Optimal mixing in recirculation zones

Coarse-scale mixing in a recirculation zone is described with a simple vortex model. Time-dependent forcing is employed to change the vortex motion and mixing properties. An optimal mixing problem is defined in which the flux across the recirculation region shall be maximized under the side-constraints of bounded vortex motion and bounded actuation. Concepts of control theory and chaotic advection are used to achieve this goal. In particular, controllability is proven with a transformation into flat coordinates. Thus, a feedforward law for the optimal trajectory and a feedback law for its stabilization are derived. Observability of the vortex motion is indicated by a dynamic observer. Mixing in the optimized flow is studied using Poincare maps. The low-frequency modulations to vortex motion are shown to substantially increase mixing in the average. Generalizations of the mathematical framework for mixing optimization are suggested for a larger class of models and flows.

[1]  Igor Mezic,et al.  Control of a vortex pair using a weak external flow , 2002 .

[2]  S. Wiggins.  Chaos in the dynamics generated by sequences of maps, with applications to chaotic advection in flows with aperiodic time dependence , 1999 .

[3]  G. Batchelor,et al.  An Introduction to Fluid Dynamics , 1968 .

[4]  S. Wiggins,et al.  An analytical study of transport, mixing and chaos in an unsteady vortical flow , 1990, Journal of Fluid Mechanics.

[5]  I. Mezić,et al.  Patchiness: A New Diagnostic for Lagrangian Trajectory Analysis in Time-Dependent Fluid Flows , 1998 .

[6]  H. Aref.  Stirring by chaotic advection , 1984, Journal of Fluid Mechanics.

[7]  Domenico D'Alessandro,et al.  Control of mixing in fluid flow: a maximum entropy approach , 1999, IEEE Trans. Autom. Control..

[8]  V. Arnold,et al.  Mathematical aspects of classical and celestial mechanics , 1997 .

[9]  H. Najm,et al.  NUMERICAL SIMULATION OF THE CONVECTIVE INSTABILITY IN A DUMP COMBUSTOR , 1991 .

[10]  Gilead Tadmor,et al.  Observer-based control of vortex motion in a combustor recirculation region , 2002, IEEE Trans. Control. Syst. Technol..

[11]  Andrew C. Poje,et al.  Universal properties of chaotic transport in the presence of diffusion , 1999 .

[12]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[13]  Arjan van der Schaft,et al.  Non-linear dynamical control systems , 1990 .

[14]  Yong Kweon Suh.  Periodic Motion of a Point Vortex in a Corner Subject to a Potential Flow , 1993 .

[15]  J. Ottino.  The Kinematics of Mixing: Stretching, Chaos, and Transport , 1989 .

[16]  G. Haller,et al.  Chaos near resonance , 1999 .

[17]  Jason L. Speyer,et al.  Skin-friction Drag Reduction Via Robust Reduced-order Linear Feedback Control , 1998 .

[18]  A. Iollo,et al.  Optimal Control of a Vortex Trapped by an Airfoil with a Cavity , 2000 .

[19]  John C. Doyle,et al.  An example of active circulation control of the unsteady separated flow past a semi-infinite plate , 1994, Journal of Fluid Mechanics.

[20]  Craig T. Bowman,et al.  Combustor performance enhancement through direct shear layer excitation , 1990 .

[21]  L. Rossi,et al.  Investigation of wall normal electromagnetic actuator for seawater flow control , 2002 .

[22]  D V Treshchëv,et al.  THE MECHANISM OF DESTRUCTION OF RESONANCE TORI OF HAMILTONIAN SYSTEMS , 1991 .

[23]  Igor Mezic,et al.  Chaotic advection in bounded Navier–Stokes flows , 2001, Journal of Fluid Mechanics.

[24]  Stephen Wiggins,et al.  Chaotic transport in dynamical systems , 1991 .

[25]  Mohamed Gad-el-Hak,et al.  Flow Control: Passive, Active, and Reactive Flow Management , 2000 .

[26]  William H. Press,et al.  Numerical recipes in C. The art of scientific computing , 1987 .

[27]  Stabilizing chaotic vortex trajectories: an example of high-dimensional control , 1996, chao-dyn/9701002.

[28]  K. Kataoka,et al.  Information Complexity of Laminar Chaotic Mixing Field Produced by Two Parallel, Rotating Cylinders , 2000 .

[29]  L. Cortelezzi.  Nonlinear feedback control of the wake past a plate with a suction point on the downstream wall , 1996, Journal of Fluid Mechanics.

[30]  Christopher K. R. T. Jones,et al.  Quantifying transport in numerically generated velocity fields , 1997 .

[31]  Mohamed Gad-el-Hak,et al.  Modern developments in flow control , 1996 .

[32]  George Haller,et al.  The geometry and statistics of mixing in aperiodic flows , 1999 .

[33]  George Haller,et al.  Finite time transport in aperiodic flows , 1998 .

[34]  Y.-C. Chen,et al.  NONLINEAR FEEDBACK CONTROL OF THE WAKE PAST A PLATE : FROM A LOW-ORDER MODEL TO A HIGHER-ORDER MODEL , 1997 .