Article 正式发布 Versions 1 Vol 28 (5) : 886-904 2019
Effect of Leading-Edge Optimization on the Loss Characteristics in a Low-Pressure Turbine Linear Cascade 前缘优化对低压涡轮直列叶栅损失特性的影响研究
: 2019 - 09 - 05
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Abstract & Keywords
Abstract: This paper presents a numerical study on the aerodynamics loss reduction characteristics after the leading-edge (LE) optimization in a low-pressure turbine linear cascade. The LE was optimized with a simple and practical method of “Class Function/Shape Function Transformation Technique” (CST). The simulation conditions, covering the whole working range, were independently determined by incidence, Reynolds number and Mach number. Quantitative loss analyses were carried out with a loss breakdown method based on volumetric integration of entropy production rates. To understand the reason of loss reduction, the local sources at different operating points were identified with entropy production rates. The results showed that LE optimization with the CST method played a positive role in decreasing the total losses, and the working range with lower loss was extended. The profile loss and the endwall loss were significantly reduced by the LE optimization, which were also verified to be the major causes of the total loss reduction by loss breakdown. The decrease of profile loss can be attributed to the boundary layer near the LE region and the boundary layer of downstream at off-design incidence. The reduction mostly came from the pressure side at negative incidence, while came from the suction side at the positive incidence. The endwall loss was decreased markedly about 2.5%–5% by the LE optimization at the incidence of ‒12°, which was 1% at the incidence of 12°. The mechanism for the endwall loss reduction at different incidences was different from each other. At negative incidence, the LE optimization diminished the corner separation vortex on the pressure side. While at positive incidence, the benefits came from three aspects, i.e., reduced suction LE separation bubbles close to the endwall, reduced passage vortex strength, and weakened shear process between passage vortex and trailing shed vortex. The loss of the downstream zone was relatively lower than that of the profile losses and the endwall losses. The effect of LE optimization on the loss of the downstream zone at different conditions was complex and it depended both on the profile boundary layer behavior at the suction trailing edge and on the passage vortex strength. 本文以低压涡轮直列叶栅为研究对象,采用数值模拟的方法对前缘优化前后全工况的叶栅损失特性进行相应研究。前缘几何构建基于一种简单的具有工程实用价值的技术——“形状变换函数技术”(CST)。研究工况由马赫数、雷诺数和攻角独立确定。损失特性的定量分析主要由熵产率体积积分的方式展开,在此基础上对叶栅流道内不同区域进行损失拆分,并进一步确定各工况条件下损失变化的局部位置和机理。结果显示:CST方法优化的前缘有利于降低叶栅总损失,使得叶栅在较宽的工作范围内具有相对较低的气动损失。其中,叶型损失和端区损失的降低是总损失降低的两个主要因素。在非设计攻角下,叶型损失降低主要来源于前缘局部区域以及下游的边界层的变化。负功角时,叶型损失降低主要发生在压力侧的边界层区域,相反,正攻角时,叶型损失的降低主要位于吸力侧的边界层区域。在攻角为-12°时,端区损失降低约2.5% ~ 5%,而攻角为12°时,端区损失降低1.0%。不同攻角条件下端区损失减小背后的机理是不同的:负功角条件下,端区损失降低的原因是前缘优化显著削弱了前缘压力侧的角区分离涡;正攻角条件下,端区损失降低的原因主要来自于三个方面:减小的吸力侧前缘分离泡,降低的通道涡的强度和削弱的通道涡和尾缘脱落涡的剪切过程。下游区域的损失变化程度相对前两者较小,并且前缘优化对下游区域损失的影响略复杂,其主要是受吸力侧尾缘边界层的特性和通道涡的强度共同决定的。
Keywords: low pressure turbine, leading edge, loss breakdown, loss audit, boundary layer, entropy production rates
1. Introduction
The design and optimization of profile are the key tasks in the turbomachinery blade design since the profile has a great effect on the aerodynamic performance. The emphasis is mainly laid on the suction side and pressure side profiles of the blade. However, the LE, as the first position of the cascade that hot gas reaches to, also has an effect on the laminar flow extended from the stagnation point. Thereby, the flow in the vicinity of the LE is also complex [1]. A small “spike” is often observed in the profile surface pressure distribution, which may result from the design process, manufacture defects and blade erosion [2]. The strength of spike is greatly determined by the LE geometry, which has a remarkable influence on the boundary layer behavior of immediate downstream. The flow separation, reattachment, transi- tion, and reverse transition around the LE, can also affect the whole profile boundary layer behavior, which is particularly critical to the aerodynamic loss and external surface heat transfer in the detailed design [3].
Bcubic polynomial coefficientsαtturbulent thermal diffusivity/m2·s-1
cnormalized reference lengthβ′turbulence model constant, 0.09
cpspecific heat at constant pressure/
γheat capacity ratio; turbulence intermittency
Hblade height/mδdistance normal to the wall/m
hspan position/mζenergy loss coefficient
iincidence/°κturbulent kinetic energy/m-2·s-2
MaMach numberthermal conductivity/W·m-1·K-1
M1index of class functionμdynamic viscosity/Pa·s
N1 (N2 , N3 )grid number in GCI calculationξtotal loss coefficient based on entropy production rates
direction vectorρdensity/kg·m-3
Ppressure sideωspecific dissipation rate/s-1
papparent order; pressure/Pavorticity/s-1
Rggas constant/J·kg-1·K-1Subscripts and Superscripts
ReReynolds number1Plane 1
Ssuction side2Plane 2
Δsspecific entropy increase/J·kg-1·K-1*stagnation quantity
strain-rate tensor/s-1̅normalized quantity; time-averaged quantity
volumetric entropy production rates/
axaxial chord
Velocity/m·s-1iindex of tensor; each fixed zone number
wtottotal pressure lossmmean quantity
x , X, y, Ycartesian coordinates/moribaseline or original
Greek Symbolsprimprimary flow
αthermal diffusivity/m2·s-1zaxial direction
The investigation of LE effects can be traced back to the 1960s on aerofoil design in external flow because the separation bubbles on the leading edge are very critical to the aerofoil stall characteristics [4-7]. In the early age, most investigations on LE flow are carried out on a flat plate. Davis [8] found that the double arc could minimize the tendency of the separation bubbles, and an elliptical leading edge should be used for a greater margin before boundary layer separation. Carter et al. [9] pointed out that the performance is largely affected by the spikes in fans, compressor and pumps. Furthermore, the study showed that the sharpest LE has the widest operating range, which is opposite to the conventional idea that the “sharpest” LE has the narrowest operating range. In recent decades, a lot of works have been done to study the LE effect on the turbomachine’s aerodynamic perfor- mance, mainly focusing on the compressor. Walraevens and Cumpsty [10] investigated the circular and elliptical shapes of LE on a single compressor aerofoil at different Reynolds number, turbulence intensity and scale. The results showed that the separation bubbles with circle LE are absent for the elliptic LE. The research of Lu et al. [11] on a circle LE with flats indicated better perfor- mance and lower manufacturing costs compared to the circle LE. Wheeler et al. [12] studied the effects of a circular and a 3:1 elliptic LE on the performance of a controlled diffusion aerofoil in a low-speed single-stage high pressure compressor. The results showed that the profile loss of circular LE is 32% higher than that of elliptic LE. The presence of wake leads to a thicker laminar boundary layer and an increase of 20% on the suction boundary loss with circular LE. Goodhand and Miller [2] concluded that the effect of LE geometry on the compressor aerodynamic performance can be ignored when the spike diffusion factor is below 0.1 over the blade’s incidence range. Liu et al. [13] conducted a numerical study on LE geometry and found that the premature transition occurs as the spike strength increasing. The variation of the performance under different Mach number and Reynolds number are similar. Song and Gu [14] investigated the influence of LE curvature on the compressor cascade performance numerically and theoretically, and revealed that the LE spike is controlled by two factors, curvature discontinuity at the junction point and the nose curvature. In addition, the spike is dominated by the curvature of junction point when the incidence is small, while mainly affected by nose curvature at large incidence. Yu et al. [15] numerically studied the effects of circular, elliptical and curvature continuous LE on the boundary layer development. The results showed that the performance is influenced at the initial development of LE boundary layer, and the suction spike can be eliminated by a curvature continuous design. Le and Liu [16] presented a numerical and experimental study about the influence of LE deformations on the boundary layer characteristics. The results showed that the boundary layer can be influenced in three aspects below: the adverse static pressure has a promoting effect on separation; pressure spike may promote the transition; and the pressure disturbance can suppress the laminar bubbles.
In the axial turbine research, the flow mechanism of transition and separation near the LE was first carried out by Hodson [3]. The effect of Reynolds number, compressibility, incidence and free-stream turbulence were considered in the experiments. Benner et al. [17, 18] carried out an experimental study on the performance of a turbine cascade with different diameters and wedge angles. The results demonstrated that, a small diameter leads to a lowprofile loss, and the cascade with a large diameter has high secondary loss due to the strong passage vortex. Mamaev [19] experimentally investi- gated the profile loss in two series of turbine cascades, including a conventional LE and a modified continuous curvature LE. The results showed that the profile loss is reduced by 0.2%–0.4% with modified LE. Bai et al.[20] explored the effects of circular and elliptical LE on the profile losses in a turbine, and the elliptical LE performed well in reducing the profile loss in a wider operating range. Zhang et al. [21] proposed a simple and practical LE redesign method based on the principle of CST on a low-pressure cascade. The method was validated to be effective in removing the LE separation bubbles and reducing the profile loss at the design point. Also, the redesigned LE, applied on a five-stage low-pressure turbine, results in a increase of 0.6% in aerodynamic efficiency. Based on the same LE modified method, Cui et al. [22] analyzed the loss reduction at design incidence, and found that the application of LE optimization leads a stage efficiency increment of 0.1%–0.5% in a high-speed aerospace vehicle low- pressure turbine rotor.
Owing to the importance of LE profile curvature on the aerodynamic performance, the implementation of LE design should attract more attention than before. Based on the previous research, the LE design method can be classified into two categories. One is named “Direct Method”. The LE profile is designed in circle or elliptic during the profile design period, which allows the curvature discontinuous at the junction point of LE and main surface, or curvature continuous with limited design space of the remainder aerofoil. The other one is implemented after the conventional profile design [23] without considering the discontinuous curvature problem at the junction point, named “Indirect Method”. The LE will be redesigned or modified in other advanced curve generation methods, such as polynomial [24], Bezier spline [25] and CST [2, 13, 21, 22], which will guarantee that the curvature is continuous at the junction point. However, a higher level of manufacturing technology is required. With recent advancements in the development of material and manufacturing technology, the “Indirect Method” is considered usable and more appropriate. Based on the above analysis, the “CST” redesign method is chosen in the present study.
To the author’s knowledge, the published literatures about the impact of LE on the loss characteristic in a turbine cascade over the whole working range is relatively few and incomplete. Most researches focus on the benefits of profile losses, while the effects on the secondary losses and the trailing wake loss are barely introduced in relevant literature. It is also important in correcting the accuracy of the loss model in turbine aerodynamics design work, especially under the off-design conditions.
Based on the above background and description, the objectives of the present study involve:
(1) To quantify the benefits of the cascade aerodynamics performance by the implementation of “CST” LE over the whole working range. The detailed operating point condition over the working range is determined by three independent variables: Reynolds number, Mach number, and incidence.
(2) To further quantify the changes in the loss at different regions by the “CST” LE, a loss breakdown method based on the volumetric integration of entropy production rates [26] is adopted. The method is imple- mented by dividing the domain into six zones and the loss reduction of each zone is calculated separately.
(3) To identify the local source of the loss reduction in profile loss, endwall zone loss, and downstream zone loss, the corresponding flow characteristics are also taken into account to analysis the reasons for the loss reduction.
2. Model and Numerical Approach
2.1   HD blade geometry and LE optimization method
In this paper, the Hodson-Dominy (HD) blade is selected as the baseline profile, which is a mature profile from the root section of a low-pressure turbine rotor and provides approximately 93° of turning. It operates at an inlet flow angle of 38.8°, exit Mach number of 0.71, and a chord-based exit Reynolds number of 2.9×105. A “spikeless” LE is obtained based on the baseline profile with CST method, which is illustrated in Fig. 1. Except for the LE part, the remaining parts are the same with the original profile. Further details about the linear cascade with “Ori” LE and “CST” LE are described in Refs. [27, 28].

