Session AM - Chaos and Fractals.
ORAL session, Sunday morning, November 23
Saratoga, SMH

[AM.001] Chaotic Mixing with a Single Stirrer

In this work, we consider a fluid in a circular container equipped with a single stirrer rotating along a circular trajectory at constant speed. The fluid flow, considered incompressible, inviscid and two-dimensional, is modeled by a point vortex model whose vortex rotates in a circular container at constant angular speed. The mixing problem is addressed by considering the Hamiltonian form of the advection equations formulated in the frame of reference moving with the vortex. The dynamics of passive fluid particles is considered using dynamical systems theory. The bifurcation diagram reveals the presence of various degenerated fixed points and homoclinic/heteroclinic orbits whose nature varies for different parameter values. By considering an initially concentrated set of marker particles and using the various structures of the phase space in the bifurcation diagram, we generate complex dynamics which, in turn, can generate efficient mixing. The latter is studied using both numerical simulations and physical experiments

[AM.002] Chaotic Mixing of Viscoelastic Flows [1]

The effect of viscoelasticity on fluid mixing is investigated by measuring stretching fields [2] in a time-periodic flow. The stretching fields label the stable and unstable manifolds, and control the mixing dynamics. The presence of polymer in the fluid produces qualitative changes in the stretching fields, compared to their viscosity-matched Newtonian counterparts at the same Reynolds number (Re). Surprisingly, higher concentrations of lines of intense stretching as well as stronger folding of these lines are observed in the viscoelastic fluid. The stable and unstable manifolds cross more frequently, indicating that chaotic mixing will be enhanced. We also compare probability distributions of stretching in the viscoelastic and Newtonian cases. The total stretching over the entire flow is larger in the polymer solution, especially at the highest Re studied. These dynamic enhancements produced by viscoelasticity grow with Re.

[1]Work supported by the NSF-DMR under Grant No. 0072203 to Haverford College.

[2] Voth, G.A., G. Haller and J.P. Gollub, 2002, Phys. Rev. Letters 88, 254501

[AM.003] Mixing Process within a Spatially Periodic Three-dimensional Flow

The chaotic mixing of fluids in the partitioned-pipe mixer (PPM) is studied by using a numerically obtained velocity field. The PPM is composed of alternately placed horizontal and vertical plates in a cylindrical duct rotating with a constant angular velocity. This rotation and an axial pressure gradient generate the flow in the PPM. We examine the dependence of the mixing performance for many axial periods on the parameters of this system. We also calculate the maximum cross-sectional stretching rate, \lambda, of the infinitesimal line segments of the fluid. Under the assumption of the Stokes flow, we obtain two types of the evolution of \lambda, and find that they are distinguished by the distance between their cross-sectional initial positions and the lines of separation, defined as the set of cross-sectional initial locations of fluid particles which move to one of the leading edges of the plates within a specified periods. It is also found that the stretching of the fluid just inside the cylindrical wall after its separation by the leading edges mainly contributes the mixing in the PPM. Furthermore, we also examine the dependence of the mixing behavior on the Reynolds number.

[AM.004] Chaotic advection and reaction-advection-diffusion systems: chemical pattern formation in a blinking vortex flow

We present results of experimental studies of pattern formation in the Belousov-Zhabotinsky (BZ) chemical reaction in a blinking vortex flow. The flow is based on the model first proposed by Aref in 1984 to demonstrate chaotic advection. The flow is generated using a magnetohydrodynamic forcing technique, and the system can be filled either with an aqueous solution (for mixing studies) or the chemicals used for the BZ reaction. The patterns that develop for the BZ system are compared with those observed for the mixing of a passive impurity in the same flow. A mixing time is defined based on the intersection of graphs of decaying striation thickness (for the mixing of a passive impurity) and growing diffusive length scales. For the BZ experiments, if the mixing time is significantly smaller than typical correlation times for the BZ oscillation, then large-scale patterns form. If the mixing time is larger, then smaller-scale patterns form. The same sort of analysis is used to describe BZ patterns in an oscillating vortex chain, a system for which the mixing is also chaotic.

[AM.005] Down-scaling approaches for estimation of drag coefficients in flows over fractal surfaces at high Reynolds number

Traditional simulations of flow over rough surfaces usually assume that subgrid-scale (SGS) surface features are much smaller than the resolved surface features, and this scale separation allows for simple relationships between drag coefficients and the size of the SGS geometry features. However, for surfaces without scale separation (e.g. rough terrain), values of the drag coefficient can vary significantly and depend on the particular geometry and flow under consideration. Our goal is to develop methodolgy to estimate drag coefficient values from resolved scales for surfaces with a wide range of surface scales. Analysis is performed using 2-D, high-Re, RANS simulations of flow over ensembles of self-affine fractal surfaces. The complex fractal boundaries are represented using a variant of the immersed boundary method (Mohd-Yusof, 1997), which has been formulated for use in high-Re simulations. Several down-scaling methods for computing the value of the surface averaged drag coefficient from resolved scale information will be presented and compared. These methods include an extension of the dynamic framework (Germano et al., 1991) to SGS geometry, and an approach based on direct measurement from resolved scales (neglecting unresolved scales). To aide in the comparison, local contributions to the drag coefficient due to certain types of geometrical features will also be considered.

