Elsevier

Physics Letters A

Volume 382, Issue 1, 5 January 2018, Pages 1-11
Physics Letters A

Visibility graph analysis of wall turbulence time-series

https://doi.org/10.1016/j.physleta.2017.10.027Get rights and content

Highlights

  • We apply the visibility algorithm to analyze turbulent channel flow time-series.

  • Temporal structures of the series are inferred by the network global metrics.

  • Peaks and irregularities of time-series are highlighted by the visibility networks.

  • Different flow dynamics along the wall-normal direction are captured by the metrics.

  • This method represents a promising tool for inhomogeneous turbulent flows analyses.

Abstract

The spatio-temporal features of the velocity field of a fully-developed turbulent channel flow are investigated through the natural visibility graph (NVG) method, which is able to fully map the intrinsic structure of the time-series into complex networks. Time-series of the three velocity components, (u,v,w), are analyzed at fixed grid-points of the whole three-dimensional domain. Each time-series was mapped into a network by means of the NVG algorithm, so that each network corresponds to a grid-point of the simulation. The degree centrality, the transitivity and the here proposed mean link-length were evaluated as indicators of the global visibility, inter-visibility, and mean temporal distance among nodes, respectively. The metrics were averaged along the directions of homogeneity (x,z) of the flow, so they only depend on the wall-normal coordinate, y+. The visibility-based networks, inheriting the flow field features, unveil key temporal properties of the turbulent time-series and their changes moving along y+. Although intrinsically simple to be implemented, the visibility graph-based approach offers a promising and effective support to the classical methods for accurate time-series analyses of inhomogeneous turbulent flows.

Introduction

One of the most challenging research topics in classical physics is represented by turbulent flows. Their great importance is evident through a number of natural phenomena (e.g., rivers, atmospheric and oceanic streams), industrial and civil applications (e.g., flow through pumps, heat exchangers, wake flows of vehicles and aircraft, wind-building interactions) in which turbulence is involved. The study of wall-bounded turbulent flows, in particular, is a very active research field, due to the great attention paid to the fluid–structure interaction. Although deeply studied from a phenomenological and theoretical point of view, the turbulence dynamics, due to their complexity, are still not fully understood [1], [2]. Nowadays, several numerical simulations and experiments are performed, providing a massive amount of spatio-temporal data that needs to be properly examined. Different approaches, mainly relying on statistical techniques, are then typically used to explore and analyze turbulent flows.

Among all the proposed techniques, time-series analysis is a broadly adopted approach to study the temporal evolution of dynamical systems, specifically those with high intrinsic complexity. Different methods, such as Fourier and wavelet transforms [3], [4], as well as nonlinear approaches [5], [6], [7], have been developed so far to extract information from time-series. However, since each method unavoidably loses some information about the temporal structure of the series analyzed, new approaches are continuously required to fill this lack. In the last decades, complex networks, by combining elements from graph theory and statistical physics [8], [9], [10], have turned out to be powerful tools to study complex systems, specifically by mapping time-series to extract non-trivial information [11]. Recently, several improvements were gained in this field and numerous advances were proposed based on different approaches [12], such as correlation [13], [14], visibility [15], [16], phase-space reconstruction [17], [18], recurrence quantification [19], [20], [21], and transition probabilities [22], [23] algorithms.

Beside the well-established applications to Internet, World Wide Web, economy and social dynamics [24], [25], growing attention has been given nowadays to the application of complex networks to fluid flows and different flow regimes have been explored, such as two-phase flows [18], [20], [26], geophysical flows [27], [28], turbulent jets [23], [29], reacting flows [30], as well as fully developed turbulent flows [31], [32] and isotropic turbulence [33], [34].

In this work, the natural visibility algorithm is exploited to investigate the spatio-temporal characterization of a fully-developed turbulent channel flow, solved through a direct numerical simulation (DNS) and available from the Johns Hopkins Turbulence Database (JHTDB) [35], [36]. Time-series of the three velocity components were analyzed at fixed spatial positions, and a single network was built at each point. In so doing, an ensemble of networks was obtained, where each network corresponds to a time-series. This novel approach allows us to capture some important aspects of the temporal structure of the signal and how these features change along the wall-normal direction. In fact, we can systematically extract information about the occurrence and temporal collocation of extreme events (i.e., peaks) and irregularities, which are fundamental features to characterize turbulent flows. The statistical tools classically adopted in turbulence, such as correlation function, higher-order statistics, structure functions, energy spectrum and probability density functions, all fail in preserving and discerning the temporal structure of a time-series (e.g., two different temporal signals can have the same probability density function or energy spectrum). The visibility approach here presented is instead able to fully inherit and point out the temporal structure of the turbulent series: the different temporal dislocation of events such peaks and fluctuations will lead, case by case, to a different network topology.

A systematic approach to highlight temporal features of the time-series through the most significant network metrics is thus proposed and discussed. Particular care is given not only to relate the network topology to the temporal structure of the series, but also paying attention to the physical interpretation of the results. New insights into how the network topology is affected by important temporal features of the mapped signal are thus provided. Specific combinations of the trend of the network metrics are able to shed light into the time-series structure. Furthermore, a qualitative correspondence between the network metrics and the flow dynamics is presented, underlying the ability of the method to identify different flow regions.

Section snippets

Database description

The data here used were extracted from a DNS of a fully developed turbulent channel flow [37], available from the JHTDB [35], [36]. The simulation is performed at Reτ=1000, where Reτ=huτ/ν is the friction velocity Reynolds number, h=1 is the half-channel height, ν=5105Ubh is the viscosity, Ub=1 is the bulk channel velocity, and uτ=5102 is the friction velocity (all physical parameters are dimensionless). Periodic boundary conditions in the streamwise (x) and spanwise (z) directions are

Relating time-series structure and network metrics

It is known that the general structure of time-series is preserved in the topology of the associated natural visibility graphs, as shown by Lacasa et al. [15] and as emerges from successive works [29], [30], [31], [32], [39], [41]. Specifically, periodic time-series are converted into regular networks, i.e. graphs where nodes have constant degrees related to the periods of the series. Fractal series, instead, convert into networks with power-law degree distributions [48]. In particular,

Results

The procedure described in the previous section is adopted to analyze the velocity time-series of the turbulent channel flow, starting from the streamwise component, u, and then considering the other velocity components, v and w.

Conclusions

In this work, the application of the natural visibility graph to time-series of a fully-developed turbulent channel flow was studied. Our attention was focused on the streamwise velocity component, u, although the other velocity components were also explored. Velocity time-series were adopted to build the corresponding networks as the velocity field is one of the most intuitive quantity to characterize a fluid flow. However, the visibility graph method can be applied to other quantities of

Acknowledgements

This work was supported by the scholarship Ernesto e Ben Omega Petrazzini, awarded by the Accademia delle Scienze di Torino, Turin (Italy). A special thank goes to the Accademia delle Scienze and the Petrazzini family. The authors would also like to thank J.G.M. (Hans) Kuerten for the fruitful discussion of the results.

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