suspended load transport_seminar nit surat

A

GRADUATE REPORT

On

SUSPENDED LOAD TRANSPORT IN ALLUVIAL STREAM FOR NON-UNIFORM

SEDIMENTSPREPARED BY

KODINARIYA ANKIT MULJIBHAI

Of

M.TECH. (Sem. III), WATER RESOURCES ENGINEERING.

(Roll No.: P10WR330)

GUIDED BY:

Dr. P.L.Patel

Dr. S.M.Yadav

DEPARTMENT OF CIVIL ENGINEERING

SARDAR VALLABHBHAI NATIONAL INSTTITUTE OF TECHNOLOGY

SURAT- 395 007, GUJARAT

2011-2012

SUSPENDED LOAD TRANSPORT IN ALLUVIAL STREAM FOR NON-UNIFORM SEDIMENTS

CERTIFICATE

This is to certify that the graduate report on “SUSPENDED LOAD TRANSPORT IN

ALLUVIAL STREAM FOR NON-UNIFORM SEDIMENTS” is prepared and presented by

Mr. KODINARIYA ANKIT M. Roll no. P10WR330 of M.Tech. (Sem. III) Water Resources

Engineering. His work is satisfactory during academic year 2011-2012.

Dr. P.L.PATEL Dr.S.M.YADAV

PROFESSOR ASSO.PROFESSORCED, SVNIT CED, SVNIT


CONTENTS

Sr.No. Description Page No.

Acknowledgement 4

1 Introduction 8

2 Literature Review 9

3 About Suspended Load Transport 9

4Concentration Distribution of The Individual Fraction in a Mixture

10

5 Suspended Load Transport of Sediment Mixtures 25

6 Conclusions 32

7 Notation 33

8 References 34


ACKNOWLEDGEMENT

I would like to take this as an opportunity to express my deep gratitude towards my guide

Dr. P.L.Patel, Professor and Dr.S.M.Yadav, Associate Professor, also my sincere thanks to our

Water Resources Engineering Section SVNIT Surat and acknowledge the help received from

them during the course of the G.R. without his tireless efforts and encouragement it would have

been difficult to complete the report. I would also like to acknowledge the cooperation and help

extended to me by other members of the SVNIT Surat, for providing me with adequate

infrastructure and the right environment for my Graduate Report.

Last but not the least a special thanks to my family and friends for providing the

necessary encouragement and mental support for completing my report.

KODINARIYA ANKIT M.M.Tech (3rd Sem.)Water Resource EngineeringP10WR330


LIST OF TABLE

Sr.No. Description Page No.

1 Range of data collected during the study 11

2 Variation of LC with M 18

3Range of data collected during the study

25

4Relation between Ls and Kramer's Uniformity Coefficient

30


LIST OF FIGURES

Sr.No. DescriptionPage No.

1 Illustration of Hiding and Exposure Effects in Case of Non-Uniform

Sediments

10

2 Size distribution of bed materials used in the investigation 11

3 Verification of Rouse's equation for concentration distribution (a = 0.2D). 15

4 Comparison of predictions from various models with measurements. 15

5 Typical dimensionless concentration profiles on log-log scale. 16

6 Predictor for concentration at 0.2D for uniform material 18

7 Non-uniform material data plotted on uniform material relation of with 19

8 Variation of with and M 19

9 Relation between and 20

10 Relation for concentration at 0.2D for non-uniform sediment 21

11 Proposed predictor for sediment distribution exponent above 0.2D 21


12 Effect of variation in Sediment distribution exponents above and below 0.2D 22

13 Relation between sediment distribution exponent over the depth on

concentration distribution

22

14 Diagram showing agreement of data using modified exponent (a = 0.2D) 23

15 Comparison of predicted and observed values of average concentration for

U.S.G.S. data

24

16 Computed and observed concentration profiles for Rhine river 25

17 Suspended Load Transport Law for Uniform Sediment 27

18 Transport Rate of Individual Fractions Plotted on Suspended Load Transport

Law for Uniform Sediment

28

19 Variation of with , and (M 0.285) 29

20 Variation of Ks with 29

21 Relation between Ks and for Various Values of M 30

22 Variation of Ks Ls with 31

23 Suspended Load Transport Law for Individual Fractions 31

1 . INTRODUCTIONSUSPENDED LOAD TRANSPORT IN ALLUVIAL STREAM FOR NON-UNIFORM SEDIMENTS

Sediment transport is the general term used for the transport of material (e.g. silt, sand, gravel

and boulders) in rivers and streams. The transported material is called the sediment load.

