Data-driven correlations for thermohydraulic roughness properties

Thermohydraulic turbulent roughness properties, namely the shift of velocity $\Delta U^+$ and temperature $\Delta \Theta^+$, account for roughness effects and their prediction is of importance in engineering applications.
Recent developments center around the application of machine learning techniques, especially neural networks. In preceding work (Yang 2023), rough surfaces are artificially generated given a specific probability density functions and power spectrums. The obtained $93$ high fidelity direct numerical simulations of a fully-developed turbulent channel flow at $Re_{\tau} \approx 800$ and $Pr=0.71$ serve as a database for training, validating and testing of neural network architectures.
The derived data-driven model, however, remains a "black-box" tool with attractive predicting capabilities.
We present a symbolic regression approach used to translate the developed neural network into an empirical correlation using understandable and simplistic statistical roughness parameters (e.g., Skewness $Sk$). %These roughness parameters (e.g., Skewness $Sk$) are computed given the roughness height map.
It employs genetic programming to construct a correlation to approximate the discretized version of the neural network. ... mehr
This version is obtained from the continuous function $f_\mathrm{MLP} : \mathbb{R}^m \rightarrow \mathbb{R}$ covered in the network by a forward pass of $4200$ artificial rough surfaces.
The workflow is implemented using the \texttt{Python} package by Cranmer 2023.

After several evaluations of the symbolic regression algorithm, the final expression to describe the neural network is chosen based on complexity, simplicity, and physical constraints. This leads to the predictive equation $k_\mathrm{r} = k_\mathrm{s}/k_{99} = ES_x (-ES_x + Sk + 2.21) + 0.819$ using a measure of shape and a measure of slope, highlighting their importance for the neural network prediction.
This expression shows superior behavior to other known correlations, as exemplarily visualized in figure 1 comparing the equation to the correlation by Forooghi 2017.
Simulations in terms of the predicted $\hat{k}_\mathrm{r}$ by the correlation for varying $ES$ and $Sk$ are shown in figure 2 as well as selected iso-skewness lines derived from the equation. The color code indicates the relative deviation of the predicted $\hat{k}_\mathrm{r}$ from the true $k_\mathrm{r}$ obtained by the simulation.
A similar procedure is performed for temperature predicting $\Delta \Theta^+$ for the considered dataset. These results will also be discussed at the conference.