Four sections are shown in Fig.15 in order to distinguish the contribution of each term to TKE in different areas. In Fig.15, a blue dashed line is used to represent the boundarylayer thickness δ.Section 1 is located upstream from the separation bubble,while Section 4 is located downstream from the separation area.Each term in the turbulent kinetic equation in Section 1 and Section 4 almost equals to zero,which is why the TKE is comparatively small upstream and downstream from the separation area. Results in Section 2 and Section 3 are shown in Figs. 16 and 17, respectively, which show that the production term,the pressure transport term,and the pressure dilatation term play important roles in TKE results.Besides,in Fig.17, the turbulent transport term also has impacts on the value and distribution of TKE in the interaction region.Fig.18 takes the same magnification factor and flow field position as Fig.9. Fig.18 shows different terms which have big effects on the TKE distribution in the separation area.

Table 3 Composition of turbulent kinetic equation.

Fig.15 Sections position in separation zone.

Fig.16 Different terms in turbulent kinetic equation on Section 2.

Fig.17 Different terms in turbulent kinetic equation on Section 3.

Fig.18 Different terms in separation area.

Fig.15 shows that upstream half of the separation bubble,the main turbulence fluctuation is located below 0.2δ. In this area, the production term and the pressure transport term dominate the turbulence fluctuation. The contour of TKE is totally similar to the production term; to put it another way,the production term causes the big TKE upstream from the separation area. Because the dissipation term nearly equals to zero in this area, the positive production term has a dominant impact on the amount of TKE and increases the turbulence. There is a strong adverse pressure gradient contributing to the reverse flow. The movement of the separation area also indicates the pressure fluctuation in the separation area. The adverse pressure gradient and pressure fluctuation result in a big value of the pressure transport term to reduce the TKE upstream half of the separation bubble.

Fig.17 indicates that the pressure transport term and the pressure dilatation term make up a large proportion among the right terms in Eq. (3). The pressure dilatation term can be used to represent the compressibility of the fluid. There is a strong adverse pressure gradient passing through the ISW and the RSW. As mentioned above, the intensity of the shock waves changes periodically. Besides, at the transonic condition,the fluid is compressive, and the compressibility is highly positively correlated with the shock waves intensity. In the shock waves area, the pressure fluctuation and shock wave intensity changes can result in a big value of the pressure transport term,and the pressure dilatation term dominates the TKE distribution.

4. Interaction between shock waves and wakes

4.1. Shock waves at trailing edge

The maximum velocity in the tapered flow passage is the sound velocity. However, the base pressure in this case is lower than that in the sonic condition. The circular trailing edge enlarges the area of flow passage, which makes it possible for the fluid to further expand and reach the supersonic condition. Fig.19 shows the instantaneous result of the shock waves near the trailing edge, and simplified wave structures are shown in Fig.20. At the connection position between the blade trailing edge arc and the pressure or suction surface,the fluid expands from sonic flow to supersonic flow. The suction-surface side and the pressure-surface side show expansion waves B1 and B2, respectively. After the connection position, the fluid detaches the surface and forms two shear layers d1 and d2.The trailing edge arc and two shear layers constitute a triangular area A where the velocity is lower compared to that of the main flow. Downstream from B1 and B2, owing to the offdesign condition of the current case, the over-expanded main flow results in a lower pressure of the fluid compared with that of the fluid in the triangular area.Then,passing through the lip shock waves b1 and b2, the pressure of the supersonic fluid increases and becomes consistent with that of the triangulararea fluid.

Fig.19 Shock waves at trailing edge.

Fig.20 Simplified view of shock waves at trailing edge.

Downstream from the blade trailing edge, physical parameters such as velocity magnitude and direction of the supersonic fluid on the pressure and suction surfaces are consistent by the influences of two main strong oblique shock waves c1 and c2. The low-energy fluid in the blade trailingedge triangle is able to form a periodic shedding vortex under the action of free sheer layers.

4.2. Influence of wakes on shock waves

When a shock wave passes through the wakes, the DGM is used to express the intensity of the shock wave. Fig.21(a)shows the instantaneous DDES result of the interaction between the wakes and the shock waves. The vortex shedding process has a big influence on the oblique shock-wave intensity. However, when the reflected shock wave passes through the wakes,there is an obvious change in the shock-wave intensity and direction like refraction. The density in the wake vortices area is relatively lower than those in other regions in the downstream flow tunnel, and the direction change of the reflected shock wave transmission is given in Fig.22.A similar experimental result also confirms that change in Fig.2334.Taub35and Jahn36have studied the refraction of a shock wave in different fluid media interfaces, while this paper gives the phenomenon that the direction is also able to change in the same fluid media, and the key influencing factor is the change of the fluid density. Besides refraction, when shock waves get close to the wake boundary, they can also be reflected, as shown in Fig.21(a).

