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On coherent structure in wall turbulence

  • A. S. Sharma (a1) and B. J. McKeon (a2)
Abstract
Abstract

A new theory of coherent structure in wall turbulence is presented. The theory is the first to predict packets of hairpin vortices and other structure in turbulence, and their dynamics, based on an analysis of the Navier–Stokes equations, under an assumption of a turbulent mean profile. The assumption of the turbulent mean acts as a restriction on the class of possible structures. It is shown that the coherent structure is a manifestation of essentially low-dimensional flow dynamics, arising from a critical-layer mechanism. Using the decomposition presented in McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), complex coherent structure is recreated from minimal superpositions of response modes predicted by the analysis, which take the form of radially varying travelling waves. The leading modes effectively constitute a low-dimensional description of the turbulent flow, which is optimal in the sense of describing the resonant effects around the critical layer and which minimally predicts all types of structure. The approach is general for the full range of scales. By way of example, simple combinations of these modes are offered that predict hairpins and modulated hairpin packets. The example combinations are chosen to represent observed structure, consistent with the nonlinear triadic interaction for wavenumbers that is required for self-interaction of structures. The combination of the three leading response modes at streamwise wavenumbers $6, ~1, ~7$ and spanwise wavenumbers $\pm 6, ~\pm 6, ~\pm 12$ , respectively, with phase velocity $2/ 3$ , is understood to represent a turbulence ‘kernel’, which, it is proposed, constitutes a self-exciting process analogous to the near-wall cycle. Together, these interactions explain how the mode combinations may self-organize and self-sustain to produce experimentally observed structure. The phase interaction also leads to insight into skewness and correlation results known in the literature. It is also shown that the very large-scale motions act to organize hairpin-like structures such that they co-locate with areas of low streamwise momentum, by a mechanism of locally altering the shear profile. These energetic streamwise structures arise naturally from the resolvent analysis, rather than by a summation of hairpin packets. In addition, these packets are modulated through a ‘beat’ effect. The relationship between Taylor’s hypothesis and coherence is discussed, and both are shown to be the consequence of the localization of the response modes around the critical layer. A pleasing link is made to the classical laminar inviscid theory, whereby the essential mechanism underlying the hairpin vortex is captured by two obliquely interacting Kelvin–Stuart (cat’s eye) vortices. Evidence for the theory is presented based on comparison with observations of structure in turbulent flow reported in the experimental and numerical simulation literature and with exact solutions reported in the transitional literature.

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Email address for correspondence: a.sharma@soton.ac.uk
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R. J. Adrian 2007 Vortex organization in wall turbulence. Phys. Fluids 19, 041301.

R. J. Adrian , K. T. Christensen & Z.-C. Liu 2000a Analysis and interpretation of instantaneous turbulent velocity fields. Exp. Fluids 29, 275290.

R. J. Adrian , C. D. Meinhart & C. D. Tomkins 2000b Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 154.

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

B. J. Balakumar & R. J. Adrian 2007 Large- and very-large-scale motions in channel and boundary layer flows. Phil. Trans. R. Soc. Lond. A 365, 665681.

P. R. Bandyopadhyay & A. K. M. F. Hussain 1984 The coupling between scales in shear flows. Phys. Fluids 27 (9), 22212228.

D. J. Benney & R. F. Bergeron 1969 A new class of nonlinear waves in parallel flows. Stud. Appl. Maths 48, 181204.

J.-L. Bourguignon & B. J. McKeon 2011 A streamwise-constant model of turbulent pipe flow. Phys. Fluids 23, 095111.

J. Carlier & M. Stanislas 2005 Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry. J. Fluid Mech. 535, 143188.

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

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.

S. Cherubini , P. de Palma , J.-C. Robinet & A. Bottaro 2011 Edge states in a boundary layer. Phys. Fluids 23 (5), 051705.

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.

K Deguchi & M Nagata 2010 Traveling hairpin-shaped fluid vortices in plane Couette flow. Phys. Rev. E 82, 056325.

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 University Press.

Y. Duguet , A. P. Willis & R. R. Kerswell 2008 Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255274.

B. Eckhardt , T. M. Schneider , B. Hof & J. Westerweel 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39 (1), 447468.

R. E. Falco 1977 Coherent motions in the outer region of turbulent boundary layers. Phys. Fluids 20, S124S132.

R. E. Falco 1991 A coherent structure model of the turbulent boundary layer and its ability to predict Reynolds number dependence. Phil. Trans. R. Soc. Lond. A 336 (1641), 103129.

B. Ganapathisubramani , E. K. Longmire & I. Marusic 2003 Characteristics of vortex packets in turbulent boundary layers. J. Fluid Mech. 478, 3546.

Q. Gao , C. Ortiz-Dueñas & E. K. Longmire 2011 Analysis of vortex populations in turbulent wall-bounded flows. J. Fluid Mech. 678, 87123.

S. C. Generalis & T. Itano 2010 Characterization of the hairpin vortex solution in plane Couette flow. Phys. Rev. E 82, 066308.

J. F. Gibson , J. Halcrow & P. Cvitanović 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243266.

M. Guala , S. E. Hommema & R. J. Adrian 2006 Large-scale and very-large-scale motions in turbulent pipe flow. J. Fluid Mech. 554, 521542.

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.

