A simplified approach to compute distribution matrices for the mapping method

Singh, Mrityunjay; Galaktionov, O.S.; Meijer, H.E.H.; Anderson, Patrick Patrick

doi:10.1016/j.compchemeng.2009.01.021

Cited by 28 publications

(35 citation statements)

References 19 publications

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“…Details of the method, with validation and comparisons with other methods, are found in Kruijt et al 19 and Singh et al 24 …”

Section: A the Mapping Matrix And Its Constructionmentioning

confidence: 99%

“…A computationally efficient way to determine these fractions is by representation of the continuous material within cells by discrete markers, as shown schematically in Figure 1. 24 The domain is subdivided into N cells that each hold M j markers, which are advected by the flow during the time interval t 0 ≤ t ≤ t 0 + t. Entries of the mapping matrix then are defined by with M ij the number of markers -and i j the corresponding fraction -fraction of the M j markers in donor cell j at t = t 0 ending up in the recipient cell i at t = t 0 + t. The constructed mapping matrix allows to compute the redistribution of any arbitrary initial concentration C 0 ∈ R N ×1 of material over the N cells during the above time interval via…”

Section: A the Mapping Matrix And Its Constructionmentioning

confidence: 99%

“…For = 10 −8 this gives n ∈ [82, 196] (T = 0.56) and n ∈ [16, 134] (T = 1.6). However, typical engineering applications are restricted to much lower precision, say = 0.1(10 −2 ), leading to n ∈ [10,24]( [21,49]) and n ∈ [2, 5]( [4,9]) for T = 0.56 and T = 1.6, respectively. This reveals that, in particular for the chaotic flow, the earlier stages of the transient mixing become within the reach of the compact method.…”

Section: A Generic Properties Of the Dominant Eigenspacementioning

confidence: 99%

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

2012

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The mapping method is an efficient tool to investigate distributive mixing induced by periodic flows. Computed only once, the mapping matrix can be applied a number of times to determine the distribution of concentration inside the flow domain. Spectral analysis of the mapping matrix reveals detailed properties of the distributive mixing as all relevant information is stored in its eigenmodes. Any vector that describes a distribution of concentration can be expanded in the complete system of linearly independent eigenvectors of the mapping matrix. The rapid decay of the contribution of each mode in the eigenmode decomposition allows for a truncation of the eigenmode expansion from the whole spectrum to only the dominant eigenmodes (characterized by a decay rate significantly lower than the duration of the mixing process). This truncated decomposition adequately represents the distribution of concentration inside the flow domain already after a low number of periods, because contributions of all non-dominant eigenmodes rapidly become insignificant. The truncation is determined independently of the initial distribution of concentration and based on the decay rates of the eigenmodes, which are inversely proportional to the corresponding eigenvalues. Only modes with eigenvalues above a certain threshold are retained. The key advantage of the proposed compact eigenmode representation of the mapping method is that it includes practically relevant transient states and not just the asymptotic one. As such the method enables an eigenmode analysis of realistic problems yet with a substantial reduction in computational effort compared to the conventional approach.

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“…Details of the method, with validation and comparisons with other methods, are found in Kruijt et al 19 and Singh et al 24 …”

Section: A the Mapping Matrix And Its Constructionmentioning

confidence: 99%

Section: A the Mapping Matrix And Its Constructionmentioning

confidence: 99%

Section: A Generic Properties Of the Dominant Eigenspacementioning

confidence: 99%

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

2012

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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%

“…The flow domain is divided into a number of cells N which are numbered sequentially and N m markers are uniformly placed inside each cell and tracked in time from t = t 0 to t = t 0 + t. If the number of markers in the donor cell j is N j = N m at t = t 0 and the number of markers found after tracking in the recipient cell i is N ij at t = t 0 + t, then the entry M ij of the mapping matrix M is calculated 36 as…”

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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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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A simplified approach to compute distribution matrices for the mapping method

Cited by 28 publications

References 19 publications

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

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

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

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