3.1 Velocity and length scales for self-similarity

Self-similarity of spectra requires suitable normalisation of $f\unicode[STIX]{x1D6F7}_{cc}$ and $f$ such that black lines in figure 3(a,b) collapse to agree well with a representative spectrum. Extensive examination of data in our experimental campaign has consistently indicated that the shape of $f\unicode[STIX]{x1D6F7}_{cc}$ at various locations is similar and only differs by the area under the curves, i.e. $\unicode[STIX]{x1D70E}_{c}^{2}$ . Hence, $f\unicode[STIX]{x1D6F7}_{cc}$ for each spectrum is non-dimensionalised by the corresponding $\unicode[STIX]{x1D70E}_{c}^{2}$ in figure 3(c,d) so that the area under each curve equals one. Normalisation of frequency requires a time scale and for single-point Eulerian measurements, a time scale is typically constructed using a velocity scale ( ${\mathcal{U}}$ ) and length scale ( ${\mathcal{L}}$ ) such that ${\mathcal{L}}/{\mathcal{U}}$ has dimensions of seconds. For a plume in a turbulent boundary layer, two relevant velocity scales are the local velocity $U$ (where the concentration is measured) and the source velocity $U_{s}:=U(z=s_{z})$ . Note that the scaling of frequency with local mean velocity ( $U$ ) has been done in previous studies such as Fackrell & Robins (Reference Fackrell and Robins1982a ), Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015). Since $U$ is a function of $z$ , the scaling of frequency with $U$ inherently has a dependence on $z$ . Such a scaling has resulted in good collapse of velocity spectra in a turbulent boundary layer (see Baidya et al. Reference Baidya, Philip, Hutchins, Monty and Marusic2017, for example). Here, $U$ is being tested as a velocity scale to see if it leads to a collapse of concentration spectra.

Figure 3(c,d) shows the normalised distributions using $f\unicode[STIX]{x1D6FF}/U$ and $f\unicode[STIX]{x1D6FF}/U_{s}$ , respectively. Light-grey and light-red lines indicate the spectra from figure 3(a,b) in normalised form at different $y$ and $z$ locations. It is evident that they have very similar shapes to that of spectra at the plume centreline (shown by solid-black and dashed-red lines) providing experimental evidence for the self-similar behaviour of concentration spectra. Although the normalised spectrum at the plume centreline is chosen as reference for comparison, it can be justifiably replaced by a concentration spectrum at any other location in the plume. Notice that both $U$ and $U_{s}$ are reasonably good candidates; however, $U_{s}$ results in marginally better overall collapse. It may be argued that other choices of velocity scale, such as $U_{\infty }$ and $U_{\unicode[STIX]{x1D70F}}$ , will also lead to collapse of spectra since the three velocity scales ( $U_{s}$ , $U_{\infty }$ and $U_{\unicode[STIX]{x1D70F}}$ ) are all constant for a given flow configuration and source height. Previous estimates of convection velocities of coherent structures in a turbulent boundary layer have shown that instantaneous motions are convected at a velocity that is of the order of the local mean velocity (e.g. del Álamo & Jiménez Reference del Álamo and Jiménez2009). Since the instantaneous transport of the scalar is due to local turbulence, the convection velocity of the scalar will be similar to that of the turbulent motions, i.e. the mean velocity, as opposed to $U_{\unicode[STIX]{x1D70F}}$ or $U_{\infty }$ . Furthermore, $U_{s}$ is a preferred choice as it is convenient to measure it in practical situations, for instance, based on the emission rate of smoke from a tall stack. On the other hand, it is much more difficult to measure $U_{\infty }$ (which might not even exist in an atmospheric flow) and $U_{\unicode[STIX]{x1D70F}}$ (or the ‘roughness length’) in the atmospheric boundary layer.

