Simulation of a curved flume bed-load experiment

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"a.#,. ,f~ri>1>

TU

Delft

Delft University ofTechnology

FacuHy of Civil Engineering DepartmentHydraulic Engineering

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part ofSTW-project;:

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Simulation of a curved flume bed-load experiment.

A.M.Talmon

report no. 5-88, June 1988

River bend morphology with suspended sediment.

Delft University of Technology Faculty of Civil Engineering Hydraulic Engineering Division

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ABSTRACT

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The mathematical model for river bend morphology, as developed by Olesen, for bed-load transport is discussed, by comparing the results with some new experimental data. The model consists of a two-dimensional depth-averaged flow model together with a sediment balance and can be used to compute the dynamic river bend morphology.

The main purpose of this investigation was to develop the data

acquisition procedures for a suspended-load experiment in a curved flume by means of an experiment with bed-load only. The experiment provided some new data to test the present morphological model.

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In this report the results of the experiment are discussed. A remarkable point .is that the experimental data show a persistent harmonic

longitudinal perturbation of the bed topography. The computations, however, show a strongly damped bed topography, even for unrealistic parameter combinations. The computed depth at the point-bar and the corresponding pool is made to correspond with the experimental data, by a significant change of the submodel for the direction of the bed-load transport and the transport model itself.

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Probable causes for the discrepancy between the data and the computation are the small width-depth ratio and the low transport rate of the

experiment (close to the initiation of sediment motion). The small width-depth ratio is an unfavourable circumstance in view of the assumptions used in the depth-averaged flow model. The low transport rate requires a modified transport model. Nonetheless a satisfying

explanation for the bed oscillation, as observed in the experiment could not be found.

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CONTENTS

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ABSTRACT

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SYMBOLS

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1.

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2.

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2.1 2.2

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3. 3.1 3.2 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.4 3.4.1 3.4.2 3.4.3

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4. REFERENCES

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TABLES FIGURES

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3 INTRODUCTION

THE LABORATORY EXPERIMENT

Design of the experiment

Results of the experiment; bed topography

NUMERI CAL SIMULATIONS The morpho1ogical model

The first approach, standard parameters Optimizing the simu1ations

Analytical calculation of prevailing wave numbers Computations with the numerical model

Accuracy of the numerical computations Stability of the numerical computations The optimized simulation

The numerical computation

Comparison of measurement and computation Discussion of model performance

CONCLUSIONS 2 4 6 7 7 8 9 9 12 13 13 19 20 23 25 25 26 27 29 31 32 35

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SYMBOLS

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a

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e f u f v

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G

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gi

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i s j k

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~ k sn

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Ln L s

I

mo

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p p P R

R

s

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Ss

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ut v

I

w

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p

ó ó s

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4 water depth

exponent in generalized transport magnitude model exponent in generalized transport direction model Chézy friction coefficient

coefficient in generalized sediment transport model shape function of velocity profile in s-direction shape function of velocity profile in n-direction

coefficient in gravitational term in sediment vector model acceleration due to gravity

channel slope

=

ü

aiR secondary flow intensity counter of grid number in n-direction dimensionless wave number in s-direction dimensionless wave number in n-direction secondary flow convection factor

wavelength in streamwise direction wavelength in normal direction

counter of grid number in s-direction counter of time step number

iteration number pressure

atmospheric pressure channel radius

streamline radius

sediment transport rate per unit width longitudinal curvilinear channel coordinate time

velocity component in s-direction velocity component in n-direction channel width

coefficient in the model for the direction of bed shear-stress angle between bed shear-stress vector and s-lines

angle between bed shear-stress and streamline 2

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V T

ç

p

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T n

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T S ~n X ~

r

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~n ~s ~t 8 8 . cr~t

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5 =~~n/ao

adaptation length of bed topography adaptation length of main flow

adaptation length of bed shear-stress

ripple factor in the Meyer-Peter

&

MUller sediment transport model

turbulent eddy viscosity

-k ~s/aO density

bed shear-stress in n-direction bed shear-stress in s-direction numerical amplification factor relaxation coefficient

angle between sediment transport vector and s-lines relative pore volume

grid dimension in n-direction grid dimension in s-direction time step

Shields parameter

critical Shields parameter

k/~ relative wave number

sub- and superscript:

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u

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_u

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depth-averaged value

value upstream of the bend (zero order unperturbated variable) first order perturbated variable

complex amplitude of variable

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1. INTRODUCTION

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

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The numerical model for the suspended sediment will, in the first stage,

be incorporated in an already existing morphological model, Olesen 1987,

which computes river bend morphology in case of bed-load transport only. Among others laboratory measurements will be used to calibrate and to validate the numerical model incorporating suspended sediment.

A bed-load experiment has been performed in order to develop data-acquisition procedures and to gain insight into model parameters to be used in the revised curved flume of the Laboratory of Fluid Mechanics. By comparison of laboratory results and numerical computation of the bed-load experiment, the suitability of the laboratory model to cali-brate and validate the numerical model can be judged.

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In this report experimental results and the numerical simulation of the bed-load experiment are reported. In chapter 2 the results of the

bed-load experiment are summarized. In chapter 3 the numerical simulation is presented. The simulation includes the analytical solution of the model and accuracy- and stability analyses. In chapter 4 conclusions are presented.

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This research is a part of the project: 'River bend morphology with suspended sediment', project no. DCT55.0842. The project is supported by the Netherlands Technology Foundation (STW).

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2. THE LABORATORY EXPERIMENT

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In 1983 the Laboratory of Fluid Mechanics (L.F.M.) curved bend flume has been revLsed. The flume has the same radius as the earlier flume, but

its width has been decreased to 0.5 m. This flume will be used to assess some of the parameters for the numerical model to be developed. The first experiment is a bed-load experiment. In this chapter the choice of model parameters will be outlined and experimental results are

summarized. In Talmon and Marsman 1988 the experimental results are given in more detail.

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2.1 Design of the experiment

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The choice of the parameters of the experiment is determined by the channel geometry, by the measuring devices and by the physics of the problem under consideration.

To simulate the flow and sediment transport of a full-scale river bend, theoretically similarity of the following parameters is desired:

Chézy :channel roughness W/aO :width-depth ratio WIR :width-radius ratio Fr :Froude number 8 :Shields number

In practice it is, however, not possible to fulfil all these similarity conditions at the same time. So scale effects might be present.

