Reynolds Number Dependence, Scaling, and Dynamics of Turbulent Boundary Layers

The past two decades (approximately 1990 to 2010) have witnessed an ever-quickening pace of new findings pertaining to the Reynolds number dependencies, scaling, and dynamics of turbulent boundary layer flows (and wall-bounded turbulent flows in general). Given this, an important objective of the present effort is to provide a review that enables researchers new to the field (e.g., graduate students) to gain an appreciation for, and an understanding of, the prevalent research themes currently under investigation. Thus, the emphasis is more on laying a contextual foundation rather than, for example, comprehensively reporting all of the research findings of the past 20 years. The review begins with a brief exposition of scaling concepts and the normalizing parameters used in exploring Reynolds number dependence. An overall focus of the effort is to describe the scaling problem in relation to the underlying behaviors of the governing transport equations. For this reason, a number of relevant equations are concisely presented. The technical challenges associated with reliably exploring Reynolds number dependence are nontrivial and are of central importance. Thus, a separate section is devoted to this topic. Similarly, since they factor importantly relative to understanding and organizing the data trends, the attributes, strengths, and weaknesses of the various theoretical approaches and models (both physical and mathematical) are briefly reviewed. The statistical data presented primarily focus on means and variances since these quantities most directly relate to the time-averaged equations. Recent results pertaining to the spatial structure of turbulent boundary layers provide a useful context for describing instantaneous dynamics, often involving coherent vortical motions and including the so-called inner/outer interaction. Overall, the cumulative evidence increasingly supports a paradigm in which the scaling behaviors of the statistical profiles stem from the existence of an internal hierarchy of motions that approach a dynamically self-similar state as the Reynolds number becomes large.

[1]  Peter Bradshaw,et al.  A note on Poisson’s equation for pressure in a turbulent flow , 1981 .

[2]  Bhimsen K. Shivamoggi,et al.  Perturbation Methods for Differential Equations , 2002 .

[3]  H. Kawamura,et al.  A DNS STUDY OF REYNOLDS-NUMBER DEPENDENCE ON PRESSURE FLUCTUATIONS IN A TURBULENT CHANNEL FLOW , 2005, Proceeding of Fourth International Symposium on Turbulence and Shear Flow Phenomena.

[4]  G. Eyink Turbulent flow in pipes and channels as cross-stream ``inverse cascades'' of vorticity , 2008, 0809.1165.

[5]  Kirit S. Yajnik,et al.  Asymptotic theory of turbulent shear flows , 1970, Journal of Fluid Mechanics.

[6]  F. Waleffe Exact coherent structures in channel flow , 2001, Journal of Fluid Mechanics.

[7]  P. Fife,et al.  On scaling the mean momentum balance and its solutions in turbulent Couette–Poiseuille flow , 2007, Journal of Fluid Mechanics.

[8]  P. Alfredsson,et al.  Pressure statistics and their scaling in high-Reynolds-number turbulent boundary layers , 2007, Journal of Fluid Mechanics.

[9]  A. J. A. Morgan,et al.  THE REDUCTION BY ONE OF THE NUMBER OF INDEPENDENT VARIABLES IN SOME SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS , 1952 .

[10]  B. McKeon,et al.  Asymptotic scaling in turbulent pipe flow , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[11]  F. Nieuwstadt,et al.  Reynolds-number-dependence of the maximum in the streamwise velocity fluctuations in wall turbulence , 1996 .

[12]  K. Winter An outline of the techniques available for the measurement of skin friction in turbulent boundary layers , 1979 .

[13]  N. Afzal Power law and log law velocity profiles in fully developed turbulent pipe flow: equivalent relations at large Reynolds numbers , 2001 .

[14]  H. H. Fernholz,et al.  The incompressible zero-pressure-gradient turbulent boundary layer: An assessment of the data , 1996 .

[15]  John Kim,et al.  Regeneration mechanisms of near-wall turbulence structures , 1995, Journal of Fluid Mechanics.

[16]  Javier Jiménez,et al.  Turbulent fluctuations above the buffer layer of wall-bounded flows , 2008, Journal of Fluid Mechanics.

