(1)
K. Blanckaert and W.H.
Graf
Laboratoire de Recherches
Hydrauliques, Ecole Polytechnique FÈdÈrale
LRH-EPFL, CH-1015 Lausanne,
Switzerland
Tel: +41-(0)21-6932375; fax:
+41-(0)21-6936767; E-mail: lrh@epfl.ch
Abstract: The secondary circulation in
open-channel bends largely determines the bed topography. It is often described
by the ¡°Rozovskii model¡± in depth-integrated flow models. Our experimental
results indicate that the ¡°Rozovskii model¡± has a tendency to overpredict
the strength of the secondary circulation and its effect on the velocity
distribution for the case of strongly curved open-channel flow. Both also
decrease with increasing curvature ratio; this is in contrast with the unique
dependence on the Chezy coefficient predicted by the ¡°Rozovskii model¡±.
Keywords: strongly curved flow, open-channel
bend, secondary circulation, experiments
The most
characteristic feature of curved flow is the helical flow pattern, also known as
secondary circulation. It advects flow momentum and redistributes transversally
the velocity and the boundary shear stress over the bend. Furthermore, the
direction of the bottom shear stress, and thus also of the sediment transport,
directly depends on the strength of the secondary circulation. As a consequence,
the evolution of the secondary circulation will largely determine the resulting
bed topography with its characteristic bar-pool formation.
Since fully-3D flow models are not
yet feasible for engineering problems concerned with the river morphology, most
often 2D depth-integrated flow models are used. By depth-integrating the flow
equations, all information on the secondary circulation is lost. However, as
mentioned above, it is essential to account for the effect of the secondary
circulation and this can only be done by providing it as input to the model.
This input is mostly based on simplified expressions for the velocity profiles,
which have been proposed by Rozovskii (1957).
In this paper, experimental data is
presented that illustrates the shortcomings of this approach for strongly curved
flows. Blanckaert (2001) presents a model for the velocity profiles in strongly
curved flows that explains the features observed in the here reported
experiments.
The distribution of the
depth-averaged downstream velocity, Us,
is governed by the depth-integrated downstream momentum equation. Assuming a
hydrostatic pressure, this equation is (Dietrich and Whiting, 1989):
(1)
where tbs is the downstream component of the bottom shear
stress, r is the water
density, g is the gravitational acceleration, h is the local flow depth, R is
the centreline radius of curvature, (1+n/R)
is a metric factor, zS is the elevation of the water surface above
the horizontal reference plane (s,n) (for which the flume-average bottom
level is chosen), tns is a component of the Reynolds tensor and vj (j=s,n,z) are the time-averaged
velocity components along the reference axes. The s-axis follows the channel centreline,
the n-axis is perpendicular to it and
points towards the outer bank and the vertical z-axis is positive in upward direction (Fig.1a). The brackets, ,
indicate depth-averaged values.
In a 2D straight
uniform flow, the bottom shear stress tbs/r is in equilibrium with the energy expenditure G
in all points (s,n) of the flume, and the momentum
equation reduces to tbs/r=G. In a 3D flow, the velocities are
non-uniformly distributed and the other terms have to be considered. The terms
T1 and T2 are shear stresses which are mainly generated by transversal velocity
gradients; they are usually of minor importance. The terms C1, C2 and C3
represent advective transport of momentum. C1 is due to downstream variation of
the flow field and drops out when the flow is completely adapted to the
curvature (
). The non-uniform velocity distribution over the channel width is mainly due
to the terms C2 and C3. These are redistribution terms that nearly cancel when
integrated over the cross-section. They represent the effect of the advective
transport of downstream momentum, rvs,
by the transversal velocities, vn,
which after depth-integration yields: . By decomposing the velocity components vj ( j=s,n) in a depth-averaged value, =Uj, and its local deviations, :
where
(2)
the velocity redistribution term can be decomposed as:
(3)
The first term, UsUn, represents a redistribution
of downstream velocity Us
by the transversal velocity Un;
it can be resolved by depth-integrated flow models. The second term, ,
represents the velocity redistribution by the secondary flow; since it depends
on the vertical distributions of and
, it is an unknown in the depth-integrated momentum equation and has to be
modelled. Often, this term is modelled using vertical profiles of and
that are derived from a simplified set of the 3D Navier-Stokes equations,
proposed by Rozovskii (1957). de Vriend (1977) gives the following solution of
these simplified equations, which will be called the ¡°Rozovskii model¡±
further on:
(4)
(5)
where fs and fn are the normalised profiles
of vs, and vn*, C is the Chezy
friction coefficient, k is the von Karman constant, h=z/h is the normalised vertical co-ordinate and h0 is the near-bottom level where fs(h0)=0. According to the ¡°Rozovskii model¡±,
the secondary circulation, , increases linearly with increasing curvature ratio,
h/R, for a given downstream velocity, Us. represents the velocity redistribution
term normalised by Us2h/R:
(6)
where . is called the velocity-redistribution coefficient
further on, while shall be
interpreted as the normalised strength of
the secondary circulation. According to the expressions, eq.4 and eq.5, both
fs and fn, - and thus also and - are unique functions of the Chezy
friction coefficient, C, and do not depend on the curvature parameter, h/R. and
calculated from the ¡°Rozovskii model¡± as a function of C are shown
in Fig.3.
