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This is to certify that the

thesis entitled

VORTICITY AND DRAG MEASUREMENTS
IN ROUGH-WALL TURBULENT BOUNDARY LAYERS
presented by

Alan Baker Folz

has been accepted towards fulfillment

’ of the requirements for

Mechanlcql
M,S, degree in Engineermg

ﬁg ggdm

Major professor

Date lOﬁGA/ q 3

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CWMHJ

 

VORTICITY AND DRAG MEASUREMENTS

IN ROUGH-WALL TURBULENT BOUNDARY LAYERS

By

Alan Baker Folz

A THESIS
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1993

 

 

 

 

 

 

 

HI
measure
bounda:
of sifted
ratio of 1
increase:
anilysn
Incas“rec
apprC‘priz
ﬁction 0
°“ the tn}
Wages, l

ABSTRACT

VORTICITY AND DRAG MEASUREMENTS
IN ROUGH-WALL TURBULENT BOUNDARY LAYERS

By
Alan Baker Folz

Hot-wire measurements using a four-wire spanwise vorticity array and drag
measurements using a ﬂoating element balance have been performed in the turbulent
boundary layers formed over two distinct rough surfaces, one a close-packed distribution
of sifted rocks and the other a series of spanwise square rods with a spacing-to-width
ratio of four. The drag results show that the skin-friction coefﬁcient of each surface
increases with Reynolds number beyond the point at which traditional rough-wall
analysis indicates skin-friction is constant. Drag coefﬁcients were then assigned to the
measured velocity profiles by matching two parameters (Reynolds number and
appropriately normalized development length) which are shown to yield a single skin-
friction coefﬁcient for a given surface. The resulting turbulence quantities, when scaled
on the friction velocity. show that different flow characteristics are generated by the two

surfaces, both of which have been categorized as 'k-type.‘

Cepyright by
Alan Baker F012

1993

I wou]
this study, tl
appreciate tl
members, Jo
reading of th
maze of 0qu
remaining co
“35. and fac1

cont‘lbllted tc

dIVCJ‘SiODal I

Acknowledgments

I would like to thank my advisor, Robert Falco, for providing the initial impetus for
this study, the unique facilities which allowed its completion, and his advice. I
appreciate the interest and scheduling ﬂexibility demonstrated by my other committee
members, I ohn Foss and David Wiggert. Dr. Foss deserves special thanks for his careful
reading of the manuscript. I owe Joe Klewicki a great debt for his guidance through the
maze of equipment at the Turbulence Structure Lab. I thank Bernd Stier, my last
remaining cohort at the TSL, for his presence and his help. Too many other students,
staff, and faculty at both the TSL and the MSU Engine Research Laboratory have
contributed to this study to name, but I appreciate all of their help, both technical and
diversional. Last, I would like to thank my parents for their unwavering support.

iv

 

 

 

listo
Insto
Lute:

Chapu

Table of Contents

List of Tables ............................................................................................................... viii
List of Figures ............................................................................................................. ix
List of Symbols ........................................................................................................... xv
Chapter 1. Introduction .................................................................................................. l
1.1. The Prevalence of Roughness ...................................................................... 1
1.2. Skin-Friction Coefﬁcient Correlations ......................................................... 2
1.3. The Physics of Turbulent Boundary Layers ................................................. 4
1.3.1. Smooth-Wall Tmbulent Boundary Layers ..................................... 4
1.3.2. Rough-Wall Turbulent Boundary Layers ...................................... 5
1.4. Characterizing Rough Surfaces .................................................................... 6
Chapter 2. Experimental Equipment and Methods ......................................................... 8
2.1. Surfaces ....................................................................................................... 8
2.1.1. Rod Roughness Construction ....................................................... 8
2.1.2. Rock Roughness Construction ...................................................... 9
2.2. Wind Tunnels ............................................................................................. 10
2.2.1. Hot-Wire Wind Tunnel ............................................................... 10
2.2.2. Drag Measurement Wind Tunnel ................................................ 11
2.3. Drag Balance .............................................................................................. 11
2.4. Drag Plates ................................................................................................. 12
2.4.1. Square Rod Roughness Drag Plate .............................................. 13
2.4.2. Rock Roughness Drag Plate ........................................................ 14
2.5. Hot-Wire Equipment .................................................................................. 14
2.6. Data Collection and Reduction Equipment ................................................. 15
Chapter 3. Drag Measurements .................................................................................... 17
3.1. Drag Estimate Options ............................................................................... 17
3.2. Drag Measurement Procedure .................................................................... 19
3.3. Drag Measurement Results ........................................................................ 20
3.3.1. Trend #1: Skin Friction Decreases with Increasing
Development Length .................................................................. 21
3.3.2. Trend #2: Skin Friction Increases with Increasing
Freestrearn Velocity .................................................................... 22
3.3.3. Low Speed Effects ...................................................................... 24

 

 

 

Chapt

Chapte

Chapter 4. Hot-Wire Measurements ............................................................................. 26

4.1. Probe Mounting and Calibration ................................................................ 26
4.2. Data Sampling ........................................................................................... 27
4.3. Square Rod Roughness Set-Up .................................................................. 28
4.4. Rock Roughness Set-Up ............................................................................ 29
Chapter 5. Data Interpretation and Analysis ................................................................. 30
5.1. Applying of to Hot-Wire Proﬁles: Matching R5 and 11/6 ............................ 30
5.1.1. R5 and ill?) for Hot-Wire Proﬁles ................................................. 31
5.1.2. R5 and x/B for Drag Measurements .............................................. 32
5.1.3. Resulting Values of Cf ................................................................. 33
5.2. Interpreting Hot-Wire Data ........................................................................ 34
5.2.1. Boundary Layer Thicknesses and Freestream Velocities .............. 34
5.2.2. Virtual Origins ............................................................................ 35
5.2.2.1. Method of Locating the Virtual Origin .......................... 36
5.2.2.2. Virtual Origin Location Results .................................... 37
5.2.3. Roughness Length, y0 ................................................................. 38
5.3. Mean Velocity Distributions ...................................................................... 38
5.4. Scaling the Roughness Function ................................................................ 39
5.5. On the Accord of yo and AU/ua; .................................................................. 40
5.6. On the Lack of Trend in 6 for the Square Rod Surface ............................... 41
Chapter 6. Turbulence Quantities ................................................................................. 44
6.1. Streamwise Fluctuations ............................................................................ 45
6.2. Discussion of the Skewness and Kurtosis Proﬁles ...................................... 47
6.3. Wall-Normal Fluctuations, Reynolds Stress,
and Correlation Coefﬁcient ........................................................................ 49
6.4. On the Small Magnitudes Measured by the X-array ................................... 50
6.5. Spanwise Vorticity Fluctuations ................................................................. 51
6.6. Concluding Remarks based on the Hot-Wire Data ..................................... 52
Chapter 7. Conclusion .................................................................................................. 54
7.1. Summary of Results ................................................................................... 54
7.2. Future Considerations ................................................................................ 56
Tables ........................................................................................................................... 58
Figures .......................................................................................................................... 60
Appendix A. Veriﬁcation of Equipment and Methods ............................................... 124

Appendix B. On the Effective Angles of the X-wires in the
Vorticity Array ..................................................................................... 125

 

App'

App:

Ampe

Appe

Appe

Lnto

Appendix C. Smoothing Low-Frequency Fluctuations in

Freestream Velocity .............................................................................. 127
Appendix D. Interpretation of R9 = 4610, Square Rod Data ....................................... 128
Appendix E. On the Accuracy of Power-Law Velocity

Proﬁle Approximations ........................................................................ 129
Appendix F. Of Interest to Future Investigators at the

Turbulence Structure Laboratory .......................................................... 132
Appendix Figures ........................................................................................................ 136
List of References ........................................................................................................ 146

 

 

 

Table

Table

List of Tables

Table 1. Statistics describing the rocks which make up the rock roughness ................... 58

Table 2. Parameters describing the boundary layers compared in this study.
Smooth-wall data are taken from Klewicki (1989); data over the
rock and square rod surfaces were taken for the present study ......................... 59

 

 

Figure

Figure
Figure
Figure
Figure
FiSure :
Figure :

“Elite 2

Figure 3.2

List of Figures

Figure 2.1 Geometry of the spanwise square rod roughness ....................................... 60
Figure 2.2. Conﬁguration of the rough surfaces in the hot-wire wind tunnel. The

rectangular border represents the test section. Flow is right to left ............ 61
Figure 2.3. Distribution of heights of the stones which compose

the rock roughness. The sample size is 160 .............................................. 62
Figure 2.4. Photograph of the rock roughness. A penny provides the scale ................. 63
Figure 2.5. Hot-wire wind tunnel. Flow is right to left ............................................... 64
Figure 2.6. Drag measurement wind tunnel. Flow is right to left ................................ 65

Figure 2.7. Photograph of the drag balance. The piezoelectric crystal is at the
lower right, and the drag plates attach to the horizontal black square
frame at the top of the picture. The bolts used for attachment are
visible. Flow is from right to left and above the drag plate (out of
the picture) ................................................................................................ 66

Figure 2.8. Spanwise vorticity probe. (a), (b), and (c) are schematic top, front,
and side views, respectively. (d) is an isometric view photograph.
Dimensions are nominal ............................................................................ 67

Figure 3.1. Skin-friction coefficient as a function of development length and
freestream velocity for ﬂow over the square rod roughness. Data
points are connected for clarity; intersections of these lines reﬂect
uncertainty in measurement, not ﬂow physics.
a) Development lengths of 38 to 180 centimeters .................................... 68
b) Development lengths of 180 to 373 centimeters .................................. 69

Figure 3.2. Skin-friction coefﬁcient as a function of development length and
freestream velocity for ﬂow over the rock roughness. Data points
are connected for clarity; intersections of these lines reﬂect
uncertainty in measurement, not ﬂow physics ........................................... 70

 

Figure 3.3

Figure5.l.

Figure 5.2.

Figure 5.3.

Harm.

F1sure 5.5.

Flgure5,6.

“ﬁn—bu

Figure 3.3. Skin-friction coefﬁcient as a function of Uoox/v and x/k based on
Schlichting's analysis. (Taken from Schlichting 1937, with symbols
changed for consistency. The original roughness heights k are
"equivalent sandgrain roughness heights") .............................................. 71

Figure 5.1. Three-dimensional plot showing Cf as a function of Umbmlv and xrlf)s
for the rock roughness. Note that Cf values were matched in the region
of x,,/5s z 30, where gradients are steep only at the lowest Reynolds
numbers. The jagged nature of the surface at low Reynolds number
is a reﬂection of measurement uncertainty, not ﬂow physics. Also,
the grid density is much greater than actual measurement density ............. 72

Figure 5.2. Three—dimensional plot showing Cf as a function of Uoth/V and x1153
for the square rod roughness. Note that Of values were matched in
the region of x116s z 30, where gradients are steep only at the lowest
Reynolds numbers ..................................................................................... 73

Figure 5.3. Skin-friction coefﬁcient as a function of development length and

U..5,/v for flow over the square rod roughness. Lines of constant

Icy/6s slant from the upper right to the lower left, qualitatively. Data

points are connected for clarity; intersections of these lines reﬂect
uncertainty in measurement, not ﬂow physics.

a) Development lengths of 38 to 180 centimeters .................................... 74
b) Development lengths of 180 to 373 centimeters .................................. 75

Figure 5.4. Skin-friction coefﬁcient as a function of development length and

wallv for ﬂow over the rock roughness. Lines of constant x1168

slant from the upper right to the lower left, qualitatively. Data points

are connected for clarity; intersections of these lines reﬂect

uncertainty in measurement, not ﬂow physics ........................................... 76

Figure 5.5. Example of the proﬁle-matching process used to derive e for the
R9 = 7330 square rod roughness proﬁle. The solid line represents
the slope imposed on the logarithmic region of the velocity proﬁle
for of = 0.0101. Note the slope mismatch for 8 at 7.62 mm ....................... 77

Figure 5.6. Difference in trend in elk with R9 for the square rod and
rock roughnesses ...................................................................................... 78

 

Figure

Figure 1

Figure 5

Figure 5

Figure 5.

I:igure 5.1

Figure 6.1

Figure 5.7. Example of the proﬁle-matching process used to derive y0 for the
R9 = 7330 square rod roughness profile. The solid line represents

ICU/1.11 = 1n(y/yo). Note that one unique value of y0 satisﬁes this
representation of the logarithmic velocity distribution. The quality

of the slope-match conﬁrms the location of the virtual origin (see

Figure 5.5) ................................................................................................ 79

Figure 5.8. Mean velocity distributions. The solid line represents the smooth-
wall logarithmic velocity distribution (AU/u1 = 0) for x = 0.40,
Co = 5.1. The roughness function AU/u.r for the R3 = 1550 rock
surface proﬁle is indicated ........................................................................ 80

Figure 5.9. Meteorological representation of the universal velocity distribution.
The solid line corresponds to ICU/111 = ln(y/y0) .......................................... 81

Figure 5.10. The roughness function scaled, on the roughness height k. The

dashed line is of the form AU/u1 = (l/x)1n(ku1/v) + B and passes

through the square rod wall data. The solid line passes through

the rock wall data. It takes on the same form for k+ > 70, and

the contour for k"' < 70 is taken from Clauser (1956) for sandgrain
roughness .................................................................................................. 82

Figure 5.11. The roughness function scaled on the error-in-origin e. The solid

line represents the correlation AU/u1 = (l/Ic)ln(su1/v) - 0.4, the
dashed line indicates the points from the square rod surface, and the
dotted line indicates the points from the rock surface ................................ 83

Figure 5.12. The roughness function scaled on the roughness length yo. The
solid line represents AU/uT = (1/K)ln(you1/v) + C0 .................................... 84

Figme 6.1. Streamwise turbulence intensities.

a) Rock and square rod surfaces .............................................................. 85
b) Rock surface. Grass' 1971 measurements over a very similar

rock surface in an open channel are included for comparison.

For Grass' ﬂow k+ = 84.7, and the Reynolds number is based

on the mean ﬂow velocity and the ﬂow depth ..................................... 86
c) Square rod surface. Antonia & Luxton's 1971 measurements

for ﬂow over a spanwise square rod surface with A. = 4 and

k+ = 76 are included for comparison. Note that Antonia &

Luxton neglected the error in origin a, so y is measured from

the roughness peaks for their data ........................................................ 87

 

Figure

Figure 6.

Figure 6,.

Figure 6.5.

Figure 6.2. Skewness of streamwise ﬂuctuations. For each rough-wall
boundary layer the increase in skewness far from the wall is due
to the increased presence of irrotational ﬂow.

a) Smooth, rock, and square rod surfaces. Smooth wall data are

from Klewicki (1989) .......................................................................... 88
b) Smooth surface (from Klewicki 1989) ................................................. 89
c) Rock surface ....................................................................................... 90
d) Square rod surface ............................................................................... 91
e) Low Reynolds number -- smooth, rock, and square rod

surfaces. Smooth wall data are from Klewicki (1989) ........................ 92
f) Middle Reynolds number -- smooth, rock, and square rod

surfaces. Smooth wall data are from Klewicki (1989) ........................ 93
g) High Reynolds number -- smooth, rock, and square rod

surfaces. Smooth wall data are from Klewicki (1989) ........................ 94

Figure 6.3. Kurtosis of streamwise ﬂuctuations.
a) Smooth, square rod, and rock surfaces. Smooth wall data

are from Klewicki (1989) .................................................................... 95
b) Smooth surface. Smooth wall data are from Klewicki (1989) ............. 96
0) Rock surface ....................................................................................... 97
d) Square rod surface ............................................................................... 98

Figure 6.4. Skewness and kurtosis of streamwise ﬂuctuations.

a) R9 = 1010, smooth wall (from Klewicki 1989) .................................... 99
b) R9 = 2870, smooth wall (from Klewicki 1989) .................................. 100
0) R9 = 4850, smooth wall (from Klewicki 1989) .................................. 101
d) R9 = 1550, rock wall .......................................................................... 102
e) R9 = 4330, rock wall .......................................................................... 103
0 R9 = 6140, rock wall .......................................................................... 104
g) R9 = 1180, square rod wall ................................................................ 105
h) R9 = 4610, square rod wall ................................................................ 106
i) R9 = 7330, square rod wall ................................................................ 107

Figure 6.5. Wall-normal turbulence intensities.
a) Smooth, square rod, and rock surfaces. Smooth wall data are

from Klewicki (1989) ........................................................................ 108
b) Smooth surface (from Klewicki 1989) ............................................... 109
c) Rock surface ...................................................................................... 110
(1) Square rod surface ............................................................................. 111

 

Figure

Figure I

Figure 6

Figure B
FiSure B.

