4.1. Mean concentration field

The analysis of the mean concentration field is a prerequisite to appreciate, understand and discuss the spatial evolution of the higher statistical moments of the concentration fluctuations. Figure 4(a) shows the downwind variation of the centreline maximum of mean concentration for the investigated source sizes and elevations. Following Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015), the concentrations are normalised as $\bar {c}^{*} = \bar {c} (u_s \delta ^2 / Q)$, where $Q$ indicates the source mass flow rate and $u_s$ the mean wind velocity at the source height $z_s$. Note that the concentration has here the dimension of mass per volume, as customary in atmospheric dispersion modelling (e.g. Panofsky & Dutton Reference Panofsky and Dutton1988; Arya Reference Arya1999). The first thing to be observed is that smaller source sizes imply initially higher concentrations, but this effect is short-lived as expected (e.g. Arya Reference Arya1999). The elevated sources, D6M and D12M, have indistinguishable centreline mean concentrations for $x / \delta \gtrapprox 0.5$, while the difference between D6 and D12 is negligible already at $x / \delta \approx 0.25$. This is due to the higher turbulence and lower mean wind speed at $z_s / \delta = 0.19$, compared with the core of the boundary layer. The situation is more complicated for the ground-level sources. Sources D6G-X and D12G become quite similar already at $x / \delta \approx 0.15$, while the mean concentrations for D6G and D12G intersect at $x / \delta \approx 0.3$, but the difference between the two persists at larger distances, with D6G having a slightly lower mean concentration.

Figure 4.

(a) The along-wind variation of the maximum of the normalised mean concentration. (b) The along-wind variation of the crosswind centreline maximum of the normalised standard deviation of the concentration. The insets show the near-field region. Note that in this and the following figures the markers on the LES data are included to help the reader to distinguish more easily the different cases one from the other and do not correspond to sampling points.

The persistence of this gap can be explained by the differences in the source centreline heights. Although both sources are located close to the ground, D12G spans a larger vertical extension than D6G. This implies that the plume released from D12G is subject to a higher overall mean wind speed, which generates a slightly faster advection. This condition makes the advection of D12G similar to that of D6G-X (which has a slightly higher elevation; see table 1). Therefore, while differences in source size disappear quite quickly, even a small difference in the source elevations ($\Delta z_s$) can result in persistent differences if $\Delta z_s$ is located in the high mean wind shear region, typical of a near-ground neutral boundary layer.

Note also that, at downwind distances $x/\delta \gtrapprox 0.6$, the ground-level sources display higher values in the mean concentration compared with the elevated sources because of the zero-flux condition at the wall.

In the following, we also compare scalar concentration results of the LES and wind-tunnel experimental results, obtained in flows with different $u_s$ and $u_{*}$. For the elevated sources, to avoid differences trivially arising due to varying advection times, we compare profiles taken at distances implying the same dimensionless advection time, $T^{*}=({x}/{u_s})({u_{*}}/{\delta })$, as suggested by e.g. Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015). Following the approach of Ardeshiri et al. (Reference Ardeshiri, Cassiani, Park, Stohl, Stebel, Pisso and Dinger2020) and looking for $T_{(exp)}^{*}=T_{(LES)}^{*}$ (where $exp$ stands here for experimental), we define therefore an equivalent along-wind distance as

(4.1)\begin{equation} x^* = \begin{cases} x & \text{(LES)}, \\ x\dfrac{u_{s(LES)}}{u_{s(exp)}} \dfrac{u_{*(exp)}}{u_{*(LES)}} & \text{(wind-tunnel experiments)}. \end{cases} \end{equation}

This approach is not used when considering plumes emitted by low-level sources, since it relies on the validity of Taylor's frozen turbulence hypothesis, $(x = u_s t)$, which is questionable in regions of strong wind shear, where the along-wind turbulence standard deviation is not negligible compared with the mean wind. Therefore, for ground-level sources, it is simply $x^*=x$.

Figure 5 shows the crosswind (panels a–c) and vertical (panels d–f) profiles of the mean concentration $\langle \bar {c} \rangle$ (including data by Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015) for $z_s = 0.19 \delta$), through the crosswind centreline for three different along-wind distances from the source. Figure 5(g,h) reports the normalised plume dispersion standard deviations in the crosswind $\sigma _y$ and vertical $\sigma _z$ directions (whose definition is given in Appendix B). As expected (e.g. Csanady Reference Csanady1973; Fackrell & Robins Reference Fackrell and Robins1982; Arya Reference Arya1999), the crosswind mean concentration profiles (figure 5a–c) are very well fitted by a Gaussian model (not shown here). The vertical profiles (figure 5d–f) are instead well modelled by a reflected Gaussian model, as evidenced by the Gaussian-D6G fit displayed in figure 5(d–f) for the near-ground source (see Appendix B for details about the reflected Gaussian formulation).

Figure 5.

Profiles of the mean concentration in the (a–c) crosswind ($z=z_s$) and (d–f) vertical direction at downwind distances (a,d) $x^*/\delta =0.36$, (b,e) $x^*/\delta =0.73$ and (c, f) $x^*/\delta =2.9$. The source size in Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015) data is $d_s=6\ {\rm mm}=0.0075\delta$ and the source elevation is $z_s/\delta = 0.19$. Panels (g,h) report the LES plume spatial standard deviation in crosswind $\sigma _y$ and vertical $\sigma _z$ directions as a function of downwind distance for both the 6.25 mm and 12.5 mm sources together with Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015) data for the 6 mm source. For ground-level sources in the vertical direction, the definition of $\sigma _z$ is explained in Appendix B, together with the definition of the Gaussian approximation for D6G.

For the downwind distances displayed in figure 5, and in agreement with what was discussed above for figure 4, the mean concentration profile is very weakly affected by a varying source size, whose influence can be detected only very close to the emission (e.g. D6M and D12M at $x^*/ \delta = 0.36$; figure 5a,d). In contrast, the source elevation significantly alters the mean concentration. Plumes emitted by sources at mid-height have a higher maximum mean concentration than those at $z_s/\delta =0.19$, at all downwind distances (figures 4a and 5a–f). This can be readily explained through a plume Gaussian model ((B3) in Appendix B): increasing $z_s$, the mean wind speed increases and, therefore, the plume travel time shortens. In this case $u_s = 4.4\ {\rm m}\ {\rm s}^{-1}$ in the middle of the boundary layer (D6M and D12M), whereas $u_s = 3.8 \ {\rm m}\ {\rm s}^{-1}$ at $z_s/ \delta = 0.19$. Furthermore, the turbulent fluctuations decrease with height (see figure 1e, f). These two aspects have a considerable impact on the plume spreads $\sigma _y$ and $\sigma _z$ (figure 5g,h) and, therefore, on $\langle c^*\rangle$. The ground-level sources present a more complex behaviour. At $x^*/\delta = 0.36$, the ground-level sources show a higher mean concentration than those of the sources at $z_s/ \delta =0.19$ but lower than the mean concentration of the sources at $z_s/ \delta =0.5$, as shown in figure 5(a,d). For downwind distances $x^*/\delta \gtrapprox 0.6$, the mean concentration of the ground-level sources are always the highest among the simulated configurations (figure 4a and 5b,c,e, f). In these cases, although the plume travel times are longer compared with the elevated sources, the ground effect has a prevalent role and produces a lower plume dispersion. The difference between the mean concentration of D6G and D12G arises, as discussed above in relation to figure 4, from the different top vertical extension of the two sources. As in previous studies (Fackrell & Robins Reference Fackrell and Robins1982; Crimaldi & Koseff Reference Crimaldi and Koseff2006), power laws are fitted to the plume standard deviations (figure 5g,h). For the source at $z_s/\delta = 0.5$, $\sigma _y \propto x^{0.86}$ and $\sigma _z \propto x^{0.85}$. For $z_s/\delta = 0.19$, $\sigma _z \propto x^{0.74}$ and $\sigma _y \propto x^{0.8}$. For near-ground sources $\sigma _z \propto x^{0.77}$, a value similar to that obtained by F&R and Crimaldi & Koseff (Reference Crimaldi and Koseff2006) and $\sigma _y \propto x^{0.67}$. This latter is due to the longer advection time and to the smaller size of near-ground turbulent structures, resulting in a more diffusive regime. The value of the exponent of the power law does not exhibit significant dependencies on the source size.