In this paper, predictions of and based on ¡°Rozovskii model¡± (eq.4
and eq.5 and shown in Fig.3) will be compared with experimental data for a
strongly curved open-channel bend (Fig4 and Fig.5). The experimental normalised
velocity profiles are calculated from the measured distributions of vs(h) and vn(h) according to eq.2, eq4 and eq.5 as:
(7)
(8)
where the flume-averaged water depth, H, will be used as an approximation
for the local water depth, h.
Experiments were
performed in a 1.3m wide laboratory flume, consisting of a 9m long straight
inflow, followed by a 193_ bend with a constant radius of curvature of R=1.7m on
the centreline, and a 5m long straight outflow. The horizontal bed was covered
by a sand with diameters in the range 1.6mm < d < 2.2mm, which has been
fixed by spraying a layer of paint on it, thus preserving the roughness of the
sediments. The vertical banks were made of Plexiglas. A ratio R/B=1.31 was
chosen, which corresponds to a very strongly curved bend; a bend is considered
strongly curved when R/B<2 to 3.
The investigated hydraulic parameters are tabulated in Fig.1c. The main
parameter of interest is the curvature ratio H/R, which was varied by testing
three different values of the flume-averaged water depth, H=10.8, 15.9 and 21.2
[cm], yielding a curvature ratio of, H/R=0.064, 0.094 and 0.125. In the paper,
the experiments will be named after their discharge: Q56, Q89 and Q104[l/s]. In
all three experiments, a similar flume-averaged velocity was chosen, U=0.4, 0.43
and 0.38[m/s], corresponding to different Fr-numbers, Fr=U/(gH)1/2=0.39,
0.35, 0.26 (A literature review has shown us that the flow field in a channel
bend is rather insensitive to the Froude number). The flume-averaged water
surface gradient and energy gradient were,
=1.41, 0.94, 0.49[¡ë] and
=1.46, 1.01, 0.54[¡ë], where ZS is the sectional-averaged water
surface elevation and is
sectional-averaged downstream velocity. The shear velocities were, u*=(gHSs)1/2=3.9, 4,0 and
3.3[cm/s] giving Chezy coefficients, C=g1/2U/u*=32, 34 and
35[m1/2/s] and the channel aspect ratios were, B/H=12.1, 8.2 and 6.1.
By moving a set of 8 acoustic limnimeters - mounted on a carriage that covers the width of the channel - along the channel, a detailed description of the water surface topography was obtained.
Velocity measurements were made with an Acoustic Doppler Velocity Profiler (ADVP), developed in our laboratory (Lemmin and Rolland, 1997). It measures simultaneously and quasi-instantaneously profiles of the three velocity components, from which the three time-averaged velocity components, , as well as the six turbulent stresses, (i, j = s, n, z), and higher order turbulent correlations can be computed. With the ADVP placed in a water-filled box on the water surface (Fig.1b), velocity profiles covering the flow depth were obtained. The measuring volumes were cylinders of (¦Ğ0.72/4)x(0.3)=0.12[cm3]. The sampling frequency was 31.25 Hz and the sampling time was 200s. Detailed information on the ADVP and estimations of the accuracy of the measurements can be found elsewhere (Hurther and Lemmin, 2000).
