Experimental investigation of vertical turbulent transport of a passive scalar in a boundary layer: statistics and visibility graph analysis

Exp erimen tal in v estigation of v ertical turbulen t transp ort of a passiv e scalar in a b oundary la y er: statistics and visibilit y graph analysis G. Iacob ello ∗ and S. Scarsoglio Dep artment of Me chanic al and A er osp ac e Engine ering, Polite cnic o di T orino, T urin, Italy M. Marro and P . Salizzoni L ab or atoir e de M ´ ec anique des Fluides et d’A c oustique, University of Lyon, CNRS UMR 5509, ´ Ec ole Centr ale de Lyon, INSA Lyon, ´ Ecul ly, F r anc e L. Ridolfi Dep artment of Envir onmental, L and and Infr astructur e Engine ering, Polite cnic o di T orino, T urin, Italy 1 Abstract The dynamics of a passiv e scalar plume in a turbulent boundary lay er is exp erimen tally in vesti- gated via vertical turbulent transp ort time-series. Exp erimental data are acquired in a rough-w all turbulen t b oundary lay er that dev elops in a recirculating wind tunnel set-up. Tw o source sizes in an elev ated p osition are considered in order to inv estigate the influence of the emission conditions on the plume dynamics. The analysis is fo cused on the effects of the meandering motion and the relativ e disp ersion of the plume w ith resp ect to its cen ter of mass. First, classical statistics are in vestigated. W e found that (in accordance with previous studies) the meandering motion is the main responsible for differences in the v ariance and intermittency , as well as the kurtosis and p o w er sp ectral densit y , betw een the t wo source sizes. On the con trary , the mean and the skewness are slightly affected by the emission conditions. With the aim to characterize the te mporal struc- ture of the turbulen t transp ort series, the visibility algorithm is exploited to carry out a complex net work-based analysis. In particular, t w o net work metrics – the a verage p eak occurrence and the assortativity coefficient – are analysed, as they are able to capture the temp oral o ccurrence of extreme ev en ts and their relative in tensit y in the series. The effects of the meandering motion and the relativ e disp ersion of the plume are discussed in the view of the net work metrics, rev ealing that a stronger meandering motion is associated with higher v alues of b oth the av erage p eak o ccurrence and the assortativity co efficien t. The net work-based analysis adv ances the level of information of classical statistics, by c haracterizing the impact of the emission conditions on the temp oral struc- ture of the signals in terms of extreme even ts (namely , peaks and pits) and their relative intensit y . In this wa y , complex netw orks provide – through the ev aluation of netw ork metrics – an effective to ol for time-series analysis of exp erimen tal data. ∗ gio v anni.iacob ello@polito.it 2 I. INTR ODUCTION The release in the atmospheric b oundary lay er of flammable or toxic substances, as w ell as the disp ersion of p ollutan ts, need to b e carefully addressed due to their significant en vi- ronmen tal and health impact. T o this end, differen t n umerical and exp erimen tal strategies ha ve b een adopted so far [ 1 ], in order to inv estigate the relation b et ween turbulence dynam- ics, release conditions and one-point probability densit y functions (PDFs) of the p ollutan t concen tration. In particular, several works inv estigated the relation b et w een statistical mo- men ts, in order to infer the corresp onding PDF [ 2 – 8 ]. F or example, Chatwin and Sul livan [ 2 ] in vestigated the relation b etw een mean and standard deviation of a passive scalar in a shear flow, while Mole and Clarke [ 3 ] found a functional dep endency b et ween the third- and fourth-order momen ts. In other works, instead, the form of the PDF of the concentration w as directly inv estigated (see, among others, [ 9 , 10 ]), with implications on the passive scalar mo delling. Although sev eral set-up configurations can b e adopted, the release of a contaminan t from a p oin t source in a turbulen t b oundary lay er provides an accurate representation of a t ypical plume disp ersion pro cess in the atmospheric b oundary la yer [ 1 ]. F rom a mo delling p erspective, Giffor d [ 11 ] describ ed the dynamics of a plume con tin uously emitted b y a p oin t- source as mainly gov erned by t w o mechanisms: the meandering motion of the instantaneou s cen ter of mass of the plume, and the relative disp ersion – i.e., spreading – of the plume with resp ect to its cen ter of mass. The plume meandering motion is due to turbulen t length scales larger than the plume size. Namely , only the large scale eddies of the turbulent flo w are able to make the plume meander in space. The tw o main parameters that affect the meandering of a dev eloping plume are the size and the distance of the source from the ground. F or an elev ated source, plumes emitted by a smaller source size are affected by a wider range of turbulent length scales (th us implying a stronger meandering motion) with resp ect to a plume emitted b y a larger source size. As shown in previous studies [ 12 , 13 ], small v ariations of the source size significan tly affect the role of meandering in the plume disp ersion, b y inducing v ariations in the (one-p oin t) concentration statistics up to streamwise distances of the order of a hundred times the source size. F or a ground-level source, instead, the plume dynamics is sligh tly affected by the source size [ 12 – 14 ] because close to the ground turbulen t length scales are t ypically of the same order of magnitude of the source size. The effect of 3 turbulen t eddies nominally smaller than the plume size, instead, is to contribute to the lo cal mixing of the concen tration field, th us promoting the relative disp ersion of the plume. In this w ork, w e address the same issue studied by F ackr el l and R obins [ 12 ] and Nir oni et al. [ 13 ] to inv estigate the spatio-temp oral dev elopment of a passive scalar plume, emitted from an elev ated p oint-source of v arying size. In particular, exp erimen tal measuremen ts of v elo cit y and passiv e scalar concen tration are p erformed in a turbulent b oundary lay er ov er a rough-wall, whic h is intended to represen t the disp ersion pro cess of a passive scalar in the atmospheric b oundary lay er. The presen t analysis highligh ts the effects of the meandering and the relativ e disp ersion on the w all-normal turbulent t ransp ort of the passiv e scalar. The turbulen t transp ort is here inv estigated as it pla ys a key role into the interpla y b et w een the turbulence v elo cit y field and the concentration of passiv e scalar. In fact, due to the presence of the ground, the extent to whic h the passiv e scalar is transp orted along the wall-normal direction is a fundamen tal asp ect for the disp ersion characterization. Sp ecifically , the role of the differen t source size on the plume dynamics, as w ell as the plume spatial ev olution along the streamwise and w all-normal co ordinates, is emphasized throughout the study . In order to inv estigate the plume dynamics, t wo approac hes are carried out: (i) we obtain statistics of w all-normal turbulent transp ort, thus enric hing the b enchmark for a disp ersing plume in a rough-wall set-up [ 12 , 13 ]; (ii) a complex net work-based analysis is p erformed, in order to adv ance the level of information of classical statistical tools, thus rev ealing non-trivial insights into the temp oral structure of the signals. Although different researc h fields ha v e tak en adv antage of netw ork science during last decades (e.g., so cial, biological, or tec hnological netw orks [ 15 ]), only recen tly complex netw orks hav e emerged as an effective framew ork also to study fluid flows. The main applications of netw ork science to fluid flows in volv e the study of tw o-phase flows [ 16 , 17 ], turbulen t jets [ 18 , 19 ], isotropic and w all-b ounded turbulence [ 20 – 22 ], reacting flows [ 23 , 24 ], Lagrangian mixing [ 25 , 26 ] and geoph ysical flows [ 27 , 28 ]. Among several techniques that hav e b een dev elop ed so far to study time-series by means of complex net w orks [ 29 ], the visibilit y graph approac h [ 30 ] was here adopted since it is a simple but p ow erful to ol to extract non-trivial