[17]  Mean momentum balance in moderately favourable pressure gradient turbulent boundary layers , 2008, Journal of Fluid Mechanics.

[18]  J. Andreopoulos,et al.  Wall-vorticity flux dynamics in a two-dimensional turbulent boundary layer , 1996, Journal of Fluid Mechanics.

[19]  Hassan M. Nagib,et al.  Variations of von Kármán coefficient in canonical flows , 2008 .

[20]  Mario J. Casarella,et al.  Spectral features of wall pressure fluctuations beneath turbulent boundary layers , 1991 .

[21]  R. Mathis,et al.  Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers , 2009, Journal of Fluid Mechanics.

[22]  Yuichi Matsuo,et al.  Direct Numerical Simulation of a Fully Developed Turbulent Channel Flow With Respect to the Reynolds Number Dependence , 2001 .

[23]  Study of the motions contributing to the Reynolds stress in high and low Reynolds number turbulent boundary layers , 2004 .

[24]  John Kim,et al.  On the structure of pressure fluctuations in simulated turbulent channel flow , 1989, Journal of Fluid Mechanics.

[25]  Ivan Marusic,et al.  Hot-wire spatial resolution issues in wall-bounded turbulence , 2009, Journal of Fluid Mechanics.

[26]  P. Fife,et al.  TIME AVERAGING IN TURBULENCE SETTINGS MAY REVEAL AN INFINITE HIERARCHY OF LENGTH SCALES , 2009 .

[27]  B. Lindgren,et al.  Evaluation of scaling laws derived from Lie group symmetry methods in zero-pressure-gradient turbulent boundary layers , 2004, Journal of Fluid Mechanics.

[28]  Martin Oberlack,et al.  Similarity in non-rotating and rotating turbulent pipe flows , 1999, Journal of Fluid Mechanics.

[29]  A. Seena,et al.  Intermediate Scaling of Turbulent Momentum and Heat Transfer in a Transitional Rough Channel , 2008 .

[30]  Neil D. Sandham,et al.  Wall Pressure and Shear Stress Spectra from Direct Simulations of Channel Flow , 2006 .

[31]  Robert M. Kirby,et al.  On the Lamb vector divergence in Navier–Stokes flows , 2008, Journal of Fluid Mechanics.

[32]  J. Foss,et al.  On the sign of the instantaneous spanwise vorticity component in the near‐wall region of turbulent boundary layers , 1990 .

[33]  P. Monkewitz,et al.  Approach to an asymptotic state for zero pressure gradient turbulent boundary layers , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[34]  A. Smits,et al.  Mean-flow scaling of turbulent pipe flow , 1998, Journal of Fluid Mechanics.

[35]  On the evolution of maximum turbulent kinetic energy production in a channel flow , 2002 .

[36]  T. Kármán,et al.  Mechanische Ahnlichkeit und Turbulenz , 1930 .

[37]  J. Wyngaard,et al.  Spatial resolution of the vorticity meter and other hot-wire arrays , 1969 .

[38]  A. D. Young,et al.  An Introduction to Fluid Mechanics , 1968 .

[39]  F. Laadhari Reynolds number effect on the dissipation function in wall-bounded flows , 2007, 1210.8172.

[40]  John F. Foss,et al.  The measurement of vorticity in turbulent flows , 1995 .

[41]  Alexandre J. Chorin,et al.  Vorticity and turbulence , 1994 .

[42]  Petar Vukoslavcevic,et al.  The velocity and vorticity vector fields of a turbulent boundary layer. Part 2. Statistical properties , 1991, Journal of Fluid Mechanics.

[43]  Viscous wall region structure in high and low reynolds number turbulent boundary layers , 1996 .

[44]  M. Bull,et al.  On the role of wall-pressure fluctuations in deterministic motions in the turbulent boundary layer , 1983, Journal of Fluid Mechanics.

[45]  Martin Skote,et al.  Reynolds Stress Budgets in Couette and Boundary Layer Flows , 2002 .

