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Closed-loop control of boundary layer streaks induced by free-stream turbulence

George Papadakis1,*, Liang Lu1, and Pierre Ricco2

  • 1Department of Aeronautics, Imperial College London, London SW7 2AZ, United Kingdom
  • 2Department of Mechanical Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom

  • *g.papadakis@ic.ac.uk
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Phys. Rev. Fluids 1, 043501 – Published 26 August, 2016

DOI: https://doi.org/10.1103/PhysRevFluids.1.043501

Abstract

The central aim of the paper is to carry out a theoretical and numerical study of active wall transpiration control of streaks generated within an incompressible boundary layer by free-stream turbulence. The disturbance flow model is based on the linearized unsteady boundary-region (LUBR) equations, studied by Leib, Wundrow, and Goldstein [J. Fluid Mech. 380, 169 (1999)], which are the rigorous asymptotic limit of the Navier-Stokes equations for low-frequency and long-streamwise wavelength. The mathematical formulation of the problem directly incorporates the random forcing into the equations in a consistent way. Due to linearity, this forcing is factored out and appears as a multiplicative factor. It is shown that the cost function (integral of kinetic energy in the domain) is properly defined as the expectation of a random quadratic function only after integration in wave number space. This operation naturally introduces the free-stream turbulence spectral tensor into the cost function. The controller gains for each wave number are independent of the spectral tensor and, in that sense, universal. Asymptotic matching of the LUBR equations with the free-stream conditions results in an additional forcing term in the state-space system whose presence necessitates the reformulation of the control problem and the rederivation of its solution. It is proved that the solution can be obtained analytically using an extension of the sweep method used in control theory to obtain the standard Riccati equation. The control signal consists of two components, a feedback part and a feed-forward part (that depends explicitly on the forcing term). Explicit recursive equations that provide these two components are derived. It is shown that the feed-forward part makes a negligible contribution to the control signal. We also derive an explicit expression that a priori (i.e., before solving the control problem) leads to the minimum of the objective cost function (i.e., the fundamental performance limit), based only on the system matrices and the initial and free-stream boundary conditions. The adjoint equations admit a self-similar solution for large spanwise wave numbers with a scaling which is different from that of the LUBR equations. The controlled flow field also has a self-similar solution if the weighting matrices of the objective function are chosen appropriately. The code developed to implement this algorithm is efficient and has modest memory requirements. Computations show the significant reduction of energy for each wave number. The control of the full spectrum streaks, for conditions corresponding to a realistic experimental case, shows that the root-mean-square of the streamwise velocity is strongly suppressed in the whole domain and for all the frequency ranges examined.

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References (48)