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The W/aO ratio should be as large as possible in the experiment. This complies with the similarity concept underlying the numerical flow model. The water depth, however, is restricted in a physical sense as

the Froude number should not be too large. From a practical point of view it was decided to allow a minimal water depth of 2 cm in the inner part of the bend which is forced upon by the bottom profile measuring device (PROVO)."

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The bed material consists of sand with a characteristic diameter of d

sO

=

0.78 mmo During pilot runs it appeared that large bed-forms developed in the whole channel when the flow velocity exceeded some value. In view of the validity of the flow model it was decided to select a flow velocity

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at which the bed remained relatively smooth, but nonetheless with a sufficient transport of bed material. It was judged by eye that this condition was fulfilled.

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Af ter various pilot runs a choice was made yielding the parameters as shown in table 1 and table 2. In figure 1 the ensemble mean water level as function of the streamwise coordinate (channel axis) is shown. The slope of the water level was i=1.2S 10-3. In figure 1 it is shown that

this slope occurs in large portions of the bend. This value yields a Chézy coefficient of 3S

mO.S/s

while the bend average slope leads to C

39 mO

.

S/s.

According to the conceptual model of Struiksma 1983, a smaller Chézy coefficient yields less damping of bed-oscilations.

The bed-load transport is rather close to the threshold of initiation of motion of the sediment. (8=0.066, 8 .t=0.047)

crL

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2.2 Results of the experiment: bed topo~raphy

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The measured bed topography is shown in figure 2. The figure shows the ensemble mean bed topography of 18 measurements. At the bend entrance the bed is relatively smooth, downstream of traverse 15 the bed consists of elongated undulations (z 1 à 1.S m wave length, z 0.05 m height at crest ) moving downstream, which resemble the alternating bar type but are only present in the outer part of the bend. By ensemble averaging the stationary part of the bed is determined. The bed topography displays a pronounced harmonic oscillation in the longitudinal

direction. In the 180° bend one point-bar at traverse 12 and three pools at traverses 12, 26 and 39 are present. The interval between the

traverses is 0.32 m at the channel centerline. The wave length in streamwise direction is

L

-4.3 m . Assuming the transverse shape to be

s

sinusoidal the wave length in transverse direction is L =1.0 m. In n

figure 4. the measured water depths at n/W --0.20, 0.0 and 0.20,

together with some calculations, are shown as a function of the streamwise co-ordinate.

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9

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3. NUMERI CAL SIMULATIONS

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First, the numerical model of Olesen, was run with standard parameters,

but the results were not satisfactory. Next dominant wave numbers of the analytical solution were calculated to indicate the optimal choice of parameters. To improve the numerical computation the accuracy and the stability of the model, for parameters related to this particular experiment, are calculated. The analytical solution, accuracy and stability analyses have been developed by Olesen. Based on experience obtained with this approach the response of the numerical model is optimized.

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Three types of calculations have been performed.

A - Numerical computations with the morphological model.

B - Calculation of dominant wave numbers of the solution, including the response of bed topography as a function of the streamwise co-ordinate

C - Calculation of the accuracy and the stability of the numerical model.

For types A and B use has been made of programs developed by Olesen. For type C software had to be written. In tables 3, 4, 5 and 6 the main parameters of the calculations are given.

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3.1 The morphological model

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The morphological model is described by Olesen 1987. The model contains two submodels; a flow model and a sediment model. First the flow model is solved, next the sediment model is solved to compute bed elevations. With the newly calculated bed elevations the flow field is recalculated and next the bed elevations. This process continues until changes in bed elevation become non-significant.

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The flow model is a so called depth-averaged model. This reduces the 3-dimensional flow field to a 2-3-dimensional mathematical model.

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The basic set of equations of the flow model in a curvilinear co-ordinate system, see figure 3, reads:

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10

au

au

₊

au

₊ uv

_1~

+

a

au

(3.1) u- _+~ ~ R

as

-v

az

as

p

az

r av av av 2

_1~

₊

_Lv

av u- ₊ ~ + ~ 1L (3.2)

as

R p an

az

r

az

aw

₊

aw

+

a

w

+

1~

(3.3) u-~ ~ g

a

s

p

az

au

+ av +

aw

+

x

0 (3.4) as an

az

R

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(3.3) is approximated by: g +

1

QQaa

=0 , hydrostatic pressure.

p z

Integration gives: p=P + pg(a - z), with P = pressure at water level Depth-averaging is performed; the depth-averaged set of equations reads:

(see 01esen 1987 and Kalkwijk and de Vriend 1980)

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_{- -}1

a

P ₊ -aü -aü uv rs ksn

a

-

isau k [u- + v-;:- + -] + - + -[-(i au) + 2-R-]- 0 p as uu

as

an

R

pa a an s (3.5)

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1ap

+ k [-av u-p an uu

as

aaü + aav + av

as

an R ü2] ₊ rn _{+ --}ksn _--

a

(

'

_{~ au}

_)

R pa a

as

s

o

(3.6)

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(3.7) in which: i s k uu

secondary flow intensity

J1

f2 dz z1.0 and k =

J1

f f dz

Zo

u sn

Zo

u v

shape functions of velocity profiles

(3.8)

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ui

Neg1ect of the ksn-_s term in equation (3.5) leads to the fol1owing

R

equation for the longitudina1 momentum:

- - r k

1ap

+ k [üau +

v4-

aun+ uv] + 2 + sn

aa

(i au)

p

as

uu

as

R pa a n s

o

(3.9)

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Stream1ine curvature is approximated by: ~ s

Substitution of eq. (3.10) in eq. (3.6) and neg1ect

1

R u

av

as (3.10) of the term due to

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secondary flow convection gives the fol1owing equation for transverse flow momentum:

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11

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(3.11)

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The boundary shear is

u -2

T = p~ u and T

s C2 n

with

S :

angle between

s stress approximated by: = pg2ü(v + ütanS ) C s the depth-averaged (3.12)

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streamline and the bed shear

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Within the mathematical model the momentum equations (3.9) and (3.11) are combined in order to calculate Ü. In the numerical

formulation, however, the T /pa term of the transverse momentum equation

n

is discarded. Based on these ü values, the v values are calculated by

-av

means of the continuity equation (3.7) with neglect of the ~ term.

