Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
3. EXPERIMENTAL SETUP! METHODOLOGY, AND PROCEDURES
Experimental studies on the bottom roughness for pure current, pure wave, and
combined wave-current Qows were conducted at the Ralph M, Parsons Laboratory
for Water Resources and Hydrodynamics at the Massachusetts Institute of
Technology, The experimental setup and procedures necessary to determine the
roughness over a rippled bed are presented in this chapter.
3.1. Ex erimental Setu
The experimental setup can be seen in Figure 3.1. Realistic wave and current
boundary layers were obtained in an existing wave fume which was modified to
accommodate combined wave-current flows. This wave flume consists of a Aume
section and two larger holding tanks on either end. The flume section is 28 meters
long, .90 meters deep, and .76 meters wide. It has glass sidewalls and is equipped
with a piston-type programmable wavemaker at one end, and a 1-on-10 sloping
absorber beach at the other end.
To perform the experiments, a current-generation system was designed and the
fume was modiGed to accommodate combined wave and current flows. The details
of the design and installation of the current generation system are summarized in
Appendix A. A 1200-gpm current was generated by recirculating the flow from the
holding tank at the downstream end of the flume back to the upstream end of the
flume. To accomplish this, a 1200-gpm pump was installed with the pump inlet
connected to the downstream holding tank of the wave flume. The pump outlet was
connected to an inlet transition structure which was located underneath the wave
flume in front of the wavemaker. The beach was perforated to allow a fairly
uniform flow of water into the downstream holding tank. The beach was also
� 49-
- 50-
covered with several sheets of plastic mesh matting to help minimize reflections,
Using the piston-type wavemaker, pure progressive second-order Stokes wa,ves
are generated. The wavemaker was controlled using a Metrabyte Dash-16 D/A
converter which was installed in an IBM XT computer. By programming this
computer, a, digital signal was generated and converted into a 0 to 5 volt analog
signal. This analog signal was converted into a -10 to 10 volt analog signal using a
reverse polar linear amplifier which was developed at MIT. With this input signal,
the piston-type wavemaker can be used to generate second-order Stokes long waves
with amplitudes up to 7 cm and periods of approximately 3 seconds. The
procedures for wavemaker operation require a, fairly detailed system calibration,
which is detailed in Section 3.3.3,
The current flowrate was controlled using a gate valve and was measured using a
delta;tube flow meter, The maximum Qowrate of 1200 gpm gives an average flume
current velocity of 17 cm/sec with a typical water depth of 60 cm. This provides
ample flexibility to study a wide range of relative current and wave intensities. The
transition structure was constructed to ensure that a. fairly uniform Qow entered the
flume such that no energy dissipator or 61ter was necessary.
With these wave and current conditions in the flume, rough turbulent boundary
layers were developed by placing triangular bars along the fume bottom. The
triangular bars are 1.5 cm high and 76 cm long. For one set of experiments, the
standard spacing experiments, the bars were placed at 10-cm intervals on the
bottom of the Qume. For another set of experiments, the bars were placed at 20-cm
intervals. Since each bar extended across the width of the Qume, these bars acted as
strip roughness elements which simulated bottom bedforms or ripples.
- 51-
In this flume, previous experiments were c depleted by Mathisen �989! and
Rosengaus �9871 in which pure attenuation measured over a movable sediment
bed. The bar height and spacing for the pres xperiments were chosen to closely
match the bedform characteristics measured under the same wavo conditions in the
previo". experiments of Mathisen �989! and Rosengaus �987!. Therefore, the
magnitudes of the fixed bed roughnesses deterniined using this experimental setup
can b~: iompared to the movable '"0 roughness determinations fr in previous
experiments. These comparisons .. i be used to understand the role of bedforms in
producing a roughness for a. moi ' bed.