Velocity profiles were measured on the centerline, with a streamwise spacing of 0.5m in the straight inflow and outflow reaches and of 15_ in the bend. In this paper, mainly these centreline measurements are exploited. Only the calculation of the coefficient as (see further, Fig.2) required detailed measurements of the 3D-velocities in cross-sections. These detailed cross-sectional measurements were made in the section with the strongest secondary circulation ¨C at 135_ for Q56, at 90_ for Q89 and at 75_ for Q104, as determined from the centerline measurements (Fig.4) ¨C as well as in the reference section m25 (Fig.1a). In these sections, 29 vertical profiles were measured, with a transversal spacing that decreases towards the banks (Fig.1b). For the Q89-experiment sections all along the flume (sections m25, m05, 30¡ã, 60¡ã, 90¡ã, 120¡ã, 150¡ã, 180¡ã, p05, p15, p25, p35, Fig.1a) have been measured in detail on the same fine grid.
Modelling the secondary flow in strongly curved channel bends (Blanckaert,
2001), a strong interdependence between the transversal non-uniformity of the
downstream velocity distribution,
, and the strength of the secondary circulation, was elaborated. The
transversal non-uniformity of the velocity distribution is well reflected by its
centreline value,
at n=0. After normalisation by Us /R at n=0, a dimensionless parameter, as, can be evaluated such as:
(9)
For the Q89-experiment, the distribution of Us(s,n) along the flume has been evaluated and
the resulting as-values are shown in Fig.2. There one observes
that as
decreases from as=0 in the straight inflow to as=-1
at about 30_ into the bend. as subsequently increases to a value
of as˜0.3
at 180_. At the bend exit, it increases to as=1 and then shows a decreasing
tendency in the straight outflow. Furthermore, the two experimental points for
the Q56 and Q104-experiments (see Fig.2) suggest a similar evolution.
For flow over a horizontal bed, the velocity
distributions are often approximated by a ¡°free-vortex¡± distribution near
the bend entry and by a ¡°forced-vortex¡± distribution near the bend exit
(de Vriend, 1981, p29-30, p213; Steffler, 1984, p30-33). In the adopted
reference system, the ¡°free-vortex¡± distribution is defined by, Us(n)=Us(n=0)(1+n/R)¨C1,
giving as=(
)/(Us/R)= ¨C1,
whereas the ¡°forced-vortex¡± distribution is defined by, Us(n)=Us(n=0)(1+n/R), giving as=(
)/(Us/R)=1. Our
experimental data thus confirm these approximations, as used in the literature.
The two types of vortices are schematically illustrated in Fig.2.
Fig.4 shows the centreline
evolution of the strength of the secondary
circulation, , for the three experiments. After being negligible in the
straight inflow, the strength of the secondary circulation starts to increase
almost linearly at the bend entry. Then a zone of almost constant maximum
strength is attained, extending from about 90_-135_ for Q56, 75_-120_ for Q89
and 75_-105_ for Q104. Subsequently, the strength decreases and it is reduced to
less than 50% of its maximum value at the bend exit. In the straight outflow, it
further decreases to attain negligible values only relatively far downstream of
the bend. Similar observations have been reported (de Vriend, 1981,
figs.65,69,73; Odgaard and Bergs, 1988, Fig.9) and explained (de Vriend, 1981,
p.216; Yeh and Kennedy, 1993, p.782) before. It is often assumed that the
strength of the secondary circulation attains an equilibrium value in long
bends, eventually after initially attaining higher values. In none of our
experiments, such an equilibrium value was observed, since the strength still
decreased considerably at the bend exit.
The ¡°Rozovskii model¡±
predicts the strength of the secondary circulation, , as a function of C, but
independent of the curvature ratio, H/R. This is in contradiction with our
experimental observation where the strength of the secondary circulation
decreases with an increasing curvature ratio. The maximum observed values on the
centreline decrease with the curvature ratio from about ˜10 for Q56
(H/R=0.064), to ˜5.5 for Q89 (H/R=0.094) and ˜4 for Q104 (H/R=0.125). The
average values over the entire bend reach (0-193_) decrease from ˜6.9 for Q56,
to ˜3.0 for Q89 and ˜2.2 for Q104. With the only exception of the maximum
value attained for Q56, these values are also lower than the value of ˜9
predicted by the ¡°Rozovskii model¡± (Fig.3). This behaviour has already
been remarked and physically explained by de Vriend (1981, Fig.5c). He shows a
similar result computed with a fully 3D flow model, which however was for the
case of laminar flow.
