Wall-bounded turbulence: structure

and passive control

Brian Jeffrey Rosenberg

A Dissertation

Presented to the Faculty

of Princeton University

in Candidacy for the Degree

of Doctor of Philosophy

Recommended for Acceptance

by the Department of

Mechanical Engineering

Adviser: Professor Alexander J. Smits

June 2015

c© Copyright by Brian Jeffrey Rosenberg, 2015.

All rights reserved.

Abstract

The nature of wall-bounded turbulence is investigated with respect to spectral charac-

teristics. As the Reynolds number approaches infinity, the k−1x overlap region does not

appear to be present, yet the k−5/3x asymptotically takes shape in the logarithmic and

wake regions. In terms of coherent structures, a new scaling is proposed that describes

the size of the very-large-scale motions throughout the flow. The corresponding sim-

ilarity to the large-scale motion scaling suggests that these two phenomena might be

interrelated. These observations appear to be robust to assumptions of a wavelength-

independent convection velocity, although it is clear that Taylors hypothesis indeed

leads to some spectral distortion, especially close to the wall.

Passive techniques to control wall-bounded turbulence are investigated with a

focus on frictional drag reduction. Surfaces coated with a thin layer of immiscible

liquid can delay the transition of a flow to a turbulent state, thereby reducing the

energy dissipation in the flow. It is hypothesized that interfacial tension between

the two fluids acts to stabilize disturbances, helping to maintain a laminar flow.

The friction reducing properties of superhydrophobic surfaces, in which micro-scale

pockets of air are trapped within surface textures, are confirmed in turbulent Taylor-

Couette flow, with drag reductions up to 11% reported. The drag reduction on the

surface decreases with increasing Reynolds number, that is, as the viscous length

scale becomes smaller compared to the surface feature size. Further, turbulent drag

reduction is measured over a textured surface infused with a second immiscible liquid,

with values of 4% in Taylor-Couette flow and 10% in boundary layer flow.

iii

Acknowledgements

Though only my name appears on the cover of this dissertation, it is a product of years

of interaction with and inspiration by a great many colleagues and friends. Therefore,

I wish to express my warmest gratitude to all those whose comments, questions, and

encouragement, both personal and academic, have left a mark on me and on this

work.

Foremost, I would like to express my gratitude to my advisor Lex Smits for creating

an environment where I had an opportunity to explore my own ideas and for providing

helpful guidance when needed. I greatly admire his spirit of adventure in terms of

research and his enthusiasm with regard to teaching. I am grateful for my committee

members, Gigi Martinelli and Elie Bou-Zeid, for their mentorship during my PhD

research and for their encouragement, insightful comments, and hard questions. I

thank Michael Schultz and Marcus Hultmark for serving as my dissertation readers

and for providing helpful feedback, which has helped me improve and clarify the final

work.

I would like to thank members of the MURI team for their friendly collabora-

tion, insightful discussions, and technical assistance the past three years: Joanna

Aizenberg, Julio Barros, Marcus Hultmark, Philseok Kim, Mughees Khan, Stefano

Leonardi, Michael Schultz, Howard Stone, and Tak Sing Wong. Within the Princeton

team, I learned a great deal from Jason Wexler, Ian Jacobi, Matthew Fu, Mohamed

Samaha, and Yuyang Fan.

In my daily work I have been blessed with a friendly group of fellow students, who

made the lab a stimulating and fun place. Thank you Leo Hellström, Jessica Shang,

Dan Quinn, Owen Williams, Anand Ashok, Peter Dewey, Keith Moored, Tristen

Hohman, Gilad Arwatz, Tyler Van Buren, Katie Hartl, Stefano Chiazza, Florian

Bremer, Florent Melchior, and Dan Floryan. Thank you to the MAE staff, especially

Jill Ray and Dan Hoffman, for all of your help and support. I am also grateful for

iv

the rest of my cohort for their camaraderie and friendship: Chris and Mary Anne

Limbach, Jeff Santner, Fred Brasz, Joe Lefkowitz, Paul Reverdy, Josh Heyne, Tat

Loon Chng, and Zhong Zheng.

Finally, to my parents, my brothers and sister, my wife Leslie, and my dog Theo,

I can’t thank you enough for your love and support.

bc

Chapter 2 of this thesis was supported under Office of Naval Research Grant 00014-

09-1-0263 (Program Manager Ron Joslin). The work presented in chapter 3 and

chapter 4 was supported by the Office of Naval Research through MURI grants

N00014-12-1-0875 and N00014-12-1-0962 (Program Manager Dr. Ki-Han Kim).

This dissertation carries T-3306T in the records of the Department of Mechanical and

Aerospace Engineering.

v

To my family.

vi

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Introduction 1

1.1 Motivation and Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Wall-bounded turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Organized motions . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 Effects of Taylor’s hypothesis . . . . . . . . . . . . . . . . . . 14

1.3 Turbulent drag reduction . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.1 Background of drag reduction methods . . . . . . . . . . . . . 18

1.3.2 Superhydrophobic and liquid-infused surfaces . . . . . . . . . 21

1.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Structure and scaling of wall-bounded turbulence 25

2.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.1 Effects of roughness . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.2 The k−1x and k−5/3x dependencies . . . . . . . . . . . . . . . . . 32

vii

2.2.3 Convection velocity effects . . . . . . . . . . . . . . . . . . . . 37

2.2.4 Scaling of the spectral peaks . . . . . . . . . . . . . . . . . . . 46

2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 Laminar and turbulent flow over liquid-coated surfaces 53

3.1 Surface preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Laminar rheometer flow over a liquid-coated surface . . . . . . . . . . 56

3.3 Flow over a liquid-coated hydrofoil . . . . . . . . . . . . . . . . . . . 62

3.4 Turbulent flow inside a liquid-coated pipe . . . . . . . . . . . . . . . . 64

3.5 Turbulent Taylor-Couette flow over a liquid-coated surface . . . . . . 66

3.5.1 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . 66

3.5.2 Characterization of the oil film . . . . . . . . . . . . . . . . . 69

3.5.3 Skin friction and critical Reynolds number . . . . . . . . . . . 70

3.6 Linear stability of stratified flow . . . . . . . . . . . . . . . . . . . . . 72

3.6.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . 72

3.6.2 Numerical methods and validation . . . . . . . . . . . . . . . 76

3.6.3 Effect of interfacial tension and thickness ratio . . . . . . . . . 78

3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4 Turbulent drag reduction of air- and liquid-impregnated surfaces 84

4.1 Turbulent Taylor-Couette flow . . . . . . . . . . . . . . . . . . . . . . 85

4.1.1 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . 85

4.1.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 89

4.2 High Reynolds number boundary layer measurements . . . . . . . . . 92

4.2.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.2.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . 94

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

viii

5 Conclusion 97

5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Bibliography 102

ix

List of Tables

2.1 Experimental conditions. Cases 1 to 8 were measured in the smooth

pipe, and Cases 9 and 10 were measured in the rough pipe. Here, pg

is the pipe gauge pressure, and y0 is the measurement location nearest

to the wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Numerical results for the complex wave speed c = cR + icI compared

to Yiantsios & Higgins (1988) [YH88]. Here, m = 5, n = 1, r = 1, and

αS = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.1 Perfluorinated oil properties at 18.0◦ C. . . . . . . . . . . . . . . . . . 87

4.2 Boundary layer properties. . . . . . . . . . . . . . . . . . . . . . . . . 94

x

List of Figures

1.1 (a) A model for the streamwise energy spectrum, φuu, where the

relevant length and velocity scales correspond to distinct regions in

wavenumber, kx. The bottom two figures are theoretical representa-

tions of the spectral behavior in (b) outer scaling and (c) inner scaling.

Figures are adapted from Perry et al. (1986). . . . . . . . . . . . . . . 5

1.2 The logarithmic variation of the mean velocity (#) and streamwise

Reynolds stress (2) with y. The white area represents the turbulent

wall region. Figure adapted from Hultmark et al. (2012b). . . . . . . 7

1.3 Hydrogen bubble visualization of coherent streamwise streaks at three

wall-normal positions in the near-wall region. From top to bottom

y+ = 2.7, y+ = 9.6, and y+ = 82. Figure adapted from Kline et al.

(1967). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 (a) Schematic view of a hairpin vortex, adapted from Theodorsen

(1952). (b) Smoke visualization at low Reynolds number in a plane

parallel (left) and perpendicular (right) to the hairpin vortex, show-

ing its characteristic vortical signature. Figure adapted from Head &

Bandyopadhyay (1981). . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Premultiplied power spectrum in pipe flow at y/R = 0.08, indicating

the VLSM peak (A) and the LSM peak (B). Figure adapted from Kim

& Adrian (1999). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

xi

1.6 Streamwise velocity fluctuations in pipe flow at R+ = 3472 measured

using a rake of hot-wire probes at a constant wall normal location

(y/R = 0.15). Meandering structures up to 20R in length can be

observed. Figure adapted from Monty et al. (2007) . . . . . . . . . . 12

1.7 Growth and alignment of hairpin vortices (yellow) into packets, which

induce a region of low streamwise momentum (blue). Figure adapted

from Adrian et al. (2000). . . . . . . . . . . . . . . . . . . . . . . . . 13

1.8 Instantaneous streamwise velocity fluctuations generated from a spa-

tial field (left) and a Taylor field (right). The dashed box highlights

differences in the largest scales. Figure adapted from Dennis & Nickels

(2008). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.9 Distortions of premultiplied spectrum due to Taylor’s hypothesis. The

dashed line corresponds to the true spectrum calculated from a spa-

tial field. The solid line corresponds to the spectrum calculated from

temporal data using Taylor’s hypothesis. The symbols correspond to

the data of Perry & Abell (1975). Figure adapted from del Álamo &

Jiménez (2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.10 (a) Flow over a superhydrophobic surface (b) Water droplet on a su-

perhydrophobic aluminum surface. . . . . . . . . . . . . . . . . . . . 21

1.11 (a) Flow over a liquid-coated surface (b) Flow over a liquid-

impregnated surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

xii

2.1 From top to bottom: Streamwise turbulence intensity distribution for

smooth pipe (filled symbols) and rough pipe (open symbols). Energy

spectra, λxTφuu(λxT ), for smooth pipe. Energy spectra for rough pipe.

