Simultaneous estimation of reference temperature and heat transfer coefficient in transient film coo

Abstract

Keywords: ransient Film Cooling; Effectiveness; Heat Transfer Coefficient; Inverse Heat Conduction Problem; Levenberg-Marquardt Algorithm

Introduction

Excessive operating temperatures in modern gas turbine engines to improve the Brayton cycle efficiency lead to frequent failure of engine components. Adequate cooling is most desirable in such operating conditions. Film cooling is a typical technique to cool the gas turbine engine components and to protect the surfaces from hot gases. Air as a coolant is issued out of small discreet holes to form a protective layer over the surface. The performance of film cooling is affected by various parameters such as shape, injection angle, blowing ratio, hole length, density ratio [1]. Numerous efforts have been carried out experimentally as well as numerically to study the effect of parameters on film cooling behavior [2]. The flat plate model of film cooling is most commonly used in many studies since it offers less geometric complexity. The film cooling performance is evaluated by knowing the reduction in the surface heat flux due to film injection. The heat flux (q'') over a film cooled surface mainly depends on the two parameters such as surface heat transfer coefficient (h) and the reference temperature (Tref) of the fluid which can be calculated according to Newton’s law cooling as [3], (1) The early works on the film cooling used steady-state experimental techniques to evaluate h and Tref [4,5]. In the case of steady-state analysis, these two parameters are evaluated independently by conducting separate experiments for each of the parameters [3]. The reference temperature is calculated by insulating the wall and measuring the adiabatic wall temperature at a steady state. In this case, the free-stream and the coolant will be at different temperatures. While evaluating the heat transfer coefficient, a constant heat flux [6,7] will be applied over the test surface using thin foil heaters and the corresponding wall temperature is measured. Subsequently, using Newton’s law of cooling, the heat transfer coefficient is evaluated by keeping the mainstream and the coolant at the same temperature. This method of evaluating heat transfer coefficient has certain limitations, such as maintaining constant heat flux over complex geometries and lateral conduction errors due to longer duration to achieve a steady-state. Also, this method evaluates only one parameter per experiment. Vedula and Metzger proposed a transient technique to estimate h and Tref in case of three temperature convection problems [8]. Later Ekkad et al. used transient liquid crystal thermography to evaluate film cooling effectiveness [9] and heat transfer coefficient [10,11]. They extended their study further using infrared thermography for transient surface temperature measurements [12]. Chen et al. [13] proposed a new data reduction technique based on the transient wall temperature data using liquid crystal thermography to evaluate film cooling parameters.

A new transient technique by applying non-homogeneous heat flux through heater foil was developed by Vogel et al. [14]. They used nonlinear regression to obtain unknown parameters by conducting multiple tests with different heat flux ratios. A major disadvantage of this method is that the applied heat flux influences the thermal boundary layer and affects the heat transfer coefficient. A dual linear regression technique based on a two-test strategy was proposed by Xue et al. for a transonic turbine cascade to determine the recovery temperature, heat transfer coefficient and effectiveness [15]. Ahmed et al. proposed a transient technique for jet impingement studies by solving a three-dimensional heat conduction equation [16]. This technique reduces the lateral conduction error but is computationally more expensive than solving a 1D heat conduction equation. A comprehensive review of recent developments in different techniques used to determine h and Tref for a film cooling problem can be found in Ekkad and Singh [17]. It has been observed that the previous studies followed two or multiple test methods to obtain various parameters. One major drawback of such techniques is the difficulty in maintaining the same aerodynamic conditions when different boundary conditions are applied. Hence it is essential to develop a method that can determine both h and Tref from a single transient test. Solutions to such problems can be obtained using the inverse heat transfer approach. The inverse heat conduction problems (IHCP) in heat transfer are related to estimating the unknown parameters such as boundary conditions, initial conditions, or the thermophysical properties using transient temperature measurements [18]. When the boundary conditions at a surface are known, it is possible to find the temperature distribution on a surface. This technique is known as the direct problem. On the other hand, in the case of inverse problems, the boundary conditions are estimated by measuring the temperature-time history [19]. The difficulty in the case of IHCPs is that the solution is illposed. Hence the solution to IHCPs is very sensitive to the errors in the measurements. Different methods have been proposed for the solution of IHCPs, such as Tikhonov’s regularization technique, Beck’s function estimation, optimization technique, Genetic algorithm, etc. [20]. Inverse solutions are majorly classified as stochastic and gradient-based methods [21]. Stochastic methods can find the global minimum but are complicated to implement and require many iterations to converge. Contrarily, the gradient-based methods are faster in convergence and simple to implement, but they may result in local minima. The difficulty associated with the gradient-based method is calculating the sensitivity coefficients accurately when the inverse problem is nonlinear. Levenberg-Marquardt Algorithm (LMA) is one of the most straightforward yet robust method used for solving inverse heat conduction problems based on optimization technique. It is a gradient-based method that combines the steepest descent

