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JOURNAL OF HYDRAULIC RESEARCH, VOL. 39, 2001, NO. 2 147
Fig. 1. Alternate bars in a channel as measured in a straight laboratory
flume Lanzoni [9]. The crests alternate between the side banks.
(The flow is from right to left.)
Height and wavelength of alternate bars in rivers: modelling vs. laboratory experi-
ments
Hauteur et longueur d’onde des bancs alternés en rivières : la modélisation faceaux
expériences de laboratoire
M.A.F. KNAAPEN, Dep. of Civil Engineering, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands.
S.J.M.H. HULSCHER, Dep. of Civil Engineering, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands.
H.J. DE VRIEND, Dep. of Civil Engineering, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands.
A. VAN HARTEN, Dep. of Management Studies, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands.
ABSTRACT
Alternate bars are large wave patterns in sandy beds of rivers and channels. The crests and troughs alternate between the banks of the channel. These
bars, which move downstream several meters per day, reduce the navigability of the river. Recent modelling of alternate bars has focused on stability
analysis techniques. We think, that the resulting models can predict large rhythmic patterns in sandy beds, especially if the models can be combined
with data-assimilation techniques. The results presented in this paper confirm this thought.
We compared the wavelength and height of alternate bars as predicted by the model of Schielen et al. [14], with the values measured in several flume
experiments. Given realistic hydraulic conditions > 2*10³, (Rthe width-to-depth ratio and Rethe Reynolds number), the predictions are in good
R Re
agreement with the measurements. In addition, the model predicts the bars measured in experiments with graded sediment. If < 2*10³, the agree-
R Re
ment between model results and measurements is lost. The wave height is clearly underestimated, and the standard deviation of the differences between
predictions and measurements increases. This questions the usefulness of small flume experiments for morphodynamic problems.
RÉSUMÉ
Les bancs alternés sont des configurations de grandes ondulations dans les fonds sablonneux des rivières et des chenaux. Les crêtes et les creux alternent
entre les rives du chenal. Ces bancs, qui se déplacent vers l’aval de plusieurs mètres par jour, réduisent la navigabilité de la rivière. La modélisation
récente des bancs alternés s’est concentrées sur les techniques d’analyse de stabilité. Nous pensons que les modèles résultant peuvent prédire les grandes
configurations régulières dans les lits sablonneux, notamment si les modèles peuvent être combinés à des techniques intégrant des données. Les résultats
présentés dans l’article confirment cette idée. Nous avons comparé la longueur d’onde et la hauteur des bancs alternés prédites par le modèle de Schielen
et al., avec les valeurs mesurées dans plusieurs expériences en canal. Des conditions hydrauliques réalistes étant données, > 2.10³ (Rétant le rapport
R Re
de la largeur à la profondeur, et Rele nombre de Reynolds), les prédictions sont en bon accord avec les mesures. En outre, le modèle prévoit les bancs
mesurés dans les expériences avec des sédiments calibrés. Si < 2.10³ , on perd l’accord entre les résultats du modèle et les mesures. La hauteur
R Re
des ondulations est nettement sous-estimée et l’écart-type des différences entre prévisions et mesures augmente. Ceci remet en cause l’utilité des
expériences à petites échelles pour les problèmes de morphodynamique.
1 Introduction
The interaction between a non-cohesive bed and the water flow-
ing over it results in interesting phenomena. Several types of
wave patterns can be seen on the bed, each caused by a different
process. The largest bedform observed in fixed-bank rivers and
channels is termed alternate bar. Alternate bars are wave patterns,
of which the crest and trough alternate between the banks of the
channel (see Figure 1). These bars move downstream at a speed
of several meters per day. Their existence reduces the navigability
and influences the water capacity of the channel. Therefore, it is
important to predict the behaviour of these bars.
The existence of alternate bars is thought to be an inherent insta-
bility of the bed-flow system. Therefore, state of the art research
into modelling of alternate bars has focused on stability analysis
techniques [2][3][4][14][16][18]. Both Colombini et al. [4] and
Schielen et al. [14] formulate a weakly non-linear model, which
allows for small temporal variations of the amplitudes of the bars.
