An efficient approach for eigenmode analysis of transient distributive mixing by the mapping method

Gorodetskyi, O Oleksandr; Speetjens, Mfm Michel; Anderson, Patrick Patrick

doi:10.1063/1.4712133

Cited by 11 publications

(26 citation statements)

References 32 publications

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“…In their 1951 paper, Spencer Several decades after this suggestion, a wealth of papers applied to different prototypical, industrial, and micro-mixers have shown that the mapping approach (i.e., the coarse-grained discretization of the action of a convective mixing field onto a discretized concentration field) provides an efficient computational tool for obtaining an accurate characterization of mixing properties in the purely kinematic case at a feasible computational cost. 15,35,36 The entries of this matrix contain the fractions of material from one part of the domain which is transferred to various parts of the domain when specific flow is applied. Figure 1 depicts how the entries of the mapping matrix are approximated.…”

Section: B Mapping Matrix Formalismmentioning

confidence: 99%

Spectral analysis of mixing in chaotic flows via the mapping matrix formalism: Inclusion of molecular diffusion and quantitative eigenvalue estimate in the purely convective limit

2012

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This paper extends the mapping matrix formalism to include the effects of molecular diffusion in the analysis of mixing processes in chaotic flows. The approach followed is Lagrangian, by considering the stochastic formulation of advection-diffusion processes via the Langevin equation for passive fluid particle motion. In addition, the inclusion of diffusional effects in the mapping matrix formalism permits to frame the spectral properties of mapping matrices in the purely convective limit in a quantitative way. Specifically, the effects of coarse graining can be quantified by means of an effective Péclet number that scales as the second power of the linear lattice size. This simple result is sufficient to predict the scaling exponents characterizing the behavior of the eigenvalue spectrum of the advection-diffusion operator in chaotic flows as a function of the Péclet number, exclusively from purely kinematic data, by varying the grid resolution. Simple but representative model systems and realistic physically realizable flows are considered under a wealth of different kinematic conditions-from the presence of large quasi-periodic islands intertwined by chaotic regions, to almost globally chaotic conditions, to flows possessing "sticky islands"-providing a fairly comprehensive characterization of the different numerical phenomenologies that may occur in the quantitative analysis of mapping matricesultimately of chaotic mixing processes. © 2012 American Institute of Physics

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Section: B Mapping Matrix Formalismmentioning

confidence: 99%

Spectral analysis of mixing in chaotic flows via the mapping matrix formalism: Inclusion of molecular diffusion and quantitative eigenvalue estimate in the purely convective limit

2012

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“…Concentration field C n admits eigenmode decomposition following,

bold-italicC_{n} = \sum_{k = 1}^{N} C_{k}^{0} h_{k} (n), h_{k} (n) = λ_{k}^{n} ν_{k},

with

{λ_{k}, ν_{k}}

the eigenvalue–eigenvector pairs of the mapping matrix and expansion coefficients

C_{k}^{0}

determined by the initial concentration distribution . The eigenmodes are ordered such that

true| λ_{1} true| \geq true| λ_{1} true| \geq \dots \geq true| λ_{N} true|

, where mass conservation dictates a trivial eigenmode

λ_{1} = 1

with associated uniform eigenfunction

ν_{1}

.…”

Section: Eigenmode Analysis Of Distributive Mixingmentioning

confidence: 99%

“…Both the Reacom and the TSE fail to accomplish a globally chaotic state, as evidenced by the existence of period‐3 elliptic islands, revealed by Poincaré sectioning (Figure ). Spectral analysis of the mapping matrix is an efficient tool for further investigating formation of coherent structures in the advective(–diffusive) transport . Eigenvectors, in fact, directly correlate with coherent structures as elliptic islands and unstable manifolds in the purely advective limit .…”

Section: Eigenmode Analysis Of Distributive Mixingmentioning

confidence: 99%

“…A key advantage of the mapping method, besides its flexibility, is that it enables spectral analysis of the advective–diffusive transport in terms of eigenmodes of the transport operator . Spectral analysis offer fundamental insights into the transport phenomena and the mapping method makes this accessible to complex flows.…”

Section: Introductionmentioning

confidence: 99%

“…However, a recent extension of the method enables incorporation of diffusion as well. [29][30][31][32] This expands the area of application tremendously.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analysis of Advective–Diffusive Transport in Complex Mixing Devices by the Diffusive Mapping Method

Gorodetskyi

Vivat

Speetjens

et al. 2015

Macro Theory & Simulations

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The present study concerns analysis of advective-diffusive transport in prototypical industrial mixing devices by the diffusive mapping method. The latter is a recent extension of the conventional mapping method for distributive mixing and enables application of this efficient technique to mixing flows, including molecular diffusion. Primary challenge is detailed numerical analysis of flow and mixing properties inside realistic systems with complex geometries. This is performed by combining the diffusive mapping method with a sophisticated computational scheme for flow simulations in domains with complex (moving) boundaries: the eXtended Finite Element Method (XFEM). The mixing properties are investigated through spectral analysis of the advectivediffusive transport by way of eigenmode decomposition of the mapping matrix. Key topic is the correlation between the evolution of concentration fields and invariant Lagrangian structures in the flow field.

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Analysis of the advection–diffusion mixing by the mapping method formalism in 3D open‐flow devices

et al. 2013

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This article extends the analysis of laminar mixing driven by a chaotic flow in the presence of diffusion to three-dimensional open-flow devices by means of the mapping-matrix method. The extended formulation of the mapping matrix recently proposed by Gorodetskyi et al. (2012) allows inclusion of the molecular diffusion in the mixing process. This provides an efficient numerical tool for understanding the interplay between a chaotic advective field and diffusion, especially for high Péclet numbers. As a prototypical open-flow device we consider the partitioned-pipe mixer. Results deriving from the application of the extended mapping method are compared with Galerkin simulations, and a close agreement is found. Short-term properties in the evolution of the concentration and the effect of axial diffusion are also addressed. © 2013 American Institute of Chemical Engineers

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An efficient approach for eigenmode analysis of transient distributive mixing by the mapping method

Cited by 11 publications

References 32 publications

Spectral analysis of mixing in chaotic flows via the mapping matrix formalism: Inclusion of molecular diffusion and quantitative eigenvalue estimate in the purely convective limit

Spectral analysis of mixing in chaotic flows via the mapping matrix formalism: Inclusion of molecular diffusion and quantitative eigenvalue estimate in the purely convective limit

Analysis of Advective–Diffusive Transport in Complex Mixing Devices by the Diffusive Mapping Method

Analysis of the advection–diffusion mixing by the mapping method formalism in 3D open‐flow devices

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