E ﬀ ects of Blade Fillet Structures on Flow Field and Surface Heat Transfer in a Large Meridional Expansion Turbine

: This paper is a continuation of the previous work, aiming to explore the inﬂuence of ﬁllet conﬁgurations on ﬂow and heat transfer in a large meridional expansion turbine. The endwall of large meridional expansion turbine stator has a large expansion angle, which leads to early separation of the endwall boundary layer, resulting in excessive aerodynamic loss and local thermal load. In order to improve the ﬂow state and reduce the local high thermal load, ﬁve typical ﬁllet distribution rules are designed. The three-dimensional Reynolds-Averaged Navier-Stokes (RANS) solver for viscous turbulent ﬂows was used to investigate the di ﬀ erent ﬁllet conﬁgurations of the second stage stator blades of a 1.5-stage turbine, and which ﬁllet distribution is suitable for large meridional expansion turbines. The inﬂuence of ﬁllet structures on the vortex system and loss characteristics was analyzed, and its impact on wall thermal load was studied in detail. The ﬁllet structure mainly a ﬀ ects the formation of horseshoe vortexes at the leading edge of the blade so as to reduce the loss caused by horseshoe vortexes and passage vortexes. The ﬁllet structure suitable for the large meridional expansion turbine was obtained through the research. Reasonable ﬁllet structure distribution can not only improve the ﬂow state but also reduce the high thermal load on the wall surface of the meridional expansion turbine. It has a positive engineering guiding value.


Introduction
One of the main objectives of efficient modern gas turbine engine design is to reduce secondary flow loss at any stage. Generally speaking, the gas turbine has a higher end-wall angle design in order to achieve high-efficiency performance. The high endwall angle is designed to reduce the large deflection angle of the blade profile and meet the requirements of the expansion ratio. Due to the large expansion angle, the end-wall boundary layer separates early, resulting in a large secondary flow loss. At the same time, the thermal load of the endwall is affected by the secondary flow and becomes more complex. Numerous articles have shown that fillet structure plays an important role in reducing secondary flow loss [1,2]. However, the fillet characteristics of the large meridional expansion turbine stators have not been fully studied publicly. Therefore, it is of great value to study the influence of fillet structure on the flow and heat transfer characteristics of the large meridional expansion turbine stators.
According to Sharma et al. [3], secondary flow loss can be as high as 30%~50% in the aerodynamic loss in the stator blade row. A lot of literature shows that the detail part plays an important role in reducing the secondary loss of flow. Thole et al. [4] found that the application of an inverted circle at the junction between the leading edge of the turbine blade and the endwall could significantly weaken the horseshoe vortex. Becz et al. [5] applied an inverted circle and a bulb structure to the leading edge of a turbine cascade with a large size, a low aspect ratio, and a large angle, and found that an inverted circle and a smaller size bulb can effectively reduce the average total pressure loss of the area, while a larger size bulb can increase the loss. It can be seen that the protruding structure of turbine blades is worthy of further study for secondary flow control. Sauer et al. [6] and Gregory et al. [7] carried out numerical simulation and experimental measurement on the blade with fillet. Results show that by modifying the leading edge, the fillet structure can reduce the secondary loss and delay the development of the passage vortex. Yan Shi et al. [8] believe that the fillet on the turbine stage is extremely effective in inhibiting the flow separation near the leading edge of the rotor blade. The mechanism of the fillet structure to reduce loss lies in that it can reduce the intensity and size of horseshoe vortexes, thus delaying the development of passage vortices [9][10][11]. It can be seen that shaping at the junction between the blade root and endwall, whether it is leading edge or surrounding the entire blade root, is very important for the influence of passage.
In order to improve turbine efficiency, increasing inlet temperature is a very important measure. This measure makes the first two stages of turbine blades bear higher temperature continuously, and the inlet temperature is obviously higher than the melting point of the metal. Therefore, blade cooling technology is critical. However, the prerequisite to achieve efficient and reliable cooling blade design is to fully study the characteristics of thermal load distribution of the channel and endwall without cooling structure. The fillet structure of turbine blades plays an active role in weakening the high-thermal load area of blades and endwalls. Shih et al. [12] carried out a numerical simulation study on the leading edge fillet structure of turbine blades, and the results showed that the leading edge fillet structure was helpful to reduce the heat transfer coefficient of the blade surface and endwall. Acharya [13] et al. studied the flow field and heat transfer results of different fillet structures, and found that the parabolic fillet contour was helpful to weaken the heat transfer of the endwall. Goldstein and Han et al. [14,15] used mass transfer measurements to reveal the effect of filet structure on blade heat transfer through experiments.
