In film cooling flows, it is important to know the temperature distribution resulting from the interaction between a hot main flow and a cooler jet. However, current Reynolds-averaged Navier–Stokes (RANS) models yield poor temperature predictions. A novel approach for RANS modeling of the turbulent heat flux is proposed, in which the simple gradient diffusion hypothesis (GDH) is assumed and a machine learning (ML) algorithm is used to infer an improved turbulent diffusivity field. This approach is implemented using three distinct data sets: two are used to train the model and the third is used for validation. The results show that the proposed method produces significant improvement compared to the common RANS closure, especially in the prediction of film cooling effectiveness.

## Introduction

Film cooling is a widely used technique to control the thermal stresses and increase the lifespan of gas turbine blades [1]. For design purposes, Reynolds-averaged Navier–Stokes (RANS) simulations are often used to solve for the temperature field in these flows, which can be used to determine the film cooling performance and blade metal temperature distribution. Past studies have shown that RANS models struggle to accurately solve for the velocity and temperature fields in film cooling configurations [2,3]. There has been considerable effort in closure strategies for the momentum equation: for example, Hoda and Acharya tested seven different models [2]. However, previous work has shown that the largest source of error in film cooling predictions is the model for the turbulent heat flux [4]. Therefore, the present work focuses solely on RANS models for the temperature equation, which reduces to the passive scalar equation in the case of a nonbuoyant flow.

The simplest model to close the turbulent transport term in the RANS scalar equation is the gradient diffusion hypothesis (GDH), which is shown in the following equation:
$u′iθ′¯=−αt∂θ¯∂xi$
(1)
ui represents the velocity in the ith direction and $θ=(T−T∞)/(Tcool−T∞)$ is the dimensionless temperature, with $T∞$ being the inlet temperature and Tcool being the coolant temperature. Primed quantities indicate turbulent fluctuations and overbars indicate time average. The turbulent diffusivity αt is a spatially varying scalar function. The standard practice in commercial codes is to calculate the eddy viscosity field νt in the momentum equations and use a fixed turbulent Prandtl number to find the αt field, usually $Prt=νt/αt=0.85$ [5].

Previous work has shown that this approach is not sufficient for film cooling flows [3,69]. He et al. [6] performed RANS simulations in a normal jet in crossflow and recommended a turbulent Schmidt number $Sct=0.2$ to better match experimental results, a value considerably lower than 0.85 (in the current paper, Sct and Prt are considered indistinguishable because temperature is assumed to behave as a passive scalar). Lakehal [7] obtained improved results for the temperature field in a film cooling geometry after considering an anisotropic diffusivity and a spatially varying Prt. The experimental measurements of Kohli and Bogard [8] suggested that Prt varied between 0.5 and 2 in an inclined jet in crossflow. Liu et al. [9] showed that the film cooling effectiveness prediction is sensitive to the choice of Prt and that allowing spanwise variations in Prt improved results. Coletti et al. [3] used magnetic resonance imaging (MRI) data and concluded that νt and αt vary spatially according to different trends, so their ratio is not uniform.

There are more complex algebraic closures of the scalar equation as alternatives to the simple GDH. The generalized gradient diffusion hypothesis of Daly and Harlow [10] and the higher order generalized gradient diffusion hypothesis of Abe and Suga [11] are two commonly cited examples. Ling et al. [12] studied these closures in the context of a discrete hole film cooling geometry and found that they indeed yield slightly more accurate turbulent transport than the GDH. However, they concluded that tuning model parameters in simpler models produces greater improvements than switching model form. In fact, their most accurate results were obtained when the GDH was used in conjunction with a turbulent diffusivity field extracted from a high-fidelity large eddy simulation (LES) [12]. Inspired by those results, the present paper seeks to use the simple GDH of Eq. (1) with a machine learning (ML) methodology to select a more appropriate αt field.

