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Chaotic mixing processes: New problems and computational issues
Chaos, Solitons & Fractals Vol. 6, pp. 425-438, 1995
Pergamon 0960-0779(94)00288-6
Copyright © 1995 ElsevierScienceLtd Printed in Great Britain. All rights reserved 0960--0779/95 $9.50 + .00
Chaotic Mixing Processes: New Problems and Computational Issues* J.M. OTTINO,
A. SOUVALIOTIS
and GUY METCALFE
Departmentof ChemicalEngineering,R. R. McCormickSchoolof Engineeringand AppliedScience NorthwesternUniversity,Evanston.Illinois60208-3120,USA Abstract - As mixing problems evolve beyond purely kinematic concerns, new issues appear. One issue is that analyses must often be based on numerical solutions of the Navier-Stokes, or more complex, equations; a second is the ability to deal with complexities involving the coupling of local and global dynamics, as occur, for example, in problems of aggregation and breakup. Both aspects are briefly considered, the bulk of the comments pertaining primarily to intrinsic limits of mixing simulations. 1. INTRODUCTION Close connection between experiments and computations places mixing squarely in the context of "experimental mathematics" [1]. Mixing arises due to repeated stretching and folding, an observation that goes back to Osborne Reynolds in 1894 [2]. Stretching and folding is, in turn, related to horseshoes and chaos, providing a nearly transparent connection between fluid mixing and chaos. During the past few years, the study of mixing has provided successful ground for the application of dynamical systems concepts leading to practical applications with direct experimental counterparts. The bulk of the developments to date has been with 2d flows, but there are now developments in volume preserving 3d flows, as well as indications that the ideas carry over to aperiodic flows and 2d turbulence. However, as realism increases, new issues appear. Non-trivial geometries or complex theology require numerical solutions of the momentum equations: there are no clean analytical solutions. A second issue is the ability (or inability) to handle the coupling 425
426
J.M. OTrlNO et al.
between local and global dynamics, as arise in problems with aggregation of particles, crystallization, chemical reaction, or drop breakup and coalescence [3]. Both issues are briefly considered, the bulk of the paper focusing on issues associated with computational errors and the limits of simulations. The paper is organized as follows: Section 2 provides a brief review of the most important kinematical issues (there is a large number of references for this material; one general reference is Ottino [4]; for reviews see [3-5-6-7]). Section 3 briefly outlines typical results obtained to-date. Section 4 focuses on new problems, involving local and global scales, such as autocatalytic processes in chaotic flows, and Section 5 sketches the nature and consequences or errors in computer simulations in mixing processes. Finally, Section 6 presents a perspective and concluding remarks. 2. KINEMATICS AND CHAOS
The study of mixing begins with the analysis of the motion due to an imposed velocity field; i.e. the study of the dynamical system drddt = v(x,t)
(1)
where v(x,t) is assumed known. The formal integration of (1) with the initial condition that x=X at t=0 gives x(t) = ¢(X,t) such that X = ¢(X,0)
(2)
i.e., x represents the position of particle X at time t. This is called the flow or motion. It is common to refer to a specific fluid particle as "particle X", when in fact we more precisely mean the fluid particle that was initially located at position X. Thus the viewpoint is purely kinematical: the dynamical system is the velocity field itself, all the fluid mechanical complexities being associated with obtaining v(x,t), a highly non-trivial step. To quantify the amount of stretch produced by the flow we follow a small material vector dX attached to the particle. The length stretch, 2L,is simply the ratio of the length at time t, dx, to the initial length, ~.=ldxl/IdXI. The orientation vector is m=dx/Idxl. The time evolution of the length stretch can be written as
Chaotic mixing processes
~- =
X
D:mm
=
ax
427
(3)
where D is the symn~tric part of the velocity gradient tensor, Vv. Steady area-preserving bounded two-dimensional flows mix poorly. If the flow is bounded, the streamlines are either closed or end at boundaries. The stretching within each closed region is poor: the stretching rate 0iX decays as 1/t - the stretching grows linearly in
time - and the efficiency of mixing decays to zero. The most studied cases of chaotic advection correspond to area-preserving flows. The onset of chaos destroys the regular structures of the steady flow and replaces them with the key structural elements of the chaotic flows: periodic points, their orbits, and their associated manifolds. A point P is said to be T-periodic if P=~p(P,nT)
(4)
for n = 1, 2, 3 .... but not for any t
3. TYPICAL RESULTS There have been a large number of theoretical and computational studies of mixing in chaotic flows. Typically, experiments and computations mark some initial fluid region
428
J.M. OTrlNO et al.
