Solution for the statistical moments of scalar turbulence

We derive a system of equations for the statistical moments of a passive scalar dispersed in a turbulent flow from the transport equation of the probability density function. We solve the system through a Green's function and we obtain a formally exact solution for the statistical moments of the passive scalar concentration. We use this solution to achieve an analytical relationship for the second moment of a passive scalar released from a point source. Comparison with wind-tunnel experiments shows that the relationship is valid also in a neutral turbulent boundary layer if the reflection onto the ground and an appropriate model for the mixing timescale are considered. This approach, combined with a suitable model for the distribution of the concentration, allows the statistics of the passive scalar to be obtained in the whole domain in a closed and ready-to-use form.

Figure 1

Example of natural and anthropogenic scalar turbulence in the atmosphere. Smoke plumes from an eruption of the Etna Volcano in Italy (a) (Credit: Sentinel-3, Copernicus ESA), and an industrial chimney (b) (Credit: Walter Baxter https://www.geograph.org.uk/photo/3220320).

Figure 2

(a) Sketch of the wind-tunnel setup showing the vortex generators and the turbulence grid at the entrance of the test section, the roughness elements on the floor, and the design of the lower level source (LLS) and the elevated source (ES) [38]. (b–d) Experimental velocity statistics within the turbulent boundary layer. Note that the $z$ axis started at the ground in Nironi et al. [38], while here starts at the source height $h_s$.

Figure 3

Mixing timescale (a) and intensity of the concentration fluctuations (b, c) along the plume axis for different source conditions. Solid (${\tau }_m^{\left(c\right)}$) and dash-dot (${\tau }_m^{\left(x\right)}$) lines refer to the present theory. Symbols are the experimental data of Nironi et al. [38] (squares), the new experiments (diamonds) and Fackrell and Robins [42] (circles). Numbers in LLS and ES acronyms refer to the source diameter in mm.

Figure 4

Vertical and transverse profiles of mean (red) and standard deviation (blue) of the concentration at increasing distance downstream of the source. Solid (${\tau }_m^{\left(c\right)}$) and dash-dot (${\tau }_m^{\left(x\right)}$) lines refer to the theory, while symbols correspond to experimental data LLS3 [38]. For continuity with Nironi et al. [38], lengths and concentration are scaled with $\delta$ and $\dot{M}{U}_{\infty }^{-1}{\delta }^{-2}$, respectively (where $U_{\infty }$ the free-stream velocity).

Figure 5

One-point one-time PDFs on the plume axis at different distances from the source. Red dotted lines are the experimental PDFs. Dashed blue lines are the $\mathrm{\Gamma }$ distributions –Eq. (14), with experimental mean and variance. Solid black lines are the $\mathrm{\Gamma }$ distributions with mean and variance from Eqs. (7) and (8).

Figure 6

Mean plume spread for the LLS3. The points are the experimental values, the solid lines are the Gaussian model Eq. (7). To account for the inhomogeneity of the velocity field close to the lower boundary, the velocity U is estimated at the height of the plume center of mass [38].

Figure 7

Mean plume spread for the elevated sources ES6. Notice that, as the mean is affected by the source diameter just in proximity of the source, there is no need to discriminate between ES3 and ES6 [38].

Figure 8

Standard deviation for the LLS3. The points are the experimental values; the solid and dash-dot lines come from Eq. (8), with ${\tau }_m^{\left(c\right)}$ and ${\tau }_m^{\left(x\right)}$, respectively.

Figure 9

Standard deviation for the elevated source ES6.

Figure 10

Standard deviation for the elevated source ES3.