The probability structure associated with a simple model of turbulent dispersion

Chatwin, P. C.; Zimmerman, William B.

doi:10.1002/(sici)1099-095x(199803/04)9:2<131::aid-env290>3.0.co;2-u

Cited by 7 publications

(9 citation statements)

References 9 publications

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“…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…”

Section: A Specific Closure Hypothesis For the Ssmtmentioning

confidence: 90%

“…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).…”

Section: The Evolution Equation For the Pdfmentioning

confidence: 99%

“…However, a particular closure hypothesis consistent with (4.7) arises from a PDF obtained in earlier work by the author with Zimmerman (Zimmerman & Chatwin [35]; Chatwin & Zimmerman [5]), and it is natural to consider this further. In the first of these papers, turbulent dispersion was modelled simplistically by supposing that the scalar disperses deterministically with no velocity field and the sole stochastic feature is that the sensor measuring C is located randomly within a domain of finite volume.…”

Section: A Specific Closure Hypothesis For the Ssmtmentioning

confidence: 99%

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

Chatwin

2002

Eur. J. Appl. Math

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The paper deals with the probability density function (PDF) of the concentration of a scalar within a turbulent flow. Following some comments about the overall structure of the PDF, and its approach to a limit at large times, attention focusses on the so-called small scale mixing term in the evolution equation for the PDF. This represents the effect of molecular diffusion in reducing concentration fluctuations, eventually to zero. Arguments are presented which suggest that this quantity could, in certain circumstances, depend inversely upon the PDF, and a particular example of this leads to a new closure hypothesis. Consequences of this, especially similarity solutions, are explored for the case when the concentration field is statistically homogeneous.

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Section: A Specific Closure Hypothesis For the Ssmtmentioning

confidence: 90%

Section: The Evolution Equation For the Pdfmentioning

confidence: 99%

Section: A Specific Closure Hypothesis For the Ssmtmentioning

confidence: 99%

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

Chatwin

2002

Eur. J. Appl. Math

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“…One question which has been the subject of a number of studies is: given an initial condition where the concentration is either zero or a constant $\theta_0$ (so the pdf consists of two $\delta$ ‐functions), how does the shape of the pdf evolve with time? Chatwin (2002) summarised the results of some of these studies and used the large‐time form of the pdf obtained by Chatwin and Zimmerman (1998) for the model of Zimmerman and Chatwin (1995) to motivate a new approach to the study of this problem. However, the result of Chatwin and Zimmerman (1998) was based on a one‐dimensional version of the model.…”

Section: Introductionmentioning

confidence: 99%

“…Chatwin (2002) summarised the results of some of these studies and used the large‐time form of the pdf obtained by Chatwin and Zimmerman (1998) for the model of Zimmerman and Chatwin (1995) to motivate a new approach to the study of this problem. However, the result of Chatwin and Zimmerman (1998) was based on a one‐dimensional version of the model. Here I examine how the number of spatial dimensions affects the result, by considering in detail some large‐time solutions in two and three spatial dimensions.…”

Section: Introductionmentioning

confidence: 99%

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

Mole

2010

Environmetrics

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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...

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A study of pollutant concentration variability in an urban street under low wind speeds

Martin¹,

Price²,

White³

et al. 2008

Atmospheric Science Letters

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The short time-scale variability in pollutant concentrations in an urban street under very low wind speed conditions and short source-receptor distance has been investigated using the inert tracer sulfur hexafluoride (SF 6 ) as a continuous point-source (release times ≥5 min), and fast detection using separation by gas chromatography coupled with a µ-electron capture detector (ECD). The results are complex but can be broadly interpreted in terms of horizontal wind speed and direction coherence. Comparisons with a simple dispersion model suggest that observed time-averaged maximum concentrations approach predicted values, whilst instantaneous maximum concentrations vary greatly and would therefore be difficult to predict.

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The probability structure associated with a simple model of turbulent dispersion

Cited by 7 publications

References 9 publications

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

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

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

A study of pollutant concentration variability in an urban street under low wind speeds

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