Classically studied coherent structures only paint a partial picture of wall-bounded turbulence

Abstract

For the last 140 years, the mechanisms of transport and dissipation of energy in a turbulent flow have not been completely understood. Previous research has focused on analyzing the so-called coherent structures, organized flow patterns characterized by their spatial coherence, lifespan and significant contribution to momentum and energy transfer. However, the connection between these structures and the flow development is still uncertain. Here, we show a data-driven methodology for objectively identifying high-importance regions. A deep-learning model is trained to predict a future state of the flow and the gradient-SHAP explainability algorithm is used to calculate the importance of each grid point. Finally, high-importance regions are computed using the SHAP data and are compared to the other coherent structures. The SHAP analysis provides an objective way to identify the regions of higher importance, which exhibit different levels of agreement with the classical structures without being completely related to any particular one.

Figures (5)

• Figure 1: Diagram showing the methodology of the paper. The methodology is divided into three main stages. The first stage, represented by the black lines, comprises a U-net deep learning (DL) model for predicting the evolution of the flow. The three components of the simulated velocity fluctuation (green blocks, $u_{\rm{t}}$ for the streamwise, $v_{\rm{t}}$ for the wall-normal and $w_{\rm{t}}$ for the spanwise velocity) in an instant $t$ are evolved through the U-net to the next time step $t+1$ separated a viscous time $t^+ = 5$ (yellow blocks, $u_{\rm{t+1}}$, $v_{\rm{t+1}}$ and $w_{\rm{t+1}}$). In the second stage (blue flow connectors), the importance of each grid node of the field, for each velocity component, is calculated by applying the expected-gradients algorithm an extension of Shapley additive explanations (SHAP values) to an infinite-player game. For evaluating the importance of each input grid point in the prediction, the mean-squared error of the predicted field is used and the simulated fields in the next step (green blocks, $u_{\rm{t+1}}$, $v_{\rm{t+1}}$ and $w_{\rm{t+1}}$) is defined as the output. The result of this expectation is the so-called SHAP values, which are represented by the light-green blocks, and represent the importance of the streamwise ($\left(\phi_u\right)_{\rm{t}}$), wall-normal ($\left(\phi_v\right)_{\rm{t}}$) and spanwise ($\left(\phi_w\right)_{\rm{t}}$) velocity fluctuations. In the final stage (purple flow connectors), the previously mentioned SHAP values are used to segment the model through a percolation analysis. As a result, the flow field can be segmented into SHAP structures. Once these structures are defined, their physical characteristics are analyzed, and their correlation with other coherent structures is evaluated for the different wall-normal distance $y^+$.
• Figure 2: Instantaneous visualization of the various coherent structures in the channel flow. The figure shows four types of coherent structures, SHAP-based structures, subfigures a) and b), Reynolds-stress structures or Q events, subfigures c) and d), streaks, subfigures e) and f) and vortices, subfigures g) and h), in half of the channel colored by wall-normal distance (purple near the wall and yellow in the mid-plane of the channel) for ${\rm Re}_\tau = 125$, subfigures a), c), e) and g), and ${\rm Re}_\tau = 550$, subfigures b), d), f) and h). In the figure, the wall is located at $y^+ = 0$ and the mid-plane of the channel is $y^+ = 125$ or $y^+ = 550$ depending on the case. In addition, we use periodic boundary conditions in $x$ and $z$ for both cases. Note that the same flow field is used for the four panels of each channel. Note that $x^+$, $y^+$ and $z^+$ define the streamwise, spanwise and wall-normal directions respectively.
• Figure 3: Joint probability density function of the streamwise velocity fluctuation, $u^+$, and the inner-scaled wall-normal distance, $y^+$, for the different types of structures. The figure presents the SHAP-based structures, subfigures a) and b), Reynolds-stress structures, subfigures c) and d), streaks, subfigures e) and f) and vortices, subfigures g) and h) for ${\rm Re}_\tau = 125$, subfigures a), c), e) and g), and ${\rm Re}_\tau = 550$, subfigures b), d), f) and h).
• Figure 4: Coincidence between the various types of structures. Percentage of coincidence of pairs of the following structures: SHAP, Q events, streaks and vortices, relative to the volume of each type of the pair for ${\rm Re}_\tau = 125$, a), and ${\rm Re}_\tau = 550$, b), along wall-normal distance $y^+$.
• Figure 5: Instantaneous coincidence between the various types of structures at various wall-normal distances, $y^+$. We show results for ${\rm Re}_\tau = 125$, subfigures a), c) and e) and ${\rm Re}_\tau = 550$, subfigures b), d) and f). The colors used for the coincidence between structures follow this code: Qs\\(streaks $\cup$ vortices), streaks\\(Qs $\cup$ vortices), (Qs $\cup$ streaks)\\ vortices, vortices \\(Qs $\cup$ streaks), (Qs $\cup$ vortices) \\ streaks, (streaks $\cup$ vortices) \\ Qs, Qs $\cup$ streaks $\cup$ vortices. In these panels, the SHAP structures are represented by the black solid lines. This figure presents a zoomed-in view of a small region on the three planes, in the streamwise, $x^+$, and spanwise, $z^+$, directions. For a representation of the whole planes the reader is referred to the Supplementary Material.