The flow and heat transfer characteristics of stator blades in large meridional expansion turbines have been fully described in previous studies [16], but the influence of fillet structure on it is still unclear. In this paper, the flow and heat transfer of the stator blades in the second stage of a large meridional expansion turbine is analyzed by numerical simulation with five different typical chamfered structures. The influence of fillet structures on inlet flow field structure, vortex development, and heat transfer characteristics was studied by steady-state three-dimensional Reynolds-Averaged Navier-Stokes (RANS) simulation. The experimental data was used for experimental validation. It is of engineering significance for the design of large meridional expansion turbine.

5-Stage Turbine in Study and Boundary Conditions
The model studied in this paper is a 1.5-stage high-pressure turbine, including guide vanes (S1), rotor blades (R1) of the first stage, and stator blades (S2) of the second stage. The model is sufficiently extended in the inlet and outlet sections. The extension is 1.5 times the average chord length. The meridional section of the model is shown in Figure 1. It can be seen from Figure 1 that the shroud and hub endwalls of the S2 blade have an obvious large expansion angle in the meridian direction. Its average shroud expansion angle (α 1 ) is 22 • and hub expansion angle (α 2 ) is 13 • . The S2 blade has completed the experimental study of aerodynamic performance under low working conditions [17]. Inlet hub/tip radius ratio (S1,R1,S2) 0.864, 0.886, 0.863 Exit hub/tip radius ratio (S1,R1,S2) 0.886, 0.868, 0.797 Pitch at midspan (mm) (S1,R1,S2) 71, 36, 61 Number of vanes (S1,R1,S2) 40, 86, 45 Tg/Tw 1.5 Tw (K) 1030

Numerical Solver and Mesh Settings
In this paper, the turbine flow field is calculated by the high-performance commercial hydrodynamics software CFX 17.0 [20]. The software adopts the finite volume method based on the finite element method and absorbs the accuracy of the numerical solution of the finite element method. In addition, the fully implicit mesh coupling method is also used to solve the continuity equation and the momentum equation at the same time, which has good convergence performance and numerical accuracy. For the numerical calculation in this paper, Reynolds-averaged Navier-Stokes equations in generalized coordinates are solved by the finite volume space dispersion technique. The spatial discrete scheme is the "high-precision" scheme of commercial software CFX, and the time discrete scheme is the second-order backward difference Euler scheme.
This paper focuses on the detailed study of the chamfered structure on the turbine flow field and wall heat transfer, so the selected turbulence model should capture the flow state near the wall as much as possible. Meanwhile, the heat transfer coefficient of the wall is mainly determined by the flow state near the wall. Therefore, capturing the flow details near the wall is conducive to the accurate simulation of wall heat load. In the research of Krishnababu et al. [21], it is shown that the Shear-Stress Transport (SST) k-ω turbulence model has a good performance in capturing the complex flows and flow details of the turbomachine. Zuckerman et al. [22,23] carried out a detailed analysis of the turbulence models of k-ε, k-ω, SST, and v 2 -f and summarized their advantages and disadvantages. The results are as follows: the SST model combines the advantages of the k-ε model Detailed design parameters and working conditions of the 1.5-stage turbine are shown in Table 1. The boundary conditions at the inlet are given as the average total pressure and the average total temperature, and that at the outlet is given as the total pressure. The inlet flow angle is 0 • . The hub/tip radius ratio of the blade and the mass flow rate at the design point are given. The number of blades in S1, R1, and S2 were 40, 86, and 45, respectively. The wall temperature boundary condition is set as isothermal, which can be obtained according to the temperature ratio of the mainstream gas and the isothermal wall (T g /T w ) is equal to 1.5. The setting of this ratio was given in detail in the papers of Imran Qureshi et al. [18] and Ameni et al. [19]. It is shown in the papers that for the heat transfer simulation of turbines, the boundary conditions of non-adiabatic are usually determined by the heat transfer between the wall surface and the main flow. However, when Nu number is used to analyze the wall heat transfer coefficient, if the wall temperature does not change dramatically, it is generally considered to be independent of the wall temperature. Periodic boundary conditions are adopted on both sides of the channel, and the wall surfaces are set as a non-slip boundary. Table 1. Design parameters and boundary conditions of the 1.5-stage turbine.

