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The engine behind (wall) turbulence: perspectives on scale interactions


Known structures and self-sustaining mechanisms of wall turbulence are reviewed and explored in the context of the scale interactions implied by the nonlinear advective term in the Navier–Stokes equations. The viewpoint is shaped by the systems approach provided by the resolvent framework for wall turbulence proposed by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in which the nonlinearity is interpreted as providing the forcing to the linear Navier–Stokes operator (the resolvent). Elements of the structure of wall turbulence that can be uncovered as the treatment of the nonlinearity ranges from data-informed approximation to analysis of exact solutions of the Navier–Stokes equations (so-called exact coherent states) are discussed. The article concludes with an outline of the feasibility of extending this kind of approach to high-Reynolds-number wall turbulence in canonical flows and beyond.

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R. J. Adrian 2007 Vortex organization in wall turbulence. Phys. Fluids 19, 041301.

J. C. del Álamo  & J. Jiménez 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205213.

W. J. Baars , K. M. Talluru , N. Hutchins  & I. Marusic 2015 Wavelet analysis of wall turbulence to study large-scale modulation of small scales. Exp. Fluids 56, 188.

S. Beneddine , D. Sipp , A. Arnault , J. Dandois  & L. Lesshafft 2016 Conditions for validity of mean flow stability analysis and application to the determination of coherent structures in a turbulent backward facing step flow. J. Fluid Mech. 798, 485504.

J.-L. Bourguignon , A. S. Sharma , J. A. Tropp  & B. J. McKeon 2014 Compact representation of wall-bounded turbulence using compressive sampling. Phys. Fluids 26, 015109.

P. Bradshaw  & Y. M. Koh 1981 A note on Poisson’s equation for pressure in a turbulent flow. Phys. Fluids 24 (771), 241258.

L. Brandt 2014 The lift-up effect: the linear mechanism behind transition and turbulence in shear flows. Eur. J. Mech. (B/Fluids) 47, 8096.

K. Butler  & B. Farrell 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids A 5, 774777.

E. J. Candes  & M. B. Wakin 2008 An introduction to compressive sampling. IEEE Signal Process. Mag. 25, 2130.

P. Chakraborty , S. Balachandar  & R. J. Adrian 2005 On the relationships between local vortex identification schemes. J. Fluid Mech. 535, 189214.

S. I. Chernyshenko  & M. F. Baig 2005 The mechanism of streak formation in near-wall turbulence. J. Fluid Mech. 544, 99131.

S. I. Chernyshenko , G. M. Cicca , A. Iollo , A. V. Smirnov , N. D. Sandham  & Z. W. Hu 2006 Analysis of data on the relation between eddies and streaky structures in turbulent flows using the placebo method. Fluid Dyn. 41 (5), 772783.

L. C. Cheung  & T. Zaki 2014 An exact representation of the nonlinear triad interaction terms in spectral space. J. Fluid Mech. 748, 175188.

D. Chung  & B. J. McKeon 2010 Large-eddy simulation investigation of large-scale structures in a long channel flow. J. Fluid Mech. 661, 341364.

C. Cossu , G. Pujals  & S. Depardon 2009 Optimal transient growth and very large scale structures in turbulent boundary layers. J. Fluid Mech. 619, 7994.

D. Dennis  & T. Nickels 2011a Experimental measurement of large-scale three-dimensional structures in a turbulent boundary layer. Part 1. Vortex packets. J. Fluid Mech. 673, 180217.

D. Dennis  & T. Nickels 2011b Experimental measurement of large-scale three-dimensional structures in a turbulent boundary layer. Part 2. Long structures. J. Fluid Mech. 673, 218244.

P. G. Drazin  & W. H. Reid 2004 Hydrodynamic Stability, 2nd edn, Cambridge Mathematical Libraries. Cambridge University Press.

S. Duvvuri  & B. J. McKeon 2015 Triadic scale interactions in a turbulent boundary layer. J. Fluid Mech. 767, R4.

S. Duvvuri  & B. J. McKeon 2016 Non-linear interactions isolated through scale synthesis in experimental wall turbulence. Phys. Rev. Fluids 1 (3), 032401(R).

B. Eckhardt , H. Faisst , A. Schmiegel  & T. M. Scheider 2008 Dynamical systems and the transition to turbulence in linearly stable shear flows. Phil. Trans. R. Soc. Lond. A 366, 12971315.

T. Ellingsen  & E. Palm 1975 Stability of linear flow. Phys. Fluids 18, 487488.

B. Farrell  & J. Ioannou 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids 5 (11), 26002609.

B. F. Farrell  & P. J. Ioannou 1998 Perturbation structure and spectra in turbulent channel flow. Theor. Comput. Fluid Dyn. 11, 237250.

B. F. Farrell  & P. J. Ioannou 2012 Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149196.

B. F. Farrell , P. J. Ioannou , J. Jiménez , N. C. Constantinou , A. Lozano-Duran  & M. A. Nikolaidis 2016 A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane Poiseuille flow. J. Fluid Mech. 809, 290315.

