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Colour of turbulence

Published online by Cambridge University Press:  05 January 2017

Armin Zare ,
Mihailo R. Jovanović Open the ORCID record for Mihailo R. Jovanović [Opens in a new window]  and
Tryphon T. Georgiou
Armin Zare
Affiliation:
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA
Mihailo R. Jovanović*
Affiliation:
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA
Tryphon T. Georgiou
Affiliation:
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN 55455, USA
*
†Email address for correspondence: mihailo@umn.edu
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Abstract

In this paper, we address the problem of how to account for second-order statistics of turbulent flows using low-complexity stochastic dynamical models based on the linearized Navier–Stokes equations. The complexity is quantified by the number of degrees of freedom in the linearized evolution model that are directly influenced by stochastic excitation sources. For the case where only a subset of velocity correlations are known, we develop a framework to complete unavailable second-order statistics in a way that is consistent with linearization around turbulent mean velocity. In general, white-in-time stochastic forcing is not sufficient to explain turbulent flow statistics. We develop models for coloured-in-time forcing using a maximum entropy formulation together with a regularization that serves as a proxy for rank minimization. We show that coloured-in-time excitation of the Navier–Stokes equations can also be interpreted as a low-rank modification to the generator of the linearized dynamics. Our method provides a data-driven refinement of models that originate from first principles and captures complex dynamics of turbulent flows in a way that is tractable for analysis, optimization and control design.

JFM classification

Flow Control: Control theory Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulent Flows
Type
Papers
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Journal of Fluid Mechanics , Volume 812 , 10 February 2017 , pp. 636 - 680
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© 2017 Cambridge University Press 

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References

Adrian, R. J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19 (4), 041301.Google Scholar
Bakas, N. A. & Ioannou, P. J. 2011 Structural stability theory of two-dimensional fluid flow under stochastic forcing. J. Fluid Mech. 682, 332–361.CrossRefGoogle Scholar
Bakas, N. A. & Ioannou, P. J. 2014 A theory for the emergence of coherent structures in beta-plane turbulence. J. Fluid Mech. 740, 312–341.CrossRefGoogle Scholar
Bamieh, B. & Dahleh, M. 2001 Energy amplification in channel flows with stochastic excitation. Phys. Fluids 13 (11), 3258–3269.CrossRefGoogle Scholar
Berkoo, G., Holmes, P. & Lumley, J. L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25 (1), 539–575.CrossRefGoogle Scholar
Bertolotti, F. P., Herbert, T. & Spalart, P. R. 1992 Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441–474.CrossRefGoogle Scholar
Bewley, T. R. & Liu, S. 1998 Optimal and robust control and estimation of linear paths to transition. J. Fluid Mech. 365, 305–349.Google Scholar
Boyd, S. & Vandenberghe, L. 2004 Convex Optimization. Cambridge University Press.Google Scholar
Bretheim, J. U., Meneveau, C. & Gayme, D. F. 2015 Standard logarithmic mean velocity distribution in a band-limited restricted nonlinear model of turbulent flow in a half-channel. Phys. Fluids 27 (1), 011702.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4 (8), 1637–1650.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids A 5 (3), 774–777.Google Scholar
Cambon, C., Godeferd, F. S., Nicolleau, F. C. G. A. & Vassilicos, J. C. 2004 Turbulent diffusion in rapidly rotating flows with and without stable stratification. J. Fluid Mech. 499, 231–255.Google Scholar