[AM.006] The Dynamics of Pulsating Flames on Annular Burners

A steady flame front on an annular burner becomes unstable to a number of different types of dynamics, as the control parameters of total flow rate and equivalence ratio are varied toward the extinction limit. The dynamics is best described in terms of events rather than modes. Some of the observed events are: counter-propagating fronts which annihilate, hot spots which move around the annulus, pairs of hot spots which suddenly appear in regions of extinction and, then, propagate in opposite directions, and trains of hot spots which begin at one point, move in opposite directions and then annihilate at the antipodal point.

[AM.007] Curvature and Torsion of Material Lines in Chaotic Flows

The deformation of a material line as it evolves in a chaotic flow is considered. Of particular interest are the curvature and torsion as a function of arclength along the line. These quantities are sufficient to define the intrinsic geometry of the line. Recent work ( J.-L. Thiffeault, (2003), in submission, arXiv:nlin.CD/0204069) has established an interesting connection between local line stretch and curvature for regions of high curvature.The present effort explores the relations between the finite-time Lyapunov exponents for the flow and curvature and torsion.

[AM.008] Transition to chaos in area preserving maps

As is well known, area-preserving maps are a characteristic aspect of Hamiltonian systems, such as those arising from the velocity field in incompressible fluids or magnetofluids. In this connection, a fundamental issue and a popular subject of numerous investigations [see, in particular, [1-4]] is the origin and physical interpretation of the onset of global chaos produced by the destruction of the last KAM curve in twist maps, such as the standard map. Despite previous attempts, several aspects of the phenomenon still escape a complete understanding and a rigorous description. Purpose of this paper is to present a critical analysis of previous approaches, with special reference to Greene's one [2]. As is well known the latter is based on the conjecture that the stochastic transition leading to the destruction of the last KAM curve in the standard map is due the linear destabilization of the elliptic points belonging to a peculiar family of invariants sets I(m,n) having all rational winding numbers and associated to the last KAM curve. These are discrete invariants sets I(m,n), the so-called rational iterates, having n points with m-periodicity. Purpose of this work is to analyze the nonlinear phenomena leading to the stochastic transition in the standard map for increasing values of the stochastic parameter and their effect on the destabilization of the invariant sets associated to the KAM curves, leading, ultimately, to the destruction of the KAM curves themselves. REFERENCES [1] B.V. Chirikov, Phys. Reports 52, 262 (1979). [2] J. Greene, J.Math.Physics 20, 1183 (1979). [3] I.C. Percival, in Nonlinear Dynamics and the beam-beam interaction, by M. Month and J.C. Herrera Eds., AIP Conference Proceedings No.67, American Institute of Physics, New York, 1979). [4] D.C. Escande and F. Doveil, Phys.Lett. 83A, 307 and 84A, 399 (1981).

[AM.009] Topological chaos in laminar flows

It has been shown that certain fluid motions have sufficient topological complexity for chaos to be `built in' to the system without regard for the details of the dynamics. The key to the analysis lies in classifying the motion of boundaries or periodic points according to the Thurston-Nielsen classification theorem. The theory provides a lower bound on the material stretch rate in the flow field, and recent investigations in Stokes flow suggest that this lower bound is physically significant. The ability of the theory to provide a useful quantitative bound on stretching without much knowledge of the dynamics makes this an attractive tool for achieving mixing enhancement in laminar flows. However, while investigations of topological chaos are increasing, the connection between the topological theory and the details of the corresponding fluid motion is still not clear. We will discuss the occurrence, characteristics, and importance of topological chaos in various laminar flows.

[AM.010] Reaction-advection-diffusion patterns in an annular chain of alternating vortices

Experiments are performed on long-range pattern formation for the Belousov-Zhabotinsky (BZ) chemical reaction in an alternating vortex chain. The system is arranged in an annular configuration to eliminate any end effects. The flow is generated by a magnetohydrodynamic technique in which a radial electrical current passing through the fluid interacts with an alternating magnetic field produced by an annular array of magnets below the fluid. The magnet array can be rotated either with a constant or an oscillating angular velocity (or a combination). If the array oscillates periodically, long-range transport is enhanced due to Lagrangian chaos, as was previously observed for a linear oscillating vortex chain. If the magnets rotate with a constant angular velocity, transport is almost ballistic, with a variance that grows quadratically with time. We are using the same flows to study traveling waves and front propagation in a reaction-diffusion-advection system – the BZ chemical reaction.

Part A of program listing