Dispersion of a Passive Scalar Fluctuating Plume in a Turbulent Boundary Layer. Part II: Analytical Modelling

Abstract

We investigate the reliability of a meandering plume model in reproducing the passive scalar concentration statistics due to a continuous release in a turbulent boundary layer. More specifically, we aim to verify the physical consistency of the parametrizations adopted in the model through a systematic comparison with experimental data. In order to perform this verification, we take advantage of the velocity and concentration measurements presented in part I of the present study (Nironi et al., Boundary-Layer Meteorol, 2015) particularly concerning estimates of the Eulerian integral length scales and the higher order moments of the concentration probability density function. The study is completed by a sensitivity analysis in order to estimate the effects of the variations of the key parameter to the model results. In the light of these results, we discuss the benefits and shortcomings of this modelling approach and its suitability for operational purposes.

1 Introduction

Fluctuating plume dispersion models are conceived to estimate the concentration statistics of a pollutant dispersing within a turbulent flow. Compared to other modelling approaches, such as micro-mixing Lagrangian models (Sawford 2004; Cassiani et al. 2005; Leuzzi et al. 2012; Amicarelli et al. 2012; Cassiani 2013) and large-eddy simulations (Xie et al. 2004; Vinkovic et al. 2006), their relatively simple formulation makes them suitable for operational purposes.

The basic principle of this modelling approach (Gifford 1959) is to split the total plume dispersion into two components: meandering and relative dispersion. The first mechanism describes the fluctuation of the plume centroid, whereas the relative dispersion drives the spreading of a plume element around its centre of mass. The two mechanisms can be treated as independent, as long as it is assumed that they are related to spatial scales separated by several orders of magnitude. As pointed out by Yee et al. (1994), and as discussed later, this hypothesis is strictly valid only close to the source or in the far field. In between there does not exist a clear spectral gap between the scales of motion contributing to the plume meandering and those associated with the relative dispersion. Despite this theoretical weakness, this modelling approach was shown to be quite robust in simulating the dispersion of a passive scalar in a variety of turbulent flows (Fackrell and Robins 1982; Sawford and Stapountzis 1986; Yee et al. 1994; Franzese 2003).

Adopting this assumption, the concentration probability density function (PDF), $p$, can be written as the convolution of the PDF of the location of the cloud instantaneous centroid, $p_m$, characterizing the large-scale random crosswind displacements of the centre of mass, and the PDF of the concentration in the meandering reference scheme $(y_m,z_m)$, $p_{cr}$,

$$\begin{aligned} p(c;x,y,z) = \int _{0}^{\infty }\int _{-\infty }^\infty {p_{cr}(c;x,y,z,y_m,z_m)\,p_m(x,y_m,z_m) \mathrm{d} y_m \mathrm{d} z_m}, \end{aligned}$$

(1)

where $c$ is the instantaneous concentration. Once $p$ is known, the moments of the concentration can be computed as,

$$\begin{aligned} \overline{c^n}(x,y,z) = \int _0^\infty {c^n p(c;x,y,z) \mathrm{d} c}. \end{aligned}$$

(2)

The practice of partitioning the concentration fluctuations depending on the their characteristic length scales was originally introduced by Gifford (1959). He proposed a model for dispersion in isotropic and homogeneous turbulence, neglecting the role of internal concentration fluctuations, so that $p_{cr}$ was parametrized by the Dirac delta function $\delta _D$,

$$\begin{aligned} p_{cr}(c;x,y,z,y_m,z_m) = \delta _D \left\{ c-\overline{c}_r(x,y,z,y_m,z_m)\right\} , \end{aligned}$$

(3)

where $\overline{c}_r$ is the mean concentration relative to the instantaneous plume centroid. The assumption of negligible internal fluctuations is reliable for short times, so far as meandering is the mechanism governing the dispersion process. It becomes unrealistic in the far field, where the role of relative dispersion overcomes that of meandering.

This relatively simple model was shown to reliably predict the main features characterizing the near-field dynamics of a fluctuating plume emitted from an elevated source. Fackrell and Robins (1982) simulated the effect of a varying source size on the intensity of the concentration fluctuation in an anisotropic and non-homogeneous velocity field. According to their analysis, this can be attributed to the different role of the meandering motion, whose intensity increases as the source size decreases, since the range of scales of the turbulent motion, that are responsible for displacing the plume centre of mass, widens. The model is also able to distinguish between the different shapes that the concentration PDF assumes according to the type of source (Sawford and Stapountzis 1986), predicting a unimodal PDF for the point source and a bimodal PDF for the line source, in agreement with experimental observations.

The role of the relative in-plume fluctuations was firstly taken into account by Yee et al. (1994) who introduced the intensity of the relative concentration fluctuations $i_{cr}$ as a new parameter, defined as the ratio between the standard deviation of the mean relative concentration, $\sigma _{cr}$, and $\overline{c}_r$. The overall statistics of the relative dispersion were then parametrized by means of a Gamma distribution,

$$\begin{aligned} p_{cr}(c;x,y,z,y_m,z_m) = \frac{\lambda ^\lambda }{\overline{c}_r\Gamma (\lambda )} \left( \frac{c}{\overline{c}_r}\right) ^{\lambda -1}\exp {\left( -\frac{\lambda c}{\overline{c}_r}\right) }, \end{aligned}$$

(4)

where $\Gamma (\lambda )$ is the gamma function and $\lambda = 1/i_{cr}^2$. A one-dimensional formulation of the model was tested against in situ measurements of concentrations (Yee et al. 1994) at different heights above the ground within an atmospheric boundary layer over a uniform flat terrain. A two-dimensional formulation of the model for homogeneous and isotropic turbulence was tested by Yee and Wilson (2000) against water-plume measurements of the first four moments of the concentration PDF of a passive scalar dispersing in grid turbulence. In order to reliably model the anisotropy and the inhomogeneity of the velocity field within a turbulent boundary layer, several authors (Reynolds 2000; Luhar et al. 2000; Franzese 2003; Mortarini et al. 2009) have reconstructed the spatial evolution of the vertical component of the plume centroid PDF, $p_{zm}$, by means of stochastic Lagrangian models simulating the trajectories of the puff centre of mass. Cassiani and Giostra (2002) have also developed a generalized approach that allows $p_{zm}$ to be computed by means of a mean concentration field, without the need for a Lagrangian particle model. All the above-mentioned models include a simple formulation of $i_{cr}$ that is assumed to depend on the longitudinal coordinate only. More complex parametrizations of $i_{cr}$ have been proposed only recently. Gailis et al. (2007) introduced a three-dimensional model of $i_{cr}$ to predict the concentration PDF within a dispersion plume in a group of obstacles. This same parametrization was used by Ferrero et al. (2013) to compute the concentration statistics of two reactive chemical species within a convective boundary layer.

In this paper, we present a formulation of the fluctuating plume model (Sect. 2) and discuss the parametrizations that render the model suitable for simulating the dispersion process within a turbulent neutral boundary layer (Sect. 3). In particular, we aim to analyze the consistency of these parametrizations in the light of the experimental characterization of the velocity field performed by Nironi et al. (2015), especially concerning the estimates of the Eulerian integral length scales (Sect. 3.1). Subsequently, the focus is on $i_{cr}$ (Sect. 3.2) that is modelled adopting two different parametrizations. Assuming $i_{cr}=i_{cr}(x)$ (Sect. 3.2.1), the formulation of the meandering model leads to an analytical solution for $p(c;x,y,z)$, whereas assuming $i_{cr}=i_{cr}(x,y,z)$ (Sect. 3.2.2) leads to a semi-analytical solution. Both formulations are compared to the experimental wind-tunnel results of Nironi et al. (2015), providing a unique dataset concerning the spatial distribution of the first four moments of the concentration PDF (Sect. 4). Finally we perform an error and sensitivity analysis of several key parameters (Sect. 5) to determine the robustness and accuracy of the model and discuss its advantages and shortcomings, as well as its suitability for operational purposes (Sect. 6).

