Complex Networks Unveiling Spatial Patterns in Turbulence
NNovember 12, 2018 12:34 Scarsoglio˙et˙al˙IJBC˙2016
International Journal of Bifurcation and Chaosc (cid:13)
World Scientific Publishing Company
COMPLEX NETWORKS UNVEILING SPATIAL PATTERNSIN TURBULENCE
STEFANIA SCARSOGLIO
Department of Mechanical and Aerospace Engineering, Politecnico di TorinoCorso Duca degli Abruzzi 24, Torino, [email protected]
GIOVANNI IACOBELLO
Department of Mechanical and Aerospace Engineering, Politecnico di TorinoCorso Duca degli Abruzzi 24, Torino, [email protected]
LUCA RIDOLFI
Department of Environmental, Land and Infrastructure Engineering, Politecnico di TorinoCorso Duca degli Abruzzi 24, Torino, Italyluca.ridolfi@polito.it
Received May 4, 2016; Revised July 18, 2016
Numerical and experimental turbulence simulations are nowadays reaching the size of the so-called big data , thus requiring refined investigative tools for appropriate statistical analyses anddata mining. We present a new approach based on the complex network theory, offering a power-ful framework to explore complex systems with a huge number of interacting elements. Althoughinterest on complex networks has been increasing in the last years, few recent studies have beenapplied to turbulence. We propose an investigation starting from a two-point correlation for thekinetic energy of a forced isotropic field numerically solved. Among all the metrics analyzed,the degree centrality is the most significant, suggesting the formation of spatial patterns whichcoherently move with similar vorticity over the large eddy turnover time scale. Pattern size canbe quantified through a newly-introduced parameter (i.e., average physical distance) and variesfrom small to intermediate scales. The network analysis allows a systematic identification of dif-ferent spatial regions, providing new insights into the spatial characterization of turbulent flows.Based on present findings, the application to highly inhomogeneous flows seems promising anddeserves additional future investigation.
Keywords : complex networks; turbulent flows; time series analysis; spatial correlation; spatiotem-poral patterns
1. Introduction
Turbulence is an important and widely investigated topic, involving everyday life in several natural phe-nomena (e.g., rivers, bird flight and fish locomotion, atmospheric and oceanic currents) and industrial ap-plications (e.g., flow through pumps, turbines, chemical reactors, and aircraft-wing tips). Although studiedfor decades [Frisch, 1995], due to its chaotic and complex nature, several important questions regarding its a r X i v : . [ phy s i c s . f l u - dyn ] J a n ovember 12, 2018 12:34 Scarsoglio˙et˙al˙IJBC˙2016 Scarsoglio et al. spatial characterization, prediction, and control remain mostly unclear [Warhaft, 2002]. In order to achievea better description of its dynamic, nowadays experimental and numerical simulations progressively pro-vide a greater amount of extremely detailed data, which need to be examined and interpreted. There istherefore an increasing urgency of refined investigative tools for appropriate statistical analyses and datamining. Different and interdisciplinary approaches, borrowed from bioinformatics to physical statistics, canhelp exploring data from a complementary and innovative perspective.In the last years, interest in complex network theory has grown enormously, as it offers a syntheticand powerful tool to study complex systems with an elevated number of interacting elements [Albert &Barab´asi, 2002; Watts & Strogatz, 1998; Newman, 2010]. By combining graph theory and statistical physics,the present approach find immediate applications to real existing networks (e.g., Word Wide Web, social,economical and neural connections) as well as in building networks from spatio-temporal data series [Costa et al. , 2011; Boccaletti et al. , 2006]. A relevant example is represented by the climate networks, wheredifferent meteorological series have been transformed into networks to disentangle the global atmosphericdynamics (see, among others, [Yamasaki et al. , 2008; Steinhaeuser et al. , 2012; Scarsoglio et al. , 2013;Sivakumar & Woldemeskel, 2014; Tsonis & Swanson, 2008; Donges