Experiments in Fluids

, 54:1580 | Cite as

Identification strategies for model-based control

  • Laurent Cordier
  • Bernd R. Noack
  • Gilles Tissot
  • Guillaume Lehnasch
  • Joël Delville
  • Maciej Balajewicz
  • Guillaume Daviller
  • Robert K. Niven
Research Article
Part of the following topical collections:
  1. Topics in Flow Control

Abstract

A reduced-order modelling (ROM) strategy is crucial to achieve model-based control in a wide class of flow configurations. In turbulence, ROMs are mostly derived by Galerkin projection of first-principles equations onto the proper orthogonal decomposition (POD) modes. These POD ROMs are known to be relatively fragile when used for control design. To understand more deeply this behaviour, a hierarchy of POD ROMs is studied and compared for a two-dimensional uncontrolled mixing layer. First, dynamical models based on POD are derived from the Navier–Stokes and the vorticity equations. It is shown that these models can exhibit for some truncation level finite-time blow-ups. A symmetrized quadratic model is then proposed to enforce the preservation of energy in the model. This formulation improves considerably the behaviour of the model, but the energy level is still overestimated. Subsequently, a nonlinear eddy-viscosity model with guaranteed stability is proposed. This model still leads to an imperfect description of the energy. Different 4D-Var identification strategies are then considered to improve the nonlinear eddy-viscosity model. It is shown that penalizing the time variation in the eddy-viscosity parameter leads to the best compromise in terms of description of the energy and of physical relevance of the eddy viscosity. Lastly, a statistical inference procedure is used to take into account the bias inherent of the sampling employed in the identification procedure. Joined probability density functions are then determined empirically for all the coefficients of the model, and a stochastic dynamical model is finally derived to study the influence on the long-term solution of different equiprobable sets of coefficients.

Notes

Acknowledgments

This work has received support from the National Agency for Research on references ANR-08-BLAN-0115-01 (CORMORED) and ANR-10-CEXC-015-01 (Chair of Excellence TUCOROM). BRN also acknowledges the University of New South Wales at the Australian Defense Force Academy for a Rector-Funded Visiting Fellowship.

