Mean flow reconstruction of turbulent flows from sparse measurements using physics-informed neural networks

Optimal aerodynamic design is guided by numerical simulation and experimental methods. For flows of engineering interest, numerical simulation is characterised by a trade-off between accuracy and tractability, whilst experimental methods are restricted by the inability to measure all spatio-temporal scales associated with turbulent dynamics. In this thesis, data assimilation (DA) methods, specifically Physics-Informed Neural Networks (PINNs), are explored, to augment low-fidelity numerical approaches (Reynolds-Averaged Navier-Stokes (RANS) equations) with sparse high-fidelity experimental data to accurately reconstruct turbulent flows. Firstly, PINNs are benchmarked for mean-flow reconstruction using sparse mean velocity measurements against variational-DA. The PINN-DA-SA outperforms adjoint-based variational-DA, attributed to discretisation and regularisation errors in the variational formulation. Introducing an empirical closure model (Spalart-Allmaras) and corrective forcing with turbulence-model augmented PINNs, increase reconstruction accuracy in high velocity gradient and separation regions. Secondly, a hierarchical PINN approach is proposed for constraining the flow physics with equations that systematically interpolate between direct numerical simulation of the Navier-Stokes (high-fidelity; resolved spectrum) and RANS (low-fidelity; modelled spectrum). We introduce PINNs constrained by the Navier-Stokes expressed in the frequency domain (Harmonic-Balanced Navier-Stokes (HBNS) equations) to resolve low-frequency harmonics and infer residual closure terms, without empirical turbulence models errors. PINN-DA-HBNS accurately reconstructs mean flow and predicts the skin-friction coefficient of the laminar circular cylinder better than RANS-constrained PINNs. Furthermore, use of PINN-DA is extended by building a neural network-based HBNS closure model, mapping low-frequency resolved quantities to residual closures. Finally, the PINN-DA formulations are used to reconstruct a turbulent, axisymmetric bluff body wake from sparse experimental velocity data. Misaligned velocity measurements significantly affect flow reconstruction accuracy, leading to the proposal of transformation-embedded PINNs, to correct spatially-misaligned measurements. Using transformation-embedded PINN-DA, the bluff body wake flow is accurately reconstructed from sparse velocity measurements, and pressure is inferred away from the wall, where it cannot be measured non-intrusively.