Overall, the good quantitative match with the measurements of Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015) and the general consistency with the data reported in Talluru et al. (Reference Talluru, Philip and Chauhan2018) for sources in the same range of elevations, show the reliability of the LES.

Finally note that, in the case of a large Schmidt number ($Sc=\nu /D)$, ground-level source releases in smooth walls get trapped in the viscous layer (Crimaldi, Wiley & Koseff Reference Crimaldi, Wiley and Koseff2002; Lim & Vanderwel Reference Lim and Vanderwel2023). This phenomenon fades out both for lower values of $Sc$ (e.g. Talluru et al. Reference Talluru, Philip and Chauhan2018) and for rough walls, and cannot therefore be simulated in our LES.

4.2. Concentration fluctuations standard deviation

Differently from the mean, the standard deviation of the concentration fluctuations ($\sigma _c^{*}$) is strongly affected by the source size for the elevated sources, whereas the effects on the ground-level sources are less evident (e.g. Fackrell & Robins Reference Fackrell and Robins1982; Thomson Reference Thomson1990; Cassiani et al. Reference Cassiani, Franzese and Giostra2005). As extensively discussed in Ardeshiri et al. (Reference Ardeshiri, Cassiani, Park, Stohl, Stebel, Pisso and Dinger2020), the generation of the fluctuations for a plume emitted by an elevated source is dominated by the early phases of plume dispersion that are characterised by the meandering motion (Gifford Reference Gifford1959) of the almost unmixed plume. We briefly recall here that the concentration fluctuations are driven by two phenomena: the meandering movement of the instantaneous plume (Gifford Reference Gifford1959; Csanady Reference Csanady1973; Cassiani & Giostra Reference Cassiani and Giostra2002; Cassiani et al. Reference Cassiani, Bertagni, Marro and Salizzoni2020) displacing the plume's centre of mass, and the dispersion (expansion) of the plume relative to the centre of mass (Sawford Reference Sawford2001; Dosio & de Arellano Reference Dosio and de Arellano2006; Franzese & Cassiani Reference Franzese and Cassiani2007; Cassiani et al. Reference Cassiani, Bertagni, Marro and Salizzoni2020).

Since turbulence is characterised by eddies with a wide range of temporal and spatial scales, if the source is small, a larger range of scales can contribute to the meandering motion of the plume compared with its relative dispersion. This implies that the concentration fluctuations increase with decreasing source size.

Moving downwind from the source, the initial difference in the (crosswind and vertical) dimension of the plumes relative to their centre of mass (i.e. the relative dispersion) becomes progressively negligible compared with the growing plume cross-section, and the source-size effect is progressively lost.

This behaviour is well confirmed for the elevated sources by figures 4(b) and 6, which show that, close to the emission point, D6 (or D6M) presents higher values of $\sigma _c^*$ compared with D12 (or D12M). These differences disappear, respectively, at $x/ \delta \approx 1.8$ for the sources at $z_s = 0.19 \delta$ and at $x/ \delta \approx 3.4$ for the sources in the middle of the boundary layer. The ground-level sources exhibit some noticeable differences in the concentration variance only in the very near-source region, and these differences rapidly vanish at $x/\delta \approx 0.4$ (see figure 4b).

Figure 6.

(a–c) Transversal and (d–f) vertical profiles of concentration fluctuations standard deviation at various downwind distances: (a,d) $x^*/\delta =0.36$ (b,e) $x^*/\delta =0.73$ and (c, f) $x^*/\delta =2.9$.

The greater persistence of the differences in $\sigma _c^*$ with the increase of the source elevation is due to several reasons: (i) the higher mean wind speed, meaning that the plume has a shorter travel time to evolve, (ii) the higher initial production of fluctuations due to meandering, and (iii) the less intense dissipation rate of the scalar variance.

The reader should note in figures 4(b) and 6 that the slight difference in elevation between D6G and D6G-X has a small influence on the value of the concentration standard deviation.

Regarding the direct quantitative comparison with the wind-tunnel experiments, similarly to what was done for the mean concentration, we limit it to the measurements for the $6\ {\rm mm}$ source at $z_s / \delta = 0.19$ in Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015). Compared with the wind-tunnel data, the simulations display a slightly higher concentration standard deviation. We believe that this is due to a combination of factors. The effective source diameter in Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015) should be considered equal to the external diameter of the pipe emitting the scalar ($8\ {\rm mm}$), rather than its internal diameter, i.e. $6\ {\rm mm}$. Furthermore, even though the gas is emitted isokinetically, the physical presence of the source in the experiments induces a wake perturbing the flow locally, an effect that is not reproduced in the LES, where the source is just a marker. Finally, the differences in the turbulent flow between Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015) and our simulations (see figures 1 and 2) are likely to contribute to the discrepancies observed in the concentration fields.

A general, although more qualitative, comparison of our results in figure 6 is possible with the measurements of Talluru et al. (Reference Talluru, Philip and Chauhan2018), showing substantial consistency in the spatial profiles of $\sigma _c^*$ for the elevated sources. For the ground-level source, the observations of Talluru et al. (Reference Talluru, Philip and Chauhan2018) at $x/\delta = 1$ showed a vertical profile similar to those observed in our simulations in figure 6, with minimal difference between the peak value and the ground value. Overall, the comparison with the wind-tunnel experiments confirms confidence in the current simulations.

In the downwind range of our simulations, the horizontal crosswind profiles of concentration fluctuations for the ground-level sources show a persistent double-peak behaviour regardless of the distance from the source location. In contrast, for the elevated sources, this behaviour is observed only in the vicinity of the source and is not visible in the range of downwind distances reported in figure 6. This feature will be further investigated in the following section.

4.2.1. Double-peak behaviour in the very near-source region

The horizontal crosswind profile of the concentration standard deviation generated by a plume shows a double-peak behaviour very close to the source ($x/\delta < 0.05$), irrespective of the source size and elevation, as illustrated in figure 7. The appropriate grid resolution of the simulations allows us to immediately capture the off-centre variance peaks very near the source location. Note that this was not possible in the previous study of Xie et al. (Reference Xie, Hayden, Voke and Robins2004b) due to the too coarse spatial discretisation. The bimodal shape of the variance profiles has seldom been observed in previous experimental studies of point sources, as operational conditions, such as the presence of the stack, could influence the very near scalar field. For elevated sources, this behaviour disappears particularly rapidly for smaller sources and is not visible already at $x / \delta = 0.15$ for both D6M and D6. For D12, the double peak disappears between $x/ \delta = 0.1$ and $x/ \delta = 0.15$ while, for D12M, it is still weakly visible at $x/ \delta = 0.2$ due to the faster advection and shorter plume travel time. For ground-level sources, the double peak appears to be persistent in the downwind range of our simulations, although initially, it is less pronounced than for the elevated sources. Figure 7 also confirms that initially the standard deviation of D6G is significantly larger than that of D12G, but this difference decreases very rapidly. For the elevated sources, the double peak is also present in the vertical direction (not shown here) and is similar to the crosswind profile since the plume development initially has an almost radial symmetry due to the similar statistics of the $v$ and $w$ velocity components. Conversely, for the ground-level sources, the double peak is not observable in the vertical section.

Figure 7.

Transversal profiles of concentration fluctuations standard deviation in the vicinity of the source: (a) $x/\delta =0.05$ (b) $x/\delta =0.10$ (c) $x/\delta =0.15$ and (d) $x/\delta =0.20$.

