ARCHIEF
An Experimental Investigation of the Structure
of
a
Turbulent Wind
over Water Waves
by
Susumu Kardki
and
En Yun Hsu
This research was sponsored by
National Science Foundation Grant GK-736
and
Office of Naval Research
Contract Nonr-225(71), NR 062-320
Distribution of this document is unlimited
s
Lab.
y.
Schèepsbouwkunde
Technische Hogeschool
DeIft
March 1968
Department of Civil Engineering Stanford University Stanford, California
AN EXPERIMENTAL INVESTIGATION OF THE STRUCTURE OF A TURBULENT WIND OVER WATER WAVES
by Susumu Karaki
and
En Yun Hsu
Technical Report No. 88
This research vas sponsored by National Science Foundation
Grant GK-736 and
Office of Naval Research Contract Nonr-225(71), NR 062-320
Reproduction in whole or in part is permitted for any purpose of
the United States Government.
Distribution of this document is unlimited.
ABSTRACT
This investigation presents experimental results from a study of the structure of a turbulent boundary layer developed by air flow over water waves in a laboratory wind-wave channel. It was demonstrated that the sheared flow region is two-layered, similar to rough wall
boundary layers. The inner layer, adjacent to the wall, is describ-able by the 'law of the wall' and the outer flow by the 'defect law.'
The data were taken in two-dimensional flow with small favorable
pres-sure gradients. The first part of the two-part investigation involved air flow over an initially still water surface on which waves developed
in response to wind excitation. In the second part, air was passed over water waves of a single frequency and varied amplitude.
The results indicated that within the range of this study the
water sürface can be classed as aerodynamically rough at wind speeds
greater than 12 fps and the flow structure of the boundary layer
cor-responds to that over a rough wall. The velocity decrement due to surface roughness is shown to be a function of the local
root-mean-square
wave height.
The efféct of wave-induced fluctuations in the air is limited
to a very close neighborhood of the wavy surface. For the specific combination of wind and wave speeds of this study it was found that
while the amplitudes of the horizontal components of the fluctuation
decreased with increasing elevation, the vertical component increased.
The latter result, however, may have been influenced by sample size in
vertical f luctuation component in the matched layer. Whereas the relative phase of horizontal and vertical components was greater than u!2 for the flow region below the matche4 layer, the -e1at-ive phase
was closer to zero ir the matched layer. Above the matched layer the phase difference gradually tended toward u!2 . The perturbation Reynolds stress, therefore, changed from negative below the matched
layer due to phase differences to positive in the matched layer. Above the matched layer thè wave-induced Reynolds stress tended toward zero
TABLE OF CONTENTS
-V-Page
1. INTRODUCTION i
1.1.
Scope of the Present Investigation ...
i
1.2. Limitations of the Study . 2
2. BACKGROUND FOR THIS INVESTIGATION 3
2.1. Introductory Remarks 3
2.2. Law of the Wall 4
2.3. Velocity-Defect Relationship 6
2.4. Influence of Flexible Boundaries 7
2.5. Wind Over Water Waves 8
2.5.1. Kinematics of the Flow 8
2.5.2. Work Done on the Waves 10
3. FACILITIES AND AN.RANGEMENT 15
3.1. Wind-Wave Channel 15 3.1.1. Geometry 15 3.1.2. Channel Characteristics 15 3.2. Special Arrangement 16 3.3. Appurtenant Apparatus 16 3.4. Instrumentation 17 3.4.Ï. Pressure 17 3.4.2. Mean Velocity 17 3.4.3. Wave Height 17 3.4.4. Turbulence 18 3.4.5. Recording Instruments 19
4. TEST PROCEDURE AND DATA REDUCTION METHODS 20
4.1. Mean Velocity 20
4.1.1. Boundary Layer 20
4.1.2. Shear Velocity 21
4.2. Turbulence 23
4.2.1. Hot Wire Response 23
4.2.2. Hot Wire Calibration 24
4.3. Data Reduction 26
TABLE OF CONTENT.S (continued) 4.3.2. Turbulence . 4.3.3.. Waie-Induced Fluctuations 5.' EXPERIMENTAL RESULTS Page 27 28 30
5.1. Turbulent Boundary Layer Over Sthooth Walls 30
5.1.1. Mean-Velocity Distributions. . . 30..
5.1.2. Turbulence Intensities . . 31
5.1.3. Skewness, Kurtosis and Intermittency . . .. . 32
5.2. Turbulent Boundary Layer Over Wind Waves 32
5.2.1. Mean-Velocity Distributions . 32
5.2.2. Turbulence Intensities ' 34
5.2.3. Skewhess, Kurtosis and 'Intermittency . . . . 35
5.2.4.. Summary of Turbulent Boundary' Layer over
Wind Waves . . . . 35
5.3. Wind Over Mechanically Generated Wave '. 36
5.3.1. Mean-Velocity Distributions . . . 36
5.3.2. Wave-Induced 'luctuations . .
. 37
6. CONCLUSIONS AND RECOENDATIONS 40
LIST OF TABLES
Number Page
5.1 Velocity Profile Data over Smooth Flat Plate 47
5.2 Velocity Profile Data over Wind-Generated Waves . . . 48
LIST OF FIGURES
2.1 Relationship of Aerodynamic Roughness to Shear
Velocity 49
3.1 Schematic Diagram of the Wind-Wave Channel 50
3.2 Wind-Wave Channel Test Section. Co-current Flow of
Air and Water Waves, Left to Right 50
3.3 Aluminum Plate Mounted in the Wind-Wave Channel . . . . 5]..
3.4 Special Apparatus for the Wind-Wave Channel Vertical Traverse Mechanism
Remote Control Panel and Position Indicator 52
3,5 Calibration Curves of Statham Pressure Transducer with
Sanborn 950 Carrier Amplifier and Harrison Mcromanometer 53
3.6
Typical Capacitance Wave Gage Calibration ...53
3.7(a) Sample Calibration of 1 mii Quartz-Coated x-wires
£/d=40
543.7(b) Sample Calibration of 0.15 mii Tungsten x-wires
2/d = 300 54
4.1 Block Diagram of Data Acquisition and Reduction
Processes 55
4.2 Typical Cross Plot for Determining u 56
4.3 Sensor Orientation Reference Diagram 57
4.4 Vector Geometry of Velocity Components 57
LIST OF FIGuRES (cOntinued)
Numbf
Page5.2 Non-Dimensional Pressure Gradiént ovet Smooth Plate . . 58
5.3(a) Veloòity Profile over Smooth Plate
Ç
= 8 fps . . . . 59 5.3(b) Velocity Profile over Smooth PlateÇ
13 f ps . . . 595.3(c) Velocity Profile over Smooth Plate.
Ç
18 fps . . . .60
5.4(a) vs. y+ Smooth Plate Station 13.4
61
5.4(b) U vs. Smooth Plate Station 23.4
61
5.4(c) vs.. y+ Smooth Plate Station 33.4
62
5.5(a) Velocity-Dèfèct Profiles Station 13.4
63
5.5(b) Velocity-Defect Profiles Station 23.4
64
5.5(c) Velocity-Defect Profiles Station 33.4
65
5.5(d) Velocity-Defect Profiles
Ç
18 fps . . .65
5.6 Power Law Fit to Outer Profile
66
5.7(a) Distribution of Turbulence
Ç
= 8 fp66
5.7(b) Distribution of Turbulence
Ç
13 fps61
5.7(c).
Distribution Of TurbulenceÇ
18 fps67
5.8(a) Distribution of Turbulent Shear Stress
Ç
= 8 fps .68
5.8(b) Distribution of Turbulent Shear Stress
Ç = .13 fps
.68
5.8(c) Distribution of Turbulent Shear Stress U 18 fps .
