ARCHIEF
Lab.
v. Scheepsbouwkunde
Technische Hogeschool
August 1968
Delft Technical Report No. 94
A Steady -State Simulation of Small
Amplitude Wind-Generated
Waves
by
Ronald
F. Ott
En Yun Hsu
Robert L. Street
This research was sponsored by
National
Science FoundationGrant GK-736
and
Office of Naval Research
underContract Nonr 2 25( 71), Task NR 062-320
Distribution of this document is unlimited.
Department of CIVIL MINTGIINTEETtING
ST.A.NFORD T_TNIVER.SITY
Department of Civil Engineering Stanford University
Stanford, California
A STEADY-STATE SIMULATION OF SMALL AMPLITUDE WIND-GENERATED WAVES
by
Ronald F. Ott
En Yun Hsu
and
Robert L. Street
Technical Report No. 94
This research was 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
The primary purpose of this investigation was to extend and to
improve the results of Zagustin, et al., on steady-state, experimental
simulation of wind blowing over small amplitude waves. Experiments were
performed in a laboratory open-channel flume in which a belt driven at a constant speed and following the shape of the wave reproduced the moving boundary conditions found at an air-water interface. Water was used as
a working fluid.
Preparatory to a description of the present results, the important features of Miles' inviscid, linearized shear-flow model are reviewed.
Then, the experimental results of Zagustin, et al., are discussed and the apparatus is described.
Two sets of new experimental results are presented. The first set
was a re-run of the Zagustin experiments in which the moving belt formed one wall of the open channel. Emphasis was placed on increasing the scope
and accuracy of the previous results. In the second set of experiments,
a cover-plate was added to the open flume to minimize the free-surface disturbances. In both sets of experiments, a pitot-static probe was used
to obtain velocity and pressure data. The measured velocity profiles in
the flow above the wavy surface were used together with Miles' theory to
obtain theoretical pressure distributions for the region near the wavy
surface. These were compared to the experimental distributions.
Within the limitations of the experimental model, the data verified Miles' inviscid mechanism of energy transfer from a shear-flow
TABLE OF' CONTENTS 10 10 11, 12 13 12, 12 v 4 13 13 .3' 8
*A; 4
13
14 $. 15 1 15 : s- 4 173.8.1,
Least-Squareg
Fit of Velocity Profiles ; 173.8.2,
Calculation of Theoretical Pressure Distribution J7. DISCUSSION OF EXPERIMENTAL RESULTS . , 6 * 6 A 1 * ;
19
4.1.
Velocity Profiles* , ,
.* . .
. ,
. # :
6 A 40 0 4" .. 4 194.2.
Pressure ProfilesAA4'
...7, ,.., 204.2.1.
Comparison with Miles TheoreticAl Distribution 20' i4,2.2.
Pressure Decay . . .* . 0
. A A ,e A W,
224 3.
Streatline Patterns . .40.44 *
w e 64 .0404 4
23 1 5. 'CONCLUSIONS, Page 11.
INTRODUCTION * o *.
** 4 -1
REWIEWOF THE EXPERIMENTS OF ZAUSTIN, ET AL4
2.1.
The, Steady-State Concept., o 42.2,
The Shear=Flow Model .* *
o, ;1 * *oL A to' 4
The Previous Experiments .
2.4,
,Discussion of Experiments in 13 74 43.
SMALL AMPLITUDE WAVE EXPERIMENTS3.1. Introduction. .
3.2.
The Apparatus 0.. 0 0,
11 0 11, 0 1_1 A3.3.
Instrumentation . io7 ;i" A Al AA 111 ,*3.4.
Special Probe Arrangement. A ,3.5.
Experiments in an Open Flume*
3.5.1.
Velocity Measurements3.5.2,
Pressure Measurements.3.6.
Experiments in a, Covered Flume . *6 4
31 63.6.1.
Flume Cover-Plate 4i4
:-AAA*6
3.6.2.
Velocity Measurements .3.6.3.
Pressure Measurements. *,
0 4 * e e4 * *
3.7. Experimental Errors r. .0 iv v 74 ot c It.3.8-
Data Reduction Methods * A 76..;i1 4, 4 0* * 25
3
3
6
TABLE OF CONTENTS (continued)
Page
REFERENCES 27
APPENDIX . Computer Program to Calculate Theoretical
TABLES
Number Page
Summary of Velocity Measurements (without cover-plate) . . 42 Summary of Pressure Measurements (without cover-plate) . . 43 Summary of Velocity Measurements (with cover-plate) 44 Summary of Pressure Measurements (with cover-plate) . . . 145
Sample Computer Printout of Least-Squares Fit to Velocity 46 Profiles
Sample Computer Printout of Theoretical Pressure
Distributions
47
ILLUSTRATIONS
Figure
Sketch of Unsteady Flow over a Small Amplitude Wave
Sketch of Steady Flow over a Small Amplitude Wave . . . . 48
Plot of B vs kyc (after Miles (1959)) 49
View of Belt Used as Moving Boundary 53
View of Beam, Sprocket and Guides 51
View of Beam with the Installed Belt 52
General View of the Channel 53
Trace of Plastic Particles Floating on the Water Surface,
from Zagustin, et al. (1966) 54
Trace of Plastic Particles Floating on the Water Surface,
from Zagustin, et al. (1966) 54
General Dimensions of Apparatus 55
View of General Probe Arrangement 56
Probe Location in Cross Section of the Channel 57
Pitot-Static Probe Dimensions 57
. 1. 3. 8« , . . . 13,
ILLUSTRATIONS (continued),
Figure Page
14.,
15.
Sample. Calibration Curves °.!
Schematic Diagram of Probe Systems
r
58
58
16. Velocity Profile Across the Channel V! LI. tot 59
17a. Velocity Profile at Station 0 . N. 60
17b. Experimental and Theoretical Pressure Distributions
60
18a. Velocity Profile at Station 4 . . .4 in 61
18b.. Experimental and Theoretical Pressure Distributions 61
19a. Velocity Profile at Station 8 .
62
19b
Experimental and Theoretical Pressure Distributions62
20a
Velocity Profile at Station 12 . 63.20b. 21a.
Experimental and Theoretical Pressure 'Distributions
Velocity Profile, at Station 16 . 4
63,
64-21b. Experimental and Theoretical Pressure Distributions
22. Static Pressure Differential Across, the Channel
65
23.
24.
General View of Cover-Plate ,
View of Plastic Test Section, . I., 4i: AI go 66 25a. Velocity Profile at Station 0 (with cover-plate)
6?
25b. Experimental and Theoretical Pressure Distributions 67
26a. Velocity Profile, at Station 4 (with cover-plate) . . 68
26b.. Experimental. and Theoretical Pressure DistributiOnS 68
27a, 27b.
Velocity Profile at Station 8 (with cover-plate) Experimental and Theoretical Pressure Distributions
69
69
28a,
2811.
Velocity
Profile
at StatiOn 12 (with. cover-plate) Experimental and Theoretical Pressure. Distributions70
70 29a. Velocity Profile at Station. 16 (with. cover-plate) *
71
29b. Experimental and Theoretical Pressure Distributions
71
ProfileILLUSTRATIONS(continued)
11
Figure
Variation of the Experimental Pressure Distribution as
a Function of Distance from Moving, .Boundary (with
11.18±.
30.
cover-plate) . :A -A' oi
72
31.
Variation of the Experimental Pressure Distribution asa Function of Distance from Moving Boundary (without
Cover-plate) ,e A A is; 73.
32.
Velocity Probe Calibration . .41'.
74
33, Experimental and Theoretical Pressure Distributions
for'
Average 1J1 and yc (without cover-plate) , .
1
75
34. Experimental and Theoretical Pressure Distributions for i
k
Average U1 and yc (with cover-plate) . . , . , . . 76,
II
35.
Streamline Pattern (without cover-plate), Vb = 3.92 ,Q...1.3 ,. . . ep 4 4 v
.
