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TP 77-4
Sediment Suspension and Turbulence in an Oscillating Flume
By Thomas C. MacDonald
TECHNICAL PAPER NO. 77-4 APRIL 1977 |
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U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING RESEARCH CENTER
Kingman Building Fort Belvoir, Va. 22060
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Department of the Army position unless so designated by other authorized documents.
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REPORT DOCUMENTATION PAGE
1. REPORT NUMBER 2. GOVT ACCESSION NO.| 3. RECIPIENT'S CATALOG NUMBER
TR. 774
4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED
SEDIMENT SUSPENSION AND TURBULENCE IN AN OSCILLATING FLUME Technical Paper
6. PERFORMING ORG. REPORT NUMBER HEL 2-39
7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s) Thomas C. MacDonald DACW7 2-71-C-0024
9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK 1 1 i H AREA & WOR Ti University of California ORK UNIT NUMBERS
Hydraulic Engineering Laboratory
Berkeley, California ~94720 D31193
CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE
Department of the Army April 1977
Coastal Engineering Research Center (CERRE-CP) 13. NUMBER OF PAGES
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Approved for public release; distribution unlimited.
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Oscillating flume Sediment suspension Turbulence Waves
20. ABSTRACT (Continue on reverse side if necesaary and identify by block number)
An experimental study measured suspended-sediment concentrations and tur- bulence above the bottom of a specially designed oscillating flume. A total of 73 concentration distributions was measured for a single fixed-bottom roughness and the same specific gravity (1.25) of sediment. Three different sediment sizes were used, 65 experiments with the same size, These experi- ments show a simple exponential distribution, except near the bottom, as previously found by other investigators. The slope of the concentration
Continued
DD en, 1473 Evrtion oF 1 Nov 65 1s OBSOLETE UNCLASSIFIED
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distribution is in the range of -5 to -15 per foot (-16 to -50 per meter) for the experiments. For the limited data on other sizes, the slope of the concentration distribution becomes more negative as fall velocity increases.
Turbuieut velucity fluctuations measured with a hot-film anemometer are normally distributed with mean zero for measurements at two elevations above the bed, well outside the viscous boundary layer. The root mean square of the velocity fluctuations decreases exponentially with distance above the
bed, and at the bed, increases approximately linearly with increase in flume velocity.
When extrapolated to typical field conditions seaward of the breaker, these experiments demonstrate the importance of fall velocity, maximum wave- induced bottom velocity, and turbulent velocity fluctuations in controlling Sediment suspension by shoaling waves. However, comparisons of data obtained with the lightweight sediment in these experiments and the probable motion of quartz sand in the field suggest that sediment suspensions caused by shoaling waves offshore of the breaker are likely to be limited.
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PREFACE
This report is published to provide coastal engineers with an analysis of data on suspensions of sediment produced by oscillatory motion in a specialized laboratory facility at the Hydraulic Engineering Laboratory (HEL), University of California, Berkeley. The work was carried out
under the coastal processes program of the U.S. Army Coastal Engineering Research Center (CERC).
The report was prepared by Dr. Thomas C. MacDonald, a former graduate student at the Hydraulic Engineering Laboratory, and now an engineer with Leeds, Hill, and Jewett, San Francisco, under CERC Contract No. DACW72- 71-C-0024. The report is a modification of report No. HEL 2-39 which was originally issued by the Hydraulic Engineering Laboratory.
The author acknowledges with sincere gratitude the active supervision and counsel of the late Professor H.A. Einstein, the advice and the opportunity to participate in this project provided by Professor J.W. Johnson, and the assistance of Professors J. Harder and L. Talbot during the experiments and in the preparation of the report. The cooperation and assistance of the staff of the Hydraulic Engineering Laboratory are
gratefully acknowledged, especially W.A. Hewett, J.C. Allison, and R.W. Cambell.
Dr. M.M. Das, former Hydraulic Engineer in the Coastal Processes Branch, and Dr. C.J. Galvin, Jr., Chief, Coastal Processes Branch, were the CERC contract monitors, under the general supervision of R.P. Savage, Chief, Research Division.
Comments on this publication are invited.
Approved for publication in accordance with Public Law 166, 79th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963.
JOHN H. COUSINS Colonel, Corps of Engineers Commander and Director
CONTENTS
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI)
SYMBORS PANDY DEESINTMTONSHieinearensciucl oneien uiebiton tone
I INH NODUWICI MONG 8 G66 ood 0 0 6 0-500 6 0.0 G0
II CONCENTRATLONE DRS TRIVBUTUONS iciereimet te lea outs i Montell tellne
iG Bo Ne 4.
5.
EXpenimentalaeAppiaralteuS) mete sihcmleelren telltemnep te ExperamentalyshroCe duels: o tenbeiiiou re) cnwel el le) RESWULCS G66 Gyo! OG) Ba 60 67 OO 06,0156 0
Experiments Using Sediments of Different Sewuilimnye: Weloesdtedles 4 (4 io 6 6 0) 66 6 Go Svc
Summary of Experimental Results. ......
III DISTRIBUTIONS OF TURBULENT VELOCITY FLUCTUATIONS.
iG Ze 3 4.
EXpendmential App aacatcUs ic inveii fai ceueiion etal el ter ie Experaimentailg pO Ceduceutiemcdich lent ciitentenceiaten ts ISSMIESo |G) 6 9 006 66 a 616 G6 6 6 6 BONS Summary of Experimental Results. ......
IV THE SUSPENDED LOAD IN OSCILLATING FLOW. .....
1.
He
Suspended-Load Theory in Unidirectional Flow
Similarities Between Oscillating and Unaidatire ctvonalGplOws tic touaey vou cenlews ermieiinen. tellne
Sediment Suspension in an Oscillating Flow . Hite Bases Concentratvonl | C7 soi. stiedrs te Wleunedle Net Transport of Sediment in the Ocean...
Additional Investigations Needed to Complete Che wSuSpPenSHonwene OiGyeaepc eo sete mrse lon eere ae
ConicluSTOnsi aaaw ce ceptors, se cecuret Kal Layee ets
IIE VNU RUS (CINE S. Gl Gio oe old) G 6.04.0 Ol Gand Glco
ARPEND XGaE Cee RaIMEN AU aD ADAGtor irene uiteletureitotilelivel Nei toinioiirswtaliits
TABLES
1 Concentration distribution data for Vg = 0.035 foot per seconds tampilttude= MOOS 5) LOO tea sc) sue) ell elle) el a pele
2 Concentration distribution data for Vg = 0.035 foot per Second wampilekeudem<am0 09 Sao Olt cious eueviieli tel) je touts ie ve
3 Concentration distribution data for Ven = 00626"and 0c 0498s footaper se cond 3) 7) 3 el oi) oie
Page
31
40
18
19
CONTENTS--Continued
FIGURES SWplsit Cantal Mreteeie Smee neue o otnrsitc airetll cle eMac cite Tem mi circus OpElcCalMmConcent ration Metetynan) owe Pele! holm) loan ema chou Yokessuppor.t oruconcenitratvom) meters ey si tei) ot le
Calibration curve for sediment diameter: OR4y/ mailsimeters So DE< 0) 49/5) madelmeters ss ee es
Examples of measured concentration distribution curves.
M versus Up (eq. 4) for amplitudes equal to or greater than 0.693 foot and sediment-settling velocity, Ve = W505) oo jer SOCOM 6 6 6B a Sls SoG 6o 68
M versus Up for optical equipment moving with the flume and stationary in space, uSing a 0.925-foot amplitude stOIe AULIL MSCS WHRSNENES 6 Sg GG) 6 6 OO Ge on dud 6.0 6c
M versus Uo for sediment-settling velocity, Vg. = O5085 SOE joe! SSCGOMG G6. 65 Bio 066d O10 oe
M versus location of optical equipment for identical PLOW CONGE TONS Veli= OL O55 oot PeRySeCOnd. js 1s 6 ie
Concentration) dasitaibutrony form muny Slay sit vey el Concentration adi sitaalbucaong tors ouia 02) vere eaimiey tons Concentration adastab talon fourm ould OSarse surements Concentration dastributdony fom) run Oia OG a) se ces ve Concentration distribution for Gun ol4 S05) s.) <tc oh
M versus U, for Vg = 0.035, 0.498, and 0.0626 foot STH SIS COM Cipree pen antennae, wien Oa tance uatilraanen eel nisl Mawel aie
Functional schematic of hot-wire bridge circuit... . Hoe ticlimssiens omvandwpcobenenieiienien te timrelnet elton onion or taille HOt tlmmanemome ter calla braltaonitamkaiijey io) en etiet ust ere) ie
Hots tiamyproverextensTonuassSemb liygisy cis wells! lepiley leuisiune
20
D2
25
26
29
Sy
34
35
36
37
38
39
41
45
46
47
49
20
21
22
23
24
29
26
27
28
CONTENTS FIGURES--Continued
Output versus velocity calibration curve for hot-film ANEMONE Cer MeaSUreMeCMESI momen emilee! lel ven Kerrey elite! ell te
Velocity. protidleracrossmcalaibmatalonnozizilen ecnn en tents Distribution of turbulent velocity fluctuations...
Velocity scale versus elevation for flume velocity, U5 S WS E58 shoe jo SSCOMG 5 6:5 6.66.6 0 06 6 6.6 6
Velocity scale versus elevation for flume velocity, Way Wo SiO stow jes SOCOM 5 5 6 6 6 6 oO 6 6 5 4 oS
Velocity scale versus elevation for flume velocity, WAS Wo JAS stooe. joer Seeemed 4 66 616 6 6 624 oa 608
Velocity scale versus elevation for flume velocity, WES UI SHOE ew SOCOM 4 GG Se abs 6 6 S08 6
Base vertical turbulent velocity scale versus flume velocity for a constant amplitude = 0.925 foot...
- par . Comparison of the theoretical and measured exponent of concentration distribution in unidirectional flow
57
58
59
60
61
63
67
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (ST) UNITS OF MEASUREMENT
U.S. customary units of measurement used in this report can be converted to metric (SI) units as follows:
EE
Multiply
by
To obtain
SS SS ee
inches
square inches cubic inches
feet
square feet cubic feet
yards Square yards cubic yards
miles square miles
knots
acres foot-pounds millibars ounces
pounds
ton, long ton, short
degrees (angle)
Fahrenheit degrees
25.4 2.54 6.452
16. 39
30. 39 0.3048 0.0929 0.0283 0.9144 0.836 0.7646
1.6093 259.0
1.8532 0.4047 1.3558 1.0197 28.35
453.6 0.4536
1.0160 0.9072 0.1745
5/9
x 1073
millimeters centimeters Square centimeters cubic centimeters centimeters meters
square meters cubic meters meters
square meters cubic meters
kilometers hectares
kilometers per hour
hectares
newton meters
kilograms per square centimeter grams
grams kilograms
metric tons metric tons radians
Celsius degrees or Kelvins!
SS eee ee lTo obtain Celsius (C) temperature readings from Farenheit (F) readings,
use formula:
Cr 705/952) To obtain Kelvin (K) readings, use formula:
K = (5/9) (F -32) + 273.15.
SYMBOLS AND DEFINITIONS exponential decay rate of the velocity scale sediment concentration calibration nozzle coefficient sediment concentration at the base elevation representative roughness diameter water depth sediment exchange coefficient output voltage of hot-film bridge output voltage of hot-film bridge with sensor in still water acceleration due to gravity crest-to-trough wave height output voltage of photoelectric cell output voltage of photoelectric cell for ambient light
output voltage of photoelectric cell with light beam passing through clear water
constant of proportionality between directional components of turbulent velocity fluctuations
wave number
amplitude of flume oscillation
length of surface wave
length scale for oscillating and unidirectional flow
exponential decay rate of sediment concentration of oscillating flow
Manning's roughness coefficient hydraulic radius
horizontal oscillating velocity
SYMBOLS AND DEFINITIONS--Continued slope of the energy gradeline velocity scale for oscillating flow
velocity scale at base elevation for oscillating flow
period of flume oscillation time effective heat transfer velocity for anemometer sensor
oscillating flow velocity
horizontal flow velocity
longitudinal component of turbulent velocity fluctuations shear velocity
peak velocity of oscillating sensor
effective heat transfer velocity
mean velocity across calibration jet
centerline velocity of calibration jet as determined from voltage measurements
sediment-settling velocity
velocity scale for unidirectional flow theory
vertical component of turbulent velocity fluctuations
angular frequency
horizontal. displacement of fluid particle in oscillating flow horizontal distance, positive in direction of wave travel
elevation above base elevation vertical distance, positive up from mean water surface
theoretical exponent of unidirectional flow concentration distribution
To
SYMBOLS AND DEFINITIONS--Continued
measured exponent of unidirectional flow concentration
distribution
vertical displacement of fluid particle in oscillating flow
thickness of the boundary layer
water density
shear stress at elevation Y
in unidirectional flow
shear stress at bed in unidirectional flow
phase angle
SEDIMENT SUSPENSION AND TURBULENCE IN AN OSCILLATING FLUME
by Thomas C. MacDonald
I. INTRODUCTION
Sediment transport by waves approaching the shore has been analyzed in two ways. Inshore of the breaker zone, the extremely complex flow patterns and turbulence resulting from the breaking waves have necessi- tated only a qualitative approach to sediment transport with quantita- tive estimates based on field measurements. Offshore of the breaker zone in relatively deep water, the problem of sediment transport can be approached in a more theoretical manner. In this zone, sediment trans- port studies are simplified because there is no turbulence in the flow field from breaking waves and the flow condition near the ocean bottom can be better estimated from linear wave theory. This report concerns sediment transport offshore of the breaker zone.
