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Linear proportional–integral control for skin-friction reduction in a turbulent channel flow

Published online by Cambridge University Press:  08 February 2017

Euiyoung Kim
Affiliation:
Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul 08826, Korea
Haecheon Choi*
Affiliation:
Department of Mechanical and Aerospace Engineering, Seoul National University, Seoul 08826, Korea
*
Also at: Institute of Advanced Machines and Design, Seoul National University, Seoul 08826, Korea. Email address for correspondence: choi@snu.ac.kr

Abstract

In the present study, we apply a proportional (P)–integral (I) feedback control to a turbulent channel flow for skin-friction reduction. The instantaneous wall-normal velocity at a sensing plane above the wall is measured as a sensing parameter, and blowing/suction is provided at the wall based on the PI control. The performance of PI controls is estimated by the change in the skin friction while varying the sensing plane location $y_{s}$ and the proportional and integral feedback gains ($\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ respectively). The opposition control proposed by Choi et al. (J. Fluid Mech., vol. 262, 1994, pp. 75–110) corresponds to a P control with $\unicode[STIX]{x1D6FC}=1$. When the sensing plane is located close to the wall ($y_{s}^{+}\lesssim 10$), PI controls result in greater skin-friction reductions than corresponding P controls. The root-mean-square (r.m.s.) sensing velocity fluctuations, considered as the control error, approach zero with successful PI controls, but do not with P controls. Successful PI controls reduce the strength of near-wall coherent structures and the r.m.s. velocity fluctuations above the wall apart from those near the wall due to the control input. The frequency spectra of the sensing velocity show that the I component of PI controls significantly reduces the energy at low frequencies, much more than P controls do. Proportional–integral controls are also applied to a linearized flow model having transient growth of disturbances. The performance of PI controls for a linearized flow model is very similar to that for a turbulent channel flow, i.e. the low-frequency components of disturbances are significantly reduced by the I component of PI controls, and the transient energy growth is suppressed more than by P controls.

