The Relationship between Skewness and Kurtosis of A Diffusing Scalar

Abstract

It has been demonstrated that in turbulent dispersion, there exists a quadratic relationship between the skewness (S) and kurtosis (K) statistics obtained from continuous, elevated sources of scalar contaminant released into both convective and stable atmospheric boundary layers. Specifically, one observes thatwhere A and B are empirically fitted constants that depend on the flow. For two reasons, this is potentially useful information in regard to modelling the probability density function (PDF) of a diffusing scalar. First, since many PDFs have a signature relationship between their skewness and kurtosis, candidate models can immediately be either accepted or rejected depending upon whether they conform to the quadratic curve that is observed experimentally. Second, if one intends to model the PDF by inverting a limited number of moments, the task is reduced when there is a functional relationship between the standardized third and fourth moments. The aforementioned relationship has been corroborated by others who have examined data over a wide range of experimental configurations. However, from one flow to another, there appears to be a non-negligible variability in the two fitting constants of the quadratic curve. In this paper we put forth a framework to help explain this phenomenon, and we also attempt to predict how these parameters vary in space and/or time. Our point is illustrated with well-resolved data from a wind-tunnel, grid-turbulence, plume experiment.

Dispersion of a Passive Scalar Fluctuating Plume in a Turbulent Boundary Layer. Part I: Velocity and Concentration Measurements

“…Their analysis was further extended by Mole and Clarke (1995), who demonstrated the existence of simple functional dependencies between second-order, third-order and fourth-order moments of the concentration PDF. Similar analyses were also conducted by Lewis et al (1997) using field data in varying stability conditions, and, more recently, by Schopflocher and Sullivan (2005) using wind-tunnel experimental data.…”

Scalar concentration reduction in a contaminant cloud

“…In Schopflocher and Sullivan (2005) it is established that the result given in (13) derives from the moment prescription given in (10); that is, the observation of (13) validates (10). In that paper, the well-controlled measurements of a plume diffusing in grid turbulence were examined, and slight variations in the values of a and b were observed and found to be consistent with the prescription given in (10).…”

“…The values of α, β, λ 3E , and λ 4E are shown in Table 1. A thorough analysis of the quadratic function relating skewness and kurtosis given in (19) to the moment expressions given in (14) is provided in Schopflocher and Sullivan (2005). Representative fits from the data are shown in Fig.…”

“…The variation of parameter values over these replications was small. In Schopflocher and Sullivan (2005), data for distances less than 10 mm downstream from the heated wire were excluded because of resolution problems (Sullivan 2004); also, data from positions greater than 2σ from the plume centerline were excluded for reasons of poor signal-to-noise ratio (Schopflocher and Sullivan 2002). The same practice will be followed herein.…”

Moment Analysis of Highway-Traffic Clearance Distribution

“…Similar techniques have been developed to address those effects in the moments of distributions of observables in various branches of physics, astronomy, biology and economy (see, e.g. [21], [22], [23], [24], [25], [26], [27], [28], [29], [30]). The paper is organized as follows.…”