2 Meandering Plume Model

We consider a source of diameter $\sigma _0$ located at coordinates ($0,y_s,z_s$) within a turbulent boundary layer of depth $\delta$. Following Yee and Wilson (2000) and Luhar et al. (2000) we assume the statistical independence of the plume meandering in the lateral and vertical directions, so that $p_m$ can be expressed as the product of two components, $p_{ym}$ and $p_{zm}$,

$$\begin{aligned} p_m(x,y_m,z_m) = p_{ym}(x,y_m)p_{zm}(x,z_m). \end{aligned}$$

(5)

We stress here that this assumption is not supported by any theoretical consideration, but is rather justified by the need for simplicity in the formulation of the model.

Since the velocity statistics are assumed to be homogeneous in the horizontal planes, the crosswind distribution of the centroid locations is Gaussian,

$$\begin{aligned} p_{ym}(x,y_m) = \frac{1}{\sqrt{2\pi }\sigma _{ym}} \exp {\left( -\frac{(y_m-y_s)^2}{2\sigma _{ym}^2}\right) }, \end{aligned}$$

(6)

where $\sigma _{ym}$ is the centroid horizontal spread.

Conversely, $p_{zm}$ requires a slightly more complex formulation in order to account for the effects of ground reflection and the non-homogeneity of the velocity statistics in the vertical direction. This is modelled by means of the following reflected Gaussian distribution (Arya 1999),

$$\begin{aligned} p_{zm}(x,z_m) = \frac{1}{\sqrt{2\pi }\sigma _{zm}} \left\{ \exp {\left[ -\frac{(z_m-z_s)^2}{2\sigma _{zm}^2}\right] } + \exp {\left[ -\frac{(z_m+z_s)^2}{2\sigma _{zm}^2}\right] }\right\} , \end{aligned}$$

(7)

where $\sigma _{zm}$ is the vertical spread of the plume centroid. In the presence of one boundary, a reflected Gaussian ensures a constant mass flux $\int _0^\infty p_{zm}\mathrm{d}z_m = 1$ through any vertical section perpendicular to the wind direction, a constraint that is not satisfied by a simple Gaussian model far from the source, which gives

$$\begin{aligned} \frac{1}{\sqrt{2\pi }\sigma _{zm}}\int _0^\infty \exp {\left( -\frac{(z_m-z_s)^2}{2\sigma _{zm}^2}\right) }\mathrm{d}z_m = \frac{1}{2}\left( 1+\text{ erf }\left( \frac{z_s}{\sqrt{2}\sigma _{zm}}\right) \right) \le 1. \end{aligned}$$

(8)

Note that different atmospheric stability conditions can alter significantly the shape of $p_{zm}$. For a detailed discussion on this topic, see Luhar et al. (2000) and Franzese (2003).

Similarly, to account for the anisotropy of the relative dispersion, we parametrize the relative mean concentration $\overline{c}_r$ as

$$\begin{aligned} \overline{c}_r = \frac{M_q}{\overline{u}_m}\,p_{yr}(x,y,y_m)\,p_{zr}(x,z,z_m), \end{aligned}$$

(9)

where $M_q$ is the mass flow rate and $\overline{u}_m=\overline{u}(\overline{z}_m)$ is the mean cloud advection velocity. This parameter, as well as all other velocity statistics—mean velocity, velocity variances and turbulent kinetic energy (TKE) dissipation rate—used in the model, is evaluated at the plume centroid $\overline{z}_m(x)$ and therefore depends solely on the downwind distance from the source. The implications of this assumption on the results are discussed in Sect. 5.2. The functions $p_{yr}$ and $p_{zr}$ are the lateral and vertical distributions of the mean concentration around the plume centroid and are modelled as,

$$\begin{aligned}&p_{yr} = \frac{1}{\sqrt{2\pi }\sigma _{yr}} \exp {\left( -\frac{\left( y-y_m\right) ^2}{2\sigma _{yr}^2}\right) }, \end{aligned}$$

(10)

$$\begin{aligned}&p_{zr} = \frac{1}{\sqrt{2\pi }\sigma _{zr}}\left\{ \exp {\left( -\frac{\left( z-z_m\right) ^2}{2\sigma _{zr}^2}\right) } + \exp {\left( -\frac{\left( z+z_m\right) ^2}{2\sigma _{zr}^2}\right) }\right\} , \end{aligned}$$

(11)

where $\sigma _{yr}$ and $\sigma _{zr}$ are the relative plume spreads around the plume’s centre of mass in the horizontal and vertical directions, respectively.

Substituting Eqs. 4, 6, 7 and 9 into Eq. 2 and solving the integral in the variable $c$, we obtain the $n$-th concentration moment as function of the position in space $(x,y,z)$,

$$\begin{aligned} \overline{c^n}(x,y,z)= & {} \int _0^\infty p_{zm} \,\mathrm{d}z_m \int _{-\infty }^\infty p_{ym}\, \mathrm{d}y_m\int _0^\infty c^n \, p_{cr}(c;x,y,z,y_m,z_m)\,\mathrm{d}c \nonumber \\= & {} \int _0^\infty p_{zm} \,\mathrm{d}z_m \int _{-\infty }^\infty \frac{1}{\lambda ^n} \frac{\Gamma (n+\lambda )}{\Gamma (\lambda )} \overline{c}_r^n \,p_{ym}\, \mathrm{d}y_m \nonumber \\= & {} \left( \frac{M_q}{2\pi \sigma _{yr}\sigma _ {zr}\overline{u}_m}\right) ^n \frac{1}{2\pi \sigma _{ym}\sigma _{zm}} \nonumber \\&\times \int _{-\infty }^\infty \exp {\left( -\frac{(y_m-y_s)^2}{2\sigma _{ym}^2}\right) } \exp {\left( -\frac{n\left( y-y_m\right) ^2}{2\sigma _{yr}^2}\right) } \mathrm{d}y_m \nonumber \\&\times \int _0^\infty \frac{1}{\lambda ^n} \frac{\Gamma (n+\lambda )}{\Gamma (\lambda )} \left\{ \exp {\left( -\frac{(z_m-z_s)^2}{2\sigma _{zm}^2}\right) } + \exp {\left( -\frac{(z_m+z_s)^2}{2\sigma _{zm}^2}\right) }\right\} \nonumber \\&\times \left\{ \exp {\left( -\frac{\left( z-z_m\right) ^2}{2\sigma _{zr}^2}\right) } + \exp {\left( -\frac{\left( z+z_m\right) ^2}{2\sigma _{zr}^2}\right) }\right\} ^n \,\mathrm{d}z_m. \end{aligned}$$

(12)

In Sect. 3.2 we provide analytical and semi-analytical solutions of Eq. 12, depending on the formulation of $\lambda =1/i_{cr}^2$.

3 Set of Model Parameters

In order to have a complete formulation of the model, we need to parametrize $\sigma _{y}$, $\sigma _{yr}$, $\sigma _{ym}$, $\sigma _{z}$, $\sigma _{zr}$, $\sigma _{zm}$, and $i_{cr}$. To this end, we take advantage of the information provided by the experimental investigation of the velocity and concentration field presented in Nironi et al. (2015). The analysis is performed for the emissions released from three different sources of varying size $\sigma _0/\delta$ and elevation $z_s/\delta$ (see Table 1).

Table 1 Diameter $\sigma _0$ and height $z_s$ of the three sources

3.1 Parametrization of the Plume Spreads

The global plume spreads are related to the spread of the plume centroid and to the relative spread by the following relations (Gifford 1959),

$$\begin{aligned} \sigma _y^2= & {} \sigma _{ym}^2 + \sigma _{yr}^2, \end{aligned}$$

(13)

$$\begin{aligned} \sigma _z^2= & {} \sigma _{zm}^2 + \sigma _{zr}^2. \end{aligned}$$

(14)

There are therefore two independent plume spread parameters that have to be set. Following Luhar et al. (2000) and Franzese (2003) we model the global spreads $\sigma _y$ and $\sigma _z$, the relative spreads $\sigma _{yr}$ and $\sigma _{zr}$, and then obtain $\sigma _{ym}$ and $\sigma _{zm}$ by means of Eqs. 13 and 14.