and track the resulting sets of orbits in the flow. In general chaotic flows generate regions of well mixed material with efficient stretching and folding and regions of unmixed material that are effectively segregated from the rest of the flow. Dye-structures in time-periodic flows evolve in an iterative fashion: an entire structure is mapped into a new structure with persistent large-scale features, but with ever finer scale features filling in during each period of the flow. Thin striations are produced at the expense of thicker ones and the average length scale decreases exponentially in time. The length stretch and striation thicknesses are inversely related. Islands of poor mixing form coherent regions that translate, periodically stretch and contract, and undergo a net rotation, but preserve their identity. The flow within islands is weakly rotational, the stretching is linear, and the rates of rotation are usually much slower than in the rest of the flow. Generally speaking the rate of generation of well-mixed areas is controlled by the unstable manifolds of the hyperbolic points belonging to the lowest order periodic points. The stretching is roughly proportional to the value of the eigenvalues and inversely proportional to the period of the point.
4. NEW PROBLEMS
A new set of issues arise in problems involving aggregation, breakup, and coalescence. The most extreme examples in terms of coupling between local and global scales are probably those involving interactions between competing autocatalytic processes and chaotic flows. The problem is how to track, visualize, and distinguish the shape and extent of dynamical manifolds and their interactions amongst the manifold tangle. Perhaps the simplest model containing elements of autocatalysis and competition is the following. A set of white particles ,W, are initially distributed uniformly over the domain and then advected by the flow. Two seed particles, green (G) and red (R), are also placed in selected regions of the flow and advected. When particles and seeds come within a distance ~i, there is an instantaneous reaction, either W + G ~ G
(5a)
W+R ~ R
(5b)
or
Chaotic mixing processes
429
Once reacted, particles become seeds and can react with other W particles. The chaotic manifolds provide the transport pathways coupling the local interactions of equations (5) to long range effects. Experimental analogues of these competing autocatalytic models are chiml crystallization of sodium chlorate [8] and autocatalytic bienzyme reactions [9]. As a model flow in which to place competing autocatalytic reactions, consider the eccentric cylinder (EC) flow, which has been extensively studied [10-11]. Reactions could also be profitably computed using even simpler mathematical flows, but the EC flow is suitable for direct experimental comparisons. In the EC geometry the fluid is contained between two cylinders with parallel but offset axes. The flow is forced by rotating the inner and outer cylinders. To produce a time-dependent velocity field (and therefore generate chaos in the flow), the cylinders rotate in a time-dependent discontinuous sequence. The "amount" of chaos is governed by the linear displacement of the cylinders and parametrized by the total angle of rotation of the outer cylinder. The calculations are for counter-rotating cylinders and a 720 ° displacement of the outer cylinder which gives a completely chaotic flow (no visible islands). The model generates a richer set of outcomes than might be expected. A green seed is initially placed at the hyperbolic point in the small gap (figure 1) and a red seed is placed anywhere else (for computational ease along the symmetry line). Depending on exactly where the red seed starts, the final results can be drastically different and counterintuitive. Figure la-c shows the results after five periods for three different red initial conditions. In figure la the green particles clearly dominate the end product. However, in figure 2b the initial position of the red seed has changed only slightly, but now the red particles dominate the end product. Figure 2c shows an intermediate case with a nearly equal end mixture of red and green. The three initial conditions differ by 3%. It is apparent that the outcome is very sensitive to the initial placement of seeds. The reason for this is that depending on just which manifold the red seeds starts, that manifold may completely (and quickly) surround the G- manifold, cutting it off from contact with the W particles; or, the G-manifold may surround the R-manifold; or, both manifolds may intertwine such that they both provide equal access to the unreacted W particles. However, the overall rate of colored seed generation is largely independent of the reaction outcome. Figure 2 shows a typical curve of the number of reacted particles versus
430
J.M. OTTINO et aL
time. Several regimes are apparent and characterized by growth rate exponents. The exponents qualitatively agree with results from simple alternating shear models. A more detailed presentation of these results will be published elsewhere. 5. COMPUTATIONAL ISSUES The most intuitively understandable definition of chaos is magnification of small errors and the impossibility of long time predictions. In light of the extreme sensitivity demonstrated in the previous example - one in which the velocity fie2d is known exactly - to what extent can we trust the results of computations? What would happen if the velocity field were known only approximately as is the case with numerically obtained velocity fields? Elements of "order within chaos" have long been recognized, though the chaos is often emphasized over the order. As we shall see what cannot be predicted is the detailed evolution of a
specific initial condition. The behavior of the system at large - the multitude
of initial conditions - is quite robust, and this is, in fact, what matters in many situations of practical interest. Mixing provides a nice example of these facts. The most stringent computation corresponds to an individual trajectory. Calculations show that small degree of error in the velocity field, order 0.1% or more, can produce large deviation in the computed trajectory of pathlines after just two or three periods. This does not mean, however, that meaningful computations are impossible. In fact, if this were taken literally, to the extent that an experiment can never be exactly duplicated - it is manifestly impossible to place a blob of dye in the same location twice experiments would be irreproducible as well. What matters is what gets computed. There are three types of errors in mixing simulations: (a) a discretization error Ax D, originating from the discretized solution of the velocity field, (b) the time integration error Ax N, appearing in the calculation of particle trajectories through the integration, and (c) the round-off error Axe, due to the finite accuracy of any computational machine. In calculating a particle trajectory, through a numerical integration method, an incremental error is introduced at every time step due to each of these three sources. Errors
Chaotic mixing processes
431
Figure 1. Initial conditions have a strong effect on the outcome of reactions in a chaotic flow. (a-c) show the state after five periods of a 720 ° chaotic flow in the eccentric cylinder geometry. The flow advects about 3500 particles (white dots when unreacted) along with red and green seeds; fi is 4% of the inner cylinder radius. Reactions occur throughout the duration of the flow. The green seed always starts at the saddle point of the steady flow. At the end of five periods (a) is klmost entirely green; (b) is almost entirely red; and, (c) is a nearly even mix of red and green. Small changes in the red initial condition lead to large changes in the final reaction product.
