Numerical analysis and design optimization on full coverage film-cooling for turbine guided vane

ABSTRACT Based on numerical simulations, the heat transfer and flow field of a turbine vane are analyzed and the film cooling is improved. The optimization objective is increasing the overall cooling effectiveness with cascade pressure loss factors staying almost unchanged. Thus, cylindrical film holes were replaced by laidback holes and V-crater holes. To analyze the effect of structural adjustment on the vane, pressure distributions, mass flow distributions, and heat transfer coefficients were investigated for internal and external cooling systems. To explain the advantages of shaped holes over cylindrical holes, the flow mechanisms, film superposition, discharge coefficients, blow ratios, and film cooling effectiveness were compared. Meanwhile, the influence of mass flow ratios and mainstream Reynolds numbers was analyzed. After optimization, at the design condition, the overall cooling effectiveness increased by 4.19% on the pressure surface and 1.78% on the suction surface. However, the cascade pressure loss factor increased by 0.26% only.


Introduction
With the development of the aviation industry, better turbine engine performance is required. To enhance the engine power ratio and cycle thermal efficiency, turbine inlet temperature has kept increasing for decades. Nevertheless, high-temperature gas can bring about thermal stress concentration and thermal ablation, resulting in a short life span, especially for turbine guided vanes (Xie et al., 2006). To keep vane temperature within the allowable limit, appropriate external film cooling and internal convective cooling systems are required (Nowak & Wróblewski, 2011). Detailedly, the internal convective heat transfer is enhanced by installing jet impingement, rib turbulators, dimples, serpentine passages, etc. The film cooling is achieved by low-temperature secondary flow ejected from film holes (Han & Rallabandi, 2010;Bunker, 2009a), which decreases the heat transfer between the hot mainstream and blade wall downstream.
Film cooling is affected by aerodynamic parameters and geometric parameters of film holes on a large scale. Regarding geometrical shape, cylindrical film holes have been utilized for a long time. Nevertheless, in higher blow ratios, film cooling effectiveness becomes lower as lift-off coolant film is generated (Bunker, 2009b). Thus, various geometric shapes of film holes are implemented mainly by adopting expansion angles CONTACT Huiren Zhu zhuhr@nwpu.edu.cn and laidback angles. The typically shaped holes are conical holes, fan-shaped holes, laidback holes, V-crater holes, and converging slot-holes (Saumweber & Schulz, 2008;Yang et al., 2015;Kalghatgi & Acharya, 2015;Saumweber & Schulz, 2012). Compared with cylindrical holes, the coolant jet of shaped holes owns lower exit momentum, weaker kidney-shaped vortexes, and higher lateral coverage, leading to higher film cooling effectiveness even under high blow ratios. Except for geometric shapes, appropriate hole arrangement on blades also enhances cooling performance (Kim et al., 2018). In addition, aerodynamic parameters such as blow ratios, turbulence intensity, and density ratio also play an important role in film cooling performance (Wang, Li, Li, Ren, & Jiang, 2019;Fu et al., 2019). Besides, the discharge coefficient is usually adopted as an evaluation index to quantify the through-flow losses of film holes. Compared with cylindrical holes, shaped holes have higher discharge coefficients (Deckker & Chang, 1965;Lichtarowicz et al., 1965;Hu Ning , 2008).
In the case of continuous film holes, film superposition can be observed downstream, which helps improve cooling performance. Sellers put forward a computational method to calculate the film cooling effectiveness of multi-row holes from that of the single hole (Sellers, 1963). The method is verified subsequently (Wang, Zhang, Shiau, & Han, 2019). In turbine vanes, full-coverage cooling effectiveness is strongly influenced by wall curvature and passage vortices. Concretely, film converges on the suction surface but diverges on the pressure surface due to passage vortices (Han et al., 2010;Mhetras et al., 2007). Different wall curvature causes different pressure gradients, leading to longer coolant streaks on the suction surface but shorter coolant streaks on the pressure surface.
Owing to a strong interaction between the internal cooling and film cooling, even a small structural adjustment can have an obvious effect on the fluid mechanism and heat transfer performance of the whole system (Xie et al., 2020;Zhou et al., 2019;Hong et al., 2009;Bunker, 2013). Thus, the overall cooling effectiveness is implemented to evaluate the overall cooling performance of the vane. To save cost, conjugate heat transfer and genetic algorithms have been widely used to optimize the cooling structure. After optimization, the turbine vane is expected to own lower wall temperature, lower coolant consumption, weaker thermal stress configuration, and longer service life (Qu et al., 2012;Nowak & Wróblewski, 2011;Verstraete et al., 2008;Girardeau et al., 2013).
In most previous studies, cooling structures are simplified and investigated in a laboratory. However, when applied in a real blade on design condition, they may have different cooling performances. Besides, affected by structure adjustment, a complicated chain reaction on the flow field may be generated and it needs deeper investigations (Bunker, 2009a). However, in a real turbine guided vane, owing to the complicated flow mechanism, it is difficult to investigate the influence of structural modification at length. Accordingly, few studies analyze the influence of film cooling adjustment on mass flow distributions.
Thus, in this study, with the help of numerical calculation, the full coverage film-cooling was analyzed and optimized on real working conditions and in real turbine vanes. The aim was to enhance the overall cooling effectiveness by replacing cylindrical holes with shaped holes. Whereas, cascade pressure loss factors were supposed to remain unchanged. The biggest feature or difficulty of this study was analyzing the influence of hole shape modification on the flow field for both film cooling and internal cooling. Therefore, changes in coolant mass flow distributions, pressure distributions, heat transfer performance, and discharge coefficients were analyzed. Meanwhile, differences in flow mechanism and film superposition between cylindrical holes and shaped holes were investigated at length. Besides, counter-rotating vortexes above the wall induced by counter-inclined showerhead holes were analyzed. To some extent, the study results and optimization process may have referential value for engineering applications and other researchers.