.
,
77
ir
36. Streamline Pattern (with cover-plate), Vb = 3.92 ,
Q= 1-3
-qv.
. ..
.',....
! ' ,,.':! t' 44 4:4m4 4:;.
i
LIST OF SYMBOLS
Amplitude of the wave
Celerity of the wave
Acceleration due to gravity Wave number, 27/X
Discharge
Wind speed at infinity
U(y) Wind velocity
U velocity,
v/To/P
1 Reference velocity, 17,/k
Relative Velocity, U(y) - c
Vb Belt velocity
Vw Water velocity
Direction in which wave propagates
Distance from moving boundary
yc Critical layer thickness
yo Streamline boundary
Dimensionless distance
zc Value of z at the critical level a Dimensionless pressure coefficient
Dimensionless pressure coefficient
Ap Difference in static pressure
6 Phase angle
von Kaman universal constant
X Wave length
Friction
-LIST OF SYMBOLS (continued)
Kinematic viscosity
Density
ACKNOWLEDGMENTS
The authors are indebted to Mr. Theodor Mogel for his suggestions concerning instrumentation and to Messrs. John Austin and Tom Wagner for their assistance in the experiments and in preparing the figures. Credit
goes to Mr. E. John Finnemore for the computer program shown in the Appendix. We are grateful to Mrs. Kay Mac Laury and Mrs. Ashby Longwell
1. INTRODUCTION
In the last decade, theoretical analyses, e.g., Miles (1957, 1959,
1960, 1962, 1967), Phillips (1957), Benjamin (1959) and Lighthill (1962),
have carefully delineated the basic mechanisms responsible for the gener-ation of waves by the wind. Phillips (1966) has given a detailed and
incisive account of these efforts. It is now accepted that the resonance
mechanism proposed by Phillips (1957) and the shear-flow mechanism pro-posed by Miles (1957) provide a reasonable qualitative description of the
initiation and growth of wind-generated gravity waves. However, the
experiments of Shemdin and Hsu (1966), Bole and Hsu (1967), and Hidy and
Plate (1965) in laboratory wind-wave facilities and those of Cox and
Snyder (1966) in the open ocean have shown that the theory and experiments do not agree quantitatively.
To verify one particular aspect of the developments by Miles,
i.e., the results of his 1957 analysis of an inviscid shear-flow over a
wavy boundary, Zagustin, et al. (1966) studied the flow of water past a
moving, wavy, yet solid boundary. The purposes of their experiments were
specifically to generate a real fluid flow that closely approximated the
theoretical model proposed by Miles (1957) and to compare the measured
pressure distributions near the boundary with those computed in accordance with Miles' theory. To precisely simulate this theory, the wave form must
remain fixed in shape and magnitude. Accordingly, wind-wave flume
exper-iments were not appropriate. Similarly, Zagustin, et al. (1966) argued
that the fixed wavy boundary experiments of Motzfeld (1937), Stanton, et
al. (1932), Thijsse (1952), Bonchkovskaya (1955), and Larras and Claria
(1960), etc., were not a correct simulation of the Miles model because
they did not permit the development of the so-called critical layer in
the flow.
The experiments of Zagustin, et al. (1966) were incomplete; how-ever, the experimental pressure profiles agreed reasonably well with those
predicted by the theory. In addition, because the experiments were run in
an open channel with the moving, wavy boundary forming one vertical wall of the channel, they experienced certain difficulties in measuring small
velocities and pressures because of the disturbances present on the free surface of the flow. As a result, they suggested a further investigation
of the areas near the critical layer and refinement of the experimental procedures.
The present work is a continuation and extension of the work of Zagustin, et al. (1966). Our objectives were to re-run and improve on
the accuracy and scope of their experiments. We undertook an analysis of
the Zagustin, et al. results; and while utilizing essentially the same experimental apparatus, we made specific changes to the measurement system and made extensive measurements to obtain detailed pressure and velocity profiles throughout the test section. Finally, a cover-plate was added
to the open channel, thereby making it a closed conduit in the region of the test section. The cover-plate removed many of the free surface
dis-turbances and made the flow at mid-depth in the channel essentially two-dimensional. Extensive measurements were made on this configuration also.
In the following sections we first summarize the principles and
results of the theoretical model upon which the experiments are based and the results of Zagustin, et al. (1966). Then the experimental apparatus
is described. Finally, the new results are cited and compared to theory
2. REVIEW OF THE EXPERIMENTS OF ZAGUSTIN, ET AL.
2.1. THE STEADY-STATE CONCEPT
The flow of a sheared fluid moving over a small amplitude,
sinu-*
soidal wave is shown in Fig. 1 . In this sketch the fixed wave form, with
amplitude a and length X , moves to the right at a constant speed or
celerity c . The velocity U of the fluid flowing over the waves varies with distance above the wave, i.e., U = U(y) , but remains constant in
time and direction.
To reduce this flow from an unsteady to a steady state, a velocity which is equal but opposite in direction to the wave celerity is superposed on the flow system. Under the assumption that for small amplitude waves the surface particle velocity variation between the crest and the trough is negligible, the wave-surface boundary is now moving to the left at a constant speed which is equal to the wave celerity c . Far from this
boundary, the velocity will be U(y) - c , while at the level where
U(y) = c the relative velocity v , defined as U(y) - c , will be equal
to zero. This level of zero velocity is called the "critical level," and the region between the moving boundary and this critical level is the "critical layer" (Miles (1957)). The critical layer thickness yc is
the distance from the boundary to the critical level. A sketch showing
the above steady-state conditions is shown in Fig. 2.
2.2. THE SHEAR-FLOW MODEL
In formulating his inviscid shear-flow model, Miles (1957) assumed
an inviscid, incompressible, two-dimensional parallel shear-flow coupled to a prescribed two-dimensional, deep-water, gravity wave train. Viscosity
and turbulence are supposed to play no essential part in the development, except that they are assumed to create and maintain a logarithmic mean velocity profile in the shear-flow. This logarithmic profile is expressed
as
U(y) = U1 ln(y/yo) (1)
*
For convenience, all figures are presented at the end of the work.
-where yo is the aerodynamic roughness height determined from extension
of the log profile to zero velocity, U(y) is the parallel shear-flow,
and U1 is the reference velocity defined by U1 =
U*/K
. Here, U* isPrandtl's shearing stress velocity and K is von Karman's universal
turbulence constant, usually taken as 0.4 . This profile is of particular significance in the energy transfer mechanism, for as Miles (1957) stated,
"... the rate at which energy is transferred to a wave of speed c is
proportional to the profile curvature
- U(y)
at the level where U = c ."The perturbations in the velocity and pressure in the shear-flow and associated with the wave motion were assumed to be two-dimensional and sufficiently small to permit the linearization of the equations of motion and application of the interface boundary conditions at the mean wave surface where y = 0 . In addition, only the component of the aerodynamic force that is in phase with the wave slope was considered to be significant;
that is, normal pressures were assumed to transfer the energy from the wind to the wave.
Through the use of the equations of motion and an assumed form for the aerodynamic pressure, the perturbed motion in the air was described
in terms of the Orr-Sommerfeld equation. Introducing dimensionless variables
into the equation, Miles formulated an aerodynamical boundary value problem,
the solution of which
would produce the values for dimensionless pressurecoefficients a and B . Miles presented an approximate solution to this problem in his 1957 paper. Later, Conte and Miles (1959) presented a
solution obtained by numerical integration of the Orr-Sommerfeld equation.
In both papers, a dimensionless wind profile parameter 2 = g yo/lq was
introduced, where g is the acceleration due to gravity. A dimensionless
wave speed c/U, was also defined, where c2 = g/k is the wave celerity
for deep-water waves and the wave number k is defined as 27/X , X
being the wave length. In Conte and Miles (1959), values of a ,
3,
and yc were tabulated for each value of the dimensionless wave speed
c/U1 and corresponding wind profile parameter
Q .