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The angle

S

between

s

the direction of the atanSs

À f

a

+ tanS = - b-Ra

s s s S

the direction of the depth-averaged velocity and

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bed shear stress is modelled by:

(3.13)

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in which

fi

is mathematically determined by the shape functions of the longitudinal en transverse velocity components. Àsf is the adaptation length of the secondary flow.

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The sediment model:

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The equation of continuity for the sediment in the curvilinear co-ordinate system reads:

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an

S tan ~ + ~s~ _ R (3.14)

as

s

as

+

a(s

tan ~) s

in which: ~ : direction of sediment transport. S tan ~

In the numerical model the sR term is neg1ected.

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The transport rate is formulated by: Ss (g)b (1 + eaa)

So

u

as

(3.15)

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12

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The direction of the bed shear stress is calculated by:

T _TI = tanó = ~

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TS u + tanó s (3.16) The direction

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tan~ = tanó + of sediment -b G

(

!

)

0

aa

o

80

an

transport is calculated by:

(3.17)

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Inspection of the numerical model, developed by Olesen 1987, leads to the conclusion that contrary to the model given at page 105, the T /pa

n term in the transverse momentum equation, the aV/R term in the

continuity equation and the S tan~/R term in the sediment continuity

s

equation have been neglected in the model.

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3.2 The first approach. standard parameters

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The first computations were performed with standard parameters. The parameters for this particular experiment are taken from Olesen 1987:

~ = 10.0 (Olesen fig. 3.3, ~ is based on theoretical computations) Àsf=0.6]ga =6.7a, adaptation length of the secondary flow (Olesen fig. 2.4, Àsfis fitted by numerical simulation of a river bend experiment)

GO =3.0, coefficient in the sediment direction model (Olesen fig. 3.3 based on experimental data)

a common used value

exponent of the transport model (the Engelund Hansen value) (based on values reported by Olesen)

Computations were performed using a numerical grid of 5 points in transverse direction and 49 points in streamwise direction. The grid covered the whole 1800 bend and small portions of the straight reaches

up- and downstream of the bend.

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bO =0.5, b =5.0, e =4.0,

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The first computation resulted in an unstable behaviour of the model. Decreasing the time-step lead to stable results. The computation showed,

however, a too small transverse slope in the major part of the bend and a very weak point-bar. Trying to improve the results the parameters have

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been varied between 1imits which were thought to be physica11y realistic.

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Various values for the b exponent of the transport model have been used (4<b<7), but no significant improvement took place. Variation of beand e did neither improve the resu1ts.

fi

was slightly altered (fi=9.7) to comply with C =35 mO.5/s. No further variation of

p

was performed whi1e

fi

follows from theoretical computations and thus shou1d not be subject to tuning. Neither Àsfwas changed because Àsffollows from an experiment under nearly the same conditions as the present experiment. GO was varied down to the lowest reported value (1.5<GO<3.0). At GO=1.5 the results improved somewhat, but not enough.

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Results of one of the first simulations is shown in figure 4, together with the experimental data. It is evident that the result is not

satisfactory. This particular simulation is referred to as AO and the parameters are given in table 3.

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3.3 Optimizin~ the simulation

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To investigate the cause of the rather poor simulation of the experi-ment, it was decided to calculate the analytical solution of the

problem. In the course of the investigation it also appeared necessary to investigate the accuracy and the stability of the model.

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3.3.1 Analytical calculation of prevailing wave numbers

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A rather thorough simplification of the problem has been made by

Struiksma 1983. The problem is simplified by dividing the channel into two adjacent straight channels, both of uniform depth. In streamwise direction sinusoidal perturbations, with amplitudes beiing a function of the streamwise co-ordinate, for the flow and the bed topography are assumed. In the analysis adaptation lengths for the flow and bed topography are introduced.

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À adaptation length of flow:

w

c

2 À =-a w 2g (3.18)

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14

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problem is given by the À À - ~) - ~ = 0 À À a a

1

(H)2 ~ 1f2 a Go following (3.19)

Àa adaptation length of bed topography: Àa=

The solution of the equation:

(3.20)

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a

O

kÀ can be expressed in i: kÀ = --2i , with: f = -&-2

w w e C ~

The solution mainly depends on the ratio of À /À , see figure 5. By

a w

comparison of the solution with the results of an extended analysis, Olesen 1987, which will be discussed further on, it is concluded that predictions are good for 0.25

<

À /À

<

5.0. For À /À

<

0.2 the model

a w a w

does not apply, because in this region the model predicts re(i)=O, which

(3.21)

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implies that harmonic oscillations of the flow and the bed are absent.

This is unrealistic. The adaptation lengths for the bed-load experiment are: À = 4.4 m, À = 0.12 m ( with GO= 3 ). The resulting ratio À /À =

w a a w

0.027 lies outside the range of validity of the model, therefore the model is not applicable to this experiment. Only a drastic reduction of GO to 0.3 for example would make the model appropriate.

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The analytical solution of Olesen includes the equations (3.5) to (3.17) of the morphological model. Some terms of minor significance are

neglected, linearization is employed and sinusoidal perturbations, in the streamwise and the normal direction, of the flow and the bed topography are assumed. The amplitudes of these perturbations are a function of the streamwise co-ordinate. The solution of the model applies also for À

/À <

0.2.

a w

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The flow model of the analytical solution is based on the depth-averaged flow model, eq. (3.5) to (3.7). Curvature of the coordinate system and secondary flow convection, however, are neglected (l/R=O, ksn=O). The depth-averaged set of equations that is used in the analytical model thus reads:

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-2

1

ap

-aü

-aû

₊ a; u ₀

as

+ u-- +

va;

c2 h p

as

1

ap

-av

₊ Tbn 0

an

+ u--p

as

p a (u-momentum) (3.22)

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(v-momentum) (3.23)

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15

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aaü

aav = a

as

+

as

(continuity) (3.24)

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This simplified set of equations is linearized by superposition of perturbations on the leading variables. For ü for instance: u-ua+ ü'. With

ü

a the value in the unperturbated situation and Ü' the

perturbation.