Fig. 1 Comparison of Ori and CST leading edge of HD profile
The principle of CST is proposed by Kulfan [29], and first applied in the application of LE redesign by Goodhand [2]. The local LE profile is determined as Eq. (1). In which, M1 is the index of class function; c is the reference length of the LE camber; S is the shape function. Since many coefficients in the construction of shape function are unknown, which are complicated and time-consuming to determine for a specified profile.
A practical method with polynomial is proposed to construct the shape function by Zhang [21], and the method is also applied in this paper. To improve the robustness and avoid the curvature discontinuity within the original camber, the reconstruction of the camber line in the present investigation is meliorated with a cubic polynomial (). The unknown coefficients of the cubic polynomial and the detailed values of the key design parameters (the radius of LE and cut length) in the CST method are the same as Ref. [22].
2.2   Operating points for simulation
Reynolds number, Mach number, incidence, and turbulence intensity are the most critical independent parameters that have a strong effect on the flow performance inside a turbine. In the present study, the free stream turbulence intensity (FSTI) is set 4%, because only the real operating environment is taken into account for evaluating the aerodynamics performance. Table 1 lists all the operating conditions in the present study. To implement the three-dimensional numerical work, the independent dimensionless values of Mach number and Reynolds number should be converted to dimensional boundary conditions. The determination of boundary conditions is followed in the allowed operating range of the transonic cascade facility of Whittle Laboratory [30]. Finally, 330 K is chosen as the inlet stagnation tempera- ture for all the simulations. To maintain the Mach number and Reynolds number to be independent, the inlet stagnation pressure and outlet static pressure are obtained in two steps. First, preliminary values are solved with one-dimensional aerodynamics equation coupled with Sutherland viscosity and total pressure loss coefficients; Second, the detailed values of pressure are determined by CFD, which is initialized with the preliminary values. The cases or the legend names at the conditions of three Reynolds numbers are labeled as “Re1”, “Re2”, and “Re3” thereinafter.
Table 1 The simulation condition of each operating point
Ma0.45, 0.65, 0.80
Re /1051.45, 2.71, 5.55
i−12, −6, 0, 6, 12
2.3   Numerical approach and grid strategy
Computational fluid dynamics (CFD) simulations are carried out with the commercial tool ANSYS-CFX to solve the three-dimensional steady Reynolds-averaged Navier–Stokes equations. The shear stress transport (SST) turbulence model is applied to predict the turbulence, and γθ transition model is chosen to capture the boundary layer transition phenomena, both of which are validated appropriate for the present investigation in Ref. [22]. Advection scheme is set to high resolution.
The convergence criteria based on the reduction of RMS (Root Mean Square) for all simulations residuals is below 1.0×10‒4,and the convergence level reaches 1.0× 10‒6 at the condition of 0° incidences. It should be noted that the domain imbalance of energy equation is controlled below 0.01%.
The computational domains are built in one passage with pitchwise periodic boundaries. In addition, all of the nodes on one side of the interface correspond in location with all of the nodes on the other side of the interface, which is achieved to reduce the interaction error and global imbalance. The inlet and the outlet positions are set at 0.8 times axial chord upstream of the LE, and two times axial chord downstream of the trailing edge (TE) respectively. The computational domain is all constructed with hexahedral cells, as exhibited in Fig. 2. The blade surface and end walls are adiabatic and no-slip. It is worth noting that, to capture the LE separation bubble, more than 20 nodes should be arranged along the bubble length [31]. Considering the boundary layer transition and turbulence model requirement on the grid distri- bution near the wall, all the simulations are guaranteed that yplus is less than 1.0, and the normal expansion ratio of the grid is set 1.1. The study of grid dependence is carried out in two aspects as discussed below.