The global spreads are parametrized according to Taylor’s statistical theory as Nironi et al. (2015),

$$\begin{aligned} \sigma _y^2= & {} \frac{\sigma _0^2}{6} + 2\sigma _v^2T_{Lv}\left\{ t-T_{Lv} \left[ 1-\exp \left( {-\frac{t}{T_{Lv}}}\right) \right] \right\} , \end{aligned}$$

(15)

$$\begin{aligned} \sigma _z^2= & {} \frac{\sigma _0^2}{6} + 2\sigma _w^2T_{Lw}\left\{ t-T_{Lw} \left[ 1-\exp \left( {-\frac{t}{T_{Lw}}}\right) \right] \right\} , \end{aligned}$$

(16)

where $\sigma _v$ and $\sigma _w$ are the standard deviations of the transverse and vertical velocity component, $T_{Lv}$ and $T_{Lw}$ are Lagrangian time scales, and $t=x/\overline{u}_m$ is the flight time. As is customary (Tennekes 1982), the Lagrangian time scales are parametrized as $T_{Lv} = {\frac{2\sigma _v^2}{C_0\varepsilon }}$ and $T_{Lw} = {\frac{2\sigma _w^2}{C_0\varepsilon }}$, where $C_0$ is the Kolmogorov constant assumed here equal to 4.5 (Nironi et al. 2015), and $\varepsilon$ is the mean dissipation rate of the TKE.

The parametrization of the relative dispersion coefficients, $\sigma _{yr}$ and $\sigma _{zr}$, has to satisfy two asymptotic conditions (for brevity we report only those of $\sigma _{yr}$, since the same conditions are imposed on $\sigma _{zr}$, see Franzese 2003; Franzese and Cassiani 2007),

$$\begin{aligned}&{ t \rightarrow 0 \qquad \sigma _{yr}^2 = (C_{r}/6)\varepsilon \left( t_s+t\right) ^3}, \end{aligned}$$

(17)

$$\begin{aligned}&{ t \rightarrow \infty \qquad \sigma _{yr}^2 \rightarrow \sigma _y^2 = 2\sigma _{v}^2T_{Lv}t,} \end{aligned}$$

(18)

where $C_r$ is the Richardson–Obukhov constant and $t_s = \left[ \sigma _0^2/(C_r\varepsilon )\right] ^{1/3}$ represents the flight time needed by a plume emitted from a virtual point source to expand to the size $\sigma _0$.

Equation 17 follows from the Richardson–Obukhov law for a finite source size (Ott and Mann 2000; Franzese and Cassiani 2007) and models the cloud spreading as a function of the flight time $t$ (Richardson 1926) and $\varepsilon$ (Obukhov 1941). Equation 18 is the Taylor’s limit for large dispersion time, which applies to $\sigma _{yr}$ (and $\sigma _{zr}$) when the meandering process becomes negligible and the relative dispersion approaches the global dispersion. The objective is to define a suitable transition between the two asymptotic behaviours. To that purpose we adopt an approach similar to that proposed by Luhar et al. (2000) and Franzese (2003). In contrast to them, we introduce the time scales $T_{my}$ and $T_{mz}$. These are needed in order to ensure that the transition from the inertial scaling to the diffusive asymptotic scaling is consistent with the main characteristics of the large-scale dynamics of the velocity field, presented in Nironi et al. (2015). The evolutions of $\sigma _{yr}^2$ and $\sigma _{zr}^2$ are then modelled as

$$\begin{aligned} \sigma _{yr}^2= & {} \frac{(C_{r}/6)\varepsilon \left( t_s+t\right) ^3}{\left\{ 1+\left[ (C_{r}/6) \varepsilon t^2/\left( 2\sigma _v^2T_{Lv}\right) \right] ^{2/5}\right\} ^{5/2}} \exp {\left[ -\left( \frac{t}{T_{my}}\right) ^2\right] } \nonumber \\&+\, \sigma _y^2\left\{ 1- \exp {\left[ -\left( \frac{t}{T_{my}}\right) ^2\right] }\right\} , \end{aligned}$$

(19)

$$\begin{aligned} \sigma _{zr}^2= & {} \frac{(C_{r}/6)\varepsilon \left( t_s+t\right) ^3}{\left\{ 1+\left[ (C_{r}/6) \varepsilon t^2/\left( 2\sigma _w^2T_{Lw}\right) \right] ^{2/5}\right\} ^{5/2}} \exp {\left[ -\left( \frac{t}{T_{mz}}\right) ^2\right] } \nonumber \\&+\, \sigma _z^2 \left\{ 1- \exp {\left[ -\left( \frac{t}{T_{mz}}\right) ^2\right] }\right\} , \end{aligned}$$

(20)

so that the spatial evolutions of $\sigma _{yr}$ and $\sigma _{zr}$ are therefore a function of the parameters $T_{my}$, $T_{mz}$ and $C_r$, whose setting is discussed in the following paragraphs.

3.1.1 The Time Scales $T_{my}$ and $T_{mz}$

The time scales $T_{my}$ and $T_{mz}$ can be thought of as thresholds beyond which relative dispersion becomes the prevalent mechanism. This occurs when the size of the relative plume exceeds that of the largest scale eddies (${\mathcal {L}}$), so that the contribution of the TKE to the fluctuations of the cloud centroid becomes negligible. It is questionable if the quantity ${\mathcal {L}}$ refers to Lagrangian statistics, as proposed by Franzese and Cassiani (2007), or to Eulerian statistics. The latter are adopted here, since we dispose of the direct measurements of the Eulerian integral length scales $L_{vv}$ and $L_{ww}$ (Nironi et al. 2015).

Both $T_{my}$ and $T_{mz}$ are assumed to be proportional to the Lagrangian time scales, $T_{my}=\alpha _{Ty} T_{Lv}$ and $T_{mz}=\alpha _{Tz} T_{Lw}$. In order to define the value of the proportionality coefficients $\alpha _{Ty}$ and $\alpha _{Tz}$, we analyze the evolution of $\sigma _{yr}$ and $\sigma _{zr}$, as given by Eqs. 19 and 20, and we compare it with the experimental estimates of $L_{vv}$ and $L_{ww}$ evaluated at source height $z_s$. In doing that, we fix the value of the Richardson–Obukhov constant $C_r=0.8$ (the sensitivity to $C_r$ is discussed in Sect. 3.1.2). The evolution of $\sigma _{yr}$ and $\sigma _{zr}$, as well as that of $\sigma _{ym}$ and $\sigma _{zm}$, is plotted in Fig. 1 for varying values of $\alpha _{Ty}=\alpha _{Tz}=1,2,3$. As expected, the plot shows that the behaviour of the dispersion coefficients depends on the source elevation. In particular the spreads due to the centre of mass, $\sigma _{ym}$ and $\sigma _{zm}$, for the elevated source ES (Fig. 1a, b) are larger than those of the low-level source LLS (Fig. 1c, d). This can be explained by two features. Firstly, in the lower part of the boundary layer the size of the most energetic eddies is smaller and, as a consequence, the effects on the dispersion due to the displacement of the plume centroid are significantly reduced. Secondly, the effects of the ground (Luhar et al. 2000; Franzese 2003) result in a more rapid damping of the plume meandering in the vertical direction (Fig. 1b, d) with respect to the transverse coordinate (Fig. 1a, c).