[46]  R. Antonia,et al.  Structure of the velocity field associated with the spanwise vorticity in the wall region of a turbulent boundary layer , 1993 .

[47]  Jiezhi Wu,et al.  Vorticity and Vortex Dynamics , 2006 .

[48]  Jonathan Morrison,et al.  Eddy structure in turbulent boundary layers , 2000 .

[49]  A. Townsend The Structure of Turbulent Shear Flow , 1975 .

[50]  Paul C. Fife,et al.  Scaling heat transfer in fully developed turbulent channel flow , 2005 .

[51]  S. Morris,et al.  Vorticity spectra in high Reynolds number anisotropic turbulence , 2005 .

[52]  Craig B. Arnold,et al.  Turbulence measurements using a nanoscale thermal anemometry probe , 2010, Journal of Fluid Mechanics.

[53]  A refined interpretation of the logarithmic structure function law in wall layer turbulence , 2006 .

[54]  R. Panton A Reynolds Stress Function for Wall Layers , 1997 .

[55]  Jerry Westerweel,et al.  Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment , 1994, Journal of Fluid Mechanics.

[56]  C. Meneveau,et al.  Effects of free-stream turbulence on rough surface turbulent boundary layers , 2009, Journal of Fluid Mechanics.

[57]  P. Moin,et al.  Reynolds-stress and dissipation-rate budgets in a turbulent channel flow , 1987, Journal of Fluid Mechanics.

[58]  P. Monkewitz,et al.  Scaling of High Reynolds Number Turbulent Boundary Layers Revisited , 2005 .

[59]  Nobuhide Kasagi,et al.  Reynolds Number Effect on Wall Turbulence: Toward Effective Feedback Control , 2002 .

[60]  Hassan M. Nagib,et al.  Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers , 2007 .

[61]  Ronald J. Adrian,et al.  Large-scale and very-large-scale motions in turbulent pipe flow , 2006, Journal of Fluid Mechanics.

[62]  Alexander Smits,et al.  Scaling of the streamwise velocity component in turbulent pipe flow , 2004, Journal of Fluid Mechanics.

[63]  J. Katz,et al.  Buffer layer structures associated with extreme wall stress events in a smooth wall turbulent boundary layer , 2009, Journal of Fluid Mechanics.

[64]  K. Sreenivasan,et al.  The wall-normal position in pipe and channel flows at which viscous and turbulent shear stresses are equal , 1999 .

[65]  K. Sreenivasan,et al.  Finite-Reynolds-number effects in turbulence using logarithmic expansions , 2006, Journal of Fluid Mechanics.

[66]  R. Adrian,et al.  Packet Structure of Surface Eddies in the Atmospheric Boundary Layer , 2003 .

[67]  G. I. Barenblatt,et al.  Scaling laws for fully developed turbulent shear flows. Part 1. Basic hypotheses and analysis , 1993, Journal of Fluid Mechanics.

[68]  A. Townsend,et al.  Equilibrium layers and wall turbulence , 1961, Journal of Fluid Mechanics.

[69]  I. Marusic,et al.  Reynolds number invariance of the structure inclination angle in wall turbulence. , 2007, Physical review letters.

[70]  R. E. Falco,et al.  Coherent motions in the outer region of turbulent boundary layers , 1977 .

[71]  Javier Jiménez,et al.  Self-similar vortex clusters in the turbulent logarithmic region , 2006, Journal of Fluid Mechanics.

[72]  Russell J. Donnelly,et al.  Pipe flow measurements over a wide range of Reynolds numbers using liquid helium and various gases , 2002, Journal of Fluid Mechanics.

[73]  Meredith Metzger,et al.  A comparative study of near-wall turbulence in high and low Reynolds number boundary layers , 2001 .

[74]  Ivan Marusic,et al.  Similarity law for the streamwise turbulence intensity in zero-pressure-gradient turbulent boundary layers , 1997 .

[75]  N. Hutchins,et al.  Some predictions of the attached eddy model for a high Reynolds number boundary layer , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[76]  R. E. Falco,et al.  Vortical motion contributions to stress transport in turbulent boundary layers , 1994 .