Fig. 2 Schematic diagram of the HD profile cascade grid
One is based on the regular analysis of regular relation of mesh nodes number and the value of parameters that concerned. Fig. 3 illustrates that five loss parameters (wtot : total pressure loss, wprof : total profile pressure loss, ξtot : total loss based on entropy production integration, ξprof : profile loss based on entropy production integration, ζtot : isentropic total energy loss) tends to be constant when the total cells number is above 5.7 million. The definition of each loss coefficient will be introduced thereinafter. The other one is the grid convergence index (GCI) study, which is used to estimate the discretization error and validate the loss audit method based on the entropy production rates, and the detailed data are listed in Table 2. In the present study, the GCI of five loss coefficients was calculated according to the procedure in Ref. [32]. Three sets of grids with N1 = 4.04 million cells, N2 = 5.71 million cells, N3 = 10.24 million cells are generated for original HD profile. The corresponding operation point is at an exit Mach number of 0.71 and Reynolds number of 2.71×105. The grid refinement factor γ21 =1.217 and γ32 =1.108. The GCI32 of the five loss parameters are all higher than GCI21, which means the discretization error reduces with the refinement of the grids. All five GCI32 values and three of the five GCI21 are less than 1%, which denotes the precision is high enough with N3 grids. The GCI32 of ξtot and wprof are slightly higher than 1.0. To minimize the whole computational costs, the scheme of medium grids number N2 is chosen for all simulations.

