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Optimal control of a separated boundary-layer flow over a bump

  • Pierre-Yves Passaggia (a1) and Uwe Ehrenstein (a2)

The optimal control of a globally unstable two-dimensional separated boundary layer over a bump is considered using augmented Lagrangian optimization procedures. The present strategy allows for controlling of the flow from a fully developed nonlinear state back to the steady state using a single actuator. The method makes use of a decomposition between the slow dynamics associated with the base flow modification and the fast dynamics, known as flapping, characterized by a large scale oscillation of the recirculation region. Starting from a steady state forced by a suction actuator located near the separation point, the base flow modification is shown to be controlled by a vanishing suction strategy. For weakly unstable flow regimes, this control law can be further optimized by means of direct–adjoint iterations of the nonlinear Navier–Stokes equations. In the absence of external noise, this novel approach proves to be capable of controlling the transient dynamics and the base flow modification simultaneously.

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Åkervik, E., Brandt, L., Henningson, D., Hœpffner, J., Marxen, O. & Schlatter, P. 2006 Steady solutions of the Navier–Stokes equations by selective frequency damping. Phys. Fluids 18, 068102.
Antoulas, A. C. 2005 Approximation of Large-scale Dynamical Systems. SIAM.
Bagheri, S., Brandt, L. & Henningson, D. S. 2009 Input–output analysis, model reduction and control of the flat-plate boundary layer. J. Fluid Mech. 620, 263298.
Bagheri, S. & Hanifi, A. 2007 The stabilizing effect of streaks on Tollmien–Schlichting and oblique waves: a parametric study. Phys. Fluids 19 (7), 078103.
Barbagallo, A., Sipp, D. & Schnmid, P. J. 2009 Closed-loop control of an open cavity flow using reduced order models. J. Fluid Mech. 641, 150.
Barkley, D., Gomes, M. & Henderson, D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 127, 473496.
Beaudoin, J.-F., Cadot, O., Aider, J.-L. & Wesfreid, J. E. 2004 Three-dimensional stationary flow over a backward-facing step. Eur. J. Mech. (B/Fluids) 23 (1), 147155.
Bewley, T. R., Moin, P. & Temam, R. 2001 DNS-based predictive control of turbulence: an optimal benchmark of feedback algortihms. J. Fluid Mech. 447, 179225.
Boujo, E., Ehrenstein, U. & Gallaire, F. 2013 Open-loop control of noise amplification in a separated boundary layer flow. Phys. Fluids 25, 124106.
Boujo, E., Fani, A. & Gallaire, F. 2015 Second-order sensitivity of parallel shear flows and optimal spanwise-periodic flow modifications. J. Fluid Mech. 782, 491514.
Boujo, E. & Gallaire, F. 2014 Manipulating flow separation: sensitivity of stagnation points, separatrix angles and recirculation area to steady actuation. Proc. R. Soc. Lond. A 470 (2170), 20140365.
Boujo, E., Gallaire, F. & Ehrenstein, U. 2014 Open-loop control of a separated boundary layer. C. R. Méc. 342 (6–7), 403409.
Cattafesta, L. N. & Sheplak, M. 2011 Actuators for active flow control. Annu. Rev. Fluid Mech. 43, 247272.
Cossu, C. & Brandt, L. 2002 Stabilization of Tollmien–Schlichting waves by finite amplitude optimal streaks in the blasius boundary layer. Phys. Fluids 14 (8), L57L60.
Cunha, G., Passaggia, P.-Y. & Lazareff, M. 2015 Optimization of the selective frequency damping parameters using model reduction. Phys. Fluids 27, 094103.
Duriez, T., Brunton, S. L. & Noack, B. R. 2017 Taming nonlinear dynamics with MLC. In Machine Learning Control–Taming Nonlinear Dynamics and Turbulence, pp. 93120. Springer.
Ehrenstein, U. & Gallaire, F. 2008 Two-dimensional global low-frequency oscillations in a separating boundary-layer flow. J. Fluid Mech. 614, 315327.
Ehrenstein, U., Passaggia, P-Y. & Gallaire, F. 2011 Control of a separated boundary layer: model reduction using global modes revisited. Theor. Comput. Fluid Dyn. 25, 195207.