Machine learning consists of a broad class of algorithms that process large amounts of data and extract patterns from them in order to make informed decisions in uncertain situations. Recently, the rapid increase in computational capability has led to a surge in the availability of large data sets from fluid mechanics simulations. In turn, this sparked interest in using machine learning techniques in the context of turbulence modeling [1318]. Tracey et al. [13] used machine learning algorithms to improve RANS models and obtain error bounds in a turbulent combustion and nonequilibrium boundary layer simulations. Tracey et al. [14] and Duraisamy et al. [15] applied machine learning algorithms to generate functional forms of turbulence models. Ling and Templeton [16] evaluated different machine learning algorithms applied to the problem of predicting regions where RANS simulations yield uncertain results. Ling et al. [17,18] predicted the Reynolds stress anisotropy in a jet in crossflow geometry using machine learning and also solved for the RANS velocity field using such anisotropy. Although the research mentioned is still in an early phase, all the authors concluded that machine learning seems a promising technique to inform turbulence modeling.

In this paper, a supervised learning algorithm is applied in an attempt to improve the RANS results of the scalar transport equation, particularly in film cooling geometries. The objective is to predict an improved αt field that can be used with Eq. (1). A total of three data sets are used, in which both RANS and high-fidelity simulations are available. The high-fidelity data are used to calibrate the machine learning model and to assess the model performance. Section 2 describes these data sets. Section 3 explains the machine learning approach. Section 4 presents the results of this study, both in terms of diffusivity and in terms of the scalar field. Finally, Sec. 5 contains the conclusions and next steps for this research.

## Computational Data Sets

To apply the machine learning framework, computational data sets of three different geometries are used: a baseline inclined jet in crossflow, a skewed inclined jet in crossflow, and a wall-mounted cube in crossflow. In all cases, a well-validated high-fidelity simulation is available in conjunction with a RANS simulation that uses the standard realizable $k−ε$ model in the momentum equation and the GDH with $Prt=0.85$ in the scalar equation. Throughout this paper, buoyancy and molecular diffusion are neglected, so the analogy between temperature and a passive scalar is used. Therefore, the scalar field will be treated as the dimensionless temperature θ and the turbulent Schmidt and Prandtl numbers are indistinguishable. The three cases are summarized in Table 1 and further described in Secs. 2.12.3.

### Baseline Inclined Jet in Crossflow.

A schematic of the baseline jet in crossflow geometry can be seen in Fig. 1. It consists of a single circular cooling hole of diameter D injecting into a square channel of side $8.62D$. The origin of the coordinate system is located at the center of the hole, such that the bottom wall is at y = 0. The hole is inclined $30deg$ with respect to the y-axis and is not angled with respect to the x-axis (the latter differentiates it from the skewed case). The Reynolds number based on the channel bulk velocity and hole diameter is $ReD=3000$. The flow is incompressible and the velocity ratio between the bulk main flow velocity and the bulk hole velocity is BR = 1. The cooling flow comes from a plenum underneath the channel and has a prescribed concentration of a scalar contaminant. Since the plenum is fed sideways (through the z-direction), the resulting flow is not expected to be perfectly symmetric about z = 0. An adiabatic boundary condition is prescribed at all the walls.

The LES of this configuration was performed by Bodart et al. [19] using CharLESx, a second-order, unstructured, finite volume solver developed at Stanford University, Stanford, CA. For the subgrid scales, it employs the Vreman model in the momentum equation and the GDH with $Sct=0.9$ in the scalar equation. A posteriori analysis showed that the subgrid viscosity and diffusivity are an order of magnitude lower than their molecular counterparts, which demonstrates that the effect of the subgrid model on the mean field is negligible. The authors thoroughly validated the simulation results against MRI data. In a near injection region ($X/D<7$ and $θ¯>0.05$), the absolute error between the LES and the experimentally measured mean scalar field was less than 10% in over 87% of the volume. In the same region, the mean streamwise velocity field agreement was within 10% of U in over 79% of the volume; the reader can find more details in Ref. [19].

The RANS simulation of this configuration was performed by Coletti et al. [3] using the software fluent. The first cell above the wall was located at $y+<1$ and they confirmed grid-independent convergence. More information can be found in Ref. [3].