Numerical Solver and Mesh Settings
In this paper, the turbine flow field is calculated by the high-performance commercial hydrodynamics software CFX 17.0 [20]. The software adopts the finite volume method based on the finite element method and absorbs the accuracy of the numerical solution of the finite element method. In addition, the fully implicit mesh coupling method is also used to solve the continuity equation and the momentum equation at the same time, which has good convergence performance and numerical accuracy. For the numerical calculation in this paper, Reynolds-averaged Navier-Stokes equations in generalized coordinates are solved by the finite volume space dispersion technique. The spatial discrete scheme is the "high-precision" scheme of commercial software CFX, and the time discrete scheme is the second-order backward difference Euler scheme. This paper focuses on the detailed study of the chamfered structure on the turbine flow field and wall heat transfer, so the selected turbulence model should capture the flow state near the wall as much as possible. Meanwhile, the heat transfer coefficient of the wall is mainly determined by the flow state near the wall. Therefore, capturing the flow details near the wall is conducive to the accurate simulation of wall heat load. In the research of Krishnababu et al. [21], it is shown that the Shear-Stress Transport (SST) k-ω turbulence model has a good performance in capturing the complex flows and flow details of the turbomachine. Zuckerman et al. [22,23] carried out a detailed analysis of the turbulence models of k-ε, k-ω, SST, and v 2 -f and summarized their advantages and disadvantages. The results are as follows: the SST model combines the advantages of the k-ε model and the k-ω model and has good simulation results for the whole complex flow. In addition, the SST model uses the variant equation to calculate the turbulent viscosity [24], so it has excellent results in simulating the adverse pressure gradient. Moreover, the accuracy of the SST model on the prediction of the wall heat transfer coefficient is also better than that of the k-ε and v 2 -f models. So, the SST turbulence model is adopted for fluid calculation in this study.
In this paper, the turbine blade passage grid is divided by IGG/Autogrid5, a preprocessing module developed by NUMECA company for generating multi-block structured grids. Figure 2 shows the grid diagram of 1.5-stage turbines with filleted S2. Figure 3 shows the topological structure of the cross-section of the S2 blade grid model. Structural hexahedral meshes are used in all the computational meshes. In order to improve the quality of the whole computational grid, O-type grids are used for blade surface and rotor blade clearance area, and H-type grids are used for inlet and outlet areas and the mainstream area. In the paper of Montomoli et al. [25], the wall heat transfer coefficient of turbine blades were studied in detail, indicating that the use of 20-layer grids near the wall can meet the requirements for the capture of boundary layer flows and wall heat transfer. Therefore, 20 meshes are used to divide the near-wall surface meshes in this paper. The final total number of cells is 13 million. In order to meet the requirements of the SST turbulence model, the average non-dimensional near-wall distance of y plus value is less than 1, so the height of the first grid of the grid model is 0.2 µm. All the computational models in this paper adopt the same topology model and grid setup. and the k-ω model and has good simulation results for the whole complex flow. In addition, the SST model uses the variant equation to calculate the turbulent viscosity [24], so it has excellent results in simulating the adverse pressure gradient. Moreover, the accuracy of the SST model on the prediction of the wall heat transfer coefficient is also better than that of the k-ε and v 2 -f models. So, the SST turbulence model is adopted for fluid calculation in this study. In this paper, the turbine blade passage grid is divided by IGG/Autogrid5, a preprocessing module developed by NUMECA company for generating multi-block structured grids. Figure 2 shows the grid diagram of 1.5-stage turbines with filleted S2. Figure 3 shows the topological structure of the cross-section of the S2 blade grid model. Structural hexahedral meshes are used in all the computational meshes. In order to improve the quality of the whole computational grid, O-type grids are used for blade surface and rotor blade clearance area, and H-type grids are used for inlet and outlet areas and the mainstream area. In the paper of Montomoli et al. [25], the wall heat transfer coefficient of turbine blades were studied in detail, indicating that the use of 20-layer grids near the wall can meet the requirements for the capture of boundary layer flows and wall heat transfer. Therefore, 20 meshes are used to divide the near-wall surface meshes in this paper. The final total number of cells is 13 million. In order to meet the requirements of the SST turbulence model, the average non-dimensional near-wall distance of y plus value is less than 1, so the height of the first grid of the grid model is 0.2 μm. All the computational models in this paper adopt the same topology model and grid setup.
For the numerical simulation in this paper, when the residual error of the momentum equation, energy equation, and turbulence model is less than 10 −5 orders of magnitude, and the difference between inlet and outlet flow is less than 0.1%, the calculation can be considered as convergence.     For the numerical simulation in this paper, when the residual error of the momentum equation, energy equation, and turbulence model is less than 10 −5 orders of magnitude, and the difference between inlet and outlet flow is less than 0.1%, the calculation can be considered as convergence.
As shown in Table 2, in order to exclude the influence of the computational grid number on the aerodynamic and heat transfer results of the numerical simulation, the author conducted grid independence verification of the total pressure loss coefficient and the shroud heat flux of the model without fillet in the paper [16]. It is pointed out that when the total number of nodes in the computational grid exceeds 11 million, the numerical results are no longer dependent on the grid number. Therefore, in order to ensure that the computational grid is no longer sensitive and can accurately predict the flow characteristics near the endwall, the number of grids used in this paper is between 12 million and 13 million.