References

  1. Artana G, Cammilleri A, Carlier J, Mémin E (2012) Strong and weak constraint variational assimilations for reduced order fluid flow modeling. J Comp Phys 231:3264–3288MATHCrossRefGoogle Scholar
  2. Aubry N, Holmes P, Lumley JL, Stone E (1988) The dynamics of coherent structures in the wall region of a turbulent boundary layer. J Fluid Mech 192:115–173MathSciNetMATHCrossRefGoogle Scholar
  3. Balajewicz M, Dowell EH, Noack BR (2013) Low-dimensional modelling of high Reynolds number shear flows incorporating constraints from the Navier-Stokes equation. Submitted to J Fluid MechGoogle Scholar
  4. Bergmann M (2004) Optimisation aérodynamique par réduction de modèle POD et contrôle optimal. Application au sillage laminaire d’un cylindre circulaire. Phd thesis, Institut National Polytechnique de Lorraine, Nancy, FranceGoogle Scholar
  5. Bergmann M, Cordier L, Brancher JP (2005) Optimal rotary control of the cylinder wake using POD Reduced Order Model. Phys Fluids 17(9):097,101:1–21Google Scholar
  6. Bogey C (2000) Calcul direct du bruit aérodynamique et validation de modèles acoustiques hybrides. PhD thesis, Ecole Centrale LyonGoogle Scholar
  7. Box G, Muller M (1958) A note on the generation of random normal deviates. Ann Math Stat 29:610–611MATHCrossRefGoogle Scholar
  8. Cavalieri A, Daviller G, Comte P, Jordan P, Tadmor G, Gervais Y (2011) Using large eddy simulation to explore sound-source mechanisms in jets. J Sound Vib 330:4098–4113CrossRefGoogle Scholar
  9. Chernick MR (2008) Bootstrap methods: a guide for Practitioners and Researchers, 2nd edn. Wiley, New JerseyGoogle Scholar
  10. Colonius T, Lele SK, Moin P (1993) Boundary conditions for direct computation of aerodynamic sound generation. AIAA J 31:1574–1582MATHCrossRefGoogle Scholar
  11. Comte P, Silvestrini J, Bégou P (1998) Streamwise vortices in Large-Eddy Simulations of mixing layer. Eur J Mech B 17:615–637MATHCrossRefGoogle Scholar
  12. Cordier L (2010) Flow control and constrained optimization problems. In: Noack BR, Morzyński M, Tadmor G (eds) Reduced-order modelling for flow control, Springer, Berlin, pp 1–76Google Scholar
  13. Cordier L, Abou El Majd B, Favier J (2010) Calibration of POD Reduced-Order models using Tikhonov regularization. Int J Numer Meth Fluids 63(2)Google Scholar
  14. D’Adamo J, Papadakis N, Mémin E, Artana G (2007) Variational assimilation of POD low-order dynamical systems. J Turbul 8(9):1–22Google Scholar
  15. Daviller G (2010) Étude numérique des effets de température dans les jets simples et coaxiaux. PhD thesis, École Nationale Supérieure de Mécanique et d’AérotechniqueGoogle Scholar
  16. Efron B (1979) Bootstrap methods: another look at the jackknife. Ann Stat 7:1–26MathSciNetMATHCrossRefGoogle Scholar
  17. Fletcher CAJ (1984) Computational Galerkin methods, 1st edn. Springer, New YorkCrossRefGoogle Scholar
  18. Galletti G, Bruneau CH, Zannetti L, Iollo A (2004) Low-order modelling of laminar flow regimes past a confined square cylinder. J Fluid Mech 503:161–170MathSciNetMATHCrossRefGoogle Scholar
  19. Gatski TB, Bonnet JP (2013) Compressibility, turbulence and high speed flow, 2nd edn. Academic Press, WalthamGoogle Scholar
  20. Gentle JE, Härdle W, Mori Y (2004) Handbook of computational statistics: concepts and methods. Springer, BerlinGoogle Scholar
  21. Getz WM, Jacobson DH (1977) Sufficiency conditions for finite escape times in systems of quadratic differential equations. J Inst Math Applics 19:377–383MathSciNetMATHCrossRefGoogle Scholar
  22. Gilbert J, Lemaréchal C (2009) The module M1QN3 – Version 3.3. INRIA Rocquencourt & Rhône-AlpesGoogle Scholar
  23. Gottlieb D, Turkel E (1976) Dissipative two-four method for time dependent problems. Math Comp 30(136):703–723MathSciNetMATHCrossRefGoogle Scholar
  24. Gunzburger MD (1997) Introduction into mathematical aspects of flow control and optimization. In: Lecture series 1997–05 on inverse design and optimization methods, Von Kármán Institute for Fluid DynamicsGoogle Scholar
  25. Hansen PC (1994) Regularization tools: a MATLAB package for analysis and solution of discrete ill-posed problems. Numer Algorith 6:1–35MATHCrossRefGoogle Scholar
  26. Hansen PC (1998) Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion. SIAM, PhiladelphiaGoogle Scholar
  27. Hayder ME, Turkel E (1993) High-order accurate solutions of viscous problem. In: AIAA paper 93-3074Google Scholar
  28. Holmes P, Lumley JL, Berkooz G, Rowley CW (2012) Turbulence, coherent structures, dynamical systems and symmetry, 2nd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  29. Joseph DD (1976) Stability of fluid motions I & II. Springer Tracts in Natural Philosophy, vol 26 & 27, Springer, BerlinGoogle Scholar
  30. Kraichnan RH, Chen S (1989) Is there a statistical mechanics of turbulence? Phys D 37:160–172MathSciNetMATHCrossRefGoogle Scholar
  31. Krstic M, Krupadanam A, Jacobson C (1999) Self-tuning control of a nonlinear model of combustion instabilities. IEEE Tr Contr Syst Technol 7(4):424–436CrossRefGoogle Scholar
  32. Ladyzhenskaya OA (1963) The mathematical theory of viscous incompressible flow, 1st edn. Gordon and Breach, New YorkGoogle Scholar