method in the neighborhood of the initial guess and uses the Gauss-Newton method near the minimum of the ordinary least squares norm [22]. LMA was successfully applied by numerous researchers to estimate thermophysical properties and boundary conditions [23,24]. Even the most modern technologies, such as neural networks, use LMA as a training algorithm [25]. Computational fluid dynamics (CFD) is one of the most effective methods for investigating complex flow phenomena in various applications [26]. A study on the flow pattern and heat transfer characteristics through CFD was investigated for a circular tube with a twisted tape insert by Al-Obaidi and Cahaer [27]. Later this study was extended using dimpled and corrugated tube surfaces [28,29]. Additionally, a numerical study on the influence of various configurations of corrugated pipes was done to conduct a heat transfer analysis, which showed that higher corrugation ratios could result in higher heat transfer rates [30–32]. Based on the numerical outcome, correlations have been developed for friction factor, Nusselt number and performance factor [33]. Further, Alhamid and Al-Obaidi used Al2O3 based nanofluid in a dimpled circular tube to investigate the hydrodynamic thermal characteristics using ANSYS Fluent [34]. Similarly, CFD techniques are successfully applied to evaluate h and Tref in case of film cooling situations [35,36] and also used to develop novel hole shapes to increase the cooling performance. Most of the numerical studies related to film cooling have considered an adiabatic surface to calculate the effectiveness. However, the velocity field might get influenced by the heat transfer between the fluid and the surface [37]. A conjugate calculation can include this effect and may lead to accurate prediction of film cooling parameters [38]. It is observed that the conjugate solution results in an underprediction of the absolute temperatures than the experimental results [39]. Hence a conjugate numerical simulation of the film cooling situation could be more realistic than the existing adiabatic film cooling assumptions. It is observed from the literature that a transient film cooling analysis can quickly predict both the heat transfer coefficient and reference temperature. Also, a significant advantage of conducting a transient study is that it can replace the two-test strategy of steady-state analysis. The existing transient studies have also followed the twotest method due to difficulties associated with the data reduction technique. Hence, developing a data reduction technique for the transient analysis of film cooling situations using a single test approach is essential. Based on the above observations, the present work aims to achieve a few objectives, such as conducting a transient analysis to evaluate boundary parameters for a film cooling problem. To develop an approach to obtain film cooling effectiveness and heat transfer coefficient using short duration transient data from a single test that is accurate and easy to

implement. Also, it is essential to study the effect of transient data on the solution approach. Hence in the present work, the feasibility of simultaneously estimating the convective heat transfer coefficient and reference temperature in case of film cooling problems through a single transient experiment is attempted. The novelty of the present work lies in the proposed data reduction method, which is based on the inverse heat conduction approach. This method can solve and evaluate multiple film cooling parameters using a single set of transient data. The proposed method utilizes the analytical solution of the transient one-dimensional semi-infinite heat conduction model as the forward solution. The inverse solution is obtained using an optimization technique known as Levenberg-Marquardt algorithm. A conjugate numerical simulation is performed in ANSYS Fluent to generate transient wall temperature data required for the analysis. The film cooling parameters, h and Tref are evaluated through an in-house MatLab code using transient data obtained from the simulations. Results from the present technique are validated with the results obtained from steady-state numerical simulation for blowing ratios of 0.5, 0.8, and 1.0. A brief discussion on the sensitivity of input data on the current solution approach in estimating film cooling parameters is also presented.

ONE Dimensional Semi-Infinite Analysis

Figure 1 shows a typical film cooling arrangement over a flat surface. The aim here is to calculate the local fluid temperature (Tref) and heat transfer coefficient (h) over the surface using the transient temperature measured at y = 0. This problem can be simplified by assuming the plate as a semi-infinite model with one-dimensional heat conduction across the thickness of the plate. But the semi-infinite assumption is valid only for a short duration such that the temperature change on the top surface (y = 0) should not affect the bottom surface temperature (i.e., at y = L, Tw = Ti) [14]. The general governing equation for one dimensional transient heat conduction through a semi-infinite flat plate with a convection boundary condition on one side can be written as [40], (2) The boundary conditions are as follows, (3)

Figure 1. Film cooling arrangement over a surface of flat plate and the one-dimensional heat transfer across the thickness of the plate.