### Skewed Inclined Jet in Crossflow.

A schematic of this configuration can be found in Fig. 2(a). The skewed jet in crossflow geometry is very similar to the baseline case described in Sec. 2.1, except that the circular cooling hole is inclined $30deg$ both with respect to the y-direction and the x-direction. The channel is also wider in the spanwise direction (8.62D × 17.3D). The skew creates a qualitatively distinct mean velocity field: while the baseline case contains a counter-rotating vortex pair downstream of injection, the secondary flow in the skewed case is dominated by a single vortex [4,20]. The Reynolds number based on the main flow bulk velocity and hole diameter is $ReD=5800$ and the blowing ratio is BR = 1. The skewed hole is fed from a plenum and contains a prescribed scalar concentration, and the walls are adiabatic.

The LES of this geometry was performed by Folkersma [20]. It was a nominally compressible simulation, but since the Mach number was low (approximately 0.2), the density variations were negligible. It also used the CharLESx solver and a posteriori results showed that subgrid scale contribution to the turbulent transport was negligible. Their results were also extensively validated against MRI data. In a near injection region ($X/D<7$ and $θ¯>0.05$), the absolute error between the LES and the experimentally measured mean scalar was less than 10% in over 81% of the volume. In the same region, the mean streamwise velocity field agreement was within 10% of U in over 92% of the volume. More details can be found in Ref. [20].

The RANS simulation of this geometry was performed in the scope of the present paper using the software fluent. The simulation domain spanned 60D in the streamwise direction, divided evenly before and after the injection point, to minimize the influence of inlet and outlet boundary conditions. At the channel inlet, a turbulent velocity profile was picked to best match the LES velocity from Ref. [20]. A slip condition was applied at the top wall, and no-slip was applied at all the other walls. A similar mesh to the one used in the baseline RANS was used in this simulation, and it can be seen in Fig. 2(b). The results were considered converged when the continuity residuals were below 2.5 × 10−2 and the other residuals were below 10−6.

### Wall-Mounted Cube in Crossflow.

Figure 3 shows a schematic of the wall-mounted cube in crossflow. A cube of side H is attached to the wall and acts as an obstacle for the incoming flow. The Reynolds number based on H and the freestream velocity is $ReH=5000$. The no-slip boundary condition for the velocity field is imposed at the wall and cube surface, and a passive scalar is released from a circular source centered on the top face of the cube. A direct numerical simulation (DNS) and a RANS simulation of this geometry were performed by Rossi et al. [21]. Their DNS was validated against experimental data: the mean and root-mean-squared of velocity were compared to hot-wire anemometer data, and the scalar concentration was compared to gas chromatograph data, and they found good agreement. More information about the configuration and the simulations can be found in Ref. [21].

## Machine Learning Approach

The proposed framework for RANS modeling consists of using a supervised learning algorithm to infer the turbulent diffusivity field αt. Supervised learning is a subset of machine learning algorithms in which the objective is to predict the value of a parameter given some information about the problem (the features). The model has to be calibrated with examples in which the correct values for both the target parameter and the features are known. These are called the training data. In the present work, the parameter that the algorithm tries to predict is the turbulent diffusivity at each cell in the computational domain and the features are RANS variables at that cell.

### Features.

The features used to predict a cell's turbulent diffusivity are exclusively local flow quantities that are calculated in a $k−ε$ RANS simulation. This is because the objective of the present approach is to improve RANS predictions in configurations for which high-fidelity data are not available. The features chosen are quantities upon which the turbulent diffusivity might depend: the RANS mean velocity gradient ($∇u¯$), the RANS mean scalar gradient as calculated by the fixed $Prt=0.85$ model ($∇θ¯$), the distance to the nearest wall (d), and the eddy viscosity calculated by the $k−ε$ model (νt). Note that if the fixed Prt model were sufficient, the only RANS variable upon which αt would depend is νt. By adding other RANS variables, the algorithm is given more information about the local velocity and concentration fields and then “learns” how to use it.

The specific features mentioned in the previous paragraph were chosen as a generalization of existing higher order models (such as the generalized gradient diffusion hypothesis and higher order generalized gradient diffusion hypothesis), since these calculate the turbulent scalar flux based on the entries of $∇u¯$ and $∇θ¯$. The distance to the wall d was added because it is expected that the presence of a surface can significantly affect the turbulent transport in ways that might not be captured by the mean gradients. The random forest (RF) algorithm used in this work and described in Sec. 3.3 has been shown to be robust to uninformative features [16,22]. This means that if some of the chosen features are not important in predicting the outcome, the performance is not significantly impacted. In the present paper, no in-depth investigation was performed to determine the effect of using different feature sets. Such investigation could be an interesting topic for future work.