Figure C.;

ﬁgure C.:

Figure 03.

Figure 6.6.

Figure 6.7.

Figure 6.8.

Figure 3.1.

Figure B.2.

Figure C. 1.

Figure C.2.

Figure C.3.

Reynolds stress <-uv>.
a) Smooth, square rod, and rock surfaces. Smooth wall data are

from Klewicki (1989) ........................................................................ 112
b) Smooth surface (from Klewicki 1989) ............................................... 113
c) Rock surface ...................................................................................... 114
d) Square rod surface ............................................................................. 115

Reynolds stress correlation coefﬁcients.
a) Smooth, square rod, and rock surfaces. Smooth wall data are

from Klewicki (1989) ........................................................................ 116
b) Smooth surface (from Klewicki 1989) ............................................... 117
c) Rock surface ...................................................................................... 118
d) Square rod surface ............................................................................. 119

Spanwise vorticity ﬂuctuations.
a) Smooth, square rod, and rock surfaces. Smooth wall data are

from Klewicki (1989) ........................................................................ 120
b) Smooth surface (from Klewicki 1989). The dashed line is a

curve ﬁt through the data. The dashed line is a curve ﬁt through

the data, (vcoz'luacz) = 0.8(yu1/v)0-5. .................................................. 121
0) Rock surface. The dashed line is a curve ﬁt through the

smooth-wall data ............................................................................... 122
d) Square rod surface. The dashed line is a curve ﬁt through

the smooth-wall data .......................................................................... 123

Effect of varying X—array correction coefﬁcients on Reynolds
stress proﬁles. Note that Klewicki's smooth-wall data are
reproduced for Cf = 1.0, CI = -1.0 ............................................................ 136

Effect of varying X-array correction coefﬁcients on spanwise
vorticity proﬁles. Note the very weak dependence on the
correction coefﬁcients ............................................................................. 137

Long-term variation in wind tunnel freestream velocity ........................... 138
Effect of the ﬁlter on velocity time-series. The long-term wind

tunnel variation is decoupled from the turbulence and removed. The
dashed line indicates the overall (14.9 minute) mean velocity for this data

point ........................................................................................................ 139
Effect of the ﬁlter on the turbulence quantities of the R9 = 1550

rock surface boundary layer.

a) Streamwise ﬂuctuations ..................................................................... 140
b) Wall-normal ﬂuctuations ................................................................... 141
c) Reynolds stress .................................................................................. 142
d) Spanwise vorticity ﬂuctuations .......................................................... 143

 

Figure 1

Figure f

Figure D.1.

Figure E.1.

Deriving appropriate Reynolds stress values for Cf = 1.0, C, = -1 .0:

The R9 = 1180 and R9 = 7 330 square rod data were analyzed for

Cf=1.0, C, = -1.0, and for Cf: 1.1, C, = -O.6. The effect of this

variation was consistent, so the same effect was assumed in deriving
Reynolds stress values for the R9 = 4610 square rod proﬁle with

Cf=1.0, CI = -1.0. This proﬁle was only calculated for Cf: 1.1,

C, = -0.6. Solid squares show the assumed values for

Cf: 1.0,C,=-1.0 .................................................................................... 144

utblv vs. R9 for smooth- and rough-wall turbulent boundary layers.

Data are from various investigators. The solid line is a curve ﬁt

through the smooth-wall data, u-rb/v = 0.76R90-92. The dashed line

is a curve ﬁt through the rough-wall data, uxblv = 0.30R91-04 ................. 145

xiv

Cf

List of Symbols

Local skin-friction coefﬁcient [ a ZTw/onoz]

Cf, C, Correction coefﬁcients for X-array of vorticity probe

Co
d

mxwfa'; ”Fwn‘mm

:1

C1

Uoo

Additive constant in the law of the wall

Displacement height (location of virtual origin relative to base of roughness, see
Figure 2.1)

Hot-wire voltage

Boundary layer shape factor [ a 6‘70]

Height of roughness elements

Kurtosis

Center-to-center spacing of spanwise square rods

Exponent in King's law calibration equation; inverse of exponent in power-law

velocity proﬁle approximation

. Root mean square

Reynolds number based on 8 [ = Dosh/v]
Reynolds number based on 0 [ a Uooe/V]
Reynolds number based on x [a UooX/V]
Skewness

Fluctuating component of streamwise velocity
Local mean velocity

Friction velocity [ a (wpﬂ/Z]

Freestream velocity

Fluctuating component of wall-normal velocity

XV

 

 

 

V Velc
w Wid
x Sire:
x, Strei
xs Sire:
y Wan
yo Rou;
Z Spar
Mimi
5 X~w
5 Bou:
5: Bou
6s Bou
6‘ Ben
AU/“t Ron
8 ER

FigL
K V01]
1 Spar
” Kim
D Flui
6 Ben
Iva Wal
‘02

V Velocity inferred by hot-wire

w Width (streamwise extent) of spanwise square rods

x Streamwise coordinate

x, Streamwise extent of rough surface, leading edge to measurement station

xs Streamwise extent of smooth wall, trip rod to rough surface

y Wall-normal coordinate, measured from the virtual origin

yo Roughness length

2 Spanwise coordinate

W

B X-wire angle deviation from 459

6 Boundary layer thickness (599)

5, Boundary layer thickness over rough surface (usually at measurement station)
6, Boundary layer thickness over smooth wall (usually at leading edge of roughness)
5* Boundary layer displacement thickness

AU/u1 Roughness function (see Figure 5.8)

8

>’

<

eras

Error in origin (location of virtual origin relative to roughness peaks, see
Figure 2.1)

von Karman's constant

Spacing ratio of spanwise square rods, Llw

Kinematic viscosity

Fluid density

Boundary layer momentum thickness

Wall shear stress

Fluctuating component of spanwise vorticity

xvi

Other
+ (Superscript) Normalized by v and u1 (e.g. y+ yudv)
< > Time-averaged quantity

Indicates an rms quantity

xvii

 

 

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decrease i
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height Wa

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spamSon,

Chapter 1

Introduction

Surface roughness extracts a great toll from society, in a very literal sense. The
cost of transportation, whether it be moving objects through ﬂuid or ﬂuid through a
conduit, is increased greatly by the drag attributable to roughness. Schlichting (1960)
relates the following episode, iwhich underscores the effect roughness can have on a ﬂow.
A 500 mm diameter water duct in the valley of the Ecker experienced a 50 percent
decrease in the mass ﬂow through it. Upon examination, this decrease was attributed to
0.5 mm high ribs which had formed at right angles to the ﬂow direction. The roughness
height was only 1/1000 of the pipe diameter, and yet the mass ﬂow was cut in half.

This study was undertaken with two goals: to document the effect of roughness
on the skin-friction coefﬁcient associated with a turbulent boundary layer, and to

contribute to the understanding of the structure of rough-wall turbulent boundary layers.

1.1. The Prevalence of Roughness

A surface need not be jagged to appear "rough" to the ﬂuid passing over it. The
general rule is that roughness has an effect on the ﬂow if it protrudes beyond the viscous
sublayer. This occurs if its roughness Reynolds number kudv exceeds ﬁve. In
technologically signiﬁcant ﬂows it is rare to ﬁnd kudv < 5 . In a low-speed flow, it may
not be cost-effective to machine a surface to this degree of smoothness, and for higher

speeds or low-viscosity working ﬂuids it may be expensive or impossible. At high

 

surface :

of a ﬂat
may be .
the plate
by the re

(Kuethe
m/s, and
scale, ar
roughne
rOUghne
Viscous
in mind
the mug

1.2. Ski

WQuId b1
l°nger h
kHOWIBd

speeds, for instance, even a "smooth" (using "smooth" in the common sense of the word)
surface may display signiﬁcant effects of roughness.

Take as an example the situation three meters downstream from the leading edge
of a ﬂat plate in a 200 mls air ﬂow, with kinematic viscosity v = 12 x 10’6 m2/s. (This
may be a ﬁrst-order model of the ﬂow over an airplane fuselage or wing.) By assuming
the plate is smooth, the local skin friction coefﬁcient Cf can be calculated approximately
by the relationship

c, =0.0592R;‘" (1.1)
(Kuethe & Chow 1986), to be 0.0017 . However, this yields a friction velocity u; of 5.8
mls, and a viscous length scale v/ue of 2.1 micrometers. With such a small a length
scale, any variation on the order of hundredths of millimeters appears to the ﬂow as
roughness. This could be a speck of dirt or a surface irregularity. If some typical
roughness height is taken to be one-tenth of a millimeter, this protrudes almost 50
viscous lengthscales from the wall, putting the ﬂow is well into the rough regime. Keep
in mind that this roughness Reynolds number is based on the smooth-wall skin-friction;
the roughness Reynolds number will be signiﬁcantly higher when an appropriate skin-

friction is used.

1.2. Skin-Friction Coefﬁcient Correlations

If the above example were continued another skin-friction coefﬁcient correlation
would be needed, because the wall is now known to be rough, so Equation 1.1 would no
longer be applicable. Rough-wall skin-friction correlations typically depend on
knowledge of roughness geometry, velocity distribution, or both.

Correlations that depend on roughness geometry include those of Betterman
(1966), Dvorak (1969), Simpson (1973), and Abou-Arab et al. (1991). Betterman relates
Cf to boundary layer thickness and roughness height and geometry. Dvorak considers the
same parameters, but quantiﬁes the effect of roughness geometry in the form of a
function f0), where A is a single parameter describing the roughness geometry. Simpson
states that Dvorak is nearly correct, but that the effect of the roughness is better described
by deﬁning A as the ratio of total surface area to roughness frontal area normal to the
ﬂow, rather than the ratio of total surface area to roughness surface area (Simpson prefers
A = 1.11: to A. = L/w, according to the conventions of Figure 2.1). Abou-Arab et al. prefer
to use Simpson's parameters, but modify the curve-ﬁt through existing data and arrive at
a different f(A).

These methods yield values for cf, but the correlations used are suspect. They
require a characterization of the roughness which is difﬁcult or impossible to deﬁne
uniquely for random roughnesses, and it is not at all clear that any given characterization
is sufﬁcient. One of these parameters (Dvorak's or Simpson's), or some other, might be
better than other single parameters for describing the roughness effect, but it is clear that
both top surface area and frontal area have some effect. To ignore either, or any other
factors such as individual element geometry (e.g. square corners versus rounded) and
interaction between element wakes, is to neglect ﬂow physics.

Koh (1992) presents an interesting method of deriving a skin-friction coefﬁcient
expression, in which he identiﬁes the equivalent roughness height with the point at which
U = 8.5%. He argues that this equivalent roughness height is identical to the physical
roughness height for fully rough ﬂow, and he concludes that of is a function of x/k (the
ratio of streamwise extent of the rough surface to the roughness height) only for fully
rough ﬂow. The results of this study disagree with both of these conclusions: the local
mean velocity does not reach 8.5u1 until well above the roughness peaks, and Cf depends
on Reynolds number beyond the point of having attained full roughness.

 

1.3. TheF

Ac<
basic goal,
process in 1

The
studied in g
lagged behi
conditional
wire anemc
physics of:

have any or

1.3. The Physics of Turbulent Boundary Layers

Accurate drag estimates are certainly one goal of turbulence research, but a more
basic goal, with greater potential beneﬁt, is the understanding of the drag production
process in turbulent boundary layers.

The smooth wall, zero pressure gradient turbulent boundary layer has been
studied in great detail over the past half-century, but our understanding of it has generally
lagged behind the rapid increase in the data base. Recently, however, techniques such as
conditional sampling, ﬂow visualization, and simultaneous ﬂow visualization and hot-
wire anemometry have shed new light on the physics of turbulent ﬂow. Insight into the
physics of turbulence is necessary both to create practical numerical simulations and to
have any chance at turbulence control.

The turbulence research community now has a relatively ﬁrm grasp of the
mechanism of ﬂuid/smooth-wall interactions, or at least most researchers have their own
conceptual understanding of this process. When the wall is rough, however, the mental
picture becomes far less crisp.

1.3.1. Smooth-Wall Turbulent Boundary Layers

Recognizable structural features are observed in smooth-wall turbulent boundary
layers, and evidence suggests that these features are critical both in propagating the
turbulence and in creating viscous drag. Some of the dominant features are: long
streamwise streaks and pockets in the near-wall region, horseshoe vortices extending up

from the wall, ring-like motions across the boundary layer, and large-scale motions in

 

which the
structure.

ICCCIII SI‘U

1.3.2. Rc

which these other features reside. For more detailed discussion of the smooth-wall
structure, see Falco's 1991 summary, the 1990 review of Kline & Robinson, and the

recent studies of Stanislas (1993) and Morrison et a1. (1992).

1.3 .2. Rough-Wall Turbulent Boundary Layers

Knowledge of rough-wall boundary layer structure is much less complete than
that of smooth-wall structure. Grass (1971) found that intermittent sweeps and ejections
exist over rock surface roughness, and that they make a large contribution to the
Reynolds stress. The ejections were found to extend over a large portion of the boundary
layer. Raupach (1981) conditionally sampled the statistics of boundary layers formed
over walls with various distributions of vertical cylinders, and found that sweeps were
the dominant contributors to the Reynolds stress in the near-wall region. Bessem &
Stevens (1984) provide evidence that hairpin vortices exist in the ﬂow over spanwise
square rods with it = 4. They also ﬁnd the inclination angle of the vortices to be similar
to that over a smooth wall. Bandyopadhyay & Watson (1988) conclude that hairpins do
exist in the ﬂow over a spanwise square rod type of rough surface. Over a three-
dimensional rough surface they believe that "necklace" vortices exist, straddling
roughness elements near their bases. Grass et al. (1991) performed detailed hydrogen
bubble visualization over several rough surfaces and found that horseshoe vortices do
exist in those ﬂows. They also found that these vortices are linked to turbulent bursts in
a manner similar to that of smooth-wall ﬂows.

Hence, it is generally accepted that horseshoe vortices, or hairpins, exist in rough
wall boundary layers, and that discrete, intense events are major contributors to the
Reynolds stress. There is as yet, however, no direct evidence for the existence of ring-

like vortical motions in such ﬂows. The near-wall structure commonly observed in

 

 

smooth 1
rough su
recogniz.

R
boundary
inner lay:
layer pro
the ansvw
Vorticity 1

surface-i1

1.4. Char

Iatenant

Where AU)
vertiCal (p,
to a 3111001]

The

smooth walls, the long streaks and pockets, probably cannot exist in the same form over
rough surfaces. Corresponding motions likely exist, but they may not take on easily
recognizable and coherent forms.

Raupach et al. (1991) point out that the outer layer structure of the turbulent
boundary layer does not seem to be dependent on the type of surface underneath, but the
inner layer structure is highly dependent on surface type. How the very different inner
layer processes result in very similar outer layers is a most basic question, and knowing
the answer would be a great stride towards understanding wall-bounded turbulence. The
vorticity measurements of the present study (see Section 6.4), however, bring the

surface-independence of the outer layer into some question.

