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Technology Reviews

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

[+] Author and Article Information
Joseph C. Klewicki

Department of Mechanical Engineering, University of New Hampshire, Durham, NH 03824joe.klewicki@unh.edu

J. Fluids Eng 132(9), 094001 (Sep 23, 2010) (48 pages) doi:10.1115/1.4002167 History: Received May 20, 2010; Revised June 23, 2010; Published September 23, 2010; Online September 23, 2010

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.

Copyright © 2010 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Schematic representation of the canonical, two-dimensional, flat plate turbulent boundary layer geometry and time-averaged axial velocity profile

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Figure 2

Variation in the friction velocity over the centerline velocity in turbulent pipe flow. Data are from McKeon (180). Reprinted from Metzger (247), copyright 2006, with permission from Elsevier.

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Figure 3

Variation in the skin friction coefficient in turbulent boundary layers with Reynolds number, from Ref. 47. Experimental data from the wind tunnels at IIT and KTH are compared with self-consistently determined zero pressure gradient skin friction correlations found in the literature. Reprinted with permission.

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Figure 4

Variation in the shape factor H in turbulent boundary layers with Reynolds number, from Ref. 47. The data from the “other experiments” are found in Refs. 248,2. Solid line is given by H=1/(1−7.135/U∞+). Reprinted with permission.

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Figure 5

(a) Inner-normalized streamwise variance profiles in a δ+≃14,000 boundary layer as derived from hot-wire sensors having l+=luτ/ν values of 22, 79, and 153 (squares, triangles, and circles, respectively). (b) Inner-normalized mean profiles derived from the same sensors. Reprinted from Hutchins (79) with permission from Cambridge University Press.

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Figure 6

Comparison of experimental (squares) and DNS (circles) based estimates of the velocity gradient variance in turbulent channel flow using a first order finite difference. These results demonstrate the attenuation effect with increasing Δy∗=Δy/η and the finite noise amplification effect in the experimental data for small Δy∗. Good agreement is exhibited between the data and the theoretical prediction of Wyngaard (line). The figure is from Antonia (73), copyright 1993 by Springer-Verlag. Reprinted with permission.

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Figure 7

Princeton Superpipe pressurized pipe flow facility, from McKeon (97). Reprinted with permission.

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Figure 8

Tower based hot-wire and sonic anemometer arrays and other instruments utilized in the 2005 SLTEST field experiments, after Metzger (103). Reprinted with permission.

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Figure 9

Schematic of the University of Melbourne HRNBLWT. Figure provided courtesy of Prof. I. Marusic.

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Figure 10

Perspective solid rendering of the large scale CICLoPE pipe flow experiment being constructed at the University of Bologna (106). This pipe has a diameter of 0.9 m and is about 130 m long. Figure provided courtesy of Prof. A. Talamelli.

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Figure 11

Schematic cut-away representation of the Flow Physics Facility (phase I version) being constructed at the University of New Hampshire. The test section of this wind tunnel has a cross section of about 6×2.7 m2 and a length of 72 m.

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Figure 12

Comparison of Eq. 60 (dashed line) with the KTH wind tunnel mean profiles of Osterlund (92) over 2530≤Rθ≤27,300. Reprinted from Ref. 136 with permission from Cambridge University Press.

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Figure 13

Schematic depiction of the continuous hierarchy of scaling layers formally admitted by the mean momentum equation. Note that an invariant form of Eq. 25 is valid on each Lβ layer of the hierarchy, from Klewicki (142).

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Figure 14

(a) Representative inner-normalized mean profiles in turbulent boundary layers and the associated layer structure: (A) viscous sublayer, (B) buffer layer, (C) logarithmic layer, and (D) wake layer. Data are from Klewicki and Falco (72). (b) Representative inner-normalized mean viscous and Reynolds shear stress profiles in turbulent channel flow, from the DNS database of Moser (196) at δ+=590.

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Figure 15

Schematic of the mean velocity profile based layer structure within the context of the framework established by an overlap layer hypothesis, after McKeon and Morrison (182)

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Figure 16

Ratio of the viscous stress gradient to Reynolds stress gradient, third to first terms in Eq. 25, in turbulent pipe flow as a function of y+ as derived from the data of Zagarola and Smits (36), from Wei (22)

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Figure 17

Schematic of the structure of the boundary layer as derived from consideration of the mean momentum balance: (I) viscous stress gradient/mean advection balance layer, (II) viscous/Reynolds stress gradient balance layer, (III) Reynolds stress gradient/viscous stress gradient/mean advection balance layer, and (IV) Reynolds stress gradient/mean advection balance layer. Adapted from Wei (22).

