An efficient approach for eigenmode analysis of distributive
mixing by the mapping method
Citation for published version (APA):
Gorodetskyi, O., Speetjens, M. F. M., & Anderson, P. D. (2010). An efficient approach for eigenmode analysis of distributive mixing by the mapping method. Poster session presented at Mate Poster Award 2010 : 15th Annual Poster Contest.
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Polymer Technology
An efficient approach for eigenmode
analysis of distributive mixing by the
mapping method
/department of mechanical engineering
O. Gorodetskyi
M. F. M. Speetjens
P. D. Anderson
Introduction
Mapping matrix can be used to describe advective mixing and all information about transport properties is contained in its eigen-modes. Eigenmode decomposition of the mapping matrix allows to obtain a physical interpretation of the mixing process.
Objective
Investigation of the truncated eigenmode decomposition of the mapping matrix. Analysis of dominant eigenvectors and coeffi-cients in the eigenmode decomposition.
Methods
The present study proposes an efficient and accurate way for the eigenmode representation of arbitrary concentration distri-butions based on orthogonalization of the truncated eigenvector basis of the mapping matrix Φ. Mapping matrix concerns purely advective transport and for any initial distribution of
concentra-tion C0, concentration Cn after n periods can be found [1],
Cn= (Φ(Φ(...(Φ
| {z }
n times
C0)...).
The mapping admits eigenmode decomposition for the whole spectrum of eigenmodes [1]. The rapid decay of eigenmodes of Φ implies that eigenmode decomposition can be limited by M
dominant modes for the concentration distribution Cm,
Cm= M X k=1 Ck0λ m kvk+ O(λ m M +1),
where {λk, vk} are eigenvalue-eigenvector pairs. By dominant
we understand eigenmodes with |λk| close to 1. Second
dom-inant eigenvector represents the structure of the flow domain, see fig. 1.
T=0.56 T=0.8 T=1.12
Fig. 1Structure of the second dominant eigenvector in the case of TPSF. Dominant eigenmodes can be divided into groups according to the region where they are non-zeros, see fig. 2.
Fig. 2Structures of dominant eigenvectors in the case of TPSF: T=0.8.
Results
Coefficients C0
k entirely depends on the initial distribution of C0
and can be found by constructing a system of equation based on orthogonalized dominant eigenvectors. This system is build only for M dominant eigenvectors and does not require computation of the whole spectrum of eigenmodes. For the analysis of the truncated eigenmode decomposition were considered two cases of TPSF: T = 0.56 and T = 1.6 and number of dominant eigen-modes M = 400 and M = 300, respectively. Fig.3 illustrates the error ε between results obtained by truncated eigenmode decomposition and mapping method.
60 80 100 120 140 10−14 10−12 10−10 10−8 10−6 10−4 ε period 20 25 30 35 40 45 10−20 10−15 10−10 10−5 ε period T=0.56 T=1.6
Fig. 3Error between results obtained by using truncated eigenmode decomposition and mapping method.
The accuracy of the truncated decomposition can be regulated by the number of dominant eigenvectors M . Fig.4 illustrates magnitudes of the coefficients in the two cases of TPSF for initial concentration when top half of the flow domain has con-centration C = 1 and bottom C = 0.
0 100 200 300 400 25 50 |C 0|k k 0 50 100 150 200 250 300 60 120 |C 0|k k T=0.56 T=1.6
Fig. 4Magnitudes of coefficients|C0 k|.
Coefficients C0
k define which eigenmodes are excited for such C0
and only these modes are needed to describe mixing.
Conclusion
Eigenmode decomposition allows to perform physical analysis of the mixing. Controlling contribution of eigenmodes, quality of the mixture can be improved. Investigation of eigenmode de-composition allows to improve the control on the mixing.
References:
[1] M. K. Singh, M.F.M. Speetjens, P.D. Anderson 2009 Eigen-mode analysis of scalar transport in distributive mixing. Phys. Fluids 21, 093601