Only the latter model also allows for slow spatial variation.
Colombini and Tubino [3] developed a fully non-linear model.
The model of Schielen et al. [14] describes behaviour, similar to
the behaviour of alternate bars in rivers. However, to predict us-
ing their model, more lengthy and error-prone mathematical deri-
vations are necessary. Data-assimilation techniques [5] may be
the solution to avoid these derivations. Further, these techniques
can generalise the results to other large rhythmic bed waves, like
sand waves and shore parallel bars in the sea. As a start, such an
approach requires the model to estimate the characteristics of the
patterns correctly.
So far, the models have hardly been tested against either field or
laboratory data. Schielen et al. [14] used their model for a qualita-
tive analysis of the phenomenon only. Colombini et al. [4] com-
pared the results of their model with data from small-scale experi-
ments, in which the width of the flume never exceeded 0.7 m.
148 JOURNAL OF HYDRAULIC RESEARCH, VOL. 39, 2001, NO. 2
run T Q i
b
h
∗
uC
z
F
r
L
b
H
b
[h] [m
3
/s] [%] [cm] [m/s] [m
1/2
/s] [-] [m] [cm]
uniform
sediment
P1801 816 0.03 0.16 7.3 0.27 25.2 0.32 11.3 8.5
P2403 260 0.047 0.21 8.3 0.38 28.9 0.42 4.5-7.5 5.0-6.0
P0404 192 0.04 0.2 7.7 0.35 27.8 0.4 4.3-8.0 6.0
P1505 28 0.03 0.45 4.4 0.45 32.2 0.69 10 7.0
P1605 24 0.02 0.5 3.3 0.4 31.6 0.71 11 7.7
P2709 24 0.045 0.51 5.7 0.53 30.7 0.7 9.7 4.5
P2809 24 0.04 0.52 5.3 0.5 30.4 0.7 10.6 4.7
P2909 24 0.045 0.52 5.6 0.53 31.3 0.72 9.5 4.4
graded
sediment
P0807 73 0.03 0.42 0.043 0.47 34.7 0.71 10.4 0.042
P0109 51 0.04 0.51 0.047 0.57 36.1 0.83 11.7 0.04
P1309 29 0.045 0.53 0.05 0.6 36.7 0.85 10.3 0.034
P2009 3 0.045 0.53 0.058 0.6 36.6 0.85 10.2 0.034
Table 1 Conditions and results of experiments on alternate bars by Lanzoni [9][10]. Here Tgives the duration of
the experiment, Qthe average water flux, ibthe longitudinal water surface slope, h*the average water
depth, uthe average flow velocity, Czthe Chezy coefficient and Fr the Froude number. Lband Hbare the
length and the height of the bars respectively
b
zi
h
u
C
∗
=(1)
The differences between their model results and the measure-
ments were imputed on the scale effects due to this small channel
width. Recently, Lanzoni [9][10] performed morphological labo-
ratory experiments in a flume of 1.5 m wide, 1 m deep and 50 m
long. The experiments in this flume, and therefore the results are
assumed more realistic.
In this paper, the wavelength and height of alternate bars as pre-
dicted by the model of Schielen et al. [14] are compared with the
length and height of alternate bars measured in several flume ex-
periments. The large flume experiments of Lanzoni [9][10] are
included in this comparison. The comparison leads to an assess-
ment of the validity of using a weakly non-linear stability analysis
to predict alternate bar behaviour.
2 Available results of laboratory experiments
The available data of experiments can be divided in 2 groups. The
first group consists of experiments carried out by Lanzoni [9][10].
These experiments are well documented. The second group con-
sists of experiments by several researchers [1][6][7][12][17].
Only little information on these experiments is available to the
authors. However, this group is much larger then the first one.
2.1 Lanzoni’s experiments
In the large straight sand flume of Delft Hydraulics, Lanzoni
[9][10] generated alternate bars, using steady flow conditions.