Chaotic mixing processes I
433
I J JI]
3.1 i0-' 0
°,-~
0
Q) Q) ~o
lO-Z
1.3
0 I
I llll
I
f
I
I0-2
I
I I I i]
I
I
I0-' t [periods]
Figure 2. Seed growth as a function of time; n is the total number of seed, i.e. the number of red plus green particles, and time is measured in periods. Several regions are apparent with each characterized by an exponent.
behave as a material line (equation (3)). The cumulative error is the sum of all the previously introduced errors. For steady two dimensional bounded flows the overall error in the location of an individual particle increases at most with t2. In chaotic flows all errors grow exponentially fast. In analytically based systems there is no discretization error and most of the blame can be placed on time integration. The time integration error, however, can be effectively controlled using higher order integration techniques and adaptive time steps and, in principle, it can be reduced to levels of round off error although the computational cost can be enormous. In general the round off error is
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J.M. OTTINO et al.
the least important although, according to the precision required it can become an overriding factor in analytically based systems [12]. There is however, a fundamental difference between the discretization and the time integration error. Once an approximate numerical solution has been obtained, the accuracy of the calculated particle trajectories is f'uted. Furthermore, the high computational cost associated with the discretized solution often prohibits the use of adequately refined meshes. Consequently if [AxDI>>IAxNI , the use of elaborate and costly integration schemes is simply a waste of time since overall error is controlled by the accuracy of the velocity field. It should be noted as well that the accuracy of the solutions cannot be assessed by a cursory evaluation of the velocity field (streamlines), as is often done in fluid mechanics. Two kinds of errors should be distinguished. In bounded flows streamlines are either closed or end on boundaries. Closed streamlines are characterized by a circulation time, the time it takes for a particle to return to its original location. Errors in the velocity field due to discretization modify the streamlines. In spatially symmetric flows the streamlines always close provided that the grid conforms to the symmetry of the geometry. In such cases the errors are normal to the streamlines, periodic function of time, and hence bounded. The largest errors, however, appear in the calculation of circulation times (they scale with the accuracy of the discretized solution). These errors are cumulative and lead to large deviations along the local direction of the striations. The differences appear primarily in the overall computed of particle paths and in the location of folds. The errors get magnified in time-periodic succession of piece-wise steady flows.
Figure (3) provides a nice illustration of these observations. Consider again the EC flow. A small line is placed on the symmetry line, close to the inner cylinder in the large gap, and advected using the 180° counter-rotating displacement protocol for five periods. Two velocity fields ar eused: (a) an analytical velocity field, and (b) a discretized finite element solution, based on a coarse grid, uniform in bi-polar coordinates. The overall appearance of the two stretched and folded lines are similar. All the major structures of the advected line appear to be faithfully reproduced and even the location of the major folds of the line appear to be close to their analytically predicted location. Nevertheless, the overall
Chaotic mixing processes
(
435
)
( (b) Figure 3. Deformation of a material line after five periods of a 180° chaotic flow in the eccentric cylinder geometry, according to (a) the exact velocity field, and, (b) a finite element solution based on a uniform in bi-polar coordinates 1 lx21 grid. Despite the overall similar appearance of the two lines, there is almost 150% difference in their overall length.