Fig. 3 Grid dependence study of HD Ori profile
Table 2 Samples calculations of discretization error
To further verify the accuracy of the computational method, numerical validation has been conducted by comparing the calculations for the HD cascade with the experiments by Hodson [3, 27]. The comparisons at different conditions are shown in Figs. 4-6. Fig. 4 shows the comparison of the total pressure losses of the cascade at the conditions of three Reynolds numbers. It should be noted that the Reynolds number of the validation work are a little different from that in Table 1. The numerical results coincide well with the experimental data, except for the results at the condition of Re = 2.71×105, that the total loss is slightly smaller than the experimental data.

Fig. 4 Comparison of CFX total pressure loss with experimental data when relative pitch = 0.556 [27]

Fig. 5 Comparison of the Mais distribution at the mid-span between experiment and 3D RANS [3]

Fig. 6 Comparison of the LE separation bubbles and suction surface flow between experiment and 3D RANS [3, 27]
Fig. 5 shows the comparison of the distribution of isentropic Mach number at mid-span between the results of CFD and that of experiments in Ref. [3]. The LE pressure spike and pressure plateau on the suction surface, typically associated with the laminar shear layer of a steady separation bubble, are predicted well compared with the experimental data. The detailed results of the flow near the profile are compared with the experiments at three conditions. The surface streamlines displayed in Fig. 6 demonstrate that the phenomenon of the LE separation bubbles and the separation flow on the suction side can be accurately captured with the numerical method presented in this paper.
3. Results and Discussion
3.1   The methods of loss breakdown and loss audit
To measure the loss reduction and identify where the loss reduction has occurred due to LE optimization, loss breakdown is carried out to investigate the effect of LE modification on the change in loss at different regions [33]. The strategy of loss breakdown is achieved by separating the domain into several fixed zones; the loss audit is calculated through the volume integration of entropy production rates in each zone. Then the loss coefficient ξi of each zone is defined in Eq. (2), which is normalized by the isentropic kinetic energy at a specified location in Fig. 7.
where is a mean temperature calculated with ; T2 is the static temperature at the Plane 2; , are the total temperature at Plane 1 and Plane 2 respectively; m1 is the mass flow at Plane 1; νi is the volume of each fixed zone; cp is the specific heat capacity at constant pressure; , p2 are the total and the static pressure at Plane 2 respectively; γ is the heat capacity ratio.
The entropy production rates consist of four aspects [34]:
(1) : entropy production rates by direct (time-averaged) mechanical dissipation;
(2) : entropy production rates by turbulent mechanical dissipation with SST k- turbulence model;
(3) : entropy production rates by heat conduction due to direct (time-averaged) temperature gradients;
(4) : entropy production rates by heat conduction due to turbulent temperature gradients with the SST turbulence model.
Where is the strain-rate tensor; k is the turbulent kinetic energy; ω is the specific dissipation rate; is the direct (time-averaged) temperature; According to SST k-ω turbulence model theory, β′=0.09; α is thermal diffusivity; αt is turbulent thermal diffusivity.
The zones for volume integration of entropy production rates include six parts, i.e., upstream zone, endwall zone, pressure side zone, suction side zone, passage zone and downstream zone. And all are illustrated in Fig. 7(b) and Fig. 7(c). All the zones for integration are located in the region between Plane 1 and Plane 2, as showed in Fig. 7(a), excluding the regions of repeating trailing wakes and inlet freestream from the whole computational domain. The hub and tip endwall zone include all the grid elements within 5% of the span near each endwall. The upstream zone starts from Plane 1 that 23.5% axial chord ahead of the leading edge, meanwhile the downstream zone before Plane 2 is 25% axial chord downstream of the TE. The pressure and suction side zones are located in the regions that both extend 2mm away from the profile surface. The remaining region is the passage zone.
Considering that the accuracy of direct volume integration of entropy production rates is sensitive to the grid resolution, further validation on the grid number of N2 = 5.71 million is carried out through comparing the total loss coefficients wtot_sw (Eq. (6)), which is transformed from the relational expression of entropy increase, against the total pressure loss wtot (Eq. (4)) and the total energy loss ζtot (Eq. (5)). The entropy increases can be transformed into total pressure loss with Eq. (6) in an adiabatic flow process, across a stationary component. Noting that the entropy increase is used to compare with that of the volumetric integration of entropy production rates in a direct way, which is obtained with the total outlet pressure in Eq. (8). Fig. 8 illustrates the comparisons at all the operating points. The ordinates of total pressure loss coefficients wtot_sw1 and wtot_sw2 in Fig. 8(c) and Fig. 8(d) are obtained through Eq. (7) and Eq. (8), respectively. Obviously, the loss coefficients wtot_sw1 of “Ori” and “CST” LE match well with the other three audit methods. Apart from the condition with the incidence of 12°, the benefit of the wtot_sw1 reduction from LE optimization is a little larger than that of wtot_sw2 , which is mainly caused by the larger region occupied by secondary flow at larger incidence. Nevertheless, as an aerodynamic loss audit parameter, the integration of entropy production rates is suitable for the present study with the grid number of 5.7 million based on the quantitative relation between ζtot , wtot and wtot_sw2 .

