Hostname: page-component-5cf477f64f-xc2pj Total loading time: 0 Render date: 2025-04-07T05:09:50.602Z Has data issue: false hasContentIssue false
  • English
  • Français

A framework for input–output analysis of wall-bounded shear flows

Published online by Cambridge University Press:  28 June 2019

Mohamadreza Ahmadi Open the ORCID record for Mohamadreza Ahmadi [Opens in a new window] ,
Giorgio Valmorbida ,
Dennice Gayme Open the ORCID record for Dennice Gayme [Opens in a new window]  and
Antonis Papachristodoulou
Mohamadreza Ahmadi*
Affiliation:
Department of Mechanical Engineering, California Institute of Technology, 1200 East California Boulevard, MC 104-44, Pasadena, CA 91125, USA
Giorgio Valmorbida
Affiliation:
Laboratoire des Signaux et Systèmes, CentraleSupélec, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91192 Gif-sur-Yvette, France
Dennice Gayme
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Lathrobe Hall 223, 3400 North Charles Street, Baltimore, MD 21218, USA
Antonis Papachristodoulou
Affiliation:
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a new framework to evaluate input–output amplification properties of nonlinear models of wall-bounded shear flows, subject to both square integrable and persistent disturbances. We focus on flows that are spatially invariant in one direction and whose base flow can be described by a polynomial, e.g. streamwise-constant channel, Couette and pipe flows. Our methodology is based on the notion of dissipation inequalities in control theory and provides a single unified approach for examining flow properties such as energy growth, worst-case disturbance amplification and stability to persistent excitations (i.e. input-to-state stability). It also enables direct analysis of the nonlinear partial differential equation rather than of a discretized form of the equations, thereby removing the possibility of truncation errors. We demonstrate how to numerically compute the input–output properties of the flow as the solution of a (convex) optimization problem. We apply our theoretical and computational tools to plane Couette, channel and pipe flows. Our results demonstrate that the proposed framework leads to results that are consistent with theoretical and experimental amplification scalings obtained in the literature.

