Applied Mathematics and Computation 218 (2011) 809–816

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

New approaches in computational dynamics of the mixing flow

Adela Ionescu ⇑, Daniela ComanUniversity of Craiova, Applied Sciences and Environment Protection, 22 December 1989 Street No. 8, G5-3-3 200724 Craiova, Dolj, Romania

a r t i c l e i n f o a b s t r a c t

Dedicated to Professor H. M. Srivastava onthe Occasion of his Seventieth BirthAnniversary

Keywords:Vortex flowStretchingFoldingTurbulent mixingRare eventInteractive plot builder

0096-3003/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.amc.2011.03.142

⇑ Corresponding author.E-mail address: adelajaneta@yahoo.com (A. Ione

Between the most mature interdisciplinary areas, computational fluid dynamics (CFD)comes recently into focus. In the same time, it becomes more and more difficult to contrib-ute fundamental research to it. However, although it remains unpredictable how CFDdevelops, it is part of what makes it an exciting and attractive discipline.

This paper aims to exhibit some part of recent work in CFD. It concerns the qualitativeapproach of the turbulent behaviour of a mixing flow in an excitable media. Studying amixing for a flow implies the analysis of successive stretching and folding phenomena forits particles, the influence of parameters and initial conditions. In the previous works,the study of the 3D non-periodic models exhibited a quite complicated behaviour. In agree-ment with experiments, there were involved some significant events, the so-called ‘‘rareevents’’. The variation of parameters had a great influence on the length and surface defor-mations. The experiments were realized with a special vortex installation, it was used awell-known aquatic algae as biologic material, and the water as basic fluid.

In the paper there are presented some features of a qualitative comparative analysis ofthe model associated to the vortex flow technology. In the computational analysis therewere used the fast analysis tools of MAPLE11 soft, in order to check the ‘‘rare events’’and to complete the statistical analysis of the behaviour of the model.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction. Flow kinematics

The mixing theory appears in an area with far from complete solving problems: the flow kinematics. Its methods andtechniques developed the significant relation between turbulence and chaos. The turbulence is an important feature of dynamicsystems with few freedom degrees, the so-called ‘‘far from equilibrium systems’’. These are widespread between the modelsof excitable media, and a recent goal is to find a consistent and coherent theory to stand up that a mixing model in excitablemedia leads to a far from equilibrium model.

After a hundred years of stability study, the problems of flow kinematics are far from complete solving. Since the begin-nings, considering the stability of laminar flows with infinitesimal turbulences was a fruitful investigation method. The non-linearity could operate in the sense of stabilizing the flow, and then the basic flow is replaced with a new stable flow, whichis considered a secondary flow. This secondary flow could be further replaced by a tertiary flow, and so on. In fact, it is abouta bifurcations sequence, and the Couette flow could be the best example in this sense.

This context becomes more difficult if the non-linearity is in the sense of increasing of the growing rate of linear unstablemodes. In fact, we are talking about strong turbulence problems, an area which still needs a lot of substance.

The problems of turbulence were recently approached in a special standpoint [2,6,7]. It concerns, on one hand, a specialvortex technology which offers a lot of applications in all fields of bio-engineering, especially in processing the polluted flu-ids, and on the other hand, the mathematical and computational models used for handling this phenomena.

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scu).

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Starting from the importance of implementation of some optimized technologies for processing the polluted fluids, thebenefits of this technology are both of scientific and technologic type [7]. It concerns, one one-hand, finding new physic-mathematical models for describing at optimal parameters the turbulent mixing created by a vorticity structure, and fromtechnological standpoint, developing the vortex technology for handling the polluting materials.

Generally, the statistical idea of a flow is represented by a map

x ¼ UtðXÞ; X ¼ Ut¼0ðXÞ ð1Þ

We say that X is mapped in x after a time t. In the continuum mechanics the relation (1) is named flow, and it is a diffeomor-phism of class Ck. Moreover, (1) must satisfy the relation:

J ¼ detðDðUtðXÞÞÞ ¼ det@xi

@Xj

� �ð2Þ

or, equivalently,

J ¼ detðDðUtðXÞÞÞ ð3Þ

where D denotes the derivation with respect to the reference configuration, in this case X. The relation (3) implies two par-ticles, X1 and X2, which occupy the same position x at a moment. Non-topological behaviour (like break up, for example) isnot allowed.

With respect to X there is defined the basic measure of deformation, the deformation gradient, F, namely [2,6]:

F ¼ ðrXUtðXÞÞT ; Fij ¼@xi

@Xj

� �ð4Þ

whererX denotes differentiation with respect to X. According to (4), F is non singular. The basic measure for the deformationwith respect to x is the velocity gradient.