1.4. Characterizing Rough Surfaces

The mean velocity distribution for the logarithmic region of a turbulent boundary

 

layer can be described by
-—q—=lln(-zl-ll)+CO—AUo (1-2)
u, x v u,

where AU/u; is called the roughness function. The roughness function corresponds to the
vertical (parallel to the U/u-c axis) displacement of the rough-wall velocity proﬁle relative
to a smooth-wall proﬁle (see Figure 5.8).

The roughness function for fully rough ﬂow can usually be given by

AU =lrn[k“*)+3, (1.3)
ll K V

1

 

 

where B is a constant for a given roughness geometry and k is the height of the
roughness elements. For certain geometries, however, the roughness function scales on

the depth of the ﬂow rather than the roughness height.

 

 

l
rough 31.
over its
mean ve
diametei
of spanv
it S 2, bj
within t]

I
describe

in similt

Perry et al. (1969) noted this difference and introduced a way of characterizing
rough surfaces. A 'k-type' surface is one for which the mean velocity proﬁle of the ﬂow
over it scales on the height of the roughness, k. A 'd-type' roughness is one for which the
mean velocity proﬁle of the ﬂow over it scales on the depth of the ﬂow, d (the pipe
diameter or boundary layer thickness). Surfaces of the 'd-type' are generally composed
of spanwise grooves with width equal to or less than their separation ((L-w) S w, or
A S 2, by the conventions of Figure 2.1). The mean ﬂow then generates stable vortices
within these grooves.

While this distinction is certainly of interest, it does not seem sufﬁcient to fully
describe the nature of a ﬂow. That 'k-type' surfaces of widely different geometries result
in similar mean velocity proﬁles is a provocative result, but to fully characterize a
surface by the mean velocity proﬁle over it is superﬁcial.

A given ﬂow -- as described by a Reynolds number, for example -- does not
simply generate motions representative of itself. The ﬂow must interact with its
surroundings to generate at least some of its characteristic motions, and it is not obvious
that these motions must be destroyed in the near-wall region. It is difﬁcult to
comprehend how radically different surfaces could interact with a flow to generate
precisely similar motions and distributions of motions. Vastly different ﬂows could
result in similar mean proﬁles, but similarity in the mean does not imply that the more
detailed structure of the ﬂows is similar.

With this in mind, two very different 'k-type' rough surfaces were selected for this
study, in hopes that by inspecting turbulence quantities differences in time-averaged

values other than velocity would emerge.

 

2.1. Surfac

0n the hour
PCFiOdiC dis
r oughness.

it was exper

Characten'st

Chapter 2

Experimental Equipment

2.1. Surfaces

In hopes of gaining a broad and general understanding of the effects of roughness
on the boundary layer quantities of interest, two distinct rough surfaces were selected: a
periodic distribution of spanwise square rods and a "uniformly random" sandgrain-type
roughness. While both surfaces have generally been categorized as 'k-type' roughnesses,
it was expected, and it is shown, that they generate markedly different flow

characteristics.

2.1.1. Rod Roughness Construction

The spanwise rods are 1.27 cm (0.50") square by 122 cm (48") long, and are
placed every 5.08 cm (2.00") along the wind tunnel wall, for a non-dimensional spacing
it = Uw = 4.0. The surface geometry is shown in Figure 2.1. The square rods were cut
from 1/2" particle board on a table saw with a rip fence. Typical variation in the width
of the cuts was less than $0.004 cm ($0.01"). The corners of the square rods were then
lightly sanded to remove burs. Base plates are 1/2" particle board cut in sheets 55.9 cm
by 121.9 cm (22" by 48"). Parallel lines, separated by 5 .08 cm (2.00"), were laid out on
the base plates, and the rods were attached along these lines with 3/4" wire brads every
15 cm (6"). Nine base plates were constructed, giving a total roughness length of 502.9

 

cm (198
slightly

The roul
in the m
in each 1
floor up
region 0:

of the re

2.1.2. R

cm (198") for the hot-wire proﬁles. The width of the roughness sheets was then reduced
slightly (by about 0.4 cm, to 121.5 cm) to avoid the sheets' wedging in the wind tunnel.
The roughness plates were constructed and stored to encourage them to bow slightly up
in the middle. The plates were then attached to the wind tunnel wall with a single screw
in each plate's center. A sheet metal ramp 30.5 cm (12.0") long brings the wind tunnel
ﬂoor up 2.54 cm (1.0") to the level of the peak of the roughness elements. A smooth
region of 13.3 cm (5 .25 ") connects the ramp to the ﬁrst rough plate. The conﬁguration

of the rough surface in the wind tunnel is shown in Figure 2.2.

2.1.2. Rock Roughness Construction

The rock roughness surface is composed of a single layer of tightly packed
peastone. The peastone is commonly available at home or garden supply stores, and the
stones have a nominal dimension of one centimeter (3/8"), but vary signiﬁcantly in both
size and shape. The stones were hand-sifted through a grate of 0.79 cm (5/16") diameter
holes punched in sheet metal. The smaller stones (those that fell through the holes) were
discarded and the larger stones were used for the rough wall. Approximately half of the
volume of peastone was discarded. The stones were then laid in the wind tunnel, packed
tightly in a single layer. For the hot-wire proﬁles, the surface covered by the rocks was
490 cm (193") long and the full width of the wind tunnel, 121.9 cm. A sheet metal ramp
30.5 cm (12.0") brought the ﬂoor of the wind tunnel up to roughly the level of some of
the higher peaks (8 mm, or 0.3"). The conﬁguration of the rough surface in the wind
tunnel is shown in Figure 2.2.

One hundred sixty (160) of the stones were measured to provide statistics
describing the rock surface. The mean height of the stones (k) was 0.6825 cm (0.2687").

This value had converged. More detailed statistics of the stone distribution are given in

 

Table l, tl

presented

2.2. Win

2.2.1. Ht

Wind tun
in Figure
controlle
88 the rel
measure]
and 110m
gradient
the 10p 9

preSSuIe

10

Table 1, their height distribution is shown in Figure 2.3, and the stone surface is

presented in Figure 2.4.

2.2. Wind Tunnels
2.2.1. Hot-Wire Wind Tunnel

Hot-wire measurements were carried out in the 17 meter (56 foot) low-speed
wind tunnel in the Turbulence Structure Laboratory at Michigan State University, shown
in Figure 2.5. This suction tunnel is in the middle of a pressure- and temperature-
controlled laboratory 30.5 x 18.3 x 6.1 meters (100 x 60 x 20 feet). The laboratory acts
as the return circuit when the radial diffuser is attached, as was the case for all hot-wire
measurements. The wind tunnel test section is 17.1 m (56') long, 1.21 m (4.00') wide,
and nominally 0.61 m (24") high. The top wall is adjusted to provide a zero pressure
gradient with smooth-wall boundary layers developing inside the tunnel. This results in
the top wall diverging from the bottom at a constant 0.25 degrees. The differential
pressure gradient, de/dx, where C, = 2p/per2, is less than :I:0.002 for a smooth wall.
(The same wall divergence was used for these rough-wall studies, which have slightly
steeper boundary layer growth on one of the wind tunnel's four walls. However,
measurements were taken roughly thirty boundary layer thicknesses downstream of the
smooth-to-rough transition, and this transition is after ten meters of smooth-wall
boundary layer development. The boundary layer growth rate at the measurement
station, therefore, is minimally different from the smooth-wall growth rate, and may
result in a very slight favorable pressure gradient.) There is no inlet contraction, and the
resulting freestream turbulence intensity is less than 0.2%. A 6.35 mm (0.25") diameter

 

threaded

more dett

2.2.2. D1

Vt
Turbulen
inlet cont
0.3 degre
732 cm 11
diameter
balance i

Shown in

2.3, 1),,“

11

threaded rod trips the boundary layer 50 cm (20") downstream of the tunnel inlet. For
more details on this wind tunnel, see Rashidnia and Falco (1987).

2.2.2. Drag Measurement Wind Tunnel

Wall drag measurements were made in the higher-speed wind tunnel at the
Turbulence Structure Lab. This wind tunnel is also a suction tunnel, however it has an
inlet contraction ratio of 13.6: 1. The top wall of the tunnel diverges hour the ﬂoor by
0.3 degrees to maintain a zero pressure gradient in the test section. The test section is
732 cm long, 122 cm wide, and 55 cm high (24.0 x 4.0 x 1.8 feet). A 1.27 cm (05“)
diameter trip rod is located 45 cm (18") into the test section. The center of the drag
balance is 499 cm (196.5") downstream of the trip rod. The tunnel conﬁguration is
shown in Figure 2.6. For more details, see Falco (1980).

2.3. Drag Balance

Drag measurements were performed in the higher speed wind tunnel at the
Turbulence Structure Laboratory, described in Section 2.2.2. This tunnel is equipped
with a Viewstar drag balance with a range of up to 200 grams force and rated accuracy of
10.1% of reading $0.02 gmf. A photograph of the drag balance is given in Figure 2.7.
The center of the drag balance is 499 cm (196.5 ") downstream of the boundary layer trip
rod. The drag balance is housed in a plywood vault to ensure that there is no mean ﬂow
around the edges of the drag plates.

The drag balance‘s principle is very simple. The test model is mounted on a

granite block, which in turn is supported by a cushion of air. The granite block is free,

 

 

within a
piezoelec
proportic
a digital 1
was used
81

dynamic
to a varyi
St
The "frict
that a ven
magnitud.
F c
Spring (a I
did 8.fter a
is referred
de‘dCe fric
mee to th
in that the;

woﬁsitt‘

24' Dram

A p]
drag balanc,
31'0qu each

12

within a very limited range, to slide horizontally. The block pulls a chain, which places a
piezoelectric crystal in tension. The voltage signal generated by the crystal is
proportional to the horizontal force applied to the test model. The output is provided on
a digital display in units of grams force, and an analog voltage signal is also provided and
was used for time-averaging the drag force.

Static calibration of the drag balance was performed over its full range, and a
dynamic veriﬁcation was executed to ensure that the balance was responding accurately
to a varying applied force.

Static calibration was performed by hanging weights from a "frictionless pulley."
The "frictionless pulley" has no moving parts, but uses a 45° support member to ensure
that a vertical gravitational force is converted to an applied horizontal force of equal
magnitude.

For the dynamic veriﬁcation weights were bounced from a soft, low damping
spring (a Slinky). The balance indicated the same mean force during the bouncing as it
did after all motion had ceased, and the time-varying output signal was a sine wave. This
is referred to as "veriﬁcation" rather than "calibration" because we were not able to rig a
device frictionless enough to give a true dynamic reading after converting a gravitational
force to the horizontal force measured by the drag balance. The readings were dynamic
in that they indicated a sine wave force input, but the minimum reading did not approach

zero as it theoretically should.

2.4. Drag Plates

A plate of each rough surface was constructed speciﬁcally for attachment to the
drag balance. The surface area of each plate was approximately 0.4 m2 (4 ft2). The gap
around each plate was roughly 2 mm (0.08"). The gap along the leading edge of each

 

 

plate wa
wind tun
touching
little effe

2.4.1. St

50.8 cm
edge of t
Roughne

13

plate was minimized by attaching a strip of very thin sheet metal on the underside of the
wind tunnel (inside the drag balance pressure chamber) to cover most of the gap, without
touching the drag plate. Small variation in the placement of this strip was found to have

little effect on the drag measurements.

2.4. 1. Square Rod Roughness Drag Plate

The drag plate for the rod roughness was 76.2 cm in the streamwise direction by
50.8 cm wide (30" x 20") to provide 15 complete cycles of roughness. The upstream
edge of the plate was a valley; the downstream edge was a peak (see Figure 2.1).
Roughness plates of 76.2 cm by 34.6 cm (30" x 13.6") were set on each side of the drag
plate to ﬁll the width of the wind tunnel. A sheet of roughness 55 .9 cm (22") long and
the width of the wind tunnel was also set downstream of the drag plate to eliminate any
sudden change in roughness adjacent to the measurement station.

A sheet metal ramp 30.5 cm (12.0") long brings the wind tunnel ﬂoor up 2.54 cm
(1.0") to the level of the peak of the roughness elements. A smooth region of 13.3 cm
(5.25") connects the ramp to the rough surface. Development length over the roughness
was measured from the start of the roughness to the center of the drag plate, so no
roughness in front of the drag plate resulted in a 38.1 cm (15") development length.
Roughness was added upstream in increments of 25.4, 30.5, and 55.9 cm (10, 12, and 22
inches; or 5, 6, and 11 roughness cycles).

 

 

 

2.4.2. Re

'11
54.6 cm v
continuou
stone was
stationary
The remai
attached.
movemen‘
t1““161. am
The last (i
deveIOpmq
Pram then
to 20 inch.
tuIlnel up 1

C0;
vorticity in

all X‘arl‘ay
but is “$131

centered be

14

2.4.2. Rock Roughness Drag Plate

The drag plate for the rock roughness was 80.0 cm in the streamwise direction by
54.6 cm wide (31.5" x 21.5 "). Great care was taken to make the rock distribution as
continuous as possible across the gap between the drag plate and the tunnel wall. No
stone was allowed to be in contact, direct or indirect, with both the drag plate and the
stationary tunnel wall. All stones along both sides of this juncture were glued down.
The remainder of the stones, both on the tunnel ﬂoor and on the drag plate, were not
attached. The density of their distribution provided sufﬁcient stability to prevent their
movement. Rocks were placed on both sides of the drag plate to ﬁll the width of the
tunnel, and the rock surface continued for 30 cm (one foot) downstream of the drag plate.
The last (furthest downstream) row of these rocks was also glued down. The minimum
development length studied, measured to the center of the drag plate, was 56 cm (22").
From there, rocks were added in increments of from 25 to 51 streamwise centimeters (10
to 20 inches). A sheet metal ramp 30.5 cm (12.0") long brought the ﬂoor of the wind
tunnel up to roughly the level of some of the higher peaks (8 mm, or 0.3").

2.5. Hot-Wire Equipment

Constant temperature anemometry was performed using a four-wire spanwise
vorticity probe constructed at the Turbulent Shear Flows I laboratory at Michigan State
University. The vorticity probe, shown in Figure 2.8, is composed of a parallel array and
an X-array. The X-array is located at the same streamwise location as the parallel array
but is displaced minimally in the spanwise direction. The center of the X—array is

centered between the parallel wires in the wall-normal direction. The spacing between

 

wires with
the arrays
Inc
active regi
diameter r.
jeweler's b
measurem.
Th1
using a cat
Wires were
(0.001 "),
The
fed throng]
mOdel 332'
TWISIatioI
hold capab
The
differential
condiu'Oner

inpUI data ‘

2.6. Data C

15

wires within each array is nominally 1.0 mm. The center-to-center spanwise spacing of
the arrays is typically around 3.5 mm.

Individual hot-wires are 3 mm long with a center active region of 1 mm. The
active region is 5 pm diameter tungsten wire; the ends are copper-plated. The length-to-
diameter ratio is nominally 200. The wires are mounted on the ends of 20 mm long
jeweler's breaches. The broach length is eight times the tapered probe head height. All
measurements were taken at an overheat ratio of 1.7.

The wires could be located vertically (wall-normal) to a precision of 0.01 mm
using a cathetometer mounted outside the wind tunnel. After initial positioning, the
wires were moved using a vertical traverse mechanism with a precision of 0.0254 mm
(0.001").

The signal output of four DISA 55M01 constant temperature anemometers was
fed through a custom-built analog signal ampliﬁer (24 channels) and two Krohn-Hite
model 3323 analog ﬁlters (two channels each). The signals were collected by a Data
Translation DT3368IDT3369 analog-to—digital subsystem with simultaneous sample and
hold capability. The Data Translation boards are housed in a PDP 11/‘23 computer.