Distinction is made between the bed load and the suspended load. The bed load characterizes

grains rolling along the bed while suspended load refers to grains maintained in suspension by

turbulence. The distinction is, however, sometimes arbitrary when both loads are of the same

material. In the case of a powder snow avalanche, the fluid phase is air and the solid phase

consists of snow particles. The dominant mode of transport is suspension. These flows are close

analogies of turbidity currents, insofar as the driving force for the flow is the action of gravity on

the solid phase rather than the fluid phase. That is, if all the particles drop out of suspension, the

flow ceases. In the case of sediment transport in rivers, it is accurate to say that the fluid phase

drags the solid phase along. In the case of turbidity currents and powder snow avalanches, the

solid phase drags the fluid phase along.

Considering a particle in suspension, the particle motion in the direction normal to the bed is

related to the balance between the particle fall velocity component and the turbulent

velocity fluctuation in the direction normal to the bed. Turbulence studies (e.g. Hinze, 1975;

Schlichting, 1979) suggested that the turbulent velocity fluctuation is of the same order of

magnitude as the shear velocity.

With this reasoning, a simple criterion for the initiation of suspension (which does not take into

account the effect of bed slope) is:

2. LITERATURE REVIEW


Samaga, B. R., Ranga Raju, K. G., and Garde, R. J (1985) state that Rouse's equation and

some of the recently developed equations for concentration distribution in the case of flow in an

open channel have been checked for their accuracy with the help of data of suspended-sediment

concentration-distribution of the individual fractions in a mixture. The agreement between

predictions and measurements is not found to be satisfactory. It is found that the sediment

distribution exponent for any size in the mixture is not constant over the whole depth of flow. A

two-layer model is accordingly suggested herein to describe the sediment distribution in the

vertical. A predictor for the reference concentration at 0.2D is also suggested. The model showed

satisfactory agreement with the measurements on the Rhine river and the Middle Rio Grande

river.

Samaga, B. R., Ranga Raju, K. G., and Garde, R. J (1986) state that Careful experiments

were conducted on alluvial beds of four sediment mixtures having different arithmetic mean

diameters and standard deviations. The suspended load transport rates of individual fractions

were measured and compared with both Einstein's and Holtroff's methods of calculation of

suspended load for individual fractions. These methods were found to be unsatisfactory in the

present range of sediment parameters. A relationship is found to exist between the dimensionless

shear stress and the suspended transport rate for uniform sediment. To make this relation

applicable to nonuniform sediments, a corrective multiplying factor for shear stress is introduced.

The dimensionless parameters which govern this correction factor are identified and a relation

for the same obtained. In this manner the relation can be applied to individual size portions of a

sediment mixture.

3. ABOUT SUSPENDED LOAD TRANSPORT

1. What is Suspended Load?

The word ‘sediment’ refers commonly to fine materials that settle to the bottom. Technically,

however, the term sediment transport includes the transport of both fine and large materials.

Suspended load is the material moving in suspension in the fluid, being kept in suspension by the

turbulent fluctuations.

2. Uniform vs. Non-uniform sediments


Sediments in nature are non-uniform in character. Coarser particles in a non-uniform

sediment bed are more exposed to flow than they would have been in uniform sediment bed

Situation is reversed for finer fractions due to their sheltering in the wakes of coarser

fractions.

Fig 3.2.1. Illustration of Hiding and Exposure Effects in Case of Non-Uniform Sediments

4. CONCENTRATION DISTRIBUTION OF THE INDIVIDUAL FRACTION IN A

MIXTURE

1. Experimental work

The experimental work was carried out in a re-circulating tilting flume 30 m long, 0.50 m deep

and 0.20 m wide located in the Hydraulics Laboratory of the University of Roorkee. The

sediment used was sand of relative density 2.65. Four mixtures having different arithmetic mean

sizes and geometric standard deviations were used in the experiments; see (Fig. 4.1.2). A 0.15 m

thick bed of sediment was laid on the flume bed in all cases.