Physical model and variable settings
A guided vane from an aero-engine turbine was selected and optimized in this paper. A basic guided vane (BAS) consisted of external film cooling and internal cooling. Figure 1 represents the basic vane internal cooling system and the flow direction of hot mainstream. Meanwhile, the external cooling and internal cooling systems are shown in Figure 2. Different from most vanes investigated in the previous study, a double-inlet cooling system was utilized in this vane.
22 columns of cylindrical film holes and trailing edge (TR) ejection comprise the film cooling system. Particularly, in the leading edge (LE), counter-inclined showered holes are utilized. The blade wall is divided into the suction surface (SS) and pressure surface (PS). For the internal cooling, the vane consists of four cooling passages named Passage 1-4. Passage 1-2 compose the front-cavity while Passage 3-4 comprise the back-cavity. Impingement cooling is installed in both front-cavity and back-cavity.
To enhance heat transfer further, ribs and dimples were installed in Target Surface 1 while ribs and protrusions were introduced in Target Surface 2. The projecting location of ribs, dimples, protrusions, film holes, and jet nozzles is shown in Figure 2(b). The diameter of jet nozzles D j is 0.9 mm. The heights/depths of ribs, dimples, and protrusions are 0.90, 0.84, and 0.50 mm respectively. The pitch ratios of ribs are 8.4 and 4.4 separately for Target Surface 1 and Target Surface 2. The jet-to-plate spacing L is 1.33 mm for both two target surfaces.
Besides, three pin-fins are implemented in the TR. The blade wall thickness and blade height are 0.85-1.6 and 76 mm respectively. The blade chord length is 54 mm while the diameter of jet nozzles is 0.9 mm.
For the convenience of description, taking the optimized blade (OPT) as an example, 22 columns of film holes and 11 columns of jet nozzles are named in Figure 3 while the chordwise direction is defined. On the SS, four columns of holes are named S1-4 along the chordwise direction while the other five columns of holes near the TR are defined S0. On the LE, 6 columns of counter-inclined showered holes are named L1-6. Besides, on the PS, 7 columns of film holes are defined P1-7. Eleven columns of jet nozzles are defined IMP 1-11 along the chordwise direction.
During the optimization procedure, conjugate heat transfer (CHT) is usually utilized for a gas turbine (Ghalandari et al., 2019). By utilizing the overall cooling effectiveness η, the overall performance of both internal and film cooling can be assessed. The main goal of the optimization was to achieve higher η. Thus, the ideal optimal structure should meet three requirements: • stronger heat transfer in internal cooling structures • no sharp increase in heat transfer on the external wall • better film cooling performance on the external wall When the hole shape gets changed, the flow condition of the external coolant film changes, leading to different pressure distributions in the cascade. Compared with cylindrical holes, shaped holes may increase pressure loss factors ψ in the blade lattice, which has an indirect negative influence on the turbine performance. Thus, in this study, changes in cascade pressure loss factors were required to be controlled within 2%.
In general, these two indexes (η and ψ) are adopted to evaluate the overall performance of the vane and determine whether the optimization is successful (Andrews et al., 1986): • higher overall cooling effectiveness on the external surface • a limited effect on cascade pressure loss factors Thus, in this research, the optimization objective function is as follows: (1) The optimization thoughts and analysis ideas are shown in Figure 4. According to the subsequent simulated results (shown in Figure 34 and Figure 26) of the BAS, for cylindrical holes (S1-4 and P1-7) on the PS and SS, jet lift-off and poor coolant lateral coverage can be observed for film cooling. The V-crater hole, a kind of outlet expansion holes, is first put forward by Bunker (Bunker, 2009b). Subsequently, V-crater holes are applied widely to the turbine vane with corresponding numerical simulations and experiments conducted (Fu et al., 2019;Wei., 2019). The research results show that with the help of the expansion angle and laidback angle, V-crater holes provide better film cooling performance than cylindrical holes. Thus, based on the previous study, the cylindrical holes (S1-4 and P1-7) are replaced by V-crater holes during the optimization. For the convenience of analysis, except for the geometric shape, the geometric parameters of the hole remain unchanged.
For counter-inclined cylindrical holes (L1-6) on the leading edge, according to the subsequent simulated results (showed as Figure 22, Figure 32, and Figure 40), jet lift-off, as well as coolant concentration appear, which deteriorate film cooling. The laid-back holes, investigated on the leading edge by Liu (Liu et al., 2012), are proved to own better wall-attached film than cylindrical holes. Thus, in this study, the cylindrical holes (L1-6) were replaced by laid-back holes.
During the optimization, the internal cooling and film hole S0 are kept unchanged. To explain the effect of hole shape modification on internal and external cooling, changes in coolant mass flow distributions, pressure distributions, heat transfer performance, and film cooling effectiveness were analyzed.   In this study, the mass flow consumption and cascade pressure loss factors are not constant, similar to the previous study (Verstraete et al., 2008) which try to get higher service life with lower coolant consumption. However, some studies keep the coolant consumption unchanged and aim at decreasing thermal stress (Qu et al., 2012), decreasing wall temperature (Nowak & Wróblewski, 2011), or decreasing pressure loss (Namgoong et al., 2008). However, the ultimate goal of all these various optimization paths is to get higher turbine service time and higher turbine efficiency (Girardeau et al., 2013).
On the LE, the schematic of cylindrical showered holes of the BAS and laidback showered holes of the OPT is illustrated in Figure 5. Except for the addition of laidback angle β, laidback holes own the same parameters as cylindrical holes.
On the pressure and suction surfaces, the schematic of cylindrical holes of the BAS and V-crater holes of the OPT is shown in Figure 6. Compared with cylindrical holes, V-crater holes own additional laidback angle β, spine-laidback angle ϕ, and laterally expansion angle ω. The hole length L 1 of V-cater holes is divided into L 2 (cylindric part) and L 3 (laidback part). Except for the aforementioned parameters, both two kinds of holes own the same geometric parameters.
During the optimization, the number of holes, location on the blade S/Ca, diameter for cylindrical part D f , hole pitch ratio J/D f , axially inclined angle α 1 , length to diameter ratio L 1 /D f , and radially inclined angle α 2 remain unchanged and are shown in Table 1.
Compared with the basic blade, the optimized blade owns different holes shape, additional laidback angle β, expansion angle ω, spine-laidback angle ϕ, length to diameter ratio for the cylindrical part L 2 /D, and length to diameter ratio for the laidback/expansion part L 3 /D. These parameters are shown in Table 2.

