THE LARGE TIME BEHAVIOUR IN A MODEL FOR CONCENTRATION FLUCTUATIONS IN TURBULENT DISPERSION

Abstract—A model for concentration fluctuation moments has previously been developed for a scalar dispersing in a turbulent flow (see Mole, Clarke & Rietzler 1997). This model assumes that the mean concentration is known as a function of space, and of time t, and that higher moments of concentration for a dispersing cloud can be determined fully by a further two parameters α(t) and β(t) (see Chatwin & Sullivan 1990a). A closure is used which enables a coupled pair of first order differential equations for α and β to be written down. Here attention is restricted to cases when the mean concentration is self-similar, with a spatial scale L(t). (It is assumed that L−1dL/dt → 0 as t → ∞, so cloud growth must be slower than exponential.) It is shown that there is a constant αs, dependent on the spatial form of the mean concentration, such that α → αs as t→ ∞ when αs> 0, and α→ ∞ when αs < 0 (and β → 0 in all cases). In the former case the asymptotic analysis shows that α − αs ∝ (L−1dL/dt)1/2 and β ∝ (L−1dL/dt)1/2. In the latter case it shows that α ∝ (L−1dL/dt)−1 and β ∝ L−1dL/dt. These results are supported by the numerical solutions for a variety of cases. Some of the corresponding results for the concentration moments are compared with experimental measurements for line sources in wind tunnels. Key word index: Turbulent diffusion, modelling concentration moments, line sources