O. Flores  & J. Jiménez 2010 Hierarchy of minimal flow units in the logarithmic layer. Phys. Fluids 22, 071704.

M. Fosas de Pando , P. J. Schmid  & D. Sipp 2015 Nonlinear model-order reduction for oscillator flows using POD-DEIM. In Proc. IUTAM 14, pp. 329336. Elsevier.

B. Ganapathisubramani , N. Hutchins , J. O. Monty , D. Chung  & I. Marusic 2012 Amplitude and frequency modulation in wall turbulence. J. Fluid Mech. 712, 6191.

D. F. Gayme , B. J. McKeon , A. Papachristodolou , B. Bamieh  & J. C. Doyle 2010 Streamwise constant model of turbulence in plane Couette flow. J. Fluid Mech. 665, 99119.

J. F. Gibson , J. Halcrow  & P. Cvitanović 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107130.

F. Gómez Carrasco , H. Blackburn , M. Rudman , A. Sharma  & B. McKeon 2016a A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator. J. Fluid Mech. 798, R2.

F. Gómez Carrasco , H. Blackburn , M. Rudman , A. Sharma  & B. McKeon 2016b Streamwise-varying steady transpiration control in turbulent pipe flow. J. Fluid Mech. 796, 588616.

N. Halko , P. G. Martinsson  & J. A. Tropp 2011 Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53 (2), 217288.

P. Hall  & S. J. Sherwin 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178205.

P. Hall  & F. T. Smith 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641666.

J. M. Hamilton , J. Kim  & F. Waleffe 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317348.

P. Holmes , J. L. Lumley  & G. Berkooz 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.

A. K. M. F. Hussain  & W. C. Reynolds 1970 The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech. 41, 241258.

N. Hutchins  & I. Marusic 2007 Large-scale influences in near-wall turbulence. Phil. Trans. R. Soc. Lond. A 365, 647664.

Y. Hwang 2015 Statistical structure of self-sustaining attached eddies in turbulent channel flow. J. Fluid Mech. 767, 254289.

Y. Hwang  & C. Cossu 2010a Amplification of coherent streaks in the turbulent Couette flow: an input–output analysis at low Reynolds number. J. Fluid Mech. 643, 333348.

Y. Hwang  & C. Cossu 2010b Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 5173.

I. Jacobi  & B. J. McKeon 2011 Dynamic roughness-perturbation of a turbulent boundary layer. J. Fluid Mech. 688, 258296.

I. Jacobi  & B. J. McKeon 2013 Phase relationships between large and small scales in the turbulent boundary layer. Exp. Fluids 54, 1481.

J. Jeun , J. W. Nichols  & M. R. Jovanović 2016 Input–output analysis of high-speed axisymmetric isothermal jet noise. Phys. Fluids 28 (4), 047101.

J. Jiménez 2012 Cascades in wall-bounded turbulence. Annu. Rev. Fluid Mech. 44, 2745.

J. Jiménez 2015 Direct detection of linearized bursts in turbulence. Phys. Fluids 27, 065102.

J. Jiménez , G. Kawahara , M. P. Simens , M. Nagata  & M. Shiba 2005 Characterization of near-wall turbulence in terms of equilibrium and ‘bursting’ solutions. Phys. Fluids 17, 015105.

J. Jiménez  & P. Moin 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213240.

M. R. Jovanović  & B. Bamieh 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145183.

G. Kawahara , M. Uhlmann  & L. van Veen 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203225.

J. Kim  & J. Lim 2000 A linear process in wall-bounded turbulent shear flows. Phys. Fluids 12 (8), 18851888.

J. C. Klewicki 2010 Reynolds number dependence, scaling, and dynamics of turbulent boundary layers. Trans. ASME J. Fluids Engng 132, 094001.

J. C. Klewicki , P. Fife , T. Wei  & P. McMurtry 2007 A physical model of the turbulent boundary layer consonant with the mean momentum balance structure. Phil. Trans. R. Soc. Lond. A 365, 823839.

M. Landahl 1967 A wave-guide model for turbulent shear flow. J. Fluid Mech. 29 (3), 441459.

J. Lehew , M. Guala  & B. J. McKeon 2011 A study of the three-dimensional spectral energy distribution in a zero pressure gradient turbulent boundary layer. Exp. Fluids 51 (4), 9971012.

L. Lu  & G. Papadakis 2014 An iterative method for the computation of the response of linearised Navier–Stokes equations to harmonic forcing and application to forced cylinder wakes. Intl J. Numer. Meth. Fluids 74 (11), 794817.

M. Luhar , A. S. Sharma  & B. J. McKeon 2014a On the structure and origin of pressure fluctuations in wall turbulence: predictions based on the resolvent analysis. J. Fluid Mech. 751, 3870.

M. Luhar , A. S. Sharma  & B. J. McKeon 2014b Opposition control within the resolvent analysis framework. J. Fluid Mech. 749, 597626.

