Explainable deep learning reveals the physical mechanisms behind the turbulent kinetic energy equation

By applying explainable deep learning to turbulent channel flow, this study reveals that near-wall turbulence is hierarchically organized with dissipation as the dominant mechanism constraining production and viscous diffusion, a structure that breaks down in the outer layer where no single classical coherent structure can represent the turbulent kinetic energy budget.

The Gist

Imagine trying to understand a massive, chaotic storm inside a pipe. For a long time, scientists have tried to predict how the energy in this swirling chaos moves around, but the math is incredibly complex, like trying to track every single raindrop in a hurricane.

This paper introduces a new way to look at that storm using a "smart camera" powered by artificial intelligence (AI). Instead of just guessing, the AI learns the rules of the storm and then explains why it behaves the way it does. Here is the story of what they found, broken down simply:

The AI Detective and the "Why"

The researchers used a special type of AI called Explainable Deep Learning. Think of this AI not just as a predictor, but as a detective that can point to a specific spot in the fluid and say, "I used this swirl of air to predict what happens next."

They trained the AI to predict five different parts of the "energy budget" of the turbulence (how energy is made, moved, and destroyed). Then, they asked the AI: "Which parts of the flow were most important for your prediction?" The AI drew a map of these important spots, which they call SHAP structures.

The Neighborhood of the Pipe Wall

The pipe has a "wall" (the metal surface) and an "outer layer" (the middle of the pipe). The AI's maps revealed two very different neighborhoods:

1. The Near-Wall Zone (The Busy City Center)
Close to the wall (within the first 30 "units" of distance), the AI found that almost all the important action happens in a very specific, crowded area.

• The "Sweep" Events: The most important structures were like high-speed cars diving down toward the curb. In fluid terms, these are called "sweeps" (fast fluid hitting the wall). They are much more important than "ejections" (slow fluid shooting away from the wall).
• The Hierarchy (The Russian Nesting Doll): This is the biggest discovery. The AI found that the structures responsible for creating energy (Production) and moving energy through the sticky fluid (Viscous Diffusion) are almost entirely inside the structures responsible for destroying energy (Dissipation).
  • Analogy: Imagine a giant, glowing net (Dissipation). Inside that net, you find smaller nets for making energy and moving it. The "Dissipation" net is the boss; it wraps around everything else. If you want to control the energy near the wall, you have to deal with this "Dissipation" net first.

2. The Outer Layer (The Open Countryside)
As you move away from the wall into the middle of the pipe, the neat, nested order breaks down.

• The "Russian nesting doll" effect disappears. The structures for making energy and destroying energy no longer overlap perfectly.
• Instead, the only things that still seem to work together are the pressure changes and the transport of energy. They overlap about 60% of the time, suggesting a looser, more scattered relationship in the middle of the pipe compared to the tight organization near the wall.

The "Old Maps" vs. The "New GPS"

For decades, scientists have used "classic" maps to understand turbulence. They looked for specific shapes like:

• Streaks: Long lines of fast or slow fluid.
• Vortices: Swirling whirlpools.
• Q-events: Specific types of intense swirling.

The researchers compared their new AI maps with these old classic maps. The result was surprising: The old maps don't match the new reality.

• Near the wall, the classic "swirls" (vortices) and "lines" (streaks) only partially explain what the AI sees.
• In the middle of the pipe, the classic structures barely match the AI's findings at all. The AI found that the old "whirlpools" are not the main drivers of the energy budget in the way we thought.

The Bottom Line

This study used AI to reveal that turbulence near a wall is organized like a strict hierarchy where energy destruction (Dissipation) is the boss, wrapping around and controlling how energy is made and moved. However, once you move away from the wall, this strict order falls apart, and the rules become much more scattered.

Most importantly, the "classic" shapes scientists have relied on for years (like specific swirls or lines) are not the full story. The AI showed us that the real mechanisms are more complex and are best understood by looking at the specific "importance maps" the AI generated, rather than relying on our old mental pictures of how turbulence works.

Technical Summary

Problem Statement
The prediction and understanding of turbulent flows remain central challenges in fluid mechanics due to the highly nonlinear and multiscale nature of turbulence. While high-fidelity numerical simulations like Direct Numerical Simulation (DNS) have advanced significantly, the detailed analysis of the physical mechanisms driving velocity field evolution remains complex and computationally expensive. Traditional approaches often focus on interpreting velocity-fluctuation fields directly. However, analyzing the individual instantaneous terms of the turbulent kinetic energy (TKE) budget—specifically production, dissipation, viscous diffusion, turbulent transport, and velocity-pressure-gradient correlation—offers a more detailed, physically grounded perspective. Each term represents a distinct physical mechanism, and their separate analysis can reveal insights inaccessible from velocity fields alone.

