Mario M. Alvarez

Experimental and computational investigation of the laminar flow structure in a stirred tank

The laminar flow structure inside an unbaffled stirred tank generated by a 6-blade radial flow impeller was characterized using flow visualization experiments, particle imaging velocimetry (PIV) experiments, and computational fluid dynamics (CFD) simulations. As expected from a previous study (Lamberto et al., 1996), the dominant flow structures in the system were two ring vortices above and below the impeller. These secondary circulation regions were segregated from the bulk of the flow and, for low impeller Reynolds numbers (5<Re<100), their positions and size were found to depend on Re and on the position of the impeller blades. The pumping capacity and circulation flow of the impeller were quantified and results indicate that the circulation flow was approximately 4 times the pumping capacity of the impeller.

Computational analysis of regular and chaotic mixing in a stirred tank reactor

Mixing in an unbaffled stirred tank equipped with a 6-blade radial flow impeller is examined computationally. Results demonstrate that the flow generated by constant impeller speed is partially chaotic. Under these conditions, the stretching experienced by fluid elements located within segregated toroidal regions of the flow increases slowly at a linear rate characteristic of regular (non-chaotic) flows. Within the bulk flow region, since the flow is chaotic, stretching increases at the expected exponential rate. The use of dynamic flow perturbations enhances mixing; when time-dependent RPM are applied, a globally chaotic flow is generated. We investigate computationally the effect on mixing performance of protocols in which the agitation speed oscillates between two steady values. Different frequencies of speed change and several RPM settings are compared. Under variable RPM schemes, the segregated torii are periodically relocated and fluid elements have the opportunity to abandon the regular regions. Under these conditions stretching increases exponentially throughout the entire flow domain. In general, mixing protocols with higher frequency of speed fluctuation produce the largest increase in stretching rates. Counter-intuitively, at a given RPM fluctuation frequency, stretching rates were higher for the lower RPM settings, either per revolutions or per unit of energy spent.

The intermaterial area density generated by time- and spatially periodic 2D chaotic flows

Abstract This paper explores in some detail the spatial structure and the statistical properties of partially mixed structures evolving under the effects of a time-periodic chaotic flow. Numerical simulations are used to examine the evolution of the interface between two fluids, which grows exponentially with a rate equal to the topological entropy of the flow. Such growth is much faster than predicted by the Lyapunov exponent of the flow. As time increases, the partially mixed system develops into a self-similar structure. Frequency distributions of interface density corresponding to different times collapse onto an invariant curve by a simple homogeneous scaling. This scaling behavior is a direct consequence of the generic asymptotic directionality property characteristic of 2D time-periodic flows. Striation thickness distributions (STDs) also acquire a time-invariant shape after a few (∼5–10) periods of the flow and are collapsed onto a single curve by standardization. It is also shown that STDs can be accurately predicted from distributions of stretching values, thus providing an effective method for calculation of STDs in complex flows.

The geometry of mixing in time-periodic chaotic flows.: I. asymptotic directionality in physically realizable flows and global invariant properties

Abstract This paper demonstrates that the geometry and topology of material lines in 2D time-periodic chaotic flows is controlled by a global geometric property referred to as asymptotic directionality . This property implies the existence of local asymptotic orientations at each point within the chaotic region, determined by the unstable eigendirections of the Jacobian matrix of the n th iterative of the Poincare map associated with the flow. Asymptotic directionality also determines the geometry of the invariant unstable manifolds, which are everywhere tangent to the field of asymptotic eigendirections. This fact is used to derive simple non-perturbative methods for reconstructing the global invariant manifolds to any desired level of detail. The geometric approach associated with the existence of a field of invariant unstable subspaces permits us to introduce the concept of a geometric global unstable manifold as an intrinsic property of a Poincare map of the flow, defined as a class of equivalence of integral manifolds belonging to the invariant unstable foliation. The connection between the geometric global unstable manifold and the global unstable manifold of hyperbolic periodic points is also addressed. Since material lines evolved by a chaotic flow are asymptotically attracted to the geometric global unstable manifold of the Poincare map, in a sense that will be made clear in the article, the reconstruction of unstable integral manifolds can be used to obtain a quantitative characterization of the topological and statistical properties of partially mixed structures. Two physically realizable systems are analyzed: closed cavity flow and flow between eccentric cylinders. Asymptotic directionality provides evidence of a global self-organizing structure characterizing chaotic flow which is analogous to that of Anosov diffeomorphisms, which turns out to be the basic prototype of mixing systems. In this framework, we show how partially mixed structures can be quantitatively characterized by a nonuniform stationary measure (different from the ergodic measure) associated with the dynamical system generated by the field of asymptotic unstable eigenvectors.

Invariant properties of a class of exactly solvable mixing transformations – A measure-theoretical approach to model the evolution of material lines advected by chaotic flows

Abstract This article analyzes the global invariant properties of a class of exactly solvable area-preserving mixing transformations of the two dimensional torus. Starting from the closed-form solution of the expanding sub-bundle, a nonuniform stationary measure μw (intrinsically different from the ergodic one) is derived analytically, providing a concrete example for which the connections between geometrical and measure-theoretical approaches to chaotic dynamics can be worked out explicitly. It is shown that the measure μw describes the nonuniform space-filling properties of material lines under the recursive action of the transformation. The implications of the results for physically realizable mixing systems are also addressed.

Self-Similar Spatio-Temporal Structure of Material Filaments in Chaotic Flows

This paper focuses on the evolution of material filaments in chaotic flows. This evolution is numerically calculated for three cases of the sine flow corresponding to a mainly regular, a mainly chaotic, and a globally chaotic case without discernible islands. In all cases, the stretched filament has an extremely non-uniform spatial distribution, with densities spanning several orders of magnitude and being the largest in the neighborhood of hyperbolic periodic points. Such spatial non-uniformity is a permanent property of time-periodic flows. As expected, the length of the filament increases exponentially in time, but due to the spatial non-uniformities in filament density, its growth is much faster than predicted by the Lyapunov exponent. The stretched filaments are self-similar in time, as revealed by their spatial structure and by the frequency distribution of the logarithm of the striation thickness, which is described by a family of curves that have an invariant shape and that can be collapsed onto a single curve by means of a simple scaling.