(162h) Rapid and Robust Optimization of Transport Limited Processes in Laminar Flows

Many fluidic processes occur in low Reynolds number (laminar) flows. As such flows are generally non-mixing, transport and dispersion of relevant species are often rate-limiting with respect to the overall process. Examples occur in a wide range of applications involving small length scales (e.g. microfluidics), small velocity scales (e.g. porous media), large viscosities (e.g. polymer, food, and mineral processing), or rheologically sensitive fluids (e.g. biological fluids and pharmaceutical processing). In such flows, species of interest are advected along streamlines of the flow, and often transverse motion is limited to comparatively slow molecular diffusion:  species distributions evolve along the streamlines of the flow but global mixing throughout the flow domain is limited.

In low Reynolds number, laminar flows viscous damping dominates, and these flows have velocity fields which are regular in the dynamical systems sense, such that the associated streamlines do not naturally promote good mixing.  However, as identified by Aref [1], while laminar flows generate regular velocity fields, laminar flows can give rise to Lagrangian chaos; i.e. particle paths generated by laminar advection can form a chaotic tangle. This chaotic advection or Lagrangian turbulence arises from the notion that the advection equation describing motion of a passive tracer

                                                                                                                 (1)

represents a dynamical system rich enough to generate Lagrangian chaos.  As such, ?good mixing? corresponds to global chaos throughout the flow domain, ensuring complete mixing to small length scales within finite time.  For incompressible fluids, (1) represents a conservative system to which the tools of Hamiltonian mechanics (Lyapunov exponents, Poincaré sections) may be applied in order to help optimize mixing.

Note that the advection equation (1) is purely kinematic; it is universal with respect to fluid rheology, flow geometry, velocity and length scales and so applies to all deforming media. Indeed, the main necessary condition for the attainment of chaos is that (1) contains at least 2.5 degrees of freedom.  There is significant scope for simple temporal perturbations to generate chaos in otherwise non-mixing laminar flows.

As all species are also subject to some level of molecular diffusion and chaotic dynamics in (1) generates extremely small length scale striations, diffusion cannot be ignored in any physical system. Hence mixing is correctly cast as scalar dispersion, as described by the advection diffusion equation

                                               (2)

where f is the concentration of scalar to be mixed, and Pe is the Peclét number quantifying the relative timescales of advection and diffusion. As species with high diffusivities naturally disperse quickly, chaotic advection is most effective for high Pe, and in the singular limit Pe→∞, the Hamiltonian system (1) is recovered. However, at finite Pe the system (2) is dissipative, so Hamiltonian mechanics and related tools no longer apply.

Judicious programming of the velocity field v in (2) can result in significant enhancement of the dissipation (mixing) rate of f. Optimized mixing rates scale as √Pe, hence improvements of multiple orders of magnitude are possible for Pe>10⁴. However, an open problem is how to optimize the advective velocity for such a goal. Given parameterization of the velocity field by a parameter set c, which contains a number of tunable flow control parameters (such as boundary motion or forcing parameters), then scalar transport varies over the multidimensional parameter space Q={c×Pe}. Clearly, direct numerical solution of (2) over this space represents an unfeasible computational problem, especially given numerical evidence [2] suggesting fractal distributions of mixing enhancement at high Pe.

Although there exist rapid methods for simulating chaotic advection with dissipation, most of these methods do not correctly treat the diffusion operator in (2). The so-called matrix mapping method [3], using a discrete map to rapidly project the scalar distribution forward at each time step, however dissipation occurs via numerical errors, which do not correspond to physical diffusion. Operator-splitting methods project the solution sequentially via advection and diffusion operators, which is shown by Giona et al [4] to generate spurious scalar transport which does not capture the complex intertwining of diffusion with chaotic advection. Likewise, with respect to optimization, given the complex and fractal nature of the enhancement rate distribution, rapid optimization techniques [5] are likely to fail over Q, and only high resolution global elucidation of the global structure of transport over Q can robustly identify optimum stirring protocols.