Laboratory and field observations indicate that, as in unidirection- al flow, sediment transport at the ocean bottom offshore of the breaker zone is of two different types--bedload and suspended load. The dis- tinguishing feature between these two types of transport is that in suspended transport the entire weight of the sediment is continuously supported by the fluid; whereas, in bedload transport the sediment rolls, skips, and jumps along the bed and therefore its weight is partially supported by the stationary bed. For moving sediment to be supported by the bed means that the regime of bedload transport is contained in a thin layer adjacent to the stationary bed, two-grain diameters thick as proposed by Einstein (1950). The majority of the sediment in motion in this area is bedload and thus most research has concentrated on the turbulent boundary layer and the oscillatory bedload rate due to wave action (Li, 1954; Manohar, 1955; Kalkanis, 1957, 1964; Abou-Seida, 1965). Sufficient advances in the theory of bedload movement in an oscillating flow have warranted studying the suspended load to deter- mine: (a) Approximately, what percentage of offshore movement is due to suspended load, i.e., a second-order approximation to total trans- port; and (b) if any of the now-predicted bedload is partially suspended load.
This investigation develops, from an empirical approach, a method for predicting the distribution of suspended-sediment concentration based on the hydraulic flow conditions; i.e., surface wave amplitude and period, depth, sediment characteristics, and bottom roughness con- ditions. Although only one bottom roughness was studied, the resulting method is general enough to be extended to other roughness conditions by additional experimentation. The suspended distributions, when used in conjunction with the bedload function of Kalkanis (1964), should give a better approximation of the total sediment transport.
As in Kalkanis' bedload function, the approach to the suspended load is based on many of the same principles proposed by Einstein (1950) in his theory of bedload and suspended-load transport in unidirectional flow. Analysis of suspended load in oscillating flow is more complicated than that of unidirectional flow because of two factors. First, in uni- directional open channel flow the entire depth of flow is turbulent and the relatively high turbulent velocity fluctuations allow the sediment exchange coefficient to be approximated by the momentum exchange coeffi- cient which can be obtained from the shear-stress distribution. In oscillatory flow this is not possible. Both laboratory and field obser- vations indicate that suspension of sediment occurs to depths considerably above the boundary layer in an area where the shear stresses due to the oscillating motion are extremely small and difficult to measure. Although it may be possible to express a sediment exchange coefficient in the boundary layer as a function of the mean shear stress, this would not provide a means of estimating the sediment exchange coefficient above the boundary layer. Therefore, a sediment exchange coefficient which is not based on a shear-stress distribution must be found.
The second factor concerns the magnitude of the turbulent velocity fluctuations. Offshore of the breaker zone where the flow velocities near the bed are low, only a small part of the wave energy is dissipated by friction at the boundary. The remaining wave energy is lost inshore by the breaking waves. Because of the relatively low intensity of tur- bulence in this offshore area, the vertical velocity fluctuations are of the same order of magnitude as the settling velocity of the sediment. Under these conditions, the sediment exchange coefficient is highly dependent on the sediment--settling velocity. Therefore, measured dis- tributions of turbulent velocity fluctuations and sediment concentrations must be used in analyzing the upward turbulent flux and downward turbulent and gravitational flux for each sediment-settling velocity.
To obtain a relationship between sediment suspension and flow hydrau- lics in an oscillating flow, concentration distributions for various flow conditions must be measured. The sediment exchange coefficient is deter- mined from these measurements. Next, the turbulent velocity fluctuation distribution with time at a constant elevation and its distribution with elevation must be measured. This measurement will yield the information necessary to describe the fluid exchange. The distribution with eleva- tion will yield a velocity scale, one of the two variables composing the sediment exchange coefficient. From the sediment exchange coefficient and the velocity scale, the second variable (the length scale or its associated time scale) can be calculated. Knowledge of these fundamental
variables of suspension as a function of the flow hydraulics should lead to a practical method of estimating the suspended-load distribution, and indicate the important variables of suspension in an oscillating flow.
The only other requirement for a solution to the suspended load is a knowledge of a base concentration as a function of flow hydraulics. The base concentration is determined from Kalkanis' (1964) bedload theory,
being the concentration at the top of the bedload layer.
Combination of the bedload and suspended load will predict the total amount of sediment in motion under specified wave and boundary conditions. By SuperimpoSing a constant unidirectional flow, such as the mass trans- port or a coastal current, the amount of sediment transport can be estimated.
In this investigation, as in many research projects, the experimental research preceded the development of a suitable method of describing sedi- ment Suspension. For this reason a description of the experiments and their results will be given first, followed by a discussion of how the results can be used to predict the sediment suspension load.
II. CONCENTRATION DISTRIBUTIONS
1. Experimental Apparatus.
Experiments and observations indicate that near the ocean bottom off- shore of the breaker zone in relatively deep water, sediment is held in suspension. This is due to the turbulence resulting from the dissipation of wave energy on the rough ocean bed. For waves with a small surface slope, where 0/dx << d/dy, the fluid motion can be approximated from linear wave theory. The equations describing the horizontal and vertical displacement of a fluid particle are given by the expressions from Lamb (1932):
ee (3) {cosh [k (y+d) ]/sinh(k d)} cos(k x - w t) (1) A = (3: {sinh [k (ytd) ]/sinh(k d)} sin(k x - w t) (2)
where d = the water depth
y = the distance from the mean water surface measured negatively downward
H = the crest-to-trough wave height KS Baia 1 = the length of the surface wave Wei=we2h/ile
T = the period of the surface wave
The corresponding velocity components are obtained by differentiation with respect to time of the above equations to give:
u= (3) w {cosh [k (y+d) ]/sinh(k d)} sin(k x - w t) (3) = (3) w {sinh [k(y+d) ]/sinh(k d)} cos(k x - wt) (4)
From these equations it is evident that the vertical component of the displacement and velocity becomes smaller as the distance from the sur- face increases. At the bottom, where y = -d, the motion degenerates into a simple harmonic oscillation in the x-direction. It is this horizontal harmonic oscillation which is of first-order importance in producing tur- bulence and suspension of sediment. The equations also indicate that the magnitudes of the horizontal displacement and velocity change slightly with depth near the ocean bottom and can therefore be considered constant in the region of sediment suspension. For example, consider a 5-foot-high wave with a wavelength of 150 feet and a period of 10 seconds in a water depth of 50 feet. Equation (3) indicates the horizontal velocity at y = -d is u = 0.393 sin(kx-wt) feet per second and that for y = -d+ 5 (S feet above the bed), the horizontal velocity is u = 0.401 sin(kx-wt). The sediment suspension measurements which will be discussed later indi- cate that the regime of measurable sediment concentrations is well with- in the bottom 5 feet of water depth for this typical wave condition. (Measurable sediment concentrations were actually found within 1 foot of ‘the bed.) The change in horizontal velocity in the bottom 5 feet of water depth is only 2 percent. Therefore, to design an experimental apparatus to simulate the flow conditions near the ocean floor, the hori- zontal flow velocity above the boundary layer can be considered constant and equal to the value given by equation (3) for y = -d. Equation (4) indicates that for the wave condition discussed above the vertical flow velocity at y = -d is zero, and at an elevation of 5 feet above the bed is 0.082 cos (kx-wt). The fact that the vertical velocity only increases slightly in the 5 feet above the bed and that its motion is symmetrical Suggests that the vertical velocity has no effect on the suspension of sediment.
With the above assumptions of a constant horizontal oscillating velocity and zero vertical oscillating velocity above the boundary layer, the turbulent flow conditions which exist near the ocean floor are easily approximated. By superimposing a constant velocity equal to that given by equation (3) for y = -d but opposite in sign, there would be no motion in the fluid above the boundary layer, the distribution of velocities in the boundary layer would be inverted, and the bed would be oscillating at
the simple harmonic given by equation (3) for y = -d.
It is now only necessary that the physical apparatus which duplicates these flow conditions contains a water depth greater than the thickness of the boundary layer. Kalkanis (1957) made velocity distribution measure- ments in a flume containing a still body of water with an oscillating
rough bed. Kalkanis (1964) showed that the velocity distribution in an oscillating flow with the same assumptions as described above can be approximated by:
u/uy = [1 + £2(Y) - 2£, (1) cos f,(Y)]# sin(wt + 6) (5)
where £,(Y) = exp(Y-103/agD), (6) a OO) = OaS(hQl 982 (7)
and sin(wt + 6) describes the variance of the velocity with time and phase angle. In equations (6) and (7), Y is the elevation above the bed measured positively upwards; a is the amplitude of bed oscillation; B = (w/2v)?; v is the kinematic viscosity; and D is the representa- tive roughness diameter. For the flume roughness conditions used in this investigation (D = 0.05 foot), and the flow velocities studied (0.67 foot <a < 2.0 feet and 2.0 seconds < T < 15.0 seconds), equation (5) indicates the boundary layer thickness is no more than a few millimeters thick.
To approximate the shear stresses and therefore the turbulence condi- tions near the ocean bed for a given water depth, wave period, and ampli- tude, a flume having a moving bed under a still body of water was used. The frame of reference, which describes the prototype fluid motion under the above assumed conditions, is then moving with the bed. It is only necessary to oscillate the bed in the harmonic motion described by the linear wave theory for y = -d. The swing flume used in this investiga- tion duplicates, on a one-to-one scale, these conditions.
The swing flume is shown in Figure 1(a). The flume bed is shaped to an arc segment of a circle with an 8.92-foot radius, a 13.33-foot chord length, and a 12-inch width. The flume, suspended from the ceiling of the laboratory, is free to rotate about its center of curvature. The flume is oscillated about its center position by a 1.5-horsepower variable-speed motor connected to a drive wheel with an eccentric arm. The eccentric arm is connected by a 10-foot connecting rod to a linkage fixed to the flume bottom. The linkage at the flume is adjustable to
allow correction of the asymmetry of motion which would result from a change in eccentricity. Variations in eccentricity and motor speed allow the amplitude and frequency of oscillation to be varied over a wide range of prototype wave conditions.
Within the flume is a stationary horizontal board, slightly less than 12 inches wide, 8 feet long, and at an elevation of 12 inches above the lowest point of the curved bottom. The board, which is separately sup- ported from the ceiling of the building and is not connected to the flume, Suppresses any standing surface waves in the flume caused by the flume motion. The fluid at this elevation must remain stationary to conform to the flow conditions described above.
Asymmetric End Roughness
Two-Dimensional Roughness Elements
So Suspension
b. Cross Section of Roughness Elements
c. Cross Section of Asymmetric End Roughness
a. Elevation View of Entire Flume Horizontal Board 10 Suppress Standing Surface Waves
Figure 1. Swing flume.
The shape of the artificial bed roughness (Fig. 1,b) was determined by experiments. A large quantity of the artificial plastic sediment used in concentration measurements was put into the smooth-bottomed flume.
The flume was oscillated at various amplitudes and periods covering the range of flow conditions to be studied. After the flume was oscillated at a constant rate for a period long enough to establish a natural bed shape, the flume was stopped and the bed dunes were measured. The bed shape was found to be approximately sinusoidal in the cross section under all flow conditions. The mean wavelength of the bed shape was 5.5 inches with a range of 4.5 to 8.0 inches; the mean wavelength-to-depth ratio was 8.0 with a range of 5.5 to 12. The fixed artificial roughness used approx- imates this shape. The artificial dunes were constructed of wood and fastened to a flexible sheet of plastic. Natural sediment with a mean diameter of 0.3 millimeter was glued to both the dunes and plastic. The plastic sheet was fixed to the flume bottom, covering the central 6.33 Hee OLe ther anc.