The generation and time evolution of the double peak in $\sigma _c$ have been studied using stochastic models and theoretically by Thomson (Reference Thomson1990, Reference Thomson1996) for instantaneous scalar releases (line source) in homogeneous turbulence with no mean advection. This case can be interpreted as a continuous plume in an approximately homogeneous turbulent flow (i.e. similar to our elevated plume dispersion cases) in the presence of mean advection by adopting Taylor's approximation to transform between plume evolution in a downwind position and time. Thomson (Reference Thomson1996) argued that the off-centre variance peaks disappear when the absolute dispersion scale becomes of the order of the source size, i.e. when the total average plume size, including the source, doubles, $\sigma _y/\sigma _s \approx 2$ in the present context, where $\sigma _s$ is the LES source size calculated as the actual plume standard deviation at the source location. Following the concentration fluctuation profiles in the along-wind direction, the approximate distances where the double-peak behaviour vanishes in the LES are reported in table 2, showing that the findings of Thomson (Reference Thomson1996) are satisfied with good approximation. Furthermore, Thomson (Reference Thomson1996) argued that the overall peak in $\sigma _c$ occurs for $\sigma _y/\sigma _s \approx 2$. Looking at figure 7, particularly for D12M, which evolves slower and is therefore better resolved at the considered time intervals, the overall peak in $\sigma _c$ slightly increases in $x/\delta = 0.05-0.1$, it remains approximately constant in the interval $x/\delta = 0.1-0.15$, and it slightly decreases for $x/\delta = 0.15-0.2$. This reasonably agrees with Thomson (Reference Thomson1996).

Table 2.

Distance $x/\delta |_{LES}$ of the disappearance of the double peak in $\sigma _c^*$ for the elevated plumes and the corresponding plume size expressed as $\sigma _y/ \sigma _s$.

For the ground-level source, the off-centre double peaks appear to be persistent in the downwind range of the simulations. In order to better understand the differences between the generation of concentration fluctuations for ground-level and elevated sources, the budget equation for the resolved scale mean scalar variance is considered:

(4.2)\begin{equation} 0 = \underbrace{-\langle\bar{u}_i \rangle \frac{\partial \langle \bar{c}''^2\rangle}{\partial x_i}}_\text{Adv.} \underbrace{- 2\langle \bar{u}_i''\bar{c}''\rangle \frac{\partial \langle \bar{c}\rangle}{\partial x_i}}_\text{Prod.} \underbrace{-\frac{\partial \langle \bar{u}_i''\bar{c}''^2 \rangle}{\partial x_i}}_\text{T.T.} - 2 \xi_{res}. \end{equation}

Here the first term on the right-hand side corresponds to the advection (Adv.), the second and third terms correspond to the production (Prod.) and turbulent transport (T.T.), respectively, and $\xi _{res}$ is the mean scalar dissipation rate computed as the residual. The factor $2$ is left explicit in analogy with what appears in the definition of the actual physical dissipation, i.e. $2 \nu _c \langle {({\partial {c''}}/{\partial x_i})({\partial {c''}}/{\partial x_i})} \rangle$, where $\nu _c$ here is the molecular diffusivity. Heinze, Mironov & Raasch (Reference Heinze, Mironov and Raasch2015) and Ardeshiri et al. (Reference Ardeshiri, Cassiani, Park, Stohl, Stebel, Pisso and Dinger2020) discussed in detail that for the numerical methods used in the PALM code, due to the numerical dissipation, other estimates of the mean scalar dissipation rate, like the equilibrium approximation for the SGS (see e.g. Sykes & Henn Reference Sykes and Henn1992; Kaul et al. Reference Kaul, Raman, Balarac and Pitsch2009) and the transfer of resolved scale scalar variance to the SGS (see e.g. Heinze et al. Reference Heinze, Mironov and Raasch2015), do not correctly represent the actual dissipation rate.Kewley (Reference Kewley1978) showed that an approximate solution of the concentration variance transport equation is possible based on a balance between the production and dissipation rates. This approximation produces a double-peak profile (see also Netterville Reference Netterville1979) with a concentration variance value of zero at the plume centreline. The turbulent transport acts to smooth this double peak towards a Gaussian-like behaviour and produces non-zero concentration variance along the plume centreline. Figure 8 shows the budget (for the 12.5 mm sources) at the elevation of maximum mean concentration in the crosswind direction. The budget terms are generically indicated as $\phi$ and reported normalised as $\phi ^{*} = \phi (\delta / u_{*}) (u_s \delta ^2 / Q)^2$. Figure 8 reveals that the production terms for both ground-level and elevated sources exhibit a clear persistent double peak. Initially, the double peak is more pronounced for the elevated sources (see figure 7 at $x/\delta$ =0.1 and 0.15) and this is in agreement with the more pronounced double-peak behaviour in the production term for the elevated sources. However, for the ground-level sources, this term remains relevant with increasing downwind distance from the source, while for the elevated sources, it becomes negligible at $x/\delta = 0.625$. The more persistent relevance for the ground-level sources is mainly due to a stronger decay of the advection term, which becomes the dominant term for the higher elevations. This behaviour is consistent with the experimental observations of F&R.

Figure 8.

Variance budget analysis of scalar concentration. Here $\phi ^{*}$ represents a generic normalised quantity in (4.2) as a function of crosswind direction ($y / \delta$) and for the $12.5$ mm source at (a) $z_s/\delta =0.003$ (D12G) (b) $z_s/\delta =0.19$ (D12) and (c) $z_s/\delta =0.5$ (D12M) and at a downwind distance of $x/\delta =0.13$. Panels (d–f) report the same quantities as (a–c) but at $x/\delta =0.625$.

4.3. Characteristics of the most energetic scales in the concentration variance power spectrum

To gain further insights into the mechanics of the scalar dispersion process, we investigate the variance-containing range of the concentration power spectrum, $\varPhi _{cc}$ (see Appendix A). First, the LES results for the elevated sources will be discussed and afterwards theoretical arguments based on a stochastic approach will be used to show the connection between the peak in the variance-containing range of $\varPhi _{cc}$ and the peak in the energy-containing range of $\varPhi _{vv}$ and $\varPhi _{ww}$ in relation to the meandering motion of the plume. The section concludes with a qualitative analysis of the shape of the variance-containing range for the ground-level sources.

The spectra are normalised by the concentration variance and smoothed using a Gaussian filter, as they would otherwise be affected by noise hindering the clear detection of the shape. We observe that, for constant downwind distances and a given source elevation, the pre-multiplied normalised spectra ($f\varPhi _{cc}/\sigma _c^2$) tend to exhibit a common shape, independent of the sampling position in the $(y,z)$ plane (figure 9). We remind the reader that the frequency of the peak in the pre-multiplied spectrum is directly related to the integral scale (see Appendix A).

Figure 9.

Pre-multiplied normalised energy spectrum of the concentration fluctuations as a function of normalised frequency for three downwind distances and two source elevations. Results are shown for (a,b,c,g,h,i) $z_s = 0.5 \delta$ and (d,e, f,j,k,l) $z_s = 0.19 \delta$. Spectra sampled at several vertical positions for $y=y_s$ (a–f) and spectra sampled at several crosswind lateral positions for $z=z_s$ (g–l).

Note that all the reported sampling points, for a given downwind distance $x$, are characterised by a mean concentration $\langle c^* \rangle (y=y_s,z) > {\langle c^* \rangle }_ {max} / 50$ or $\langle c^* \rangle (y,z=z_s) > {\langle c^* \rangle }_ {max} / 50$, which identifies a distance of approximately two standard deviations, on the axis, from the local mean plume centreline. At a greater distance from the plume centreline, the sampled time series are too short (given the high level of fluctuations) to obtain reliable spectra.

The common shape of the most energetic part of the pre-multiplied normalised concentration fluctuation spectra was originally noted in Talluru et al. (Reference Talluru, Philip and Chauhan2019).

The results in panel ( f) ($z_s/\delta = 0.19$) at $y=y_s$, $z / \delta = 0.024$, $x/\delta =2.51$ show that the spectra do not follow the common curve at this low elevation, as the plume has a large vertical extension (see e.g. figure 5g,h) and the time series is sampled close to the ground, where the turbulence characteristics are significantly different from those in the source region, as reported in figure 1. This anticipates that the spectra computed for the ground-level sources cannot be invariant.