69
5.9(a) Kurtosis añd Intermittency at Station 33.4 overSmooth Plat Plate U= 18 fps
705.9(b) Skewness Factor Distribution over Smooth Flat Plate U = 18 fps . . . .
. 70
5.10 Pressure Gradient over Wind-Generated Waves
71
5.11. Non-Dimensional Pressure Gradient over Wind-Generated
Wavès . .
.. 71
5.12(a) Velocity Profile above Wind-Generated Waves J = 8 f ps
72
5.12(b) Velocity Profile above Wind-Generated WavesÇ
13 fps72
5.12(c) Velocity Profile above Wind-Generated Waves Um = 18 fps73.
+
...
,,.. .Number
5.14 Water Surface Texture for
Station 13.4 Station 23.4 Station 33.4
+
+
5.15 U vs. y Wind-Generated Waves 5.20 Aerodynamic Roughness vs. 5.21(a) 5.21 (b) 5.21(c) 5.22 5.23(a) 5.23(b) 5.23(c) 5..24(a) 5.24(b) 5.24(c)LIST OF FIGURES (continued)
Distribution of Turbulence Distribution of Turbulence Distribution of Turbulence Distribution of Shear Stress Distribution of Shear Stress Distribution of Shear Stress
Power Law Fit to Profiles on Outer Layer
U=l3fps
. . . . 755.19 Velocity Decrement Relationship to and
+
aU= 8fps
Ç
= 13 fpsÇ
= 18 fpsU= 8 fps
U = 13 fpsÇ
= 18 fps Page = 8 fps 74 7677
775.16 Water Surface Texture for
Ç
= 13 fpsStation 13.4 Station 23.4
Station 33.4 75
5.17 U vs. Wind-Generated Waves U = 18 fps 76
5.18 Water Surface Texture for = 18 fps
Station 13.4 Station 23.4 Station 33.4
Velocity-Defect Profiles, Wind-Generated Waves
U= 8fps
Velocity-Defect Profiles, Wind-Generated Waves U = 13 fps
Velocity-Defect Profiles, Wind-Generated Waves
Ç
= 18 fps 78 78 79 7980
80
8181
82
82
LIST OF FIGURES (continued)
NumL
Page\
.25(a). Kurtosis and Intermittency at Station 33.4; BoundaryLayer Development over Wind Waves
Ç =
.18 fps . . . . 835.25(b) Skewness Factor Distribution over Wind Waves
83
5.26 Velocity Profiles over Smooth Flat Plate with 3.5 in.
Drop below Intake
Ç
= 7.5 fps84
5.27 Velocity Profiles over Smooth Flat Plate with 3.5 in.
Drop below Intake
Ç
= 12.5 fps85
5.28 Velocity Profiles over Smooth Flat Plate with 3.5 in.
Drop below Intake
Ç
= 17.5 fps 865.29 Velocity Profiles over Mechanically Generated Waves
with Frequency = 0.76 cps .
...87
5.30: Characteristics of Mechanically Generated Waves In
the Wind-Wave Channel
88
5.31 Variation of < > with x and y
a=O.25in.
f=O.76cps
89
5.32 Variation of < > with. x and y
a = 0.25 in. f = 0.76 cps 90
5.33 Variation of
< t> with
x and ya0.25in.
f=O.76cps
91
5.34 Variation of < > with x and y
a = 0.50 in. f 0.76 cps
...
92
5.35 Variation of < > with x and y a = 0.50 in. f = 0.76 cps
5.36 Variation of < V >
with x
and ya = 0.50 in. f = 0.76 éps
94
5.37 Variation of < 1T > with x and y
a = 0.75 in. f = 0.76 cps . . . .. .
95
5.38 Variation of < > with x and y
a0.75in.
f=0.76cps
965.39 Variation, of < > with x and . y
LIST OF FIGURES (continued)
Number Page
5.40(a) Phase Relationship of Wave-Induced Fluctuations
a=O.25in.
f=O.76cps
98
5.40(b) Amplitude of Wave-Induced Fluctuations
a=O.25in.
f=O.76cps
98
5.40(c) Distribution of < > with ky
a0.25in.
f=O.76cps
98
5.41(a) Phase Relationship of Wave-Induced Fluctuations
a=O.SOin.
f=0.76cps
5.41(b) Amplitude of Wave-Induced Fluctuations
a=0.5Oin.
f=0.76cps
5.41(c) Distribution of < > with ky
a=0.5Oin.
f=0.76cps
5.42(a) Phase Relationship of Wave-Induced Fluctuations
a0.75in.
f=0.76cps
5.42(b) Amplitude of Wave-Induced Fluctuations
a=0.75in.
f=O.76cps
5.42(c) Distribution of < > with ky a = 0.75 in. f = 0.76 cps99
99
99
100
100
100
LIST OF SYMBOLS
a Wave amplitude (in.)
A Constant in hot wire respönse equation
-Am Coefficient for perturbation Reynolds stress
B Constant in hot wire response equation
c Wave celerity (fps)
C Constant in law of thé wall, (4.9)
d Hot wire sensor diameter .(in.)
D Constant in velocity-decrement relationship
e Base of natural logarithm
e Also without ambiguity, output voltage fluctuatjôn
E Hot wire anemometer output voltage
F Kurtosis
F Mömentum flux
w
-g Gravitational acceleration (fr/sec2)
h Water depth in wind-wave channel (in.)
H Wave height (in.)
k Wave number (ft')
Roughness height (in.)
K Correction coefficient for hot wire anemometer
Length of hot wire (in.)
Nu Nusselt number
p Static pressure at a point (psi)
Wave-induced pressure fluctuation (psi)
Rd Reynolds number based on wire diameter
'Y'
LIST OF SYMBOLS (continued)
u Fluctuation of velocity in x-direction, could include turbulent and perturbation components
u' Turbulent fluctuation in x-dïrection (fps)
Horizontal component of wave-induced fluctuation. (fps)
rms value of tur.bulent u' component (fps)
u Shear velocity
JT/P
(fps)Total instantaneous velocity (fps) Temporal mean velocity
Temporal mean velocity above boundary layer (fps)
U/uk
Fluctuation of velocity in y-direction Turbulent fluctuation in y-direction (fps)
Vertical component of wave-induced fluctuation (fps)
rms value of turbulent y' component (fps)
Coordinate direction of wave propagation (horizontal) Vertical coordinate direction (in.)
Aerodynamic roughness height determined from extension of log profile to zero velocity (in.)
Mean distance to matched layer (in.) yu/\)
Coordinate direction normal to x along wave crest
Flow angle to hot wire normal (deg)
Deviation of wire from 45° with probe axis (deg) Deviation of wire from 45° with probe axis (deg)
Ratio of pressure force to wall shear force U U
U'
U+ Vy
V A V Xy
+
y
z a ala2
LIST OF SYMBOLS. (continued)
Geometric angle f or vèlócity vector diagram (deg)
y Intérmittency
y' Geometric angle in velocity vector diagram (deg)
Boundary. layer thicknessat 0.99 U (in.)
m
Thickness of matched layer (in.)Displacement thickness'(in.)
E Adjustment to measured. y (in.)
Wave profile
e Momentum thickness (in o)
e Geométric angle in velocity vector diagram (deg)
X
Krtnn constant 0.41,p Fluid dynamic viscosity
Fluid kinematic viscosity
p Fluid density .
Standard deviatión of wave heights
T Wall shear (psf) .