Candès, E. J. & Plan, Y. 2010 Matrix completion with noise. Proc. IEEE 98 (6), 925–936.CrossRefGoogle Scholar
Candès, E. J. & Recht, B. 2009 Exact matrix completion via convex optimization. Found. Comput. Math. 9 (6), 717–772.Google Scholar
Chen, K. K., Tu, J. H. & Rowley, C. W. 2012 Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J. Nonlinear Sci. 22 (6), 887–915.CrossRefGoogle Scholar
Chen, Y., Jovanović, M. R. & Georgiou, T. T. 2013 State covariances and the matrix completion problem. In Proceedings of the 52nd IEEE Conference on Decision and Control, pp. 1702–1707.Google Scholar
Chernyshenko, S. I. & Baig, M. F. 2005 The mechanism of streak formation in near-wall turbulence. J. Fluid Mech. 544, 99–131.Google Scholar
Chevalier, M., Hœpffner, J., Bewley, T. R. & Henningson, D. S. 2006 State estimation in wall-bounded flow systems. Part 2. Turbulent flows. J. Fluid Mech. 552, 167–187.Google Scholar
Clark, N. R. & Vassilicos, J. C. 2011 Kinematic simulation of fully developed turbulent channel flow. Flow Turbul. Combust. 86 (2), 263–293.Google Scholar
Constantinou, N. C., Farrell, B. F. & Ioannou, P. J. 2014a Emergence and equilibration of jets in beta-plane turbulence: applications of stochastic structural stability theory. J. Atmos. Sci. 71 (5), 1818–1842.Google Scholar
Constantinou, N. C., Lozano-Durán, A., Nikolaidis, M.-A., Farrell, B. F., Ioannou, P. J. & Jiménez, J. 2014b Turbulence in the highly restricted dynamics of a closure at second order: comparison with DNS. J. Phys.: Conf. Ser. 506, 012004.Google Scholar
Cortelezzi, L. & Speyer, J. L. 1998 Robust reduced-order controller of laminar boundary layer transitions. Phys. Rev. E 58 (2), 1906–1910.Google Scholar
Cossu, C., Pujals, G. & Depardon, S. 2009 Optimal transient growth and very large-scale structures in turbulent boundary layers. J. Fluid Mech. 619, 79–94.CrossRefGoogle Scholar
Del Álamo, J. C. & Jiménez, J. 2003 Spectra of the very large anisotropic scales in turbulent channels. Phys. Fluids 15 (6), 41–44.CrossRefGoogle Scholar
Del Álamo, J. C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.Google Scholar
Del Álamo, J. C. & Jiménez, J. 2009 Estimation of turbulent convection velocities and corrections to Taylor’s approximation. J. Fluid Mech. 640, 5–26.Google Scholar
Del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500 (1), 135–144.Google Scholar
DelSole, T. 1996 Can quasigeostrophic turbulence be modeled stochastically? J. Atmos. Sci. 53 (11), 1617–1633.Google Scholar
DelSole, T. 2000 A fundamental limitation of Markov models. J. Atmos. Sci. 57 (13), 2158–2168.2.0.CO;2>CrossRefGoogle Scholar
DelSole, T. 2004 Stochastic models of quasigeostrophic turbulence. Surv. Geophys. 25 (2), 107–149.Google Scholar
DelSole, T. & Farrell, B. F. 1995 A stochastically excited linear system as a model for quasigeostrophic turbulence: Analytic results for one-and two-layer fluids. J. Atmos. Sci. 52 (14), 2531–2547.Google Scholar
DelSole, T. & Farrell, B. F. 1996 The quasi-linear equilibration of a thermally maintained, stochastically excited jet in a quasigeostrophic model. J. Atmos. Sci. 53 (13), 1781–1797.Google Scholar
Durbin, P. A. & Reif, B. A. P. 2000 Theory and Modeling of Turbulent Flows. Wiley.Google Scholar
Elliott, F. W. & Majda, A. J. 1996 Pair dispersion over an inertial range spanning many decades. Phys. Fluids 8 (4), 1052–1060.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993a Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5 (6), 1390–1400.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993b Perturbation growth in shear flow exhibits universality. Phys. Fluids A 5 (9), 2298–2300.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993c Stochastic dynamics of baroclinic waves. J. Atmos. Sci. 50 (24), 4044–4057.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993d Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids  A 5 (11), 2600–2609.