For a, v and l/R similar expressions are substituted.

s

The governing set of equations for the analytical model becomes:

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-2

1ap'

-

aü'

ua

-

,

a' - _{+ ua - +} L- (2':.1 -) a p

a

s

as

c

2 aa ua aa

1ap'

-

av' _L ua v'= a - + _{ua -} + p an

as

c

2 aa

aa'

aü' av'

a ua -

_as

+ aa -

_as

+ aa _an (u-momentum) (3.25)

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(v-momentum) (3.26)

,

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(continuity) (3.27)

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The fluctuating pressure is eliminated by cross differentiation of the equations for

ü

and v momentum. Subtracting both differentiated

equations leads to an equation in which the dependent variables are: Ü',v'and a'. The fluctuating components at coordinate s, n, are modelled by harmonie perturbations. The fluctuating components at eoordinate s,n,

are modelled by harmonie perturbations.

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a' a'

la

-a _aa+ a sn sn sn a ü' Ü'

lü

-

i (

slaak

+

n/aa~)

(3.28) u ua+ u exp sn sn sn a

-

_{v' v'}

_lü

-v v sn sn sn a in whieh:

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k =

2~aa/Ls'

dimensionless wave number in streamwise direetion

~=

2~aa/Ln

'

dimensionless wave number in transverse direetion

The phase shift between the variables is aeeounted for in the complex amplitudes. Damping or growth of these harmonie perturbations in streamwise direetion is aceomplished by the imaginary part of k:

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16

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im(k»O damping, im(k)<O growth. Inserting these harmonic perturbations into the governing set of flow equations gives: (see also Olesen

1987,eq. (4.25) and (4.26»

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,

-

_{u (3.29)}

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v

-

(3.30)

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In which: i = k/~ =L /L_{--b2 n s} ,ratio of wave lengths € =

g/(C ~)

The wave number ~ in transverse direction is assumed to posses no imaginary part, which means that an undamped oscillation of the variables in transverse direction is assumed. This means that im(i»O characterizes exponential damping of oscillations in streamwise direction, while im(i)<O characterizes exponential growth of oscillations.

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The set of equations of the sediment model (3.14) to (3.17) are processed the same way as the equations of the flow model. The angle

a

between the bed shear stress and the depth-averaged streamlines,

s

however, is not neglected. The equation for continuity of the sediment expressed in complex amplitudes reads:

,

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(3.31)

see also Olesen 1987 eq. (4.29).

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By inserting the complex amplitudes of the flow model eq.(3.29) and (3.30) in the continuity equation of the sediment (3.31) the analytical solution of the morphological model can be computed. Based on the chosen C,

p

,

Àsf' GO' band e parameters the complex ratio of wave numbers can be solved. For this purpose the resulting equation is sorted with

respect to i to obtain a cubic type of expression. The result is:

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17

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(3.32)

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~sf +

,

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This equation is somewhat different from Olesen 1987 (4.30), but hisealeulations were based on equation (3.32).

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i is ealeulated by eomputing the 6 complex roots of this eubie equation.

4 roots are purely imaginair. The 2 remaining roots have equal imaginary

parts but opposite real parts.

These two roots are of primary interest while the real part represents the ratio of wave lengths of the harmonie solution and the imaginary part represents the damping of oseillations.

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The refined analysis is used to ealeulate i . In the experiment i is

estimated to be: re(i)=0.23, im(i)=0.02 +-100%

The ealeulations are summarized in table 4. The response of bed topo

-graphy along the outer part of the bend, at n=0.5 W, is given in figure

8. Small values of re(i) represent large wave lengths in streamwise direction of the harmonie part. Small values of im(i) represent little

damping.

Caleulation BO ,with parameters equivalent to eomputation AO, shows a

strongly damped response and a too large value for the harmonie part;

re(i)-0.42, im(i)-0.30. Based on this result it is eoneluded that

para-meters are to be varied in order to optimize ealeulations. To optimize ealeulations the following parameters are varied:

C, ~, GO' band e.

The ehannel roughness, Chézy eoeffieient, has been varied in order to

assess roughness effects. This eoeffieient is governed by mean flow

velocity, averaged water depth and water surface slope. As these

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18

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parameters are all known, this parameter is not allowed to deviate too much from the va1ue obtained from the measured data. Chézy coefficients of 21.8 and 49, as 1isted in tab1e 4, are to be considered as physica11y unrea1istic, and serve on1y to judge the response of the model.

The ~ and GO parameter are varied in order to obtain a 1arger transversa1 bed-s1ope.

The va1ues of band e parameters are increased because the transport rate of the experiment is close to the thresho1d of motion. From a

1inearization of the Meyer-Peter

&

Mü11er transport formu1a, 01esen 1987 has ca1cu1ated band evalues for sma11 va1ues of 8, see figure 6 (for a smooth bed ~=1). 01esen a1ready conc1uded that the e parameter has

1itt1e effect on the solution, but a1so noticed that an increased va1ue stabi1izes the numerical computation. It is anticipated that an

increased va1ue of b, the exponent in the transport model, wi11 increase the transversal bed-s1ope, because for a given lateral velocity gradient an increased transport gradient wi11 resu1t in a 1arger transversa1 slope.

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From these ca1cu1ations it is conc1uded that the solution is 1arge1y governed by the fo11owing parameters, see a1so tab1e 4 and figure 7.

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variation resu1t

re(i) im(i)

aa

(downs an

C up up up

b/GO up down down

GO/~ up up up down

tream of point-bar)

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Because of the comp1icated structure of the model, these effects can notbe exp1ained easy, however, some resu1ts can be qua1itative1y exp1ained. The C coefficient affects both wave 1ength and damping. An increased roughness yie1ds longer waves and 1ess damping.

The b/GO ratio affects wave 1ength and damping.

Increasing b ,and a1so increasing e according to Meyer Peter Muller, leads to 1arger wave lengths and 1ess damping, see ca1cu1ations B6 and B7.

The model of Struiksma a1so prediets these damping effects of osci11ations, even though the parameters are outside the range of

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1

19

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applicability of the model. The imaginary part of the solution of equation (3.20) can be written as:

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_{im(k) [}3-b g ~2 (aO)2G

1

1

--2--2 + 2 W 0 a C (3.33)

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,

For b>3 equation (3.33) shows a decrease of im(k) (less damping) for decreasing C and increasing b/GO.