Fig. 1

Modelled transverse and vertical dispersion coefficients varying $\alpha _T$ ($C_0 = 4.5$ and $C_r = 0.8$): a, b ES 3 source; c, d LLS source. Solid line $\alpha _T=1$, dash line $\alpha _T=2$, dash-dot line $\alpha _T=3$. The black-dotted lines represent reference values of the Eulerian integral length scales $L_{vv}$ and $L_{ww}$ at source height, as estimated by Nironi et al. (2015)

The aim here is to define $\alpha _{Ty}$ and $\alpha _{Tz}$ so that $\sigma _{ym}$ and $\sigma _{zm}$ do not exceed $L_{vv}$ and $L_{ww}$, respectively, (Fig. 1) as $\sigma _{yr}$ and $\sigma _{zr}$ attain their asymptotic values. The choice of the most appropriate $\alpha _{Ty}$ and $\alpha _{Tz}$ is somehow arbitrary. In order to constrain the degree of freedom of the model in the parameter space, we impose $\alpha _{Ty}=\alpha _{Tz}=\alpha _T$, a condition that may not be appropriate for any source configurations (as a crosswind line source). Adopting these criteria yields $\alpha _T = 2$. It is worth noting that, in the present case study, the original formulation proposed by Luhar et al. (2000) and Franzese (2003), i.e. without the corrective terms involving $T_{my}$ and $T_{mz}$ in Eqs. 19 and 20, leads to a slower increase of $\sigma _{yr}$ and $\sigma _{zr}$ with the distance from the source, implying an unphysical growth of $\sigma _{ym}$ and $\sigma _{zm}$ to values exceeding the Eulerian scales. Finally, we point out that the characteristic length scales of the meandering and relative dispersion processes are well-separated only very close to the source and in the far field (where $\sigma _{my},\sigma _{mz} \rightarrow 0$). However, as expected (Fig. 1), this hypothesis is not verified in an intermediate region that actually covers most of the investigated domain, both for the ES and LLS cases.

3.1.2 Richardson–Obukhov Constant $C_r$

Values of $C_r$ are affected by a significant uncertainty. In homogeneous and isotropic turbulence, estimates obtained with direct numerical simulations are approximately in the range $0.4 < C_r < 0.8$ (Ishihara and Kaneda 2002; Boffetta and Sokolov 2002; Biferale et al. 2005), and in non-homogeneous and anisotropic turbulence the range is further widened. Franzese (2003) adopted $C_r = 1.4$ in a convective boundary layer whereas Mortarini et al. (2009) assumed $C_r = 0.06$ in a plant canopy.

In order to emphasize the influence on the model due to the Richardson–Obukhov constant variations, in Fig. 2 we plot ${\frac{\sigma _{ym}}{\sigma _y}}$, ${\frac{\sigma _{zm}}{\sigma _z}}$, ${\frac{\sigma _{yr}}{\sigma _y}}$, and ${\frac{\sigma _{zr}}{\sigma _z}}$ as a function of the downwind distance from the source, for values of $C_r$ spanning a range consistent with the literature data, i.e. $C_r=0.4{-}1.2$. The effects of the variations of $C_r$ are significantly reduced compared to those of $\alpha _T$, plotted in Fig. 1. Nevertheless, the influence of $C_r$ is non-negligible in an intermediate region (see Fig. 2), from $x/\delta \approx 0.2$ up to $x/\delta \approx 1.5$, within which it can appreciably affect the model results. According to Franzese and Cassiani (2007), the quantity $C_r / C_0$ should be fixed and equal to $1/11$. Since we adopted $C_0=4.5$ this leads to $C_r \approx 0.4$. However, for reasons that will be made clear in the next paragraph (Sect. 3.2), this value was not consistent with the formulation of the model for $i_{cr}$. We have therefore assumed $C_r=0.8$ which implies a ratio $C_r/C_0 \approx 0.17$. Note however that this value is quite close to the theoretical value suggested by Franzese and Cassiani (2007) compared to those presented in previous studies (Franzese 2003 imposed $C_r/C_0 \approx 0.5$ and Mortarini et al. 2009 assumed $C_r/C_0 \approx 1/33$).

Fig. 2

a $\sigma _{yr}/\sigma _y$, $\sigma _{ym}/\sigma _y$, and b $\sigma _{zr}/\sigma _z$, $\sigma _{zm}/\sigma _z$ vs $x/\delta$ for ES 3 varying $C_r=0.4,0.8,1.2$ ($C_0= 4.5$ and $\alpha _T = 2$); solid line $C_r=0.4$, dash line $C_r=0.8$, dash-dot line $C_r=1.2$

3.2 Parametrization of the Intensity of Relative Concentration Fluctuations

The determination of the spatial evolution of the intensity of relative concentration fluctuations $i_{cr}$ is a key aspect in the formulation of the meandering models. The dependence of this parameter on the flow dynamics and emission conditions however has been rarely characterized either experimentally or numerically. As far as we know, the only attempt to measure this parameter is that of Gailis et al. (2007), who studied the dispersion of a passive scalar with optical measurement techniques within both a turbulent boundary layer and an obstacle array. Given this lack of information, most of the meandering plume models found in the literature (Luhar et al. 2000; Yee and Wilson 2000; Franzese 2003; Mortarini et al. 2009) adopt quite simple models for $i_{cr}$, which is generally assumed to be dependent on the $x$-coordinate only. This assumption however can significantly deteriorate the numerical results in the far field. As shown by Mortarini et al. (2009), to avoid this lack of accuracy of the model it is then necessary to impose an unphysical growth of $i_{cr}$ for increasing distance from the source.

In what follows, we consider two different formulations of the model. In the first case we consider $i_{cr}=i_{cr}(x)$, in the second case $i_{cr}=i_{cr}(x,y,z)$. Both formulations are set in order to ensure the physical consistency of the model with respect to the evolution of the intensity of the concentration fluctuations, $i_c=\sigma _c/\overline{c}$, determined experimentally in the wind-tunnel experiments presented in Nironi et al. (2015).

3.2.1 1-D Model of $i_{cr}$

In the case of $i_{cr}=i_{cr}(x)$, Eq. 12 can be solved analytically, leading to

$$\begin{aligned} \overline{c^n}(x,y,z)= & {} \left( \frac{M_q}{2\pi \sigma _{yr}\sigma _{zr}{ \overline{u}_m}}\right) ^n \frac{\sigma _{yr}}{\left( n\sigma _{ym}^2+\sigma _{yr}^2\right) ^{0.5}} \frac{\sigma _{zr}}{\left( n\sigma _{zm}^2+\sigma _{zr}^2\right) ^{0.5}} \frac{1}{\lambda ^n} \frac{\Gamma (n+\lambda )}{\Gamma (\lambda )} \nonumber \\&\times \exp {\left[ -\frac{n(y-y_s)^2}{2\left( n\sigma _{ym}^2+\sigma _{yr}^2\right) }\right] } \sum _{k=0}^n \left\{ {n \atopwithdelims ()k} \exp \left[ -\frac{(n-k)(z-z_s)^2}{2(n\sigma _{zm}^2+\sigma _{zr}^2)}\right] \right. \nonumber \\&\times \left. \exp \left[ -\frac{k(z+z_s)^2}{2(n\sigma _{zm}^2+\sigma _{zr}^2)}\right] \exp \left[ -\frac{(2z)^2k(n-k)}{2(n\sigma _{zm}^2+\sigma _{zr}^2)} \frac{\sigma _{zm}^2}{\sigma _{zr}^2}\right] \right\} , \end{aligned}$$

(21)

where ${\displaystyle {n \atopwithdelims ()k}}$ is the binomial coefficient.

With some algebra, Eq. 21 can be rearranged in order to illustrate the relation between $i_{cr}$ and $i_c$ on the plume centreline ($y=y_s,z=z_s$),

$$\begin{aligned}&i_{cr}^2 = \left( i_c^2 +1\right) F_c -1, \end{aligned}$$

(22a)

$$\begin{aligned} F_c= & {} \frac{\sigma _{yr}\sqrt{2\sigma _{ym}^2+\sigma _{yr}^2}}{\sigma _y^2} \frac{\sigma _{zr}\sqrt{2\sigma _{zm}^2+\sigma _{zr}^2}}{\sigma _z^2} \nonumber \\&\times \left\{ 1 + 2\exp \left( -\frac{2( z_s)^2}{(2\sigma _{zm}^2+\sigma _{zr}^2)}\right) \exp \left( -\frac{2( z_s)^2}{(2\sigma _{zm}^2+\sigma _{zr}^2)} \frac{\sigma _{zm}^2}{\sigma _{zr}^2}\right) \right. \nonumber \\&+\left. \exp \left( -\frac{4( z_s)^2}{(2\sigma _{zm}^2+\sigma _{zr}^2)}\right) \right\} ^{-1} \left\{ 1 + \exp \left( -\frac{2( z_s)^2}{\sigma _z^2}\right) \right\} ^2, \end{aligned}$$

(22b)

where the evolution of $F_c$ is fully determined by the parametrization (Sect. 3.1) of the plume spreads (total, relative and centroid position). For all release conditions $i_c$ is larger than $i_{cr}$ close to the release point, where the meandering process is significant both for the ES and LLS cases. Moving away from the source, the relative dispersion becomes the prevalent mechanism and $i_{cr} \rightarrow i_c$.

