## Introduction

In the modern gas turbine engines of today, the high-pressure turbine (HPT) blades are exposed to harsh aerothermal environments, with the temperatures of the working fluid in excess of the metal melting point temperature. The need for effective cooling of the metal surface is crucial to maintaining the component life as well as the safe operation of the engine. One such area of cooling that is a challenge is the trailing-edge slot (also known as the pressure-side bleed), as shown in Figure 1. The aerodynamic requirements dictate the trailing-edge be tapered to increase the aerodynamic efficiency, which prevents internal cooling solutions. Instead, the cooling is provided by injecting the coolant fluid out of the slot, forming a film on the metal cutback surface. The interaction of this film with the extraneous working fluid leads to interesting flow dynamics, producing either wall-jet or wall-wake type flows. The flow physics for the trailing-edge slot has received attention both experimentally (Kacker and Whitelaw, 1969, 1971) and numerically (Medic and Durbin, 2005; Joo and Durbin, 2009; Haghiri and Sandberg, 2020) in order to understand how effective the slot fluid is in shielding the metal surface, quantified by a non-dimensional adiabatic wall effectiveness

where *a posteriori* has shown to improve the flow prediction remarkably over the commonly used closures but improvement of the turbulent stress tensor or heat flux vector prediction does not always lead to equally improved mean flow predictions. In that vein, the GEP algorithm was modified by Zhao et al. (2020) to include contributions of the velocity field into the closure development process to build more robust turbulence stress closures for high-pressure and low-pressure turbine wake flow predictions. This was achieved by running RANS calculations within a loop during the algorithm's execution to develop turbulence stress closures that improved the flow prediction instead of just the turbulence stress. The modification of the algorithm in this way was labelled as CFD-driven machine learning (or “looped” training) while the conventional use of the GEP is denoted as data-driven machine-learning (or “frozen” training).

Thus, in the present study the CFD-driven machine learning approach of Zhao et al. (2020) is adapted to develop closures for the turbulent heat-flux, where RANS calculations for the temperature field are incorporated into the training and used as a target for improvement. High-fidelity datasets of three blowing ratios for a given lip thickness geometry from Haghiri and Sandberg (2020) are used to lead the model development and the closures obtained from this looped training are compared with the high-fidelity data, the conventional frozen trained closures in addition to the commonly used gradient diffusion hypothesis closure.

## Numerical setup

### Reference datasets

The highly-resolved large eddy simulation (LES) datasets of a fundamental trailing edge slot, performed by Haghiri and Sandberg (2020), are used in the present study as the reference for sake of model development. The data was produced using an in-house multi-block structured compressible Navier–Stokes solver HiPSTAR (Sandberg, 2015). Figure 2 shows a schematic of the flow configuration. At the inlet boundary, a cold wall jet and a plane hot co-flow with different speeds are specified, separated by a flat plate with thickness *t*. The inlet mean velocity profiles were adapted from the experimental study of Whitelaw and Kacker (1963). In the LES, only mean profiles were prescribed at the inlet boundary and no fluctuations were added. All walls were set to no-slip boundaries and adiabatic temperature conditions were used. The wall adapting local eddy viscosity (WALE) model (Nicoud and Ducros, 1999) was used for modelling of the sub-grid scale stresses. Table 1 summarises the details of the three cases. Interested readers can refer to the original paper of Haghiri and Sandberg (2020) for more details on the flow configuration and numerical setup.

### RANS-based scalar flux closures

Even with the recent rise of computational power, LES is prohibitively restrictive within an industrial design context, ensuring that the dependence on RANS continues. Unfortunately, RANS is known to suffer from two major sources of modelling errors: Reynolds stress and turbulent heat-flux closures. In the present study, the focus is placed only on the latter and it is assumed that the momentum field (velocity) and other inputs to heat-flux models are known accurately (from time-averaged LES in this case). The mean temperature transport equation for an incompressible flow is given (in non-dimensional form) below:

where

In the above, *t* are the spatial coordinate vector and time, respectively. In addition, *T* is the temperature, *Re* is the Reynolds number and *Pr* is the Prandtl number. It should also be noted that “asterisk” and “overbar” represent dimensional and time-averaged variables, respectively. The use of the incompressible framework here is justified due to the low Mach number (<0.3) in the freestream and the jet flows. Furthermore, the buoyancy effects are negligible due to the small temperature differences between the freestream and the jet flows, implying that the energy equation can be decoupled from the momentum equations (Sandberg et al., 2018). As such, the temperature field can be solved passively.

The turbulent heat-flux is commonly modelled using the gradient diffusion hypothesis (GDH), assuming that the heat-flux terms are in the direction of their corresponding maximum mean temperature gradient

where the diffusivity

## The machine-learning framework

To perform the optimisation in the present study, we use gene-expression programming (GEP), which is a symbolic regression algorithm returning mathematical equations (Ferreira, 2001). The biggest advantage of using GEP over other algorithms is the portability and transparency of returning an equation, allowing for inspection of the generated terms and interpretation of the optimisation outputs. In GEP (drawing an analogy with Darwin's survival-of-the-fittest theory), an initial population of candidate solutions is randomly generated at generation step *a priori* and *a posteriori* (Haghiri and Sandberg, 2020). Interested readers can refer to Koza (1994) and Ferreira (2001) for introductory information of the GEP algorithm, whereas the details of using GEP in turbulence modelling was specified in detail in Weatheritt and Sandberg (2016).