insigh ts in to the non-linear pro cess from which the time-series are obtained [ 29 ]. The pap er is organized as follows. Section I I includes the description of exp erimen tal set-up and the measurement tec hniques (Section I I A ), as w ell as the data pre-pro cessing (Section I I B ). The statistical analysis of the wall-normal turbulent transp ort is rep orted 4 FIG. 1. Sk etch of the TBL set-up; the plume is illustrated in green, while the horizontal dash-dot line refers to the source axis. The sym b ols are defined in the main text. Two large-scale eddies of (Eulerian) characteristic size λ are also depicted as rotating arrows. in Section I I I . Typical statistical quan tities are ev aluated, suc h as the first four moments (i.e., the mean v alue, the standard deviation, the skewness and kurtosis), the p o wer spectral densit y and the intermittency factor. The complex netw ork analysis is shown in Section IV : the main concepts of complex netw ork and visibility graph are outlined in Section IV A , while the metrics definition and interpretation is discussed in Section IV B . Two netw ork-metrics are inv estigated – i.e., the a verage p eak o ccurrence and the assortativity co efficien t [ 21 ] – with the aim to c haracterize the temp oral structure of the turbulen t transp ort time-series. The av erage p eak o ccurrence and the assortativity co efficient are here selected as they are able to highlight the temp oral structure of extreme ev ents (i.e., p eaks and pits) and their relativ e intensit y in a time-series. The results of the visibilit y-netw ork approach are shown in Section IV C , while the conclusions are dra wn in Section V . I I. METHODS A. Exp erimen t set-up and measuremen ts A neutrally-stratified atmospheric turbulen t b oundary lay er (TBL) was generated in a recirculating wind tunnel of the Lab oratoire de M ´ ecanique des Fluides et d’Acoustique at the ´ Ecole Centrale de Ly on, in F rance. The set-up and the measurement to ols are the same as that adopted b y Nir oni et al. [ 13 ] (see Appendix A ). Ho wev er, measuremen ts were p erformed in a wind tunnel which smaller than that used by Nir oni et al. [ 13 ], with a w orking section that is 9 m long, 1 m wide, and 0 . 7 m high. A ro w of Irwin spires [ 31 ] were placed at the b eginning of the test section, while cubic roughness elements with size h r = 0 . 02 m were 5 uniformly displaced on the flo or. As a result, a TBL of free-stream velocity u ∞ = 4 . 94 m/s and thic kness δ = 0 . 314 m was generated, with δ ev aluated as the w all-normal co ordinate where the mean v elo cit y u = 0 . 95 u ∞ (see a sketc h of the set-up in Fig. 1 ). The Reynolds n umber of the exp eriment w as ev aluated as Re δ = δ u ∞ /ν ≈ 1 . 034 × 10 5 ( ν = 1 . 5 × 10 − 5 m 2 / s is the kinematic viscosit y of air), whic h guarantees a w ell-dev elop ed rough turbulent flo w [ 32 ]. In this work, the stream wise, span wise and wall-normal directions are indicated as ( x, y , z ), resp ectively , and the origin of the axes is at the wall in corresp ondence to the outlet section of the source (see the sk etch in Fig. 1 ). A mixture of air and a passive scalar w as con tin uously ejected from a metallic L-shap ed tub e. Due to its densit y similar to air, Ethane (C 2 H 6 ) was used as a passive tracer. The passiv e scalar source w as lo cated at a stream wise distance from the b eginning of the w orking section x s /δ ≈ 17 . 5 and at a w all-normal height h s /δ ≈ 0 . 24. Tw o internal diameter configurations w ere considered (see Fig. 1 ): D = 0 . 003 m (i.e., D/δ ≈ 9 . 55 × 10 − 3 ) and D = 0 . 006 m (i.e., D /δ ≈ 1 . 91 × 10 − 2 ). In the following, these tw o configurations are referred to as D3 and D6, resp ectiv ely . The Ethane-air mixture do es not substantially introduce or subtract momen tum from the flo w field at the source. This condition is referred to as isokinetic [ 13 , 14 ], namely the source velocity , u s , of the mixture equals the lo cal mean v elo cit y , u , at the source heigh t, u s ≡ u ( z = h s ) ≈ 3 . 37 m/s. In order to hav e isokinetic conditions, the total mass flow rate, M t = ρu s π D 2 / 4, w as imposed as M t /ρ ≈ 86 l / h for D3 and M t /ρ ≈ 344 l / h for D6, where ρ = ρ air = ρ C 2 H 6 is the density of the air-Ethane mixture. F urthermore, to consider the recirculation of Ethane-air in the wind tunnel, the background concen tration (whic h increases linearly with time) w as subtracted from the recorded time-series. The stream wise and vertical velocity time-series, u and w , were acquired by means of a X-prob e hot-wire anemometer (HW A), while concentration time-series, c , were recorded with a fast Flame Ionization Detector (FID) [ 33 ] (for further details on the instrumen ts see App endix A ). The acquisition time w as set equal to T = 180 s while the num b er of recorded data is N T = 1 . 8 × 10 5 . Measuremen ts were p erformed at different lo cations along the three Cartesian directions, ( x, y , z ). Sp ecifically , data w ere recorded at x/δ = { 0 . 325 , 0 . 650 , 1 . 30 , 2 . 60 , 3 . 90 } in the streamwise direction. F or each x/δ lo cation, one-p oint measuremen ts w ere taken along the v ertical (i.e., at fixed y /δ ) and transversal (i.e., at fixed z /δ ) directions. T ransv ersal profiles of concen tration and v elo cit y are obtained at z = h s , and at span wise locations ranging in the interv al y /δ = [ − 0 . 6 , 0 . 6]. On the other hand, 6 v ertical profiles are obtained at y /δ = 0 at v arious w all-normal lo cations ranging in the in terv al z /δ = [0 . 096 , 0 . 828] (the limits dep end on the estimated size of the plume at a giv en x/δ ). Due to the crucial role play ed by the w all-normal direction that represents the direction of spatial inhomogeneity of the flow, here w e fo cus on measuremen ts tak en at y /δ = 0, namely in the ( x, z )-plane normal to the wall and passing through the source axis (Fig. 1 ). W e refer to the ne ar field and the far field as the streamwise lo cations closest and farthest from the source, resp ectiv ely . Therefore, the near and far fields corresp ond to lo cations x/δ = 0 . 325 and x/δ = 3 . 90, resp ectiv ely , while x/δ = 1 . 30 is considered as an in termediate lo cation. B. Data pre-pro cessing Eac h time-series was normalized b y a reference v alue, which is u ∞ for the v elo cit y com- p onen ts (measured in m/s) and ∆ c = M e / ( ρu ∞ δ 2 ) for the passiv e scalar concentratio n (measured in ppm), where M e is the mass flow rate of Ethane. Therefore, in this work we indicate as c , u and w the normalized concentration, stream wise and wall normal v elo cit y , resp ectiv ely . T o take in to account the presence of random instrumental noise on c – that pro duces negativ e concentration v alues – we prepro cessed the concen tration data as c ( x, z ; t i ) = 0 , if c ( x, z ; t i ) < , (1) where  = | min x,z [min i [ c ( x, z ; t i )]] | is the absolute v alue of the minimum amplitude of all concen tration series for a given D . In other words, we set equal to zero all concen tration v alues that are smaller (in mo dulus) than the maximum amplitude of negativ e v alues in the series. This pre-pro cessing op eration is reasonably v alid as the v alues of  are tw o orders of magnitude lo w er than the a v erage concen tration v alues, and three orders of magnitude lo wer than the maxim um c v alues. F urthermore, since in this w ork w e fo cus on the v ertical passiv e scalar flux, w 0 c 0 , the Reynolds decomp osition was p erformed for velocity and concentration time-series as w 0 = w − w and c 0 = c − c , where w and c are the time-av erages of w and c , resp ectiv ely . Finally , w e estimated the vertical p osition, h ∗ s , of the actual axis of the plume, for b oth D3 and D6, as the z co ordinate of maxim um c ( z ) v alue (see App endix B for more details). In fact, since the plume dev elops in a turbulen t b oundary lay er, it is affected by the mean 7 FIG. 2. V ertical profiles of the mean v alue of wall-normal transp ort w 0 c 0 (a-c), and the standard deviation σ w 0 c 0 (d-f ). The profiles are plotted at three stream wise locations, in the near field ( x/δ = 0 . 325), in the far field ( x/δ = 3 . 90), and at an in termediate lo cation ( x/δ = 1 . 