At this point, it is important to discuss the relevance of $U_{s}$ as an appropriate velocity scale for scaling concentration spectra. In the present study, the measurements are limited to $s_{x}/\unicode[STIX]{x1D6FF}\leqslant 4$ and the maximum plume width in this region is found to be less than 10 % of the boundary layer thickness. This implies that the investigation here is limited to narrow plumes, and so, it is expected that the growth of the plume is dictated by the source conditions, namely the mean velocity and the turbulence intensity at the source location (Fackrell & Robins Reference Fackrell and Robins1982b ; Nironi et al. Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015). This is the reason why we observe a good collapse when $U_{s}$ is used as a velocity scale. However, in the far-field region, where the plume is thoroughly mixed across the entire boundary layer, it is possible that plume development is governed by a wide range of velocity scales present in the flow. The lack of such data in the literature to test self-similarity of concentration spectra in the far-field region provides scope for future study.

Figure 4.

Comparison of normalised spectra of concentration at the plume centreline using (a) dimensional frequency; three different length scales – (b) half-plume width $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},z}$ ; (c) boundary layer thickness $\unicode[STIX]{x1D6FF}$ and (d) integral length scale $l_{u-int}$ and source velocity $U_{s}$ as velocity scale for four source distances, $s_{x}/\unicode[STIX]{x1D6FF}=0.5$ , 1, 2 and 4, and at a fixed height, $s_{z}/\unicode[STIX]{x1D6FF}=0.1$ . Note that the data of Fackrell & Robins (Reference Fackrell and Robins1982a ) and Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015) used here are digitally extracted from the respective papers.

The choice of ${\mathcal{L}}=\unicode[STIX]{x1D6FF}$ in figure 3(c,d) also requires further discussion on the appropriate length scale to be employed in scaling frequency. Three different length scales are investigated in figure 4 for the plume released at $s_{z}/\unicode[STIX]{x1D6FF}=0.1$ and four different streamwise locations ( $s_{x}/\unicode[STIX]{x1D6FF}$ ). Here we also include data from Fackrell & Robins (Reference Fackrell and Robins1982a ) and Nironi et al. (Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015), where a localised passive scalar plume was released in turbulent boundary layers over rough surfaces. We note that data from Fackrell & Robins (Reference Fackrell and Robins1982a ) show a lower amplitude compared to other data, which is likely due to the limitations of the sensor employed in the 1980s.

Figure 4(a) shows the abscissa in dimensional frequency in Hz, whereas we test three different length scales in figure 4(b,c,d), respectively, the plume (variance) thickness in the vertical direction $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},z}$ (where variance reduces to half its maximum value), the boundary layer thickness $\unicode[STIX]{x1D6FF}$ and the velocity integral length scale $l_{u-int}:=\int R_{uu}(x)\,\text{d}x$ (where, $R_{uu}$ is the normalised autocorrelation function of the streamwise velocity). Here, $l_{u-int}$ is calculated by integrating the autocorrelation function $R_{uu}(x)$ in the interval $0\leqslant x/\unicode[STIX]{x1D6FF}\leqslant 20$ following the approach of Talluru et al. (Reference Talluru, Philip and Chauhan2018). Note that here we are testing variance half-width, rather than the plume width based on mean concentration (traditionally used in the atmospheric science community) as a length scale, because the focus of this study is the scaling of concentration spectra, which is directly linked to the variance (cf. (1.1)). It is clear from figure 4(b) that $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},z}$ is not an appropriate length scale; in fact, none of the ‘plume’ length scales (which were sometimes used in the past, e.g. Hanna & Insley Reference Hanna and Insley1989) are suitable for the scaling purposes. This is perhaps not surprising, since it known that passive plume length scales are dictated by the background velocity field length scales (e.g. Gifford Reference Gifford1959; Talluru et al. Reference Talluru, Philip and Chauhan2018). For our data figure 4(a,c,d) shows good collapse. This is because, for the present study, $\unicode[STIX]{x1D6FF}$ is a constant, and it is known that $l_{u-int}$ is proportional to $\unicode[STIX]{x1D6FF}$ in the outer part of a TBL (e.g. Hutchins & Marusic Reference Hutchins and Marusic2007). Inclusion of other data from rough-wall TBL, however, shows that $l_{u-int}$ is a more appropriate length scale (cf., figure 4 d). In rough-wall flows the roughness can introduce its own length scales that need not be proportional to $\unicode[STIX]{x1D6FF}$ . In fact, as a practical advantage for $l_{u-int}$ , in atmospheric flows it is more convenient to measure $l_{u-int}$ because it depends on the local flow conditions, and can be measured with relative ease compared to $\unicode[STIX]{x1D6FF}$ that requires the knowledge of the entire BL. In the rest of the paper, however, we continue to use $\unicode[STIX]{x1D6FF}$ because of its constancy, and because both $\unicode[STIX]{x1D6FF}$ and $l_{u-int}$ show almost identical results.