". '. Wave Measurement and Resolution
Determination of the bc roughness experienced. by waves propagating over a
rough . for t. =se experime«is was completed analys energy dissipation
hich .,is determined from measurements of wave attenuation. Determination of
the .;e attei ion first required accu '. determination of the characteristics of
all significant;;- ve compoiient, hroug'. the Our The use of wave gauges to
...easure wave perties and thi icedures necessary for resolution of the incident
ve are pres ..",ted in this secticn,
9 '~.1. 'Vave Measurement
W'ave characteristics were r. ~sured using conductivity wave gauges. The wave
iuges and tve gauge controi 'l were ma:facti by the Danish Hydraulic
Institute. 4 sketch of a typical wave gauge is showi; i: Figure 3.2.
The wave gauge consists of a fi which can be set on a trolley which rides on
guiderails on the tops of the flume walls. The wave gauge frame includes a movable
� 52-
Point gage suooort tor eaae orcatibrating wave gage
Mount icr =oint aage
ptgure 3.2 Wave gauge setup from Dean and Dalrymple, 1984!
� 53-
vertical arm with a vernier scale. The electronic connections are housed in a, plastic
plate which is attached to the vertica' arm. This plastic plate also holds a thin tube
which is extended into the water suan thai .wo thir, nichrome wires with a spacing
of,25 cm are extended vertically across the water interface. The wave gauges
prod» voltage signal which is proportional to the conductivity between the two
nichr wires .h depends on t, level of the water between the wires. The
output signal from the wave gauge control unit is converted to a digital signal by a
Metrabyte Dash-16 D/A converter installed in a Compuadd 320 computer.
Before each experiment, wave gauges were calibrated by sampling voltage
output for diffe~~,t surface displacements and defining a regression curve relating
voltage to surface displacement, Before calibrations, the pump was turned on so
tha,t water was completely mixed in the flume. Therefore, the conductivity of the
»-: .er was uniform and the calibration completed at one location was valid for any
Ii.ation along the flume. A typical wave gauge calibration is shown in Figure 3.3.
At a specific location, digital records were ained by programming the Dash16
A /D board to sample the real-time voltag» - tput N times at discrete time intervals
of uniform duration. This provided a reco... of digital values which could r; e
between -2047 and 2048 and were proporti~- ' to the output volta of the wave
g' =ge. which c old range from -10 to +10 volts!. These N data points were then
converted to a time record of surface profile ele .tions by applying the calibration
curve.
To .::termir..- '..he characteristics of the various wave components in the flume, a
Fa.st Fourier Transform FFT! algorithm was used to convert the time record of
surface elevations into a frequency record consisting of amplitudes and phases. For
convenience in using a Fast Fourier Transform FFT! algorithm, the total points
sampled per record, N, was chosen to be an integral power of two typically 2048
� 54-
points were sampled for wave measurements!, In additio», the total sampling time
T,, �, was chosen to equal an integral number of wave cycles typically, Tszpp was
set equal to 20 T, where T is the wave period!. This ensured that the wave
frequency would exactly equal one of the discrete frequencies or "spikes"! resolved
by the FFT algorithm so that the energy determined by the FFT would be
minimally affected by leakage into adjacent frequencies.
Records of amplitudes and phases werc,obtained at approximately 2" miformly
spaced intervals along the length of the flume. These records were used to obtain
the variation i mplitude ~ g~ as a function ni x. Using the record of ~ qi vs. x, the
various wave .nponents which exist in the flume can be resolved.
3.2.2. Resolution of the Incident Wave
Procedu. ~ for resolvin, -rious wave components of pure waves we-- 'eveloped
by Rosengaus �987!. For -.tudy, the measuring procedure defined as the
Reference Measuring Method . MM! in Rosengaus �987! was modified to include
combined wave-current flows.
At the first harmonic for a combined wave-current flow in the flume, an absolute
radian frequency, a�will be f. ced by the wavemaker motion. An incident radian
frequency, ~�may be defined as
This is the frequency of the incident wave relative to a mme of reference which is
moving with the current. Similarly, a reflected radian frequency, a�may be defined
- 56-
>r � ada + kr U � 2!