Experimental Validation of Stator Blades (S2)
In order to verify the reliability of the numerical calculation method and grid generation in this paper, aerodynamic and heat transfer tests were carried out according to the existing open experimental results. In the experimental results published so far, there is no experimental data about the transonic expansion turbine at large meridional. Therefore, this paper verifies the accuracy of numerical calculation by comparing the Mach number distribution and Nu number distribution results of similar blade shapes under the condition of transonic velocity. This verification has been discussed in the previous paper [16]. In the paper of Imran Qureshi et al. [18], sufficient experimental studies on aerodynamic and heat transfer related to MT1 turbine blades have been carried out, and the data are worth believing. The blade profile and turning angle of the MT1 turbine are very similar to those of the S2 blade studied in this paper, and both of them work under transonic conditions. Therefore, the verification of numerical simulation in this paper uses the experimental data made by Imran Qureshi et al. [18] on MT1 turbines. Since the S2 blade concerned in this paper is characterized by large meridional expansion, the isentropic Mach number of the blade near the shroud endwall and Nu distribution of the shroud endwall are compared. In the verification and simulation of the MT1 turbine, the grid model, topology, and turbulence model set are consistent with the turbine model set in this paper. Figure 4 shows the comparison of isentropic Mach number between experimental data and numerical simulation results at the position of 90% blade height. Figure 5 shows the Nu distribution of the circumferential mean of the upper endwall. It can be seen from Figure 4 that the numerical simulation method adopted in this paper has a good effect on the aerodynamic simulation of the near-endwall of the turbine blade. Figure 5 shows that the difference between numerical data and experimental data is 9.62% in the simulation of heat transfer. The numerical simulation method can accurately predict the heat transfer trend of the shroud. The difference between the leading edge and the trailing edge is theoretically reasonable. Therefore, the heat transfer prediction in this paper can meet the research needs. It can be seen from the above analysis that the numerical simulation method and the grid model adopted in this paper are accurate and reasonable considering the errors caused by the experiment.

Fillet Configuration
Six different fillet structure models of S1 are designed in this paper based on the parameters fillet radius (R) and included an angle (θ). The specific design parameters are shown in Table 3. As shown

Fillet Configuration
Six different fillet structure models of S1 are designed in this paper based on the parameters fillet radius (R) and included an angle (θ). The specific design parameters are shown in Table 3. As shown in geometric design Figure 6 for fillet, a chamfering circle is not completely tangent to the endwall

Fillet Configuration
Six different fillet structure models of S1 are designed in this paper based on the parameters fillet radius (R) and included an angle (θ). The specific design parameters are shown in Table 3. As shown in geometric design Figure 6 for fillet, a chamfering circle is not completely tangent to the endwall but has a minimum θ to rearrange itself in the wall. When the fillet and endwall are completely tangent, it will cause a sharp decline in the grid orthogonality near the endwall, which can lead to producing a very big error in the process of numerical simulation. For this reason, θ is set to 10 • in this article. Therefore, in the setting of the grid structure, the fillet is determined by two parameters. When the R changes along the axis, there are five representative trends, that is, the radius size is constant along the axis, monotone minus, monotone increase, first decrease then increase, first increase then decrease. Therefore, this paper designs five types of fillet and non-fillet blades, which are circular structures at the root and the top. At the same axial position, the radius of the suction pressure surface is the same, that is, the distribution is symmetric. At present, there is no uniform regulation on the limitation of the blade fillet radius, but due to the limitation of channel size, the radius cannot be enlarged indefinitely. According to the existing literature, the largest radius of the fillet in the blade is in the research of the Becz et al. [5], and in the turbo blade, the maximum radius is about 25% of the width of the throat. When the radius of the blade root rounding continues to be enlarged, in practical applications, the flow loss of the blade will rise sharply, which is unfavorable to the aerodynamic performance of the engine. Therefore, the sum of the radius of the suction surface and pressure surface at the same axial position shall not exceed 50% of the minimum width of the cascade channel. That is, the radius size relationship is determined by the following formula: where, the subscripts ps, ss, min, and z respectively represent the pressure surface, suction surface, minimum value, and an axial position, and t represents the channel width of the cascade.

