Intermittency of passive-scalar decay: Strange eigenmodes in random shear flows

Vanneste, Jacques

doi:10.1063/1.2338008

Cited by 14 publications

(14 citation statements)

References 39 publications

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“…The result (41) can be refined by noting that Laplace's method applied to (39) leads to the expectation of a term of the form exp(−q A − A * , A − A * ), where A * is the maximiser in (40) and ·, · is some scalar product (both A * and ·, · depend onê). Carrying out the expectation yields a factor q −D/2 , where D is the dimension of the support of the measure.…”

Section: Large-|q| Asymptoticsmentioning

confidence: 99%

“…Since the literature on fluid mixing literature makes extensive use of white-in-time velocity fields as an alternative to renewing flows, it would also be desirable to develop methods for the efficient evaluation of generalised Lyapunov exponents in the context of linear stochastic differential equations [see 38, for recent analytical results]. Finally, we note that the methods discussed in this paper apply to large matrices (d ≫ 1) and so could be employed to study the large-deviation statistics of discretised infinite-dimensional systems as arise, for instance, in the problem of passive scalar decay [39].…”

Section: B Three-dimensional Sine Mapmentioning

confidence: 99%

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Estimating generalized Lyapunov exponents for products of random matrices

Vanneste

2010

Phys. Rev. E

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We discuss several techniques for the evaluation of the generalized Lyapunov exponents which characterize the growth of products of random matrices in the large-deviation regime. A Monte Carlo algorithm that performs importance sampling using a simple random resampling step is proposed as a general-purpose numerical method which is both efficient and easy to implement. Alternative techniques complementing this method are presented. These include the computation of the generalized Lyapunov exponents by solving numerically an eigenvalue problem, and some asymptotic results corresponding to high-order moments of the matrix products. Taken together, the techniques discussed in this paper provide a suite of methods which should prove useful for the evaluation of the generalized Lyapunov exponents in a broad range of applications. Their usefulness is demonstrated on particular products of random matrices arising in the study of scalar mixing by complex fluid flows.

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Section: Large-|q| Asymptoticsmentioning

confidence: 99%

Section: B Three-dimensional Sine Mapmentioning

confidence: 99%

Estimating generalized Lyapunov exponents for products of random matrices

Vanneste

2010

Phys. Rev. E

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“…[1][2][3][4][5][6][7][8][9][10] In general, the evolution of a scalar field is governed by linear advectiondiffusion equations, which allows an analysis in terms of eigenmodes. Such eigenmode analyses reveal the exponential decay of the scalar field, perhaps modulated by oscillations if the eigenvalues are complex.…”

Section: Introductionmentioning

confidence: 99%

Eigenmode analysis of scalar transport in distributive mixing

2009

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In this study, we explore the spectral properties of the distribution matrices of the mapping method and its relation to the distributive mixing of passive scalars. The spectral (or eigenvector-eigenvalue) decomposition of these matrices constitutes discrete approximations to the eigenmodes of the continuous advection operator in periodic flows. The eigenvalue spectrum always lies within the unit circle and due to mass conservation, always accommodates an eigenvalue equal to one with trivial (uniform) eigenvector. The asymptotic state of a fully chaotic mixing flow is dominated by the eigenmode corresponding with the eigenvalue closest to the unit circle (“dominant eigenmode”). This eigenvalue determines the decay rate; its eigenvector determines the asymptotic mixing pattern. The closer this eigenvalue value is to the origin, the faster is the homogenization by the chaotic mixing. Hence, its magnitude can be used as a quantitative mixing measure for comparison of different mixing protocols. In nonchaotic cases, the presence of islands results in eigenvalues on the unit circle and associated eigenvectors demarcating the location of these islands. Eigenvalues on the unit circle thus are qualitative indicators of inefficient mixing; the properties of its eigenvectors enable isolation of the nonmixing zones. Thus important fundamental aspects of mixing processes can be inferred from the eigenmode analysis of the mapping matrix. This is elaborated in the present paper and demonstrated by way of two different prototypical mixing flows: the time-periodic sine flow and the spatially periodic partitioned-pipe mixer.

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“…This is quantified by showing that the decay rate decreases to zero like κ 2/5 in the presence of a small noise, and is explained in terms of pseudomodes. We note that the pseudomodal behaviour may be relevant to the situation recently examined by Vanneste (2006) where the shear flow translates randomly in time rather than steadily.…”

Section: αmentioning

confidence: 70%

Fast scalar decay in a shear flow: modes and pseudomodes

Vanneste

Byatt-Smith

2007

J. Fluid Mech.

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The decay of a passive scalar in a sinusoidal shear flow translating in the cross-stream direction at a constant speed u is studied in the limit of small diffusivity κ. The decay rate, obtained by solving an eigenvalue problem, is found to tend to a non-zero constant as κ→0 when u is of order κ1/2. This result, establishing that fast decay is possible in shear flows, is fragile however: because of the existence of pseudomodes, the addition of a small noise leads to decay rates that decrease to zero with κ as κ2/5.

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Intermittency of passive-scalar decay: Strange eigenmodes in random shear flows

Cited by 14 publications

References 39 publications

Estimating generalized Lyapunov exponents for products of random matrices

Estimating generalized Lyapunov exponents for products of random matrices

Eigenmode analysis of scalar transport in distributive mixing

Fast scalar decay in a shear flow: modes and pseudomodes

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