Fig.21 Instantaneous results of DDES and URANS.

4.3. Influence of shock waves on wakes

As shown in Fig.24,wake vortices detach from the triangular downstream area and begin to dissipate. After that, the core region of vortices named vortex tubes will change from twodimensional structures to three-dimensional structures shown in Fig.4(a).

When a wake vortex passes the reflected shock wave c1,under the action of a strong adverse pressure gradient, wake vortex structures have been completely broken before passing through c1. The vortex tubes have been stretched and twisted owning to the marked differences of pressure and velocity near the c1 area. Downstream from c1, the deterministic vortex structures have been completely broken into many random small-scale structures. When the small-scale structures arrive at the second shock wave c2, they have been basically dissipated and disappeared quickly.Downstream from c2,the flow field tends to be uniform.

Fig.22 Process of shock wave direction change from wake area.

Fig.23 A similar experimental result of shock wave direction change.34

Fig.24 Effect of a shock wave on a wake.

Based on the rich flow field information of the DDES result, the above interaction process and mechanism can be analyzed. Compared with the RANS result in Fig.21(b), the DDES result can capture fine vortex structures,vortices breakdown, the shock waves propagation path, and their direction change.These are extremely useful for the analysis of the wake and the wake vortices dispassion,as well as the changes of the shock wave intensity and direction. The RANS method in these kinds of calculation misses these details,and correspondingly, it will miss these flow mechanisms.

4.4. Modal analysis by POD method

A proper orthogonal decomposition method is able to obtain the dominant factors of the flow field unsteadiness.37,38102 unsteady results containing more than two wake vortex shedding periods are adopted in the POD process. Mode 1 represents the time-averaged flow field, and its energy gains ascendancy over the other modes which represent flow field unsteady characteristics. Subtracting Mode 1, the relative energy and the cumulative energy of the remaining modes are shown in Fig.25(a) and (b), respectively. The relative energy reduces exponentially with the POD mode, which indicates that the unsteady turbulence characteristics are dominated by the first several modes. The cumulative energy quickly converges to 1,and the first 50 modes contain the main unsteady flow information. Among these modes, the first 5 modes almost contain a cumulative energy of 50%; hence,the first 5 modes will be analyzed in detail in this paper.

Fig.25 POD eigenvalues results about the first 50 modes.

Fig.26 shows different modes which contain flow field main unsteadiness characteristics. The contour is the result of the spanwise vorticity ωx, while the arrows are velocity vectors,and their colors represent their velocity magnitudes.The main flow structures in these modes are also the main resources of flow unsteadiness. In the four modes, there are large magnitudes of velocity vectors behind the oblique shock waves and downstream from the trailing edge.Meanwhile,in these areas,a large value of the spanwise vorticity represents the oblique shock waves and the wake vortices. It indicates that the main unsteady factors in this transonic turbine blade are the strong shock waves and the wake vortices. Fig.26(a) and (b) show that without the influence of the shock waves, the blade wake vortices shed from the trailing edge almost in a classic Von Karman street pattern. However, as previously mentioned,after the reflected shock wave, the deterministic large-scale unsteady wake vortices break down into small-scale random structures. In this process, the flow field unsteady characteristics have been weakened.

Generally, there are similar results shown in Mode 2 and Mode 3, but specifically, the reflux arrows marked by a red dashed circle in Mode 2 show that the weak shock waves on the blade suction surface have a big effect on the flow field unsteadiness, while in Mode 3 which takes comparatively lower relative energy, the reflux arrows in the rectangular dash-line area indicate that the separation bubble has a limited influence on flow unsteadiness characteristics. Mode 4 and Mode 5 take smaller relative energy, and there are so many random small-scale structures in these two modes. Similar to Mode 2 and Mode 3,in Mode 4 and Mode 5,the unsteadiness characteristics mainly occur downstream from the oblique shock waves and the wake.The weak shock waves on the blade suction surface still have big influences in Mode 4 and Mode 5.

5. Loss analysis

5.1. Trailing edge area loss analysis

The viscosity and the irreversibility of the heat transfer are the sources of entropy generation. The second law of thermodynamics can be used to evaluate the entropy loss rather than the first law of thermodynamics. The EGR has been used to analyze flow loss in SWBLI and SWWVI areas, and the formula is as follows39–41:

Fig.26 POD modes for main unsteadiness characteristics.

