Limitations of Second Order Modeling of Passive Scalar Diffusion

Abstract

The problem of modeling scalar variance from an elevated source is discussed at length. From a simple model, it is shown that observed behavior is entirely a result of the growth of the instantaneous plume width relative to the mean plume width. From a model valid for large sources, we suggest a simple explanation for the apparent dependence of asymptotic levels of variance on source size. Pope (1983) has pointed out that second order models of passive scalar transport do not retain the superposability of the primitive equations. Explanations for this, and possible consequences are explored. It is shown how second order models for the rapid terms, the time scales of which are determined by the mean velocity (rather than inertial spectral transfer), can be constructed in a superposable manner (Shih, 1984). Difficulties are described in simultaneously satisfying requirements imposed on the cross-dissipation by realizability and by superposability. We present computations of the data of Warhaft (1984) using a model in which the transport is based on first principles, and much of the rest of the model satisfies realizability. The model is able to reproduce the data of Warhaft satisfactorily if the initial plume width is larger than the Kolmogorov microscale. Calculations are carried out of a plume superposed on a background; satisfactory results are achieved when the background level is low, despite the lack of formal superposability, leading to the conclusion that this may not be a serious problem.