An idealised model of turbulent dispersion: one rectangular pulse initial condition

Abstract

SUMMARYWe examine an idealised model of turbulent dispersion which was introduced by Zimmerman and Chatwin. It involves deterministic diffusion in a periodic one-dimensional domain, for a specified initial concentration field. A stochastic element is introduced by sampling at random across the domain, equivalent to random advection by a velocity which does not vary spatially. We consider initial conditions consisting of a single rectangular pulse, displaced a distance X 0 from the centre of the domain. We present numerical results for the dependence of the variance, skewness, kurtosis and probability density function (pdf) of concentration on X 0 and on time, and we derive the large time asymptotic forms for these quantities. At large time, the pdf is inevitably bimodal, with the peaks at the smallest and largest concentrations. To allow for a range of possible distances between strands of high concentration, as would be expected in a real turbulent flow, we suggest a new model pdf for concentration, obtained by averaging over different values of X 0 . For a uniform distribution of X 0 values, we show that this gives a unimodal concentration pdf at large time, with the peak at the mean concentration.

An idealised model of turbulent dispersion: two rectangular pulse initial condition

“…The detailed behaviour depends in a complex way on the pulse locations, but as a pulse tends towards one of the boundaries there is a tendency for the timescale of the evolution of the moments to increase by a factor of 4. This is the same behaviour that was found by Mole and Yeun (2007) for the one pulse case, and is essentially the consequence of pulses effectively merging together at the boundary. The consequent doubling of the spatial scale leads to a quadrupling of the time scale, because of the form of the diffusion Equation (1).…”

“…Since it is the strands of highest concentration which are being considered, a first approximation to p θ 0 (θ) would be the generalised Pareto distribution (see for example Mole et al, 1995Mole et al, , 2008. As shown by Mole and Yeun (2007) for the case of a distribution of inter-strand distances, the large time pdf will in general no longer be bimodal, and a range of shapes are possible. The effect of variations in the strand width from one strand to another could be investigated in the two pulse model, but it would not be possible to include this in the one pulse model.…”

“…Incorporating a distribution of strand separations and concentrations using the approach of Mole and Yeun (2007) is not straightforward for the two pulse case. This is because each individual case Copyright contains two pulse locations and two concentrations, and they cannot be related directly to the overall distributions of separations and concentrations.…”

“…Mole and Yeun (2007) examined the effect of variations in spacing by considering an initial pulse located off-centre in the finite domain used in the model. For a given pulse location this only gave two spacings, but by averaging over the cases for different pulse locations a distribution of spacings could be incorporated.…”

“…One of the main aims of the present study is to gain more of this understanding, in particular of the impact of some features which are present in real dispersion. Mole and Yeun (2007), henceforth to be referred to as MY, examined the model in some detail, and extended it to the case when the initial condition is an off-centre rectangular pulse. In this case new behaviour results because of the interaction with the closest boundary.…”