Singular PDFs of a dispersing scalar in turbulence

When a finite quantity of scalar of uniform concentration C-m is injected at time t = 0 into a turbulent flow containing no scalar, the initial PDF (probability density function) of the concentration of the scalar is a weighted sum of two delta functions. Because of molecular mixing, p(C) must eventually become a single delta function. In an earlier paper (Chatwin, Eur. J. Appl. Math. 13 (2002) 95-108). it was suggested that the evolution from the initial state could take place with pC having modes at its end-points that merged into one another as time increased. This is different from what has normally been supposed. The present paper extends an examination of two families of PDFs that first appeared in Kowe and Chatwin (J. Eng. Math. 19 (1985) 217-231); one of these was obtained again in two separate investigations (Chatwin, Eur. J. Appl. Math. 13 (2002) 95-108; Chatwin and Zimmerman, Environmetrics 9 (1998) 131-138). Here the two families are extended, but in a preliminary and idealized way, by allowing an isotropic spatial dependence and by including the diffusion and advection terms, present in the evolution equation for pC, but assumed to vanish identically in Chatwin ( Eur. J. Appl Math. 13 (2002) 95-108). It is shown that there is consistency, thereby supporting a more detailed investigation using DNS.