The asymmetric end roughnesses shown in Figure 1(c) were used to eliminate secondary currents in the central measuring section of the flume. Proper placement of the asymmetric roughness elements depends on the flow conditions of the flume. The optimum placement of the roughness elements for a given flow condition was determined by dropping potassium dichromate crystals through holes in the horizontal wave suppressent board and observing the movement of the dye streaks. When no transverse move- ment of the dye streaks in the central part of the flume were observed, the roughness elements were considered to be in the optimum location.
A lightweight, black plastic material with specific gravity of 1.25, which had been crushed and sieved, was used as artificial sediment in the experiments. The grain diameter of the sediments was uniform, bracketed by two consecutive sieve sizes of approximately the same diameter as the natural sediment glued to the flume bed. Only a small quantity of the sediment was used in the flume during an experiment in order to limit the deposition of sediment which would alter the flume bottom geometry.
When the sediment was first put into the flume it was found that air bubbles adhered to the sediment particles, thereby changing its settling velocity. To eliminate this buoyancy effect, deaerated water was used. The water was deaerated in a 60-cubic foot-capacity tank located on the wall of the laboratory at an elevation above the swing flume, heated by a 5-kilowatt immersion heater to a temperature of 90° Fahrenheit, and then cooled to room temperature. The deaerated water was transported to the swing flume by gravity through a hose to reduce air entrainment.
The optical concentration meter used in the experiments was developed by Das (1971) for measuring tm sttu concentrations in laboratory flumes. The equipment consists of a light source, a beam collimator, a receiving unit, and necessary recording units. A collimated beam of light, 8.5- millimeter average diameter, is projected through the glass walls of the flume to a duo-photodiode mounted on the opposite side of the flume.
The photoelectric cell produces a signal which is proportional to the amount of light received which, in turn, is proportional to the amount of light blocked out by suspended sediment. The signal from the photo- electric cell is transmitted, amplified, and recorded on an analog paper chart recorder and an analog-to-digital data acquisition system set to sample at the rate of 58 samples per second. The light source and receiver are shown in Figure 2.
The light source and receiver are mounted on a rigid, aluminum yoke support (Fig. 3) held by friction brackets; by loosening four thumbscrews the elevation of the support can be varied from below the flume bottom to above the flume top while maintaining precise alinement of the optical equipment. This flexibility allows calibration of the optical equipment and flume concentration measurements to be made without removing: the equipment from the supports. The yoke support and brackets are pinned to the laboratory ceiling on the same axis as the flume so that the optical equipment can swing with the flume or be held stationary in space.
2. Experimental Procedure. a. Settling Velocity of the Sediment. The sediment used in the. filxsit
series of experiments was the material which passed the 0.495-millimeter sieve and was retained on the 0.417-millimeter sieve. The fall velocity of this sediment was measured in a 2-inch-diameter glass cylinder filled with deaerated water at a temperature of 72° Fahrenheit. The time re- quired for 220 particles chosen at random to fall 8.59 inches was measured with a stopwatch. The average settling velocity (V,), the range of velocities, and the standard deviation were then calculated to be 0.035, 0.0213 to 0.0532, and 0.0059 foot per second, respectively.
b. Calibration of Optical Concentration Meter. Calibration of the optical equipment was done in a 0.5- by 0.5- by 1.0-foot clear, plastic
calibration tank placed on the top of the swing flume. The 1.0-foot dimension of the tank was positioned parallel to the axis of the light source and receiver; i.e., the same width as the swing flume. The tank was then filled with a measured quantity of deaerated water. A small, measured amount of cleaned sediment was added to the tank and the sediment-water mixture was stirred mechanically to give a uniform sus- pended concentration. The collimated light beam and receiver were posi- tioned and a 5-second record of the voltage from the receiver was recorded on the analog chart recorder and magnetic tape. Uniformity of concentra- tion in the tank was checked by measuring the fraction of light blocked by the sediment at various locations in the cross section of the tank. The mechanical stirrer was stopped and the sediment allowed to settle. Records were then made of the voltage from the light beam passing through clear water and the voltage of the ambient light. The fraction of light passed was then calculated as the ratio of the voltage with sediment in suspension to the difference between the voltages of the beam through clear water and the ambient light. Measured amounts of sediment were
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added to the calibration tank and the procedure repeated until the range of the optical equipment was covered. Measurable concentrations ranged from 0).1 to 2.5 grams per liter. The logarithm of the fraction of light passed versus the concentration in grams per liter was plotted to give the calibration curve shown in Figure 4. The least squares, best fit equation for this curve is:
GS (COsSeS), tan /G, =I) 5 (8) where C = concentration (in grams per liter) I = mean voltage from the receiver when the light beam is passed
through the sediment-water mixture
I, = mean voltage from the receiver when the light beam is passed through clear water
I' = mean voltage from the receiver when the light beam is turned off
The calibration curve, which was checked periodically during the experi- ments, did not change.
In setting up the calibration experiments, it was found that a change in the focus of the optical equipment would change the resulting calibra- tion curve. To eliminate this problem a brace was made to hold the equip- ment in focus. A check of the focus was made during each concentration measurement by a wire-screen filter which, when placed in the light beam, blocked out a known and constant amount of light. If a change in focus was detected by the filter measurements, the concentration measurements were not used.
c. Concentration Distribution Measurements. For concentration dis- tribution measurements in the flume, a datum elevation of the optical equipment had to be established. The mean elevation of the crest of the artificial dunes was used as the base elevation and was determined by a scale fixed to the flume and a pointer fixed to the optical equipment brace.
The desired period and eccentricity were selected and the flume link- age adjusted to give a symmetric oscillation. The flume was filled with deaerated water to the elevation of the wave suppressent board (12 inches above the lowest point of the bed), and the asymmetric end roughnesses were positioned. Depending on the flow conditions chosen, 100 to 500 grams of sediment was cleaned and deposited in the flume. The flume was then oscillated until the distribution of sediment in the flume was at equilibrium. The equilibrium condition was determined by periodically measuring the concentration of sediment at a fixed point in space until the concentration did not change with time.
2 |
T’)] , FRACTION OF LIGHT PASSED
Te/ (eli
r
04
C=-0.585 In [I/(Ip-1)]
OF2
0.02
0 0.4 0.8 lee 1.6 220 24 C, CONCENTRATION (gf™') Figure 4. Calibration curve for sediment diameter:
0.417 millimeter < D < 0.495 millimeter,
Vg = 0.035 foot per second.
22
The concentration distribution in the vertical was then measured for the flow condition selected. Again, for each elevation, voltage records were made for the sediment-laden water, clear water, ambient light, and clear water plus the filter. If the optical equipment was held stationary in space, the records for clear water and clear water plus filter were made while the flume was oscillated at such a long period that no sediment was in suspension. The time interval of the record for sediment-laden water, clear water, and clear water plus filter was always an integer multiple of the flume oscillation period. This procedure automatically allowed any irregularities in the transmissibility of the flume windows to be compensated for when calculating the fraction of light passed.
The output signal of the photoelectric cell, which was recorded on Magnetic tape at the rate of 58 samples per second, was not constant With time, due to instantaneous concentration fluctuations. Because the concentration is related to the logarithm of the output voltage, it was necessary to calculate the concentration for each sample and then average
the concentrations over the period of record to determine the true mean concentration.
The concentration measurements were usually started as near the bound- ary as possible, about 0.5 centimeter above the crest of the roughness elements. The flume was stopped, the elevation of the optical equipment was raised and recorded, and new measurements were taken. This procedure was followed until an elevation was reached at which the concentration was too low for the optical equipment to measure. The optical equipment was then lowered in a stepwise fashion until near the boundary to obtain concentration measurements at intermediate elevations. In this manner,
7 to 15 concentration measurements were obtained to describe the concen- tration distribution for one flow condition.
3. Results.
Sixty-five concentration distribution curves were obtained of the form,
C=C, exp(MyY) , (9) where M = slope of the curve (in feet 1) C, = concentration at the base elevation (in grams per liter)
C = concentration of sediment (in grams per liter); average of the concentrations calculated from equation (8) for each sample of record during the period
Y = elevation above the crest of the artificial roughness (in feet)
23
The variable used to describe the flow condition is the flow velocity; U,, defined as: U,= @L/T , (10)
where L is the amplitude (in feet) and is equal to one-half the arc length which passes a fixed point during one-half cycle, and T is the period of oscillation (in seconds). The flow conditions studied ranged from a minimum of U, = 0.235 foot per second (minimum to allow suspension of sediment) to a maximum of U, = 1.18 feet per second (maximum allowed by flume construction). The amplitudes and periods used in these experi- ments ranged from 0.235 to 1.60 feet and 1.65 to 15.16 seconds, respec- tively. Figure 5 shows some typical concentration distribution curves that were obtained.
For the 65 different flow conditions studied, the base concentration, C, (eq. 9), could not be correlated to any of the hydraulic parameters but depended on the amount of sediment in the flume. For different flow conditions the sediment in the flume would be distributed differently along the bottom of the flume, thereby giving a different and uncontrol- lable base concentration.
The 65 experiments also showed that two different laws exist in two ranges of conditions governing sediment suspension. For amplitudes of 0.693 foot and larger, the concentration distribution is determined by the flow velocity alone. For amplitudes less than 0.693 foot, the con- centration distribution is a function of the amplitude relative to the wavelength of the artificial roughness.
Thirty-six of the 65 concentration distribution curves were deter- mined using amplitudes of 0.693, 0.770, 0.925, 1.25, and 1.60 feet. For these five amplitudes, it was found that the slope of the distribution curve is a function of the flow velocity, independent of amplitude. Figure 6 graphically illustrates the relationship between M, the slope of the concentration distribution curve of equation (9), and U,, the variable of equation (10) used to describe the flow conditions of the flume. These data are also tabulated in Table 1 which gives the periods, amplitudes, and use of the optical equipment. The least squares, best fit equation for this relationship is:
Ye atIS ee GSS) Uy. (11)
where U, is in feet per second, and M is the slope of the concentration distribution curve (in feet™!). There is no statistical evidence to indi-
cate that this relationship is significantly different from a higher order
polynomial.
Results of experiments by Shinohara, et al. (1958) confirm the above results. They found the same linear relationship between the logarithm of concentration and elevation and qualitatively determined that as the inten- sity of the flow increased the slope of the concentration distribution curve, M, became flatter.
24
518.01 518.02 518.03 713.03 612.01
CONCENTRATION RATIO
Cie
0 0.1 ©:2 0:3 0.4 O'S 0.6 ONG Y, ELEVATION ABOVE CREST OF ROUGHNESS ELEMENTS (ft)
Figure 5. Examples of measured concentration distribution curves;
sediment-settling velocity, V, = 0.035 foot per second.
(29)
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26
Table 1. Concentration distribution data for Vs = 0.035 foot per second, amplitude > 0.693 foot.
Slope, Variance in d(1nC)/d|least squares Remarks (Eta) curve
-5.60 0.0182 optics
-7.99 0.0184 stationary
-11.28 0.0196
-13.91 0.0236
-15.39 0.0136
optics stationary
optics stationary
optics moving optics moving optics moving optics moving
optics moving
Not calculated.
27
The same results were obtained more recently by Kennedy .and Lacher (1972). Their investigation examined the behavior of the mean sediment concentration and the periodic sediment concentration fluctuations. The experiments were conducted in a stationary flume with a fixed bed on which a limited amount of loose sediment was distributed. Turbulence for sedi- ment suspension was caused by surface waves generated by a wave generator. The sediment concentrations were measured with optical equipment which incorporates the same theoretical principles as the equipment used in the swing-flume experiments but of a much smaller size. The smaller size and configuration of the equipment allowed measurements very near the bed and sampled a much smaller volume of flow.
