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

TL;DR

Dissipation is revealed as the dominant organizing mechanism of near-wall turbulence, constraining production and viscous diffusion within a single structural hierarchy that breaks down in the outer layer.

Abstract

In this work, we investigate the physical mechanisms governing turbulent kinetic energy transport using explainable deep learning (XDL). An XDL model based on SHapley Additive exPlanations (SHAP) is used to identify and percolate high-importance structures for the evolution of the turbulent kinetic energy budget terms of a turbulent channel flow at a friction Reynolds number of $Re_τ= 125$. The results show that the important structures are predominantly located in the near-wall region and are more frequently associated with sweep-type events. In the viscous layer, the SHAP structures relevant for production and viscous diffusion are almost entirely contained within those relevant for dissipation, revealing a clear hierarchical organization of near-wall turbulence. In the outer layer, this hierarchical organization breaks down and only velocity-pressure-gradient correlation and turbulent transport SHAP structures remain, with a moderate spatial coincidence of approximately $60\%$. Finally, we show that none of the coherent structures classically studied in turbulence are capable of representing the mechanisms behind the various terms of the turbulent kinetic energy budget throughout the channel. These results reveal dissipation as the dominant organizing mechanism of near-wall turbulence, constraining production and viscous diffusion within a single structural hierarchy that breaks down in the outer layer.

Figures (5)

• Figure 1: Workflow diagram of the study, including data generation, neural-network training, SHAP analysis, and structure identification.
• Figure 2: Joint probability density functions (JPDFs) of the SHAP values based on the wall-normal distance and the streamwise velocity fluctuation for the production (top-left), dissipation (top-right), viscous diffusion (bottom-left) and turbulent transport (bottom-right).
• Figure 3: (Top) Volume of SHAP structures of $P_k$ (solid red), $\varepsilon_k$ (solid green) and $V_k$ (solid blue). Intersection of SHAP structures of $P_k \cap \varepsilon_k$ (dashed yellow), $P_k \cap V_k$ (dashed magenta) and $\varepsilon_k \cap V_k$ (dashed cyan). Volume of the triple intersection $P_k\cap \varepsilon_k \cap V_k$ in dashed black. (Bottom) Percentage of coincidence of $P_k\cap \varepsilon_k \cap V_k$ normalized by the volume of SHAP structures of $P_k$, $\varepsilon_k$, $V_k$, $P_k \cap \varepsilon_k$, $\varepsilon_k \cap V_k$ and $\varepsilon_k \cap V_k$, with colors corresponding to the reference set shown in the top panel.
• Figure 4: (Top) volume of SHAP structures of $T_k$ (solid red), $P_k$ (solid green) and intersection of SHAP structures of $T_k \cap P_k$ (dasehd yellow). (Bottom) Coincidence percentage of the volume intersection of $T_k \cap P_k$ compared to the volume of SHAP structures of $T_k$ (solid line), $P_k$ (dashed line).
• Figure 5: Coincidence of classical structures with SHAP structures of $\varepsilon_k$ (top) and $T_k$ (bottom). Solid lines represents the percentage compared to SHAP structures of $\varepsilon_k$ (top) and $T_k$ (bottom). Dashed lines represent the percentage compared to the classical structures. Colors are for intersection with Q event structures (green), streaks (blue), vortices (yellow) and SHAP of velocity fluctuation structures (red).