Numerical method
With the help of the CFD method, the complex flow and heat transfer could be analyzed (Ez Abadi et al., 2020). In this paper, to assess the cooling performance of each structure, the CHT process and AD (adiabatic) process were executed for each case to obtain enough data. As illustrated in Figure 7, for the CHT procedure, fluid grids and solid grids comprise the computation domain. In addition, in the CFX-Pre, 'conservative heat transfer flux' is set for the interface between solid and fluid domains. Besides, during the simulation process, heat radiation is not considered. For the AD case, the calculation domain is only made up of fluid meshes while 'adiabatic' is set for the domain walls. In the AD process, film cooling effectiveness was calculated by wall adiabatic temperature. During the CHT process, overall cooling effectiveness, pressure loss factors, and discharge coefficients are directly worked out. Meanwhile, to calculate the Nusselt number, the CHT process is utilized to work out heat flux, wall adiabatic temperature, and wall temperature. In the upcoming sections, these parameters can be utilized to investigate the heat transfer, cooling performance, and flow field.
Based on the finite volume method software ANSYS CFX, simulation. Taking the CHT process as an example, boundary conditions are shown in Figure 8(a). At the design condition, non-uniform total temperature (Figure 8(b)) and mass flow rate (M = 1.85 kg/s) comprise the inlet mainstream condition while the outlet mainstream condition is the non-uniform static pressure (Figure 8(c)). In detail, Table 3 gives corresponding data about the inlet and outlet mainstream flow condition. As a double-inlet cooling structure is utilized in this vane, the coolant is comprised of two parts which are defined as Coolant 1 and Coolant 2, corresponding to front-cavity and back-cavity separately. The total temperature (T t = 818.7K) and total pressure make up the inlet coolant condition. Besides, for the fluid domain, the symmetry planes are set as transitional periodicity. Meanwhile, for fluid and solid domains, heat transfer models are set as 'total energy' and 'thermal energy' separately. The mainstream turbulence intensity is 10%. Regarding the solver control, the residual target is 10 −9 .
In the solid and fluid domain, the material is set as Nickel-base alloy and air ideal gas respectively. The temperature affects the thermal conductivity (Liu & Xue, 2014) while the correlation is illustrated in Eq.11-12.
The mass flow ratio is defined as follows. M c and M g represent the mass flow rate of coolant and mainstream separately.
To research the influence of mass flow rate and mainstream Reynolds number (based on blade chord width and mainstream inlet condition), in this study, by adjusting the inlet mainstream mass flow rate and inlet coolant pressure, four working conditions were chosen. The working conditions and corresponding parameters are shown in Table 4. Besides, the design condition is at   To illustrate the Mach number distributions, take the BAS as an example, calculation results at the midspanwise blade height are shown in Figure 9. The mainstream Mach number is in the range of 0.13 ∼ 0.91. For the vane configuration, the highest Mach number locates near the trailing edge (Ma = 0.62). For film holes, the Mach number of the inner flow ranges from 0.18 ∼ 0.36. Based on the jet nozzle diameter (0.9 mm), the Mach number, as well as the average jet flow Reynolds number, are illustrated in Table 5.
The heat transfer coefficient is calculated by T aw , Q, and T w , which represent adiabatic wall temperature, heat flux, and wall temperature respectively.
Local Nusselt number on the internal wall is defined as follows. l is the characteristic length that represents the   diameter of the impingement hole and is taken as 0.9 mm.
The discharge coefficient is defined as follows. P s and P t represent the hole outlet static pressure and inlet total pressure respectively. Besides, ρ and A are the fluid density inside orifices and the total flow area of orifices separately. For shaped holes, the discharge coefficient is based on the cylindrical cross-section at the hole inlet.
The cooling performance of a whole system is evaluated by Overall cooling effectiveness. Here, T c and T g are separately set as 818.7 K and 1840K, which represent the static temperature of coolant and mainstream.
To assess the wall-attached condition for coolant film, adiabatic cooling effectiveness is utilized and defined as follows.
The blow ratio of film holes is defined as follows. ρ in and ρ g represent the density of the inlet of hole and mainstream respectively while v in and v g represent the velocity of the inlet of coolant and mainstream separately.
The density ratio DR, defined as follows, is the ratio of the coolant density ρ c to mainstream density ρ g .
When the pressure loss factor f is based on the inlet and outlet mainstream, it is named the cascade pressure loss factor ψ. u out , P in , and P out separately represent the outlet velocity, inlet pressure, and outlet pressure.

Mesh independence validation
Mesh generating software ANSYS ICEM was adopted to generate grids for calculation domains. To conduct the independence validation, taking the case of the whole domain of the basic blade (BAS), three different numbers of grids (155, 130, 92 million) were chosen on the working condition (MFR = 11.59% ∼ 18.34%, Re = 556337). It is obvious that on the external wall, the pressure values are similar for different numbers of grides. However, regarding the heat flux and overall cooling effectiveness distributions, a huge difference emerges for coarse meshes. Meanwhile, for fine grids and moderate grids, the data curves nearly overlap in all figures.
In conclusion, to use computing resources wisely, the moderate mesh was selected to make calculations, which is constituted by the fluid domain (98 million) and solid   domain (32 million). The fluid grids are shown in Figure  11(a). Specifically, tetrahedral meshes are utilized in the far region while prismatic meshes are utilized near the wall. Figure 11(b) shows the dimensionless wall distance y+ distribution on the external surface. The area average value is 1.05, which meets the requirements of the turbulence model.

