Acta Polytechnica https://doi.org/10.14311/AP.2023.63.0430 Acta Polytechnica 63(6):430–438, 2023 © 2023 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague PREDICTION OF TEMPERATURE FIELD DISTRIBUTION IN A GAS TURBINE USING A HIGHER ORDER NEURAL NETWORK Jan Pařez∗, Patrik Kovář, Adam Tater Czech Technical University in Prague, Faculty of Mechanical Engineering, Center of Aviation and Space Research, Technická 4, Prague 6, Czech Republic ∗ corresponding author: jan.parez@fs.cvut.cz Abstract. This paper presents the prediction of temperature field distribution in a single annular section using an artificial neural network (ANN). This temperature distribution is non-uniform on the outer tube due to continuous natural convection and radiation caused by the homogeneous steady-state heating of the inner tube, which represents the hot gas flow path through the turbine. The outer tube represents the case of a gas turbine. This temperature is important for the electronic components attached to the engine or the overall engine deformation. The presented approach allows for a quick estimation of the temperature distribution without the need to perform time consuming computational fluid dynamics (CFD) simulations. This can greatly accelerate the design and development of gas turbines. A machine learning approach is applied to an extensive set of CFD simulations under different operating conditions and geometry setups. Keywords: Gas turbine cooling, heat transfer, artificial neural network, natural convection. 1. Introduction The cooling process and temperature distribution of solid parts in gas turbines is critical to their reliable operation and maximum performance. High tempera- tures in the combustion chambers and hot section of the engine can adversely affect the lifespan of parts and accessories, trend/health monitoring, and electronic control system components. Therefore, it is impor- tant to design and implement cooling systems that ensure optimal temperature conditions and protect key components attached to the engine case. These components have maximum temperature limits, and exceeding these limits can cause damage and affect engine operation. Nakayama [1] introduced the issue of maximum temperature for electronic devices attached to gas turbines. A significant problem is the temperature sensitivity of the gas turbine rotor, which affects the dynamic behaviour of the engine, or in the worst case, rotor thermal lock. Uneven cooling can cause shaft deflection known as rotor thermal bow. Many papers have addressed the problem of rotor thermal deformation. Chatterton [2] presented the results of a diagnostic method of multiple linear regression models for temperature field prediction and analysis that can be very useful for maintenance prediction. Uneven temperature distribution during post shut- down cooling on the engine casing has been studied by Yu et al. [3], resulting in deformation of the ro- tor system. Different geometry and environmental settings on the temperature field distribution were ex- perimentally verified. The temperatures cause struc- tural deformation and cause mass non-uniformities, resulting in vibrations. Peng [4] presented extensive experimental measurements with heat distribution on the rotor during cooling and thermal deflection pre- diction. A computationally intensive simulation was performed by Padilla [5] using the large eddy simula- tion (LES) for the transient and turbulent flow region. Improvement of the numerical prediction of RANS using the natural convection heat transfer method in gas turbines was also presented by Pilkington [6] who also performed measurements on a large gas turbine test rig. In his work, he compared several models and proposed a modification of the GGDH model, which he called GGDH+, that takes into account the buoyancy effects on the turbulent heat flux and is comparable to the LES results. There are several heat management methods of cool- ing gas turbines. These include internal cooling, where a coolant (typically air or liquid) is directly supplied to cavities within the turbine’s internal structure [7], and external cooling, which utilises the flow of coolant around the surface of turbine components [8]. Another technique is film cooling, where a thin layer of coolant is applied to the surface of the components to reduce their temperature [9]. These methods are combined and optimised to achieve the best cooling effect and minimise thermal stress on turbine compo- nents. In addition, there are areas within the engine where spontaneous natural convection occurs. This has a significant effect on the temperature distribution on the engine case. Earlier research by Pařez et al. [10–12] was also focused on a numerical study of the effect of heat transfer in a double annulus. Methods to improve the uniformity of the temperature field were proposed and the numerical model was experimentally verified on several geometries