Using Physics Informed Neural Network (PINN) and Neural Network (NN) to Improve a $k-ω$ Turbulence Model

This paper presents a hybrid $k-\omega$ turbulence model that integrates Physics Informed Neural Networks (PINN) and standard Neural Networks (NN) to correct the underprediction of turbulent kinetic energy by improving the turbulent diffusion term and recalibrating model coefficients, resulting in accurate flow predictions across various channel and boundary layer configurations.

The Gist

The Big Picture: Fixing a Broken Weather Forecast

Imagine you are trying to predict the weather. You have a very smart computer model that is great at predicting where the wind blows and how fast it moves. However, this model has a major flaw: it consistently underestimates how much "energy" or "churn" is in the air. It thinks the air is calmer than it actually is.

In the world of engineering, this "air" is fluid (like water in a pipe or air over a car wing), and the "churn" is called Turbulent Kinetic Energy ($k$).

The author, Lars Davidson, has a model (the $k-\omega$ model) that is excellent at predicting the wind speed but terrible at predicting the energy. His goal? To fix the energy prediction without breaking the wind speed prediction.

The Problem: The "Recipe" is Missing Ingredients

Think of the turbulence model as a recipe for baking a cake.

• The Ingredients: The model uses constants (numbers like 2, 0.5, etc.) to mix the ingredients.
• The Result: The cake (the wind speed) tastes perfect. But the texture (the energy) is too dry and crumbly.

The author looked at the "recipe" and realized the problem wasn't the mixing speed, but the diffusion. In physics, diffusion is like how a drop of ink spreads out in water. The model wasn't spreading the energy correctly. It was keeping the energy too localized, like a drop of ink that refuses to spread.

The Solution: A Two-Step "AI" Fix

To fix this, the author used two types of Artificial Intelligence: PINN and NN.

Step 1: The "Physics Detective" (PINN)

First, the author used a Physics Informed Neural Network (PINN). Think of this as a detective who knows the laws of physics perfectly but needs to solve a specific mystery.

• The Mystery: "What is the correct way for energy to diffuse (spread) in this specific pipe?"
• The Clue: The author had a "gold standard" dataset from a super-accurate simulation called DNS (Direct Numerical Simulation). This is like having a high-definition video of the real fluid, showing exactly how the energy spreads.
• The Investigation: The PINN looked at the gold standard video and figured out the exact mathematical rule for how the energy should spread. It didn't just guess; it learned the rule by forcing itself to obey the laws of physics while matching the video.
• The Result: The PINN created a new, perfect "diffusion rule" (a new Prandtl number).

Step 2: The "Translator" (Neural Network)

Here is the tricky part. The PINN's rule was perfect for the specific pipe it studied, but it was written in a very complex, specific language (it depended on the distance from the wall in that specific pipe). If you tried to use that rule on a different pipe or a car wing, it would fail.

So, the author needed a Translator.

• The Job: He trained a standard Neural Network (NN) to look at the PINN's complex rule and learn to mimic it using simple, universal inputs (like "how fast is the flow?" and "how far are we from the wall?").
• The Analogy: Imagine the PINN wrote a poem in a very specific dialect. The NN is a translator that learns to say the same thing in a universal language that anyone (any computer simulation) can understand.

The "Balancing Act"

There was one major catch. When the author fixed the energy ($k$), the math said the "viscosity" (how thick the fluid feels) would change. If the viscosity changed, the wind speed prediction would break, and the model would fail.

To prevent this, the author added two "damping knobs" (mathematical functions) to the recipe.

• Knob 1: Adjusts how fast energy is destroyed.
• Knob 2: Adjusts how fast the "swirliness" ($\omega$) is destroyed.

He tuned these knobs so that even though the energy ($k$) went up to the correct level, the ratio between energy and swirliness stayed the same. This kept the wind speed prediction perfect while fixing the energy.

The Results: Does It Work?