M. R. Head & P. R. Bandyopadhyay 1981 New aspects of turbulent boundary-layer structure. J. Fluid Mech. 107, 297337.

L. H. O. Hellström , A. Sinha & A. J. Smits 2011 Visualizing the very-large-scale motions in turbulent pipe flow. Phys. Fluids 23, 011703.

N. Hutchins & I. Marusic 2007a Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 128.

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

N. Hutchins , J. P. Monty , B. Ganapathisubramani , H. C. H. Ng & I. Marusic 2011 Three-dimensional conditional structure of a high-Reynolds number turbulent boundary layer. J. Fluid Mech. 673, 255285.

T. Itano & S. C. Generalis 2009 Hairpin vortex solution in planar Couette flow: a tapestry of knotted vortices. Phys. Rev. Lett. 102, 114501.

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. Jeong & F. Hussain 1995 On the identification of a vortex. J. Fluid Mech. 285, 6994.

R. R. Kerswell 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18 (6), R17.

K. C. Kim & R. J. Adrian 1999 Very large-scale motion in the outer layer. Phys. Fluids 11, 417422.

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.

J. H. Lee & H. J. Sung 2011 Very-large-scale motions in a turbulent boundary layer. J. Fluid Mech. 673, 80120.

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.

I. Marusic , R. Mathis & N. Hutchins 2010 High Reynolds number effects in wall turbulence. Intl J. Heat Fluid Flow 31, 418428.

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

R. Mathis , I. Marusic , N. Hutchins & K. R. Sreenivasan 2011 The relationship between the velocity skewness and the amplitude modulation of the small scale by the large scale in turbulent boundary layers. Phys. Fluids 23, 121702.

B. J. McKeon , J. Li , W. Jiang , J. F. Morrison & A. J. Smits 2004 Further observations on the mean velocity distribution in fully developed pipe flow. J. Fluid Mech. 501, 135147.

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

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

C. D. Meinhart & R. J. Adrian 1995 On the existence of uniform momentum zones in a turbulent boundary layer. Phys. Fluids 7 (4), 694696.

A. Meseguer & L. N. Trefethen 2003 Linearized pipe flow to Reynolds number $1{0}^{7} $. J. Comput. Phys. 186, 178197.

J. P. Monty , J. A. Stewart , R. C. Williams & M. S. Chong 2007 Large-scale features in turbulent pipe and channel flows. J. Fluid Mech. 589, 147156.

S. C. Morris , S. R. Stolpa , P. E. Slaboch & J. Klewicki 2007 Near-surface particle image velocimetry measurements in a transitionally rough-wall atmospheric boundary layer. J. Fluid Mech. 580, 319338.

T. Mullin 2011 Experimental studies of transition to turbulence in a pipe. Annu. Rev. Fluid Mech. 43 (1), 124.

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

A. E. Perry & M. S. Chong 1982 On the mechanism of wall turbulence. J. Fluid Mech 119, 173217.

A. E. Perry & I. Marusic 1995 A wall-wake model for the turbulence structure of boundary layers. Part 1. Extension of the attached eddy hypothesis. J. Fluid Mech. 298, 361388.

C. C. T. Pringle , Y. Duguet & R. R. Kerswell 2009 Highly symmetric travelling waves in pipe flow. Phil. Trans. R. Soc. Lond. A 367, 457472.

S. C. Reddy , P. J. Schmid & D. S. Henningson 1993 Pseudospectra of the Orr–Sommerfeld equation. SIAM J. Appl. Maths 53 (1), 1547.

S. K. Robinson 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601639.

P. Schlatter & R. Örlü 2010 Quantifying the interaction between large and small scales in wall-bounded turbulent flows: a note of caution. Phys. Fluids 22, 051704.

P. Schlatter , R. Örlu , Q. Li , G. Brethouwer , J. H. M. Fransson , A. V. Johansson , P. H. Alfredsson & D. S. Henningson 2009 Simulations of spatially evolving turbulent boundary layers up to $R{e}_{\theta } = 4300$. Intl J. Heat Fluid Flow 31, 251261.

F. T. Smith & R. J. Bodonyi 1982 Amplitude-dependent neutral modes in the Hagen–Poiseille flow though a circular pipe. Proc. R. Soc. Lond. A 384, 463489.

C. R. Smith , J. D. A. Walker , A. H. Haidari & U. Sobrun 1991 On the dynamics of near-wall turbulence. Phil. Trans. R. Soc. Lond. 336, 131175.

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

C. D. Tomkins & R. J. Adrian 2005 Energetic spanwise modes in the logarithmic layer of a turbulent boundary layer. J. Fluid Mech. 545, 141162.

J. M. J. den Toonder & F. T. M. Nieuwstadt 1997 Reynolds number effects in a turbulent pipe flow for low to moderate Re. Phys. Fluids 9, 33983409.

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

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

H Wedin & R. R. Kerswell 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333371.

X. Wu , J. R. Baltzer & R. J. Adrian 2012 Direct numerical simulation of a $30R$ long turbulent pipe flow at ${R}^{+ } = 685$ : large- and very large-scale motions. J. Fluid Mech. 698, 235281.

Y. Wu & K. T. Christensen 2006 Population trends of spanwise vortices in wall turbulence. J. Fluid Mech. 568, 5576.

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