The GO/~ ratio affects the wave length, the damping and the transverse bed slope. Increasing ~ only leads to less damping and a larger

transversal bed slope, see calculations B3 and B4. One computation showed that the effect of variation of ~ on re(i) can be compensated by

variation of e, see calculations B8 and BlO.

The effect of the GO/~ ratio on the transversal slope downstream of the point-bar is explained as follows: Downstream of the point-bar, where oscillations of bed topography are absent or very small, no significant variation of variables in downstream direction takes place. Here the bed material is moving in concentric circles. In this case; tan~O, v/ü«l and atana

/as=o.

With these approximations equations (3.13), (3.16) and

s (3.17) reduce to:

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(3.34)

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From this equation it is evident that an increase of GO/~ yields a smaller

aa/an.

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To optimize parameters with respect to the i value and

aa/an,

the parameter combination of calculation BlO seems to be best of the parameter sets examined. Recalling the estimated wave length ratio of experiment: re(i)=0.23 im(i)-0.02 +-100% , the i values of BlO are: re(i)=0.19, im(i)=0.094.

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Next, on basis of these calculations, computations with the numerical model are performed in combination with calculations of accuracy and stability of the model.

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20

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3.3.2. Computations with the numerical model

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The computations with the numerical model are summarized in table 3.

Computations Al to A4 are based on physical parameter combinations

following from the analytical model. High values of b resulted in

unstable computations.( A2 and A3 ). At this stage it was not clear

whether the grid dimensions and the time-steps used, allowed optimal

response of the numerical model. To this purpose accuracy and stability

of the model were computed.

Computations A5 to A9 are based on parameter combinations following from

the analytical model and on the calculation of optimal grid dimensions

and time steps. However, to obtain a stabIe computation the time-step in

the numerical model often had to be taken about a factor 10 smaller than

predicted by the stability analyses. Accuracy and stability are

discussed in par. 3.3.3 and 3.3.4.. From the accuracy analyses grid

dimensions of Àn=0.125 mand Às-0.32 m appeared to be sufficient. Using

these Àn, ÀS and conservative Àt, computations with b~20 remained

un-stabIe although the stability analyses indicated stable computations.

Computation with b=15 resulted, after 500 time-steps of 14 s in a large pool at 1450 which continues to become deeper.( the calculation has not

reached an equilibrium state ).

Computations with b=lO finally resulted in a stabIe solution (A7 to A9).

Compared with preceeding computations, A9 gives the best results. This

is discussed in par. 3.4.

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3.3.3. Accuracy of the numerical computations

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The accuracy analysis of the numerical model is reported by Olesen 1985.

The results of the accuracy analysis given by Olesen re late to

common encountered parameter combinations. Because parameters of the bed-Ioad experiment deviate significant from these common conditions it

is uncertain if the results of Olesen 1985 apply to this case too.

Therefore it was necessary to calculate accuracy of the model for

parameter combinations corresponding to this experiment.

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21

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The accuracy analysis is basicly a comparison of the magnitude of flow variables in the discretized model and in the analytical model. In these models the ljR terms are neglected and equations are linearized.

Consequentely this model, from a theoretical point of view, applies to the flow and bed-topography in a straight channel with small

disturbances of variables. However, Olesen 1987 has shown that the model also yields satisfactory results in mildly curved channels.

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The flow model, which is used in the accuracy analysis, is given by equations (3.7), (3.9) and (3.11). ljR terms and the T term in the

n

transverse momentum equation are neglected. The

ü

and v momentum

equations, in discretized form, are combined to yield an equation which solves ü (see Olesen 1982b eq.13). Next

v

is computed by use of the continuity equation. By subsequent iteration the flow variables are calculated.

A steady perturbation of bed and flow variables is introduced in the discretized flow model:

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a = _a

O

+ a' a' ja

-

a sn sn sn 0

-

_{ü' ü' jü}

-U u

O

+ u sn sn sn 0 (m IJ.s IJ..n exp i _a

_O

k + j _a

_O

~) (3.35)

-

_v'

_v'

_jü

-v v sn sn sn 0

1

1

_{1 w}

1

_{with: s m t:.s, n = j IJ.n} R R' R' sn sn sn R

-

- -

_{and ljR are dimensionless complex amplitudes and IJ.s}and IJ.n

a, u, v are

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grid dimensions. The perturbations are stationary.

Inserting these perturbations in the equation which combines u and v momentum, yields an equation for the complex amplitude of the u component.

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-

u

p (3.36)

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in which:

€

= k IJ.sja

o

and ~ = ~ IJ.nja

o

Ü and (ljR) = complex amplitudes at iteration number p.

p p

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22

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From the discretized continuity equation, Olesen 1982b eq. 29, follows the equation for the complex amplitude of the v component:

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_

v

p (3.37)

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From the discretized equation of streamline curvature, Olesen 1985 eq. 28, follows:

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(3.38)

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Combining the three above equations gives a differential equation for the streamline curvature:

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(3.39) with:

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(3.40)

F2

= iZ

W

/::"n(l /::,.s+ 3 1\

2 2

À uS w i2tan(çj2)]sin2(çj2)cos<{);2 cos '1 [itan(çj2) + ~s]-l (3.41)

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The solution of equation (3.17), with ljR = 0 as initial condition

reads:

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(3.42)

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At this stage the ratios of the complex amplitudes

ü jä,

_p

v

_{p _ _}

jä

(ljR) jä_{p_} -in the discretized model are known. The ratio of amplitudes uja and v/a of the analytical solution are given by equations (3.29) and (3.30).

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The amplitude of the streamline curvature is:

1

_

i1riv

R

(3.43)

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23

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By disregarding the phase shift between discretized and analytic

solution, the ratio of complex amplitudes is to be used as a measure of accuracy. For computations AO to A9 lü I/Iül, Iv I/lvi_{p p} , I<l/R) I/Il_p /RI have been calculated over a range of 0.05 < i < 2.0. Maxima of these ratios lie in the range 0.05 < i < 0.5. In these computations several

parameters are varied. From these computations it is concluded that the parameters which affect the accuracy are C, a and ~s. In computations AO

,A2 and A5 to A9 these parameters are the same, which leads to the same results of the accuracy analyses. In table 5 the minimum and maximum values of the ratios lü I/Iül, Iv I/lvi, I(l/R) I/Il/RI within the range

p p p

0.05 < re(2)< 0.5 and im(2)=0 are given at iteration p=lOO,. This is the range of interest while the experiment is characterized by re(2)=0.23.

