Dispersion of passive tracers in a velocity field with non-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>δ</mml:mi></mml:math>-correlated noise

Castiglione, Patrizia; Crisanti, A.

doi:10.1103/physreve.59.3926

Cited by 16 publications

(21 citation statements)

References 17 publications

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“…For a steady square integrable velocity field, i.e. a quasibounded velocity field, the effective shear-induced diffusivity increases with the Lagrangian time scale (Castiglione & Crisanti 1999), although a complete expression for the effective diffusivity's dependence upon time and τ L was not considered. Numerical simulations of particle dispersion in steady and time-dependent Taylor-Green flow show that the effective diffusivity increases with increasing τ L (Pavliotis & Stuart 2005;Pavliotis, Stuart & Zygalakis 2007).…”

Section: Shear Dispersion With a Non-zero Lagrangian Time Scalementioning

confidence: 99%

The effect of a non-zero Lagrangian time scale on bounded shear dispersion

Spydell

Feddersen

2011

J. Fluid Mech.

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Previous studies of shear dispersion in bounded velocity fields have assumed random velocities with zero Lagrangian time scale (i.e. velocities are$\delta $-function correlated in time). However, many turbulent (geophysical and engineering) flows with mean velocity shear exist where the Lagrangian time scale is non-zero. Here, the longitudinal (along-flow) shear-induced diffusivity in a two-dimensional bounded velocity field is derived for random velocities with non-zero Lagrangian time scale${\tau }_{L} $. A non-zero${\tau }_{L} $results in two-time transverse (across-flow) displacements that are correlated even for large (relative to the diffusive time scale${\tau }_{D} $) times. The longitudinal (along-flow) shear-induced diffusivity${D}_{S} $is derived, accurate for all${\tau }_{L} $, using a Lagrangian method where the velocity field is periodically extended to infinity so that unbounded transverse particle spreading statistics can be used to determine${D}_{S} $. The non-dimensionalized${D}_{S} $depends on time and two parameters: the ratio of Lagrangian to diffusive time scales${\tau }_{L} / {\tau }_{D} $and the release location. Using a parabolic velocity profile, these dependencies are explored numerically and through asymptotic analysis. The large-time${D}_{S} $is enhanced relative to the classic Taylor diffusivity, and this enhancement increases with$ \sqrt{{\tau }_{L} } $. At moderate${\tau }_{L} / {\tau }_{D} = 0. 1$this enhancement is approximately a factor of 3. For classic shear dispersion with${\tau }_{L} = 0$, the diffusive time scale${\tau }_{D} $determines the time dependence and large-time limit of the shear-induced diffusivity. In contrast, for sufficiently large${\tau }_{L} $, a shear time scale${\tau }_{S} = \mathop{ ({\tau }_{L} {\tau }_{D} )}\nolimits ^{1/ 2} $, anticipated by a simple analysis of the particle’s domain-crossing time, determines both the${D}_{S} $time dependence and the large-time limit. In addition, the scalings for turbulent shear dispersion are recovered from the large-time${D}_{S} $using properties of wall-bounded turbulence.

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Section: Shear Dispersion With a Non-zero Lagrangian Time Scalementioning

confidence: 99%

The effect of a non-zero Lagrangian time scale on bounded shear dispersion

Spydell

Feddersen

2011

J. Fluid Mech.

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“…Markovian process driven by a colored noise (Ornstein-Uhlenbeck) in the Langevin equation, as already described in [39]. The Lagrangian approach has also been followed in [40] to find exact expressions for the particle eddy diffusivity in shear or Gaussian flows.…”

Section: Equationsmentioning

confidence: 99%

“…Let us firstly prove the rewriting (5) of the full terminal velocity, by exploiting the definition of the phase-space density as an average of Dirac delta's on every random factor: (39) Let us remind that both μ(t) and ν(t) are white noises, meaning that the values assumed at a certain time instant are completely uncorrelated from the ones assumed at a time instant immediately following. Moreover, by invoking causality, one infers that the instantaneous values of the noises at time t can influence the particle dynamics only at future times, but not computed at t itself.…”

