oRANS: Online optimisation of RANS machine learning models with embedded DNS data generation

TL;DR

This framework provides a scalable route to physics-informed machine learning closures, enabling data-adaptive reduced-order models that generalise across flow regimes without requiring large precomputed training datasets.

Abstract

Deep learning (DL) has demonstrated promise for accelerating and enhancing the accuracy of flow physics simulations, but progress is constrained by the scarcity of high-fidelity training data, which is costly to generate and inherently limited to a small set of flow conditions. Consequently, closures trained in the conventional offline paradigm tend to overfit and fail to generalise to new regimes. We introduce an online optimisation framework for DL-based Reynolds-averaged Navier--Stokes (RANS) closures which seeks to address the challenge of limited high-fidelity datasets. Training data is dynamically generated by embedding a direct numerical simulation (DNS) within a subdomain of the RANS domain. The RANS solution supplies boundary conditions to the DNS, while the DNS provides mean velocity and turbulence statistics that are used to update a DL closure model during the simulation. This feedback loop enables the closure to adapt to the embedded DNS target flow, avoiding reliance on precomputed datasets and improving out-of-distribution performance. The approach is demonstrated for the stochastically forced Burgers equation and for turbulent channel flow at $Re_τ=180$, $270$, $395$ and $590$ with varying embedded domain lengths $1\leq L_0/L\leq 8$. Online-optimised RANS models significantly outperform both offline-trained and literature-calibrated closures, with accurate training achieved using modest DNS subdomains. Performance degrades primarily when boundary-condition contamination dominates or when domains are too short to capture low-wavenumber modes. This framework provides a scalable route to physics-informed machine learning closures, enabling data-adaptive reduced-order models that generalise across flow regimes without requiring large precomputed training datasets.

Figures (14)

• Figure 1: Schematic of the coupled RANS-DNS framework. The primary RANS-ML domain provides boundary conditions and forcing (mean velocity, turbulent kinetic energy, Reynolds stresses) to the embedded DNS/LES region, which in turn supplies high-fidelity data to update the RANS closure and boundary conditions. This coupling enables consistent training of turbulence models on statistically representative flow fields.
• Figure 2: Solutions to the stochastically forced Burgers equation with embedded domains of size (left) $L_e/L_0=1/4$ and (right) $L_e/L_0=1/8$. The panels compare the true means $\bar{u}_\textrm{ref}$,$k_\textrm{ref}$ with online-trained model predictions (in-sample and out-of-sample), and no-model baseline. The results show that the trained model reproduces the true statistics more accurately than the no-model baseline, including in out-of-sample settings.
• Figure 3: Learned turbulence closure functions for stochastically forced Burgers turbulence. The oRANS-ML trained mixing length $\ell_m^{\mathrm{NN}}$ (black) and turbulence model coefficients $C_\mu^{\mathrm{NN}}$ (red) and $C_D^{\mathrm{NN}}$ (blue) are shown, with standard constants for Navier--Stokes turbulence (Pope, 2000) indicated by dotted lines ($C_\mu=0.55$, $C_D=C_\mu^3$). The learned parameters deviate from the classical values and vary spatially, reflecting adaptation to the flow.
• Figure 4: Relative adjoint gradient error as a function of the finite-difference perturbation size $\epsilon$. For small $\epsilon$, floating-point round-off error dominates, while for large $\epsilon$, finite-difference truncation error dominates. This characteristic curve, with low minimum error, confirms the correctness of the adjoint implementation.
• Figure 5: Comparison of RANS closure models at $Re_\tau=180$. Top row: mean velocity $u^+$ (left) and turbulent kinetic energy $k^+$ (right). Bottom row: specific dissipation rate $\omega^+$ (left) and objective function $J$ decay during training (right). Results are shown for DNS, the default $k$--$\omega$ model, a parametric closure (equation \ref{['eq:parametric']}), and both local (equation \ref{['eq:local']}) and global (equation \ref{['eq:global']}) feature neural-network closures. The neural closures closely reproduce DNS statistics and converge effectively, in contrast to the default and parametric baselines; note that $J$ is optimised only over $u$ and $k$, and $\omega^+$ is included only for reference.