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In this work, we propose using an ensemble Kalman method to learn a nonlineareddy viscosity model, represented as a tensor basis neural network, fromvelocity data. Data-driven turbulence models have emerged as a promisingalternative to traditional models for providing closure mapping from the meanvelocities to Reynolds stresses. Most data-driven models in this category needfull-field Reynolds stress data for training, which not only places stringentdemand on the data generation but also makes the trained model ill-conditionedand lacks robustness. This difficulty can be alleviated by incorporating theReynolds-averaged Navier-Stokes (RANS) solver in the training process. However,this would necessitate developing adjoint solvers of the RANS model, whichrequires extra effort in code development and maintenance. Given thisdifficulty, we present an ensemble Kalman method with an adaptive step size totrain a neural network-based turbulence model by using indirect observationdata. To our knowledge, this is the first such attempt in turbulence modelling.The ensemble method is first verified on the flow in a square duct, where itcorrectly learns the underlying turbulence models from velocity data. Then, thegeneralizability of the learned model is evaluated on a family of separatedflows over periodic hills. It is demonstrated that the turbulence model learnedin one flow can predict flows in similar configurations with varying slopes.

Ensemble Kalman method for learning turbulence models from indirect observation data