The flume is 1.5 m wide, 1 m deep and 50 m long. The
bathymetry can be measured over 43.8 m. During the experi-
ments, all flow characteristics were controlled. Both the water
depth and the flow velocity were held constant at the required
values. The sand leaving the flume was weighed and subse-
quently fed back at the upstream end of the flume, evenly spread
over the width. The experiments were divided into two series. In
the first series the sediment was uniform (d50 = 481 µm, d90 = 710
µm, ρs= 2.65 g/cm³). In the second series graded sediment was
used (d50 = 262 µm, d90 = 3210 µm, ρs= 2.65 g/cm³). Table 1
summarises the conditions of the experiments using both uniform
and graded sediment.
A water-level indicator and a profile indicator measure the water
level and the bed profile respectively. In bursts of 4 to 6 minutes,
these measurements were taken over three longitudinal sections,
one in the middle and the other two at 0.20 m from each bank.
The interval between two successive measurements depended on
the bed form celerity. At the end of an experiment, more mea-
surements of the bathymetry were performed at 0.40 and 0.60
meter from each wall. The sediment transport rate was deter-
mined from the immersed weight of the sediment collected at the
end of the flume. The Chezy coefficient was estimated from the
measured water depth using:
where h*is the average bottom depth, ibis the bed slope, and uis
the average flow velocity.
The measurements of the water level and the bed level can be
used to determine the water depth: h*=ζ–z
b, in which zband ζ
are the bed-level and the water level, respectively. Figure 2 shows
an example of three resulting longitudinal bed profiles.
For each section, the average longitudinal bed slope (ib1,ib2 and
ib3) is calculated using linear regression. The noise, caused by
JOURNAL OF HYDRAULIC RESEARCH, VOL. 39, 2001, NO. 2 149
Fig. 2. Longitudinal bed profile measured during experiment P1801.
Top down is subsequently shown: the profiles along a section
near the left bank, in the middle and near the right bank of the
flume.
Name y
∗
#d
sρsuh
∗
[m] [-] [mm] [g/cm3] [m/s] [c m]
Ashida 0.5 13 1 2.65 0.30-0.50 2.0-3.5
Jaeggi 0.3 13 0.52 2.65 0.25-0.50 1.6-2.8
0.3 23 3 1.45 0.15-0.40 1.6-4.1
0.3 11 1.8 2.65 0.35-0.50 1.4-4.1
0.3 1 4 2.65 0.37 2.44
Kinoshita 0.132 4 0.38 2.65 ±0.30 0.8-1.1
0.132 3 0.76 2.65 0.35-0.45 0.6-1.2
0.132 14 1.24 2.65 0.30-0.50 0.4-1.2
0.132 5 1.7 2.65 0.30-0.50 0.7-1.7
Muramoto 0.55 8 0.99 2.65 0.35-0.75 2.1-4.5
0.25 3 0.99 2.65 0.35-0.60 1.3-2.0
Sukegawa 0.15 5 2.3 2.65 0.40-0.60 0.6-1.9
0.15 6 0.45 2.65 0.20-0.50 0.6-1.5
0.31 8 2.3 2.65 0.40-0.65 1.7-3.7
0.31 9 0.45 2.65 0.30-0.60 1.2-2.5
0.3 9 3.55 2.65 0.60-0.90 1.0-4.2
Table 2 Conditions during several small-scale flume experiments. Here
y*is the width of the flume, #is the number of experiments,
with the range of the flow velocity and the water depth given by
uand h*, respectively. dsand ρsare the size and the density of
the sediment, respectively.
Fig. 3. Sketch explaining the definitions in the model (after Schielen et
al. [14]).