Wave-indúced shear stress (psf) Phase angle of air wave (radians) Wäve-induced vorticity perturbation
AC KNOWLEDGENES
Appreciation is expressed to colleagues and members of the staff, Professors R. L. Street and B. Perry and Mr. T. R. Mogel for their helpful suggestions in the course of this research and in the
prep-aration of this report. Special appreciation is reserved for T. R. Mogel for his many suggestions concerning problems of instrumentation
and for design and assembly of special instruments and mechanisms essential to the successful conduct of this research. We are grateful
to Mr. P. Whitford, G. Schueller, and P. Jackson for their assistance
in computations and in preparing the figures of this report, and to
Mrs. Kay Mac Laury and Mrs. Ashby Longwell for typing the various
1. INTRODUCTION
The question of how wind action on water surfaces generates waves has attracted much interest in the past several decades. From a scientific viewpoint, we seek satisfactory explanation of wind-wave energy transfer processes. The problem is complex and involves detailed knowledge about the flow regions on both sides of the interface.
The large ntmber of publications concerning this subject gives evidence that the wind-wave relationship is difficult to explain physically and mathematically. More detailed experimental observations are evidently needed to determine some of the prominent characteristics of the flow and to gain better understanding of the mechanisms involved.
1.1. SCOPE OF THE PRESENT INVESTIGATION
This study is concerned with air flow above water surfaces. Tur-bulent boundary layers over rough solid surfaces, with and without pressure
gradients, exhibit two characteristic layers. The inner layer or flow region closest to the wall, where turbulence intensity is largest, follows the 'law of the wall,' in which mean velocity varies logarithmically with normal distance from the wall-. Typically this layer occupies less than 20 per cent of the total boundary layer thickness. The outer layer follows the 'defeòt law' which in fact also describes the velocity distribution through the logarithmic layer. The flow in the inner layer is completely turbulent while flow in the outer layer exhibits wake-like intermittent turbulence. The boundary layer over a smooth solid wall has, in addition to the inner and outer layers, a very thin 'laminar-like' region called the viscous sublayer adjacent to the wall. The sublayer, if it exists over water surfaces, is not within the scope of this investigation.
A logarithmic velocity distribution is commonly assumed to exist over water waves. Its existence seems to be strongly supported by obser-vations over oceans and land-locked bodies of water. The thickness of this region and hence of the inner layer of the boundary layer in natural conditions is difficult to establish, but investigators have, conveniently,
their range of interest. The'vàlidity of a logarithitic distribution has been questioned (cf., Stewart (1961, 1967)) on the basis that momentum exchange with the waves must surely affect the mean-velocitypröfile. Others only cbncedé uncertaintyabout the profile (cf., Miles (1957,
1967)). It seems appropriate therefore to investigate the boundary layer
over water waves in more detail, with particular attention to the region close to the water surface.
1.2. LIMITATIONS OF THE STtJDY
Invéstigätioñ into the structure of turbulent bounda-y layers over water wavés is limited n this study to small amplitude waves. To
estáblish a basis for comparison, data were assembled from boundary layers over a smooth plate within the wind-wave channel rather than to refer to the published information
which
are regardedas
èstablished standards. This was done primarily tO delineate facility characteristics, especiallythe géometric influênce Of the air intake on the boundary layer. Velocity and"urbulencemeansürements were made over waves generated by the air flow over an initially still water surface (wind waves) and over mechani-cally generated waves (mechanical waves) superposed on the. wind waves. The mechañical waves were of low frequency (0.76 cps) and had small
amplitudes. .
The wind speed was varied
f rom8
to 18 fps, and conditions were assuméd to be statistically stationary during measurements. The testsection was lm1ted to the upstream 40 ftof the. channel in regions where
two-dimensional flow existed and where, because the cross-sectional flow
areas were relatively constant, favorable pressure gradients existed.
- Data were taken with a fixed probe; accordingly, measurements could be made only down to the, level of the highest wave crest. This applies to both wind and mechanical waves. Determination of average
values at a given elevation (described in Ch 4) was limited to finite
time intervals, generally less than 2 minutes, because of the calibratØn drift of the hot wire ánemometer. The dr.if t of the anemometer posed a serious limitation to 'the study. '
2. BACKGROUND FOR THIS INVESTIGATION
2.1. INTRODUCTORY REMARKS
The sheared velocity profile over water waves is commonly given in the form:
u(x)
:i yo
where u is the shear velocity defined by
T
= pU
(2.1)
(2.2)
in which
t
is shear at the boundary and p is fluid density. Theempirical constant X is generally assumed to be 0.4 and y is the "aerodynamic roughness" of the sea surface. An impressive number of investigators have taken data over oceans and lakes, and their results seem to verify this profile shape. The significant results were sum-marized by Roll (1965). On the other hand, Eq. (2.1) apparently lacks universality, for a given aerodynamic roughness height fails to lead to a unique value of u for given wind speed, and further, y cannot
be determined easily for a given boundary geometry. It is instructive to note the wide variations in results of many investigators for
y0
as a function of u as depicted graphically in Fig. 2.1. Individua], studiesseem to present results which command a special section of the figure. Shemdin (1967) suggested that the technique of measuring mean velocities with a probe in fixed position above a wavy water surface requires cor-rection for the effects of shifting streamlines and of wave-induced
perturbations. Utilizing various wind speeds over mechanically generáted waves, he f:irst conducted experiments with fixed amplitude and varying
frequency from 0.6 to 1.2 cps, then fixed the frequency at 0.75 cps and varied the amplitude from approximately 1 to 3 in. The result was that in both cases he found aerodynamic roughness y to decrease with
in-creasing u up to about 1.5 fps and then increase with further increase
results of the present study are indicated on the graph.
2.2. LAW OF THE WALL
To establish the foundation for the first part of this investi-gation we look to turbulent boundary layers over smooth and rough rigid walls. It has been experimentally established that a turbulent boundary layer over a solid surface develops two dominant layers. The layer close to the wall behaves according to the law of the wall (cf., Clauser (1956), Coles,(1960)), that is,
for smooth walls, and
- ln + C
-u(x)
,( y1
yu(x)
u(x)X
nfor rough walls, where U/u is the velocity decrement expressed by
Thus, if is indeed the same for Eqs. (2.4) and (2.5), they may be combined to form
____ =
ln+ (C - D)
.(26)
u(x)
x k1In the equations above,
x
is the Kármán 'constant,' C and D are additive experimental constants, k1 is roughness height, and is the kinematic viscosity of the flowing fluid. For economy in writing it is customary to adopt the notation:u* U , (2.7) (2.4) +, C (2.3) k1u = - in + D. (2.5) u X V
yu*
- y+ , (2.8)
and
kiu
(2.9)
The velocity decrement is the reduction in velocity due to surface roughness and is represented by a vertical displacement of the logarithmic
relationship in a vs. y+ representation. The magnitude of X is
apparently not a constant, ranging in value from 0.38 to 0.44. In this respect the universality of the law of the wall is somewhat impaired.
The difference between Eqs. (2.. 1) and (2.6) is evidently only an
additive term, which under special conditions could be included in y
It is appropriate therefore that y in Eq. (2.1) is called aerodynamic
roughness height while k1 is a geometric roughness height. It is not
immediately evident why universality is achieved with one (Eq. (2.6)) but not the other.
One of the subtle differences between Eqs. (2.1) and (2.6) is the
manner in which the normal distance y is referenced. Ordinarily y is
the distance from a geometric mean surface. In Eq. (2.1) therefore, y
is measured from the smooth surface, while in Eq. (2.6) y is the distance
from an adjusted level which must be located for a given roughness height, type and distribution. For a smooth wall no question of the origin for
y arises, but for a rough wall the origin could be taken anywheré from
the highest point of a given roughness element to the lowest point. Appropriately no vertical adjustment has ever exceeded the range between the top and bottom of roughness elements. It should be noted, then, that profiles can be forced to fit the logarithmic profile of Eq. (2.6) by adjustment of origin for y . Despite this, universal correlation of experimental data with and y+ is strongly emphasized. The experi-mental investigations of Moore (1951) with square bars, Hama (1954) with screen roughness, Nikuradse with sand grains (described by Schiicting
(1960)), and Liu, Kline and Johnston (1966) with square bars are some that support the logarithmic distribution of the inner layer over rough
surfaces. Away from the wall, in the bulk of the boundary layer outer 80 per cent),, neither Eq. (2.3) nor (2.6) can describé the
profile, and another empirical relation is given for U(y) and
the velocity-defect law (o alternativaly the 'law of the wake,'
(1,960)).
