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

The dynamics of a passive scalar plume in a turbulent boundary layer is experimentally investigated via vertical turbulent transport time series. Experimental data are acquired in a rough-wall turbulent boundary layer that develops in a recirculating wind tunnel setup. Two source sizes in an elevated position are considered in order to investigate the influence of the emission conditions on the plume dynamics. The analysis is focused on the effects of the meandering motion and the relative dispersion of the plume with respect to its center of mass. First, classical statistics are investigated. We found that (in accordance with previous studies) the meandering motion is the main factor responsible for differences in the variance and intermittency, as well as the kurtosis and power spectral density, between the two source sizes. On the contrary, the mean and the skewness are slightly affected by the emission conditions. With the aim to characterize the temporal structure of the turbulent transport series, the visibility algorithm is exploited to carry out a complex network-based analysis. In particular, two network metrics—the average peak occurrence and the assortativity coefficient—are analyzed, as they are able to capture the temporal occurrence of extreme events and their relative intensity in the series. The effects of the meandering motion and the relative dispersion of the plume are discussed in view of the network metrics, revealing that a stronger meandering motion is associated with higher values of both the average peak occurrence and the assortativity coefficient. The network-based analysis advances the level of information of classical statistics by characterizing the impact of the emission conditions on the temporal structure of the signals in terms of extreme events (namely, peaks and pits) and their relative intensity. In this way, complex networks provide—through the evaluation of network metrics—an effective tool for time-series analysis of experimental data.

Figure 1

Sketch of the TBL setup; the plume is illustrated in green, while the horizontal dash-dotted line refers to the source axis. The symbols are defined in the main text. Two large-scale eddies of (Eulerian) characteristic size $\lambda$ are also depicted as rotating arrows.

Figure 2

Vertical profiles of (a)–(c) the mean value of wall-normal transport $\overline{w^{\prime }{c}^{\prime }}$ and (d)–(f) the standard deviation ${\sigma }_{w^{\prime }{c}^{\prime }}$. The profiles are plotted at three streamwise locations, in the near field ($x/\delta =0.325$), in the far field ($x/\delta =3.90$), and at an intermediate location ($x/\delta =1.30$), for the two source diameters, $D=3$ mm and $D=6$ mm. The wall-normal coordinate of the source axis, $h_s$, is illustrated as a horizontal dotted line, while the plume axis height, $h_s^{*}$, is displayed as a blue (red) dashed line for the source D3 (D6).

Figure 3

Vertical profiles of (a)–(c) the skewness of wall-normal transport ${\mathcal{S}}_{w^{\prime }{c}^{\prime }}$, in linear-linear scale, and (d)–(f) the kurtosis ${\mathcal{K}}_{w^{\prime }{c}^{\prime }}$, in log-linear scale. The profiles are plotted at three streamwise locations, in the near field ($x/\delta =0.325$), in the far field ($x/\delta =3.90$), and at an intermediate location ($x/\delta =1.30$), for the two source diameters, $D=3$ mm and $D=6$ mm. The wall-normal coordinate of the source axis, $h_s$, is illustrated as a horizontal dotted line, while the plume axis height, $h_s^{*}$, is displayed as a blue (red) dashed line for the source D3 (D6).

Figure 4

Normalized spectral density, $E^{*}$, as a function of the normalized wavenumber, ${\kappa }^{*}$, of the wall-normal turbulent flux, $w^{\prime }{c}^{\prime }$. Spectra are evaluated at the source axis ($z=h_s$), for the two source diameters, $D=3$ mm and $D=6$ mm, at three streamwise locations: (a) in the near field ($x/\delta =0.325$), (b) at an intermediate location ($x/\delta =1.30$), and (c) in the far field ($x/\delta =3.90$).