Turbulence model
To ensure the calculation accuracy, experiments were conducted based on geometric parameters of the blade profile. Meanwhile, the realizable k − ε model, SST k − ω model, and SST k − ω with transition γ − θ model were chosen to perform the validation. The laterally average Nusselt number distribution and experiment facility are illustrated in Figure 12.
Specifically, Figure 12(a) illustrates a steady-state experiment apparatus. Polyether-ether-ketone was used to manufacture the test piece (blade BAS). To reduce cost, the internal cooling system was omitted while the blade was magnified by 1.6 times, which was convenient to test. Accordingly, to keep the same as the test piece, the computation model was also simplified. By installing thermocouples in the blade external wall, the wall temperature was measured. The operational use time was more than five seconds while the air source volume was 140m 3 . The specific heat capacity, thermal conductivity, and density were separately C =1157.61 J/(kg · K), λ =0.29 W/(m · K), and ρ =1310.5 kg/m 3 . Regarding the mainstream, the gas thermal conductivity, Reynolds number, density were separately 0.0478 W/(m · K), 556337 W/(m · K), and 1.165 kg/m 3 . Besides, the inlet total temperature, mass flow rate, turbulence intensity were 300 K, 0.9124 kg/s, and 9.8% separately.
At the first stage, the blade was heated to a specific temperature. Subsequently, the blade was cooled when the mainstream was ejected through the cascade. When the penetration depth was smaller than the half wall thickness, the test piece was treated as a semi-infinity onedimensional model. Table 6 illustrates the measurement uncertainty for the mass flow, inlet Reynolds number, cascade pressure ratio, inlet Mach number, and the outlet isentropic Mach number.
Regarding the SST transition model, on the suction surface, the data from calculations are lower than the experiment results. Owing to a complex aerodynamic parameter variation and a thermodynamic property variation, it is quite difficult to obtain accurate computation results, especially when it comes to flow transition. Thus, at S/Ca =−0.7 ∼ −0.4, a huge deviation (approximately 25%) can be observed. Subsequently, after going through flow transition (S/Ca =−1.1 ∼ −1.0), the data of both calculations and experiments decrease. Besides, at S/Ca =0.6 ∼ 1.0 (trailing edge) and S/Ca =−0.2 ∼ 0.1 (leading edge), the Nusselt number is overpredicted by 15% by the simulation. As for the realizable k − ε and SST k − ω models, a huge deviation appears.
Obviously, by utilizing the Gamma Theta transition model in the SST k − ω model, the calculation accuracy is increased significantly. By combining two transport equations with the turbulence model (Langtry et al., 1987), the Gamma Theta transition model is widely used and proved reliable in the previous research (Ke & Wang, 2016;Mangani et al., 2010). Specifically, one equation is adopted to assess the intermittency coefficient distribution. The other one is derived for a special parameter, which represents a critical Reynolds number based on boundary-layer momentum thickness. To sum up, the SST k − ω with transition γ − θ model is the most accurate and is adopted in the simulation.

Pressure distribution
As the mass flow distributions for cooling structures depend on pressure distributions of mainstream and internal cooling passages on a large scale, the influence of structural modification on pressure distribution was investigated in the chordwise direction (located at twothird of the vane height) and height direction.
Firstly, taking the BAS for example, the influence of mainstream Re numbers on pressure distributions is shown in Figure 13. Similar regularity of mainstream pressure distributions at different Re can be observed, although the pressure descends obviously near the PS and LE at lower mainstream Re. Furthermore, the spanwise average pressure near the external wall is dotted in Figure 14. Combined with Figure 13, along the chordwise direction, the mainstream pressure decreases firstly and then fluctuates at S/Ca = −1.2 ∼ −0.65. Subsequently, the pressure keeps continuously increasing and peaks at the maximum value at S/Ca = 0.06. Afterward, the pressure keeps decreasing on the PS. Affected by the impingement effect of the mainstream, the pressure on the leading edge and nearby pressure surface is influenced by mainstream Re strongly. Thus, the pressure is higher at a higher Reynolds number, especially in the range of S/Ca = −0.4 ∼ 0.8. In other intervals, however, the pressure difference between the two Re numbers is negligible.
For the sake of observation on internal cooling passages, the scale range in Figure 13 is changed as depicted in Figure 15.
Taking the BAS as an example, the influence of MFR is illustrated in Figure 15(a-b). At MFR = 11.59%, the pressure difference between the outlet and inlet is small for the showered hole. It can be deduced that the effusion condition of L1-6 is poor. With the MFR increasing, the pressure difference between the outlet and inlet for all holes increases, leading to better effusion condition of the coolant. Figure 15(b), 15(c) illustrate the influence of mainstream Re. With Re declining, a pronounced pressure drop can be observed on the mainstream, especially near the LE and PS. Meanwhile, to remain the MFR unchanged, lower coolant pressure is required at lower Re.   Figure 15(b), 15(d) reveal the influence of structural modification at MFR = 18.34%, Re = 556337. Both two blades own nearly the same pressure distributions for mainstream (serves as film hole outlets) and Passage 2-3 (serves as inlets of jet nozzles, L-6, and P1-6). Compared with the BAS, however, the OPT possesses lower pressure in Passage 1 and Passage 4 (serves as outlets of jet nozzles and inlets of S0-3, TR, and P7). A higher pressure difference between inlets and outlets of holes means a higher mass flow rate or better effusion condition. Thus, in the front-cavity, compared with the BAS, the OPT allocates more mass flow to the IMP7-11 and S1-3, leading to less coolant flowing through S4, L1-6, and P1-4. In the back cavity of the OPT, the TR, S0, and P7 are allocated with less coolant, resulting in a higher mass flow rate for P5-6.
To better explain the influence of the MFR and Re on the internal cooling passage, taking Passage 2 of the BAS as an example, the area average pressure loss factor distributions along the height direction is dotted in Figure  16. As a double-inlet cooling system is utilized, on all working conditions, the lower and upper coolant confluence around the center vane height, leading to high factors near the end wall region and low values at around h/H = 0.5. With the MFR increasing or Re increasing, the location of the lowest point of pressure loss factor approaches the upper side, resulting in higher factors near the upper side and lower factors near the lower side.

Discharge coefficient
The discharge coefficient C d , influenced by hole shapes, mass flow distributions, and pressure distributions, was used to quantify the through-flow losses or effusion condition of film holes. The ratios of discharge coefficients of the OPT to those of the BAS (Cd OPT /Cd BAS ) are shown in Figure 17. According to the analysis above ( Figure  15(b, c)), compared with the BAS, the OPT owns lower coolant pressure around the leading edge. This phenomenon is obvious at MFR = 11.59%, meaning that the pressure difference between the mainstream and coolant is very low, especially for L1-6 of the OPT. Accordingly, the coolant flow is blocked at the hole inlet, leading to uneven coolant distributions in the front-cavity. Thus, compared with the BAS, the discharge coefficient in OPT is much lower for L3-6 but higher for P1-2. With the MFR increasing, the poor effusion of L1-6 of the OPT gets narrow. Thus, the gap between the two vanes narrows. At MFR = 15.39%, except for the L3 and P3, discharge coefficients are higher in shaped holes and lower in cylindrical holes. With the MFR increasing, the influence of MFR and mainstream Re on the ratio Cd OPT /Cd BAS becomes smaller.

