Linear stability analysis of purely elastic travelling wave solutions in pressure driven channel flows

Recent studies of pressure-driven flows of dilute polymer solutions in straight channels demonstrated the existence of two-dimensional coherent structures that are disconnected from the laminar state and appear through a sub-critical bifurcation from infinity. These travelling-wave solutions were suggested to organise the phase-space dynamics of purely elastic and elasto-inertial chaotic channel flows. Here, we consider a wide range of parameters, covering the purely-elastic and elasto-inertial cases, and demonstrate that the two-dimensional travelling-wave solutions are unstable when embedded in sufficiently wide three-dimensional domains. Our work demonstrates that studies of purely elastic and elasto-inertial turbulence in straight channels require three-dimensional simulations, and no reliable conclusions can be drawn from studying strictly two-dimensional channel flows.Comment: 10 pages, 5 page

Using Machine Learning to predict extreme events in the Hénon map

Machine Learning (ML) inspired algorithms provide a flexible set of tools for analyzing and forecasting chaotic dynamical systems. We here analyze the performance of one algorithm for the prediction of extreme events in the two-dimensional H\'enon map at the classical parameters. The task is to determine whether a trajectory will exceed a threshold after a set number of time steps into the future. This task has a geometric interpretation within the dynamics of the H\'enon map, which we use to gauge the performance of the neural networks that are used in this work. We analyze the dependence of the success rate of the ML models on the prediction time $T$ , the number of training samples $N_T$ and the size of the network $N_p$. We observe that in order to maintain a certain accuracy, $N_T \propto exp(2 h T)$ and $N_p \propto exp(hT)$, where $h$ is the topological entropy. Similar relations between the intrinsic chaotic properties of the dynamics and ML parameters might be observable in other systems as well.Comment: 9 pages, 12 figure

Explaining wall-bounded turbulence through deep learning

Despite its great scientific and technological importance, wall-bounded turbulence is an unresolved problem that requires new perspectives to be tackled. One of the key strategies has been to study interactions among the 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 in a turbulent channel is used to predict the velocity field in time through a convolutional neural network. 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 quantifying the importance of the Reynolds-stress structures, finding a connection between these structures and the dynamics of the flow. The process, based on deep-learning explainability, has the potential to shed light on numerous fundamental phenomena of wall-bounded turbulence, including the objective definition of new types of flow structures

Effects of anisotropy on the geometry of tracer particle trajectories in turbulent flows

Using curvature and torsion to describe Lagrangian trajectories gives a full description of these as well as an insight into small and large time scales as temporal derivatives up to order 3 are involved. One might expect that the statistics of these properties depend on the geometry of the flow. Therefore, we calculated curvature and torsion probability density functions (PDFs) of experimental Lagrangian trajectories processed using the Shake-the-Box algorithm of turbulent von K\'arm\'an flow, Rayleigh-B\'enard convection and a zero-pressure-gradient turbulent boundary layer over a flat plate. The results for the von K\'arm\'an flow compare well with previous experimental results for the curvature PDF and numerical simulation of homogeneous and isotropic turbulence for the torsion PDF. Results for Rayleigh-B\'enard convection agree with those obtained for K\'arm\'an flow, while results for the logarithmic layer within the boundary layer differ slightly, and we provide a potential explanation. To detect and quantify the effect of anisotropy either resulting from a mean flow or large-scale coherent motions on the geometry or tracer particle trajectories, we introduce the curvature vector. We connect its statistics with those of velocity fluctuations and demonstrate that strong large-scale motion in a given spatial direction results in meandering rather than helical trajectories

Effects of anisotropy on the geometry of tracer particle trajectories in turbulent flows

Using curvature and torsion to describe Lagrangian trajectories gives a full description of these as well as an insight into small and large time scales as temporal derivatives up to order 3 are involved. One might expect that the statistics of these properties depend on the geometry of the flow. Therefore, we calculated curvature and torsion probability density functions (PDFs) of experimental Lagrangian trajectories processed using the Shake-the-Box algorithm of turbulent von Kármán flow, Rayleigh-Bénard convection and a zero-pressuregradient boundary layer over a flat plate. The results for the von-Kármán flow compare well with experimental results for the curvature PDF and numerical simulation of homogeneous and isotropic turbulence for the torsion PDF. For the experimental Rayleigh-Bénard convection, the power law tails found agree with those measured for von-Kármán flow. Results for the logarithmic layer within the boundary layer differ slightly, we give some potential explanation below. To detect and quantify the effect of anisotropy either resulting from a mean flow or large-scale coherent motions on the geometry or tracer particle trajectories, we introduce the curvature vector. We connect its statistics with those of velocity fluctuations and demonstrate that strong large-scale motion in a given spatial direction results in meandering rather than helical trajectories