(2)
() ( )
()
0z-hUz-
tbb =+⋅∇+
∂
∂∗
ζζ rr
(3)
gUU
tψζ r
rrrr =∇+∇⋅+
∂
∂U
(4)
hz−
vuv
C−,gi)}
hz−
vuv
C−b
22
d
b
22
d
+
+
+
+
+
=∗∗ ζζ
ψb
r
small-scale ripples and dunes, is filtered out of the bottom pro-
files, using the moving-frame averaging method [13]. (The size
and shape of the windows is unknown.) Finally, the large-scale
bar characteristics are estimated. The bar height is defined as the
difference between the maximum and the minimum bed eleva-
tion, within a bar unit. Note that the filtering of the measurements
results in low estimates of the bar height, since it reduces the ex-
tremes. The bar length is estimated using a spectral analysis [13]
of the filtered bottom profile Figure 2 shows that only a few
large-scale bars exist in the flume. Therefore, the accuracy of the
height and length estimates of the bars is limited. Table 1 presents
the characteristics of the observed alternate bars.
2.2 Small scale experiments by others
In the past, several small-scale flume experiments were carried
out. Here the experimental results of Ashida [1], Jaeggi [6],
Kinoshita [7], Muramoto [12], Sukegawa [17] are relevant. In
these experiments, the width of their flumes varied from 13 to 55
cm. In the experiments, the water depth, the surface slope, and the
flow velocity were measured. The information of the sediment
used is limited: only the mean diameter and the density
characterise the sediment. Table 2 summarises the conditions un-
der which these experiments were run.
3 The amplitude evolution model of Schielen et al. [14]
3.1 Basic equations
To describe the behaviour of alternate bars in rivers, the
morphodynamic model of Schielen et al. [14], which is an exten-
sion of Colombini et al. [4], will be used. The model of Schielen
et al. [14] is discussed briefly below.
Schielen et al. [14] considered a uniform, shallow-water flow in
a straight, infinitely long channel, with a uniform, mild slope: ib
<< 1. The banks are assumed to be non-erodible and the bottom
sediment to be non-cohesive. (See the sketch in Figure 3.) The
flow is described using the mass balance and the depth-averaged
St. Venant equations:
in which represents the forcing and friction mechanism:
ψ
r
Here gis the acceleration due to gravity and zbis the elevation of
the disturbed bed relative to the undisturbed bed. ζis the eleva-
tion of the disturbed free surface, with respect to the undisturbed
150 JOURNAL OF HYDRAULIC RESEARCH, VOL. 39, 2001, NO. 2
∂
∂x,∂
∂y
(5)
2
z
d
C
Cg
=
(6)
.z-
|U|U
|U|S
b
b
∇= r
r
r
rγσ
(7)
0
S
t
b
z=⋅∇+
∂
∂
(8)
0|S0,|v0,|S0,|v
yyy0yy0y
====
∗∗
==
(9)
,0,0,0)(u)z,,v,(u b0000 ∗
=
ζ
(10)
()
dkyf
ek
tikx
0
0ω
εφφ +
∞
∫
+=
R=y*
h*
Fig. 4. Example of a neutral curve, dividing the region with a stable
basic state from the region with an unstable basic state. R is the
width-to-depth ratio; k gives the wave number of the perturba-
tions.
(11)
()
r1RR
2
c
ε+=
(12)
ε+=
1c
kkk
(13)
c.c.h.o.t)y'cos(),A(ht)y,(x,z
t'x'ki
cc
++πτχε=
ω+
∗b
e
water level h*.=(u,v) is the flow velocity vector in xand y-
U
r
direction and the operator is defined as
r
∇
.
Finally,
is the drag coefficient, in which Czis the Chezy coefficient.
The sediment is transported as bed-load, which is modelled as:
The sediment mass balance yields:
where = (Sx,Sy) is the sediment transport, in volume per unit of
S
time, in xand y-direction. Furthermore, the non-linearity of the
sediment transport with respect to the flow bis limited to 2 < b<
7. The downhill preference of the sediment transport is accounted
for by γ> 0. Both parameters are dimensionless. The sediment
transport proportionality parameter σdepends on the sediment
properties and includes the effect of the porosity of the bed.