length of the line in Figure 3b is almost two and a half times the length of the line in Figure 3a. Furthermore, the pointwise error (i.e. the distance between two particles with the same initial position) is of the order of the system itself. For some of the initial conditions, this distance is larger than two times the diameter of the inner cylinder, even after three periods. Discretization manifests itself in several other ways. For example, the effect of the discretization error in Poincar6 sections affects the shepe and size of periodic islands. Chains of alternating elliptic and hyperbolic points are most seriously affected and some K.AM surfaces may appear broken. Small scale features are not expected to survive and perfunctory visual observations may lead to erroneous conclusions. Additional periodic points, which may not survive upon mesh refinement, might be created as well. Changes
436
J.M. O'VI'INOet
al,
in the location of periodic points are affected by the character of the point. Simulations and a linearized analysis suggest that the error in the location of a hyperbolic point in a symmetric area preserving map is, asymptotically, inversely proportional the value of its largest eigenvalue. Consequently, among hyperbolic points, the weakest ones are most affected (eigenvalues closest to one). These, fortunately, are the ones leading to the weakest spreading. On the other hand if the objective is to capture details of the large scale structure, rather than capturing a single trajectory, a 1% discretization error in the velocity field may be acceptable. The fact that computing lots of trajectories at once produces an accurate picture of the system at large might be grounded (heuristically) on the shadowing lemma and its extensions [13-14]. This seemingly encouraging result should be interpreted with care though. A discretization error affects all parts of the flow uniformly. Other sources of less uniformly distributed "errors" might lead to large variations in the advected patterns. Consider, for instance, a small change in the velocity field, due to viscoelasticity, when the flow first deviates from the Newtonian limit [15]. A 2-3% variation in the velocity field is enough to produce large-scale variations in advection patterns; for example, a large island present in the viscoelastic case might mot be present in the Newtonian case and viceversa. A more detailed presentation of all the results presented in this section will be published elsewhere. 6. CONCLUDING
REMARKS
The very fact that a flow is chaotic impinges on what can be computed. In turn what has to be computed depends on needs. Consider for example the case of passive particles convected by the flow without Brownian motion that bond irreversibly when their distance becomes less than d. The establishment of uniformity relies on a balance between the rate of coagulation or particle collisions, which depends on the square of coagulation distance d, and the rate of stirring. The kinetics of coagulation becomes tractable if the spatial distribution of clusters is uniform. Efficient stirring creates conditions for a mean field description governed by Smoluchowski's equations even though there is no molecular diffusion [16]. The case of two highly reactive species A and B dissolved in a common solvent undergoing an infinitely fast reaction A+B-->P is an example where such an approach runs into difficulties. The reaction is controlled by the amount of area created
Chaotic mixing processes
437
between reactants and necessitates detailed tracking of the stretched and folded interface were the diffusion controlled reaction occurs. In this case the physics is governed by the behavior at smallest scales and the important processes o¢,';ur within striations [17]. The impossibility of computing detailed pictures would seem to point out the need for statistical or coarse-grained descriptions. It is however, the element of order that makes a statistical approach difficult. Two kinds of tools are available and they have been used to a limited extent. The first tool is single-parameter scaling [18], which sits somehow in the broader context of multiplicative processes [19], the second is multifractal descriptions. [20-21]. It is apparent that there is a need for creative solutions to attack the multitude of problems suggested by stirring and mixing.
Acknowledgments I would like to thank the organizers of the First International Conference on Complex Systems in Computational Physics, Drs. G. Marshall and L. Lam, for the opportunity to present this work. This research has been supported by the Depami~ent of Energy, Office of Basic Energy Sciences. REFERENCES
* Presented at the First International Conference on Complex Systems in Computational Physics, Buenos Aires, Argentina. To be published in a special issue of Chaos,
Solitons, and Fractals. 1 j. Horgan. The death of proof, Scient. Amer., 269(4), 92-103 (1993). 2 O. Reynolds. Study of fluid motion by means of coloured bands, Nature, 50, 161-164 (1894). 3 J.M. Ottino. Unity and diversity in mixing: stretching, diffusion, breakup; and aggregation in chaotic flows, Phys. Fluids A, 5, 1417-1430 (1991). 4 J.M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge: Cambridge University Press (1989, reprinted 1990). 5 H. Aref. Chaotic advection of fluid particles, Phil. Trans. Roy. Soc. Lond. A333, 273281 (1991). 6 J.M. Ottino. Mixing, chaotic advection, and turbulence, Ann. Revs Fluid Mech., 22, 207-254 (1990).