at different temperature settings. Proper temperature distribution and effective cooling 430 https://doi.org/10.14311/AP.2023.63.0430 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 63 no. 6/2023 Prediction of temperature distribution in a gas turbine using a HONN are crucial for achieving long life and reliability of gas turbines. Therefore, a considerable attention is given to the design and optimisation of cooling systems to minimise the risk of overheating and ensure optimum operating temperatures. This increases the perfor- mance and lifespan of turbines and ensures reliability in various industrial applications. However, the problem of predicting non-uniform temperature fields arises from their computational complexity. Therefore, it is necessary to find an effi- cient tool for fast and accurate prediction of temper- ature distribution in different engine modes, which is the scope of this paper. An extensive set of CFD simulations under different operating conditions and geometrical setups was performed in order to obtain a training data set for the machine learning approach. A higher order neural network (HONN) was chosen as the approximator for the task. Its advantage is that it has fewer degrees of freedom and therefore fewer parameters to optimise during the learning process. Other advantages include faster convergence and more accurate results due to its non-linear nature as it was presented in [13–15]. 2. Task The case presented is considered to be the temperature field distribution at steady-state engine operation. The temperature field distribution depends only on the temperature in the current flow path due to operating conditions. A uniform temperature distribution in the flow path is considered, thus giving a boundary condition of constant temperature at the inner edge of the inner tube. The geometry of this case is then simplified to a single annulus between the inner and outer tube as it can be seen in Figure 1. Due to the constant temperature at the edge of the inner tube, a steady-state temperature distribution is sought on the outer tube. Figure 1. Natural convection in aircraft engine. 2.1. Geometry The computational geometry is based on a 2D cross- section of a turboprop engine. It consists of an inner tube representing the flow path boundary of the run- ning engine, labelled Tube 1 in Figure 2, and an outer engine case, labelled Tube 2 in Figure 2. The outer diameter D2 is constant for all the calculated cases. The inner diameter D1 is varied according to the val- ues in Table 1. The thickness of the outer tube t2 is chosen in the range given in Table 1. The thickness of the inner tube is assumed to be constant t1 = 1 mm. Figure 2. Scheme of the physical domain with its parameters. Dimension Value Unit D2 400 [mm] t2 {1, 2, 3} [mm] D1/D2 {0.5, 0.6, 0.7, 0.8, 0.9} [1] TD1 {773.15, 873.15, 973.15} [K] Table 1. Operating parameters. 2.2. Operational conditions The operating conditions and engine modes define the temperature field for the worst case, namely the tur- bine section. The temperatures are based on steady- state operating conditions in the three engine modes generalised and rounded to the nearest hundred Cel- sius degrees. The first mode is the idle mode, the next is the cruise mode and the last is the maximum take-off mode. The steady-state uniform temperature values for the flow path are marked as TD1 in Table 1 and remain constant throughout the calculation. The temperature response of the temperature dis- tribution on the outer case of the engine is marked as TD2 . Referring to previous considerations, the de- sired function represented by neural network N (•) is assumed in the form N (•) ≈ TD2 = f(D1/D2, t2, TD1 , φ). (1) 2.3. Computational methods and domain description In this section, we focus on computational techniques, the specific domain under consideration, meshing, and the associated spatial discretisation error. All calcula- tions were carried out using the Ansys® Fluent soft- ware, using fully structured meshes generated through the blockMesh utility available in the OpenFOAM® 431 J. Pařez, P. Kovář, A. Tater Acta Polytechnica package. The computational mesh is shown in Fig- ure 3. Figure 3. Computational mesh with detail on the refinement near walls. In order to effectively model turbulent flow in practi- cal engineering scenarios, the Navier-Stokes equations necessitate the implementation of Reynolds and Favres averaging. This procedure results in the formula- tion of the unsteady Reynolds-averaged Navier-Stokes (URANS) equations, as elucidated by Wilcox [16]. These URANS equations encompass the Reynolds stress tensor, necessitating a supplementary closure. Consequently, we adopt the two-equation Generalised K − ω (GEKO) turbulence model [17]. These equa- tions can be written as ∂ρ ∂t + ∇ · (ρ #–u) = 0, ∂(ρ #–u) ∂t + ∇ · (ρ #–u ⊗ #–u) + ∇p − ∇ · #– #–τ = ρ #–g , ∂(ρE) ∂t + ∇ · ((ρE + p) #–u − #– #–τ · #–u) = −∇ · #–q , (2) where the heat flux #–q is composed of the conduction part #–q c and the radiation part #–q r. To address the ra- diation part, we introduce an extra radiative transfer equation into the simulation. Given that the system pertains to surfaces that are gray and diffuse, the heat exchange between these surfaces depends on factors such as their dimensions, separation distance, and orientation. Moreover, we consider processes like ab- sorption, emission, and scattering of radiation. In this context, we implement the discrete ordinates (DO) radiation model [18]. This results in a system of seven partial differential equations. Four of these equations come from system (2), another two equations modelling the transport of turbulent kinetic energy and the specific dissipation of turbulent kinetic energy come from the turbulence model and the last equation models the radiation. To finally close the governing equations modelling the natural convection as discussed earlier, we can use a selection of equations of state. One such approach is the Boussinesq approximation [19], which postulates that density is contingent upon temperature in the following manner ρ ≈ ρ0 + ∂ρ ∂T ∣∣∣∣ T0 (T − T0). (3) In certain cases, this method loses its validity, par- ticularly when dealing with substantial temperature differentials where linearisation becomes inadequate. In such instances, a more precise approach is employed to ascertain the relationship between the density and the temperature. An appropriate closure formula for describing natural convection with significant temper- ature gradients, as outlined in the work by Mayeli [20], involves the utilisation of the weakly compressible gas equation of state, which is defined as follows ρ = pref RT , (4) where pref = 101 325 Pa is the reference pres- sure, which stays constant throughout the simulation. Therefore, density ρ is only a function of temperature. This equation of state was used in all simulations. For the solution of the equations governing natural convection including turbulence and radiation as de- scribed earlier, we opted for the finite volume method. Within this Ansys® Fluent framework, we have cho- sen the pressure-based approach using the Rhi-Chow numerical flux [21], over the density-based approach which uses numerical fluxes from AUSM [22], HLL [23] or rotated-hybrid [24] Riemann solver families. Next, a “coupled”. scheme was applied to handle the pressure-velocity coupling. Gradients were calculated using the cell-based least squares approach. Through- out the simulations, all spatial discretisation schemes were configured to achieve second-order accuracy. Fur- thermore, we employed a second-order implicit pseudo transient time formulation. All simulations were per- formed until steady states were reached. The schematically pictured computational task can be seen in Figure 4. This representation is misleading in terms of dimensions. In reality, the side of the square is approximately 12 times greater than the large diameter of the outer tube. Tubes’ thicknesses are also enlarged in Figure 4. Figure 4. Schematic computational domains with marked boundary conditions. 432 vol. 63 no. 6/2023 Prediction of temperature distribution in a gas turbine using a HONN The whole computational task is composed of four subdomains. Two of them, coloured blue, model air as a fluid. The other two, coloured in grey, model steel tubes as a solid. These subdomains are connected with coupled walls and the heat conduction through the tubes is fully modelled. The square-shaped subdo- main represents the atmosphere with inlet and outlet boundary conditions. The variables prescribed at the inlet were the total pressure, temperature, and angle of the velocity vector, 101 325 Pa, 293.15 K, and per- pendicular to the boundary, respectively. The variable prescribed at the outlet was the static pressure with a value of 101 325 Pa. Constant temperature TD1 was prescribed at the inner edge of the inner tube. Zero velocity magnitude was also specified on all walls. The distribution of the temperature and velocity magnitude field calculated using CFD in the single annulus due to natural convection and radiation is shown in Figure 5 and Figure 6. Figure 5. Temperature field distribution for case D1/D2 = 0.5, t2 = 1 mm, TD1 = 973.15 K. Figure 6. Velocity magnitude field distribution for case D1/D2 = 0.5, t2 = 1 mm, TD1 = 973.15 K. 2.3.1. Grid convergence study Analysing mesh convergence in a simulation is a straightforward approach to gauge the level of dis- cretisation error within a numerical simulation. This method entails conducting the simulation on progres- sively finer meshes. As the mesh becomes more refined, spatial discretisation errors should gradually diminish, asymptotically approaching zero, except for negligible computer rounding errors. For the scope of this study, it is necessary to des- ignate a specific scalar quantity. In this