This is the frequency of the reQected wave relative to a, frame of reference moving
with the current.
These frequency definitions may be used to determine the wave numbers of
interest. The absolute wave number is defined by:
>a~ = gka tanh kah � 3!
An incident wave number is obtained from:
= gk; tanh k h �.4!
and a rejected wave number from
~,~ = gk, tanh k�h �.5!
Since the wavelength is defined as
k = 2'/L �.6!
these equations indicate that incident waves are stretched out due to the effects of
the current while the reQected waves are shortened,
'gi = ag cos k,x � a~t + P;! �.7!
For the reQected wave which is propagating against the current, q, is represented
- 57-
These definitions can be used to characterize the surface profile in the Qume.
For the incident wave which is propagating with the current, the surface profile, g;,
is represented by
by:
7!~ = Rg. cos krx + s~t + P,!
Therefore, at the first harmonic, the surface profile is given by:
g = gi + 7/r = ai cos kix Hat + 4i! + ar cos krx + slat + 4r! �.9!
+ ~r + aiarcos i+ r x +
.assuming a,- c a,, the result can be rewritten as
~ q~ � a; + a,cos[ k;+k,!x + Pr-Pi!]
where a beat length may be determined from
Finally, by allowing for a linear decay in incident wave amplitude,
7/ ~ a jp mix + a~os[ k;+k,!x + Pr- P!]
where a;0 is the amplitude at the wavemaker x =- 0! and mt is the total wave
attenuation slope in cm/cm!. Thus, the incident wave amplitude in the flume is
given by
ai = aia -mtx
By taking the absolute value and manipulating the trigonometric functions in a
manner similar to that detailed in Rosengaus �987!, an equation characterizing the
>t harmonic amplitude variation ale ..g the length of the Quine can be written as.
Iri the limit of au extremely weak current, k; equals k, and Equation 3.13
approaches the result obtained by Rosengaus i987! for pure waves with no current,
in which the first harmonic beat length is z'/k,. For the first harmonic, since k; is
less than k~, and k� is larger than k~, the beat length for a combined wave-current
flow turns out to be quite similar to that for a pure wave flow,
For the second harmonic variation for a combined wave-current flow, the radian
frequency for the bound harmonic associated with the incident wave is 2~; and the
wave number is 2k;. The radian frequency for the free harmonic is given by
> i = 2za � kr'U
Thus, the wave number for the second free harmonic, ki-;, is defined by:
{~fj!' � gkf; tanh kr,h
Therefore, at the second harmonic, the bound wave, Tjh which is propagating with
the current may be represented by
ab cos{2k jx 2 dgt Pb!
while the free harmonic wave, rg, which is propagating with the current can be
written as:
Qf = af cos kr;x � 2+~t � Pr!
Therefore, the second harmonic surface profile variation may be represented by:
+ gf j: aq cos{2kx - 2zat - Pb! + ar cos{krx � 2aat � Pf!
Taking the absolute value of Equation 3.19 and manipulating the trigonometric
functions in a manner similar to that detailed in Rosengaus �987!, an equation
- 59-
characterizing the second harmonic amplitude variation along the length of the
flume is obtained to be:
a +a, + abafcos fj jx+ h f �.20!
Finally, assuming af < aq and approximating the result following the procedures of
Rosengaus �987!, ~ q can be written as:
LQ + af cos kfj-2k;!x � itig-4'f!!
where a beat length may be defined as:
I.>, = 27.-/ 'kf;-2k;! �.22!
3.2 3. Resolution of Spectral Waves
The procedures for resolving the various components for monochromatic waves
can easily be applied for use with spectral waves which are generated in the flume,
For a wave spectrum with a fiiL'.e number of components, the analysis discussed in
3.2.2 is simply applied to each component of interest.
