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

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T=".63 sec

a. First harmonic

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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!

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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

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