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7 j. M. Ottino, F. J. Muzzio, M. Tjahjadi, J. G. Franjione, S. C. Jana, and H. A. Kusch. Chaos, symmetry, and self-similarity: exploiting order and disorder in mixing processes, Science, 257, 754-760 (1992). 8 D.K. Kondepudi, R.J. Kaufman, and N. Singh. Chiral symmetry breaking in sodium chlorate crystallization. Science, 250, 975-976 (1990). 9S. Cortassa, H. Sun, J.P. Kernevez and D. Thomas. Pattern formation in an immobilized bienzyme system. Biochem J., 269, 115-122 (1990). 10 P.D. Swanson and J.M. Ottino. A comparative computational and experimental study of chaotic mixing of viscous fluids, J. Fluid Mech., 213, 227-249 (1990). 11 j. Chaiken, R. Chevray, M. Tabor, and Q.M. Tan. Experimental study of Lagrangian turbulence in Stokes flow, Proc. Roy. Soc. Lond. A408, 165-174 (1986). 12 J.G. Franjione and J.M. Ottino. Feasibility of numerical tracking of material lines and surfaces in chaotic flows, Phys. Fluids., 30, 3641-3643 (1987). 13 S.M. Hammel, J.A. Yorke and C. Grebogi. Do numerical orbits of chaotic dynamical processes represent true orbits?, J. Complexity, 3, 136-145 (1987). 14 C. Grebogi, S.M. Hammel, J.A. Yorke and T. Sauer. Shadowing of physical trajectories in chaotic dynamics: Containment and refinement. Phys. Rev. Lett., 65, 1527-1530 (1990). 15 T.C. Niederkorn and J.M. Ottino. Mixing of viscoelastic fluids in time-periodic flows, J. Fluid Mech., 256, 243-268 (1993). 16F.J. Muzzio and J.M. Ottino. Coagulation in chaotic flows, Phys. Rev. A, 38, 25162524 (1988). 17 F.J. Muzzio and J.M. Ottino. Evolution of a lamellar system with diffusion and reaction: a scaling approach, Phys. Rev. Lett., 63, 47-50 (1989). 18F.J. Muzzio, P.D. Swanson, and J.M. Ottino. The statistics of stretching and stirring in chaotic flows, Phys. Fluids A, 5, 822-834 (1991). 19 S. Redner. Random multiplicative processes: an elementary tutorial. Amer. J. Phys. 58, 267-273 (1990). 20 K.R. Sreenivasan. Fractals and multifractals in turbulence. Ann. Revs. Fluid Mech., 23, 539-600 (1991). 21 F.J. Muzzio, C. Meneveau, P.D. Swanson, and J.M. Ottino. Scaling and multifractal properties of mixing in chaotic flows. Phys. Fluids A, 4, 1439-1456 (1992).
Pergamon 0960-0779(94)00288-6
Copyright © 1995 ElsevierScienceLtd Printed in Great Britain. All rights reserved 0960--0779/95 $9.50 + .00
Chaotic Mixing Processes: New Problems and Computational Issues* J.M. OTTINO,
A. SOUVALIOTIS
and GUY METCALFE
Departmentof ChemicalEngineering,R. R. McCormickSchoolof Engineeringand AppliedScience NorthwesternUniversity,Evanston.Illinois60208-3120,USA Abstract - As mixing problems evolve beyond purely kinematic concerns, new issues appear. One issue is that analyses must often be based on numerical solutions of the Navier-Stokes, or more complex, equations; a second is the ability to deal with complexities involving the coupling of local and global dynamics, as occur, for example, in problems of aggregation and breakup. Both aspects are briefly considered, the bulk of the comments pertaining primarily to intrinsic limits of mixing simulations. 1. INTRODUCTION Close connection between experiments and computations places mixing squarely in the context of "experimental mathematics" [1]. Mixing arises due to repeated stretching and folding, an observation that goes back to Osborne Reynolds in 1894 [2]. Stretching and folding is, in turn, related to horseshoes and chaos, providing a nearly transparent connection between fluid mixing and chaos. During the past few years, the study of mixing has provided successful ground for the application of dynamical systems concepts leading to practical applications with direct experimental counterparts. The bulk of the developments to date has been with 2d flows, but there are now developments in volume preserving 3d flows, as well as indications that the ideas carry over to aperiodic flows and 2d turbulence. However, as realism increases, new issues appear. Non-trivial geometries or complex theology require numerical solutions of the momentum equations: there are no clean analytical solutions. A second issue is the ability (or inability) to handle the coupling 425
426
J.M. OTrlNO et al.