[77]  Hassan M. Nagib,et al.  Experimental investigation of coherent structures in turbulent boundary layers , 1991, Journal of Fluid Mechanics.

[78]  A. Perry,et al.  A wall-wake model for the turbulence structure of boundary layers. Part 2. Further experimental support , 1995, Journal of Fluid Mechanics.

[79]  Jean-Luc Redelsperger,et al.  The structure of the near neutral atmospheric surface layer as observed during the CASES'99 experiment , 2004 .

[80]  A. G. Hansen Similarity analyses of boundary value problems in engineering , 1964 .

[81]  Richard M. Lueptow Transducer resolution and the turbulent wall pressure spectrum , 1995 .

[82]  J. M. Allen Experimental Study of Error Sources in Skin-Friction Balance Measurements , 1977 .

[83]  M. Metzger,et al.  Statistical structure of the fluctuating wall pressure and its in-plane gradients at high Reynolds number , 2008, Journal of Fluid Mechanics.

[84]  Javier Jiménez,et al.  On the structure and control of near wall turbulence , 1994 .

[85]  Jinhee Jeong,et al.  On the identification of a vortex , 1995, Journal of Fluid Mechanics.

[86]  Reza Sadr,et al.  Scaling the near-wall axial turbulent stress in the zero pressure gradient boundary layer , 2001 .

[87]  Alexander Smits,et al.  Further observations on the mean velocity distribution in fully developed pipe flow , 2004, Journal of Fluid Mechanics.

[88]  P. Fife,et al.  On the logarithmic mean profile , 2009, Journal of Fluid Mechanics.

[89]  R. Antonia,et al.  On the measurement of lateral velocity derivatives in turbulent flows , 1993 .

[90]  Ronald J. Adrian,et al.  Spanwise structure and scale growth in turbulent boundary layers , 2003, Journal of Fluid Mechanics.

[91]  J. C. Vassilicos,et al.  Stagnation point von Kármán coefficient. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[92]  Ellen K. Longmire,et al.  Characteristics of vortex packets in turbulent boundary layers , 2003, Journal of Fluid Mechanics.

[93]  J. Eaton,et al.  Reynolds-number scaling of the flat-plate turbulent boundary layer , 2000, Journal of Fluid Mechanics.

[94]  K. Christensen,et al.  Spatial signatures of retrograde spanwise vortices in wall turbulence , 2007, Journal of Fluid Mechanics.

[95]  Ellen K. Longmire,et al.  Investigation of large-scale coherence in a turbulent boundary layer using two-point correlations , 2005, Journal of Fluid Mechanics.

[96]  K. Sreenivasan,et al.  CICLoPE—a response to the need for high Reynolds number experiments , 2009 .

[97]  Carl D. Meinhart,et al.  Vortex organization in the outer region of the turbulent boundary layer , 2000, Journal of Fluid Mechanics.

[98]  M. Gad-el-Hak,et al.  Recent developments in scaling of wall-bounded flows , 2006 .

[99]  M. V. Dyke,et al.  An Album of Fluid Motion , 1982 .

[100]  F. Waleffe On a self-sustaining process in shear flows , 1997 .

[101]  Yanhua Wu,et al.  Population trends of spanwise vortices in wall turbulence , 2006, Journal of Fluid Mechanics.

[102]  Ronald L. Panton,et al.  Review of Wall Turbulence as Described by Composite Expansions , 2005 .

[103]  R. A. Antonia,et al.  Low-Reynolds-number effects on near-wall turbulence , 1994, Journal of Fluid Mechanics.

[104]  Peter Bradshaw,et al.  ‘Inactive’ motion and pressure fluctuations in turbulent boundary layers , 1967, Journal of Fluid Mechanics.

[105]  Jason Monty,et al.  Large-scale features in turbulent pipe and channel flows , 2007, Journal of Fluid Mechanics.

[106]  S. Balachandar,et al.  Spanwise growth of vortex structure in wall turbulence , 2001 .

[107]  Luciano Castillo,et al.  Zero-Pressure-Gradient Turbulent Boundary Layer , 1997 .

