Convection of Microstructure and Related Problems

McLaughlin, David W.; Papanicolaou, George; Pironneau, Olivier

doi:10.1137/0145046

Cited by 124 publications

(120 citation statements)

References 12 publications

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“…Exploiting the scale separation in the dynamics we derive, using the multiscale technique [4], an effective diffusive equation for the macrodynamics, the calculation of the effective diffusivity second-order tensor is reduced to the solution of one auxiliary partial differential equation [5], [6], [8].…”

Section: Pacs Number(s): ;mentioning

confidence: 99%

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

Castiglione

Crisanti

1999

Phys. Rev. E

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(Physical Review E, in press)The diffusive properties in velocity fields whose small scales are parameterized by non δ-correlated noise is investigated using multiscale technique. The analytical expression of the eddy diffusivity tensor is found for a 2D steady shear flow and it is an increasing function of the characteristic noise decorrelation time τ . In order to study a generic flow v, a small-τ expansion is performed and the first correction O(τ ) to the effective diffusion coefficients is evaluated. This is done using two different approaches and it results that at the order τ the problem with a colored noise is equivalent to the white-in-time case provided by a renormalization of the velocity field v →ṽ depending on τ . Two examples of 2D closed-streamlines velocity field are considered and in both the cases an enhancement of the diffusion is found.

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Section: Pacs Number(s): ;mentioning

confidence: 99%

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

Castiglione

Crisanti

1999

Phys. Rev. E

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“…The earliest effort in studying homogenization of the incompressible Euler equations was by McLaughlin, Papanicolaou, and Pironneau (MPP for short) in 1985 [6]. One of the key observations in MPP's work is that the oscillation is convected by the mean velocity field.…”

Section: Introductionmentioning

confidence: 99%

“…The homogenization theory of the Euler equations with oscillating data was first studied by McLaughlin-Papanicolaou-Pironneau (MPP for short) in 1985 [6]. To construct a·multiscale expansion for the Euler equations, they made an important assumption that the oscillation is convected by the mean flow: E E E where W(t,X,T,y), UI(t,x,T,y), q(t,X,T,y) and pI(t,x,T,y) are assumed to be periodic in y and T with zero mean, and the phase () is convected by the mean velocity field u:…”

Section: Formulationmentioning

confidence: 99%

Multiscale computation of isotropic homogeneous turbulent flow

Hou¹,

Yang²,

Ran³

2006

Inverse Problems, Multi-Scale Analysis, and Effective Medium Theory

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ABSTRACT. In this article we perform a systematic multi-scale analysis and computation for incompressible Euler equations and Navier-Stokes Equations in both 2D and 3D. The initial condition for velocity field has multiple length scales. By reparameterizing them in the Fourier space, we can formally organize the initial condition into two scales with the fast scale component being periodic. By making an appropriate multiscale expansion for the velocity field, we show that the two-scale structure is preserved dynamically. Moreover, we derive a well-posed homogenized equation for the incompressible Euler equations in the Eulerian formulations. Numerical experiments are presented to demonstrate that the homogenized equations indeed capture the correct averaged solution of the incompressible Euler and Navier Stokes equations. Moreover, our multiscale analysis reveals some interesting structure for the Reynolds stress terms, which provides a theoretical base for establishing an effective LES type of model for incompressible fluid flows.

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“…In the case of zero or weak mean divergence free flow, the large scale behavior is well understood and has been rigorously analyzed in the framework of homogenization by McLaughlin, Papanicolaou, Pironneau, 6 Avellaneda and Majda, 7 Ma-jda and Kramer, 8 E, 9 and many others. We refer to Ref.…”

Section: Introductionmentioning

confidence: 99%

Effective velocity for transport in heterogeneous compressible flows with mean drift

Attinger

Abdulle

2008

Physics of Fluids

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Solving transport equations in heterogeneous flows might give rise to scale dependent transport behavior with effective large scale transport parameters differing from those found on smaller scales. For incompressible velocity fields, homogenization methods have proven to be powerful in describing the effective transport parameters. In this paper, we aim at studying the effective drift of transport problems in heterogeneous compressible flows. Such a study was done by Vergassola and Avellaneda in Physica D 106, 148 ͑1997͒. There, it was shown that for static compressible flow without mean drift, impacts on the large scale drift do not occur. We will first discuss the impact of a mean drift and show that static compressible flow with mean drift can produce a heterogeneity driven large scale drift ͑or ballistic transport͒. For the case of Gaussian stationary random processes, we derive explicit results for the large scale drift. Moreover, we show that the large scale or effective drift depends on the small scale diffusion coefficients and thus on the molecular weights of the particles. This study could be applied to weight-based particle separation. Numerical simulations are presented to illustrate these phenomena.

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Convection of Microstructure and Related Problems

Abstract: Abstract. We study how two flows that vary spatially in two widely separated scales evolve under the dynamics of Euler's equations. We use a multiple scale approach which we motivate by studying some simpler model problems.

Cited by 124 publications

References 12 publications

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

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

Multiscale computation of isotropic homogeneous turbulent flow

Effective velocity for transport in heterogeneous compressible flows with mean drift

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