between local and global dynamics, as arise in problems with aggregation of particles, crystallization, chemical reaction, or drop breakup and coalescence [3]. Both issues are briefly considered, the bulk of the paper focusing on issues associated with computational errors and the limits of simulations. The paper is organized as follows: Section 2 provides a brief review of the most important kinematical issues (there is a large number of references for this material; one general reference is Ottino [4]; for reviews see [3-5-6-7]). Section 3 briefly outlines typical results obtained to-date. Section 4 focuses on new problems, involving local and global scales, such as autocatalytic processes in chaotic flows, and Section 5 sketches the nature and consequences or errors in computer simulations in mixing processes. Finally, Section 6 presents a perspective and concluding remarks. 2. KINEMATICS AND CHAOS
The study of mixing begins with the analysis of the motion due to an imposed velocity field; i.e. the study of the dynamical system drddt = v(x,t)
(1)
where v(x,t) is assumed known. The formal integration of (1) with the initial condition that x=X at t=0 gives x(t) = ¢(X,t) such that X = ¢(X,0)
(2)
i.e., x represents the position of particle X at time t. This is called the flow or motion. It is common to refer to a specific fluid particle as "particle X", when in fact we more precisely mean the fluid particle that was initially located at position X. Thus the viewpoint is purely kinematical: the dynamical system is the velocity field itself, all the fluid mechanical complexities being associated with obtaining v(x,t), a highly non-trivial step. To quantify the amount of stretch produced by the flow we follow a small material vector dX attached to the particle. The length stretch, 2L,is simply the ratio of the length at time t, dx, to the initial length, ~.=ldxl/IdXI. The orientation vector is m=dx/Idxl. The time evolution of the length stretch can be written as
Chaotic mixing processes
~- =
X
D:mm
=
ax
427
(3)
where D is the symn~tric part of the velocity gradient tensor, Vv. Steady area-preserving bounded two-dimensional flows mix poorly. If the flow is bounded, the streamlines are either closed or end at boundaries. The stretching within each closed region is poor: the stretching rate 0iX decays as 1/t - the stretching grows linearly in
time - and the efficiency of mixing decays to zero. The most studied cases of chaotic advection correspond to area-preserving flows. The onset of chaos destroys the regular structures of the steady flow and replaces them with the key structural elements of the chaotic flows: periodic points, their orbits, and their associated manifolds. A point P is said to be T-periodic if P=~p(P,nT)
(4)
for n = 1, 2, 3 .... but not for any t
3. TYPICAL RESULTS There have been a large number of theoretical and computational studies of mixing in chaotic flows. Typically, experiments and computations mark some initial fluid region
428
J.M. OTrlNO et al.
and track the resulting sets of orbits in the flow. In general chaotic flows generate regions of well mixed material with efficient stretching and folding and regions of unmixed material that are effectively segregated from the rest of the flow. Dye-structures in time-periodic flows evolve in an iterative fashion: an entire structure is mapped into a new structure with persistent large-scale features, but with ever finer scale features filling in during each period of the flow. Thin striations are produced at the expense of thicker ones and the average length scale decreases exponentially in time. The length stretch and striation thicknesses are inversely related. Islands of poor mixing form coherent regions that translate, periodically stretch and contract, and undergo a net rotation, but preserve their identity. The flow within islands is weakly rotational, the stretching is linear, and the rates of rotation are usually much slower than in the rest of the flow. Generally speaking the rate of generation of well-mixed areas is controlled by the unstable manifolds of the hyperbolic points belonging to the lowest order periodic points. The stretching is roughly proportional to the value of the eigenvalues and inversely proportional to the period of the point.
4. NEW PROBLEMS
A new set of issues arise in problems involving aggregation, breakup, and coalescence. The most extreme examples in terms of coupling between local and global scales are probably those involving interactions between competing autocatalytic processes and chaotic flows. The problem is how to track, visualize, and distinguish the shape and extent of dynamical manifolds and their interactions amongst the manifold tangle. Perhaps the simplest model containing elements of autocatalysis and competition is the following. A set of white particles ,W, are initially distributed uniformly over the domain and then advected by the flow. Two seed particles, green (G) and red (R), are also placed in selected regions of the flow and advected. When particles and seeds come within a distance ~i, there is an instantaneous reaction, either W + G ~ G
(5a)
W+R ~ R
(5b)
or
Chaotic mixing processes
429
Once reacted, particles become seeds and can react with other W particles. The chaotic manifolds provide the transport pathways coupling the local interactions of equations (5) to long range effects. Experimental analogues of these competing autocatalytic models are chiml crystallization of sodium chlorate [8] and autocatalytic bienzyme reactions [9]. As a model flow in which to place competing autocatalytic reactions, consider the eccentric cylinder (EC) flow, which has been extensively studied [10-11]. Reactions could also be profitably computed using even simpler mathematical flows, but the EC flow is suitable for direct experimental comparisons. In the EC geometry the fluid is contained between two cylinders with parallel but offset axes. The flow is forced by rotating the inner and outer cylinders. To produce a time-dependent velocity field (and therefore generate chaos in the flow), the cylinders rotate in a time-dependent discontinuous sequence. The "amount" of chaos is governed by the linear displacement of the cylinders and parametrized by the total angle of rotation of the outer cylinder. The calculations are for counter-rotating cylinders and a 720 ° displacement of the outer cylinder which gives a completely chaotic flow (no visible islands). The model generates a richer set of outcomes than might be expected. A green seed is initially placed at the hyperbolic point in the small gap (figure 1) and a red seed is placed anywhere else (for computational ease along the symmetry line). Depending on exactly where the red seed starts, the final results can be drastically different and counterintuitive. Figure la-c shows the results after five periods for three different red initial conditions. In figure la the green particles clearly dominate the end product. However, in figure 2b the initial position of the red seed has changed only slightly, but now the red particles dominate the end product. Figure 2c shows an intermediate case with a nearly equal end mixture of red and green. The three initial conditions differ by 3%. It is apparent that the outcome is very sensitive to the initial placement of seeds. The reason for this is that depending on just which manifold the red seeds starts, that manifold may completely (and quickly) surround the G- manifold, cutting it off from contact with the W particles; or, the G-manifold may surround the R-manifold; or, both manifolds may intertwine such that they both provide equal access to the unreacted W particles. However, the overall rate of colored seed generation is largely independent of the reaction outcome. Figure 2 shows a typical curve of the number of reacted particles versus
