Enhancing One-Equation Turbulence Models for Delta Wings by Gene Expression Programming

This paper presents a computational-fluid-dynamics-driven machine learning framework to enhance Reynolds-averaged Navier–Stokes turbulence models for complex delta wing flows at high Reynolds numbers and transonic speeds. Building upon the coupled Spalart–Allmaras model framework, this study represents the first application of an interpretable machine learning framework to such complex three-dimensional vortex-dominated external aerodynamic applications. The key contribution lies in applying gene expression programming to learn optimal anisotropy tensor corrections for delta wing flows, where the Boussinesq approximation is extended with nonlinear stress–strain terms and coupled to a redefined production term in the transport equation for turbulent viscosity. Trained on a single-delta wing configuration, the proposed approach is validated on double- and triple-delta wing geometries under challenging transonic flow conditions, demonstrating significant improvements in the prediction of aerodynamic coefficients and showcasing both accuracy and robustness. The use of gene expression programming enables the automated discovery of models, generating explicit, physically interpretable expressions that provide deeper insight into vortex-dominated flow physics and facilitate generalization across related flow regimes. While the present results are specific to delta wing configurations and remain most accurate near the training conditions, they demonstrate the potential of physics-informed machine learning as a promising direction for advancing turbulence modeling in such complex three-dimensional flow regimes.