As Eq. 22b clearly shows, the model of $i_{cr}$ depends on the distribution of $i_{c}$, which therefore requires an independent estimate. To overcome this problem, Yee et al. (1994) and Yee and Wilson (2000) have set $i_{cr}$ by fitting models of the form of Eq. 22b to the experimental estimates of $i_{c}$. Following this same approach, we then turn to the experimental values of $i_c(x,y_s,z_s)$ collected by Nironi et al. (2015). By substituting these data in Eq. 22b we determined $i_{cr}$ at six different distances from the source. Since two asymptotic bounds have to be satisfied at source location ($x\rightarrow 0$) and in the very far field ($x\rightarrow \infty$), the following rational curve was used to fit the $i_{cr}$ estimates,

$$\begin{aligned}&i_{cr} \rightarrow 0 \,\,\qquad \text{ at } \,\, \frac{x}{\delta } \rightarrow 0, \end{aligned}$$

(23a)

$$\begin{aligned}&i_{cr} \rightarrow i_{c} = \frac{\sigma _c}{\overline{c}} \ne 0 \qquad \text{ at } \,\, \frac{x}{\delta } \rightarrow \infty , \end{aligned}$$

(23b)

$$\begin{aligned}&i_{cr} = x_{ad}\frac{p_1x_{ad}^2+p_2x_{ad}+p_3}{x_{ad}^3+q_1x_{ad}^2+q_2x_{ad}+q_3}, \end{aligned}$$

(23c)

where $x_{ad}=x/\delta$. The values of the parameters in Eq. 23c for the three cases considered are computed with the method of least-squares and are summarized in Table 2.

Table 2 Coefficients used in Eq. 23c to calculate $i_{cr}$

As discussed in Sect. 3.1.2, the values of $i_{cr}$ close to the source depend significantly on the choice of $C_r$ (with variations of order 30 %). In particular, for $C_r=0.4$, we found unphysical negative values of $i_{cr}$. We have therefore excluded this value, even though it is supported by the theoretical analysis proposed by Franzese and Cassiani (2007), and adopted $C_r=0.8$ instead.

Figure 3 shows a comparison at increasing distances from the source between the experimental values of $i_{c}$, measured on the plume centreline, and the values of $i_{cr}$ computed through Eq. 23c. In both ES and LLS cases, $i_{cr}$ exhibits an initial growth and thereafter decreases monotonically to an asymptotic value, which is reached far from the source. As the meandering motion weakens, $i_{cr}$ correctly approaches $i_{c}$. In the ES case the meandering influence disappears later ($x/\delta \approx 2$, Fig. 3a, b) in accordance with the model of $\sigma _{ym}$ and $\sigma _{zm}$ (Fig. 1). Conversely, in the LLS case the fluctuation of the plume centre of mass is damped very rapidly by the presence of the ground and $i_{cr} \rightarrow i_{c}$ at $x/\delta \approx 1$ (Fig. 3c). In both cases, the model leads to the same asymptotic value of the relative concentration fluctuations (at $x/\delta \rightarrow \infty$, $i_{cr} \rightarrow 0.35$).

Fig. 3

Experimental values of $i_{c}$ (symbols) and $i_{cr}$ computed through Eq. 23c (lines) vs $x/\delta$ at the plume centreline: a ES 3, b ES 6, c LLS

It is worth mentioning that as $i_{cr}\rightarrow i_c$ the intermittency in the core of the plume is suppressed. Note that according to the analysis performed in Sect. 5.3 and Sect. 5.4 in Nironi et al. (2015), this actually takes place at distances ($x/\delta \approx 3.75$ for the ES) that are larger than those predicted by the present model ($x/\delta \approx 2.5$ for the ES, see Fig. 3a, b). In this sense, the model is not fully consistent with the experimental data in this intermediate region between the near and the far field.

As shown in Fig. 3, the model reproduces a dependence of $i_{cr}$ on the source size and elevation. The dependence on $z_s$ is due to the inhomogeneity of the velocity field, whereas the influence of $\sigma _0$ is not easily explained. According to this model, the increased intensity $i_c$ observed for the smaller source size is due partially to the increased intensity of the meandering motion and partially to an increased intensity of the relative concentration fluctuations. It is questionable if this trend represents the real physics of the phenomenon or if it has to be attributed to a fictitious effect related to the formulation of the model. The answer to this question, however, can be given only through a direct estimate of $i_{cr}$ by means of experiments or direct numerical simulations.

3.2.2 3-D Model of $i_{cr}$

By means of experimental measurements, Gailis et al. (2007) showed that the lateral and vertical profiles of $i_{cr}$ exhibit large variations on the y–z plane. Based on these measurements, they proposed to parametrize $i_{cr}$ as a function of the mean relative concentration $\overline{c}_r$,

$$\begin{aligned} { i_{cr}^2(x,y,z) = [1 + i_{cr0}^2(x)]\left[ \frac{\overline{c}_r(x,y,z)}{\overline{c}_r(x,y_m,z_m)} \right] ^{-\zeta (x)}- 1}, \end{aligned}$$

(24)

where, for each transverse section corresponding to a given distance $x$, $i_{cr0}$ is the minimum of $i_{cr}$, $\zeta$ is a shape parameter depending on the longitudinal coordinate and $\overline{c}_r(x,y_m,z_m)$ is the mean relative concentration evaluated at the instantaneous plume centroid. The same parametrization was assumed by Ferrero et al. (2013) in a convective boundary layer. We adopt here a similar formulation, which we slightly modify by introducing two shape parameters, $\zeta _y$ and $\zeta _z$, to take into account the effects of anisotropy in the $y$ and $z$ directions,

$$\begin{aligned} i_{cr}^2= & {} (1 + i_{cr0}^2) \left\{ \exp \left( -\frac{(y-y_m)^2}{2\sigma _{yr}^2}\right) \right\} ^{-\zeta _y} \nonumber \\&\times \left\{ \exp \left( -\frac{(z-z_m)^2}{2\sigma _{zr}^2}\right) + \exp \left( -\frac{(z+z_m)^2}{2\sigma _{zr}^2}\right) \right\} ^{-\zeta _z} \nonumber \\&\times \left\{ 1 + \exp \left( -\frac{(2z_m)^2}{2\sigma _{zr}^2}\right) \right\} ^{\zeta _z} - 1 , \end{aligned}$$

(25)

where the longitudinal evolution of $i_{cr0}$ remains the same as that defined in the previous paragraph (Sect. 3.2.1).

