Suspended-load experiments in a curved flume, run no. 4

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~!k

TU

Delft

Delft Universlty of Technology

Fecuhy of CMI Engln .. rlng Department Hydraulic Engineering

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part of: STW-project;

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Suspended-load experiments in a curved flume, run no. 4

A.M. Talmon and J. de Graaff

report no. 3-90, August 1990

River bend morphology with suspended sediment.

Delft University of Technology Faculty of Civil Engineering Hydraulic Engineering Division

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3

ABSTRACT

A laboratory experiment in a 180 degree curved flume with a mobile bed and suspended sediment transport is described. The flow is steady. The median sediment diameter (160 ~m) is larger than in the preceeding

experiments, run no. 1..3 (90 ~m).

The bed topography is measured by means of a profile indicator. The bed topography is characterized by a slowly damped oscillation of the

transverse bed slope. Downstream of the bend entrance a pool and a submerged point-bar are present, here the radial bed slope is maximal. Further downstream the transverse bed slope decreases and converges to a constant slope (constant in main flow direction), here the bed

topography is axi-symmetrical. The topography resembles that of run no. 2 but has a more pronounced axi-symmetrical region.

Suspended sediment concentrations are determined by the method of siphoning and by optical measurement. In the region

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CONTENTS ABSTRACT 1. 2. 2.1. 2.2. 2.2.1. 2.2.2. 2.2.3. 2.2.4. 2.2.5. 2.3. 5 INTRODUCTION

THE LABORATORY EQUIPMENT

The flume

Measuring equipment Discharge measurement

Slope and depth measurements

Concentration measurement by siphoning Optica1 measurement of concentration Temperature measurement

Measuring procedures

3. FLOW AND SEDIMENT CONDITIONS

3.1. 3.1.1. 3.1. 2. 3.2. 4. RESULTS 4.1. 4.1.1. 4.1. 2. 4.2. 4.2.1. 4.2.2. 4.2.3. 4.2.4. The sediment Sieve curve Fa11 velocity Flow conditions Depth measurements Mean depth

Bed form statistics

Concentration measurements Mean concentration

Curve fit of equilibrium concentration profile Depth-averaged concentrations

The concentration field at cross-sections 35 and 40

page 3 11

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13 13 13 14 14

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17 18 18 19 19

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25 27

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30 32 33

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34 35

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5. DISCUSSION 5.1. Introduction 5.2. The Z parameter

5.3. Percentage suspendedtransport 5.4. Transport formulae

5.5. Bed-shearstress and sediment transport 5.6. Adaptation lengths

5.7. The bed topography

5.8. Concentrationsin cross-section35 and 40

6. CONCLUSIONS

REFERENCES

APPENDIX A Ensemble averagedwater depth data APPENDIX B Concentrationdata

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7

LIST OF TABLES page

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3.la 3.lb 4.1 4.2 5.1

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5.2

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5.3

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Measured parameters 17 Calculated parameters 17

Parameter sets of the equilibrium concentration profile 20 Depth averaged concentrations in the 180 degree bend 22 Fraction of suspended sediment transport, in cross-section 1

by method 1 26

Fraction of suspended sediment transport, in cross-section 1

by method 2 27

The mobility parameter B 31

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LIST OF FIGURES

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1 2 3 4

Layout, Laboratory of Fluid Mechanics curved flume Sieve curve of sediment

probability density distribution of fall velocity Longitudinal water level slope

5 Contour lines of the relative water depth a/aO 6 Longitudinal profile of the water depth

7a..l Water depth in cross-direction

8a probability distribution of bed level Bed form height in cross-section 30-45 8b

9a..i Concentration profiles

10 Curve fit of equilibrium profile

lla..b Concentration profiles at cross-section 35 llc ..d Concentration profiles at cross-section 40

lle Iso-concentration contours at cross-section 35 and 40

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8

LIST OF SYMBOLS

a local ensemble mean water depth

alocal fluctuation of bed level

aO mean water depth of cross-section 1 to 5

(in earlier reports: mean depth at cross-section 1)

a complex amplitude of bed oscillation

A critical mobility number

B mobility parameter; B - T_cr/(~T)

c local concentration

c concentration at reference level

r

c local depth averaged concentration

c tr ctrb c trs C

total transport concentration; ctr- Qs/~ 10-3

transport conc. of bed-load; ëtrb=ss bed/(üaO) 10-3

transport conc. of suspended-load; ctrs=Ss sus/(uaO) 10-3

parameter in Ackers White formula

C Chézy coefficient, with d=aO; C = ü/J(di)

d a representative water depth

D gr D g D P

dimensionless grain diameter; D - D (~g/v2)1/3

gr 50

geometric mean grain diameter; D -J(D84/D16)

g

grain size for which p% of the grains is smaller than D

P

Dso median grain size

D sedimentation diameter

s

F grain Froude number

g

FgO critical grain Froude number

F grain mobility number

gr

Fr Froude number, with d=aO; Fr = ü/J(gd)

G coefficient in gravitation term

H

depth of the flume

i water surface slope

k complex wave number

~ wave number in transversal direction

k secondary flow convection factor

sn

L arc length of the bend

c

L length scale of adaptation of concentration

cs

m parameter in Ackers White formula

n parameter in Ackers White formula

n coordinate in transverse direction

p wetted perimeter

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[m ]

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[m] [m ]

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[

-

] [

-

]

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[

-

] [gil]

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[gil] [gil]

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[gil] [gil]

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[gil] [

-

] [mo.s/s]

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[m ] [

-

]

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[m] [

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]

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rml [m ]

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[

-

] [

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] [

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]

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[

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] [

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]

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[m] [

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]

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[l/m] [l/m]

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[

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] [m] [m]

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[

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] [

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]

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[m] [m]

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r u

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r c R c R g s 9 water discharge sediment discharge

profile function of the velocity profile profile function of the concentration profile radius of curvature ofaxis of flume

3 grain Reynolds number; Rg - J(gD50)/v coordinate in streamwise direction

[m

3

/s]

[gIs] [

-

] [

-

] [m] [

-

] [m] S transport rate of suspended sediment, per unit width, in s-direc.

s sus

[g/m/s]

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S transport rate of suspended sediment, per unit width, in n-direc.

n sus

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u u

total transport rate, per unit width water temperature

local depth averaged mean flow velocity

overall averaged mean flow velocity: u - ~(WaO) critical depth averaged velocity

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w

s

W width of the flume

bed friction velocity, based on C u* -

(u)g)/C

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z r z surface level s

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f3

f3

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fall velocity of sediment

the

Z

parameter:

Z -

ws/(f3~u*) reference level

ratio of exchange coefficients of sediment and momentum coefficient in the bed shear-stress direction model von Karman constant

adaptation length of concentration adaptation length of bed level

adaptation length of bed shear-stress adaptation length of velocity

p ~ efficiency factor density of water; p -1000 kg/m3

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r , r cr

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11 tm v_tc density of sediment; p -2650 kg/m3 s gradation of sediment; total drag

effective grain-shear stress; .,.'- ~.,. critical bed-shear stress

turbulent diffusion coefficient of momentum turbulent diffusion coefficient of mass

[gIm/sj [g/m/s] lOC] fm/sj [mIs] [mis] fm/sj [m] .> .-[mIs

1.

-:» [

-

] [ m] [m] [

-

] [

-

] [

-

] [m] [m] [m] [m ] [

-

] [kg/m3] [kg/m3] [

-

] [N/m2] [N/m2] [N/m2] [m2

/s]

[m2/s]

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11 1. INTRODUCTION

The project at hand is directed towards the computation of river bend morphology in case of alluvial rivers transporting a significant part of

their bed material in suspension.

