Turbulence model augmented physics informed neural networks for mean
  flow reconstruction
Experimental measurements and numerical simulations of turbulent flows are
characterised by a trade-off between accuracy and resolution. In this study, we
combine accurate sparse pointwise mean velocity measurements with the
Reynolds-Averaged Navier-Stokes (RANS) equations using data assimilation
methods. Importantly, we bridge the gap between data assimilation (DA) using
Physics-Informed Neural Networks (PINNs) and variational methods based on
classical spatial discretisation of the flow equations, by comparing both
approaches on the same turbulent flow case. 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 by 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.
Yusuf Patel, Vincent Mons, Olivier Marquet, Georgios Rigas
# Turbulence model augmented physics informed neural networks for mean flow reconstruction

###### Abstract

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-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 \(Re=5600\).

## I Introduction

Whilst the Navier-Stokes equations describe accurately the evolution of fluid flow in space and time, their direct numerical simulation (DNS) is intractable in turbulent regimes. As a compromise, industrial simulation is dominated by Reynolds-Averaged Navier-Stokes simulations (RANS), which govern the time-averaged flow quantities instead of unsteady time-varying values. Whilst solutions of the RANS equations are computationally tractable, they are also less accurate due to their under-determined nature and the lack of exact turbulence closure models [1]. On the other hand, experimental methods are often more expensive, with extra costs such as test part manufacture, and require access to laboratory facilities (i.e. wind tunnels). Error from sources, such as imperfect test parts and sensor noise can also make measurements less reliable while measurement tools can also be limited. Particle Image Velocimetry (PIV) is often used to measure flow velocity but its resolution may be hardware restricted and limited to specific planes of interest (2D planes in a full 3D field), whilst pressure measurements are difficult to obtain in the bulk flow without being intrusive.

Data assimilation methods have enabled the augmentation of low-fidelity numerical simulations with high-fidelity experimental data for correcting turbulence RANS models, such as with the parameter correction [2], addition of corrective terms [3] and development of new models [4]. The goal of this work is to approach the inverse problem, where a set of flow observations (data) is used to discover the factors which caused them. This contrasts the forward problem, where factors (boundary conditions) are defined, and the resultant effects of the system are calculated. By leveraging sparse high-fidelity time-averaged measurements, which obey the governing RANS equations, we extract information not present in the original dataset: interpolate missing data by super-resolving the flow fields (such as coarse PIV fields); and infer missing fields (such as pressure and Reynolds stresses).