Difference between the smooth and rough pipe spectra. (a) Smooth

pipe at R+ = 37, 690 and transitionally rough pipe at R+ = 36, 676.

(b) Smooth pipe at R+ = 98, 190 and fully rough pipe at R+ =

100, 530. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Streamwise velocity spectra at y/R = 0.001, 0.005, 0.01, 0.05, 0.1, 0.5,

1.0. (a) R+ = 1, 985; (b) R+ = 37, 690 . . . . . . . . . . . . . . . . . 31

2.3 (a) Kolmogorov-scaled spectra at pipe centerline for all Reynolds num-

bers (cases 1-8). (b) Kolmogorov-scaled spectra at R+ = 37, 690 for

y/R =0.05, 0.07, 0.1, 0.2, 0.5, 0.7, 1.0. . . . . . . . . . . . . . . . . . 33

2.4 Premultiplied energy spectrum in the turbulent wall region. Top: outer

coordinates. Bottom: inner coordinates. (a) R+ = 1, 985 at y/R =

0.061, 0.072, 0.085, 0.1 [y+ = 120, 140, 160, 190]. (b) R+ = 3, 334

at y/R = 0.037,0.044, 0.052, 0.061, 0.072, 0.085, 0.1 [y+ = 120, 140,

170, 200, 240, 280, 330]. (c) R+ = 20, 250 at y/R = 0.022, 0.026,

0.031, 0.037, 0.044, 0.052, 0.061, 0.072, 0.1 [y+ = 450, 540, 630, 750,

890, 1050, 1240, 1460, 2030]. (d) R+ = 98, 190 at y/R = 0.011, 0.013,

0.019, 0.026, 0.037, 0.052, 0.072, 0.1 [y+ = 1090, 1300, 1850, 2600,

3640, 5090, 7090, 9840]. The dashed line in (c) and (d) represents

kxφuu/u2τ = −β log[kxy/(2πα2)] with α = 2 and β = 0.2 (del Álamo

et al., 2004a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5 Large-scale structures in wall-bounded turbulence. Flow is left to right. 37

xiii

2.6 Solid: Wavenumber energy spectra obtained using a wavenumber de-

pendent convection velocity. Dashed: Wavenumber energy spectra ob-

tained using the local mean velocity as the convection velocity. y/R =

0.037, 0.044, 0.052, 0.061, 0.072, 0.085, 0.10. (y+ = 520, 620, 730, 860,

1020, 1200, 1410). Top: outer scaling. Bottom: inner scaling. . . . . . 41

2.7 TOP: Contours of two-dimensional premultiplied spectrum, kxTkzΦ/u2τ ,

at y/R = 0.1 and ReD = 1.5 × 105. Dashed line denotes λxT ∝ λ1/2z .

BOTTOM: one-dimensional premultiplied spectrum at y/R = 0.1

and ReD = 1.5 × 105. Square symbols are obtained by integrating

the two-dimensional spectrum over all λz. The line represents the

single-point temporal spectrum converted to the spatial domain using

Taylor’s hypothesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.8 Pipe flow data at y/R = 0.1 and R+ = 3.5 × 103 from Bailey &

Smits (2010) (a) The convection velocity distribution as a function of

streamwise and spanwise wavelength. White represents Ul and black

represents 〈U〉. (b) The shaded area represents the two-dimensional

premultiplied spectrum, shown in figure 2.7, above a threshold of 10%

of the maximum value. The dashed lines represent contours of the

mapping factor J(λxT , λz) from 0.85 to 1 in increments of 0.3. The

solid lines represent isolines of constant λx. . . . . . . . . . . . . . . 44

2.9 Pipe flow data at y/R = 0.1 and R+ = 3.5× 103 from Bailey & Smits

(2010). The dashed line represents the temporal spectrum converted to

the spatial domain using Taylor’s hypothesis. The solid line represents

the true spatial spectrum obtained using the correction of del Álamo

& Jiménez (2009). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

xiv

2.10 Peak locations in inner coordinates. Filled symbols: y/R < 0.1, Open

symbols: y/R > 0.1. (�) R+ = 1, 985,(�) R+ = 3, 334, (4) R+ =

5, 412, (◦) R+ = 10, 481, (O) Re+ = 20, 250, (/) R+ = 37, 690, (.)

R+ = 68, 371, (?) R+ = 98, 190. . . . . . . . . . . . . . . . . . . . . . 46

2.11 Peak locations in outer coordinates. Filled symbols : y+ > 67, Open

symbols : y+ < 67. (�) R+ = 1, 985,(�) R+ = 3, 334, (4) Re+ = 5, 412,

(◦) Re+ = 10, 481, (O) R+ = 20, 250, (/) R+ = 37, 690, (.) R+ =

68, 627, (?) R+ = 98, 190. The red points denote the lower wavenumber

peaks assuming the convection velocity is spatially uniform and equal

to the bulk velocity. The shaded region, obtained from the survey

of Wu et al. (2012b), represents the approximate scatter of previous

datasets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.12 Contour maps showing variation of streamwise pre-multiplied spec-

tra with wall-normal distance. The dashed lines trace the locus of

peaks shown in figures 2.10 and 2.11. The white cross represents

y+ = 0.23R+2/3 (Hultmark et al., 2012a), the black cross represents

y+ = 3.9R+1/2 (Mathis et al., 2009), and the blue cross represents

y+ = 400 (Vallikivi, 2014). (a) R+ = 1, 985 (b) R+ = 3, 334 (c)

R+ = 5, 412 (d) R+ = 20, 250 (e) R+ = 37, 690 (f) R+ = 98, 190. . . . 50

3.1 (a) Schematic of a liquid-coated surface. (b) Scanning electron micro-

scope image of the boehmite nanostructure. . . . . . . . . . . . . . . 54

3.2 Rheometer geometry for a liquid-coated surface in a parallel-plate flow.

Axisymmetry is assumed about the z-axis, and the length scales are

distorted to make relevant dimensions more visible. . . . . . . . . . . 56

3.3 Film thickness on a liquid-coated substrate (inferred by weight) com-

pared to spin-coating theory. Here, µo ≈ 150 × 10−3 Pa · s and the

sample is spun at 500 RPM for a varying length of time. . . . . . . . 57

xv

3.4 Drag reduction over an oil-coated surface as a function of oil film thick-

ness, h, for µw/µo = 1/150 (a) Parallel-plate flow with water as the

test liquid (b) Taylor-Couette flow with 75:25 wt% glycerol:water as

the test liquid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5 Rheometer geometry for a parallel-plate configuration where the inter-

face is deformed by centrifugal pressure gradient, adapted from Jacobi

et al. (2015). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.6 The effect of viscosity ratio (µw/µo) on the drag reduction measured in

a parallel-plate rheometer. Here, µo ≈ 30 Pa ·s and the viscosity of the

aqueous phase is adjusted by using varying concentrations of glycerol

to water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.7 (a) NACA0018 hydrofoil in a high-speed water channel. The flow is

left to right and the velocity is measured using a Pitot-static tube (b)

Apparatus for measuring drag force. . . . . . . . . . . . . . . . . . . . 62

3.8 Drag coefficient of liquid-coated hydrofoil with µw/µo ≈ 1/150 com-

pared to an untreated airfoil (control). The drag contribution from the

struts is included in the measurement. . . . . . . . . . . . . . . . . . 64

3.9 Moody diagram for a liquid-coated pipe with µw/µo ≈ 1/150. . . . . . 65

3.10 (a)Taylor Couette apparatus and relevant geometric parameters (b)

Reflected light spectrum off a smooth aluminum surface (control) and

the same surface coated with a 6 µm oil film (liquid-coated). . . . . . 67

3.11 (a) Microstructure of the polished aluminum test surface (b) Scanning

electron microscope image of the boehmite nanostructure. . . . . . . 68

xvi

3.12 (a) Azimuthal distribution of the oil film at different heights, x/H =

0.25 ( ), x/H = 0.5 (�), and x/H = 0.75 (N) for h ≈ 19µm and

h ≈ 5µm obtained using reflectometry. −− corresponds to the mean

film thickness inferred from a weight measurement (b) Drainage dy-

namics at x/H = 0.5 measured using reflectometry and compared with

theory (Jeffreys, 1930). . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.13 (a) Drag reduction in turbulent Taylor-Couete flow at Re = 5000 for

varying film thickness and viscosity ratio (b) Critical Reynolds number

for varying film thickness and µw/µo = 1/30. The inset is a typical

profile of torque with increasing Reynolds number . . . . . . . . . . . 71

3.14 Two-fluid stratified Poiseuille flow. . . . . . . . . . . . . . . . . . . . 73

3.15 50 most unstable eigenvalues of the two-fluid Orr-Sommerfeld equa-

tions for m = 1, n = 1, r = 1, and S = 0. (a) α = 1.02 and Re = 2000

(stable) (b) α = 1.02 and Re = 5772 (at criticality). . . . . . . . . . . 78

3.16 Neutral stability curves for m = 1, n = 1, r = 1, and S = 0. The

dashed lines denote the critical Re and α for single fluid Poiseuille flow

(Orszag, 1971). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.17 Neutral stability curves for m = 1, n = 1, r = 1, and varying S. (a)