In this report an experiment is described which will serve to calibrate and test morphological models for river bend flow with suspended

sediment. The experiment is performed in the curved flume of the Laboratory of Fluid Mechanics. It is the fourth of a number of

successive runs with suspended sediment transport. The steady state bed topography and local concentrations of suspended sediment are measured.

In chapter 2 the laboratory equipment is described briefly. In chapter 3 the properties of the sediment and the overall flow conditions are

given. In chapter 4 the results of the measurements of bed topography and concentration are reported. In chapter 5 the results are discussed, attent ion is being paid to implications regarding the mathematical and numerical simulation of the experiment. In chapter 6 the conclusions are presented.

This research is a part of the project: 'River bend morphology with suspended sediment', project no. DCT59.0842. The project is supported by the Netherlands Technology Foundation (STW).

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12 2. LABORATORY EQUIPMENT

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2.1 The flurne

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The layout of the LFM curved flume is shown in figure 1. Water is purnped from an underground reservoir to an overhead tank and led to the flurne. The water discharge is controlled by a valve in the supply pipeline. Sand is supplied to the model 2 m downstream of the entrance of the flurne.The sand supply is effectuated by three holes of 3.15 mm diameter and four smaller ones of 1.6 mm, in the bottom of a container located 0.5 m above the water surface.

After passing the tailgate of the flume, by which the water level is

adjusted, the water pours in a settling tank. After passing this tank

the water flows back into the underground reservoir.

The dimensions of the flume are:

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inflow section length

outflow section length

arc length of the bend radius of the bend 11.00 m 6.70 m L = 12.88 m c R = 4.10 m c W 0.50 m H 0.30 m

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width of the flume depth of the flurne

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The bottom of the flurneis made of glass and the side walls are made of perspex.

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2.2 Measuring eguipment

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2.2.1 Discharge measurement

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The discharge is controlled by a valve in the supply pipeline.

The discharge is measured by a volumetrie method. A 150 liters barrel is partly filled during about 20 seconds at the downstream end of the

flume. The volume is measured and divided by the filling time.

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2.2.2 Slope and depth measurements

--------

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The measurements of the bottom and water level are performed with an electronic profile indicator (PROVO). From these measurements the longitudinal slope of the water level and the local depth are

calculated. This device is traversed in cross-sectional direction. In each cross-section 9 equidistant measuring points are used. The carriage

in which the PROVO is mounted is also traversed in longitudinal direction. In longitudinal direction 48 cross-sections are situated, they are indicated in figure 5. The distance between these cross-sections at the flume axis is 0.32 m. The profile indicator is

continuously moved in cross-sectional direction, this is achieved by specially developed electronic hardware. The position of the profile indicator is measured electronically. The carriage is moved manually in longitudinal direction.

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2.2.3 Concentration measurement by siphoning

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Concentration measurements by siphoning have been taken throughout the specific lokations of the bend: at cross-sections 35 and 40

(iso-concentration curves) and at cross-section 1 (curve-fit) are

measured. The sediment concentration is determined from samples siphoned by a tube-pipette of stainless steel (outside diameter 5 mm, inside

diameter 3 mm) shaped much like a pitot tube. The tip of the sampler is flattened in order to minimize the vertical extended of the measuring volume.

To prevent sand to accumulate in the plastic tube it is necessary to increase the sampling velocity. This yields a non-isokinetic sampling velocity slightly higher then the local flow velocity. This does not

seriously affect measurements (Talmon and Marsman, 1988). Measuring periods of about 45 minutes are employed.

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2.2.4 Optical measurement of concentration

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The optical concentration meter OPCON has only used at cross-section 1 near the bed. Although, according to the manual, concentrations are

14

within the measurement range, an electronic drift complicates the application use of the OPCON. Consequently a zero concentration adjustment is made prior to each (45 min) measurement.

The sensitivety of the OPCON is obtained by calibration. Suspended sediment gathered by the siphoning method is used: E = 1.07 c, c[g/l],

E[V] at output 10x amplifier

2.2.5 Temperature measurements

--

--

--

-

-

--

---

-

--

--

---Temperatures are measured by inserting a thermometer into the flow near

the downstream end of the flume. The water temperature during the

measurements was 20

±

0.5

°c

.

2.3 Measuring procedures

The flume is partly filled up with sand. The thickness of the sand bed at the entrance of the flume is 0.16 m, at the exit the bed thickness is about 0.09 m.

The sand supply is measured daily. The sand settled in the settling

tank is gathered at regular intervals (about 32 hours) and is weighed

under water. The results are converted to equivalent weights of dry

sand. The supply rate is adjusted such that the supply rate and the discharge rate balance.

The water surface slope in longitudinal direction is measured daily. Af ter about 220 hours of flow, measurement of the bed topography and the

concentration are started when steady conditions are established. At that stage no significant changes of the water surface slope and differences between in and outflow of sand are measured.

The stationary bed topography is obtained by ensemble averaging of 10

measuring sessions. A measuring session consists of a water level - and

a bed level measurement. The water level and the bottom are measured

during flow conditions. One measuring session takes about one hour. The

average time interval between the first 4 sessions is about 5 hours.

Time lapse between sessions 4 and 5 is 85 hours (during which the

concentration measurements were performed.). The interval between the

last 6 sessions is also 5 hours.

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15

Each session consists of 2

*

48 cross-sectional traverses (one bed and one water level measurement). Within a cross-section 9 measuring points are used. The data are digitized and stored at alocal data-acquisition system which uses a HP1000 mini computer. Next, the data are processed by a central main frame IBM computer of the Delft University. From the mean water level in each cross-section the longitudinal slope is

determined. Comparing the results of each measuring session, only local differences in the water level slope are noticed.

Sediment concentration profiles are taken at the cross-section numbers 1, 35, and 40, (see figures 9 and 11).

In a vertical, depending on the local water depth, 15 to 40 samples are taken. The samples are siphoned into buckets. With a measuring time of 45 minutes about 9 liters water are gathered. The sample is weighed to determine the volume. Then the water is separated from the sediment. The sediment is weighed under water with an electronic balance (Mettier PE 360). Weights are read with an accuracy of 10 mg. The results are converted to equivalent weights of dry sand.

Near the bed it is difficult to apply this method, because of propagating bed forms.

The OPCON is applied only in cross-section 1 . With this apparatus concentration measurements somewhat closer to the sediment bed are possible. The data are send to the integrator, or if very close to the bed, the signal is fed to the HP1000 and next to the main frame IBM computer of the Delft University. A numerical program is used to

calculate the mean concentration. The program ignores samples which are affected by the presence of ripples (ripples occasionally block the OPCON's light beam).

16

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3. FLOW AND SEDIMENT CONDITIONS

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3.1 The sediment

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3.1.1 Sieve curve

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The sediment used in this experiment has a partiele diameter of

about two times greater than used in the previous experiments:

run 1 to run 3. At the end of the present experiment sediment samples were collected from three different sourees: the sand supply container,

the upper layer of the sediment bed and sediment which is transported in

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suspension. Figure 2 shows the cumulative probability distributions of

the grain sizes of these sediment samples. Characteristic grain

diameters are: DlO[~ml D16[~ml D50[~ml D84[~ml D90[~ml D_g[~ml u_g bed layer 133 145 181 218 229 178 1.50 supply conto 118 124 162 201 211 158 1.62 suspended sed.: 115 121 159 197 211 154 1.63

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The quantity D is defined as the grain size for which p % of the total p

mixture volume is smaller then D .

P

The geometrie mean diameter is defined by: Dg= J(D84D16)

The gradation of the sediment is defined by: ug= D84/D16

These results indicate that some grain sorting has taken place during

the course of the experiments. The sediment of the bed layer has a relatively large amount of course particles.

