Source-linked AI summary
Physics-informed neural networks for solving Reynolds-averaged Navier$\unicode{x2013}$Stokes equations
Hamidreza Eivazi, Mojtaba Tahani, Philipp Schlatter, Ricardo Vinuesa
TL;DR
The paper addresses solving the underdetermined incompressible turbulent RANS equations without specifying a turbulence model. It trains PINNs with governing-equation residuals and boundary data, obtaining excellent laminar predictions and very good turbulent-flow accuracy, including Reynolds-stress components.
Problem
Averaging makes the RANS equations underdetermined, so traditional solvers require turbulence-modeling assumptions to close them.
Method
PINNs use spatial coordinates as inputs, flow quantities as outputs, automatic differentiation to construct RANS residuals, and boundary data for supervised training.
Results
Across laminar and four turbulent-flow cases, PINNs produce excellent predictions, including pressure for the laminar case and Reynolds-stress components for turbulent flows.
Takeaways & Limitations
The reported cases support using PINNs to solve incompressible turbulent RANS flows without a specific turbulence model and with data restricted to domain boundaries.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper applies PINNs to solve incompressible turbulent RANS equations without a specific turbulence model, using boundary data and governing equations. This approach builds on PINNs’ use for solving complex PDE systems.
- Background: PINNs are machine-learning methods used to solve and identify partial differential equations.They have been applied to forward and inverse problems across several fluid-mechanics settings.
- Research gap: The RANS equations are underdetermined after averaging, so traditional solvers require modeling assumptions to close the system.The paper addresses this underdetermination with boundary data and the governing equations during neural-network training.
- Approach: The proposed PINN uses spatial coordinates as inputs and mean velocities, pressure, and Reynolds-stress components as outputs.Automatic differentiation constructs the RANS equations from output derivatives with respect to the inputs.
- Training data: Only domain-boundary data are used for training, while the governing-equation residual provides an unsupervised learning signal inside the domain.The Reynolds number is specified through the governing equations, and reference quantities come from DNS or well-resolved LES.
2 Methodology
The methodology represents RANS flow quantities with a fully connected neural network and enforces the equations through automatic differentiation. Training combines boundary supervision with governing-equation residuals evaluated at interior and boundary points.
- Network formulation: A fully connected neural network maps spatial coordinates x and y to velocity, pressure, and Reynolds-stress components.The outputs are U, V, P, u2, uv, and v2, while automatic differentiation supplies derivatives for continuity and momentum equations.
- Training data: Only domain-boundary data are used as the supervised training dataset.The boundary data constrain the PINN predictions while the equations guide the solution within the domain.
- Loss function: The total loss combines the governing-equation residual loss Le with the boundary-condition loss Lb.Le measures RANS residuals, whereas Lb measures discrepancies between boundary data and PINN predictions.
- Residual evaluation: RANS residuals are computed at selected interior points together with points on the domain boundaries.Weighting coefficients may balance different loss terms and accelerate convergence.
3 Results
PINNs are evaluated on laminar and four turbulent-flow cases using boundary data together with governing-equation residuals. The predictions agree well with reference data across velocity, pressure, and Reynolds-stress quantities, including adverse-pressure-gradient and separated flows.
- Test cases: The study evaluates PINNs on a Falkner–Skan laminar boundary layer and four turbulent cases: ZPG, APG, NACA4412 airfoil, and periodic hill flows.The turbulent cases use reference data from DNS or well-resolved LES, while the laminar reference is analytical.
- Method: PINNs use boundary data for supervised learning while enforcing RANS or Navier–Stokes governing-equation residuals inside the domain.The framework represents mean-flow variables and Reynolds-stress components with neural-network outputs and evaluates residuals at interior points.
- ZPG turbulent boundary layer: 1.02%, 4.25%, and 6.46% are the reported errors for U, V, and uv, respectively, in the ZPG turbulent boundary layer.The comparison also includes H12, cf, and inner-scaled U+ and uv+ profiles at Reθ = 2500, 4000, and 5500.
- APG turbulent boundary layer: 0.28% and 7.96% are the lowest and highest reported errors, associated with U and uv, respectively, in the APG turbulent boundary layer.The predictions are compared with reference H12, cf, and inner-scaled profiles at Reθ = 1623, 2138, and 2588.
4 Conclusions
The study introduces PINNs for solving incompressible turbulent RANS flows without a specific turbulence model, combining boundary data with governing-equation residuals. It demonstrates the approach on laminar and multiple turbulent-flow cases.
- PINNs solve incompressible turbulent RANS equations without any specific turbulence model or assumption.
- Boundary data, including Reynolds-stress components, are combined with RANS governing equations to guide learning.
- The method is evaluated on Falkner–Skan, zero-pressure-gradient and adverse-pressure-gradient boundary layers, NACA4412 flow, and the periodic hill.
- Laminar boundary-layer predictions remain accurate under adverse pressure gradients, including pressure inferred from velocity data supplied only on boundaries.