Section: A1 Proof Of the Expression For The Terminal-velocity Corrementioning

confidence: 99%

Settling velocity of quasi-neutrally-buoyant inertial particles

Afonso

Gama

2017

Comptes Rendus. Mécanique

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We investigate the sedimentation properties of quasi-neutrally buoyant inertial particles carried by incompressible zero-mean fluid flows. We obtain generic formulae for the terminal velocity in generic space-and-time periodic (or steady) flows, along with further information for flows endowed with some degree of spatial symmetry such as odd parity in the vertical direction. These expressions consist in space-time integrals of auxiliary quantities that satisfy partial differential equations of the advection-diffusion-reaction type, which can be solved at least numerically, since our scheme implies a huge reduction of the problem dimensionality from the full phase space to the classical physical space. r é s u m é Nous étudions les propriétés de sédimentation de particules inertielles dotées de flottabilité quasi neutre et transportées par un écoulement incompressible à moyenne nulle. Nous obtenons des formules génériques pour la vitesse terminale dans des écoulements en général périodiques en espace et en temps (ou statiques), avec d'ultérieures informations disponibles pour les écoulements dotés de symétries spatiales spécifiques, telles qu'une parité négative dans la direction verticale. Ces expressions consistent en des intégrales spatio-temporelles de quantités auxiliaires qui obéissent à des équations aux dérivées partielles du type advection-diffusion-réaction. Ces dernières peuvent être résolues au moins numériquement, car notre procédure implique une forte réduction de la dimensionnalité du problème, de l'espace des phases complet à l'espace physique classique.

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“…For example, when the randomness of the velocity field is due to turbulence, the average tracer velocity equals the fluid mean velocity, while the effective, or eddy, diffusivity is the time integral ͑when it is finite͒ of the fluid velocity autocorrelation function ͑see Monin and Yaglom ͓1͔, and references therein͒. More recently, this problem was studied also by Biferale et al ͓2͔ and by Castiglione et al ͓3,4͔, who analyzed standard and anomalous transport in incompressible flow using multiscale techniques. Similar results are found in the related process of convection of a passive tracer by a fluid flowing through a random medium, even if, as mentioned above, in this case the randomness of the flow is provided not by turbulence ͑in fact, the fluid flow can very well be laminar͒, but by its interaction with the dispersed particles ͑see Refs.…”

Section: Introductionmentioning

confidence: 99%

Heat and mass transport in nonhomogeneous random velocity fields

Mauri

2003

Phys. Rev. E

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The effective equation describing the transport of passive tracers in nonsolenoidal velocity fields is determined, assuming that the velocity field U(r,t) is a function of both position r and time t, albeit remaining locally random. Assuming a strong separation of scales and applying the method of homogenization, we find a Fickian constitutive relation for the coarse-grained particle flux, as the sum of a convective part, V(E)c, and a diffusive term, -D(s). Inverted Delta c, where V(E) is the Eulerian mean tracer velocity, c the average particle concentration, and D(s) the effective diffusivity. The latter can be written as D(s)(r,t)=D(0)I+D(r,r,t), where D0 is the molecular diffusivity, I the unit dyadic and D(r(1),r(2),t) the cross diffusion dyadic. Conversely, the Eulerian mean velocity V(E)(r,t) is the sum of the microscale mean tracer velocity V(r,t) and a particle drift velocity, V(d)(r,t)=-[(delta/delta r(2)).D(T)(r,r(2),t)](r(2)=r), which depends on the nonhomogeneity of the velocity field at the macroscale. The microscale mean particle velocity, in turn, is the sum of the mean fluid velocity and the ballistic tracer velocity, which is due to the local nonuniformity of the concentration field and is therefore structurally different from the tracer drift velocity. In the limit of large Peclet numbers, D(s) coincides with the self-diffusion dyadic, as it measures the local temporal growth of the mean square displacement of a tracer particle from its average position. In this case, the motion of a tracer particle is a random process in the manner of Stratonovich, where the smoothly varying mean tracer velocity equals the microscale mean tracer velocity and the fluctuating term is described through the cross diffusion dyadic D(r(1),r(2),t).

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Dispersion of passive tracers in a velocity field with non-δ-correlated noise

Cited by 16 publications

References 17 publications

The effect of a non-zero Lagrangian time scale on bounded shear dispersion

The effect of a non-zero Lagrangian time scale on bounded shear dispersion

Settling velocity of quasi-neutrally-buoyant inertial particles

Heat and mass transport in nonhomogeneous random velocity fields

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