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3.1.2 Fall velocity

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The fall velocity of the suspended sediment is determined in a settling

tube. This is a device to determine the fall velocity distribution of particles in a sample. At the lower end of the settling tube the

sediment partieles accumulate on a very sensitive weighing device. A cumulative weight distribution of the sample as a function of the

measuring time is obtained. This distribution is converted into the fall velocity distribution of the sample using the height of the settling tube (Slot and Geldof, 1986).

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A sample is extracted from the supply container and samples of suspended sediment are siphoned at cross-section 5. These are siphoned at the centerline 10, 30 and 50 mm below the water level. The sediment is gathered during 30 hours. The samples are dried and split into amounts that can be used in the settling tube.

Figure 3 shows the probability distribution of the fall velocity of sediment originating from the supply container.

The mean fall velocity, at 200C, of sediment originating from the supply container is: w - 0.0172 m/s. The mean fall velocity, at 200C, of

s

suspended sediment is: w - 0.0172 m/s. At higher temperatures the fall

s

velocity increases; 2% per °C. The sedimentation diameter is:

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D -

159 ~m. (Slot, 1983) s

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3.2 Flow conditions

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The flow conditions are given in table 3.la and 3.lb. The values of parameters determined by measurement are given in table 3.la. The values

of parameters obtained by calculation are given in table 3.lb. The Vanoni and Brooks (1957) correction method for side wall effects is not

applied because the parameters are hardly affected (W/aO

>

5).

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Table 3.la Measured parameters Table 3.lb Calculated parameters 0.0098 3 - V(WaO) - 0.28 [m/s]

~-

[m /s] u - 0.50 [m] _ -3 - 0.40 [g/l] W Ctr-(Qs/~)lO aO 0.070 [m] C - ü/J(aOi) 20.0 [mO.5/s] i 2.8 10-3[_] Fr - ü/J(gaO) 0.34 [

-

] D50- 160 [Jjm]

e

- aOi/(6.D50) 0.74 [

-

] 17.2 -3 0 (üjg)/C 0.044 [m/s] w 10 [m/s] (20 C susp.) u = s * Q_s - 3.9 [g/s] Ds - 159 [Jjm] (susp.,sec. 3.1.2) T 20 [OC] Z - ws/({3ltu*) 0.61 (sec. 4.2.2)

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The bed forms are classified as ripples (Talmon and de Graaff, 1989)

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4. RESULTS

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4.1 Depth measurements

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4.1.1 Mean depth

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The ensemble relative water depth of the la measuring sessions are tabulated in appendix A. Figure 5 shows the ensemble averaged contour line map of the relative water depth (normalized with the mean water depth of cross-section 1 to 5). The contour lines are drawn at intervals of ba/aa = 0.2. The relative depth, at 0.3 W, 0.5 Wand 0.7 W, as a

function of longitudinal distance is depicted in figure 6. Figures 7a to 71 show the ensemble averaged flow depths of each cross section.

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A maximum of the transverse bed slope occurs at cross sections 12 to 14. Downstream of this maximum the slope converges to a constant value

independent of the longitudinal coordinate.

The bed topography of the bend is characterised by a damped oscillation of the transverse bed slope in downstream direction. In the region of cross section 30 to 45 the bed is axi-symmetrical.

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4.1.2 Bed form statistics

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-

---The bed consists of bed forms moving downstream. ---The height of the bed forms is a significant fraction of the flow depth. These bed forms cause a significant form drag. This is reflected in the low Chézy value; C z 20 mO.s/s. The large dimensions of the bed forms also affect the choice of reference level, i.e. the level above which the sediment is

considered to be transported as suspended load and below which the sediment is considered to be transported as bed-load transport.

To guide the choice of reference level the probability distribution of bed form height is calculated. This is achieved as follow: In a selected

region of the flume, the data of all individuallocal depth measurements is gathered and normalized with their local ensemble averaged value: a'/a. (at each location la data points are available.)

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Two regions have been selected, each possessing local ensemble averaged water depths about equal to aO·

The inflow section, cross section 1 to 5; 450 data points

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_{The centerline of the channel} 480 data points

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The probability distribution of the water depth of both regions are shown in fig. 8a. Both distributions are very similar. These

distributions, assuming steady state of the bed, equal the bed form height distributions. In fig. 8a also the 5% and 10% exceedance levels

of bed form height are indicated. These are within the range: 0.15a to 0.20a.

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The bed form statistics are also calculated in the region of cross-section 30 to 45, fig. 8b. The results are similar to that of run no. 2 and run no. 3.

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4.2 Concentration rneasurernents

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4.2.1 Mean concentration

----

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The mean concentrations are tabulated in appendix B.

The figure 9 show the concentration profiles of the cross-sections 1 and the figures lla to lld shows the concentration profiles of respectively the cross-sections 35 and 40.

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4.2.2. Curve fit of equilibrium concentration profile

---I

The straight reach prior to the bend entrance serves to estab1ish flow and sediment conditions which are in equilibrium with the local

conditions, i.e. the flow and concentration fields are independent of streamwise coordinates. The length of this reach is sufficient (Ta1mon and Marsman, 1988).

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To establish the va1ues of parameters of the concentration vertica1 at equilibrium conditions the measurements in the straight reach are used

(cross-section 1).

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The Rouse concentration profile is fitted with the measurements. This profile is based on the parabolical function for the turbulent exchange

coefficient over the vertical.

The parameters of the concentration vertical are: the choice of reference height z_rja

the concentration at reference height c_r

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The Z parameter, wsj(P~u*)

The concentration profile is given by:

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(4.1)

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Curve fitting has been performed with the aid of a computer program which, given z , estimates the Zand c parameters of eq.(4.l). A least

r r

squares method is employed. Results are given in table 4.1. About 5% of the time the bed form height is larger than 0.20a see fig. Sa. The root mean square value of the bed form height is 0.12a. Therefore a reference height of

>

0.15a should be appropriate. The curve fit of the

:::::

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concentration data at cross-section 1 is given in fig. 10, a reference height of zrjaO=0.15 is applied. The relevant parameters are given in

table 4.1.

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Table 4.1 Parameter sets of the equilibrium concentration profile c [gjl] r Z [-]

ë

[gjl]

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0.15 0.20 0.79 0.64 0.61 0.61 0.25 0.20

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cross-sec. 1

The estimated Z parameter of the concentration vertical is: Z=0.6l. The standard deviation is: GZ- 0.09. The reference concentration will vary with the choice of reference level. The depth-averaged concentration

given in table 4.1 is the integral of the concentration curve eq. (4.1)

, section 4.2.3.

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21

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4.2.3. Depth-averaged concentrations

----------------------------

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The results of the experiment will be used to test depth-averaged

mathematical modeIs. To that purpose depth-averaged values of concentration have to be computed. The depth-averaged value of the concentration is defined by:

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_{c -} _1_ a-z r c dz (4.2)

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z r

local flow depth

reference level, close to the bed with: a

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The choice of reference level is uncertain. This level will be locatednear the top of the bed forms. Concentration measurements below zja

<

"'"

0.10 were troubled by the presence of bed forms. Consequently depth-averaged concentrations have been computed for z_rja - 0.10, 0.15 and 0.20.

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The depth-averaged concentration of a vertical is computed by:

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c - _1_ jmax jmax b j-l c. J (4.3)

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with j the number of measurements above z

max r

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For a very large number of data points, uniformly distributed over the depth, the summation series converge to the definition (4.2). The available number of data points is, however, limited. Measurements are taken with a vertical increment in vertical direction of 5 mmo At each x,y,z location two or more measurements have been performed.

