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

Advancements in deep learning (DL) have shown great promise in accelerating and enhancing the accuracy of flow physics simulations. However, these models are often constrained by the availability of high-fidelity training data, which is computationally expensive to generate across a representative range of physical conditions. These limitations significantly restrict the model's predictive performance beyond the physics represented by the training dataset's scope.

An online optimisation method for DL closure of Reynolds-Averaged Navier-Stokes (RANS) models which seeks to address the challenge of limited high-fidelity datasets is presented. Training data is dynamically generated by direct numerical simulation (DNS) of the governing equations over a subset of the domain, and single and two-equation DL-RANS models are optimised online, which in turn provide the boundary conditions for the DNS. The DNS conversely provides the target mean velocity and turbulence statistics profiles, improving the predictive capability of the RANS model. Through this coupled online training, the unresolved dynamics are appropriately captured while avoiding overfitting, allowing the reduced-order model to effectively generalize over the remainder of the flow.

The efficacy of the approach is demonstrated on canonical turbulent channel flow, and it is shown that both relatively small DNS domains and inflow lengths are needed for improvements to the low-order modelling and for the reproduction of full periodic domain flow statistics. We present results for the channel flow at $Re_\tau=180, 395, 590$ with varying domain lengths. Accuracy of the online-optimized RANS models is compared against offline-optimized models and models with reference parameters from the literature, demonstrating significantly improved performance, particularly for out-of-sample predictions.