Physics-informed neural networks for solving Reynolds-averaged Navier$\unicode{x2013}$Stokes equations (2107.10711v1)

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.

Physics-Informed Neural Networks for Reynolds-Averaged Navier--Stokes Equations

The paper investigates the application of Physics-Informed Neural Networks (PINNs) to solve the Reynolds-averaged Navier--Stokes (RANS) equations for incompressible turbulent flows. Unlike traditional methods, this approach does not rely on specific models or assumptions for turbulence. Instead, it utilizes boundary data and the inherent dynamics captured by the RANS equations to train the neural networks. This method leverages the universal function approximation property of neural networks and is particularly notable for addressing the underdetermined nature of RANS equations, which poses challenges due to the loss of information during averaging.

Methodology and Implementation

The researchers employed a Fully Connected Neural Network (FNN) structure with inputs consisting of spatial coordinates and outputs comprising mean-flow quantities such as velocity components, pressure, and Reynolds-stress components. The training leverages Automatic Differentiation (AD), and loss functions are formed by considering both the residuals of the RANS equations and the discrepancies at domain boundaries. This hybrid supervised-unsupervised learning paradigm is implemented using the TensorFlow framework.

The authors systematically applied their methodology to several test cases, including:

• Falkner–Skan Boundary Layer: Demonstrating PINNs' capability on laminar flows with pressure gradients, achieving error margins below 1%.
• Zero-Pressure-Gradient (ZPG) and Adverse-Pressure-Gradient (APG) Boundary Layers: Utilizing reference DNS or LES datasets for validation, the paper showed high accuracy in velocity and Reynolds-stress profiles, with relative errors below 8% for most flow characteristics.
• NACA4412 Airfoil and Periodic Hill Cases: These represented more complex turbulent flows, with substantial accuracy in PINN predictions for both mainstream and Reynolds-stress components.

Results and Implications

The comparative results across different flow scenarios indicate the robustness and potential of PINNs as a general tool for solving RANS equations. The test cases revealed that PINNs could resolve complex flow features such as strong separations and reattachment points, critical for aerospace and hydrodynamic applications. Specifically, the relative error kept below 11.36% in most tested scenarios underscores the practical validity of this approach.

The work thus affirms the feasibility of utilizing PINNs in practical engineering applications where the complexity and high Reynolds numbers would traditionally require conventional turbulence modeling assumptions. The authors' contribution suggests a potential reduction in computational cost and possibly more adaptive solutions that could dynamically adjust to varying fluid conditions.

Future Directions

The promising results open several pathways for subsequent work. Further exploration into extending PINNs for three-dimensional turbulent flows and coupling with automatically tuned hyperparameters could enhance the models' predictive power. Additionally, integrating PINNs with other neural network architectures and applying them to more varied PDE systems might spur further advancements in computational fluid dynamics. The broader implications also include the possibility of leveraging data-driven techniques to refine simulations in fields related to fluid dynamics, significantly intersecting machine learning with classical physics applications.