30), for the t wo source diameters, D = 3 mm and D = 6 mm. The wall-normal co ordinate of the source axis, h s , is illustrated as a horizon tal dotted line, while the plume axis height, h ∗ s , is displa yed as a blue (red) dashed line for the source D3 (D6). shear and b y the source w ake. While the latter is mainly presen t v ery close to the source, the mean shear acts at any streamwise lo cation and tends to tilt the plume axis tow ards the w all. As a consequence, the wall-normal co ordinate of the plume axis, h ∗ s , along z /δ is not exactly at z = h s , but it decreases do wnstream from the source. Although the v alues of h ∗ s for D3 and D6 should b e differen t, this is true only in the near field, i.e. where the differences b et ween the plumes emitted by D3 and D6 are the strongest. 8 I I I. ST A TISTICAL ANAL YSIS OF THE PLUME D YNAMICS Previous works (e.g., [ 12 , 13 ]) fo cused on the influence of the source size on the one-p oin t concen tration statistics, at v arying distance from the source. Namely , it was shown that, while the mean concen tration profiles are almost unaffected by the source conditions, the higher-order statistics (v ariance, sk ewness, kurtosis) show a high sensitivit y on the source size, ev en at large distances from the release p oin t [ 13 ]. In a similar wa y , we examine here the statistics of vertical turbulen t transp ort, w 0 c 0 . W e show the vertical profiles of the statistics by fo cusing on the effect of the source size in the t wo configurations D3 and D6, corresp onding to source diameters D /δ ≈ 9 . 55 × 10 − 3 and D /δ ≈ 1 . 91 × 10 − 2 , resp ectiv ely . A. Mean and standard deviation of turbulent transp ort Fig. 2 sho ws the v ertical profiles of mean and standard deviation v alues of the (nor- malized) turbulen t flux, w 0 c 0 . The profiles are rep orted at three representativ e stream wise lo cations, i.e. in the near field ( x/δ = 0 . 325), in the far field ( x/δ = 3 . 90), and at an inter- mediate lo cation ( x/δ = 1 . 3). The vertical profiles of w 0 c 0 – namely the total mass transp ort – tend to collapse for the t wo configurations D3 and D6, as sho wn in Fig. 2 (a-c). This b eha viour is more evident in the far field than in the proximit y of the source, as the dep en- dence of the mean concentration on the source size D rapidly v anishes downstream from the source. In fact, the plume sizes – i.e. the transversal and wall-normal spread of the plume – become muc h more larger than D with increasing x/δ due to the relative disp ersion (e.g., see Fig. 1 ); as a result, the effect of D on w 0 c 0 b ecomes negligible for x/δ  0. The v ertical turbulen t transp ort is zero at the plume axis, h ∗ s (as an effect of the mean shear), namely where the vertical concentration gradient is minimum (see Fig. 10 in App endix B ). Ab o ve and b elo w the plume axis, instead, the mean v alue of turbulent flux is non-zero (see Fig. 2 (a-c)): for z > h ∗ s , w 0 c 0 is p ositiv e while, for z < h ∗ s , w 0 c 0 is negative. Ab ov e the source axis, the passiv e scalar is mainly carried out up wards by p ositiv e w 0 fluctuations; on the other hand, b elo w the source axis the passive scalar is mainly transp orted do wnw ards by nega- tiv e w 0 fluctuations. In particular, the maximum/minim um v alue of w 0 c 0 corresp onds to the maxim um of the mean concentration gradien t (this is also evident by using the Boussinesq appro ximation w 0 c 0 ∼ − ∂ c/∂ z [ 1 ]). 9 FIG. 3. V ertical profiles of the skewness of wall-normal transp ort S w 0 c 0 (a-c), in linear-linear scale, and the Kurtosis K w 0 c 0 (d-f ), in log-linear scale. The profiles are plotted at three stream wise lo cations, in the near field ( x/δ = 0 . 325), in the far field ( x/δ = 3 . 90), and at an intermediate lo cation ( x/δ = 1 . 30), for the t wo source diameters, D = 3 mm and D = 6 mm. The wall-normal co ordinate of the source axis, h s , is illustrated as a horizontal dotted line, while the plume axis heigh t, h ∗ s , is displa yed as a blue (red) dashed line for the source D3 (D6). As shown in Fig. 2 (d-f ), the effect of the source size for an elev ated source is instead m uch more eviden t for the standard deviation, σ w 0 c 0 , rather than for the mean v alues – in analogy with what is observed in the concen tration statistics [ 12 , 13 ] – even at large distances from the source. This is a consequence of the stronger meandering motion of the plume emitted by the smallest source size, D3, which pro duces more v ariabilit y in the series and the corresp onding high intermittency in the dynamics of vertical turbulent transp ort (see Section I I I C ). The maximum difference of standard deviation b etw een D3 and D6 is presen t close to the plume axis in the near field, and such difference strongly decreases 10 b y moving do wnstream tow ards the far field due to the w eakening of the meandering and the strengthening of the relative disp ersion. By mo ving in the wall normal direction, σ w 0 c 0 decreases as the plume intensit y v anishes aw a y from the source axis. B. Sk ewness and kurtosis of turbulent transp ort The b eha viour of the higher order momen ts is here in v estigated b y fo cusing on the sk ew- ness, S w 0 c 0 , and the kurtosis, K w 0 c 0 , of the vertical turbulent flux. F ormally , they are defined as the normalized third- and fourth-order cen tral momen ts, namely S w 0 c 0 = ( w 0 c 0 − w 0 c 0 ) 3 /σ 3 w 0 c 0 and K w 0 c 0 = ( w 0 c 0 − w 0 c 0 ) 4 /σ 4 w 0 c 0 , resp ectiv ely . Fig. 3 sho ws the sk ewness and the kurtosis as a function of z /δ for three stream wise lo cations (as for the mean and standard deviation sho wn in Fig. 2 ). The b eha viour of the sk ewness (Fig. 3 (a-c)) is similar for the tw o source configurations D3 and D6 at an y x/δ . In particular, S w 0 c 0 ≈ 0 at the plume axis, while the sk ewness is nega- tiv e/p ositiv e b elo w/ab o v e the plume axis, because b elo w and ab ov e h ∗ s the v ertical turbulen t transp ort is mainly down wards (i.e., w 0 c 0 < 0) and upw ards (i.e., w 0 c 0 > 0), resp ectiv ely . As sho wn in Fig. 3 (d-f ), the kurtosis v alues are greater than three (whic h corresp onds to nor- mal distribution), thus implying that the PDFs of w 0 c 0 are fat-tailed distributions (under some circumstances, the PDFs can w ell fitted b y a Gamma distribution [ 13 ]). In particu- lar, K w 0 c 0 is minimum at the plume axis at eac h stremwise lo cation, as the plume develops around z = h ∗ s and extreme even ts (with resp ect to w 0 c 0 ) are less probable to appear; on the con trary , a wa y from the plume axis extreme w 0 c 0 v alues are more probable, as the signals are m uc h more in termittent. Differen tly from the skewness, the kurtosis profiles for D3 and D6 are different in the near field (see Fig. 3 (d)), and the difference of K w 0 c 0 progressiv ely reduces to wards the far field (see Fig. 3 (f )). This implies that the meandering affects the b eha viour of the standard deviation but also the b eha viour of the kurtosis: the v alues of K w 0 c 0 for D3, in fact, are higher than the v alues of K w 0 c 0 for D6, namely extreme v alues are more probable for D3 than for D6. 11 FIG. 4. Normalized sp ectral densit y , E ∗ , as a function of the normalized wa ven umber, κ ∗ , of the w all-normal turbulent flux, w 0 c 0 . Spectra are ev aluated at the source axis ( z = h s ), for the tw o source diameters, D = 3 mm and D = 6 mm, at three streamwise lo cations: (a) in the near field ( x/δ = 0 . 325), (b) at an in termediate lo cation ( x/δ = 1 . 30), and (c) in the far field ( x/δ = 3 . 