[Paper Review] Turbulence model augmented physics informed neural networks for mean flow reconstruction

Yusuf Patel, Vincent Mons+2 more|arXiv (Cornell University)|Jun 1, 2023

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes a physics-informed neural network (PINN) augmented with the Spalart-Allmaras (SA) turbulence model—called PINN-DA-SA—for high-accuracy mean flow reconstruction from sparse experimental or simulation data. By combining RANS constraints with sparse mean velocity measurements, the method reduces mean velocity reconstruction error by up to 73% compared to baseline PINNs and outperforms variational data assimilation due to avoidance of discretization errors.

ABSTRACT

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.

Motivation & Objective

• To address the trade-off between accuracy and resolution in experimental and numerical turbulent flow measurements.
• To improve mean flow reconstruction accuracy by integrating sparse high-fidelity mean velocity measurements with physics constraints from the RANS equations.
• To enhance flow reconstruction in regions with high gradients and flow separation by incorporating the Spalart-Allmaras (SA) turbulence model as a physical prior.
• To compare the proposed PINN-DA-SA framework against variational data assimilation, identifying advantages in error reduction and numerical stability.

Proposed method

• The method employs Physics-Informed Neural Networks (PINNs) trained to satisfy the Reynolds-Averaged Navier-Stokes (RANS) equations as soft constraints, using sparse pointwise mean velocity measurements for data assimilation (DA).
• A baseline PINN-DA approach is first applied, constraining the network with RANS equations and sparse data without turbulence model closure.
• The PINN-DA-SA framework further integrates the Spalart-Allmaras (SA) turbulence model as an additional physical constraint to improve accuracy in complex flow regions.
• The pressure field is reconstructed by solving the RANS momentum equations, with an integration constant determined via a single pressure measurement point in post-processing.
• The method avoids discretization errors inherent in variational data assimilation by directly enforcing PDEs in a continuous weak form via neural network loss functions.
• High-fidelity direct numerical simulation (DNS) data of a turbulent periodic hill at Re=5600 is used to generate ground-truth mean velocity, pressure, and Reynolds stress fields for training and validation.

Experimental results

Research questions

• RQ1 Can a PINN constrained by RANS equations and sparse mean velocity measurements reconstruct turbulent mean flows more accurately than a standard RANS solver with the Spalart-Allmaras model?
• RQ2 Does augmenting the PINN with the SA turbulence model significantly improve reconstruction accuracy, particularly in regions with high velocity and pressure gradients or flow separation?
• RQ3 How does the PINN-DA-SA approach compare to variational data assimilation in terms of reconstruction error and robustness across varying data resolution levels?
• RQ4 To what extent does the PINN-DA-SA method generalize to complex 3D turbulent flows, such as the periodic hill case, when only sparse 2D velocity measurements are available?

Key findings

• The PINN-DA-SA method reduces mean velocity reconstruction error by up to 73% compared to the baseline PINN-DA approach when using coarse measurement grids.
• The inclusion of the SA turbulence model significantly improves reconstruction accuracy in regions with strong gradients and flow separation, where baseline PINN-DA struggles.
• PINN-DA-SA outperforms variational data assimilation across all tested data resolutions, achieving lower absolute reconstruction error due to avoidance of discretization errors.
• The PINN-DA-SA method successfully reconstructs the pressure field with highest error in regions of high pressure gradient, and the integration constant is corrected using a single pressure measurement point in post-processing.
• In laminar cylinder flow at Re=150, the PINN-DA-Baseline already outperforms variational DA, with a maximum absolute error of 0.11 compared to 0.30, and better wake length prediction.
• The PINN-DA-SA framework generalizes well to turbulent flows, accurately capturing physical features such as symmetric wakes and recirculation zones even with sparse data.