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1994 A theory for the statistical equilibrium energy spectrum and heat flux produced by transient baroclinic waves. J. Atmos. Sci. 51 (19), 2685–2698.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1995 Stochastic dynamics of the midlatitude atmospheric jet. J. Atmos. Sci. 52 (10), 1642–1656.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1998 Perturbation structure and spectra in turbulent channel flow. Theor. Comput. Fluid Dyn. 11, 237–250.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2003 Structural stability of turbulent jets. J. Atmos. Sci. 60 (17), 2101–2118.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2007 Structure and spacing of jets in barotropic turbulence. J. Atmos. Sci. 64 (10), 3652–3665.Google Scholar
Farrell, B. F. & Ioannou, P. J. 2012 Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149–196.Google Scholar
Fazel, M.2002 Matrix rank minimization with applications. PhD thesis, Stanford University.Google Scholar
Fransson, J. H. M., Talamelli, A., Brandt, L. & Cossu, C. 2006 Delaying transition to turbulence by a passive mechanism. Phys. Rev. Lett. 96 (6), 064501.Google Scholar
Fung, J. C. H., Hunt, J. C. R., Malik, N. A. & Perkins, R. J. 1992 Kinematic simulation of homogeneous turbulence by unsteady random Fourier modes. J. Fluid Mech. 236, 281–318.Google Scholar
Fung, J. C. H. & Vassilicos, J. C. 1998 Two-particle dispersion in turbulent-like flows. Phys. Rev. E 57 (2), 1677–1690.Google Scholar
Georgiou, T. T. 2002a Spectral analysis based on the state covariance: the maximum entropy spectrum and linear fractional parametrization. IEEE Trans. Autom. Control 47 (11), 1811–1823.Google Scholar
Georgiou, T. T. 2002b The structure of state covariances and its relation to the power spectrum of the input. IEEE Trans. Autom. Control 47 (7), 1056–1066.Google Scholar
Goodwin, G. C. & Payne, R. L. 1977 Dynamic System Identification: Experiment Design and Data Analysis. Academic.Google Scholar
Guala, M., Metzger, M. & McKeon, B. J. 2011 Interactions within the turbulent boundary layer at high Reynolds number. J. Fluid Mech. 666, 573–604.Google Scholar
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 98, 149.Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.Google Scholar
Henningson, D. S. & Reddy, S. C. 1994 On the role of linear mechanisms in transition to turbulence. Phys. Fluids 6 (3), 1396–1398.Google Scholar
Herbert, T. 1997 Parabolized stability equations. Annu. Rev. Fluid Mech. 29 (1), 245–283.Google Scholar
Hœpffner, J. 2005 Modeling flow statistics using convex optimization. In Proceedings of the 2005 European Control Conference and the 44th IEEE Conference on Decision and Control, pp. 4287–4292.Google Scholar
Hœpffner, J., Brandt, L. & Henningson, D. S. 2005a Transient growth on boundary layer streaks. J. Fluid Mech. 537, 91–100.CrossRefGoogle Scholar
Hœpffner, J., Chevalier, M., Bewley, T. R. & Henningson, D. S. 2005b State estimation in wall-bounded flow systems. Part 1. Perturbed laminar flows. J. Fluid Mech. 534, 263–294.Google Scholar
Hœpffner, J., Naka, Y. & Fukagata, K. 2011 Realizing turbulent statistics. J. Fluid Mech. 676, 54–80.Google Scholar
Högberg, M., Bewley, T. R. & Henningson, D. S. 2003a Linear feedback control and estimation of transition in plane channel flow. J. Fluid Mech. 481, 149–175.Google Scholar
Högberg, M., Bewley, T. R. & Henningson, D. S. 2003b Relaminarization of Re 𝜏 = 100 turbulence using linear state-feedback control. Phys. Fluids 15 (11), 3572–3575.Google Scholar
Högberg, M. & Henningson, D. S. 2002 Linear optimal control applied to instabilities in spatially developing boundary layers. J. Fluid Mech. 470, 151–179.Google Scholar
Hoyas, S. & Jiménez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to Re 𝜏 = 2003. Phys. Fluids 18 (1), 011702.CrossRefGoogle Scholar