[108]  Ronald J. Adrian,et al.  Hairpin vortex organization in wall turbulencea) , 2007 .

[109]  E. Krause,et al.  Comparative measurements in the canonical boundary layer at Reδ2≤6×104 on the wall of the German–Dutch windtunnel , 1995 .

[110]  A. Smits,et al.  Azimuthal structure of turbulence in high Reynolds number pipe flow , 2008, Journal of Fluid Mechanics.

[111]  Paul C. Fife,et al.  Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows , 2005, Journal of Fluid Mechanics.

[112]  F. Clauser Turbulent Boundary Layers in Adverse Pressure Gradients , 1954 .

[113]  Y. Tsuji,et al.  Peak position of dissipation spectrum in turbulent boundary layers. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[114]  Tie Wei,et al.  Multiscaling in the Presence of Indeterminacy: Wall-Induced Turbulence , 2005, Multiscale Model. Simul..

[115]  John Kim,et al.  DIRECT NUMERICAL SIMULATION OF TURBULENT CHANNEL FLOWS UP TO RE=590 , 1999 .

[116]  Alexandre J. Chorin,et al.  Self-similar intermediate structures in turbulent boundary layers at large Reynolds numbers , 2000, Journal of Fluid Mechanics.

[117]  M. S. Chong,et al.  A general classification of three-dimensional flow fields , 1990 .

[118]  Ivan Marusic,et al.  Large-scale influences in near-wall turbulence , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[119]  Ronald J. Adrian,et al.  Statistical evidence of hairpin vortex packets in wall turbulence , 2001, Journal of Fluid Mechanics.

[120]  J. Lumley,et al.  A First Course in Turbulence , 1972 .

[121]  G. I. Barenblatt Scaling: Self-similarity and intermediate asymptotics , 1996 .

[122]  P. Alfredsson,et al.  Effects of imperfect spatial resolution on measurements of wall-bounded turbulentbx shear flows , 1983, Journal of Fluid Mechanics.

[123]  P. Moin,et al.  Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer , 2009, Journal of Fluid Mechanics.

[124]  Scott C. Morris,et al.  Near-surface particle image velocimetry measurements in a transitionally rough-wall atmospheric boundary layer , 2007, Journal of Fluid Mechanics.

[125]  Ivan Marusic,et al.  Simultaneous orthogonal-plane particle image velocimetry measurements in a turbulent boundary layer , 2006, Journal of Fluid Mechanics.

[126]  Alexander Smits,et al.  High–Reynolds Number Wall Turbulence , 2011 .

[127]  M. S. Chong,et al.  A theoretical and experimental study of wall turbulence , 1986, Journal of Fluid Mechanics.

[128]  Martin Wosnik,et al.  A theory for turbulent pipe and channel flows , 2000, Journal of Fluid Mechanics.

[129]  Martin Skote,et al.  Studies of turbulent boundary layer flow throughdirect numerical simulation , 2001 .

[130]  W. Cabot,et al.  Group analysis, direct numerical simulation and modelling of a turbulent channel flow with streamwise rotation , 2006, Journal of Fluid Mechanics.

[131]  B. McKeon,et al.  The near-neutral atmospheric surface layer: turbulence and non-stationarity , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[132]  W. George,et al.  Is there a universal log law for turbulent wall-bounded flows? , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[133]  Ronald L. Panton,et al.  Self-Sustaining Mechanisms of Wall Turbulence , 1997 .

[134]  Javier Jiménez,et al.  The autonomous cycle of near-wall turbulence , 1999, Journal of Fluid Mechanics.

[135]  Ivan Marusic,et al.  Streamwise turbulence intensity formulation for flat-plate boundary layers , 2003 .

[136]  J. Foss,et al.  Statistical structure of turbulent-boundary-layer velocity–vorticity products at high and low Reynolds numbers , 2007, Journal of Fluid Mechanics.

[137]  Meredith Metzger,et al.  Viscous sublayer flow visualizations at Rθ≂1 500 000 , 1995 .

[138]  Luciano Castillo,et al.  Similarity Analysis for Turbulent Boundary Layer with Pressure Gradient: Outer Flow , 2001 .

