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

TL;DR

This work addresses the challenge of predicting instantaneous scalar concentrations in elevated point-source plumes within turbulent boundary layers, going beyond mean and RMS statistics by validating a two-parameter gamma distribution $\mathcal{P}(\tilde{C})=\frac{1}{\Gamma(k)\theta^{k}}\tilde{C}^{k-1} e^{-\tilde{C}/\theta}$ as a unified model for all concentration statistics. Using high-frequency, long-duration experimental data, the authors show that $k$ and $\theta$ can be inferred from the mean $C$ and RMS $\sigma_c$ (with $k_0=(C_0/\sigma_{c,0})^2$ and $\theta_0=\sigma_{c,0}^2/C_0$), and that $k(z)/k_0=\exp[-(\ln 2)\xi_C^2]$ while $\theta(z) \approx \theta_0$, yielding Gaussian-like similarity in higher-order statistics. The framework accurately predicts skewness, kurtosis, and the 99th percentile $\tilde{C}_{99}/\sigma_c$ across the plume, and extends to higher-order moments up to eighth order, validating analytic expressions for $\overline{c^n}$ and standardised moments. Practically, this gamma-based approach provides a robust, parametric basis for estimating concentration tails and risk-relevant thresholds, and offers a rigorous benchmark for numerical simulations of scalar transport in plumes, with applicability to buoyant plumes as well.

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.

Figures (10)

• Figure 1: A schematic of the experimental setup used to study the dispersion of neutral and buoyant scalar plumes in a turbulent boundary layer.
• Figure 2: Normalised profiles of (a) mean concentration and (b) root-mean-square(RMS) of concentration fluctuations. The black solid lines in (a) and (b) are the Gaussian model described by equations \ref{['eq.CmeanGauss']} and \ref{['eq.CrmsGauss']}.
• Figure 3: Comparison of distributions for the concentration PDF on log -- linear axes and linear -- linear axes for $S_z/\delta= 0.32$, $S_x/\delta=2$. (a,d), (b,e), and (c,f) correspond to measurements at $\xi_C \approx -1$, $\xi_C \approx 0$, and $\xi_C \approx 1$, respectively. Symbols as per table \ref{['tb.config']}. Solid black line (—): gamma distribution, solid red line (—): log-normal distribution, dashed blue line (- -): normal distribution, dashed yellow line (- - -): Weibull distribution, dashed-dotted magenta (- $\cdot$ -): beta distribution, and dotted green line (…): generalised Pareto distribution (GPD). (a-c) are on log-linear axes, and (d-f) are on linear-linear axes. The vertical dotted, dashed-dotted, and dashed lines indicate the 95th, 99th, and 99.9th percentile of the PDF.
• Figure 4: Pre-multiplied probability density functions of the instantaneous concentration, $\tilde{C}$. (a-b) $S_z/\delta$ = 0.32, $S_x/\delta$ = 2, $\rho_s/\rho_\infty\approx1$. (c-d) $S_z/\delta$ = 0.32, $S_x/\delta$ = 1, $\rho_s/\rho_\infty\approx 0.17$. The solid black line represents the PDF on the plume centreline, the red lines are PDFs above the centreline, and the blue lines are the PDFs below the centreline. The dashed black lines are the best-fit gamma distributions.
• Figure 5: (a) Fitted shape parameter for the measurements at the centreline ($k_0$) vs. the statistical prediction of equation \ref{['eq.GammaTheta']}. (b) Variation of fitted shape parameter $k(z)$ relative to the magnitude at centreline versus normalised distance from the plume centreline $\xi_C$. (c) Scale parameter evaluated at the centreline ($\theta_0$) vs. the statistical prediction of equation \ref{['eq.GammaK']}. (d) Shape parameter, $\theta$, of the gamma distribution vs. normalised distance from the plume centreline $\xi_C$. Symbols as defined in table \ref{['tb.config']}.