For each value of Q and
c/U1 the aerodynamic pressure
distri-bution over the wavy surface can be determined from the expression ,
p = - p a "Lq k
(a2
(32)1/2 cos(kx - 0) , (2)where
is
the phase shift of the pressure distribution relative to the wave form and in the direction of wave propagation. This phase angle canbe determined by the expression
tan 8 = - (6/a) . (3)
In a later paper, Miles (1959) extended his previous analysis by presenting results based on a more accurate solution of the Orr-Sommerfeld equation (Conte and Miles (1959)). He imposed the interface boundary
con-dition at the wave surface, rather than at the mean surface, and included the dominant viscous term in the complete Orr-Sommerfeld equation. He
found that imposing the boundary condition at this location had no effect
on the end results; also, the viscous effects in the air just above the water surface were found to be small compared with the effect due to normal pressures. He concluded that his model for energy-transfer from
a parallel shear-flow to deep-water gravity waves gave a total energy transfer in order-of-magnitude agreement with observation.
Miles (1959) also included other results which are significant.
Defining a dimensionless distance to the critical level
zc = kyc , he presented an expression which relates 0 to
zc
U1\2
zc =
QL-
exp(E-) .ul
In addition, noting that 0 versus
zc is essentially independent of
Q for
zc < 2 , he presented a figure which gives the coefficient B
as a function of
zc . This figure is reproduced as Fig. 3 of this study.
In Appendix B of his paper, Miles (1959) showed that for a small z,
the coefficient a could be obtained directly from
a = -
[(/7z)
-
62]1/2 . (6)(5)
.
-Since the papers cited above, Miles has presented, as have others,
improvements to his basic shear-flow model. Miles (1960) combines the
inviscid shear-flow model with Phillips' resonance model. This
combin-ation formed a stochastic theory which can be applied over a complete spectrum of waves. Miles (1962) investigated the viscous effects on
capillary and short gravity waves. Miles (1967) added the wave-induced
perturbations in the turbulent Reynolds stresses for momentum transfer to his model. At present, what can be considered his combined model
pro-vides a reasonable description of the physical phenomena between wind and small amplitude gravity water waves.
2.3. THE PREVIOUS EXPERIMENTS
Since the Miles-Benjamin theory suggests the importance of the existence of the critical layer in the energy transfer from wind to wave,
it is evident that the theoretical conditions leading to formation of a layer must be correctly reproduced in order to simulate Miles' (1957)
shear-flow model. Zagustin, et al. (1966) argued that the previous fixed
wave form models failed to reproduce correctly these conditions, in that a critical layer was not formed above the wave. Using the assumption by
Miles (1957) that the variation in particle velocity between the crest and trough of a small amplitude wave is small, Zagustin, et al. used a moving belt apparatus to establish the correct boundary conditions. In
their experiments a belt capable of moving at a constant speed and following the prescribed shape of a wave was placed vertically in a flume. Water
was used as the working fluid.
The belt was a U.S.S. Cyclone, "Flex-Deck"-type, flat wire mesh covered with a rubber sheet to provide the required smooth surface (Fig.
4). The rollers at each edge of the belt ran in guide tracks that set
the shape for the belt to follow, while cast iron sprockets transmitted the driving force to the belt from a variable speed electric motor. The
guides and sprockets were supported by a light I-beam (Fig. 5). In turn,
the I-beam was held by three metal supports which allowed for the leveling of the wave guides. The I-beam was placed in the depressed center section
in place, with the belt installed, is shown in Fig. 6. A Venturi meter
was used to monitor the discharge into the stilling basin at the upstream end of the flume, while a tailgate, located at the downstream end, pro-vided for regulation of water depth. A general view of the apparatus is
given in Fig. 7.
Sinusoidal waves with lengths of two and three feet and respective belt speeds of 3.2 and 3.92 fps were investigated. These speeds are the
celerities of deep-water gravity waves with the corresponding wave lengths.
Discharges of 0.9 and 1.3 cfs, which produced mean water velocities of 0.8 and 1.15 fps respectively, were used for each wave length. Pressure
measurements were taken every 1/16 of the wave length in each case. Two pitot-static probes were used to register the pressure dif-ference between the undisturbed pressure near the straight wall and the pressure near the moving wall. Vertical and horizontal pressure and
velocity profiles revealed that the most desirable location for the probes was 0.35 ft below the water surface, in a water depth of 0.75 ft. The
stationary probe was located 1.5 in. from the fixed wall, while the second probe was kept a constant 0.2 in. from the moving boundary for all pres-sure meapres-surements.
Velocity measurements were taken across the channel at the same depth as the static pressure measurements, and for each particular dis-charge these profiles were measured at the crest and trough of each wave.
Logarithmic profiles were fitted to these measured velocity distributions with a resulting large systematic deviation occurring in the vicinity of
the critical level where the mean velocity is zero. It was argued that
these large deviations were caused by the combination of extremely low velocities near the critical layer and the large shearing action in the critical layer. As a result of these effects, the logarithmic profiles
were fitted to the measured profiles only in that region above the critical level. Using these fitted profiles, values of kyc and S
(Miles (1959)) were obtained and the theoretical pressure distributions
calculated.
With regard to the experimental pressure distributions, Zagustin, et al. remarked that the surface effects in the open flow made it
extremely difficult to obtain good measurement resolution. Even though
screens were installed along the straight wall to damp out these effects,
it was noted that in some cases the dynamic fluctuations in the static
pressure readings were as large as 50 percent of the reading.
In spite of the stated limitations, Zagustin, et al. point out
that the magnitudes of the measured pressure profiles agree reasonably well with the theoretical distributions; however, the phase shift in most of the experimental distributions was less than the theory predicted.
They attribute this difference in phase shift to viscous effects not
accounted for in the Miles (1957) theory. This differential in shift does,
they argue, support Benjamin's (1959) conclusion that viscosity tends to retard the phase shift in the pressure distribution.
To achieve qualitative verification of the Lighthill (1959) pre-diction of a stationary vortex located over the wave crest, Zagustin, et al. also made flow visualization experiments by taking time exposures of plastic balls moving in the flow. The experiments were performed on the
sixth wave, of 3-ft length, in the wave train. The belt speed was reduced
to 50 percent of the value used in the pressure measurements described
above. In addition, to increase the thickness of the critical layer, the average velocity was reduced to 75 percent of that used for the pressure measurements. These reductions were justifiable because the mechanism proposed by Miles does not require that the moving boundary have the celerity of the equivalent deep-water wave. No pressure or velocity
measurements were taken at the above conditions; therefore, no detailed analysis of the patterns were attempted. However, it did appear that the
resulting flow patterns (Figs. 8 and 9) displayed the "cat's-eye flow pattern" hypothesized by Lighthill (1962) in his physical interpretation of Miles' energy transfer mechanism.
2.4. DISCUSSION OF EXPERIMENTS
Although the measured pressures seem to agree reasonably well with the theoretical predictions, the analysis by Zagustin, et al. (1966) to obtain
U1 and yc from measured velocity profiles suggests further
mentioned in Section 2.2., Miles (1957) points out that the rate of energy transfer is proportional to the profile curvature at this region.
By considering only the velocity measurements above the critical level,
they obtained lower yc and higher U1 values than those that would
have been obtained if the velocities around U = c were accounted for.
As a result, their theoretical maximum pressures were lower than they would have been if the total profile had been considered. In the present
study we investigated the region mentioned above and have determined U1
and yc in terms of the entire logarithmic-portion of the velocity
distribution.
The flow visualization experiments did show the presence of vortex patterns above the wave. However, since no detailed pressure or velocity
measurements were made at the reduced belt speed and mean flow velocity of the visualization experiments, it was difficult to obtain more than a qualitative analysis of these patterns. To investigate the flow pattern
in the present study, we derived streamline patterns by integrating the
velocity profiles used to obtain U1 and yc . The belt speed and mean flow velocity were the same as used for the pressure measurements.