90). C. Sp ectra and intermittency factor Fig. 4 shows the normalized p ow er sp ectral densit y , E ∗ = E δ /σ w 0 c 0 , of the signals w 0 c 0 , as a function of the normalized w av en um b er κ ∗ = κδ , where κ = 2 π /λ is the w av enum b er, λ = u/f is a characteristic turbulen t length scale (see Fig. 1 ), f is the frequency and u is the (lo cal) mean streamwise v elo cit y . Spectra are plotted along the source axis, namely for z = h s . Since the (instan taneous) plume size, λ z , dep ends on the source size and the spatial lo cation, the relation b etw een λ z and turbulent length scales, λ , affects the b ehaviour of the sp ectra. In particular, λ z is smaller for D3 than for D6 in the near field, due to the spatial pro ximity of the source; by mo ving downstream, instead, λ z increases and the difference of λ z b et w een D3 and D6 diminishes. In the near field (Fig. 4 (a)), the difference of sp ectral densit y b et ween D3 and D6 is larger at small w av enum b ers than at high wa ven um b ers. In fact, turbulent length scales, λ , larger than the (instantaneous) plume size, λ z , contribute to the (instantaneous) plume meandering motion in the w all-normal direction (e.g., see the sk etch in Fig. 1 ). Therefore, since in the near field λ z , D3 < λ z , D6 , the differences of E ∗ b et w een D3 and D6 at low κ ∗ are more evident, b ecause the plume for D3 is affected b y a wider range ( λ z < λ < λ max , or equiv alently , κ ∗ min < κ ∗ < 2 π δ /λ z ) of turbulent scales. On the other hand, at high w av enum b ers (namely , small turbulen t length scales), the sp ectral densit y for the t w o source sizes tends to coincide, as turbulen t scales λ < λ z only promote the 12 FIG. 5. (a) Example of the in termittent b eha viour of the concen tration signal (first 50 v alues) measured at x/δ = 1 . 30 and at the source axis. Blue and red data corresp ond to an upw ard and do wnw ard transp ort, resp ectively . The corresp onding v alues of the intermittency factor for the sho wn time in terv al are also reported. (b) In termittency factors of the passive scalar concen tration, γ c , and wall-normal turbulent flux, γ + and γ − , as a function of x/δ along the source axis (i.e., y /δ = 0 and z = h s ). disp ersion of the plume. The large scale fluctuations – induced b y a wider range of turbulen t scale in the near field – progressiv ely weak en to wards the far field (see Fig. 4 (b)-(c)), so that the intensit y of sp ectral density decreases with x/δ and approaches the same b eha viour at all wa v enum b ers for D3 and D6. In fact, for increasing x/δ , the plume size increases (i.e., λ z → λ max ) and the range of scales for which λ > λ z decreases. In other words, all turbulen t scales tend to con tribute to the relativ e disp ersion of the plume in the far field. Finally , it should b e noted that sp ectra of v ertical turbulen t transp ort normalized b y its v ariance do not sho w a self-similar b eha viour along the streamwise direction. This is in contrast to what has b een recently rep orted for concentration series, which sho w a self-similar b eha viour in the range 0 . 5 ≤ x/δ ≤ 4 [ 34 ]. The last parameter in vestigated is the in termittency factor. F or the concen tration series, an in termittency factor, γ c , can b e defined as the fraction of non-zero concentration v alues, where small γ c v alues correspond to highly in termitten t series [ 13 ]. In other words, γ c is the fraction of time in whic h the passiv e scalar is measured. In a similar w ay , here w e define the intermittency factor for the v ertical turbulen t transp ort as the fraction of time in which the passive scalar is transp orted upw ards, γ + = prob [ w 0 > 0 , c 6 = 0], and down wards, γ − = prob [ w 0 < 0 , c 6 = 0], with ( γ + + γ − ) = γ c (b y definition) and prob [ • , • ] indicating the joint 13 probabilit y . In the definition of γ + and γ − , the v elo cit y fluctuations, w 0 , imp ose the sign to the fluxes while the concentration discriminates b et w een the presence ( c 6 = 0) or the absence ( c = 0) of the plume. Therefore, although the o v erall in termittency is go v erned by the concen tration field (i.e., c 6 = 0 or c = 0), the v elo cit y comp onen t introduces a directionality for the intermittency (namely w 0 > 0 or w 0 < 0). F or example, the in termittency factor for the p ortion of the concentration signal sho wn in Fig. 5 (a) is γ c = 0 . 44, b ecause there are 22 non-zero v alues of c ( t i ) out of 50 v alues. Among the 22 non-zero v alues, 14 observ ations corresp ond to an up ward motion (i.e., γ + = 14 / 50 = 0 . 28), while 8 observ ations corresp ond to a down w ard motion (i.e., γ + = 8 / 50 = 0 . 16). Fig. 5 (b) sho ws that, at the source axis, the vertical transp ort is more intermitten t do wnw ard ( w 0 < 0) than upw ard ( w 0 > 0), namely γ − < γ + for b oth D3 and D6. Consis- ten tly with the results shown in Fig. 2 and Fig. 3 , at the source axis the passive scalar is mainly transp orted upw ards (as the plume axis lies b elo w the source axis). Consequently , the fraction of time in which the passiv e scalar is transp orted upw ards, γ + , results to b e greater than the fraction of time in which the passiv e scalar is transp orted do wnw ards, γ − . F urthermore, the intermittency factor is alwa ys smaller for the source diameter D3 than for D6, whether γ c , γ + or γ − is considered. This v alidates the fact that meandering motion is stronger for the plume emitted b y a smaller source, inducing higher intermittency in the signals. More in detail, in the near field, the v alues of in termittency factor for D3 and D6 are differen t while they approac h the same v alue in the far field, as the effect of the meandering is replaced by the relative disp ersion of the plume. Although a strong meandering motion is presen t in the near field, the intermittency factors do not monotonically increase with x/δ b ecause of the effect of the source proximit y in the near field (as mentioned in Section I II A for the mean turbulen t transp ort). Therefore, a minim um v alue of the intermittency is found at x/δ ≈ 1 . 3, which is also found b y Nir oni et al. [ 13 ] in the case of γ c . IV. VISIBILITY-NETW ORK ANAL YSIS OF TURBULENT TRANSPOR T In this section, we presen t the results of the analysis of the vertical turbulen t transp ort b y means of the visibilit y graph approach. The netw ork analysis is here p erformed to adv ance the lev el of information of classical statistical analysis (see Section II I ), th us providing a ric her picture of the plume dynamics via turbulent transp ort inv estigation. Differently from 14 classical statistics to ols, differen t temp oral arrangements of the same time-series generate differen t visibilit y net works (e.g., a sh uffled series maintains the same statistics of the origi- nating series, but exhibits a differen t temp oral structure and visibility-net work). Since the net work metrics ev aluated from the visibility graph approach are able to characterize the temp oral structure of the time-series, they carry high-order and nonlinear information of the signal [ 21 ]. As a result, the visibility graph approac h is prop osed to shed ligh t on the temp oral structure – in terms of extreme even ts and their relative intensit y – of the turbulen t transp ort time-series. A. Concepts and definitions A complex netw ork is formally defined as a graph that sho ws non-trivial top ological features [ 15 ]. Netw orks are made up of N entities called no des in terconnected by a set of L links , and they are commonly represented b y an adjac ency matrix , which is defined as A ij =    1, if { i, j } ∈ L , with i 6 = j, 0, otherwise , (2) where i, j = 1 , ..., N and L is the set of L links. Therefore, the entries A ij tak e in to accoun t the presence of a link b etw een eac h pair of no des. In this w ork, w e considered eac h connection to b e undirected (i.e., A ij = A j i ) and unw eighted, resulting in to binary and symmetrical adjacency matrices. In order to inv estigate the time-series of turbulent transp ort, we employ ed the natural visibilit y graph (NVG) metho d [ 30 ], whic h is a widely exploited technique to map time-series in complex netw orks. This metho d was firstly prop osed b y L ac asa et al. [ 30 ], who show ed that the resulting visibility-net work is able to inherit imp ortan t features of the mapp ed time-series. According to the algorithm, each datum of a time-series, s i ≡ s ( t i ), is mapp ed in a no de of the net work and a link b et ween t w o no des, ( i, j ), is established if: s k < s j + ( s i − s j ) t j − t k t j − t i , (3) for all t k b et w een t i and t j (or analogously ∀ k , i < k < j ). Therefore, b y construction, eac h no de i is alwa ys linked to its immediately closest no des, namely j = i ± 1. F rom Eq. ( 3 ) it follows that the natural visibilit y algorithm satisfies a con v exity criterion, so that 15 FIG. 6. Examples of tw o interv als of time-series, ( t i , s i ), and corresp onding visibility net works, sho wing different temp oral structures. No des and links are depicted as red dots and green lines, resp ectiv ely . (a) First 10 out of 10 5 observ ations of a series extracted from a uniform probabilit y distribution in the in terv al [0 , 1]. (b) First 20 out of 10 5 observ ations of a series extracted from a uniform probability distribution in the interv al [0 , 1], with p eriodic spikes (ev ery 14 instan ts, t i ) uniformly distributed in the interv al [0 , 100]. The v alues of s i in the vertical axis are not shown due to the inv ariance of visibilit y algorithm to affine transformations. (subsets of ) no des that form a conv ex series (e.g., a b o wl-lik e series) are fully link ed with eac h other. In this work, each net work has N = N T = 1 . 