The raw RANS variables mentioned before need to be processed before they are used in the machine learning framework to ensure the model is dimensionally consistent and produces a Galilean invariant turbulent diffusivity. To guarantee the former, the features are nondimensionalized by the following local scales: $∇u¯$ is divided by $(k/ε)$, as suggested by Pope [23], $∇θ¯$ is divided by $(k3/2/ε)$, d is divided by $(ν/k)$, which creates a turbulence Reynolds number, and νt is divided by ν. k and ε are the kinetic energy and dissipation values predicted by the RANS equations, and ν is the fluid's kinematic viscosity. The resulting turbulent diffusivity αt is nondimensionalized by the global scale UD, where U and D are the relevant velocity and length scales of the problem. Other possible nondimensional forms were tried, but the ones reported yielded the best overall results.

Since the output of the machine learning algorithm should be a Galilean invariant scalar, Ling et al. [24] suggest that this property be enforced by preprocessing the features and extracting an invariant basis. The idea behind this is to add physical meaning to the predictions: if the same flow were studied under a different coordinate system, the raw RANS variables would change, but the predicted αt should not. Therefore, a polynomial basis that is invariant to coordinate transformations is extracted from the dimensionless velocity gradient tensor and the dimensionless scalar gradient vector as described in Smith [25]. This set of invariant scalars is then used together with the dimensionless forms of d and νt as the features fed into the supervised learning algorithm, for a total of 19 features. If S and R are the symmetric and antisymmetric components of $∇u¯$, and w is a vector (in this case, $w=∇θ¯$), then the features are shown in Table 2. More information on how they are obtained can be found in Ref. [24].

### Extracting the LES Turbulent Diffusivity.

To calibrate the model, training data in which the turbulent diffusivity is known are needed. High-fidelity, time-resolved simulations are used for this purpose. From the mean concentration gradient and turbulent transport term, which are directly available in a DNS or LES, the turbulent diffusivity can be inferred by assuming that the GDH of Eq. (1) holds, according to the following equation:
$αt,LES=−u′iθ′¯∂θ¯∂xi∂θ¯∂xj∂θ¯∂xj$
(2)

where $αt,LES$ is the turbulent diffusivity inferred from the high-fidelity simulation (either DNS or LES) and the terms on the right-hand side are the statistics calculated from the high-fidelity simulation at that cell. Equation (2) assumes that there is an isotropic turbulent diffusivity that relates the mean gradient and the turbulent transport. This assumption is not necessarily valid and would still cause model form error. However, Ling et al. [12] showed that using $αt,LES$ in a RANS simulation dramatically increased the accuracy of the results. So, $αt,LES$ as calculated by Eq. (2) is considered to be the correct value that is used to train the machine learning model.

### Algorithm.

There are several supervised learning algorithms that can be applied to the problem of predicting a continuous variable such as the turbulent diffusivity. These include support vector regressors, decision trees, and neural networks. Bishop [26] provides a good introduction on the subject. Ling and Templeton [16] tested different algorithms applied to turbulence modeling and concluded that RF had the “best combination of good performance and easy implementation.” Random forests are capable of nonlinear decision boundaries and are robust against overfitting and against the inclusion of nonimportant features. Also, preprocessing to force the data to fall within a specified range is not required, as it is for some other algorithms. Finally, RFs are relatively inexpensive to train when compared with neural networks [16]. Therefore, to demonstrate the framework proposed in the present work, the random forest algorithm is employed.

Random forests consist of an ensemble of binary decision trees [27]. A diagram with an example binary decision tree is shown in Fig. 4. It consists of a collection of rules (a feature with an associated threshold) that are used to classify an example. When a prediction is needed for a set of features, the decision tree follows its rules starting from the top (the root) until it reaches an endpoint (the leaves), where a value is assigned. To construct a binary decision tree from a set of training examples, typically a greedy algorithm is used to assign rules that maximize the local information gain sequentially to each node, starting from the root.

