Calculating effective diffusivities in the limit of vanishing molecular diffusion

Pavliotis, Grigorios A.; Stuart, Andrew M.; Zygalakis, Konstantinos C.

doi:10.1016/j.jcp.2008.10.014

Cited by 26 publications

(46 citation statements)

References 34 publications

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“…Moreover, for non-zero τ L , the large-time shear-induced diffusivity D ∞ S depends upon √ τ L . This leading-order dependency of D ∞ S on τ L was also found for sinusoidal shear flow (Pavliotis et al 2009). The first two and four terms of (6.7b) reasonably reproduce the D ∞ S /D ∞ T (dashed curves in figure 4).…”

Section: Asymptotic Large-time Shear-induced Diffusivitysupporting

confidence: 73%

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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: Asymptotic Large-time Shear-induced Diffusivitysupporting

confidence: 73%

“…As the imposed velocity is U(y), numerical problems occurring when the Langevin equations are time stepped for large Pe using an Euler method were not observed (e.g. Pavliotis et al 2009). The Lagrangian time scale is fixed at τ L /τ D = 0.1, which results in a substantially increased figure 4).…”

Section: Effect Of Initial Distributionmentioning

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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“…The graph was computed using a Monte-Carlo simulation using 10,000 realizations. The numerical method used is based on [16,17], and is described in Section 5.…”

Section: Numerical Results In the Critical Regimementioning

confidence: 99%

“…Equation (1.1) is stiff, and when A is large the standard Euler-Maruyama method requires a time step which is too small to be practically useful. We instead use the method developed in [16]. We conclude the paper, with a description of the numerical method and a new, simpler, analysis of it based on [23], which extends the theory of modified equations [20] to more general numerical integrators than the Euler-Maruyama method.…”

Section: Introductionmentioning

confidence: 99%

“…The double limit as both the flow amplitude A → ∞ and the number of cells L 2 → ∞ was recently studied [9], one observes a sharp transition between the homogenization and averaging regimes occurring at A ≈ L 4 . This paper numerically studies a few theoretically unresolved aspects of this problem when both A and L are large that were left open in [9] using the numerical method devised in [16]. Our treatment of the numerical method uses recent developments in the theory of modified equations for numerical integrators of SDEs [23].…”

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confidence: 99%

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Numerical Studies of Homogenization under a Fast Cellular Flow

Iyer¹,

Zygalakis²

2012

Multiscale Model. Simul.

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Abstract. We consider a two dimensional particle diffusing in the presence of a fast cellular flow confined to a finite domain. If the flow amplitude A is held fixed, and the number of cells L 2 → ∞, then problem homogenizes, and has been well studied. Also well studied is the limit when L is fixed, and A → ∞. In this case the solution averages along stream lines. The double limit as both the flow amplitude A → ∞ and the number of cells L 2 → ∞ was recently studied [9], one observes a sharp transition between the homogenization and averaging regimes occurring at A ≈ L 4 . This paper numerically studies a few theoretically unresolved aspects of this problem when both A and L are large that were left open in [9] using the numerical method devised in [16]. Our treatment of the numerical method uses recent developments in the theory of modified equations for numerical integrators of SDEs [23].

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Pathwise accuracy and ergodicity of metropolized integrators for SDEs

Bou-Rabee

Vanden‐Eijnden

2009

Comm Pure Appl Math

118

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Metropolized integrators for ergodic stochastic differential equations (SDEs) are proposed that (1) are ergodic with respect to the (known) equilibrium distribution of the SDEs and (2) approximate pathwise the solutions of the SDEs on finitetime intervals. Both these properties are demonstrated in the paper, and precise strong error estimates are obtained. It is also shown that the Metropolized integrator retains these properties even in situations where the drift in the SDE is nonglobally Lipschitz, and vanilla explicit integrators for SDEs typically become unstable and fail to be ergodic.

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Calculating effective diffusivities in the limit of vanishing molecular diffusion

Cited by 26 publications

References 34 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

Numerical Studies of Homogenization under a Fast Cellular Flow

Pathwise accuracy and ergodicity of metropolized integrators for SDEs

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