JFM classification

Flow Control: Control theory Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 873 , 25 August 2019 , pp. 742 - 785
Check for updates
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahmadi, M., Valmorbida, G. & Papachristodoulou, A. 2015 A convex approach to hydrodynamic analysis. In 2015 54th IEEE Conference on Decision and Control (CDC), pp. 7262–7267.Google Scholar
Ahmadi, M., Valmorbida, G. & Papachristodoulou, A. 2016 Dissipation inequalities for the analysis of a class of PDEs. Automatica 66, 163–171.Google Scholar
del Alamo, J. C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.Google Scholar
Alexakis, A. & Doering, C. R. 2006 Energy and enstrophy dissipation in steady state 2d turbulence. Phys. Lett. A 359 (6), 652–657.Google Scholar
Bamieh, B. & Dahleh, M. 2001 Energy amplification in channel flows with stochastic excitation. Phys. Fluids 13 (11), 3258–3269.Google Scholar
Bobba, K. M., Bamieh, B. & Doyle, J. C. 2002 Highly optimized transitions to turbulence. In Proceedings of the 41st IEEE Conference on Decision and Control, vol. 4, pp. 4559–4562.Google Scholar
Bovet, D. & Crescenzi, P. 1994 Introduction to the Theory of Complexity. Prentice-Hall.Google Scholar
Boyd, S. & Vandenberghe, L. 2004 Convex Optimization. Cambridge University Press.Google Scholar
Bullock, K. J., Cooper, R. E. & Abernathy, F. H. 1978 Structural similarity in radial correlations and spectra of longitudinal velocity fluctuations in pipe flow. J. Fluid Mech. 88, 585–608.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4 (8), 1637–1650.Google Scholar
Chernyshenko, S.2017 Relationship between the methods of bounding time averages. Preprint, arXiv:1704.02475.Google Scholar
Chernyshenko, S., Goulart, P., Huang, D. & Papachristodoulou, A. 2014 Polynomial sum of squares in fluid dynamics: a review with a look ahead. Phil. Trans. R. Soc. Lond. A 372 (2020), 20130350.Google Scholar
Chesi, G., Tesi, A., Vicino, A. & Genesio, R. 1999 On convexification of some minimum distance problems. In 5th European Control Conference, Karlsruhe, Germany.Google Scholar
Childress, S., Kerswell, R. R. & Gilbert, A. D. 2001 Bounds on dissipation for Navier–Stokes flow with Kolmogorov forcing. Physica D 158 (1), 105–128.Google Scholar
Choi, M. D., Lam, T. Y. & Reznick, B. 1995 Sums of squares of real polynomials. In Proceedings of Symposia in Pure Mathematics, vol. 58, pp. 103–126. American Mathematical Society.Google Scholar
Curtain, R. F. & Zwart, H. J. 1995 An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, vol. 21. Springer.Google Scholar
De Branges, L. 1959 The Stone–Weierstrass theorem. Proc. Am. Math. Soc. 10 (5), 822–824.Google Scholar
Doering, C. R. & Constantin, P. 1994 Variational bounds on energy dissipation in incompressible flows: shear flow. Phys. Rev. E 49, 4087–4099.Google Scholar
Doering, C. R. & Foias, C. 2002 Energy dissipation in body-forced turbulence. J. Fluid Mech. 467, 289–306.Google Scholar
Doering, C. R. & Gibbon, J. D. 1995 Applied Analysis of the Navier–Stokes Equations, Cambridge Texts in Applied Mathematics, vol. 12. Cambridge University Press.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Fantuzzi, G. 2018 Bounds for Rayleigh–Bénard convection between free-slip boundaries with an imposed heat flux. J. Fluid Mech. 837, R5.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids A 5 (11), 2600–2609.Google Scholar
Gahlawat, A. & Peet, M. M. 2017 A convex sum-of-squares approach to analysis, state feedback and output feedback control of parabolic PDEs. IEEE Trans. Autom. Control 62 (4), 1636–1651.Google Scholar
Gayme, D. F., McKeon, B. J., Bamieh, B., Papachristodoulou, A. & Doyle, J. C. 2011 Amplification and nonlinear mechanisms in plane Couette flow. Phys. Fluids 23 (6), 065108.Google Scholar
Gayme, D. F., McKeon, B. J., Papachristodoulou, A., Bamieh, B. & Doyle, J. C. 2010 A streamwise constant model of turbulence in plane Couette flow. J. Fluid Mech. 665, 99–119.Google Scholar
Goluskin, D. & Fantuzzi, G. 2019 Bounds on mean energy in the Kuramoto–Sivashinsky equation computed using semidefinite programming. Nonlinearity 32 (5), 1705.Google Scholar
Goulart, P. J. & Chernyshenko, S. 2012 Global stability analysis of fluid flows using sum-of-squares. Physica D 241 (6), 692–704.Google Scholar
Grant, M., Boyd, S. & Ye, Y.2008 CVX: Matlab software for disciplined convex programming.Google Scholar
Grossmann, S. 2000 The onset of shear flow turbulence. Rev. Mod. Phys. 72, 603–618.Google Scholar
Gustavsson, L. H. 1991a Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241–260.Google Scholar
Heins, P. H., Jones, B. Ll. & Sharma, A. S. 2016 Passivity-based output-feedback control of turbulent channel flow. Automatica 69, 348–355.Google Scholar
Hill, D. J. & Moylan, P. J. 1980 Dissipative dynamical systems: basic input–output and state properties. J. Franklin Inst. 309 (5), 327–357.Google Scholar
Huang, D., Chernyshenko, S., Goulart, P., Lasagna, D., Tutty, O. & Fuentes, F. 2015 Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application. Proc. R. Soc. Lond. A 471 (2183), 20150622.Google Scholar
Hutchins, N. & Marusic, I. 2007 Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.Google Scholar
Joseph, D. D. 1976 Stability of Fluid Motions. Springer.Google Scholar