Hutchins, N. & Marusic, I. 2007a Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.Google Scholar
Hutchins, N. & Marusic, I. 2007b Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.Google Scholar
Hwang, Y. & Cossu, C. 2010a Amplification of coherent streaks in the turbulent Couette flow: an input-output analysis at low Reynolds number. J. Fluid Mech. 643, 333–348.Google Scholar
Hwang, Y. & Cossu, C. 2010b Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 51–73.Google 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, 157–184.Google Scholar
Jovanović, M. R.2004, Modeling, analysis, and control of spatially distributed systems. PhD thesis, University of California, Santa Barbara.Google Scholar
Jovanović, M. R. 2008 Turbulence suppression in channel flows by small amplitude transverse wall oscillations. Phys. Fluids 20 (1), 014101; (11 pages).Google Scholar
Jovanović, M. R. & Bamieh, B. 2001 Modelling flow statistics using the linearized Navier–Stokes equations. In Proceedings of the 40th IEEE Conference on Decision and Control, pp. 4944–4949.Google Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.Google Scholar
Jovanović, M. R. & Georgiou, T. T. 2010 Reproducing second order statistics of turbulent flows using linearized Navier–Stokes equations with forcing. In Bulletin of the American Physical Society.Google Scholar
Jovanović, M. R., Schmid, P. J. & Nichols, J. W. 2014 Sparsity-promoting dynamic mode decomposition. Phys. Fluids 26 (2), 22 pages.Google Scholar
Keating, A., Piomelli, U., Balaras, E. & Kaltenbach, H.-J. 2004 A priori and a posteriori tests of inflow conditions for large-eddy simulation. Phys. Fluids 16 (12), 4696–4712.Google Scholar
Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383–417.Google Scholar
Kim, J. & Hussain, F. 1993 Propagation velocity of perturbations in turbulent channel flow. Phys. Fluids A 5 (3), 695–706.Google Scholar
Kim, J. & Lim, J. 2000 A linear process in wall-bounded turbulent shear flows. Phys. Fluids 12 (8), 1885–1888.Google Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.Google Scholar
Klebanoff, P. S. 1971 Effect of free-stream turbulence on the laminar boundary layer. Bull. Am. Phys. Soc. 16 (11), 1323.Google Scholar
Klebanoff, P. S., Tidstrom, K. D. & Sargent, L. M. 1962 The three-dimensional nature of boundary-layer instability. J. Fluid Mech. 12, 1–34.CrossRefGoogle Scholar
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30 (04), 741–773.Google Scholar
Klingmann, B. G. B. 1992 On transition due to three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 240, 167–195.Google Scholar
Kraichnan, R. H. 1959 The structure of isotropic turbulence at very high Reynolds numbers. J. Fluid Mech. 5 (04), 497–543.Google Scholar
Kraichnan, R. H. 1971 An almost-Markovian Galilean-invariant turbulence model. J. Fluid Mech. 47 (03), 513–524.Google Scholar
Kwakernaak, H. & Sivan, R. 1972 Linear Optimal Control Systems. Wiley-Interscience.Google Scholar
Landahl, M. T. 1975 Wave breakdown and turbulence. SIAM J. Appl. Math. 28, 735–756.Google Scholar
Lee, K. H., Cortelezzi, L., Kim, J. & Speyer, J. 2001 Application of reduced-order controller to turbulent flows for drag reduction. Phys. Fluids 13 (5), 1321–1330.Google Scholar
Lee, M. J., Kim, J. & Moin, P. 1990 Structure of turbulence at high shear rate. J. Fluid Mech. 216, 561–583.Google Scholar
Lehew, J., Guala, M. & McKeon, B. J. 2011 A study of the three-dimensional spectral energy distribution in a zero pressure gradient turbulent boundary layer. Exp. Fluids 51 (4), 997–1012.Google Scholar
Lieu, B. K., Moarref, R. & Jovanović, M. R. 2010 Controlling the onset of turbulence by streamwise traveling waves. Part 2: direct numerical simulations. J. Fluid Mech. 663, 100–119.CrossRefGoogle Scholar
Lin, F. & Jovanović, M. R. 2009 Least-squares approximation of structured covariances. IEEE Trans. Autom. Control 54 (7), 1643–1648.Google Scholar