The freestream velocity was measured with an MKS Baratron type 398
differential pressure transducer connected to a Prandtl tube and an MKS type 270B signal
conditioner. The signal conditioner produces an analog voltage signal which is used as

input data during hot-wire calibration.

2.6. Data Collection and Reduction Equipment

Data were collected on PDP 11/23 computer running on the RSX-l 1M operating
system. The system is limited by a 40 megabyte hard drive, which requires data transfer
after data for ﬁve points in space have been collected (based on the sample sizes used for

 

the rough-
temporaril
and writter
environme
statistics, V

extracted f

16

the rough-wall proﬁles presented in this study). Data are transferred and stored
temporarily on a PDP 11/7 3 operating under RSX-11M+, then they are demultiplexed
and written to a Sun SPARCstation 2 in one step. The data are then reduced in the Sun
environment. The voltage series are converted to velocity series, and then turbulence
statistics, velocity gradients, and other values of interest including spanwise vorticity are

extracted from the four velocity series.

 

3.1. DragI

Chapter 3

Drag Measurements

3.1. Drag Estimate Options

Accurately estimating the skin friction over a rough surface is a notoriously
difﬁcult procedure. Estimates are usually based on one of four general methods:
momentum balance, some variation of the Clauser plot technique, measurements of the
form drag around individual roughness elements, or extrapolating the Reynolds stress to
the wall. The reliability and feasibility of these methods were considered to be
insufﬁcient for this study, so direct drag measurements were taken. This is in line with
the conclusion of Acharya & Escudier (1983) that "a direct measurement of the wall
shear stress is important in all but the simplest of boundary-layer ﬂow situations."

The momentum balance technique relies upon simplifying the momentum integral
equation for steady, two-dimensional, incompressible ﬂow. This yields cf = 2(d0/dx).
However, any deviation from two-dimensionality of the ﬂow seriously inhibits the
accuracy of this technique. Regardless, to take complete proﬁles at enough streamwise
locations and with sufﬁcient care to ensure reliability in the gradient d0/dx would take
extraordinary effort. More typically, the momentum thickness has been calculated at
four or ﬁve streamwise locations and these data have been curve ﬁt, often to the ﬁrst
order and often with signiﬁcant scatter. Gradient information is then extracted from this
curve ﬁt. Pineau et al. (1987) report that this method of estimating Cf gives unreliable
results, and previous studies at the Turbulence Structure Laboratory (Rashidnia 1985,

17

 

 

 

 

Signor 198
streamwise

Ind
surface. T1
boundary 1
coefﬁcient
Virtual orig
the measun
810136; this:
The rough:
muShtless;
53-) The i
satiSfY the :
Antonia &
Combinatic

Fur
r0d l'Ollghn
she” Reg
impracticaj

It v,
Value of tht
known, am

18

Signor 1982) have yielded inconsistent values for the momentum thickness and its
streamwise gradient in both wind tunnels used in this study.

Indirect techniques for estimating cf lose their viability for ﬂow over a rough
surface. The universal representation of the mean velocity proﬁle of a rough-wall
boundary layer requires knowledge of two parameters in addition to the skin friction
coefﬁcient: a, the location of the virtual origin, and AU/u—e, the roughness function. (The
virtual origin is the y-location at which an imaginary smooth wall would have to be for
the measured velocity proﬁle to yield a logarithmic velocity distribution with appropriate
slope; this is discussed more thoroughly in Section 5.2.2 and is illustrated in Figure 2.1.
The roughness function is a measure of the velocity defect caused by the surface
roughness; this is discussed more thoroughly in Section 5.4 and is illustrated in Figure
5.8.) The interdependence of e and Of allows many combinations of these quantities to
satisfy the slope requirement of the universal velocity distribution (see, for example,
Antonia & Luxton 1971). Then AU/u1 can be found, but is different for each
combination of e and Cf.

Furuya et al. (1976) report that form (pressure) drag measurements on spanwise
rod roughness underestimate the total drag by 30%, due to the unaccounted for viscous
shear. Regardless, form drag measurements of the rock surface studied would be
impractical.

It was felt for this study that to assume u? is identically equal to the maximum
value of the Reynolds stress is unjustiﬁed. Over rough surfaces this is certainly not
known, and over smooth surfaces Reynolds number dependence has been observed.

 

des
the
len;

pla1

dev
(4 t
dev
50 s
suff

Stati

Volt
RVer

com

19

3.2. Drag Measurement Procedure

The rough surfaces were conﬁgured in the drag measurement wind tunnel as
described in Section 2.4. For each roughness the transition ramp from the smooth wall to
the roughness is identical to that used in the hot-wire experiments. The development
length is deﬁned as the distance from the start of the roughness to the center of the drag
plate.

For each roughness the streamwise development length was varied, and for each
development length the freestream velocity was varied from approximately 1.0 to 25 mls
(4 to 80 ft/s) in about 12 increments evenly spaced on a logarithmic scale. For each
development length and velocity the horizontal force on the drag plate was averaged over
50 seconds and an average local drag coefﬁcient was calculated. Fifty seconds was
sufﬁcient time for roughly 500 boundary layer thicknesses to pass the measurement
station at a freestream velocity of 1.0 m/s.

The Viewstar drag balance is equipped with a digital readout which has an analog
voltage output. This analog output was fed through a DISA type 52830 true integrator to
average the drag over 50 seconds. The freestream velocity is derived from a Prandtl tube
connected to an MKS Baratron type 398 differential pressure transducer and an MKS
type 270B signal conditioner.

To test the repeatability of the measurements and to verify that 50 seconds is
sufﬁcient time for the average drag value to converge, 15 independent drag
measurements were made for each of two different velocities over the square rod
roughness with 206 cm (81 ") of development length. With the freestream velocity
9.48 m/s (31.1 ft/s), the standard deviation in the average local drag coefﬁcient of was
0.000108, for an uncertainty of :I:l .04%. With the freestream velocity 1.85 mls
(6.07 We) the standard deviation in of increases to 0.000424, for an uncertainty of
t3.62%.

 

 

installed in
gradient m

3.3. Drag 1

rod roughn-
Figures 5.3

two clear 9

2) after hit

20

The streamwise pressure gradient was not measured with the rough surfaces
installed in the drag tunnel. It is not known what effect, if any, a non-zero pressure

gradient may have had on these measurements.

3.3. Drag Measurement Results

The results of these drag measurements are shown in Figures 3.1 for the square
rod roughness and Figure 3.2 for the rock roughness. (These same data are shown in
Figures 5.3 and 5.4 with the velocity axis transformed to an approximate R5.) There are
two clear trends: 1) Cf decreases with increasing roughness development length, and
2) after hitting a minimum at some intermediate velocity, of increases with increasing
freestream velocity. At low speeds other interesting behavior is exhibited, and a
difference between the two surfaces may be detected.

The following is an explanatory note in regards to the data presentation.
Schlichting's (1937) curves (shown in Figure 3.3 and discussed in Section 3.3.2) indicate
that for a fully rough boundary layer ﬂow Cf is a function of erx/v and x/k only. The
present data, however, cannot be converted into the form of Schlichting's presentation,
because the origin of the boundary layer does not coincide with the leading edge of the
roughness. The same problem applies to determining the ratio x/k. Because of this, it is
impossible to say exactly where these data fall relative to Schlichting's curves. If the x-
origin is taken to be midway between the smooth-wall origin and the leading edge of the
roughness, then the dips shown in the present data occur at higher Rx than those shown in
Schlichting's curves. In order to bring the dips in the present data in line with
Schlichting's curves, the x in the ratio x/k would have to be interpreted as the streamwise
distance from the boundary layer origin (x = x, + x3), and the x in the Reynolds number
UooX/V would have to be interpreted rough-wall development length (x = x,).

 

Regs
beyond the l
MRX = 0'

3.3.1. Trent

The :'
Luxton (197
133’” thicknt
examined th
being the sa;
Surdy provic‘
x, is decreas
their 0011cm:
The form dr
element and
(1969) f0, (1.
thickness at

21

Regardless of the interpretation of x, the present data show Cf increasing well

beyond the Reynolds number at which Schlichting's curves are characterized by
ace/8R, = 0.

3.3.1. Trend #1: Skin Friction Decreases with Increasing Development length

The illustrated effect of development length is contrary to the result of Antonia &
Luxton (1971): "...the skin friction appears to adjust rapidly, within 3 or 4 boundary
layer thicknesses, to the new rough wall boundary condition." Antonia & Luxton
examined the response to a step change in roughness, with the rough surface geometry
being the same as the spanwise square rods of the present study. However, the present
study provides up to 5083 of development length, and, while the rate of change of c, with
x, is decreasing, the asymptote has not yet been reached. Antonia & Luxton arrived at
their conclusion by measuring the form drag of roughness elements, not the total drag.
The form drag method involves analyzing the control volume around a single roughness
element and assuming some quantities negligible and others equal. See Perry et al.
(1969) for details on this method. Antonia & Luxton also measured the momentum
thickness at several stations, but their data do not illustrate any clear second-order trend,
so a line with constant slope was drawn through their data of 0 vs. x, again suggesting a
constant of according the momentum integral method.

Support for the present result can be found in Acharya & Escudier (1986), who
measured wall-drag with a ﬂoating element as part of a surface which simulated the
roughness found on turbomachinery blading. They found of to decrease for the entire
length of their test surfaces, or almost three meters. Their boundary layer had a
momentum thickness growing, roughly, from one to 7 mm, so it can be safely assumed

that their ratio of ﬁght, is much greater than that of the present study. Acharya &

 

 

Escudier (1
(increasing
measuremt
full extent
conclude tl
spanwise s
Bandyopac
decreasing
matching 1]
trend.

The
increasgs v
Shear over

Iw must de

22

Escudier (1983) also used a ﬂoating element to show the decrease of c, with increasing x
(increasing R9 in their presentation). Schetz & Kong (1981) made direct drag
measurements with a ﬂoating element in a sandpaper surface and found deﬂax < 0 for the
full extent studied. Moore (1951) used the momentum integral method, but was able to
conclude that Of decreased across the entire length of the surfaces studied, which were
spanwise square rods with geometry identical to those of the present study.
Bandyopadhyay (1987), using the square rod geometry of the present study, shows of
decreasing with x. He used both the momentum integral equation and a velocity proﬁle
matching method, and the two sets of results differ widely but show the same general
trend.

The present result should be expected, because the boundary layer thickness
increases with increasing rough surface development length. This distributes the total
shear over a greater distance 6, so (barring an abrupt change in the velocity distribution)

1,, must decrease. It is not clear whether this effect is fully responsible for Trend #1.

3.3.2. Trend #2: Skin Friction Increases with Increasing Freestream Velocity

This result is in sharp contrast to commonly accepted engineering wisdom.
Nikuradse (1950, original work 1933) showed that, after a transition region, c, is
independent of Reynolds number in a fully-developed pipe flow. The skin ﬁiction is a
function only of the ratio of pipe radius to roughness height. Schlichting ( 1937, original
work 1936) extrapolated this result to external ﬂows, arriving at the conclusion that the
local skin friction coefﬁcient is independent of the streamwise Reynolds number UooX/V
(again, beyond a transition region), and is a function only of the ratio of the development
length to roughness height, x/k. These results, shown in Figure 3.3, are consistent with

Trend #1 above, as Schlichting shows Cf decreasing with increasing x/k, and therefore

23

decreasing with development length for a given roughness. However, this conclusion
also implies that of remains constant for a given x/k as Uoo increases. The present results
show that this is not the case. Schlichting's result is a product of calculations only; no
actual measurements were made. To this author's knowledge, no such drag
measurements in a rough-wall boundary layer had been made prior to this study, which
did independently vary both of Schlichting's parameters, erx/v and x/k.

One possible explanation for this surprising result was that the ﬂoating element
drag balance was not returning quickly after large positive excursions. This possibility
was eliminated after the dynamic veriﬁcation, which showed the average force applied
by a bouncing weight was the same as that applied if the weight were still.

There is limited support for the present conclusions in the literature. Acharya &
Escudier (1986) show a plot of C, vs. R9 (their ﬁgure 21), and on this plot are two curves
for ﬂow above the same turbomachinery-type surface but at different freestream
velocities. The higher-speed ﬂow has a higher Cf across the length of the ﬂow. (In their
plot, increasing R9 indicates increasing development length.) Acharya & Escudier
(1983) demonstrate this trend again in the same form for ﬂow over a cast surface (their
ﬁgure 4). No explanation or discussion is given in either case.

Moore (1951) examined turbulent ﬂow over three different sizes of spanwise

square rods of the geometry used in this study. He writes:

"For ﬂow in the fully rough zone, lines of constant x/k
should be horizontal and independent of the absolute size
of the roughness if the analytical considerations discussed
previously are correct. The experimental lines for x/k are
essentially horizontal, but an offset of about 10 percent
occurs with the change in the size of the roughness.
Careful checking of the experimental procedure failed to
disclose a reason for the offset of these curves." (p. 26)

The analytical considerations referred to by Moore are those of Schlichting which show
that Uoox/v and Uook/v (or UooX/V and x/k) are the only parameters relevant to Cf for a

 

 

fully r01
32 or Fi
be expre
the lead‘
means 0
values 0
roughne
Ueex/v.
ﬁrst poi
velocity
had char
through.
velocitie

Moore c

3.3.3. 1

24

fully rough ﬂow. To translate Moore's data into the form presented in Figures 3.1 and
3.2 or Figures 5.3 and 5.4 would require much supposition, and the present data cannot
be expressed in terms of erx/v because the boundary layer origin does not coincide with
the leading edge of the roughness. The following discussion is provided as an alternative
means of understanding Moore's result. Take any one of Moore's data points, with given
values of Jill: and Ueex/v. If, for the second point, Moore doubled the size of his
roughness, he would have to double x and halve Uoo to keep the same values of x/k and
erx/v. This, according to Schlichting's analysis, should yield the same value Cf as the
ﬁrst point. If, for a third point in the fully rough region, he then increased the freestream
velocity while holding x and k constant, Cf should again remain constant, as only erx/v
had changed. This did not happen; of increased from points 2 to 3. This trend holds
throughout Moore's data, which considered three sizes of roughness, three freestream
velocities, and ﬁve or six streamwise stations for each Dee and k It is impressive that

Moore could achieve such precision in of using the momentum integral method.

3.3.3. Low-Speed Effects

Both surfaces tested show interesting behavior at low freestream velocities. It
should be noted here that the boundary layer was tripped upstream of the leading edge of
each rough surface, and ﬂow was fully turbulent throughout these tests.

Over the rock surface, there is a sharp increase in c, from the ﬁrst to the second
point tested for each development length. As the velocity continues to increase, of
decreases to a minimum, then increases for the duration of the tests. This minimum
tends to become less distinct as the development length increases, and the minimum
occurs at lower speeds with increased deve10pment length. This allows for the

possibility of the minimum Cf occurring at a constant Uoox/v.

 

(We
confuses the
Mumsdm
developmen
layer only a
smmummm

Unlike the r:

25

Over the square rod surface, scatter in the data at some of the middle velocities
confuses the trends somewhat. However, for all except the two shortest development
lengths, the same sharp increase in c, at low velocities is exhibited. The two shortest
development lengths display the opposite trend, but these have allowed the boundary
layer only a few 5 to adjust to the new surface. Again, cf reaches a local minimum at
some middle velocity, and this velocity decreases with increasing development length.

Unlike the rock surface, the minimum remains distinct for all development lengths.

 

4.1. Probe

Th1
Figure 2.8.
Of Adjustec'
was consta
but PFObe 2
"000113th :
the same fr
311d the Spa
and they di

“sedwas 0.