For spectral waves generated in these experiments, the primary non-linear
components which exist in the flume are associated with second harmonics of the
wave components generated and nonlinear interactions between different
components. For example, for these experiments, five component spectra were
generated. Thus, the primary radian frequencies for these components may be
� 60-
In the limit of an extremely weak current, this result approaches the result obtained
by Rosengaus �987! for pure waves. For pure waves, the beat length is still given
by Equation 3,22 although the values for kF and kj are different since these wave
numbers are modified in presence of a current.
defined as ~~, ~2, ~q, ~4, and ~5. Second harmonic components are expected at 2w>,
2+2 2'� 2z4, and 2 as, The discrete frequencies were chosen to ensure that the
second harmonics for any of the components would not interact with any of the five
primary coinponents with frequencies of ai, ez. a3, a~, or ss. In addition, however,
other components will appear due to nonlinear interactions between components of
different frequency. For example, a> and a> will interact to yield frequencies at
llJ/+ a2 ~ and ai- ~2 ~ . The primary frequencies for each of the components were
chosen to ensure that they did not interfere with these non-linear frequencies.
Despite the careful choice of frequencies for each of the components, these non-
linear interactions will result in energy transfers between different components along
the length of the flume. These energy transfers will be included in the measured
total attenuation slopes for each of the spectral components, Accounting for the
effects of these non-linear energy transfers is discussed in Section 3,5.2.
3,3, Wave Generation
3,3,1. Generation of Stokes Waves
Estimates of bottom roughness experienced by waves can be obtained from
measurements of wave attenuation, Using wave attenuation to determine bottom
roughness requires the generation of pure second-order Stokes progressive waves
using a piston-type wavemaker. Because of non-linear components which arise due
to the wavemaker forcing, the wa,vemaker motion must be carefully controlled.
Madsen �971! showed that progressive Stokes second-order periodic waves of
permanent form may be generated in a wave flume of uniform depth by prescribing
a wavemaker motion of the form:
� 61-
'K lal1 3 nil�.23!
where is the wa~ aker displacement, a,nd ni is d~.fined by:
�.24
� is the amplitude of the wavemaker motion, which can be related to the wave
amplitude by:
ani
tanTn El! �.25!
3.3,2. Generatioii of Spectral Waves
For the spectral wav neration, the linear theory noted above may be used by
implementing the theory oi linear superposition. To simulate a wa,ve spectrum, a
finite number of discrete frequencies is defined and a representative monochromatic
wave is chosen for each .. Thus, the surface profile would be represented by:
.,= 5 a;cos k; � ': -0;!i= 1 �.26!
where one set of randoinly chosen phases can be used to simulate one realization of
the spectrum. Generation of the spectrum can be accomplished by superimposing
� 62-
The first, rrn of Equation 3,23 refl~'.ts linearized wavemaker theory as
leveloped by Biesel a.nd Suquet �9.' "he cond term of Equation 3.23
represents a correction to the linearized wavernaker motion to eliminate the
.eneration of a free component which ises at the second harmonic frequency due
to the wavemaker f ..ing. Previous ex~erirnents b' Mathisen �989! and Rosengaus�987! have verified the removal of this second free harmonic by operating thewavema.ker in accordance with Equation 3.23,
the wavemaker motion for each one of the components:
t! =.~ ' t} � 27!
Ideally, an accurate simulation of a, wave spectrum would require many
components with many realizations. However, due to the size limitations associated
with the wave flume, only a limited number of components can be generated.
Therefore, for each simulation, five wave components are defined with equal
amplitudes and with the frequencies set such that they model the energy of a wave
spectrum.