Kennedy and Locher's (1972) experiments in mean sediment concentra- tion distribution were limited to a wave height of 0.24 foot, a wave period of 1.0 second, and a mean water depth of 0.82 foot. The mean sediment concentration was measured at various elevations along five evenly spaced verticals in the flow. The spacing of the verticals was selected to cover one wavelength of the bed dune shape. A total of 78 data points was measured, and when the logarithm of the mean concentra- tion was plotted against elevation above the bed a well-defined linear relationship of C = C, exp(-36.5 Y) was obtained. This relationship is identical to that obtained in the swing-flume experiments, with the ex- ception of the high rate of decay of sediment concentration, (-36.5). Kennedy and Locher used a quartz sediment of 0.14-millimeter mean dia- meter in their experiments. The settling velocity of this sediment was not reported; it was probably about 0.050 foot per second, which is 43 percent greater than that of the principal plastic sediment (settling velocity of 0.035 foot per second) used in the swing-flume experiments. Swing-flume measurements, using sediments of different settling velocities, are discussed later in this section. For a given flow velocity a higher rate of decay of concentration is expected, as settling velocity increases.
Much of the analysis of data by Kennedy and Locher dealt with the periodic sediment concentration fluctuations. Although no specific con- clusions were obtained regarding these fluctuations, the data did indi- cate the fluctuations were only apparent near the bed (within about 0.05 foot of the bed). This explains the lack of periodicity in concentration fluctuations for the swing-flume measurements, in that 0.05 foot is near the lower limit where the much larger optical equipment of the swing flume could be used.
Similar exponential concentration curves were obtained for the two different methods used in simulating sediment suspension in an oscillat- ing flow; i.e., an oscillating flow over a fixed bed as used by Shinohara, et al. (1958) and Kennedy and Locher (1972), and an oscillating bed under a ''stationary" body of water as used in the swing-flume experiments.
Figure 7 shows the relationship between M and Uj, _ for two condi-
tions. One set of data represents the condition that the optical equip- ment is stationary in space; the other data points are for the optical
28
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equipment moving with the flume. All other variables in these experiments were the same. The amplitude of oscillation was 0.925 foot. There is no statistically significant difference in the results, indicating uniform suspension along the flume. None of the measurements for large amplitudes (greater than 0.925 foot) (Table 1) were made with the optical equipment moving with the flume. The reason for this is at large amplitudes the optical equipment would move into an area of the fluid where secondary circulations due to the asymmetric roughness elements may exist and there- fore not give a representative concentration.
The remaining 29 of the 65 experiments were made using amplitudes of 05255), 08465) and On6i7) foots | Mabile) 2 as) a\stabullation o£ the data jvand Figure 8 shows the results of the 65 experiments. As shown in Figure 8, there is a great deal of scatter in the data for small amplitudes; there- fore, only qualitative conclusions have been made. In general, the smaller the amplitude the smaller the slope of the concentration distri- bution curve.
Experiments, movies, and photos were used to determine why the small amplitudes do not obey the flow velocity relationship of the larger ampli- tudes given in equation (11). Based primarily on visual observation, the following explanation is hypothesized. For small amplitudes, the distance of travel of the flume bottom during a half cycle is not great enough for the boundary layer to fully develop during each stroke and to become tur- bulent. Only at the end of the half cycle when the acceleration forces cause separation is sediment thrown into suspension. Separation only occurs at the downstream face of the artificial dunes. The observed sus- pension pattern when a 0.235-foot amplitude of oscillation was used con- sisted of plumes of suspension separated by areas of zero concentration. These plumes were accentuated because, on the return, the half-cycle separation at the downstream face occurred such that the succeeding burst of sediment was thrown into approximately the same region of fluid as in the first half cycle; i.e., the amplitude of motion was about equal to a multiple of the wavelength of the dune shape. When the amplitude was increased to 0.465 foot, a more fully developed boundary layer was attained. In this case, separation occurred over a somewhat longer distance of travel but still less than the wavelength of the dune shape because less deceleration force was required. The suspension pattern was the same as with the 0.235-foot amplitude but much less distinct; the plumes were wider and overlapping. Finally, with the 0.693-foot amplitude, separation occurred over a distance equal to or greater than the wavelength of the dunes, and a uniform longitudinal concentration was attained.
Special experiments were conducted to verify the above hypothesis. The vertical distribution of concentration at various horizontal loca- tions in the fluid was measured to determine if the concentration distri- bution varied. If the above hypothesis is correct, the concentration distribution should vary along the horizontal and in a regular manner determined by the shape of the bed dunes. In addition, if suspension
30
Table 2. Concentration distribution data for V, = 0.035 foot per second, amplitude < 0.693 foot.
Curve Period | Amplitude Uo Slope Variance in d(1nC)/d | least squares Remarks No. (s) (ft) (ft/s) (ft-!) curve 621.01 Zeelell 0.617 Mg 70 -11.15 0.0511 optics OZ 02 Dolls ORG, OaTT, = 6.5 0.0459 stationary 621.03 4.60 0.617 ORS 37 -13.10 0.0134 621.04 6.58 0.617 ORS75 -15.12 0.00268 621.05 9.38 0.617 0.263 -10.95 OF0233
628.01} 9.40 ORION 0.263 optics 628.02 |] 6.57 0.617 0.376 moving 628.03} 4.59 0.617 0.538
629. optics 629. moving 629.
629,
Te optics plays moving “ae
lols:
plea:
ples
Wakako
802. 802. 802. 802. 802.
optics moving
0. 0. 0 optics 0.465 0.585 -14.10 0.0239 stationary 0.465 0.585 -12.82 0.0208
0.465 0.585 Sol Se 0.0152
0.465 0.585 -14.00 0.0447
3|
M,SLOPE OF CONCENTRATION DISTRIBUTION CURVE (ft7')
| oO
nm fo)
I De) on
ds fo)
Figure 8.
Uo
M versus U, for sediment-settling velocity,
Vg
AMPLITUDE = 0.693 ft AMPLITUDE = 0.617 ft AMPLITUDE = 0.465 ft AMPLITUDE = 0.235 ft
04 0.6 0.8 1.0 ine , FLUME VELOCITY (ft/s)
= 0.035 foot per second.
B72
for small amplitudes is a function of location, the shape of the concen- tration distribution curve for a particular vertical may not be the same flume velocity, U,, but with a large amplitude. The following is a description of the experiments.
The flume was adjusted to a 0.465-foot amplitude and a 3.18-second period was selected. A special brace with attached scale was construc- ted to hold the optical equipment stationary in space, and therefore stationary relative to the fluid. This brace allowed relocation of the optical equipment to any desired new location along its arc while main- taining the freedom to change the elevation of the optical equipment relative to the bed of the flume. An initial location of the optical equipment was chosen and a vertical concentration distribution measured. The optical equipment was then relocated to a new position and another concentration distribution was measured. This procedure was repeated until five concentration distribution curves were obtained. Each was at a different horizontal location relative to the position, at a fixed phase, of the flume bed, but with identical flow conditions of the flume. The horizontal range of the five distribution curves was about equal to one wavelength of the bottom roughness shape. The results of these ex- periments (Fig. 9) indicate that the distribution of concentration is not uniform along the flume bed for the 0.465-foot amplitude studied. Figure 9 also indicates that the rate of sediment concentration decay is less in the areas between the dunes and greatest at the crest of the dunes. This implies that the plumes of suspension are above the trough of the bed shape, where it would be expected if the sediment is thrown upward and downstream from the downstream face of the dunes.
Figures 10 to 14 are the concentration distribution curves measured for the locations indicated in Figure 9. Although not conclusive, these data indicate that the vertical distribution of sediment in the plumes does not conform to the exponential distribution of equation (9). The relationships shown in Figures 11 to 14 display a slight, but statisti- cally significant curvature and were obtained at locations in the plume; whereas, Figure 10 shows no curvature and was obtained at a location between plumes. This could be explained by the existence of periodic stationary eddies, one located on each side of the plume axis. The eddies would sweep sediment into the base of the plume axis giving a relatively high concentration and carry sediment out of the top of the plume, causing the concentration to be lower than would be predicted in a randomly tur- bulent flow field.
The small amplitudes studied represented rare prototype conditions ;
therefore, all subsequent experiments were limited to amplitudes greater than 0.617 foot.
4. Experiments Using Sediments of Different Settling Velocities.
Two different sediments of the same black plastic were selected and investigated to determine the effect of settling velocity on the
39
M, SLOPE OF CONCENTRATION DISTRIBUTION CURVE (ft7')
al 1)
ale iS
als D
814.03
Numbers opposite the points indicate curve numbers
flume is reversing its direction of motion
Faigumey O)
M versus location of optical equipment for identical
flow conditions, V, = 0.035 foot per second.
34
C/Cy , CONCENTRATION RATIO
0.4
Or
0.04
002
Jn Cll o=(-17.20) ¥
0.05 0.1 OziS 0.2 Y, ELEVATION ABOVE CREST OF ROUGHNESS ELEMENTS (ft)
Figure 10. Concentration distribution for run 814.01.
35
C/C,, CONCENTRATION RATIO
0.4
O.2
0.04
0.02 O
In C/Cy =(-14.08) Y
°
0.05 0.1 0.15 0.2 Y, ELEVATION ABOVE CREST OF ROUGHNESS ELEMENTS (ft)
Figure 11. Concentration distribution for run 814.02.
36
C/Cy , CONCENTRATION RATIO
In. C/C,=(-12.83) ¥ oO
OZ
0 0.05 OE 0.15 0.2 Y, ELEVATION ABOVE CREST OF ROUGHNESS ELEMENTS (ft)
Figure 12. Concentration distribution for run 814.03.
Sih
Cee, CONCENTRATION RATIO
0.2}
0.02
O 0.05 Y, ELEVATION ABOVE
bn. C/Cg=(-15.70) Y
O.| Oxl5 0.2 CREST OF ROUGHNESS ELEMENTS (ft )
Figure 13. Concentration distribution for run 814.04.
38
CONCENTRATION RATIO
Ln C/C, =(-14.00) ¥ {e)
0.05 0.1 0.15 Ore Y, ELEVATION ABOVE CREST OF ROUGHNESS ELEMENTS (ft)
Figure 14. Concentration distribution for run 814.05.
SHS)
concentration distribution. The first of these sediments was the material which passed the 0.701-millimeter sieve and was retained on the 0.589- millimeter sieve. Settling-velocity measurements of 225 randomly chosen particles determined the mean settling velocity, the velocity range, and the standard deviation as 0.0626, 0.0387 to 0.113, and 0.0131 foot per second, respectively. The second sediment was the material which passed the 0.589-millimeter sieve and was retained on the 0.495-millimeter sieve. Settling-velocity measurements of 240 particles of this material deter- mined the mean settling velocity, the velocity range, and the standard deviation to be 0.0498, 0.0285 to 0.0754, and 0.00885 foot per second, respectively.
Table 3 gives the experimental results obtained for the two sediment types and Figure 15 shows how these results compare with the results for sediment with a settling velocity of 0.035 foot per second. As expected, for sediment with a higher V,, the concentration of sediment decreases with elevation above the bed at a higher rate. Only this qualitative conclusion was obtained. Not enough data were obtained to define quanti- tatively the relationship between Vg and M.
Table 3. So eaeae eae distribution data for Vs = 0.0626 and 0.0498 foot per second.
Curve Period Amplitude No. [eS CGB se Re Na ORES (&t/s)
0.0626 a olneeee ey ee
Variance in least squares curve
Slope, d(inc)/@
Ge")
0.0327 0.0154 0.00497 0.000966
= 0.0498 ft/s
0.00095 0.00698 0.00378 0.00177
The interesting indication of these measurements is that the increase in the rate of decay with elevation of the concentration for a sediment of higher settling velocity is not as great as predicted from the O'Brien (1933) equation for concentration equilibrium conditions. Consider the concentration equilibrium equation used in unidirectional flow for sus- pended sediment:
Glvs) E(de/dy)) = 0.) (12)
40
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(s/s) ALIDOTSA awnd4 ‘°n COs =k) COs = CO e210
S/4} SCO'O = “A YO4 3ANND SquvNods LSV3aq
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¢'0
¢ 0
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4
where
C = concentration V, = settling velocity of the sediment particle E = sediment exchange coefficient (which here is assumed equal to
the momentum exchange coefficient in unidirectional flow) Y = elevation above the bed
Inserting the expressions for C and (dC/dY) obtained from equation (9) into equation (12) and solving for E yield:
E = Wait 0 (13)
In unidirectional flow and for relatively low sediment concentrations,
it is assumed that the sediment exchange coefficient is equal to the momentum exchange coefficient, is a function of the fluid motion only, and is independent of the particle-settling velocity. For this assump- tion to be valid in oscillating flow, the slope of the concentration dis- tribution curve must be directly proportional to the settling velocity. Although Figure 15 indicates M is proportional to V,, it does not indicate direct proportionality. This discrepancy is discussed in Section IV.