To close the model at both side walls (y=0,y=y
*) boundary
conditions have to be defined:
The model defined by the Equations (2),(3),(6),(7) and the bound-
ary conditions (8) allows for a solution, describing uniform flow
over a plane sloping bed:
(Note that u*is the uniform flow velocity, not the shear velocity.)
3.2 Stability analysis
The starting point of every stability analysis should be a physi-
callyrelevant,exactsolutiontothe mathematical model. Equation
(9) represents such a basic solution. This basic state is perturbed
by small-amplitude, periodic bed waves:
in which = (u,v,ζ,zb) and denotes the basic state given by
φ
φ0
Equation (9). The complex morphological wave frequency
ω(k,R,b,γ,Cd) is related to the morphological wave number k, the
width-to-depth ratio
and the model parameters Cd,band γ. If the growth rate (given by
the real part of ω) is negative, the basic state is stable: the pertur-
bationsdecrease inamplitude anddisappear.Under otherphysical
circumstances, the basic state is unstable: the disturbances start to
grow, forming a rhythmic pattern.
Schielen et al. [1] substituted the perturbation (10) into the model
Equations (3)-(7), linearised for small perturbations. Thus, they
found a relationship between the complex wave frequency and the
wave number. Using this relationship, one can determine whether
the basic state is stable or unstable (see Figure 4). As can be seen
in Figure 4, a whole range of waves has positive growth rates.
However, for each condition the alternate bars have only one
wavelength. More information is needed to find this wavelength.
To derive information about the shape and the behaviour of the
bars, Schielen et al. [14] had to consider the non-linear terms. In
agreement with Colombini et al. [4], they assumed a weakly non-
linear regime:
with ε<< 1 and some unknown –1 < k1<1.(Rc,kc) is the first
combination of the width-to-depth ratio Rand the wave number
kto give an unstable basic state for increasing R(the lowest point
above the neutral curve in Figure 4). Assumption (11) means, that
the width-to-depth ratio is just large enough to give growing per-
turbations.
This approach led to the following wave [14]:
JOURNAL OF HYDRAULIC RESEARCH, VOL. 39, 2001, NO. 2 151
(14)
0A|A|
A
A
A
2
3
2
2
21
=+
∂
∂
++
∂
∂α
ξ
αα
τ
(15)
t'
2
ετ=
(16)
t'x'
k
νεχ+=
(17)
u
hig
C
2
b
d
∗
∗
=
(18)
50
s
z
2
d1-C
u
=∗
ρ
ρ
θ
(19)
()
3
50
s
2
3
cdg1--13.3S
=ρ
ρ
θθµ
(20)
θµ
θc
−1
3
b≈
(20)
4
1
c
0.75
=θµ
θ
γ
Fig. 5. The predicted wave length plotted against the measured values
(a) for all experiments and (b) for the experiments with >
R Re
2000. The dotted lines show the 20% error boundaries.
in which εis defined by Equation (11). kcand ωcare the critical
wave number and the critical frequency, respectively. Further-
more, (x’,y’,t’) are the dimensionless co-ordinates
.
y*h*
σu
b
,t
y*
y
,
y*
x
(
(
The cosine in Equation (13)models the alternation of the crest,
h.o.t. denotes the higher order terms (O (ε2)) and c.c. means the
complex conjugates. The amplitude A(ξ,τ) follows from:
This equation is known as the Ginzburg-Landau equation. Here,
τis the morphological time and ξis the morphological co-ordi-
nate in a frame moving with the group velocity νbof the bars:
The parameters αi(i=1,2,3) in Equation (14) are complex func-
tions of the drag coefficient Cdand the transport parameters band
γand σ.
3.3 Model parameters
To calculate the wave length and wave height using Equation
(13), we need values for Cd,band γ. Unlike small bedforms, al-
ternate bars do not influence the bed roughness. (During the ex-
periments the depth h*, slope iband flow velocity u*do not change
significantly, while the alternate bars develop.) Therefore, the
drag coefficient Cdcan be estimated using Equations (1) and (5):
Accurate values of exponent bare unknown. However, using
Chezy’s law:
(where θis the shields stress) and the commonly used formula by
Meyer-Peter and Müller [11], describing transport of sediment:
one can derive that:
where θcis the critical Shields parameter and µthe bed-form or
efficiency factor.