8 × 10 5 no des, corresp onding to the recorded data v alues of eac h v elo cit y and concentration series. The most imp ortan t no des ( hubs ) for a visibility-net w ork are asso ciated with p ositive p eaks in the series, b ecause v ery high v alues are more likely to see other no des (i.e., hubs ha ve a b etter visibility). Fig. 6 sho ws t wo simple examples of time-series (illustrated as blac k stems) mapped into visibility net works, where no des and links are depicted as red dots and green lines, resp ectiv ely . It is w orth highligh ting that the visibility criterion emphasizes the p ositiv e p eaks, but not the negative ones. Consequently , when the series mainly display pits (i.e., negativ e p eaks) instead of p ositiv e p eaks, it is p ossible to exploit the Eq. ( 3 ) to build visibilit y netw orks from the complementary series, − s i . The comparison of the metrics extracted from the original series, s i and its opp osite, − s i , allows one to characterize the p eak-pit asymmetry in the series [ 35 ], namely if p eaks are mainly p ositiv e or negativ e. The main adv antage of the visibility algorithm relies on the fact that it do es not require an y a priori parameter. How ev er, the NVG is in v arian t under rescaling and translation of b oth horizon tal and v ertical axes (i.e., affine transformations) [ 30 ]. This p eculiar feature of NV G is crucial, and it must alw ays b e taken into accoun t when the netw ork structure is 16 in vestigated. In fact, the inv ariance to affine transformations implies that tw o time-series with different mean and standard deviation v alues but with similar temp oral structure, are mapp ed into the same visibilit y net work. If the analysis should b e sensitiv e to affine transformations of the series, the inv ariance of NVG represents a dra wback of the metho d. On the other hand, if the fo cus is on the temp oral structure of the series (as in this work), the inv ariance is a p oten tial b enefit. In fact, it is p ossible to exploit the NVG to analyse series without a pre-pro cessed normalization of the series. Finally , it should b e noted that the condition in Eq. ( 3 ) still holds for non-uniform sampling or time-series with missing v alues [ 29 ], whic h is an adv an tage when incomplete data measurements are a v ailable. B. Net w ork metrics In order to characterize the structure of complex net w orks, several metrics ha v e been prop osed so far [ 15 ]. In a previous work on the inv estigation of velocity time-series in a turbulen t channel flow, three metrics were exploited [ 21 ]: the transitivity , the mean link- length and the degree cen trality . Among these three metrics, the transitivity and the mean link-length are able to highlight the presence of small v ariations in the series and the o ccur- rence of p eaks, resp ectiv ely . On the other hand, the degree centralit y cannot b e uniquely related to a sp ecific temp oral feature, since the degree accoun ts for b oth the recurrence of p eaks and the presence of small v ariations in the series [ 21 ]. Therefore, to extract informa- tion on the c haracterization of extreme even ts of the plume dynamics we here exploited t wo sp ecific net work metrics: the me an link-length , and the assortativity c o efficient . The mean link-length, d i , was prop osed by Iac ob el lo et al. [ 21 ] as a lo cal measure to c haracterize the temp oral o ccurrence of p eaks in a series, defined as d i = 1 k i N X j =1 | t j − t i | A ij , (4) where k i = P j A ij is the degree of a no de i , that is the num b er of no des link ed to i . The a verage v alue o v er all no des of the mean link-length is then h d i = P i d i / N , and it represen ts a characteristic temp oral distance b et ween t wo visible data in a series. A high v alue of the a verage mean link-length, h d i , indicates that p eaks rarely o ccur in a series, since p eaks prev ent visibilit y b et w een data that are far from each other [ 21 ]. Therefore, large d i v alues 17 corresp ond to h ubs in the netw ork and p eaks in the series, as p eaks activ ate long-range links (i.e. farther temp oral horizons ). A s mentioned ab o ve, although the degree is usually adopted as the metric to characterize hubs, the mean link-length reveals to b e a more reliable metric than degree for p eaks characterization in visibility net works. F or example, in a visibilit y- net work built on a fully conv ex series (e.g., a b o wl-like series with a minimum v alue), each no de has maximum degree v alue equal to N − 1 due to the full conv exity of the series, but the no de corresp onding to the minim um v alue do es not represen t a p eak. It is also worth noting that while the kurtosis is an estimation index of extreme v alues in a PDF, the mean link-length quan tifies the a v erage temp oral distance betw een extreme ev en ts (while the PDF is in v arian t to the temp oral structure of the signal). In this w ork, in order to b etter highligh t the cases in whic h p eaks frequently app ear (i.e., low h d i v alues), we introduce the aver age p e ak o c curr enc e , φ = h d i − 1 , corresp onding to a c haracteristic frequency of p eaks in a series. It must b e emphasized that we refer to p eaks as the lo cal (or global) highest p ositiv e v alues in the series [ 21 ]. This does not necessarily imply that p eaks also correspond to outliers, namely very large v alues with resp ect to a lo cal subset of data; on the contrary , outliers t ypically corresp ond to p eaks. F or example, in Fig. 6 (a) outliers are not presen t and p eaks corresp ond to no des i = { 2 , 4 , 7 , 9 } ; in Fig. 6 (b), instead, p eaks corresp ond to outliers, namely no des i = { 2 , 15 } . Therefore, the a verage p eak o ccurrence, φ , is sensitiv e to the app earance of p eaks in time (horizontal separation), but it do es slightly take in to accoun t the relative intensit y of p eaks compared to all the other v alues in the series. T o address this issue, we also inv estigated the assortativit y co efficien t, r , whic h is the Pearson correlation co efficien t of the degree of the no des at the ends of each link [ 36 ], namely r = P i P j >i A i,j L k i k j − h P i P j >i A i,j 2 L ( k i + k j ) i 2 P i P j >i A i,j 2 L ( k 2 i + k 2 j ) − h P i P j >i A i,j 2 L ( k i + k j ) i 2 , (5) where ( i, j ) are the end-no des of eac h link l ∈ L . P ositiv e r v alues are obtained when nodes are linked with other no des of similar degree: in this case, the netw ork is said assortative . On the con trary , the netw ork is said disassortative if r < 0, or non-assortative if r = 0. As for the correlation co efficient, r ranges in the interv al [ − 1 , 1]. In particular, a negativ e r v alue means that high degree no des (i.e., no des with more visibility) tend to b e more link ed with lo w degree no des (i.e., no des with less visibilit y), rather than with other high degree no des. When p eaks are fo cused, the assortativity co efficien t quantifies the extent to which 18 p eaks (whic h are exp ected to hav e more visibility) are more prominen t with resp ect to small fluctuations (which are exp ected to ha v e less visibility). Highly p ositiv e v alues of r indicate that p eaks are sligh tly pronounced with resp ect to the other v alues in the series (e.g., Fig. 6 (a)), while strongly negative v alues of r indicate a substantial presence of outliers (e.g., Fig. 6 (b)). In other w ords, r is a measure of the v ertical separations in the series, that is ho w in tense p eaks are with resp ect to the other data in the time-series. As a rule of th umb, r = 0 discriminates b et w een the prominence of p eaks (i.e, r > 0) and the prominence of outliers (i.e., r < 0). If the net work is non-assortativ e (i.e., r ≈ 0), in general neither p eaks nor outliers are exp ected to b e prominent. Ho wev er, if tw o signals are compared, it can b e inferred that the signal sho wing r ≈ 0 is more lik ely to ha ve outliers or peaks than the signal with r > 0 or r < 0, resp ectiv ely . T o summarize, the t wo net work metrics, φ and r , are here selected to characterize the temp oral structure of the series in terms of horizontal and v ertical separations, resp ectiv ely . F or instance, in Fig. 6 (a), φ = 0 . 