3.2 Self-similarity of spectra across the plume cross-section ( $y$ - $z$ plane) at a particular frequency

In the previous section § 3.1, we discussed the ‘usual kind of self-similarity’ of spectra that is studied in most plume investigations. There, the typical aim is to scale the spectral amplitude and frequency appropriately so as to collapse all the available spectra data onto to a ‘universal’ curve, which can then be employed for prediction if we know the scaling variables (such as $\unicode[STIX]{x1D70E}_{cc}$ for amplitude, and $\unicode[STIX]{x1D6FF}$ or $l_{u-int}$ and $U_{s}$ for frequency). In this section, we discuss self-similarity of a slightly different kind, one that we realised almost serendipitously from the three-dimensional figure 3(a,b). We refer back to the blue lines on the rear panels in figure 3(a,b), which are distributions of $\unicode[STIX]{x1D6F7}_{cc}$ as a function of $y$ or $z$ at three arbitrary frequencies: 6, 60 and 150 Hz. These curves are replotted in figure 5(a,b) after normalising by the respective maxima, i.e. $f_{i}\unicode[STIX]{x1D6F7}_{cc}^{\ast }=f_{i}\unicode[STIX]{x1D6F7}_{cc}/\text{max}(f_{i}\unicode[STIX]{x1D6F7}_{cc}),i=6$ , 60 and 150 Hz. It is clearly evident that $f_{i}\unicode[STIX]{x1D6F7}_{cc}^{\ast }(x,y,z)$ collapses on to a single curve and has a Gaussian behaviour. The (grey) dashed lines are Gaussian curves plotted in figures 5(a,b), and they are indistinguishable from the data.

Figure 5.

(a,b) Blue curves shown in figure 3(a,b) are replotted here after normalising by the maxima ( $f_{i}\unicode[STIX]{x1D6F7}_{cc}^{\ast }$ , $i=6$ , 60 and 150 Hz). Grey dashed lines are the Gaussian curves and the vertical dot-dashed lines represent the plume centreline.

The above similarity is equally valid at all frequencies such that both $f\unicode[STIX]{x1D6F7}_{cc}(x,y,z)$ and $\unicode[STIX]{x1D70E}_{c}^{2}(x,y,z)$ follow the same spatial distribution. For a fixed $x$ and $z$ , the similarity model can be written as,

(3.1) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{f\unicode[STIX]{x1D6F7}_{cc}(f,y)}{\unicode[STIX]{x1D70E}_{c}^{2}(y)}}=g(f)\quad \Rightarrow \quad {\displaystyle \frac{f\unicode[STIX]{x1D6F7}_{cc}(f,y_{1})}{\unicode[STIX]{x1D70E}_{c}^{2}(y_{1})}}={\displaystyle \frac{f\unicode[STIX]{x1D6F7}_{cc}(f,y_{2})}{\unicode[STIX]{x1D70E}_{c}^{2}(y_{2})}}, & & \displaystyle\end{eqnarray}$$