The author tested this new "Super Model" (called $k-\omega$-PINN-NN) in three scenarios:

1. Flow in a Pipe (Channel Flow): The model nailed it. It predicted the wind speed and the energy perfectly, matching the high-definition "gold standard" data.
2. Flow over a Flat Plate (Airplane Wing): Great results. The energy prediction was much better than before, though slightly too high at the very peak.
3. Flow over a Hill (Complex Terrain): This is the hardest test. The model did a fantastic job predicting the wind speed and the chaotic swirling behind the hill. However, it slightly overestimated the energy in some spots.

The "Secret Sauce" for the Future

At the end of the paper, the author offers a bonus. He showed that the complex Neural Network "Translator" could be replaced with a simple algebraic equation (a formula you could write on a piece of paper).

• Why does this matter? Most commercial engineering software (like the kind used by Boeing or car manufacturers) can't easily run complex AI code. They can, however, easily run a simple math formula.
• The Analogy: It's like taking a complex, high-tech recipe that requires a robot chef and simplifying it into a list of instructions that a home cook can follow.

Summary

The author took a model that was good at speed but bad at energy.

1. He used PINN to figure out the exact physics of how energy should spread, based on perfect data.
2. He used a Neural Network to translate that complex physics into a simple, universal rule.
3. He added balancing knobs to ensure the speed prediction didn't break.
4. The result is a new model that predicts both speed and energy much better than before, and he even provided a simple formula version so engineers can use it in their everyday software.

Technical Summary

1. Problem Statement

The standard Wilcox $k-\omega$ turbulence model (and most two-equation RANS models) successfully predicts mean velocity profiles, turbulent shear stress, and turbulent viscosity in simple flows like fully-developed channel flows and flat-plate boundary layers. However, it consistently underpredicts the turbulent kinetic energy ($k$).

Analysis comparing the terms in the $k$-equation against Direct Numerical Simulation (DNS) data reveals that while the production ($P_k$) and dissipation ($\varepsilon$) terms are reasonably accurate, the turbulent diffusion term is poorly modeled. Previous attempts to improve these models using PINN have focused on optimizing constant coefficients (e.g., $\alpha, \beta, \sigma_k$), but this paper argues for a more dynamic approach: replacing specific model terms with functions derived from data to correct the diffusion mechanism without degrading the mean flow prediction.

2. Methodology

The proposed methodology, termed the $k-\omega$-PINN-NN model, involves a three-stage process:

A. PINN Formulation for Turbulent Diffusion

The authors reformulate the $k$-equation for fully-developed channel flow into an Ordinary Differential Equation (ODE) for the turbulent viscosity ($\nu_{t, PINN}$):
$$\frac{d}{dy}\left( (\nu + \nu_{t, PINN}) \frac{dk}{dy} \right) + P_k - \varepsilon = 0$$

• Inputs: $k$, $P_k$, and $\varepsilon$ are taken directly from DNS data.
• Unknown: $\nu_{t, PINN}$ is the variable to be solved.
• Process: A Physics Informed Neural Network (PINN) solves this ODE to find a $\nu_{t, PINN}$ that minimizes the residual error against the DNS diffusion term.
• Result: This yields a corrected turbulent Prandtl number for the $k$-equation: $\sigma_{k, PINN} = \nu_t / \nu_{t, PINN}$.

B. Consistency Constraints (Preserving Mean Flow)

Modifying the $k$-equation to increase $k$ would naturally alter the turbulent viscosity ($\nu_t = k/\omega$), which would degrade the already accurate mean velocity prediction. To prevent this:

1. The model enforces that the new $\nu_t$ must equal the standard Wilcox $\nu_t$.
2. This requires modifying the specific dissipation rate ($\omega$) and the dissipation term in the $k$-equation.
3. Two new damping functions are introduced:
   • $C_{k, PINN}$: Modifies the dissipation term in the $k$-equation.
   • $C_{\omega2, PINN}$: Modifies the destruction term in the $\omega$-equation.
     These functions are derived analytically using DNS data to ensure the modified equations reproduce the correct $k$ and $\omega$ profiles while maintaining the original $\nu_t$.