Within the first iterations, p<lO, these ratios are smaller.

Calculations at p=200 and p=lOO give identical results. The calculated ratios are close to 1.

Morphological computations are very sensitive to calculated flow v eloci-ties because transport rates are strongly non-linear functions of the flow velocity. At 2=0.25, ~n=0.125 and ~s=0.32 (all computations except C3 and C4) the lü I/Iül ratios are equal to: lü 1/lül=1.00 . On basis of

p p .

these results it is concluded that the grid dimensions of ~n=0.125 mand ~s=0.32 are sufficient to assure accurate numerical computation.

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3.3.4. Stability of the numerical computations

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Stability of the numerical model is calculated by inserting time dependent harmonic perturbations in the numerical model. Like in the accuracy analysis the l/R and the Tn term in the v-momentum equation are

neglected. In contrast to the calculation of accuracy, the bed

topography is allowed to change in time. Each subsequent iteration of the model uses the latest calculated bed topography. Based on the grid dimensions, the time-step and the physical parameters of the problem it is calculated whether these perturbations tend to grow or decay in time.

In case these perturbations grow in time, the computation will be

unstable. The following perturbations are substituted in the discretized equations:

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1

R

number of the morphologieal model.

amplifieation factor.

The real part of ~ represents a time-dependent harmonie perturbation of

n

the numerical model superimposed on spaee-dependent harmonie

perturbations with wave numbers k and ~. The imaginary part of ~n is responsible for growth or deeay of these time dependent perturbations.

Stability of the numerical model requires: im(~ )

<

0 n

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_a

-

_{aO+ a' a' /a} sn sn sn 0 u uO+

ü

'

_{ü' /ü}

sn sn sn 0

-v'

v

'

/ü

v sn sn sn 0

1

1

1

_\ol R R' R' sn sn sn in whieh: 0 time-step ~n numerical

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24

-a exp i(m ~s k + j ~n ~ - 0 ~t ~ ) (3.44) aO aO -0 n

-u

-v

The solution is given by Olesen 1985 and Olesen 1987. However, the formulae are different in both reports. The solution has been reealeulated.

The solution reads:

S a

-exp(-i~ ~t)- 1 = ~t_Q[i_Q ~ bsin(e/2)Y

n 2 \ol uS

-"o

a

. aO \ol

f3

-+

~u-

Aneos(e/2)(1 + ---)sin(~):

w u Àsf A u

i2 ~s ~ tan(e/2) a a

O

w a

O

aO W 2 aO \ol 2

+

(w-

~s) (eos(e/2) - 1)2e +

(w-

~n) (cos

r

n) - 1)2G)

The ratios of complex amplitudes u/a, v/ä and (l/R)/ä are calculated by the set of equations (3.36), (3.37) and (3.42). It is assumed that sufficient iterations of the flow model have been performed, so that these ratios are only dependent on the discretization error. In that case: (1/R)/ä=F2/Fl. Using this value ü/ä and v/ä are calculated. The amplification within one time-step is given by:

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a(t+~t)a(t) = eim(~ ~t)n

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(3.45)

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25

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The time-steps used in computation A5 to A9 are based on predictions by eq. (3.46). The amplification is also calculated for the other numerical simulations. Results are summarized in table 6. Within the range of interest, 0.05 <re(i)< 0.5, calculations indieate stabie computations. Amplification is smaller than the critical value of: a(t+~t)/a(t)-l.OO Caleulations up to re(i)=2.0 also indieate stable computations for the reported time-steps. Time-steps much larger than the reported values often lead to a(t+~t)/a(t»l.OO For eomputation A9, for instanee, a stable solution is also predieted for a time-step of 10 times the value in A9. using a time-step 100 times the value in A9 leads to the

prediction of an unstable model.

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Whenrun witthe stabie moh larger time-steps, the numerical model becomes unstablerphologieal eomputations, as reported in table 3., are although a stabie computation is predieted for these time-steps. This discrepancy is illustrated by eomputations A3, A4 and A6. These

computations are unstable despite the predietion of stabie eomputation. It is to be concluded that this stability analysis ean not be used as a conservative prediction of stability for this partieular experiment. This will probably be due to the neglect of l/R terms and the

linearization employed in the analytical model.

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3.4. The optimized simulation

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It has been tried to optimize the simulation with the aid of the analytical solutions, the aeeuraey and the stability analyses. It is eoneluded, however, that results of these solutions and analyses are not very good eorresponding with the numerieal eomputations. Although the analyses are useful in guiding the choiee of optimal parameters, the best suited parameters are found by performing many computations with the numerieal model. Computation A9 is considered to give the best

result of all eomputations. Computations A7 and AB have been made in the course of optimizing the eomputations preceeding A9. Parameters used in these cases are ehosen such that the amplitude of first pool and point-bar are maximized.

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26

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3.4.1. The numerical computation

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In performing the numerical computations with the morphological model the following parameters have been varied: C,

p,

GO' band e. The constant in Àsf model for the adaptation of the secondary flow, the

constant k of the secondary flow term and the sediment direction model ~r/2

G=GO(8/80) (except A6) have not been varied.

In the computations it is necessary to optimize the

P/GO

ratio to im-prove the transversal bed slope. This is effected by decreasing GO'

The -1/2 exponent in the sediment direction model (3.17) has been chosen because this value will result in the most pronounced effect of 8 on the direction of sediment transport. The effect of this choice will be a small angle between the direction of the sediment transport and the wall shear stress at the inner side of the bend. At the inner side the

transport is very small, making the influence of this choice of minor importance for the locally calculated bed-slope. At the outer side of the bend the calculated lateral bed slope will be enhanced, compared to calculations with higher exponents.