230 and r = 0 . 22, while in Fig. 6 (b) φ = 0 . 156 and r = − 0 . 19. These v alues are ev aluated as describ ed in the caption of Fig. 6 . Accordingly , φ is more reliable to detect the o ccurrence of p eaks in the series, while r is able to discern b et w een p eaks (in which r is generally p ositiv e) and outliers (in whic h r is generally negative) in a time-series. F or example, large v alues of φ indicate that the corresp onding series has man y p eaks, whic h app ear as outliers only if r decreases to wards negativ e v alues. Accordingly , the t wo metrics, φ and r , should b e analysed in pairs in order to infer the temp oral structure of the series ev aluated at different spatial lo cations. C. Spatio-temp oral in vestigation of w 0 c 0 through the netw ork metrics The results from the application of the visibility algorithm to the time-series of turbulent transp ort, w 0 c 0 , are rep orted in this Section. The tw o configurations of source diameter, D3 and D6, are display ed for different downstream lo cations, x/δ , and at v arious wall-normal co ordinates, z /δ . The b eha viours of the av erage p eak o ccurrence, φ , and the assortativity coefficient, r , are sho wn in Fig. 7 (a) and Fig. 7 (b), resp ectiv ely , as a function of the vertical co ordinate, z /δ . In general, φ and r hav e their maxim um v alues close to the plume axis, h ∗ s (see horizontal dashed lines in Fig. 7 ): this means that, around the plume axis, peaks frequently o ccur 19 FIG. 7. V ertical profiles of the a verage p eak o ccurrence φ [Hz] (a), and the assortativit y co efficien t r (b), based on signals of wall-normal turbulen t transp ort, w 0 c 0 . The profiles are plotted from the near field ( x/δ = 0 . 325) to the far field ( x/δ = 3 . 90), for the tw o source diameters, D = 3 mm and D = 6 mm. The w all-normal co ordinate of the source axis, h s , is illustrated as a horizontal dotted line, while the plume axis heigh t, h ∗ s , is display ed as a blue (red) dashed line for the source D3 (D6). The insets at x/δ = 0 . 325 show t wo zo oms around the plume axis. and the vertical separation b et ween them and the other data in the series is weak (since r > 0). This b ehaviour of the metrics is due to the fact that the plume is mainly lo cated around the source axis while it meanders and dev elops downstream. The difference in the features of temp oral structure of the w 0 c 0 can b e observ ed in Fig. 8 , whic h sho ws segments of time-series of w 0 c 0 measured at x/δ = 0 . 325 for D3 (Fig. 8 (a-c)) for D6 (Fig. 8 (d-f )). 20 Sp ecifically , high v alues in the signals are muc h more frequent (i.e., high φ ) and m uc h less prominen t (i.e., high r ) around the plume axis (see Fig. 8 (b,e)) than aw a y from it. The effect of the source size, D , on the metrics is clearly visible in the near field (i.e., x/δ = 0 . 325) and also at intermediate streamwise distances (i.e., up to x/δ = 1 . 30) around the plume axis, where the metrics for D3 ha ve smaller v alues than the metrics for D6. The p eaks tend to app ear more as outliers (i.e., lo wer r ) and they occur less frequen tly (i.e., lo w er φ ) for the smallest source size D3 (see Fig. 8 (b)) than for D6 (see Fig. 8 (e)). Since a plume emitted from a smaller source diameter is affected b y a wider range of turbulent scales, its meandering motion is more intense. A strong meandering motion implies a high v ariabilit y (i.e., standard deviation) and a large fraction of small data v alues (i.e., strong intermittency), whic h are captured as different v alues b etw een D3 and D6 of the net work metrics, φ and r . It should b e noted that the netw orks corresp onding to lo cations around the plume axis do not show large negativ e r v alues, as the higher v ariability in the signals preven t a strong separation b et ween small v alues and p eaks (which w ould giv e r < 0). In the far field (i.e., x/δ = 2 . 60 − 3 . 90), the v ertical profiles of the tw o metrics tend to collapse for b oth source diameter configurations, D3 and D6. This b eha viour is a consequence of the increase of the plume size b y moving downstream, due to the relative disp ersion. In the far field the plume size exceeds the largest turbulent scales and the mixing of the passive scalar is then fully regulated by the relativ e disp ersion rather than by the meandering. Therefore, by moving do wnstream around the plume axis, the av erage p eak o ccurrence alw ays decreases with x/δ , since large concentration v alues (and in turn high turbulent transport) are less probable to app ear in the far field b ecause of the plume w eakening. On the other hand, for z ≈ h ∗ s , the assortativity co efficien t first decreases (reaching a minim um at ab out x/δ = 1 . 30) and then increases with x/δ . The b eha viour of the assortativit y co efficien t is still a consequence of the in terplay b et ween the reducing meandering motion and increasing disp ersion of the plume as it evolv es downstream. It is w orth noting that the minim um v alue of r along the source axis is found at x/δ ≈ 1 . 3, that is the streamwise lo cation where the plume reaches the ground and all the intermittency factors are minimum (see also Fig. 5 (b)). By fo cusing on the effect of the wall normal co ordinate, z /δ , at a given streamwise lo cation, the tw o metrics tend to decrease b y moving a wa y from the plume axis in the wall normal direction. Lo w v alues of φ and r indicate that, from the p oint of view of the temp oral structures of the series, p eaks o ccur less frequently and suc h p eaks app ear as outliers (e.g., 21 FIG. 8. Time-series of v ertical turbulen t transp ort in the near field, x/δ = 0 . 325, for the source diameter D3 (a-c) and D6 (d-f ). Signals are plotted at three w all-normal coordinates, namely abov e the source axis at z /δ = 0 . 303 (panels (a) and (d)), at the source axis h s /δ = 0 . 240 (panels (b) and (e)), and b elo w the source axis at z /δ = 0 . 175 (panels (c) and (f )). F or comparison purp oses, the time-series are normalized as ( w 0 c 0 ) ∗ =  w 0 c 0 − w 0 c 0  /σ w 0 c 0 . see Fig. 8 (a,d) and Fig. 8 (c,f ) for series ab o ve and b elow the plume axis, resp ectiv ely). Therefore, the temp oral structure of the signals b ecomes spik e-like moving a w ay from the plume axis along the z direction (see Fig. 8 (a,c) for D3 and Fig. 8 (d,f ) for D6). Additionally , the metrics rapidly decrease with z /δ in the near field (as exp ected), since the plume rapidly v anishes in the v ertical direction; in the far field, instead, the metrics gently decrease with z /δ from z = h ∗ s , as the plume size increases downstream due to the relativ e disp ersion. It is worth noting that, in the near field, the difference of a v erage peak o ccurrence, φ , b et w een D3 and D6 is larger b elo w than ab o ve the plume axis (e.g., see the first-left panel of Fig. 7 (a)). Therefore, the b eha viour of φ in the near field indicates that p eaks in the series of D6 are more frequen t than for D3 b elo w the plume axis. This b eha viour can b e seen, for instance, by lo oking at the signals shown in Fig. 8 (c,f ). Since in the near field the effect of 22 the source size is still notable, the plume for D6 is larger and more tilted tow ards the w all than for D3 (i.e., h ∗ s is smaller for D6). As a consequence, at a giv en v ertical co ordinate b elo w the plume axis (e.g., at z /δ = 0 . 