The spectrum selected for generation is a JONSWAP spectrum modified for
finite depth. The JONSWAP spectrum was initially developed during the Joint
North Sea Wave Project JONSWAP} by Hasselmann et al. �973!. This narrow-
- 63-
Using these principles, simulation of a particular wave spectrum can then be
accomplished by defining n components with discrete frequencies and amplitudes for
generation. Two methods may be used to simulate a wave spectrum, One method
is to choose equally spaced frequency components and then define amplitudes to
provide an energy distribution which would model the spectrum of interest. In this
method, one or two components would comprise the majority of the energy of the
spectrum. The second method is to choose the frequency such that all wave
components have an equal energy. Therefore, the amplitudes of each of the
components would be approximately the same. In this case, more frequency
components would be located near the frequency associated with the peak spectral
density. Since the amplitudes of components are all equal and relatively large, the
attenuation for all components can be measured more accurately. Therefore, for
this study, wave spectra are simulated using the second method.
banded spectrum, which is generally considered to provide an excellent
characterization of typical real-world ocean wav~ spectra, is given by
�.28!
where p is a peak enhancement factor, o is a spectral width factor. '.",d Epw is a
Pierson-Moskowitz spectrum. The Pierson-Moskowitz spectrum is defined by:
where a� is the radian frequency a ociated with t" oeak energy and uis the
Phillips .. ant. Following Kit ~rodskii et al .i75!, Graber {1984! defined a
finite dep. JOVSK .' spectrum .~
S. = q ~!E �.30!
where qi ~'1 is obtained from:
ln this equation:
� 32!
and y is o~ .ned from:
y tanh ~~ g! = l �.33!
� 64-
Tl.e parameter;s which may be varied in these equations included a peak
enhancement factor, p, a spectral widt .~tor, o, and the Phillips constant, a. The
peak enhancement factor was taken to be 3.3. For simplicity, a single value of 0.08
was used for the spectral width factor.
The final parameter, a, was varied to set the total energy associated with the
spectrum. The total spectral energy, E,, may be defined in terms of a
representative monochromatic wave using:
pl > 1i=i
�.34!
3 3.3. Calibration of the Wavemaker System
To operate the wavemaker system such that its movement satisfied Equation
3 23, the wa,vemaker was calibrated. This involved development of a transfer
function which relates the wavemaker input to the actual wavemaker displacement.
Since the wavemaker is a linear system, the wavemaker transfer function was
defined in terms of an amplitude and phase. Thus, the amplitude of the transfer
function defines the magnification or reduction of the actual output wavemaker
displacement relative to the input wavemaker displacement, and the phase of the
transfer function represents the lag of the output behind the input. More
specifically, input wavernaker motion is represented by:
and the desired output motion by:
In this case, the tra.nsfer function is defined as
� 65-
A typical value for o was 0.0015. With a set at .0015, the energy of the spectrum
was set equal to that of a 6-cm monochromatic wave with a frequency of
2.39 rad/sec. More details on the spectral characteristics are summarized in Section
3.7 Experiment design!.
�,37!
and the phase as:
�.38!
To get the proper wavemaker motion, a modified wavemaker motion is defined by:
�.39!
If proper v; s are used for th,amplitude H ~! and phase ~! of the transfer
function, and Equation 3.39 is used as the wavemaker input, the desired wavemaker
displacement described by Eq» ' on 3.23 is obtained.
l!etermination of this trans;er function requires calibration of the wavemaker
system. Tl; .nplitude and phase of the transfer function, ~ H ~ and p, depend on
the frequerii .nd amplitude ~ ~vemaker motion. Hr.;er, the previous
wavemaker calibrations by Rosengaus �987! and Mathisen �989! showed the
transfer function to be strongly dependent on the wavemaker frequency and only
.weakly dependent on the amplitude of wavemaker motion. Therefore, the transfer
function can be represented by characterizing the dependence of the amplitude and
phase on wavemaker frequency.
For these experiments, transfer function values were calibrated for the erst and
second harmonic components of monochromatic waves with period' of 2 "4. 2.63,
and 2.89 seconds. With a depth of t"" m, the, ective relative wavelengths kh!
>7. Th choices of kh were selected tok .iese three cases were .75,,63,
� 66-
allow investigation of the theory of Trowbridge and Madsen {1984!, which predicts a
positive wave-iriduced mass transport for a kh of .75 and a negative mass transport
for a kh of .57.
