Higher order moments of scalar within a plume in a turbulent boundary layer

Abstract

This study examines the statistical nature of instantaneous scalar concentration in an elevated point-source plume (neutral or buoyant) dispersing within a turbulent boundary layer. Using high-frequency long-duration experimental measurements, we extensively validate the gamma distribution as the appropriate probability density function of concentration, particularly at large scalar magnitudes. The two-parameter gamma distribution is shown to capture the PDF at all locations across the plume. The classical similarity of the mean and root-mean-square (RMS) concentration, often expressed through a Gaussian form, is recovered through similarity of the scale and shape parameters of the gamma distribution. In addition, statistics of extreme events, such as the 99th percentile of the instantaneous concentration signal, are also well predicted, and their observed invariance near the plume centreline is reasoned. Further, similarity is observed for the third- and higher-order central moments and standardised central moments from the experimental data. The framework of the gamma distribution is also analytically extended to higher-order statistics. The experimental data are in good agreement with the predicted central moments up to the eighth order. The results emphasise the importance of achieving statistical convergence for the intermittent concentration signal, directly influenced by finite sampling times in a measurement. A secondary result is obtained for the ratio of plume half-widths based on the mean and the RMS concentration to be $1/\sqrt{2}$, consistent with experimental observations. The results establish the gamma distribution as a consistent and unified model for all scalar concentration statistics in elevated point source plumes within a turbulent boundary layer.