5. Summary of Experimental Results.
The following is a summary of the results of the concentration meas- urements and a brief discussion of their limitations.
a. The vertical distribution of sediment concentration can be ex- pressed by equation (9). Sediment is held in suspension by the random motion of turbulence which is generated in the boundary layer and is transported by diffusion upward while decaying continuously because of viscosity. As shown later in Section III, the turbulence intensity decays rapidly with elevation above the bed. The upper elevation to which the turbulence can diffuse (the water surface in the prototype and the wave suppressent board in the flume) is larger than the eleva- tion at which the turbulence intensity decays to extremely small values; therefore, this upper boundary can be approximated as being at infinity. Under these conditions it is reasonable that the empirical results given by equation (9) indicate an exponential decay (as is commonly found in a diffusion process), and only become zero at Y = infinity. As expected, the concentration distribution in oscillating flow is different than in unidirectional flow. In unidirectional flow the turbulence is distrib- uted between the bed and the water surface, the turbulence intensity is Significant at the water surface. Because the water surface is not effec- tively at infinity, the distribution of both the turbulence and the sedi- ment concentration would be different than in oscillating flow.
42
b. The base concentration, C, (eq. 9), for flume measurements is a function of the sediment charge in the flume and therefore could not be correlated to flow hydraulics. As shown by Einstein (1950), a flow is only capable of transporting a limited amount of sediment of a given size. This limiting capacity is determined by the flow velocity, sediment characteristics, and roughness of the boundary. In addition, the flow will only transport this capacity rate if there is a sufficient supply of sediment available. Otherwise, the transport rate will be reduced by the ratio of the supply rate to the capacity rate. Because the capacity transport rate for a given flow is determined by the proba- bility of a particular sediment particle being subjected to sufficient hydraulic forces to move it, there must be some particles in the bed that are not in motion at any instant of time. Had there been enough sediment in the flume to satisfy the flow's sediment transport capacity, the measured Co, could have been correlated to flow velocity. Unfortunately, under those conditions some sediment must be loosely deposited on the flume bed, thereby changing the fixed-bed geometry and roughness. There- fore, only the flow's capacity to transport sediment of a specific size can be estimated. This estimate must use the capacity base concentration calculated from Kalkanis' (1964) bedload equation and not the base concen- trations measured in this investigation.
canines thesrangerotetiliow velocities W0.2foot per second i<sU-¥< il feet per second, for amplitudes of oscillation equal to or greater than 0.693 foot, and for V, = 0.035 foot per second, the slope of the exponen- tial distribution of sediment concentration is a function of flow veloc- ity only. The slope, M, can be approximated from the flow velocity, U>, by equation (11). This equation is only a best fit empirical rela- tionship and cannot give reasonable approximations of M for Ug values very far outside the stated range. This becomes apparent when substitut- ing iniarlarge-valuevof WU, 330 exe, 2.0lneet per second, andjicaleulating M. The result would be a positive value for M; i.e., the concentration of sediment increases with elevation which is not reasonable. The limit- ing value of M for extremely large values of U, should be zero, or uniform concentration of sediment throughout the depth. At the other extreme, equation (11) gives a value of M = -18.45 feet! for Uz = 0 foot per second which is also not reasonable. But, for low values of U,, any continuous function is not expected to give a correct relationship since at some point the flow changes from turbulent to laminar. In laminar flow there is no turbulence and therefore no sediment suspension. As Up is increased, the flow, at some velocity, suddenly changes from laminar to turbulent and just as suddenly the suspended-sediment concentration changes from zero to some positive value. Therefore, the relationship expressed by equation (11) becomes invalid at some small value of U)-
d. For the flow velocities studied, the sediment-settling velocity has a significant effect on the slope of the concentration distribution curve. Not enough data were obtained to define the relationship between V, and M, but the data did yield the qualitative relationship that for constant U,, M decreases (or becomes a larger negative value) with in- creasing Vs. As discussed earlier, if the sediment exchange coefficient
43
is equal to the momentum exchange coefficient, as is assumed for unidirec- tional flow, the slope, M, should be directly proportional to V,. The experimental results for oscillating flow did not indicate direct propor- tionality. Therefore, the sediment exchange coefficient for oscillating flow given by equation (13) is, as yet, an undetermined function of Vz. and U,. A discussion in Section IV indicates why this result is expected.
III. DISTRIBUTIONS OF TURBULENT VELOCITY FLUCTUATIONS
ils Experimental Apparatus.
Successful measurements of turbulent velocity fluctuations in fluids have been made using either constant current or constant temperature hot-film anemometers. A constant temperature, quartz-coated hot-film sensor, model number 6010 made by Thermo-Systems Incorporated, Minneapolis, Minnesota, was used in this investigation. The sensor was connected to a 1050 series anemometer also made by Thermo-Systems Incorporated. The anemometer uses a bridge and feedback system to maintain a constant resis- tance and therefore a constant temperature of the sensor. Any change that affects the heat transfer between the sensor and the environment is reflec- ted in the voltage output of the bridge. This output voltage is amplified and recorded on magnetic tape. The record of voltage fluctuations is then converted by use of a calibration curve to a record of velocity fluctua- tions. A schematic of the hot-film bridge is shown in Figure 16. The hot-film sensor and probe are shown in Figure 17.
Calibration of the hot-film anemometer was done in the calibration tank (Fig. 18) which was divided into two chambers, a fore chamber and a calibration chamber. The fore chamber contained two wire screens to ensure a uniform velocity distribution. The two chambers were connected by a 1-inch-diameter nozzle located at the midpoint of the partition Separating the two chambers, 5.25 inches above the bottom of the tank.
An overflow was located at the downstream end of the calibration chamber, 4.75 inches above the nozzle. Water was supplied from the deaeration tank, the rate controlled by a 0.25-inch needle valve. The flow into the calibration chamber was a submerged jet. The probe with sensor was held at the downstream face of the nozzle by clamps connected to a point- gage assembly. The point-gage assembly was used to raise and lower the sensor known amounts to obtain velocity measurements across the diameter of the nozzle. The water collected at the overflow in a measured time was weighed to determine the flow rate and mean nozzle jet velocity.
Measurements of the turbulent velocity distributions were conducted in the swing flume. Only minor modifications to the flume apparatus were necessary to accommodate the anemometer equipment.
As shown in Figure 17, the sensor is extremely delicate and there-
fore cannot be used in flows containing solid particles. For this reason the flume had to be cleaned of all the black plastic sediment used in
44
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TO DEAERATION TANK
\/4" NEEDLE VAL a VALVE
CREST OF OVERFLOW
Pech;
| | | | | | 1 | | nozzle i! |
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ELEVATION VIEW
OVERFLOW COLLECTOR
WIRE SCREENS
PLAN VIEW
Figure 18. Hot-film anemometer calibration tank.
AT
concentration measurements. To ensure that the natural sediment glued to the artificial dunes did not.break loose and damage the sensor, the surface of the dunes was sprayed with a thin film of plastic. The plas- tic was thin enough to not alter the surface roughness and strong enough to hold the sediment in place.
To accommodate the anemometer equipment, the yoke support used in concentration measurements had to be removed from the support frame (Fig. 3). The hot-film probe assembly (Fig. 19) was attached to the support frame at a position midway across the flume. The elevation of the sensor could be changed by loosening two friction clamps and reposi- tioning the probe holder in relation to a scale fixed to the assembly.
A 6-inch-square opening was cut into the wave suppressent board to allow the sensor to be lowered to the flume bottom.
2. Experimental Procedure.
The hot-film sensor is extremely sensitive to both water temperature and water quality. Therefore, it was necessary to determine a new cali- bration curve each day that velocity measurements were made in the flume. Water temperature readings made during calibration measurements were com- pared to readings made during flume measurements and did not vary. It was necessary to use deaerated water in both the calibration and flume measurements to prevent air bubbles from adhering to the hot-film sensor, thereby either burning out the sensor or altering its heat transfer char- acteristics. Periodically, during both calibration and flume measurements a record was made of the base voltage; i.e., the voltage for the sensor in Still water. These voltage readings were then averaged to obtain a mean base voltage which is needed in calculating the calibration curve.
Both days that velocity measurements were made with the anemometer, a 5-percent overheat of the sensor was used. This ensured uniform sensi- tivity of the sensor and base voltage of the measurements. The conditions for measurements on both days were so similar that the’two calibration curves could not be distinguished. This allowed all the calibration data to be used for one curve and only one equation used to convert voltage into velocity.
a. Calibration of the Hot-Film Sensor. The hot-film sensor was positioned at the center and as close to the downstream face of the cali- bration nozzle as possible (approximately one-eighth inch). A high flow rate through the flume was obtained by fully opening the needle valve. This flow rate was allowed to continue until a steady flow through the calibration tank was established (about 5 minutes). The flow through the nozzle was monitored by the continuous voltage output of the hot-film bridge displayed on the anemometer equipment. When the voltage readings became constant with time the flow rate was steady. A 24-second magnetic tape record was then made at the rate of 117 samples per second. Simul- taneous with the voltage record, a flow rate measurement was made. The flow rate was determined by timing the period required to collect 10 to
48
\ FRICTION CLAMPS TO ALLOW ASSEMBLY TO BE MOVED VERTICALLY
CLAMP
ELECTRICAL LEADS TO ANEMOMETER
BOA SEE DETAIL "A"
LeOety PROBE HOLDER
A. DETAILS OF CLAMPS
PROBE HOLDER
TIP SUPPORT 12"
PROBE HOLDER 13" (approx.)
SENSOR HOLDER
Figure 19. Hot-film probe extension assembly.
49
15 pounds of water at the overflow of the calibration tank. The water was weighed on a scale to the nearest 0.05 pound. The flow through the flume was then reduced by partially closing the needle valve and a second set of measurements made. The procedure was repeated until five voltage- velocity points were measured for a calibration curve.
The results of the two calibration curves are shown in Figure 20. The velocity range measured was from 0.0374 to 0.580 foot per second. The relationship between velocity and voltage was:
Ine(E' - E,) = (0.422) ing(Uz) + 7.147 , (14)
where E' is proportional to the output voltage of the hot-film bridge, and E9 is proportional to the mean output voltage for the sensor in still water. U,, the actual velocity of the fluid, was calculated by dividing the mean velocity determined from flow rate measurements by the nozzle coefficient, C,. The log-log relationship shown in Figure 20 is consistent with the theoretical results from the manufacturer and the results by Das (1968). Because magnitudes of the voltage readings for a given flow velocity are dependent on the water temperature, water quality, overheat percentage, and amplification of the sensor output signal, no attempt was made to compare quantitatively the measured calibration curve ‘with other published curves.
b. Determination of the Nozzle Coefficient. The sensor was posi- tioned close to the downstream face and at the lower edge of the nozzle. After establishing a high, constant flow rate through the nozzle, the voltage from the hot-film bridge was recorded. Without interrupting the flow, the sensor was then raised a small measured amount and a second voltage record made. This procedure was repeated until the sensor was at the upper edge of the nozzle. The velocity of flow was estimated for each location from a calibration curve similar to Figure 20 but not cor- rected by a nozzle coefficient. The velocity profile across the diameter of the nozzle for the high flow rate is shown in Figure 21(a). Because of the high flow rate, the water surface elevation in the deaeration tank was lowering a Significant amount, thereby decreasing the flow rate through the calibration tank. This is the reason for the lower measured velocity at the top of the nozzle. The mean velocity was determined by integrating the velocity profile across the jet. The nozzle coefficient was then calculated from:
ay sg! F 2 C= Ore) Chee CR. Ohes (15) where Cy = nozzle coefficient Vaan = mean velocity across the jet
(dia.) jo4 = diameter of the jet
(dia.). = diameter of the nozzle (1 inch) Ue ) = velocity determined from voltage measurements
50
O |2 Jan. 1973 data @ 19 Jan. 1973 data
dn (E-Eq) = (0.422) dn Us + 7.147
Ue, FLUID VELOCITY (ft/s)
SO) (Ou aclom cla Cou acley "Merere (E-E,), HOT-FILM BRIDGE OUTPUT MINUS BASE OUTPUT
Figure 20. Output versus velocity calibration curve for hot-film anemometer measurements.
