Joint Learning of Reynolds-Stress and Heat-Flux Closures for Jet Flows

Reynolds-averaged Navier–Stokes methods are widely used for predicting jet flows in industrial applications. However, their accuracy is often limited by turbulence closures for the Reynolds stress and heat flux. In this work, we introduce the ensemble Kalman method to jointly learn optimal Reynolds-stress and heat-flux closures from indirect sparse data of mean velocity and temperature. The Reynolds-stress closure is represented using the tensor-basis neural network (TBNN) augmented with an additional correction term to capture vortex stretching effects, and the heat-flux closure is represented using the TBNN for scalar flux. The ensemble-based joint learning strategy captures the physical coupling between velocity and temperature fields and is more effective than the conventional approach that sequentially learns the Reynolds-stress and heat-flux closures. The method is used to learn turbulence closure models using centerline velocity and temperature data in a subsonic heated jet case. The learned model is then generalized to subsonic unheated and near-sonic heated jets, showing its generalization capability across different Mach numbers and temperature ratios. Furthermore, the model is tested on multiple geometric configurations, including twin round jets and jets from triangular and square nozzles, demonstrating its satisfactory generalization capability to different nozzle shapes unseen during training within the free-jet regime.