To calibrate the transfer function for a particular frequency, the amplitude and
phase of the wavemaker input was defined using a wavemaker control program.
The voltage signal from the D/A board was sampled and a Fast Fourier Transform
was applied to obtain the amplitude and phase of the wavemaker input signal, The
wavetnaker output was determined using the voltage output of the paddle
displacement transducer, V, which was related to the actual wavemaker
displacement, , by:
�.40!C = .94861 � 4,5138 V
where C is in cm and V is in volts, This output signai was also sampled and a Fast
Fourier Transform applied to obtain the amplitude and phase of the output signal.
� 67-
The ratio of the output amplitude to the input wavemaker displacement yielded
the amplitude of the transfer function, ~ H . The difference between the output and
input phase yielded an accurate estimate of the phase. This information was
recorded for each of the six frequencies of interest, The Gnal transfer function
amplitudes which were determined by the calibration for each of the six frequencies
are shown in Figure 3.4a. The associated transfer functions for the phases are
shown in Figure 3.4b. Values for the amplitude and phase of the transfer function
were essentially the same as the values determined in a previous wavemaker
calibration by Ochoco �990!, From these figures it is seen that the amplitude of
the transfer function increases while the phase lag decreases as the frequency of the
wavemaker motion decreases within the range of interest. To allow generation of
spectral waves these calibrated transfer function values were incorporated into the
wave generation programs by making use of Newton's Divided Difference
Polynomials to define general relationships covering a range of frequencies.
i 3
u 8�
:6 I-
IH I >! 7T sec!
a. Amplitude of transfer function
A
T sec!
b. Phase of transfer function
Figure 3.4 Wavemaker transfer function
- 68-
3.3.4, Removal of Free Harmonic
To ensure that the wavemaker could be used to generate progressive second-
order Stokes waves, preliminary tests were completed. Various tests included wave
periods of 2,24, 2,63, and 2.89 seconds. The tests showed that the wavemaker
motion prescribed in Sections 3.3.1 through 3,3,2 required modifications to remove a
free second-harmonic wave component which arose due to the effects of the
transition inlet. The results of these tests are discussed in this section.
3.3.4.1. Pure waves
To test the use of the transfer function values for operating the wavemaker, a
pre!iminary experiment with a period of 2.63 seconds was completed in which a
cover was placed over transition inlet which is associated with the current
generation system, With a uniform still water depth throughout the length of the
Qume, wavemaker operation using the theory and calibrations discussed in Sections
3.1 though 3.3 was tested. Wave characteristics were measured using the
measurement, procedures discussed in Section 3.2. The results of this preliminary
experiment are plotted in Figures 3.5a and 3.5b. As can be seen in Figure 3.5a, the
first harmonic amplitude variation could be accurately fit by applying the
measuring method, Thus, the first harmonic components could be resolved, In
addition, as shown in Figure 3.5b, the curve fit to the second harmonic amplitude
variation indicated that the free second harmonic component was much smaller than
the bound second harmonic component. Therefore, essentially pure Stokes waves
were generated in the flume. This result verified the use of the wavemaker
calibration and operation. for a flume with uniform depth,
� 69-
I
|
~ I ~ i ~I
0
~ mj
a. First harmonic
C0 10 20
Y Ill!b. Second harmonic
Figure 3.5 Pure wave amplitude variation with no step using standard wavemakercorrection T = 2.63 sec!
- 70-
Since the final experiments include an open transition inlet, more preliminary
experiments were completed with the transition inlet uncovered. For this
experiment, waves were generated using the standard wavemaker correction
discussed in Section 3.3.1. In this case, waves which propagate into the test section
first pass over the transition inlet. This transition effectively serves as a trough
which extends from a distance of 80 cm from the wavemaker to 190 cm from the
wavemaker, with the trough bottom located approximately 60 cm below the flume
bottom. The results of this experiment are shown in Figure 3.6a and 3.6b. In this
case, the erst harmonic amplitude is lower than the amplitude which was obtained
when the transition inlet was covered. In addition, the second harmonic amplitude
variation shown in Figure 3.6b indicates that a significant second harmonic was
present in the flume. Clearly, the trough had a significant effect on the waves
propagating above it.