[139]  John Foss,et al.  Springer Handbook of Experimental Fluid Mechanics , 2007 .

[140]  J. Klewicki,et al.  Flow field properties local to near-wall shear layers in a low Reynolds number turbulent boundary layer , 2004 .

[141]  M. R. Head,et al.  New aspects of turbulent boundary-layer structure , 1981, Journal of Fluid Mechanics.

[142]  M. Hites,et al.  Scaling of high-Reynolds number turbulent boundary layers in the National Diagnostic Facility , 1997 .

[143]  Eric S. Winkel,et al.  High-Reynolds-number turbulent boundary layer friction drag reduction from wall-injected polymer solutions , 2009, Journal of Fluid Mechanics.

[144]  Rabia Nacereddine,et al.  An interactive bypass transition mechanism in wall-bounded flows , 2008, Journal of Fluid Mechanics.

[145]  Fazle Hussain,et al.  Coherent structure generation in near-wall turbulence , 2002, Journal of Fluid Mechanics.

[146]  David R. Dowling,et al.  Bubble-induced skin-friction drag reduction and the abrupt transition to air-layer drag reduction , 2008, Journal of Fluid Mechanics.

[147]  P. Spalart Direct simulation of a turbulent boundary layer up to Rθ = 1410 , 1988, Journal of Fluid Mechanics.

[148]  K. A. Flack,et al.  Velocity-defect scaling for turbulent boundary layers with a range of relative roughness , 2005 .

[149]  C. Bender,et al.  Matched Asymptotic Expansions: Ideas and Techniques , 1988 .

[150]  Ivan Marusic,et al.  On the role of large-scale structures in wall turbulence , 2001 .

[151]  Ivan Marusic,et al.  Evidence of very long meandering features in the logarithmic region of turbulent boundary layers , 2007, Journal of Fluid Mechanics.

[152]  A. Smits,et al.  Development of NSTAP: Nanoscale Thermal Anemometry Probe , 2005 .

[153]  P. Davidson,et al.  The logarithmic structure function law in wall-layer turbulence , 2006, Journal of Fluid Mechanics.

[154]  Carl D. Meinhart,et al.  On the existence of uniform momentum zones in a turbulent boundary layer , 1995 .

[155]  M. Metzger LENGTH AND TIME SCALES OF THE NEAR-SURFACE AXIAL VELOCITY IN A HIGH REYNOLDS NUMBER TURBULENT BOUNDARY LAYER , 2006, Proceeding of Fourth International Symposium on Turbulence and Shear Flow Phenomena.

[156]  On the asymptotic similarity of the zero-pressure-gradient turbulent boundary layer , 2008, Journal of Fluid Mechanics.

[157]  R. Panton Scaling and correlation of vorticity fluctuations in turbulent channels , 2009 .

[158]  J. Klewicki,et al.  Spanwise vorticity structure in turbulent boundary layers , 1996 .

[159]  P. Fife,et al.  A physical model of the turbulent boundary layer consonant with mean momentum balance structure , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[160]  J. D. Li,et al.  Experimental support for the attached-eddy hypothesis in zero-pressure-gradient turbulent boundary layers , 1990, Journal of Fluid Mechanics.

[161]  P. Guest,et al.  Evaluations of the von Kármán constant in the atmospheric surface layer , 2006, Journal of Fluid Mechanics.

[162]  Mesoscaling of Reynolds Shear Stress in Turbulent Channel and Pipe Flows. , 2005 .

[163]  Seong‐Ryong Park,et al.  The influence of instantaneous velocity gradients on turbulence properties measured with multi-sensor hot-wire probes , 1993 .

[164]  H. L. Evans,et al.  Laminar Boundary-Layer Theory. , 1968 .

[165]  Yiannis Andreopoulos,et al.  Vorticity, strain-rate and dissipation characteristics in the near-wall region of turbulent boundary layers , 1997, Journal of Fluid Mechanics.

[166]  J. Klewicki Velocity–vorticity correlations related to the gradients of the Reynolds stresses in parallel turbulent wall flows , 1989 .

