O.S. Galaktionov

An adaptive front tracking technique for three-dimensional transient flows

An adaptive technique, based on both surface stretching and surface curvature analysis for tracking strongly deforming fluid volumes in three-dimensional flows is presented. The efficiency and accuracy of the technique are demonstrated for two- and three-dimensional flow simulations. For the two-dimensional test example, the results are compared with results obtained using a different tracking approach based on the advection of a passive scalar. Although for both techniques roughly the same structures are found, the resolution for the front tracking technique is much higher. In the three-dimensional test example, a spherical blob is tracked in a chaotic mixing flow. For this problem, the accuracy of the adaptive tracking is demonstrated by the volume conservation for the advected blob. Adaptive front tracking is suitable for simulation of the initial stages of fluid mixing, where the interfacial area can grow exponentially with time. The efficiency of the algorithm significantly benefits from parallelization of the code

Analysis of mixing in three-dimensional time-periodic cavity flows

A method to locate periodic structures in general three-dimensional Stokes flows with time-periodic boundary conditions is presented and applied to mixing cavity flows. Numerically obtained velocity fields and particle tracking schemes are used to provide displacement and stretching fields. From these the location and identification of periodic points can be derived. The presence or absence of these periodic points allows a judgement on the quality of the mixing process. The technique is general and efficient, and applicable to mixing flows for which no analytical velocity field is available (the case for all three-dimensional flows considered in this paper). Results are presented for three different mixing protocols in a three-dimensional time-periodic cavity flow, serving as an accessible test case for the methods developed. A major result is that periodic lines are obtained for these three-dimensional flows. These lines can be complex in geometry and their nature can change along a line from hyperbolic to elliptic. They can serve as practical criteria in the optimization of three-dimensional mixing processes

Analysis and optimization of Kenics static mixers

Abstract The mapping approach is applied to study the distributive mixing in the, widely industrially used, Kenics static mixer. The flexibility of the mapping method makes it possible to study and compare thousands of different mixer layouts and perform optimization with respect to macroscopic homogenization efficiency and interface generation. In the paper two different designs of the mixer are investigated. The conventional mixer with sequentially different twisted blades and adesign where the twist direction is maintained constant. In both cases the blade twist angle is varied. Recommendations are given in the choice of the design of the mixer dependent on the desired structure of mixing.

A global, multi-scale simulation of laminar fluid mixing : the extended mapping method

Abstract We present a global, multi-scale model of fluid mixing in laminar flows, which describes the evolution of the spatial distribution of coarse-grain concentration and interfacial area in a mixture of two fluids with identical viscosity with no interfacial tension. This results in an efficient computational tool for mixing analysis, able to evaluate mixing dynamics and identify mixing problems such as dead zones (islands), applicable to realistic mixing devices. The flow domain is divided into cells, and large-scale variations in composition are tracked by following the cell-average concentrations of one fluid, using the mapping method developed previously. Composition fluctuations smaller than the cell size are represented by cell values of the area tensor which quantifies the amount, shape, and orientation of the interfacial area within each cell. The method is validated by comparison with an explicit interface tracking calculation. We show examples for 2D, time-periodic flows in a lid-driven rectangular cavity. The highly non-uniform time evolution of the spatial distribution of interfacial area can be determined with very low computational effort. Cell-to-cell differences in interfacial area of three orders of magnitude or more are found. It is well known that, for globally chaotic flows, the microstructural pattern becomes self-similar, and interfacial area increases exponentially with time. This behavior is also captured well by the extended mapping method. The present calculations are 2D, but the method can readily be applied in 3D problems.

Chaotic fluid mixing in non-quasi-static time-periodic cavity flows

Abstract Fluid mixing in a two-dimensional square cavity with a time-periodic pulsating lid velocity is studied. A spectral element technique for spatial discretization is combined with a continuous projection scheme for temporal discretization to obtain a numerical representation of the non-quasi-static velocity field in the cavity. It is well known that mixing in a cavity with a steady lid velocity results in linear mixing of fluid inside the cavity. Here, it is shown that superposition of a pulsating component on the steady lid velocity can lead to chaotic mixing in the core of the cavity. An extra steady motion of the opposite cavity wall, resulting in a small perturbation to the original flow, causes the chaotically mixed region to be spread over almost the whole cavity. Poincare and periodic point analysis reveal the main characteristics for these transient time-periodic flows, and elucidate the details and properties of the chaotic mixing in these flows.

A mapping approach for three-dimensional distributive mixing analysis

Abstract A mapping approach is proposed for the numerical simulation of distributive mixing in three-dimensional laminar flows. The method is based on a spatial discretization of the locally averaged concentration of fluid components in the mixture (the so-called “coarse grain density”). A distribution matrix, that describes the changes in component concentration, is composed. The proposed method makes it possible to rapidly predict the (short- and long-term) mixing performances, and to compare a large number of different mixing protocols in an efficient way. The consistency and accuracy of this algorithm is validated by comparing the results obtained on grids with different spatial resolution, and by comparison with front tracking results. The technique is evaluated in a prototype mixing flow in a cubic cavity, generated by sliding opposite walls. Different mixing protocols are compared quantitatively, and result in optimal mixing protocol parameters.