Fig. 4.1.2 Size distribution of bed materials used in the investigation

After the required sediment was placed in the flume and the flume set to the desired slope, water

at a predetermined rate was allowed into the flume. The sediment-water mixture at different

elevations at this section was collected in containers using a Pitot tube ensuring that the velocity

through the tube during sampling was the same as the flow velocity at that elevation. The

samples so collected were dried and analysed for the size distribution by means of a Visual-

Accumulation tube. Knowing the size distribution of the sample and its concentration at a

particular level, the concentration of individual fractions at that level was determined and hence

the concentration distribution of each fraction obtained. The experiments were repeated for other

discharges, slopes and sediments. The temperature of water was also recorded. No pronounced

bed undulations occurred in these experiments. The range of data collected during the study is

given in Table 4.1.1.


Table 4.1.1. Range of data collected during the study

sediment ArithmeticMean

da

mm

DischargeQX 103

m3/s

Flowdepth

D,M

AverageVelocity.

Um/s

Slope

5x103

totalload

qt

N/ms

M1 0.57 5.58-14.64 0.06-0.11 0.49-0.73 5.05-6.87 0.94-4.59

M2 0.55 7.54-14.36 0.07-0.10 0.53-0.78 4.96-6.93 1.39-6.46

M3 0.42 6.26-13.38 0.06-0.09 0.60-0.76 5.42-6.93 1.90-5.77

M4 0.25 8.14-13.46 0.06-0.09 0.61-0.75 4.49-6.10 1.81-4.48

2. Verification of existing methods

Rouse proposed the sediment distribution equation as

……………………………………(1)

HereCa =the concentration at Y= aC = the concentration at any level YZ0 =the sediment distribution exponent

The theoretical value of Z0 is given as , where is the fall velocity of particle of size d, is the shear velocity and K is Karman constant, which was taken to be 0.4.

Itakura and Kishi argued that the value of K for sediment-laden flows is not different from its value for clear-water flow. They wrote the concentration distribution equations as

…………………………..(2)

Here

= specific weight of sediment = specific weight of fluid= average volumetric concentration of sediment

Qf = mass density of fluid

Willis proposed a model for concentration distribution based on error function distribution of diffusivity. He assumed and distribution of as


…………………………….(3)

where = the momentum transfer coefficient = the sediment transfer coefficient

P1 = the normalised depth variable defined as

The concentration distribution was then derived as

….. …………………………..(4)

where is the value of at Y=a.He observed that the use of measured concentration at Y=0.2D as Ca gave good agreementbetween measured and computed suspended loads for 0.1 mm sand.

Antsyferov and Kos'yan proposed a two-layer model for the distribution of sediment. They argued that the sediment transfer coefficient differs considerably from the momentum transfercoefficient close to the boundary. Introducing an additive term for below Y/D = 0.20, theyexpressed the sediment distribution equation as

…….(5)

where u0 is the fluid velocity close to the bottom given by

U being the average velocity of flow and v the kinematic viscosity of the fluid.

McTigue proposed a mixture theory using a two-layer model. He considered a fluid and a solid phase and wrote down the mass and momentum balance equations for both phases. Using turbulent diffusion equation he wrote

for …………………………………….(6)where a = 0.2D and K1 =0.11. He also wrote


for ……………………………………………….. (7)

where K2 = 0.35

All the above models were checked for their accuracy with the help of the present data. The values of were calculated as , where is the mass density of the fluid and to is the average shear stress on the bed obtained from the measured depth and slope of uniform flow. The reference concentration was taken as C1, the concentration at Y= 0.2D. Calculations of C/C1

were carried out at all depths and for all the eight fractions of the mixture using these methods and these values compared with the observed values of C/C1. A typical plot of observed values of C/C1 with those computed from Rouse's equation is shown in (Fig.4.2.3.). (Fig. 4.2.4) shows bar diagrams for the different methods indicating the percentage of data lying within certain band of error. It can be seen that Rouse's equation and the equation of Itakura-Kishi, Antsyferov-Kos'yan and Willis give almost the same errors in spite of the rather more involved derivation in the latter three cases. About 56% of the data lie within, 40% of the predicted values in the case of Rouse's method and the corresponding percentages are about 59, 54 and 58 in case of Itakura-Kishi, Antsyferov- Kos'yan and Willis methods respectively. The corresponding value is 69% for McTigue's model. The equation of Navntoft showed better agreement and 73% of the data lie within 40% of the predicted values. But the use of his equation necessitates knowledge of the values of Y/D where the concentration is zero. Determination of this value in case of the coarser fractions of a mixture is not easy, since these tend to move in significant amounts only close to the bed.