Fig. 7 The different loss breakdown zones for entropy production rate integral calculation

Fig. 8 The comparison of loss in different definitions at the condition of Ma = 0.65
3.2 Influence on total loss reduction and loss proportion
Firstly, a global review of the impact of LE optimi- zation on the change in total losses is described in Fig. 9. It is found that the performance of the cascade is improved in a certain amount under all conditions, especially at the condition of nonzero incidences. The LE optimization with the CST method can significantly maintain the flow in a state of lower loss over a wider operating range. It should be noted that the relation between the total losses and the Reynolds number varies with Mach number and incidence. The total loss with “Ori” LE gets higher when the Reynolds number increases from 2.71×105 to 5.55×105 at zero incidence when the Mach number is less than 0.8. Unlike the results of “Ori” LE, the total losses are significantly reduced under the conditions with higher Reynolds number, because of the delay of the transition on the suction surface with “CST” LE. The effects of Reynolds number, Mach number, and incidence on profile loss are provided in Ref. [3]. More numerical information about the boundary layer behavior of HD profile at different Reynolds number can be seen in Ref. [22]. Fig. 9 also demonstrates that the benefit reduces as the Mach number increases from 0.45 to 0.80 when the other two independent variables remain unchanged.
The cases with Ma=0.65 are chosen for the quantitative study, considering the similar tendency of the relation of total loss and incidence or total loss and Reynolds number at different Mach number. It should be noted that the benefits are weakened as the Mach number increases. Fig. 10 presents the loss reduction percentage of each zone relative to the corresponding total loss of “Ori” LE, and the loss of each zone is calculated through Eq. (2). Obviously, the relative loss reductions of the pressure side, the suction side, and the endwall zones are the mostly affected three ones among the six loss components.
Regarding profile loss, most loss reduction comes from the pressure side at negative incidence condition, while comes from the suction side at positive incidence condition. At the incidence of ‒12°, the relative loss reduction of the pressure side and the endwall are approximately 3.5%–10% and 2.5%–5% respectively. At the incidence of ‒6°, the relative loss reduction of the pressure side is 2.5%, and the percentage of loss reduction seems not to be influenced by the Reynolds number. At the incidence of +6°, the loss reduction of suction side increases by 16%–26% as the increase of Reynolds number. While at the incidence of +12°, the loss reduction of the suction side decreases nearly by 10%–17% compared with that at +6° incidence.
Regarding the relative loss reduction of endwall zone, it decreases markedly by 2.5%–5.0% with the LE optimization at the incidence of ‒12°, while decreases by 1.0% at the incidence of +12°. To summarize, the profile zone and the endwall zone are the two main sources of the total loss reduction. To further identify the local position of the loss reduction, the flow mechanism in these two regions needs to be paid more attention to.
Secondly, the loss proportion of each zone () is calculated at different conditions. In the present study, the profile loss is more than 50% of the total loss and about 2 times of the endwall loss. Considering the similar tendency of the loss proportion against the Mach number, the cases at the Mach number of 0.65 are chosen for further investigation. A bar chart is created to visualize the contribution of each loss component to the total loss in Fig. 11. The incidence has a more significant influence on the loss proportion of each zone compared with Mach number and Reynolds number. As the increase of incidence, the loss proportion of the pressure side reduces, and the suction side loss on the contrary decreases. Meanwhile, the loss proportion of endwall reduces as the increase of the incidence, especially at the conditions of positive incidence.
In addition, the absolute difference of loss proportion of each zone () is shown in Fig. 12. Obviously, the difference between the “CST” and “Ori” cascade can be found variation with the incidence and Reynolds number. Overall, the changes in the loss proportion of endwall are all enlarged with the “CST” LE, which is unlike the behavior of the two sides loss. At the nonzero incidence, the loss proportion on the suction side and the pressure side changes inversely, i.e., the pressure side loss proportion is decreased with the suction side increased at the negative incidence, and vice versa. At the incidence of +6°, it is interesting to find that the increase of the difference of downstream loss proportion is about 6%, which is just a little less than that of the endwall. The reason of this phenomenon will be discussed in the later section, since the downstream loss is not as high as the endwall loss at other incidences.

