AI and PINNs in CFD Turbulence Closure Modeling

Computational fluid dynamics relies heavily on solving the Navier-Stokes equations to predict fluid behavior, yet resolving every turbulent eddy at microscopic scales remains computationally prohibitive for practical engineering. This article explores how artificial intelligence, specifically physics-informed neural networks, is revolutionizing fluid simulations by accurately learning sub-grid scale turbulence closure models. By combining high-fidelity simulation data with fundamental physical laws, these advanced machine learning frameworks bridge the gap between computational efficiency and physical accuracy, paving the way for faster and more reliable aerodynamic, hydrodynamic, and meteorological predictions.

The Challenge of the Navier-Stokes Equations and Turbulence

The Navier-Stokes equations govern the motion of fluids and are foundational to computational fluid dynamics (CFD). In turbulent flows, energy cascades from large, geometry-dependent eddies down to microscopic scales where viscosity dissipates it as heat.

To capture every scale directly via Direct Numerical Simulation (DNS), the computational grid must be finer than the smallest turbulent eddies. For high Reynolds number flows, such as air flowing over a commercial aircraft, DNS requires computational power that exceeds the capabilities of modern supercomputers.

To bypass this limitation, engineers use Large Eddy Simulation (LES) or Reynolds-Averaged Navier-Stokes (RANS) methods. LES resolves the large, energy-containing eddies on a coarser grid while filtering out the smaller, sub-grid scale (SGS) eddies. However, because the small eddies still influence the larger flow structures, their effects must be approximated. This introduces unknown terms into the filtered Navier-Stokes equations, specifically the sub-grid scale stress tensor. Defining these unknown terms using known resolved variables is known as the turbulence closure problem.

Traditional Sub-Grid Scale Models vs. Artificial Intelligence

Historically, CFD has relied on empirical or mathematical approximations to close the Navier-Stokes equations. Models like the Smagorinsky-Lilly model assume a simplified relationship between the resolved strain rate and the sub-grid stresses, using a concept called eddy viscosity. While computationally cheap, these traditional models often fail in complex scenarios involving flow separation, transitional flows, or strong pressure gradients because they cannot capture non-local and non-linear energy transfers, such as backscatter (where energy flows backward from small scales to large scales).

Artificial intelligence offers a data-driven alternative. By training deep neural networks on high-resolution DNS data that has been artificially filtered to mimic a coarse LES grid, AI can learn the exact mathematical mapping between the resolved large-scale flow features (like velocity gradients) and the unclosed sub-grid scale stresses. Unlike simple algebraic models, deep learning architectures can process complex spatial and temporal patterns, capturing the intricate physics of the energy cascade.

The Role of Physics-Informed Neural Networks (PINNs)

Purely data-driven AI models have a major flaw in CFD: they do not inherently understand physics. A standard neural network might produce predictions that fit the training data well but violate conservation of mass, momentum, or energy when deployed in an actual simulation, leading to numerical instability and simulation failure.

Physics-Informed Neural Networks (PINNs) solve this problem by embedding the governing physical equations directly into the AI training process. A PINN modifies the neural network's loss function—the mathematical measure of error the network tries to minimize. Instead of only penalizing the network for deviating from the DNS training data, the PINN loss function includes residual terms derived directly from the Navier-Stokes equations.

To successfully learn sub-grid turbulence closure models, PINNs enforce specific physical constraints:

• Conservation Laws: The network is penalized if its predicted sub-grid stresses result in a violation of mass conservation (continuity) or momentum conservation.
• Galilean Invariance: The physics of turbulence do not change if the coordinate system is moving at a constant velocity or rotated. PINNs are structured—often using tensor basis neural networks—so that the inputs and outputs automatically respect rotational and translational symmetries.
• Realizability and Dissipation: The neural network is constrained to ensure that the turbulent kinetic energy remains positive and that the net transfer of energy accurately reflects physical dissipation, preventing the simulation from blowing up due to artificial energy generation.

How the Training and Deployment Workflow Operates

The integration of PINNs into CFD for sub-grid modeling generally follows a two-stage process:

1. A Priori Training (Offline): Researchers run high-fidelity DNS on smaller, canonical fluid problems (such as decaying isotropic turbulence or channel flow). This data is filtered to create pairs of coarse-grid resolved variables (inputs) and exact sub-grid stress tensors (outputs). The PINN is trained on this dataset while simultaneously evaluating the partial differential operators of the Navier-Stokes equations via automatic differentiation to ensure physical compliance.
2. A Posteriori Deployment (Online): Once trained, the PINN replaces the traditional closure model inside a standard CFD solver. At each time step of a coarse-grid LES, the solver passes the resolved velocity gradients to the PINN. The PINN rapidly infers the sub-grid scale stresses and feeds them back into the Navier-Stokes solver, which then advances the fluid flow to the next time step.

By uniting the universal approximation capabilities of deep learning with the strict mathematical boundaries of fluid mechanics, PINNs allow CFD solvers to run on coarse, computationally inexpensive grids while retaining the high accuracy of fine-grid simulations.