Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-19T17:10:13.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Feedback control for unsteady flow and its application to the stochastic Burgers equation

Published online by Cambridge University Press:  26 April 2006

Haecheon Choi ,
Roger Temam ,
Parviz Moin  and
John Kim
Roger Temam
Affiliation:
Permanent address: Université de Paris-Sud, Laboratoire d'Analyse Numérique, Bâtiment 425, 91405 Orsay, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

Mathematical methods of control theory are applied to the problem of control of fluid flow with the long-range objective of developing effective methods for the control of turbulent flows. The procedure of how to cast the problem of controlling turbulence into a problem in optimal control theory is presented using model problems through the formalism and language of control theory. Then we present a suboptimal control and feedback procedure for general stationary and time-dependent problems using methods of calculus of variations through the adjoint state and gradient algorithms. This suboptimal feedback control procedure is applied to the stochastic Burgers equation. Two types of controls are investigated: distributed and boundary controls. The control inputs are the momentum forcing for the distributed control and the boundary velocity for the boundary control. Costs to be minimized are defined as the sum of the mean-square velocity gradient inside the domain for the distributed control or the square velocity gradient at the wall for the boundary control; and in both cases a term was added to account for the implementation cost. Several cases of both controls have been numerically simulated to investigate the performances of the control algorithm. Most cases considered show significant reductions of the costs. Another version of the feedback procedure more effective for practical implementation has been considered and implemented, and the application of this algorithm also shows significant reductions of the costs. Finally, dependence of the control algorithm on the time-discretization method is discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 253 , August 1993 , pp. 509 - 543
Copyright
© 1993 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abergel, F. & Temam, R. 1990 On some control problems in fluid mechanics. Theor. Comput. Fluid Dyn. 1, 303.Google Scholar
Abergel, F. & Temam, R. 1992 Optimal control of turbulent flows. In Optimal Control of Viscous Flows (ed. S. Sritharan). SIAM (to appear.)
Banach, A. S. & Baumann, W. T. 1990 Gain-scheduled control of nonlinear partial differential equations. In Proc. 29th IEEE Conf. on Decision and Control, pp. 387–392.
Barbu, V. 1992 H[vprop ] boundary control with state feedback; the hyperbolic case. Preprint.
Bechert, D. W. & Bartenwerfer, M. 1989 The viscous flow on surfaces with longitudinal ribs. J. Fluid Mech. 206, 105.Google Scholar
Bensoussan, A. & Temam, R. 1972 Equations aux dérivées partielles stochatiques non linéaires (I). Israel J. Maths 11, 95.Google Scholar
Bensoussan, A. & Temam, R. 1973 Equations stochastiques du type Navier–Stokes. J. Funct. Anal. 13, 195.Google Scholar
Bushnell, D. M. & McGinley, C. B. 1989 Turbulence control in wall flows. Ann. Rev. Fluid Mech. 21, 1.Google Scholar
Byrnes, C. I. & Gilliam, D. S. 1991 Boundary feedback design for nonlinear distributed parameter systems. In Proc. 30th IEEE Conf. on Decision and Control, pp. 2340–2342.
Cantwell, B. J. 1981 Organized motion in turbulent flow. Ann. Rev. Fluid Mech. 13, 457.Google Scholar
Chambers, D. H., Adrian, R. J., Moin, P., Stewart, D. S. & Sung, H. J. 1988 Karhunen–Loéve expansion of Burgers' model of turbulence. Phys. Fluids 31, 2573.Google Scholar
Choi, H., Moin, P. & Kim, J. 1992 Turbulent drag reduction: studies of feedback control and flow over riblets. Rep. TF-55. Department of Mechanical Engineering, Stanford University, Stanford, CA.
Finlayson, B. A. 1972 The Method of Weighted Residuals and Variational Principles. Academic.
Foias, C. & Tannenbaum, A. 1989 Weighted optimization theory for nonlinear systems. SIAM J. Control Optim. 27, 842.Google Scholar
Gunzburger, M. D. (ed.) 1992 Proc. Conference on Control of Flows. Inst. for Math. and Its Appl., University of Minnesota, November 1992.
Gunzburger, M. D., Hou, L. & Svobodny, T. P. 1990 A numerical method for drag minimization via the suction and injection of mass through the boundary. In Stabilization of Flexible Structures (ed. J. P. Zolesio). Springer.
Gunzburger, M. D., Hou, L. & Svobodny, T. P. 1991 Analysis and finite element approximations of optimal control problems for the stationary Navier–Stokes equations with Dirichlet controls. Math. Model. Numer. Anal. 25, 711.Google Scholar
Gunzburger, M. D., Hou, L. & Svobodny, T. P. 1992 Boundary velocity control of incompressible flow with an application to viscous drag reduction. SIAM J. Control Optim. 30, 167.Google Scholar
Kang, S., Ito, K. & Burns, J. A. 1991 Unbounded observation and boundary control problems for Burgers equation. In Proc. 30th IEEE Conf. on Decision and Control, pp. 2687–2692.
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133.Google Scholar
Ladyzhensakaya, O. A. 1963 The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach.
Lions, J. L. 1971 Optimal Control of Systems Governed by Partial Differential Equations. Springer.
Luenberger, D. 1973 Introduction to linear and nonlinear programming. Addison-Wesley.
Sritharan, S. S. 1991a Dynamic programming of the Navier–Stokes equations. Systems Control Lett. 16, 299.Google Scholar
Sritharan, S. S. 1991b On the nonsmooth verification technique for the dynamic programming of viscous flows. IMA Preprint 850.
Sritharan, S. S. 1992 Optimal control theory of viscous flows. In Optimal Control of Viscous Flows (ed. S. Sritharan). SIAM (to appear).
Sritharan, S. S. (ed.) 1992 Optimal Control of Viscous Flows. SIAM (to appear.)
Sritharan, S. S., Ou, Y.-R., Burns, J. A., Ladd, D., Hendricks, E. & Nossier, N. 1991 Active control of viscous flow past a cylinder: theory, computation and experiment. Bull. Am. Phys. Soc. 36, 2626.Google Scholar
Temam, R. 1984 Navier–Stokes Equations, 3rd revised edn. North-Holland.
Temam, R. 1991 Navier–Stokes equations: theory and approximations, chapter I. Preprint 91-16, Institute for Scientific Computing and Applied Mathematics, Bloomington, Indiana. Also to appear in the Handbook of Numerical Analysis (ed. P. G. Ciarlet & J. L. Lions). North-Holland.
Walsh, M. J. 1983 Riblets as a viscous drag reduction technique. AIAA J. 21, 485.Google Scholar