  1. S. Bagheri, Computational hydrodynamic stability and flow control based on spectral analysis of linear operators, Arch. Comput. Methods Eng. 19, 341 (2012).
  2. S. Bagheri and D. Henningson, Transition delay using control theory, Philos. Trans. R. Soc. London A 369, 1365 (2011).
  3. S. Bagheri, D. Henningson, J. Hoepffner, and P. Schmid, Input-output analysis and control design applied to a linear model of spatialy developing flows, Appl. Mech. Rev. 62, 020803 (2009).
  4. J. Kim and T. R. Bewley, A linear systems approach to flow control, Annu. Rev. Fluid Mech. 39, 383 (2007).
  5. T. T. Medjo, R. Temam, and M. Ziane, Optimal and robust control of fluid flows: some theoretical and computational aspects, Appl. Mech. Rev. 61, 010802 (2008).
  6. L. Baramov, O. R. Tutty, and E. Rogers, H∞ control of non-periodic two-dimensional channel flow, IEEE Trans. Control Sys. Tech. 12, 111 (2002).
  7. T. R. Bewley and S. Liu, Optimal and robust control and estimation of linear paths to transition, J. Fluid Mech. 365, 305 (1998).
  8. J. McKernan, G. Papadakis, and J. F. Whidborne, Linear and non-linear simulations of feedback control in plane Poiseuille flow, Int. J. Num. Mech. Fluids 59, 907 (2009).
  9. B. Bamieh, F. Paganini, and M. A. Dahleh, Distributed control of spatially invariant systems, IEEE Trans. Automatic Control 47, 1091 (2002).
  10. M. Högberg and D. S. Henningson, Linear optimal control applied to instabilities in spatially developing boundary layers, J. Fluid Mech. 470, 151 (2002).
  11. M. Chevalier, J. Hoepffner, E. Akervik, and D. S. Henningson, Linear feedback control and estimation applied to instabilities in spatially developing boundary layer, J. Fluid Mech. 588, 163 (2007).
  12. P. Cathalifaud and P. Luchini, Algebraic growth in boundary layers: optimal control by blowing and suction at the wall, Eur. J. Mech. B-Fluids 19, 469 (2000).
  13. P. Cathalifaud and T. R. Bewley, A noncausal framework for model-based feedback control of spatially developing perturbations in boundary-layer flow systems. Part I: Formulation, Sys. Control Lett. 51, 1 (2004).
  14. P. Cathalifaud and T. R. Bewley, A noncausal framework for model-based feedback control of spatially developing perturbations in boundary-layer flow systems. Part II: Numerical simulations using state feedback, Syst. Control Lett. 51, 15 (2004).
  15. A. Bryson and Y. Ho, Applied Optimal Control (Taylor and Francis, Philadelphia, 1975).
  16. R. F. Stengel, Optimal Control and Estimation (Dover, New York, 1994).
  17. B. Moore, Principal component analysis in linear systems: Controllability, observability and model reduction, IEEE Trans. Autom. Control 26, 17 (1981).
  18. C. Rowley, Model reduction for fluids using balanced proper orthogonal decomposition, Int. J. Bifur. Chaos 15, 997 (2005).
  19. S. Bagheri, L. Brandt, and D. Henningson, Input-output analysis, model reduction and control of the flat plate boundary layer, J. Fluid Mech. 620, 263 (2009).
  20. O. Semerano, S. Bagheri, L. Brandt, and D. S. Henningson, Feedback control of three-dimensional optimal disturbances using reduced-order models, J. Fluid Mech. 677, 63 (2011).
  21. J. Pralits and P. Luchini, Riccati-less optimal control of bluff-body wakes, in Proc. 7th IUTAM Symposium on Laminar-Turbulent Transition Stockholm, Sweden, 2009, edited by P. Schlatter and D. Henningson (Springer, Dordrecht, 2010), pp. 325–330.
  22. O. Semerano, J. Pralits, C. Rowley, and D. Henningson, Riccati-less approach for optimal control and estimation: an application to two dimensional boundary layers, J. Fluid Mech. 731, 394 (2013).
  23. S. J. Leib, D. W. Wundrow, and M. E. Goldstein, Effect of free-stream turbulence and other vortical disturbances on laminar boundary layer, J. Fluid Mech. 380, 169 (1999).
  24. P. Davidson, Turbulence: An Introduction for Scientists and Engineers (Oxford University Press, Oxford, 2004).
  25. M. Goldstein, Response of the pre-transitional laminar boundary layer to free-stream turbulence-Otto Laporte Lecture, Bull. Am. Phys. Soc. 42, 2150 (1997).
  26. M. E. Goldstein, Unsteady vortical and entropic distortions of potential flows around arbitrary obstacles, J. Fluid Mech. 89, 433 (1978).
  27. M. E. Goldstein, The evolution of Tollmien-Schlichting waves near a leading edge, J. Fluid Mech. 127, 59 (1983).
  28. H. Schlichting, Boundary Layer Theory (McGraw-Hill, New York, 1985).
  29. A. N. Gulyaev, V. E. Kozlov, V. R. Kuznetson, B. I. Mineev, and A. N. Sekundov, Interaction of a laminar boundary layer with external turbulence, Fluid Dyn. 24, 700 (1989).
  30. P. Ricco, The pre-transitional Klebanoff modes and other boundary layer disturbances induced by small wavelength free-stream vorticity, J. Fluid Mech. 638, 267 (2009).
  31. D. Bender and A. Laub, The linear-quadratic optimal regulator for descriptor systems, IEEE Trans. Automatic Control 32, 672 (1987).
  32. J. Hunt, A theory of turbulent flow round two dimensional bluff bodies, J. Fluid Mech. 61, 625 (1973).
  33. P. Ricco and X. Wu, Response of a compressible laminar boundary layer to free-stream vortical disturbances, J. Fluid Mech. 587, 97 (2007).
  34. M. Green and D. Limebeer, Linear Robust Control (Prentice Hall, Englewood Cliffs, NJ, 1995).
  35. M. Högberg, T. R. Bewley, and D. S. Henningson, Linear feedback control and estimation of transition in plane channel flow, J. Fluid Mech. 481, 149 (2003).
  36. F. L. Lewis, D. L. Vrabie, and V. L. Syrmos, Optimal Control, 3rd ed. (John Wiley and Sons, New York, 2012).
  37. P. Luchini and A. Bottaro, Adjoint equations in stability analysis, Annu. Rev. Fluid Mech. 46, 493 (2014).
  38. P. Schmid and D. Henningson, Stability and Transition in Shear Flows (Springer, New York, 2001).
  39. P. Ricco, J. Luo, and X. Wu, Evolution and instability of unsteady nonlinear streaks generated by free-stream vortical disturbances, J. Fluid Mech. 677, 1 (2011).
  40. D. Wundrow and M. Goldstein, Effect on a laminar boundary layer of small-amplitude streamwise vorticity in the upstream flow, J. Fluid Mech. 426, 229 (2001).
  41. M. Choudhari, Boundary layer receptivity to three-dimensional unsteady vortical disturbances in the free stream, AIAA Paper 96-0181, 1996.
  42. X. Wu and M. Choudhari, Linear and non-linear instabilities of a Blasius boundary layer perturbed by streamwise vortices. Part 2. Intermittent instability induced by long-wavelength Klebanoff modes, J. Fluid Mech. 483, 249 (2003).
  43. X. Wu and J. Luo, Linear and non-linear instabilities of a Blasius boundary layer perturbed by streamwise vortices. Part 1. Steady streaks, J. Fluid Mech. 483, 225 (2003).
  44. B. Farrell and P. Ioannou, Stochastic forcing of the linearized Navier-Stokes equations, Phys. Fluids A 5, 2600 (1993).
  45. M. Jovanovic and B. Bamieh, Componentwise energy amplification in channel flows, J. Fluid Mech. 534, 145 (2005).
  46. J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed. (Dover, New York, 2001).
  47. C. Canuto, M. Hussaini, A. Quarteroni, and T. Zhang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Springer, New York, 2007).
  48. P. Moin and J. Kim, On the numerical solution of time-dependent viscous incompressible fluid flows involving solid boundaries, J. Comp. Phys. 35, 381 (1980).

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Phys. Rev. Fluids 1, 043501– Published 26 August, 2016
  • Received 12 January 2016

©2016 American Physical Society

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