S ∈ [0, 0.4](b) S ∈ [0.5, 0.6] . . . . . . . . . . . . . . . . . . . . . . . . 80

3.18 Effect of interfacial tension, S, and thickness ratio, n, on the critical

Reynolds number Recrit. . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.19 Growth rates of the viscous (interfacial) instability for a linear Couette

flow, based on the parallel-plate rheometer measurements described in

section 3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

xvii

4.1 (a) Schematic of the Taylor-Couette apparatus and relevant geomet-

ric parameters. (b) Surface topography of the 106 µm pitch threaded

cylinder obtained using confocal microscopy near x/H = 0.5 (c) Sur-

face profiles near x/H = 0.25 (−), x/H = 0.5 (−−), and x/H = 0.75

(...). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2 Water with a trace amount of black dye demonstrates the wetting be-

havior of the textured surface, where the darker band in the center of

the image shows the wetting produced by a single drop of water. (a)

Hydrophilic after surface cleaning; (b) Superhydrophobic after fluori-

nation treatment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3 Skin friction variation for the control case. # measured torque data;

data corrected for end effects. ——- laminar flow, equation 4.2; - -

- - - turbulent flow, data from Taylor (1936). . . . . . . . . . . . . . . 89

4.4 Drag reduction comparison between superhydrophobic and two liquid-

infused cases. � air-filled superhydrophobic surface (µw/µ0 ≈ 50)

; liquid-infused surface (µw/µ0 ≈ 1/1.5); N liquid-infused surface

(µw/µ0 ≈ 1/30). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.5 Experimental setup, adapted from Schultz & Flack (2007). . . . . . . 92

4.6 Liquid-infused surface comprised of ∼ 150µm transverse grooves. Flow

is left to right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.7 Reynolds shear stress profiles for the liquid-infused and control surfaces. 95

5.1 Micro-fabricated surface. The feature size is about 25× 100 µm. Flow

is bottom-left to top-right . . . . . . . . . . . . . . . . . . . . . . . . 100

xviii

Chapter 1

Introduction

1.1 Motivation and Goals

Turbulent flows are ubiquitous in nature and industry, for instance the plume of a

volcano or the wake of a ship. Beautiful and complex, these flows contain a disparate

range of eddying motions that exhibit unsteady, three-dimensional behavior as they

interact with each other. Understanding and modeling the behavior of turbulent

flows is relevant to a host of environmental and industrial applications. For example,

the whirling and mixing motion in the planetary boundary layer is responsible for

exchanging oxygen and carbon dioxide between the Amazon rainforest and New York

City, which is necessary for both locations to thrive.

The frictional force that a flow exerts on a solid surface, for example the drag on a

submarine, has tremendous implications in industry, especially in the transportation

sector. If the flow has sufficiently high momentum or sufficiently low viscosity, the

fluid motion near the wall can become turbulent, which drastically increases the

friction on the surface. This effect (often unfavorable unless the goal is to slow down)

is due to the conversion of mechanical energy, for example the propulsive energy of a

submarine, into heat by the transfer of energy from large turbulent length scales to

1

smaller and smaller scales, where it is eventually dissipated by viscous action. Near-

wall turbulence also has benefits, of course, such as enhanced mixing of mass and

heat, for example in a heat exchanger.

Frictional drag accounts for about 50% of commercial aircraft and surface ship

drag, 70% of drag for most underwater bodies, and nearly all the pumping power for

long distance pipelines (Bushnell, 1983). Reducing friction, in addition to enabling

longer range and increased speed for a vehicle, would have a significant economic

impact. For example, a 20% reduction in fuselage skin friction drag for a civilian

aircraft fleet translates to fuel savings of about one billion dollars per year (Bushnell,

1983).

Wall-bounded turbulent flows have been studied for more than a century, yet there

is still a limited understanding of the underlying physics, which makes modeling

difficult. While the subject has been rigorously studied in terms of statistics and

other mathematical tools of stochastic processes, perhaps the most insight into the

fundamental behavior of wall-bounded turbulence has been gained by visualizing and

studying coherent organized structures in the flow. In this regard, understanding

the structure of the flow as well as the associated length scales, time scales, and

mechanisms might guide methods to control the turbulence in a way that lowers

friction.

To that end, this thesis focuses on two main themes: (1) the structure of wall-

bounded turbulent flows and (2) novel passive ways to reduce turbulent friction over

a surface.

2

1.2 Wall-bounded turbulence

1.2.1 Scaling

For canonical wall-bounded turbulent flows (fully developed pipes and channels and

zero pressure gradient boundary layers), there exist distinct regions in space where

the flow is governed by different physical effects. The region closest to the wall is

dominated by viscosity whereas the region further from the wall is dictated by inertia

of the turbulent motions.

In this view, the relevant length scale in the near-wall region is the viscous length,

δv ≡ ν/uτ , and the relevant velocity scale is the friction velocity uτ =√

(τw/ρ), where

τw is the wall shear stress and ρ and ν are the density and kinematic viscosity of the

fluid, respectively. The relevant length scale in the outer region is the macroscopic

scale of the shear layer R, that is, the radius of the pipe, the half-width of the channel,

or the boundary layer thickness. The velocity scale in this region is also the friction

velocity since the motions are affected by the level of shear present at the wall.

For such canonical flows, the ratio of the outer and viscous (inner) length scales

R+ = Ruτ/ν, called the friction Reynolds number or sometimes the Kármán number,

characterizes the flow and physically represents the ratio of the largest length scales

to the smallest length scales in the flow. If there is a large disparity between these

two length scales, in other words if the Reynolds number is high, there exists a third

intermediate region in space, ν/uτ � y � R, where both scalings apply called the

turbulent wall region. Applying overlap arguments to the mean velocity profile yields

a logarithmic dependence with wall-normal distance y in this region, which is why

it is typically referred to as the logarithmic layer. Indeed, the mean velocity profile

collapses near the wall when scaled with inner variables, collapses far from the wall

when scaled with outer variables, and follows a logarithmic dependence in the overlap

region (Zagarola & Smits, 1998).

3

The near-wall region, which for pipe flow spans from the wall to y+ ≈ 800, can be

further divided into the viscous sublayer, located below y+ ≈ 5, and the buffer layer,

which spans up to y+ ≈ 30− 50. Here, the ‘+’ convention represents scaling with the

inner variables, that is, u+ = u/uτ and y+ = yuτ/ν. For pipe flow, the logarithmic

region spans from y+ ≈ 800 to y/R ≈ 0.12 (Hultmark et al., 2012b), and the outer

region extends beyond y/R ≈ 0.12.

The nature of wall-bounded turbulence in the asymptotic limit of infinite Reynolds

number remains an important question, specifically because its understanding is crit-

ical to predicting many engineering and environmental flows. A particular question

is the behavior of the turbulence spectrum at high Reynolds numbers. For canoni-

cal wall-bounded turbulent flows (pipes, channels and boundary layers), scaling laws

can be derived for the energy spectrum in the turbulent wall region by using dimen-

sional analysis in conjunction with overlap arguments. In the turbulent wall region,

the turbulence spectrum is classically divided into a low wavenumber range where

motions scale with R, an intermediate wavenumber range where motions scale with

wall-normal distance y, and a high wavenumber range where motions scale with the

Kolmogorov length η (Perry et al., 1986), shown in figure 1.1(a).

Consider the streamwise velocity fluctuations, u, taken to be with respect to the

local mean velocity U . At a sufficiently high Reynolds number, an overlap of the

R-scaled and y-scaled regions is expected to form where

φuu(kxR)

u2τ=

A1kxR

, (1.1)

or

φuu(kxy)

u2τ=

A1kxy

, (1.2)

referred to as the k−1x law (Perry et al., 1986), shown schematically in figure 1.1(b-c).

Here, kx is the streamwise wavenumber, which is related to a streamwise wavelength

4

(‘outer’ flow scaling)

(‘inner’ flow scaling)

(Kolmogorov scaling)

Overlap region I

€

φuu kxR( )uτ2 = g1 kxR( )

€

φuu kx y( )uτ2 = g2 kxy( )

€

φuu kxη( )ϑ 2

= g3 kxη( )

€

kxη = M

€

kxy = N

€

kxy = P

€

kxR = F

Overlap region II

€

kx

Motions that are independent of viscosity Viscosity dependent

motions

(a)

€

φuu kxR( )uτ2

€

kxRφuu kxR( )uτ2

€

kxR

€

kxR

(b)

€

kxyφuu kx y( )uτ2

€

φuu kx y( )uτ2

€

kxy

€

kxy

(c)

Figure 1.1: (a) A model for the streamwise energy spectrum, φuu, where the relevantlength and velocity scales correspond to distinct regions in wavenumber, kx. Thebottom two figures are theoretical representations of the spectral behavior in (b)outer scaling and (c) inner scaling. Figures are adapted from Perry et al. (1986).

5

λx by kx = 2π/λx, φuu is the power spectral density of the streamwise velocity

fluctuations, and A1 is a universal constant.