Chapter 4

Hot-Wire Measurements

4.1. Probe Mounting and Calibration

The vorticity probe was conﬁgured as described in Section 2.5 and shown in
Figure 2.8. The two probe bodies were mounted in a probe-holder and never remounted
or adjusted for the duration of the rough-wall studies, so the pitch angle (in the x-y plane)
was constant for all proﬁles. The yaw angle (in the x-z plane) could have varied slightly,
but probe alignment was quite unambiguous, and any variation would have been
accounted for in calibration. No wires broke during the collection of data presented here;
the same four wires were used throughout. The spacing of the wires in the parallel array
and the spanwise spacing of the parallel and X-arrays were both measured periodically,
and they did not move noticeably once mounted. The y-separation of the parallel array
used was 0.96 mm (0.038"); the center-to-center z-separation of the parallel and X-arrays
was 3.58 mm (0.141").

The probe bodies were aligned with the freestream, so that each of the X-wires is
nominally at 45° to the ﬂow, but they cannot be constructed consistently such that this is
a precise measurement. Correction factors for deviation from 45 ° have been used in the
past at the Turbulence Structure Laboratory (most recently by Klewicki 1989), but the
procedure used to derive these factors (Cf and C,) depends on perfect stability in both the
freestream velocity and the data acquisition electronics. Such was not the situation for
this study, so, for reasons which are presented and justiﬁed in Appendix B the X-array
was taken to be ideal, with each wire inclined at 45° to the mean ﬂow.

26

 

 

 

C1
pairs to a
V is the k
deviation
ranged fr«
performe
unaccept
(roughly
inevitable

F
Pressure

Squares c”

27

Calibration curves for the wires were derived by ﬁtting sets of {velocity, voltage}
pairs to a King's Law type equation: E2 = A + BV“, where E is the hot-wire voltage and
V is the known velocity. A, B, and n are the constants that minimize the sum of squares
deviation of the measured points from the curve ﬁt. The value for n is typically 0.45 and
ranged from 0.36 to 0.56 for the 24 calibrations used in this study. Calibrations were
performed before and after each proﬁle was taken, and if the calibration drift was
unacceptable the data were not used. The time between calibrations was signiﬁcant
(roughly eighteen hours for the low Reynolds number ﬂows), so some drift was
inevitable.

For the initial calibration the velocity was deduced from the Prandtl tube and
pressure transducer. Then the member of the parallel array with the lower sum of
squares deviation was used as the reference velocity for the other wires. This helps to
eliminate errors in Bu/By due to mismatched calibrations.

The ampliﬁer channels were adjusted prior to calibration so that the channels
could be differentiated. The data acquisition equipment would on occasion "ﬂip"
channels during an acquisition period (see Appendix F). The only way to detect this was
to know the relative magnitudes of the four channels prior to data collection. This offset
between channels was minimal (typically 200 mV), so as not to lose resolution in the

digitizing process.

4.2. Data Sampling

Data were sampled at 500 Hz for the low speed (0.6 m/s) ﬂows, 1000 Hz for the
middle (1.8 m/s) ﬂows, and 2000 Hz for the highest Reynolds number (Uoo=3.0 m/s)
ﬂows. The low-pass cutoff frequency was set at one-half of the sampling rate for all

ﬂows. 1.792 x105 values were collected for each point in space (448,000 samples per

 

channel). A
Klewicki &
rms. value
statistics ex'

5% converg

4.3. Square

28

channel). According to the convergence criteria set forth by Klewicki (1989) (also in
Klewicki & Falco 1991), this quantity is easily sufﬁcient to guarantee convergence of all
r.m.s. values to within 3% of their true value, probably ensures fourth moments of all
statistics except velocity-vorticity products within 5%, and gives a very solid start toward

5% convergence of third moments.

4.3. Square Rod Roughness Set-Up

The roughness was put into the wind tunnel in the 55.9 cm by 121.9 cm (22" x
48") sections described in Section 2.1.1, and a ramp and smooth region (also described in
Section 2.1.1) brought the tunnel floor up to the level of the tops of the roughness
elements. Each rough plate was attached to the tunnel ﬂoor with a single screw in its
center.

Tests were run to verify that the development length of the roughness was
sufﬁcient for fully developed ﬂow. The rough plates were put into the tunnel one at a
time, and after each plate was ﬁxed in place the transition ramp and smooth region were
placed at the leading edge of the roughness. A single hot-wire was kept at the
measurement station, 3.8 cm (1.5") above the roughness peaks, which places the probe in
a plateau region of streamwise turbulence intensity. For each development length data
were sampled at 2000 Hz for two minutes in a 3 m/s (10 ft/s) freestream ﬂow. The hot-
wire was not calibrated, so r.m.s. voltage ﬂuctuations were interpreted directly as
turbulence intensity. The r.m.s. voltage had converged by the time 7 of the 9 plates were
in place.

Five full roughness cycles (25.4 cm, or approximately 1.05) were downstream of
the measurement station. This yielded 477.5 cm (188") of rough-wall development

length (leading edge of roughness to measurement station) when all nine plates were in

 

place. The
should be 1
parallel arr
described i

4.4. Rock

29

place. The four-wire vorticity array was centered above a roughness element. (This
should be kept in mind when examining the near-wall data.) The lower wire of the
parallel array was located relative to the roughness peaks using the cathetometer

described in Section 2.5.

4.4. Rock Roughness Set-Up

The rocks which compose the rock roughness were placed in the wind tunnel by
hand and were not attached to the wind tunnel wall. The rocks were placed in a single
layer and tightly packed so that each was surrounded by neighbors. Conspicuous peaks
were avoided, but the rocks were generally allowed to lie in whatever orientation came
naturally. A ramp brought the floor up to roughly the level of the roughness peaks, as
described in Section 2.1.2.

Tests were run to verify that the development length of the roughness was
sufﬁcient for fully developed ﬂow. The same test described in Section 4.3 was run, for
the ﬁrst time when the rock surface had 178 cm (70") of development length. At that
point, the r.m.s. voltage ﬂuctuation was already within 2% of its ﬁnal value. The r.m.s.
voltage continued to converge until a total of 470 cm (185") of rocks were placed
upstream of the measurement station. Twenty-ﬁve centimeters (1.08) of rock surface
were placed downstream of the measurement station.

To locate the lower wire of the parallel array, a machined metal block was placed
on the wind tunnel floor, and the cathetometer was used to position the wire relative to
the block. This required removing and replacing the rocks in the immediate vicinity of
the vorticity array for each of the rock surface proﬁles. Because of this, near-wall
conditions are different for each of the rock surface proﬁles. This should be considered

when examining the near-wall data.

 

 

Thi
from the he

5.1. Appl)

Dre

be “Signa
roughness I

the bOunda
these differ

Sch
funCtion 0f

Chapter 5

Data Interpretation and Analysis

This chapter describes the methods used to glean mean boundary layer parameters

from the hot-wire velocity proﬁles and presents the results of these analyses.

5.1. Applying of to Hot-Wire Proﬁles: Matching R5 and x/B

Drag coefﬁcients were measured directly in the drag tunnel, but values of of must
be assigned to the boundary layer velocity proﬁles measured in the hot-wire tunnel. The
roughness development length in the hot-wire tunnel was greater than in the drag tunnel,
the boundary layers were thicker, and mean velocities, in general, were lower. Due to
these differences, the critical parameters for a cf-matching process must be identiﬁed.

Schlichting's classical analysis argues that the local skin ﬁiction coefﬁcient is a
ﬁmction of only RK and the ratio of development length to roughness height:

c=c,(.U_,.%) (5.1)

The results given in Section 3.3 show that this conclusion is incorrect. The data,
however, do support the hypothesis that Of is a function of a Reynolds number based on
the boundary layer thickness and a ratio of development length to boundary layer
thickness. The boundary layer thickness of interest in the Reynolds number term is 6,,
because this is the value relevant at the measurement station. The development length

ratio, however, carries information about the interaction of the boundary layer with the

30

 

roughness 0
roughness, )1
boundary 1a;
was chosen '

Therefore, t]

31

roughness over the extent of the rough surface, and 6 is growing for the duration of the
roughness, x,. A single value of x,/6 is needed for the ct-matching procedure, so the
boundary layer thickness at the leading edge of the roughness, 6,, is used. This value
was chosen because this is the 5 on which the roughness acts with maximum effect.
Therefore, the relationship used for cf-matching is

U 5 x
= -°1—',-4 . 5.2
01 Ci[ v 5) ( )

1

5.1.1. R5 and x/B for Hot-Wire Proﬁles

The boundary layer thickness in the hot-wire tunnel is known quite accurately at
both the leading edge of the roughness (63) and at the measurement station (5,). 6, was
measured in these studies. 63 was calculated via the momentum integral equation by

assuming a power-law velocity proﬁle of the form

1

i=9)", (5.3)

The values of n in the exponent were taken from the shape factors of Klewicki's (1989)

smooth-wall proﬁles at similar freestream velocities:

H=§—=n+2, (5.4)
6 n

 

Klewicki's data were also plotted against sets of lines representing ideal power-law
proﬁles, and the shape factor method yielded the same values. The values of n chosen
were 4.9 for the 0.6 m/s proﬁles, 5.5 for the 1.8 m/s proﬁles, and 6.0 for the 3.0 m/s

proﬁles.

with A four
(1985) trac]
U00 3 3.0 II

5.1.2. R5;

No
boundary 1

wider and
0f the exp
the data 0.
in good a,

V:
sde- B:

the drag 1

in Place:

The Vail]
in the hc
thicknes

20'“ er

32

The momentum integral equation then yields

-2

5 = HF“)? (5.5)
V

 

with A found by knowing Klewicki's boundary layer thickness at one station. Rashidnia
(1985) tracked the smooth-wall boundary layer development in the hot-wire tunnel at
Uoo z 3.0 mls, and Equation 5 .5 yields values of 68 in good agreement with his.

5.1.2. R5 and x/6 for Drag Measurements

N o velocity proﬁles were measured in the drag wind tunnel. The smooth-wall
boundary layer thicknesses 68 were derived in the manner described in Section 5.1.1. A
wider and more complete range of Reynolds number was covered, however, so the value
of the exponent n was varied continuously. Appropriate values of n were gleaned from
the data of Winter & Gaudet (1973), and the calculated boundary layer thicknesses were
in good agreement with the data of Signor (1982) taken in the drag tunnel.

Very little is known about the ﬂow's behavior at the smooth-rough junction in this
study. Because of this, the average boundary layer growth rate over the rough surfaces in
the drag tunnel was assumed to be 2.2 times that of the same ﬂow were the roughness not
in place:

6. =(6.),, +2.2[(6.),, ~(6.),,]. (5.6)

The value 2.2 was arrived at by observing the difference in boundary layer growth rates
in the hot-wire tunnel. For example, at 3.0 m/s in the hot-wire tunnel, the boundary layer
thickness 68 is 14.84 cm at x8 = 10 m, corresponding to the start of the roughness. If the
smooth wall extended to the measurement station, the boundary layer would grow to
20.11 cm, for a growth of 5.27 cm over the last 4.78 m. With the square rod roughness

 

 

 

 

 

 

in place. tr
growth of
growth rat!
growth rat
proﬁles, a1

It i
not critica

seethatm

5.13. Re:

Sk

were deri‘
52 show -
*1/53 the 1
d0“ deﬁr

A:
generated

whiCh shc
RBynords

to 10we, 11

Proﬁle. T.
Eastman]
49.1%. T]

33

in place, the boundary layer thickness 6, is 25.9 cm at the measurement station, for a
growth of 11.1 cm over the last 4.78 m, the length of the roughness. The boundary layer
growth rate over the roughness can then be approximated as 11.1/5.27, or 2.11, times the
growth rate over a smooth wall. This calculation was averaged for the six hot-wire
proﬁles, and the value 2.2 was selected.

It is realized that this is an extremely rough approximation, but its imprecision is
not critical. The hot-wire tunnel ratio x,,/6s that is being matched is roughly 30. We will
see that mismatching this value slightly is not critical to the crmatching process.

5.1.3. Resulting Values of of

Skin-friction coefﬁcients for the velocity proﬁles measured in the hot-wire tunnel
were derived by matching Uoo6llv and x,/68, as described in Section 5.1. Figures 5.1 and
5.2 show three-dimensional surface plots of c, as the dependent variable and Um6,/v and
x,/6s the independent variables. These ﬁgures demonstrate that this matching process
does deﬁne a single value of Of for each rough surface.

Actual matching of these values was done by consulting tables of computer-
generated tables of R5r and x,,/6s for the drag tunnel ﬂows and then consulting Figures
5.3 and 5 .4, (The tables allow for accurate interpolation between points on the ﬁgures.)
which show the original data of Figures 3.1 and 3.2 with the velocity axes transformed to
Reynolds numbers U.»6,Iv. Note that lines of constant x1163 slope down from upper right
to lower left, because 63 decreases with increasing Uoo.

As an example, consider the derivation of c, for the R9 = 6140 rock roughness
proﬁle. The boundary layer thickness at the measurement station was 25.1 cm, and the
freestream velocity Uoo was 2.96 mls, resulting in a Reynolds number R5I( = Uoobr/V) of
49,100. The roughness development length (x,) was 470 cm, and the boundary layer

thickness

Xr/E’s of 31
this comb
xrz 170 c
more rapiu
skin-fricti
uncertaint
accurate)

due to est
general, h
because 0
gradients

34

thickness at the leading edge of the roughness (68) was 15.14 cm. This yields a ratio
x,,/6s of 31. Going now to Figure 5.4, and keeping in mind that tables pinpointed x,/6$,
this combination of R5 and x1163 is satisﬁed only at the point where R5r = 49,100 and

x, z 170 cm. (This corresponds to U00 :3 8 mls, and note that in this region Of changes
more rapidly with development length than with Reynolds number.) This yields a local
skin-friction coefﬁcient of = 0.0060, with (roughly) vertical error bars introduced by the
uncertainty in the development length ratio (due to estimates of 68, which are quite
accurate) and horizontal error bars introduced by the uncertainty in the Reynolds number
due to estimates of 6,, which are less accurate. (The error bars are not shown.) In
general, however, the longer horizontal error bars do not introduce too much imprecision
because of the shallow gradient with R5. At low freestream velocities, however, the

gradients are steep and uncertainties are more signiﬁcant.

5.2. Interpreting Hot-Wire Data

This section describes assumptions made and methods used in determining mean

length scales and ﬁ'eestream velocities of the boundary layer ﬂows.

5.2.1. Boundary Layer Thicknesses and Freestream Velocities

As described in Appendix F, the motor driving the hot-wire wind tunnel did not
maintain a steady speed, and this resulted in poor mean velocity proﬁles, especially at
low speeds. In order to determine the boundary layer thicknesses and U00, proﬁles were
plotted in dimensional coordinates (U vs. y), and faired curves were drawn through the
data by hand. The bias of experience was used to compensate for bad points: if a point

35

did not represent the long-term mean of the wind tunnel, it was not weighted heavily.
(The freestream velocity was not sampled continuously during these tests, but it was
monitored continuously on a digital display, and the motor's shaft RPM was monitored
periodically.) Different faired curves were analyzed for each proﬁle, and weighted
average values were selected. This method yielded the values of 6, 6*, 6, and U00 given
in Table 2. Boundary layer thicknesses are measured from the virtual origin (see Section
5.2.2) in all cases. Note that there was very little ambiguity in the mean velocity data of
the 3.0 mls proﬁles, the 1.8 mls proﬁles required somewhat more interpretation, and the
0.6 mls mean velocities leave all mean values open to some question. The 0.6 mls rock
surface proﬁle was by far the worst, so the mean velocity proﬁles presented in this case

(those shown in Figures 5.8 and 5.9) are faired curves.

5.2.2. Virtual Origins

The skin friction coefﬁcient Cf deﬁnes the slope of the logarithmic region of the
mean velocity proﬁle (see Clauser 1954). The virtual origin (or zero-plane displacement)
of a rough-wall boundary layer is the y-location at which a smooth wall would have to be
to give a logarithmic velocity distribution of the appropriate slope as deﬁned by cf. For a
roughness element of height k, the "error in origin" a is the distance from the element
peak to the virtual origin, and the "displacement height" dis the distance from the
element base to the virtual origin: a + d = k (see Figure 2.1).