instance, we have selected the volume average of temperature on the outer tube, denoted as T , to serve this pur- pose. We generated three progressively finer meshes, maintaining a refinement ratio of r = 2. This means that the finer mesh contains twice as many cells as the coarser mesh in both spatial directions. Table 2 provides a comprehensive listing of all three meshes, along with their respective cell counts. Mesh label “1” “2” “3” Normalised mesh spacing 4 2 1 Number of cells 53 504 214 016 856 064 Table 2. Parameters of meshes generated and used in convergence study. Given that the parameter r remains constant, we can determine the order of convergence O shown in Equation (5) using Roache’s method, as described in [25], by directly utilising three solutions in the following manner O = ln ( T1−T2 T2−T3 ) ln(r) . (5) Once the convergence order has been assessed, we can proceed to apply Richardson extrapolation, as elucidated by Zlatev et al. [26], to the two most refined meshes. This method leverages these meshes to derive an estimate for the “exact” solution at zero mesh spacing, represented as Equation (6) Text = T3 + T3 − T2 rO − 1 . (6) As per Roache’s guidance mentioned in [25], the Grid Convergence Index (GCI) stands out as a favoured means of approximating discretisation er- rors. It quantifies the percentage by which the com- puted value deviates from the asymptotic numerical value, essentially delineating an error margin for the solution’s proximity to the asymptotic value. Addi- tionally, it provides insight into how the solution might alter with further grid refinement. A diminutive GCI value suggests that the computation resides within the asymptotic range. The computation of GCI can be performed using Equation (7) GCI = FS|ϵ| rO − 1 · 100, ϵ = Tfiner − Tcoarser Tfiner . (7) 433 J. Pařez, P. Kovář, A. Tater Acta Polytechnica In accordance with Wilcox’s recommendation [16], a safety factor denoted as FS is advised to be set at FS = 1.25 when employing three or more grids. From the utilisation of three meshes, we can derive two Grid Convergence Indexes, denoted as GCI12 and GCI23. The results of the mesh convergence study are listed in Table 3. These results were obtained for the case D1/D2 = 0.5, TD1 = 973.15 K and t1 = 1 mm. T1 T2 T3 687.569 685.805 685.274 O GCI12 GCI23 1.73 1.4 % 0.4 % Table 3. Results of the mesh convergence study. Based on these results, the mesh labeled as “1” was chosen for further computations as the discretisation error is sufficiently low. Thus, the error band is ap- proximately ± 9.6 K around the T1 value. 2.4. Temperature fields In order to obtain the training data set for neural network and replace experimental measurement, vari- ous numerical simulations with different geometrical setups and boundary conditions on the inner tube were performed as listed in Table 1. Trends of the temperature field distributions around the outer tube were observed on both inner and outer surfaces. The setup with ratio D1/D2 = 0.8, t2 = 1 mm and TD1 = 873.15 K is shown in Figure 7. Temperature distribution is a function of the angle φ where 0 indicates the highest point of the investigated geometry (see Figure 2). Figure 7. Results of the CFD: temperature field dis- tribution observed at the surfaces of the inner tube. To simplify the task, some assumptions were ac- cepted. Firstly, the temperature is assumed to be symmetrical along the vertical axis, thus the data with corresponding angular coordinate are averaged. And secondly, due to small temperature differences on the inner and outer edge of the outer tube and due to error band arising from the CFD discretisation error (see Table 3), it is assumed that the influence of the inner tube thickness can be neglected, so that temperatures on these edges are also averaged. Part of the training data set, representing one batch of D1/D2 ratios and boundary conditions TD1 pre- scribed on the inner tube permutations with constant thickness t2, is shown in Figure 8. The directions of individual parameter value growth are indicated by ar- rows, while the resulting part of the training data set is visualised against the number of the sample. Each cluster of data represents a single CFD simulation. As one can see, the behaviour is non-linear, which is a suitable task for the neural network learning. Figure 8. Part of the training data set with accepted simplifications. 2.5. CFD experimental validation The accuracy of the proposed numerical CFD model was verified and validated experimentally using an experimental setup. The time dependence of the tem- perature fields was observed and compared with the computational model. The results of this experiment and validation were presented by authors in [10, 12]. The same CFD model was used in the current cal- culation. A good agreement between the experiment and the numerical model was achieved with an error of the order of a few percent. 