3. SMALL AMPLITUDE WAVE EXPERIMENTS
3.1. INTRODUCTION
In Sections 2.1. and 2.2., it was emphasized that the
Miles-Benjamin theory confirms that the moving boundary conditions existing in the steady-state picture of wind-generated waves play a major role in the energy transfer from the wind to wave. In particular, the critical layer
thickness
yc has considerable effect on the pressure distribution along
the wave. In Section 2.3. the apparatus used by Zagustin, et al. to
simulate the moving boundary conditions was reviewed. Unlike the fixed
boundary models, this apparatus did generate a critical layer with a finite thickness.
The small amplitude wave experiments described below are a contin-uation of the Zagustin, et al. (1966) experiments with particular emphasis now being placed on the region around the critical layer. The apparatus
used in the present experiments is essentially the same as that described in Section 2.3. with the exception of the specific changes discussed below.
3.2. THE APPARATUS
The general dimensions of the facility, along with a detailed description of the test section, are shown in Fig. 10. The test section
Is the four-foot long area described in Detail A of the figure. It can
be seen in this detail that the wave used throughout the experiments had an amplitude a of 0.75 in. and a wave length X of 3.0 ft. Because of
the limitations of our variable speed motor, the maximum speed at which the belt could be driven was Vb = 3.92 fps . This corresponds to the celerity of a deep-water gravity wave with a length of 3.0 ft. A greater
belt speed would have been desirable because it would have produced a thicker critical layer. The amplitude a = 0.75 in. was selected because
it gave a ratio a/X = 0.0208 , i.e., the wave was a small amplitude wave.
This ratio, in turn, gave a reasonably large pressure variation along the wave with a minimum surface disturbance in the open channel for the given
flow. The discharge Q = 1.3 cfs was also held constant throughout the
without overflowing because of the downstream control.
In the test section, two pitot-static probes made by the United
Sensor Corporation were used to make all velocity and pressure measure-ments. One probe was fixed 1.5 in. from the straight boundary, while
the other could be moved to any distance from the moving boundary by a traversing mechanism. This motorized traversing and position indicator
mechanism could position and give the horizontal location of the pitot-static probe relative to the moving boundary to ± 0.01 in. A special
mechanism adapted to the device allowed for the rotation and alignment of the probe in the vertical axis. The traversing mechanism and fixed probe
were supported above the test section on an aluminum carriage which could be moved longitudinally along the full length of the test section. This
combination of the carriage and traversing mechanism allowed for the accurate placement of the movable probe at any location in the test section. The traversing mechanism and carriage are shown in Fig. 11.
The test section was located on the fifth wave in a series of seven waves
along the channel. The fifth wave was used because the flow was fully
developed before it reached that section of the channel. The test section
included the region from the upstream crest (Station 0) to downstream crest (Station 16) of the three foot wave (see Fig. 10).
3.3. INSTRUMENTATION
A cross section of the probe arrangement in the channel can be seen in Fig. 12. A typical probe is sketched in Fig. 13. These probes
were connected by a series of three-way stopcocks to a Pace model P90 differential pressure transducer, with a range of ± 0.037 psi. The
trans-ducer was coupled to a Sanborn Recording Oscillograph, model 650, with a carrier-amplifier system. The transducer was calibrated after each major
series of experiments by a Harrison micromanometer which has a stated resolution of ± 0.0001 in. of water. Calibration curves that relate
pres-sure to output voltage (or galvanometer deflection) are given in Fig. 14.
The calibration curves displayed no significant change in the time span of this study.
and/or static pressure from the movable probe and static pressure from the stationary probe. These combinations permitted velocity readings to
be taken from the movable probe or pressure differential readings between the static pressures at the straight wall and at the moving wall (see
Fig. 15).
3.4. SPECIAL PROBE ARRANGEMENT
Also tied into the probes by means of stopcocks was a small reser-voir that was kept at an elevation of about 15 ft above the apparatus.
This device, shown in the schematic diagram of the probe setup (Fig. 15), allowed for the flushing of air out of the probes, lines, and transducer.
The surge of water through the lines during flushing also cleaned out
foreign particles that may have deposited in the dynamic end of the movable probe. Before each run heated water was placed in the small reservoir,
and the hot water was washed through the lines, probes and transducer.
As pre-heating the water removed the dissolved air, the water was deficient in air when it cooled in the lines before the run. This deficit caused
any small air pockets that were present in the lines to be re-dissolved into the solution. This process proved very effective in removing air
bubbles from the lines.
3.5. EXPERIMENTS IN AN OPEN FLUME
3.5.1. Velocity Measurements
Velocity measurements were made at Stations 0, 4, 8, 12 and 16.
At these stations, velocity profiles were taken over y-distances ranging from 0.2 in. to at least six critical layer thicknesses from the moving boundary. To investigate the cross-channel velocity distribution,
measure-ments were taken completely across the channel at Stations 0, 8 and 16 and are shown in Fig. 16. Care was taken to insure that the probe was always
pointed in the direction of the flow. This required rotation of the probe
by 1800 at the critical level. In addition, the distance from the moving
boundary was checked after each rotation. All velocity measurements taken
3.5.2. Pressure Measurements
Pressure measurements were taken every 1/16 of the wave length.
At each station the movable probe was located 0.2 in. from the moving boundary. Care was taken to insure that the probe was always oriented
into the flow. This was accomplished by turning the probe in the flow
until the static pressure difference was a maximum between the fixed and moving probe. Each run consisted of measuring one pressure reading at
each of the 17 stations along the wave. Thus, during one run a complete
pressure distribution along the wave was obtained. Three of these profiles
were taken, and they are plotted on each of the Figs. 17b-21b. The range
of the values at a given point is an indication of the experimental errors present in the three otherwise identical runs.
To study the change in the pressure profile as a function of dis-tance from the moving boundary, pressure measurements were taken from 0.2 in. through the critical layer at Stations 0, 2, 4, 6, 8, 10, 12, 14 and
16. The measurements at Stations 0, 8 and 16 were continued across the
channel to the fixed probe, to study the pressure decay with distance from the wave. These cross-channel profiles are plotted in Fig. 22.
All the above pressure measurements made in the open flume are summarized in Table 2.
3.6. EXPERIMENTS IN A COVERED FLUME
3.6.1. Flume Cover-Plate
To suppress the surface disturbances encountered in the experiments in the open flume, a cover was built and installed. This cover-plate made
the channel a closed conduit in the region of the test section, thereby making the flow at mid-depth essentially two-dimensional. The plate
covered three wave lengths upstream and two and a half wave lengths down-stream of the test section (Fig. 10). The upstream length was sufficient
to allow full development of the flow in the region of the test section.
Half-inch marine plywood was cut so that it fit snugly between the straight wall and the moving boundary (Figs. 10 and 12). Special fittings were
between the cover and the channel bottom was set at 9.0 in. (Fig. 12).
A
round transition on the upstream end of the plate insured a smooth entranceof the flow into the test section. A general view of the cover is shown
in Fig. 23.
The 4-ft test section in the cover-plate had the general
dimen-sions shown in Fig. 10. This section was constructed out of half-inch
plexiglass, which allowed for a visual check of the probes in the flow. Slots were cut in the plexiglass at each station to allow for transverse
movement of the probes. In addition, the plexiglass plate was also split
longitudinally. This allowed the two plexiglass halves
to be slid over one another, facilitating access to the probes. An aluminum T section was
countersunk in the split to add strength to the plexiglass sections.
Countersunk screws with wing nuts held the section in place. Pressure
sensitive tape was used to seal all the slots that the probes did not
occupy. This insured that the surface exposed to the flow was completely
smooth. Figure 24 shows the plexiglass section with all slots taped,
except Station 8 where the probe is in place.
