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

Abstract

In the idealised model of turbulent dispersion introduced by Zimmerman and Chatwin (1995), the probability density function (pdf) of concentration becomes bimodal at large times, with peaks at the smallest and largest concentrations. That model only has one spatial dimension, but here I extend the model to two and three spatial dimensions. I also extend to two and three dimensions the pdf calculation method that was given by Mole and Yeun (2007). I use this method to derive large-time analytical solutions for the pdf, and to show that in two and three dimensions the weight of the pdf shifts from extreme concentrations towards the mean concentration. In the typical three-dimensional case, the large-time pdf is unimodal with the peak at the mean concentration.The model that Zimmerman and Chatwin (1995) introduced was an idealised model of turbulent dispersion. Their intention was not to produce a realistic practical model of turbulent dispersion, but rather to examine a 'toy' model which retained only some of the features of real turbulent dispersion. Their hope was that some of the results of this model would be robust, and would give insight into the possible behaviour in real turbulent dispersion. Even if this hope is not justified, the model does allow the effect of individual features of real dispersion to be investigated.Their model involved deterministic molecular diffusion in a finite one-dimensional domain, with no-flux conditions at the boundaries. This is equivalent to diffusion in an unbounded periodic domain. Randomness was introduced by assuming that measurements of concentration were made at random positions distributed uniformly within the domain. This is equivalent to advection past the measurement position by a velocity field which is spatially uniform, but varies randomly from one realisation to another. Thus, the action of a real turbulent velocity field in stretching and twisting a cloud or plume is not captured in the model. Zimmerman and Chatwin (1995) took the initial condition to be a rectangular pulse of non-zero concentration centred at the midpoint of the domain. In the equivalent unbounded domain one can view this as a model of the diffusion of regularly spaced strands of non-zero concentration. Chatwin (2002) argued that the physical dimension of the model box was of the order of the Taylor microscale (see, for example, pp. 65-66 of Tennekes and Lumley (1972)). So the interpretation would be that the model is inappropriate for the development close to the source. Any useful deductions from it should rather be applied to positions sufficiently far from the source for the cloud or plume to have been stretched out into many small-scale strands.Furthermore, any universal characteristics to be deduced from the model would probably come from its large-time developments, once non-zero concentrations have been built-up outside the strands through molecular diffusion, and once transient sensitivity to the fine details of the initial field has gone. The relevant timescale for this development is...

Concentration Statistics for Conservative Solute Transport in River Estuaries

“…Nevertheless, all developed methodologies have their limitations which have already been covered in the discussion section of Chapter 5. The near-field approximation or the zero molecular diffusion approximation becomes less applicable after relevant distance from the source (as covered by Section 5.4.4), but one might argue that most of the area of interest does fall in the near-field zone defined in this manner [105]. The application of developed analytical approach requires a river dominated and stratified estuaries since otherwise certain sea processes (e.g.…”

“…In most environmental flows, the Peclet number is very large, indicating that turbulent advection acts on a much shorter timescale than molecular diffusion. The ratio of these timescales can be estimated by P e −1 if the length scales for the velocity and concentration fields are comparable [105]. This approach was previously implemented in the development of concentration moments in contaminant clouds such as in the atmosphere [140] or in a groundwater medium [4,5].…”

“…with values between 0 and C 0 . For a hypothetical case of no molecular diffusion (e m = 0), as described by [25,105], the contaminant is pulled into ever-thinning sheets and strands by advection, but they always remain at the initial source concentration (C 0 ). That behavior is represented by a two-state distribution (e.g.…”

“…(3.7), it is expected to use the same methodology for obtaining one by inversion of its lower moments. In Section 2.3.1, Beta distribution is represented as a convenient and rewarding way of obtaining the PDF from its lower moments [105][13] [133], [124][140]. In Fig.…”

“…That surface predominately represents intermittency, since intermittency is the probability that a fluid at certain (x, t) is coming from the source. This definition of intermittency was introduced by [28], and it conveys valuable information about the effect of the velocity field [105]. The behavior of intermittency, as the probability of a fluid element coming from the source, can be seen in all three Figs.…”