scholarly journals Dispersion in the large-deviation regime. Part 2. Cellular flow at large Péclet number

2014 ◽  
Vol 745 ◽  
pp. 351-377 ◽  
Author(s):  
P. H. Haynes ◽  
J. Vanneste
Keyword(s):  
Peclet Number ◽  
Eigenvalue Problems ◽  
Large Deviation ◽  
Effective Diffusivity ◽  
Rate Function ◽  
Homogenization Theory ◽  
Péclet Number ◽  
Stagnation Points ◽  
Scalar Dispersion ◽  
Scalar Concentration

AbstractA standard model for the study of scalar dispersion through the combined effect of advection and molecular diffusion is a two-dimensional periodic flow with closed streamlines inside periodic cells. Over long time scales, the dispersion of a scalar released in this flow can be characterized by an effective diffusivity that is a factor$\mathit{Pe}^{1/2}$larger than molecular diffusivity when the Péclet number$\mathit{Pe}$is large. Here we provide a more complete description of dispersion in this regime by applying the large-deviation theory developed in Part 1 of this paper. Specifically, we derive approximations to the rate function governing the scalar concentration at large time$t$by carrying out an asymptotic analysis of the relevant family of eigenvalue problems. We identify two asymptotic regimes and, for each, make predictions for the rate function and spatial structure of the scalar. Regime I applies to distances$|\boldsymbol {x}|$from the scalar release point that satisfy$|\boldsymbol {x}|= O(\mathit{Pe}^{1/4} t)$. The concentration in this regime is isotropic at large scales, is uniform along streamlines within each cell, and varies rapidly in boundary layers surrounding the separatrices between adjacent cells. The results of homogenization theory, yielding the$O(\mathit{Pe}^{1/2})$effective diffusivity, are recovered from our analysis in the limit$|\boldsymbol {x}|\ll \mathit{Pe}^{1/4} t$. Regime II applies when$|\boldsymbol {x}|=O(\mathit{Pe}\, t/{\rm log}\, \mathit{Pe})$and is characterized by an anisotropic concentration distribution that is localized around the separatrices. A novel feature of this regime is the crucial role played by the dynamics near the hyperbolic stagnation points. A consequence is that in part of the regime the dispersion can be interpreted as resulting from a random walk on the lattice of stagnation points. The two regimes overlap so that our asymptotic results describe the scalar concentration over a large range of distances$|\boldsymbol {x}|$. They are verified against numerical solutions of the family of eigenvalue problems yielding the rate function.

scholarly journals Dispersion in the large-deviation regime. Part 1: shear flows and periodic flows

2014 ◽  
Vol 745 ◽  
pp. 321-350 ◽  
Author(s):  
P. H. Haynes ◽  
J. Vanneste
Keyword(s):  
Shear Flows ◽  
Eigenvalue Problems ◽  
Large Deviation ◽  
Effective Diffusivity ◽  
Homogenization Theory ◽  
Large Deviation Theory ◽  
Periodic Flows ◽  
Scalar Concentration ◽  
Cellular Flows ◽  
Deviation Theory

AbstractThe dispersion of a passive scalar in a fluid through the combined action of advection and molecular diffusion is often described as a diffusive process, with an effective diffusivity that is enhanced compared with the molecular value. However, this description fails to capture the tails of the scalar concentration distribution in initial-value problems. To remedy this, we develop a large-deviation theory of scalar dispersion that provides an approximation to the scalar concentration valid at much larger distances away from the centre of mass, specifically distances that are$O(t)$rather than$O(t^{1/2})$, where$t \gg 1$is the time from the scalar release. The theory centres on the calculation of a rate function characterizing the large-time form of the scalar concentration. This function is deduced from the solution of a one-parameter family of eigenvalue problems which we derive using two alternative approaches, one asymptotic, the other probabilistic. We emphasize the connection between the large-deviation theory and the homogenization theory that is often used to compute effective diffusivities: a perturbative solution of the eigenvalue problems in the appropriate limit reduces at leading order to the cell problem of homogenization theory. We consider two classes of flows in some detail: shear flows and periodic flows with closed streamlines (cellular flows). In both cases, large deviation generalizes classical results on effective diffusivity and captures new phenomena relevant to the tails of the scalar distribution. These include approximately finite dispersion speeds arising at large Péclet number$\mathit{Pe}$(corresponding to small molecular diffusivity) and, for two-dimensional cellular flows, anisotropic dispersion. Explicit asymptotic results are obtained for shear flows in the limit of large$\mathit{Pe}$. (A companion paper, Part 2, is devoted to the large-$\mathit{Pe}$asymptotic treatment of cellular flows.) The predictions of large-deviation theory are compared with Monte Carlo simulations that estimate the tails of concentration accurately using importance sampling.