Fig. 9 Total loss variations along with Reynolds number, Mach number, and incidence (based on entropy production rates integration)

Fig. 10 The loss reduction percentage of each zone at the condition of Ma = 0.65
In Fig. 12, each side difference of the loss proportion is affected remarkably at nonzero incidence, which is unlike the loss variation as discussed above in Fig. 10. In respect to the suction side, both the loss and loss proportion are decreased significantly at the positive incidence. However, at negative incidence, the variations are smaller than that of positive incidences. In respect to the endwall loss at negative incidence, a slight increment in the loss proportion difference is shown in Fig. 12, while the relative loss is reduced obviously. As to the cases at positive incidences, the difference in loss proportion can be found to be increased significantly with only a little reduction in the relative loss.
To investigate the loss distribution further in two- dimension, Fig. 13 provides the comparison of the spanwise loss distribution before and after the LE optimization at the incidences of ‒12°, 0°, and +12°. Clearly, the location of the loss reduction varies in the spanwise direction at different incidences, i.e., the reduction of total loss is mainly located close to the endwall and mid-span region at the incidence of ‒12°. while located at the regions of passage vortex and mid-span for the incidence of +12°.

Fig. 11 The proportion of each loss component to the total loss at Ma=0.65

Fig. 12 The absolute difference of the proportion of each zone at the condition of Ma = 0.65

Fig. 13 The comparison of the spanwise loss distribution before and after the LE optimization
As discussed above, the profile and endwall loss are both influenced significantly by the LE optimization over the whole working range, which in general are the two major losses in turbomachines. The reasons for the reduction of these two kinds of losses will be discussed below respectively.
3.3   Influence on profile loss
Fig. 14 illustrates the comparison of the profile loss between the cases with the “Ori” and “CST” LE over the whole working range. Overall, the tendency of the changes in profile loss is similar to that of the total loss in Fig. 9, because the profile loss is the largest one among the six loss components. The reduction amount of the profile loss due to LE optimization, to some extent, is decreased with the increase of Mach number. Concerning the relation of total pressure loss to the incidence, the ranges of working state with low-loss are extended at all the Reynolds numbers.
Fig. 15 presents the relative loss changes in the pressure and suction side ( ). It can be found that the profile loss reduction is mainly attributed to the pressure side at negative incidence, and the pressure side profile loss reduces by 20%–40% at least. While, at positive incidence, the profile loss reduction is mainly attributed to the suction side, and the loss of suction side is reduced by 20%–30% at least, except that at the condition of Ma =0.8 and i=12°.
To ascertain the location of the profile loss reduction, the difference in the local entropy production rates () of the mid-span section are conducted at the incidences of ‒12°, 0°, and 12°, as shown in Fig. 16. At incidence of ‒12°, the local reduction of is mostly distributed near the pressure surface of the front half of blade, and a small part located at the cascade expansion section after the throat. It should be noted that the region around the LE of the suction side is found with increased. The reason is that the separation bubble size is diminished and the transition is delayed by the “CST” LE, and thus leads to a higher quadratic term of the velocity gradient at outer region of the boundary layer. At zero incidence, the local is increased in a finite region close to the LE surface, which is covered by a region with the decreased. However, the region affected is relatively smaller compared with the cases in the other incidences. At 12° incidence, the reduction of is located in a dramatic region that starts from the LE point to the TE some distance away from the suction surface, and the region is even extended into the downstream zones. In Fig. 17, the separation flow on the LE of the suction side surface is suppressed and the transition is delayed, which results in a fuller velocity profile on the surface in a long distance. Then, a relatively large region with the increased exists in the front half of cascade close to the suction surface.

Fig. 14 Profile loss variations with Reynolds number, Mach number, and incidences

Fig. 15 The relative change of the pressure and suction side of profile loss (ps: pressure side; ss: suction side)
3.4   Influence on endwall loss and secondary flow
Fig. 18 presents the change in the endwall loss at different conditions. It can be found that the “CST” LE has little influence on the endwall loss at the incidence of ‒6° to +6°, which is not as obvious as that on the profile loss. However, the difference of the endwall loss between

Fig. 16 The difference contour of entropy production rates due to LE optimization at mid-span (Ma=0.65)

