Adjoint inversion-informed machine learning for optimizing turbulent Prandtl number in supercritical flows: A case study of hydrogen fuel flows under cross conditions

Developing accurate turbulence models for supercritical flows is challenging due to the flawed assumption that the turbulent Prandtl number (Pr _t) is constant and typically set to 0.85. This paper proposes an adjoint-based inversion method to optimize the full-spatial Pr _t by computing temperature gradient differences between RANS and LES data. Firstly, the inversion method, implemented in the SU2 solver, calculated temperature discrepancies across 10⁸ grid points. Secondly, the principal flow features were identified using Pearson correlation analysis as inputs of a new hybrid neural network based on radial basis function (RBF) and multi-layer perceptron (MLP) methods, namely RBF-MLP. This network was designed to achieve precise nonlinear approximation to align RANS results from the inverted Pr _t model with LES data. Finally, the hydrocarbon fuel flows were used to train and test the new Pr _t model across various channels and flow directions. The constant- Pr _t model showed significant deviations from LES data, particularly near walls, while the adjoint-based Pr _t model closely matched LES trends. In rectangular channels, the adjoint-based model had a maximum error of 2% under 0.189 MW/m² and 5% under 0.341 MW/m². In circular channels, prediction errors rose with heat flux, but the adjoint-based Pr _t model's maximum error was only 4%, compared to 20% of the conventional model. In vertical channels, the adjoint-based Pr _t model maintained an error within 4.2%, while the constant- Pr _t model had a 14.29% error. Across all test cases, the adjoint-based model demonstrated superior accuracy, reducing the mean absolute relative error (e _R) to 1.1–12.5% and the root mean-square relative error (e _S) to 1.3–16.4%, compared to the conventional model's e _R=5.8–68.7% and e _S=8.3–68.7%. In summary, this adjoint-based inversion offers significant advantages in developing turbulence models by capturing nonlinear, complex relationships from large datasets. It also extends universality to diverse flow cases, and supports reliable turbulence modeling for supercritical flows.