Characterization of stationary mixing patterns in a three-dimensional open Stokes flow: spectral properties, localization and mixing regimes

2009 ◽  
Vol 639 ◽  
pp. 291-341 ◽  
Author(s):  
M. GIONA ◽  
S. CERBELLI ◽  
F. GAROFALO
Keyword(s):  
Finite Length ◽  
Peclet Number ◽  
Scaling Laws ◽  
Eigenvalue Problems ◽  
Operating Conditions ◽  
Creeping Flow ◽  
Scaling Analysis ◽  
Dominant Eigenvalue ◽  
Péclet Number ◽  
The Real

This article analyses stationary scalar mixing downstream an open flow Couette device operating in the creeping flow regime. The device consists of two coaxial cylinders of finite length Lz, and radii κ R and R (κ < 1), which can rotate independently. At relatively large values of the aspect ratio α = Lz/R ≫ 1, and of the Péclet number Pe, the stationary response of the system can be accurately described by enforcing the simplifying assumption of negligible axial diffusion. With this approximation, homogenization along the device axis can be described by a family of generalized one-dimensional eigenvalue problems with the radial coordinate as independent variable. A variety of mixing regimes can be observed by varying the geometric and operating parameters. These regimes are characterized by different localization properties of the eigenfunctions and by different scaling laws of the real part of the eigenvalues with the Péclet number. The analysis of this model flow reveals the occurrence of sharp transitions between mixing regimes, e.g. controlled by the geometric parameter κ. The eigenvalue scalings can be theoretically predicted by enforcing eigenfunction localization and simple functional equalities relating the behaviour of the eigenvalues to the functional form of the associated eigenfunctions. Several flow protocols corresponding to different geometric and operating conditions are considered. Among these protocols, the case where the inner and the outer cylinders counter-rotate exhibits a peculiar intermediate scaling regime where the real part of the dominant eigenvalue is independent of Pe over more than two decades of Pe. This case is thoroughly analysed by means of scaling analysis. The practical relevance of the results deriving from spectral analysis for fluid mixing problems in finite-length Couette devices is addressed in detail.

The effective diffusivity of ordered and freely evolving bubbly suspensions

2018 ◽  
Vol 840 ◽  
pp. 215-237 ◽  
Author(s):  
Aurore Loisy ◽  
Aurore Naso ◽  
Peter D. M. Spelt
Keyword(s):  
Numerical Simulations ◽  
Peclet Number ◽  
Random Media ◽  
Chemical Species ◽  
Effective Diffusivity ◽  
Passive Scalar ◽  
Solid Particles ◽  
Péclet Number ◽  
Mechanical Dispersion ◽  
Fixed Beds

We investigate the dispersion of a passive scalar such as the concentration of a chemical species, or temperature, in homogeneous bubbly suspensions, by determining an effective diffusivity tensor. Defining the longitudinal and transverse components of this tensor with respect to the direction of averaged bubble rise velocity in a zero mixture velocity frame of reference, we focus on the convective contribution thereof, this being expected to be dominant in commonly encountered bubbly flows. We first extend the theory of Kochet al.(J. Fluid Mech., vol. 200, 1989, pp. 173–188) (which is for dispersion in fixed beds of solid particles under Stokes flow) to account for weak inertial effects in the case of ordered suspensions. In the limits of low and of high Péclet number, including the inertial effect of the flow does not affect the scaling of the effective diffusivity with respect to the Péclet number. These results are confirmed by direct numerical simulations performed in different flow regimes, for spherical or very deformed bubbles and from vanishingly small to moderate values of the Reynolds number. Scalar transport in arrays of freely rising bubbles is considered by us subsequently, using numerical simulations. In this case, the dispersion is found to be convectively enhanced at low Péclet number, like in ordered arrays. At high Péclet number, the Taylor dispersion scaling obtained for ordered configurations is replaced by one characterizing a purely mechanical dispersion, as in random media, even if the level of disorder is very low.

Dispersion in fixed beds

1985 ◽  
Vol 154 ◽  
pp. 399-427 ◽  
Author(s):  
Donald L. Koch ◽  
John F. Brady
Keyword(s):  
Peclet Number ◽  
Molecular Diffusion ◽  
Volume Fraction ◽  
Effective Diffusivity ◽  
Ensemble Averaging ◽  
Péclet Number ◽  
Physical Mechanisms ◽  
Molecular Diffusivity ◽  
Fixed Beds ◽  
Effective Diffusivities