Fig.4.2.3. Verification of Rouse's equation for concentration distribution (a = 0.2D).


Fig. 4.2.4 Comparison of predictions from various models with measurements.

3. New Approach Willis, McTigue and Antsyferov-Kos'yan argued that the convenient reference concentration is that at Y= 0.2D. (Fig. 4.3.5) shows a typical plot of concentration versus Y on log-log paper. It can be seen from this figure that there seems to be a break in the concentration profile for two sizes of the mixture at (D - Y)/ Y= 4.0 or Y= 0.2D. In fact a break at this elevation was observed in the velocity profile also in most cases. It was therefore felt that a two-layer model in which the concentration at Y=0.2D is chosen as the reference concentration will be an appropriate one for sediment distribution in the vertical.

The concentration distribution equation of Rouse can be written in terms of Q as

…………………………………(8)

In the light of the variation seen in (Fig. 4.2.5) it is hypothesised that the values of Z would be different below and above Y=0.2D.

Thus

Z=ZU in the region SUSPENDED LOAD TRANSPORT IN ALLUVIAL STREAM FOR NON-UNIFORM SEDIMENTS

Z=ZL in the region

An analysis concerning the importance of the various relevant parameters indicated that

………………………………(9)

Fig.4.3.5 Typical dimensionless concentration profiles on log-log scale.

………………………………..(10)

and

………………………………..(11)

where

= fall velocity of sediment of size di

= fall velocity of the arithmetic size da

M = Kramer's uniformity coefficient =

= the theoretical value of the sediment distribution exponent for size da i.e.


Here Pi is the percentage of sediment of size di in the mixture.

4. Reference concentration

A relation was found to exist between and C1 for uniform material as shown in

(Fig.4.2.6). Here is the shear velocity corresponding to grain roughness. The correlation

between C1 and was inferior to that between C1 and and hence the latter was

preferred. This suggests that

Fig. 4.4.6 Predictor for concentration at 0.2D for uniform material


Fig. 4.4.7 Non-uniform material data plotted on uniform material relation of with

Fig. 4.4.8 Variation of with and M


the concentration is dependent on the grain shear stress rather than on the total shear stress. Data of 0.18 mm sand collected by Barton and Lin and 0.1 mm sand and 0.04 mm sand collected by Laursen were used to establish the above relation, since the material used by them was of fairly uniform size. This relation was used as the basic curve to study the effect of non-uniformity on reference concentration. When data of individual fractions were plotted on this curve, in general, sizes, smaller than da showed less concentration and sizes larger than da showed higher concentration than that for uniform material; see (Fig. 4.4.7) where ib is the fraction by weight of any size di in the bed mixture. This decrease is due to the sheltering-cum-exposure effects in case of bed load motion which would eventually affect suspended load transport and interference effects in case of suspension of sediment mixtures. Thus a factor can be introduced

to predict the concentration of individual fractions; see (Fig. 4.4.7). Here is the shear velocity which would give the same concentration as that of uniform material of the same size in question. as defined above was determined for various size fractions and plotted against

for different M values; see (Fig. 4.4.8). It was observed that there was a systematic increase in with increase in for each value of M. Examination of (Fig. 4.4.8.) showed that a unique relation was possible between and where Lc is a coefficient dependent on M as shown in (Table 4.4.2.).

Table 4.4.2. Variation of LC with M

M LC

0.23 0.50

0.30 0.75

0.40 0.90

0.60 1.00

The variation of with is shown in (Fig. 4.4.9.) Thus, if the effective shear velocity

is computed as , (Fig. 4.4.5) can then be used to compute the reference concentration for

individual fractions. The parameter was computed for each size fraction and plotted

against in (Fig. 4.4.10.). The data showed good agreement with the relation derived for uniform material.