In a random forest, each decision tree is constructed by taking a random sample, with replacement, of all the training examples (a process called “bagging”). In the training stage, each tree used all available features to recursively determine the optimal splitting thresholds for the branches based on minimizing the variance across child nodes. The overall prediction of the RF is an average of the predictions of each individual tree. A single tree can be prone to overfitting, but constructing an ensemble is done to minimize this problem while retaining good accuracy. More information on random forests can be found in Ref. [22].

The main hyperparameter that must be chosen is the number of trees in the ensemble, Ntrees. Usually, a larger ensemble of trees provides more accuracy and lower variance between realizations. However, there are diminishing returns as the size of the forest grows. Since the computational cost for training and making predictions scales linearly with Ntrees, there is a trade-off between computational cost and accuracy. A convergence study was performed to determine a good number of trees for this particular application. For different values of Ntrees, a RF was trained on the skewed and cube data sets and used to predict the turbulent diffusivity in the baseline case (see more details in Sec. 4). Since each realization of the algorithm is different (because decision trees are built with a random subset of the data), this process was repeated 30 times for each value of Ntrees, and an error metric was calculated according to Eq. (3). Figure 5 shows the mean and standard deviation of this metric across the 30 trials. Based on these results, a value of $Ntrees=1000$ was chosen because it provides high accuracy, while the very low standard deviation suggests that different instances will predict very similar outcomes. In the remainder of the paper, $Ntrees=1000$ is used.

## Results

To demonstrate the proposed machine learning approach, the skewed and the cube data sets are used for training the RF, which in turn is tested on the baseline data set. The machine learning routines were coded in Python using the Scikit-learn open source library [28]. The high-fidelity data sets and the extracted turbulent diffusivity are linearly interpolated onto the respective RANS meshes. The features are calculated and the cells of the skewed and cube data sets are simultaneously used to calibrate the random forest model. Not all points are used: only the cells where the two-norm of the dimensionless mean scalar gradient from both the DNS/LES and RANS is larger than $10−5$ are used for training and testing. This is because in regions of low gradient, Eq. (2) could yield near singular results and the GDH would predict negligible scalar transport anyway. Also, two small regions in the LES of the skewed case (one close to injection and one near the outlet) had poorly converged values of $u′iθ′¯$, which caused the extracted $αt,LES$ to be orders of magnitude higher than what was observed in the rest of the flow. These two regions were removed from the training set to eliminate the error introduced by these noisy labels. Overall, around 250 k cells in the cube case and 400 k cells in the skewed case are used for training, and the algorithm is tested on around 475 k cells of the baseline case.

Running on a desktop computer, with one core and 32GB of RAM, it took 6.7 h to train a RF with $Ntrees=1000$, K = 19 features, and $M=650k$ training examples. However, this is a fixed cost because the model needs to be calibrated just once and then it can be applied across different simulations. The prediction step is considerably cheaper: predicting the diffusivity in 475 k cells, once the algorithm was trained, took around 1 min on the same desktop computer. In the current framework, this step needs to be performed just once per RANS simulation, so the computational cost of prediction is negligible. Louppe [22] shows that in the average case, training a random forest has complexity $O(Ntrees·K·M· log2M)$, while testing has a complexity $O(Ntrees· log M)$ per prediction. Here, log indicates a base-2 logarithm, K is the number of features, and M is the number of training cells that were used to build the forest. These scalings allow one to estimate the computational cost of this approach in a different scenario. Also, it is important to note that the random forest algorithm is embarrassingly parallel, since it consists of an ensemble algorithm.

### Turbulent Diffusivity Prediction.

Figure 6 shows contours of the nondimensionalized turbulent diffusivity in the baseline inclined jet in crossflow test set, including the prediction from the machine learning algorithm after it was trained on the skewed and cube cases. In Fig. 6(a), the turbulent diffusivity extracted from the baseline LES according to Eq. (2) is shown. This is the field that the machine learning algorithm is trying to replicate. One interesting feature is that it contains areas where $αt,LES$ is negative, which implies that these are regions of counter gradient transport. This behavior hints that underlying assumptions of the GDH, such as turbulent length scales being smaller than the length scales over which the mean temperature changes [29], are not valid in those regions.