The common shape of the normalised pre-multiplied concentration variance spectra for elevated sources was motivated by Talluru et al. (Reference Talluru, Philip and Chauhan2019), arguing that it is controlled by large-scale flow properties driving the plume meandering. Our data in figure 9 show that the shape of the pre-multiplied normalised spectra changes with the downwind distance, including a shift of the spectral peak towards lower frequencies and, consequently, a change in the integral time scales. This behaviour is consistent with the literature on plume dispersion, which relates concentration fluctuations to the meandering of the plume, as it also expands through a relative dispersion process with increasing downwind distance (e.g. Hanna Reference Hanna1986; Hanna & Insley Reference Hanna and Insley1989; Sawford Reference Sawford2001; Ardeshiri et al. Reference Ardeshiri, Cassiani, Park, Stohl, Stebel, Pisso and Dinger2020; Cassiani et al. Reference Cassiani, Bertagni, Marro and Salizzoni2020) and will be investigated in the next section.

4.3.1. Analysis of the spectra using velocity conditioned stochastic time series for elevated sources

Following the evidence of our LES data and the customary interpretation of concentration fluctuations by meandering and relative dispersion, we provide here a quantitative explanation of the relation between the velocity and concentration time scales. We remind the reader that the spectral peak is directly related to the time scale (e.g. Kaimal & Finnigan Reference Kaimal and Finnigan1994), as explained in Appendix A.

Our analysis is based on the ideas of Cassiani et al. (Reference Cassiani, Franzese and Albertson2009) that the concentration fluctuations time series at a fixed point in space, and therefore its spectrum, are characterised by two time scales, one related to the meandering of the expanding plume, which can be directly connected to the time scales of crosswind and vertical velocities through a conditional average meandering model, and the other to the scalar dissipation rate. For a slender plume, and using Taylor's hypothesis, the mean concentration field from an elevated source is well described by a Gaussian model ((B3) in Appendix B) when neglecting any ground effect. A good approximation for the elevated source plume is also the assumption of homogeneous anisotropic turbulence, with the turbulence statistics sampled at the source location. This assumption allows us to describe the standard deviations of the plume spread in the vertical and crosswind directions according to Taylor (Reference Taylor1922) theory:

(4.3)\begin{equation} \left. \begin{gathered} \sigma_{y}^2 = \sigma_s^2 + \sigma_v^2 T_{Lv}^2 \left[2\frac{t}{T_{Lv}} - 2\left(1-\exp\left(-\frac{t}{T_{Lv}}\right) \right) \right],\\ \sigma_{z}^2 = \sigma_s^2 + \sigma_w^2 T_{Lw}^2 \left[2\frac{t}{T_{Lw}} - 2\left(1-\exp\left(-\frac{t}{T_{Lw}}\right) \right) \right], \end{gathered} \right\} \end{equation}

where $\sigma _s$ is the source size, and $T_{Lv}$ and $T_{Lw}$ are here the crosswind and vertical Lagrangian integral time scales, respectively (e.g. Arya Reference Arya1999). For elevated sources, these equations can be fit to the LES plume dispersion variances (e.g. Ardeshiri et al. Reference Ardeshiri, Cassiani, Park, Stohl, Stebel, Pisso and Dinger2020).

Therefore, we use here the semi-analytical approach originally proposed in Sawford (Reference Sawford2004) and Cassiani et al. (Reference Cassiani, Franzese and Albertson2009) to analyse dispersion from a line source in grid turbulence but extending it here to a point source with two-dimensional dispersion. Following Cassiani et al. (Reference Cassiani, Franzese and Albertson2009) a stochastic model for the concentration time series incorporating meandering can be defined as

(4.4a)$$\begin{gather} {\rm d}v ={-} \frac{v}{T_{v}} \,{\rm d}t + \left(\frac{2 \sigma_v^2}{T_{v}} \right)^{1/2} \,{\rm d}W_v, \end{gather}$$

(4.4b)$$\begin{gather}{\rm d}w ={-} \frac{w}{T_{w}} \,{\rm d}t + \left(\frac{2 \sigma_w^2}{T_{w}} \right)^{1/2} \,{\rm d}W_w, \end{gather}$$

(4.4c)$$\begin{gather}{\rm d}c ={-} \frac{c-\left\langle c\mid v,w \right\rangle}{T_s}\,{\rm d}t + g(c,T_s^{{-}1},\left\langle\left\langle c\mid v,w \right\rangle \mid c \right\rangle, f_c) \,{\rm d}W_c, \end{gather}$$

where $\textrm {d}W_{v}$, $\textrm {d}W_w$ and $\textrm {d}W_c$ are independent Wiener process (see e.g. Pope Reference Pope2000). The use of a simple Langevin model for the velocity components allows for an unambiguous relationship between the spectral peaks and the time scales $T_{v}$ and $T_{w}$. These time scales are selected here from the spectral peak frequency at the plume source elevation, as identified in figure 2(h,i), using the relation $fm_{i} = 1/(2 {\rm \pi}T_{i})$ (e.g. Kaimal & Finnigan Reference Kaimal and Finnigan1994, see also Appendix A). Equation (4.4c) is the most general formulation of the concentration time series evolution proposed in Cassiani et al. (Reference Cassiani, Franzese and Albertson2009), but here we significantly simplify this model by neglecting the stochastic diffusion term (function $g$ above) because it is not necessary for our analysis. The diffusion term allows the correct fluctuations of concentration far downwind from the source where the plume meandering is negligible (Cassiani et al. Reference Cassiani, Franzese and Albertson2009), while it is unimportant close to the source where the meandering dominates. The interested reader may find the explicit formulation of the diffusion term in Cassiani et al. (Reference Cassiani, Franzese and Albertson2009). Nonetheless, it is instrumental for the present analysis to underline that the explicit form contains only one time scale $T_s$, the same time scale contained in the drift (relaxation) term in (4.4c). In the drift term, the instantaneous concentration relaxes towards $\langle c\mid v,w \rangle$, which is the local mean concentration conditioned over the crosswind and vertical velocity components. At short travel time from the source location, $T_s \rightarrow 0$ (Cassiani et al. Reference Cassiani, Franzese and Albertson2009) and the instantaneous concentration in (4.4c) can be significantly simplified as

(4.5)\begin{equation} c= \left\langle c\mid v,w \right\rangle. \end{equation}

Sawford (Reference Sawford2004) demonstrated that this definition of the instantaneous concentration corresponds to the concentration generated by a meandering plume model where the relative dispersion ($\sigma _r$) is modelled as $\sigma _{ry}^2 = \sigma _s^2 + \sigma _y^2(1-\rho _{vy}^2)$ for the crosswind direction, and $\sigma _{rz}^2 = \sigma _s^2 + \sigma _z^2(1-\rho _{wz}^2)$ for the vertical direction. The terms $\rho _{vy}$ and $\rho _{wz}$ represent the correlation between the velocity and the displacement $\Delta$ of a marked fluid particle passing through the source (Sawford Reference Sawford2004) and are defined as

(4.6)\begin{equation} \left. \begin{gathered} \rho_{vy} = \frac{\left\langle v \Delta y \right\rangle}{ \sigma_v \sigma_y}=\frac{1}{ 2 \sigma_v \sigma_y}\frac{{\rm d}\sigma_y^2}{ {\rm d}t},\\ \rho_{wz} = \frac{\left\langle w \Delta z \right\rangle}{ \sigma_w \sigma_z}=\frac{1}{ 2 \sigma_w \sigma_z}\frac{{\rm d} \sigma_z^2}{ {\rm d}t}. \end{gathered} \right\} \end{equation}

The analytical formulation of the conditional mean concentration can be obtained as (Sawford Reference Sawford2004)

(4.7) \begin{align} \left\langle c(y,z,t) \mid v,w \right\rangle &= \frac{Q}{u_s 2{\rm \pi} ((\sigma_s^2 + \sigma_y^2 (1-\rho_{vy}^2))^{(1/2)} ) ((\sigma_s^2 + \sigma_z^2 (1-\rho_{wz}^2))^{(1/2)} )}\nonumber\\ &\quad \times\exp\left(-\frac{(y-\rho_{vy} v \sigma_y / \sigma_v )^2}{2 (\sigma_s^2 + \sigma_y^2 (1-\rho_{vy}^2)) } \right) \exp\left(-\frac{(z-\rho_{wz} w \sigma_z / \sigma_w )^2}{2 (\sigma_s^2 + \sigma_z^2 (1-\rho_{wz}^2)) } \right). \end{align}

The stochastic model defined by (4.4a,b), (4.5), (4.7) and (4.6) was applied using the LES flow data sampled at the source elevation, $z_s / \delta = 0.5$, namely the velocity statistics and the time scales obtained from the spectral peaks, $fm$, as shown in figure 2(h,i): $T_w=1/(2{\rm \pi} fm_w)=0.14\delta /u_{\infty }$, $T_v=1/(2 {\rm \pi}fm_v)=0.17\delta /u_{\infty }$. The resulting spectra are plotted in figure 10 for the elevated source plume, $z_s = 0.5 \delta$, for which the hypotheses underlying this stochastic model are better respected. The $v$, $w$ and $c$ spectra at the downwind distance $x = 0.16 \delta$ are reported for different vertical and lateral positions, respectively, in panels (a,b). We are aware that the overall shape of the LES spectrum cannot be perfectly reproduced by a simple diffusive model, but the peaks in the velocity component spectra correspond, by construction, exactly to those extracted from figure 2(h,i) at the source elevation.

