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

TL;DR

This work targets the Wilcox $k-ω$ turbulence model's underprediction of turbulent kinetic energy by integrating physics-informed learning with data-driven coefficient calibration. A PINN is used to solve an ODE for the turbulent viscosity in the $k$ equation, enabling a $y/δ$-dependent diffusion correction via $ u_{t, ext{PINN}}$, while two additional terms, $C_{K, ext{NN}}$ and $C_{ ext{ω2,NN}}$, adjust the $ω$ equation. A Neural Network then maps to three coefficients $\sigma_{k, ext{NN}}$, $C_{k, ext{NN}}$, and $C_{ ext{ω2,NN}}$ as functions of nondimensional inputs $rac{ar{|u'v'|}_{tot}}{u_τ^2}$ and $rac{ν_t}{y u_τ}$, enabling consistent turbulence predictions across channel flow, flat-plate boundary layers, and hill flows. The resulting $k-ω$-PINN-NN model achieves excellent velocity and $k$ profiles when validated against DNS across multiple $Re_τ$ and geometries, demonstrating a practical route to physics-constrained machine learning for RANS closures.

Abstract

The Wilcox $k-ω$ 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 modeling of the turbulent diffusion is improved by making the turbulent diffusion constant, $σ_k$, a function of $y/δ$ ($y$ and $δ$ denote wall distance and boundary-layer thickness, respectively). The $k$ equation is turned into an ordinary differential equation for the turbulent viscosity which is solved using Physics Informed Neural Network (PINN). A new coefficient, $C_{K,NN}(y/δ)$, is added to the dissipation term and another, $C_{ω2,NN}(y/δ)$, to the destruction term in the $ω$ equation. Finally, all three coefficients -- $σ_{k,NN},$,$C_{K,NN}$ and $C_{ω2,NN}$ -- are made functions of two new input parameters, $\overline{|u'v'|}_{tot}/u_τ^2$ and $ν_t/(y u_τ)$, in a Neural Network (NN) model. The new turbulence model, called the $k-ω$-PINN-NN model, is shown to produce excellent velocity and turbulent kinetic profiles in channel flow at $Re_τ= 550$, $2\, 000$, $5\, 200$ and $Re_τ= 10\, 000$ as well as in flat-plate boundary layer flow. The $k-ω$-PINN-NN model is also used for predicting the flow over a periodic hill and the agreement with DNS is very good. The Python PINN and NN scripts and the Python CFD codes can be downloaded (Davidson, 2025a).

Figures (18)

• Figure 1: Fully-developed channel flow. $Re_\tau = 5\, 200$. Solid lines: $k-\omega$; dashed lines:DNS moser:15.
• Figure 2: Schematic of Simple NN
• Figure 3: Prediction by PINN compared to DNS and the Wilcox $k-\omega$ model. $Re_\tau = 5\, 200$.
• Figure 4: CFD predictions of Eq. \ref{['ODE-k']} with different $\sigma_{k,max}$.
• Figure 5: $C_{k,PINN}$ and $C_{\omega 2,PINN}$ vs. $y/\delta$.