et al. , 2009]).In turbulence, few and very recent network-based approaches have been proposed to characterizepatterns in two-phase stratified flows [Gao & Jin, 2009; Gao et al. , 2013, 2015a,b,c,d, 2016], turbulent jets[Shirazi et al. , 2009; Charakopoulos et al. , 2014], as well as reacting [Murugesan & Sujith, 2015] and fullydeveloped turbulent flows [Liu et al. , 2010; Manshour et al. , 2015]. Most of them focused on temporaldata measured in different spatial locations and, by means of the visibility algorithm [Lacasa et al. , 2008]or recurrence plots [Donner et al. , 2011; Marwan et al. , 2009], converted each time series into a network.Because of the promising results so far obtained and the potentiality of the network tools, turbulencenetworks certainly merit further investigation.We here proposed a complex network analysis on a forced isotropic turbulent field solved through directnumerical simulation (DNS), available from the Johns Hopkins Turbulence Database (JHTDB) [Li et al. ,2008; Perlman et al. , 2007]. Differently to what was carried out so far, we did not transform each temporalseries into a network but constructed a single global network from spatio-temporal data. The network wasbuilt starting from a two-point correlation for the turbulent kinetic energy computed over all the couples ofthe selected nodes. In so doing, a unique monolayer network was obtained, whose nodes partially overlap thenumerical grid cells and whose links are active if the distance and statistical interdependence between twonodes satisfy suitably chosen constraints [Donges et al. , 2009]. Correlation-based networks [Donner et al. ,2011; Yang & Yang, 2008] is probably the most used way of applying network science techniques to timeseries, with examples ranging from financial markets [Caraiani, 2013] to brain activity [Stam & Reijneveld,2007]. However, to the best of our knowledge, the application of correlation networks to spatio-temporalturbulent data has not been analyzed to date.Once the network was built, different topological features were analyzed. The degree centrality turnedout to be the most meaningful parameter, suggesting the onset and evolution of spatial patterns whichcoherently move with similar vorticity over the large eddy turnover time scale. A new network metric hereintroduced (i.e., average physical distance) is able to indicate the spatial scale of the turbulent patterns,ranging from small to intermediate scales.
2. Methods2.1.
Johns Hopkins Turbulence Database Description
The forced isotropic turbulence field here used was solved by means of a DNS over 1024 nodes and isavailable from the JHTDB [Li et al. , 2008; Perlman et al. , 2007]. Velocity ( u, v, w ), vorticity ( ω x , ω y , ω z ),and pressure ( p ) fields were computed over a cube of dimension 2 π x 2 π x 2 π . A forcing term was addedto the Navier-Stokes equations so that the total kinetic energy does not decay and, after a transient range,the field can be considered statistically stationary. Once this state was reached, 1024 frames of data wererecorded (time-step=0.002), lasting about one large-eddy turnover time, T L . Energy was injected by keepingthe total energy constant, so that only the integral scale is influenced by the forcing, while the intermediateand the dissipative ranges are not involved. Some statistical characteristics are here given together with aovember 12, 2018 12:34 Scarsoglio˙et˙al˙IJBC˙2016 Complex networks unveiling spatial patterns in turbulence brief physical recall: • Taylor microscale, λ = 0 . • Taylor-scale Reynolds number, Re λ = ( u rms λ ) /ν = 433, is the ratio between inertial and viscous forcesat the Taylor scale, λ ( u rms is the root-mean-square velocity and ν is the kinematic viscosity); • Kolmogorov time scale, τ k = 0 . η = 0 . • integral scale, L = 1 . • large eddy turnover time, T L = 2 .
02, is the time scale over which the largest eddies develop.The JHTDB provides an accurate multi-terabyte and comprehensive data archive, which has been widelyexploited for testing modeling [Li et al. , 2009], structural properties [Lawson & Dawson, 2015], experimentaldata [Fiscaletti et al. , 2014] and statistical analyses [Mishra et al. , 2014].