Schemes Parameter Setting Value (mm) Code
Case1 Case5  According to the limitation of formula (2), R is calculated to be less than 6 mm, and the maximum radius is about 1/2 of the thickness of the inlet boundary layer. In this paper, according to the actual processing requirements, the minimum R is selected as 8% t, and the value is equal to 2 mm. Figure 7 shows the distribution diagram of 6 typical fillets (each form has numerous distributions, but the mechanism of the influence on the flow field is similar, which will be analyzed below and the reasons will be given). The curve represents the projection of the inverted circle contour on the meridian plane, the abscissa is axial, the starting point and ending point correspond to the leading edge (LE) and trailing edge (TE) of the S2 respectively, and the ordinate is radial. Figure 8 shows the three-dimensional (3D) schematic diagram of 6 cases, which is conducive to a better understanding of their distribution rules.  Figure 7 shows the distribution diagram of 6 typical fillets (each form has numerous distributions, but the mechanism of the influence on the flow field is similar, which will be analyzed below and the reasons will be given). The curve represents the projection of the inverted circle contour on the meridian plane, the abscissa is axial, the starting point and ending point correspond to the leading edge (LE) and trailing edge (TE) of the S2 respectively, and the ordinate is radial. Figure  8 shows the three-dimensional (3D) schematic diagram of 6 cases, which is conducive to a better understanding of their distribution rules.

Aerodynamic Analysis of S2 with Different Fillet Structures
The formation of the horseshoe vortex (HV) at the leading edge of the blade is first affected by the fillet structure. Figure 9 is the limiting flow diagram at the hub of the S2 leading edge, showing the position and structure of HV. Results show that the HV at the leading edge is only affected by R of the leading edge fillet and not by the downstream structure of the fillet. The fillet influences the radial development of HV at the leading edge. With the increase of R at the leading edge, HV develops upward along the fillet and is closer to the blade. This change is due to the concavity of the fillet surface, which results in a low acceleration of the flow. This reduces the intensity of the adverse pressure gradient applied by the leading edge of the blade, which is the main driving force for the formation of HV. In addition, it leads to a decrease in the shape of the passage vortex (PV), which will be discussed in the next section. The formation of the horseshoe vortex (HV) at the leading edge of the blade is first affected by the fillet structure. Figure 9 is the limiting flow diagram at the hub of the S2 leading edge, showing the position and structure of HV. Results show that the HV at the leading edge is only affected by R of the leading edge fillet and not by the downstream structure of the fillet. The fillet influences the radial development of HV at the leading edge. With the increase of R at the leading edge, HV develops upward along the fillet and is closer to the blade. This change is due to the concavity of the fillet surface, which results in a low acceleration of the flow. This reduces the intensity of the adverse pressure gradient applied by the leading edge of the blade, which is the main driving force for the formation of HV. In addition, it leads to a decrease in the shape of the passage vortex (PV), which will be discussed in the next section. For the static pressure coefficients (Cps) at different blade heights, the load distribution and aerodynamic performance of the blades can be well expressed. The blade profile load distribution of the turbine blade directly affects the intensity of downstream secondary flows (Marchal and Sieverding et al. [26] and Benner et al. [27]). Figure 10 shows the pressure distribution of the 1%, 50%, and 99% blade height of S2, in which the positions of 1% and 99% blade height are located on the fillet structure. The magnitude of the adverse-pressure region of the suction surface is represented by h, and a histogram is drawn. Combined with Figure 10a-c, it can be seen that the fillet structure only affects the pressure distribution near the endwall but has little impact on the middle of the blade height. Therefore, the For the static pressure coefficients (C ps ) at different blade heights, the load distribution and aerodynamic performance of the blades can be well expressed. The blade profile load distribution of the turbine blade directly affects the intensity of downstream secondary flows (Marchal and Sieverding et al. [26] and Benner et al. [27]). Figure 10 shows the pressure distribution of the 1%, 50%, and 99% blade height of S2, in which the positions of 1% and 99% blade height are located on the fillet structure. The magnitude of the adverse-pressure region of the suction surface is represented by h, and a histogram is drawn. Combined with Figure 10a-c, it can be seen that the fillet structure only affects the pressure distribution near the endwall but has little impact on the middle of the blade height. Therefore, the research in this paper focuses on the vicinity of the endwall. Combined with Figure 10a,c, it can be seen that fillet mainly affects the pressure distribution of the front half of the blade suction and the trailing edge of the pressure surface. When R = 25% t of the leading fillet radius is compared with R = 8% t, the adverse-pressure region decreases by about 50%. So the large leading fillet radius is helpful to improve the flow state of the suction surface. Near the trailing edge of the pressure surface, the small fillet radius (R = 8% t) is used to reduce the fluctuation, so as to improve the outlet flow state of the near endwall.  The relative total pressure loss coefficient is defined as: where, in t p , is the average relative total pressure at the inlet of S2, t p is local relative total pressure, and out s p , is the average static pressure at the outlet of S2.