Figure 10.

Pre-multiplied and normalised energy spectrum of velocity components and concentration obtained with the stochastic model for the source located at $z_s=0.5\delta$. (a,b) Vertical and lateral variations of the spectrum at $x = 0.16 \delta$. (c) Vertical variation of the spectrum at $x = 2.51 \delta$. The different dashed lines correspond to different positions for the concentration time series stochastic model with respect to the plume centre. Panels (d–f) report the corresponding LES spectra for reader convenience.

Figure 10 shows a significant shift towards higher frequencies in the concentration spectra compared with the velocity components $(u,w)$ in the stochastic theoretical model. This modelled behaviour is consistent with that observed in the LES data, i.e. comparing figure 2(h,i) (at the source plume elevation) with figure 9(a,g). The LES scalar spectra are also reported in figure 10(d–f) to facilitate comparison with the stochastic model. The results from the stochastic model show a very weak dependence of the spectra on the position in the crosswind plane, i.e. they do not show a complete overlap (irrespective of the sampling position in the crosswind plane) of the concentration spectra close to the source location ($x = 0.16 \delta$). We note that at this location the meandering dominates the fluctuations and the conditional mean is a very good approximation of the instantaneous scalar field, allowing the conditional mean model to capture about $90\,\%$ of the actual concentration variance. A very weak dependence on the position in the crosswind plane cannot be ruled out also in the LES data in figure 10(d,e).

At larger distances from the release point, the conditional mean model for the meandering process becomes less precise. Furthermore, the fraction of actual variance that can be explained by the meandering of the plume decreases, while the concentration fluctuations generated by the entrainment process due to relative dispersion become more significant. Nonetheless, it is instructive to examine what happens to the spectra of fluctuations generated by the meandering process as the plume increases its relative dispersion. For this reason, the stochastic model was applied at the distance $x = 2.51 \delta$, although the model is able to explain only a small fraction of the actual concentration variance at this location. We observe that the concentration spectral peak shifts to $1 < f \delta /u_{\infty } < 2$ (figure 10c), which is consistent with the LES data reported in figure 9(c) and in figure 10( f). However, the spectra generated by the LES display a better collapse onto a unique curve, while the stochastic model based on the conditional mean approximation shows a small but clear shift of the spectra depending on the vertical sampling position. We can argue that the fluctuations due to relative dispersion, which are not accounted for by the meandering model, enhance the similarity in the concentration spectra since they are associated with a unique time scale (Cassiani et al. Reference Cassiani, Franzese and Albertson2009), $T_s$ in (4.4). The scalar time $T_s$ becomes increasingly relevant as one moves downwind from the source. However, the exact value of $T_s$ is not known a priori and requires a model itself. Cassiani et al. (Reference Cassiani, Franzese and Albertson2009) considered $T_s \propto T_m \propto T_{\phi } = \sigma _c^2 / \xi$, where $T_m$ is the Lagrangian mixing time scale usually considered proportional to the scalar dissipation rate time scale $T_{\phi }$ ($\xi$ is the scalar dissipation rate defined here as the residual in (4.2) for the special case of the LES simulation) as extensively reviewed in e.g. Cassiani et al. (Reference Cassiani, Bertagni, Marro and Salizzoni2020). Therefore, to further investigate the level of similarity in the spectrum, we calculated the scalar dissipation rate time scale $T_{\phi }$ from the LES data, under the assumption that $T_s$ and $T_{\phi }$ are proportional. Figure 11 shows $T_\phi$ as a function of both crosswind and vertical positions for the plume released at $z_s = 0.5\delta$ at varying distances from the source locations. The dissipation rate time scale is approximately independent of the $y$ and $z$ coordinates (for a fixed value of $x/\delta$) and increases with the downwind distance as expected. The time scale $T_s$ should exhibit the same behaviour, assuming that it is proportional to $T_{\phi }$, as hypothesised in Cassiani et al. (Reference Cassiani, Franzese and Albertson2009).

Figure 11.

Dissipation time scale $T_\phi$ scaled by $\delta / u_{\infty }$ for the source at $z_s = 0.5 \delta$. (a) Variation along the vertical direction for four downwind distances. Panel (b) is the same as (a) but for the crosswind direction. Lateral and vertical coordinates are normalised for the plume standard deviations.

Summarising, for a given position, the concentration time series are influenced by the conditional mean $\langle c\mid v,w \rangle$, which generates a spectral peak (and corresponding time scale) that depends weakly on the sampling point for a fixed downwind distance, and on the time scale $T_s$ that seems to be constant in the $(y,z)$ plane. As the plume travel time increases, both time scales grow and the concentration progressively switches from a direct dependence on the conditional mean to a dependence mediated by the time scale $T_s$, which enhances the similarity of the spectrum.

This discussion supports the evidence that the spectral peak in the concentration spectrum (and the related time scales) must necessarily shift towards lower frequencies as the downwind distance increases, being, however, independent (or weakly dependent) of the sampling position in the $(y,z)$ plane (for a fixed travel time). Moreover, such a shift is directly linked to the relative dispersion process and, therefore, to the actual (instantaneous) plume size.

There are some aspects that deserve further discussion. At first glance, one might be tempted to relate the dissipation time scale $T_\phi$ directly to $T_s$ or even to the spectral peak observed in the LES by a relation of the form $fm_{\alpha } \cong 1/(2 {\rm \pi}T_{\alpha })$. However, $T_s$ and the spectral peaks in figure 10 are related to Eulerian time series in the presence of a mean wind field, while $T_\phi$ is a physical time scale similar to the turbulent flow time scale $E/\epsilon$, where $E= 1/2 (\sigma _u^2(z) + \sigma _v^2(z) + \sigma _w^2(z))$ is the turbulent kinetic energy and $\epsilon$ is its dissipation rate (e.g. Pope Reference Pope2000). These physical time scales are unrelated to the mean advecting wind and are indeed more similar to Lagrangian integral scales. In the presence of a mean wind field, the relation between the Eulerian and the Lagrangian time scales ($T_E$ and $T_L$) involves the turbulence intensity, $E^{1/2}/\langle u \rangle$, with $T_E \propto T_L E^{1/2} /\langle u \rangle$ (e.g. Arya Reference Arya1999; Cassiani et al. Reference Cassiani, Franzese and Albertson2009). For a neutral boundary layer, the turbulence intensity at the considered source and plume elevation is ${\ll }1$, and we may expect that the time scale corresponding to the concentration time series is much shorter compared with that of the dissipation process, i.e. $T_s \ll T_{\phi }$. The aforementioned scalar time scales are generally assumed to exhibit a very weak dependence on the Schmidt number (Fox Reference Fox2003), which is usually neglected in scalar dispersion models (see Cassiani et al. Reference Cassiani, Bertagni, Marro and Salizzoni2020).

4.3.2. LES spectra of concentration fluctuations for ground-level sources

The concentration spectra of a ground-level source (figure 12) present a strong dependence on the vertical position. Their shape changes significantly with elevation, and the spectral peak of $\varPhi _{cc}$ shifts from low to high frequency. This reflects the high vertical inhomogeneity. In figure 12 the vertical lines bound the location of the spectral peak of $f\varPhi _{uu}$ for $z/\delta > 0.011 \delta$ above the sharp transition at $z/\delta \approx 0.01$, which moves the peak from the high frequencies to the low frequencies (see also figure 2g,d).

