Using Physics Informed Neural Network (PINN) and Neural Network (NN) to Improve a $k-\omega$ Turbulence Model

Abstract

l 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 modeling 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, nu_{t,PINN}, which is solved using PINN. A new turbulent Prandtl number is then computed as sigma_{k} = nu_{t}/nu_{t,PINN} where nu_t = k/omega.To compensate for the new, larger turbulent kinetic energy, three coefficients in the new k-omega model are computed using three NN models. The new turbulence model, called the k-omega-PINN-NN model, is shown to produce excellent velocity, skin friction and turbulent kinetic profiles in channel flow at Re_tau = 2 000, 5 200 and Re_tau = 10 000 as well as in flat-plate boundary layer flow (slightly too large a k for the latter case). The k-omega-PINN-NN model is also used for predicting the flow over a periodic hill and the agreement with DNS is very good. At the end of the Conclusions, we give an example on how a NN model can be replaced with a Python symbolic regression (pySR); the latter may conveniently be imported in commercial CFD codes. All Python PINN, NN and pySR scripts as well as the Python CFD code can be downloaded (Davidson, 2025a).