The fillet structure initially affects the development of HV at the leading edge of the blade. Therefore, Figure 11 shows the development charts of HV and PV at the leading edge and channel under the six chamfered structures. The HVPS and HVSS of the S2 are clearly visible, where they flow downstream and form a strong PV after the intersection of the suction surface. According to Figure  11b,c,e, the leading edge radius R = 25% t is conducive to the reduction of HVPS and PV. With the The relative total pressure loss coefficient is defined as: where, p t,in is the average relative total pressure at the inlet of S2, p t is local relative total pressure, and p s,out is the average static pressure at the outlet of S2. The fillet structure initially affects the development of HV at the leading edge of the blade. Therefore, Figure 11 shows the development charts of HV and PV at the leading edge and channel under the six chamfered structures. The HV PS and HV SS of the S2 are clearly visible, where they flow downstream and form a strong PV after the intersection of the suction surface. According to Figure 11b,c,e, the leading edge radius R = 25% t is conducive to the reduction of HV PS and PV. With the increase of R in the middle of the channel, the attachment position of HV SS on the suction surface moves up and its strength decreases. Therefore, a larger R is adopted in the middle of the channel, which is conducive to improving the local flow of the suction surface. In order to illustrate the influence of the fillet structure on shed vortices and passage vortices, Figure 12 shows the entropy-increase isoline at 110% axial chord length of S2. As can be seen from Figure 12a-f, the fillet mainly affects the flow loss at the outlet near the hub but has little impact on the other parts of the blade. Compared with the case without fillet in Figure 12a, the large radius fillet (R = 25% t) significantly increases the shed vortex (SV) loss at the outlet near the hub. The SV loss at the hub position at the outlet can be significantly reduced by using the appropriate small radius fillet (R = 8% t) at the blade trailing edge. Therefore, the appropriate small radius fillet at the outlet of the blade is helpful for improving the flow state. In order to quantitatively compare the loss changes at the outlet from the numerical perspective, Figure 13 shows the entropy-increase curves after the circumference average. While the fillet structure mainly affects the loss near the hub, it also affects the loss of SV and PV at the exit. As can be seen from the partial magnification, the loss of the SV and PV is increased by the enlargement of the fillet radius. At the same time, fillet structures also affect the radial position of the vortices. The loss near the shroud at the exit was reduced by about 8% by using large radius fillet (R = 25% t) at the trailing edge. However, it increased the loss near the hub by 9%. Figure 11. Horseshoe vortex imposed with relative total pressure loss coefficient.
In order to illustrate the influence of the fillet structure on shed vortices and passage vortices, Figure 12 shows the entropy-increase isoline at 110% axial chord length of S2. As can be seen from Figure 12a-f, the fillet mainly affects the flow loss at the outlet near the hub but has little impact on the other parts of the blade. Compared with the case without fillet in Figure 12a, the large radius fillet (R = 25% t) significantly increases the shed vortex (SV) loss at the outlet near the hub. The SV loss at the hub position at the outlet can be significantly reduced by using the appropriate small radius fillet (R = 8% t) at the blade trailing edge. Therefore, the appropriate small radius fillet at the outlet of the blade is helpful for improving the flow state. In order to quantitatively compare the loss changes at the outlet from the numerical perspective, Figure 13 shows the entropy-increase curves after the circumference average. While the fillet structure mainly affects the loss near the hub, it also affects the loss of SV and PV at the exit. As can be seen from the partial magnification, the loss of the SV and PV is increased by the enlargement of the fillet radius. At the same time, fillet structures also affect the radial position of the vortices. The loss near the shroud at the exit was reduced by about 8% by using large radius fillet (R = 25% t) at the trailing edge. However, it increased the loss near the hub by 9%.

Heat Transfer Analysis of S2 with Different Fillet Structures
The heat transfer of the wall surface can be expressed by the Nusselt number (Nu). Nu is the dimensionless heat transfer coefficient. This working with Salvadori Simone et al. [28] and Imran Qureshi et al. [18] research defined the same, is defined as: where, the parameter . q is the heat flux, C is the average chord length of blades, k T01 is the thermal conductivity of the air referring to the average total inlet temperature.