to within experimental uncertainty, where, $g(f)$ is some arbitrary function of frequency only (or a reference spectrum from figure 3 d), $y_{1}$ and $y_{2}$ are two arbitrary spanwise locations and a similar expression holds in the $z$ -direction. Importantly, this shows that at each frequency, the concentration energy ( $\unicode[STIX]{x1D6F7}_{cc}$ ) is distributed across the $y$ and $z$ -directions and is proportional to $\unicode[STIX]{x1D70E}_{c}^{2}$ at that location. This is a strong indication that the meandering nature of the plume, which is the reason for the variance distribution to be Gaussian (Gifford Reference Gifford1959), is also likely to be responsible for imposing a uniform bandwidth of frequencies in concentration spectra at all locations across the plume. Thus, it is important to note here that the observed self-similar behaviour of concentration spectra is applicable only for a narrow meandering passive plume. Further, it should be noted that the measurements acquired in this study are for $0.5\unicode[STIX]{x1D6FF}\leqslant s_{x}\leqslant 4\unicode[STIX]{x1D6FF}$ , where $s_{x}$ is the distance between the point source and the measurements. It is known that a Gaussian shape for the concentration variance is not valid close to the plume source, particularly for $s_{x}<0.5\unicode[STIX]{x1D6FF}$ (Sawford & Sullivan Reference Sawford and Sullivan1995), and as such, the model presented here should be deemed not applicable in this region.

A detailed mechanism for the instantaneous meandering and resulting self-similarity is still not completely understood; however, the experimental observations strongly support the meandering nature of the plume. Although the results presented in this study are for slender plumes (i.e. the plume size is nominally less than 10 % of boundary layer thickness), we believe that the analysis is equally valid when the source size is varied as long as it generates a narrow meandering plume. From previous studies (Fackrell & Robins Reference Fackrell and Robins1982a ; Nironi et al. Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015), it is well known that meandering is caused when the source size is typically smaller than the integral length scale of the background turbulence. And in all such cases, the self-similar behaviour of concentration spectra can be expected.

4.1 Model parameters

Figure 6(a,b) plots the normalised concentration variance as a function of $\unicode[STIX]{x1D709}_{y}=y/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},y}$ and $\unicode[STIX]{x1D709}_{z}=(z-s_{z})/\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},z}$ , respectively for the plume released at different heights. A representative Gaussian profile, in accordance with the meandering plume model (e.g. Gifford Reference Gifford1959; Weil Reference Weil and Fernando2012), for the normalised variance profiles is also plotted in figure 6 and is written as:

(4.3a,b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D70E}_{c}^{2}(y)}{\unicode[STIX]{x1D70E}_{c,ref}^{2}}=\exp \left\{-\text{ln(2)}\left(\frac{y}{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},y}}\right)^{2}\right\};\quad \frac{\unicode[STIX]{x1D70E}_{c}^{2}(z)}{\unicode[STIX]{x1D70E}_{c,ref}^{2}}=\exp \left\{-\text{ln(2)}\left(\frac{z-s_{z}}{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},z}}\right)^{2}\right\}. & & \displaystyle\end{eqnarray}$$

Here, ln(2) is the logarithm of 2. It is used in the above form of Gaussian equation so that $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},y}$ or $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},z}$ represents the half-plume width in the concentration variance (i.e. at $y=\unicode[STIX]{x1D6FF}_{c,y},\unicode[STIX]{x1D70E}_{c}^{2}(y)=(1/2)\unicode[STIX]{x1D70E}_{c,ref}^{2}$ , and similarly in the $z$ -direction). Such a representation was previously used by Fackrell & Robins (Reference Fackrell and Robins1982a ). Although the data in figure 6 are in good agreement with (4.3), some differences are observed in $\unicode[STIX]{x1D70E}_{c}^{2}(y)$ for a ground level source (shown by ○ symbols in the figures). The distinct behaviour of $\unicode[STIX]{x1D70E}_{c}^{2}(y)$ is more clearly seen in figure 2(b) due to less clutter, where two peaks (one on either side of the plume centreline) are observed in the variance profile. Hence, a simple Gaussian model is not adequate to describe such a behaviour. This will be further discussed in a later section. For all other sources, (4.3) is a good fit to the empirical data. Note that (4.2) essentially requires a single parameter, that is, the plume half-width. The variation of plume half-widths $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},y}$ and $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},z}$ are plotted as a function of source height $s_{z}$ in figure 7(a) as filled and open symbols, respectively. Indeed, $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},z}$ depends on $s_{x}$ and it is different from the half-plume width based on the mean concentration ( $\unicode[STIX]{x1D6FF}_{C}$ ). Typically the mean concentration profile is relatively easier to predict/calculate with confidence. If one adopts a standardised distribution (probability density functions) for concentration fluctuations (such as gamma functions (Nironi et al. Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015)), as typically done in practical applications, one can then easily translate the variance to mean and vice versa and thus work with either of $\unicode[STIX]{x1D6FF}_{C}$ or $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E}}$ . However, the aim of estimating $\unicode[STIX]{x1D6FF}_{C}$ or $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E}}$ as a function of source distance, $s_{x}/\unicode[STIX]{x1D6FF}$ is beyond the scope of this study.