The following points are made to offer an alternate way of understanding the
physical meaning of the virtual origin. Thom (1971) demonstrated experimentally that
"the zero-plane displacement of an aerodynamically rough surface can be identiﬁed with
the level of action of the drag on its elements." Jackson (1981) then showed analytically

that the location of the virtual origin is an expression of the distribution of the drag force

on the rou;
force. Thi

Per
ﬂow and ti

The resulti

leaves sigr
this is prol
mean drag
graphical r
as30¢“.rt1tine:

5.2.2.1. L

36

on the roughness, while the roughness length yo is an expression of the magnitude of this
force. This interpretation is accepted in the present study.
Perry et al. (1969) suggest that e is a measure of the interaction between the mean

ﬂow and the roughness, and use 8 as the critical parameter in an expression for AU/m.

The resulting empirical correlation of Perry et al.,

AU =5.7srog(8“* )-0.4. (57)
u V

1

 

 

leaves signiﬁcant scatter in the data (see Figure 5.11). Jackson's analysis indicates that
this is probably not appropriate scaling, as e is related to the effective location of the

mean drag, not its magnitude. It will be shown here that by using straightforward
graphical methods yo (which can be expressed identically in terms of AU\u«;) can be

ascertained with sufﬁcient accuracy to allow the above correlation to be bypassed.

5.2.2.1. Method of Locating the Virtual Origin

The graphical method of Perry & Joubert (1963) was used to measure a. The
skin-friction coefﬁcient is known, so a unique value of 8 satisﬁes the slope of the
logarithmic velocity distribution. This value was found by a trial-and-error slope

matching process, matching the logarithmic region of the mean velocity proﬁle to a line

i=l‘lf—L1n(__yU~)+A, (5.8)
U“, x 2 v

where Cf is known and A is chosen so magnitudes are matched. (8 is chosen so that the

of the form

data fall parallel to the line of Equation 5.8, then A is chosen so that the data and the line
coincide.) The wall-normal coordinate y is measured from the virtual origin, so 8 is

 

varied until
Figure 5.5.

0th
precision tr
layers prod
achieve go«
the lack of

the graphic

5.2.2.2. V

37

varied until slopes are matched. An example illustrating this process is provided in
Figure 5 .5.

Other investigators have found that elk cannot be measured with sufﬁcient
precision to regard it as anything other than a constant. However, the thick boundary
layers produced in the hot-wire tunnel allow large enough physical roughness scales to
achieve good precision in elk. The accuracy of these measurements is demonstrated by
the lack of ambiguity is selecting yo values (Section 5.2.3). If e is not selected precisely,

the graphical method of ﬁnding yo will clearly show a slope mismatch.

5.2.2.2. Virtual Origin Location Results

The two rough surfaces displayed distinct and opposite Reynolds number trends.
For the square rod surface, 2 increased with increasing Reynolds number (the virtual
origin moved closer to the wall), for the rock surface, a decreased with increasing
Reynolds number (the virtual origin moved away from the wall). For the square rod
surface, elk increased from 0.30 to 0.60, while for the rock surface elk decreased from
0.81 to 0.37. These trends are illustrated in Figure 5 .6.

Some investigators (e.g. Cowan 1968) have argued that elk can be considered a
constant. This would eliminate one unknown and allow the Clauser plot (with slope-
matching only) to yield of. Antonia & Luxton (1971), using spanwise square rods of
A = 4, assumed elk = 0.64, independent of both development length and Reynolds
number. They used the same method described here, but slope-matched to a skin-friction
coefﬁcient derived from the form drag method, which indicated that c, adjusted to the
new surface condition within a few boundary layer thicknesses. It has been shown here,

however, that Of continues to evolve far beyond the point at which the mean ﬂow is

considered "fully developed" in terms of turbulence quantities. Because Cf and e. are

interdepend'
is not Chang

same extent

5.2.3. Rout

Nov

velocity dis

can be use
mutate ﬁl
Figure 5.7.

Val
from 0.15 1

38

interdependent, this leads to the conclusion that if the shape of the mean velocity proﬁle

is not changing but Cf is, then 8 must vary with streamwise development length for the

same extent that Cf does.

5.2.3. Roughness Length. Yo

Now knowmg both Cf and e, the meteorological expression for the logarithmic
velocity distribution

3:11:1[1] (5.9)

u K ya

can be used to ﬁnd yo. This was again done graphically, trial-and—error, and resulted in
accurate ﬁts through the logarithmic region for all proﬁles. The method is illustrated in
Figure 5.7.

Values of yolk increased with Reynolds number for both of the surfaces tested,

from 0.15 to 0.18 for the square rod surface and from 0.041 to 0.074 for the rock surface.

5.3. Mean Velocity Distributions

The mean velocity distributions are scaled on the friction velocity and plotted in
Figures 5 .8 and 5 .9. Figure 5.8 presents the data in the standard engineering form of
Equation 1.2, while Figure 5.9 presents the same data in the meteorological form of
Equation 5 .9. The y-coordinate is measured from the virtual origin, von Karman's
constant K is taken to be 0.40, and Co, the additive constant in the logarithmic law, is
taken to be 5.1.

._ -“'"-“.

 

fairing-in 11

also faired

5.4. Scalir.

39

The R9 = 1550, rock surface boundary layer velocities shown are the result of a

fairing-in by hand. The other proﬁles shown are original data, but these proﬁles were
also faired in order to deﬁne U00, 6, 6*, and 0 (as described in Section 5.2.1).

5.4. Scaling the Roughness Function

The value of the roughness function is derived by direct measurement off the plot
of Figure 5.8. Because both rough surfaces are 'k-type', AU/uT should scale on the height
of the roughness, k. This is shown in Figure 5.10. Note that the ﬂows over the square
rods are fully rough, and they scale quite well. The ﬂows over the rock surface also
scale well, but the lower two Reynolds numbers are transitionally rough (5 < kuTIv < 70),
so their roughness functions fall below the straight line predicted for fully rough ﬂows.
Their deviation is in agreement with that shown in the data compiled by Clauser (1956).

The roughness function is plotted against eudv in Figure 5.11, and the correlation
of Perry et al. (1969, Equation 5.7) is indicated. The scatter is large, comparable to that
of other investigators. It is, however, interesting to note that each of the two rough
surfaces behave individually as though they may scale on e.

The roughness function is plotted against yoqu in Figure 5.12. The relationship
of Equations 5.12 and 5.13 should be identically satisﬁed, if the length scale yo can be
derived with precision. Note the minimal scatter, indicating that yo can be accurately
determined. Because y0 depends on the location of the virtual origin, this lends
credibility to the values of a determined in Section 5 .2.2.

5 .5. On the Accord of yo and AU/uT

If the engineering expression for the rough-wall logarithmic velocity distribution

 

 

 

2. ._. lm[_>l‘;)+co _ AU (5.10)
u, K v a1

is equated to the meteorological expression
51:1 31). (5.11)
M, 1‘ ya

the resulting equation can be written as
AU = lm(ﬂ]+co, (5.12)

u1 K v

or alternatively,

y. =-v-exrv[K[AU -C.)]- (5.13)
“1 “1

Equations 5.10 and 5.11 carry the same information, as both are derived based on the
assumption of the existence of a logarithmic region with slope dependent on the skin-
friction coefﬁcient. However, the accuracy with which the independently calculated
values of AU/u—c and y0 satisfy Equations 5.12 and 5.13 makes a statement about the
precision with which the length scale y0 can be determined.

The roughness functions AU/u1 and the roughness lengths ya for the six measured
rough-wall proﬁles are plotted against each other in Figure 5.12. Their scatter about the
line

 

AU =11n[22’i]+c, (5.12)
u K V

1

is minimal. This tells us that for a given velocity proﬁle with known values of of and 8,

yo can be determined with good accuracy.

41

A brief note on the upcoming terminology. In the proﬁle-matching process used
(Section 5.2.2.1), any number of {Cf, 8} pairs will satisfy the slope requirement imposed
by cf. If this is satisﬁed, the pair is deemed "appropriate." However, there is only one
true value of Cf, and therefore only one true {Cf, a} pair. This pair is deemed "correct."
There are many appropriate but incorrect {cf, 8} pairs.

The accord of the "independently" derived values of y0 and AUIu-g does not imply
that the chosen pair {cf, 8} is the correct pair. The magnitudes of the roughness length
and roughness function will necessarily correspond if they are chosen carefully, but
during this process it will become clear that the slope-matching is not accurate. The
mismatch is due to an inappropriate {c9 8} pair and is especially clear when ﬁnding yo

graphically.
Solving Equation 5.10 for AU/ua; and equating with Equation 5 . 12 yields
m(y_r'_)_m(2£.)-.,£ (5.14)
v v u1

which allows a non-graphical method for ﬁnding yo. The difference between ln(y+) and
KU/u-g is a constant in the log region, and this constant is equal to 1n(yo+).

That yo can be found with such accuracy as demonstrated in Figures 5.7, 5.9, and
5.12 renders empirical relationships of the type of Equation 5.7, which relates the
roughness function to a length scale, extraneous. If AU/ua; is to be expressed in terms of
a measured length scale, the length scale should be y0 and the expression is Equation
5.12.

5.6. On the Lack of Trend in 6 for the Square Rod Surface

The momentum thickness 0 is signiﬁcantly smaller in the lowest Reynolds

number run over square rods than for the two other proﬁles. Because all three flows are

42

in the fully rough regime, this is contrary to expectations. (The roughness Reynolds
number ku1lv is 40.9 for the 0.6 mls proﬁle. This is smaller than the 70 v/ua; required for
the fully rough regime according to the classical analysis of Nikuradse, but more recent
studies (Dvorak 1969) which consider roughness geometry show that k+ > 30 establishes
full roughness for spanwise square rods of A = 4.) The observed difference in 0 is much
more than can be explained by the uncertainty introduced in fairing the velocity proﬁles.

It is proposed that at this low ﬂow speed and large physical roughness, the
frequency Uoo/I: (12.3 Hz) allows the ﬂuid time to enter the cavities between rods, at
, which point the ﬂuid may ﬁnd a path of least resistance by traveling spanwise between
the rods. This would explain two of the surprising characteristics of this flow, the small
values of 0 and elk.

The large value of of assumed for this ﬂow implies a large momentum thickness
if a two-dimensional ﬂow is assumed. (Recall that of decreases with development length,
so it was even higher upstream of the measurement station.) If spanwise ﬂow is
established between the rods, however, the ﬂuid will eventually eject and carry its
spanwise velocity component away from the wall. This organized three-dimensional
motion would void the applicability of the momentum integral method of ﬁnding Cf and
would explain the otherwise contradictory phenomena of small 0 and large Cf.

The value elk of this ﬂow is smaller than that found by any other investigator
using this roughness geometry, but this ﬂow also has by far the smallest frequency Uoo/Ia.
Recall that a small value of 8 places the virtual origin near the peaks of the roughness,
and that the virtual origin is the level of action of the mean drag. This is compatible with
the idea of spanwise ﬂow between the rods, as this ﬂow would establish a new pseudo-
surface above the wind tunnel wall, allowing the mean level of streamwise no-slip to
move up. The "smoother" wall (somewhat like a 'd-type' surface in that it has motions
with the scale of the grooves in the grooves) thus created is not necessarily inconsistent

with the high skin-ﬁiction coefﬁcient of this ﬂow, because the ﬂuid which is trapped

1
l
1

between ele

elements.

 

43

between elements must have its streamwise momentum abruptly removed by those

elements.

Chapter 6

Turbulence Quantities

This chapter presents some of the data acquired by the spanwise vorticity probe in
the six boundary layers studied: three Reynolds numbers over each of the two rough
surfaces. The streamwise velocity component statistics are derived from the lower wire
of the parallel array, wall-normal statistics and Reynolds stress are derived from the X-
array. Fluctuating spanwise vorticity,
is calculated by making use of the difference in u-series of the parallel array, and Taylor's
hypothesis is used to transform the v time-series into a time-varying spatial gradient.

Note that the probe was centered above the roughness elements for the spanwise
square rod surface. This brings the no—slip condition closer to the probe than implied by
the y-coordinate location for the near-wall points. Also note that the rock smface in the
immediate vicinity of the probe was different for each boundary layer, and for the ﬁrst
two points of the middle Reynolds number the center of the probe was below the average
peak of the rocks.

The lowest Reynolds number ﬂow over each surface has been ﬁltered as
described in Appendix C, so that long-term deviation from the mean is not interpreted as

a turbulent ﬂuctuation.

45
6.1. Streamwise Fluctuations

The r.m.s. quantities u' are scaled on inner variables and shown in Figure 6.1.
The measurements of Grass (1971) over a rock surface and Antonia & Luxton (1971)
over a spanwise square rod surface with 7t = 4 are included for comparison. The present
proﬁles are generally in line with the data of most other investigators, though the peak
values of the present study are generally slightly lower. This is probably due to the
slightly larger friction velocities assigned to these data than assumed in other similar
studies. The lowest Reynolds number square rod data are certainly affected by this. The
lowest Reynolds number rock data show considerable scatter near the wall, but this is
probably primarily due to local effects from the random surface.

In order to gain more insight from the streamwise ﬂuctuations, third and fourth
moments will be examined.

Skewness is a measure of the shape of a probability distribution:
< u3 >

(u' )3 '

S(u) reﬂects the relative intensities of positive and negative ﬂuctuations. If positive

S(u) =

 

(6.2)

ﬂuctuations are generally more intense but less frequent than negative ﬂuctuations, this
will result in a positive skewness.

The present skewness proﬁles are presented in Figure 6.2. Klewicki's smooth-
wall skewness data, which are in good agreement with the measurements of other
investigators (see Klewicki 1989) are included for comparison. Two major differences
between the smooth- and rough-wall proﬁles are worthy of note.

The ﬁrst difference lies in the skewness magnitudes. For comparable Reynolds
number (the same Rx), the skewness is consistently greater for the rough-wall boundary
layers than for the smooth-wall boundary layers. (This effect is illustrated in Figures

6.2e, f, and g.) This indicates that the relative contribution to u' of motions causing

46

negative u fluctuations is smaller in a rough-wall boundary layer than in a smooth-wall
boundary layer.

The second difference is in the shapes of the skewness proﬁles. The smooth-wall
proﬁles all have a characteristic dip: the skewness hits a local minimum around y+ z 30,
increases out to y+ z 100, then decreases through the remainder of the boundary layer.
This clip is missing in the rough-wall proﬁles, most of which decrease monotonically
from the wall outward. The reason for the dip's absence is not that the near-wall data
have been missed because of the small viscous length scales and large roughness
elements; the shape of the rough-wall proﬁles corresponds well to the smooth wall in
both the nearest-wall and wake regions.

The S(u) results of Andreopoulos & Bradshaw (1981) for ﬂow over a sandpaper
roughness show the same wall-normal shift as the present data: the large positive
skewness values near the wall are closer to the wall for a smooth wall than a rough wall.
Their data points are too far apart, though, to either support or contradict the observation
of the decreasing "clip" with increasing roughness.

Kurtosis, or ﬂatness, is a measure of the contribution of the tails of a probability
distribution:

<u‘>

(u' )‘ '
Flatness indicates the length of the tails in a probability distribution, so large ﬂatness

K(u)=

 

(6.3)

indicates that large magnitude ﬂuctuations make a signiﬁcant contribution to u'. The
present kurtosis proﬁles are presented in Figure 6.3. Klewicki's smooth wall kurtosis
data are included for comparison. The magnitudes of the kurtosis proﬁles are similar for
all three surfaces, indicating that the relative contribution of intense motions (large

magnitude ﬂuctuations) to u' is similar for smooth- and rough-wall ﬂows.