Figure 5

(a) Example of the intermittent behavior of the concentration signal (first 50 values) measured at $x/\delta =1.30$ and at the source axis. Blue and red data correspond to upward and downward transport, respectively. The corresponding values of the intermittency factor for the shown time interval are also reported. (b) Intermittency factors of the passive scalar concentration, ${\gamma }_c$, and wall-normal turbulent flux, ${\gamma }^{+}$ and ${\gamma }^{-}$, as a function of $x/\delta$ along the source axis (i.e., $y/\delta =0$ and $z=h_s$).

Figure 6

Examples of two intervals of time series, $(t_i,s_i)$, and corresponding visibility networks, showing different temporal structures. Nodes and links are depicted as red dots and green lines, respectively. (a) First 10 out of ${10}^5$ observations of a series extracted from a uniform probability distribution in the interval $[0,1]$. (b) First 20 out of ${10}^5$ observations of a series extracted from a uniform probability distribution in the interval $[0,1]$, with periodic spikes (every 14 instants, $t_i$) uniformly distributed in the interval $[0,100]$. The values of $s_i$ in the vertical axis are not shown due to the invariance of the visibility algorithm to affine transformations.

Figure 7

Vertical profiles of (a) the average peak occurrence $\varphi$ (in hertz), and (b) the assortativity coefficient $r$, based on signals of wall-normal turbulent transport, $w^{\prime }{c}^{\prime }$. The profiles are plotted from the near field ($x/\delta =0.325$) to the far field ($x/\delta =3.90$), for the two source diameters, $D=3$ mm and $D=6$ mm. The wall-normal coordinate of the source axis, $h_s$, is illustrated as a horizontal dotted line, while the plume axis height, $h_s^{*}$, is displayed as a blue (red) dashed line for the source D3 (D6). The insets at $x/\delta =0.325$ show two zooms around the plume axis.

Figure 8

Time series of vertical turbulent transport in the near field, $x/\delta =0.325$, for the source diameters (a)–(c) D3 and (d)–(f) D6. Signals are plotted at three wall-normal coordinates, namely, above the source axis at $z/\delta =0.303$ [(a) and (d)], at the source axis $h_s/\delta =0.240$ [(b) and (e)], and below the source axis at $z/\delta =0.175$ [(c) and (f)]. For comparison purposes, the time series are normalized as ${\left(w^{\prime }{c}^{\prime }\right)}^{*}=(w^{\prime }{c}^{\prime }-\overline{w^{\prime }{c}^{\prime }})/{\sigma }_{w^{\prime }{c}^{\prime }}$.

Figure 9

Vertical profiles of velocity statistics normalized with the friction velocity, $u_{*}$. (a) Mean velocity defect, $\left(u_{\infty }-\overline{u}\right)$. (b) Standard deviations of the longitudinal, ${\sigma }_u$, and vertical, ${\sigma }_w$, velocity component, depicted in black and red, respectively. (c) Reynolds stress, $-\overline{u^{\prime }{w}^{\prime }}$. Symbols: $\circ$, $x/\delta =0.65$; $\square$, $x/\delta =1.30$; $•$, $x/\delta =2.60$; $◊$, $x/\delta =3.90$.

Figure 10

Vertical profiles of the mean concentration value, $\overline{c}$, at different streamwise locations, for the two source diameters, $D=3$ mm and $D=6$ mm. The profiles obtained from the reflected Gaussian distribution of Eq. (B1) are also shown. The wall-normal coordinate of the source axis, $h_s$, is illustrated as a horizontal dotted line, while the plume axis height, $h_s^{*}$, is displayed as a blue (red) dashed line for the source D3 (D6).

Figure 11

Vertical profiles of the average peak occurrence, $\varphi$, and the assortativity coefficient, $r$, for the top and bottom visibility. The profiles are plotted for (a) D3 and (b) D6, in the near field ($x/\delta =0.325$), at an intermediate location ($x/\delta =1.30$), and in the far field ($x/\delta =3.90$). The wall-normal coordinate of the source axis, $h_s$, is illustrated as a horizontal dotted line, while the plume axis height, $h_s^{*}$, is displayed as a dashed horizontal line for both D3 and D6.