On the Spectral Theory of Turbulence and Atmospheric Dispersion
A new theoretical model has been developed to predict the transport of atmospheric pollutants as they disperse in the environment. The proposed model is the first successful effort to theoretically predict the dispersion coefficient under all six Pasquill stability classes. This new approach is based on the spectral theory of turbulence and Higbie's penetration theory. It is assumed that the fundamental mechanism of mass transfer under the influence of turbulence is similar to the process of molecular diffusion: i.e., the turbulent dispersion coefficient is dependent on the molecular properties. It is also assumed that the rate of diffusion is related to the rate of energy dissipation in the flow field. During this investigation it was realized that there are at least two independent time scales that require consideration in turbulent diffusion modeling. The smaller of the two, the dissipative time scale, is related to the rate of energy dissipation and hence the state of turbulence. On the other hand, a much larger dispersive time scale determined by the size of the largest eddy also plays a critical role in the phenomenon of dispersion. The Penetration Theory is used to obtain an expression for the mass transfer coefficient that is used in conjunction with a dispersive time scale to obtain a final expression for the dispersion coefficient. The dispersive time scale used in this development is based on Gifford's time-dependent velocity autocorrelation function. Therefore, the final expression for the dispersion coefficient contains the two time scales and the molecular diffusivity. In support of Gifford's conclusion, this research has also found that the dispersive time scale is of the same order of magnitude as the Coriolis parameter. The dissipative time scale is the only parameter in the model that requires adjustment to account for the change in atmospheric stability. The new model demonstrates the variability in the diffusion characteristics of various gases and hence points to a possible solution to the long-standing problem of heavy gas dispersion. The new model provides consistent agreement for the horizontal dispersion under all six Pasquill stability classes with the well known Pasquill-Gifford empirical correlation. The new model was also tested against the empirical correlation proposed by Hage et. al., and it was observed to be faithfully follow the correlation for the entire range. Subsequently, the model was extended to predict vertical dispersion coefficients by assuming that the only additional factor governing dispersion in the vertical direction was the net effect of buoyancy and gravity. Buoyancy is assumed to either increase (for an unstable condition) or decrease (for a stable condition) the dissipative time scale. The new vertical dispersion model works well for all six stability classes. The numerical values of the vertical dissipative time scale for each stability class were found to be very close to the respective time scale in the horizontal direction. The extension of the model for vertical dispersion further supports its validity. Review of the pertinent literature on boundary layer effect indicated that at near ground levels the time scales exhibit a very sharp profile. This observation explains the reason for non-Gaussian behavior of ground-level emissions.
S. Usman, "On the Spectral Theory of Turbulence and Atmospheric Dispersion," University of Cincinnati, Jan 1997.
Mining and Nuclear Engineering
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