2.3. VELOCITY-DEFECT RELATIONSHIP
Turbulent boundary layers with constant pressures have been found experimentally to have a consistent profile shape that can be expressed in terms of
U(y) - U
u*
= f(y/cS) , (2.10)
in which U is the free stream velocity and f represents an arbitrary
'fùnction. Even when the boundary surface is rough, the relationshjp eems
to be valid (cf., Clauser (1956) Fig.. 3), and provided the boundary layer thickness is large in comparison, say, to roughness height, the rela-tionship is insensitive to arbitrary adjustments for y
An important generalization of the defect "law" was made by
Clauser (.1956) to a class of turbulent boundary layets he defined as being in 'equilibrium.' Specifically, boundary layer flows, in which pressure añd shear forces at. the boundary are maintained in a constant ratio, are defined to be in equilibrium. The ratio can be expressed ás
t.
dx (2.1.1)o
where.- is a scaled pressure force and
t
is the resistive forcedx ' o
-per unit area. Clauser (1956> proposed that the displacement thicknes
should be used as the characteristic dimension . Mellor and Gibson
(1966) used the concept of equilibrium (which is synonymous wit'l profile similarity) to analytically solve the two-dimensional boundary layer equation. Their computed results agree with data taken by Clauser for posi.tive pressure gradients. Later.,, Herring and Norbury (1967)
6
-(say the velocity
y called
experimentally established two equilibrium f lows with negative pressure gradients ànd compared them favorably with computed results of Mellor and Gibson.
The significance of these investigations is that the defect "law,t' which was developed from experimental observations, is linked fundamentally
to the equatiOns of motion (albeit for equilibrium flows). This lends considerable stature to Eq. (2.10) as a universal expression for velocity distribution in thé outer flow region.
2.4. INFLUENCE OF FLEXIBLE BOUNDARIES
However convincing the experimental data have been in presenting a universal picture for a large class Of turbulent, rigid-wall, boundary layer flows, we lack detailed knowledge about the basic turbulent mechan-isms in the flow. Thus, when an additional variable is introduced to the flow we cannot predict the effect with great assurance. Such, for instance,
is the case when we introduce a flexible boundary to the flow. After
Kramer (1957, 1960) reported that significant drag re4uction (nearly 50
per cent) was achieved in experiments with compliant boundaries, several
independent investigations were made to determine if changes in turbulent
structure resulted because of the presencé of the flexible boundary.
Lauf er and Maestrello (1963) constructed a special wind tunnel with various flexible test walls and measured pressure drops through the tunnel. They terminated the investigatioti on an inconclusive note as they foundno definitive effect on the flow attributable to flexible walls. Similarly,
Dinkelacher (1966) reported experiments with flexible wall pipes in which
he compared power spectra of pressure fluctuations with rigi4 wall
measure-ments. The comparisons were indistinct. If changes were manifest they were undetectable by the comparisons. Most of the information available to date on turbulent f lOw over flexible boundaries lacks convincing evidence of significant influence from boundary oscillations. The boundaries con-sidered in most cases have been relatively stiff, though compliant; hence, the effect of boundary-induced fluctuations could well be negligible for such surfaces. For sizable wall fluctuation developed, for instance, over
turbulence could be significant. The results of a storm over the ocean are quite familiar,, and it, is not, difficult to imagine strong coupling between the wind and waves.
2.5.., WIND OVER WATER WAVES
Early efforts to explain the coupling of wind to water waves dates back to the nineteenth.century, starting with the Keivin-Helmholtz insta-bility theory (Lamb (1945)). The theoretical model assumes uniform
velocity distributions in the air and water flows with a discontinuity at the water surface. Both air and water are assumed inviscid and irrota-tional. . The wind speed necessary to.generate waves according to the theory far exceeds the observed.wind speeds that create water waves. Jeff ries'
(19.24, 1.925) sheltering.model assumes that a pressure difference between the windward and leaward sides of a wave is in phase with the wave slope, and the amount of energy input to the waves is dependent on an
experi-mentally determined "sheltering coefficient." Unfortunately, measurements
of these-pressure differences were not obtainable for water waves and sheltering coefficients determined from stationaryrigid wave models gave unrealistically large estimates for wave growth.
The first realistic calculation of momentum flux from the ar to the waves was accomplished by Miles (1957). His model involved inviscid parallel shear flow with, a prescribed velocity profile over a small-amplitude two-dimensional water wave. The effects of turbulence and viscosity were assumed negligible. Improvements to this basic model were made by Benjamin (1959) and by Miles (1962) by including viscous effects and most recently by Bryant (1966), Phillips (1966) and Miles (1967) by adding the turbulent fluctuations of the wind stream. The combined model
is con" idered to be the most appropriate to date that describes the
physical phenomena between wind and small amplitude gravity water waves.
We shall' therefore discuss the. pertinent details, on subsequent pages.
2.5.1. Kinematics of the Flow
The wavy sea surface, altho'ugh random at á point iñ time,' can 'be pictured as being made up of many Fourier components. A particular com-ponent may' be selected by translating the frame of reference with the
celerity c of the component waves (cf., Phillips (1966) p. 89ff.).
Let the direction of propagation be denoted by the x axis, the direction
normal to x and in the horizontal plane by the z àxis, and the vfeical
direction by the y axis. The mean kinematic features of this component wave may then be observed by averaging in the z-direction. This averaging process is denoted by the symbols < > . The surface displacement, which in a stationary frame is expressible by
fl(x,t)
= a(k,t)e1 , (2.12)becomes, in a moving frame, simply
< n(x) > = a cos kx , (2.13)
where k is the wave number, and the wave shape can be regarded as fixed.
We assume that the air waves induced by this surface wave are then periodic with the same fundamental frequency.
The velocity of the wind above the waves is, for a stationary observer, expressed by
(x,y,z,t) = ii(x,y,z) + u'(x,y,z,t) + ix,y,z,t) , (2.14a)
where T is temporal mean, ' is the turbulent fluctuation and is
the wave-induced perturbation. The temporal mean velocity is deter-miñed by
ÌY=lim2T
Udt
4. 4.
-T
With respect tö the convected frame and donsidering the wind to
be in the x-direction only, Eq. (2.14a) can be reduced to
< (x,y) > = (x,y) - c + < (x,y) > (2.15a)
because
< u'(x,y)
> E
O (2.15b)and
< ÏT(x,y) >
=ii(x,y) . (2.15c)2.5.2. Work Done on the Waves
In the convected reference frame the mean flux of momentum to the particular wave component is the çorrelation of surface pressure and wave
slope (Phillips (1966) p. 94)
F = < p > < Bn/Dx >
w
owhich in view, of x-wise independence is also expressible as
F = - < n > <
w
owhere, the overbar is now regarded as the average over the wave length of the selected componentj. In Eq. (2.16), p = + p' , where ' is the
perturbation pressure due to wave motion and
p'
is the turbulentpres-sure:fluctuat,ion. The subscript o refers to the level y = O . An
expression for the gradient of pertUrbation pressure is found from the mómentum equation for the 'air motion above the wave, which in cartesian
tensor notation is
Du. Du. 32u.