Fig.27 EGR in weak shock waves areas.

5.2. Loss analysis at shock wave and boundary layer interaction area

Fig.28 takes the same magnification factor and flow field position as Fig.9. In Fig.28, there is a large EGR in the RSW area,the shock wave intersection area,and the boundary layer area. The reason and proportion about losses in the shock wave areas and near-wall areas have been analyzed in Section 5.1. Compared with the upstream area of interaction,there is an intermittent area in the distribution of losses marked as C downstream.

Region A represents the mainstream, so both the surface viscous effect and the boundary layer separation have less influences on Region A. Therefore, the loss in Region A is small.In Fig.11,passing through the Mach stem,the fluid outside the separation bubble changes from supersonic flow to subsonic flow directly.Downstream from the intersection area,the fluid turns back into a large positive streamwise pressure gradient flow, and expands quickly. As a result, the velocity here changes greatly, and the irreversible aerodynamic loss is large, which results in a large EGR in Region B. Region D is strongly affected by the viscosity effect, and hence the aerodynamic loss is large. Region C is located between Region B and Region D, downstream from the separation bubble, so the wall viscosity effect of Region C is smaller than that of Region D,and the velocity and the streamwise pressure gradient magnitude of Region C are much lower than those of Region B. Thus, Region C has a smaller EGR.

Fig.28 Total entropy generation rate of interaction area.

Fig.29 Loss distribution in wake area.

5.3. Loss analysis in shock wave and wake interaction area

Fig.29 shows the time-averaged EGR symbolizing the loss in the shock wave and wake interaction area.In the respect of the wake,the EGR is large,but owing to the existence of the shock wave, the wake is quickly dissipated. In the vicinity of and downstream from the shock wave, the proportion of the loss caused by the wake is very small. In the respect of the shock wave, the reflected shock wave intensity is weakened by the process of reflection.At the same streamwise position,passing through the wake,the left shock wave intensity is weaker than that on the right side, and the EGR of the left shock wave is lower too. Therefore, on one hand, the vortex structures breakup is helpful to reduce the flow field loss, and on the other hand, the shock wave intensity attenuation is also conducive to reduce the flow field loss.

6. Conclusions

In this paper, the DDES method is used to obtain a highlyloaded turbine cascade’s instantaneous and time-averaged flow field structures. Results show that a hybrid RANS/LES method can obtain detailed flow structures and has its advantages over the RANS method. Based on detailed flow structures, flow mechanisms in a transonic cascade have been analyzed, and conclusions are as follows:

(1) In transonic flow fluid, there are strong oblique shock waves and weak shock waves simultaneously. The strong oblique shock waves that origin from the blade trailing edge in the flow passage can induce boundary layer separation. In turn, the presence of a separation bubble can increase the incident angle of the oblique shock wave on the blade suction surface, resulting in Mach reflection. In the separation region, the existences of shock waves and the upstream half of the separation bubble can cause a large value of TKE.In the upstream half of the separation bubble, the production term and the pressure transport term play dominant roles in the influence of TKE, while in the shock wave intersection area,the pressure transport term and the pressure dilatation term dominate the TKE distribution.

(2) A wake can cause shock wave reflection and refraction on the boundary of the wake and change the shock wave intensity. Compared with those upstream from the shock wave, the large-scale vortex structures are broken and dissipative,and they disappear quickly downstream,which makes the flow field tend to be uniform.The POD method can be used to analyze transonic flow mechanisms.The shock waves and the wake vortices dominate the flow field deterministic unsteadiness in POD Mode 2.

(3) Furthermore,the total irreversible loss can be separated into two parts,i.e.,the viscosity loss and the irreversible heat transfer loss. The viscosity loss plays a dominant part in the transonic flow field. The SWBLI changes the loss distribution near the boundary layer.The interaction between shock waves and wakes is beneficial to reduce the flow field loss. There are several weak shock waves in the transonic flow field including weak normal shock waves and lip shock waves.The weak shock waves are able to have big effects on flow unsteadiness characteristics and can increase the irreversible loss.

This paper is instructive for TKE,POD,and EGR analysis in highly-loaded turbines in SWBLI and SWWVI areas. In future work,finer experimental results will be needed to verify the simulation results.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 51876098). Professor J. P. Gostelow kindly provided the blade geometry.