3. Neural network approach From a mathematical point of view, processing the information within neuron consists of two separate mathematical operations [27]. The first, synaptic operation, contains weights of the synapse, which 434 vol. 63 no. 6/2023 Prediction of temperature distribution in a gas turbine using a HONN represents storage of knowledge and thus the memory from previous knowledge. The second is somatic operation, which provides various mathematical operations, such as thresholding, non-linear activation, aggregation, etc. The neural output of the unit ỹ is then scalar as it is indicated in Figure 9 and expressed by following Equation (8) ỹ = σ(s). (8) Let us assume N -th order neural unit, then the product of a synaptic operation can be written as Equation (9). s = w0x0 + n∑ i=1 wixi + n∑ i=1 n∑ j=i wijxixj + . . . + n∑ i1=1 · · · n∑ iN =iN−1 wi1i2...in xi1xi2 . . . xin , (9) where x0 = 1 denotes the threshold and n stands for the length of the input feature vector. Neural unit   x1 ... xi ... xn   Inputs   w1 ... wi ... wn   Synapse ∑ σ(s) Soma w0 Threshold Learning algorithm s ỹ ∈ R Figure 9. Neural network: single neural unit. Since the desired outputs are predetermined, the process of machine learning is termed as supervised learning. This involves learning a function that con- nects input to output using a cost function #–e . As we can see, the neural output is strongly depen- dent on the neural memories represented by vector of the weights # – W . So, to help the neural unit learn, the information processing needs to be structured appro- priately. Batch Levenberg-Marquardt algorithm for updating weights [27] is utilised in present work # – W = # – W + ∆ # – W , (10) where ∆ # – W T = − ( #– #– J T #– #– J + 1 µ #– #– I )−1 #– #– J T #–e . (11) Coefficient µ is the learning rate, #– #– I is nw × nw identity matrix, nw the number of weights and #– #– J represents n × nw Jacobian matrix. Usually, the training data set is divided into three subsets. The first, training set, which serves for learn- ing and updating weights. The second is the validating set. After each epoch of the learning algorithm, error estimation is performed on this subset in order to avoid neural unit overfitting. The training continues until the validating error increases. The third part is called the testing set, which measures the error after learning is terminated. According to previous considerations, the desired function is assumed in a form of Equation (1). The neural network was assembled by five neurons with third order synaptic operation and bipolar sigmoid activation in the first layer and single second order neuron with linear activation in the output layer as it can be seen in Figure 10. Error propagation through the network is performed by Gupta et al. [28] using multilayer backpropagation algorithm.   (D1/D2, t2, TD1 , φ)1 ... (D1/D2, t2, TD1 , φ)i ... (D1/D2, t2, TD1 , φ)n   Inputs NU 1 1 ...... NU 5 1 1st layer NU 1 2 2nd layer   (TD2)1 ... (TD2)i ... (TD2)n   Outputs Figure 10. Neural network: shallow neural network. 4. Results All samples in the obtained data set (5 805 in total) were normalised using the minmax method within a range of ⟨0; 1⟩ Learning rate was set to a constant value µ = 0.2 and the total number of epochs was set to 1 000. In Figure 11, it can be seen that the validation error reached its minimum at 48-th epoch with a test error of 2.1338e-5 in the norm space. Figure 11. Results: progress of the learning. 435 J. Pařez, P. Kovář, A. Tater Acta Polytechnica Figure 12. Results: comparison of neural outputs against training data set in the sample space. Figure 13. Results: comparison of neural outputs against testing data set obtained using LHS. Weights set in this learning epoch were evaluated as the best approximation of the task before any possible overfitting. There is a graphical comparison of the neural output against the training data set shown in Figure 12. It can be seen that the neural outputs are in good agreement with the training data set. The mean squared error (MSE) reached 1.548 in the sample space, i.e. not norm space. Latin hypercube sampling (LHS) was used to gener- ate the data set. The strategy chosen was 16 samples with a maximum sum of distances in the sample space [29]. A comparison of the neural outputs against the testing data obtained using LHS is shown in Figure 13. Figure 14. Results: comparison of neural outputs against testing data set - the best match. Figure 15. Results: detailed comparison of neural outputs against testing data set - the worst match. The MSE for the testing data was 2.41. In Figure 14, there is a graphical comparison of the obtained prediction using neural network. It can be seen that the trend of the temperature field copies the trend of the temperature distribution obtained using a standard CFD approach. The maximum difference between the prediction and the true distribution is 2.1 K, which is in a sufficient error band, according