Mass flow distribution
Changes in pressure distribution between hole inlets and outlets influence the effusion condition for jet nozzles, film holes, and TR, leading to different mass flow allocations for these cooling structures. Due to the weak correlation of the fluid between the front-cavity and back-cavity, it is appropriate to analyze mass flow distributions separately. Owing to similar regularities for the BAS and OPT, firstly, taking the BAS as an example, the influence of the MFR and mainstream Re is illustrated in Figure 18.
In the front-cavity, all coolant enters mainstream after flowing through film holes. The coolant can be divided into two branches: one branch flows through   S1-3 ultimately after discharging through jet nozzles IMP 7-11 while one branch directly ejects through the other film holes. At MFR = 11.59%, on account of poor effusion condition for L1-6, more than half of the coolant ejects from S1-3. Therefore, the flow distribution is very uneven at MFR = 11.59%. With the MFR increasing or mainstream Re decreasing, the effusion condition of L1-6 gets improved, resulting in a lower percentage for S1-3 and IMP7-11. Thus, the proportion of coolant ejecting from the S4, L1-6, and P1-4 keeps increasing. Besides, at high MFR, the influence of MFR and mainstream Re becomes weaker.
For the back-cavity, as all coolant flows out of TR, S0, and P5-7, the percentage of these structures adds up to 100%. The coolant discharging from the IMP 1-6 flows ultimately through P7, TR, and S0. As much as 68.5 ∼ 74.3% of the coolant flows out of the TR while only 8.3 ∼ 8.8% of coolant ejects through the S0. With the MFR increasing or Re decreasing, less coolant flows through the TR and IMP1-6 while the proportion of S0 remains basically unchanged, leading to more coolant allocated to P5-7.
To investigate the effect of structural modification on mass flow allocation, based on the BAS, changes on mass flow distribution after optimization are shown in Figure  19.
For the front-cavity in the OPT at MFR = 11.59%, compared with the BAS, more mass flow is allocated to IMP7-11 and S1-3, resulting in a lower percentage of mass flow for the S4, L1-6, and P1-4. On account of poor effusion condition for the L1-6, the cooling performance of the LE exacerbates further. When the MFR increases to 15.39%, the effusion condition of film holes gets improved, especially for the L1-6. Accordingly, the mass flow distribution difference between the OPT and BAS narrows. With the MFR increasing to 18.34% or Re decreasing, the influence on the difference between BAS and OPT is limited.
For the back-cavity in the OPT, compared with BAS, less coolant is allocated to S0, TR, and P7, resulting in a larger percentage of mass flow for P5-6. The result of the overall effect is more mass flow being allocated to P5-7 in the OPT. However, the influence on the mass flow distribution of IMP1-6 is negligible. With MFR increasing or Re decreasing, the increasing extent of P5-7 decreases while the influence on S0, TR, and  IMP1-6 keeps basically unchanged. Thus, the mass flow allocation difference between the OPT and BAS becomes narrows.
According to the analysis above, it can be concluded that the influence of decreasing mainstream Re is similar to the effect of increasing the MFR, which both can narrow the mass flow distribution difference between the BAS and OPT.

Blow ratio
Jet lift-off usually occurs at high blow ratios (or high mass flow ratios), which is detrimental to film cooling effectiveness, especially for cylindrical holes. However, influenced by film superposition, the full-coverage film cooling effectiveness still increases with the MFR increasing even if jet lift-off has occurred on several film holes . In the turbine vane, the flow condition of each column of holes is different. Thus, to investigate the flow condition in-depth, the blow ratio of every single column of holes is discussed in this section.
Firstly, owing to a similar law, taking the BAS as an example, the influence of MFR and Re on the blow ratio of every single column of film holes is presented in Figure  20. S1-3 and P7 possess the highest blow ratio in all working conditions. At MFR = 11.59%, just like the explanation about Figure 17, as most of the coolant is allocated to S1-3, the blow ratio of S4, L1-6, and P1-4 is very low. Compared with P5-6, P7 owns a lower pressure inlet but also a much lower pressure outlet, leading to higher blow ratios. With the MFR increasing, blow ratios of all holes increase, especially for S4, L1-6, and P1-4. However, the increasing extent for P5-7 is limited, which means the MFR has a stronger influence on holes of the frontcavity than that of the back-cavity. At MFR = 18.34%, with mainstream Re decreasing, the blow ratios remain basically unchanged for most holes.
To analyze the influence of structural modification on blow ratios, the ratios of blow ratios of the OPT to that of the BAS (BR OPT /BR BAS ) are shown in Figure 21. The regularity is similar to Cd OPT /Cd BAS distributions.  Exorbitant blow ratios lead to lift-off of coolant while excessively low blow ratios mean insufficient coolant distributions. Compared with the BAS, the shaped hole in the OPT owns a larger exit area, leading to lower blow ratios for most film holes. However, on account of different mass flow distributions and pressure distributions as discussed above, OPT owns higher blow ratios for S1, S4, L1, and P5-6 at different working conditions. Notably, at MFR = 11.59%, blow ratios for L3-6 are much lower for the OPT, which means the OPT owns worse cooling performance on the LE. Owing to uneven mass flow distribution for the OPT at MFR = 11.59, % S1-3 and P2-4 own higher blow ratios than that at higher MFR. With the MFR increasing, the blow ratio difference between the OPT and BAS narrows, especially for L1-6. With the MFR increasing, the influence of MFR and mainstream Re on the ratio BR OPT /BR BAS becomes smaller.

Flow mechanism of shaped holes
For S1-4 and P1-7, the OPT adopts V-crater holes while the BAS adopts cylindrical holes. For L1-6, the OPT adopts laid-back holes while the BAS adopts cylindrical holes. The flow characteristic of film holes is not only influenced by hole shapes but also affected by mass flow allocation and pressure distributions. The pressure gradient on the SS and PS is different. Thus, the shaped holes will be classified into three categories: L1-6 (laidback holes), S1-4 (V-crater holes on the SS), and P1-7 (V-crater holes on the PS). Each category of film holes owns different flow mechanisms and will be discussed in turn.