47

6.2. Discussion of the Skewness and Kurtosis Proﬁles

This discussion depends upon one premise: positive u ﬂuctuations are generally
caused by motion of higher-speed ﬂuid to a region of lower local mean velocity.
Acceptance of the premise allows positive u ﬂuctuations to be associated with wallward
(henceforth "downward") moving ﬂuid, and negative u ﬂuctuations can be associated
with upward moving ﬂuid. A negative skewness then suggests that the more intense
ﬂuctuations are generally associated with upward moving ﬂuid. The converse also
follows: positive skewness indicates that the intense ﬂuid motions are downward. In
each case, continuity demands that the more prevalent trend is counter to the violent
motions.

Examining the S(u) data (Figure 6.2) within the framework of the premise, the
larger skewness magnitudes of the rough-wall ﬂows (compared to the smooth-wall
ﬂows) suggest that at least one of two possibilities has occurred: either (1) the relative
contribution to u' of the upward moving ﬂuid ("ejections") has decreased, or (2) the
relative contribution to u' of the downward moving ﬂuid ("sweeps") has increased.

If (1) has happened, then the contribution to u' of large magnitude ﬂuctuations must
decrease, so K(u) would decrease. If (2) has happened, then the contribution to u' of
large magnitude ﬂuctuations must increase, so K(u) would increase. The kurtosis
magnitudes, however, remain generally similar. This suggests that both (1) and (2) have
happened to some degree.

Now the premise will be applied to the smooth-wall skewness dip and its
disappearance in rough-wall ﬂows. If the skewness dip is missing because of stronger
positive ﬂuctuations (downward motions) counteracting the intense negative ﬂuctuations,
K(u) in the dip region would be higher. This is difﬁcult to interpret, because the nearest-
wall data do indicate larger ﬂatnesses in the rough-wall ﬂows, but the sense of these

increases is the same as that observed in the very near wall region of the low R9 smooth-

48

wall ﬂow and the wall-normal shift of the skewness proﬁles. However, the skewness dip
in the smooth-wall ﬂows is accompanied by an increase in the kurtosis. (S(u) and K(u)
are plotted together for each proﬁle in Figure 6.4.) This increased kurtosis reﬂects an
increase in the contribution of large magnitude ﬂuctuations to u', and it can be assumed
that the same mechanism causes both the skewness dip and the kurtosis increase. This
kurtosis increase is absent in most of the rough-wall boundary layers. The exception is
the low R9 rock roughness boundary layer. This transitionally rough ﬂow shows S(u)
hitting a local minimum and K(u) hitting a local maximum at y" z 70. The middle
Reynolds number rock roughness boundary layer also shows evidence of these extrema
occurring at y+ z 70. These extrema occur in the smooth-wall ﬂows around y"' z 40.
The shape of the streamwise skewness proﬁle could be a good way of identifying
the effect of surface roughness on a ﬂow. None of the three square rod proﬁles or the
high R9 rock proﬁle show any sign of the smooth-wall skewness dip. However, the low
Reynolds number rock surface proﬁle does show evidence of the smooth-wall skewness
dip, but this is also the most smooth-like rough-wall proﬁle shown. The roughness
Reynolds number for this ﬂow, kudv, is 16.1. As this is essentially a sandgrain type
surface, k+ > 70 is required for "fully rough" ﬂow. The middle R9 rock surface ﬂow also
ﬂattens out a bit at the beginning of the logarithmic region, and this ﬂow has k+ = 47.0,
which categorizes it as another "transitionally rough" ﬂow. The other four proﬁles are
all "fully rough," and all show S(u) monotonically decreasing from the wall outward.
Here is another way of analyzing the meaning of the shapes of the skewness
proﬁles. Near the wall the mean velocity is low, so the only large ﬂuctuations must be
positive, so S(u) must be positive near the wall. Near the freestream the only large
ﬂuctuations must be negative, so S(u) must be negative near y = 6. In between, S(u)
must pass through zero, and the simplest proﬁle would show S(u) monotonically
decreasing. However, in the smooth-wall boundary layer S(u) crosses zero three times

and has several inﬂection points. The explanation for this lies in the strong structure of

49

the turbulence, with different motions dominant in different regions. In the rough-wall
boundary layer S(u) crosses zero only once and has fewer inﬂection points.

The lack of these distinctive features in the rough-wall boundary layers suggests a
more "generic" ﬂow through the logarithmic region, or a weakening (or total
transformation) of some motion which is dominant in smooth-wall boundary layers. If
the structure of the smooth-wall turbulent boundary layer in the range 20 < y+ < 100 is
modelled as a general trend of wallward-moving ﬂuid with the occasional violent
ejection (as suggested by the negative S(u) in this region), then the rough-wall turbulent
boundary layer in this range must be considered more symmetric in that the frequency
and intensity of wallward motions and ejections are more similar.

The speculative nature of this section can be eliminated by further analysis of the
data. Quadrant analysis will be most helpful and will clarify the degree of validity of the

premise.

6.3. Wall-Normal Fluctuations, Reynolds Stress, and Correlation Coefﬁcient

Wall-normal ﬂuctuations are sealed with inner variables and shown in Figure 6.5,
and again Klewicki's smooth-wall proﬁles are presented for comparison. The boundary
layers over both rough surfaces show a Reynolds number trend, with the peak in
normalized v' increasing with Reynolds number. This effect is also evident in Klewicki's
data.

The Reynolds stress <-uv> is scaled on wall variables and shown in Figure 6.6.
As Klewicki found for ﬂow over a smooth wall, the peak of the normalized Reynolds
stress increases with Reynolds number for the rock surface ﬂows. This trend does not
hold for the square rod surface; the maximum normalized Reynolds stress is slightly
greater for the middle R9 ﬂow than for the highest R9 flow. More noteworthy, however,

50

is the very low peak in <-uv> for the low R9 ﬂow over the square rod roughness. The
peak is around 0.31u12.

The correlation coefﬁcient, <-uv>/u'v', also hits a plateau around 0.30 for the low
R9 square rod surface boundary layer (see Figure 6.7), much lower than for any of the
other ﬂows in the present study. This is in contrast to the smooth wall study of Klewicki,
which showed a higher correlation coefﬁcient for the low R9 ﬂow than for the other two.

The correlation coefﬁcients are generally somewhat lower than those measured
by Klewicki, but the rock surface shows a trend similar to that displayed in Klewicki's

data in that the low R9 ﬂow has a noticeably higher correlation coefﬁcient than the two

higher speed ﬂows.

6.4. On the Small Magnitudes Measured by the X-array

The small measured values of v', <-uv>, and <-uv>/u'v' can be attributed to three
factors .

First is the relatively high friction velocities used for normalizing. Because u, is
squared in normalizing <-uv>, the effect of choosing a large u, is especially pronounced
her .

Second, X-wires have been found to signiﬁcantly underestimate Reynolds stress
in high turbulence intensity ﬂows (Tutu & Chevray 1975, and others). The turbulence
intensities near the wall in this rough-wall study were quite high, u'lU being over 0.70 for
the nearest-wall point in the low R9 square rod roughness proﬁle, and not dropping
below 0.30 until y+ > 70 for that boundary layer. For the high R9 square rod roughness
proﬁle, u'IU was near 0.50 for the nearest-wall point, and dropped below 0.30 around
y+ = 300. Tutu & Chevray indicate that X-wires could underestimate u', v', <-uv>, and

the correlation coefﬁcient, while overestimating the mean velocity U. It should be noted,

51

however, that agreement between the X-arrays and the straight wires was good for the
streamwise statistics, though the X-arrays did generally underestimate u' near the wall.
Also, values do not increase strikingly as local turbulence intensities decrease. For
example, looking at the high R9 rock surface proﬁle, the maximum value of <-uv>/u.;2 is
only 05, and this occurs in a region with u'lU :3 0.2.

Third, studies (Perry et al. 1987, Acharya & Escudier 1987) indicate that X-
arrays with apex angles of 90° (as used in this study) underestimate v' and <-uv> when
making measurements in rough-wall boundary layers. They ﬁnd that apex angles of
around 120° produce slightly larger and more accurate results.

All three of these factors have probably played some role in reducing the
magnitudes of the rough-wall data in the plots of Figures 6.5, 6.6, and 6.7. However,
even in combination, they do not explain the Reynolds number trends observed or the
extremely low magnitudes measured.

One result of this analysis is that it is risky to derive the skin-friction coefﬁcient
by extrapolating the Reynolds stress to the wall.

6.5. Spanwise Vorticity Fluctuations

The r.m.s. spanwise vorticity ﬂuctuations, (02', are normalized on inner variables
and plotted in Figure 6.8. The smooth-wall data of Klewicki lend support to the
hypothesis of a universal vorticity distribution. The two higher R9 ﬂows over the rock
surface fall roughly in line with Klewicki's data, but the ﬂows over the square rod surface
show a distinct Reynolds number trend.

The lowest Reynolds number over the rock surface shows higher than expected
normalized spanwise vorticity, if universal scaling applies. However, this is the proﬁle
which has very signiﬁcant long-term variation in Ugo. The ﬁlter described in Appendix

52

C has been applied to the data for all quantities shown, but this ﬁlter only removes
variations in the mean; it does not smooth Bulay. Because of this, the calculated values
of a); are unrealistically high: as the mean velocity varies, so does arr/3y. Therefore,
long-term variation in Bu/By, which is not representative of turbulence, is treated as such
when calculating (oz by Equation 6.1. The time-series data have not yet been
reevaluated, but when away is ﬁltered in a manner similar to that described in Appendix
C, the low R9 rock surface (01' results will decrease in magnitude. It is entirely possible
that the normalized (02' data will then fall in line with the smooth-wall data and the other
rock surface proﬁles.

This issue does not apply to the square rod data, which shows a clear Reynolds
number trend in (02' when sealed in inner variables. The degree of tunnel variation was
less than for the low R9 rock data, and Reynolds number dependence is displayed
between all three proﬁles. Also of note is the plateau in (oz for the lowest R9 square rod
proﬁle. This plateau extends out to y+ :3 100 before descending. The two higher speed
ﬂows over the square rods show similar or; distributions in terms of shape, and the outer
portion of the low R9 ﬂow declines similarly. The magnitude of normalized (0;,

however, decreases steadily with increasing R9 for ﬂow over the square rods.

6.6. Concluding Remarks based on the Hot-Wire Data

The data presented thus far in this chapter all indicate that the rock surface is
more "smooth-like" in nature than the spanwise square rod surface. The higher-order
statistics of the streamwise ﬂuctuations suggest this, and the spanwise vorticity proﬁles
show a clear difference between surfaces. Trends in wall-normal ﬂuctuations and

Reynolds stress are similar for smooth and both rough surfaces. The Reynolds stress

53

correlation coefﬁcient indicates similarity between the smooth and rock surfaces, but a
clear difference in the square rod surface is apparent.

This result is intuitively appealing, because the rock surface seems more "smooth-
like" in nature than the square rod surface, because it is random. It will not act to
reinforce any scale or frequency, because it has no repetitive scale of its own. The square
rod surface, however, has a cyclic nature of ﬁxed geometry, which may encourage
certain scales and reinforce certain frequencies. If we assume that characteristic scales of
ﬂow structure vary with Reynolds number, then as Reynolds number changes different
forms of structure may be encouraged or inhibited by the periodic nature of the square
rods. Perhaps a natural bursting frequency is stiﬂed by the given rod geometry at a given
ﬂow speed, channeling turbulent energy into alternative forms, as proposed in Section
5.6.

The skewness and kurtosis proﬁles presented in Section 6.2 and discussed in
Section 6.3 suggest that some turbulent motion which dominates the near-wall statistics
in a smooth-wall boundary layer is weaker in a rough-wall boundary layer.

Further examination of the data is certainly warranted. Higher-order moments of
wall-normal ﬂuctuations, Reynolds stresses, and spanwise vorticity will be calculated and
individual events will be studied. Simple ﬂow visualization will clarify many of the
questions raised here, and quadrant analysis will remove any ambiguity from the

discussion of Section 6.3.

Chapter 7

Conclusion

7.1. Summary of Results

Contrary to the classical analysis of Schlichting, the local skin-friction coefﬁcient
0f incfeases with Reynolds number beyond the transition region shown in Figure 3.3
(Schlichting's curves). For this study, drag force was measured over two rough surfaces
while varying both Reynolds number and rough-wall development length. The results

show that Of for a given surface is dependent on these two parameters.

The location of the virtual origin was found to be Reynolds number dependent,
but the sense of this dependence is different for the two surfaces studied. With
increasing Reynolds number, the virtual origin moves toward the base of the roughness
for the spanwise square rod roughness but toward the peak of the roughness for the rock
surface.

Numerous researchers cite the uncertainty in estimating the location of the virtual
origin as a major limitation to the interpretation of their data. This uncertainty, however,
is not a necessary consequence of studying rough-wall turbulent boundary layers; it is a
consequence of the scale of the boundary layers and roughnesses usually studied. The

present study used a 250 mm thick boundary layer and roughness elements 7 and 13 mm
high. With these scales, establishing e. is straightforward. For a given value of Of, a

54

55

unique value of 8 satisﬁes the logarithmic velocity distribution. The accord of yo and

AU/uq; supports the accuracy of e.

Skewness and kurtosis proﬁles of the streamwise ﬂuctuations, when taken in
combination, suggest that some motion which dominates the near-wall statistics of the
smooth-wall turbulent boundary layer is weaker in the rough-wall turbulent boundary

layer.

Spanwise vorticity ﬂuctuations were measured in boundary layers at three
Reynolds numbers over each of two rough surfaces. When scaled on inner variables, it is
likely that the vorticity proﬁles over the rock surface scale in the same manner that
smooth-wall vorticity proﬁles do. However, the vorticity proﬁles over spanwise square

rods show a clear Reynolds number dependence.

Reynolds number trends in wall-normal ﬂuctuations and Reynolds stress are
generally similar over both of the rough surfaces and a smooth wall: maximum inner-
normalized values increase with Reynolds number. Reynolds stresses, however, are very

low, especially over the square rod surface. This result is also reﬂected in the Reynolds

stress correlation coefﬁcient.

The present data suggest that extrapolating the Reynolds stress to the wall is not a
valid method of deriving the skin-friction coefﬁcient, especially at low Reynolds
numbers. The experimental methods of this study probably do act to systematically
reduce measured turbulence quantities, but the qualitative conclusions reached have not
been affected by experimental method.

56

The "dip" in the skewness of streamwise ﬂuctuations which is characteristic of
smooth-wall turbulent boundary layers (near the inner side of the logarithmic region)

decreases with increasing roughness effect.

7.2. Future Considerations

Much of the speculation in this thesis will be eliminated by ﬂow visualization
experiments, which are planned. The character of the low Reynolds number ﬂow over
the square rods is of particular interest. Light sheet visualizations in the x-y plane are
also planned. These ﬁlms will be studied in hopes of identifying characteristic motions
and scales in the boundary layer; the character of the ﬂow over the two rough surfaces
will be compared and contrasted; and Reynolds number dependencies will be examined.

The results of this study suggest that the virtual origin can be located quite
precisely for a known skin-friction coefﬁcient. Because researchers have had limited
success in generating correlations for Cf, it may be appropriate to tabulate elk values for a
variety of surfaces, Reynolds numbers, and development lengths. Then, with 8 taken as a
known value, Cf could be derived by a proﬁle-matching method. This would allow
slope-matching only, but in the present study slope-matching was sufﬁcient, as slope
mismatches were readily apparent.

Existing hot-wire data requires much more analysis. Higher order statistics of
wall-normal ﬂuctuations, Reynolds stresses, and vorticity ﬂuctuations should be
calculated and quadrant analysis should be performed. The spanwise vorticity
ﬂuctuations must be recalculated with anomalous Bulay ﬂuctuations removed, as

described in Section 6.4 and Appendix C.

