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

Chatwin, P. C.

doi:10.1017/s0956792501004727

Cited by 11 publications

(12 citation statements)

References 29 publications

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“…The pdf was found to be bimodal at all times, with the peaks at the smallest and largest concentrations, and with these peaks moving towards each other and merging as T → ∞. (See Chatwin (2002) for discussion of the large time behaviour in a general context.) This bimodal structure is inevitable with this initial condition and the boundary condition Equation (2), regardless of the value of γ and the details of the pdf.…”

Section: Zimmerman and Chatwin's Numerical Results Zc Obtained Numermentioning

confidence: 99%

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

Mole

Ring

2007

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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.

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Section: Zimmerman and Chatwin's Numerical Results Zc Obtained Numermentioning

confidence: 99%

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

Mole

Ring

2007

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“…where φ 0 = φ max is the maximum mean value in a transversal/vertical section (in the centerline for a jet or a wake, and on the wall for a boundary-layer plume), and r = φ /φ 0 . From comparison with several experimental results, Chatwin andSullivan (1990, 1993) proposed 1 < α < 1.5 and 0 < β < 1. Equation 33 is particularly attractive because from the mean concentration field one may readily obtain higher-order moments in an expeditious way.…”

Section: Empirical Models For the Statistical Moments Of Concentrationmentioning

confidence: 84%

“…In fact, although φ = 0 is a physically plausible value, it is in contrast with the description of scalar mixing according to the advection-diffusion equation, Eq. 2 (Chatwin andSullivan 1989, 1993). Furthermore, the intermittency is not a precisely measurable quantity since its experimental value depends on a threshold that must be necessarily related to the instrument used for measuring the concentration (e.g., Fackrell and Robins 1982b) or, in case of odours, to the neurobiology of the olfactory system (Celani et al 2014).…”

Section: Analytical Pdf Modelsmentioning

confidence: 99%

“…Similarly, a threshold is also required in numerical simulations due to the presence of molecular and numerical diffusivities. Chatwin andSullivan (1989, 1993) proposed an alternative definition of intermittency that is based on the representation of the concentration in a single realization neglecting the dissipative effects of molecular diffusivity, i.e., γ = φ /φ s where φ s is the unique value of the concentration at the marking source. Yet, the validity of this definition seems justified only in the very early phases of dispersion.…”

Section: Analytical Pdf Modelsmentioning

confidence: 99%

“…Yee and Chan ( 1997 ) derived a clipped-gamma distribution assuming ad hoc closures for what they called pseudo-diffusion and pseudo-dissipation terms. Chatwin ( 2002 ) presented empirical considerations to support the idea that, under simplifying assumptions, the PDF is inversely proportional to the mixing process by molecular diffusion. As Chatwin debated, this relation may serve in the modelling of the mixing term in the PDF transport equation.…”

Section: Phenomenologically-based Approachesmentioning

confidence: 99%

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Concentration Fluctuations from Localized Atmospheric Releases

Cassiani

Bertagni

Marro

et al. 2020

Boundary-Layer Meteorol

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We review the efforts made by the scientific community in more than seventy years to elucidate the behaviour of concentration fluctuations arising from localized atmospheric releases of dynamically passive and non-reactive scalars. Concentration fluctuations are relevant in many fields including the evaluation of toxicity, flammability, and odour nuisance. Characterizing concentration fluctuations requires not just the mean concentration but also at least the variance of the concentration in the location of interest. However, for most purposes the characterization of the concentration fluctuations requires knowledge of the concentration probability density function (PDF) in the point of interest and even the time evolution of the concentration. We firstly review the experimental works made both in the field and in the laboratory, and cover both point sources and line sources. Regarding modelling approaches, we cover analytical, semi-analytical, and numerical methods. For clarity of presentation we subdivide the models in two groups, models linked to a transport equation, which usually require a numerical resolution, and models mainly based on phenomenological aspects of dispersion, often providing analytical or semi-analytical relations. The former group includes: large-eddy simulations, Reynolds-averaged Navier–Stokes methods, two-particle Lagrangian stochastic models, PDF transport equation methods, and heuristic Lagrangian single-particle methods. The latter group includes: fluctuating plume models, semi-empirical models for the concentration moments, analytical models for the concentration PDF, and concentration time-series models. We close the review with a brief discussion highlighting possible useful additions to experiments and improvements to models.

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

Mole

2010

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

Cited by 11 publications

References 29 publications

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

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

Concentration Fluctuations from Localized Atmospheric Releases

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

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