During all experiments, a thin film of water was allowed to flow over the entire cover-plate. This insured that the underside of the plate
was under a slight pressure, which in turn reduced the effect of air pockets that may have formed under the plate. Also, before any
measure-ments were made, all significant air bubbles were removed from under the
plexiglass. In both the velocity and pressure measurements, the probes
were located 4.5 in. below the plate (Fig. 12). This placed the probes in
the region of maximum horizontal velocity in the conduit.
3.6.2. Velocity Measurements
Velocity profiles were taken at Stations 0, 4, 8, 12 and 16. The
measurements ranged from 0.2 to 4.5 in. from the moving boundary. Again,
care was taken to rotate the probe 180° at the critical level and to check
the distance from the wave. A summary of velocities with distance from the
moving boundary is presented in Table 3. The velocity profiles are plotted in Figs. 25a-29a.
3.6.3. Pressure Measurements
The pressure measurements with the cover-plate were made in the same manner as those made without it. That is, the movable probe was
positioned 0.2 in. from the moving boundary and measurement was made at each station along the wave. The fixed probe was maintained 1.5 in. from
the straight wall at each station (Fig. 11). Two complete sets of
measure-ments were taken along the wave and are presented in each of the Figs. 25b-29b. Again, the range of values at a given point is an indication of
the experimental error in the two otherwise identical runs. To investigate
the pressure distribution along the wave as a function of distance from the wave, pressure measurements were also made at various distances from the moving boundary. Profiles were taken from 0.2 to 4.5 in. from the
moving boundary at Stations 0, 4, 8, 12 and 16. At each of the remaining
stations, profiles were measured from 0.2 to 1.0 in. Pressure
distribu-tions along the wave are given as funcdistribu-tions of distance from the wave in Fig. 30. The comparable result for the open flume is shown in Fig. 31.
3.7. EXPERIMENTAL ERRORS
In the open flume experiments, the same types of errors arose that Zagustin, et al. (1966) experienced in their experiments. These errors
were caused primarily by surface disturbance effects on the pressure readings. The surface disturbances resulted from the belt motion, and
their relative effect on the pressure reading diminished as the flow-rate was increased. Recall that a flow of Q = 1.3 cfs was used throughout
all experiments because it was the maximum flow the facility could handle without modification to the downstream outlet works. Also, three layers
of screens were attached along the straight wall to help damp out reflec-tions of surface undulareflec-tions. Even with these precautions, surface
dis-turbances caused a noticeable effect on the pressure and velocity measure-ments. To filter out high frequency disturbances induced by mechanical
vibrations, a 500 mfd capacitor was attached in parallel with the galvano-meter in the Sanborn 650. Despite the measures taken above, the
fluctu-ations around the mean value of the reading were about 20 percent without the cover-plate. When the cover-plate was used to make the flume a closed
flow, the fluctuations in the pressure readings were reduced to half of
those without the plate. To insure a representative reading, pressure
fluctuations were registered for two minutes at each point. The mean
value of the reading was taken from these readings by drawing a best-fit line by eye on the recording paper.
To insure that a compatible relationship between the shape and phase of the measured pressure distributions along the moving boundary was obtained, the pressure probe location relative to the belt was
main-tained at 0.2 in. during measurements with and without the cover-plate.
This distance was more than twice the probe diameter and so was large enough to avoid effects due to the presence of the solid boundary. In
addition, this distance places the probe in a region where the pressure change with distance from the moving wall is small (see Fig. 22 for an indication of the pressure variation with distance from the wall in the open flume).
As mentioned in Section 3.4., errors from the formation of air bubbles in the probes and lines were kept at a minimum by periodic flushing
of the system. After each flushing, the system was given adequate time to
restabilize before a new reading was attempted.
The flow in the experiments had a finite depth and width and, in the case of the open flume, a free surface. As a result, the experimental
flow was neither infinite nor fully two-dimensional, in contrast to the theory of Miles (1957). However, all the velocity and pressure measure-ments were taken at one depth where the flowfield was expected to be most
nearly plane. In addition, the flow developed quickly over the moving boundary, which tends to make the effect of the finite length of the test
section negligible. The maximum variation in channel width was ± 0.75 in., which caused a small variation in the average flow-velocity along the wave.
This variation would be a maximum of 8 percent between the crest and
,
trough if the flume water surface remained plane. However, it can be seen
in Fig. 16 that the actual variation in velocity is small in our open-flume
experiments. This would indicate that the free surface does adjust its level to compensate for the variations in channel width. In the closed
it appears that the errors due to finite channel width and depth are relatively small.
Small errors in velocity measurements did occur in the region
close to the critical level. Two causes of these errors were the extremely
low velocities near the critical level and the large shearing action in the critical layer. These errors are discussed below in more detail. To
help alleviate these errors, two readings were taken at each point in the region of the critical level and longer time spans were used to determine the mean for each reading.
3.8. DATA REDUCTION METHODS
3.8.1. Least-Squares Fit of Velocity Profiles
The velocity was determined for each data point from the difference in dynamic and static pressure at the movable pitot-static probe. The
velocity profiles at each station were taken from 0.2 in. to at least 4.5 in. from the moving boundary. The portion of the profile that was
con-sidered logarithmic was from 0.2 in. to a distance where the measured
velocities decreased or remained constant with distance from the moving boundary (Fig. 16). In most cases, this distance was about 3.5 in. The
velocity values were tabulated, digitized and fed into a computer, and a
least-squares-fit straight line was calculated and plotted through these data points on semi-logarithmic plots (Figs. 17a-21a and 25a-29a). The
intersection of the least-squares-fit line with the relative velocity value v = 0 gave the value for yc . In most cases, this value agreed
very well with the measured critical layer thickness. The value of the
reference velocity U1 was derived from the slope of the least-squares
line by using Eq. (1). The values of U1 and yc are listed on the
above plots. A sample computer print-out of the least-squares-fit line,
yc and U1 is given in Table 5.
3.8.2. Calculation of Theoretical Pressure Distribution
Using the six equations of Section 2.2. and a computer program (given in the Appendix) developed from the Conte and Miles (1959) paper, we calculated the theoretical pressure distributions predicted by Miles.
These distributions were then compared to the measured pressure distribu-tions described in Secdistribu-tions 3.5.2. and 3.6.3.
Having obtained the values of U1 and yc for a particular
station by the least-squares method, we used Eqs. (4) and (5) to compute
zc and Q . For a given set of values of Q ,
zc and c/U, , we
cal-culated 8 using the program mentioned above. The range of 8 values for these experiments is depicted in Fig. 3. Since
zc is less than
unity for the range of these experiments, a can be evaluated directly
from Eq. (6). The phase shift angle was calculated from Eq. (3). It
follows that the theoretical pressure distribution over the full wave length is given by Eq. (2). The theoretical distributions are plotted in
Figs. 17b-21b for the open flume and in Figs. 25b-29b for the closed flume experiments. Note that each theoretical distribution is uniquely
deter-mined by the parameters of the velocity profile used. Accordingly, the
theoretical pressure distribution along the wave determined by the velocity profile at Station 0 (Figs. 17a and b) is different from that determined by the velocity profile at Station 4 (Figs. 18a and b). The experimental
4. DISCUSSION OF EXPERIMENTAL RESULTS
4.1. VELOCITY PROFILES
In the velocity profiles taken without the cover-plate, there is
a noticeable discontinuity at the critical level where U - c = 0 , i.e.,
v = 0 (cf., Figs. 17a-21a). This discontinuity seems to be caused by
fluctuations in the extremely small velocities near the critical level.
The velocity fluctuations then appear as fluctuations in the reading of the static-dynamic pressure difference obtained from the pitot-static
probe. In some instances these fluctuations were as large as the mean
difference between the static and dynamic pressures. However, one would
expect that the fluctuations would be averaged out by our data recording technique and that no discontinuity would appear in the velocity profile.
