A posteriori cross-case parametric learning of the Spalart–Allmaras model for separated flows

The paper presents a computational fluid dynamics driven parametric learning approach for the Spalart–Allmaras Reynolds-averaged Navier–Stokes model, focused on the class of separated flows. A key feature of the approach is a two-stage learning procedure: training on a physically representative problem that encapsulates the key flow mechanisms of the target class (separation at a sharp edge, shear mixing, recirculation, and reattachment), followed by strict cross-validation on independent flows, each isolating specific mechanisms to test the model's generalization ability. Closure coefficients were learned using gradient-free algorithms via an OpenFOAM–Python coupling. The obtained set of coefficients eliminates the qualitative discrepancies of the standard model on the training case (a square cylinder on a wall). Cross-validation on benchmark cases from the turbulence modeling resource database confirms that the learned model preserves the log-law of the wall, correctly reproduces the mixing layer development and the recirculation zone length behind a backward-facing step, and qualitatively captures the stall onset on an airfoil with increasing angle of attack. Physical analysis shows that the identified parameters restore the balance between the turbulent shear layer and the recirculation zone. The obtained parameter values lie within the “family of acceptable solutions” originally defined by Spalart and Allmaras during calibration on free shear flows, confirming the physical consistency of the results. The developed approach can be extended to other closure models and flow classes.