These trough effects can be qualitatively explained in terms of linear theory for
progressive waves propagating over a trough. For these experiments, the
wavelength for the first harmonic ranged from 500 to 700 cm, which is somewhat
longer than the depth of the trough. Therefore, as the incident wave crosses over
t,he downward step of the trough, a significant portion of the wave energy will be
transmitted and a smaller portion will be reflected back towards the wavemaker.
The reflected portion will propagate back to the wavemaker where it is reflected. It
then propagates in the forward direction and passes over the trough. The
transmitted portion of the original incident wave propagates through the deeper
trough section and then passes over the upward step A portion of this transmitted
component continues propagating into the test section, while another portion
propagates back through the trough towards the initial downward step.
Superposition of all of the surface profiles of the various transmitted and reflected
� 71-
components results in an altered surface profile due to the differences in amplitude
and phase of the various components. Therefore, the amplitude and phase of the
wave train which propagates into the test section is modified relative to t,he
amplitude and phase of the wave train incident to the trough,
Vor a free second harmonic component propagating over the trough, the same
physical argument as that presented in the preceding paragraph would apply.
Again, the amplitude and phase of the free second harmonic wave which propagates
into the test section would be modified relative to the amplitude and phase of the
second harinonic wave train incident on the step. However, the wave lengths of the
free second harmonic components range from approximately 190 cm to 250 cm,
which is smaller relative to the depth of the step. The trough will have a much
smaller effect on the second harmonic components.
Therefore, t,he important effect of the trough is to change the amplitude and
phase of the first harmonic relative to amplitude and phase of any second free
harmonic. This trough effect can be eliminated by modifying the amplitude and
phase of the second harmonic wavemaker motion relative to that of the first
harmonic wavemaker motion. This is a modi6cation to second harmonic correction
factor proposed by Madsen �971!.
To estimate this additional modification, a simple theory was applied. While
this theory provided some verification of the trough effects described above, it did
not model the trough hydraulics with sufficient accuracy to enable theoretical
removal of the second free harmonic in the flume. Accurate determination of the
amplitude and phase of this modification would be extremely involved due to the
complex hydraulics in the transition region.
� 73-
iefore, the modifications to the wavemaker motion were essentially
:eted by an organized trial and error methodology. For this methodology,
,ments were completed in which the second-harmonic wavemaker input was
i d. Additional revisions to the wavemaker input were determined by
ving the second-harmonic surface profile variation. Experiments were
dieted with different phases used for second-harmonic wavemaker motion. In
ianner, the phase was obtained which optimized removal of the free second
inic. !nce the phase which optimized second-harmonic removal was obtained,
amplitude of the second-harmonic wavemaker motion was varied until the free
i-harmonic was adequately removed in the flume.
,- modifying both the amplitude and phase of the second harmonic of the
naker motion, the free second harmonic was effectively removed in the flume,
hown by Fight" ' 3.7a and 3.7b. More specifically, comparison of Figure 3.7b
Figure 3.6b c.'«y indicates that; e second free harmonic has been essentially
~ated in the test section. The modifications necessary to remove the second
~nic for each of the three pure wave cases are summarized in Table 3.1. The
;tude change represents the ratio r2/ i.i where |,'gi is the final wavemaker
ction factor and ii is the initial wavemaker correction factor. The phase
;e represents the difference I'it<2 � Pfi! when /f2 is the final phase of the
maker correction and Pii is the initial phase of the wavemaker correction
� Pi i! represents a phase lead!.