The velocity fluctuations arise from two sources. First, small
surface disturbances may induce subsurface fluctuations. Addition of the
cover-plate would reduce the size of this source of disturbance. Second,
through the viscous and turbulent transfer action in the critical layer,
there is a high shearing action in the layer. This action is particularly
evident near the critical level where the mean velocity is zero.
Accord-ingly, the flow direction can be expected to reverse itself randomly in the region of the critical level as a result of the shear- and surface-disturbance-induced fluctuations.
A flow reversal has a serious impact on the apparent reading of a pitot-static probe of the type used here. To investigate this impact, a
probe was placed in a known flow and aligned in the proper or positive direction. The difference between static and dynamic pressures was
re-corded, and then the probe was reversed in the flow. Again, the pressure
difference was recorded. By carrying out this procedure for a number of
flow-rates, we were able to plot a probe calibration curve (Fig. 32). It
is seen that for a given velocity (pressure) the recording galvanometer
deflection is relatively very large for the probe-into-the-flow position.
Accordingly, in a fluctuating velocity field, the reading will be zero on the average only when the fluctuations are primarily in the negative direction (against the back of the probe). As we approach the critical
level, where v = 0 , from either side, the probe (which has been oriented
into the dominant flow direction) will give undue weight to positive flow.
In particular at the point where v = 0 , the probe will indicate a
posi-tive velocity. As a result, a discontinuity appears in our measured
velocity profiles where v should be equal to zero.
With the cover-plate in place, the surface source of velocity fluctuations was minimized. This reduced, but did not remove, the profile
discontinuity, indicating that the shearing action near the critical level is a significant source of direction-reversing velocity fluctuations.
Since the pitot-static probe was reversed when it passed through the critical level, the measured points to each side of the level tend to have equal, but opposite-in-sign, errors induced by the velocity
fluctua-tions. In the process of data reduction (see Sec. 3.8.1.) least-squares-fit straight lines were calculated for the presumed-logarithmic portion of each velocity profile (which included the critical level). The
least-squares differences were associated with the velocities in a data set of velocities versus distance from the moving boundary. Because of the
dis-tribution of the probe error near the critical level, the least-squares fit tended to balance out the velocity profile errors near v = 0 by
weighing the errors on both sides of this region equally, thus, averaging out their effect. In addition, the majority of the points used to establish the logarithmic profile lie outside the region where the probe error due to fluctuations is significant. This tends to decrease the effect of the few
points near the critical level. Note, however, that we are using the entire
measured velocity profile in our present analysis in contrast to Zagustin, et al. (1966) who used only points outside the critical layer.
4.2. PRESSURE PROFILES
4.2.1. Comparison With Miles' Theoretical Distribution
The procedure used to calculate the theoretical pressure distri-butions was discussed in Section 3.8.2. Given the values of U/ and yc
as input data, a computer program (see the Appendix) calculated the
program showing the distribution is given in Table 6. The distributions
predicted on the basis of the local profile at various stations are shown
as
solid lines in Figs. 17b-21b for experiments in the open flume and in Figs. 25b-29b for experiments with the cover-plate in place.In the experiments without the cover-plate, the theoretical dis-tributions predicted for the velocity profiles at the two crests are es-sentially the same; that is, they have the same magnitude and phase shift.
This is expected because the flow is fully developed before it reaches the test section. For example, we see in Fig. 16 that the cross-channel
velocity profiles at the two crests are very similar. The magnitude and
phase shift of the theoretical pressure distribution at the crests are larger than at any other location on the wave. This is a result of the
combination of a large U1 and yc at these stations. The theoretical
distributions at Stations 4 and 12 are also very similar to each other.
The combination of a small
U1 and a large yc at Station 4 seems to
balance with a large U1 and a small yc at Station 12 to give
essen-tially the same theoretical pressure distribution. A combination of small
U and a small
yc at Station 8 produced a pressure distribution with 1
the minimum magnitude.
With the cover-plate on, the experimental pressure displayed less scatter than in the open flume experiments. In addition, with the
cover-plate on, the pressure measurements on the leeward side of the wave are slightly lower than in the open flume experiments. The difference could
be explained by the fact that the pressure measurements with the cover-plate on were taken in the region where the flow was fully two-dimensional and in the region of maximum horizontal velocity. This insured the
maxi-mum negative pressure readings. However, in the case of the open flume,
the free surface made it difficult to precisely determine a fully two-dimensional flow region in which the flow velocity was a maximum. As a
result, the pressure measurements may not have been taken in the region of maximum horizontal velocity.
In most cases, the theoretical pressure distribution displays a larger phase shift in the downwind direction than the experimental dis-tributions. This difference is probably due to viscous effects not
accounted for in the theory. That is, as mentioned above, Miles assumes
the air flow to be inviscid, incompressible and two-dimensional. Viscosity
acts only to maintain the logarithmic mean-velocity profile. However,
this difference in phase shift does coincide with Benjamin's (1959) con-clusion that the general effect of viscosity would be to retard the phase shift in the pressure distribution (cf., Zagustin, et al. (1966)).
It is clear from the above that the velocity profile changes sub-stantially from station to station along the wave and that this in turn affects the theoretical pressure distribution predicted from one local profile. It would seem reasonable to remove the profile variation by
averaging and to predict a "best" pressure distribution from the average profile parameters.
Accordingly, to investigate this effect of the local velocity pro-files on the theoretical pressure distribution, the values of U1 and
yc were averaged for Stations 0 and 16. These new values were then
averaged with the U1 and yc values taken from Stations 4, 8 and 12.
The theoretical pressure distribution associated with these average values is plotted in Fig. 33, for velocity profiles taken without the cover-plate,
and in Fig. 34 for those taken with the cover-plate in place. In both
cases, these distributions lie somewhere between the theoretical distribu-tions of Stadistribu-tions 4 and 12. The phase shift and magnitudes of the
experi-mental profiles agree reasonably well with the theoretical profiles pre-dicted by Miles.
4.2.2. Pressure Decay
Miles (1957) and Lighthill (1962) point out that within the approx-imations of Prandtl's boundary layer theory, the pressure variations across
the critical layer region can be assumed small. This assumption is
veri-fied by the results shown in Figs. 22, 30 and 31.
In Figs. 30 and 31, pressure distributions were plotted for equal distances from the moving boundary. The dashed lines in Fig. 31 were used
when insufficient data was taken at that distance to warrant a precise profile. In both figures, it can be seen that the pressure distribution
remains essentially constant through the critical layer, which was about
one inch thick, and starts to die out or decay with greater distance from
the critical layer. In Fig. 22, this decay can be seen more clearly. In
these plots, it can be seen that the static pressure difference between
the fixed and moving probe decays in an exponential fashion with distance
from the moving boundary. This would coincide with the Lighthill (1959) argument that pressure gradients decrease above the wave in an exponential manner.
4.3. STREAMLINE PATTERNS
The streamline patterns for the experiments with and without the cover-plate are shown in Figs. 35 and 36, respectively. These patterns
were sketched in order to obtain a qualitative picture of the flow
char-acteristics in the logarithmic region of the velocity profiles. To obtain
these streamlines, the velocity profiles at Stations 0, 4, 8, 12 and 16
were integrated to produce a set of distance points between each of which
0.05 cfs passed. These points of equal flow were plotted at each of the
above stations. A distorted scale was used to facilitate examination of
the region near the critical layer. A dashed line was placed through the
points where v = 0 . Starting from the moving boundary, we connected
points of equal flow by smooth lines that were guided between points by the contour of the wave. It was assumed that Lighthill's "cat's-eye
patterns" did exist around the critical layer. The Zagustin, et al.