The effect of secondary flow convection is modelled on basis on

experience of Olesen. The secondary flow convection is formulated by the following term in the combined momentum equation of the flow model, see

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eq. (3.9):

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in which i is secondary flow intensity: i

s s

-a

ua

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At the inner and the outer wall the secondary flow convection is zero.

In these computations this has been effected by modeling i as a

s

parabolic function of the tranversal coordinate. Instead of an parabolic function an higher order function can also be selected, however the effect on the solution of the model is of minor significance.

Olesen concluded that k should be taken twice the theoretical value to sn

obtain the best results. In these computations k =0.90 is selected, sn

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circa twice the theoretical value.

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27

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3.4.2. Comparison of measurement and computation

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The best results are obtained with computations A7, A8 and A9. In A7 the model for the direction of bed-load transport is slightly altered. The GO coefficient is modelled independent of local bed shear-stress. By comparison of the results with A8 no significant differences are noticed.

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The results of the computations A8 and A9 are given in figures 8, and 9. The measurements are also given in these figures. Between the results of these computations only small differences are noticed. Between the computations and the measurement considerable differences are present.

The water depth at the location of the point-bar is computed reasonable weil. The depth of the corresponding pool is also weil computed. However the harmonic oscillation of bed topography downstream of this location is absent in the computations.

Near the bend entrance the bed topography computed by the numerical model responds more quickly than in the measurements. In the model adaptation of the secondary flow is accounted for (Àsf=6.7a).

Computations with a value for Àsf two times greater display nearly the same results. The amplitude of bed response is largest in computation A9. This has been achieved by a smaller GO value. Computation A9 is considered to give the best response of the numerical model for this particular experiment.

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3.4.3. Discussion of model performance

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The use of the Olesen morphological model on this specific problem is discussed.

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Simulation with the Olesen morphological model of this particular bed-load experiment is very difficult. In the first approach the model, using standard parameters, gives a far too small amplitude of the bed oscillations. As the transport rate was close to the initation of motion some of the parameters have to be modified. To improve the transversal slope, the parameters of the model for the sediment direction have to be

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28

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changed. By changing the parameters of the transport and the sediment

direction models the depth at the first pool and point-bar can be improved. However, a too drastic change of parameters of the transport model leads to instability of the numerical model. The finally chosen parameters give a response of the model which is characterized by a too quick response of bed topography near bend entrance and absence of the harmonic part.

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Numerical solutions, stability analyses and analytic solutions

give contradictory results. For the parameters used in the optimized computation A9 the analytical solution gives a wave length twice the measured one. When the numerical model is unstable the stability analysis predicts a stabie model. It is to be concluded that for this specific experiment the stability analysis and analytic solution are less appropriate.

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It is questionable whether the model formulation is appropriate for this particular experiment. The flow model is based on the similarity concept for flow velocity. In the derivation of the equations for this model it is also assumed that the streamlines and the co-ordinate system nearly coincide.

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The transport rate is near to the threshold of motion, this is a very complicating facto

In order to improve the computations a GO value is used which lies outside the range of reported experimental values. It is questionable whether this GO value is representative for this particular experiment or whether it is an artifact only to improve the computations.

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more favourable experimental parameters have to be chosenIn order to obtain data better to be computed by the numerical model,. The water-depth has to be decreased to give a more favourable width-water-depth ratio, necessary for a good performance of the flow model. The transport stage has to be increased, because close to initiation of motion the

prediction of the transport model is uncertain owing to the strong non-linearity of the physical process.

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29

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4. CONCLUSIONS

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Results of a bed-load laboratory experiment and its numerical

computa-tion are compared. The experiment shows a persistent harmonic

oscillation of the bottom topography. Using standard parameters in the

numerical computation, important differences between the experiment and

the computation are noticed. The transverse bottom slope throughout the whole bend is underpredicted and the harmonie oscillation of the bottom topography is absent.

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The numerical simulation is optimized with respect to the physical and

the numerical parameters to improve the results. Different models are

used for the direction of the bed shear stress, the transverse bed slope

and different exponents are used in the transport model, while the grid dimensions and the time-step are varied also.

In the process of optimizing the simulation of this particular

experiment a simple variation of the model parameters of the numerical

model is not sufficient. It is useful to calculate analytically the

prevailing wave length and the damping of bottom topography oscillations. The numerical accuracy as function of the physical

parameters and the grid dimensions is estimated by computation. Also

the stability of the numerical model with respect to the time-step is predicted by computation.

The theoretical models on which these calculations are based upon have

been reported by Olesen. Some typographic errors are present in his

formulae. By theoretical investigation these models have been

reformulated, and the resulting revised formulae are presented.

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The optimized computation shows at the location of the first pool and point-bar a bottom topography which fairly agrees with the measurements. Further downstream, however, the harmonic oseillation is absent.

In the computation the coefficient GO in the transverse bed-slope model is taken significantly smaller than the recommended value. The exponent b of the generalized transport formula is taken twice the common used

value (b=5 Engelund Hansen).

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30

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Using these parameters, however, the analytical calculation shows a

pre-vailing longitudinal wave with a wave length twice the measured one. The analytical calculation also shows an harmonie oscillation.

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It has not been succeeded to explain the strong harmonie oscillation of

the bed by numerical simulation. A probable cause could be the small width-depth ratio, which causes deviations of the underlying similarity concept. It is thought, however, that the most important drawback of the experiments lies in the transport rate of sediment.

The transport rate is near the initiation of motion, 8~0.06. At this Shields number the transport rate is a very non-linear function of the

velocity. According to a linearization of the Meyer-Peter

&

Müller formula an exponent of b~25 should be taken in the transport model. Using this very large exponent yields to instability of the numerical model, even though the stability analyses indicate stabie computations.

This demonstrates the drawback of these analyses, curvature of the

channel is neglected and linearization of the equations is employed. The discrepancy between the measured wave lengths and the analytical

solution is also attributed to this problem. However in case of more appropriate conditions these theoretical calculations can be used as a powerful tooi to guide the final choice of optimal grid dimensions and time-step.

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Due to the limitations of the numerical and the theoretical models the parameters of the experiment will have to be adapted to comply with the validity of the model formulation. To obtain better agreement between experiments in the L.F.M. curved flume and these modeis, the transport rate is to be increased and, if possible, the width-depth ratio should be increased also.

