Skip to main content
SpringerLink
Log in

Vortex control in channel flows using translational invariant cost functionals

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

The use of translation invariant cost functionals for the reduction of vortices in the context of shape optimization of fluid flow domain is investigated. Analytical expressions for the shape design sensitivity involving different cost functionals are derived. Channel flow problems with a bump and an obstacle as possible control boundaries are taken as test examples. Numerical results are provided in various graphical forms for relatively low Reynolds numbers. Striking differences are found for the optimal shapes corresponding to the different cost functionals, which constitute different quantification of a vortex.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abergel, F., Temam, R.: On some control problems in fluid mechanics. Theor. Comput. Fluid Dyn. 1(6), 303–325 (1990)

    Article  MATH  Google Scholar 

  2. Bach, M., Constanda, C., Hsiao, G.C., Sandig, A.M., Werner, P.: Analysis, Numerics and Applications of Differential and Integral Equations. Research Notes in Mathematics Series, vol. 379. Addison Wesley/Longman, Reading (1998)

    MATH  Google Scholar 

  3. Blackburn, H.M., Mansour, N.N., Cantwell, B.J.: Topology of fine-scale motions in turbulent channel flow. J. Fluid Mech. 310, 269–292 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boisgérault, S., Zolésio, J.P.: Shape derivative of sharp functionals governed by Navier-Stokes flow. In: Jäger, W., Necǎs, J., John, O., Najzar, K., Stará, J. (eds.) Partial Differential Equations: Theory and Numerical Solution. Research Notes in Mathematics, pp. 49–63. Chapman & Hall/CRC, London (1993)

    Google Scholar 

  5. Chenais, D.: On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52, 189–219 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chong, M.S., Perry, A.E., Cantwell, B.J.: A general classification of three-dimensional flow fields. Phys. Fluids 2(5), 765–777 (1990)

    Article  MathSciNet  Google Scholar 

  7. Delfour, M.C., Zolésio, J.P.: Shapes and Geometries: Analysis, Differential Calculus, and Optimization. Society for Industrial and Applied Mathematics, Philadelphia (2001)

    MATH  Google Scholar 

  8. Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  9. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24. Pitman, Massachusetts (1985)

    MATH  Google Scholar 

  10. Gunzburger, M.: In: Gunzburger, M.D. (ed.) Flow Control. IMA, vol. 68. Springer, Berlin (1995)

    Chapter  Google Scholar 

  11. Haller, G.: Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Phys. Fluids 13(11), 3365–3385 (2001)

    Article  MathSciNet  Google Scholar 

  12. Haller, G.: An objective definition of a vortex. J. Fluids Mech. 525, 1–26 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Haslinger, J., Mäkinen, R.A.E.: Introduction to Shape Optimization: Theory, Approximation, and Computation. Society for Industrial and Applied Mathematics, Philadelphia, USA (2003)

    MATH  Google Scholar 

  14. Henrot, A., Privat, Y.: What is the optimal shape of a pipe? Arch. Ration. Mech. Anal. 196(1), 281–302 (2009)

    Article  MathSciNet  Google Scholar 

  15. Heywood, J.G., Rannacher, R., Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 22(5), 325–352 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hintermüller, M., Kunisch, K., Spasov, Y., Volkwein, S.: Dynamical systems based optimal control of incompressible fluids. Int. J. Numer. Methods Fluids 4, 345–359 (2004)

    Article  Google Scholar 

  17. Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluids Mech. 285, 69–94 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaplan, W.: Ordinary Differential Equations. Addison-Wesley, Reading (1967)

    Google Scholar 

  19. Kunisch, K., Vexler, B.: Optimal vortex reduction for instationary flows based on translation invariant cost functionals. SIAM J. Control Optim. 46(4), 1368–1397 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Martinec, D.: Lecture notes on continuum mechanics. [pdf document] retrieved from lecture notes online. Web site: http://geo.mff.cuni.cz/vyuka/martinec-continuummechanics.pdf

  21. Moubachir, M., Zolésio, J.: Moving Shape Analysis: Applications to Fluid Structure Interactions. Chapman and Hall, London (2006)

    Book  MATH  Google Scholar 

  22. Sokolowski, J., Zolésio, J.P.: Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer, Berlin (1992)

    MATH  Google Scholar 

  23. Temam, R.: Navier Stokes Equations: Theory and Numerical Analysis. Studies in Mathematics and Its Applications, vol. 2. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, 4040, Linz, Austria

    H. Kasumba

  2. Institute of Mathematics and Scientific Computing, Karl-Franzens University, 8010, Graz, Austria

    K. Kunisch

Authors
  1. H. Kasumba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Kunisch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Kunisch.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kasumba, H., Kunisch, K. Vortex control in channel flows using translational invariant cost functionals. Comput Optim Appl 52, 691–717 (2012). https://doi.org/10.1007/s10589-011-9434-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-011-9434-y

Keywords

Search

Navigation