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

TL;DR

The paper addresses the challenge of limited-data turbulence modeling in RANS by introducing a few-shot, physically restorable symbolic regression approach based on a normalized general effective-viscosity hypothesis. It formulates a normalized tensor-basis representation, defines invariants and extra features, and trains PySR to predict normalized tensor coefficients while fixing the first term to preserve baseline behavior; the workflow is integrated into a CFD framework (TCAE) using DNS-derived inputs. The SR5T model, trained on 3D periodic hill DNS slices, generalizes to diverse flows including ZPG plates, NACA0012, and NASA Rotor 37, often outperforming the baseline $k$-$\omega$-SST and the SR3T variant, with key improvements arising in near-wall and high-shear regions and the ability to revert to baseline behavior in physically regimes. The work demonstrates a promising path for data-efficient turbulence modeling with explicit physical interpretability and restoration capabilities, potentially enabling more reliable extrapolation to engineering-scale flows.

Abstract

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 constitutes 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. The normalized general effective-viscosity hypothesis was proposed in our previous study. Specifically, we first formalize the concept of few-shot data-driven turbulence models. Second, we train our symbolic regression turbulence models using only 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.

Figures (20)

• Figure 1: The core algorithmic workflow of PySR. Within each population, expressions evolve continuously through mutation, crossover, simplification, and constant optimization. Concurrently, individuals migrate across different populations.
• Figure 2: The framework of our symbolic regression-based turbulence model.
• Figure 3: Parameterized period hill geometries with different $\alpha$.
• Figure 4: Contour plots of the normalized tensor basis coefficients from prior results.
• Figure 5: Streamlines from various simulation results of the test sets. The contours represent $U_x/U_b$.