Figure 12.

Pre-multiplied normalised energy spectrum of the concentration fluctuations as a function of normalised frequency for the ground-level source D6G at three downwind distances. Spectra sampled at several vertical positions for $y=y_s$. The vertical blue lines bound the location of the spectral peak of $f\varPhi _{uu}$ for $z/\delta \gtrapprox 0.011$.

We observe that the frequency location of the peak at low vertical positions (figure 12) is in a very similar location when compared with the peak in the spectrum of the along-wind velocity fluctuations $\varPhi _{uu}$ for $z/\delta > 0.01$ (figure 2g). Moving vertically towards the plume edges the spectral peak location in $\varPhi _{cc}$ shifts to the high frequencies that are similar to what was previously investigated for the elevated sources. This feature is related to the meandering motions and is linked to the spectra of the $v$ and $w$ velocity components, as discussed above in § 4.3.1.

This peak transition from low to high frequencies occurs at higher vertical positions as the distance from the source location increases. Such behaviour suggests that this shift is not related to the flow properties but to the plume dispersion and its vertical size. As a first approximation, we estimate the transition at $z\approx 2 \sigma _z$ from the ground.

We speculate that away from the ground, where the entrainment process creates plume filaments, the fluctuations are dominated by meandering motions in the directions orthogonal to the mean wind flow ($(y,z)$ plane). On the contrary this process is negligible for the plume main body (defined here as $z \lessapprox 2 \sigma _z$) from the ground and the fluctuations are instead generated by the actions of the along-wind turbulence, and therefore, the peak is at a lower frequency.

4.4. Higher-order central moments and scalar p.d.f.

Analysis of the high-order statistics focuses on the skewness and kurtosis of the concentration. Furthermore, we also report the spatial evolution of the fluctuation intensity $i_c=\sigma _c / \langle c \rangle$, since it is generally a key parameter for modelling the scalar p.d.f. (e.g. Cassiani et al. Reference Cassiani, Bertagni, Marro and Salizzoni2020). When comparing the downwind evolution of the LES data with experiments, the equivalent downwind distance ($x^*/\delta$) is considered for the elevated source (4.1). This transformation is not suitable for sources close to the ground and, therefore, in that case, $x/\delta$ is used. Nevertheless, note that the downwind variation of these statistics is almost negligible for the ground-level sources and the differences in the plume mean advection time have no effect. Close to the releasing point, the fluctuation intensity is higher for the smaller elevated sources ($D6M$ and $D6$) compared with the corresponding larger ones ($D12M$ and $D12$) (figure 13a). These differences are reduced downwind and vanish at $x^*/\delta \approx 3$ for both $D6$ and $D12$. The difference is larger and more persistent between $D6M$ and $D12M$ (figure 13b,c). This behaviour reflects what we observed for $\langle c^*\rangle$ and $\sigma _c^*$ in figures 5 and 6: 6.25 mm elevated sources show higher scalar variance with respect to 12.5 mm sources, whereas the mean concentrations do not exhibit any relevant differences (figure 5). In figure 13(c) the downwind variability of $i_c$ is compared with the experimental data of Fackrell & Robins (Reference Fackrell and Robins1982), Xie et al. (Reference Xie, Hayden, Voke and Robins2004b), Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015). For consistency with the measurements, figure 13(c) does not report the values of $i_c$ collected at the plume centreline, but we use the ratio of the maximum of $\sigma _c$ and $\langle \bar {c} \rangle$ along the vertical direction, i.e. ${\max (\sigma _c)}/{\max (\langle \bar {c} \rangle )}$. For the ground-level sources, $i_c$ from the LES stays almost constant (equal to 0.5) and is minimally affected by the source size only for $x/\delta \lessapprox 0.5$. This behaviour is in good agreement with the experiments. The small elevation difference between D6G and D6G-X shows some effect for $x/\delta < 0.25$. Comparison of the concentration fluctuation intensity with the data of Xie et al. (Reference Xie, Hayden, Voke and Robins2004b), for the source at $z_s /\delta = 0.5$, shows a good agreement between the LES results and the experiment only for $x^*/\delta >1.5$. However, for $x^*/\delta <1.5$, the experiment of Xie et al. (Reference Xie, Hayden, Voke and Robins2004b) shows lower values of fluctuations also when compared with the F&R experiments with larger sources and lower elevations, and this should not be the case. For the sources at $z_s/\delta = 0.19$, a satisfactory agreement is visible for the LES when compared with both the experimental data of Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015) and F&R. In general, for the elevated sources, we can state that: (i) the difference in the results close to the source between the LES and the experimental values is comparable to the difference among the experiments, and (ii) far from the source, $x/\delta \approx 3.75$, the values from the experiments and the LES are in acceptable agreement for both the elevated source positions.

Figure 13.

Intensity of concentration fluctuations $i_c$ at source elevation and as a function of crosswind distances for $x^*/\delta = 0.36$ (a) and $x^*/\delta = 1.45$ (b). Intensity of concentration fluctuations expressed as ${\max (\sigma _c)}/{\max (\langle \bar {c} \rangle )}$ as a function of downwind distance (c). Panels (d,e) are the same as (a,b) but for the skewness, $Sk$. Panel ( f) is the along-wind variation of $Sk$ around the position of the maximum mean concentration; see text for full details. Panels (g–i) are the same as (d–f) but for the kurtosis, $Ku$. Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015) data in ( f,h) are on the plume centreline for a source size equivalent to D6 in the LES.

In order to describe the shape of the p.d.f., the skewness $Sk={\langle ( \bar {c}-\langle \bar {c} \rangle )^3 \rangle } / \sigma _c^{3}$ and the kurtosis $Ku={\langle ( \bar {c}-\langle \bar {c} \rangle )^4 \rangle } / \sigma _c^{4}$ are also useful statistics and quantify, respectively, the symmetry and the weight of the tail of the concentration p.d.f. (see also Nironi et al. Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015; Ardeshiri et al. Reference Ardeshiri, Cassiani, Park, Stohl, Stebel, Pisso and Dinger2020).

Figure 13(d–f) shows the crosswind and along-wind variation of skewness for different source sizes and source elevations. More precisely, the downwind variation reports the minimum value of the statistics in a small area surrounding the position of maximum mean concentration. This value allows one to better appreciate the along-wind variation of the statistics although it may underestimate the actual value of it by as much as about $25\,\%$ for the skewness and $50\,\%$ for kurtosis (Ardeshiri et al. Reference Ardeshiri, Cassiani, Park, Stohl, Stebel, Pisso and Dinger2020). This approach is used because the value exactly in the position of maximum was affected by fluctuations that hindered the detection of the along-wind variation (see Ardeshiri et al. Reference Ardeshiri, Cassiani, Park, Stohl, Stebel, Pisso and Dinger2020). First, the negative value of $Sk$ (figure 13f) near the source location should be noted, that, to our knowledge, have not been reported by any previous studies. Nevertheless, negative values of $Sk$ must be expected on physical grounds near the source since the undiluted plume, with an almost constant concentration, meanders. Close to the plume centreline, this meandering initially creates short intervals of near-zero concentration alternated with relatively longer periods of near-maximum concentration, and this is associated with a negatively skewed p.d.f. Generally, for the elevated sources, the smaller the source, the higher (less negative or more positive) the skewness at comparable downwind distances. This difference becomes smaller towards the crosswind plume edge and at distances very far away from the source location.

For the ground-level emissions (D6G, D12G, D6G-X), the source size has almost no effect on $Sk$ already at $x/\delta \approx 0.5$. This is similar to what was observed for $i_c$, although the differences are generally enhanced in $Sk$. Even the ground-level sources initially show a period of negative $Sk$ driven by the same mechanism as discussed above. The crosswind variation of $Sk$ clearly displays an almost constant (actually slightly concave) interval around the plume centreline, where $Sk$ stays close to its minimum. This interval increases its extension while the plume expands, and at $x/\delta = 1.45$ it extends over the interval $y/\sigma _y \approx \pm 0.5$.