Complex network metrics
The network measures used in the present work are here summarized [Albert & Barab´asi, 2002; Boccaletti et al. , 2006]. A network is defined by a set V = 1 , ..., N of nodes and a set E of links { i, j } . We assumethat a single link can exist between a pair of nodes. The adjacency matrix , A : A ij = (cid:40) , if { i, j } / ∈ E , if { i, j } ∈ E, (1)accounts whether a link is active or not between nodes i and j . The network is considered as undirected,thus A is symmetric, and no self-loops are allowed ( A ii = 0).The normalized degree centrality of a node i is defined as k i = N (cid:80) j =1 A ij N − , (2)and gives the number of neighbors of the node i , normalized over the total number of possible neighbors( N − K i = k i ( N −
1) as the (non-normalized) degree centrality.The eigenvector centrality , measuring the influence of the node i in the network, is given by x i = 1 λ (cid:88) k A ki x k , (3)with A ki the adjacency matrix and λ its largest eigenvalue [Newman, 2010]. In matrix notation, we canwrite: λx = xA, (4)where the centrality vector x is the left-hand eigenvector of the adjacency matrix A associated with theeigenvalue λ , which is the largest eigenvalue in absolute value.The local clustering coefficient of a node is C i = e (Γ i ) K i ( K i − , (5)ovember 12, 2018 12:34 Scarsoglio˙et˙al˙IJBC˙2016 Scarsoglio et al. where e (Γ i ) is the number of links connecting the vertices within the neighborhood Γ i , and K i ( K i − / i . The local clustering coefficient represents the probability that tworandomly chosen neighbors of a node are also neighbors.The betweenness centrality of a node is BC k = (cid:88) i,j (cid:54) = k σ ij ( k ) σ ij , (6)where σ ij are the number of shortest paths connecting nodes i and j , while σ ij ( k ) represents the numberof shortest paths from i to j through node k . If node k is crossed by a large number of all existing shortestpaths (i.e. if BC k is large), then node k is reputed an important mediator for the information transport inthe network. Modularity Q is a measure of the structure of networks, detecting the presence of communities/modules[Newman & Girvan, 2004]. Q is defined, up to a multiplicative constant, as the fraction of the edges thatfall within the given groups minus the expected such fraction if edges were distributed at random. A highmodularity degree (roughly above 0.3) indicates a strong division of the network into clusters [Newman,2006]. Q can be mathematically quantified as Q = 14 m (cid:88) ij (cid:18) A ij − K i K j m (cid:19) s i s j , (7)where A ij is the adjacency matrix, ( K i K j ) / (2 m ) is the expected number of edges between nodes i and j if edges are placed at random, m is the total number of links in the network, s is a membership variableconsidering that the graph can be partitioned into two communities ( s i = 1 if node i belongs to community1, s i = − / (4 m ) is merely conventional.In the end, we introduce a new metric which is related to the reciprocal physical distance of the networknodes. The neighborhood physical distance , L i , of a node i is the averaged physical distance between node i and its neighborhood Γ i : L i = (cid:80) j ∈ Γ i l ij K i , (8)where l ij is the physical distance between node i and its neighbor j , K i is the degree centrality of node i . Building the network
To build the network, we considered a spherical subdomain with center C = (391 , , r =0 .
24. For all the nodes inside this sphere we computed the kinetic energy time series, E = ( u + v + w ) / R ij . Alinear Pearson correlation was adopted, as it is one of the simplest possible metrics to quantify the level ofstatistical interdependence between the temporal series. To avoid results biased by the network geometry,a link between nodes i and j exists if the following conditions are simultaneously satisfied: • | R ij | > τ , where τ is a suitable threshold; • At least one between nodes i and j lies inside the reference sphere with radius r = 0 .
12 and center C ; • The physical distance between nodes i and j is less or equal to r = 0 . i within the reference sphere had a well-defined region of influence (a spherewith radius 0.12 and centered in the node i itself) where links with other nodes can occur. The region ofovember 12, 2018 12:34 Scarsoglio˙et˙al˙IJBC˙2016 Complex networks unveiling spatial patterns in turbulence influence had the same size for all nodes, so that every node within the reference sphere experienced thesame number of potential links.The size of the reference sphere ( r = 0 .
12) is linked to the Taylor scale, λ = 0 . r ≤ λ ) where the noisy links are not present. Theturbulent field is isotropic as a consequence of the DNS geometry and boundary conditions imposed, thusno preferential directions can be detected. Moreover, since the forcing to keep the total energy constantacts on bigger scales (wavenumber | w | ≤ C . To test the sensitivity of the results, another domain portion was then analyzed, namely aspherical subdomain with radius r = 0 .
24 centered in C (cid:48) = (530 , , τ , was a non-trivial aspect of building the network and had to take intoaccount the goal of both evidencing strong spatial correlations and managing an appropriate number ofnodes. The influence of the threshold has been deeply analyzed in climate networks [Donges et al. , 2009].The threshold τ = 0 . τ values is reported in the Results and Discussionsection.The network is composed by N = 128785 nodes and m = 80920781 links, indicating with N int = 31343the cardinality of nodes inside the reference sphere and with N ext the number of nodes outside of it( N = N int + N ext ). The edge density, ρ ( τ ), is defined as ρ ( τ ) = n ( τ ) N ( N − − N ext ( N ext − , (9)where n ( τ ) is the number of active links when the absolute value of R ij is above the threshold τ for thetwo-point correlation. The denominator accounts for the total number of links of the network, excludinglinks between purely external nodes (links between internal and external nodes are allowed). The edgedensity, ρ , is the ratio between active links above a given threshold τ and the total number of possiblelinks. For the chosen threshold, ρ = 2 . · − . The combined bidimensional edge density, ρ ( τ, l ), isintroduced as ρ ( τ, l ) = n ( τ, l ) l , (10)where l ∈ (0 , .