2. Combined wave-current flows
nce the modificatioiis were completed to verify the removal of the second
onic for pure waves, combined wave-current flows were tested. For the
iined wave-current flow, the wavemaker input was modified using the same
- 74-
, ~
/
~ ~i ~
x m!
a. First harmonic
~ ~ ~ ~ ~ 0
10
x m!
b. Second harmonic
Figure 3.7 Pure wave amplitude variation with step using pure wave modificationto wavemaker correction T = 2.63 sec!
� 75-
Table 3.1
Modifications to Wavemaker Second Harmonic Correction Factorto Account for Effects of Trough
Waves with 16-cm s currentAmplitude
modificationPhase
modification
r2/ f i! �r~-Ai!
0.0 1.00
5
0.5
Pure waves
Amplitude PhaseT modification modification
2.24 0.0 2.40
2.63 0.5 1.25
2.89 0.5 0.43
0. 50
0. 17
modifiications used to remove the free harmonic when pure waves propagate over the
trough. The current setting for these experiments was 16 cm/sec. The results of
this experiment are shown in Figures 3.8a and 3.8b. As can be seen in Figure 3.8b,
despite the modified wavemaker input, a free harmonic appears when a current is
flowing through the inlet transition. While this second harmonic is not as
pronounced as the second harmonic shown in Figure 3,6b, additional modifications
are required due to the presence of a current in the flume.
'1'hese additional effects due to the current can again be explained in terms of the
propagation of linear waves over a trough, In this case, the current enters through
the trough. As the waves propagate over the trough they will experience a current
which increases from 0 at the downward step adjacent the wavemaker, to the full
current velocity as the waves propagate into the test section. To some extent, the
reflection and transmission characteristics of wave components passing over the
trough will be similar to the characteristics for the pure waves case. However,
because of the current, the relative phase relationships of the various reflected and
transmitted components will change, As waves pass from still water into a current,
a portion of the wave energy flux will be transmitted by the current, which changes
the portion of energy which is reflected and transmitted at the step. The second
harmonic components will also be changed due to the presence of the current. The
effects of the current on the the second harmonic will differ from effects of the
current on the first harmonic component. These effects will depend on the velocity
of the current, Because of these effects of the current, the amplitude and phase of
the second harmonic wavemaker motion must be modified from the amplitude and
phase used for the pure waves.
Again, while a simplified model was developed to characterize these additional
current effects, the model could not be used to adequately remove the free harmonic
� 77-
T=".63 sec
a. First harmonic
201510
b. Second harmonic
Fjgure 3 8 Wave amplitude variation with step and current, using pure wavemodification to wavemaker correction T = 2.63 sec; U = 16 cm/s!
� 78-
0
0
x rn j
x m!
in the flume. Therefore, modifications to remove the second free harmonic for
combined wave-current Qows was completed by trial and error, The final results
obtained after completing this trial and error procedure are shown in Figures 3.9a
and 3,9b. The removal of the second free harmonic is shown in Figure 3.9b. Of all
efforts at removing the second harmonic, t,his is certainly the most successful, As is
shown in Figure 3.9b, the second free harmonic was effectively removed. The final
modifications necessary to remove the second harmonic for each of the three wave-
current cases are summarized in Table 3,1. With the modifications to the
wavemaker input for pure waves and combined wave current Bows, the
experimental conditions necessary for accurate determination of bottom roughness
were verified.
3,4, Velocit Measurement and Processin
Characterization of roughnesses experienced by currents required determination
of velocity profiles in the flume. Fluid velocities in the near-bottom region were
measured using laser doppler velocimetry. Using this experimental technique, time
records of the instantaneous velocity were obtained at various locations in the
flume. These time records were used to obtain vertical profiles of fluid velocity in
the flurne. The procedures for obtaining these velocity profiles are discussed in this
section,
3.4.1. Velocity Measurement
Velocity measurements were obtained using a low-powered �0 rnW} DISA 55L2
one-axis laser-doppler anemometer operating in the forward-scattering mode with
associated traversing equipment. This laser was used in the experiments of Ochoco
�990! in which velocity measurements were taken in osciHatory flow to study
� 79-