The depth-averaged concentration data, for z ja - 0.10, 0.15, 0.20 and

r

0.25 are given in table 4.2. According to this procedure the depth-averaged concentration in the entrance section assuming z ja_r -0.15, is c

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- 0.27 gjl.

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22

Table 4.2 Depth-averaged concentrations in the 180 degree bend

cross-sec. no. c (1/4 W) c (2/4 W) c (3/4 W) 1 0.2588 0.3190 0.3433 reference level at: 35 0.1706 0.3365 0.3040 z la = 0.10 40 0.2181 0.2968 0.3391 r 1 0.2348 0.2880 0.3011 reference level at: 35 0.1706 0.3365 0.2906 z la - 0.15 40 0.2181 0.2968 0.3236 r 1 0.2348 0.2880 0.3011 reference level at: 35 0.1706 0.3340 0.2575 z la - 0.20 40 0.2181 0.2759 0.3047 r 1 0.2152 0.2658 0.2906 reference level at: 35 0.1706 0.3134 0.2248 z la - 0.25 40 0.2181 0.2623 0.2803 r

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4.2.4 The concentration field at cross-sections 35 and 40

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----

---The bed topography of the experiment is more damped than in the preceeding experiment: run no. 3. In run no. 2 cross-section 40 was

considered axi-symmetrical. Extended measurements of the concentration field at cross sections 35 and 40 are performed. The bed-topography indicates that the axi-symmetrical case has been reached at these locations. The concentration fields could probably be considered as close to axi-symmetrical.

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In fig. lla to lid the concentration verticals of cross sections 35 and

40 are given. The concentration verticals have been measured at 1/8,

2/8, 3/8, 4/8, 5/8, 6/8 and 7/8 of the channel width.

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An iso-concentration contour representation of the concentration field

at cross-sections 35 and 40 is given in figure lle. The contour plot is made by linear interpolation between the data points. The contour

interval is 0.10 gil.

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The lowest concentrations are found in the inner part of the bend. In

the upper part of the flow up to Y - 0.7 W the concentrations remain

almost constant in transverse direction (a slight increase is noticed).

In the region Y

>

0.7 W the concentrations decrease with Y. The near bed

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23

concentration in the inner part of the bend is circa 1/2 of the near bed concentration in the mid region of the bend.

24

5 DISCUSSION

5.1. Introduction

The general purpose of the experiment is to provide data on which numerical and ana1ytica1 morpho1ogical models, inc1uding suspended

sediment transport, can be calibrated and verified.

Important input parameters of morpho1ogical mode1s are:

The percentage of suspended sediment transport The shape of the equilibrium concentration profile A transport formula

These subjects are discussed in sections 5.2, 5.3, 5.4 and 5.5.

Adaptation lengths of flow, bed level and concentration are calculated

in sec. 5.6. The bed topography is discussed in sec. 5.7. A1so a

mathematical approximation of the bed topography is given. The

concentration field at cross-sections 35 and 40 is discussed in sec.

5.8.

5.2. The Z parameter

Curve fitting of the concentration profile prior to bend entrance yields a Z parameter of 0.61 (sec. 4.2.2.). The Z parameter is defined by: Z WS/(~KU*).The

Z

parameter is a measure of the ratio of the downward

flux by the fall velocity wand the upward flux by turbulent diffusion.

s

Turbulent diffusion of sediment is modelled by:

vtc ~ v ,with v turbulent diffusion of momentum

tm tm

v turbulent diffusion of mass (sediment) tc

It is generally accepted that the turbulent diffusion coefficient of mass is greater than of momentum (Csanady 1973). Consequently ~>1. In

the experiment, upstream of the bend entrance the bed shear velocity is equal to

u*-

0.044 mis while the fal1 velocity of the suspended sediment

is: w _ 0.0172 mis (from the supply container). This yields ~ = 1.6

s

·

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~

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-

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25

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Based on a large data set van Rijn (1982) has calculated ~ by fitting the data with concentration verticals which are based on a parabolical-constant profile for the turbulent diffusion coefficient vtc· (The present curve fitting is based on a parabolical profile for v_tc). For ws/u*- 0.0172/0.044 - 0.39 van Rijn reports effective ~ values of 0.7

and 1.8 for the experiments of Coleman (1970).

Hinze (1959) reports values of the turbulent Prandtl number Pr_turb- l/~ of 0.65 to 0.72 (~-1.4 to 1.5) for various measurements on the

distribution of heat and matter in pipe flow and two-dimensional channels.

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_{5.3. Percentage of suspended sediment transport}

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The percentage of suspended sediment transport upstream of the bend is an important physical parameter in the experiment.

The division between bed and suspended load transport is somewhat arbitrary and is effected by the choice of reference level. The amount of suspended sediment transport per unit width is defined by:

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z S s s sus

z

r Two methods u c dz (5.1)

will be employed to estimate the suspended sediment

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transport:

1 _ Based on curve fitting of the concentration profile upstream of the bend entrance. By integration of the product of the mathematical

functions of u and c, over the suspended load region, the suspended sediment transport is calculated.

2 - Based on an estimate of the depth-averaged concentration, multiplied by the depth-averaged velocity.

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26

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The suspended sediment transport rate per unit width is equa1 to:

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Method 1 u c

_J

Zs r r_{u c}dz = z r shape 1 (aO-z

)ü

ë

J

r r

dr

-

(a -z

)ü

ë

Q r 0 uc 0 r s

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S_{s sus} (5.2) with: r ,r u c

The total transport rate per unit width is equa1 to:

functions of velocity and concentration

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(5.3)

in which: c

tr

the transport concentration defined by eq.(5.3)

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The resu1ts for zr/aO= 0.15 (using the mean concentration determined by

the curve fit of the concentration, tab1e 4.1) are given in tab1e 5.1

are given in tab1e 5.1.

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Tab1e 5.1 Fraction of suspended sediment transport in cross section 1,

by method 1

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Z=0.61 c [g/l] Ss sus/Stot 0.25 0.53 0.20 0.40

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Q = 1 s

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Method 2

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The suspended sediment transport per unit width is approximated by:

S

=

1

__

J

Zs u dz

s sus z-z

s r z

r

The depth-averaged concentration c is computed by the method outlined in

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Zs c dz

=

z r

I

(z -z ) s r u c (5.4)

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subsection 4.2.3. Dividing the suspended sediment transport by the tota1 sediment discharge at channe1 exit, yie1ds the fraction of suspended

sediment transport (tab1e 5.2).

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27

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Table 5.2 Fraction of suspended sediment transport, in cross section 1, by method 2

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z la r 0.15

o

20 ë[g/l] 0.27 0.27 Ss sus/Stot 0.57 0.54

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Both methods involve some disadvantages.

Method 1 is based on curve fitting of the concentration profile. This fitting will be affected by the non-homogeneous distribution of

measuring points in the vertical. Consequently the integral of the concentration profile will be affected also, even though by integrating the profile all points in the vertical are weighed equally.

Method 2, which yields a rough estimate of the depth-averaged

concentration, favours the region were many measuring points are taken. In computing the depth integrated suspended transport the shape of the concentration and velocity profiles are neglected.

For z la=0.15 the percentage of suspended sediment transport is 50..60%. r

For z la-0.20 the percentage of suspended sediment transport is 40 ..50%. r

It is concluded that the percentage of suspended transport in the experiments is within the range: 40..60 % .

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_{5.4 Transport formulae}

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To simulate the experiment numerically or analytically a transport formula is necessary to predict concentration and sediment transport rates. In this section the overall transport rate of the experiment is compared with some transport formulae known from literature. It is common practice to express the total sediment transport rate in the transport concentration: ëtr-

Qs/~

(Stot- ëtrü aO

[

gim

i

s

]).