12] is the physical distance between two nodes, n ( τ, l ) is the number of active links abovethe threshold τ and at a fixed l . The combined bidimensional edge density, ρ , is the ratio of active linksabove a given threshold τ at a fixed distance l and the total number of potential links at the same distance l . A graphical representation of ρ ( τ, l ) is reported in Fig. 1, where high density values are found for smallphysical distances, confirming that at τ = 0 . ρ → l → τ represents the link length distribution.To summarize, ρ evaluates the density of active links independently of their physical lengths, while ρ isthe link density as function of the length.The network analysis presented in the following section is focused on the set of internal nodes, N int , ofthe reference sphere. External nodes, which are part of the network but only exploited to evaluate linksbetween internal and external nodes, are not shown.ovember 12, 2018 12:34 Scarsoglio˙et˙al˙IJBC˙2016 Scarsoglio et al.
Fig. 1. Combined bidimensional edge density, ρ ( τ, l ).Fig. 2. Normalized degree centrality, k . (left) 3D perspective (values higher than 70 % of the maximum value are reportedwith points, the the rest of the network is transparently colored). (center) 2D section on the z = 512 plane. (right) Cumulativedegree distribution function, P cum ( k ), with the exponential and uniform fittings of the data and the corresponding coefficientsof determination, R . A semi-log graph is adopted.
3. Results and Discussion
The properties of the turbulence network are here discussed. The degree centrality was first analyzed,evidencing regions with high values which are clearly distinguishable from the rest of the network. InFig. 2 (left panel) the highest values (above 70 % of the maximum value) are highlighted through a 3Dperspective, while the other values are transparently colored. A 2D diametral section on the plane z = 512 isalso displayed, reporting all k i values (central panel). In the right panel, the cumulative degree distributionfunction, P cum ( k ) = (cid:80) ∞ k (cid:48) = k p ( k (cid:48) ), is shown in a linear-log plot. The degree distribution is adequately fittedby an exponential distribution for low k values ( k < . et al. , 2002; Deng et al. , 2011]. The right-tail has a qualitative downward behavior [Dunne et al. ,2002], with a decay which is faster than an exponential but slower than an uniform distribution. Moreover,the network presents a rich-club effect [Boccaletti et al. , 2006], i.e. high degree vertices connect one to eachother.ovember 12, 2018 12:34 Scarsoglio˙et˙al˙IJBC˙2016 Complex networks unveiling spatial patterns in turbulence z = 512. Other network properties, such as the eigenvector centrality, the local clustering coefficient and thebetweenness centrality, are reported and compared with the degree centrality in Fig. 3, as sections of theplane z = 512. The eigenvector centrality (panel b) carried the same information of the degree centrality(a), confirming the presence of distinct spatial regions with high correlation. To this end, it should be notedthat a completely random flow field would result in a highly disconnected network, which in turn wouldentail a spotted distribution for the centrality indexes, with the most part of values close to zero. Thelocal clustering coefficient (panel c) was poorly related to the degree centrality, as in general happens inspatial networks [Boccaletti et al. , 2006]. The betweenness centrality (panel d) presented quite low valuesand a spotted distribution over the section, which weakly correlates to the degree centrality. No sources ofinhomogeneity and anisotropy were present in the field, thus there are no preferential pathways transportingthe information. This translated into a spotted distribution, with significantly high BC gradient values,which is scarcely informative from the point of view of pattern formation.The modularity value of the present network is about Q = 0 .
