[Paper Review] Physics-informed neural networks for solving Reynolds-averaged Navier$\unicode{x2013}$Stokes equations

Hamidreza Eivazi, Mojtaba Tahani+2 more|arXiv (Cornell University)|Jul 22, 2021

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes a physics-informed neural network (PINN) framework to solve the Reynolds-averaged Navier-Stokes (RANS) equations for incompressible turbulent flows without turbulence modeling assumptions. By leveraging boundary data and enforcing the RANS equations via automatic differentiation, the method achieves high accuracy—under 3% error for mean velocity and up to 28.18% error for Reynolds-stress components—demonstrating strong performance across laminar and turbulent flow cases including boundary layers, airfoils, and periodic hills.

ABSTRACT

Physics-informed neural networks (PINNs) are successful machine-learning methods for the solution and identification of partial differential equations (PDEs). We employ PINNs for solving the Reynolds-averaged Navier$\unicode{x2013}$Stokes (RANS) equations for incompressible turbulent flows without any specific model or assumption for turbulence, and by taking only the data on the domain boundaries. We first show the applicability of PINNs for solving the Navier$\unicode{x2013}$Stokes equations for laminar flows by solving the Falkner$\unicode{x2013}$Skan boundary layer. We then apply PINNs for the simulation of four turbulent-flow cases, i.e., zero-pressure-gradient boundary layer, adverse-pressure-gradient boundary layer, and turbulent flows over a NACA4412 airfoil and the periodic hill. Our results show the excellent applicability of PINNs for laminar flows with strong pressure gradients, where predictions with less than 1% error can be obtained. For turbulent flows, we also obtain very good accuracy on simulation results even for the Reynolds-stress components.

Motivation & Objective

• To develop a data-driven framework for solving RANS equations without relying on explicit turbulence models or closures.
• To investigate whether PINNs can accurately simulate incompressible turbulent flows using only boundary data and the governing PDEs as constraints.
• To evaluate the method's performance on canonical turbulent flow configurations with complex flow features such as separation and reattachment.
• To assess the accuracy of PINN predictions for both mean-flow quantities and Reynolds-stress components in the absence of training data inside the domain.
• To demonstrate the feasibility of PINNs for solving underdetermined systems arising from Reynolds averaging in turbulent flow simulations.

Proposed method

• A fully-connected neural network (FNN) takes spatial coordinates (x, y) as inputs and predicts mean-flow variables: streamwise and wall-normal velocities (U, V), pressure (P), and Reynolds-stress components (⟨u²⟩, ⟨uv⟩, ⟨v²⟩).
• Automatic differentiation (AD) is applied to compute spatial derivatives of the network outputs to construct the residual of the RANS equations, which serves as an unsupervised loss.
• Supervised learning loss is computed only on domain boundaries using measured data for velocity, pressure, and Reynolds-stress components.
• The total loss function combines supervised loss on boundaries and unsupervised residual loss of the RANS equations over a set of collocation points inside and on the boundaries.
• The Reynolds number is enforced through the governing equations, and the model is trained end-to-end using gradient-based optimization.
• The framework is applied to four test cases: Falkner–Skan boundary layer (laminar), zero-pressure-gradient (ZPG) and adverse-pressure-gradient (APG) turbulent boundary layers, NACA4412 airfoil, and periodic hill.

Experimental results

Research questions

• RQ1 Can PINNs accurately solve the RANS equations for incompressible turbulent flows without any turbulence model or closure assumption?
• RQ2 How well can PINNs predict mean-flow and Reynolds-stress components when only boundary data are used for training?
• RQ3 To what extent can PINNs capture complex flow features such as flow separation and reattachment in turbulent flows like over a periodic hill?
• RQ4 How does the accuracy of PINN predictions compare to high-fidelity DNS or LES data across different flow configurations?
• RQ5 What is the relative contribution of boundary data versus residual enforcement of the RANS equations in achieving accurate solutions?

Key findings

• For the laminar Falkner–Skan boundary layer with strong pressure gradients, PINNs achieved predictions with less than 1% error, demonstrating high accuracy even in challenging laminar cases.
• In the zero-pressure-gradient (ZPG) turbulent boundary layer, the average relative ℓ₂-norm error across all predicted variables was 3.91%, indicating strong predictive capability.
• For the adverse-pressure-gradient (APG) boundary layer, the average relative ℓ₂-norm error was 3.60%, showing robust performance under flow separation conditions.
• On the NACA4412 airfoil, PINN predictions for streamwise velocity had less than 3% error, with good agreement in flow structures and pressure distribution.
• For the periodic hill case at Re_b = 2,800, the PINN model accurately predicted the separated flow region and reattachment point, with a relative ℓ₂-norm error of 2.77% for streamwise velocity.
• The highest error was observed for the ⟨u²⟩ component (28.18%), indicating that Reynolds-stress components remain more challenging to predict accurately, though still within acceptable bounds for many engineering applications.