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Figure 18

Schematic representation of the self-sustaining process (SSP), associated with the model for the near-wall cycle as put forth by Waleffe and Kim (153)

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Figure 19

(a) Top view and (b) side view of streamwise vortices generated by the streak transient growth of linearly stable sublayer streaks as described by Schoppa and Hussain (150). Isosurfaces of ωx at levels +0.6ωx∣max and −0.6ωx∣max are (dark) shaded and hatched, respectively; contours of u at y+=20 are shaded to indicate the low-speed streak. Reprinted from Schoppa and Hussain (150), with permission from Cambridge University Press.

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Figure 20

Sketch of a representative attached eddy delineating its key geometric and kinematic features. Note that in this figure, δ refers to the height of the eddy and that the wall-normal direction is given by z. Reprinted from Perry and Marusic (111) with permission from Cambridge University Press.

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Figure 21

Conceptual scenario of nested packets of hairpinlike vortices growing up from the wall. These packets align in the streamwise direction and coherently add together to create large zones of nearly uniform streamwise momentum. Large scale motions in the wake region ultimately limit their growth. Smaller packets move more slowly because they induce faster upstream propagation. Reprinted from Adrian (117) with permission from Cambridge University Press.

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Figure 22

Schematic of the boundary layer scales that factor importantly in the physical model of Hunt and Morrison (169). Reprinted from Hunt and Morrison (169), copyright 2000, with permission from Elsevier.

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Figure 23

Schematic representation of some of the dynamical attributes of the model for the turbulent boundary layer based on the structure of the mean momentum balance (143). Layer numbers are the same as those identified in Fig. 1. Note that the position of Tmax and ymax+ varies like δ+. Note also that the attached eddies in this case are similar but not identical to Townsend’s prescription.

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Figure 31

(a) Maximum value of the turbulence kinetic energy production in channel flows plotted versus Reynolds number. (b) Positions of the maximum values of turbulence kinetic energy production and mean viscous stress gradient in channel flows plotted as a function of Reynolds number. Equations 4,10,11,12 in these figures are empirical relations given by Laadhari (192). Reprinted with permission from Ref. 192, copyright 2002, American Institute of Physics.

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Figure 32

(a) Well-resolved measurements of the inner-normalized streamwise velocity intensities for 1×103≲Rθ≲5×106. Data are from laboratory facilities and the SLTEST site. Open symbols are from Ref. 72. Solid symbols are from Ref. 74. Dashed line is the prediction of the similarity formulation of Ref. 89. (b) Peak values of u′+ versus Reynolds number for studies in which l+≤10. Logarithmic curve fits do not incorporate the high Reynolds number estimate from the SLTEST site. Figures are adapted from Metzger and Klewicki (77); also see the studies by Marusic and Kunkel (89,200).

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Figure 33

Data of Fig. 3 made dimensionless via the empirically determined mixed normalization proposed by De Graaff and Eaton (74). Figure adapted from Metzger (197).

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Figure 39

Channel flow mesonormalized profile of the deviation of the Reynolds stress from its maximal value, Tm, versus mesonormalized distance from the peak position of the Reynolds stress ym at various Rτ, from Ref. 138. Note that in this figure, ε=1∕(δ+)1∕2, i.e., the inverse of the quantity indicated.

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Figure 40

(a) Representative profiles of −ρuv=uv¯/(u′v′) in turbulent boundary layers for a range of Reynolds numbers. (b) Turbulent boundary layer values of −ρuv evaluated where uv¯ attains its maximal value and plotted as a function of Reynolds number. Curve fit does not incorporate the high Reynolds number data from the SLTEST site. Symbols and Reynolds numbers are given in Table 3. Figure adapted from Priyadarshana and Klewicki (207).

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Figure 41

Inner-normalized mean velocity gradient profiles in turbulent channel flow at various Reynolds numbers. Data are from the DNS database of Hoyas and Jimenez (90).

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Figure 42

(a) Estimates of the inner-normalized wall-normal vorticity intensities in turbulent boundary layers at various Reynolds numbers. (b) Estimates of the inner-normalized spanwise vorticity intensities in turbulent boundary layers at various Reynolds numbers. Symbols and Reynolds numbers are given in Table 3. The figure is adapted from Priyadarshana (201).