Fig. 4.4.9 Relation between and

Fig. 4.4.10 Relation for concentration at 0.2D for non-uniform sediment


Fig. 4.4.11 Proposed predictor for sediment distribution exponent above 0.2D

The procedure for computation of reference concentration can thus be outlined as below.

1. Divide the bed material into various size fractions and find ib for each fraction.

2. Read the value of Lc corresponding to the known value of M from (Table 4.4.2.).

3. Compute the value of for any desired size.

4. Read the value of for the known value of from (Fig. 4.4.9.).

5. Knowing the value of Lc, compute ,

6. Compute

7. Read the value of corresponding to the known of from (Fig. 4.4.10).

8. Repeat steps 3-7 for different sizes and compute C1, of each fraction and hence C1 of the

mixture

5. Exponent of concentration distribution

By plotting the concentration profiles in log-log form the values of ZU and ZL were determined for all the runs for all fractions. The location of the lines of content indicated a

unique relation between and . This relationship is shown in (Fig. 4.5.11).

The value of decreases from a little over unity to about 0.10 as increases

from 102 to 104. No effect of M on this relationship was noticed. Based on the hypothesis that ZL


Fig. 4.5.13 Effect of variation in sediment Fig. 4.5.12 Relation between sediment distribution exponent over the distribution exponents above and below depth on concentration distribution0.2D (ZU and ZL)

should be related to ZU, a plot of was prepared as shown in (Fig. 4.5.12). It can be seen that, for the finer fractions (characterized by low ZU values), ZL is greater than Z0 at high values of Z0, i.e. in case of coarse material. In other words, departure from the theoretical distribution are pronounced below Y= 0.2D in case of coarse fractions, whereas such is the case above Y= 0.2D in case of fine material. The effect of difference between ZL and ZU is shown in (Fig. 4.5.13). It may be seen from here that a value of ZL less than ZU implies smaller concentration close to the bed than obtained in case of a constant exponent ZU for the whole depth. Since ZL < ZU for large values of ZU or for coarse fractions, it means that the sediment distribution equation with an exponent fitting the data in the upper portion of the flow tends to give concentrations which are too large in the lower portion of flow in case of coarse material. Alternatively, if one uses the measured or estimated concentration near the bed along with a value like ZU the concentrations are under predicted over the whole depth in case of coarse material and over predicted in case of fine material.

Using the above values of C1, ZU and ZL for individual fractions the relative concentration is computed and compared with the observed values in (Fig. 4.5.13). About 88% of the data lie within 40% of the predicted values, thereby showing that the proposed method is more accurate than the ones discussed earlier in the paper. The validity of the above method was firstly checked with the help of the USGS data of 0.28 mm and 0.45 mm sands. Plane bed, dune bed and anti-dune bed data were chosen at random covering a wide range of concentration. The concentration distribution and hence the average concentration was determined for each fraction.


Fig. 4.5.14 Diagram showing agreement of data with equation (1) using modified exponent (a = 0.2D)

The sum of the average concentrations of individual fractions gave the average concentration of the mixture. These values are compared with the observed values in (Fig. 4.5.14). The comparison is satisfactory, barring a few data of the anti-dune regime of high concentration.


Fig.4.5.15 Comparison of predicted and observed values of average concentration for U.S.G.S. data


Fig. 4.5.16 Computed and observed concentration profiles for Rhine river

5. SUSPENDED LOAD TRANSPORT OF SEDIMENT MIXTURES

1. EXPERIMENTAL WORK The Experiment work is done as earlier mention

Table 5.1.3. Range of data collected during the study

Sediment ArithmeticMean

da

mm

Geometric standard deviation

Flowdepth

D,m

AverageVelocity.