The evolution of the shape parameters $\zeta _y$ and $\zeta _z$ with $x$ has been modelled in order to ensure consistency with the main feature characterizing the plume relative dispersion. Close to the source the size of the cloud relative to the plume centroid, $\sigma _{yr}$ and $\sigma _{zr}$, is smaller than the Eulerian integral length scale and mixing with the ambient air is due to eddies whose size ranges from the Kolmogorov scale to that of the cloud itself. We can then expect an efficient mixing within the core of the plume, leading to an almost uniform $i_{cr}$ with respect to $y$ and $z$ directions. Conversely, in the far field $\sigma _{yr}$ and $\sigma _{zr}$ approach respectively $\sigma _{y}$ and $\sigma _{z}$ and exceed significantly the Eulerian integral length scale, so that mixing with ambient air is due to eddies smaller than the plume. The intermittency of the entrainment of ambient air within the plume produces high fluctuations of the relative concentration at the edges of the plume, that are progressively reduced approaching the core. In this case, $i_{cr}$ depends on $y$- and $z$-coordinates and the form of its transverse and vertical profiles varies with downwind distance from the source and tends to a self-similar behaviour in the far field. This tendency can be reproduced by modelling the shape parameters with a sigmoid function, viz.

$$\begin{aligned} \zeta = \frac{\alpha _1}{\alpha _2+\exp \left( -\alpha _3\frac{x}{\delta }\right) }. \end{aligned}$$

(26)

Close to the source location $i_{cr}(x) \approx i_{cr0}(x)$ and, therefore, $\zeta _y$ and $\zeta _z$ should assume values close to zero. The values of $\alpha _1$ and $\alpha _2$ have been set in order to fit the self-similar profile of $i_{c}$ observed experimentally in the LLS case (see Fig. 4a) at large distance from the source. The values of $\alpha _3$ drive the transition between the asymptotic states corresponding to the near and far fields. The spatial extent of this transition is small for the LLS plume, since the profiles of $i_{c}$ rapidly attain self-similarity, and larger for the ES plume. Therefore, moving away from the source, $\zeta _y$ and $\zeta _z$ increase and tend to different values, giving an $i_{cr}$ that is shaped differently in the transverse and vertical directions. The values of the coefficients $\alpha _1$, $\alpha _2$ and $\alpha _3$ adopted in our model are reported in Table 3 and the resulting downwind variations of $\zeta _y$ and $\zeta _z$ are plotted in Fig. 4b.

Fig. 4

a Self-similarity of the vertical profiles of $i_c$ in the far field, experimental data for the LLS from Nironi et al. (2015); b shape parameters $\zeta _y$ and $\zeta _z$ vs $x/\delta$

Table 3 Sigmoid function coefficients

Substituting Eqs. 25 and 26 in Eq. 12, we obtain an analytical solution in $y_m$ and an integral in $z_m$, that has to be solved numerically. The moments of the concentration are then given by the following relation,

$$\begin{aligned} \overline{c^n}(x,y,z)= & {} \left( \frac{M_q}{2\pi \sigma _{yr}\sigma _{zr}\overline{u}_m}\right) ^n \int _0^\infty \sum _{j=0}^{n-1}\left\{ a_j\left[ (i_{cr0}^2 + 1) \left( 1 + \exp \left( -\frac{4 z_m^2}{2\sigma _{zr}^2}\right) \right) ^{\zeta _z} \nonumber \right. \right. \\&\times \left. \left( \frac{1}{\sqrt{2\pi }\sigma _{zr} \,p_{zr}}\right) ^{\zeta _z}\right] ^j \frac{\sigma _{yr}}{\sqrt{(n-j\zeta _y)\sigma _{ym}^2+\sigma _{yr}^2}} \nonumber \\&\times \left. \exp \left( -\frac{(n-j\zeta _y)(y-y_s)^2}{2((n-j\zeta _y)\sigma _{ym}^2 + \sigma _{yr}^2)}\right) \right\} \left( \sqrt{2\pi }\sigma _{zr} \,p_{zr}\right) ^n \,p_{zm} \, \mathrm{d}z_m \end{aligned}$$

(27)

where $a_j$ are the coefficients of the polynomial

$$\begin{aligned} P(x) = \sum _{j=0}^{n-1} a_{j}x^j = \prod _{k=1}^n{\left[ (n-k)x-(n-k-1)\right] } \end{aligned}$$

(28)

computable through Vieta’s formulae. The values of the coefficients $a_j$ for the first four concentration moments are reported in Table 4.

Table 4 $a_j$ coefficients evaluated through Eq. 28 for the first four concentration moments

3.2.3 Asymptotic Behaviour

At large distance from the source ($x/\delta \rightarrow \infty$, $t/T_L\rightarrow \infty$) the relative dispersion becomes the only mechanism characterizing the dispersion process, so that,

$$\begin{aligned}&i_{cr} \rightarrow i_c, \end{aligned}$$

(29a)

$$\begin{aligned}&\sigma _{ym} \rightarrow 0, \quad \sigma _{yr} \rightarrow \sigma _{y}, \end{aligned}$$

(29b)

$$\begin{aligned}&\sigma _{zm} \rightarrow 0, \quad \sigma _{zr} \rightarrow \sigma _{z}. \end{aligned}$$

(29c)

In these conditions, the centroid PDF $p_m$ tends to a Dirac delta distribution and the PDF of the global dispersion ($p$) is equal to the relative concentration PDF, that assumes the following formulation,

$$\begin{aligned}&p_m \rightarrow \delta _D, \end{aligned}$$

(30)

$$\begin{aligned}&p_{cr} \rightarrow p = \frac{\lambda ^\lambda }{\overline{c}\Gamma (\lambda )} \left( \frac{c}{\overline{c}}\right) ^{\lambda -1} \exp \left( -\frac{\lambda c}{\overline{c}}\right) \end{aligned}$$

(31)

with $\lambda = 1/i_{cr}^2 \rightarrow 1/i_c^2$.

This formulation of the model is in agreement with one of the main findings of the experimental investigation presented in Nironi et al. (2015), i.e. that the PDF of the concentration can be modelled with high accuracy by a Gamma distribution.

4 Comparison with Experimental Results

We finally test the agreement of the fluctuating plume model with the wind-tunnel measurements of the concentration statistics, carried out by Nironi et al. (2015). A first qualitative analysis concerns the form of the concentration PDF at the plume centreline, presented for both formulations of $i_{cr}$ in Sect. 3.2. As an example, in Fig. 5 we report a comparison between the modelled and experimental PDFs at two distances from the source. Close to the source the meandering mechanism prevails (see Fig. 1) and the form of the PDF is similar to a negative exponential distribution (Fig. 5a). In the far field, the meandering becomes negligible with respect to the relative dispersion. The intermittency within the plume is damped, and the shape of the PDF is similar to a log-normal distribution with a short tail, as shown in Fig. 5b. In both cases the model captures the plume dynamics well and accurately reproduces the shape of the concentration PDF.

Fig. 5

Experimental and modelled PDF on the mean plume centreline varying with the distance from the source location for ES 6 at $y/\delta =0$, $z/\delta =z_s/\delta$: a $x/\delta =0.625$, b $x/\delta =5.0$

To perform a more accurate analysis of the model reliability and to quantify the errors in the predictions, we focus on the profiles of the first four moments around the mean, referred to as $m_i$, with $i=1,2,3,4$. These can be computed from Eq. 2 using the following relations (Monin and Yaglom 1971),

$$\begin{aligned} m_1= & {} \overline{c}, \end{aligned}$$

(32a)

$$\begin{aligned} m_2= & {} \sigma _c^2 = \overline{c^2} - \overline{c}^2, \end{aligned}$$

(32b)

$$\begin{aligned} m_3= & {} \overline{c^3} -3\overline{c^2}\overline{c} + 2\overline{c}^3, \end{aligned}$$

(32c)

$$\begin{aligned} m_4= & {} \overline{c^4} -4\overline{c^3}\overline{c} +6\overline{c^2}\overline{c}^2 - 3\overline{c}^4. \end{aligned}$$

(32d)

In the analysis we apply the following normalization:

$$\begin{aligned} m_i^* = {m_i}^{1/i}\frac{u_{\infty }\delta ^2}{M_q}, \end{aligned}$$

(33)

where $u_\infty$ is the velocity at the top of the boundary layer.

It is worth noting that in what follows (as well as in the analysis presented in Sect. 5), the computed values of the first two moments, the mean and the standard deviation, are actually the results of a best fit of the models given by Eq. 21 and Eq. 27 (with $n=1$ and $n=2$) to the experimental data, obtained by tuning the model parameters, as discussed in Sect. 3. A real comparison between model and experiments is therefore performed only for $m_3^*$ and $m_4^*$, whose estimates can be considered as fully independent from the experimental observations.