31, and twenty-eight communities weredetected through the Newman algorithm [Newman & Girvan, 2004; Newman, 2006], where Q = (cid:80) c =1 q ( c )and q ( c ) is the modularity of a single community. Modularity is not uniformly distributed over the commu-nities (Fig. 4, left panel), as the last eight modules have q values close to zero, while the first community hasthe highest q value (0.055), which is about 18% of the total modularity value, Q . Nodes belonging to theovember 12, 2018 12:34 Scarsoglio˙et˙al˙IJBC˙2016 Scarsoglio et al. community with the highest modularity q are reported in Fig. 4 (right panel, red points). This communityis the largest in terms of cardinality and detects a cluster of nodes which are physically close one to eachother, representing a wide coherent region sharing the same properties. Moreover, high degree centralityvalues are usually found for nodes belonging to high-order communities. In particular, about 86% of thenodes highlighted in Fig. 2 (left panel) falls within the first eight communities. The latest communitiesare instead less populated with nodes having medium to low degree centrality values. From the presentfindings, nodes seem to be partitioned into communities based on their reciprocal physical distance and ontheir connection to high centrality nodes. Fig. 4. (left) Modularity distribution over the twenty-eight communities, q ( c ). (right) Nodes belonging to the communitywith the highest modularity value ( q = 0 . The most meaningful parameter turned out to be the degree centrality together with the eigenvectorcentrality, both direct measures of the importance of a node in the network. In order to interpret thenetwork results in terms of physical properties of the turbulent field, we considered the highest degreecentrality node (node HDC, k = 7 . · − , coordinates (385,401,508)) and another with very low degreecentrality (node LDC, k = 3 . · − , coordinates (372,387,510)). For both nodes we evaluated theirneighborhoods (Γ( HDC ) = 10180 and Γ(
LDC ) = 393) and the average physical distance ( L HDC = 7 . · − , L LDC = 2 . · − ). In Fig. 5 (left) HDC and LDC nodes are shown together with the respectiveneighborhoods. We then considered nodes A and B at an intermediate physical distance 6 . · − (10grid cells) from nodes HDC and LDC, respectively. Nodes A and B have normalized degree centralityvalues k = 4 . · − and k = 1 . · − , respectively. We evaluated the temporal series of the vorticitymodulus ( | ω | = (cid:113) ω x + ω y + ω z ) for the two pairs of nodes, (HDC-A) and (LDC-B). The couple (HDC-A)presented a strong temporal correlation for | ω | ( R = 0 .
92) and the two time series showed values closeone to the other. The couple (LDC-B) had a much weaker correlation for | ω | ( R = 0 .
68) and the twotime series often reached very different specific values (Fig. 5, right). The behaviour of the pairs (HDC-A)and (LDC-B) is representative of high degree centrality and low degree centrality regions, since analogouscomparisons were found for many other couples of nodes. In Table 1 examples of couples of nodes showingthe mentioned behaviours are shown. Thus, we can say that high degree centrality values indicate regionswith the same instantaneous vorticity, that is turbulent patterns coherently moving over the time scale T L .Moreover, there is a direct correlation between the degree centrality, k , and the average physical distance, L , of a node. L gives the order of magnitude of the spatial patterns identified by the k i distribution. Forovember 12, 2018 12:34 Scarsoglio˙et˙al˙IJBC˙2016 Complex networks unveiling spatial patterns in turbulence node HDC, L = 7 . · − , for node LDC, L = 2 . · − , meaning that the size of the patterns rangesbetween the dissipative scale and the Taylor microscale. Fig. 5. (left) High degree centrality (HDC) and low degree centrality (LDC) nodes shown together with their neighborhoods(Γ(
HDC ) in red, Γ(
LDC ) in blue). Nodes A and B are at a distance 6 . · − from nodes HDC and LDC, respectively( l HDC,A = l LDC,B = 6 . · − , 10 grid cells). (right) Time series of the vorticity modulus | ω | are shown for the pairs(A-HDC) and (B-LDC) with the corresponding correlation coefficient, R .Table 1. Examples of couples of nodes belonging to high and low degree centralityregions. Nodes Distance RHigh HDC =(382,405,515), A =(382,405,507) 8 grid cells 0.96degree centrality HDC =(389,378,522), A =(389,390,522) 12 grid cells 0.93region HDC =(376,403,509), A =(392,403,509) 16 grid cells 0.94Low LDC =(375,396,520), B =(375,388,520) 8 grid cells 0.58degree centrality LDC =(403,388,497), B =(403,388,509) 12 grid cells 0.61region LDC =(374,390,521), B =(390,390,521) 16 grid cells 0.49 As mentioned in the Methods section, the turbulent energy field is fundamental to characterize thenetwork. We checked a posteriori that from a localized information - such as the energy time series ata fixed point of the field - the network is able to infer the spatial behaviour of the surroundings, whichinvolves velocity gradients, i.e., the vorticity field. Building the network from the vorticity field would haveintroduced spatial variations, by requiring a higher order information and leading to analogous results interms of network.
Sensitivity Analysis
In the end, we performed a sensitivity analysis of the results regarding the reference sphere ( r = 0 . C = (391 , , τ , for the link activation. Then, anothernetwork based on a different reference sphere ( r = 0 .