430
J.M. OTTINO et aL
time. Several regimes are apparent and characterized by growth rate exponents. The exponents qualitatively agree with results from simple alternating shear models. A more detailed presentation of these results will be published elsewhere. 5. COMPUTATIONAL ISSUES The most intuitively understandable definition of chaos is magnification of small errors and the impossibility of long time predictions. In light of the extreme sensitivity demonstrated in the previous example - one in which the velocity fie2d is known exactly - to what extent can we trust the results of computations? What would happen if the velocity field were known only approximately as is the case with numerically obtained velocity fields? Elements of "order within chaos" have long been recognized, though the chaos is often emphasized over the order. As we shall see what cannot be predicted is the detailed evolution of a
specific initial condition. The behavior of the system at large - the multitude
of initial conditions - is quite robust, and this is, in fact, what matters in many situations of practical interest. Mixing provides a nice example of these facts. The most stringent computation corresponds to an individual trajectory. Calculations show that small degree of error in the velocity field, order 0.1% or more, can produce large deviation in the computed trajectory of pathlines after just two or three periods. This does not mean, however, that meaningful computations are impossible. In fact, if this were taken literally, to the extent that an experiment can never be exactly duplicated - it is manifestly impossible to place a blob of dye in the same location twice experiments would be irreproducible as well. What matters is what gets computed. There are three types of errors in mixing simulations: (a) a discretization error Ax D, originating from the discretized solution of the velocity field, (b) the time integration error Ax N, appearing in the calculation of particle trajectories through the integration, and (c) the round-off error Axe, due to the finite accuracy of any computational machine. In calculating a particle trajectory, through a numerical integration method, an incremental error is introduced at every time step due to each of these three sources. Errors
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Figure 1. Initial conditions have a strong effect on the outcome of reactions in a chaotic flow. (a-c) show the state after five periods of a 720 ° chaotic flow in the eccentric cylinder geometry. The flow advects about 3500 particles (white dots when unreacted) along with red and green seeds; fi is 4% of the inner cylinder radius. Reactions occur throughout the duration of the flow. The green seed always starts at the saddle point of the steady flow. At the end of five periods (a) is klmost entirely green; (b) is almost entirely red; and, (c) is a nearly even mix of red and green. Small changes in the red initial condition lead to large changes in the final reaction product.
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Figure 2. Seed growth as a function of time; n is the total number of seed, i.e. the number of red plus green particles, and time is measured in periods. Several regions are apparent with each characterized by an exponent.
behave as a material line (equation (3)). The cumulative error is the sum of all the previously introduced errors. For steady two dimensional bounded flows the overall error in the location of an individual particle increases at most with t2. In chaotic flows all errors grow exponentially fast. In analytically based systems there is no discretization error and most of the blame can be placed on time integration. The time integration error, however, can be effectively controlled using higher order integration techniques and adaptive time steps and, in principle, it can be reduced to levels of round off error although the computational cost can be enormous. In general the round off error is
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the least important although, according to the precision required it can become an overriding factor in analytically based systems [12]. There is however, a fundamental difference between the discretization and the time integration error. Once an approximate numerical solution has been obtained, the accuracy of the calculated particle trajectories is f'uted. Furthermore, the high computational cost associated with the discretized solution often prohibits the use of adequately refined meshes. Consequently if [AxDI>>IAxNI , the use of elaborate and costly integration schemes is simply a waste of time since overall error is controlled by the accuracy of the velocity field. It should be noted as well that the accuracy of the solutions cannot be assessed by a cursory evaluation of the velocity field (streamlines), as is often done in fluid mechanics. Two kinds of errors should be distinguished. In bounded flows streamlines are either closed or end on boundaries. Closed streamlines are characterized by a circulation time, the time it takes for a particle to return to its original location. Errors in the velocity field due to discretization modify the streamlines. In spatially symmetric flows the streamlines always close provided that the grid conforms to the symmetry of the geometry. In such cases the errors are normal to the streamlines, periodic function of time, and hence bounded. The largest errors, however, appear in the calculation of circulation times (they scale with the accuracy of the discretized solution). These errors are cumulative and lead to large deviations along the local direction of the striations. The differences appear primarily in the overall computed of particle paths and in the location of folds. The errors get magnified in time-periodic succession of piece-wise steady flows.