Figure 6(b) shows the turbulent diffusivity field that typical RANS codes would use, given by $αt,RANS=(Cμk2/ε)/Prt$, with $Cμ=0.09$ and $Prt=0.85$. Comparing this field to Fig. 6(a) helps explain the problems that such codes have in solving for $θ¯$. It is clear that in the region close to injection (especially for $x/D<2$) and in a layer underneath the jet (y/D around 0.5), the fixed Prt model greatly overestimates the turbulent diffusivity, resulting in inaccurate scalar fields predictions.

Figure 6(c) presents the turbulent diffusivity field that was calculated by the random forest, $αt,ML$. Note that there are still differences between this field and the one shown in Fig. 6(a). For example, the RF fails to predict the regions of negative diffusivity. However, it shows significant improvement over the RANS diffusivity field close to injection and around the jet, regions where the fixed Prt model was particularly inappropriate.

For quantitative comparison between the fields in Figs. 6(b) and 6(c), one can define an integral error metric. A possible choice is the sum of the error magnitudes in each cell normalized by the sum of the magnitudes of the LES diffusivity, as shown in the following equation:
$errorα=∑|αt−αt,LES|∑|αt,LES|$
(3)

For each turbulent diffusivity field ($αt,RANS$ and $αt,ML$), the error can be calculated by Eq. (3). The sum is evaluated for all cells in the RANS mesh of the baseline case where the two-norm of the dimensionless mean scalar gradient from the LES and RANS is larger than $10−5$. By this metric, the RANS diffusivity has an error of 1.21, while the machine learning diffusivity achieves 0.59, an improvement of over 50%. The error is still relatively high, but since the improvement is concentrated in regions of the flow with high gradients, such as near injection and close to the wall, using $αt,ML$ to solve the scalar equation could conceivably generate significant improvements.

### Forward Propagation.

The current machine learning framework is designed to predict a turbulent diffusivity field. However, the ultimate goal is to improve the predictions of the mean temperature field $θ¯$. Therefore, the best way to assess the performance of the RF prediction shown in Fig. 6(c) is to use that field in a RANS code and compare its temperature prediction against the LES results. This is the forward propagation step.

Since the passive scalar assumption is used, the continuity and momentum equations do not need to be solved again; it is enough to take their results as a prescribed mean velocity field to be used in the scalar equation. Assuming the mean fields are steady, using the GDH of Eq. (1) to close the turbulent transport term, and neglecting molecular diffusion produces the Reynolds-averaged advection–diffusion (RAAD) equation shown in the following:
$∂∂xi(ui¯θ¯)=∂∂xi(αt∂θ¯∂xi)$
(4)

Note that with a known mean velocity field and turbulent diffusivity, Eq. (4) can be directly solved for the dimensionless mean temperature field. To test the potential of the machine learning approach, Eq. (4) is solved in the baseline geometry using the RANS velocity field and each of the three turbulent diffusivity fields shown in Fig. 6. The RANS velocity does not exactly match the true velocity field from the LES [3]. But, in an arbitrary geometry in which RANS predictions are desired, a high-fidelity velocity field typically would not be available and the RANS velocity field might be the only option. Thus, to simulate the predictive capabilities in this scenario, the RANS velocity field is chosen. The diffusivity fields were marginally preprocessed before being used in the equation. The LES and ML diffusivities were made non-negative to guarantee convergence and a moderate positive value ($αt/(UD)=0.01$) was assigned for the blanked region of Fig. 6 in all three cases. The latter is done because the LES diffusivity field was not extracted and the machine learning algorithm was not applied in regions of low mean temperature gradient. Note that the value of αt in those regions is not important because the GDH applied there would predict negligible turbulent transport regardless.

Equation (4) is solved using an in-house code with a uniform and structured mesh. It uses a finite volume method with second-order central differencing for the diffusion term and first-order upwinding for the convective term. The resolutions in the streamwise and spanwise direction are similar to the ones in the RANS simulation of Coletti et al. [3] ($dx/D=dz/D=0.1$), and the wall-normal resolution was picked to match theirs at $y/D=1$ ($dy/D=0.05$). This results in the first cell above the wall being at approximately $y+=10$. Despite its simple numerical methods and coarse mesh, this solver was picked because of the ease of prescribing a velocity and turbulent diffusivity fields. The grid is still sufficient to resolve the main features of the mean velocity and temperature, so the results are useful to determine whether the machine learning algorithm can improve the temperature predictions.