The measured transport concentration is equal to: ëtr- 0.40 gil.

The sediment transport in the experiment is about 1/4 of the transport in run no. 2 which has a comparable bed topography.

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The transport formulae of Engelund and Hansen (1967)(1973), Brownlie (1981) and Van Rijn (1984c) will be evaluated, Ackers and White.

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28

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These formu1ae are often emp10yed outside their range of app1icabi1ity, yie1ding reasonab1e resu1ts. The Ackers White and Brown1ie formu1ae are based on data sets which inc1ude data of laboratory f1umes with fine

sediments.

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1-f with

e

= di ,~- S 3' ~DsO J(~gD ) (s.6a) or: (s.6b)

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The predicted transport concentration is: c_tr - 1.07

g

il

(for DsO the va1ue of the supp1y container is used)

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The Ackers White formu1a reads:

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c =

tr

(5.7)

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with: F 1 un ( u )l-n= 0 58

gr J(~gDsO)

*

J32

log

(10aO/DsO)

.

A -

0.23

/J

D

+ 0.14 0.254 gr n - 1.00 - 0.56 log D = 0.660 gr m

9

.

66

/

D

+1.34 = 3.73 gr 2 C = 10(2.86 log Dgr- log Dgr- 3.52) 0.00705 Dgr-

Dso

(

~g

/

v2

)

1/3

4

.

05

According to White (1972) the formu1a is fitted to data for which no side wa11 correction method has been emp1oyed, i.e. d=aO' This yie1ds a

transport concentration equa1 to: c_tr- 0.36

gil

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The Brown1ie formu1a reads:

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ë

-7115 (F _ F )1.978 iO.6601 (

rb/D50)-0.3301

tr g gO [mg/1] (5.8)

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with: F

-

u grain Froude number J(~gDsO)

g

F -

4.5

96

e

0.52

9

3

.-0.1405 -0.1606 _{critica1 grain Froude number}

1 a

gO cr

(10)-7.7 y g

e

- 0.22 Y + 0.06 critica1 Shie1ds number cr

<J~

R )-0.6 Y =

3g

R

-

J(

g

D

s

O)lv

grain Reyno1ds number

g

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29

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rb - 0.070 [m], hydraulic radius related to the bed according to

Vanoni and Brooks (1957), here aO is used.

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Prediction with this formula yields: c_tr 0.21 gil

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The Van Rijn (1984c) formulae read:

u-ucr 2.4 1 2

ctrb- -, O.OOs(J(<YAD))

(

DsO/aO)

·

Er" 50 (s.9a) bed-load:

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(s.9b) totalload:

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with: d*-

DsOJ(~g/v2)

ucr-0.19 Ds~·l log (12rb/(3D90))- 0.248 mis

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The transport predicted with these formulae is:

ë

tr= O.

This is caused by a small difference between u - u_cr~0.032 mis.

Exept for Ackers

&

White none of these transport formulae predicts the actual transport concentration of the experiment.

The difference between the experiment and predicted by Ackers

&

White formula is about 10X. Like run3 the Brownlie formula underpredicts the transport concentration by a factor 0.5.

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Prediction of the ratio of suspended-load and total-load ean be

accomplished by the equations of Van Rijn eq.(s.9a,b). Due, however, to u

>

u this is impossible.

cr

Van Rijn (1984b) has calculated the ratio of suspended-load and total-load of measurements reported by Guy et.al. (1966). It is noticed that

for

u*/ws

>

3 more than sOX suspended-load is present. This is in

accordance with the results of the experiment:

u*/w

_s

-

2.6, SISt_{s sus 0}t=

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0.50

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The performance of the transport formulae with regard to this experiment is comparable to the performance of the formulae in case of the

preceeding suspended load experiments.

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30

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5.5. Bed-shear stress and sediment transport

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In case of a dune covered bed the bed resistance consist of bed shear stress (friction drag) and of a pressure gradient generated by the dunes

(shape drag). The total drag (which actually consist of friction and

shape drag) is defined by: T-pgai

The process of sediment transport is caused by the shear stress acting on the grains. The shear stress related to sediment transport is given

by: T'=~T

in which: ~ - efficiency factor

T'- effective grain-shear stress

T - total drag.

To initiate sediment transport the shear stress has to exceed a critical

value: T

cr

In the experiment ~ is unknown.

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One of the reasons of the poor performance of the transport forrnulae could be caused by the relatively high resistance ( Cz20 mo.s/s). The data on which the transport formulae have been developed generally relate to less resistance ( C>30 mo.s/s ). The transport forrnulae

z

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implicitly, or explicitly, contain the ratio of friction and total drag. This ratio could differ under the present conditions (the relatively

large bed form height is quite exceptional). Consequently the effective

grain-shear stress will differ also.

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In the following sediment transport related parameters ~ and 0_cr are estimated with the aid of some empirical formulae known from literature.

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The transport formulae which incorporate the critical bed-shear stress

are generally proportional with:

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~T-T

~O-O

(l-B)b ( cr)b ( cr)b T 0 B cr cr or: (F -F O)b u - u _b (ç_

JO)b

cr b (J(~D

»

- (l-JB) g g ₅₀ Jg (s.lOa)

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(s.lOb)

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T

1'n Wh1·ch.·B - ~T~ ' mob'1 1ty pl' arameter (s.lOc)

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31

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Three methods are used to estimate B. The methods are:

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1)- The set of transport formulae by Van Rijn (1984c), eq.(5.9a,b), is used to relate the total transport concentration c_tr and the B parameter. Substitution of the calculated c_tr value yields B. 2)- The bed load transport formula by Van Rijn (1984a), eq.(5.l0) is

used to relate the bed-Ioad transport concentration and the B parameter. Substitution of the calculated ctrb value yields B.

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c_trb- ~_{a u} 0.053 J(t.g)

[

gil

]

(5.11)

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3)- Arelation to estimate the critical Froude grain number by Brownlie

(1981) is used.

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F _ 4.596

e

0.5293 i-0.1405 gO cr -0.1606 (J g (5.12)

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This relation has been obtained by Brownlie by manipulation of an empirical function which was derived to predict the flow depth.

(The Brownlie depth prediction for this experiment is 140 % too large). With the aid of eq.(5.l0b) B is calculated.

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According to the Shields diagram the critical Shields number of the

sediment is:

e

-

0.06. cr

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The methods are applied to the data of run no. 4. The resu1ts are given in tab1e 5.3. A median grain diameter of dSO= 160 pm is applied.

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Tab1e 5.3 The mobi1ity number B

run no. 4 method 1 method 2 method 3

B 0.28 0.34 0.36

p (at

e

-0.06) 0.29 0.24 0.23

cr

remark 50 % susp. depth prediction

trans~. 65 % too sma11

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The third method, Brownlie's method, is c10sely related to Brownliewater depth prediction. Considering the large error in the depth 's

32

prediction the estimate of ~ is questionable. The results of the first two methods are comparable. The ~ parameter is calculated by eq.(5.l0c).

The ~ parameter of the 90 ~m experiments, run no. 1 to 3, is within the range: 0.3

< ~ <

0.4. For run no. 4 the ~ value is within the range:

0.25

< ~

<

0.3. The van Rijn (1984a) model for ~, which is applied in the Van Rijn transport formulae, yields a distinct result: ~ = (C/C,)2=

(20/60)2= 0.11. These results indicate that the estimate of ~,

implicitly or explicitly contained in the transport formulae, could be erroneous.

The estimated value of ~ indicates that about 30% of the total drag is

available for sediment transport.