Type
Papers
Copyright
© 2017 Cambridge University Press 

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References

Abergel, F. & Temam, R. 1990 On some control problems in fluid mechanics. Theor. Comput. Fluid Dyn. 1, 303325.CrossRefGoogle Scholar
Bewley, T. R. & Liu, S. 1998 Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365, 305349.CrossRefGoogle Scholar
Bewley, T. R. & Moin, P. 1994 Optimal control of turbulent channel flow. In Active Control of Vibration and Noise (ed. Wang, K. W., Von Flotow, A. H., Shoureshi, R., Hendricks, E. W. & Farabee, T. W.), vol. DE75. ASME.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids 4, 16371650.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids 5, 774777.CrossRefGoogle Scholar
Chang, Y., Collis, S. S. & Ramakrishnan, S. 2002 Viscous effects in control of near-wall turbulence. Phys. Fluids 14, 40694080.CrossRefGoogle Scholar
Chevalier, M., Hœpffner, J., Åkervik, E. & Henningson, D. S. 2007 Linear feedback control and estimation applied to instabilities in spatially developing boundary layers. J. Fluid Mech. 588, 163187.CrossRefGoogle Scholar
Choi, H., Moin, P. & Kim, J. 1994 Active turbulence control for drag reduction in wall-bounded flows. J. Fluid Mech. 262, 75110.CrossRefGoogle Scholar
Choi, H., Temam, R., Moin, P. & Kim, J. 1993 Feedback control for unsteady flow and its application to the stochastic Burgers equation. J. Fluid Mech. 253, 509543.CrossRefGoogle Scholar
Choi, K.-S., DeBisschop, J.-R. & Clayton, B. R. 1998 Turbulent boundary-layer control by means of spanwise-wall oscillation. AIAA J. 36, 11571163.CrossRefGoogle Scholar
Choi, K.-S., Jukes, T. & Whalley, R. 2011 Turbulent boundary-layer control with plasma actuators. Phil. Trans. R. Soc. Lond. A 369, 14431458.Google ScholarPubMed
Chung, Y. M. & Talha, T. 2011 Effectiveness of active flow control for turbulent skin friction drag reduction. Phys. Fluids 23, 025102.CrossRefGoogle Scholar
Collis, S. S., Joslin, R. D., Seifert, A. & Theofilis, V. 2004 Issues in active flow control: theory, control, simulation, and experiment. Prog. Aerosp. Sci. 40, 237289.CrossRefGoogle Scholar
Das, P. K., Mathew, S., Shaiju, A. J. & Patnaik, B. S. V. 2016 Energetically efficient proportional–integral–differential (PID) control of wake vortices behind a circular cylinder. Fluid Dyn. Res. 48 (1), 015510.CrossRefGoogle Scholar
Deng, B.-Q., Xu, C.-X., Huang, W.-W. & Cui, G.-X. 2014 Strengthened opposition control for skin-friction reduction in wall-bounded turbulent flows. J. Turbul. 15 (2), 122143.CrossRefGoogle Scholar
Doyle, J. C., Glover, K., Khargonekar, P. P. & Francis, B. A. 1989 State-space solutions to standard H2 and H control problems. IEEE Trans. Autom. Control 34, 831847.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Turbulence suppression by active control. Phys. Fluids 8, 12571268.CrossRefGoogle Scholar
Franklin, G. F., Powell, J. D. & Emami-Naeini, A. 1994 Feedback Control of Dynamic Systems. Addison-Wesley.Google Scholar
Hammond, E. P., Bewley, T. R. & Moin, P. 1998 Observed mechanisms for turbulence attenuation and enhancement in opposition-controlled wall-bounded flows. Phys. Fluids 10, 24212423.CrossRefGoogle Scholar
Högberg, M., Bewley, T. R. & Henningson, D. S. 2003 Relaminarization of Re 𝜏 = 100 turbulence using gain scheduling and linear state-feedback control. Phys. Fluids 15 (11), 35723575.CrossRefGoogle Scholar
Hu, H. H. & Bau, H. H. 1994 Feedback control to delay or advance linear loss of stability in planar Poiseuille flow. Proc. R. Soc. Lond. A 447, 299312.Google Scholar
Iwamoto, K., Suzuki, Y. & Kasagi, N. 2002 Reynolds number effect on wall turbulence: toward effective feedback control. Intl J. Heat Fluid Flow 23, 678689.CrossRefGoogle Scholar
Jeong, J. & Hussain, F. 1995 On the identification of vortex. J. Fluid Mech. 285, 6994.CrossRefGoogle Scholar
Jeong, J., Hussain, F., Schoppa, W. & Kim, J. 1997 Coherent structures near the wall in a turbulent channel flow. J. Fluid Mech. 332, 185214.CrossRefGoogle Scholar
Joshi, S. S., Speyer, J. L. & Kim, J. 1997 A systems theory approach to the feedback stabilization of infinitesimal and finite-amplitude disturbances in plane Poiseuille flow. J. Fluid Mech. 332, 157184.CrossRefGoogle Scholar
Joshi, S. S., Speyer, J. L. & Kim, J. 1999 Finite dimensional optimal control of Poiseuille flow. J. Guid. Control Dyn. 22, 340348.CrossRefGoogle Scholar
Jung, W. J., Mangiavacchi, N. & Akhavan, R. 1992 Suppression of turbulence in wall-bounded flows by high-frequency spanwise oscillations. Phys. Fluids A 4 (8), 16051607.CrossRefGoogle Scholar
Kasagi, N., Suzuki, Y. & Fukagata, K. 2009 Microelectromechanical systems-based feedback control of turbulence for skin friction reduction. Annu. Rev. Fluid Mech. 41, 231251.CrossRefGoogle Scholar
Kim, E., Choi, H. & Kim, J. 2016 Optimal disturbances in the near-wall region of turbulent channel flows. Phys. Rev. Fluids 1, 074403.CrossRefGoogle Scholar
Kim, J. 2003 Control of turbulent boundary layers. Phys. Fluids 15, 10931105.CrossRefGoogle Scholar
Kim, J. 2011 Physics and control of wall turbulence for drag reduction. Phil. Trans. R. Soc. Lond. A 369 (1940), 13961411.Google ScholarPubMed
Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383417.CrossRefGoogle Scholar
Kim, J. & Lim, J. 2000 A linear process in wall-bounded turbulent shear flows. Phys. Fluids 12, 18851888.CrossRefGoogle Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.CrossRefGoogle Scholar
Lee, C., Kim, J., Babcock, D. & Goodman, R. 1997 Application of neural networks to turbulence control for drag reduction. Phys. Fluids 9, 17401747.CrossRefGoogle Scholar
Lee, C., Kim, J. & Choi, H. 1998 Suboptimal control of turbulent channel flow for drag reduction. J. Fluid Mech. 358, 245258.CrossRefGoogle Scholar
Lee, J. 2015 Opposition control of turbulent wall-bounded flow using upstream sensor. J. Mech. Sci. Technol. 29 (11), 47294735.CrossRefGoogle Scholar
Lee, K. H., Cortelezzi, L., Kim, J. & Speyer, J. L. 2001 Application of reduced-order controller to turbulent flows for drag reduction. Phys. Fluids 13, 13211330.CrossRefGoogle Scholar
Lim, J. & Kim, J. 2004 A singular value analysis of boundary layer control. Phys. Fluids 16, 19801988.CrossRefGoogle Scholar
Min, T., Kang, S. M., Speyer, J. L. & Kim, J. 2006 Sustained sub-laminar drag in a fully developed channel flow. J. Fluid Mech. 558, 309318.CrossRefGoogle Scholar
Nakanishi, R., Mamori, H. & Fukagata, K. 2012 Relaminarization of turbulent channel flow using traveling wave-like wall deformation. Intl J. Heat Fluid Flow 35, 152159.CrossRefGoogle Scholar
Nita, C., Vandewalle, S. & Meyers, J. 2016 On the efficiency of gradient based optimization algorithms for DNS-based optimal control in a turbulent channel flow. Comput. Fluids 125, 1124.CrossRefGoogle Scholar
Pamiés, M., Garnier, E., Merlen, A. & Sagaut, P. 2007 Response of a spatially developing turbulent boundary layer to active control strategies in the framework of opposition control. Phys. Fluids 19, 108102.CrossRefGoogle Scholar
Quadrio, M. & Ricco, P. 2004 Critical assessment of turbulent drag reduction through spanwise wall oscillations. J. Fluid Mech. 521, 251271.CrossRefGoogle Scholar
Quadrio, M., Ricco, P. & Viotti, C. 2009 Streamwise-travelling waves of spanwise wall velocity for turbulent drag reduction. J. Fluid Mech. 627, 161178.CrossRefGoogle Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209238.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Sharma, A. S., Morrison, J. F., Mckeon, B. J., Limebeer, D. J. N., Koberg, W. H. & Sherwin, S. J. 2011 Relaminarisation of Re 𝜏 = 100 channel flow with globally stabilising linear feedback control. Phys. Fluids 23 (12), 125105.CrossRefGoogle Scholar
Son, D., Jeon, S. & Choi, H. 2011 A proportional–integral–differential control of flow over a circular cylinder. Phil. Trans. R. Soc. Lond. A 369, 15401555.Google Scholar