Laid-back hole on leading edge
In the first place, taking the L3 as an example, the influence of the MFR, mainstream Re, and hole shape on the flow mechanism is analyzed. Avoiding repeated explanations, three typical working conditions are selected. Firstly, in the spanwise section, the temperature distributions are depicted in Figure 22. For the convenience of explanation, the near-wall region is divided into two kinds of areas: BET (the area located between holes) and ABV (the area located above holes).
At MFR = 11.59%, owing to poor effusion condition, the coolant coverage for both two kinds of holes is poor, especially in the BET. Compared with the BAS, as the mass flow distribution is more inadequate for the OPT, the coolant temperature inside holes is higher owing to strong mixing with hot mainstream. Thus, at MFR = 11.59%, cooling performance is worse for the OPT. With the MFR increasing, mass flow allocation is sufficient enough, leading to better wall-attached coolant film. Besides, a local high-temperature region appears in the BET upstream of holes. With Re decreasing, the area of the local high-temperature region declines. At high MFR, the OPT owns a smaller local high-temperature region than the BAS.
For Figure 23, at MFR = 11.59%, hot mainstream directly impinges on the BET. Impeded by the mainstream, a low-velocity region is generated inside laidback holes. At MFR = 18.34%, the local high-tempera ture region in the BET is induced by the lift-off of coolant upstream of holes. Compared with cylindrical holes, laidback holes offer better wall-attached jet flow, leading to better cooling performance. Similar conclusions can also be confirmed by the experiment (Liu et al., 2012).

V-crater Hole on pressure surface
As the front-cavity and back-cavity own similar flow structures for shaped holes, taking P3-4 as an example, pressure distributions on the chordwise section are illustrated in Figure 24. Near the concave pressure surface, the pressure gradient owns a component towards the mainstream, which results in detached jet flow. From streamlines distributions in Figure 25, for the BAS, the jet flow detaches from the wall downstream of P3 even at low MFR. With MFR increasing, worse wall-detached coolant film can be observed. However, compared with cylindrical holes, V-crater holes own lower blow ratios because of a larger exit area. Besides, owing to the laidback angle and laterally expanded angle, the jet flow owns a lower normal velocity component and a higher spanwise velocity component. Thus, V-crater holes provide better wall-attached jet flow and coolant spanwise coverage, resulting in better cooling performance. No pronounced jet lift-off can be found even at MFR = 18.34%. Meanwhile, a larger low-velocity recirculation zone can be found inside the V-crater hole.
For temperature distributions in Figure 26, V-crater holes own better coolant coverage downstream of P3.
Owing to better wall-attached jet flow and coolant spanwise coverage, the low-temperature coolant film is thinner for V-crater holes. With the MFR increasing, the advantage of V-crater holes over cylindrical holes becomes more distinct (Kalghatgi & Acharya, 2015).
Notably, owing to a negligible effect of mainstream Reynolds numbers, the counters for MFR = 18.34%, Re = 511229 are not shown repeatedly.

V-crater Hole on suction surface
Taking the S1 and S3 as an example, pressure distributions are shown in Figure 27. Different from the concave pressure surface, the pressure gradient owns a component towards the convex suction surface, leading to a better wall-attached film. Thus, from the streamlines in Figure 28, no pronounced jet lift-off can be found for both holes even at MFR = 18.34%. However, owing to lower blow ratios, lower normal velocity components, and higher laterally velocity components, V-crater holes provide better wall-attached coolant film.     From temperature distributions in Figure 29, the coolant coverage is poor downstream of S3 for cylindrical holes. However, better coolant coverage can be found in V-crater holes. With MFR increasing, the advantage of V-crater holes over cylindrical holes increases.

Influence on film superposition
To investigate the influence of hole shapes on film superposition and flow field above the wall, taking P4 as an example, three spanwise sections located downstream of P4 are selected as depicted in Figure 30.   Firstly, to research the flow field near the wall, taking the BAS as an example, the pressure distributions superposed by streamlines at S/D = 0 and S/D = 10 are shown in Figure 31. There is a pair of counter-rotating vortexes (CRV) located above the wall at the center of the blade height. Besides, along the streamwise, the pressure declines while the CRV keeps moving away from the blade wall.
The forming mechanism of the CRV is displayed in Figure 32. Due to the counter-inclined holes (L1-6) on   the leading edge, a confluence region of lower and upper coolant branches can be found at the center vane height, leading to the formation of the CRV. From Figure 31(b), the CVR lifts coolant film away from the wall but entrains hot mainstream neat the wall, which will strengthen the mixing of coolant and mainstream. Thus, the CVR has a negative influence on film cooling performance.
To investigate the influence of hole shapes, temperature distributions of the BAS and OPT are compared in Figure 33. For the OPT, as V-crater holes and laidback holes provides better wall-attached and lateral coverage coolant film, the height of the CRV becomes lower and the coolant is more evenly distributed near the wall. Compared with the BAS, the CRV is lower, smaller, and weaker for the OPT, which enhances film cooling performance and makes more use of coolant.

