Three Control Methods for Time-Dependent Fluid Flow

We summarize some of the recent developments in the field of optimal and suboptimal control strategies for time-dependent incompressible Navier–Stokes equations. Besides the Newton method,reduced order modeling (ROM) and receding horizon control strategies are described. All approaches are applied to compute controls for the unsteady flow around a cylinder in two spatial dimensions.

[1]  L. Sirovich.  Turbulence and the dynamics of coherent structures. III. Dynamics and scaling , 1987 .

[2]  Bartosz Protas,et al.  Theoretical and Computational Study of the Wake Control Problem , 1999 .

[3]  S. Ravindran,et al.  A Reduced Basis Method for Control Problems Governed by PDEs , 1998 .

[4]  R. Temam,et al.  Feedback control for unsteady flow and its application to the stochastic Burgers equation , 1993, Journal of Fluid Mechanics.

[5]  Petros Koumoutsakos.  Active control of vortex–wall interactions , 1997 .

[6]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[7]  L. S. Hou,et al.  Optimization Problems for the Navier-Stokes Equations with Regular Boundary Controls , 1993 .

[8]  Kazufumi Ito,et al.  A Dissipative Feedback Control Synthesis for Systems Arising in Fluid Dynamics , 1994 .

[9]  E. Bänsch.  An adaptive finite-element strategy for the three-dimensional time-dependent Navier-Stokes equations , 1991 .

[10]  M. Heinkenschloss.  Formulation and Analysis of a Sequential Quadratic Programming Method for the Optimal Dirichlet Boundary Control of Navier-Stokes Flow , 1998 .

[11]  Karl Kunisch,et al.  On suboptimal control strategies for the Navier-Stokes equations , 1998 .

[12]  E. Sachs,et al.  Trust-region proper orthogonal decomposition for flow control , 2000 .

[13]  Omar Ghattas,et al.  Optimal Control of Two- and Three-Dimensional Incompressible Navier-Stokes Flows , 1997 .

[14]  Rolf Rannacher,et al.  ARTIFICIAL BOUNDARIES AND FLUX AND PRESSURE CONDITIONS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS , 1996 .

[15]  Manservisi Sandro.  Optimal Boundary and Distributed Controls for the Velocity Tracking Problem for Navier-Stokes Flows , 1997 .

[16]  Sanjay S. Joshi,et al.  A systems theory approach to the feedback stabilization of infinitesimal and finite-amplitude disturbances in plane Poiseuille flow , 1997, Journal of Fluid Mechanics.

[17]  Sivaguru S. Ravindran,et al.  Proper Orthogonal Decomposition in Optimal Control of Fluids , 1999 .

[18]  Max D. Gunzburger,et al.  Analysis and Approximation of the Velocity Tracking Problem for Navier-Stokes Flows with Distributed Control , 2000, SIAM J. Numer. Anal..

[19]  Max Gunzburger,et al.  The Velocity Tracking Problem for Navier--Stokes Flows with Bounded Distributed Controls , 1999 .

[20]  R. Temam,et al.  On some control problems in fluid mechanics , 1990 .

[21]  K. Kunisch,et al.  Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition , 1999 .

[22]  K. Kunisch,et al.  Instantaneous control of backward-facing step flows , 1999 .

[23]  Rolf Rannacher,et al.  Finite Element Methods for the Incompressible Navier-Stokes Equations , 2000 .

[24]  D Joslin Ronald,et al.  A Methodology for the Automated Optimal Control of Flows including Transitional Flows , 1996 .

[25]  C. Fabre,et al.  Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems , 1996 .

[26]  Max D. Gunzburger,et al.  The Velocity Tracking Problem for Navier-Stokes Flows With Boundary Control , 2000, SIAM J. Control. Optim..

[27]  R. Temam.  Navier-Stokes Equations , 1977 .

[28]  L. Hou,et al.  Boundary Value Problems and Optimal Boundary Control for the Navier--Stokes System: the Two-Dimensional Case , 1998 .

[29]  L. S. Hou,et al.  Dynamics for Controlled Navier--Stokes Systems with Distributed Controls , 1997 .

[30]  J. Coron,et al.  On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions , 1996 .

[31]  R. Rannacher,et al.  Benchmark Computations of Laminar Flow Around a Cylinder , 1996 .

[32]  Kazufumi Ito,et al.  Optimal Controls of Navier--Stokes Equations , 1994 .

[33]  K. ITO,et al.  Reduced Basis Method for Optimal Control of Unsteady Viscous Flows , 2001 .

[34]  Ernst Heinrich Hirschel,et al.  Flow Simulation with High-Performance Computers II , 1996 .

[35]  L. S. Hou,et al.  Computations of Optimal Controls for Incompressible Flows , 1998 .

[37]  S. Sritharan.  Optimal control of viscous flow , 1998 .

[38]  Karl Kunisch,et al.  Second Order Methods for Optimal Control of Time-Dependent Fluid Flow , 2001, SIAM J. Control. Optim..

[39]  Petros Koumoutsakos,et al.  VORTICITY FLUX CONTROL FOR A TURBULENT CHANNEL FLOW , 1999 .

[40]  L. Hou,et al.  Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls , 1991 .

[41]  Roger Temam,et al.  A general framework for robust control in fluid mechanics , 2000 .

[42]  W. Hager,et al.  Optimal Control: Theory, Algorithms, and Applications , 1998 .

[43]  L. S. Hou,et al.  Dynamics and Approximations of a Velocity Tracking Problem for the Navier--Stokes Flows with Piecewise Distributed Controls , 1997 .

[44]  Martin Berggren,et al.  Numerical Solution of a Flow-Control Problem: Vorticity Reduction by Dynamic Boundary Action , 1998, SIAM J. Sci. Comput..

[45]  R. Goodman,et al.  Application of neural networks to turbulence control for drag reduction , 1997 .

[46]  Parviz Moin,et al.  Optimal feedback control of turbulent channel flow , 1993 .

[47]  A. Fursikov,et al.  Exact boundary zero controllability of three-dimensional Navier-Stokes equations , 1995 .

[48]  Kazufumi Ito,et al.  Asymptotic Properties of Receding Horizon Optimal Control Problems , 2002, SIAM J. Control. Optim..

[49]  R. Glowinski,et al.  Incompressible Computational Fluid Dynamics: On Some Finite Element Methods for the Numerical Simulation of Incompressible Viscous Flow , 1993 .

[50]  Max Gunzburger,et al.  Analysis and approximation for linear feedback control for tracking the velocity in Navier–Stokes flows , 2000 .

[51]  Haecheon Choi,et al.  Suboptimal feedback control of vortex shedding at low Reynolds numbers , 1999, Journal of Fluid Mechanics.

[52]  Patrick Justen.  Optimal Control of Thermally Coupled Navier-Stokes Equations in Food Industry , 1998, Universität Trier, Mathematik/Informatik, Forschungsbericht.

[53]  Wr Graham,et al.  OPTIMAL CONTROL OF VORTEX SHEDDING USING LOW-ORDER MODELS. PART I-OPEN-LOOP MODEL DEVELOPMENT , 1999 .

[54]  Belinda B. King,et al.  Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations , 2001 .

[55]  Haecheon Choi,et al.  Suboptimal control of turbulent channel flow for drag reduction , 1998, Journal of Fluid Mechanics.

[56]  Roger Temam,et al.  DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms , 2001, Journal of Fluid Mechanics.