The reference temperature in Equation 3, Tref, is the mixture temperature of the mainstream and coolant over the film cooled surface, and Tw is the temperature of the plate. The initial condition is given by,

through an iterative technique with nonlinear least squares regression.

To evaluate film cooling unknown boundary parameters, h and Tref, existing in Equation 6, one of the robust optimization technique known as the Levenberg-Marqurdt algorithm (LMA) is used in the present study. LMA is an iterative approach that evaluates the optimum values of unknown parameters by minimizing the variance between the measured and the estimated temperatures. Firstly the inverse problem needs to be constructed by defining an objective function (S) defined as the summation of the squared difference between the measured and the estimated temperatures [18], which can be written as Equation 8,

The solution for Equation 2 using the prescribed initial and boundary conditions is given by [40], (6) where α and k are the thermal diffusivity and thermal conductivity of the plate material respectively. In case of jet in cross-flow situations such as film cooling, a coolant stream is injected into the mainstream as represented in Figure 1. Hence the fluid temperature in the downstream of injection is a mixture of two fluid streams. This mixture temperature is unknown and has to be evaluated, which is attributed as reference temperature (Tref). The reference temperature is highly localized with respect to the location and depends on the mixing phenomena of hot and cold fluids. In case of film cooling arrangement, Tref is represented in terms of a non-dimensional quantity known as effectiveness (η), which is defined as [4], (7) where Tc and Tm are the temperatures of the coolant and mainstream, respectively. Equation 6 has two unknowns (h and Tref) which necessitates one more additional equation to evaluate the two parameters. Since only one equation is available, the solution for this problem can be obtained

Levenberg-Marqurdt Algorithm

(8) where Y and T are the measured and estimated temperatures at a particular location (x/d) and the subscript i = 1,2,3,4....I. The estimated temperature values are obtained by solving Equation 6. The term P refers to the unknown parameters, Tref and h in the present case. The objective function S is in the form of ordinary least squares norm. In order to minimize the objective function, the derivatives of Equation 8 with respect to the unknown parameters need to be evaluated and equated to zero [18]. (9)

Equation 9 can be represented in the form of matrix notation as follows, (10) where,

that the oscillations due to ill-conditioning will be damped, and the solution tends to the steepest descent method. As the iteration progresses, the value of λ will be reduced, and the solution follows the Gauss-Newton method. Stopping criteria for the iterations is given by [18],

sitivity coefficient, which is defined as the rate of change of the dependent variable with respect to the unknown parameters. In the present work, the sensitivity coefficients are evaluated by the direct differentiation of Equation 6 with respect to h and Tref. The expressions obtained to calculate the sensitivity coefficients are as follows, (11)

The inverse problem becomes nonlinear if the Jacobian has a functional dependency on the unknown parameters. The solution to Equation 10 is obtained through an iterative procedure. Here the vector T(P) is linearized using the Taylor series of expansion with respect to iteration κ as [18], (13) Substituting the Equation 13 in Equation 10 and after some rearrangement yields the following expression for the unknown parameters, P [18]: (14) Equation 14 is known as Gauss-Newton method, which is an approximation of the Newton’s method. When |JT J| in Equation 14 becomes very small or zero, the inverse problem becomes ill-conditioned, and the parameter estimation is impossible. This difficulty has been overcome by introducing a positive scalar known as damping parameter (λ) to Equation 14 as [18], (15) The Equation 15 is known as Levenberg-Marquardt method. The term Ω is a diagonal matrix which is calculated as follows [18], (16) The value of the damping parameter, λ, used in Equation 15 is made large during the initial iterations so

(17) (18) The flow chart of the in-house MatLab code to find the unknown film cooling parameters using the present IHCP technique is given in Figure 2.

