Using physics informed neural network (PINN) and neural network (NN) to improve a k-ω turbulence model

Journal article, 2026

The Wilcox $ k-\omega $ k-omega turbulence model predicts turbulent boundary layers well, both fully-developed channel flows and flat-plate boundary layers. However, it predicts too low a turbulent kinetic energy. This is a feature it shares with most other two-equation turbulence models. When comparing the terms in the k equations with DNS data it is found that the production and dissipation terms are well predicted but the turbulent diffusion is not. In the present work the poor modelling of the turbulent diffusion is improved using Physics Informed Neural Network (PINN) and Neural Network (NN). The k equation is turned into an ordinary differential equation for the turbulent viscosity in the k equation, $ u _{t,PINN} $ nu t,PINN, which is solved using PINN. A new turbulent Prandtl number is then computed as $ \sigma _{k,PINN} = u _{t}/ u _{t,PINN} $ sigma k,PINN=nu t/nu t,PINN where $ u _t = k/\omega $ nu t=k/omega. Hence, the turbulent Prandtl number, $ \sigma _{k,PINN} $ sigma k,PINN, is determined using PINN, followed by the use of DNS data for estimating $ C_{k,PINN} $ Ck,PINN and $ C_{\omega 2,PINN} $ C omega 2,PINN which appear in the destruction terms in the k and omega equation, respectively. Neural networks are then used to generalise these results and thus construct a turbulence model. All Python PINN, NN and pySR scripts as well as the Python CFD code can be downloaded [Davidson. Using physical informed neural network (PINN) and neural network (NN) to improve a $ k-\omega $ k-omega turbulence model: python CFD code and PINN script. In: Division of fluid dynamics. Gothenburg: Dept. of Mechanics and Maritime Sciences, Chalmers University of Technology; 2025].