Um/s

Slope

5x103

Suspended load

qt

N/ms

M1 0.57 5.58-14.64 0.06-0.11 0.49-0.73 5.05-6.87 0.46-0.89

M2 0.55 7.54-14.36 0.07-0.10 0.53-0.78 4.96-6.93 0.33-1.22

M3 0.42 6.26-13.38 0.06-0.09 0.60-0.76 5.42-6.93 0.36-1.40

M4 0.25 8.14-13.46 0.06-0.09 0.61-0.75 4.49-6.10 0.57-1.30

2. Approach


The suspended load is customarily calculated by first determining the velocity and concentration distribution curves and then integrating the "Cu" curve over the depth D-2d i. Although this is a rational approach to the problem, the ultimate accuracy is dependent on the accuracy in the prediction of velocity and concentration distributions. The present state of knowledge is such that, with the uncertainties in the Karman constant and the exponent for sediment concentration distribution, a high degree of accuracy cannot be attained in the prediction of u-y and C-y profiles.

A relation for suspended load transport of uniform material is the first step towards the study of the effect of non-uniformity. Vittal, et al. argued that the turbulence within the standing eddy behind a ripple or a dune helps in throwing material into suspension. Consequently, they felt that at least a large fraction of the form drag may aid in suspended load transport. Engelund also related the average suspended load concentration to the total shear stress. Accordingly, the functional relationship for suspended load qs of uniform material was written as

……………………………………………..(12)

where ……………………………………..(13)

and ………………………………………………...(14)

Uniform material data of 0.18 mm sand collected by Barton and Lin and those of 0.1 mm and 0.04 mm sands collected by Laursen were used to establish the variation between Q s and . The standard deviation of the material used by them was close to unity and hence these sands were considered to be uniform. A plot of Qs versus was prepared and is shown in (Fig.5.2.17). The data indicate a unique relation between these parameters (for dune and plane beds), which is

……………………………………………...(15)

The scatter band for Qs ranging from half to twice the predicted value is also shown on this figure. Following a line of analysis similar to that for bed load, one can define a coefficient for non-uniform sediment as follows = shear stress required to give same suspended load as in case of uniform sediment / actual shear stress or

……………………………………………………(16)


Fig.5.2.17 Suspended Load Transport Law for Uniform Sediment

Following Misri, et al. the functional relationship for can then be written as

…………………………………………..(17)

in which is the critical shear stress for the arithmetic mean size determined from Shields' criterion. The dimensionless shear stress for each size was plotted against the dimensionless transport parameter Qs in all cases. A typical plot for mixtures M2 and M3 is shown in (Fig.5.2.18). Here

………………………………………………(18)

It is seen from (Fig.5.2.18) that in general, the fractions finer than the arithmetic mean size da

show transport rates smaller than those given by the relation for uniform material, while coarser fractions show higher transport rates. The coefficient represents the cumulative effect of:

(1) Sheltering- exposure on the bed load transport rate and hence on the reference concentration; and (2) The interference of one size on the other in the processes of entrainment and suspended load movement

which would affect the value of sediment distribution exponent.


Fig.5.2.18 Transport Rate of Individual Fractions Plotted on Suspended Load Transport Law for Uniform Sediment

Since the lines for different fractions are not truly parallel to the lines for uniform material, the values of are dependent on apart from and M. The values of were calculated for all the data as indicated above. In accordance with Eq. 17 the variation of was studied with

for various values of for a constant value of M. A systematic variation of with and was observed, as may be seen from (Fig.5.2.19). The parameter decreases

with increase in for given values of M and . A careful study of (Fig.5.2.20) indicates that a unique relation can be established between and Ks for a particular value of M, where Ks is a coefficient depending on


Fig.5.2.19 Variation of with , and (M 0.285)

Fig.5.2.20 Variation of Ks with


Fig.5.2.21 Relation between Ks and for Various Values of M

Table 5.2.4 Relation between Ls and Kramer's Uniformity Coefficient

Kramer's uniformity

coefficient M

LS

<0.20 0.80

0.30 0.90

0.40 0.96

>0.5 1.0


Fig.5.2.22 Variation of Ks Ls with

Fig.5.2.23 Suspended Load Transport Law for Individual Fractions

as shown in (Fig.5.2.20). The data are of limited range in (Fig.5.2.20) and extrapolation as shown by the dotted line is justified by subsequent check with some field data. The variation of

Ks with for different values of M is shown in (Fig.5.2.21). The parameter Ks

decreases with an increase in in all cases. A unique relation was established between Ks

Ls and , where Ls is a coefficient dependent on M. The relation between Ls and M is shown in (Table 5.2.4). The variation of Ks Ls with for all the data is shown in (Fig.5.2.22). The data plot neatly around a single curve as shown in this figure. see (Fig.5.2.23). It may be noticed that field data from the Snake river are also plotted (Fig.5.2.23). The value of

for the river was smaller than those in the case of the present laboratory data and the extrapolation of (Fig.5.2.20) was used in the calculations for this river.


