The probability structure associated with a simple model of turbulent dispersion

Abstract

The paper derives the probability density function of concentration for a simple model of the turbulent diffusion process presented by Zimmerman and Chatwin (1995). Relationships with other work are discovered, and some implications for future research are assessed. © 1998 John Wiley & Sons, Ltd.

An idealised model of turbulent dispersion: two and three spatial dimensions

“…Chatwin and Zimmerman (1998) showed that in the one‐dimensional model the large‐time pdf has unbounded peaks at the smallest and largest concentrations. Here I have shown that in two and three dimensions this is not always the case.…”

“…In two dimensions the archetypal examples give either two unbounded peaks at intermediate concentrations (Example 1 with $k>1$ ) or a single unbounded peak at the mean concentration (Example 1 with $k=1$ , or Example 2). These are plotted, together with the one‐dimensional case of Chatwin and Zimmerman (1998), in Figure 10. They are all symmetric about the mean concentration, which corresponds to a normalised concentration of zero.…”

“…The normalisation ensures that in all cases the concentration range is [−1, 1] and the area under the curve is 1. The one‐dimensional case of Chatwin and Zimmerman (1998) (solid curve), the two‐dimensional cases of Example 1 (dashed) and Example 2 (long‐dashed) and the three‐dimensional cases of Example 3 (dotted) and Example 4 (dash‐dotted) are shown…”

Some remarks on modelling the PDF of the concentration of a dispersing scalar in turbulence

“…It follows that for similarity solutions there must be constants α and β such thaṫ 5) where the minus signs are chosen because, then, positive values of α and β allow Θ 1 and Θ 2 to tend to zero as T →∞to attain the limiting delta-function distribution in equation (3.4). Then equation (6.4) becomes…”

“…Since p C is a PDF, it follows that 4) and the (ensemble) mean concentration E {C}, where E {·} denotes the expected value, is defined by 5) with analogous equations for the variance and higher moments -see Mole et al [18]. The PDF p C (q; x,t) obeys an evolution equation determined from equation (2.1).…”