Interpreted machine learning in fluid dynamics: Explaining relaminarisation events in wall-bounded shear flows

Machine Learning (ML) is becoming increasingly popular in fluid dynamics. Powerful ML algorithms such as neural networks or ensemble methods are notoriously difficult to interpret. Here, we introduce the novel Shapley Additive Explanations (SHAP) algorithm (Lundberg & Lee, 2017), a game-theoretic approach that explains the output of a given ML model, in the fluid dynamics context. We give a proof of concept concerning SHAP as an explainable AI method providing useful and human-interpretable insight for fluid dynamics. To show that the feature importance ranking provided by SHAP can be interpreted physically, we first consider data from an established low-dimensional model based on the self-sustaining process (SSP) in wall-bounded shear flows, where each data feature has a clear physical and dynamical interpretation in terms of known representative features of the near-wall dynamics, i.e. streamwise vortices, streaks and linear streak instabilities. SHAP determines consistently that only the laminar profile, the streamwise vortex, and a specific streak instability play a major role in the prediction. We demonstrate that the method can be applied to larger fluid dynamics datasets by a SHAP evaluation on plane Couette flow in a minimal flow unit focussing on the relevance of streaks and their instabilities for the prediction of relaminarisation events. Here, we find that the prediction is based on proxies for streak modulations corresponding to linear streak instabilities within the SSP. That is, the SHAP analysis suggests that the break-up of the self-sustaining cycle is connected with a suppression of streak instabilities.Comment: 29 pages, 10 figure

Data for "Interpreted machine learning in fluid dynamics" Lellep et al., Journal of Fluid Mechanics, 2022

This dataset is the supplementary data for the publication https://arxiv.org/abs/2102.05541. Machine Learning (ML) is becoming increasingly popular in fluid dynamics. Powerful ML algorithms such as neural networks or ensemble methods are notoriously difficult to interpret. Here, we introduce the novel Shapley additive explanations (SHAP) algorithm (Lundberg & Lee, Advances in Neural Information Processing Systems, 2017, pp. 4765–4774), a game-theoretic approach that explains the output of a given ML model in the fluid dynamics context. We give a proof of concept concerning SHAP as an explainable artificial intelligence method providing useful and human-interpretable insight for fluid dynamics. To show that the feature importance ranking provided by SHAP can be interpreted physically, we first consider data from an established low-dimensional model based on the self-sustaining process (SSP) in wall-bounded shear flows, where each data feature has a clear physical and dynamical interpretation in terms of known representative features of the near-wall dynamics, i.e. streamwise vortices, streaks and linear streak instabilities. SHAP determines consistently that only the laminar profile, the streamwise vortex and a specific streak instability play a major role in the prediction. We demonstrate that the method can be applied to larger fluid dynamics datasets by a SHAP evaluation on plane Couette flow in a minimal flow unit focussing on the relevance of streaks and their instabilities for the prediction of relaminarisation events. Here, we find that the prediction is based on proxies for streak modulations corresponding to linear streak instabilities within the SSP. That is, the SHAP analysis suggests that the break-up of the self-sustaining cycle is connected with a suppression of streak instabilities.Lellep, Martin; Linkmann, Moritz. (2022). Data for "Interpreted machine learning in fluid dynamics" Lellep et al., Journal of Fluid Mechanics, 2022, [dataset]. https://doi.org/10.7488/ds/3450

Effects of anisotropy on the geometry of tracer particle trajectories in turbulent flows

Using curvature and torsion to describe Lagrangian trajectories gives a full description of these as well as an insight into small and large time scales as temporal derivatives up to order 3 are involved. One might expect that the statistics of these properties depend on the geometry of the flow. Therefore, we calculated curvature and torsion probability density functions (PDFs) of Lagrangian trajectories obtained from experimental data using the Shake-the-Box algorithm. We analyse three datasets, turbulent von Kármán flow, Rayleigh-Bénard convection and a zero-pressure-gradient (ZPG) boundary layer over a flat plate. The results for the von Kármán flow compare well with experimental results for the curvature PDF and numerical simulation of homogeneous and isotropic turbulence for the torsion PDF. For Rayleigh-Bénard convection, the power law tails found agree with those measured for von Kármán flow. Results for the logarithmic layer within the boundary layer differ. To detect and quantify the effect of anisotropy either resulting from a mean flow or large-scale coherent motions on the geometry of tracer particle trajectories, we introduce the curvature vector. We connect its statistics with those of velocity fluctuations and demonstrate that strong large-scale motion in a given spatial direction results in meandering rather than helical trajectories. For the turbulent boundary layer, this is commensurate with the current understanding of turbulent superstructures