5.6 Adaptation lengths

In order to formulate mathematically the interaction of flow and sediment adaptation lengths of flow velocity, bed level and

concentration have been defined: Struiksma et.al. (1986) and Olesen (1987). These adaptation lengths are defined as follows:

adaptation length of flow:

C2 _(5.l3a)

À =- aO

w 2g

adaptation length of bed level: À =

L

(H

)2

1

a (5.13b)

s 2 aO G

1f

adaptation length of concentration: À_c:::::aü/ws (5.l3c)

in which: G - coefficient of the gravitational term in the bed-load

sediment direction model

The adaptation lengths for flow and bed level in the experiment are:

À w À s The 1.43 m = 0.24 m (for G-l.5) = 0.52 m (for G=0.7)

adaptation length of concentration depends mainly on the choice of

boundary condition for the concentration at reference level (Talmon, 1989b). The adaptation length depends further on the value of the Z parameter, the reference height and the Chézy value. The adaptation

lengths are calculated based on the assumption of a logarithmic velocity profile and a Rouse distribution for the concentration. To this purpose

software which is used in Talmon (1989b) has been employed. Curve fitting of the concentration profile yields: Z = 0.61

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The Chézy value of the experiment is about: C = 20 mO.Sjs

The reference height should be chosen near the top of the dunes,

consequently z will be in the range: 0.1 < z ja <0.2, (fig. 8ab)

r r

Taking into account these ranges, the adaptation length of the concentration becomes approximately:

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In case of the gradient condition:

À - 0.2 m

c

À - 0.5 m

c

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In case of the concentration condition:

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_{5.7 Bed topography}

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The stationary bed topography in the 180 degree bend is depicted in fig.

5. A maximum of the transverse bed slope occurs at cross sections 12 to 14. At this location a point-bar is present in the inner part of the bend. A pool is present in the outer part of the bend. Further

downstream from cross-sections 20 to the end of the bend no significant change of this slope occurs. The bed topography of the bend is

characterized by a damped oscillation of the transverse bed slope in downstream direction. The bed topography is comparable with run no.2

(Talmon, 1989a).

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An analytical approximation of the bed topography can be formulated by:

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, A iks A i~n

g - (ae - ilal)e

aO A

with: a complex amplitude (including a phase shift of the harmonie

(5.14)

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s coordinate in streamwise directiooscillation with regard to the bennd entrance)

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n coordinate in transverse direction

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~_

wfW

wave number in transverse direction k complex wave number

A i~n

The - ilal e term yields the axi-symmetrical bed topography

(sinusioidal). Fitting equation (5.14) to the measured bed topography

(cross section 12...45) yields:

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2w

re(k)- ~4- 0.75 im(k)

=

0.21

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34

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im(k) and a are difficult to estimate, consequently the accuracy is limited. These results indicate a wave length of oscillation of 8.4 m,

-1

and 63% damping (e ) at s = 5 m.

These results are close to the wave length and damping of run no 2. (difference: z 20 I).

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5.8 Concentrations in cross-sections 35 and 40

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The concentration data at cross-sections 35 and 40 is given in fig. lla and fig. lld. The iso-concentration contour line representation, in fig.

lle, will be used to discuss the relevant physics.

In a straight reach the balance is between vertical turbulent diffusion

and the fall velocity, while boundary conditions determine the concentration levels.

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In the inner part of the bend the concentrations are expected to be low because of smaller bed shear stresses. The results depicted in fig. lle confirm this. In the outermost part of the bend, beyond YjW-0.8, the concentrations decrease as well in the upper as the lower part of the flow. The decrease of concentrations could be caused by an additional secondary flow (Taylor-Gortler) cel due to the presence of the convex

wall. The same effect is noticed in run no. 2 and 3.

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35

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6 CONCLUSIONS

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The bed topography and sediment concentrations have been measured in a 180 degree curved flume. The median diameter of the sediment is 160 ~m.

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The main features of the experiment are:

The stationary bed topography displays over- and undershoot effects due to the abrupt change of curvature at the bend entrance. The bed topography is characterized by a damped oscilation of the

transverse bed slope. The water depth is nearly the same, but the sediment transport rate in run no. 4 is 3.5 times larger.

The bed topography is similar to that of run no.2 (90 ~m sediment).

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The following parameters characterize the experiment. The Chézy value is about: C - 20 mo,s/s

With the aid of curve fitting the Z parameter of the equilibrium concentration profile is estimated to be: Z ~0.6l

Due to the exaggerated bed form dimensions the reference height

should be chosen within: 0.1

<

z

la

<

0.2

r

The percentage suspended sediment transport is about 50 % .

,---..

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In view of an analytical and numerical simulation of the experiment the

following has been investigated:

Transport formulae yield divergend results.

Adaptation lengths of flow velocity, bed level and concentration

have been calculated.

The measured bed topography is approximated by an analytical expression incorporating harmonie oscillation and damping.

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The experiments run no. 2 and 4 provide data on the axi-symmetrical region of suspended sediment transport in river bends. The difference is the median sediment diameter. The transverse bed slope in run no. 4 is smaller than in run no. 2, nearly a factor 2.

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36 REFERENCES

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Ackers, P. and W.R. White, 1973, Sediment transport: a new approach and ana1ysis, Journalof the Hydrau1ics Division, ASCE, vol. 99,

no. HY11, pp. 2041-2060

Brown1ie, W.R., 1981, Prediction of flow depth and sediment discharge in

open channe1s, W.M. Keck Laboratory of Hydrau1ics and Water Resources, Ca1ifornia Institute Of Techno1ogy, Pasadena Ca1ifornia, rep. no. KH-R-43A

Co1eman, N.L., 1970, F1ume studies of the sediment transfer coefficient Water Resources, Vol 6, no 3.

Csanady, G.T., 1973, Turbulent diffusion in the environment, D. Reidel Publishing Co., Dordrecht, the Netherlands

Delft Hydrau1ics, 1986, Optica1 concentration meter, modelOPCON,

Technica1 manua1

Enge1und, F. and E. Hansen, 1967, A monograph on sediment transport in a11uvia1 streams, Teknisk For1ag, Copenhagen, Denmark, pp. 62 Guy, H.P., D.B. Simons and E.V. Richardson, 1966, Summary of a11uvia1

channe1 data from f1ume experiments, 1956-1961, Geo1ogica1 Survey Professional Paper 462-1, Washington, D.C. pp. 93 01esen, K.W., Bed topography in sha110w river bends

Doctoral thesis Delft University of Technology, 1987 (also: ISSN 0169-6548 Communications on Hydraulic and Geotechnica1 Engineering, Delft University of Technology, Faculty of Civil Engineering)

Rijn, L.C. van, 1984a, Sediment transport, part I: bed load transport,

Journalof Hydrau1ic Engineering, Vol IlO, no. 10, pp. 1431-1456

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Rijn, L.C. van, 1984b, Sediment transport, part 11: suspended load transport, Journalof Hydraulic Engineering, Vol 110, no. 11, pp. 1613-1641

Rijn, L.C. van, 1984c, Sediment transport, part 111: bed form and

alluvial roughness, Journalof Hydraulic Engineering, Vol 110, no. 12, pp. 1733-1754

Rijn, L.C. van, 1987, Mathematical modelling of morphological processes in the case of suspended sediment transport

Doctoral thesis Delft University of Technology, 1987 (also: Delft Hydraulics Communication no. 382)

37

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Slot, R.E. and H.J.Geldof, 1986, An improved settling tube system for sand. ISSN 0169-6548, Communications on Hydraulics and Geotechnical Engineering, Delft University of Technology,

