Identifying regions of importance in wall-bounded turbulence through explainable deep learning

TL;DR

A deep learning approach is proposed to evaluate the importance of various types of coherent structure in the flow, to uncover main mechanisms of wall-bounded turbulence and develop techniques for its control.

Abstract

Despite its great scientific and technological importance, wall-bounded turbulence is an unresolved problem in classical physics that requires new perspectives to be tackled. One of the key strategies has been to study interactions among the energy-containing coherent structures in the flow. Such interactions are explored in this study for the first time using an explainable deep-learning method. The instantaneous velocity field obtained from a turbulent channel flow simulation is used to predict the velocity field in time through a U-net architecture. Based on the predicted flow, we assess the importance of each structure for this prediction using the game-theoretic algorithm of SHapley Additive exPlanations (SHAP). This work provides results in agreement with previous observations in the literature and extends them by revealing that the most important structures in the flow are not necessarily the ones with the highest contribution to the Reynolds shear stress. We also apply the method to an experimental database, where we can identify completely new structures based on their importance score. This framework has the potential to shed light on numerous fundamental phenomena of wall-bounded turbulence, including novel strategies for flow control.

Figures (10)

• Figure 1: Conceptual map of the workflow employed in this study.(Top-left) Instantaneous Reynolds-stress (Q) events identified in a turbulent channel. Four different kinds of structures exist based on the quadrant analysis Lozano2014: outward interactions (purple), ejections (blue), inward interactions (green) and sweeps (yellow). (Top-right) Total contribution, $\Phi_e / \Phi_T$, (left column) and total contribution per unit volume, $\Phi_e^v / \Phi_T^v$, (right column) of each event type to the U-net prediction. Their definition and implications are discussed in the Results section. (Bottom) Workflow comprising three steps: 1) a U-net is used to predict the next instantaneous flow field (time $t_{i+1}$) based on the current one ($t_i$); 2) the structures evolve, so some may dissipate in the next field (yellow), others may be convected (rest of colors), and some may even merge into larger ones (not shown); 3) calculation of the contribution of each structure (gray shade) to the prediction of the next field. The error on the prediction of the flow field of the U-net in $t_i$ with respect to the simulated flow in $t_{i+1}$ is used to determine the importance of every single structure. In this way, it is possible to rank the various structures in terms of their relative importance to predict the next instantaneous field. The workflow is performed on the full three-dimensional data but shown on a vertical slice of the turbulent channel here for simplicity.
• Figure 2: Comparison of ground truth and prediction for streamwise velocity fluctuations. We show a representative horizontal slice at $y^+=12$ for a single instantaneous field, where left and right columns represent the lower and upper channel walls. (Top) simulated $u$ velocity field, (middle) predicted velocity field and (bottom) relative error between the two previous fields. The subscripts $s$ and $p$ correspond to the fields in the reference simulation and the prediction, respectively.
• Figure 3: Instantaneous visualization of the turbulent structures.This Figure shows (views A) the type of turbulent structure, (views B) the SHAP values and (views C) the SHAP values divided by the volume of the corresponding structures. The three-dimensional perspective is presented in images A3, B3, and C3. The side view of the turbulent channel (left) highlights the more influential structures (views A2 and B2). The most important structures per unit of volume are highlighted in views A1 and C1. Note that the highest SHAP values are obtained for large wall-attached ejections, while the moderate-size wall-attached ejections and sweeps exhibit the highest influence per unit of volume. The dashed line marks $y^+=20$, which was used in previous studies Lozano2012 to separate wall-attached and wall-detached structures. The visualization is presented for half of the channel in all the subfigures.
• Figure 4: Magnitude of the SHAP values (left) and SHAP values per unit of volume (right) of the structures for different turbulent events as a function of their volume, expressed in inner units.The SHAP values determine the importance of the various turbulent structures, i.e. the most relevant structures exhibit a higher magnitude of the SHAP value. High-volume ejections are the most important structures for the predictions, while wall-detached structures, mainly medium-size ejections, exhibit a high importance per volume. These structures are often associated with a low Reynolds stress and therefore their importance is typically not identified by the methods based on contribution to the Reynolds-stress profile. Note that the distinction between wall-attached and wall-detached structures is presented in the Supplementary Material.
• Figure 5: Magnitude of the SHAP values as a function of the fractional contribution to the total Reynolds shear stress (left) and same quantities scaled with the structure volume (right).The left figure shows a clear relationship between the SHAP values and the contribution to the Reynolds stress; this correlation is connected with the structure size. The right panel shows some differences, since the highest SHAP values are obtained for structures which do not have the highest fractional contribution to the total Reynolds shear stress. In this panel we highlight different regions: region A (yellow) is a band containing most of the structures, region B (purple) contains the structures with the highest fractional contribution to the total Reynolds shear stress and region C (blue) contains the structures with the highest SHAP per volume. Note that the distinction between wall-attached and wall-detached structures is presented in the Supplementary Material.