Advertisement

Flow, Turbulence and Combustion

, Volume 65, Issue 3–4, pp 299–320 | Cite as

Numerical Approaches to Optimal Control of a Model Equation for Shear Flow Instabilities

  • Markus Högberg
  • Martin Berggren
Article

Abstract

We investigate two different discretization approaches of a model optimal-control problem, chosen to be relevant for control of instabilities in shear flows. In the first method, a fully discrete approach has been used, together with a finite-element spatial discretization, to obtain the objective function gradient in terms of a discretely-derived adjoint equation. In the second method, Chebyshev collocation is used for spatial discretization, and the gradient is approximated by discretizing the continuously-derived adjoint equation. The discrete approach always results in a faster convergence of the conjugate-gradient optimization algorithm. Due to the shear in the convective velocity, a low diffusivity in the problem complicates the structure of the computed optimal control, resulting in particularly noticeable differences in convergence rate between the methods. When the diffusivity is higher, the control becomes less complicated, and the difference in convergence rate reduces. The use of approximate gradients results in a higher sensitivity to the degrees of freedom in time. When the system contains a strong instability, it only takes a few iteration to obtain an effective control for both methods,even if there are differences in the formal convergence rate. This indicates that it is possible to use the approximative gradients of the objective function in cases where the control problem mainly consists of controlling strong instabilities.

shear flow optimal control adjoint equations discrete adjoint exact gradient 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Axelsson, O., Iterative Solution Methods. Cambridge University Press, Cambridge (1996).Google Scholar
  2. 2.
    Berggren, M., Numerical solution of a flow-control problem: Vorticity reduction by dynamic boundary action. SIAM Journal on Scientific Computing 19(3) (1998)829–860zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Berggren, M., Glowinski, R. and Lions, J.L., A computational approach to controllability issues for flow-related models. (II): Control of two-dimensional, linear advection-diffusion and Stokes models. International Journal of Computational Fluid Dynamics 6(4) (1996)253–274Google Scholar
  4. 4.
    Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A., Spectral Methods in Fluid Dynamics. Springer-Verlag, Berlin (1988).Google Scholar
  5. 5.
    Dautray, R. and Lions, J.L., Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5: Evolution Problems I. Springer-Verlag, Berlin (1992).Google Scholar
  6. 6.
    Gad-el-Hak, M., Modern developments in flow control. Applied Mechanics Review 49(7) (1996)365–379Google Scholar
  7. 7.
    Glowinski, R., Numerical Methods for Nonlinear Variational Problems. Springer-Verlag, New York (1984).Google Scholar
  8. 8.
    Glowinski, R. and He, J., On shape optimization and related issues'. In: Borggaard, J., Burns, J., Cliff, E. and Schreck, S. (eds), Computational Methods in Optimal Design and Control, Proceedings of the AFOSR Workshop on Optimal Design and Control, Arlington, Virginia. Birkhäuser, Boston (1998) pp.151–179Google Scholar
  9. 9.
    Glowinski, R. and Lions, J.L., Exact and approximate controllability for distributed parameter systems. Acta Numerica(1994)269–378Google Scholar
  10. 10.
    Golub, G.H. and Van Loan, C.F., Matrix Computations. The Johns Hopkins University Press, Baltimore, MD (1989).Google Scholar
  11. 11.
    Hanifi, A., Schmid P.J. and Henningson D.S. Transient growth in compressible boundary layer flow. Physics of Fluids 8(3) (1996)826–837zbMATHMathSciNetCrossRefADSGoogle Scholar
  12. 12.
    Hestenes, M.R. and Stiefel E., Methods of conjugate gradient for solving linear systems. Journal Research National Bureau of Standards 45 (1952)409–436MathSciNetGoogle Scholar
  13. 13.
    Högberg, M., Berggren, M. and Henningson, D.S., Numerical investigation of different discretization approaches for optimal control. Technical Report TN 1999–74, FFA, The Aeronautical Research Institute of Sweden, P.O. Box 11021, S-161 11 Bromma, Sweden (1999).Google Scholar
  14. 14.
    Lions, J.L. and Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, Vol. 2. Springer-Verlag, Berlin (1972).Google Scholar
  15. 15.
    Metcalfe, R.W.,Boundary layer control: A brief review. In:Wagner, S., Periaux, J. and Hirschel, E. (eds), Computational Fluid Dynamics '94, Invited Lectures of the Second European CFD Conference, Stuttgart, Germany. John Wiley & Sons, New York (1994) pp. 52–60.Google Scholar
  16. 16.
    Reshotko, E., Boundary layer instability, transition and control. AIAA Paper 94–0001 (1994).Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Markus Högberg
    • 1
  • Martin Berggren
    • 2
  1. 1.Department of MechanicsRoyal Institute of TechnologyStockholmSweden
  2. 2.FFA, the Aeronautical Institute of Sweden, and Department of Scientific ComputingUppsala UniversityUppsalaSweden

Personalised recommendations