Fig. 17 Flow mechanism near the LE region before and after CST LE optimization
“Ori” LE and “CST” LE is significant. As the Mach number increases, the difference always decreases, which increases when the incidence is ±12°. Moreover, the benefits can be neglected when Ma = 0.80 and i = +12°.
It can be inferred that the loss generation mechanism of the endwall zone is quite distinct from each other at different incidence, considering the loss proportion of each zone in Section 3.2 and the spanwise distribution of the total pressure loss in Fig. 12. As the incidence plays a dominant role in the endwall loss, the normalized entropy production rates at the axial planes (Cz = 0–0.98) within 10% height are displayed in Fig. 19, thus the spatial distribution of the endwall loss sources is obtained. It should be kept in mind that the loss integration for the endwall zone is only processed within 5% cascade height, and the contours of an additional 5% height are for the assistant analysis.
At zero incidence, the change in entropy production rates is found to exist on the axial planes of the regions “B” (Cz = 0.3–0.5), which implies that the corner vortex (CV) of pressure side is affected by the “CST” LE. However, the region is too small to make a great impact on the endwall loss reduction.
At the incidence of ‒12°, LE corner separation vortex on pressure side (LECSVp), located in the region of “A1” (Cz = 0.3–0.5), is suppressed significantly by the “CST” LE, thus results in a thinner region of “A2” (Cz = 0.9–0.98), which makes a great contribution on the reduction of endwall loss.
At the incidence of +12°, the region of the pressure side in the endwall zone is not affected by the LE optimization. In fact, the endwall loss is mainly influenced on the suction side including three positions. The first one is located near the profile and the endwall when Cz=0.15–0.4. Both the area and the normalized entropy production rates are reduced significantly, especially on the suction surface. This is because that the boundary layer of the suction side (BLs) is thinner than that of the “Ori” LE. The second one is located in the region of “C”, where the horse vortex of the pressure side (HVp) impinges on the suction surface. The interaction process between the HVp and the mainstream flow nearby is enhanced by the “CST” LE, leading to an increase in the area with high normalized . The third one is located in the region of “D” (Cz=0.7–0.98), where is mainly occupied by the passage vortex (PV). The area counted for the 5% height endwall zone with high normalized is decreased significantly. It can be attributed that the loss induced by interaction between the PV and mainstream is weakened.

Fig. 18 The loss at endwall zone before and after the LE optimization

Fig. 19 The contours of the normalized entropy production rates at the end of 10% height (cax is axial chord length)
Actually, the location of PV is beyond the region of the 5% blade height. To further explore the effect of the LE on the aerodynamics performance, the change of the mixing between the PV and the mainstream is deserved to clarify due to that the PV is the most significant secondary flow in a turbine. Fig. 20 illustrates the features of secondary flow. The kinetic energy of secondary flow (SKE) is used to measure the strength of the secondary flow, which is defined in Eq. (8). CSKE is the coefficient of secondary kinetic energy normalized by the outlet dynamic pressure head (shown on the Plane “C”), i.e., Eq. (9). Where, represents the direction vector of the primary flow referring to pitch-average velocity of the mid-span. To further demonstrate the PV structure based on the surface streamlines, Fig. 20 also superimposes the sectional spin vector [35] lines on the contour of the entropy production rates at the Plane “S”. The direction of the sectional spin vector is determined by the vector of , in which represents the normal vector of the view section; is the local vorticity; is the corresponding velocity vector.
As shown in Fig. 20, the region occupied by the secondary flow is obviously enlarged at the positive incidence. With the increase of incidence from ‒12° to 12°, the spanwise size of the passage vortex on the Plane “C” is enhanced to some extent. However, the changes in the CSKE distribution of PV can be neglected at the incidence of −12° and 0°.
Concerning the condition at +12° incidence, it is noticeable that the size of secondary flow with high CSKE is decreased by the LE optimization, which demonstrates that the strength of PV is weakened. Furthermore, it is clear that the source with high loss is located at a shear layer after the TE of the cascade (marked as “PTS” in Fig. 20). And the shear layer is formed between the fluids near the suction surface, which has a significant spanwise velocity driven by the PV, and the fluid on the pressure side (marked as “PPS” in Fig. 20) has a small spanwise velocity.

Fig. 20 The secondary flow features at the Plane “C” and Plane “S” before and after the LE optimization
To further investigate what has impacted on the endwall flow and the PV strength that leads to a reduction of the normalized on the Plane “S” at +12° incidence, Fig. 21(a) illustrates the comparison of surface streamlines and normalized secondary velocity to gain a further insight. According to the flow patterns on the suction surface, the PV initiates close to the three-dimensional separation point (3DSP) on the corner, just located in the region of the LE separation bubble. It is known that the location of 3DSP is primarily affected by the LE separation bubble on the profile and the strong crossflow on the endwall, which will further affect the PV strength. In the present study, the 3DSP location is delayed approximately 4% of the curvilinear distance (S) on the suction surface with “CST” LE, which is 8%S downstream of the LE. The separation bubbles of “Ori” LE start at 2%S from the LE and reattach at 10%S for “CST” LE, while start at 1.3%S and ends at 12%S for “Ori” LE.
Moreover, due to the stronger cross-pressure gradient and exist of the LE separation bubbles, the flow pattern on the endwall is unlike that of the common sense, i.e., the two primary legs of the horseshoe vortex (HV) merge at an obvious position during the growth process. In the present investigation, the pressure side leg (HVp) consists of multi-dispersed groups of vortices, and the primary leg impinges on the suction surface some distance downstream the reattachment point. Besides, the process of suction side leg (HVs) that approaches on the suction surface is clearly influenced and promoted with the reduction of the LE separation bubble length (SBL). In addition, the interaction between the HVp and the HVs is slightly enhanced with the “CST” LE, while less contributes are made on the inviscid rotation effect of the PV in counterclockwise direction. It can be inferred reasonably that both of these two phenomena result in the decrease of the PV strength with the “CST” LE.
Fig. 21(b) illustrates the comparisons of the normalized secondary velocity distribution with both the LE configurations at Plane 2. Usec is the local norm of the secondary velocity (); U2 is the mass averaged velocity of the Plane 2. Clearly, the area occupied by the PV is reduced, and the magnitude of secondary velocity in the core part is also decreased. Thus, the strength of PV is decreased that leads to the reduction of entropy production rates in the PTS (shown in Fig. 20).
In addition, considering the spatial distribution of the zone for loss breakdown, the passage loss consists of a small portion of profile loss and secondary flow loss, which is caused by the PV accounted in the passage. Fig. 19 illustrates the loss variation of the passage zone with the “Ori” and “CST” LE. The tendency of passage loss change is similar to that of the profile loss at different conditions. Because the passage loss accounted takes only a small part of the total losses. Except for the conditions at high incidences, the benefits from “CST” LE are all slight. At +12° incidence, the passage loss increases steeply, because the loss will generate more by the suction profile loss and secondary flow in the “passage” region that beyond the limited 2 mm.