12 and centered in C (cid:48) = (530 , , τ = 0 .
9, networks for two different values were analyzed, τ = 0 .
85 and τ = 0 .
95. In Fig. 6 thenormalized degree centrality on the plane z = 512 is reported for the three τ values. In Table 2, sometopological and spatial features of the three networks are compared. As τ decreased, the number of activelinks, m , increased at a faster rate than the size of the network (total number of nodes, N ), while theovember 12, 2018 12:34 Scarsoglio˙et˙al˙IJBC˙2016 Scarsoglio et al.
Fig. 6. Normalized degree centrality, k i , on the z = 512 plane. (left) τ = 0 .
85. (center) τ = 0 .
9. (right) τ = 0 .
95. Differentcolor scales are adopted. cardinality N int does not change in any case. As a consequence, the degree centrality averagely increasedand the high k i regions were more spatially expanded for decreasing τ values. The average physical distance, L , for the nodes HDC = (385 , , LDC = (372 , , τ , similarlyto what happens for the degree centrality. Despite the specific values assumed and the qualitative changesinduced by the three threshold values, the spatial pattern detection is essentially independent from thechoice of the threshold. Weighted networks, though computationally more expensive, can be adopted infuture work as they can made results more robust against threshold variations. Table 2. Topological and spatial features of the networkswith τ = 0 . , . , .
95. ¯ k = (cid:80) Ni =1 k i /N is the mean degreecentrality of the network, ¯ L = (cid:80) Ni =1 L i /N is the averagedphysical distance computed as mean value of the network. HDC = (385 , , LDC = (372 , , τ = 0 . τ = 0 . τ = 0 . N m k . · − . · − . · − ¯ L . · − . · − . · − L HDC . · − . · − . · − L LDC . · − . · − . · − A different spherical subdomain with radius r = 0 .
24 centered in C (cid:48) = (530 , , τ = 0 . C and C (cid:48) is about 1.93, that largely exceeds the integral scale, L = 1 . C and C (cid:48) are far enough so that the two influence regions do not physically overlap. The new referencesphere has radius r = 0 .
12 and center C (cid:48) = (530 , , C and this results into a newnetwork having different cardinality and topology.In Fig. 7 the network results in terms of normalized degree centrality, eigenvector centrality, average physicaldistance, and local clustering coefficient are reported on a 2D section of the plane z = 475. In Table 3,structural properties of the networks centered in C and C (cid:48) are given for comparison. Despite the differentshape assumed by the network metrics, results are of the same order of magnitude of those observed in thenetwork centered in C = (391 , , Complex networks unveiling spatial patterns in turbulence C (cid:48) = (530 , , z = 475.Table 3. Topological features of the networks centered in C = (391 , , C (cid:48) = (530 , , k = (cid:80) Ni =1 k i /N isthe mean degree centrality of the network, ¯ L = (cid:80) Ni =1 L i /N is theaveraged physical distance computed as mean value of the network,¯ C = (cid:80) Ni =1 C i /N is the global clustering coefficient. C = (391 , , C (cid:48) = (530 , , N m k . · − . · − ¯ L . · − . · − ¯ C . · − . · −
4. Conclusions
In the present work, the complex networks instruments were applied to analyze a forced isotropic turbulentfield. Differently to recent literature studies which transformed each time series into a different network,here a single global network was built from spatio-temporal data following a two-point correlation approachcarried out for all the pairs of selected nodes. The kinetic energy time series of the grid cells was chosen todefine a monolayer network. A link between two nodes is active if the distance and statistical interdepen-dence between two nodes are above suitably selected thresholds. Degree centrality, k , and average physicaldistance, L , were the best metrics able to quantify the spatial dynamics. High degree centrality regionsovember 12, 2018 12:34 Scarsoglio˙et˙al˙IJBC˙2016 REFERENCES evidenced spatial patterns coherently moving with similar vorticity over the large eddy turnover time scale.An indication of the spatial size of these regions was suggested by the average physical distance, varyingfrom small scales up to the Taylor microscale.The network analysis allowed us to handle big data and systematically identify different spatial regions.This goal would not have been so easily feasible without the use of the network metrics, which synthesizedin a single framework a huge amount of detailed information. The proposed approach can suggest newinsights into the spatial characterization of turbulent flows and, based on present findings, the applicationto highly inhomogeneous flows - such as compressible or wall flows - seems to be promising and is worthadditional future investigation.
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