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Figure 43

Inner-normalized vorticity intensity data in turbulent channel flow at various Reynolds numbers: (a) streamwise, (b) wall normal, and (c) spanwise. (d) Inner-normalized wall values of the streamwise and spanwise vorticity intensities as derived from the y+=0 data of (a) and (c) along with the peak data point from Fig. 4. Channel flow data are from DNS database of Hoyas and Jimenez (90).

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Figure 44

Properties of the near-wall spanwise vorticity field for 103≲δ+≲106: (a) probability of observing negative ω̃z as a function y+; (b) two-point spanwise vorticity correlation, ωzωz¯(Δy+), for lower probe at y+≃3. Figures are adapted from Refs. 77,155.

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Figure 24

Plots of the Reynolds stress from the channel flow DNS of Iwamoto (249) for four different Reynolds numbers based upon the bulk mean velocity and the channel width. The data fit is for a two term logarithmic expansion, as described by Sreenivasan and Bershadskii (174). Reprinted from Sreenivasan and Bershadskii (174) with permission from Cambridge University Press.

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Figure 58

Example of the signals derived from the horizontal array of sonic anemometers used in the experiments of Hutchins and Marusic (206). The x-axis is reconstructed using Taylor’s hypothesis employing an advection velocity equal to the local mean velocity. Shading shows only negative u fluctuations (see gray scale). Reprinted from Hutchins and Marusic (206) with permission from Cambridge University Press.

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Figure 59

Vertical derivative of premultiplied cospectra, ϕuv, for (a) δ+=3815, (b) δ+=5884, and (c) δ+=7959. Note the positive peak in the low-wavenumber region for y/δ=0.15. Reprinted from Guala (205) with permission from Cambridge University Press.

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Figure 25

Mean flow properties of smooth-wall turbulent pipe flow as derived from the Princeton Superpipe experiments of McKeon (97): (a) Pitot tube-based inner-normalized mean velocity profiles plotted in standard semilogarithmic form; (b) friction factor versus Reynolds number as determined via pressure drop measurements. Used with permission.

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Figure 26

Development of the boundary layer wake parameter, Π99≈κ/2(0.99U∞+−κ−1 ln(δ99+)−B), as a function of Reynolds number. For these data the values κ=0.384 and B=4.173 were employed. Used with permission from Nagib (47).

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Figure 27

Inner-normalized mean velocity profiles and logarithmic indicator function for selected experiments from the IIT and KTH wind tunnels. The channel flow data are from the DNS of Jimenez and Hoyas (198). Used with permission from Nagib (47).

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Figure 28

Estimated values for the von Kármán constant (coefficient), κ, for zero pressure gradient turbulent boundary layers and fully developed turbulent pipe and channel flows. The high Reynolds number values are estimated by κtbl=0.384, κp=0.41, and κc=0.37. Reprinted with permission from Nagib and Chauhan (50), copyright 2008, American Institute of Physics.

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Figure 29

Characteristic lengths of the continuum of scaling layers in turbulent channel flow as a function of y+ for δ+=547, δ+=934, and δ+=2003. Horizontal lines denote approximate position at which the characteristic length scale distribution deviates from a linear y dependence. Curve fit of the δ+=2003 profile is given by W=0.6247y++5.61 for 118≤y+≤667. The figure is adapted from Klewicki (142).

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Figure 30

Development of the time scale ratio, Ω=(δ99/uτ)/(x/U∞), as a function of Reynolds number in turbulent boundary layers. Used with permission from Nagib (47).

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Figure 34

Representative inner-normalized wall-normal velocity intensities from boundary layer flows plotted over a range of Reynolds numbers. Symbols and Reynolds numbers are given in Table 3. Figure is from Priyadarshana (201).

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Figure 35

Inner- and outer-normalized wall-normal velocity variance profiles. ○, δ+=689(188); ◇, δ+=23013(250); △, δ+=1335; ◁, δ+=2217; ▽, δ+=5813; ▷, δ+=13,490(74); solid symbols δ+≃2.3×106 or 3.1×106(200). Solid, dashed, and dot-dashed lines are smooth-wall similarity formulations given in Kunkel and Marusic (200) that are also valid for rough walls in the outer region. Lighter dashed lines are similarity formulation for δ+=3.1×106 if boundary-layer thickness is 50% larger or smaller. Note that the wall-normal direction is denoted by z in this plot. Reprinted from Kunkel and Marusic (200) with permission from Cambridge University Press.

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Figure 36

(a) Representative inner-normalized spanwise velocity intensities plotted over a range of Reynolds numbers. (b) Peak values of w′+ as a function of Reynolds number. Logarithmic curve fit does not incorporate the high Reynolds number estimate from the SLTEST site data. Symbols and Reynolds numbers are given in Table 3. Figure is adapted from Priyadarshana (201).