Finally, the slope coefficient γcan be estimated using the relation
derived by Sekine and Parker [15]:
4 Comparison between theoretical and experimental results
Given the conditions of the experiments described in section 2,
the model of Schielen et al. [14] predicts the wave length and
wave height of alternate bars. These predictions are compared
with the measured values. Figure 5(a) and Figure 6(a) show the
results of the wavelength and wave height predictions respec-
tively.
The model predicts the characteristics of the larger bars accu-
rately, but the predictions of the smaller sized bars are not satis-
factory. These smaller bars are measured in the small-scale flume
experiments. Figure 7 shows, that the errors in the predictions are
related to the parameter . Here is the Reynolds num-
R Re
Re=h u
**
v
ber, with νthe kinematic viscosity. More generally, used parame-
ters, like the Reynolds number or the Froude number, give no
relation with the errors.
The relation to the parameter can be explained from the as-
R Re
sumptions in the model of Schielen et al. [14]. They assume that,
in zeroth order (O(1)), the dissipation is small compared to the
advection: Re>> 1. In first order i.e. O(ε) they assume that the
total dissipation is dominated by the vertical dissipation: the
width-to-depth ratio R>1. In the model, the effect of the horizon-
tal dissipation is neglected. This assumption is valid if >> 1
R Re
Horizontal dissipation distributes the friction from the boundaries
to the inner domain. The horizontal dissipation distributes the
friction generated at the side-banks, while the vertical dissipation
distributes the friction generated at the bed. Using the reasoning
in the above, Schielen et al. [14] neglect the boundary effects near
the banks.
Due to the horizontal dissipation of the friction with the side
152 JOURNAL OF HYDRAULIC RESEARCH, VOL. 39, 2001, NO. 2
Fig. 6. The predicted wave height plotted against the measured values
(a) for all experiments and (b) for the experiments with >
R Re
2000. The dotted lines show the 20% error boundaries.
Fig. 7. Mean of the absolute errors in the wave height predictions as a
function of the dimensionless parameter , exp denotes the
R Re
measured values.
walls, the flow velocity in the boundary layer near these walls is
smaller thanthe mean velocity. The model of Schielen et al. [14]
assumes uniform flow velocities over the channel width. Conse-
quently, the model locally underestimates the drag coefficient.
Finally, in the model the bar height is proportional to the drag
coefficient. Thus, the bar height, which is measured near the
walls, is underestimated. Apparently, the effect of the boundary
layer is significant to the bar height if < 2*10³ (Figure 7)
R Re
This value has to be regarded as an empirical result. So far, we
can not derive this number from theoretical arguments.
In agreement with Colombini et al. [4], the authors assume that
this is an artefact of the small channel width in the experiments.
The dynamics in the small flume experiments differ significantly
from the dynamics in the field, where in general > 2*10³.
R Re
Therefore, one has to consider, whether the results of small-scale
flume experiments are useful in case of morphodynamic prob-
lems.
If > 2*10³, the model estimates both wave length and wave
R Re
height accurately, as can be seen in Figure 5(b) and Figure 6(b).
(The standard deviation is of the same order as the measurement
noise, which is merely caused by small bed forms. Even in the
strong non-linear cases, in which the width-to-depth ratios are
large compared to the critical values, the results are good. This is
striking, since the model is based on the assumption that the
width-to-depth ratio is close to its critical value. This means that
the model is useful outside the theoretical restrictions.
In the Lanzoni experiments, the predicted wave heights are high
compared to the measurements. However, the errors are small and
may be caused by the filter method used by Lanzoni. As stated in
section 2.1 the moving-average method results in low estimates
of the wave height.
In the experiments with larger flumes, the predicted wavelengths
coincide with the measured values. Most errors are less than 10%
of the values, which is small compared to the measurement noise.