Film cooling effectiveness
The film cooling effectiveness is used to evaluate the coolant coverage condition for film holes. Contours of film cooling effectiveness are shown in Figure 34. For better explanations, the blade surface is divided into PSS, LE, and PPS, corresponding to where S0-4, L1-6, and P1-7 locate respectively.
At MFR = 11.59%, owing to insufficient mass flow supply, film cooling effectiveness ε of both two vanes decreases on the LE. The advantage of laid-back holes is not obvious. Meanwhile, the counter-rotating vortexes (CRV) are weak and have a limited effect on the film superposition on the PPS. As the pressure gradient is beneficial to coolant coverage on the PSS but detrimental to the PPS, PSS owns longer high-ε coolant streaks downstream of film holes. Due to the low-pressure outlet near the bottom, the coolant deviates to the hub, especially on the PSS. Compare with the BAS, the V-crater holes of OPT enhance the lateral coolant coverage, resulting in higher ε and a larger high-ε region on the PPS and PSS.
At MFR = 15.39%, on account of film superposition, although jet lift-off occurs for the BAS, the ε still gets improved with MFR increasing. On the LE, the advantage of laid-back holes over cylindrical holes becomes pronounced, especially on the downstream of L5-6. For the BAS, owing to a stronger effect of CVR on the PPS at high MFR, the coolant accumulates on the center of blade height, resulting in lower ε near the shroud and hub (especially downstream of L6) but higher ε near the center blade height. Compared with BAS, the CVR gets suppressed for the OPT, leading to more evenly distributed film cooling effectiveness.
With MFR increasing to 18.34%,ε keeps rising for both two blades. However, the effect of CVR gets strengthened, leading to stronger coolant accumulation, especially for the BAS. The influence of Re decreasing is similar to that of MFR increasing. At MFR = 18.34%, the counters under different Re numbers are similar. With mainstream Re decreasing, the cooling effectiveness increases, however, the influence is limited.  The laterally average film cooling effectiveness is dotted in Figure 35. For the convenience of presentation, the data of the BAS and OPT are shown in two graphs respectively. For both blades, compared with the PPS, the maximum value is higher while the influence of MFR is less distinct on the PSS. With MFR increasing, the increasing extent of ε decreases. Compared with BAS, except for the LE at MFR = 11.59%, the OPT improves film cooling performance on the whole. At MFR = 18.34, the curves at different Re nearly overlap. Besides, the influence of optimization on s/Ca = −0.6 ∼ −1.2 is limited.
After optimization, the enhancement on area average ε based on the BAS is shown in Figure 36. For the whole vane, the film cooling effectiveness increases by around 6.2% on all working conditions. With MFR increasing from 11.59% to 15.39%, the enhancement increases for the SS but decreases for the PS. However, with MFR increasing from 15.39% to 18.34%, the enhancement keeps relatively constant for the PS and SS. With Re decreasing, the enhancement gets slightly improved. Compared with the SS, the improvement on the PS is higher on all working conditions.

Heat transfer coefficient
In film cooling technology, the coolant film isolates the blade wall from the hot mainstream, which weakens heat transfer and limits the wall temperature within the allowable limit. The introduction of shaped holes is a double-edged sword. On the one hand, shaped holes provide better wall-attached and more evenly distributed coolant film, resulting in better protection for blade walls and lower heat transfer coefficients h. On the other hand, a larger hole exit area and lateral coverage area strengthen the blending of coolant and mainstream, leading to higher heat transfer coefficients. Thus, to investigate the comprehensive effect of shaped holes on heat transfer coefficients, contours are compared in Figure 35.
At MFR = 11.59%, for the BAS, the highest h locates on the LE and shroud near the TR. The h locating downstream of holes is higher than that locating between downstream of holes. Meanwhile, the PSS owns a larger low-h region than the PPS. Compared with BAS, the OPT owns a larger high-h area on the PPS but a smaller high-h area near the LE, hub, and shroud. Besides, the area of the lowh region is similar for both two blades on the PPS.
With MFR increasing, the h increases on the whole area for both two blades. However, at MFR = 18.34%, there is a long low-h strip region on the PPS, especially for the OPT. With mainstream Re declining, except for the appearance of a long low-h strip near the shroud, the influence of Re on the BAS is negligible. Nevertheless, for the OPT, with Re decreasing, the long low-h strip becomes larger while a long high-h strip forms nearby on the shroud of the PPS. Except for the shroud, h decreases on the whole area of the OPT.
The introduction of shaped holes also influences the internal cooling. Regarding the internal cooling system, higher heat transfer performance is beneficial to decrease the wall temperature. Owing to limited length, internal cooling is not discussed in detail. Taking the suction surface as an example, Figure 38 illustrates the Nusselt number distributions at MFR = 15.39%, Re = 556337. As   has been discussed in Figure 19, after optimization, the mass flow rate allocated to jet nozzles IMP 7-11 and IMP 1-6 increases by 3.9% and 0.8% separately, which helps enhance heat transfer. Thus, after optimization, a pronounced improvement in heat transfer can be observed for Target surface 1. However, little change in the Nusselt number distribution can be found for Target surface 2.
After optimization, changes in area average h based on the BAS are shown in Figure 39 and Table 7. Compared with the BAS, the introduction of shaped holes augments heat transfer on the external wall on the whole. With MFR increasing, the increase extent decreases, which means the influence of MFR and Re is limited at high MFR. The enhancement of heat transfer of the external SS is higher than that of the external PS. For the internal cooling system at Re = 556337, the introduction of shaped holes decreases heat transfer on the internal PS but increases heat transfer on the internal SS, although, the change is negligible. However, with Re decreasing to 511229, the OPT owns lowerh for both internal PS and SS.