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Figure 37

Inner- and outer-normalized Reynolds shear stress profiles. Symbols same as in Fig. 3. Reprinted from Kunkel and Marusic (200) with permission from Cambridge University Press.

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Figure 38

Profiles of the inner-normalized Reynolds stress, T+=−uv¯+, from turbulent pipe and channel flows at various Rτ=δ+: (a) versus y+ and (b) versus y/δ from Wei (138)

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Figure 45

Inner-normalized surface vorticity flux intensities versus δ+; (∂ωz/∂y)′+, ▽; (∂ωx/∂y)′+ △. Open symbols are from Andreopoulos and Agui (87), while the solid symbols are from Klewicki (86).

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Figure 46

Temporal correlation between the surface ∂p/∂x(t) and p(t+Δt) at δ+≃1×106, from Klewicki (86)

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Figure 47

Premultiplied power spectra of v, Ψ(v+) (solid dark gray), premultiplied power spectra of ωz, Ψ(ωz+)) (dashed medium gray), and premultiplied cospectra of v+ and ωz+, k+Λ(v+ωz+) (solid light gray) at Rθ=2870 ((a) y+≃ym+/2 and (b) y+≃2ym+) and at Rθ≃4×106 ((c) y+≃ym+/2 and (d) y+≃2ym+). Adapted from Priyadarshana (201).

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Figure 48

(a) Wall-pressure variances as a function of δ+, from various sources. ○, channels; △, pipes; ▽, boundary layers. Closed symbols are from the study of Jimenez and Hoyas (198). Other symbols are from the data compilation of Hu (215). (b) Pressure variance profiles. – - –, δ+=547; – –, δ+=934; —, δ+=2003. Lines without symbols are the full pressure. ○, fast pressure; △, slow pressure. Reprinted from Jimenez and Hoyas (198) with permission from Cambridge University Press.

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Figure 49

(a) Root-mean-square static pressure, normalized using inner variables: △, Rθ=7420; ◻, Rθ=8920; ▽, Rθ=10,500; ◇, Rθ=12,100; ×, Rθ=15,200 data from Ref. 66. (b) Root mean square averaged in the logarithmic region plotted with ○ versus the Reynolds number. Solid symbols indicate the peak of p′+ obtained by DNS: ●, Skote (251); ◼, Moser (196); ▲, Spalart (188); ▼, Eggels (252); ◆, Abe (253). Solid and dashed lines are best fits for experiments and DNS, respectively. For both p′+∝Rθ0.24. Dash-dotted line corresponds to the relation pmax′+=0.5ρu′+2. Reprinted from Tsuji (66) with permission from Cambridge University Press.

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Figure 50

(a) Compilation of inner-normalized wall-pressure intensities from wall-flow studies as a function of δ+. (b) Inner-normalized boundary layer wall-pressure power spectra, — —, δ+=2010(216); - - - -, δ+=3822(66); ▲, δ+=4956(254); ——, δ+≃1×106. Adapted from Klewicki (86).

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Figure 52

(a) Galilean-decomposed velocity field in the streamwise wall-normal plane at δ+=2350. (b) Detected vortices bearing negative and positive ωz calculated from the velocity field in (a). The dashed line highlights the tent-line interface of the visualized vortex packet. Reprinted from Natrajan (213) with permission from Cambridge University Press.

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Figure 53

The ((a) and (b)) Scaling of Πp and Πr with Rτ1.17 and Rτ1.5, respectively, versus y/δ. ((c) and (d)) Scaling of Πp+ and Πr+ with Rτ−0.5 and Rτ−0.64, respectively, versus y+. Channel flow data are given by open symbols, and the boundary layer data are given by solid symbols. Reprinted from Wu and Christensen (212) with permission from Cambridge University Press.

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Figure 54

Fraction of retrograde (positive ω̃z) vortices in the boundary layer versus Rτ at (a) y=0.2δ and (b) y=0.75δ. Reprinted from Wu and Christensen (212) with permission from Cambridge University Press.