175 in Fig. 8 ) the temp oral structure of the series for D6 (e.g., Fig. 8 (f )) is less spik e-like than for D3 (e.g., Fig. 8 (c)). It should b e emphasized that the b eha viour of the netw ork metrics along the wall-normal direction, z /δ , is consisten t with the outcomes rep orted in Ref. [ 14 ], which pro vides a phenomenological description of the velocityscalar interaction. Finally , we remark that the visibilit y algorithm emphasizes the presence of p ositiv e p eaks in the signals. How ever, as sho wn in Fig. 8 , there is an asymmetry b etw een p ositiv e and negativ e extreme v alues in the series, esp ecially aw a y from the plume axis. Therefore, the analysis of φ and r can also b e carried out by applying the visibility algorithm to the signals − w 0 c 0 , th us highlighting the temp oral o ccurrence and relative intensit y of the negative extreme v alues (i.e., pits) of vertical turbulen t transp ort. W e found that the metrics obtained from w 0 c 0 and − w 0 c 0 coincide at the plume axis, h ∗ s , while they are different aw a y from it. In particular, ab ov e the plume axis the v alues of φ and r are higher for the series − w 0 c 0 than for w 0 c 0 , that is p eaks are less frequen t and app ear more as outliers than pits; on the con trary , the opp osite b eha viour is found b elo w the plume axis (see App endix C for more details). V. DISCUSSION AND CONCLUSIONS The disp ersion of a passive scalar in a turbulent boundary la yer is exp erimen tally in ves- tigated via wall-normal turbulent transp ort time-series, w 0 c 0 , for t wo elev ated source sizes. The plume dynamics is analysed b y means of classical statistical to ols and through a com- plex net work-based approach. The statistical analysis reveals that the mean v alue and the sk ewness of w 0 c 0 are not substantially affected b y the source size, while the standard devia- tion and the kurtosis are sensitive to the emission conditions. The plume meandering – that is mainly active in the near field of the source – is the main resp onsible for the differences in the statistics: a stronger meandering motion pro duces higher v ariability (i.e., standard deviation) and more extreme ev ents (i.e., kurtosis) for the smallest source size than for the largest source size. F ar from the source, the meandering motion strongly reduces its in- tensit y and the relativ e disp ersion turns out to b e the principal mechanism affecting the 23 plume dynamics. In the far field, therefore, the statistics approach the same v alues for the t wo source sizes. The effect of the meandering on turbulent transp ort is also highligh ted b y the p o w er sp ectral density and the in termittency factor. Larger v alues of p o w er sp ectral densities are obtained at small w av en um b ers for the smallest source size, as a consequence of a wider range of turbulent scales affecting the plume dynamics. Additionally , the stronger meandering motion asso ciated with the plume emitted by a smaller source size induces a more intense intermittency in the signals, namely a low er fraction of time for whic h the passiv e scalar is measured and transp orted. As for the statistical moments, the main differ- ences of p ow er sp ectral densit y and intermittency b etw een the tw o source sizes are large in the source proximit y and v anish in the far field. In this wa y , b y inv estigating the vertical turbulen t transp ort time-series, w e extend the b enc hmark of (one-p oint) statistics of Nir oni et al. [ 13 ] and F ackr el l and R obins [ 12 ] on the dynamics of a passiv e scalar plume emitted in a rough-wall turbulen t b oundary lay er. With the aim to adv ance the lev el of information of classical statistics, a complex net work analysis is also carried out b y exploiting the visibilit y algorithm. W e focused on t w o net work metrics: the a verage p eak o ccurrence and the assortativity co efficien t, which c haracterize the temporal structure of w 0 c 0 in terms of o ccurrence and relative in tensity of (p ositiv e) p eaks, resp ectiv ely . The net work metrics are significantly affected b y the measurement lo cations. Specifically , at each stream wise co ordinate, the temp oral structure of the signals at the plume axis is made up of p eaks that frequently appear in time (i.e., high v alues of the a verage peak o ccurrence), while p eaks app ear less frequently a w ay from the plume axis (i.e., small v alues of the av erage p eak o ccurrence). The assortativity co efficien t indicates that the relativ e intensit y of p eaks (with resp ect to all the other v alues in the signal) is smaller at the plume axis (p ositiv e assortativity) rather than aw ay from it (negative assortativit y): b elo w and ab o v e the plume axis, therefore, p eaks app ear infrequen tly and as outliers. A t the b orders of the plume, indeed, the turbulent transp ort is reduced b ecause of the presence of small concentrations (due to the relative disp ersion of the plume), th us outliers sp oradically app ear in the corresp onding time-series of turbulen t transp ort. By fo cusing on the effect of the source size, the metrics at the plume axis for the smallest source (D3) display low er v alues than the metrics for the largest source (D6). This means that the temp oral structure of the w 0 c 0 signals for D3 (i.e., for a stronger plume meandering) is characterized by p ositiv e p eaks that app ear more as outliers and are less frequent than 24 for D6 (i.e., for a weak er plume meandering). Suc h difference b etw een D3 and D6 is eviden t in the pro ximit y of the source, while it decreases for increasing streamwise co ordinates. Therefore, the unequal intensit y of the meandering motion b et w een D3 and D6 differently affects the temp oral structure of turbulen t transp ort series in the near field. Since a strong meandering motion induces a high intermittency in the series, it is exp ected that tw o p eaks of turbulen t transport are more unlikely to app ear close in time (as there m ust b e large fraction of time in which the passiv e scalar is not measured). F urthermore, the high v ariability (i.e., standard deviation) asso ciated with the meandering motion implies that the relative in tensity of p eaks with resp ect to the other v alues in the signal increases (i.e., extreme ev ents app ear more as outliers), b ecause the time-series b etw een tw o p eaks is made up of v alues that are differen t from the p eaks (otherwise the standard deviation v alues w ould b e small). T o summarize, the resp ectiv e influence of the meandering motion and the relative disper- sion for different source sizes, as well as the plume weak ening with the stream wise distance from the source, is fully captured by the netw ork metrics. Sp ecifically , the a verage p eak o ccurrence and the assortativity co efficien t are able to highlight the differen t temporal struc- tures of the series – in terms of p eaks o ccurrence and their relativ e in tensity – at different spatial lo cations. In this wa y , the visibility netw ork-based approac h enriches the analysis of the plume dynamics carried out via classical statistics, revealing significant information ab out the temporal structure of the measured signals. In fact, the netw ork metrics are able to discern b et w een p eaks and outliers and their frequency in the signals, which cannot b e easily deduced from classical statistics. As a result, the visibility algorithm can b e ex- ploited as an effectiv e to ol for the time-series analysis of exp erimental data, by highligh ting non-trivial features of the main mechanisms affecting the plume dynamics. A CKNO WLEDGMENTS P . Salizzoni ac knowledges funding from the ”R´ egion Auvergne-Rhˆ one-Alp es” for the ”SCUSI” pro ject. The authors would also lik e to express their gratitude to Patric k M´ ejean and Horacio Correia for the technical supp ort in the wind tunnel exp erimen ts. 25 App endix A: Characterization of the concen tration and v elo cit y field V elo cit y time-series w ere acquired b y means of a ± 45 ◦ X-prob e hot-wire anemometer (HW A) w orking at a constan t temp erature, which allo ws for the sim ultaneous measure- men ts of t w o velocity comp onents. The prob e was calibrated by exploiting a Pitot tube that measures a reference v elo cit y (calibration in ya w w as not p erformed). In particular, the calibration v elo cities w ere decomp osed into the longitudinal and transversal v elocity comp onen ts by adopting a ya w correction with constan t co efficien ts K 2 1 = K 2 2 = 0 . 0225 [ 37 ]. The exp erimen tal error at a fixed reference lo cation was appro ximately ± 2% for the mean and the standard deviation. FIG. 9. V ertical profiles of velocity statistics normalized with the friction velocity , u ∗ . (a) Mean v elo cit y defect, ( u ∞ − u ). (b) Standard deviations of the longitudinal, σ u , and vertical, σ w , ve- lo cit y comp onen t, depicted in black and red, resp ectiv ely . (c) Reynolds stress, − u 0 w 0 . Sym b ols: x/δ = 0 . 