Methodology
This study employs Explainable Deep Learning (XDL) to investigate the physical mechanisms governing TKE transport in a turbulent channel flow at a friction Reynolds number of $Re_\tau = 125$. The workflow involves four main stages:

1. Data Generation: A DNS of an incompressible turbulent channel flow is conducted using the spectral code LISO. The computational domain is $(8\pi h, 2h, 3\pi h)$, and the data is normalized in wall units. Instantaneous turbulent budget terms are calculated from the velocity field.
2. Deep Learning Model: A U-net architecture is trained to predict the future evolution of each individual TKE budget term based on the instantaneous velocity field. The model uses a time interval of $\Delta t^+ = 5$ and is trained on 10,000 snapshots (with 20% reserved for validation) using an RMSprop gradient-descent algorithm to minimize mean-squared error until a relative error below 1% is achieved.
3. Explainability (SHAP): To interpret the model's predictions, SHapley Additive exPlanations (SHAP) are applied. Specifically, a gradient-SHAP methodology is used to quantify the spatial importance of each grid point in the input velocity-fluctuation field for the output TKE term. This generates three-dimensional importance fields ($\phi_u, \phi_v, \phi_w$) for each snapshot.
4. Structure Identification: The SHAP importance fields are processed using a percolation algorithm to identify coherent "SHAP structures." These are defined as regions where the local SHAP norm exceeds a threshold determined by a hyperbolic hole parameter $H$. These identified structures are then compared against classical flow structures (intense Q events, streaks, vortices) and the SHAP structures of the velocity-fluctuation field itself.

Key Results
The analysis yields several critical findings regarding the spatial organization and hierarchy of turbulence:

• Near-Wall Dominance: The most important structures for all TKE budget terms are predominantly located in the near-wall region ($y^+ < 30$), with peak density around $y^+ \approx 15$.
• Event Types: Positive streamwise velocity fluctuations ($u^+ > 0$) near the wall are more important than negative ones, indicating that sweep-type events (Q4) dominate over ejection-type events (Q2) in explaining TKE budget mechanisms.
• Hierarchical Organization in the Viscous Layer: In the region $y^+ < 30$, a clear hierarchical organization is observed. The SHAP structures relevant for production and viscous diffusion are almost entirely contained within the structures relevant for dissipation. Specifically, approximately 99% of production and viscous diffusion structures intersect with dissipation structures below $y^+ < 10$. This suggests that dissipation acts as the dominant organizing mechanism, constraining production and viscous diffusion within a single structural hierarchy.
• Breakdown in the Outer Layer: This hierarchical organization breaks down in the outer layer. Here, only velocity-pressure-gradient correlation and turbulent transport SHAP structures persist. These two terms exhibit a moderate spatial coincidence of approximately 60%, indicating a strong coupling between pressure effects and energy redistribution in the outer layer, but a more distributed and less tightly coupled nature compared to the near-wall region.
• Limitations of Classical Structures: None of the classical coherent structures (Q events, streaks, vortices) are capable of representing the mechanisms behind the various TKE budget terms throughout the entire channel. While Q events are largely contained within dissipation structures near the wall, and streaks show some embedding very close to the wall, classical structures fail to capture the dominant mechanisms for $y^+ \ge 15$. The similarity with vortices is notably low (<50%) in most relevant regions.

Significance and Claims
The paper claims that this work introduces a novel methodology to identify the most relevant flow structures for TKE budget terms, bridging data-driven models with classical wall-turbulence physics. By linking SHAP importance maps to physical processes, the study provides a physically consistent interpretation of the TKE budget that complements existing spectral frameworks.

The primary significance lies in revealing dissipation as the dominant organizing mechanism of near-wall turbulence, which constrains production and viscous diffusion within a single structural hierarchy. This finding suggests that targeting dissipation structures could inherently influence other near-wall turbulent processes, offering potential insights for flow control interventions. Furthermore, the results demonstrate that SHAP is capable of extracting precise and meaningful information from extremely complex, highly nonlinear governing equations, providing a framework to extend the classical analysis of wall-bounded turbulence. The study concludes that while classical structures offer some insight near the wall, they are insufficient for describing the mechanisms behind TKE budget terms across the full channel, necessitating the data-driven, explainable approach presented here.