Lumley, J. L. 2007 Stochastic Tools in Turbulence. Dover.Google Scholar
Majda, A. J., Timofeyev, I. & Eijnden, E. V. 1999 Models for stochastic climate prediction. Proc. Natl Acad. Sci. 96 (26), 14687–14691.Google Scholar
Majda, A. J., Timofeyev, I. & Eijnden, E. V. 2001 A mathematical framework for stochastic climate models. Commun. Pure Appl. Maths 54, 891–974.Google Scholar
Malik, N. A. & Vassilicos, J. C. 1999 A Lagrangian model for turbulent dispersion with turbulent-like flow structure: Comparison with direct numerical simulation for two-particle statistics. Phys. Fluids 11 (6), 1572–1580.Google Scholar
Malkus, W. V. R. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1 (5), 521–539.Google Scholar
Marusic, I., Monty, J. P., Hultmark, M. & Smits, A. J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.Google Scholar
Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J. Fluid Mech. 628, 311–337.Google Scholar
Matsubara, M. & Alfredsson, P. H. 2001 Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149–168.Google Scholar
McComb, W. D. 1991 The Physics of Fluid Turbulence. Oxford University Press.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
McKeon, B. J., Sharma, A. S. & Jacobi, I. 2013 Experimental manipulation of wall turbulence: a systems approach. Phys. Fluids 25 (3), 031301.Google Scholar
Mezić, I. 2005 Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41, 309–325.Google Scholar
Mezić, I. 2013 Analysis of fluid flows via spectral properties of Koopman operator. Annu Rev. Fluid Mech. 45 (1), 357–378.Google Scholar
Moarref, R.2012, Model-based control of transitional and turbulent wall-bounded shear flows. PhD thesis, University of Minnesota.Google Scholar
Moarref, R. & Jovanović, M. R. 2010 Controlling the onset of turbulence by streamwise traveling waves. Part 1: Receptivity analysis. J. Fluid Mech. 663, 70–99.Google Scholar
Moarref, R. & Jovanović, M. R. 2012 Model-based design of transverse wall oscillations for turbulent drag reduction. J. Fluid Mech. 707, 205–240.Google Scholar
Moarref, R., Jovanović, 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
Moarref, R., Sharma, A. S., Tropp, J. A. & McKeon, B. J. 2013 Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels. J. Fluid Mech. 734, 275–316.Google Scholar
Moin, P. & Moser, R. 1989 Characteristic-eddy decomposition of turbulence in a channel. J. Fluid Mech. 200 (41), 509.Google Scholar
Monin, A. S. & Yaglom, A. M. 1975 Statistical Fluid Mechanics, vol. 2. MIT.Google Scholar
Monty, J., Hutchins, N., Ng, H. C. H., Marusic, I. & Chong, M. S. 2009 A comparison of turbulent pipe, channel and boundary layer flows. J. Fluid Mech. 632, 431–442.Google Scholar
Monty, J. P. & Chong, M. S. 2009 Turbulent channel flow: comparison of streamwise velocity data from experiments and direct numerical simulation. J. Fluid Mech. 633, 461–474.Google Scholar
Monty, J. P., Stewart, J. A., Williams, R. C. & Chong, M. S. 2007 Large-scale features in turbulent pipe and channel flows. J. Fluid Mech. 589, 147–156.Google Scholar
Moser, R. D., Kim, J. & Mansour, N. N. 1999 DNS of turbulent channel flow up to Re 𝜏 = 590. Phys. Fluids 11 (4), 943–945.Google Scholar
Noack, B. R., Morzyński, M. & Tadmor, G. 2011 Reduced-order Modelling for Flow Control, CISM Courses and Lectures, vol. 528. Springer.Google Scholar
Orszag, S. A. 1970 Analytical theories of turbulence. J. Fluid Mech. 41 (02), 363–386.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Pujals, G., García-Villalba, M., Cossu, C. & Depardon, S. 2009 A note on optimal transient growth in turbulent channel flows. Phys. Fluids 21 (1), 015109.Google Scholar
Recht, B., Fazel, M. & Parrilo, P. A. 2010 Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52 (3), 471–501.Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.Google Scholar
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Maths 53 (1), 15–47.Google Scholar