-.-2+
_2_iDP
+
iDt Uj Xj P 3x
DxDx
By introducing the fluctuating component.o,f velocity and pressure, and afteraveraging in the z-direction, it can be shown by suitable rearrange-ment (cf., Phillips (1966) p. 94ff.) that
10
-(2.16)
(2.17)
_-<
/x>=(UO)
c)
<u>
in which it is noted that
o
=
U(0) -
<n/x
(2.20)Substituting Eq. (2.19) into Eq. (2.17) gives
< n > < r1/x > = O
F =F +F
w i t F = - p < > - p < utZ > <> + p
w
o o < utv' >\n>
(2.19) (2.21) since (2.22) (2.23)in which F1 is the contribution of the wave-induced Reynolds stress, and is the additional contribution because of the presence of turbu-lence in the wind stream. Although Phillips ((1966) pp. 94, 95) gives arguments which imply that Ft should be negligible, Miles (1967) suggests that this be accepted with reservation because of the lack of experimental information.
The wave-induced Reynolds stress may be regarded with another physical interpretation (cf., Phillips (1966) p. 95ff., Miles (1967)).
The product of vorticity perturbation induced by the wave motion and
ver-tical component of the wave-induced velocity may be expressed as
+---< u'2>
+L<
'' >
o y o
The effect of turbulence is clearly seen to be expressed by the last two terms of Eq. (2.21), and in view of this, the momentum flux could then be considered to consist of two component parts
because by hypothesis
= o =
y
-DxID?i
Qv=t - -
\Dy
DxIf Eq. (2.26) is integrated vertically over the interval
(y,)
, the result is= - p <
(y) > = p. j dy
(2.28)which states that the wave-induced Reynolds stress at a l.eel above the
wave is equál to.the integral of the horizOntal component of the vortex
force above the point. In particular
rO) =-p
dy (2.29)The wave-induced Reynolds stress derived from the invscid laminar model (Miles (1957)) is given by
ÍtI(Y)
T =- T LU.' (y).
.12 -.(224) (2.26) (2.27) (2.30) By cofltinuity y so that =() +
22)
(2.25)where the primes represent derivatives with respect to y and the
sub-script m refers to the matched layer. Phillips ((1966) p. 96) shows
the relationship of Eq. (2.30) to Eq. (2.29) by considering 2 to be the wave-induced perturbation of the mean vorticity
k21
U"(y)
w (2.31)where the thickness is the displacement of a mean streamline about
w
its mean level y due to the wave-induced motion. He also shows that
(yi) cos[kx + c(y)] = Y - Yl
k[U(y1) - c]
w
Multiplying Eq. (2.31) by and averaging along x gives
-
Ç
U"7
klTJ-cI
The portion of the wave-induced shear derived at the matched layer is calculated from
tT
-
U"(y \ (yp in mì
\mi m
If the thickness of the matched layer is estimated to be
in
[4
(Y) iin k LT' (y) I
(2.35)
JT7
then by substitution of Eq. (2.35) into Eq. (2.34) we find
=A
inmLkU?
JYY
so that (2.32) (2.33) (2.34)14 -p4 r-tt
w k.
LU'
(2.36)m
A cmparison of Eq. (2.36) to Eq. (2.3Ö) shows the constänt of proportion-ality is evidently equal to r , which agrees with the inviscid laminar model. Miles (1967) found through detailed investigation of the equations of motion in the matched layer that, with linearized approximations to the air waves induced by the water iaves,
Am = Tr[i ± O(ka)] (2.37)
Accordingly, for small ka , the contribution to T through the matched Ïayer is given correctly to first order by Eq. (2.30).
Contributions to the wave-induced Reynolds stress from regions outside the matched layer can only be hypothesized at the present time.
More xperimental information must be provided before quantitative
calcu-lations can be made. Furthermore, establishment of the size of the con-tributions to the total momentum flux in Eq. (2.21) by the turbulence terms requires experimental information.
Much has been accomplished in understanding and mathematically formu1at.ng some aspects of the coupling o wind and waves, albeit for small amplitude waves. Much more needs to be accomplished., with particular demand for expérimental data concerning turbulence and wave perturbat-ion in the vicinity of the wave surface. The difficulty of obtaining such data in natural environments strongly suggests laboratory experiments that have better control of the many different variables in the problem.
3. FACILITIES AND ARRANGEMENT
3.1. WIND-WAVE CHANNEL
The laboratory facility utilized in this study has been described in adequate detail in a number of previous reports [Hsu (1965), Colonell (1966), Shemdin and Hsu (1966), Bole and Hsu (1967), Camfield and Street
(1967)1. It will be useful, however, to introduce some geometric
dimen-sions and characteristic channel behavior.
3.1.1. Geometry
The channel, which in this report is sometimes referred to as a flume to avoid redundancy, has cross-sectional dimensions of 6.3 ft high
by 3 ft wide and a test section of about 65 ft in length. The
cross-sectional area varies by about one per cent along the test section because of fabrication tolerances. The schematic diagram in Fig. 3.1 gives the overall arrangement. Other structural details are described by Hsu (1965).
Water depth in the channel is variable. For this study two levels of water surf àce were selected. The first level was even with the air intake floor and the second 3.5 in. lower. The drop in water surface was necessary to permit mechanical generation of waves without disturbanceby the air intake transition. It may be useful to visualize the channel as being half full of water where the air passage height is about equal to water depth.
An overall photographic view of the flume is shown in Fig. 3.2. The wave and air flow are co-currentand left to right in the photograph.
The clear glass walls on both sides along the entire test section especially
facilitate experimental measurements.
3.1.2. Channel Characteristics
The cross-sectional distribution of air flow in the channel reported
by Hsu (1965) indicates that flow is two-dimensional to the geometric center
of the channel flow cross-section for substantially the entire test length.
The two-dimensional flow width narrows however to about 6 in. at Station
60 (see Fig. 3.1 for reference to stations). Colonell (1966) found that
character of an aroused seaa 'Thus there is some assurance that the texture of the water surface, in so far as air flow above it is concerned, is
similar to that iñ nature.
Dynamic f èatures of the channel, particularly with respect to mechanically generated waves; are reported by Bole and Hsu (1967). The
wave reflection coefficient for waves at a f requericy of 1 cps was found to be aboutj 5 per cent. Although specific information for wind waves is not available, it is expected that the ref lectión coefficient ould be less because the waves are of smaller amplitude and higher frequency.
Surface currents in the flume for wind speeds n a range which ±nciides
those used in the present study were reported to amount to 2 per cent of core velocity. Setup In the channel due to tangential shear and pressure differences wast about .1 in. at core velocities of 40 fps and muth less for the maximùm speed near 20 fps used in this investigation.
3.2.
SPECIAL ARNGNT
A smooth aluminum plat.e (constructe4 for other studies in the channel) was suspended in the flume at water level to constitúte thé flat surface. The total suspendéd length was 40 ft measured from the intake. Construction details of the plate are given by Camfield and Street (1967).
Joints between individual plates and between the plates and walls were taped smooth with pressure sensitive tape 1. 5 mil thick. The arrangement is depicted schematically in Fig. 3.1, with än internal photographic view of the plates given in Fig. 3.3.
3.3. APPURTENA1T APPARATU S
A motorized traversing an4 position indicator enabled emote probe
operation in a vertical plane during measurement. The, signif.icant feature
of this mechànism is a bi-directional motor attached to a f-inely threaded rOd through a gear reduction bx. A multi-notched disc f ixed, to the rod indicates vertical traverse position electro-mechanically through a micro-switch. esolution of position is controllable by changing the notchd
discó In this investigation vertical position was measured, to ±0.01 in.,:
whith is
consistent
with féatures Of the overall facility. A photograph 16-of the mechanism is shown in Fig. 3.4 for reference, along with a view -of the remote control panel and position indicator. Only the longitudinal centerline of the channel was traversed in this study.