In the overlap of the y-scaled and η-scaled regions, energy is transferred by inertial

mechanisms, and so φuu is a function only of kx and the rate of energy transfer given

by the dissipation rate �. If it is further assumed that in the turbulent wall region the

local dissipation rate is equal to the local production rate P (that is � = P = u3τ/κy),

then

φuu(kxy)

u2τ=

K0κ2/3(kxy)5/3

, (1.3)

referred to as the k−5/3x law or the inertial subrange (Perry et al., 1986). Here, K0 is

the Kolmogorov constant (≈ 0.5), and κ is the von Kármán constant (≈ 0.40).

By integrating this theoretical spectrum over all wavenumbers, Perry et al. (1986)

derived a logarithmic variation in the streamwise Reynolds stress, u2, given by

u2

u2τ= B1 − A1ln

y

R− V (y+), (1.4)

where the overbar denotes a temporal or spatial average at a particular wall distance,

B1 is a constant that is characteristic of the macroscopic flow geometry and V (y+) is

a Reynolds number dependent viscous correction term that decreases with y+, where

y+ = yuτ/ν. The term V (y+) accounts for the energy contained in the dissipation

region of the spectrum at finite Reynolds numbers. The logarithmic term comes

from the integration of the k−1x region from kx = a/R to kx = b/y, where a and b

are universal constants. While the k−5/3x dependence has been widely observed in

experiments, the k−1x dependence has only been observed at high Reynolds number

over a limited spatial extent in the laboratory (Nickels et al., 2005), although it is

commonly observed in the atmospheric surface layer (see, for example, Katul et al.,

1995). The behavior of this k−1x region is an important question because it plays a

significant role in many turbulence models, particularly Townsend’s attached-eddy

6

10

15

20

25

30

35

101 102 103 104 105

1

2

3

4

5

6

7

8

10–3 10–2 10–1 100

0

9

Figure 1.2: The logarithmic variation of the mean velocity (#) and streamwiseReynolds stress (2) with y. The white area represents the turbulent wall region.Figure adapted from Hultmark et al. (2012b).

framework (Townsend, 1976; Perry & Chong, 1982; Perry & Li, 1990; Marusic et al.,

1997; Marusic & Kunkel, 2003).

While a logarithmic variation in the mean velocity profile has been extensively

measured over the past several decades, a logarithmic variation for the streamwise

turbulent fluctuations (equation 1.4) was first measured only recently by Hultmark

et al. (2012b) in pipe flow, who showed that for R+ > 10000 the mean and streamwise

streamwise turbulence exhibit this functional relationship in the same region in space

(see figure 1.2). Marusic et al. (2013) showed that this logarithmic dependence was

also found in boundary layers at a sufficiently high R+, with the same slope found

for pipe flows.

7

Figure 1.3: Hydrogen bubble visualization of coherent streamwise streaks at threewall-normal positions in the near-wall region. From top to bottom y+ = 2.7, y+ = 9.6,and y+ = 82. Figure adapted from Kline et al. (1967).

1.2.2 Organized motions

Turbulent wall-bounded flows consist of a medley of organized coherent motions, and

their shape, length/time scales, and mechanism can be described based on where they

are in relation to the wall. The four distinct organized motions are the near-wall quasi-

streamwise vortices, hairpin vortices, large-scale motions (LSM), and very-large-scale

motions (VLSM) (Smits et al., 2011a).

8

Near-wall region

The turbulent motion near the wall is affected strongly by viscosity and therefore the

inner variables, δv and uτ , characterize the length and time scales of the processes in

this region.

Using hydrogen bubble visualization, Kline et al. (1967) observed long coherent

streaks near the wall, shown in figure 1.3. The streaks are the signatures of quasi-

stationary longitudinal counter-rotating vortices with typical radii of 30δv and average

spanwise spacings of 100δv, which form alternating low-speed streaks (where fluid is

drawn upward from the wall) and high-speed streaks (where high-momentum fluid is

entrained from above).

With increasing wall distance, the low-speed streaks undergo instability and even-

tually break up. This process can be described in three stages: (1) low-speed streak

lift-off for 0 < y+ < 10, (2) streak oscillation and growth for (8 < y+ < 12), and (3)

streak eruption for 10 < y+ < 30 (Kim et al., 1971). The violent and intermittent

bursting process, and the resulting inrush of high momentum fluid from above (called

a sweep), accounts for most of the turbulence production in the shear layer (Black-

welder & Haritonidis, 1983).

Further from the wall, characteristic horseshoe-shaped vortex filaments are found,

inclined 45◦ in the downstream direction, called hairpin vortices, first observed by

Theodorsen (1952) (figure 1.4(a)). Hairpin vortices exhibit a minimum height and

transverse dimension of about 100δv (Head & Bandyopadhyay, 1981), similar to that

of the near-wall streaks, suggesting that the two structures might be interconnected.

In figure 1.4(b), smoke visualization in a 45◦ inclined plane reveals the loop-like

structure of the hairpin vortex, while a plane perpendicular to it reveals the the

vortical motion of the hairpin vortex which circulates fluid about the loop through

the legs (Head & Bandyopadhyay, 1981).

9

(a)

(b)

Figure 1.4: (a) Schematic view of a hairpin vortex, adapted from Theodorsen (1952).(b) Smoke visualization at low Reynolds number in a plane parallel (left) and per-pendicular (right) to the hairpin vortex, showing its characteristic vortical signature.Figure adapted from Head & Bandyopadhyay (1981).

Outer region

In the outer region, hairpin vortices are often observed to be aligned into packets,

called large-scale motions (LSM), which are approximately 2− 3R long and 1− 1.5R

wide. These hairpin packets are also identified as turbulent bulges at the edge of

boundary layers (Robinson, 1991), and they are characterized by a region of low-

momentum fluid induced by the hairpin legs and contribute significantly Reynolds

shear stress (Adrian et al., 2000).

The nature of the turbulence spectrum has also revealed the importance of co-

herent motions with streamwise length scales much longer than the characteristic

shear layer thickness, the very-large-scale motions (VLSM). Kim & Adrian (1999)

first inferred their presence in pipe flow by observing a peak in the streamwise veloc-

ity spectrum at very long wavelengths, shown in figure 1.5. These long, meandering

10

Figure 1.5: Premultiplied power spectrum in pipe flow at y/R = 0.08, indicating theVLSM peak (A) and the LSM peak (B). Figure adapted from Kim & Adrian (1999).

features consist of regions of low streamwise momentum fluid, flanked by regions of

higher momentum fluid (figure 1.6), and have been shown to contain 40 − 65% of

the total turbulent kinetic energy and 30 − 50% of the Reynolds shear stress (Kim

& Adrian, 1999; Guala et al., 2006; Hellström & Smits, 2014). They also become

increasingly energetic at higher Reynolds number (Balakumar & Adrian, 2007), and

so understanding their behavior is critical to predicting high Reynolds number flows.

The origin of the VLSM remains unclear. For example, Kim & Adrian (1999) sug-

gested that the VLSM are formed by the quasi-streamwise alignment of the LSM. In

contrast, del Álamo et al. (2006) proposed that the VLSM could be formed by linear

or nonlinear processes. More recently, Hellström & Smits (2014) used proper orthog-

onal decomposition (POD) in turbulent pipe flow and found that the most energetic

modes have frequencies associated with structures that are 10− 20D long, indicative

of the VLSM. However, POD analysis in the cross plane and a streamwise-radial plane

revealed that the most energetic modes are only 2 − 3R long, representative of the

LSM (Hellström, 2015). This observation led Hellström (2015) to hypothesize that

11

2

(a)

(b)

1

02

00

5

10

15

20

25

30

–2

3

2

1

0

–1

–2

–3

uU�

zR

sR

2

0

0 5 10 15 20 25 30–2

x/R

y/R

x–R

Figure 1.6: Streamwise velocity fluctuations in pipe flow at R+ = 3472 measuredusing a rake of hot-wire probes at a constant wall normal location (y/R = 0.15).Meandering structures up to 20R in length can be observed. Figure adapted fromMonty et al. (2007)

the large-scale structure in pipes consists only of LSM and that the VLSM signature,

observed as peaks in the energy spectrum or visually as long meandering structures,

is associated with the time scale of their formation.

Additionally, it has been suggested recently that the pipe, channel and boundary

layer flows may behave quite differently with regard to these motions (Monty et al.,

2009; Smits et al., 2011a). Pipe and channel flow measurements have revealed VLSM

of typical length 20R (Monty et al., 2007), while boundary layer measurements have

shown superstructures (as they are called in boundary layers) of typical length 6δ

(Hutchins & Marusic, 2007).

Whereas the VLSM persist well into the outer layer of pipe flow (Bailey & Smits,

2010), the superstructures in boundary layers appear confined to the logarithmic

region (Monty et al., 2009). Furthermore, boundary layer experiments (Hutchins &

12

Figure 1.7: Growth and alignment of hairpin vortices (yellow) into packets, whichinduce a region of low streamwise momentum (blue). Figure adapted from Adrianet al. (2000).

Marusic, 2007) and DNS channel-flow simulations (del Álamo & Jiménez, 2003) reveal

a footprint of these outer-scaled structures within the inner-scaled near-wall region,

and their interaction may be described in terms of an amplitude modulation of the

small scales (Marusic et al., 2010). Recent pipe-flow experiments by Hultmark et

al. (2010, 2012a), however, show that the near-wall peak in streamwise turbulence

intensity is Reynolds number independent, or only very weakly dependent, suggesting

that in pipe flows this particular outer-inner layer interaction may not be as strong,

or absent altogether.