It must be noted that the crosswind variation of $Sk$ is generally parabolic for elevated sources, while it clearly shows inflection points for the ground-level sources and a hyperbolic behaviour at the edges. As discussed above in § 4.2.1, the mechanisms that create fluctuations are different near the ground, but we do not have an explanation for this different behaviour.

The behaviour of $Ku$ moving among source sizes and elevations is generally similar to that of the skewness discussed above, apart from the obvious lack of the initial negative phase and the considerably higher values. Here $Ku$ is generally affected by a higher statistical uncertainty, especially visible with the increase in downwind and crosswind position. For the ground-level sources, remarkably, the crosswind interval with an almost constant value around the plume centreline ($Ku \approx 3$) extends over a larger interval compared with $Sk$. The interval extension at $x/\delta = 1.45$ is $y/\sigma _y \approx \pm 1$. Also, the crosswind change of $Ku$ outside this middle interval shows a rather different behaviour, being it parabolic. The difference in the shape of the curve between elevated and ground-level sources is also less pronounced for $Ku$ compared with $Sk$.

Near the centreline, the skewness and kurtosis of the ground-level sources for $x/\delta \gtrapprox 0.25$, irrespective of the source size, is $Sk\approx 0.3$ and $Ku \approx 3$, which clearly indicates that the ground-level source p.d.f. is very similar to a normal distribution. This is discussed in more detail next.

Figure 14 shows the shape of the concentration p.d.f. body at the plume centreline for three selected distances. Notably, at $x/\delta = 0.16$, the p.d.f. for D12M has a negative $Sk$ (see figure 13f) with a shape characterised by two peaks. The two distinct peaks stem from the source emitting a top-hat undiluted plume of concentration $c_s$, which meanders. Therefore, in the very near-source region, the concentration time series must display mainly two values, $c\approx 0$ and $c \approx c_s$. The p.d.f. observed in figure 14(a) also nicely displays the merging of the two peaks due to the effect of mixing.

Figure 14.

Concentration p.d.f. on the plume centreline at three downwind distances and for selected source cases.

A more quantitative view of the p.d.f. evolution can be sought in figure 15. This figure shows $Sk$ and $Ku$ as a function of $i_c$ and $i_c^2$, respectively. The continuous red dot-dashed line represents the behaviour of the Gaussian p.d.f., $Sk = 0$, $Ku = 3$. The continuous grey line represents the behaviour of the gamma p.d.f.,

(4.8)\begin{equation} p(\chi) = \frac{\kappa^\kappa}{\varGamma(\kappa)} \chi^{\kappa-1}\exp(-\kappa\chi), \end{equation}

with $\varGamma (\kappa )$ the gamma function, $\kappa =i_c^{-2}$ and $\chi =c/\langle c\rangle$, for which $Sk = 2i_c$, $Ku = 1.5 Sk^2 + 3 = 6i_c^2+3$, (see e.g. Nironi et al. Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015; Marro et al. Reference Marro, Salizzoni, Soulhac and Cassiani2018; Ardeshiri et al. Reference Ardeshiri, Cassiani, Park, Stohl, Stebel, Pisso and Dinger2020). Obviously, neither of these p.d.f.s can predict the values of $Sk < 0$ and $Ku < 3$ observed in the LES data close to the plume centreline (panels a,b), particularly for the elevated sources.

Figure 15.

Skewness, $Sk$ (a,c) and kurtosis, $Ku$ (b,d) on the position of maximum mean concentration (a,b) and at $2\sigma _y$ (c,d) in the crosswind direction, as a function of $i_c$ (for $Sk$) and $i_c^2$ (for $Ku$), for different source sizes and elevations. The lines represent the Gaussian (red dot-dashed) and gamma (grey continuous) p.d.f.s.

Moving downwind from the source, $i_c$ initially increases (ascending path of $i_c$) and the data for the elevated sources tend towards the gamma p.d.f. Such a statistical model becomes a very reliable representation of the LES data after the peak in $i_c$ (descending path of $i_c$). This last point was already noted by Ardeshiri et al. (Reference Ardeshiri, Cassiani, Park, Stohl, Stebel, Pisso and Dinger2020), but it is extended here to different source elevations and sizes.

We underline that on the centreline, in the ascending phase of $i_c$, the values of $Sk$ and $Ku$ decrease as the source height increases. Actually, the ground-level sources have the highest $Sk$ and $Ku$ for a given $i_c$. Although these differences are small, this feature indicates that the p.d.f.s are not exactly the same, and the rate of convergence towards the gamma p.d.f. is related to the source elevation. The source size does not significantly influence the relations $i_c, Sk$ and $i_c^2, K_u$, for $Sk>0$ and $Ku>3$, and, therefore, the p.d.f. shape does not change with varying source size once the early phase with negative $Sk$ is passed.

The behaviour of D6G-X on the centreline (green crosses in figure 15a,b) is rather interesting. This is not exactly a ground-level source, and its values of $Sk$ and $Ku$ initially tend to rise, similarly to D6 and D6M, but the strong turbulent dissipation acting on this plume does not allow $i_c$ to rise enough to observe the transition to a gamma p.d.f. typical of D6-12 and D6M-12M. Instead, the shape of the p.d.f. evolves quickly towards the Gaussian p.d.f. This suggests that there must be a threshold elevation for which it is possible to reach values of $i_c$ that are high enough to allow the p.d.f. to attain the gamma shape. Generally, it is clear that for the simulated ground-level sources (D6G, D12G, D6G-X) on the plume centreline, the Gaussian p.d.f. is a good representation. Moreover, based on figure 13(e,h), the validity of the Gaussian approximation for the p.d.f. generated by ground-level sources extends to an area around the plume centreline, which at $x/\delta = 1.45$ is about $y/\sigma _y \approx \pm 0.5$.

Far away from the centreline, at $2 \sigma _y$, figure 15(c,d) shows that both for ground and elevated sources, $Sk$ and $Ku$ assume values that are generally close to those predicted by a gamma distribution for all $i_c$. Moving towards the plume edge, $i_c$ increases, and this compresses the evolution of the p.d.f. towards the gamma shape also during the ascending phase of $i_c$. However, the best agreement between the gamma p.d.f. model and the LES data is again observable only in the descending phase starting from the peak of $i_c$ for the elevated sources.

When considering the whole picture, it is clear that the gamma distribution provides very accurate values only in the descending phase of $i_c$ and only for sources at sufficiently high elevations. However, the gamma p.d.f. progressively becomes a better approximation of the concentration distribution as $i_c$ increases towards high values, in both downwind and crosswind directions. This occurs for both ground and elevated sources, but it is actually a quicker process for the ground-level sources.

It is worth mentioning here that in Ardeshiri et al. (Reference Ardeshiri, Cassiani, Park, Stohl, Stebel, Pisso and Dinger2020) it was found that the gamma distribution provided an accurate model in the descending phase of $i_c$, irrespective of the grid resolution. The effect of a degraded grid resolution was a lower value of $i_c$, $Sk$ and $Ku$, similar to the effect of an increase in the source size.

A quantity of interest related to the concentration p.d.f. is the peak concentration. Fackrell & Robins (Reference Fackrell and Robins1982) defined the peak concentration as the value of concentration which is exceeded only $1\,\%$ of the time, $c_{99}$, and empirically found that the ratio $c_{99} / \sigma _c \approx 4.5$ and, more generally, between 4 and 5 for many different plume positions. For an exponential p.d.f., we have that $p(c) = \lambda \exp (-\lambda c)$ with the mean equal to $1/ \lambda$ and $i_c = 1$, $Sk = 2$ and $Ku = 9$. For this p.d.f., $c_{99}/ \sigma _c = 4.605$ can be calculated analytically. For the gamma distribution, we calculated the ratio $c_{99}/ \sigma _c$ for varying values of $i_c$ in table 3. This shows that previous findings of 4–5 well agree with the gamma distribution over a range $i_c = 0.5$–6 but are not accurate for lower or upper values of $i_c$ that are observed for very aged plumes (low values) or near the plume edges (high values).

Table 3.