Overall cooling performance
The overall cooling effectiveness η is used to evaluate the cooling performance of the whole blade. Contours of η are shown in Figure 40. At MFR = 11.59%, for the BAS, the lowest η mainly locates near the LE, shroud, and hub. Compared with PPS, a larger high-η area locates downstream of film holes for the PSS. Compared with the BAS, the OPT owns higher η on the PSS and PPS. However, due to insufficient coolant distribution, η on the LE (s/Ca = −0.05 ∼ 0.10) and near the TR (s/Ca = 0.88 ∼ 1.00) drops after optimization.
With MFR increasing, except for the region near the hub and shroud, the η increases on the whole area. Owing  to adequate mass flow allocation, the advantage of laidback holes appears, leading to a better cooling performance on the LE for the OPT. For the BAS, affected by strong CRV generated by L1-6, the high-η area concentrates on the center of the blade height on the PPS. Thus, the cooling performance near the shroud and hub deteriorates. Compared with the BAS, shaped holes of the OPT suppress the effect of the CRV, leading to higher η near the shroud and hub, especially on the downstream of L6 at MFR = 18.34%.  At MFR = 18.34%, theη contours of different Re numbers are similar. With Re decreasing, the influence of CRV becomes more distinct. The influence of Re decreasing is similar to the effect of MFR increasing.
The laterally average overall cooling effectiveness is dotted in Figure 41. For the BAS, the highest η value locates on the PSS while the lowest η locates near the LE and TR. With MFR increasing, the η climbs while the increasing extent decreases. Besides, at MFR = 18.34%, the curves at different Re nearly overlap, meaning that the influence of Re is limited. Compared with the BAS, the OPT owns lower η on the LE at MFR = 11.59%. However, the OPT owns a higher η on the whole on all working conditions. The area average overall cooling effectiveness is shown in Figure 42. For both two blades, with MFR increasing, the increasing extent decreases. Besides, PS owns higher η than the SS on all working conditions. At MFR = 11.59% ∼ 18.34%, the introduction of shaped holes increases η of the PS from 0.71 ∼ 0.80-0.73 ∼ 0.84 while increases η of the SS from 0.70 ∼ 0.77-0.71 ∼ 0.78.
After optimization, the enhancement on area average η based on the BAS is shown in Figure 43. The improvement on the PS (2.80 ∼ 4.60%) is higher than that on the SS (0.85 ∼ 1.89%). From what has been discussed above, for the external cooling system, the OPT owns higher film cooling effectiveness than the BAS. Besides, with the MFR increasing, the enhancement decreases first and then remains relatively unchanged. The OPT owns higher heat transfer coefficients on the external  As the structural modification on the external cooling affects pressure distributions of the cascade passage, the cascade pressure loss factors ξ and changes on pressure loss factors ( ξ )/ξ BAS based on the BAS are illustrated in Figure 44. For both two blades, with MFR increasing, ξ increases. However, at MFR = 18.34%, with mainstream Re decreasing, ξ decreases. Compared with the BAS, the OPT owns a higher ξ . However, after optimization, the increase in ξ is limited (0.26 ∼ 0.85%).
As the change in cascade pressure loss factors and overall cooling effectiveness meets the requirements of the optimization (function (1)), the design optimization is successful. After optimization, changes in cascade pressure loss factors, film cooling effectiveness, overall cooling effectiveness, and wall temperature are concluded in Table 8.
It is worth noting that in this study, the adverse effect on internal cooling did not lead to a decrease in overall cooling performance. There are mainly two reasons. One significant reason is that with the help of impingement cooling, ribs, dimples, and protrusions, the heat transfer of the internal wall is high enough. In this situation, the contribution of internal cooling to overall cooling effectiveness remains relatively unchanged (Zhou et al., 2020). Thus, the influence of internal cooling is limited while the contribution of the improved film cooling could be more obvious. On the other hand, the changes in the internal cooling are small, with only less than 5% at MFR = 11.59% ∼ 18.34%, Re = 556337. The heat transfer is even enhanced on the internal suction surface.

Conclusions
Conjugate heat transfer was adopted in this article to analyze the flow field and heat transfer characteristics. Based on the basic blade (BAS), the film cooling was optimized by replacing cylindrical holes with laidback holes and V-crater holes for the optimized blade (OPT). The simulation was performed at the mass flow ratio MFR = 11.59% ∼ 18.34% and mainstream Reynolds number Re = 511229 ∼ 556337 (corresponding internal jet flow Reynolds number Re = 17444 ∼ 23974).
This study lacks experiments about the overall cooling effectiveness and film cooling for the two blades, which is the direction deserving striving for in the future. Moreover, after optimization, at the design condition, the enhancement on the cooling performance near the trailing edge is not enough, hence, the arrangement of film holes may need further improvement in the future.
The main conclusions are as follows.
(1) (3) discharge coefficient: At MFR = 11.59%, the discharge coefficients, adopted to quantify the effusion condition of holes, decrease by 2% ∼ 77% for five showered holes on the leading edge but increase for other film holes on the whole after optimization. At the design condition (MFR = 15.59%, Re = 556337), after optimization, the discharge coefficients change by −2.3% ∼ 4.1% for showered holes and increase by 1.1% ∼ 39.5% for most other film holes, which means the effusion condition gets improved. With the MFR increasing, the difference between the OPT and BAS narrows. (4) Blow ratio: with the MFR increasing from 11.59% to 18.34%, blow ratios increase for every column of film holes, especially for showered holes on the leading edge (increase by 4.9% ∼ 38.6% for each hole). The influence of Reynolds numbers is negligible. After optimization, blow ratios decrease for most shaped holes. With the MFR increasing, the difference between the OPT and BAS narrows. (5) Influence on film superposition: owing to counterinclined showered holes on the leading edge, a pair of counter-rotating vortexes is induced above the wall at the center blade height, which is detrimental to coolant coverage and leads to coolant concentration. After optimization, counter-rotating vortexes are weakened by shaped holes, leading to better film cooling performance. (6) Film cooling effectiveness: due to different pressure gradient directions, the suction surface owns longer high-effectiveness streaks downstream of holes than the pressure surface. Lower Reynolds numbers or higher MFR lead to higher film cooling effectiveness. Compared with the BAS, the shaped holes of OPT provide better wall-attached coolant film and improve lateral coolant coverage. At the design condition, after optimization, the film cooling effectiveness ε increases by 9.84% on the pressure surface and 3.27% on the suction surface. Meanwhile, the film cooling effectiveness is more evenly distributed. (7) Heat transfer coefficient: after optimization, heat transfer coefficients decrease on the leading edge and shroud but increase on the other area. After optimization, at the design condition, the heat transfer coefficient h increases by 7.56% on the external pressure surface and 13.19% on the external suction surface. However, the influence on heat transfer of the internal wall is negligible.
(8) Overall cooling effectiveness: with the MFR increa sing or Reynolds numbers decreasing, the overall cooling effectiveness η increases while the increasing extent decreases. After optimization, at the design condition, the overall cooling effectiveness exacerbates on the trailing edge owing to insufficient coolant allocation but increases on the other area. At the design condition, the overall cooling effectiveness increases by ( η)/η BAS =4.19% on the pressure surface (η BAS =0.79) and ( η)/η BAS =1.78% on the suction surface (η BAS =0.75), corresponding to 33.6 and 13.7 K temperature drop respectively. At MFR = 18.34%, the advantage of the optimized vane over the basic vane is maximized with an increase of 3.17% on the whole. The influence of the mainstream Reynolds number is limited. (9) Cascade pressure loss factor: with the MFR increasing or Reynolds numbers increasing, cascade pressure loss factors increase. After optimization, the cascade pressure loss factor increases by 0.26% only at the design condition, which is within the acceptable range. Adiabatic wall c C o o l a n tfl o w g M a i n s t r e a mfl o w s S t a t i c t T o t a l

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was supported by the National Science and Technology Major Project [2017-III-0003-0027].