[167]  J. D. Li,et al.  Wall turbulence closure based on classical similarity laws and the attached eddy hypothesis , 1994 .

[168]  R. Adrian,et al.  On the relationships between local vortex identification schemes , 2005, Journal of Fluid Mechanics.

[169]  C. R. Smith,et al.  The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer , 1983, Journal of Fluid Mechanics.

[170]  R J Adrian,et al.  Large- and very-large-scale motions in channel and boundary-layer flows , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[171]  A. Smits,et al.  Wall-bounded turbulent flows at high Reynolds numbers: Recent advances and key issues , 2010 .

[172]  P. Fife,et al.  Overview of a Methodology for Scaling the Indeterminate Equations of Wall Turbulence , 2005 .

[173]  Michel Stanislas,et al.  Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry , 2005, Journal of Fluid Mechanics.

[174]  Alexandre J. Chorin,et al.  New Perspectives in Turbulence: Scaling Laws, Asymptotics, and Intermittency , 1998, SIAM Rev..

[175]  S. J. Kline,et al.  Quasi-coherent structures in the turbulent boundary layer. I - Status report on a community-wide summary of the data , 1990 .

[176]  Kwing-So Choi,et al.  Accurate measurements of local skin friction coefficient using hot-wire anemometry , 2002 .

[177]  Donald Coles,et al.  THE YOUNG PERSON'S GUIDE TO THE DATA , 1968 .

[178]  M. S. Chong,et al.  On the mechanism of wall turbulence , 1982, Journal of Fluid Mechanics.

[179]  Javier Jiménez,et al.  Reynolds number effects on the Reynolds-stress budgets in turbulent channels , 2008 .

[180]  J. Morrison,et al.  The interaction between inner and outer regions of turbulent wall-bounded flow , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[181]  K. Mcnaughton Attached Eddies and Production Spectra in the Atmospheric Logarithmic Layer , 2004 .

[182]  Javier Jiménez,et al.  Scaling of the velocity fluctuations in turbulent channels up to Reτ=2003 , 2006 .

[183]  Jiezhi Wu,et al.  Interactions between a solid surface and a viscous compressible flow field , 1993, Journal of Fluid Mechanics.

[184]  J. Wallace,et al.  Near-surface turbulence in the atmospheric boundary layer , 2010 .

[185]  S. K. Robinson,et al.  Coherent Motions in the Turbulent Boundary Layer , 1991 .

[186]  H. Stephani Differential Equations: Their Solution Using Symmetries , 1990 .

[187]  B. Ganapathisubramani Statistical structure of momentum sources and sinks in the outer region of a turbulent boundary layer , 2008, Journal of Fluid Mechanics.

[188]  Javier Jiménez,et al.  What are we learning from simulating wall turbulence? , 2007, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[189]  G. I. Barenblatt,et al.  A mathematical model for the scaling of turbulence. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

[190]  Fazle Hussain,et al.  Coherent structures near the wall in a turbulent channel flow , 1997, Journal of Fluid Mechanics.

[191]  S. Lele Vorticity form of turbulence transport equations , 1992 .

[192]  L. Prandtl 7. Bericht über Untersuchungen zur ausgebildeten Turbulenz , 1925 .

[193]  N. Afzal Fully developed turbulent flow in a pipe: an intermediate layer , 1982 .

[194]  Joseph Klewicki,et al.  On accurately measuring statistics associated with small-scale structure in turbulent boundary layers using hot-wire probes , 1990, Journal of Fluid Mechanics.

[195]  P. Monkewitz,et al.  High Reynolds Number Turbulent Boundary Layers Subjected to Various Pressure-Gradient Conditions , 2006 .

[196]  P. Davidson,et al.  A simple model for the streamwise fluctuations in the log-law region of a boundary layer , 2009 .

[197]  P. Dimotakis The mixing transition in turbulent flows , 2000, Journal of Fluid Mechanics.

[198]  N. Afzal,et al.  Analysis of turbulent pipe and channel flows at moderately large Reynolds number , 1973, Journal of Fluid Mechanics.

