Particle acceleration in turbulent flows: A class of nonlinear stochastic models for intermittency

Abstract

The problem of modeling the velocity and acceleration of inertial particles in turbulent flows is discussed. Particular attention is focused on the modeling of the particle Lagrangian velocity increment, especially, but not exclusively, in the case in which only the low frequencies of the carrier turbulent flow field are available. The need for suitable models arises in the simulation of particle laden flows by the means of new computational techniques such as large-eddy simulation. For this, stochastic differential equations, sde, are often introduced, though there is a lack of clarity in how such models should deal with the experimental observed far from Gaussian statistics, intermittency, and heavy tailed probability density function for particle acceleration. It is well known that Langevin-type equations are not capable of reproducing such features. It is first shown how the stochastic model for the particle Lagrangian velocity increments is far from being a Langevin equation, and it is characterized by nonlinear drift and diffusion; the statistical characteristics of this first model are shown to be in qualitative agreement with experimental findings. These results suggest an improved model for the particle dynamics based upon a more general family of nonlinear sde; the family, which is generated by a single parameter, includes both the Langevin equation and the first model as special cases. An analysis of the statistical properties of the new sde shows that the model is capable of accurately reproducing the strong deviations from Gaussianity observed in recent experiments.

Large-eddy simulation of particle-laden turbulent flows

“…Now it is known -see the earlier discussion -that particle accelerations at the small scales exhibit a probability density function which deviates strongly from Gaussianity. By suitable choices of τ t , Bini & Jones (2007) have shown how a family of stochastic processes can be generated that account for this. If τ t is made to depend on the random variable v p itself then, depending on the dependence, different asymptotic forms for the PDF of particle velocity increment can be obtained.…”

“…The approach followed involves a two-way coupling in which a probabilistic approach, adopted for the dispersed phase, explicitly accounts for subgrid-scale dispersion effects on the droplets. In Bini & Jones (2007), a model for the LES unresolved fluctuations upon particle dynamics is formulated and it is shown how the model is capable of reproducing the experimentally observed heavy-tailed probability distribution of particle acceleration. In the present work, this model is applied in LES and the applicability of various alternative formulations, such as those of Miller & Bellan (2000) and Okong'o & Bellan (2004), are also reviewed and discussed.…”

“…These latter two scales are related to the time taken for a particle to cross a notional 'unresolved eddy'. Following the reasoning of Bini & Jones (2007), it is to be noted that whilst the former time scale will result in a leptokurtic behaviour of the small-scale acceleration PDF, because of the appearance of term |v p | 1/2 in the denominator, the latter choice will give rise to particle acceleration PDFs that are Gaussian. 3.2.…”

“…If τ t is made to depend on the random variable v p itself then, depending on the dependence, different asymptotic forms for the PDF of particle velocity increment can be obtained. Bini & Jones (2007) constructed a stochastic process which is able to reproduce the desired behaviour of the PDF of particle acceleration without over-imposing the PDF shape and which correctly attributes high levels of probability to extreme acceleration events. A time scale given by:…”

“…Mordant et al 2001;La Porta et al 2002;Voth et al 2002), provide evidence that particles are likely to experience accelerations which are many times higher than the acceleration standard deviation with a probability significantly larger than that corresponding to a Gaussian distribution. The problem of modelling sub-filter particle accelerations has been addressed by Bini & Jones (2007) where a stochastic model, capable of reproducing the observed far from Gaussian behaviour, has been constructed.…”