(1966) flow visualization experiments supported this assumption (Figs. 8
and 9). Using Lighthill's physical interpretation of Miles' theory, we knew that the low pressure region was located in the center of the "cat's-eye pattern," while a high pressure region was located in the region where
the vortices meet. Reading from the experimental pressure distributions for each flow, one can see that the high pressure area is located at
Station 10 and the low at Station 2. Using this feature of the theory and
experiment, the center of the vortex and the region where the vortices
meet were located. Also known was the fact that the vortices distribute themselves essentially symmetrically around the low pressure area. Using
these facts, the estimated points where the streamlines crossed the
The large critical layer at Station 4, as well as the small criti-cal layer at Station 12, can be seen in the sketched patterns. The
vari-ance in the thickness of the critical layer perhaps can be attributed to the effect of a pressure gradient on boundary layer growth. In Figs. 30
and 31, it can be seen that the pressure distribution at Station 4 has a positive gradient and at Station 12 it has a negative gradient. As a
result of the positive, or favorable gradient, the boundary layer is the thickest at Station 4. Since the critical layer is directly affected by
the boundary layer, it also is the thickest at Station 4. As a result of
the negative, or adverse gradient, the boundary layer is diminished and the critical layer becomes thinner at Station 12. The streamline patterns
at Station 0 and Station 16 appear to be essentially the same. This
con-sistency supports the argument that the flow is fully developed at the test section.
5. CONCLUSIONS
The primary objective of this experimental investigation was to
extend and to improve the experimental data obtained by Zagustin, et al.
(1966) on steady-state simulation (in an open flume) of small amplitude
wind-generated waves. Extensive velocity and pressure measurements in the flow-field, particularly in the region of "critical level," were obtained to gain further insight into Miles' inviscid mechanism of trans-fer of energy from wind to water waves. In addition, measurements of
velocities and pressures were repeated with a cover-plate installed to minimize free-surface disturbances.
It has been found that the addition of the cover-plate over the
open flume reduces the free-surface disturbances and improves the quality
of the data. Fluctuations in pressure readings with the cover-plate in place were about one half of those in an open flume. The data agree with
the previous data obtained by Zagustin, et al. It may be concluded that
the free surface had a negligible effect in so far as the steady-state
simulation of wind blowing over progressive, small amplitude waves is
concerned.
It was determined that the critical layer thickness varies along
the wave and that this variation is correlated with the local pressure
gradient. As a result, the values of the logarithmic velocity profile
parameters U1 and
yc vary along the wave. The calculated theoretical
pressure distributions, based on the measured local values of U1 and yc at the wave crest and wave trough, had the largest and the smallest
amplitudes, respectively. The values of U1 and
yc measured at
inter-mediate stations along the wave yield calculated pressure amplitudes
lying between those at the crest and trough. However, most significantly, the theoretical pressure distribution based on the average values of U1
and yc along the wave agrees reasonably well with the experimental dis-tribution for the case of the open flume, as well as for the flume with a cover-plate in place.
Finally, the singular behavior of the velocity distribution in the region of critical level deserves further experimental investigation.
It would be extremely desirable to investigate the structure of this
region by employing hot-wire anemometry in a steady-state simulation of wind waves such as that accomplished here.
REFERENCES
Benjamin, T. B. (1959): Shearing flow over a wavy boundary, J. Fluid Mech., 6, pp. 161-205.
Bole, J. B. and E. Y. Hsu (1967): Response of Gravity Water Waves to Wind Excitation, Dept. of Civil Eng. Tech. Rept. No. 79, Stanford Univ., Stanford, Calif.
Bonchkovskaya, T. V. (1955): Wind Flow Over Solid Wave Models, Akademia Nauk SSSR, Morskoi Gidrofizicheskil Institute, 6, pp. 98-106.
Conte, S. D. and J. W. Miles (1959): On the numerical integration at the Orr-Sommerfeld equation, J. Soc. Indust. Appl. Math., 7,
pp. 361-366.
Hidy, G. and E. Plate (1965): Wind Action on Water Standing in a Labor-atory Channel, NCAR Manuscript No. 66, Boulder, Colo.
Larras, J. and A. Claria (1960): Wind Tunnel Research on Relative Wave and Wind Action, La Houille Blanche, No. 6, pp. 647-677.
Lighthill, M. J. (1962): Physical interpretation of the mathematical theory of wave generation by wind, J. Fluid Mech., 14,
pp. 385-398.
Miles, J. W. (1957): On the generation of surface waves by shear flows, J. Fluid Mech., 3, pp. 185-204.
Miles, J. W. (1959): On the generation of surface waves by shear flows, J. Fluid Mech., 6, pp. 568-582.
Miles, J. W. (1960): On the generation of surface waves by turbulent shear flows, J. Fluid Mech., 7, pp. 469-478.
Miles, J. W. (1962): On the generation of surface waves by shear flows. Part 4, J. Fluid Mech., 13, pp. 433-448.
Miles, J. W. (1967): On the generation of surface waves by shear flows. Part 5, J. Fluid Mech., 2, p. 417.
Motzfeld, H. (1937): Die turbulente Stromung an welligen Wanden, Z. angew. Math. Mech., 17, pp. 193-212.
Phillips, O. M. (1957): On the generation of waves by turbulent wind, J. Fluid Mech., 2, pp. 417-445.
Phillips, O. M. (1966): Dynamics of the Upper Ocean, Cambridge Univer-sity Press.
Shemdin, 0. H. and E. Y. Hsu (1966): The Dynamics of Wind in the Vicinity of Progressive Water Waves, Dept. of Civil Eng. Tech. Rept. No. 66, Stanford Univ., Stanford, Calif.
Snyder, R. L. and C. S. Cox (1966): A field study of the wind generation of ocean waves. Sears Foundation, J. Marine Res., 24(2),
pp. 141-178.
Stanton, T. E., D. Marshall and R. Houghton (1932): The Growth of Waves on Water Due to the Action of the Wind, Proc. Roy. Soc., A, 137, pp. 283-293.
Thijsse, J. Th. (1952): Growth of Wind-Generated Waves and Energy Transfer, Gravity Waves, Nat. Bur. Stand., Circular No. 521, Washington, D. C., pp. 281-287.
Zagustin, K., E. Y. Hsu, R. L. Street and B. Perry (1966): Flow over a Moving Boundary in Relation to Wind-Generated Waves, Dept. of Civil Eng. Tech. Rept. No. 60, Stanford Univ., Stanford, Calif.
APPENDIX
COMPUTER PROGRAM TO CALCULATE THEORETICAL PRESSURE DISTRIBUTIONS
This appendix presents the essential parameters needed as input data to a computer program that uses numerical integration of the Orr-Sommerfeld equation to solve for B and a . The program is based on
Conte and Miles (1959) and was developed for the IBM model 7090 by E. John Finnemore at Stanford University. For this study we have
modi-fied the program for use on an IBM model 360/67 computer. In addition,
we have added routines that use the CALCOMP x-y plotter to plot directly the pressure distributions presented in this text.
The program, of which a listing is given below, is written in Fortran IV and requires approximately one minute to compile and execute for one typical run. It consists of a main program and one subprogram
INTGRT.
PROGRAM VARIABLES
Input Variables
EPSILON (always positive) = w is the inte-gration limit used to bound the numerical Integration calculation away from the singu-larity in the Orr-Sommerfeld equation at
w =0 (E=1x10-5 ).
RNVDW 1/Aw , w = integrating step ( Aw = 1/64 ).
IDWPL Awt/Aw (must be an even integer multiple of
1/Aw) = 4.0 .
Number of stations on pressure plot.
NUMPAT Number of data sets or number of pressure plots.
WA Wave amplitude (inches).
WL Wave length (feet).
WC Wave celerity (ft/sec).
QCFS Water discharge (cfs).
Ul The reference velocity obtained from the slope of the logarithmic velocity profile.
YC The critical layer thickness. . ...
Important results calculated in the program are listed in the
sample printout in Table 6.
The subroutine INTGRT performs the numerical integration by the method of Runge and Kutta and is accurate to the order of (406.
The basic wave data is read in; Q , WO ( WO = - (WC/U1) ) and
zc are calculated by methods outlined in Section 3.8.2. The program
then uses the methods of Conte and Miles (1959) to calculate 6 and a .