S|
(ft/s)
U,, VELOCITY OF FLUID
(ft/s)
FLUID
Ue, VELOCITY OF
(a) Flow rate velocity = 0.567 ft/s
Measured velocity = OOO atts
Mean velocity = 0.487 ft/s
C,= 0.99
Measured velocity =OLOW Se fits NS Mean velocity
= 0.0668 ft/s \
\ \ \ i) \ \ \
Calibration nozzle
0 0.4 0.8
DISTANCE ALONG THE DIAMETER OF CALIBRATION NOZZEE)> Cin)
Figure 21.. Velocity profile across calibration nozzle.
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The nozzle coefficient for the high velocity (0.569 foot per second) was then calculated to be 0.99. This set of measurements was repeated for
a low flow rate. The results of these measurements are shown in Figure 21(b). The nozzle coefficient for the low velocity (0.072 foot per sec- ond) was calculated'to be 0.98. The nozzle coefficient for intermediate velocities was assumed to be a linear interpolation between the two above values.
c. Flume Measurements of Turbulent Velocity Distributions. The
velocity, which is of significance in attempting to describe the sus- pension of sediment in an oscillating flow, is the vertical component of velocity fluctuations due to turbulence, v'. When the swing flume is operating, the only velocities which exist in the fluid are: (a) the three directional components of velocity fluctuations caused by turbu- lence, and (b) the oscillating flow contained in the boundary layer.
The boundary layer extends only a few millimeters above the bed of the flume. Sediment is in suspension at an elevation considerably above the upper limit of the boundary layer. Therefore, the flow regime of interest has no measurable mean or periodic velocities, only the random motions caused by the turbulence diffusing upward from the bed. The problem, then, is to determine only the vertical component of the veloc- ity fluctuations.
Das (1968) developed a method of measuring v' in a still body of water with an oscillating rough bed. This method involved imparting an oscillating motion to the sensor in the vertical direction. If only measurements made during the peak velocity of the sensor are considered, then the total velocity affecting heat transfer from the sensor is:
v2 = (V + vt)? + ut? + wi2 - (16) where Ve = total velocity affecting heat transfer Vv = peak velocity of the sensor; known from the period and amplitude of oscillation Vie = vertical component of the turbulent velocity fluctuations u' and w' = turbulent velocity fluctuations in the remaining two
directions Dividing this equation by v2 yields: CES We GNM ee ICON Ca De. (a7)
If the sensor oscillation is such that V >> v', u', and w', then the above equation can be approximated by:
CENA Osean” (18)
53
This equation is sensitive to only v'. Experiments by Das were conduc- ted in a stationary flume in which only a horizontal bottom plate was oscillated to produce turbulence. The results showed some promise for
the method.
When Das' method of measuring v' was tried in the swing flume, it was not successful because of excessive vibrations of the sensor. These vibrations were mainly due to: (a) the long holder required to extend the sensor to the flume bottom, and (b) attaching this holder to the support frame which was indirectly subjected to the vibrations from the flume motion. The velocities of the sensor due to vibrations were greater than the velocities of the turbulent flow; therefore, no com- ponent of turbulent velocity could be distinguished.
An approximation to v!' had to be obtained based on the following assumptions. It was assumed that at a given elevation the root-mean- square value of the three components of turbulent velocity fluctuations is proportional to each other. It was also assumed that the heat con- vected from the sensor due to velocities in the direction parallel to the sensor axis was insignificant compared to the heat convected by velocities perpendicular to the axis. This assumption is justified in that the hot-film has directional properties making the maximum sensi- tivity at right angles to the flow. Also, the aspect ratio (length- diameter) of the sensor is such that its properties approach those of an infinite wire where there is no effect of a longitudinal velocity. Based on these assumptions, the effective velocity causing heat convec- tion is, as an average:
Uss=eiivic Ke yas) (19)
e
where Ug is the velocity corresponding to the output voltage of the hot-film bridge, K' is the constant of proportionality between the vertical component and one of the horizontal components of turbulent
velocity fluctuation, and v' is the vertical component of the turbulent velocity causing heat convection from the sensor. It is apparent then that the sensor must be placed in the flume with its axis horizontal.
The magnitude of the vertical component of velocity fluctuation can then be calculated and is, as an average:
ECs TE (20)
Although equation (20) is only an approximation, the assumptions used do not. affect the basic relationships (a) between the root-mean-square
54
value of v' and elevation above the bed, and (b) between the root- mean-square value of v' at a fixed elevation and the flow velocity, Up. The assumptions also allow an approximation of the absolute magnitude of the root-mean-square value of v'. The horizontal component of turbulent velocity fluctuation is probably on the same order of magnitude as the vertical component and therefore, for qualitative analysis, the value
of K"' in equation (20) can be approximated as equal to unity.
The procedures used in measuring velocity distributions in the flume were as follows. A period and an amplitude of flume oscillation were selected and the flume linkage adjusted to give a symmetric motion.
After the flume was filled with deaerated water to the elevation of the wave suppressent board, the asymmetric roughness elements were adjusted to eliminate secondary currents in the central part of the flume. The sensor was then placed in the flume as near the bottom as possible and its elevation recorded. The flume was started and the motion allowed to continue until equilibrium flow conditions were established. A record
of the hot-film bridge output voltage was made on magnetic tape, the length of which was an integer multiple of the flume oscillation period. The flume was stopped and the sensor elevation raised for a new measure- ment. The procedure was repeated until an elevation was reached at which the velocity fluctuations were too small to be accurately measured with the anemometer. The sensor was then lowered in a stepwise manner to obtain velocity measurements at intermediate elevations. In this manner, 10 to 13 velocity-elevation measurements were obtained to give a velocity distribution for the flow condition used. The period of the flume was changed and the measurements repeated to give a second velocity distribu- tion. In all, four velocity distributions were obtained for four differ- ent flow conditions.
3. Results.
The purpose of the velocity measurements was to obtain the following three relationships needed for an analysis of the suspended-load equation: (a) An approximation of the magnitude of the root-mean-square value of v' versus flow velocity, Up; (b) the distribution of the root-mean-square value of v' versus elevation above the bed; and (c) the distribution with time of v' at a constant location in space. Results pertaining to the third unknown listed above will be discussed first.
Two sets of data were analyzed to determine the distribution with time of v'. The period and amplitude of flume oscillation for both sets of data were 10.48 seconds and 0.925 foot, respectively. In both cases, the length of record analyzed was 10.48 seconds (1,224-voltage samples). One set of data was taken at an elevation of 0.168 foot above the crest of the artificial dunes, the other 0.209 foot above.
The data were analyzed in the following manner. For each voltage sample recorded, the effective heat transfer velocity, U,, was calcu- lated from equation (14). The velocities were ordered and percentages
595
equal to or less than various selected velocities calculated. These percentages were divided by two and plotted against velocity on normal probability paper. The percentages were divided by two in order to adjust for the fact that the anemometer measured the absolute effective velocity without regard to its direction; i.e., the velocities in each range were composed of an equal number of negative and positive veloci- ties, thereby giving twice the percentage of actual positive velocities. As shown in Figure 22(a, b), these plots approximate straight lines and the 50-percent velocity is zero, thereby indicating that the distribu- tion of the turbulent velocity fluctuations is approximately normal with a mean of zero. Similar data by Das (1968) give the same results.
The above result suggests that the standard deviation, s, of the normal distribution (which equals the root-mean-square velocity) be used as the velocity scale describing turbulence intensity for a given eleva- tion.
The data were then analyzed to determine the distribution of s with respect to elevation. For each sample of a voltage record, the effective heat transfer velocity was calculated from equation (14). From the velo- city record, the root-mean-square effective velocity was calculated. Knowing this velocity the velocity scale for the elevation at which the record was made was calculated from:
s = U,/v2 , (21)
where s is the velocity scale and is equal to the standard deviation of v' for the elevation of the record and flow conditions of the flume, Us is the root-mean-square effective heat transfer velocity of the record, and v2 comes from equation (20) when K' is assumed equal to unity.
The velocity scale was plotted against elevation on semilogarithmic paper to give the relationships shown in Figures 23 to 26. In general, these relationships can be expressed by:
Si =a Sn exp CA NY) (22)
where s, is the value of the velocity scale (in feet per second) at the
elevation of the crest of the artificial bed dunes, A is the slope of the exponential curve (in feet-!), and Y is the elevation (in feet) above the crest of the bed dunes. The flume flow conditions for which a velocity scale-elevation distribution was measured were Up = 0.353, 0.510, 0.748, and 0.930 foot per second.
Comparison of the four velocity-elevation distributions revealed that, for the range of flow conditions studied and bed roughness used, the slope of the exponential relationship was constant. This implies that the in- tensity of the turbulence decreases in a manner which is independent of the flow velocity generating the turbulence. The constant rate of velo- city decay appears to apply to elevations near the bed, within 1.5 centi- meters as measured with the hot-film sensor.
56
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O 0.05 0.10 0.15 0.20 0.25 Y, ELEVATION ABOVE CREST OF BED DUNES (ft)
Figure 23. Velocity scale versus elevation for flume velocity, U5 = 0.353 foot per second.
58
s, VELOCITY SCALE (ft/s)
Oo ~ o@ 02
O 0.05 0.10 0.15 0.20 OXZS Y, ELEVATION ABOVE CREST OF BED DUNES (ft)
Figure 24. Velocity scale versus elevation for flume velocity, U, = 0.510 foot per second.
39
s= 0.1160 exp (-10.86: Y)
S| VELOCITY SCALE (ft/s)
Oo ~ o o2
O 0.05 0.10 OAS 0.20 0725 Y,, (ELEVATION: ABOVE: CREST OF BED DUNES 4( ft,
Figure 25. Velocity scale versus elevation for flume velocity, U, = 0.748 foot per second.
60
i 9 8 7 s = 0.138 exp(-I0.67 - 5
s, VELOCITY SCALE
o ~ wo oD
O 0.05 0.10 Ors 0.20 O3Z5 Y, ELEVATION ABOVE CREST OF BED DUNES (ft)
Figure 26. Velocity scale versus elevation for flume velocity, Up = 0.930 foot per second.
6)
The base velocity, So (eq. 22), was found to be a function of the
flume flow conditions. The relationship between s, and U, is shown in Figure 27. From the limited data it was only possible to determine an approximate mathematical relationship between s, and U,. This relationship is:
See (0.0885) UR OOS57., (23)
which, from boundary considerations, only gives approximate s, values in the range of experimental values of Up).
4. Summary of Experimental Results.
The following is a summary of the results of the turbulent velocity fluctuation measurements and a brief discussion of their limitations.
a. The velocity fluctuations caused by turbulence are, for a con- stant elevation above the bed and a constant flow velocity, approximately normally distributed with a mean of zero. The standard deviation of the distribution, s, is used as the velocity scale to measure turbulence intensity at any elevation. Most turbulent velocity fluctuation measure- ments give approximately Gaussian results, although it is known that except for isotropic turbulence, the distribution cannot be Gaussian.
b. For the range of flow conditions studied and the bed roughness used, the velocity scale can be expressed by equation (22). The exponen- tial nature of this relationship was discussed in Section II, paragraph 4(a). The relationship also conforms to the boundary conditions of the flow. As expected, the turbulence intensity assumes a limiting value, Sp, at the ocean bottom (Y = 0). This limiting value of turbulence in- tensity is determined, in some manner, by the flow velocity, UL: As
the turbulence diffuses upward its intensity decays because of viscosity. The body of fluid into which the turbulence diffuses is, by comparison, extremely large; therefore, the empirical relationship is expected to indicate that the turbulence intensity decays to zero at an infinite distance from the bed.
c. The slope, A, of the exponential distribution of the velocity scale is constant with respect to elevation and constant throughout the range of flow conditions studied. It was found to be -10.57 feet !. This result is not surprising since the rate of turbulence intensity decay for the oscillating flow conditions measured is determined by vis- cosity. Therefore, for fluids of the same viscosity and density, the rate of decay should be constant and independent of flow velocity.
d. The base velocity scale, s, (eq. 22), is a function of the flow velocity and can be approximated by equation (23). This equation is a best fit relationship of the empirical data and does not apply for flow velocities outside the measured range. This becomes obvious by
letting Up = 0 foot per second and finding sy = 0.0557 foot per second.
