Mathematics

This work aims to improve RANS simulations using data-assimilation (DA) and machine-learning (ML) methods. The focus is on compressible and supersonic flow configurations, which are known to be challenging cases for traditional RANS models. In light of this, we present a comprehensive approach to solve inverse problems related to the compressible Reynolds averaged Navier-Stokes (RANS) equations, discretized using high-order numerical methods. An adjoint-based strategy is employed to minimize a cost functional through optimization of control parameters within a discontinuous Galerkin (DG) framework. The methodology demonstrates effectiveness in correcting discrepancies between low-fidelity and high-fidelity simulations across various flow scenarios, including turbulent and compressible flows in supersonic regimes, with different degrees of DG discretization and different type of measurements. Furthermore, the study investigates the design and impact of input features (IFs) in ML models for turbulence closure, addressing key challenges related to invariance and normalization. The introduction of a novel a posteriori formulation enhances rotational symmetry and conservation properties in the field inversion (FI) step. The influence of IFs has been rigorously examined for both artificial neural networks (ANNs) and tensor basis neural networks (TBNNs). For the latter, a compact tensor basis is proposed, resulting in a substantial reduction in both the number of tensor components and the associated IFs. Results indicate significant improvements in model accuracy and generalization capabilities, especially in complex compressible shear-flow with shock waves. These findings underscore the potential of advanced optimization and ML techniques in enhancing the performance of RANS simulations, paving the way for future research in computational fluid dynamics (CFD).