Deep Learning-based Algebraic Reynolds Stress Closures for RANS simulations of Turbulent Flows

Turbulence is ubiquitous in engineering, yet direct simulation is prohibitively expensive. The Reynolds-averaged Navier-Stokes (RANS) equations provide computational savings exceeding ten orders of magnitude but introduce unclosed terms requiring modelling (the closure problem). Existing machine-learning (ML) closures suffer distribution shift when trained offline on high-fidelity data, and ML models that bypass the governing equations often have limited capability to generalise. We develop a physics-derived deep learning closure for RANS, the Deep Algebraic Reynolds Stress Model (DARSM), which can be trained on one or two flow cases and generalise across an order of magnitude in Reynolds number, to unseen geometries, and across flow regimes. A neural network maps flow invariants to empirical parameters in an implicit algebraic Reynolds stress equation, derived from the Reynolds stress transport equations under the weak-equilibrium assumption, imposing significant physics-based structure on the ML closure. End-to-end optimisation through the governing PDEs and the coupled implicit closure eliminates distribution shift, but both unrolled and implicit automatic differentiation fail on the stiff coupled solver. We derive adjoint equations that exploit the solver's implicit-explicit structure to enable computationally efficient optimisation. On the canonical square-duct and periodic-hill benchmarks, DARSM reduces test velocity error as compared to the baseline RANS by $2$-$4\times$ across Reynolds number, geometries, and flow regimes, with peak case-level reductions of $12\times$. The model trained on attached, anisotropy-dominated flows (square duct) accurately generalises without retraining to separated flows (periodic hills), a regime change in the underlying turbulence physics. DARSM also outperforms five established ML methods: offline training, tensor-basis neural networks, field-inversion machine learning, DeepONets, and physics-informed neural networks.