C. Neural Network Generalization

The functions $\sigma_{k, PINN}$, $C_{k, PINN}$, and $C_{\omega2, PINN}$ are currently dependent on the wall-normal coordinate ($y/\delta$), limiting them to simple channel flows. To generalize the model for complex, recirculating flows:

• Training: Three separate Neural Networks (NN) are trained to predict these three coefficients.
• Targets: The outputs are the values of $\sigma_{k, PINN}$, $C_{k, PINN}$, and $C_{\omega2, PINN}$ derived in the previous step.
• Inputs: The NNs use two non-dimensional, flow-independent input parameters:
  1. $\tau_{tot} / u_\tau^2$ (Total stress ratio)
  2. $\nu_t / (y u_\tau)$ (Viscosity ratio)
• Implementation: These NNs are embedded into the CFD solver (pyCALC-RANS), replacing the constant coefficients in the standard $k-\omega$ equations.

3. Key Contributions

1. Hybrid PINN-NN Framework: The paper introduces a novel workflow where PINN is used to "reverse-engineer" the correct physics (diffusion terms) from DNS data, and NNs are used to map these corrections to general flow variables, avoiding the need for $y/\delta$ dependency.
2. Correction of Turbulent Kinetic Energy: The model successfully corrects the chronic underprediction of $k$ in the $k-\omega$ model without sacrificing the accuracy of the mean velocity profile or skin friction.
3. Symbolic Regression Alternative: The authors demonstrate that the trained NNs can be replaced by Symbolic Regression (pySR) to generate algebraic equations (e.g., Eq. 21). This is a significant contribution for practical CFD, as algebraic equations are easily implementable in commercial codes, whereas importing PyTorch NN models is often difficult.
4. Open Source: All Python scripts for PINN, NN, pySR, and the CFD solver are made available.

4. Results

The model was validated against DNS data for three distinct flow configurations:

• Fully-Developed Channel Flow ($Re_\tau = 550, 2000, 5200, 10000$):

  • Velocity: Excellent agreement with DNS (slightly over-predicted at the lowest $Re_\tau$).
  • Turbulent Kinetic Energy ($k$): Dramatic improvement over the standard model; the new model captures the magnitude and shape of the $k$ profile accurately.
  • Coefficients: The NN-predicted coefficients ($\sigma_{k, NN}, C_{k, NN}, C_{\omega2, NN}$) were consistent across Reynolds numbers, validating the choice of input parameters.
  • Note: Slow oscillations in the coefficients during iteration were observed and mitigated using exponential weighted averaging.

• Flat-Plate Boundary Layer ($Re_\theta = 4500$):

  • Skin friction and velocity profiles were well-predicted.
  • The peak turbulent kinetic energy was slightly over-predicted, but the profile shape was significantly better than the standard model.

• Periodic Hill Flow ($Re = 10,600$):

  • Velocity: The $k-\omega$-PINN-NN model showed much better agreement with DNS than the standard model, particularly in the separation and reattachment regions.
  • Shear Stress: Improved prediction, though both models still over-predicted Reynolds shear stress in the outer region (a known limitation of Boussinesq-based models).
  • Coefficients: In this complex flow, the NN predicted nearly constant values for the coefficients in the outer region ($\sigma_{k, NN} \approx 1.21$, etc.). Interestingly, using these constant values in the hill flow yielded similar results, but they failed in channel flows, highlighting the necessity of the input parameter dependency for generalizability.

5. Significance

This work addresses a fundamental limitation in RANS modeling: the inability of standard two-equation models to predict turbulent kinetic energy accurately while maintaining correct mean flow. By leveraging PINN to extract the "true" diffusion physics from DNS and using NNs to generalize these corrections, the author creates a model that is both physically consistent and computationally efficient.

The transition from NN to Symbolic Regression (pySR) is particularly significant for the CFD community. It bridges the gap between high-fidelity machine learning corrections and the practical constraints of commercial software, offering a pathway to integrate data-driven improvements into standard engineering solvers without requiring complex neural network libraries.