Joseph, D. D. & Hung, W. 1971 Contributions to the nonlinear theory of stability of viscous flow in pipes and between rotating cylinders. Arch. Rat. Mech. Anal. 44 (1), 1–22.Google Scholar
Jovanović, M. R.2004 Modeling, analysis, and control of spatially distributed systems. PhD thesis, University of California, Santa Barbara, CA.Google Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.Google Scholar
Khalil, H. K. 1996 Nonlinear Systems. Prentice-Hall.Google Scholar
Kim, K. J. & Adrian, R. J. 1999 Very large scale motion in the outer layer. Phys. Fluids 11, 417–422.Google Scholar
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741–773.Google Scholar
Lasserre, J. B. 2009 Moments, Positive Polynomials and Their Applications. Imperial College Press.Google Scholar
Löfberg, J. 2004 YALMIP: a toolbox for modeling and optimization in MATLAB. In Proceedings of the 2004 IEEE International Conference on Robotics and Automation. IEEE.Google Scholar
Lumer, G. & Phillips, R. S. 1961 Dissipative operators in a Banach space. Pacific J. Math. 11 (2), 679–698.Google Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.Google Scholar
Moarref, R., Jovanovic, M. R., Tropp, J. A., Sharma, A. S. & McKeon, B. J. 2014 A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization. Phys. Fluids 26 (5), 051701.Google Scholar
Motzkin, T. S. 1965 The arithmetic-geometric inequality. In 1967 Inequalities Symposium, Wright-Patterson Air Force Base, OH, pp. 205–224.Google Scholar
Nesterov, Y. & Nemirovskii, A. 1994 Interior-point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics.Google Scholar
Papachristodoulou, A., Anderson, J., Valmorbida, G., Prajna, S., Seiler, P. & Parrilo, P. A.2013 SOSTOOLS: sum of squares optimization toolbox for MATLAB. arXiv:1310.4716, available from http://www.eng.ox.ac.uk/control/sostools.Google Scholar
Parrilo, P.2000 Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology.Google Scholar
Payne, L. E. & Weinberger, H. F. 1960 An optimal Poincare inequality for convex domains. Arch. Rat. Mech. Anal. 5 (1), 286–292.Google Scholar
Peixinho, J. & Mullin, T. 2006 Decay of turbulence in pipe flow. Phys. Rev. Lett. 96, 094501.Google Scholar
Prajna, S., Papachristodoulou, A. & Wu, F. 2004 Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach. In 5th Asian Control Conference, vol. 1, pp. 157–165. IEEE.Google Scholar
Pujals, G., Garćia-Villalba, M., Cossu, C. & Depardon, S. 2009 A note on optimal transient growth in turbulent channel flows. Phys. Fluids 21 (1), 015109.Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.Google Scholar
Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and the law of resistance in parallel channels. Phil. Trans. 935 (51), 18830029.Google Scholar
Reznick, B. 2000 Some concrete aspects of Hilbert’s 17th problem. Contemp. Maths 253, 251–272.Google Scholar
Rollin, B., Dubief, Y. & Doering, C. R. 2011 Variations on Kolmogorov flow: turbulent energy dissipation and mean flow profiles. J. Fluid Mech. 670, 204–213.Google Scholar
Rubin, S. G. & Khosla, P. K. 1977 Polynomial interpolation methods for viscous flow calculations. J. Comput. Phys. 24 (3), 217–244.Google Scholar
Van der Schaft, A. 2017 L 2 -gain and Passivity Techniques in Nonlinear Control. Springer.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39 (1), 129–162.Google Scholar
Schmid, P. J. & Henningson, D. S. 1994 Optimal energy density growth in Hagen–Poiseuille flow. J. Fluid Mech. 277, 197–225.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Serrin, J. 1959 On the stability of viscous fluid motions. Arch. Rat. Mech. Anal. 3, 1–13.Google 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.Google Scholar
Shivakumar, S. & Peet, M. M.2018 Computing input–output properties of coupled PDE systems. Preprint, arXiv:1812.05081.Google Scholar
Sontag, E. D. 2008 Input to state stability: basic concepts and results. In Nonlinear and Optimal Control Theory (ed. Nistri, P. & Stefani, G.), Lecture Notes in Mathematics, vol. 1932, pp. 163–220. Springer.Google Scholar
Sontag, E. D. 2013 Input to state stability. In Encyclopedia of Systems and Control (ed. Baillieul, J. & Samad, T.), pp. 1–14. Springer.Google Scholar
Sturm, J. F. 1999 Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Method. Softw. 11, 625–653.Google Scholar
Tang, W., Caulfield, C. P. & Young, W. R. 2004 Bounds on dissipation in stress-driven flow. J. Fluid Mech. 510, 333–352.Google Scholar
Taylor, G. I. 1923 Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. R. Soc. Lond. A 223 (605–615), 289–343.Google Scholar
Tillmark, N. & Alfredsson, P. H. 1992 Experiments on transition in plane Couette flow. J. Fluid Mech. 235, 89–102.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.Google Scholar
Vassilicos, J. C. 2015 Dissipation in turbulent flows. Annu. Rev. Fluid Mech. 47 (1), 95–114.Google Scholar
Vazquez, R. & Krstic, M. 2008 Control of Turbulent and Magnetohydrodynamic Channel Flows: Boundary Stabilization and State Estimation. Springer Science & Business Media.Google Scholar
Weihs, D. 1975 On the polynomial approximation of boundary-layer flow profiles. Appl. Sci. Res. 31 (4), 253–266.Google Scholar
Willems, J. C. 1972 Dissipative dynamical systems part I: general theory. Arch. Rat. Mech. Anal. 45 (5), 321–351.Google Scholar
Willems, J. C. 2007 Dissipative dynamical systems. Eur. J. Control 13 (23), 134–151.Google Scholar