A macroscopic equation of mass conservation is obtained by ensemble-averaging the basic conservation laws in a porous medium. In the long-time limit this ‘macro-transport’ equation takes the form of a macroscopic Fick's law with a constant effective diffusivity tensor. An asymptotic analysis in low volume fraction of the effective diffusivity in a bed of fixed spheres is carried out for all values of the Péclet number ℙ = Ua/Df, where U is the average velocity through the bed. a is the particle radius and Df is the molecular diffusivity of the solute in the fluid. Several physical mechanisms causing dispersion are revealed by this analysis. The stochastic velocity fluctuations induced in the fluid by the randomly positioned bed particles give rise to a convectively driven contribution to dispersion. At high Péclet numbers, this convective dispersion mechanism is purely mechanical, and the resulting effective diffusivities are independent of molecular diffusion and grow linearly with ℙ. The region of zero velocity in and near the bed particles gives rise to non-mechanical dispersion mechanisms that dominate the longitudinal diffusivity at very high Péclet numbers. One such mechanism involves the retention of the diffusing species in permeable particles, from which it can escape only by molecular diffusion, leading to a diffusion coefficient that grows as ℙ2. Even if the bed particles are impermeable, non-mechanical contributions that grow as ℙ ln ℙ and ℙ2 at high ℙ arise from a diffusive boundary layer near the solid surfaces and from regions of closed streamlines respectively. The results for the longitudinal and transverse effective diffusivities as functions of the Péclet number are summarized in tabular form in §6. Because the same physical mechanisms promote dispersion in dilute and dense fixed beds, the predicted Péclet-number dependences of the effective diffusivities are applicable to all porous media. The theoretical predictions are compared with experiments in densely packed beds of impermeable particles, and the agreement is shown to be remarkably good.

NUMERICAL SCHEMES OBTAINED FROM LATTICE BOLTZMANN EQUATIONS FOR ADVECTION DIFFUSION EQUATIONS

2006 ◽  
Vol 17 (11) ◽  
pp. 1563-1577 ◽  
Author(s):  
SHINSUKE SUGA
Keyword(s):  
Lattice Boltzmann ◽  
Peclet Number ◽  
Eigenvalue Problems ◽  
Velocity Model ◽  
Diffusion Equations ◽  
Numerical Schemes ◽  
Péclet Number ◽  
Velocity Models ◽  
Advection Diffusion ◽  
The Stability

Stability and accuracy of the numerical schemes obtained from the lattice Boltzmann equation (LBE) used for numerical solutions of two-dimensional advection-diffusion equations are presented. Three kinds of velocity models are used to determine the moving velocity of particles on a squre lattice. A system of explicit finite difference equations are derived from the LBE based on the Bhatnagar, Gross and Krook (BGK) model for individual velocity model. In order to approximate the advecting velocity field, a linear equilibrium distribution function is used for each of the moving directions. The stability regions of the schemes in the special case of the relaxation parameter ω in the LBE being set to ω=1 are found by analytically solving the eigenvalue problems of the amplification matrices corresponding to each scheme. As for the cases of general relaxation parameters, the eigenvalue problems are solved numerically. A benchmark problem is solved in order to investigate the relationship between the accuracy of the numerical schemes and the order of the Peclet number. The numerical experiments result in indicating that for the scheme based on a 9-velocity model we can find the parameters depending on the order of the given Peclet number, which generate accurate solutions in the stability region.

Approximate analysis of the containment of contaminated sites prior to remediation

2000 ◽  
Vol 42 (1-2) ◽  
pp. 319-324 ◽  
Author(s):  
H. Rubin ◽  
A. Rabideau
Keyword(s):  
Steady State ◽  
Peclet Number ◽  
Dimensionless Parameter ◽  
Contaminated Sites ◽  
Unsteady State ◽  
Péclet Number ◽  
Vertical Barrier ◽  
Time Period ◽  
Barrier Performance ◽  
Vertical Barriers

This study presents an approximate analytical model, which can be useful for the prediction and requirement of vertical barrier efficiencies. A previous study by the authors has indicated that a single dimensionless parameter determines the performance of a vertical barrier. This parameter is termed the barrier Peclet number. The evaluation of barrier performance concerns operation under steady state conditions, as well as estimates of unsteady state conditions and calculation of the time period requires arriving at steady state conditions. This study refers to high values of the barrier Peclet number. The modeling approach refers to the development of several types of boundary layers. Comparisons were made between simulation results of the present study and some analytical and numerical results. These comparisons indicate that the models developed in this study could be useful in the design and prediction of the performance of vertical barriers operating under conditions of high values of the barrier Peclet number.

Convective diffusion toward the sphere in laminar flow around the sphere

1979 ◽  
Vol 44 (4) ◽  
pp. 1218-1238
Author(s):  
Arnošt Kimla ◽  
Jiří Míčka
Keyword(s):  
Laminar Flow ◽  
Velocity Profile ◽  
Peclet Number ◽  
Convective Diffusion ◽  
Péclet Number ◽  
Stability Of The Solution

The problem of convective diffusion toward the sphere in laminar flow around the sphere is solved by combination of the analytical and net methods for the region of Peclet number λ ≥ 1. The problem was also studied for very small values λ. Stability of the solution has been proved in relation to changes of the velocity profile.

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