57

The drag data show abrupt changes in Of at low Reynolds number. More data
should be collected to clarify the functional dependence of C, on Reynolds number at low
Reynolds number.

TABLES

Table 1.

Statistics describing the rocks which make up the rock roughness.

58

 

 

 

 

 

 

 

 

 

 

Average Height (= k) 6.825 mm (0.2687 '9
Maximum Height 11.887 mm @4683
Minimum Height 3.683 mm (0.145")
Median Heigh_t 6.464 mm (0.2545 ")
Average Absolute Deviation 1.3668 mm @05381")
Standard Deviation 1.6708 mm (0.06578")
Skewness 2.214
Stones per Unit Area 1.3158 stones/cm2
(8.4890 stonesfmchz)
Average Stone Volume 0.3071 cm3/stone

 

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APPENDICES

124

Appendix A. Veriﬁcation of Equipment and Methods

To be sure that all components of the hot-wire and computer systems (including
the operator) were working properly, a complete smooth-wall proﬁle was conducted at a
speed already studied in the same wind tunnel by Klewicki (1989), roughly 1.8 mls for
Re=2900. All statistics, including spanwise vorticity and higher order statistics such as
skewness and ﬂatness, match reasonably. The only difference worthy of note is
boundary layer thickness. Klewicki's smooth-wall boundary layers were consistently
thinner than those measured prior to this rough wall study. This is probably due to a
minor overhaul of the wind tunnel performed in the summer of 1991.

Some of the smooth-wall data are compared to Klewicki's results in Figures 8.1
and 3.2.

125

Appendix B. On the Effective Angles of the X-wires in the Vorticity Array

In previous work with the four-wire vorticity probe used for this study, the
effective angles of the X-wires have been calculated by subjecting the array to a flow at a
known yaw angle and comparing the apparent velocities seen by the slanted wires to
those seen with the probe body aligned with the freestream. This process yields
correction coefﬁcients Cf and C,, where the subscripts denote the forward and rearward
slanting wires (see Lovett 1982, but note the correct relationship between angles and
correction coefﬁcients below). However, for this study, the unsteadiness in the
electronics and freestream velocity (see Appendix F) rendered calculation of these
coefﬁcients meaningless. Because the scatter in measured values of Cf and C, was so
wide, there was no evidence that the wires were anything other than ideal; that is, each
wire inclined at 45 ° to the mean flow. Visually, the X-array appeared perfect. The wires
were quite straight, perpendicular to each other, and not obviously inclined at anything
other than 45 ° to the probe body. They remained in this situation when heated and
subjected to a cross flow. For these reasons, there was no better guess than to say the
wires were ideal, with Cf=1.0 and C,=-—l.0.

The smooth-wall data taken to verify methods and equipment were now used to
conﬁrm the validity of these estimates for the wire alignments. The data were ﬁrst
reduced using "best guess" coefﬁcients of Cf = 1.10 and CI = -0.60, which correspond to
the wires' effective angles being inclined from 45° by 2.7° and 14.3° respectively, for the
forward and rearward slanting wires. The inclination angles are related to the
coefﬁcients by

l-C,
1+C, '

 

65, = and

126

1+C,
1-C,’

 

561 =

where 65 indicates a wire's deviation from 45 ° in radians. These values were the average
of several results, but the scatter was very large. Absolute values of both Cf and C, fell
on both sides of 1.0. With these values of the correction coefficients, the new data did
not produce results that matched Klewicki's. However, with the assumption that 65f =
66, = 0 (a "perfect" X-array), Klewicki's results were reproduced. The effects of varying
the correction coefﬁcients and comparison with Klewicki's data are shown in Figures 8.1
and B .2.

The data presented in this thesis were all taken with the same four hot-wire array.
This was not only convenient; it was critical because of the inability to accurately
measure the X-array correction factors Cf and C,. For all of the data presented in this
thesis, Cf was taken to be 1.0 and C, to be -1.0, unless it is stated otherwise.

It should be noted that, in studying the effects of varying values of Cf and C,, it
was found that streamwise quantities, including skewness and ﬂatness, are only very
slightly dependent on the X—array correction coefﬁcients. This eliminates the possibility
of matching X-array u-statistics to those measured by the parallel array as a method of
discovering appropriate values of Cf and C,. Spanwise vorticity is only slightly more
dependent on the correction factors, but wall-normal measurements and especially
Reynolds stress are highly dependent on Cf and C,. Reynolds stress and spanwise
vorticity are illustrated in Figures B.1 and 3.2.

127

Appendix C. Smoothing Low-Frequency Fluctuations in Freestream Velocity

As explained further in Appendix F, the hot-wire wind tunnel did not maintain a
constant freestream velocity at low speeds. For the 0.6 mls runs, data were collected for
roughly ﬁfteen minutes at each point, and during the course of this ﬁfteen minutes,
signiﬁcant variation in the freestream velocity could often be observed. Figure C.1
shows the minute-to-minute variation in the average freestream velocity for one point.

This variation has a signiﬁcant effect on the turbulence statistics, especially the
streamwise quantities. To separate real ﬂow physics from effects of the wind tunnel
ﬂuctuations, the 0.6 mls proﬁles were ﬁltered. The data reduction scheme was modiﬁed
by sending the velocity time series through a 1001 point moving average ﬁlter
(essentially a high-pass filter). The average measured velocity for each 15 minute period
was deﬁned as the average for each moment within that period. The deviation of each
point from its local 1001 point average was then added to the 15 minute average to
generate a new velocity time series. The change in a velocity time series is illustrated in
Figure C.2.

The size of the ﬁlter was selected after running data through 501 point, 1001
point, and 2001 point ﬁlters, and ﬁnding the effect of each ﬁlter to be very similar. This
was expected, because at 0.6 mls data were sampled at 500 Hz, so a 1001 point ﬁlter
allows roughly two boundary layer thicknesses before and after the point of interest to be
included in each average.

The effects of this ﬁlter on turbulence quantities are shown in Figures GB, for the

low Reynolds number boundary layer over the rock surface.

128

Appendix D. Interpretation of Re=4610, Square Rod Data

The data for the middle Reynolds number over the square rod roughness was
stored on a hard disk that encountered difﬁculties. The data were corrupted and
irretrievable. At that time, the data had been reduced only once, with the X-array
correction coefﬁcients €51.10, C,=»0.60. In order to assign values to turbulence
quantities for the more appropriate values Cf=l.0, C,=-l.0, the trend in the effect of that

difference over the square rod surface was studied. The turbulence quantiﬁes of the high

 

and low Reynolds number ﬂows over the square rods were graphed for each
interpretation (Cf=l.10, C,=-0.60 and Ct=l.0, C,=-1.0), and the trend in the change in
statistics was applied to the middle Reynolds number square rod ﬂow. This procedure is
illustrated in Figure DJ for the Reynolds stress, and was also used to derive the proﬁles
for a); and v' for the Re=4610 boundary layer. Note that the correction coefﬁcients have

relatively little effect on the spanwise vorticity ﬂuctuations.

 

129

Appendix E. On the Accuracy of Power-Law Velocity Proﬁle Approximations

The scaling methods presented in Falco's 1991 paper rely primarily on three
relationships: the scaling of w; on inner variables, the Reynolds number trend of the
length scale trend of the vortex ring-like structures (Cy/6 vs. Re), and the Reynolds
number-Reynolds number relationship u16/v vs. Re. If rough-wall boundary layers are to
be sealed in a similar fashion, corresponding rough-wall relationships must be
established. This thesis presents data on possibilities for (02 scaling, and ﬂow
visualization studies are planned with hopes of establishing a relationship corresponding
to the smooth-wall Cy/6 vs. Re. The third component relationship is documented
through a literature survey.

The data of several investigators are presented in Figure E.1, showing smooth-
and rough-wall trends in u16/v vs. Re. These trends introduce an interesting issue: the
smooth-wall relationship can be derived approximately by making a power-law velocity
proﬁle assumption, but this derivation predicts a shallower slope for the rough-wall u16/v
vs. Re curve. The experimental data in Figure 13.1 suggest the opposite.

The derivation: assume a power-law velocity proﬁle of the form

1

i=(l)", (13.1)

 

 

U,” 6
andusethe fact that
C!
1 = 7U... (E2)
Then
14,6 0 Um6
= —’ (13.3)
v 2 v

 

 

130

and use the result from Hinze (1975) that a power-law velocity proﬁle yields

Jigga)? 1114)
2 v ' °

where B is a constant. Plugging expression E.4 into 13.3 yields

“15 = 3(U“5 )5. (5.5)

Then using the power-law relationship for momentum thickness to total boundary layer

 

 

 

 

 

 

 

thickness,
0 n
_ = E6
6 (n+1)(n+2) ( )
and plugging into E5,
1116 = B( (n + l)(n + 2) U,6)n+1 (13.7)
v n v
or
14,6 4'-
V = D(Ro)n+1, (E8)
where
D = Bl: (n + l)(n+2)]n+1’ (E.9)
n
which is a constant for n = constant. Equation E8 is the form of the smooth-wall
relationship used by Falco (1991),
“‘5 = 0.7612,”? (3.10)
v

The value of 11 most often used to approximate a smooth-wall velocity proﬁle is
seven, which yields an exponent in equation 13.8 of 0.875, in fairly good agreement with

equation BID. The exponent 11 also increases with Reynolds number, which causes both

131

the value of [n/(n+l)] and the "constant" D to increase. The data accumulated by Falco
permit, and perhaps encourage, the interpretation that the slope increases with Reynolds
number.

For a rough wall, however, the data do not conform to the prediction. The
exponent n for a rough-wall velocity proﬁle is considerably less than seven. Best-ﬁt
values of n vary from 3.0 to 4.5 for the velocity proﬁles presented in this thesis, with 11
generally decreasing with increasing Reynolds number for a fully rough ﬂow. This
predicts a much shallower slope of the curve u16/v vs. R9 for a rough wall than for a
smooth wall. The data show the opposite. For an equation of the form of 13.8, the value
of the exponent that best ﬁts the accumulated data base is 1.04.

This analysis leads to the conclusion that a power-law ﬁt of rough-wall velocity
proﬁles omits critical information. The most suspect link in the above analysis is
equation E.4, which relates Of to a Reynolds number for a given power-law proﬁle.
Relationships of this kind are especially weak for rough-wall boundary layers, even with
the shape of the-velocity proﬁle taken into account.

 

 

132

Appendix F. 01‘ Interest to Future Investigators at the Turbulence Structure
Laboratory

F.l Space Heater

The space heater in the northwest corner of the lab must be off during all hot-wire
data collection periods. If the heater is cycling on and off, it produces a distinct,
reproducible, and easily traceable temperature change cycle. Hot air is sucked the length
of the lab into the tunnel inlet and back down through the test section, and radiative
effects are noticeable almost immediately near the wall in the test section. (The wall is

painted black for ﬂow visualization experiments.)

F.2 Channel Flipping

The digital data acquisition system, as it is currently set up, occasionally ﬂips
channels. (The effect is channel ﬂipping, the cause is the skipping of a channel in one
sampling pass.) The frequency of these flips increases with sampling frequency.
Channel ﬂipping occurs in roughly 5% of the data sets sampled at 500 Hz and roughly
20% of the data sets sampled at 2000 Hz. There are two distinct types of channel ﬂips.
If channel ﬂipping occurs, the 10241st point is always skipped. This is the ﬁrst point of
the third buffer, and all data thereafter are shifted by one column. Also, any number of
points in the first line of data may be skipped. If there is a skip in the first line, then
there is exactly one more skip between line 1 and point 10241. No skipping occurs after
point number 10241.

133

After much investigation, the cause of this channel ﬂipping remains a mystery.
Data sets which were flipped cannot be rectiﬁed because calibration is channel-
dependent. That is, the voltage readings for a given input will vary slightly from channel
to channel. The solution is to offset the channel voltages so that the appropriate columns
can be recognized. Then each of the ﬁrst few buffers can be quickly averaged, and any
channel ﬂipping will be evident. If channel ﬂipping did occur, simply throw out the data

and repeat the point.

F.3 Unsteadiness in Electronics

The X-array calibration coefﬁcients Cf and C, cannot be measured with
repeatability because of unsteadiness in the data acquisition electronics. (The long-term
unsteadiness in the wind tunnel's freestream velocity only compounds the problem.) For
a given quantity being measured (zero velocity, for example), the channels will vary
from sample to sample, but will remain constant during a given sample, and the channels
will vary in unison. The variation is only a few millivolts, and each "bin" in the AID
conversion is 2.442 millivolts, so it appears to be a matter of indecisiveness in choosing
the initial bins. Once chosen, a constant analog input is properly converted to a constant
digital signal. Any variation during the run is an accurate representation of a varying
input.

This problem, too, was investigated thoroughly, but no satisfactory explanation
nor solution was found. The result is a bias in the mean velocities measured at each
point, but this bias is small, and higher order statistics are not signiﬁcantly affected.
(The range of zero to 3 mls covers roughly 1000 mV; the range of zero to 0.6 mls covers
roughly 500 mV. The velocity-voltage relationship is not linear, but 2 "bins" in the AID

134

card, or 4.882 mV, corresponds to the order of 0.01Uoo at 0.6 mls and 0.005Uoo at 3
mls.)

F.4 Unsteadiness in Wind Tunnel

The steadiness of the motor driving the hot-wire wind tunnel has degraded. It is
always difﬁcult to measure a smooth velocity proﬁle at Uoo=0.6 mls, but the current state
of the motor makes the proﬁle's jaggedness quite pronounced. The Eaton motor, with
Eaton Dynamatic 4000 controller, has a rated steadiness of :l:l% of full speed. It still
does perform in this range, so it is not broken and therefore cannot be repaired.
However, the :I:l% of full scale variation in shaft RPM translates to a :l:10% variation in
freestream velocity at 0.6 mls. (Freestream velocity is proportional to shaft RPM, and
top speed is roughly 6 mls, or 1000 RPM.) The performance for a given speed can be
optimized by adjusting the potentiometers inside the controller: zero adjust, maximum
speed, time constant, acceleration rate, and velocity damping. This is a nial-and-error
process, and the wind tunnel must be monitored for at least two hours at each

combination of settings to get a good idea of its performance.

F.5 Data Loss in Transfer

Hot-wire data are collected on a PDP 11/23 and transferred for temporary storage
on a PDP 11ﬂ3. All data appear to arrive intact on the PDP 11/7 3. However, some ﬁles
will not transfer from the PDP 11ﬂ3 to the SPARCstation 2 on which the data are
reduced. Usually three points out of a 25 to 30 point proﬁle will fail to transfer on the
ﬁrst attempt. If these ﬁles are copied into identical ﬁles, two of the three new ﬁles will

135

usually transfer. The third set of data, or one point per proﬁle, is usually lost. N o

explanation is available.

F.6 AID Sampling Order

The AID system described in Section 2.6 insists upon sampling sequential
channels beginning with channel 0. If ﬁve channels are being sampled, for example,
they must be channels 0 to 4, not 1 to 5 or any other ﬁve channels. If sampling is not
initiated at channel zero, every channel has a characteristic behavior: a sudden voltage
boost, then a steady drift downward. The boost-drift cycle is synchronized between
channels, and is roughly 10 Hz. The boost is on the order of 10 to 20 mV, but is not a

constant. As long as sampling is initiated at channel 0, this behavior does not occur.

F.7 DISA Integrator

The DISA type 52B 30 true integrator used to average the ﬂuctuating drag force is
accurate in that its output is a repeatable function of its input, but the correspondence is
not one-to-one. The integrator was calibrated several times during the period that drag
measurements were being taken, and the calibration curve did not change. Its output
ranged from well over 100% of its average input for input of a few millivolts, down to
just below 98% of its input for a one to two volt input. The calibration is accurate to a
third decimal place.

 

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LIST OF REFERENCES

146

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147

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152

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