A few-shot and physically restorable symbolic regression turbulence model based on normalized general effective-viscosity hypothesis

Turbulence is a complex, irregular flow phenomenon ubiquitous in natural processes and engineering applications. The Reynolds-averaged Navier-Stokes (RANS) method, owing to its low computational cost, has become the primary approach for rapid simulation of engineering turbulence problems. However, the inaccuracy of classical turbulence models in predicting complex turbulence remains the main drawback of the RANS framework. With the rapid development of data-driven approaches, many data-driven turbulence models have been proposed, yet they still suffer from issues of generalizability and accuracy. In this work, we propose a few-shot, physically restorable, symbolic regression turbulence model based on the normalized general effective-viscosity hypothesis. Few-shot indicates that our model is trained on limited flow configurations spanning only a narrow subset of turbulent flow physics, yet can still outperform the baseline model in substantially different turbulent flows. Physically restorable means our model can nearly revert to the baseline model in regimes satisfying specific physical conditions, using only the symbolic regression training results. Specifically, we first formalize the concept of few-shot data-driven turbulence models. Second, we train our symbolic regression turbulence models using only time-averaged direct numerical simulation (DNS) data for three-dimensional periodic hill flow slices. Third, we evaluate our models on periodic hill flows, zero pressure gradient flat plate flow, NACA0012 airfoil flows, and NASA Rotor 37 transonic axial compressor flows. One of our symbolic regression turbulence models consistently outperforms the baseline model, and we further demonstrate that this model can nearly revert to baseline behavior in certain flow regimes. This restoration behaviour also provides a physically interpretable perspective on the transferability of the learned model and may help guide future analysis and the selection of symbolic regression turbulence closures.