After defining the basic deformation of a material filament and the corresponding relation for the area of an infinitesimalmaterial surface, we can define the basic deformation measures: the length deformation k and surface deformation g, with therelations:

k ¼ ðC : MMÞ1=2; g ¼ ðdet FÞ � ðC�1 : NNÞ1=2 ð5Þ

with C (=FT � F) the Cauchy–Green deformation tensor, and the vectors M, N – the orientation versors in length and surfacerespectively, defined by:

M ¼ dXjdXj ; N ¼ dA

jdAj ð6Þ

Very often, in practice is used the scalar form of (5), namely:

k2 ¼ Cij �Mi � Nj;g2 ¼ ðdet FÞ � C�1ij �Mi � Nj ð7Þ

We say that the flow x = Ut(X) has a good mixing if the mean values D(lnk)/Dt and D(lng)/Dt are not decreasing to zero, forany initial position P and any initial orientations M and N.

Thus, there is defined [5] the deformation efficiency in length, ek = ek(X,M, t) of the material element dX, as:

ek ¼Dðln kÞ=Dt

ðD : DÞ1=2 6 1 ð8Þ

and similarly, the deformation efficiency in surface, eg = eg(X,N, t) of the area element dA: in the case of an isochoric flow (thejacobian equal 1), we have:

eg ¼Dðln gÞ=Dt

ðD : DÞ1=2 6 1 ð9Þ

where D is the deformation tensor, obtained by decomposing the velocity gradient in its symmetric and non-symmetric part,namely:

rv ¼ DþX ð10Þ

D ¼ rv þ ðrvÞT

2; the deformation tensor ðsymmetricÞ

X ¼ rv � ðrvÞT

2; the spinal tensor ðnonsymmetricÞ

The deformation tensor F and the associated tensors C, C�1, form the fundamental quantities for the analysis of deforma-tion of infinitesimal elements. In most cases the flow x = U t(X) is unknown and has to be obtained by integration from theEulerian velocity field. If this can be done analytically, then F can be obtained by differentiation of the flow with respect to

A. Ionescu, D. Coman / Applied Mathematics and Computation 218 (2011) 809–816 811

the material coordinates X. The flows of interest belong to two classes: flows with a special form of rv and flows with aspecial form of F. The second class is of very large interest, as it contains the so-called Constant Stretch History Motion –CSHM flows.

2. The analysis of the mixing behaviour in 3D excitable media

2.1. Computational features

The mathematical modelling has matched the experiments [2,5].Numeric simulation of 3D multiphase flows is currently in study, approaching different computational implements. In the

mathematical framework, the flow complexity implies the following three stages:

� modelling the global swirling streamlines;� local modelling of the concentrated vorticity structure;� introducing the elements of chaotic turbulence.

The mathematical model associated to the vortex phenomena is the 3D version of the widespread isochoric two-dimen-sional flow [6]

_x1 ¼ G � x2

_x2 ¼ K � G � x1; �1 < K < 1; G > 0

�ð11Þ

namely the following model is studied [2]:

_x¼G � x2

_x2 ¼ K � G � x1; �1 < K < 1; c ¼ const:_x3 ¼ c

8><>: ð12Þ

where for the third component, the axis z it was taken the rotation velocity, with a constant value, namely c.A lot of tests were realized for this model, both analytical and numerical. The statistical cases were very few, about 60.

There were approached various computational standpoints for the mixing phenomena, starting with the analysis of the mix-ing efficiency [2,3] for 3D model, and continuing with some phase-portrait analysis for 2D models [4].

In what follows it is approached another computational standpoint, in order to achieve more features for unifying this mix-ing theory. Namely, using specific tools of MAPLE11 soft, it is developed a phase-portrait analysis for 3D model in order to com-pare the performance of specific but different tools: the ‘‘phase-portrait’’ plot builder and the discrete – numeric plot builder.

The phase-portrait builder is a plot builder which realizes the phase-portrait for a differential equations system [1]. It is afast procedure, based on specific numeric methods for approximating the solution of the studied differential system. For thepresent aim it was chosen the classical method of Euler, in fact ‘‘forward Euler’’ method.

Fig. 1. Phase-portrait for the mixing model, the 1st parameter case.

812 A. Ionescu, D. Coman / Applied Mathematics and Computation 218 (2011) 809–816

From the parameters standpoint, there were chosen approximately the same values like in the analysis with discrete-nu-meric plot builder. The difference consists in choosing some classical initial conditions and varying the parameter KG. Inwhat follows, in the Figs. 1–5, there are presented some significant of the main studied cases. The parameter cases are la-belled on the figures Here follow Figs. 1–5.

2.2. The perturbed model analysis

Recently, it was extended the computational analysis for the mixing flow using new computational and numeric appli-ances [3].

The phase-portrait analysis brings new features concerning the influence on parameters on the model behaviour. To-gether with the mixing-efficiency analysis, it should complete the panel of the mixing flow behaviour in the context of afar from equilibrium model.

