High concentrations of a passive scalar in turbulent dispersion

Abstract

In problems involving the dispersion of hazardous gases in the atmosphere, the distribution of high concentrations is often of particular interest. We address the modelling of the distribution of high concentrations of a dispersing passive scalar at large Péclet number, concentrating on the case of steady releases. We argue, from the physical character of the small-scale processes, and from the statistical theory of extreme values, that the high concentrations can be fitted well by a Generalized Pareto Distribution (GPD). This is supported by evidence from a range of experiments. We show, furthermore, that if this is the case then the ratios of successive high-order absolute moments of the scalar concentration are linearly related to the reciprocal of the order. The linear fit thus obtained allows the GPD parameters to be determined from the moments. In this way the moments can be used to deduce the properties of the high concentrations, in particular the maximum possible concentration θmax = θmax(x). We argue, on general physical grounds, that θmax/C0 (where C0 = C0(X) is the centreline mean concentration, and X is the downstream distance from the source) decreases to zero very far from the centreline, but that the decrease takes place on a length scale much larger than the mean plume width (because it is controlled by the relatively slowly acting molecular diffusion, rather than the fast turbulent advection). Thus, over the distances for which accurate measurements can be made, we expect θmax/C0 to be approximately constant throughout the plume cross-section. On the centreline, we argue that θmax/C0 increases downstream from the source, reaches a maximum and then decreases, ultimately tending to 1 far downstream. In support of these deductions we present results for some high-quality data for a steady line source in wind tunnel grid turbulence. Finally, we apply to this problem some existing models for the relationships between moments. By considering the behaviour far from the centreline in these models, and linking the moments to the high concentrations, we derive relationships between the model parameters. This allows us to derive an expression for θmax/C0 which depends on a total of 5 parameters, and (weakly) on C/C0 (where C = C(x) is the local mean concentration). Comparison with the data is encouraging. We also discuss possible methods for modelling the spatial variation of these 5 parameters.

Concentration Fluctuations in Smoke Plumes Released Near the Ground

“…The simple relationship (7) resulting from the single concentration level model has been generalized (Mole and Clarke 1995;Schopflocher and Sullivan 2005;Mole et al 2008) to…”

“…For the idealized "two-level" model for the release, where the concentration is either zero or C m , the effects of molecular diffusivity are particularly important (Zimmerman and Chatwin 1995;Mole et al 2008). The skewness (4) attains negative values in the case λ 2 − λ 2 2 − 8λ 3 λ 1 /9 < (4/3) λ 2 1 α < λ 2 + λ 2 2 − 8λ 3 λ 1 /9, with λ 2 2 ≥ 8λ 3 λ 1 /9, while also (a) (b) Fig.…”

Large-Eddy Simulation of Dispersion from Line Sources in a Turbulent Channel Flow

“…Consequently, both c rms−max /c max and measures of the relative extreme concentrations (i.e. maximum possible concentration normalised by the mean, Xie et al, 2007;Mole et al, 2008) are expected to be lower for line than for point sources.…”

Experimental Measurements of Expected Mass Fraction in a Contaminant Plume

“…3. Mole et al (2008) provide a discussion of the widely observed representation of a substantial range of the highest concentration values by a generalized Pareto density function. They observed that there was little variation of the parameters, including the maximum concentration, over a plume cross-section.…”