Logarithmic region

In the logarithmic region, where ν/uτ � y � R, the only relevant length scale is

distance from the wall y. This scaling led Townsend (1976) to propose that the

turbulent motion here consists of self-similar motions that have streamwise lengths

proportional to their wall-distance (heights), and that they must be attached to the

wall since their properties scale with their distance from it. This idea is supported by

13

the current view of the LSM evolution, illustrated in figure 1.7. In this picture, the

LSM form at the wall, grow in the streamwise direction at a mean angle of about 12◦

to form a ramp-like structure, and then finally detach from the wall, which initiates

the formation of a new LSM (Adrian, 2007; Hellström, 2015).

1.2.3 Effects of Taylor’s hypothesis

Taylor’s hypothesis of frozen eddies convecting with the local mean flow remains a

powerful approximation in turbulence research, particularly in experiments where

measurements of the complete wavenumber spectrum are not feasible (Taylor, 1938;

Smits et al., 2011a). Typically, turbulence measurements are resolved in time at a

fixed location in the laboratory frame, for example by using a hot-wire anemometer,

and so Taylor’s approximation allows temporal structure to be interpreted as spatial

structure, provided the eddies do not evolve significantly in the time it takes them

to pass over the sensing element. The convection velocity of the eddies, c, serves as

the bridge between the two domains, whereby an inferred wavenumber kx is related

to a measured frequency ω by kx = ω/c. Typically, c is taken to be the local mean

velocity for all scales of motion.

In wall-bounded flows, however, it is clear that eddies of different scale will move

at different convection velocities (Spina et al., 1991; Cogne et al., 1993; Goldstein &

Smits, 1994; Delo et al., 2004). The smallest eddies, for example, are likely to move at

the local mean velocity, though discrepancies have been noted near the wall (y+ . 20),

where the convection velocities are increasingly larger than the mean, and so small

eddies near the wall have a range of c (Geng et al., 2015). The large-scale structures,

on the other hand, will maintain spatial organization across a significant wall-normal

extent as they convect downstream, and over this extent the mean velocity can vary

appreciably. The choice of convection velocity for these motions therefore becomes

ambiguous. Moreover, using the local mean velocity as the convection velocity causes

14

the wavenumber of the same eddy to depend on the wall-normal position, which will

tend to distort the inferred wavenumber spectrum.

The validity of Taylor’s hypothesis in turbulent shear flows has long been in ques-

tion. For example, Lin (1953) used the Navier-Stokes equations to show that Taylor’s

approximation is accurate only if the turbulence intensity is low, viscous forces are

negligible, and the mean shear is weak. Wills (1964) was perhaps the first to experi-

mentally demonstrate the wavenumber dependence of the convection velocity, doing

so in a turbulent jet via two-point space-time velocity correlations. Perry et al. (1986)

theoretically assessed the error in using Taylor’s approximation by simulating a wall-

bounded flow using the attached eddy framework, which models the turbulence as

a hierarchy of representative coherent structures (Townsend, 1976; Perry & Chong,

1982). They computed the “true” wavenumber spectrum by spatially Fourier decom-

posing the velocity fields of stationary hierarchies and compared it with the “false”

wavenumber spectrum generated using Taylor’s approximation where each hierarchy

was allowed to convect at the mean velocity of its geometric center. The effect of Tay-

lor’s approximation on the spectrum was to artificially shift energy from the largest

scales to the smaller scales.

More recently, Dennis & Nickels (2008) used time-resolved particle image velocime-

try in a turbulent boundary layer to compare instantaneous spatial velocity fields with

pseudo-spatial velocity fields generated using Taylor’s approximation. The main fea-

tures between the two velocity fields showed remarkable similarity, although subtle

differences were evident in the largest motions resolved in their field of view, about 6

boundary thicknesses in extent, highlighted by the dashed box in figure 1.8. It should

be noted that their measurements were limited to the edge of the logarithmic region,

and we would expect differences to be more pronounced closer to the wall where there

is a larger disparity between the local mean velocity and the convection velocities of

the range of structures that contribute to the wall motions.

15

–0.20 –0.15 –0.10 –0.05 0 0.05 0.10 0.15 0.20

6

xδ

Spatial field: σ = 0.06δ

5

4

3

2

1

0 1 2z/δ

3

6

xδ

Taylor field: σ = 0.06δ

5

4

3

2

1

0 1 2z/δ

3

Figure 1.8: Instantaneous streamwise velocity fluctuations generated from a spatialfield (left) and a Taylor field (right). The dashed box highlights differences in thelargest scales. Figure adapted from Dennis & Nickels (2008).

In a comparison between spatial spectra computed from the channel flow simula-

tion of del Álamo et al. (2004b) and spatial spectra obtained by converting frequency

spectra to the spatial domain using Taylor’s approximation, Monty & Chong (2009)

observed slight differences in the near-wall and logarithmic regions at the largest

scales. Most notable was a reduction of energy in the simulated spectra around the

low wavenumber peak, which is thought to be associated with the very-large-scale

motions (VLSM) (Adrian, 2007; Monty et al., 2007).

This observation led them to propose a formulation for a wavenumber dependent

convection velocity which yielded better agreement between the experiments and sim-

ulations. A similar “shouldering” in the VLSM range was observed by del Álamo &

Jiménez (2009), who used a direct numerical simulation (DNS) to compare actual

spatial spectra and pseudo-spatial spectra inferred from temporal DNS data assum-

ing the local mean velocity as the convection velocity, shown in figure 1.9. They

argued that Taylor’s approximation aliases large-scale energy to shorter apparent

wavelengths, thereby creating an artificial VLSM peak, which challenges the bimodal

16

0

0.5

1.0

kΦ+ uu

10–2 100 102

λx/h

Figure 1.9: Distortions of premultiplied spectrum due to Taylor’s hypothesis. Thedashed line corresponds to the true spectrum calculated from a spatial field. Thesolid line corresponds to the spectrum calculated from temporal data using Taylor’shypothesis. The symbols correspond to the data of Perry & Abell (1975). Figureadapted from del Álamo & Jiménez (2009).

nature of the spectrum typically observed in time-resolved experiments (Moin, 2009).

They also showed that while the small scales away from the wall convect at the local

mean velocity, the largest scales convect at a more uniform velocity that is propor-

tional to the bulk velocity. Similar disparities between true and Taylor spectra have

also been noted by Chung & McKeon (2010) and Wu et al. (2012a).

The spectral shouldering at lower wavenumber plays an important role in assessing

the k−1x scaling law, which is predicted to occur in the turbulent wall region (Perry

et al., 1986). The dependence is based on an overlap argument between the inner

and outer wavenumber regions, which scale with y and R, respectively. In the overlap

region, neither length scale is valid and so the only relevant parameters are kx and the

characteristic velocity scale for the turbulence, uτ , which leads to the k−1x relation,

discussed in section 1.2.1. This argument does not consider the variation of convection

velocity with wavenumber, which would be relevant in a time-resolved measurement

made by a stationary observer, and therefore only pertains to the relative motions of

17

the turbulence, not the absolute motions (see for example the dimensional analysis

of Perry & Abell (1975) who considered a form of the spectrum proposed by Wills

(1964) that takes into account the convection velocity of each Fourier mode). It is

possible, therefore, that a time-resolved measurement can mask a k−1x region if one

exists.

1.3 Turbulent drag reduction

1.3.1 Background of drag reduction methods

Turbulent drag reduction has been intensively studied for several decades due to its

significant industrial and economic impact. Numerous techniques have shown promise

to reduce drag in turbulent flows. For example, injecting polymers into a flow extracts

energy from the turbulence through elastic interactions (Lumley, 1969), introducing

bubbles into a water flow by injection (Ceccio, 2010) or electrolysis (McCormick &

Bhattacharyya, 1973) lowers the density and viscosity near the wall, and wall cooling

in air generates a buoyant force that stabilizes turbulence (Arya, 1975).

While these methods, and others, show promise for reducing friction in realistic

engineering flows, they are generally complex, expensive, and need to be tuned to

the flow of interest. In this section, we discuss passive techniques for turbulent drag

reduction, that is, techniques in which only the properties of the surface are changed.

Riblets

Perhaps the most convincing technology for passive drag reduction in turbulent flow

is the riblet surface, which consists of microscopic striations oriented in the stream-

wise direction. While riblet surfaces have shown promise in engineering application

(they may have allowed the United States to win an olympic medal as well as an

America’s Cup in 1987), the height and spacing of these grooves must be tuned to

18

the viscous length scale in the turbulent flow, which means that these surfaces are

limited to only a narrow Reynolds number range. Another practical limitation of

riblets is their yaw sensitivity. If, for example, they are not aligned rather closely

with the surface streamlines, they can act as roughness elements. From a biological

perspective, shark skins closely resemble riblets, which has inspired the design and

investigation of similar biomimetic surfaces (Bechert et al., 2000).

For riblet surfaces in turbulent flow, Walsh (1982) reports net drag reductions up

to approximately 8% as well as reduced rms turbulence intensity near the surface.

Experimentally, the optimal riblet surface is a sawtooth shape with a height of 8−12

viscous units and transverse spacing of 15− 20 viscous units, although spacings less

than 30 viscous units give some drag reduction.