  33. Lehnasch G, Delville J (2011) A stochastic low-order modelling approach for turbulent shear flows. In: Seventh symposium on turbulent shear flow phenomena, OttawaGoogle Scholar
  34. Lodato G, Domingo P, Vervisch L (2008) Three-dimensional boundary conditions for direct and large-eddy simulation of compressible viscous flows. J Comp Phys 227(10):5105–5143MathSciNetMATHCrossRefGoogle Scholar
  35. Lugt H (1996) Introduction to vortex theory. Vortex Flow Press, PotomacGoogle Scholar
  36. Navon I (2009) Data assimilation for numerical weather prediction: a review. Data Assimilation Atmos Ocean Hydrol Appl 18(475):326Google Scholar
  37. Noack BR, Eckelmann H (1994) A global stability analysis of the steady and periodic cylinder wake. J Fluid Mech 270:297–330MATHCrossRefGoogle Scholar
  38. Noack BR, Afanasiev K, Morzyński M, Tadmor G, Thiele F (2003) A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J Fluid Mech 497:335–363MathSciNetMATHCrossRefGoogle Scholar
  39. Noack BR, Papas P, Monkewitz PA (2005) The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J Fluid Mech 523:339–365MathSciNetMATHCrossRefGoogle Scholar
  40. Noack BR, Schlegel M, Ahlborn B, Mutschke G, Morzyński M, Comte P, Tadmor G (2008) A finite-time thermodynamics of unsteady fluid flows. J Non Equilibr Thermodyn 33:103–148MATHGoogle Scholar
  41. Noack BR, Morzyński M, Tadmor G (eds) (2011) Reduced-order modelling for flow control. No. 528 in CISM Courses and Lectures, Springer, BerlinGoogle Scholar
  42. Papadakis N (2007) Assimilation de données images : application au suivi de courbes et de champs de vecteurs. PhD thesis, Université Rennes IGoogle Scholar
  43. Perret L, Collin E, Delville J (2006) Polynomial identification of POD based low-order dynamical system. J Turbul 7:1–15MathSciNetCrossRefGoogle Scholar
  44. Protas B, Styczek A (2002) Optimal rotary control of the cylinder wake in the laminar regime. Phys Fluids 14(7):2073–2087CrossRefGoogle Scholar
  45. Rempfer D (2000) On low-dimensional Galerkin models for fluid flow. Theoret Comput Fluid Dyn 14:75–88MATHCrossRefGoogle Scholar
  46. Rempfer D, Fasel FH (1994) Dynamics of three-dimensional coherent structures in a flat-plate boundary-layer. J Fluid Mech 275:257–283CrossRefGoogle Scholar
  47. Rowley CW, Mezić I, Bagheri S, Schlatter P, Henningson D (2009) Spectral analysis of nonlinear flows. J Fluid Mech 645:115–127CrossRefGoogle Scholar
  48. Schlegel M, Noack BR, Comte P, Kolomenskiy D, Schneider K, Farge M, Scouten J, Luchtenburg DM, Tadmor G (2009) Reduced-order modelling of turbulent jets for noise control. In: Numerical simulation of turbulent flows and noise generation: results of the DFG/CNRS Research Groups FOR 507 and FOR 508, Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), Springer, Berlin, pp 3–27Google Scholar
  49. Schlegel M, Noack BR, Jordan P, Dillmann A, Gröschel E, Schröder W, Wei M, Freund JB, Lehmann O, Tadmor G (2012) On least-order flow representations for aerodynamics and aeroacoustics. J Fluid Mech 697:367–398MATHCrossRefGoogle Scholar
  50. Schmid PJ (2010) Dynamic mode decomposition for numerical and experimental data. J Fluid Mech 656:5–28MathSciNetMATHCrossRefGoogle Scholar
  51. Sirisup S, Karniadakis G (2004) A spectral viscosity method for correcting the long-term behavior of POD models. J Comp Phys 194:92–116MathSciNetMATHCrossRefGoogle Scholar
  52. Titaud O, Vidard A, Souopgui I, Le Dimet F (2010) Assimilation of image sequences in numerical models. Tellus A 62(1):30–47CrossRefGoogle Scholar
  53. Ukeiley L, Cordier L, Manceau R, Delville J, Bonnet JP, Glauser M (2001) Examination of large-scale structures in a turbulent plane mixing layer. Part 2. Dynamical systems model. J Fluid Mech 441:61–108CrossRefGoogle Scholar
  54. Venturi D, Wan X, Karniadakis G (2008) Stochastic low dimensional modeling of random laminar wake past a circular cylinder. J Fluid Mech 606:339–367MathSciNetMATHCrossRefGoogle Scholar
  55. Wang Z, Akhtar I, Borggaard J, Iliescu T (2012) Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison. Comput Methods Appl Mech Engrg 237-240:10–26MathSciNetCrossRefGoogle Scholar
  56. Wei M (2004) Jet noise control by adjoint-based optimization. PhD thesis, University of Illinois at Urbana-ChampaignGoogle Scholar
  57. Wiener N (1948) Cybernetics or control and communication in the animal and the machine, 1st edn. MIT Press, BostonGoogle Scholar
  58. Wu JZ, Ma HY, Zhou MD (2006) Vorticity and vortex dynamics, 1st edn. Springer, BerlinCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Laurent Cordier
    • 1
  • Bernd R. Noack
    • 1
  • Gilles Tissot
    • 1
  • Guillaume Lehnasch
    • 1
  • Joël Delville
    • 1
  • Maciej Balajewicz
    • 2
  • Guillaume Daviller
    • 1
  • Robert K. Niven
    • 3
  1. 1.Institut Pprime, CNRS – Université de Poitiers – ENSMA, UPR 3346, Département Fluides, Thermique, Combustion, CEATPoitiers CedexFrance
  2. 2.Mechanical Engineering and Materials SciencesDuke UniversityDurhamUSA
  3. 3.School of Engineering and Information TechnologyThe University of New South Wales at ADFACanberraAustralia

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