M. Luhar , A. S. Sharma  & B. J. McKeon 2015 A framework for studying the effect of compliant surfaces on wall turbulence. J. Fluid Mech. 768, 415441.

J. B. Marston , G. P. Chini  & S. M. Tobias 2016 Generalized quasilinear approximation: application to zonal jets. Phys. Rev. Lett. 116, 214501.

I. Marusic , R. Mathis  & N. Hutchins 2010 Predictive model for wall-bounded turbulent flow. Science 329, 193196.

S. A. Maslowe 1986 Critical layers in shear flows. Annu. Rev. Fluid Mech. 18 (1), 405432.

R. Mathis , N. Hutchins  & I. Marusic 2009 Large-scale amplitude modulation of the small-scale structures of turbulent boundary layers. J. Fluid Mech. 628, 311337.

B. J. McKeon  & A. S. Sharma 2010 A critical layer model for turbulent pipe flow. J. Fluid Mech. 658, 336382.

B. J. McKeon , A. S. Sharma  & I. Jacobi 2013 Experimental manipulation of wall turbulence: a systems approach. Phys. Fluids 25, 031301.

A. Meseguer  & L. N. Trefethen 2003 Linearized pipe flow to Reynolds number 107 . J. Comput. Phys. 186, 178197.

M. M. Metzger  & J. C. Klewicki 2001 A comparative study of near-wall turbulence in high and low Reynolds number boundary layers. Phys. Fluids 13, 692701.

R. Moarref , M. R. Jovanović , A. S. Sharma , J. A. Tropp  & B. J. McKeon 2014 A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization. Phys. Fluids 26, 051701.

R. Moarref , A. S. Sharma , J. A. Tropp  & B. J. McKeon 2013 Model-based scaling and prediction of the streamwise energy intensity in high-Reynolds number turbulent channels. J. Fluid Mech. 734, 275316.

J. P. Monty , N. Hutchins , H. C. H. Ng , I. Marusic  & M. S. Chong 2009 A comparison of turbulent pipe, channel and boundary layer flows. J. Fluid Mech. 632, 431442.

M. Nagata 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519527.

V. K. Natrajan , Y. Wu  & K. T. Christensen 2007 Spatial signatures of retrograde spanwise vortices in wall turbulence. J. Fluid Mech. 574, 155167.

J. S. Park  & M. D. Graham 2015 Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech. 782, 430454.

S. B. Pope 2000 Turbulent Flows. Cambridge University Press.

O. Reynolds 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. R. Soc. Lond. 174, 935982.

W. C. Reynolds  & A. K. M. F. Hussain 1972 The mechanics of an organized wave in shear flow. Part 3. Theoretical models and comparisons with experiment. J. Fluid Mech. 54, 263288.

P. J. Schmid 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129162.

P. J. Schmid  & D. S. Henningson 2001 Stability and Transition in Shear Flows. Springer.

W. Schoppa  & F. Hussain 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57108.

A. S. Sharma  & B. J. McKeon 2013b On coherent structure in wall turbulence. J. Fluid Mech. 728, 196238.

A. S. Sharma , I. Mezić  & B. J. McKeon 2016a On the correspondence between Koopman mode decomposition, resolvent mode decomposition, and invariant solutions of the Navier–Stokes equations. Phys. Rev. Fluids 1 (3), 032402(R).

C. M. de Silva , N. Hutchins  & I. Marusic 2016 Uniform momentum zones in turbulent boundary layers. J. Fluid Mech. 786, 309331.

L. Sirovich , K. S. Ball  & L. R. Keefe 1990 Plane waves and structures in turbulent channel flow. Phys. Fluids A 2 (12), 22172226.

A. J. Smits , B. J. McKeon  & I. Marusic 2011 High Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43, 353375.

K. M. Talluru , R. Baidya , N. Hutchins  & I. Marusic 2014 Amplitude modulation of all three velocity components in turbulent boundary layers. J. Fluid Mech. 746, R1.

V. L. Thomas , B. F. Farrell , P. J. Ioannou  & D. F. Gayme 2015 A minimal model of self-sustaining turbulence. Phys. Fluids 27, 105104.

A. A. Townsend 1961 Equilibrium layers and wall turbulence. J. Fluid Mech. 11, 97120.

L. N. Trefethen , A. E. Trefethen , S. Reddy  & T. A. Driscoll 1993 Hydrodynamic stability without eigenvalues. Science 261, 5121.

F. Waleffe 1991 The nature of triad interactions in homogeneous turbulence. Phys. Fluids 4 (2), 350363.

F. Waleffe 1995 Transition in shear flows: non-linear normality versus non-normal linearity. Phys. Fluids 7, 30603066.

F. Waleffe 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883900.

F. Waleffe 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93102.

F. Waleffe 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15 (6), 15171534.

X. Wu  & P. Moin 2008 A direct numerical simulation study on the mean velocity characteristics in pipe flow. J. Fluid Mech. 608, 81112.

A. Zare , M. R. Jovanović  & T. T. Georgiou 2017 Colour of turbulence. J. Fluid Mech. 812, 636680.

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