According to the flow performance analysis, as the shroud expansion angle of the S2 is larger, the flow separation caused by it is more serious. Therefore, Figure 14 shows the Nu distribution of the S2 shroud. The maximum thermal load area (a) of the shroud is located at the leading edge of the blade, which is caused by the saddle point of the airflow at the leading edge. In the rear part of the suction surface, due to the separation of airflow, there is a minimum thermal load area (b). The relatively high thermal load area is located at the rear part of the pressure side (c). In the trailing edge part, due to the influence of trailing edge shedding vortexes, there is a local high thermal load area (d). Combined with Figure 14a-f, it can be easily seen that the existence of the fillet structure can significantly eliminate the maximum thermal load area (a). Whether it increases the thermal load at the leading edge fillet will be discussed in the next section. Fillet structure has little effect on regions (b) and (c). It can be seen from Figure 14c,f that appropriate small radius fillet (R = 8% t) at the trailing edge can eliminate the local high thermal load area (d). However, if the fillet radius of the trailing edge is too large (R = 25% t), the heat transfer coefficient of the area (d) will be significantly increased. The heat transfer of the wall surface can be expressed by the Nusselt number (Nu). Nu is the dimensionless heat transfer coefficient. This working with Salvadori Simone et al. [28] and Imran Qureshi et al. [18] research defined the same, is defined as: where, the parameter is the heat flux, C is the average chord length of blades, kT01 is the thermal conductivity of the air referring to the average total inlet temperature. According to the flow performance analysis, as the shroud expansion angle of the S2 is larger, the flow separation caused by it is more serious. Therefore, Figure 14 shows the Nu distribution of the S2 shroud. The maximum thermal load area (a) of the shroud is located at the leading edge of the blade, which is caused by the saddle point of the airflow at the leading edge. In the rear part of the suction surface, due to the separation of airflow, there is a minimum thermal load area (b). The relatively high thermal load area is located at the rear part of the pressure side (c). In the trailing edge part, due to the influence of trailing edge shedding vortexes, there is a local high thermal load area (d). Combined with Figure 14a-f, it can be easily seen that the existence of the fillet structure can significantly eliminate the maximum thermal load area (a). Whether it increases the thermal load at the leading edge fillet will be discussed in the next section. Fillet structure has little effect on regions (b) and (c). It can be seen from Figure 14c,f that appropriate small radius fillet (R = 8% t) at the trailing edge can eliminate the local high thermal load area (d). However, if the fillet radius of the trailing edge is too large (R = 25% t), the heat transfer coefficient of the area (d) will be significantly increased. In order to further study the influence of fillet structure on the thermal load distribution on the blade surface, Figure 15 shows the Nu distribution of the S2 suction surface with different fillet structures. Since it has little influence on the thermal load on the pressure surface, the Nu distribution on the pressure surface is not given. It can be clearly seen from Figure 15a that the high thermal load area of the blade is located on the suction surface near the shroud, which is jointly caused by the In order to further study the influence of fillet structure on the thermal load distribution on the blade surface, Figure 15 shows the Nu distribution of the S2 suction surface with different fillet structures. Since it has little influence on the thermal load on the pressure surface, the Nu distribution on the pressure surface is not given. It can be clearly seen from Figure 15a that the high thermal load area of the blade is located on the suction surface near the shroud, which is jointly caused by the passage vortex and the upstream leakage vortex. As can be seen from Figure 15b,c,e, the high thermal load area (PV region) on the suction surface can be significantly weakened by the use of large radius fillet (R = 25% t) at the leading edge of the blade. And the SV separation position will be postponed correspondingly. However, the fillet with a small radius (R = 8% t) at the leading edge does not reduce the high thermal load area (PV region). The variation laws of the fillet along the chord length and the radius of the fillet at the trailing edge have little influence on the distribution of thermal load on the blade surface. passage vortex and the upstream leakage vortex. As can be seen from Figure 15b,c,e, the high thermal load area (PV region) on the suction surface can be significantly weakened by the use of large radius fillet (R = 25% t) at the leading edge of the blade. And the SV separation position will be postponed correspondingly. However, the fillet with a small radius (R = 8% t) at the leading edge does not reduce the high thermal load area (PV region). The variation laws of the fillet along the chord length and the radius of the fillet at the trailing edge have little influence on the distribution of thermal load on the blade surface. In order to more accurately show the influence of the fillet on the heat transfer coefficient distribution of the blade surface, Nu distribution of three blade sections (90% span, 50% span, and 10% span) is drawn in Figure 16. Through the analysis of Figure 16a-c, it can be concluded that fillet mainly affects Nu distribution of the suction surface near the shroud and hub of the blade, but has little influence on the middle part of the blade. The maximum thermal load on the blade surface is located at the leading edge of the S2, 2.5 times of the average heat load. As can be seen from Figure  16a, at the position of 90% span sections of the S2, the thermal load on the suction surface is significantly higher than that on the pressure surface, which is twice as high. The