3.4. INSTRUMENTATION
3.4.1. Pressure
A Pace model P9OD differential pressure transducer with a range of ±0.03 psid and a Statham model PM197, ±0.01 psid range transducer were
used. A Sanborn model 950, 2.4 KHz carrier-amplifier was coupled to the
Statham transducer and a 20 KHz Sanborn model 650 carrier-amplifier system was used with the Pace. Roth transducers were calibrated by a Harrison micromanometer, which indicated differential. head with a resolution of ±0.0001 in. of water. Linear relationships between output voltage and differential pressure resulted. Figure 3.5 shows typical calibration curves. A number of calibrations were made through the period of
inves-tigation. All were found to agree. withIn 2 per cent.
3.4.2. Mean Velocity
Temporal mean velocities were calculated from measurements made with a probe consisting of separate static and stagnation tubes placed about 1 in. apart lateral to the flow. The static probe was positioned slightly above and downstream of the stagnation probe so as not to inter-fere with the stagnation flow field. The outside diameter of the probes was 0.1 in. with an opening diameter of 0.05 in. The pressure transducers were used to indicate differential pressures registered by the stagnation and static tubes.
3.4.3. Wave Height
Details of the wave gage instrumentation were given by Colonell
(1966). The capaçitance wave gage is presently being used as a standard
gage. It is basically responsive to changes in wire capacitance due to
changes in immersion depth. Output voltage in most instances is linear with immersion length as .is evident in the calibration curves of Fig. 3.6.
3.4.4. Turbulence
Two standard Thermo-Systems model 1010 constant-temperature ho wire anemometers were used to measure turbulent fluctuations. Cross wires were sed exclusively in this investigation. Liarizers were not avail-able, and perhaps it was. because of this that techniques of data reduction were made adaptable to digital computation. Although our personal acquain-tance with analog linearization of hot wires is limited, it is believed that linearization with present-day digital computers is easier and oper-ationally more dependable. For individual tudes jt is also believed to be economically competitive. Details of data reductjon are discussed in Chapter 4.
A zero-suppression,. d-c amplifier combination .wàs the only addi-tional equipment utilized with the hot wire anemometer at measurement
time. To gain maximum accuracy in analysis-, the d-c component was
sup-pressed and the fluctuating or a-c signal wasamp14fie before recording. Stab]Le. amplifier-suppression units were. designed and constructed at the
laboratory. . . .
Two types of hot wire sensors were tried. Thé first was a quartz-coated, i mil diameter sensor with a length-to-diameter ratio ( Lid ) of
40. The second was a tungsten 0.15 mil sensor with an Lid ratio of 300.
The data presented in this investigation were taken with the second,
smaller diameter sensor. .
A tpicäl calibration curve of the quartz-coated sensor is given
in Fig. 3.7(a). A càlibratiòn curve for the tungsten sensor is also pre-sented for comparison in Fig. 3.7(b.). It is to be noted that the quartz sensor responded non-linearly in the range shown, while the tungsten sénsor was linear as required by Kings' Law. It is believed that part of the
non-linearity for the former sensor arises from large end effects because of the small Lid ratio. The effect could be redùced by ïncreaing L/d
tf measurements are to be meaningful, however., there is a practical limit to the wire length that should be used for X-wires in a sheared-flow region.. The diameter cannot be materially reduced for quartz-coated sénsors because of construction details. While the non-linearity is a troublesome
char-acteristic to analog linearization, it is of no serious consequence in
-digital analysis. The linear response of the tungsten wires of course offers no difficulty in data reduction.
There was a markedly longer period of calibration stability for the quartz-coated sensor compared to the tungsten wires in the environment of the wave channel. Small dust particles caused drift in the calibration of the tungsten wires while the quartz wires were stable over many hours of use. Because of drift, calibrations were made frequently and dif f
er-ences between successive calibrations were adjusted linearly with time
when they occurred. A drift of greater than 5 per cent was considered cause to void the measurement involved.
3.4.5. Recording Instruments
The principal recording nstrument was a seven-channel FR-1100
Ampex FM tape recorder. All recordings were at 15 ips with amplifier characteristics responsive to frequencies less than 2.4 KHz. The Sanborn 950 and 650 recorders were used as monitors for the pressure transducers, while the hot wire signals were monitored af ter amplification with an oscilloscope as a basis for regulation of input voltage to the tape recorder.
4. TEST PROCEDURE. AND DATA REDUCTION METHODS
4.1. MEAN VELOCITY.
Measurements of mean velocities were made with tbe
tag.näiOn-tatic probe at threê lengthwise or x-stations over the flat plate and wind waves. The three stations were 13.4, 23.4, and 33.4 ft (see. Fig. 3.1
for orientation).. Thr.ee wind .speeds of approximately 8, 13 and 1.8 fps
were selected for the flat plate and wind waves and a low speed of approx-imately 8 fps was used with the mechànicai waves to keep the level of the matched layer measureably high. The differential pressure of the velocit.y
probe was sensed by a pressure transducer, amplified appropriately, and recorded with the FM tape recorder. Thé analog record was then converted to digita]. representation with the analog-to-digital (A-D) conversion facilities of the Stanford Hybrid Computer Laboratory. The conversion equipment involved was described .by Colonell (1966). Data acquisition and conversion processes are depicted in the. block diagram of Fig. 4.1.
4.1.1. Boundary Layer
The boundary layer thIckness 6 is defined as the value of y
where the local mean velocity is equál to 0.99 U . The distance y was
measured from the surface of the plate and. from mean water level for water waves. Measurements of y above a water surface with waves were convert-ed to sôme adjusted values y and 6 was then based upon
a The
prò-cedure used to determine
a is described in the next section.
The usual indicators of profile shape were. calculated from measured velocities by d.ig:ital computation using finite increments and central differencing .f
velocities. The appropriate definitions for the quantities will be listed here for convenience of reference:
1. Displacement thickness
-Momentum thickness
e=fL(1_L)dY
(4.2)Defect thickness (Clauser (1956))
o'
IJ -
u* J 1- = - ln - + C
u*lt
V d(y/iS) . (4.3) (2.3)Although some controversy exists about the constants X and C , as
previously mentioned in Chapter 2, these constants were assumed to be
0.41 and 4.9, respectively, from Clauser's (1956) results. The above transcendental equation for u was solved iteratively for each measured
U at elevation y . If a logarithmic region exists (close to the boundary)
then u would be independent of y through that region and its value
could be readily selected graphically. This procedure is often referred to as 'cross-plotting' (cf., Schraub and Kline (1965), Liu et al. (1966)). A typical graphical result is plotted in Fig. 4.2. Cross-plotting was the
only technique used for determining boundary shear in this investigation.
Direct shear measuring instruments, while workable for rigid surfaces,
are not available for an undulating water surface with propagating waves.
4. Integral defect shape factor (Clauser (1956))
f
¡U-G=J
Jd(y/)
(4.4)o
*
i
4.1.2. Shear Velocity
Determination of shear velocity was based on the law of the wall.
The momentum integral method was not considered appropriate becaue of the difficulty of. establishing gradients for boundary layer growth. This diff:Lculty arose partly because positioning, the instri*ents along the
channel necessitated stopping and restarting the wind-creating fan o the
channel each time, and partly because of inaccuracies .ássociated with the calculatiOn of longitudinal gradients of t:he mothentum t ekness from pro-f iles that were limited by the height opro-f wave crests.
