A Machine Learning Augmented RANS Approach for Rapid and Accurate Simulations of Periodic Unsteady Flows

Periodic unsteady flows are common in many scientific and engineering problems, such as vortex shedding and propeller wakes. The Reynolds-Averaged Navier–Stokes (RANS) approach is widely used in practical design problems due to its computational efficiency. However, the unsteady RANS (URANS) for periodic flow simulations remains costly, as it requires small time steps and long simulations spanning many flow periods. In many cases, only time-averaged quantities are of interest, making time-resolved URANS inefficient. To address this challenge, we propose a machine-learning-based method that enables rapid and accurate simulations of periodic unsteady flows at a cost comparable to steady-state RANS simulations. Drawing inspiration from Reynolds stress modeling, we represent the effect of large-scale, periodic flow variations as an additional unsteady stress term in the momentum equation. We then solve an inverse problem to find the optimal values of the unsteady stress term to minimize a composite objective function, which consists of: (1) the discrepancy between the augmented steady-state RANS solution and the time-averaged URANS reference, and (2) the residuals of the augmented RANS equations. To generalize the unsteady stress model, we will perform the inversion for various geometries and flow conditions, then train a neural network model to map local flow features to the optimal unsteady stress. We will demonstrate the proposed method using the vortex shedding problem over a cylinder.