maximum thermal load can be reduced by about 28% by using the large radius fillet (R = 25% t) at the leading edge of the S2. In addition, a large radius fillet (R = 25% t) designed in the middle of the blade chord length can reduce the heat load on the suction surface. At the trailing edge of the blade, the fillet structure has no obvious influence on the heat load distribution. As can be seen from Figure 16c, fillet structure has little influence on the maximum heat load near the hub region but has a great influence on its In order to more accurately show the influence of the fillet on the heat transfer coefficient distribution of the blade surface, Nu distribution of three blade sections (90% span, 50% span, and 10% span) is drawn in Figure 16. Through the analysis of Figure 16a-c, it can be concluded that fillet mainly affects Nu distribution of the suction surface near the shroud and hub of the blade, but has little influence on the middle part of the blade. The maximum thermal load on the blade surface is located at the leading edge of the S2, 2.5 times of the average heat load. As can be seen from Figure 16a, at the position of 90% span sections of the S2, the thermal load on the suction surface is significantly higher than that on the pressure surface, which is twice as high. The maximum thermal load can be reduced by about 28% by using the large radius fillet (R = 25% t) at the leading edge of the S2. In addition, a large radius fillet (R = 25% t) designed in the middle of the blade chord length can reduce the heat load on the suction surface. At the trailing edge of the blade, the fillet structure has no obvious influence on the heat load distribution. As can be seen from Figure 16c, fillet structure has little influence on the maximum heat load near the hub region but has a great influence on its fluctuation on the suction surface. Similarly, large radius fillet (R = 25% t) in the middle of blade chord length is conducive to reducing the thermal load near the hub.   Table 4 shows the influence of fillet structure on the overall aerodynamic performance parameters and thermal load parameters of the S2 blade. For the loss coefficient, Cp, no matter what form of fillet structure is adopted, it will be conducive to the improvement of the overall flow state of the S2 with a large meridional expansion endwall. In the case of R3, it can be seen that the leading edge is filleted with an appropriate large radius (R = 25% t) and the trailing edge is filleted with an appropriate small radius (R = 8% t), so as to minimize the loss coefficient of the S2 and reduce 0.8% for the case without fillet. By comparing cases R2 and R3, it can be concluded that a small-radius fillet at trailing edge is conducive to improving the flow state. Correspondingly, by comparing cases R2 and R4, it can be concluded that a large-radius fillet at the leading edge is conducive to improving the flow state. The total thermal load can be reflected by the area-averaged heat flux. It can be seen that the thermal load of shroud and blade decreases with the fillet structure. Area-averaged shroud heat flux is reduced by 3.1% by using the fillet distribution of R3. Use case R2's fillet radius distribution, reducing the area-averaged blade heat flux by 2.7%. Therefore, it can be concluded that the use of the large leading edge radius fillet and the small trailing edge radius fillet is conducive to the improvement of the flow state and the reduction of the overall thermal load of the large  Table 4 shows the influence of fillet structure on the overall aerodynamic performance parameters and thermal load parameters of the S2 blade. For the loss coefficient, C p , no matter what form of fillet structure is adopted, it will be conducive to the improvement of the overall flow state of the S2 with a large meridional expansion endwall. In the case of R3, it can be seen that the leading edge is filleted with an appropriate large radius (R = 25% t) and the trailing edge is filleted with an appropriate small radius (R = 8% t), so as to minimize the loss coefficient of the S2 and reduce 0.8% for the case without fillet. By comparing cases R2 and R3, it can be concluded that a small-radius fillet at trailing edge is conducive to improving the flow state. Correspondingly, by comparing cases R2 and R4, it can be concluded that a large-radius fillet at the leading edge is conducive to improving the flow state. The total thermal load can be reflected by the area-averaged heat flux. It can be seen that the thermal load of shroud and blade decreases with the fillet structure. Area-averaged shroud heat flux is reduced by 3.1% by using the fillet distribution of R3. Use case R2's fillet radius distribution, reducing the area-averaged blade heat flux by 2.7%. Therefore, it can be concluded that the use of the large leading edge radius fillet and the small trailing edge radius fillet is conducive to the improvement of the flow state and the reduction of the overall thermal load of the large meridional expansion turbine. Nomenclature Romans and Greeks Nu = Nusselt number p = pressure, kPa q = heat flux, kW/m 2 T = temperature, K T aw = adiabatic wall temperature, K T g = gas temperature, K T w = isothermal wall temperature, K u = velocity, m/s T 01 = inlet temperature, K Ω = rotation tensor S = strain tensor C = vane midspan true chord k = thermal conductivity p t,out = average relative total pressure at the outlet, kPa p t,in = average relative total pressure at the inlet, kPa p t = local relative total pressure, kPa p s,out = average static pressure at the outlet, kPa C ps = static pressure coefficient C p = total pressure loss coefficient R = fillet radius Yp = relative total pressure loss coefficient Subscripts, Superscripts, Abbreviations R1

Comparison of Aerodynamic and Heat Transfer Performance of S2
= first rotor S1 = first stator S2 = second stator Exp = experimental PS, SS = pressure side, suction side CFD = Computational Fluid Dynamics HTC = Heat Transfer Coefficient SST = shear stress transport TLV = tip-leakage vortex SV = shed vortex RANS = Reynolds-Averaged Navier-Stokes LE = leading edge TE = trailing edge