The law of the wall for 1ough walls was used to determine over water surfaces. A visual fit o the velocity profile was used in guiding the calculations. The procedure requires some explanation. in
Eq. (2.3), for rough walls, the origin for normal distance y usually
requires some adjustment to locate a mean surface. This problem does not exist for water waves because the mean surface is the mean water level. However, as was noted earlier, the esséntiâl fact is that the adjusted amount c is determined by forcing the profile to fit a logarithmic line
close to the wall. Thus c loses some of the convenient interpretation
as a purely geometric adjustment for mean surface. Kinematic and dynamic effects included in the velocity U(y) are reflected in the rathér arbi-tráry ordinate adjústment. From this view, adjustthents to y might be justified for velocity profiles over water waves to account for sothe dynamic influence of wave-induced perturbations, particularly in the inner layer.
The velocity decrement U/u* expressed by Eq. (2.4) is a function of representative roughness height k1 . The surface roughness exhjbited by water waves is not constant either with respect to time or space. In
a statistical sense, however, the waves do provide àn effective roughness
which can be characterized by a mean roughness height, say k The
ordinate adjustment c and k. could, in the first estimate, be related
to the standard deviation of localwave heiglts p , for. a is the mos
significant measure of the probability density function of wave heights.
Both k and c were interpreted to be.local effects on the flow profile
w.
and expressed in simple proportion to. a. as,
22
and
c=C2a
(4.6)Thus
= y.+ C
(4.7)in which y is the distance from mean water level. In Eqs. (4.5) and
(4.6) C1 and C2 are constants. Although the correction is seldom
greater than k for rough walls, this limitation was not imposed here
since it was assumed that the dynamic effects of the oscillating
propa-gating wall were included in . The value of u,, was calculated by
cross-plotting using Eq. (2.4) and compared to a visual fit of a log line
to the lower portion of the velocity profile adjusted for c
4.2. TURBULENCE
4.2.1. Hot Wire Response
The heat transfer from the hot wire to the surrounding fluid medium can be expressed by (Hinze (1959))
Nu = A' + B(Rd)n (4.8)
where Nu is the Nusselt number, A'. and B' are constants dependent
upon the physical properties of the wire and the fluid, and Rd is the
Reynolds number based on wire diameter. When the energy loss is converted in terms of electrical power we have
E2 = (A" + B" u) f(T , T - T
s s a (4.9)
where E is output voltage, T is wire temperature, Ta is ambient
temperature and U is the "instantaneous" speed of fluid flow. The
exponent n is approximately 0.5. If the temperature of the wire and the differential with ambient fluid temperature are constant, then Eq. (4.9)
reduces to
E2 = (Ai + B11J') , (4.10)
where A1 and B1 are constants appropriately redefined.
Equation (4.10) is an expression for wires placed torma]. to the
flow. For yawed wires, the effective cooling velocity Ue is found to
be the component of U ñotmal to the wire. Usua1l it is assumed that U U cos a , which is referred to as the cosine law of cooling.
Champagne (1965) suggests that the cosine law requires modification by a corrective amount (K sin a)2 so that
1.J
U(cosa + K2sin2a)
(4.11)where c is the angle between flow direction and the normal to the wire
and
K = K(Z/d) . (4.12)
Champagne found that K varies from 0.20 for 9/d = 200 to zero for
9/d = 600 . According to Eq. (4.11), then, the deviation of U from the cosine cooling law for a 45° and
K =0.1
is much less than the accuracy of measurement, although Champagne shows that the correction applied to calculation of fluctuat-ing components could be appreciable.4.2.2. Hot Wire Calibration
Calibration of the hot w-ire sensors was made in the core of uniform
f 1w region of the wind-wave channel, imediately before data recording and
at approximately one-half hour intervals until the final calibration at the
termInation of the experimental run. If the difference between any two
successive calibrations was greater than 5 per cent, the intervening data
were invalidated. Differences less than 5 per cent were linearly adjusted with time. For a straight line calibration as in Fig. 3.7(b), line4r
adjustient was simple. For curved calibrations adjustment was only slightly more difficult.
-and
and
Calibration of thé x-wires used n this study will be described in detail because the technique applied is believed to be widely applicable and is readily adaptable to digital computers. There are three important geometric degrees of freedom to consider in using the x-wire probe, namely,
the angle of the probe axis with respect to méan flow direction and the angles of the two wires with respect to the probe axis. Insofar as possible the latter angles are 45° and the two wires are perpendicular. However, small deviations ct and a2 may occur because of manufacturing tolerances. Figure 4.3 shows the sensor orientation to flow direction.
Two successive calibrations were made before each run with the probe rotated 1800 about the probe axis between calibrations. From the
two calibrations, theangles , , and a2 were determined by trial,
since a unique relationship is demanded forj coincidence of the calibrations at both rotated positions with a comon angle
Let the instantaneous wind velocity be denoted by U and the
com-ponents normal to the two wires by UNA and UNB , where A and B
identify the particular inclined wire and N refers to a normal direction. The angles and which enable calculation of UNA and UNB are
= 45 + - a1
= 45 - + a2
for the position of the probe shown in Fig. 4.3.. In the rotated probe position the angles are
_y'i =45 - - a1
=45++a2
Initially it was supposed that the angles , a1 and a2 were zero. -The two calibrations for the wires in the two positions should be the same. Let it be supposed tiatboth wires showed differences.
Calibrations for ,one of the wires were thei brought into coincidence by introducing a small correction since it is the comon angle for the two wires. Now, depending upon whether the prôbe was pointed upward or downward, a small correction to or a2 would be made to bring tie remaining wire calibrations into coincidence. Generally the corrections were smaller than 2 degrees.
4.3. DATA REDUCTION
4.3.1. Fluctuations
It is normally necessary to assume that the hot wire bridge voltage is linear with velocity in thé range of the fluetuatiön about a mean
velocity so that the magnitude of the fluctuation can be calculated from
u
£
EU
,26
-(4.13)
where P is a constant, e (dE) is the output voltage fluctuation and
u (du) is the turbulent velocity fluctuation. LinearIzation is not
required when
data
reduction is accomplished on the digital computer. Thecalibration curve may be used directly to determine 'instantaneous' U
for each output E . The fluctuating vélocity at the instant of time con-cerned Is then
uTJ-
-according to definition.
Velocity fluctuations in two directions may be determined from data taken with x-wires. The two normai velocities, U and tJNB
are determined directly from the output voltages of the hot wire anemometer and the respective calibration curves. The Instantaneous velocity vector
may then be calculated from angles determined in the calibration procedure
previOusly described and from simple geometric relationships defined in
Fig. 4.4. The angles y1 and are known from , a]. and
a2
and O' can be calculated from known velocities, and angles :in Fig. 4.4.determined. The temporal mèan velocity U is calculated from either UNA or UNB , or preferably both. Checks in calculation are readily
made from knowledge that temporal means of the. fluctuations must bero and the mean velocity calculated from both wires must be the same.
This
data reduction technique can obviously be extended quite readily tothree wire probes for sImultaneous measurements of fluctuations in threeco-ordinate directions.
4.3.2. Turbulence
The usual expressions for turbulence intensity may be calculated in a discrete sense at a point in space and time after having determined the fluctuating components of velocity. The most complete inormation
would be the probability density functions of u and y and the joint
probability density function for u and y . Such computations for sig-nificant numbers of elevations at the various stations involve appreciable amounts of computer time; thus, for this study only the indicators of the probability density functions at spatial points were calculated in the usual sense. The first moment of the distribution is by definition zero. The second, third and fourth moments, or variance, skewness and kurtosis, respectively, were calculated.
The square root of. the variance, or the root-mean-square (rms) value, is called the intensity of turbulence. Skewness is an indicator
of non-symmetry of the probability distribution and signifies whether
positive or negative fluctuations occur in the flow (cf., Liu et al. (1966)). It is defined to be
S=
(7)3 2
(4.14)Positive S indicates predominantly positive fluctuations while negative
S indicates negative fluctuations.
The flatness factor (kurtosis) is used in turbulence measurements to indicate intermittency, i.e., the ratio of time that the flow is
in which Ft is the factor for the fully turbulent region and F is the kurtosis at the given elevation.