Peak concentration $c_{99}/\sigma _c$ estimated from gamma p.d.f. for different values of $i_c$.

4.5. Effects of concentration threshold on intermittency factor and in-plume intensity of concentration fluctuations

The intermittency factor of a concentration signal is usually simply defined as the probability of non-zero concentration at a given position in time. However, zero concentration is not measurable in experiments due to limitations of the instrumentation, in simulation due to numerical diffusion/noise, and even theoretically is not fully justifiable due to the diffusivity coefficient appearing in the advection–diffusion equation (e.g. Chatwin & Sullivan Reference Chatwin and Sullivan1990). Therefore, this probability is more correctly redefined based on a threshold $\varGamma _t$:

(4.9)\begin{equation} \gamma_c\left(\boldsymbol{x},t \right) = \mathrm{P}\left(c^*\left(\boldsymbol{x},t \right) > \varGamma_t \right) = \int_{\varGamma_t}^{\infty} {\rm p.d.f.}(c^*) \,{\rm d}c^*. \end{equation}

Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015) chose the threshold $\varGamma _t = 1$, i.e. a value of 1 for the dimensionless concentration, $c^*$, and argued that the choice is rather arbitrary. Here, we perform a sensitivity analysis varying the value of this threshold in order to show the effect on the intermittency factor. Figure 16 shows the value of $\gamma _c$ on the position of maximum mean concentration with the downwind variation obtained by sampling the time series at $x/\delta =$ 0.17, 0.32, 0.64, 0.95, 1.25, 1.9, 2.5, 3.75. Figure 16 demonstrates quite clearly the effect of the selected threshold on $\gamma _c$ and also the dependence on the source size and source elevation. The intermittency factor is generally higher for larger sources and for lower elevations. This is in agreement with the observed concentration fluctuations since an inverse relation between $i_c$ and $\gamma _c$ is expected. For the sources at $z_s = 0.19 \delta$, the experimental values of Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015) for $\varGamma _t = 1$ and F&R (the threshold value was not reported in their paper) are included. The LES results with $\varGamma _t = 1$ agree quite well with Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015), especially near the source, while far away the experiments show a larger $\gamma _c$. The LES results with $\varGamma _t = 3$ somewhat agree with F&R especially for D12, while the experimental values of F&R for $d_s = 0.0075 \delta$ show far lower values of the intermittency factor.

Figure 16.

Intermittency factor plot as a function of the concentration threshold ($\varGamma _t$) and along-wind distance at the position of maximum mean concentration for the sources at $z_s/\delta = 0.19$ (a) and $z_s/\delta = 0.5$ (b).

The dependence of $\gamma _c$ on the threshold $\varGamma _t$ is clear. In figure 17 the dependence of $\gamma _c$ in a wide range for the dimensionless concentration threshold is explored. The exact values of $\varGamma _t$ used in the figure are $10^{-7}, 10^{-6}, 10^{-5},$ $10^{-4}, 10^{-3}, 10^{-2},$ $0.05, 0.1, 0.5, 1, 3$ and $9$. For the ground-level sources, along the plume centreline, the intermittency factor remains close to unity (figure 17c) and does not change significantly irrespective of the threshold chosen in the considered range. The behaviour is quite different for the elevated sources (figure 17a), where, for $\varGamma _t <3$, the intermittency factor shows an along-wind variation with a minimum between $1< x/\delta <1.5$ corresponding approximately to the position of the maximum of the fluctuation intensity. Most importantly, for the elevated sources, a strong dependence on the threshold value is observed even at low values of $\varGamma _t$, as already noted in figure 16. In the crosswind direction, the variation of $\gamma _c$ is generally more pronounced. For elevated sources (figure 17b), a clear dependence on the lateral position is observed, and an overall strong dependence on the threshold values is found everywhere. Conversely, for the ground-level sources, the intermittency factor does not show significant variations when the lateral distance is close to the plume centreline, irrespective of the concentration thresholds (figure 17d). Approaching the plume edges, for sufficiently high values of $\varGamma _t$, a dependence emerges. In general, figure 17 shows that the intermittency is clearly linked to the selected threshold, and the sensitivity to the threshold increases where $i_c$ is high. According to our LES data, it is therefore a somewhat arbitrary parameter.

Figure 17.

Intermittency factor ($\gamma _c$, a–d) and in-plume intensity of concentration fluctuations ($i_p$, e–h) for 6.25 mm source at $z_s = 0.19 \delta$, D6 (a,b,e,f) and at ground level, D6G (c,d,g,h). Variables are plotted as a function of $\varGamma _t$ and along-wind distance (a,e,c,g) at the position of maximum mean concentration and as a function of $\varGamma _t$ and crosswind ($y$) position at source elevation for $x/\delta =0.32$ (b, f,d,h).

We will now investigate the influence of the threshold on the so-called in-plume intensity of concentration fluctuations, i.e. the ratio of the mean and standard deviation of concentration fluctuations when the concentration values below the threshold are excluded (e.g. Wilson Reference Wilson1995), $i_p ( \boldsymbol {x},t ) = \sigma _c^* / \langle \bar {c}^* \rangle$ for $c^* (\boldsymbol {x},t ) > \varGamma _t$. It is worth noting that the experimental evaluations of $i_p$ and their empirical relationship with $i_c$ are used in the definition of simple models of concentration fluctuations (e.g. Wilson Reference Wilson1995). Figure 17(e–h) shows the dependence of $i_p$ on the threshold values. The along-wind variation for an elevated and a ground-level source at the position of maximum mean concentration is shown in (e,g). The crosswind variation, at $x/\delta =0.32$ and $z=z_s$, is shown in ( f,h) for elevated and ground-level sources, respectively.

The in-plume intensity of concentration fluctuations decreases as $\varGamma _t$ increases for both elevated and ground-level sources. This is mainly due to a rise in the mean concentration. The crosswind variation of $i_p$ decreases with increasing threshold (figure 17f,h) for suitably high values of $\varGamma _t$. Therefore, previous findings showing $i_p^2$ always lower than about $2.0$ (see Wilson Reference Wilson1995) may be linked to the specific threshold used in these studies. However, the results in figure 17 show that in the threshold range considered here the intermittency factor and $i_p$ remain almost constant for ground-level sources over a limited distance from the plume centreline $| (y-y_s) | < \sigma _y$.

Based on previous experimental findings, Wilson (Reference Wilson1995) recommended for practical applications the empirical relation $i_p^2=2i_c^2 / (2+ i_c^2)$ proposed by Wilson & Zelt (Reference Wilson and Zelt1990). This relation is compared with our results in figure 18. According to our LES data, Wilson & Zelt (Reference Wilson and Zelt1990)'s relation is somewhat adequate for the ground-level source (D6G) at moderate values of fluctuation intensity, $i_c < 2$. As discussed above, this value can be found near the centreline for ground-level sources. At higher values of $i_c$, the sensitivity to the threshold is too high to provide a unique relation between $i_p$ and $i_c$, and the relation proposed by Wilson & Zelt (Reference Wilson and Zelt1990) can be justified only based on suitably high threshold values. For the elevated sources, the Wilson & Zelt (Reference Wilson and Zelt1990) relation is not accurate at any value of $i_c$, but it could be slightly modified to fit our data at low values of $i_c (< 2)$. At higher values of $i_c$, the sensitivity to the threshold is again too high to ensure a unique relation between $i_p$ and $i_c$. This analysis suggests that semi-empirical models of concentration fluctuations based on the intermittency factor and in-plume concentration fluctuations may be accurate for ground-level sources but only over a limited crosswind distance from the plume centreline where $i_c < 2$.

Figure 18.

Scatter plot of in-plume concentration fluctuation intensity, $i_p$, and overall concentration fluctuation intensity, $i_c$. Different symbols refer to different thresholds, $\varGamma _t$, used in calculating $i_p$. Panel (a) is for the 6.25 mm source in the middle of the boundary layer (D6M), panel (b) for the source at $z_s = 0.19 \delta$ (D6), and panel (c) for the near-ground-level source (D6G). The continuous line is the Wilson & Zelt (Reference Wilson and Zelt1990) empirical relation $i_p^2=2i_c^2 / (2+ i_c^2)$.