The mapping method for mixing optimization - Part II : transport in a corotating twin-screw extruder

Abstract The application of the mapping method [1, 2] on fully three dimensional flows with dynamic boundaries is presented. The computational domain is a (three dimensional) section of a particular type of screw element in a closely intermeshing corotating twin screw extruder. Screw elements include transport, different kneading, and counterconveying zones, for each of which a mapping matrix has to be constructed. In the future, different screw assemblies could be studied by combining distinct mapping matrices, much like an actual screw is assembled from distinct modules. In this paper, the principle and applicability of the mapping method is demonstrated by the analysis of a fully filled transporting section of the corotating twin screw extruder. Concentration and residence time distributions can be computed straightforwardly and the volume average intensity of segregation is chosen as a characteristic mixing measure to quantitatively describe distributive mixing.

The Mapping Method for Mixing Optimization Part I: The Multiflux Static Mixer

Abstract The development of the mapping method [1, 2] and its application to viscous flow in a (multiflux) static mixer [3] is discussed. The mapping method describes the repeated transport of fluid from one domain to another and computes the concentration distribution, yielding a unique mixing measure, like e.g. the intensity and/or scale of segregation [4,5], the residence time distribution and structure parameters like e.g. the area tensor [6]. In the multiflux static mixer, the domain is a (two dimensional) cross-section of the flow channels at the beginning and end of a single element. Since static mixers are build of a sequence of similar elements, we only need to determine one mapping matrix (a computationally expensive operation). To simulate the complete mixer, the mapping matrix is repeatedly applied (a computationally cheap operation). Three different geometries are analyzed. It is shown that an approach like the mapping method is a prerequisite in order to obtain meaningful results after a number of mixer elements. Moreover, it enables optimization of industrial mixers.

Structure development during chaotic mixing in the journal bearing flow

Laminar mixing in the two-dimensional time-periodic Stokes flows between eccentric cylinders [journal bearing flow (JBF)] is studied using the extended mapping method [Galaktionov et al., Int. J. Multiphase Flows 28, 497 (2002)] with the emphasis on the material stretching, e.g., the interface generation abilities, of the flow. With this flexible and computational advantageous method both, the macroscopic material transport and the evolution of the microstructure can be described. It enables a convenient way for studying the material stretching in the flow and moreover, it provides spatial distribution of locally averaged stretching values instead of pointwise statistics, which was typical for previous studies [Liu et al., AIChE J. 40, 1273 (1994); Muzzio et al., Phys. Fluids A 3, 822 (1991)]. The results clearly indicate how the total amount of stretching generated by the flow depends on the parameters of the flow protocol, and that this is not just proportional to the work done on the system, as was suggested earlier in Muzzio et al., Phys. Fluids A 3, 822 (1991). It was found that when self-similar patterns are established, distinctive zones in the flow, which we call “microstructural demixing zones,” are observed, where interfaces are contracted during a typical period of the mixing process. Spatial nonuniformity of stretching in chaotic flows calls for additional mixing measures that reflect the nonuniformity of self-similar stretching patterns, created by time-periodic mixing flows.Laminar mixing in the two-dimensional time-periodic Stokes flows between eccentric cylinders [journal bearing flow (JBF)] is studied using the extended mapping method [Galaktionov et al., Int. J. Multiphase Flows 28, 497 (2002)] with the emphasis on the material stretching, e.g., the interface generation abilities, of the flow. With this flexible and computational advantageous method both, the macroscopic material transport and the evolution of the microstructure can be described. It enables a convenient way for studying the material stretching in the flow and moreover, it provides spatial distribution of locally averaged stretching values instead of pointwise statistics, which was typical for previous studies [Liu et al., AIChE J. 40, 1273 (1994); Muzzio et al., Phys. Fluids A 3, 822 (1991)]. The results clearly indicate how the total amount of stretching generated by the flow depends on the parameters of the flow protocol, and that this is not just proportional to the work done on the system, as was sug...

Symmetry of periodic structures in a 3D mixing cavity flow

Symmetry concepts are applied to analyze three-dimensional time-periodic mixing Stokes flows in a cubic domain. Usage of simple map algebra, developed by Franjione et al. [Phys. Fluids A 11, 1772–1783 (1989); Philos. Trans. R. Soc. London, Ser. A 338, 301–323 (1992)], allows us to reveal the symmetry in the arrangement of periodic points in such a flow. Symmetries reveal essential features of the underlying physics, and with manipulation of symmetries mixing protocols can be made efficient over the entire flow domain.