Figure (3) provides a nice illustration of these observations. Consider again the EC flow. A small line is placed on the symmetry line, close to the inner cylinder in the large gap, and advected using the 180° counter-rotating displacement protocol for five periods. Two velocity fields ar eused: (a) an analytical velocity field, and (b) a discretized finite element solution, based on a coarse grid, uniform in bi-polar coordinates. The overall appearance of the two stretched and folded lines are similar. All the major structures of the advected line appear to be faithfully reproduced and even the location of the major folds of the line appear to be close to their analytically predicted location. Nevertheless, the overall
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( (b) Figure 3. Deformation of a material line after five periods of a 180° chaotic flow in the eccentric cylinder geometry, according to (a) the exact velocity field, and, (b) a finite element solution based on a uniform in bi-polar coordinates 1 lx21 grid. Despite the overall similar appearance of the two lines, there is almost 150% difference in their overall length.
length of the line in Figure 3b is almost two and a half times the length of the line in Figure 3a. Furthermore, the pointwise error (i.e. the distance between two particles with the same initial position) is of the order of the system itself. For some of the initial conditions, this distance is larger than two times the diameter of the inner cylinder, even after three periods. Discretization manifests itself in several other ways. For example, the effect of the discretization error in Poincar6 sections affects the shepe and size of periodic islands. Chains of alternating elliptic and hyperbolic points are most seriously affected and some K.AM surfaces may appear broken. Small scale features are not expected to survive and perfunctory visual observations may lead to erroneous conclusions. Additional periodic points, which may not survive upon mesh refinement, might be created as well. Changes
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in the location of periodic points are affected by the character of the point. Simulations and a linearized analysis suggest that the error in the location of a hyperbolic point in a symmetric area preserving map is, asymptotically, inversely proportional the value of its largest eigenvalue. Consequently, among hyperbolic points, the weakest ones are most affected (eigenvalues closest to one). These, fortunately, are the ones leading to the weakest spreading. On the other hand if the objective is to capture details of the large scale structure, rather than capturing a single trajectory, a 1% discretization error in the velocity field may be acceptable. The fact that computing lots of trajectories at once produces an accurate picture of the system at large might be grounded (heuristically) on the shadowing lemma and its extensions [13-14]. This seemingly encouraging result should be interpreted with care though. A discretization error affects all parts of the flow uniformly. Other sources of less uniformly distributed "errors" might lead to large variations in the advected patterns. Consider, for instance, a small change in the velocity field, due to viscoelasticity, when the flow first deviates from the Newtonian limit [15]. A 2-3% variation in the velocity field is enough to produce large-scale variations in advection patterns; for example, a large island present in the viscoelastic case might mot be present in the Newtonian case and viceversa. A more detailed presentation of all the results presented in this section will be published elsewhere. 6. CONCLUDING
REMARKS
The very fact that a flow is chaotic impinges on what can be computed. In turn what has to be computed depends on needs. Consider for example the case of passive particles convected by the flow without Brownian motion that bond irreversibly when their distance becomes less than d. The establishment of uniformity relies on a balance between the rate of coagulation or particle collisions, which depends on the square of coagulation distance d, and the rate of stirring. The kinetics of coagulation becomes tractable if the spatial distribution of clusters is uniform. Efficient stirring creates conditions for a mean field description governed by Smoluchowski's equations even though there is no molecular diffusion [16]. The case of two highly reactive species A and B dissolved in a common solvent undergoing an infinitely fast reaction A+B-->P is an example where such an approach runs into difficulties. The reaction is controlled by the amount of area created
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between reactants and necessitates detailed tracking of the stretched and folded interface were the diffusion controlled reaction occurs. In this case the physics is governed by the behavior at smallest scales and the important processes o¢,';ur within striations [17]. The impossibility of computing detailed pictures would seem to point out the need for statistical or coarse-grained descriptions. It is however, the element of order that makes a statistical approach difficult. Two kinds of tools are available and they have been used to a limited extent. The first tool is single-parameter scaling [18], which sits somehow in the broader context of multiplicative processes [19], the second is multifractal descriptions. [20-21]. It is apparent that there is a need for creative solutions to attack the multitude of problems suggested by stirring and mixing.
Acknowledgments I would like to thank the organizers of the First International Conference on Complex Systems in Computational Physics, Drs. G. Marshall and L. Lam, for the opportunity to present this work. This research has been supported by the Depami~ent of Energy, Office of Basic Energy Sciences. REFERENCES
* Presented at the First International Conference on Complex Systems in Computational Physics, Buenos Aires, Argentina. To be published in a special issue of Chaos,
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