Faculty of Civil Engineering, rep. no. 86-12,

Struiksma, N.; K.W. Olesen, C. Flokstra and H.J. de Vriend, 1985, Bed deformation in alluvial channel bends. IAHR, Journalof Hydraulic Research, vol. 23, no. 1, pp. 57-79

Talmon, A.M., 1989a, Suspended-load experiments in a curved flume, run no.2, Delft Univ. of Techn., Dept. Civil Eng., rep. no. 4-89 Talmon, A.M., 1989b, A theoretical model for suspended sediment

transport in river bends, ISSN 0169-6548, Communications on Hydraulic and Geotechnical Engineering, Delft University of Technology, Faculty of Civil Engineering, rep. no. 89-5 Talmon, A.M. and J. de Graaff, 1989, Suspended-load experiments in a

curved flume, run no.3, Delft Univ. of Techn., Dept. Civil Eng., rep. no. 7-89

Talmon, A.M. and E.R.A. Marsman, 1988, Suspended-load experiments in a curved flume, run no.l, Delft Univ. of Techn., Dept. Civil Eng., rep. no. 8-88

Vanoni, V.A. and N.H. Brooks, 1957, Laboratory studies of the roughness and suspended load of alluvial streams, Rep. no. E-68,

Publication no. 149, California Institute of Technology Pasadena, California, pp. 121

White, W.R., 1972, Sediment transport in channels: a general function, rep. INT 104, Hydraulics Research Station, England

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Al

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Appendix A: Ensemble averaged water depths.

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In this appendix the ensemble averaged relative water depths of the 10

measurements are tabulated.

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Re1ative mean water dep th a/aO· (aO - 0.070 m.)

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fromside of bendinner CSOl eS02 eS03 eS04 eS05 eS06 eS07

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0.05 1.04 0.93 1.02 1.01 l.04 1.03 0.90 0.10 1.05 0.91 0.99 1.06 1.13 1. 03 1.00 0.15 1.01 0.91 0.95 0.99 1.08 0.99 0.96

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0.20 0.94 0.99 0.99 0.98 0.97 1.01 0.93 0.25 0.98 0.99 0.98 1.01 0.96 1.06 0.98 0.30 0.99 1.07 1.01 0.98 0.98 1.05 0.98

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0.35 1.03 l.03 1.03 0.99 0.97 1.02 1.03 0.40 1.07 0.99 0.98 1.02 0.95 0.96 1.05 0.45 1.05 0.96 0.97 1.00 0.97 1.01 1.07

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from inner

side of bend eS08 eS09 eS10 eS11 eS12 eS13 eS14

I

0.05 0.87 0.71 0.69 . 0.62 0.53 0.57 0.58 0.10 0.88 0.81 0.74 0.68 0.68 0.60 0.65

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0.15 0.96 0.90 0.86 0.74 0.73 0.76 0.67 0.20 1.01 0.99 0.92 0.86 0.86 0.79 0.74 0.25 1.01 1.08 1.03 1.04 1.12 1. 02 0.99

I

0.300.35 01.05.99 1.111.08 1.201.19 1.111.18 l.15l.19 1.161. 24 1.231.30 0.40 l.13 l.13 1.29 1.35 l. 39 1. 37 1.34

I

0.45 1.23 1.24 1.43 1.46 l.50 1.48 1.41 from inner

I

side of bend eS15 eS16 eS17 eS18 eS19 eS20 eS21

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0.05 0.65 0.58 0.67 0.65 0.70 0.68 0.74 0.10 0.67 0.72 0.71 0.64 0.78 0.75 0.77 0.15 0.68 0.74 0.80 0.71 0.77 0.82 0.84

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0.20 0.85 0.82 0.87 0.82 0.85 0.86 0.87 0.25 1.06 0.95 0.97 0.93 0.97 1.00 0.96 0.30 1.18 1.09 1.10 1.08 l.04 1.05 1.11 0.35 1.20 1.16 1.16 1.10 1.03 1.03 1.15

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0.40 1.30 1.29 1.20 1.25 1.17 1.15 1.20 0.45 1.39 1.39 1.32 1.34 1. 30 1.28 1.19

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I

A2

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I

Re1ative mean water dep th a/aO' (a -0.070m.)

I

from inner

I

side of bend CS22 CS23 CS24 CS25 CS26 CS27 CS28 0.05 0.77 0.80 0.76 0.69 0.73 0.72 0.69

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0.10 0.81 0.77 0.81 0.82 0.74 0.85 0.70 0.15 0.92 0.86 0.82 0.78 0.77 0.81 0.77 0.20 0.92 0.88 0.88 0.84 0.83 0.89 0.87

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0.25 0.93 1.04 1.01 1.06 0.96 1.04 1.07 0.30 1.03 1.04 1.04 1.07 0.96 1.11 1.15 0.35 1.04 1.09 1.06 1.07 0.98 1.14 1.16

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0.40 1.15 1.21 1.19 1.14 1.11 1.18 1.10 0.45 1.18 1. 25 1.24 1.20 1.27 1.27 1.24

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from inner side of bend CS29 CS30 CS31 CS32 CS33 CS34 CS35

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0.05 0.72 0.79 0.75 0.78 0.72 0.67 0.71 0.10 0.73 0.77 0.80 0.79 0.78 0.73 0.73

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0.15 0.84 0.82 0.79 0.92 0.79 0.86 0.80 0.20 0.84 0.84 0.91 0.92 0.87 0.92 0.88 0.25 0.97 1.00 1.05 1.04 1.04 1.01 1.03 0.30 1.00 1.05 1.08 1.08 1.09 1.08 1.14

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0.35 1.05 1.04 1.07 1.07 1.14 1.09 1.10 0.40 1.16 1.18 1.05 1.09 1.15 1.13 1.18 0.45 1.28 1.34 1.17 1.23 1.19 1.16 1.21

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from inner side of bend CS36 CS37 CS38 CS39 CS40 CS41 CS42

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0.05 0.75 0.67 0.70 0.66 0.68 0.79 0.70

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0.10 0.69 0.75 0.76 0.80 0.78 0.74 0.80 0.15 0.73 0.82 0.85 0.81 0.87 0.80 0.82 0.20 0.86 0.85 0.91 0.86 0.85 0.89 0.94

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0.25 1.03 0.98 0.99 1.12 1.02 1.01 1.02 0.30 1.09 1.12 1.13 1.13 1.05 1.05 1.13 0.35 1.11 1.13 1.17 1.12 1.06 1.05 1.12 0.40 1.21 1.20 1.24 1.18 1.20 1.09 1.11

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0.45 1.27 1.23 1.37 1.28 1.31 1.20 1.21

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A3

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I

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Relative mean water depth a/aO· (a - 0.070 m.)

from inner side of bend CS43 CS44 CS45 CS46 CS47 CS48 0.05 0.66 0.73 0.76 0.73 0.80 0.92 0.10 0.78 0.79 0.72 0.79 0.84 0.91 0.15 0.86 0.82 0.82 0.81 0.85 0.86 0.20 0.93 0.96 0.89 0.93 0.94 0.94 0.25 1.05 1.04 1.01 1.04 1.03 0.99 0.30 1.12 1.12 1.09 1.14 1.12 1.01 0.35 1.13 1.15 1.13 1.15 1.13 0.99 0.40 1.19 1.09 1.20 1.21 1.06 1.05 0.45 1.28 1.18 1.25 1.23 1.07 1.04

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BI

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Appendix B: Concentration data

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Cross section I,

I

location Mean Distance

Concen-in cross- water beneath tration

direction depth water

I

surface

[y,IW1

[

mm1

[mm1

[g/1

1

218

70 5 0.080

I

10 0.060 0.090 15 0.160

I

20 0.100 0.150 25 0.250 0.200 0.130 30 0.150 0.230 0.220

I

35 0.430 0.270 0.160 40 0.340 0.260 0.220 45 0.370 0.220 50 0.430

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55 0.600 0.280 60 l.070 1.610 0.810 65 0.620

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70 l.450

4/8

70 5 0.050 0.080

I

10 0.110 15 0.140 0.100 20 0.180 25 0.240 0.200 0.170 .>

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-~ 30 0.290 0.260 35 0.300 0.420 0.240 40 0.360 0.340

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45 0.430 50 0.660 0.480 55 0.7l0 2.340 60 l.620 0.940

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65 0.990 70 l.360 1.910 6/8 70 5 0.050

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10 0.120 15 0.100

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20 0.180 25 0.190 0.150 30 0.150 0.340 0.270 35 0.240 0.210

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40 0.420 0.440 0.230 45 0.360 50 0.670 0.410 0.700

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55 0.490 60 0.750 1.670 0.740 65 0.880

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70 0.980

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I

I

B2

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Cross section 35.