The dissipative nature of (2) gives rise to a contraction of phase space, such that under normal conditions, solutions to (2) take the form of so-called ?strange eigenmodes? [6, 7] which are superposed exponentially decaying spatio-temporal patterns which govern the homogenization of f. As the slowest decaying eigenmode is also the most regular, the majority of the initial variance in f is mapped to this dominant eigenmode. As such, within a short time, the decay rate of the dominant eigenmode dictates the rate of scalar dispersion in the flow domain, and this metric is exact in the asymptotic limit of complete mixing. Therefore, quantification of the dominant strange eigenmode associated with the advection diffusion operator L in (2) for a given point in Q corresponds to quantification of the dispersion rate and pattern for the corresponding flow field and Peclét number.

By exploiting both the symmetries inherent to chaotic flows and the dissipative nature of the advection diffusion equation (2), in this paper we present a rapid method which accurately resolves the dominant strange eigenmodes over the parameter space Q. As these eigenmodes quantify both the mixing rate and dominant mixing pattern, the rate and nature of mixing over this space can be rapidly and accurately resolved, facilitating laminar flow optimization and exploitation of the potential benefits offered by chaotic advection. Mathematical justification and computational details of this method are described in a separate paper [2], and in this paper we briefly describe the method and focus on applications of the method to various flow scenarios.

To achieve global resolution of Q, we assume that the all of the flows over the parameter space c can be composed from a small set of so-called steady ?base flows? v_i. This is not as restrictive as it may first appear as the linear nature of laminar flows renders symmetries inherent to such flows, and this method acts to exploit these symmetries. By applying the mapping operations of scaling, superposition, reflection and reorientation to the base flows, a much greater set of composite steady flows can be generated. By applying transient amplification to this set of steady flows, the full set of transient flows across c can be constructed. The main point is that the full set of transient flows v(c) can be composed from a small set of steady base flows v_i via mappings and transient amplification.

By representing the advection operator for each base flow through an appropriate spectral expansion over the physical flow domain, the building blocks of a spectral representation (denoted L') of the advection-diffusion operator L in (2) can be constructed. Although spectral expansion of the base flows advection operator may be computationally expensive, this only needs to be performed once for each base flow. Subsequent composition of L' in spectral space is quite inexpensive, and so the operator can be rapidly composed over the entire parameter space Q. The dominant eigenvector and eigenvalue of L' correspond respectively to a spectral approximation to the dominant strange eigenmode and its decay rate. As such, construction of L' and solution of the dominant eigenproblem over Q generates a map of both the mixing rate and dominant mixing pattern over the entire parameter space.

These results allow simple identification of the optimal velocity field v(c) for a particular value of Pe to achieve chaotic advection and hence optimization of transport-limited processes. We show examples of application of the method to several industrially relevant flows, and illustrate how the method may be applied to a wide variety of flow scenarios.

 [1] Aref, H., 1984, Stirring by chaotic advection. J. Fluid Mech. 143:1-21.

[2] Lester, D. R., Metcalfe, G., Rudman, M. and Blackburn, H., 2008, Global parametric solutions of scalar transport, J. Comp. Phys., 227, 3032-3057.

[3] Galaktionov, O. S., Anderson, P. D., Peters, G. W. M., and Meijer, H. E. H., 2002, Mapping approach for 3D laminar mixing simulations: application to industrial flows, Int. J. Num. Methods. Fluids., 40, 345-351.

[4] Giona, M., Adrover, A., Cerbelli, S., 2005, On the use of the pulsed-convection approach for modeling advection-diffusion in chaotic flows ? A prototypical example and direct numerical simulations, Physica A, 348, 37-73.

[5] Gibout, S., Le Guer, Y., Schall, E., 2006, Coupling of a mapping method and a genetic algorithm to optimize mixing efficiency in periodic chaotic flows, Comm. Nonlin. Sci. Num. Sim., 11, 413-423.

[6] Liu, W. and Haller, G., 2004, Strange Eigenmodes and Decay of Variance in the Mixing of Diffusive Tracers, Physica D, 188, 1-39

[7] Pierrehumbert, R. T., 1994, Tracer microstructure in the large-eddy dominated regime, Chaos, 4, 6, 1091-1110