Using a and 6 , the program then calculates the theoretical pressure
distribution above the wave by the methods of Section 3.8.2.
Sample Q and WO values from the table in Conte and Miles
(1959) were read into the program and the resulting 6 and a values were checked against those listed in the paper. In all cases the 6
values checked to i 0.01. In addition, in the pressure distribution
calculations, the computed values of 6 were checked against the 6
versus
zc from Miles (1959) (Fig. 3).
They agreed very well. The
a values calculated by the program method generally agreed with those
listed in Conte and Miles (1959). However, in some cases they disagreed
by ± 0.1.
In this study zc < 1 . Therefore, a was obtained directly
CALL STRTP1(10) 122 READ(5,50) E. ANV1J6, 106,PL, N 5O FORMAT( 20X, E10.2, F10.1, (10, 15) RD6PL = IDWPL DwPLuS = PIMPL/RNVDw 144 961 = 1.0/RNVOw OH? = OW1 - E E2 E*F 13 L2*E PI = 3.14159 UNDFLO = 0.1**35 CVP(LC = 10.0**35
NAMELIST / (ASTI / F, oWl, DWPLUS
WRITE(6,LIST1) NUMCUT = 1
RFAC(5,52) NUmDAT
NEW PAGE
PROGRAM ,,HICH USES MILES THEORY TO CALCULATE THE PRESSURE DISTRIBUTF-IN ABOVE A MOVING WAVE. FORM, USING THE THICKNESS CF THE CRITICAL LAYER (YC) AND THF CHARACTERISTICS IF THE LCGAkITE1C PROFILE (U1).
NOTATIGN
HA HAVE AFRLITUDE (INCHES)
wL = WAVE LENGTH (FEET)
wC = wAVF CELERITY (ET/SEC)
Wk WAVE NUMRER
QCFS = WATER DISCHARGE ICES)
Dl FOUND FRCM SLOPE OF SEMILOG PLOT OF EXP VELOCITIES
YC = CRITICAL THICKNESS FOUND FROM SAME PLOT (INCHES)
LC = 1 AT SINGULARITY (6=0)
WC = 4 AT Z=0 (HERE REQ. IWOI.ET.12 - SEE S.103)
= EPSILON (ALWAYS PuSITIVE) (= WBAR)
Dv. = DELTA
DwPLUS = DELTA 6+
RNVDW = 1/0w
Eli = PH1(SUB-1)
FIIX = SEkIES EXPANSION FOR Eli AT w = +CR- E FlIv = PHI(SU0-1) PRIME (DERIVATIVE W.R.T. W)
= SERIES EXPANSICN FOR FII AT w = +E
FI1XN = SERIES EXPANSION FCR FII AT W = -E
IDwPL = D6PLUS/C6 (+VE INTEGER)
(60 = HO/OW (-VI INTEGER)
NUmBER OF STATIONS ON THE PRESSURE PLOT NuMDAT = NUmBER OF DATA SETS
NUM)'LT = NUMBER OF EXPERIMENTAL PRESS PLOTS IN DATA SET
V REPRESENTS PRIME (DERIVITIVE W.R.T. w)
IMPLICIT PEAL*R(A-H, u-Z),INTEGEk*,,(I-N)
R(-AL*4 P(15)),PS(150),01(19).DS1(19) ,D2(19),DS2(19). 03(19),DS3(19),D4(19),OS4(19)
COMMON 6, OH, IC?, Y1, Y3, F11, FI3 SUBR
C. C, =
-FIIXP -= =52 FURMAT(12) 1000 RL4D(5,51) wA, WL, WC, CFS, Ul, YC 51 FORMAT( 6F10.4) YCFT = YC / 12.0 WN = 2.0 * 3.1416/WL LC = WN * YCFT w0
- (WC /
01) CMEGA = 70 * (W(')**2 * OEXP(w0) WO = WO * 64.0 (WC = 40 RwO = IWOTEMP = DABS (0 - RWO)
IF (TF.P.(17.0.5) IWO = IWO + I RWO = IWO WO = RWO/RNVCW 188 702 = 1C*10 W03 = WO*4,40*WO WPITE(6,60) 60 FORmA7(11,1, 13IH
/
WRITE(6,61) WO, OMEGA, ZC, YCFT
61 FORMAT(IHO, 8HFUR WO =, F6.2, 10H OMEGA =, F7.4, I4X, 7H...ZC =, F9.5,7H YCFT =,F6.4 ///)
AT SINGULAPITY PLUS AND MINUS EPSILON
200 71 = ZC2*E2/6.0 12 = ZC2*E3*7.0/35.0 73 = ZC2*(31.0+Z02+2C2)*E2*E2/ 240.0 74 = 7C2*(333.0+101.)*ZC2)*E2*E3 /5400.0 211 FI1XP = F*(1.0
+ LI
+ 72 Z3 + 74)FI1XN = -FIIXP 2.0*112 + 14)*E
FIIVXP = 1.0 + 3.04,71 + 4.0*72 + 5.0*73 + 6.0*74 = FIIVXP - 6.0*72 - 12.0*14 22? 75 = (1.0 - ZC2)*E/2.0 Z6 = (3.0- 20.0*ZC2)*E2/36.0 Z7 = (6.0 - 137.0*ZC2 - 18.0*ZC2 *702)*E3/432.0 Z8 (45.0/1896.0 + 179.0*ZC2/ 632.0 - LC2*2C2)*74.0*E2* E2/900.0 233 FI3XP = -1.0 + L*(Z5 + 76 r 77 Z8)
FI3XN = FI3X12 - 2.0*176 + Z8)*E
FI3VXP = 2.0*75 + 1.0*Z6 + 4.0*77 5.0*Z8 FI3VXN FI3VXP - 4.0*Z5 - 8.0*Z7 -- - - --- -FIIVXN - -= -+ -+ -
-NUMERICAL INTEGRATION FOR VI, Y3, Ni, FI3 FROM W=-E TO WO
Y1V = DYI/DW = YI-F11(1/w-ZC2*EXR(2W)) OE 1 (1)
FIIV = DFII/OW = YI OF 2 (1)
Y3V = DY3/DW = Y3-FI3(1/W-ZC2*EXPI2W1)
+F11(1+1/W)/W-2Y1/W OF I (2/
FI3V = DFI3/DW - Y3 DE 2 (2)
C
NOTE..
EXP(X), X -Gr;,. 68.029692 NOT ALLOWED
KCUNT1 = 0 300 KOUNT1 = KOUNI1 + 1 If-(KCUNTI .CT. 1) GO IC 333 311 W = -E Vi = FIIVXN Y3 = FI3VXN FII = FI1XN FI3 = F13XN 322 DW = -DW2 GO TO 344 333 OW = -DWI 344 CALL INTGRT 355 KTCHK1 = -WO*RNVOW*1.2 IF(KOUNT1 .GT. KTCHK1) GO TO 911
IF( W .LT. WO-OW/2.0 .AND. W .GT. wO+DW/2.0 I GO TG 400
GC TO 300 HENCE, AT WO 400 YIWO = Y1 FIlw0 =
FII
Y3w0 = Y3 FI3W0 = FI3 wATwO = W KNTIWC = KOUNT1 FOR FI2W0..FI2 = FIl*LOGW + FI3
C AT WO, A IS -VE, SO FI2 = FI2R + H*F12I COMPLEX
WHERE FI2R = FI1*LOGI-W1 + FI3 REAL PRT
FI2I -P1*FII [MAO PRT
THIS IS 4CCCUNTE0 FOR IN THE SOLUTION (REF 900) FOR A AND
411 ROGWO = OLOC(-WO)
FI2Rw0 = FII*RCCWO + FI3 REAL PRT
NUMERICAL INTEGRATION FOR Y1,
Y3. Ni, FI3
FROM W=+E TO W+C
=