Numerical Evaluation Of Transient Temperature Data

The transient surface temperature data required for the present IHCP technique is obtained from a three-dimensional numerical simulation using ANSYS Fluent. The geometrical parameters of the computational domain are specified following the experimental work of Sinha et al. [41]. Figure 3 shows a half section of the computational domain with the dimensions normalized using the film cooling hole diameter, d. The computational domain is divided into a solid region made up of film cooled plate with an inclined hole and a fluid region which contains a mainstream, a plenum chamber, and the fluid inside the film hole. Mainstream at higher temperatures enters and exits the domain through the inlet and outlet boundary surfaces as shown in Figure 3. At the inlet of the mainstream, velocity and temperature values are specified, and zero relative pressure boundary condition is set at the outlet. The coolant enters through the bottom surface of the plenum chamber and exits through the film hole into the mainstream. The coolant inlet boundary condition is specified with a mass flux having a particular temperature.

Table 1. Input conditions used in the present simulation Parameter

Method

At first, the numerical solution results are validated against the experimental results of Sinha et al. (1991). A steady-state simulation is carried out for blowing ratios of 0.5 and 1.0. Figure 5 compares the centerline effectiveness obtained from the steady-state simulation with the

experimental results of [41] for M = 0.5 and 1.0. There is a close agreement with the experimental results that exists up to x/d = 15 with an average error of around 5%. However, the deviation gradually increases further downstream (x/d > 15), resulting in an average error of 19%. At M = 1.0 the deviation is observed to be around 8% up to x/d = 30. The experimental result for M = 1.0 shows a sudden drop in the effectiveness values at x/d = 2.5 and increases slightly till x/d = 7 and then decreases further downstream. This is due to jet separation and reattachment, which occurs at blowing ratio of 1. In film cooling, the jet-mainstream interaction creates a distinctive flow feature known as counter rotating vortex pair (CRVP). The lift-off of the coolant jet is mostly caused by the presence of CRVP, whose strength rises as the blowing ratio increases. Hence, exact prediction of film cooling behaviour in numerical simulation becomes extremely difficult. It appears in the present simulation, the jet separation and the reattachment phenomena are not captured precisely. However, an overall film cooling behaviour is roughly apprehended. Previous numerical works have also observed the downstream deviation of numerical results from the experimental results. One primary reason is attributed to the isotropic turbulence viscosity assumption causing an overprediction in the computed centerline effectiveness [36]. Also, it was reported that the sources of errors could be the possibility

Figure 5. Validation of the present numerical results with the literature.

of jet skewness [35], lateral conduction errors during the experiments [47], use of tetrahedral meshes and inappropriate wall functions [48]. A validation of the present IHCP method is performed using the experimentally obtained h and Tref values from Chen et al. (2001) [13] for M = 0.5. The values of h and Tref from [13] are used to generate transient temperature from one-dimensional semi-infinite solution given in Equation

6. A random noise of ±1 K is introduced to the transient

wall temperature generated using Equation 6 to obtain noisy data. Then from the present IHCP technique, the effectiveness and heat transfer coefficients were estimated using the noisy data. Figure 6(a) shows the transient temperature generated from the solution of direct problem at x/d = 10 along with externally added random noise of ±1 K. Figure 6(b) and Figure 6(c) shows the comparison of estimated parameter values with corresponding experimental values. It was found that the present IHCP technique has estimated the parameters accurately with an average error of less than 1% from the noisy data. It is observed that higher noise levels increase the deviation in the estimated

parameters. When a noise of ±5 K is introduced to the input data, the estimated parameters have deviated with more than 10% error. Linear Fit Method When the transient surface heat flux data is available, it is possible to estimate the unknown film cooling parameters using the linear relation between the wall temperature and the surface heat flux data. The linear relation can be obtained using Newton’s law of cooling as [49], (20) This equation can be rewritten as [49], (21) The Equation 21 is in the form of y = mx + c , where the slope, m = 1/h and the y-intercept is Tref . By plotting transient heat flux vs wall temperature both heat transfer

Figure 6. (a) Transient temperature data generated from the forward solution with additional random noise of ±1 K, comparison of estimated parameters using noisy data (b) effectiveness, (c) heat transfer coefficient.

coefficient and the reference temperature can be obtained by finding the slope and the y-intercept respectively. Figure 7 shows the linear relation between wall temperature and heat flux at different x/d by plotting q'' (t) vs Twall(t).