to [30] and performed convergence grid study. The worst case in terms of matching the testing data set is shown in Figure 15. It is the case when the ratio D1/D2 = 0.826 and the temperature at the inner tube was set to TD2 = 868.39 K. 436 vol. 63 no. 6/2023 Prediction of temperature distribution in a gas turbine using a HONN The maximum difference can be observed at the top of the inner tube, i.e. at φ = 0◦, where the difference between the prediction and the data obtained through CFD reached 4.3 K, which is up to the measurement error limits described in [30] but still within the error band established by GCI. The oscilation in the area φ = ⟨30; 60⟩° is not cor- rectly captured bu the prediction differs from the CFD data by less than 2.0 K which is also within the sufficient error band. 5. Conclusion In order to eliminate time-consuming CFD simulations and accelerate the design of aircraft engines, this paper presents an approach to modelling the temperature field distribution on the outer tube in a single annulus, representing an aircraft engine section, using a higher order neural network. Based on a gas turbine section, various geometrical setups were prepared for the CFD simulations to obtain a training data set. Moreover, three engine steady-state modes, standing for idle, cruise and maximal take-off, were considered. A grid convergence study was performed on a chosen case in order to detect error band arising from the CFD. A large data set was obtained after comprehensive set of CFD simulations. Some simplifications, based on the results of the convergence study, were accepted as described in Section 2.4. A higher order artificial neural network architecture was also described. The learning was successful with a training error, measured by MSE, of 1.548 in the sample space, which is a sufficiently low error given the error band given by the CFD simulations. The results of the learning were also tested on the testing data set, which consisted of sixteen cases generated using Latin hypercube sampling. The MSE performed on the testing data set reached 2.41, which is also within the error band and is also covered by the measurement uncertainty of the thermocouple as described in [30]. Further work should aim at more complex geometry, more accurate CFD simulations and transient flow cases. The next step is to use finite element methods for deformation analysis and to develop a coupled solver that will allow faster analysis, and thus faster design of the device. List of symbols φ Angle [◦] D1 Inner tube diameter [mm] TD1 Inner tube temperature [K] t1 Inner tube thickness [mm] D2 Outer tube diameter [mm] TD2 Outer tube temperature [K] t2 Outer tube thickness [mm] t Time [s] T Volume average of temperature [K] ρ Density [kg m−3] O Order of convergence [1] p Static pressure [kg m−1s−2] pref Reference static pressure [kg m−1s−2] E Specific total energy [m2 s−2] R Specific gas constant [m2 s−2] r Refinement ratio [1] #–u Velocity [m s−1] #–q Heat flux [m s−1] #–q r Radiation heat flux [m s−1] #–q c Conduction heat flux [m s−1] #–g Gravitational acceleration [m s−2] #–e Cost function [1] #–#– I Identity matrix [1] #–#– J Jacobian matrix [1] n Length of input feature vector [1] nw Number of weights [1] x0 Neural threshold [1] ỹ Neural output [1] # – W Vector of the weights [1] σ(·) Activation function [1] µ Learning rate [1] Acknowledgements The authors acknowledge support from the ESIF, EU Operational Programme Research, Development and Edu- cation, and from the Centre of Advanced Aerospace Tech- nology (CZ.02.1.01/0.0/0.0/16_019/0000826), Faculty of Mechanical Engineering, Czech Technical University in Prague. 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TH0906. 438 https://doi.org/10.1016/j.ijheatmasstransfer.2019.06.074 https://doi.org/10.1016/j.ijheatmasstransfer.2019.06.074 https://doi.org/10.14311/AP.2023.63.0065 https://doi.org/10.29008/ETC2021-523 https://doi.org/10.1111/j.1749-6632.2001.tb05864.x https://doi.org/10.1111/j.1749-6632.2001.tb05864.x https://doi.org/10.1177/14680874231167206 https://doi.org/10.1088/1757-899X/1190/1/012002 https://doi.org/10.1051/epjconf/202226401027 https://doi.org/10.1017/S0022112095211388 https://doi.org/10.1016/j.icheatmasstransfer.2021.105316 https://doi.org/10.1016/j.icheatmasstransfer.2021.105316 https://doi.org/10.2514/3.8284 https://doi.org/10.1006/jcph.1993.1122 https://doi.org/10.1137/1025002 https://doi.org/10.1016/j.cam.2023.115129 https://doi.org/10.1515/9783110533002 https://doi.org/10.4018/978-1-4666-2175-6.ch006 https://doi.org/10.4018/978-1-4666-2175-6.ch006 https://doi.org/10.2307/1269769 Acta Polytechnica 63(6):430–438, 2023 1 Introduction 2 Task 2.1 Geometry 2.2 Operational conditions 2.3 Computational methods and domain description 2.3.1 Grid convergence study 2.4 Temperature fields 2.5 CFD experimental validation 3 Neural network approach 4 Results 5 Conclusion List of symbols Acknowledgements References