The similarity between the optimal riblet size and the dimensions of the low-

speed streaks, approximately 30 viscous units in transverse extent with spacings of

100 viscous units (Kim et al., 1971), leads to speculation that the riblets reduce drag

by inhibiting oscillation of the streaks, thereby mitigating the bursting process. This

drag reduction mechanism has been comprehensively demonstrated through direct

numerical simulation (Chu & Karniadakis, 1993; Choi et al., 1993).

Compliant coatings

The idea of using compliant coatings for turbulent drag reduction originates from

Kramer (Kramer, 1960), who was inspired by the apparent exceptionally fast swim-

ming speed of dolphins, which is sometimes referred to as “Gray’s paradox” since the

power the animals exert to overcome drag cannot be reconciled with their muscular

power (Gray, 1936). It should be noted that the current interpretation of drag reduc-

tion in dolphins lies in their streamlined body shape and behavioral mechanisms such

porpoising and drafting (Fish, 2006). By coating a torpedo with an artificial skin in-

spired by the dolphin, Kramer measured a 50% reduction in drag. His hypothesis was

19

that the compliance of the skin, due to its viscoelastic properties, absorbs energy from

pressure oscillations and damps turbulence perturbations helping to relaminarize the

flow or delay the onset of turbulence.

Theoretical treatments of compliant coatings also support this idea. Linear sta-

bility analyses of boundary layer flow show significant stabilization of Tollmien-

Schlichting instabilities and a delay in the transition to turbulence by use of compliant

coatings (Benjamin, 1960; Carpenter & Garrad, 1985).

Further experiments, however, show conflicting results. For example, Blick &

Walters (1968) measured a substantial drag reduction using a flexible surface under

a turbulent boundary layer, but under identical conditions, McMichael et al. (1980)

measured no significant change. More recently, Choi et al. (1997) performed turbulent

flow measurements over a viscoelastic silicon rubber and found a 7% friction reduc-

tion in addition to weaker intensities of fluctuating wall-pressure. Other experimental

studies, however, report an increase in drag due to the presence of hydroelastic in-

stabilities on the surface (Hansen & Hunston, 1974; Gad-el Hak et al., 1984). With

respect to turbulence structure, Lee et al. (1993) found that the surface motion mod-

ulated the flow field increasing the spanwise spacing of the near-wall streaks and

elongating their spatial coherence. The numerical simulations of Xu et al. (2003), on

the other hand, demonstrated little change in turbulence behavior.

Semenov (1991) proposed that a successful viscoelastic coating for drag reduction

requires (1) quick attenuation of free surface vibrations, (2) wall-deformations less

than 6 viscous units, and (3) natural frequencies of the wall that have a large phase-

frequency region of favorable interaction with the turbulence. Yet, Xu et al. (2003)

report that physically realistic materials do not meet these requirements. Due to

the conflicting nature of the literature, the current status of compliant walls for drag

reduction is probably well described by Bushnell (1983) as “murky”.

20

(air%filled)+superhydrophobic+surfaces+

liquid!

air!

(a) (b)

Figure 1.10: (a) Flow over a superhydrophobic surface (b) Water droplet on a super-hydrophobic aluminum surface.

1.3.2 Superhydrophobic and liquid-infused surfaces

Modifying the texture and wetting behavior of a surface can have important conse-

quences for drag reduction. For example, superhydrophobic surfaces (figure 1.10(a)),

where pockets of air are trapped inside micro- or nano-scale features on a non-wetting

solid surface, have received much attention over the past decade as they have been

shown to reduce wall shear stress in laminar channel flows (Ou et al., 2004; Joseph

et al., 2006) and rheometer flows (Choi & Kim, 2006; Srinivasan et al., 2013) by intro-

ducing a partial slip at the air/water interface (Lauga et al., 2007). Drag reduction

using superhydrophobic surfaces has also been demonstrated in turbulent flows, nu-

merically (Martell et al., 2009; Park et al., 2013) and experimentally (Daniello et al.,

2009; Rothstein, 2010; Park et al., 2014), although the degree of success has been

variable (Aljallis et al., 2013). In such surfaces, sustaining drag reduction hinges

on the retention of air in the surface features. However, the air pockets fail when

using complex liquids such as crude oil (Wong et al., 2011), when they are under

high hydrostatic pressure or under intense turbulent pressure fluctuations (Bocquet

& Lauga, 2011; Poetes et al., 2010), and under high shear rates due to dissolution of

vapor into the working liquid (Samaha et al., 2012). Unless the impregnated vapor

21

liquid!!!

oil!

(a)

liquid!!!

oil!

Slippery!Liquid-Infused!Porous!Surface!(SLIPS)!

(b)

Figure 1.11: (a) Flow over a liquid-coated surface (b) Flow over a liquid-impregnatedsurface.

is replenished, for example by electrolytic methods (Lee & Kim, 2011), the drag re-

ducing properties will be lost or, in turbulent flow, may lead to a drag increase due

to roughness effects if the features are large enough.

An alternative to maintaining these stable air pockets is to infuse a second liquid

in the surface features. These liquid/liquid systems, which demonstrate omniphobic

properties and robustness to pressure, will be stable as long as the two liquids are

immiscible, the impregnating liquid preferentially wets the substrate compared to the

working liquid, and the interfacial tension inhibits destabilizing body forces (Wong

et al., 2011; Lafuma & Quéré, 2011; Smith et al., 2013). Recent rheometer measure-

ments over a liquid infused surface indicate a drag reduction of up to 16% in laminar

flow when the impregnating fluid is two orders of magnitude less viscous than the

working fluid, with the drag benefit weakening for more viscous lubricants (Solomon

et al., 2014).

Here, we consider two variations of a liquid-infused surface:

• Liquid-coated surface: Here, a coherent film overlies the substrate and the

texture has no hydrodynamic impact on the flow, as shown in figure 1.11(a). The

texture only serves to increase the effective area of the surface which facilitates

complete wetting of the substrate by the film.

22

• Liquid-impreganted surface: Here, the impregnated liquid is trapped and

circulates within the micro-texture, as shown in figure 1.11(b). There will always

be a thin film overlying the micro-texture, however, here it is much thinner than

the characteristic feature size.

For both the liquid-coated and liquid-impregnated surface, a potential drag reduc-

tion mechanism is turbulence damping effects by interfacial tension, similar to com-

pliant surfaces. Additionally, recirculation of the lubricant in the liquid-impregnated

surface may impart a non-zero velocity at the interface, which would lower near-wall

velocity gradients.

1.4 Outline of the thesis

The principal aims of this thesis are two-fold. The first aim is to elucidate the scaling,

structural, and statistical behavior of very high Reynolds number turbulent pipe flow.

The second aim is to test air- and liquid-impregnated or coated surfaces for turbulent

drag reduction applications and to investigate their performance experimentally and

numerically. The organization of the remaining chapters is as follows:

• Chapter 2 examines the scaling laws for the streamwise spectrum and eluci-

dates the nature of the VLSM over a very large Reynolds number range and in

different roughness regimes, using the experimental pipe flow data of Hultmark

et al. (2012b). We also critically examine the role of Taylor’s hypothesis in dis-

torting the spectral characteristics. This chapter forms the basis of Rosenberg

et al. (2013), which is published in the Journal of Fluid Mechanics.

• Chapter 3 contains experimental and numerical results on laminar and turbu-

lent flow over liquid-coated surfaces. Laminar flow experiments are performed

in rheometers (parallel-plate and Taylor-Couette), and the turbulent measure-

ments are performed over a hydrofoil, in pipe flow, and in Taylor-Couette flow.

23

The section on thin-film distortion in rheometers contains excerpts from an

article currently submitted to a peer-reviewed journal.

• Chapter 4 contains experimental results on turbulent flow over air- liquid-

impregnated surfaces. We show, for the first time, drag reduction in turbulent

flow over a liquid-infused surface. The results of the Taylor-Couette measure-

ments are the basis for a paper that is currently submitted to a peer-reviewed

journal. Also shown are preliminary results for an air- and liquid-impregnated

surface in a high Reynolds number boundary layer flow.

24

Chapter 2

Structure and scaling of

wall-bounded turbulence

In this chapter, well-resolved streamwise velocity spectra are reported for smooth-

and rough-wall turbulent pipe flow over a large range of Reynolds numbers. The

turbulence structure far from the wall is seen to be unaffected by the roughness, in

accordance with Townsend’s Reynolds number similarity hypothesis. In addition,

the energy spectra within the turbulent wall region follow the classical inner and

outer scaling behavior. While an overlap region between the two scalings and the

associated k−1x law are observed near R+ ≈ 3000, the k−1x behavior is obfuscated at

higher Reynolds numbers due to the evolving energy content of the large scales (the

very-large-scale motions, or VLSM). The inertial subrange begins to develop around

R+ > 2000 in the wake and and logarithmic regions where the spectra appear to

asymptotically approach the characteristic k−5/3x law, however slight deviations from

this behavior suggest that viscous effects are still present.

We apply a semi-empirical correction (del Álamo & Jiménez, J. Fluid Mech.,

vol. 640, 2009, pp. 5-26) to the experimental data to estimate how Taylor’s frozen

field hypothesis distorts the pseudo-spatial spectra inferred from time-resolved mea-

25

surements. While the correction tends to suppress the long wavelength peak in the

logarithmic layer spectrum, the peak nonetheless appears to be a robust feature of

pipe flow at high Reynolds number.