### Frozen training process

The data-driven closure for the turbulent heat-flux in the frozen approach is developed by adopting the gradient-diffusion hypothesis by minimising the cost function

which is the mean-squared error of Equation 4 where *V* represents the points within the training region. The training region refers to the spatial extent of the domain from where the data points are picked to undergo the training. Therefore, the target for optimisation is a non-dimensional parameter *non-constant* turbulent Prandtl number with a functional dependence on the velocity and temperature gradients,

where, considering the present statistically two-dimensional flow, the function

The invariants used in this study represent an integrity basis (Smith, 1965) to ensure the coefficient

A box domain training region was chosen for developing this closure, where the wall normal extent ranged from

### Looped training process

The looped training process discussed here is an adaptation of the method described in Zhao et al. (2020), extended to scalar regression. While the cost function in the previous section was obtained by computing the error difference between the reference and model heat-fluxes, the cost function evaluation here is performed by running RANS calculations in the loop to obtain the difference between the reference and the model. A key advantage of running RANS in the loop is the ability to modify the quantity used to evaluate the cost function. Thus, the looped training process can be used to find a turbulent heat-flux closure that improves the temperature field, instead of the turbulent heat-flux as was considered in the frozen training process. An example of the cost function used here is given by

## Results and discussion

### Grid independence

Since the looped training relies on running multiple RANS calculations simulataneously and iteratively, the runtime of these RANS can require large computational resources and dramatically increase the model development time. Thus, it is crucial to have as cheap a RANS setup as possible for the training. This is attained by coarsening the grid resolution without compromising the solution accuracy. A grid independence test is performed on the frozen GEP trained case (referred to as “Frozen”), i.e. with

Figure 4 shows the adiabatic wall effectiveness

### Effect of cost function

With the availability of RANS calculations within the looped training process, a clear advantage over the frozen training is the ability to choose any quantity as a cost function to develop the closure for the heat-flux. Using the BR = 1.26 case as the reference for model development, two cost functions are tested, the temperature distribution along the wall, i.e.

It is evident from the figure that looped training leads to an improvement over frozen training using both cost functions. The use of the streamwise temperature profiles as a cost function improves the temperature prediction everywhere except for points near the wall. The wall temperature cost function performs better for points near the wall, leading to an excellent match with the

### Generalisability

Based on the excellent improvements obtained with the wall temperature closure (Equation 11) for the BR = 1.26 case, its potential is also tested on separate flow conditions to evaluate its generalisability. Blowing ratios of 0.86 and 1.07 are run with this closure and the

There is a slight overprediction in

The results show that the

### Multidata training

The ability of running RANS in the loop to develop a closure leads to an interesting possibility, i.e. developing closures for multiple datasets at once. A short run with 100 generations using 6 RANS per generation, with 2 RANS per BR and a long run with 400 generations using 36 RANS per generation, with 12 RANS per BR were performed with the resulting closures given by:

where the superscriptsNevertheless, the results with the looped training show that the closures developed do perform better than their counterparts from frozen training, are shorter in their mathematical extent and generalise well across other flow conditions.

## Conclusions

In this study, a novel symbolic scalar regression machine-learning algorithm was applied to improve the prediction of the wall-temperature in a fundamental trailing-edge slot. The prediction of the temperature is improved by developing a scalar mathematical expression to represent a non-constant turbulent Prandtl number within the gradient diffusion hypothesis for the turbulent heat-flux. The conventional use of the machine-learning algorithm, also known as the “frozen” training process, obtained this expression by minimising the error between the true turbulent heat-flux from high-fidelity data and the model expressions generated by gene-expression programming (GEP). In the novel approach, i.e. the “looped” training process, the scalar expression is derived by conducting RANS calculations in a loop within GEP. This leads to promoting the expressions which minimise the error between the true temperature field from the high-fidelity data and the RANS solution. Thus, the closure for the turbulent heat-flux which improves the temperature field prediction can be obtained instead of the closure that improves the turbulent heat-flux prediction. This approach also allows for the use of any type of cost function and for the largest blowing ratio of 1.26, two such functions were tested, with the one based on the wall-temperature value leading to the best prediction. The resulting closure from the looped approach was also found to be mathematically succinct compared with its frozen counterpart and improved upon the prediction of the wall-temperature significantly further. The closure showed that the effective turbulent Prandtl number was required to be even smaller (∼0.32) than the frozen closure's value of 0.42 to realise improvements and generalised well upon application to the other blowing ratios. Finally, the looped approach was also demonstrated to build closures by combining datasets from multiple blowing ratios, which led to further improvements in terms of the generalisation.