The two outliers in the Lanzoni experiments (P2403,P0404) can
be explained (see [8]). In one case, the power spectrum is double
peaked. The wavelength of the peak that is disregarded by
Lanzoni [9] coincides with the predicted values. In the other case,
there is no clear alternate bar pattern. The outliers in the small-
scale flume experiments are related to a high Froude number (Fr
> 1). Schielen et al. [14] assume a Froude number < 1.
The model predicts the wavelength and height of the alternate
bars equally well in experiments with graded sediment as in ex-
periments with uniform sediment. Apparently, the grading of sed-
iment has no significant influence on the behaviour of the alter-
nate bars. Therefore, it can be concluded, that the sediment trans-
port parameters capture the effect of the sediment composition
sufficiently.
5 Conclusions
The model of Schielen et al. [14] predicts the wavelength and
height of alternate bars accurately in flume experiments if >
R Re
2*10³. For these experiments, the errors can be considered as
white noise. The standard deviation is of the same order as the
measurement noise, which is merely caused by small-scale bed
forms, such as dunes and ripples.
There are no differences between the results in the experiments
using uniform sediment [9] and the experiments using graded
sediment [10]. The model parameters capture the effects of the
sediment composition sufficiently.
However, if < 2*10³, the measured heights are significantly
R Re
larger than predicted. The model used neglects the dissipation
near the side banks. This results in an underestimation of the al-
ternate bar heights. The dynamics in the small-scale flume experi-
ments differ significantly from the dynamics in the field. This
questions the usefulness of small flume experiments for
morphodynamic problems.
We conclude, that the model predicts the basic characteristics of
alternate bars well, even far from critical conditions. This justifies
further research into predicting alternate bar behaviour, using the
model of Schielen et al. [14] in combination with data-assimila-
tion.
6 Acknowledgement
The authors thank the Dutch Organisation for Scientific Research
JOURNAL OF HYDRAULIC RESEARCH, VOL. 39, 2001, NO. 2 153
symbol description unity
A scaled alternate bar amplitude [-]
Cddrag coefficient [-]
CzChezy coefficient [m½/s]
FrFroude number [-]
Hbalternate bar height [m]
Lbalternate bar length [m]
Q water flux [m3/s]
R width-to-depth ratio [-]
ReReynolds number [-]
S sediment transport vector [m3/s]
T duration of experiment [s]
U flow velocity vector [m/s]
b on linearity of sediment transport [-]
d grain diameter [µm]
g gravity constant [m/s2]
h water depth [m]
iblongitudinal free surface slope [-]
k morphological wave number [-]
t time [s]
u,v longitudinal and transverse flow velocity [m/s]
x,y,z longitudinal, transverse and vertical position [m]
y
∗
channel width [m]
zbelevation disturbed bed [m]
φState vector of the morphological problem [-]
ΨForcing and friction term in the flow equations [-]
α1exponential amplitude growth coefficient [-]
α2horizontal amplitude variation coefficient [-]
α3non-linear amplitude decay coefficient [-]
εsmall parameter [-]
γdownhill preference of sediment transport [-]
µbed form factor [-]
νbgroup velocity of the alternate bars [-]
ψforcing of the flow [m/s2]
ρdensity [g/cm3]
σsediment transport proportionality [-]
τmorphological time [-]
θShields parameter [-]
ξmorphological length [-]
ζelevation disturbed free surface [m]
ωmorphological wave frequency [m/s2]
subscripts
50 median
90 90\% is smaller
ccritical value
sof sediment
xlongitudinal component
ytransversal component
∗
average value
0basic solution
bf the river bed
superscript
' scaled value
Table 3. List of symbols, subscripts and superscripts
(NWO) for funding this work through the ’Non Linear Systems’
priority program (under project number 620-61-349). It is linked
to the PACE-project in the framework of the EU-sponsored Ma-
rine Science and Technology Program (MAST III), under con-
tract number MAS3-CT95-0002.
7 Symbols
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