3. Procedure for Computation

The following stepwise procedure is recommended for the computation of suspended load

transport rates of different fractions in a mixture.

1. Divide the bed material into various size fractions and find ib for each fraction.

2. Compute the average shear stress, .

3. Compute the critical shear stress corresponding to the arithmetic mean diameter da from

Shields' criterion.

4. Compute the value of and read the value of Ks from (Fig.5.2.20).

5. Read the value of Ls corresponding to the known value of M from (Table 5.2.4).

6. Read the value of Ks Ls for the value of , corresponding to any desired size from

(Fig.5.2.22).

7. Compute the value of knowing values of Ls and Ks.

8. Compute the value of ; and read the value of Qs corresponding to the computed

value of from (Fig.5.2.23).

9. Compute the transport rate of the fraction from the relation in Eq. 18 for the known value of

ib.

10. Repeat steps 6-9 for other sizes.

11. Compute the total suspended load transport rate as the sum of the rates of individual

fractions, i.e. .

6. CONCLUSION

It has been shown that predictions from the relations of Rouse, Itakura-Kishi, Willis, Antsyferov-

Kos'yan are not in good agreement with the measurements of concentration distribution on

alluvial beds of non-uniform sediments. The agreement is slightly better in case of McTigue's

and Navntoft's methods. The actual exponents of concentration distribution for individual

fractions are different from their theoretical values. In case of coarser fractions, the departure

from the theoretical concentration distribution is more pronounced below Y= 0.2D, whereas such


is the case above Y= Q.2D for the finer fractions. The methods for determination of

concentration distribution were checked with some field data of the Rhine river and the Middle

Rio Grande river and found to result in predictions in good agreement with the measurements.

A unique relation is found to exist between and Qs for suspended load transport of uniform

material. This relation can be used to predict the transport rate of individual fractions in a

mixture by introducing a correction factor . has been related to , and M. The

relation has shown fair agreement with the data of the Snake river.

7. NOTATION

A2, B2 empirical constantsa reference levelC concentration in ppm by weight at any level YC1 concentration in ppm by weight at Y = 0.2 DCa concentration in ppm by weight at Y= aD depth of flowD* dimensionless parameter in Itakura-Kishi modelda arithmetic mean size of the mixturedi size of particleib fraction by weight of any size d, in the mixtureK Karman constantLc a correction coefficient dependent on MM Kramer's uniformity coefficientP1 normalized depth variable in Willis' modelP1a value of P1 at Y=aU average flow velocityuo fluid velocity close to the bottom as defined by Antsyferov-Kos'yanu* shear velocity

shear velocity based on shear stress calculated on the basis of grain resistanceZ0 theoretical sediment distribution exponent =

theoretical exponent for the size ZU sediment distribution exponent above Y=0.2DZL sediment distribution exponent below Y=0.2D

sediment transfer coefficient momentum transfer coefficient specific weight of sediment specific weight of fluid mass density of fluid

kinematic viscosity of fluid dimensionless grain shear stress =


grain shear stress fall velocity of a particle fall velocity of a particle of size da

fall velocity of a particle of size di correction factor

8. REFERENCES

1. Samaga, B. R., Ranga Raju, K. G., and Garde, R. J, (1985). Concentration Distribution of

Sediment Mixtures in open-channel flow, Journal of Hydraulic Research, Vol.23, No.5,

467-483

2. Samaga, B. R., Ranga Raju, K. G., and Garde, R. J.,(1986). Suspended Load Transport of

Sediment Mixtures, J. Hydraul. Eng., 112, Page 1019–1035

3. Garde R.J. and Ranga Raju K.G.(2006). Mechanics of Sediment Transportation and

Alluvial Stream Problems, 3rd Edition, New Age Int.(P)Ltd., Pg.79

4. Garde R.J. (2006), River Morphology, New Age Int.(P)Ltd., Pg.120


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