Moment approximations for probability density functions

et al. 2010

The sudden release of a quantity of gas into the atmospheric boundary layer produces a contaminant cloud. The expected mass fraction function provides a relatively simple measure of the contaminant concentration values found within the cloud and represents the ensemble-averaged fraction of the conserved release mass found at the different contaminant concentration intervals as the cloud evolves. The plume generated by a line source in grid turbulence is used to investigate the expected mass fraction function as it applies to scalar concentration values found on a typical line normal to the plume axis. Simultaneous particle image velocimetry and planar laser induced fluorescence are used to measure velocity and concentration fields, respectively. The measured expected mass fraction functions are observed to be approximately self-similar when concentration values are normalized by the centreline mean concentration. The moments of the expected mass fraction function are observed to be simply related to the centreline moments of the probability density function of scalar concentration. Arguments based on a source fluid, non-source fluid decomposition of the scalar probability density function are used to explain these observations. The results are compared with the theoretical and experimental results established for a line source of scalar in grid turbulence.

Section: Discussionmentioning

confidence: 99%

Section: Experimental Set-upmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Experimental Measurements of Expected Mass Fraction in a Contaminant Plume

Sarathi

Gurka

et al. 2010

“…It is expected that a good approximation of the PDF and EMF can be found by using a limited number of the lower ordered moments (Derksen and Sullivan 1990, Lewis and Chatwin 1996, Schopflocher and Sullivan 2002. The procedure here will be to select a parametric expression for the EMF and determine the value of the parameters from these moments.…”

Section: Introductionmentioning

confidence: 98%

Scalar concentration reduction in a contaminant cloud

Schopflocher

2006

The expected mass fraction (EMF) is used to quantify concentration values in a contaminant cloud. Low-ordered moments of the probability density function (PDF) of concentration are integrated and inverted to approximate the EMF. A simple prescription for the distributed moments of the PDF is shown to be related to a set of relationships between normalized moments. These relationships appear to be very robust and approximately universal across measured values from laboratory clouds and plumes in the atmospheric boundary layer over different stability classes. Also, these relationships have the important advantage that they can be validated from isolated fixed-point measurements. In this paper, we verify our results with data from a plume in grid turbulence. The EMF for these experiments is shown to be well approximated by a simple Beta function.

“…Even in the absence of a parametric model, and with an incomplete set of measured moments, one can still obtain an accurate representation of the density (using an orthogonal polynomial expansion, for example). In practice, the first three or four sample moments are enough to capture the bulk characteristics of the PDF (Derksen and Sullivan, 1990). If one wishes to retrieve the more subtle features of the probability distribution, higher-order moments are required.…”

Section: Introductionmentioning

confidence: 99%

The Relationship between Skewness and Kurtosis of A Diffusing Scalar

Schopflocher

2005

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.