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Figure 55

(a) Three-dimensional plot of the average velocity field conditioned to the tall attached clusters. The mesh is an isosurface of the pdf of the vortex positions and contains 57% of the data. The volume surrounding the cluster is the isosurface ⟨u′⟩+=0.3, where the angle brackets denote the conditional average. The shaded volume downstream of the cluster is the isosurface ⟨u′⟩+=−0.1. The vector plots represent (⟨v⟩,⟨w⟩) in the indicated planes. The scale of the arrows in the downstream plane has been magnified by a factor of 1.7 to facilitate their visualization. (b) A magnification of the surroundings of the average position of the clusters, including a vector plot of (⟨v⟩,⟨w⟩) in the plane x=0. The longest arrow measures 0.5uτ. The Reynolds number of the flow is δ+=934. Reprinted from del Alamo (226) with permission from Cambridge University Press.

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Figure 56

Wall-normal profiles of total streamwise velocity show regions where the velocity is nearly constant. The solid black vertical lines indicate the x location of each profile. The wall-normal profiles of streamwise velocity are overlaid on a map of ∂ũ/∂y, where the light-gray and dark-gray shaded areas correspond to regions where 0.01<∂ũ/∂y<0.03 and 0.03<∂ũ/∂y, respectively. The regions where ∂ũ/∂y is large often coincide with the boundaries of the momentum zones. Black lines are used to denote zone boundaries. Reprinted with permission from Meinhart and Adrian (230), copyright 1995, American Institute of Physics.

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Figure 57

Illustrative example of the large scale structure of hairpin vortex packets at Rθ=7705. The solid lines are contours of constant streamwise velocity at 70% and 94% of the freestream velocity. The velocity field in the lower plot has a convection (advection) velocity Uc=0.79U∞ subtracted, and gray levels indicate swirling strength. The upper plot of the inset region A has Uc=0.76U∞ subtracted, and gray levels indicate fluctuating spanwise vorticity. Reprinted from Adrian (117) with permission from Cambridge University Press.

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Figure 60

(a) Inner-normalized sublayer streak spacing versus Reynolds number. Low Reynolds number data are from Smith and Metzler (255), while the high Reynolds number data are from Klewicki (243). (b) Inner-normalized sublayer pocket widths and durations versus Reynolds number. Figures are adapted from Klewicki (243) and Metzger (244), respectively.

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Figure 61

The (i) Contour maps showing the variation in one-dimensional premultiplied spectra with wall-normal position and (ii) the relationship between these contour maps and the mean velocity and streamwise normal stress profiles: (a) Rτ=2630 and (b) Rτ=7300. The y-axis shows length scale in both inner (left) and outer (right) scaling. The x-axis shows wall-normal position for both inner (lower) and outer scaling (upper). The gray scale shows the magnitude of kxϕuu/uτ2. (+) denote “inner” and “outer” energy sites (white) λx+=1000, z+=15 and (black) λx=6δ, z=0.06δ. Dot-dashed lines show kx−1 limits λx=15.7z (and z+=100 and λx=0.3δ) for plot (b). Profiles of (●) turbulence intensities and (−) mean velocity. Dotted line shows U+=z+. Dashed lines and (+) symbols denote z+=15 and z=0.06δ. Dot-dashed line shows Eq. 51 (where κ=0.41 and B=5.0). Solid line intensity profile shows the prediction from the formulation of Marusic and Kunkel (89). In this figure, z is the wall-normal coordinate. Reprinted from Hutchins and Marusic (206) with permission from Cambridge University Press.

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Figure 62

Reynolds number comparison of the correlation coefficient between the large scale component and the filtered envelope of the small-scale component associated with the streamwise velocity amplitude modulation. Reprinted from Mathis (78) with permission from Cambridge University Press.

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Figure 51

(a) Mean radius profiles and (b) mean vorticity profiles of detected eddy structures: ▽, (x,y) plane at Rθ=7500; ◆, upstream slanted cross-stream (z,u) plane at Rθ=7500; ◻, (z,u) plane at Rθ=10500; ▲, (z,u) plane at Rθ=13500; ○, (z,u) plane at Rθ=19,000; – – –, ωz′+ profile; ——, Ωz profile. Reprinted from Carlier and Stanislas (104) with permission from Cambridge University Press.

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Figure 63

Wall-normal location at which the degree of amplitude modulation is zero (R=0) versus Reynolds number (filled symbols): (a) with inner length scaling (z+); (b) with outer length scaling (z/δ). Lines represent the location of the middle of the log layer, corresponding to (solid) 100<z+<0.15Rτ and (dashed) based on Reynolds-number-dependent boundaries KRτ1/2<z+<0.15Rτ (here, K=1.2). The open symbol is the estimated location of the outer u-spectral peak measured at SLTEST in 2005 by Metzger and McKeon. Reprinted from Mathis (78) with permission from Cambridge University Press.

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