4.1 Elevated Source

Firstly, we analyze the longitudinal profiles of the ratio between the values of the concentration statistics ($\sigma _c$, $m_3^{1/3}$, and $m_4^{1/4}$) and mean concentration $\overline{c}$ at source height, where the estimates of the two models of $i_{cr}$ do not differ one from the other. As Fig. 6 shows, the model provides accurate estimates in the whole domain investigated here ($0 \le x/\delta \le 5$), for the two elevated sources considered. In particular, the model reproduces well the influence of the source size in the near field, that progressively vanishes as the meandering motion becomes less effective at displacing the plume centre of mass. The concentration statistics then tend to the same asymptotic values in the far field, independently of the source conditions.

Fig. 6

ES case: concentration statistics vs $x/\delta$ at $y/\delta =0$, $z/\delta =z_s/\delta$: a $\sigma _c/\overline{c}$, b $m_3/\overline{c}$, c $m_4/\overline{c}$

A further analysis concerns the transverse profiles of the concentration statistics at source height. As an example we show in Fig. 7a, c, e, and g a comparison between experiments and model predictions at $x/\delta = 0.625$, $z/\delta =z_s/\delta$. At this distance from the source, the two formulations of the $i_{cr}$ provide almost identical values since $i_{cr}=i_{cr0}$ in both cases. The varying source diameter does not affect the profiles of mean concentration (Fig. 7a), whereas it significantly influences the profiles of the higher order moments (Fig. 7c, e, g). The model results are in excellent agreement with the experimental observations.

Fig. 7

ES case: comparison between experimental and modelled transverse profiles of the concentration statistics at source height and at $x/\delta = 0.625$: a $\overline{c}^*$, c $\sigma _c^*$, e $m_3^*$, g $m_4^*$, and at $x/\delta = 3.75$: b $\overline{c}^*$, d $\sigma _c^*$, f $m_3^*$, h $m_4^*$. Blue circles experimental values for ES 3, red crosses experimental values for ES 6, blue solid line and red dash line solutions provided by Eq. 27, blue dash line solutions provided by Eq. 21. Note that at $x/\delta = 0.625$ the differences between the solutions computed by means of Eq. 21 and Eq. 27 are negligible

As we proceed downwind from the source, the results show a slight deterioration, which is only partially corrected by adopting a 3-D model of $i_{cr}$. In Fig. 7b, d, f, and h we show a comparison between experimental and analytical results for the ES 3 emission, computed with both the 1-D and the 3-D models of $i_{cr}$ (Eq. 25) at $z/\delta =z_s/\delta$ and $x/\delta = 3.75$, where the influence of the source size has become negligible. The model provides quite accurate estimates of the concentration statistics, even though the spreads of the simulated profiles of $m^*_3$ and $m^*_4$ are narrower than the experimental profile.

4.2 Low-Level Source

As for the ES case, the modelled profiles of the second-, third- and fourth-order moments of the concentration as function of the $x$-coordinate at $y=0$ and $z=z_s$ (Fig. 8) present a fairly good agreement with the experimental data, both close to the source and in the far field.

Fig. 8

LLS case: concentration statistics vs $x/\delta$ at $y/\delta =0$, $z/\delta =z_s/\delta$: a $\sigma _c/\overline{c}$, b $m_3/\overline{c}$, c $m_4/\overline{c}$

Even in this case the predictions performed through 1-D and 3-D formulations of $i_{cr}$ provide good results of the concentration statistics close to the source, given the $\zeta _y$ coefficient is next to zero up to $x/\delta \approx 0.5$ (see Fig. 4). This is shown in Fig. 9a, c, e, and g, where we have plotted the transverse profiles of the concentration statistics at the source height.

Fig. 9

LLS case: comparison between experimental and modelled transverse profiles of the concentration statistics at the source height and at $x/\delta = 0.625$: a $\overline{c}^*$, c $\sigma _c^*$, e $m_3^*$, g $m_4^*$, and at $x/\delta = 3.75$: b $\overline{c}^*$, d $\sigma _c^*$, f $m_3^*$, h $m_4^*$. Circles experimental values, solid line solutions provided by Eq. 27, dash line solutions provided by Eq. 21. Note that at $x/\delta = 0.625$ the differences between the solutions computed by means of Eq. 21 and Eq. 27 are negligible

In the far field, the model with $i_{cr}=i_{cr}(x)$ is not able to reproduce the profiles of the concentration statistics, even qualitatively. As Fig. 9d, f, and h show, the model fails to reproduce the off-centreline peaks and the transverse profiles keep a Gaussian shape. In order to improve the model prediction it is then necessary to assume $i_{cr}=i_{cr}(x,y,z)$. The results of the model are however less accurate than in the ES case. In the far field, the model underestimates the mean concentration peak on the plume axis (Fig. 9b), due to the discrepancies between the modelled $\sigma _y$ (Eq. 15) and its experimental values (Nironi et al. 2015). The transverse profiles of the higher order moments show the emergence of off-centreline peaks, that are particularly marked for the third order moment. The model captures these tendencies qualitatively but its quantitative predictions show significant discrepancies with the experimental data (Fig. 9d, f, h).

5 Error and Sensitivity Analysis

Finally, we evaluate the reliability and robustness of the fluctuating plume model by means of:

• estimates of its global accuracy, computed through a comparison between the measured and computed concentration statistics;

• a Monte-Carlo analysis providing the sensitivity of the solutions to variations of the key parameters.

5.1 Errors

To investigate the reliability of the model we need to quantify the gap between the measured, $(m_i^*)_{exp}$ and the computed $(m_i^*)_{mod}$ values of the moments of the concentration. To this end we define the relative error as

$$\begin{aligned} RE_i = \sqrt{\frac{\int _{}^{} \left[ (m_i^*)_{mod}-(m_i^*)_{exp}\right] ^2\mathrm{d}s}{\int _{}^{} \left[ (m_i^*)_{exp}\right] ^2\mathrm{d}s}}, \end{aligned}$$

(34)

with $i=1,2,3,4$, $\mathrm{d}s = \mathrm{d}y, \mathrm{d}z$ and where $i$ is the moment number.

The analysis is performed considering the cases of $i_{cr}$ parametrized by the 1-D model (Eq. 23c) and the 3-D model (Eq. 25). Figure 10a–c show that the relative error associated to the mean concentration $RE_1$ takes low values across the whole domain. Conversely $RE_2$, $RE_3$ and $RE_4$ computed for 1-D $i_{cr}$ model are bounded close to the release point, but they increase significantly away from it.

Fig. 10

Relative error of the transversal profiles vs $x/\delta$ for 1-D model of $i_{cr}$: a ES 3, b ES 6, c LLS; and 3-D model of $i_{cr}$: d ES 3, e ES 6, f LLS

The relative errors evaluated for 3-D $i_{cr}$ model are reported in Fig. 10d–f, where $RE_1$ is not plotted, since it does not depend on the formulation of $i_{cr}$. $RE_i$ of the transversal profiles are bounded in all the cases, except for $RE_3$ in the LLS case (Fig. 10f). This is due to the particular shape of the experimental profile of $m_3^*$ at $x=5\delta$, that exhibits significant off-centreline peaks. As shown in Fig. 9f, the model reproduces this behaviour only qualitatively, but it fails in quantifying the centreline values of $m_3^*$.

In the light of this analysis, we can however conclude that, by adopting a suitable parametrization of $i_{cr}$, the model reproduces the statistics of the concentration field produced by a fluctuating plume with good accuracy.

5.2 Sensitivity Analysis

In order to discuss the reliability of the model for operational purposes, we analyze its sensitivity to several key parameters, whose evaluation is potentially affected by non-negligible errors. Our analysis focuses on two main features. Firstly, we discuss the uncertainties related to the reference value of the vertical coordinate at which are evaluated the velocity statistics used to compute the model parameters. Secondly, we analyze the errors induced by uncertainties in the parametrizations of $i_{cr}$.