Location Mean Distance

Concen-I

directin cross-ion waterdepth beneathwater tration

surface

I

[yjW]

[mm]

[

mm

]

[

g/l

]

1/8 50 5 0.040 0.040

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1015 00..080160 0.0300.090 20 0.070 0.140

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2530 00.250.270 0.1600.140 35 1.010 0.300 40 0.230

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2/8 54 5 0.040 0.050 10 0.060 0.070

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1520 0.1200.100 0.1200.120 25 0.230 0.180

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30 0.160 0.150 0.210 35 0.250 0.350 40 0.230 0.270 0.360 50 0.330

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3/8 60 5 0.020 0.050 10 0.070 0.100

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1520 0.1600.140 0.0700.170 0.0900.130 25 0.150 0.250 0.260 0.170 30 0.190 0.250 0.240 0.280

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35 0.300 0.560 0.370 0.250 0.380 40 0.400 0.390 0.280 0.380 45 1.190 0.610 0.340

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5055 1.2500.910 0.8001.040

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4/8 72 5 0.050 0.060 10 0.110 0.090 15 0.160 0.120 20 0.170 0.200 0.170

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25 0.230 0.280 30 0.250 0.340 0.240 0.260 0.240 35 0.410 0.400 0.440 0.460

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4045 00.600.310 00..340590 0.460 0.400 0.510 50 0.560 1.480 0.640

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55 2.400 0.930 60 2.330 0.410 65 3.020

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5/8 79 5 0.090 0.040 10 0.090 0.090 15 0.090 0.190

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2025 00.290.160 0.1900.160 30 0.350 0.330 0.260 0.250 35 0.240 0.430 0.380 0.390

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40 0.370 0.490 0.460 0.360 45 0.730 0.500 0.590

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B3 50 0.640 0.610

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55 0.690 1.090 60 0.950 1.750

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65 2.200 1.260 0.700 75 4.680 6/8 80 5 0.030 0.040

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10 0.070 0.070 15 0.080 0.110 20 0.130 0.120

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25 0.140 0.110 30 0.170 0.210 35 0.210 0.160 0.180 0.270 0.180 40 0.270 0.200 0.310

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45 0.310 0.260 0.360 50 0.380 0.400 0.340 55 0.490 0.510 0.410

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60 0.630 0.600 0.490 65 1.020 0.750 0.890 70 3.890 3.560 0.760

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80 5.120 2.220 7/8 84 5 0.020 0.010 0.000 10 0.020

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15 0.030 0.020 0.020 20 0.040 25 0.030 0.050 0.040

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30 0.060 35 0.050 0.070 0.060 0.060 40 0.120 0.080 45 0.090 0.190

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50 0.220 0.200 0.200 55 0.140 0.250 0.130 0.280 60 0.370 0.310 0.340

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65 0.410 0.680 0.220 0.220 70 1.490 0.500 4.720 75 2.270 0.420 3.770

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80 0.910 85 1.250 1.680 Cross section 40.

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Location Mean Distance Concen -in cross- water beneath tration direction depth water

I

surface

[y!W]

[mm]

[

mm

]

[

g

/

l

]

1/8 49 5 0.030 0.050

I

10 0.090 0.080 15 0.080 0.070

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20 0.120 0.140 25 0.110 0.120 30 0.230 0.210

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35 0.160 0.180 40 0.760 0.830 2/8 58 5 0.020 0.050

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10 0.130 0.050 15 0.070 0.110 20 0.190 0.150

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25 0.120 0.180

I

B4

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30 0.300 0.270 35 0.220 0.250

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_3/8 60 405 00..500160 00.050.880

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10 0.070 0.050 15 0.100 0.120 20 0.110 0.140 25 0.170 0.250 0.180

I

3035 00..250230 00.390.210 0.240 40 0.420 0.350 0.330

I

45 0.650 50 0.460 55 2.160 2.760 60 1.470

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4/8 72 5 0.070 0.040 10 0.090 0.140

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1520 00.100.140 0.1000.260 0.170 25 0.220 0.250 0.180

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30 0.210 0.410 0.150 0.260 0.280 35 0.310 0.250 0.330 40 0.290 0.140 0.350 0.460 0.560 0.220 45 0.480

I

5055 0.2300.700 0.540 0.550 0.350 60 1.060 0.330 0.930

I

70 0.470 ".- --5/8 74 5 0.140 0.090

-

---I

10 0.150 0.120 15 0.160 0.220 20 0.220 0.210 25 0.330 0.250

I

3035 0.3100.420 0.3300.320 00..300310 40 0.490 0.380 0.420 0.530

I

45 0.330 0.420 50 0.750 0.720 55 0.590 0.500 60 0.680 1.410 1.030

I

65 1.360 1.060 70 3.360 2.950 75 1.660

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6/8 79 5 0.040 0.030 10 0.060 0.050

I

15 0.080 0.130 20 0.100 0.110 25 0.120 0.230 30 0.220 0.150 0.170

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4035 0.2100.340 00.400.310 0.0.130060 0.330 0.300 0.210 45 0.450 0.780

I

50 0.480 0.410 0.450 0.560 55 0.780 0.450 60 2.430 0.720 1.050 0.620

I

65 3.560 0.930 70 0.850 3.640 1.280

B5 7/8 90 5 0.020 10 0.020 0.020 0.020 15 0.030 20 0.040 0.040 0.030 25 0.050 30 0.050 0.040 35 0.060 40 0.080 0.070 0.110 0.070 45 0.190 0.200 0.170 50 0.150 0.140 55 0.280 0.260 0.210 60 0.220 0.220 0.180 65 0.690 0.760 0.380 70 0.290 1.810 0.320 75 0.910 0.580 0.640 80 2.010 0.410

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E

0 0

E

...:

-

0 [' u:j

-,..-E

o M Ó

---

---

---W= 0.50

m \

--

-LAYOUT, LFM

CURVED

FLUME

DELFT

UNIVERSITY

OF TECHNOLOGY

FIG. 1

~ c

o

-

Ol C

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I

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I

I

I

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I

I

I

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I

I

I

99.99 99.9

/,

suspended sediment

I

I

bed layer 99

II

98

/1

95

II

90

Ij

80

/

/

/1

50

II

II

I

I

I

,..

_/

1/

al s:: 20

.~

1/

.-....

al bi) Ol 10 ~

/1

s:: al <J

,..

al 5

P-I /

2

/

1

I

0.1 q 0.01 0.05 0.10 0.20 0.50 1.00 grain size in mm

-S

I

EVE

C

U

RVES

OF

SE

DI

MENT

FIG. 2