Experimental measurements and numerical simulations of turbulent flows are
characterised by a trade-off between accuracy and resolution. In this study, we
bridge this gap using Physics Informed Neural Networks (PINNs) constrained by
the Reynolds-Averaged Navier-Stokes (RANS) equations and accurate sparse
pointwise mean velocity measurements for data assimilation (DA). Firstly, by
constraining the PINN with sparse data and the under-determined RANS equations
without closure, we show that the mean flow is reconstructed to a higher
accuracy than a RANS solver using the Spalart-Allmaras (SA) turbulence model.
Secondly, we propose the SA turbulence model augmented PINN (PINN-DA-SA), which
outperforms the former approach - up to 73% reduction in mean velocity
reconstruction error with coarse measurements. The additional SA physics
constraints improve flow reconstructions in regions with high velocity and
pressure gradients and separation. Thirdly, we compare the PINN-DA-SA approach
to a variational data assimilation using the same sparse velocity measurements
and physics constraints. The PINN-DA-SA achieves lower reconstruction error
across a range of data resolutions. This is attributed to discretisation errors
in the variational methodology that are avoided by PINNs. We demonstrate the
method using high fidelity measurements from direct numerical simulation of the
turbulent periodic hill at Re=5600