Convection Enhanced Diffusion for Periodic Flows

Fannjiang, Albert; Papanicolaou, George

doi:10.1137/s0036139992236785

Cited by 212 publications

(267 citation statements)

References 19 publications

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“…A that was first shown formally in [8] and later proved in [11,24,28]. The same scaling may be obtained from the formal predictions V A ∼ A 1/4 for the front speed V A in a cellular flow [1,2,3,32] -this implies that the front width is of the order A 1/4 .…”

Section: Introductionsupporting

confidence: 71%

Quenching of reaction by cellular flows

Fannjiang

Kiselev

Ryzhik

2006

GAFA, Geom. funct. anal.

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We consider a reaction-diffusion equation in a cellular flow. We prove that in the strong flow regime there are two possible scenario for the initial data that is compactly supported and the size of the support is large enough. If the flow cells are large compared to the reaction length scale, propagating fronts will always form. For the small cell size, any finitely supported initial data will be quenched by a sufficiently strong flow. We estimate that the flow amplitude required to quench the initial data of support L 0 is A > CL 4 0 ln(L 0 ). The essence of the problem is the question about the decay of the L ∞ norm of a solution to the advection-diffusion equation, and the relation between this rate of decay and the properties of the Hamiltonian system generated by the two-dimensional incompressible fluid flow.

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Section: Introductionsupporting

confidence: 71%

Quenching of reaction by cellular flows

Fannjiang

Kiselev

Ryzhik

2006

GAFA, Geom. funct. anal.

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“…Similar dependence has been carefully documented previously in the context of passive scalar transport in which the measure of the unbounded fluid streamlines were shown to play an important role in the ensuing effective large scale mixing. [39][40][41][42] In the current context, we will see that the PDF behavior of the transported field will demonstrate subtleties even for pure shear layers with the size of the shear-free zone affecting the width of the distribution. The PDF considered here is well-defined even for single functions f ; an ensemble of fields f is not required to compute PDF f (z).…”

Section: Probability Density Functions For Scalar Fieldsmentioning

confidence: 99%

Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

2007

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The probability density function (PDF) for a decaying passive scalar advected by a deterministic, periodic, incompressible fluid flow is numerically studied using a variety of random and coherent initial scalar fields. We establish the dynamic emergence at large Péclet numbers of a broad-tailed PDF for the scalar initialized with a Gaussian random measure, and further explore a rich parameter space involving scales of the initial scalar field and the geometry of the flow. We document that the dynamic transition of the PDF to a broad tailed distribution is similar for shear flows and time-varying non-sheared flows with positive Lyapunov exponent, thereby showing that chaos in the particle trajectories is not essential to observe intermittent scalar signals. The role of the initial scalar field is carefully explored. The long time PDF is sensitive to the scale of the initial data. For shear flows we show that heavy-tailed PDFs appear only when the initial field has sufficiently small-scale variation. We also connect geometric features of the scalar field with the shape of the PDFs. We document that the PDF is constructed by a subtle balance between spatial regions of strong and weak shear in conjunction with the presence of

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“…This justifies the application of Lemma 2.7 in this context. Since we know from [6] that ∇χ 1 L 2 = O(A 1/4 ), we immediately obtain…”

Section: Lemma 41 Implies Thatmentioning

confidence: 92%

“…The equation for ψ 1 stems from the well known behaviour of the corrector χ 1 . We know from [6,9,19] that χ 1 grows rapidly in a boundary layer of width O(1/ √ A) and decreases linearly in the cell interior. That is, for i = 1, 2, away from the boundary layer.…”

Section: Lemma 41 Implies Thatmentioning

confidence: 99%

From homogenization to averaging in cellular flows

Iyer

Komorowski

Novikov

et al. 2014

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A, in a twodimensional domain with L 2 cells. For fixed A, and L → ∞, the problem homogenizes, and has been well studied. Also well studied is the limit when L is fixed, and A → ∞. In this case the solution equilibrates along stream lines.In this paper, we show that if both A → ∞ and L → ∞, then a transition between the homogenization and averaging regimes occurs at A ≈ L 4 . When A ≫ L 4 , the principal Dirichlet eigenvalue is approximately constant. On the other hand, when A ≪ L 4 , the principal eigenvalue behaves likeσ(A)/L 2 , whereσ(A) ≈ √ AI is the effective diffusion matrix. A similar transition is observed for the solution of the exit time problem. The proof in the homogenization regime involves bounds on the second correctors. Miraculously, if the slow profile is quadratic, these estimates can be obtained using drift independent L p → L ∞ estimates for elliptic equations with an incompressible drift. This provides effective sub and super-solutions for our problem.

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Convection Enhanced Diffusion for Periodic Flows

Abstract: Abstract.We study the in uence of convection by periodic or cellular ows on the e ective di usivity o f a passive scalar transported by the uid when the molecular di usivity is small. The ows are generated by t wo-dimensional, steady, divergence-free, periodic velocity elds.

Cited by 212 publications

References 19 publications

Quenching of reaction by cellular flows

Quenching of reaction by cellular flows

Dynamics of probability density functions for decaying passive scalars in periodic velocity fields

From homogenization to averaging in cellular flows

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