The method described earlier is used to solve the RAAD equation and the results are shown in Fig. 7. Figure 7(a) shows the LES scalar field from Bodart et al. [19]. This is assumed to be the correct mean temperature field and is used to assess the other three calculations. Figures 7(b)7(d) present results of the RAAD equation solved with the three different turbulent diffusivity fields.

Comparing the LES field of Fig. 7(a) with the field calculated using $αt,RANS$ in Fig. 7(b) demonstrates the difficulties that fixed Prt models have in film cooling geometries. In reality, at BR = 1, the jet separates after injection and the temperature field stays concentrated within it, in a characteristic kidney shape observed in the right frame of Fig. 7(a). This causes a low adiabatic effectiveness at around $x/D=2$. However, the diffusivity obtained from $Prt=0.85$ is too high close to injection and between the jet and the wall, as observed in Fig. 6(b). This diffuses the temperature toward the wall too quickly, which causes the simulation to overestimate the adiabatic effectiveness and fail to predict the kidney shape at $x/D=2$.

Figure 7(c) is the result of using the turbulent diffusivity field extracted directly from the high-fidelity LES of the baseline geometry. Here, using the wrong turbulent diffusivity field is eliminated as a source of error; the discrepancies between Figs. 7(a) and 7(c) are caused from model form error, from using the RANS velocity field, and from numerical discretization. As expected, it yields improved results when compared to the field solved for using the RANS diffusivity field. The adiabatic effectiveness, for example, is qualitatively closer to the correct one: particularly, it captures the jet separation and a region of low effectiveness immediately downstream of injection.

The results of Fig. 7(d) show the temperature field calculated from the turbulent diffusivity field inferred by the machine learning algorithm, $αt,ML$. It shows remarkable improvement over the RANS diffusivity field. The adiabatic effectiveness is considerably more accurate, and the separation region at around $x/D=2$ is well captured. The kidney shape of the temperature field at $x/D=2$ is not as clearly resolved as in Fig. 7(a), but it is much closer than what was inferred from the $Prt=0.85$ calculation. Surprisingly, some of the qualitative features mentioned above are better captured in Fig. 7(d) than in Fig. 7(c). This is probably due to errors in the ML diffusivity field negating part of the other sources of error in the simulation mentioned in the previous paragraph and causing a more appropriate temperature field in certain regions.

The plot of Fig. 8 shows the variation of the spanwise-averaged adiabatic effectiveness along the streamwise direction in the baseline case. This consists of the dimensionless temperature field $θ¯$ evaluated at the first cell above the wall, then averaged from $Z/D=−0.75$ to $Z/D=0.75$. The results obtained from solving Eq. (4) with the typical RANS and ML diffusivity fields are compared to the LES results of Bodart et al. [19]. As it was previously seen in Fig. 7, solving the RAAD equation with the machine learning diffusivity field drastically improves the adiabatic effectiveness prediction over the fixed Prt model used in commercial RANS codes. In the ML line, the low scalar concentration at around $X/D=2$ is captured much better (even though the lowest point is shifted downstream and the recovery was overestimated) and the behavior at $X/D>10$ is also significantly more accurate than in the fixed Prt model.

For quantitative comparison, an integral error metric is defined to assess the difference between each field and the LES field. It consists of an average of the magnitude of the difference between the temperature calculated and the LES temperature at each cell and is shown in the equation below
$errorθ¯=1N∑|θ¯−θ¯LES|$
(5)

where the sum is performed over all cells in a particular region of interest, N is the number of cells in that region of interest, and $θ¯LES$ is the LES mean temperature in that cell.