In the logarithmic region, the streamwise wavelength of the VLSM peak scales

approximately with distance from the wall, which is in contrast to boundary layers,

where the superstructures have been shown to scale with boundary layer thickness

throughout the entire shear layer. Moreover, the similarity in the streamwise wave-

length scaling of the large- and very-large scale motions supports the notion that the

two are physically interdependent.

2.1 Experiments

Measurements were performed in the Princeton/ONR Superpipe, which uses com-

pressed air to achieve a large range of Reynolds numbers (Zagarola, 1996; Zagarola

& Smits, 1998). The smooth pipe, originally used by Zagarola & Smits (1998), has an

average inner radius of R = 64.68 mm and is hydraulically smooth for all Reynolds

numbers reported here. The rough pipe, originally used by Langelandsvik et al.

(2007), is a commercial steel pipe with an average inner radius of R = 64.92 mm and

a surface roughness of krms = 5 µm.

Streamwise velocity fluctuations were acquired at 48 wall-normal positions for

each test case. In the smooth pipe, measurements were taken for a Reynolds number,

ReD = 〈U〉D/ν, ranging from 81 × 103 to 6.0 × 106, where D = 2R and 〈U〉 is the

area-averaged velocity. This corresponds to a friction Reynolds number, R+ = Ruτ/ν,

ranging from 1985 to 98190 (Cases 1-8 in table 2.1). Measurements in the rough

pipe (cases 9-10) were taken at ReD = 2.0 × 106 and 5.6 × 106, corresponding to

R+ = 36676 and 100530. The lower Reynolds number case in the rough pipe is in

26

Cas

eRe D

R+

p g[a

tm]〈U〉

[m/s

]ν uτ

[µm

]`

[µm

]`+

y 0[µ

m]

y+ 0

k+ rms

181×

103

1,98

50

9.48

3360

1.8

140.

40.

002

146×

103

3,33

40.

6710

.119

603.

114

0.74

0.01

324

7×

103

5,41

22.

408.

4012

605.

014

1.2

0.01

451

2×

103

10,4

815.

439.

376.

260

9.7

142.

30.

025

1.1×

106

20,2

5010

.810

.53.

260

18.8

144.

40.

056

2.1×

106

37,6

9022

.510

.51.

760

35.0

148.

20.

097

4.0×

106

68,3

7145

.910

.30.

9530

31.7

3537

0.16

86.

0×

106

98,1

9069

.710

.60.

6630

45.5

2843

0.23

92.

0×

106

36,6

7613

.316

.31.

860

33.9

2816

2.82

105.

6×

106

100,

530

34.3

19.2

0.65

6092

.928

447.

74

Tab

le2.

1:E

xp

erim

enta

lco

ndit

ions.

Cas

es1

to8

wer

em

easu

red

inth

esm

oot

hpip

e,an

dC

ases

9an

d10

wer

em

easu

red

inth

ero

ugh

pip

e.H

ere,p g

isth

epip

ega

uge

pre

ssure

,an

dy 0

isth

em

easu

rem

ent

loca

tion

nea

rest

toth

ew

all.

27

the transitionally rough regime while the higher Reynolds number case is in the fully

rough regime (k+rms > 5).

Two of the main obstacles in acquiring accurate high Reynolds number hot-wire

data in the laboratory are poor spatial and temporal resolution (see for example

Ligrani & Bradshaw, 1987; Hutchins et al., 2009; Smits et al., 2011b). Here we

have used a novel Nano-Scale Thermal Anemometry Probe (NSTAP), with a sensing

element measuring 60 µm long by 2 µm wide by 100 nm thick in lieu of a conventional

hot wire to reduce spatial and temporal filtering of energy contained in small-scale

motions. For the highest two Reynolds number cases in the smooth pipe, a 30 µm

long sensing element was used to further reduce filtering effects.

The design, fabrication, and validation of the NSTAP are described by Bailey et al.

(2010) and Vallikivi et al. (2011). The non-dimensional sensor length `+ = `uτ/ν was

between 1.8 and 92.9 for the different Reynolds numbers. For most of the cases under

consideration here, the `+ of this probe was small enough that spatial filtering effects

are confined to the high-wavenumber dissipation region. However, for the highest

Reynolds numbers, `+ was large enough to cause some filtering of more energetic low

wavenumber eddies close to the wall, but we expect the severity of this effect to be

inversely proportional to the inner-scaled wall distance (Smits et al., 2011b), so that

this effect is negligible for most of the analysis in this study.

The measurement point closest to the wall was 14 ± 4 µm for the smooth pipe

cases with a 60 µm NSTAP, 28 ± 4 µm for the smooth pipe cases with a 30 µm

NSTAP, and 28± 4 µm for the rough pipe cases. The resolution of the traverse was

±0.5 µm, with an accuracy of 3µm/m. The frequency response for the NSTAP in

still air was always greater than 150 kHz, increasing to more than 300 kHz at the

highest Reynolds number. The data were sampled at 300 kHz and filtered at 150 kHz

using an analog 8-pole Butterworth filter. Further experimental details can be found

in Hultmark et al. (2012b).

28

102

103

104

2

4

6

8

u2+

λxT

+

λxT

φuu

/uτ

2

(smooth)

102

103

104

103

104

105

106

0.2

0.4

0.6

0.8

1

1.2

y/R

λxT

+

λxT

φuu

/uτ

2

(rough)

102

103

104

103

104

105

106

y+

λxT

+

λxT

φuu

/uτ

2

(smooth−rough)

102

103

104

103

104

105

106

−0.2

−0.1

0

0.1

0.2

(a)

102

103

104

2

4

6

8

u2+

λxT

+

λxT

φuu

/uτ

2

(smooth)

102

103

104

103

104

105

106

0.2

0.4

0.6

0.8

1

1.2

y/R

λxT

+

λxT

φuu

/uτ

2

(rough)

102

103

104

103

104

105

106

y+

λxT

+

λxT

φuu

/uτ

2

(smooth−rough)

102

103

104

103

104

105

106

−0.2

−0.1

0

0.1

0.2

(b)

Figure 2.1: From top to bottom: Streamwise turbulence intensity distribution forsmooth pipe (filled symbols) and rough pipe (open symbols). Energy spectra,λxTφuu(λxT ), for smooth pipe. Energy spectra for rough pipe. Difference betweenthe smooth and rough pipe spectra. (a) Smooth pipe at R+ = 37, 690 and transition-ally rough pipe at R+ = 36, 676. (b) Smooth pipe at R+ = 98, 190 and fully roughpipe at R+ = 100, 530.

29

2.2 Results and Discussion

Here we are particularly interested in the behavior of the spatial spectrum of stream-

wise turbulent kinetic energy. In order to convert the measured frequency spectra

into the spatial wavenumber domain, Taylor’s frozen field hypothesis (Taylor, 1938)

was used with the local mean velocity Ul as the convection velocity. Herein, the

subscript T will be used to denote quantities that have been inferred from Taylor’s

hypothesis. For example, the streamwise Taylor wavelength λxT = 2πUl/ω, where ω

is the temporal angular frequency. Implications of Taylor’s hypothesis are examined

in section 3.3.

2.2.1 Effects of roughness

The influence of wall roughness on the structure of the turbulence is shown in fig-

ure 2.1. According to Townsend’s Reynolds number similarity hypothesis (Townsend,

1976), the only effect of the roughness on the outer layer of the flow is to change the

wall shear stress, that is, uτ . Therefore, if we nondimensionalize the energy spec-

tra by uτ , the outer layer structure should be independent of the roughness itself.

Figure 2.1(a) compares the streamwise turbulence intensities, u2+ = u2/u2τ , and the

premultiplied energy spectra, λxTφuu(λxT ), for a smooth pipe at R+ = 37, 690 and a

transitionally rough pipe at R+ = 36, 676 where

u2 =

∫ ∞−∞

λxTφuu(λxT ) d log(λxT ). (2.1)

Figure 2.1(b) compares a smooth pipe flow at R+ = 98, 190 and a fully rough pipe

flow at R+ = 100, 530.

For both Reynolds numbers, the streamwise turbulence intensities match quite

well beyond the location of the outer peak, which is in accordance with Townsend’s

hypothesis. Nearer to the wall, however, the profiles deviate slightly, with the smooth

30

10−2

100

102

10−6

10−4

10−2

100

kxT

y

φuu/u

τ2

−1

−3

1

5

increasingy/R

(a)

10−2

100

102

10−6

10−4

10−2

100

kxT

y

φu

u/u

τ2

1

increasingy/R

5

−1

−3

(b)

Figure 2.2: Streamwise velocity spectra at y/R = 0.001, 0.005, 0.01, 0.05, 0.1, 0.5,1.0. (a) R+ = 1, 985; (b) R+ = 37, 690

pipe profile consistently above that of the rough pipe. To examine structural differ-

ences introduced by the roughness, we next look at the energy spectrum, shown in

the following two subplots in figure 2.1 for the smooth and rough pipe, respectively.

While the energy spectra have a very similar shape and structure between the smooth

and rough cases, their subtle differences are highlighted in the bottom subplot, which

represents the difference between the two spectral maps. This profile, upon integra-

tion over all wavelengths, is equal to the difference between the smooth and rough

intensity profiles. We see that for wall distances beyond the outer peak, the small

differences between the smooth and rough variance profiles are attributed to a broad

band of wavelengths. This is likely due to systematic uncertainty in the measurements

rather than motions generated by wall roughness. Structural differences between the

two pipes closer to the wall likewise appear to be broadband in nature, however,

further detailed analysis should be limited because these data are affected by spatial

filtering, which may be of different magnitude for the smooth and rough pipes. Due to

the spectral similarities between the two pipes, all following discussions are confined

to the smooth pipe cases, but it should be noted that the same trends can be seen

for the rough pipe data.