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REFERENCES

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Olesen, K.W. 1982a

Influence of secondary flow on meandering of rivers

Rep. 1-82, Lab. Fluid Mech., Dept. Civil Engrg., Delft Univ.

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Olesen, K.W. 1982b

Introduction of streamline curvature into flow computation for shallow river bends, Rep. 5-82, Lab. Fluid Mech., Dept. Civil Engrg., Delft Univ.

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OleseFlow and bed topography computation for shallow channel bendsn, K.W. 1985 Communications on Hydraulics 85-1, Dept. Civil Engrg., Delft Univ.

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Olesen, K.W. 1987

Bed topography in shallow river bends

Dissertation: Lab. Fluid Mech., Dept. Civil Engrg., Delft Univ.

also as: Communications on Hydraulics and Geotechnical Engineering 87-1, Dept. Civil Engrg., Delft Univ.

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Struiksma, N.Point bar initiation in bends of alluvial rivers with dominant bed load1983 transport

TOW rep R657-XVllj308-PART 111, Delft Hydr. Lab.

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Bed deformation in curved alluvial channels.Struiksma, N., Olesen, K.W., Flokstra, C., Vriend, H.J. De 1985 J. Hyd. Res., vol. 23, no 1, p.57

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Talmon, A.M. Marsman, E.R.A. 1988 A curved flume bed-load experiment.

Rep.4-88 , Lab. Fluid Mech., Dept. Civil Engrg., Delft Univ.

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tab1e 1. Parameters of the bed-1oad experiment.

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channe1 width W 0.5 [m]

charme1 radLus R 4.1 [m]

mass flow Q 0.0115 [m3Is]

water depth at entrance aO 0.07 [m]

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depth-averaged velocity u 0.33

[

mis

]

longitudina1 slope i 1.25 10-3

d10 0.7110-3[m3] sediment diameters d50 0.7810-3[m3]

-3 3

dgO 0.85 10 [m]

sediment mass flow Q d 2.3 [kgjh] dry

san -7 2

(per unit width: So =8.0 10 [m

Is

]

(at r=0.4))

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tab1e 2. Dominant parameters of the experiment

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C (=u/~ ) 35 mO.5/s W

_I

aO 7.14 W

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R 0.122

8 ( _aOi

I(IJ.

dgO) ) 0.062

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tab1e 3. Type A: Numerical computations with the morpho1ogica1 model

No Chezy

W/aO

_f3

6.s 6.n 6.t 6.t*SO b e _GO [m] [rn ] [s] [m2] A9 35 7.14 9.7 0.32 0.125 138 l.le-4 10 4 0.5 A8 35 7.14 9.7 0.32 0.125 138 l.le-4 10 4 0.6 A7 35 7.14 9.7 0.32 0.125 138 l.le-4 10 8 0.6 [G=G

o]

A6 35 7.14 9.7 0.32 0.125 14 1.le-5 15 10 0.6 unstab1e A5 35 7.14 9.7 0.32 0.125 400 3.2e-4 5.5 4 0.6 A4 35 7.14 9.7 0.16 0.125 200 l.6e-4 5.5 4 0.6 A3 35 7.14 9.7 0.16 0.125 53 4.2e-5 26 1l.5 l.15 unstab1e A2 35 7.14 9.7 0.32 0.125 28 2.2e-5 26 1l. 5 l.15 unstab1e Al 49 8.1 10.5 0.32 0.125 103 8.2e-5 7 2 2 AO 35 7.14 9.7 0.32 0.125 200 l.6e-4 5.5 4 3.0

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tab1e 4. Type B: Ca1cu1ation of dominant wave numbers

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input parameters: dominant wave number

No Chezy W/aO

_f3

_Àsf/h b e _GO re(.2) im(.2)

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_{B11 35 7.14 9.7 0 10 4 0.5 0.084 0.039}

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BlO 35 7.14 18 0 26 4 2.3 0.19 0.094 B9 35 7.14 18 0 5.5 4 1.2 0.23 0.11 B8 35 7.14 9.7 0 26 11.5 2.3 0.19 0.16

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B7 35 7.14 9.7 0 26 11.5 1.15 0.11 0.039 B6 35 7.14 9.7 0 10.4 3.7 1.15 0.19 0.14

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B5B4 352l.8 78.1.14 98.01.7 00 106.4 1.53.7 01.15.6 00..087087 0.050.074 B3 21.8 8.1 3 0 6 1.5 1.15 0.18 0.11

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B2 49 8.1 10.5 0 20 8.5 4 0.30 0.23 B1 49 8.1 10.5 0 7 2 2 0.30 0.19

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BO 35 7.14 9.7 0 5.5 4 3 0.42 0.30

tab1e 5. Type C: Accuracy of the numerical model

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No corr. lü I/Iül_p Iv I/lvi_p 1(1/R)pl/11/RI lü I/Iül_p

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with ( range: 0.05 <re(,e)< 0.5 ) at re (,e)=0 .25

min. max. min. max. min. max.

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C9 A9 0.90 1.02 0.99 1.00 0.94 1.00 1.00 C8 A8 0.90 1.02 0.99 1.00 0.94 1.00 l.00

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C7C6 A7A6 0.900.90 1.021.02 0.990.99 1.001.00 0.940.94 1.001.00 l.00l.00 CS A5 0.90 1.02 0.99 1.00 0.94 1.00 l.00

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C4 A4 0.95 1.03 l.00 1.00 1.00 l.00 l.01 C3 A3 0.95 l.03 l.00 l.00 l.00 l.00 l.01

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C2Cl A2Al 0.910..90 1.021.01 00..9999 1.001.00 00.94.94 1.001.00 l.001.00

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34

tab1e 6. Type C: Stabi1ity of the numerical model

No corresponding a(t+.6.t)/a(t) .6.t[s

1

with (range:

o .

05<re

0

)<0.5)

C9 A9 0.997 <1.000 138 C8 A8 0.997 <1.000 138 C7 A7 0.993 0.999 138 C6 A6 0.999 <1.000 14 C5 A5 0.989 0.996 400 C4 A4 0.995 0.998 200 C3 A3 0.996 <1.000 53 C2 A2 0.998 <1.000 28 Cl Al 0.994 0.995 103

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