Table 3 shows the error calculated according to Eq. (5) in three distinct regions of interest of the baseline geometry, with their respective definitions. The errors are not high in absolute terms because the cells are uniformly spaced and the regions of interest include portions, where $θ¯$ is low, so any reasonable RANS closure would get results similar to the LES field. However, the numbers on Table 3 can still be assessed relative to each other. As Fig. 7 suggested, both the LES and ML diffusivities produce improvements over the RANS diffusivity calculated with $Prt=0.85$, with the LES diffusivity producing higher gains than the ML field as expected. In the most critical regions of interest, the improvements are more noticeable and the difference between LES and ML diffusivity fields is relatively smaller. In the injection region, the ML diffusivity produced almost 30% improvement, while in the wall region it produced over 60% improvement over the RANS field. Combined with the qualitative improvements seen in Fig. 7, this shows that machine learning approaches have potential to significantly improve RANS turbulence closures.

## Conclusion

A machine learning approach to improve turbulent mixing models was proposed, with special interest to film cooling geometries. The method consists of using a closure with a simple form for the turbulent heat flux, the gradient diffusion hypothesis, and then using a supervised learning algorithm to better determine the parameter of this model, the turbulent diffusivity αt. This approach was demonstrated using three data sets: a baseline jet in crossflow, a skewed jet in crossflow, and a wall-mounted cube in crossflow. The last two were used to train the machine learning model, while the baseline case was used to test it. Calculations of the temperature field using the machine learning diffusivity showed significant qualitative and quantitative improvements over the usual $Prt=0.85$ closure. In particular, predictions of the temperature at the wall (the adiabatic effectiveness) were remarkably better. While the typical RANS model was shown to drastically overestimate the centerline effectiveness, the data-driven model generated accurate adiabatic effectiveness predictions. These results give hope that such data driven models could enable robust film cooling design optimization driven by predictive simulations.

The proposed framework is general in scope and could also be applied to build data-driven closures in other configurations. The machine learning model can be trained in a few geometries where high-fidelity data for velocity and concentration exist (like the three data sets presented in this paper) and then applied to predict the turbulent diffusivity in a related geometry. A RANS $k−ε$ simulation of a given geometry is needed to produce the features for the supervised learning algorithm, which would then output a predicted turbulent diffusivity field for that geometry. Finally, a second RANS simulation would need to be performed to solve for the temperature field, with the constraint that it should use the turbulent diffusivity field produced by the machine learning algorithm to close the temperature equation.

Future work includes training and testing this turbulent diffusivity model across a broader set of film cooling flows and even other classes of turbulent flows. This poses the question of how closely related flows used as training sets need to be to the target flow to guarantee improved answers. In the present work, attempts were made to train only on the cube and only on the skewed data sets and then test on the baseline case; the former produced worse results than the $Prt=0.85$ closure, while the latter produced temperatures virtually as accurate as the ones presented in Sec. 4. This hints that training data sets must indeed contain similar flow features as the cases where predictions are sought if good results are desired. Also, more complex closures can be used in conjunction with machine learning: the move to an anisotropic diffusivity could produce significant improvement. Finally, a careful study to understand how the machine learning algorithm utilizes the features to make decisions could reveal important physics underlying turbulent mixing, which, in turn, could inspire the development of new physics-based models.

## Acknowledgment

Pedro M. Milani was supported by a Stanford School of Engineering Fellowship. Julia Ling was supported by the Laboratory Directed Research and Development program at Sandia National Laboratories, a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under Contract No. DE-AC04-94AL85000. SAND2016-11832 C.

## Nomenclature

• BR =

blowing ratio in a jet in crossflow configuration

•
• d =

distance to the nearest wall

•
• D =

hole diameter in the jet in crossflow geometries

•
• H =

cube height in the cube in crossflow geometry

•
• k =

turbulent kinetic energy

•
• Prt =

turbulent Prandtl number $νt/αt$

•
• $tr()$ =

the trace of a tensor

•
• ui =

ith component of the velocity

•
• U =

velocity scale (either bulk velocity or freestream velocity)

•
• ν =

fluid kinematic viscosity

•
• αt =

turbulent diffusivity

•
• $αt,LES$ =

diffusivity extracted from the high-fidelity simulation

•
• $αt,ML$ =

diffusivity predicted by the machine learning algorithm

•
• $αt,RANS$ =

diffusivity predicted by the fixed $Prt=0.85$ model

•
• ε =

turbulent dissipation rate

•
• θ =

dimensionless temperature $(T−T∞)/(Tcool−T∞)$

•
• νt =

eddy viscosity calculated by RANS, $Cμk2/ε$

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