In the results presented in Sect. 4, velocity statistics, i.e. $\overline{u}_m$, $\sigma _v$, $\sigma _w$ and $\varepsilon$, were evaluated at the plume centroid $z=\overline{z}_m$, which varies with the distance from the source. A simpler approach consists of estimating these same quantities at a fixed reference height, generally the source elevation $z_s$. As an example, we show in Table 5 the differences of these velocity statistics in the far field ($x/\delta = 3.75$) for the LLS emission, as computed at $z_s$ and $\overline{z}_m$ (in this case $\approx 2.5 z_s$). As shown in Table 5 the two parameters that exhibit higher variations are $\overline{u}_m$ and $\varepsilon$. Comparisons of the concentration statistics computed at $x/\delta = 3.75$ for the LLS case, and adopting these two different sets of input data, are plotted in Fig. 11a–c. Results clearly show that these variations in the input data affect significantly the model performances and produce differences in the standard deviation and the third- and fourth-order moments of the concentration exceeding 100 %.

Fig. 11

Sensitivity analysis for the LLS case. Influence of the plume centroid on the vertical profiles of the concentration statistics, $x/\delta = 3.75$, $y=0$: a $\sigma _c^*$, b $m_3^*$, c $m_4^*$; circles experiments; solid line model with $\overline{u}(\overline{z}_m)$, $\sigma _v(\overline{z}_m)$, $\sigma _w(\overline{z}_m)$, $\varepsilon (\overline{z}_m)$; black dash line model with $\overline{u}({z}_s)$, $\sigma _v({z}_s)$, $\sigma _w({z}_s)$, $\varepsilon ({z}_s)$. Influence of an uncertainty of $\pm 10~$% in the estimate of the coefficients of $i_{cr}$ (Table 2) on the longitudinal profiles of the concentration statistics at $y/\delta =0$, $z/\delta =z_s/\delta$: d $\sigma _c/\overline{c}$, e $m_3/\overline{c}$, f $m_4/\overline{c}$. Influence of an uncertainty of $\pm 10$ % in the estimate of the coefficient $\zeta _z$ (Table 3) on the vertical profiles of the concentration statistics at $y/\delta =0$, $x/\delta =0.625$: g $\sigma _c^*$, h $m_3^{*}$, i $m_4^{*}$. The uncertainty is evaluated as the ratio between the standard deviation and the mean value and it is represented by red dash lines

Table 5 Turbulent velocity field at $\overline{z}_m$ and $z_s$

As a second step, we investigate the sensitivity on the parametrization of $i_{cr}$ (Eq. 23c). Since the spatial distribution of the concentration fluctuations downwind the source is highly influenced by the emission conditions (Nironi et al. 2015), i.e. source size and elevation, the determination of a suitable longitudinal profile of $i_{cr}$ represents actually the main modelling challenge. To test the influence of uncertainties in the parametrization of $i_{cr}$ we performed a Monte-Carlo simulation, assuming that the coefficients in Eq. 23c are normally distributed, with averages values given by the reference values reported in Table 2 and a standard deviation corresponding to 10 % of the average. These turn out to be a maximum close to the source (${\approx }15~\%$) and a minimum far from it (${\approx }10~\%$). The variations of the second-, third- and fourth-order moments are shown in Fig. 11d–f and attain a maximal value ${\approx }20~\%$.

The same analysis was performed for the parameters $\zeta _y$ and $\zeta _z$ (see Eq. 25) characterizing the evolution of $i_{cr}$ in the transversal and vertical directions. These were assumed to be normally distributed, with a standard deviation equal to $10~\%$ of the mean values reported in Table 3. The resulting uncertainties in the vertical profiles of the high order concentration statistics at $x/\delta = 0.625$ for the LLS case are plotted in Fig. 11g–i. These show that variations of these parameters have little influence on $\sigma _c^*$ (Fig. 11g) and $m_3^{*}$ (Fig. 11h) whereas they can induce significant variations in the $m_4^{*}$ profiles (Fig. 11i), especially at the plume edges.

6 Discussion and Conclusion

We have investigated the reliability of a meandering plume model to simulate the dispersion of a passive scalar emitted within a neutral turbulent boundary layer. Following most authors who presented a similar model (Sawford and Stapountzis 1986; Yee et al. 1994; Luhar et al. 2000; Cassiani and Giostra 2002; Franzese 2003), we base its formulation on two main assumptions. The first is that the dispersion of the plume centre of mass and that of the tracer particles around it are statistically independent. The second is that both dispersion processes are given by motions in the vertical and transversal plane that are decoupled one from the other. The first assumption is actually verified only close to the source and in the far field. The second cannot be strictly justified on some physical basis. It is rather adopted since it allows simplification of the model formulation.

A fluctuating plume model requires the setting of several parameters. According to our formulations, these are $C_0$ and $C_r$, the Kolmogorov and Richardson–Obukhov constants, respectively, $\alpha _T$, required to evaluate the time scales $T_{my}$ and $T_{mz}$ characterizing the intensity of the meandering downwind the source, and the parameters needed to model the spatial evolution of the intensity of the relative concentration fluctuations $i_{cr}$ in the longitudinal direction and in the transversal planes ($\alpha _1$, $\alpha _2$ and $\alpha _3$). All these parameters were set by systematically comparing our model results to the experimental data presented in Nironi et al. (2015), in particular concerning the Eulerian integral length scales, the total plume spreads $\sigma _y$ and $\sigma _z$ and the spatial distribution of $i_c$. Furthermore, the information provided by this experimental dataset was used here to verify the consistency of the model formulation, as well as of the procedure adopted to tune its parameters, with the main features characterizing the physics of the dispersion process.

We have then tested the performances of the model by comparing its solutions with experimental profiles of the higher-order moments of the concentration PDF (Nironi et al. 2015). The comparison shows that, despite the theoretical weakness of some of its basic assumptions, once properly set the governing parameters, the meandering model is able to predict the concentration statistics with a suitable accuracy and to simulate the effects due to the source size and elevation. In the near field, a good accuracy of the results could be achieved assuming a constant $i_{cr}$ on the $yz$-planes. The good agreement between model simulations and experimental data persists even at larger distances from the source, where the assumption of the statistical independence of the meandering and the relative dispersion, as well as of the vertical and transversal motions, is far from being verified. Finally, in the far field, as the relative dispersion becomes the only relevant dispersion mechanism, the 1-D model results progressively deteriorate. To reliably reproduce profiles of the higher-order moments, the model requires a more complex formulation of $i_{cr}$, which takes account for its variability along the transverse and vertical directions.

In view of the application of the model for operational purposes, we have finally tested its sensitivity to the variations of several key parameters. The analysis shows that the model performances can be significantly affected by varying the reference values of the height from the ground at which the velocity statistics are estimated. Furthermore, the model is shown to exhibit also a strong sensitivity on the parametrization of $i_{cr}$, especially in the near field.

This sensitivity of the model to parametrization of $i_{cr}$ represents, in our opinion, the main limitation for its use for operational purposes, since the spatial distribution of $i_{cr}$ is strictly linked to that of $i_{c}$. As widely discussed in Nironi et al. (2015), the evolution of $i_{c}$ is highly influenced by the conditions at the emissions. These include the source elevation and diameter and the emission velocity, as well as other features characterizing the source design that can affect the flow dynamics in the wake of the source, where most of the production of the concentration variance takes place. All these aspects influencing the plume dynamics in its initial phase of growth can not realistically be fully characterized when applying the model to atmospheric emissions. As a consequence, estimates of the higher-order concentration statistics in this near-field region can be affected by significant uncertainties.

Given these uncertainties of the model results in real case studies and in the light of the findings presented in Nironi et al. (2015), a last remark can be made. As long as the Gamma distribution was shown to reliably model the concentration PDF (independently of the emission conditions), it is actually questionable if, for operational purposes, the concentration statistics deserve to be computed by a meandering plume model, that requires several input parameters, rather than with more simple semi-empirical models. These should be formulated in order to provide solely estimates of $\overline{c}$ and $i_{c}$, the two independent quantities needed to fix the form of the Gamma distribution, which can be subsequently used to compute the higher-order moments.