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

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 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 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.

Validation of symmetry-induced high moment velocity and temperature scaling laws in a turbulent channel flow

The symmetry-based turbulence theory has been used to derive new scaling laws for the streamwise velocity and temperature moments of arbitrary order. For this, it has been applied to an incompressible turbulent channel flow driven by a pressure gradient with a passive scalar equation coupled in. To derive the scaling laws, symmetries of the classical Navier-Stokes and the thermal energy equations have been used together with statistical symmetries, i.e., the statistical scaling and translation symmetries of the multipoint moment equations. Specifically, the multipoint moments are built on the instantaneous velocity and temperature fields other than in the classical approach, where moments are based on the fluctuations of these fields. With this instantaneous approach, a linear system of multipoint correlation equations has been obtained, which greatly simplifies the symmetry analysis. The scaling laws have been derived in the limit of zero viscosity and heat conduction, i.e., Re_{τ}→∞ and Pr>1, and they apply in the center of the channel, i.e., they represent a generalization of the deficit law, thus extending the work of Oberlack et al. [Phys. Rev. Lett. 128, 024502 (2022)0031-900710.1103/PhysRevLett.128.024502]. The scaling laws are all power laws, with the exponent of the high moments all depending exclusively on those of the first and second moments. To validate the new scaling laws, the data from a large number of direct numerical simulations (DNS) for different Reynolds and Prandtl numbers have been used. The results show a very high accuracy of the scaling laws to represent the DNS data. The statistical scaling symmetry of the multipoint moment equations, which characterizes intermittency, has been the key to the new results since it generates a constant in the exponent of the final scaling law. Most important, since this constant is independent of the order of the moments, it clearly indicates anomalous scaling.

A deep-learning approach for reconstructing 3D turbulent flows from 2D observation data

Turbulence is a complex phenomenon that has a chaotic nature with multiple spatio-temporal scales, making predictions of turbulent flows a challenging topic. Nowadays, an abundance of high-fidelity databases can be generated by experimental measurements and numerical simulations, but obtaining such accurate data in full-scale applications is currently not possible. This motivates utilising deep learning on subsets of the available data to reduce the required cost of reconstructing the full flow in such full-scale applications. Here, we develop a generative-adversarial-network (GAN)-based model to reconstruct the three-dimensional velocity fields from flow data represented by a cross-plane of unpaired two-dimensional velocity observations. The model could successfully reconstruct the flow fields with accurate flow structures, statistics and spectra. The results indicate that our model can be successfully utilised for reconstructing three-dimensional flows from two-dimensional experimental measurements. Consequently, a remarkable reduction in the complexity of the experimental setup and the storage cost can be achieved.

Turbulence Statistics of Arbitrary Moments of Wall-Bounded Shear Flows: A Symmetry Approach

The calculation of turbulence statistics is considered the key unsolved problem of fluid mechanics, i.e., precisely the computation of arbitrary statistical velocity moments from first principles alone. Using symmetry theory, we derive turbulent scaling laws for moments of arbitrary order in two regions of a turbulent channel flow. Besides the classical scaling symmetries of space and time, the key symmetries for the present work reflect the two well-known characteristics of turbulent flows: non-Gaussianity and intermittency. To validate the new scaling laws we made a new simulation at an unprecedented friction Reynolds number of 10 000, large enough to test the new scaling laws. Two key results appear as an application of symmetry theory, which allowed us to generate symmetry invariant solutions for arbitrary orders of moments for the underlying infinite set of moment equations. First, we show that in the sense of the generalization of the deficit law all moments of the streamwise velocity in the channel center follow a power-law scaling, with exponents depending on the first and second moments alone. Second, we show that the logarithmic law of the mean streamwise velocity in wall-bounded flows is indeed a valid solution of the moment equations, and further, all higher moments in this region follow a power law, where the scaling exponent of the second moment determines all higher moments. With this we give a first complete mathematical framework for all moments in the log region, which was first discovered about 100 years ago.

Influence of geometrical parameters on the linear stability of a Bénard-Marangoni problem

A linear stability analysis of a thin liquid film flowing over a plate is performed. The analysis is performed in an annular domain when momentum diffusivity and thermal diffusivity are comparable (relatively low Prandtl number, Pr=1.2). The influence of the aspect ratio (Γ) and gravity, through the Bond number (Bo), in the linear stability of the flow are analyzed together. Two different regions in the Γ-Bo plane have been identified. In the first one the basic state presents a linear regime (in which the temperature gradient does not change sign with r). In the second one, the flow presents a nonlinear regime, also called return flow. A great diversity of bifurcations have been found just by changing the domain depth d. The results obtained in this work are in agreement with some reported experiments, and give a deeper insight into the effect of physical parameters on bifurcations.

Codimension-three bifurcations in a Bénard-Marangoni problem

This Brief Report studies the linear stability of a thermoconvective problem in an annular domain for relatively low (~1) Prandtl (viscosity effects) and Biot (heat transfer) numbers. The four possible patterns for the instabilities, namely, hydrothermal waves of first and second class, longitudinal rolls, and corotating rolls, are present in a small region of the Biot-Prandtl plane. This region can be split in four zones, depending on the sort of instability found. The boundary of these four zones is composed of codimension-two points. Authors have also found two codimension-three points, where some of the former curves intersect. Results shown in this Brief Report clarify some reported experiments, predict new instabilities, and, by giving a deeper insight into how physical parameters affect bifurcations, open a gateway to control those instabilities.

Basic networks: definition and applications

We define basic networks as the undirected subgraphs with minimal number of units in which the distances (geodesics, minimal path lengths) among a set of selected nodes, which we call seeds, in the original graph are conserved. The additional nodes required to draw the basic network are called connectors. We describe a heuristic strategy to find the basic networks of complex graphs. We also show how the characterization of these networks may help to obtain relevant biological information from highly complex protein-protein interaction data.

Bifurcation diversity of dynamic thermocapillary liquid layers

We study, from the theoretical point of view, instabilities appearing in a liquid layer, where a dynamic flow is imposed through a nonzero temperature gradient at the bottom. Experimentally many interesting dynamical behaviors have been discovered in this system. In this Brief Report we prove that the basic solution can display great richness of bifurcations which are controlled by heat related parameters. Different kinds of spatially extended and localized structures appear, which are both stationary or oscillatory. These last ones can present amazing patterns such as squares or spirals. Also competing solutions at codimension-two bifurcation points have been found: stationary radial rolls with different wave numbers, radial rolls with hydrothermal waves, and hydrothermal waves with different wave numbers. Remarkably our results recover many features of numerous reported experiments, predict new instabilities, and by giving a deeper insight into how physical parameters contribute to bifurcations, open a gateway to control those instabilities.