Effectiveness Laterally averaged effectiveness estimated from the present IHCP technique for blowing ratios of 0.5, 0.8 and 1.0 are shown in Figure 8. The jet direction is mostly determined by the blowing ratio, which influences the distribution of the coolant and there by affecting the effectiveness. At low blowing ratio (M = 0.5), the η- reduces gradually along the length. As M increases, the coolant jet issues out with more momentum, causing low values of η-. It can be noted from the figure that the rate of reduction of η- is less for M = 1 than M = 0.5. At M = 1, η- drastically reduces for x/d < 15 and thereafter appears to have a negligible variation along x/d. The estimated η- values are compared with the steadystate simulation results for corresponding blowing ratios. In steady-state analysis, the film cooling surface is insulated while evaluating its effectiveness. The adiabatic wall temperature yields the local fluid temperature at steady-state. Figure 8 also presents the η- obtained using the method of linear fit as discussed in the previous section. It is observed that η- values estimated using the present IHCP technique are in good agreement with steady-state simulation results

Figure 8. Comparison of laterally averaged effectiveness obtained using different techniques for (a) M = 0.5, (b) M = 0.8 and (c) M = 1.0.

linear fit method. Average deviation between the steadystate and the present IHCP solution is around 6% - 7% for all blowing ratios. The estimated values are in close agreement till x/d ≈ 7 with an error less than 5% and slightly increases thereafter. Results obtained from the IHCP solution are almost coincides with the output of liner fit method. The deviation between the linear fit method and steady-state analysis is around 8% for M = 0.5 and 0.8, but slightly increases to 12% for M = 1. One major disadvantage of the linear fit method is that it requires transient heat flux values along with wall temperature to compute the parameters. The accurate measurement of surface heat flux is very challenging, hence the linear fit method might produce results with higher uncertainty. Sensitivity Analysis To avoid the ill-conditioned problem, the values of sensitivity coefficients must have large magnitudes [18]. Figure 9 shows the sensitivity coefficient values for reference temperature and heat transfer coefficient with time. The sensitivity coefficients can be calculated either by direct differentiation or numerical differentiation. If the forward solution is simple, then direct differentiation is possible. In the present work analytical solution to the one-dimensional heat conduction equation is used as a forward model. Instead, the sensitivity coefficients may have to be evaluated using numerical differentiation if a three-dimensional governing equation is considered. However, the numerical differentiation is computationally expensive and less accurate than direct differentiation. In such situations, the use of other optimization techniques which are independent of Jacobian evaluation can be implemented. In the present case, the values of JTref and Jh are calculated using Equation 11 and 12, respectively. Both JTref and Jh values are increasing with time, but JTref values are lower than Jh values. This indicates that the estimated temperature is less sensitive to change in reference temperature than the heat transfer coefficient, which makes the estimation of effectiveness more difficult than h. An immediate conclusion is that the least square error between the exact and estimated values for heat transfer coefficient would be less than the effectiveness. Also, from Figure 9, it is noted that the magnitudes of sensitivity coefficients are very small during the initial time. Hence considering transient data for a concise duration may lead to inaccurate parameter estimation. It can be observed from the Equation 11 and 12 that JTref is dependent only on estimated h, but Jh is a function of both Tref and h values indicating both the parameters are linearly dependent. The determinant of JT J is plotted with time in Figure 10 to compare the effect of sampling rate. Transient data is sampled with a time interval of 0.1, 1, 5 and 10 seconds. The figure indicates a steep rise in the magnitude of |JT J| up to around 20 seconds and gradually increases further. Hence the transient data considered below this point would increase the uncertainty in the parameter estimation. One

Figure 9. Sensitivity coefficients for Tref and h with time at x/d = 10 for M = 1.0.

Figure 10. Determinant of sensitivity coefficients with time at x/d = 10 for M = 1.0. of the significant observations from the figure is that higher values of |JT J| can be obtained by increasing the rate of sampling. Hence it can be concluded that, even with a short duration of the transient experiment, the parameter estimation uncertainty can be reduced using a higher sampling rate. Heat Transfer Coefficient In the case of film cooling, the heat transfer coefficient is normalized by the heat transfer coefficient over a flat plate with no coolant injection (h0). Therefore, it is essential to calculate the heat transfer coefficient for a non-film cooling flat plate case. A steady-state numerical simulation is performed for this particular case by specifying a surface heat flux (q'') of 1000 W/m2 throughout the plate. The plate dimensions are kept as that of the solid domain

as mentioned in the Figure 3. In addition, h0 is calculated using the Nusselt number correlation for turbulent boundary layer over a flat plate with constant heat flux boundary condition as [46], (22) The fluid properties such as ρ, μ, and Pr in case of correlation are evaluated at an average temperature using the property table available in Incropera et al. (2006) [46]. The average fluid temperature is calculated using free-stream temperature and average temperature of the wall subjected to constant heat flux boundary condition obtained from steady-state simulation. Now, heat transfer coefficient values for non-film cooling flat plate case are estimated from the present IHCP technique and validated with the steady-state numerical simulation results and the values obtained from the correlation given in Equation 22. Therefore a conjugate transient