4.3.3. Wave-Induced Fluctuations
Calculation of z-averagéd quantities in a laboratory facility can be accomplished in the following manner. A stationary process is assumed to become established after some period of time with mechanically generated waves in the channel. With the aid of the ergodic hypothesis, phase
averages (indicated by the symbol < > ) are made at fixed elevation from
a continuous record of many waves which pass by a fixed probe to give a z-average. The velocity field, assumed to be linearly composed, may then be separated. If a two-dimensional velocity field is represented by
U(x,y,t) = ÏY(y) + u' (x,y,t) + l(x,y) , (4.17)
where U(x,y,t) is instantaneous or total velocity, IY(y) is temporal mean, u'(x,y,t) is turbulent fluctuation and (x,y) is perturbation
due to the mechanical wave, then
< U(x,y,t) > =
ii(y) + <
T(x,y) > , (4.18) that Is,< u'(x,y,t) > O . (4.19)
Averaging Eq. (4.18) over the period of the wave at a fixed eleva-tion provides that
28 -F t Y =-;- (4.16) r
F-(;7) 2 (4.15)< U(x,y,t)
> E
U(y) , (4.20)or that
< (x,y) > O . (4.21)
With these definitions the wave-induced perturbations and turbulent fluctuations of the velocity field may be separated; hence, the perturba-tion and turbulent Reynolds stresses can be identified. The product of the vertical and horizontal fluctuating velocity components at a point is given by
< uy > = < (u' +)(v' +V) >
(4.22)so that, by phase averaging,
< Uy
> = < > + < u'v' > (4.23)In Eq. (4.23) the product < > = < >< V > ; hence, the turbulent Reynolds stress may be determined by subtracting the wave-induced Reynolds
stress from the total quantity.
Certain test procedures directly related to phase averaging are worthy of mention. In Fig. 4.1 it will be noted that there is a standard
frequency input to tape recording at time of measurement. This signal (a square wave with some minimum peak voltage time interval) is used to command the multiplexer of the A-D system and serves two purposes. First,
for low frequency signals, it serves to prevent frequencyaliasing from
imperfections in tape transport motor drives both at record and playback
time. Secondly, it provides a means of referencing the velocity field to
the water wave phase for each channelof the hot wire anemometer; correct products may then be formed in computing the perturbation and turbulent Reynolds stresses.
5. EXPERIMENTAL RESULTS
5.1. TURBULENT BOUNDARY LAYER ÒVER SMOOTH WALLS
Turbulent boundary layers that developed over the smooth solid. surface in the wind-wave flume exhibited a two-layer structure character-istic of such sheared flows. The velocity distribution in the lower layer followed the waillaw and the outer flow region could be described by the defect law. The dist-ributions of turbulence intensities theasured at three stations and three wind speeds were similar.
5.1.1. Mean-Velocity Distributions
The free stream velocity (equivalently the core velocity or Ç ) increased slightly from station to station under the influence of a mildly favorable pressure gradient. The measured gradients are shown in Fig. 5.1 in dimensional form and non-dimensionalized with dynamic pressure in Fig.
5.2.. The mean-velocity distributions at successive stations are given in
Table 5.1 and shown in Figs. 5.3(a), (b) and Cc) for nominal wind speeds
of 8, 13 and 18 fps, respectively. All of the data were taken with a stagnation-static probe in traverses through the central vertical plane of the channel.
The shear velocity was calculated for each wind speed and station
by use of the wall law with Clauser's suggested constants as explained in
the previous chapter. The correlations of U with y+ based on these
calcúlated values are shown in Figs. 5.4(a), (b) and Cc) and those for
the defect parameter (U - U)/u vs. y/6 are shown in Figs. 5.5(a), (b) and (c) for each longitudinal station. The defect correlations -in Fig. 5.4(d) are for a wind speed of 18. fps at the three stations. The boundary layer thickness was identified through the equality
U(s) = 0.99 U
+.
+The U vs. y correlations show reasonable fit to the logar-ithmic profile in the region 40 y+ < 300 . This corresponds
approxi-mately to 0.02 < y/6 < 0.20 . The logarithmic profile coincides with
the part of the boundary layer termed the inner layer. Beyond the log region, in t-he outer layer of the boundary layer, the mean-velocity dis-tribution is described by the universal defect law. The defect profile,
-which presumably is more directly applicable to equilibrium boundary layers, probably applies well to these data because of the very mild pressure gradients. As noted on Figs. 5.5(a) to
Cc),
the values for the defect shape parameter G defined by Eq. (4.4) are all in a range 6.1 to 6.8.The mean-velocity profile is often expressed in the form
(5.1)
which is commonly called the power-law profile. This relationship is shown to hold reasonably well for y/5 > 0.25 in Fig. 5.6 where n is
approximately 5.4.
5.1.2. Turbulence Intensities
Distributions of horizontal and vertical turbulence intensities through the boundary layer are given in non-dimensional form normálized with u in Figs. 5.7(a), (b) and (c). Although data are scattered, the distributions are substantially similar. A solid cürve drawn in Fig. 5.7(a) represents data of Klebanof f (1954) and is shown for comparison with present results. The scatter in the present data comes from two sources. First, within a given profile, the accuracy of measuring tur-bulence intensities is about ±10 per cent, particularly in regions near the wall where turbulent fluctuations are large and mean-velocity gradient
is steep (cf., Champagne (1966)). Second, determination of shear velocity by cross-plotting is accurate to about 5 per cent. The uncertainties are noticeable in the profiles, as the variations between distributions and points in the profile are randomly scattered with respect to station and wind speed.
The distributions of turbulent shear stréss are shown in Figs. 5.8(a) to (c). The value of t was obtained from u by the relation-ship
There is considerable scatter of data and it is evjdent that the ratio of turbulent shear stress to wall shear r0 approaches i .at.the wall for some profiles only. From the previous paragraph we know that an error of ±1Q per cent could be present in. calculated t0 . In addition, because
the x-wire probe occupies a finite spatial height within the steep mean-velocity gradient, larger values of mean mean-velocity and' smaller values of
the fluctuating components are sense4. This combination could result in smaller values of measured Reynolds stress. It is significant that in no instance does a measured value of rt/T0 exceed 1, indicating that
measured values perhaps do indeed tend to be low. The distributions could be normalized but they would then lose sorne of their value as a comparative
reference. It was considered more useful to. retain turbulent stress dis-tributions in the männer shown with the difficulties indicated.
5.1.3. Skewness, Kurtosis and Intermittency
The thitd and fourth moments of the probability density function,
S and' F respectively, for turbúlent fluctuations u and y are pre-sented in Figs. 5.9(a):and (b) for a wind speed. of 18 fps at Station 33.4. The variation In kurtosis interpreted in. accordance with Eq. (4.16) yields intermittency y . Intermittency calculated from F for u fluçtuations only is shown in Fig. 5.9(a). The negative skewness factor in Fig. 5.9(b)
indicates that the intensity of turbülence , u in the outer layer of the
boundary layer has strong negative spikes of turbulent fluctuations while
the turbulence intensity y coñtaiùs po'sitve. spikes of turbulence. The
kurtosis and skewness factors calculated f rpm other turbulence profiles were similar to t-hose above.,
5.2. - TURBULENT BOUNDARY LAYERS OVER WIND WAVES
5.2.1. Mean-Velocity Distributions
The pressure gradients for nominal wind speeds of 8, 13 and 18 fps are given in Fig. 5.10 and non-dimensionally in Fig. 5.11. The gradients are considered mild. The pressure gradients are substantially independent of flow Reynolds number. This.was not tr:ue for the smooth plate which
dsplayed decreasing drag effect (mä.11er non-dimensional gradient,