65 ◦ ; x/δ = 1 . 30  ; x/δ = 2 . 60 • ; x/δ = 3 . 90 ♦ . A fast Flame Ionization Detector (FID) was used to p erform concentration measurements. The FID system uses a sampling tub e that is 0 . 3 m long, p ermitting a frequency resp onse of the instrumen t to ab out 400 Hz. The calibration w as carried out twice a day by setting ethane-air concen trations equal to 0, 500, 1000 and 5000 ppm. A linear relation holds b et w een ethane concen tration and tension resp onse, with slop e v ariations (i.e., the sensitivity v ariations of the instrumen t) of ab out ± 3%, depending on the am bient conditions. The error in the first four momen ts of the concen tration due to all the uncertain ties in the experimental c hain, was estimated to b e up to 4 . 5%. By exploiting measuremen ts p erformed on different da ys of distant w eeks, the first t wo moments of the concen tration are affected by an error 26 of 2% in the far field and 3% in the near field (this increase is due to uncertainties in the source flo w control system in the near field). F or the third and fourth momen ts of the concen tration, the error rises up to 4 . 5% b oth in the near- and far-field. In order to ev aluate one-p oin t turbulent fluxes, simultaneous measuremen ts of velocity and concentration are necessary at the same spatial lo cation, implying that HW A and FID systems ha ve to b e sync hronized and sufficiently close in space. The optimal distance b e- t ween the HW A and FID w as found to b e 5 mm in the span wise direction, in order to av oid lo cal perturbations induced b y the measuring system in the flow field. The coupling HW A- FID do es not require signal re-sampling or filtering, because b oth HW A and FID ha ve a constan t sampling frequency (equal to 1000 Hz), so that their resp onses are contin uous and regular. The main features of the velocity field are sho wn in Fig. 9 . Sp ecifically , the vertical profile of the mean velocity defect is displa yed in Fig. 9 (a), while standard deviations of the v elo cit y comp onen ts, σ u and σ w , and the Reynolds stress, u 0 w 0 , are sho wn in Fig. 9 (b) and Fig. 9 (c), resp ectively . The vertical profiles – measured at differen t streamwise lo cations in the range x/δ ∈ [0 . 65 , 3 . 90] – are in go od agreement with v alues rep orted in literature [ 13 ], and collapse rather well b oth for the mean flo w (Fig. 9 (a)) and the v elo cit y fluctuations (Fig. 9 (b-c)). Sp ecifically , from the Reynolds stress profile we are able to estimate the friction velocity as u ∗ =  − u 0 w 0  0 . 5 = 0 . 209 m/s, by a veraging the profiles of u 0 w 0 in the region close to the wall [ 13 ]. App endix B: Plume axis estimation In this section, the v ertical p osition, h ∗ s , of the actual axis of the plume is estimated for b oth D3 and D6 as the z co ordinate of maxim um c ( z ) v alue. In fact, the maxim um v alue of the mean concen tration, h ∗ s , along z /δ is not exactly at z = h s , but it decreases downstream from the source mainly due to tw o factors: the effect of the mean shear ∂ u/∂ z (where u is mean streamwise velocity), and the source wak e. Fig. 10 shows the v ertical profiles of the mean concen tration, c , for the tw o source config- urations, D3 and D6, at differen t streamwise lo cations, x/δ . In order to extract a reliable v alue of the wall-normal co ordinate of maximum c , w e fitted the vertical profiles of mean concen tration by adopting a Gaussian distribution with total reflection on the ground [ 1 ], 27 FIG. 10. V ertical profiles of the mean concen tration v alue, c , at different stream wise lo cations, for the t wo source diameters, D = 3 mm and D = 6 mm. The profiles obtained from the reflected Gaussian distribution of Eq. ( B1 ) are also sho wn. The wall-normal co ordinate of the source axis, h s , is illustrated as a horizon tal dotted line, while the plume axis height, h ∗ s , is displa yed as a blue (red) dashed line for the source D3 (D6). defined as c f it ( x, z ) = M e /ρ 2 π σ y σ z u adv " exp − ( z + h ∗ s ) 2 2 σ 2 z ! + exp − ( z − h ∗ s ) 2 2 σ 2 z !# (B1) where u adv is the mean streamwise velocity at the plume center of mass, h ∗ s is the vertical co ordinate of the plume axis, while σ y and σ z are the transv ersal and w all-normal (a v erage) spread of the plume, respectively . The fitting function rep orted in Eq. ( B1 ) is the most suited distribution to repro duce the vertical mean concentration profiles [ 1 ]. In particular, here σ y , σ z and h ∗ s are adopted as free parameters of the fitting pro cedure. Although the v alue of h ∗ s can also b e set as constant (as an appro ximation) and equal to h s , in this work w e explicitly used h ∗ s as a free parameter in the Eq. ( B1 ) to highligh t the effect of the mean shear on the plume. By doing so, h ∗ s is not fixed but dep ends on the source size b y mo ving do wnstream, with h ∗ s ≤ h s . In Fig. 10 , the v alues of h ∗ s are shown as horizontal dashed and dot-dashed lines for D3 and D6, resp ectively . It is also worth noting that the v alues of the v ertical co ordinate of the plume axis, h ∗ s , are b etter iden tified b y the parameters of the fitting pro cedure than b y lo cating the maximum (exp erimen tal) c v alue. This issue is crucial in the far field (i.e., far downstream from the source), where h ∗ s > 0 whereas the 28 maxim um (exp erimen tal) c v alue is lo cated very close to the flo or (i.e., ( z /δ ) c max → 0, due to the plume disp ersion and the effect of the ground reflection of the plume). App endix C: Peak-pit asymmetry in turbulen t transp ort signals In this Section, w e inv estigate the asymmetry b et w een p eaks and pits in the temp oral structure of vertical turbulent transp ort time-series. This analysis is motiv ated b y the fact that the passiv e scalar can b e transp orted b y turbulence upw ards ( w 0 > 0) and do wnw ards ( w 0 < 0). FIG. 11. V ertical profiles of the a verage p eak o ccurrence, φ , and the assortativit y co efficien t, r , for the top- and b ottom-visibilit y . The profiles are plotted for D3 (a) and D6 (b), in the near field ( x/δ = 0 . 325), at an in termediate lo cation ( x/δ = 1 . 30), and in the far field ( x/δ = 3 . 90). The w all-normal co ordinate of the source axis, h s , is illustrated as a horizontal dotted line, while the plume axis heigh t, h ∗ s , is displa yed as a dashed horizontal line for b oth D3 and D6. T o address this issue from the net work p ersp ectiv e, in Fig. 11 w e show the b eha viour of φ and r obtained from the series w 0 c 0 (top-visibilit y) and from the series − w 0 c 0 (b ottom- visibilit y). The in tersection b et w een the metrics from the top- and b ottom-visibilit y accu- rately corresp onds to the vertical lo cation of the actual plume axis, h ∗ s , for b oth the source configurations D3 and D6. In fact, in the plume axis the v ertical turbulent transp ort is 29 exp ected to b e symmetrical up wards and down w ards (e.g., see Fig. 8 (b,e)). On the other hand, the p eak-pit asymmetry is significan tly observ ed in the metric v alues ab ov e and b elow the plume axis, h ∗ s . In particular, the metrics from the b ottom-visibilit y are alwa ys smaller than the metrics from the top-visibility when the region b elo w the plume axis is fo cused, while the opp osite b eha viour is found ab o ve the plume axis. The assortativity co efficient, r , for the b ottom-visibilit y do es not reach strong negativ e v alues for z > h ∗ s , in b oth cases D3 and D6. This implies that there is not a strong vertical separation b et w een pits and the other data v alues in the series of − w 0 c 0 , i.e. extreme negative v alues are unlik ely to app ear for z > h ∗ s (as shown in Fig. 8 (a,d)). In fact, since most of the passiv e scalar is presen t around the plume axis, for z > h ∗ s v ertical turbulent transp ort is mainly upw ards while for z < h ∗ s it is mainly do wnw ards. In order to understand why the av erage p eak o ccurrence, φ , for the b ottom-visibilit y is larger than for the top-visibility for z > h ∗ s , w e recall that the visibilit y algorithm is insensitive to absolute intensit y of the data in the series. 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