Generalization of K Theory for Turbulent Diffusion. Part II: Spectral Diffusivity Model for Plume Dispersion

Abstract

Further development of the spectral turbulent diffusivity concept is presented with the aim of obtaining an Eulerian dispersion model applicable for multiple interacting sources. The theory is applied for studies of plume dispersion in a field of a homogeneous and stationary turbulence. A continuous plume is considered as consisting of an infinite number of expanding puffs. The puffs' center of mass fluctuates following the long-wave range of the turbulent velocity fluctuation spectrum. The center-of-mass fluctuations are assigned to phases of the Fourier coefficients of the concentration distribution. The standard deviation of the velocity of the phase fluctuations is dependent on the wave vector of the Fourier coefficient. Time-averaging results in a spectral phase diffusivity coefficient. It is shown that the rate of growth and the center-line concentration obtained by the spectral diffusivity model are in agreement with results predicted by the Lagrangian statistical theory. For a narrow plume, it is shown that the plume width is proportional to the time of travel, while for a narrow puff, the 3/2-power dependence is found. For a narrow distribution, the concentration shape deviates, however, from a Gaussian shape, in contradiction to results of the statistical theory. It is shown that only two external parameters are required in the spectral turbulent diffusivity model. These are the long-wave range diffusivity coefficient K0 and the wave vector km of the most energetic turbulent eddies. An Eulerian integro-differential transport equation is the final result of the model. This equation can also be used for dispersion in case of space- and time-dependent parameters. We suggest a procedure for a direct experimental test of the spectral turbulent diffusivity concept.