31

2.2.2 The k−1x and k−5/3x dependencies

To investigate power-law scalings, figure 2.2 shows the streamwise velocity spectra

plotted in log-log form for R+ = 1, 985 and 37, 690 at multiple wall-normal positions.

For both Reynolds numbers, an approximate k−1x region is observed in the outer

part of the turbulent wall region (y/R = 0.05 and 0.1), although it is somewhat

more extensive at the lower Reynolds number. For the lower Reynolds number case

no k−5/3x region can be seen, but for the higher Reynolds number case, one can see

approximate k−5/3x behavior for y/R > 0.05.

To examine the inertial subrange in more detail, figure 2.3(a) shows the spectra

plotted in Kolmogorov scaling at the pipe centerline over the entire Reynolds number

range examined here. The dissipation rate was computed using an isotropic estimate

� = 15ν

∫ ∞0

k2xφuu(kx)dkx, (2.2)

and the Kolmogorov length is given by η = (ν3/�)1/4. We see that an apparent

inertial subrange just begins to form at the lowest Reynolds number, R+ = 1, 985, and

there are two decades of approximate k−5/3x behavior at the largest Reynolds number,

terminating around kxTη = 0.08. Further analysis by Vallikivi (2014) revealed that

the power law in this region more closely follows k−3/2x than k

−5/3x , which is consistent

with the notion that there are still viscous effects present and that these effects are

only completely absent in the inertial subrange at infinite Reynolds number (Gamard

& George, 2000; McKeon & Morrison, 2007).

To examine the spatial extent of the inertial subrange, the Kolmogorov-scaled

spectra at several wall-normal distances at R+ = 37, 690 are shown in figure 2.3(b).

Data closer to the wall are not shown due to errors in the computed dissipation rates

introduced by spatial filtering. The k−3/2x behavior persists well into the turbulent

wall region, and the dissipative region in the spectrum exhibits a universal behavior

32

10−4

10−2

100

10−2

100

102

104

106

kxT

η

φu

u/(

εν5)1

/4 increasing R+

−3

5

(a)

10−4

10−2

100

10−2

100

102

104

106

kxT

η

φu

u/(

εν5)1

/4

increasing y/R

5

3

−2

−3

(b)

Figure 2.3: (a) Kolmogorov-scaled spectra at pipe centerline for all Reynolds numbers(cases 1-8). (b) Kolmogorov-scaled spectra at R+ = 37, 690 for y/R =0.05, 0.07, 0.1,0.2, 0.5, 0.7, 1.0.

33

for all wall distances shown. A value of K0 = 0.51 for the Kolmogorov constant was

found using a least squares fit to the data of figure 2.3 in the inertial subrange.

Now we consider the k−1x region. The data in the logarithmic region are shown

in pre-multiplied form kxφuu in figure 2.4, so that a k−1x region should appear as

a plateau. For the two lower Reynolds numbers shown in figures 2.4(a) and (b),

the low and high wavenumber regions collapse reasonably well using the outer and

inner scalings, respectively. At R+ = 1, 985, the lowest Reynolds number, we see no

evidence for a k−1x shoulder region. The two spectral peaks, thought to be associated

with the LSM and the VLSM (Kim & Adrian, 1999), both have a similar magnitude

of A1 ≈ 0.8, shown by the horizontal dashed line, which could be confused with a

true k−1x region. At R+ = 3, 334, however, a narrow region of about a half decade

in extent appears where the outer and inner scalings both collapse the spectra and

a true k−1x law holds with A1 ≈ 0.8, which is consistent with the value reported by

Nickels et al. (2005). This value for A1 is also in agreement with the measurements

of Perry & Abell (1977) at a similar Reynolds number, and explains why a log law

with A1 = 0.8 fits their u2+ data reasonably well in this Reynolds number range. The

wavenumber bounds of the k−1x region, approximately kxTR > 2 and kxTy < 0.4, are

demarcated in figure 2.4(b) by the vertical dashed lines.

With increasing Reynolds number, however, the overlap region and the plateau

vanish. For the two higher Reynolds number cases shown in figures 2.4(c) and (d),

we see clearly that the low wavenumber region scales with R and the intermediate

wavenumber region scales with y. In fact, at the largest Reynolds number the spectra

collapse in y−scaling for the entire high wavenumber range which demonstrates the

small contribution to the energy content of the Kolmogorov-scaled viscous motions,

that is, V (y+) → 0. What is not evident, however, is the presence of an overlap

region where R and y scalings both collapse the data. Morrison et al. (2004), who also

observed this “incomplete similarity”, argued that while inner variables, y and uτ , may

34

10−2

10−1

100

101

102

103

0

0.5

1

kxT

R

kxTR

φuu/u

τ2

10−3

10−2

10−1

100

101

102

0

0.5

1

kxT

y

kxTyφ

uu/u

τ2

increasingy/R

increasingy/R

(a)

10−2

10−1

100

101

102

103

0

0.5

1

kxT

R

kxTR

φuu/u

τ2

10−3

10−2

10−1

100

101

102

0

0.5

1

kxT

y

kxTyφ

uu/u

τ2

increasingy/R

kxT

R=2

increasingy/R

kxT

y=0.4

(b)

10−2

10−1

100

101

102

103

0

0.5

1

1.5

kxT

R

kxTR

φuu/u

τ2

10−3

10−2

10−1

100

101

102

0

0.5

1

1.5

kxT

y

kxTyφ

uu/u

τ2

increasingy/R

increasingy/R

(c)

10−2

10−1

100

101

102

103

0

0.5

1

1.5

kxT

R

kxTR

φuu/u

τ2

10−3

10−2

10−1

100

101

102

0

0.5

1

1.5

kxT

y

kxTyφ

uu/u

τ2

increasingy/R

increasingy/R

(d)

Figure 2.4: Premultiplied energy spectrum in the turbulent wall region. Top: outercoordinates. Bottom: inner coordinates. (a) R+ = 1, 985 at y/R = 0.061, 0.072,0.085, 0.1 [y+ = 120, 140, 160, 190]. (b) R+ = 3, 334 at y/R = 0.037,0.044, 0.052,0.061, 0.072, 0.085, 0.1 [y+ = 120, 140, 170, 200, 240, 280, 330]. (c) R+ = 20, 250 aty/R = 0.022, 0.026, 0.031, 0.037, 0.044, 0.052, 0.061, 0.072, 0.1 [y+ = 450, 540, 630,750, 890, 1050, 1240, 1460, 2030]. (d) R+ = 98, 190 at y/R = 0.011, 0.013, 0.019,0.026, 0.037, 0.052, 0.072, 0.1 [y+ = 1090, 1300, 1850, 2600, 3640, 5090, 7090, 9840].The dashed line in (c) and (d) represents kxφuu/u

2τ = −β log[kxy/(2πα2)] with α = 2

and β = 0.2 (del Álamo et al., 2004a).

35

properly scale the spectrum over a range of wavenumbers, these wavenumbers may

simply be too small for there to be simultaneous scaling with outer variables, R and

uτ . Another possible explanation, suggested by del Álamo et al. (2004a), is that the

velocity of large wall-attached motions does not scale with uτ since wall permeability

limits their contribution to the Reynolds shear stress. Another prediction of del

Álamo et al. (2004a) occurs in the region y < λx < 10y (0.63 < kxy < 6.3), in which

isotropy considerations led them to propose a logarithmic correction to the k−1x law

given by kxφuu/u2τ = −β log[kxy/(2πα2)] with α = 2 and β = 0.2. This prediction

is tested at higher Reynolds numbers, shown by the dashed line in figure 2.4(c) and

(d), with reasonable agreement in the specified wavenumber range.

Perhaps the most intriguing result for this data set was shown by Hultmark et al.

(2012a) who, for the first time, observed the predicted logarithmic variation of u2+

given by equation 1.4, with A1 = 1.27, called the Townsend-Perry constant (Marusic

et al., 2013). Yet, the k−1x law that should accompany it, according to the spectral

arguments of Perry et al. (1986), appears to be missing. Furthermore, in these spectral

arguments the Townsend-Perry constant offers a connection between the forms of the

streamwise energy spectrum and the turbulence intensity, as given by equations 1.2

and 1.4, because A1 represents the level at which the k−1x region has a plateau as

well as the slope of the logarithmic term in u2+. Since the k−1x law observed at

R+ = 3, 334 plateaus at approximately A1 = 0.8, it therefore has no relation to

the log law observed in the turbulence intensity. In order to be consistent with the

theory, the overlap and plateau in the spectrum would need to be significantly higher,

which is not evident, even at higher Reynolds number. From these observations, it

is doubtful that the log law in the streamwise Reynolds stress observed by Hultmark

et al. (2012a) can be explained using the k−1x overlap argument advanced by Perry

et al. (1986). Hultmark (2012), for example, offers a derivation of this log law that

does not invoke such spectral arguments.

36

λB = 2π/kB

λA = 2π/kA

y

c(kB) ≈ Ul (δB)

sensor

c(kA) ≈ Ul (δA)

δB

δA

Figure 2.5: Large-scale structures in wall-bounded turbulence. Flow is left to right.

2.2.3 Convection velocity effects

To assess the distortion associated with the use of Taylor’s approximation qualita-

tively, consider the large-scale structures depicted in Figure 2.5. We assume that

each eddy maintains its spatial structure as it convects dow