[199]  R. E. Falco,et al.  A coherent structure model of the turbulent boundary layer and its ability to predict Reynolds number dependence , 1991, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[200]  Paul C. Fife,et al.  Stress gradient balance layers and scale hierarchies in wall-bounded turbulent flows , 2005, Journal of Fluid Mechanics.

[201]  Ahmed Naguib,et al.  Characterization of the Pressure Fluctuations Under a Fully Developed Turbulent Boundary Layer , 1998 .

[202]  S. Balachandar,et al.  Mechanisms for generating coherent packets of hairpin vortices in channel flow , 1999, Journal of Fluid Mechanics.

[203]  G. Karniadakis,et al.  Reynolds stress analysis of EMHD-controlled wall turbulence. Part I. Streamwise forcing , 1997 .

[204]  Günter Schewe,et al.  On the structure and resolution of wall-pressure fluctuations associated with turbulent boundary-layer flow , 1983, Journal of Fluid Mechanics.

[205]  M. V. Zagarola,et al.  A new friction factor relationship for fully developed pipe flow , 2005, Journal of Fluid Mechanics.

[206]  L. Perret,et al.  Vortical structures in the turbulent boundary layer: a possible route to a universal representation , 2008, Journal of Fluid Mechanics.

[207]  Tomoaki Itano,et al.  Interaction between a large-scale structure and near-wall structures in channel flow , 2005, Journal of Fluid Mechanics.

[208]  J. Garratt The Atmospheric Boundary Layer , 1992 .

[209]  Alan R. Kerstein,et al.  One-dimensional turbulence: model formulation and application to homogeneous turbulence, shear flows, and buoyant stratified flows , 1999, Journal of Fluid Mechanics.

[210]  B. J. McKeon,et al.  A critical-layer framework for turbulent pipe flow , 2010, Journal of Fluid Mechanics.

[211]  Peter Bradshaw,et al.  Spatial resolution and measurement of turbulence in the viscous sublayer using subminiature hot-wire probes , 1987 .

[212]  Mohamed Gad-el-Hak,et al.  Reynolds Number Effects in Wall-Bounded Turbulent Flows , 1994 .

[213]  Jason Monty,et al.  Developments In Smooth Wall Turbulent Duct Flows , 2005 .

[214]  P. Monkewitz,et al.  Comparison of mean flow similarity laws in zero pressure gradient turbulent boundary layers , 2008 .

[215]  I. Marusic,et al.  Study of the near-wall-turbulent region of the high-Reynolds-number boundary layer using an atmospheric flow , 2006, Journal of Fluid Mechanics.

[216]  R. Mathis,et al.  Comparison of large-scale amplitude modulation in turbulent boundary layers, pipes, and channel flows , 2009 .

[217]  B. Cantwell,et al.  Introduction to Symmetry Analysis , 2002 .

[218]  P. A. Lagerstrom,et al.  Matched Asymptotic Expansions , 1988 .

[219]  R. Falco,et al.  New results, a review and synthesis of the mechanism of turbulence production in boundary layers and its modification , 1983 .

[220]  A. Smits,et al.  Scaling of the wall-normal turbulence component in high-Reynolds-number pipe flow , 2007, Journal of Fluid Mechanics.

[221]  Ivan Marusic,et al.  A wall-wake model for the turbulence structure of boundary layers. Part 1. Extension of the attached eddy hypothesis , 1995, Journal of Fluid Mechanics.

[222]  P. Moin,et al.  The minimal flow unit in near-wall turbulence , 1991, Journal of Fluid Mechanics.

[223]  W. Willmarth,et al.  Reynolds-number effects on the structure of a turbulent channel flow , 1989, Journal of Fluid Mechanics.

[224]  P. Bradshaw,et al.  Bursts and the law of the wall in turbulent boundary layers , 1992, Journal of Fluid Mechanics.

[225]  M. Oberlack A unified approach for symmetries in plane parallel turbulent shear flows , 2001, Journal of Fluid Mechanics.

[226]  H. Schlichting Boundary Layer Theory , 1955 .