Figure 11. Heat transfer coefficient for a flat surface with no coolant injection. numerical simulation is conducted and using the transient wall temperature data, h0 values are estimated by employing the present IHCP approach. The linear fit method is

Figure 12. Comparison of normalized laterally averaged heat transfer coefficient obtained using different techniques for (a) M = 0.5, (b) M = 0.8 and (c) M = 1.0.

also used to determine the h0. Figure 11 compares the heat transfer coefficient values obtained using the above mentioned methods. The results of the various methodologies demonstrate an excellent agreement with each other, with an average deviation of less than 5%. Figure 12 shows the normalized laterally averaged heat transfer coefficient values estimated using the present IHCP algorithm for film cooling case for M = 0.5, 0.8, and 1.0. The figure also includes h̅ values obtained from the steadystate simulation and the linear fit method for comparison. A general trend of heat transfer coefficient in film cooling is that it increases with blowing ratio due to higher turbulence generated with an increase in injected coolant mass flux. Also, injection of coolant into the mainstream augments local turbulence levels. Hence, the steady-state simulation exhibits a higher heat transfer rate near the film hole exit and gradually reduces to a more or less constant value along the downstream. The estimated h̅ from the IHCP algorithm follows the steady-state simulation results, but it appears to have slightly deviated in the immediate downstream of the jet exit (x/d < 15). Further downstream (for x/d > 15), the laterally averaged heat transfer coefficient values from all the methods closely agree with each other. The average deviation between the estimated heat transfer coefficient using IHCP and steady-state simulation is observed to be around 4% for all the blowing ratios studied. Results from the linear fit method also followed a similar trend as the IHCP solution with an error of around 3%. In the region immediately downstream of injection, a maximum deviation of around 9% is observed between the IHCP results and steady-state simulation output. One primary cause of such an anomaly in the current transient approach might be the conjugate heat transfer solution employed in the present simulation. As the coolant passes through the film hole, the plate temperature surrounding the hole might reduce, causing less heat transfer coefficient values near the film hole exit. As a result, it can be concluded that the current method may be more realistic than a steady-state analysis since it includes the effect of heat transfer across the film hole and the plate.

Conclusion

The present paper demonstrated a novel approach to simultaneously estimating the thermal boundary parameters of film cooling based on an inverse heat conduction technique. The solution methodology adopted in the present work employs an optimization technique to evaluate film cooling effectiveness and heat transfer coefficient appearing in the analytical solution of the transient one-dimensional semi-infinite model. The transient wall temperature data at the film cooling surface is the only input required for this method and is generated through a three-dimensional numerical simulation for a cylindrical hole film cooling arrangement. The estimated values from the present IHCP method showed good agreement having

a slight deviation of around 7% for the effectiveness and 4% for the heat transfer coefficient. The estimated h values from the present IHCP solution were underpredicted by about 6% for x/d less than 5, where the effect of coolant flow through the film hole prevails. A significant advantage of this technique is that only up to 100 seconds of transient temperature data is sufficient to evaluate both η and h using a single test. One major limitation of the current technique is the accurate calculation of sensitivity coefficients. Also, it can be concluded from the sensitivity analysis that a higher sampling rate while measuring the transient temperature data would reduce the estimation uncertainty. Overall, it can be ascertained from the present work that the IHCP technique can be successfully implemented to simultaneously estimate the effectiveness and heat transfer coefficient for transient film cooling analysis.

Nomenclature

d Diameter of film cooling hole, mm h Heat transfer coefficient, W/m2K J Jacobian k Thermal conductivity, W/mK L Thickness of the plate, mm M Blowing ratio P Parameter q'' Heat flux, W/m2 S Objective function T Temperature, K t Time, s U Velocity, m/s x, y, z Coordinates Y Measured data Greek symbols α Thermal diffusivity, m2/s η Effectiveness λ Damping parameter Subscripts c Coolant f Film cooling i Initial m Mainstream ref Reference temperature w Wall

Abbreviations

CRVP Counter rotating vortex pair IHCP Inverse heat conduction problem LMA Levenberg-Marquardt algorithm

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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