62
(ft/s)
Uy, FLUME VELOCITY
OLS)
0.8
OL
0.6
Sq = 0.0885 U,+ 0.0557
0:5
0.4
OS ©1877 = .O109 0. | O.) | 0.12 OAS 0.14 Sy — VERTICAL COMPONENT OF TURBULENT VELOCITY FLUCTUATIONS AT THE CREST OF THE BED ROUGHNESS ELEMENTS (ft/s)
Figure 27. Base vertical turbulent velocity scale versus flume velocity for a constant amplitude = 0.925 foot.
63
In the range of flow velocities where equation (23) is valid, the calcu- lated so is an approximation of a base velocity at an arbitrary eleva- tion, the crest of the bed dunes. To apply the experimental results to
a real situation, the constants in equation (23) must be adjusted to give So values at the elevation of the top of the bedload layer. This eleva- tion depends on the grain diameter of the sediment being considered and on the bed geometry; therefore, no attempt was made to express Ss, at the bedload elevation.
IV. THE SUSPENDED LOAD IN OSCILLATING FLOW 1. Suspended-Load Theory in Unidirectional Flow.
The suspended-load theory for unidirectional flow and the available field data to test the theory supply valuable insight to some problems which exist in determining the suspended load in oscillating flow. For this reason, Einstein's (1950) suspended-load theory and field data from the Missouri and Atchafalaya Rivers are presented.
The suspension theory in unidirectional flow is based on an equilib- rium equation for mass flux across a unit horizontal area in the flow. Assume the unit horizontal area is at elevation Y. Across this area fluid is being exchanged by the vertical component of the random motion of fluid particles caused by turbulence. From continuity, the picture of fluid exchange can be simplified by assuming that through one-half of the unit area, fluid is moving upward with an average velocity of v; through the other half area the fluid is moving down with an average velocity -v. If the exchange occurs over an average distance of 1, it can be assumed that the downward-moving fluid originates, as an average, from an eleva- tion Y + 1/2 le while the upward-moving fluid originates from Y - 1/2 le. The important assumption is made that the fluid preserves, during its exchange), the propertiesvot ithe) fluid iat ats pointiof or1vein. | bi. the concentration of sediment at elevation Y is C and the sediment has a settling velocity of V,, the equilibrium equation for sediment flux is given by:
[c - Fle (ac/ay)| (3) (ava) + [c ce (ac/ay)| (3) (-v-Vz) = 0. (24) This equation reduces to: CV, + sl, v (dc/d¥) = 0. (25)
To solve this equation the term, 1/2 fv omust ibevevaluated? eathuaisi as normally done by equating this term to the corresponding term in a similar equation of momentum exchange; i.e., the sediment exchange coefficient is
64
assumed equal to the momentum exchange coefficient. Assuming that shear due to viscosity may be neglected, compared with that due to momentum transport, the depth, d, may be introduced:
t= t) [(d-Y)/d] = Sv 0 {[u - $16 (au/av)| i lu tole (au/ayy|} , (26) where
To = shear stress at the bed; equal to goRS
t = shear stress at elevation, Y R = hydraulic radius (in feet)
S = energy slope
g = acceleration due to gravity o = density of the water
u = horizontal flow velocity
Using the logarithmic formula based on von Karman's (1934) similarity law for the distribution of flow velocity, du/dY may be calculated:
du/dY = (1.0/0.4) (u,/Y) , (27)
where u, is the shear velocity and equal to Gone. Substituting this
value into equation (26) and solving for 1/2 1, v yield:
Fle v = (-0.4) Yu, (d-Y)/d . (28)
Using this value in equation (25), separating variables and introducing the abbreviation:
Zia VE CORA uy) (29)
the result can be integrated from a to Y. The solution is: (C/o Ned Wi nayy duane! 2 (30)
It has been found that equation (30) gives the correct form of the distribution function, but the value of the exponent Z _ given by equa- tion (29) does not always agree with the exponent that fits the measured data. Let Z' be the exponent which best fits the data. It was found that torshagh) values o£ Z)/CZ>10)),) 2" was) stenifacantliy, less. Asi iZ) jas
65
reduced, the difference between Z and Z' decreased and finally when Z assumed values less than unity, which is normally the case, the dif- ference between Z and Z'! was small enough such that use of equation (29) allows accurate results. The relationship between Z andi Ze ass shown in Figure 28 (Einstein and Chien, 1954).
2. Similarities Between Oscillating and Unidirectional Flow.
The similarities between oscillating flow and unidirectional flow with low shear velocities are pronounced. It was found that the concen- tration distribution in oscillating flow (from equation 9) could be ex- pressed by:
€/€, = exp(MY) , (31) where M was found in Section II to be (from equation 13): M= -V,/E , (32)
E is the sediment exchange coefficient. The coefficient, M, which defines the rate at which the concentration decays with elevation, behaves in a manner similar to Z of the unidirectional flow theory. In oscillat- ing flow, as in unidirectional flow, the value of M fitting the experi- mental results was different than the value which would be predicted from equation (32), assuming E independent of V,. Figure 15 and Table 3 show that for the four concentration distribution curves obtained when
V, was increased from 0.035 to 0.0626 foot per second, the absolute value of M increased, as an average, by the factor 1.19. When V, was in- creased from 0.035 to 0.0498 foot per second, the absolute value of M increased, as an average, by 1.13. If the exchange coefficient, E, of equation (32) were a function of the flow hydraulics only and independent of the sediment-settling velocity, then the average increase in the abso- lute value of M would have been 0.0626/0.035 (= 1.8) and 0.0498/0.035 (= 1.42), respectively. Therefore, it can be concluded that the sediment exchange coefficient, E, is a function of both Ve) sand) Up; vandathac M, for a constant flow velocity, is not directly proportional to Vg. This conclusion agrees with Einstein and Chien (1954) that in unidirec- tional flow for high values of settling velocity relative to the flow shear velocity the sediment exchange coefficient cannot be accurately approximated by the momentum exchange coefficient.
The difference between the sediment and momentum exchange coefficients depends on the relative magnitudes of the sediment-settling velocity and the turbulence intensity. If the sediment-settling velocity is small compared to the turbulence intensity, the two coefficients are approxi- mately equal; when Vz = 0, the two coefficients are identical. As V, becomes relatively larger, the difference between the two coefficients increases. In unidirectional flow the turbulence intensity is usually very large compared to V, and the theory is accurate under most situa-
tions. Unfortunately, this is usually not the case in oscillating flow.
66
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Examining results b, c, and d of Section III, the range of the root-mean- Square valluenofmuy CS) y's Os O00V25< 1S <1 0)10/5) footy pers second amine settling velocities of the sediments used in the experiments were
Vg = 0.035, 0.0498, and 0.0626 foot per second. V, is then, the same order of magnitude as the velocity scale. As discussed earlier, the sediment used in the experiments had a specific gravity of 1.25. There- fore, the settling velocity of natural sediment would be even larger compared to turbulence intensity in the approximate prototype flow con- ditions of the experiments.
For illustration, a typical oscillatory flow condition of this in- vestigation will be approximated as a quasi-steady unidirectional flow and compared to the field data of Figure 28. This is possible because the time scale for oscillation is much greater than the time scale of the turbulence. For example, the average flow had a period of 6 seconds and an amplitude of about 1 foot, Up = 0.667 foot per second. A time scale of turbulence can be defined as S/S where 6 is the thick- ness of the boundary layer and s, is the base vertical velocity fluc- tuation. If 6 is defined as the distance above the bed at which the boundary layer oscillation velocity is equal to 99 percent of the free- stream velocity (velocity given by linear wave theory for y = -d),
6 can be calculated from equation (5). This calculation indicates that
6 is equal to or less than 0.05 foot and from Figure 27, So is 0.11 foot per second. Therefore, the time scale for the turbulence is 0.45 second
as compared to a 6-second time scale for the oscillation. To calculate the theoretical Z value for the quasi-steady unidirectional flow it is necessary to determine a mean flow shear velocity. The flow shear veloc- ity 1s given by:
ie
De GES) (33) where R is the hydraulic radius (in feet), g is the acceleration due to gravity (in feet per second squared), and S is the energy slope of the quasi-steady flow. The energy slope is obtained from Manning's equation by using the root-mean-square flow velocity (= 0.707 2 Il L/T),
and estimating values of the roughness coefficient, n, and the hydrau- lic radius, R. The expression for the energy slope is:
a See= 1 (n/ RO Pei (Sal Alin. (34) Substituting equation (34) into equation (33) yields: Big eS (GY i Ge RO OME) 5 (35)
Using the above expression for u, in equation (29), the Z value for the quasi-steady flow becomes:
A (Me) TF ROOMS GLB. an ih) < (36)
68
For the average flow conditions, Vg is 0.035 foot per second, T is 6 seconds, and L is 1.0 foot. Based on the bottom roughness shape and
the hydraulic radius, n is approximately 0.015 foot2:167. The hydrau- lic radius is estimated from the flow geometry as about equal to unity.
Since Z is proportional to R_ to the 0.167 power, and therefore tends to unity, there is probably not much error in using this estimate. Using these values in equation (36) yields Z = 2.1. In Figure 28, this value of Z is well into the range where the sediment exchange coefficient is Significantly different from the momentum exchange coefficient.
Because not enough data were obtained to define a relationship be- tween M and V, and because the sediment exchange coefficient could not be expressed as a function of the shear-stress distribution, no attempt was made to derive a theoretical relationship for the concentra- tion distribution as was done in Einstein and Chien (1954) for unidirec- tional flow.
3. Sediment Suspension in an Oscillating Flow.
This investigation was done to determine the behavior of sediment suspension in an oscillating flow and present a method by which the sus- pended load could be approximated from flow hydraulics. It is apparent from the field measurements of unidirectional flow (Fig. 28), and the results of this investigation for oscillating flow that the mechanism by which sediment is held in suspension is complex and not fully understood. For this reason, the following method for estimating the suspended load in oscillating flow as a function of the flow hydraulics is based on the general turbulent mixing length theory first proposed by O'Brien (1933). His derivation is as follows:
There is a continuous up and down motion of fluid across any horizon- tal plane caused by the turbulent vertical velocity fluctuations. This exchange motion is capable of transporting suspended matter. Consider a horizontal reference section of unit area at a distance Y from the bed. The transfer of sediment in the vertical direction from the region of high concentration to a region of low concentration through this unit section will be -1/2 1, v dC/dY, where 1, is the mixing length for the sediment exchange, v denotes the exchange discharge through the unit area due to vertical velocity fluctuations, and C is the concentration of suspended sediment with settling velocity, V,, at elevation Y. How- ever, a continuous settling of particles through the unit area at a rate of C Vg exists. A statistical equilibrium condition is given by equa- tion (25). This equation is identical to the equation derived by Einstein (1950), but without the assumption that the origin of the sediment is the same as the origin of the fluid and without assuming any distribution of fluid exchange.
The mixing length theory incorporates all of the factors which affect
sediment suspension in two artificial variables, the velocity scale and the length scale. The velocity scale for oscillating flow was measured
69
directly in this investigation and is given by equation (22). The sedi- ment exchange coefficient which is the product of the length and velocity scales was also measured directly and is given by equation (13). From these two equations, the corresponding length scale can be calculated as:
ale =H GV Mesyy) extn (An Yo) (37)
For the limited flow conditions investigated, all the variables of equations (13), (22), and (37) have been found as a function of U,, where U, is a function of the surface wavelength, period, and water depth. The variable A was found to be a constant (= -10.57 feet-!), and M and s, are graphically given in Figures 6 and 27, respectively. These expressions for the velocity and length scales are physically reasonable. Not only is the base turbulent velocity a function of the surface wave intensity (Fig. 27), but the exchange length tends to small values as the ocean bed is approached, indicating that no sediment should be exchanged across the bed surface.
4. The Base Concentration, C,.
With the exception of the base concentration, C,, all the variables needed to describe the suspended load as a function of flow hydraulics have been discussed. The following is only a brief discussion of the base concentration. Kalkanis (1964) provides a complete mathematical derivation.
In oscillating flow, as in unidirectional flow, sediment transport
is by two different types: (a) Bedload transport, and (b) suspended-load transport. As discussed in Section I, the thickness of the bedload layer is about two-grain diameters. Therefore, for both prototype and experi- mental sediments, the bedload is contained in the boundary layer described in Section II. The theory proposed by Kalkanis to predict the amount of bedload transport and the concentration of sediment is general and only requires knowledge of the surface wave characteristics, the water depth, the bed sediment characteristics, and statistical parameters which have
been found experimentally. Because the thickness of the bedload layer
is small, the concentration of sediment in this layer is assumed constant and equal to Cy. The concentration, Co, for the bedload is the base concentration to be used for the suspended load.