Fig. 21 The comparison of streamlines and normalized secondary velocity contour between “Ori” and “CST” LE at incidence of +12°
3.5   Influence on downstream loss
Fig. 23 illustrates the loss variation of the downstream zone before and after the LE optimization. In general, the downstream losses are relatively smaller than the profile losses and the endwall losses. The implement of “CST” LE does not seem to have an unambiguous effect on the downstream loss at the conditions of Re =1.45×105 with +6° incidence. However, it shows that the “CST” LE is a little harmful to the cascade performance.
To further analysis these phenomena, first, the mid- span differences of entropy production rates can be reviewed in Fig. 16. Clearly, when the incidence is less than +12°, the downstream loss reduction due to “CST” LE configuration is mainly concentrated in the limited region just downstream the TE within the wake shear layer. While, at +12° incidence, the source of loss reduction consists of three parts: enhanced wake shear layer downstream the trailing edge, extended region into the downstream came from the suction profile loss, and the region occupied by the PV in the downstream, which is also presented with the distribution of entropy production rates of trailing shed vortex, marked as “TEV” in Fig. 20. It can be inferred that the downstream loss is the results of multiple factors, which depends upon the upstream flow characteristics.
Second, to analyses the downstream performance deterioration are at +6° incidence. Fig. 24 and Fig. 25 are carried out to demonstrate the suction surface streamlines, entropy production rate contour of the profile and Plane 2, respectively. The boundary layer state on the suction surface in Fig. 24(a) is significantly influenced with LE configurations, i.e., the LE separation bubbles (LESB) are eliminated, and the transition point is greatly delayed to the expansion section after the throat with the “CST” LE, followed by the boundary layer separation and reattachment. Whereas, the suction boundary layer of “Ori” LE is almost completely turbulent from the LE. Although, both of the two boundary layers at the TE of the suction surface are turbulent, the velocity profiles are significantly distinct from each other because the turbulent boundary layers initiation and growth are in different external environment as shown in Fig. 24(b).

Fig. 22 The loss at passage zone before and after the LE optimization

Fig. 23 The comparison of downstream loss at the whole working conditions between “Ori” and “CST” LE

Fig. 24 The suction surface flow characters at incidence of +6°

Fig. 25 Comparison of the Plane 2 normalized s contour between “Ori” and “CST” LE at the incidence of +6°
The boundary layer with “Ori” LE is thicker than that of “CST” LE, which results in a lower velocity gradient and lower entropy production rates close to the surface. And downstream loss is increased in mid-span.
On the other hand, Fig. 25 shows the comparison of the normalized entropy production rates on Plane 2. It can be concluded that the secondary loss component is affected slightly. The entropy production rates values around the PV core region is increased, while the size of the PV region is reduced to some extent. Then the total downstream losses are generated by the PV strength.
Above all, the downstream zone loss depends both on the profile boundary layer state at the suction TE and the PV strength, both of which are mainly determined by the detailed operating point conditions.
4. Conclusions
The present study focuses on the benefits and loss variation character gained from the blade leading edge optimization over the whole working range with a simple and practical method of CST. A detailed quantitative loss analysis on different regions was presented with a loss breakdown method based on volumetric integration entropy production rates, which was validated to be appropriate. In addition, the local loss sources at different operating points were identified with the entropy production rates.
In general, LE optimization with the CST method plays a positive role in reducing the total cascade loss, and the working range with lower loss of the low-pressure turbine is extended. Both of the profile zone and the endwall zone are the main sources of loss reduction.
Compared with Mach number and Reynolds number, the incidence has a more significant influence on the loss proportion of each zone. Profile loss is the largest source term of the total loss in the loss reduction, and the reduction mostly comes from the pressure side at negative incidence, while comes from the suction side at the positive incidence. The locations of profile loss reduction are not only concentrated near the LE region but also located in the downstream boundary layer at off-design incidence.
Endwall loss is also affected by LE optimization, which is more obvious when the incidence gets larger. The relative endwall loss is decreased markedly by 2.5% –5% by the LE optimization at the incidence of −12°, and only decreased by 1.0% at the incidence of 12°.
Downstream loss is relatively smaller than the profile loss and the endwall loss. The influence of the LE optimization on the downstream loss at different conditions is complex.
It should be noted that the upstream wake was not taken into account presently, and that will be investigated in the following research.
This research is sponsored by the Project NO.51576037 supported by National Natural Science Foundation of China (NSFC).
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WANG Songtao*
TANG Xiaolei
WEN Fengbo*
WANG Zhongqi
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Published: Sept. 5, 2019 (Versions1
Journal of Thermal Science