As a side point, we compare the plume half-widths with profiles of spanwise and wall-normal velocity variance ( $\unicode[STIX]{x1D70E}_{v}^{2}$ and $\unicode[STIX]{x1D70E}_{w}^{2}$ ) measured in our previous study (Talluru et al. Reference Talluru, Philip and Chauhan2018) in figure 7. Note that the velocity variances are scaled by an arbitrary factor. We find that there is a good agreement between plume half-widths and scaled variance profiles in the respective directions. This result suggests that the plume spread is governed by local turbulence levels at the source height, and thus for practical applications, it might be possible to estimate the plume half-width using turbulence intensities of the background flow which are relatively easier to measure than concentration profiles. The relationship between (Lagrangian) turbulence intensities and the plume width in the respective directions has been previously derived using analytical arguments (Taylor Reference Taylor1922); however, it has not been extensively tested against experimental data. In this regards, the current study provides clear support to the theoretical understanding via the data of plume half-widths in the transverse and vertical directions at eight source locations in a high Reynolds number turbulent boundary layer in figure 6(a). Furthermore, $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},y}$ is greater than $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},z}$ for $s_{z}/\unicode[STIX]{x1D6FF}\leqslant 0.33$ , beyond which the values are quite similar, a behaviour similar to the relative magnitude of $\unicode[STIX]{x1D70E}_{v}^{2}$ and $\unicode[STIX]{x1D70E}_{w}^{2}$ near the wall, where $\unicode[STIX]{x1D70E}_{v}^{2}$ is relatively large compared to $\unicode[STIX]{x1D70E}_{w}^{2}$ (e.g. see Talluru et al. Reference Talluru, Baidya, Hutchins and Marusic2014; Baidya et al. Reference Baidya, Philip, Hutchins, Monty and Marusic2017, for most recent high-resolution measurements of transverse and lateral velocity fluctuations in a high Reynolds number turbulent boundary layer). A similar reasoning explains the transverse plume width being approximately 25 % larger than the vertical plume width in a uniformly sheared flow (Vanderwel & Tavoularis Reference Vanderwel and Tavoularis2014). The results presented in figure 7(a) clearly support the common modelling practice of using local turbulence at the source height to model plume dispersion in the cross-wind and vertical directions.

Figure 7.

The two inputs for the spectrum model. (a) Half-plume widths ( $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},y}/\unicode[STIX]{x1D6FF}$ , $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E},z}/\unicode[STIX]{x1D6FF}$ ); (b) normalised pre-multiplied spectra of concentration fluctuations at the reference location for all source heights. Solid black and dashed grey lines in (a) represent the variance profiles of spanwise velocity, $\unicode[STIX]{x1D70E}_{v}^{2}$ and wall-normal velocity $\unicode[STIX]{x1D70E}_{w}^{2}$ , respectively.