Flinois, T. & Colonius, T. 2015 Optimal control of circular cylinder wakes using long control horizons. Phys. Fluids 27, 087105.
Gallaire, F., Marquillie, M. & Ehrenstein, U. 2007 Three-dimensional transverse instabilities in detached boundary-layers. J. Fluid Mech. 571, 221233.
Gautier, N. & Aider, J.-L. 2013 Control of the separated flow downstream of a backward-facing step using visual feedback. Proc. R. Soc. Lond. A 469 (2160).
Gautier, N. & Aider, J.-L. 2014 Feed-forward control of a perturbed backward-facing step flow. J. Fluid Mech. 759, 181196.
Geering, H. P. 2007 Optimal Control with Engineering Applications. Springer.
Haller, G. 2004 Exact theory of unsteady separation for two-dimensional flows. J. Fluid Mech. 512, 257311.
Huang, S.-C. & Kim, J. 2008 Control and system identification of a separated flow. Phys. Fluids 20 (10), 101509.
Hwang, Y., Kim, J. & Choi, H. 2013 Stabilization of absolute instability in spanwise wavy two-dimensional wakes. J. Fluid Mech. 727, 346378.
Jordi, B. E., Cotter, C. J. & Sherwin, S. J. 2014 Encapsulated formulation of the selective frequency damping method. Phys. Fluids 26 (3), 034101.
Joslin, R. D., Gunzburger, M. D., Nicolaides, R. A., Erlebacher, G. & Hussaini, M. Y. 1995 A self-contained, automated methodology for optimal flow control validated for transition delay. ICASE 95, 64.
Kim, J. & Bewley, R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383417.
Lanzerstorfer, D. & Kuhlmann, H. C. 2012 Global stability of the two-dimensional flow over a backward-facing step. J. Fluid Mech. 693, 127.
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2014 Self-consistent mean flow description of the nonlinear saturation of the vortex shedding in the cylinder wake. Phys. Rev. Lett. 113 (8), 084501.
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2015 A self-consistent model for the saturation dynamics of the vortex shedding around the mean flow in the unstable cylinder wake. Phys. Fluids 27 (7), 074103.
Marquillie, M. & Ehrenstein, U. 2002 Numerical simulation of a separating boundary-layer flow. Comput. Fluids 31, 683693.
Marquillie, M. & Ehrenstein, U. 2003 On the onset of nonlinear oscillations in a separating boundary-layer flow. J. Fluid Mech. 490, 169188.
Marxen, O. & Henningson, D. S. 2011 The effect of small-amplitude convective disturbances on the size and bursting of a laminar separation bubble. J. Fluid Mech. 671, 133.
Noack, B. R., Konstantin, A., 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, 335363.
Passaggia, P.-Y. & Ehrenstein, U. 2013 Adjoint based optimization and control of a separated boundary-layer flow. Eur. J. Mech. (B/Fluids) 41, 169177.
Passaggia, P.-Y., Leweke, T. & Ehrenstein, U. 2012 Transverse instability and low-frequency flapping in incompressible separated boundary layer flows: an experimental study. J. Fluid Mech. 703, 363373.
Rodríguez, D., Gennaro, E. M. & Juniper, M. P. 2013 The two classes of primary modal instability in laminar separation bubbles. J. Fluid Mech. 734, R4.
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115127.
Ruhe, A. 1984 Rational krylov sequence methods for eigenvalue computation. Linear Algebr. Applics. 58, 391405.
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 528.
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.
Semeraro, O., Pralits, J. O., Rowley, C. W. & Henningson, D. S. 2013 Riccati-less approach for optimal control and estimation: an application to two-dimensional boundary layers. J. Fluid Mech. 731, 394417.
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333358.
Turton, S. E., Tuckerman, L. S. & Barkley, D. 2015 Prediction of frequencies in thermosolutal convection from mean flows. Phys. Rev. E 91 (4), 043009.
Weldon, M., Peacock, T., Jacobs, G. B., Helu, M. & Haller, G. 2008 Experimental and numerical investigation of the kinematic theory of unsteady separation. J. Fluid Mech. 611, 111.
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