Using Co as the suspended-load concentration incorporates a small error in the total suspended load. As indicated previously, the bedload is contained in the boundary layer. The distribution of sediment concen- tration in this area is unknown. Because the distance between the top of the boundary layer and the top of the bed layer is small, extension of the exponential suspension distribution down to the bed layer would incur only a minor error in the total amount of sediment in motion.
70
5. Net Transport of Sediment in the Ocean.
The initiation of sediment movement in the ocean is by wave action. Since the waves are approximately linear, the wave-induced fluid motion is a symmetric oscillation causing an equally symmetric movement of the sediment. This motion ordinarily cannot cause net transport of sediment, but it does suspend sediment so that currents superimposed on the oscil- lating velocity will cause a net transport of the sediment. Examples of unidirectional currents in the ocean are the longshore current caused by waves attacking the coastline at an acute angle and the secondary currents caused by the coastline geometry. These two examples indicate that deter- mination of the transporting current shouid be by field measurement.
6. Additional Investigations Needed to Complete the Suspension Theory.
The suspended-load theory is by no means complete. The relationship between the rate of decay of sediment concentration, M, and sediment- settling velocity, Vs, is needed. Figure 15 indicates that settling velocity has a significant affect on the distribution of sediment con- centration, but there are not enough data to indicate the relationship. Additional measurements are required to quantitatively determine the relationship between M and V, for a constant Up.
Another variable affecting the suspended-load theory and not studied in this investigation is the bed roughness. It is probable that a change in bed roughness would affect the intensity of turbulence at a given elevation. Also, as discussed in Section II, the amplitude of oscilla- tion relative to the wavelength of the bed roughness has an affect on the distribution of suspended sediment. Determination of how the bed rough- ness affects sediment suspension requires a great deal of experimentation; however, the results of this investigation will supply some guidelines which would reduce the amount of experimental work required.
7. Conclusions.
These experiments in an oscillating flow simulating wave motion at the ocean floor provide results which, in some cases, substantiate pre- vious results and provide new information on the behavior of sediment suspension. It should be stressed that the conclusions of this investi- gation are confined to the rather limited range of the variables studied.
The conclusions obtained from sediment concentration measurements are:
a. The relationship between the mean sediment concentration and elevation above the bed is exponential. This conclusion is based on 65 concentration distribution relationships covering a wide range of proto- type flow conditions and each composed of numerous point concentration measurements. Typical examples of this relationship are shown in Figure 5 and the basic data substantiating this conclusion are given in the appendix. Although this relationship was determined using an artificial
TI
. sediment with a settling velocity less than found in the prototype, results of experiments by other investigators using prototype sediments resulted in the same exponential relationship. In addition, the bed roughness used by other investigators varied; therefore, this conclusion does not appear to be limited to the single-bed roughness used in this investigation.
b. Using the above conclusion and the O'Brien's (1933) equation for continuity of sediment exchange results in a sediment exchange coefficient which is independent of elevation above the bed. Behavior of the rate of sediment concentration decay with elevation, which is related to the sedi- ment exchange coefficient by equation (13), was found to be a function of the flow velocity causing the suspension and the settling velocity of the sediment. A linear relationship between the flow velocity and the sedi- ment concentration decay rate was found for the constant sediment-settling velocity used in the majority of the experiments. This relationship is shown in Figure 6. The relationship between flow velocity and the sedi- ment concentration decay rate for other sediments studied in this investi- gation is shown in Figure 15. The limited data indicate a possible linear relationship for the different sediment-settling velocities. There is not enough data to determine the relationship between the concentration decay rate and settling velocity for a constant flow velocity. Only qualita- tive conclusions can be obtained from the eight concentration distribution Measurements shown in Figure 15. For a constant bed roughness and flow velocity, a higher sediment-settling velocity results in a higher sedi- ment concentration decay rate. The limited data consistently indicate that the concentration decay rate is not proportional to the settling velocity to the first power; i.e., they are not directly proportional. This and equation (13) imply that the settling velocity is an important variable influencing the sediment exchange coefficient. Therefore, in oscillating flows the sediment exchange coefficient cannot be accurately approximated by the momentum exchange coefficient as is commonly done in unidirectional flow analysis. No experiments were conducted in this investigation to determine how the above relationships would change with a change in bed roughness.
The conclusions obtained from measurements of the turbulent velocity fluctuations are:
a. The distribution of turbulent velocity fluctuations at a constant elevation in an oscillating flow was found to be approximately normal with a mean of zero. This relationship was determined from distribution analy- ses of measurements made at two elevations above the bed, both of which were above the boundary layer described in Section II. Results of these analyses are shown in Figure 22.
b. The relationship between the root--mean-square turbulent velocity fluctuation and elevation above the bed was found to be exponential. This conclusion is based on measurements of distributions made for four differ- ent flow velocities, all using an amplitude of oscillation of 0.925 foot and covering approximately the same range of prototype flow velocities as
Ue
studied in the concentration measurements. Results of these measurements are shown in Figures 23, 24, 25, and 26, and are tabulated in the appen- dix, Table A-5. This relationship was valid for elevations of approxi- mately 0.04 foot above the crest of the bed dunes. The relationship below this elevation, which would be in the boundary layer, was not determined. The turbulent velocity fluctuation distribution was only measured for the single-bed roughness described in Section II.
c. Based on the four turbulent velocity fluctuation distributions described above, it was concluded that the rate of turbulent velocity decay with elevation above the bed is independent of both the elevation and the flow velocity generating the turbulence. The exponential decay rate, determined from a least squares curve fitting of the data, for the four distributions ranged from -10.38 to -10.86 feet-1, with a mean of -10.57 feet™! and a variance of 0.05 foot 2.
d. The relationship between the flow velocity and the root-mean- Square turbulent velocity fluctuation at zero elevation (calculated from the empirical relationships) is shown in Figure 27. A linear relation- ship is indicated. However, this relationship, which is far from con- clusive, is based on only four data points with a significant amount of scatter. The qualitative conclusion that the turbulence intensity at zero elevation becomes larger with greater flow velocities is not only indicated by the data but is logical.
Ue)
LITERATURE CITED
ABOU-SEIDA, M.M., ''Bed Load Function Due to Wave Action," Technical Report No. HEL-2-11, Hydraulic Engineering Laboratory, University of California, Berkeley, Calif., 1965.
DAS SMM ce WExtended Application of a Single Hot-Film Probe for the Measurement of Turbulence in a Flow Without Mean Velocity," Technical Report No. HEL-2-20, Hydraulic Engineering Laboratory, University of California, Berkeley, Calif., 1968.
DAS, M.M., ''Mechanics of Sediment Suspension Due to Oscillatory Water Waves,"' Technical Report No. HEL-2-32, Hydraulic Engineering Laboratory, University One Calattornilam sBenkele yen Calera, Olly
EINSTEIN, H.A., ''The Bed-Load Function for Sediment Transportation in Open Channel Flows,'' Technical Bulletin No. 1026, U.S. Department of Agriculture, Soil Conservation Service, Washington, D.C., 1950.
EINSTEIN, H.A., and CHIEN, N., "Second Approximation to the Solution of the Suspended Load Theory,'' M.R.D. Sediment Series No. 3, Institute of Engineering Research, University of California, Berkeley, Calif., 1954.
KALKANIS, G., "Observation of Turbulent Flow Near An Oscillating Wall," ‘M.S. Thesis, University of California, Berkeley, Calif., 1957.
KALKANIS, G., ''Transportation of Bed Material Due to Wave Action," TM-2, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Washington, D.C., Feb. 1964.
KENNEDY, J.F., and LOCHER, F., ''Sediment Suspension by Water Waves," Waves on Beaches, Academic Press, New York, 1972.
LAMB, H., Hydrodynamics, Dover Publications, New York, 1932.
LI, H., "Stability of Oscillatory Laminar Flow Along A Wall,'' TM-47, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., Aug. 1954,
MANOHAR, M., "Mechanics of Bottom Sediment Motion Due to Wave Action," TM-75, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, DiGue Junegl9S5.
O'BRIEN, M.P., "Review of the Theory of Turbulent Flow and its Relation to Sediment Transportation," Transacttons, Amertcan Geophysical Union,
Apr. 1933, pp. 487-491.
SHINOHARA, K., et al., "Sand Transport Along A Model Sandy Beach by Wave Action," Coastal Engineering in Japan, Vol. 1, 1958.
74
VON KARMAN, T., "Turbulence and Skin Friction," Journal of the Aeronautical DCLECHCES NOM le NO) Jans L954) ppe) l= 20)
BIBLIOGRAPHY BIJKER, E.W., ''Littoral Drift Computations on Mutual Wave and Current Influence, Delft University of Technology, Department of Civil Engi- neering, Delft, The Netherlands, 1971.
EATON, R.O., "Littoral Processes on Sandy Coasts," Proceedings of the First Conference on Coastal Engineering, 1950.
EINSTEIN, H.A., ''A Basic Description of Sediment Transport on Beaches," Waves on Beaches, Academic Press, New York, 1972.
EINSTEIN, H.A., and LI, H., "The Viscous Sublayer Along A Smooth Boundary," Transactions, American Soctety of Civil Engineers, Vol. 123, 1958, pp. 293-317.
HENDERSON,’ F.M., Open Channel Flow, Macmillan, New York, 1971.
JOHNSON, J.W., “Sand Transport by Littoral Currents," Proceedings of the Fifth Hydraulte Conference, 1953.
MILNE-THOMSON, L.M., Theoretical Hydrodynamics, Macmillan, New York, 1969.
SUTHERLAND, A.J., “Proposed Mechanism for Sediment Entrainment by Turbu- lent Flows,'' Journal of Geophysical Research, Vol. 72, No. 24, 1967.
WIEGEL, R.L., Oceanographical Engineering, Prentice-Hall, Englewood Cliffs, N.J., 1964.
ins
APPENDIX
EXPERIMENTAL DATA
7?
Table A-1. Basic data, concentration distribution measurements, Ve = 0.035 foot per second, amplitude 2 0,693 foot.
Curve No. | Concentration‘ Elevation Curve No. | Concentration Curve No. | Concentration | Elevation Curve No. | Concentratlon | Elevation (Uo, ft/s) (s/1) (cm) (Up, ft/s) (8/1) (Up, £t/s) (8/1) (cm) (Ug, ft/s) (g/1)
717.01 (0.392)
602.06 (0.292)
602.07 (0.235) 612.01 (0.422) 612.02 (0.504)
612.03 (0.600)
2.64
518.01 (1.00) 518.02 (0.681)
518.03 (0.470)
717.02 (0.328)
HPuubaun SROSGHEEOE
$18.04 (0.394)
518,05 (0.339) - 602.01 (1.157) 602.02 (0.737)
602.03 (0.504) 602.04 (0.420)
602.05 (0.348)
717.04 (0.667)
euro
RU Sauunsd SaFTRSES
anon
» hy
BRB
am Be
L, any . NeO@eSC VME SSeS] Ree BK ES] SRE
CNA@O HUE
721.04 (0.352)
OP nur neune
SBllRs
ON ARON YUE
@
S WUIwvaaan Beaurbauan SSSSesee
78
Tahle A-2. Basic data, concentration distribution measurements, Vg = 0.035 foot per second, amplitude < 0.693 foor.
Curve No. Curve No. | Concentration | Elevation Curve No. {Concentration | Elevation (Uo, £t/s) (Uo, £t/s) (g/1) (Uo, ft/s) (g/1)
802.04 r
(0.247)
mH OSC CORN
814.01 (0.585)
NE COCO ONN
Seeneusyss
814.02
(0.585) 711.07 (0.890)
629.04 (0.448)
hBabousvonvo
BNSURAeUMO] CONAVEaUNHOS Deo wo Ora Seng
eeveuag Uneee oe
BRU BaAVIYUEE Hone RWoRo eae Ab emer
Beooa
621.05 (0.263) 628.01 (0.263)
628.02 (0.376)
CeNUNeEH oO] NUeUAan
NooCCOnN
AWreouwe S28
RN EUDeUHS
814.04 (0.585)
KuNnneaeno
Lavovoveawvo BORE EGane
OREN UNE OS
aPeSeseey
RNASE O
802.03 (0.363) 814.05
(0.585)
BLES Sesse
bm aovo00
& HUUNAaNS
BOR Rhee ee
SRSLSSRs lel Rsegss
CRN Un RRO
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(4700-0-1L£-Z2MOVa
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