The second input to the model is the concentration spectrum measured at a reference location, which is the plume centreline in this study. Normalised spectra at the reference location for eight source heights is shown in figure 7(b). It is found that non-dimensional spectra are similar to each other for source heights in the fully turbulent region, i.e. $s_{z}/\unicode[STIX]{x1D6FF}\leqslant 0.33$ . Notably, there is a common bandwidth of frequencies and the spectral peak occurring at $f\unicode[STIX]{x1D6FF}/U_{s}\approx 2$ . This trend is slightly different for sources in the outer region ( $s_{z}/\unicode[STIX]{x1D6FF}=0.5,0.67,\unicode[STIX]{x1D6FF}_{99}/\unicode[STIX]{x1D6FF}$ ), for which the peak shifts to a lower frequency, consistent with the trend observed by Baidya et al. (Reference Baidya, Philip, Hutchins, Monty and Marusic2017) in velocity spectra. A notable exception is the ground level source, where the reference location does not match with the plume centre.

Figure 8.

Contour maps of pre-multiplied energy spectra ( $f\unicode[STIX]{x1D6F7}_{cc}/\unicode[STIX]{x1D70E}_{c,ref}^{2}$ ) of concentration fluctuations plotted as a function of $f\unicode[STIX]{x1D6FF}/U_{s}$ and (a–d) $z/\unicode[STIX]{x1D6FF}$ ; (e–h) $y/\unicode[STIX]{x1D6FF}$ . Each panel shows concentration spectra for plume released at different source heights. Horizontal dashed lines indicate the source location. Coloured contours and solid-grey lines indicate experimental data; dashed-black lines indicate prediction using (4.3). Eight levels of $f\unicode[STIX]{x1D6F7}_{cc}/\unicode[STIX]{x1D70E}_{c,ref}^{2}$ from experiments are indicated, while the dashed-lines represent the prediction at same levels.

4.2 Ground level source

As evident from the discussion on model parameters, the ground level source exhibits different characteristics in comparison to elevated sources, namely the variance distribution and the reference spectrum. Revisiting the results plotted in figures 2(b) and 6(a), it is observed that the ground level plume has two distinct peaks in its variance distribution along $y$ . This result is consistent with several past experimental studies on jets and plumes (Becker, Hottel & Williams Reference Becker, Hottel and Williams1967; Birch et al. Reference Birch, Brown, Dodson and Thomas1978; Fackrell & Robins Reference Fackrell and Robins1982a ; Pitts & Kashiwagi Reference Pitts and Kashiwagi1984). Based on these observations, Chatwin & Sullivan (Reference Chatwin and Sullivan1990) presented a two parameter ( $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FD}$ ) model that predicted the off-axis peak behaviour of $\unicode[STIX]{x1D70E}_{c}^{2}(y)$ in the near-field region of a dispersing scalar from a uniform source. Later, Sawford & Sullivan (Reference Sawford and Sullivan1995) developed the model further by taking into account of the source distribution as,

(4.4) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}_{c}^{2}(y)=\unicode[STIX]{x1D6FD}^{2}C_{0}^{2}\unicode[STIX]{x1D712}(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D712}), & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D712}=C(y)/C_{0}$ and $C_{0}$ is the maximum mean concentration on the plume axis. In the model, $\unicode[STIX]{x1D6FC}$ is concerned with the evolution of the plume that is controlled by large eddies of the velocity field and hence it depends on source geometry, type of flow and development distance. On the other hand, $\unicode[STIX]{x1D6FD}$ accounted for the in-plume dissipation. Using this model, Sawford & Sullivan (Reference Sawford and Sullivan1995) demonstrated that for $\unicode[STIX]{x1D6FC}<2$ , there is an off-axis maximum in the variance profile. In the current study, the values of $\unicode[STIX]{x1D6FC}=1.3$ and $\unicode[STIX]{x1D6FD}=0.78$ are obtained by fitting the variance profile to (4.4) and the fit is shown as a dashed line in figure 2(b).