Reed, H. L., Saric, W. S. & Arnal, D. 1996 Linear stability theory applied to boundary layers. Annu. Rev. Fluid Mech. 28 (1), 389–428.Google Scholar
Reynolds, W. C. & Hussain, A. K. M. F. 1972 The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech. 54 (2), 263–288.CrossRefGoogle Scholar
Reynolds, W. C. & Tiederman, W. G. 1967 Stability of turbulent channel flow with application to Malkus’s theory. J. Fluid Mech. 27 (2), 253–272.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23 (1), 601–639.Google Scholar
Rowley, C. W. 2005 Model reduction for fluids using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15 (3), 997–1013.Google Scholar
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127.Google Scholar
Schlatter, P. & Örlü, R. 2010 Assessment of direct numerical simulation data of turbulent boundary layers. J. Fluid Mech. 659, 116–126.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.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
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.Google Scholar
Sharma, A. S. & McKeon, B. J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238.Google Scholar
Sillero, J. A., Jiménez, J. & Moser, R. D. 2013 One-point statistics for turbulent wall-bounded flows at Reynolds numbers up to 𝛿+ ≈ 2000. Phys. Fluids 25, 105102.CrossRefGoogle Scholar
Sillero, J. A., Jiménez, J. & Moser, R. D. 2014 Two-point statistics for turbulent boundary layers and channels at Reynolds numbers up to 𝛿+ ≈ 2000. Phys. Fluids 26, 105109.Google Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. Part I: coherent structures. Q. Appl. Maths 45, 561–571.Google Scholar
Smith, C. R. & Metzler, S. P. 1983 The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer. J. Fluid Mech. 129, 27–54.Google Scholar
Smits, A. J., McKeon, B. J. & Marusic, I. 2011 High-Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43, 353–375.Google Scholar
Tadmor, G. & Noack, B. R. 2011 Bernoulli, Bode, and Budgie (Ask the Experts). IEEE Contr. Syst. Mag. 31 (2), 18–23.Google Scholar
Taylor, G. I. 1938 The spectrum of turbulence. Proc. R. Soc. Lond. A 164, 476–490.Google Scholar
Thomas, V. L., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2015 A minimal model of self-sustaining turbulence. Phys. Fluids 27 (10), 105104.Google Scholar
Thomas, V. L., Lieu, B. K., Jovanović, M. R., Farrell, B. F., Ioannou, P. J. & Gayme, D. F. 2014 Self-sustaining turbulence in a restricted nonlinear model of plane Couette flow. Phys. Fluids 26 (10), 105112; (17 pages).Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883–900.Google Scholar
Weideman, J. A. C. & Reddy, S. C. 2000 A MATLAB differentiation matrix suite. ACM Trans. Math. Softw. 26 (4), 465–519.Google Scholar
Westin, K. J. A., Boiko, A. V., Klingmann, B. G. B., Kozlov, V. V. & Alfredsson, P. H. 1994 Experiments in a boundary layer subjected to free stream turbulence. Part 1. Boundary layer structure and receptivity. J. Fluid Mech. 281, 193–218.Google Scholar
Wu, X. & Moin, P. 2009 Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer. J. Fluid Mech. 630, 5–41.Google Scholar
Zare, A., Chen, Y., Jovanović, M. R. & Georgiou, T. T. 2016a Low-complexity modeling of partially available second-order statistics: theory and an efficient matrix completion algorithm. IEEE Trans. Autom. Control; doi:10.1109/TAC.2016.2595761; arXiv:1412.3399.Google Scholar
Zare, A., Jovanović, M. R. & Georgiou, T. T. 2015 Alternating direction optimization algorithms for covariance completion problems. In Proceedings of the 2015 American Control Conference, pp. 515–520.Google Scholar
Zare, A., Jovanović, M. R. & Georgiou, T. T. 2016b Perturbation of system dynamics and the covariance completion problem. In Proceedings of the 55th IEEE Conference on Decision and Control, pp. 7036–7041.Google Scholar
Zhou, K., Doyle, J. C. & Glover, K. 1996 Robust and Optimal Control. Prentice-Hall.Google Scholar