Fig. 2. Phase-portrait for the mixing model, the 2nd parameter case.

Fig. 3. Phase-portrait for the mixing model, the 3rd parameter case.

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Let us consider a modified version of (12), namely taking into account a perturbation in the first two components. Themodified model is:

_x1 ¼ G � x2 þ x1

_x2 ¼ K � G � x1 þ x2; �1 < K < 1; c ¼ const:_x3 ¼ c

8><>: ð13Þ

Few tests of the phase-portrait behaviour of (13) were performed, taking into account the same values for the parametersG and K as in (12), in order to get a more comprehensible comparative analysis of the models (12) and (13). Few of the impor-tant plots are exhibited in Figs. 6–9 that follow. Each parameter case is labelled on the figure. Here follow Figs. 6–9.

Fig. 4. Phase-portrait for the mixing model, the 4th parameter case.

Fig. 5. Phase-portrait for the mixing model, the 5th parameter case.

Fig. 6. Phase-portrait for the perturbed model, the 1st simulation case.

Fig. 7. Phase-portrait for the perturbed model, the 2nd simulation case.

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3. Qualitative results: remarks

Taking a first sight of the graphics, it is obvious that both in initial and perturbed case, the phenomenon is not linear; thereare relatively linear cases but also non-linear cases.

In fact it is about two types of phenomena for the same mixing model, and this is extremely important: on one hand, inthe phase-portrait analysis it is very important to notice that when modifying the parameters the behaviour is going to beperiodic for the same time units (Figs. 3, 5), and on the other hand, in the perturbed model analysis, when modifying theparameters the phenomenon becomes hard to control. All the Figs. 6–9 show this, therefore it can be matched with so-called‘‘rare events’’ [2,5], corresponding to the breakup of the simulation of the basic mixing model.

It is important to notice that for a little perturbation of the model, the behaviour is strongly changed, from an initial peri-odic trend to a far from control behaviour. It is exhibited in Fig. 3 comparing to Fig. 4 and Fig. 1 versus Fig. 2. This is very goodpointed out by taking into account the graphical ‘‘scene’’ as [y(t),z(t)] and the line trend like exp(t/2)[1].

Fig. 8. Phase-portrait for the perturbed model, the 3rd simulation case.

Fig. 9. Phase-portrait for the perturbed model, the 4th simulation case.

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Thus, it must be noticed that a little perturbation has a consistent influence on the model, going into a far from equilibriummodel. If there is annexed the irrationality of the length/surface versors values (an appliance used since the beginning of themixing study [2–5], a globally panel is obtained, with random distributed events. This time–space context consolidates thebasic statement that the turbulent mixing flows must be approached as chaotic systems. This is in fact regaining the idea ofa system/model high sensitive to initial conditions.

It is obvious that testing more values/parameters sets remains basic. If for 3D case, 60 statistical cases were enough forproving the special rare events, it is expected an acceptance testing of the consistency with the interdisciplinary area. Thus,the issues of repetitive phenomena (present in both analysis types above), give rise to achieve some appliances of chaoticdynamical systems. Also a next aim is testing more MAPLE11 appliances, both from numeric and graphic standpoint. Cumu-lating these features would produce numeric models, giving new research directions on the behaviour in excitable media.This would be a next aim.

References

[1] A.C. Hindmarsh, R.S. Stepleman (Eds.), Odepack, a Systemized Collection of ODE Solvers, Amsterdam, North Holland, 1983.[2] A. Ionescu, The structural stability of biological oscillators. Analytical contributions, Ph.D. Thesis, Polytechnic University of Bucarest, 2002.

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[3] A. Ionescu, M. Costescu, Computational aspects in excitable media. The case of vortex phenomena, Int. J. of Computers, Communications and Control,vol. I, Suppl. issue: Proceedings of ICCCC2006, 2006, pp 280–284.

[4] A. Ionescu, M. Costescu, The influence of parameters on the phaseportrait in the mixing model, International Journal of Computers, Communications andControl, vol. III, Suppl. issue: Proceedings of IJCCCC2008, 2008, pp. 333–337.

[5] A. Ionescu, Recent challenges in turbulence: computational features of turbulent mixing, Recent advances in Continuum Mechanics (Proceedings of the4th IASME/WSEAS Int. Conference on Continuum Mechanics), Mathematics and Computers in Science and Engineering, Cambridge, UK, 2009, pp. 29–38.

[6] J.M. Ottino, The Kinematics of Mixing: Stretching, Chaos and Transport, Cambridge University Press, 1989.[7] S.N. Savulescu, Applications of multiple flows in a vortex tube closed at one end, Internal Reports, CCTE, IAE (Institute of Applied Ecology) Bucarest,

1998.