A second important difference between ground level and elevated sources is the location of the reference spectrum used for predicting spectra across the plume in the vertical direction. In our analysis, it is observed that the concentration spectrum on the plume axis for a ground level source was limited to a smaller bandwidth of frequencies, which is consistent with the behaviour of streamwise velocity spectra in the near-wall region. In order to capture the full bandwidth of frequencies, we chose the geometric mid-point of the log region, i.e. $z_{ref}^{+}=\sqrt{(3.6\sqrt{\unicode[STIX]{x1D6FF}^{+}})~(0.2\unicode[STIX]{x1D6FF}^{+})}=0.85{\unicode[STIX]{x1D6FF}^{+}}^{(3/4)}$ (Morrill, Philip & Klewicki Reference Morrill, Philip and Klewicki2017) as the appropriate reference location for a ground level source. With this modification, the model is able to yield better predictions of the concentration spectra across the entire plume (cf. figure 8 a in the following section).

4.3 Spectrum predictions

With the model parameters established, the $c$ -spectra can now be predicted at all locations in the plume using (4.2), where the only inputs are a reference $f\unicode[STIX]{x1D6F7}_{cc}$ from figure 7(b) and the concentration variance profile, defined using plume half-width in (4.3). We recall that data are collected for eight different $s_{z}/\unicode[STIX]{x1D6FF}$ and four $s_{x}/\unicode[STIX]{x1D6FF}$ locations; however, for brevity only limited measurements will be discussed here. Figure 8(a–d) shows concentration spectra varying with wall distance for four different source heights as coloured contours, whereas the predicted spectra are indicated by dashed-black lines. Together, these data cover the source heights across the entire boundary layer (buffer region, log region and the outer region). The horizontal dot-dashed line in each panel marks the reference location. Note that the vertical size of each panel proportionally indicates the plume size relative to the TBL thickness. Overall, the predictive model contours agree very well with measurements for all source heights in figure 8(a–d). The principal effect of varying source height, i.e. the frequency shift and location of the peak, is suitably accounted for by the model in figure 8(b–d), whereas, for the ground level source, the peak in the two-dimensional spectrum map is not accurately predicted. The influence of the wall on spectra of concentration very close to the surface requires further investigation.

The model is also tested by comparing the predicted spectra with measurements in the spanwise direction, and these are shown in figure 8(e–h). Again, the predictive model is able to capture the variation of $c$ -spectra in the spanwise direction for the plume at four source heights. As highlighted before, the distribution of variance for a ground level source along $y$ is not Gaussian (cf. figure 2 a). In this case, the analytical model for concentration variance (see (4.4)) proposed by Sawford & Sullivan (Reference Sawford and Sullivan1995) is used as an input to the model for predicting $c$ -spectra. With this refinement, the model is able to accurately represent the two peaks on either side of the plume axis in figure 8(e).

Finally, the model is also employed to predict $c$ -spectra at a fixed height but varying source distance $s_{x}$ in figure 9(a–d). As a representative case, $s_{z}/\unicode[STIX]{x1D6FF}=0.25$ is considered here and prediction results at other source heights are comparable. Note that here the streamwise variation of concentration variation is also needed, i.e. of the form $\unicode[STIX]{x1D70E}_{c,ref}^{2}=A(s_{x}/\unicode[STIX]{x1D6FF})^{B}$ . It is found that $A=663.5$ and $B=-2.25$ in our study, and a similar power law can be readily obtained from plume models or standard experiments (e.g. Nironi et al. Reference Nironi, Salizzoni, Marro, Mejean, Grosjean and Soulhac2015). Similar to figure 8, the contour maps of measured $c$ -spectra in figure 9 are compared against predicted contour lines for four different streamwise locations. There is an excellent agreement between model predictions and the measured values for source distances up to $4\unicode[STIX]{x1D6FF}$ . In this representation, it is clearly seen that the plume grows in the streamwise direction and the model works well if the streamwise development of $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D70E}}$ is appropriately accounted for.

Figure 9.

Same as figure 8. Each panel shows concentration spectra for plumes released at different source distances, $s_{x}/\unicode[STIX]{x1D6FF}=0.5$ , 1, 2 and 4, for a log-region source, $s_{z}/\unicode[STIX]{x1D6FF}=0.25$ .