Fig.5 shows, for the same three
experiments, the centreline evolution of the velocity-redistribution coefficient, . For all cases, this
coefficient is negligible in the straight inflow. At the bend entry it increases
and attains maximum values at about 60_ in the bend. After reaching these
maximum values, it drops to nearly zero values at the bend exit. In the straight
outflow the term is positive for the experiments Q56 and Q89, whereas it is
slightly negative for Q104. Similar to , the centreline evolution of does not attain an equilibrium value in
the bend.
The ¡°Rozovskii model¡±
predicts the coefficient as a
function of C, but independent of the curvature ratio, H/R. Our experimental
observations suggest that decreases
with increasing curvature ratio. The average values over the entire bend reach
(0-193_) decreased from ˜0.156 for Q56 (H/R=0.064), to ˜0.152 for Q89
(H/R=0.094) and ˜0.075 for Q104 (H/R=0.125). The maximum values on the
centreline were ˜0.25 for Q56, ˜0.29 for Q89 and ˜0.20 for Q104. In all
cases, these values are lower than the value of ˜0.44 predicted by the ¡°Rozovskii
model¡± (Fig.3). An even stronger over-prediction by the ¡°Rozovskii
model¡± has been observed for the case of flow in an open-channel bend over a
developed bar-pool topography (Blanckaert, 2001).
Blanckaert (2001) presents a model
for the downstream velocity and the secondary circulation that explains the
observed centreline evolution of and and the discrepancies with the ¡°Rozovskii
model¡±.
The secondary circulation in
open-channel bends largely determines the bed topography. Depth-integrated flow
models, used in engineering practice, cannot resolve the secondary circulation
and information on it has to be provided as input to these models. The effect of
the secondary circulation on the velocity distribution is accounted for by the
velocity-redistribution coefficient, , in the depth-integrated flow models,
while the strength of the secondary circulation is defined as, . Both are often
described according to the ¡°Rozovskii model¡±, which gives analytical
expressions for the normalised profile of the downstream velocity, fs, as well as for the
normalised profile of the secondary circulation, fn. This paper reports on an
experimental investigation on strongly curved open-channel flow (Fig.1), used to
evaluate the ¡°Rozovskii model¡±.
The strength of the secondary
circulation, (see Fig.4), and the
velocity-redistribution coefficient, (see
Fig.5), increase at the bend entry. After reaching a maximum value, they
decrease and do not attain an equilibrium value in the bend. (see Fig.4) and (see Fig.5) are over-predicted by the
¡°Rozovskii model¡± (see Fig.3). Furthermore,
and show a decrease with
increasing curvature ratio, H/R, which is in contradiction with the unique
dependence on the Chezy coefficient C predicted by the ¡°Rozovskii model¡±.
It can be concluded that the ¡°Rozovskii model¡± does not correctly
represent the secondary circulation and its effect on the velocity distribution
for strongly curved open-channel flows.
Blanckaert (2001) presents a model for the flow in strongly curved
channel bends that explains and predicts the behaviour observed in the reported
experiments. This model depends on the transversal non-uniformity of the
downstream velocity, which has been parametrised by the normalised transversal
velocity gradient on the centreline, as=(
)/(Us/R). In the reported
experiments (see Fig.2), as decreases from as=0
in the straight inflow to as=-1 just downstream of the bend
entry. In the bend, as increases
slightly, only to increase strongly near the bend exit to values of as=1.
This research is being sponsored by
the Swiss National Science Foundation under grant Nr.2100-052257.97/1. D.
Hurther and U. Lemmin are acknowledged for help with the ADVP instrument.
Blanckaert, K. (2001). discussion on: Bend-flow simulation using 2D depth-averaged model, by. J. Hydr. Eng. , ASCE, Vol.127(No.2), (scheduled for publication) .
Blanckaert, K. (2001). A model for flow in strongly curved channel bends XXIX IAHR-congress, Beijing, China (submitted for publication) .
de Vriend, H. J. (1977). A mathematical model of steady flow in curved shallow channels. J. Hydr. Res. ,IAHR, 15(1), 37-54.
de Vriend, H. J. (1981). Steady flow in shallow channel bends. Rep. No. 81-3, Comm. on Hydraulics, Dept. Civ. Eng., Delft Univ. Techn., Delft.
Dietrich, W.E. and Whiting, P. (1989). Boundary shear stress and sediment transport in river meanders of sand and gravel. River Meandering Eds. S. Ikeda and G. Parker. Washington, AGU. 1-50.
Hurther, D. and Lemmin, U. (2000). A correction method for turbulence measurements with a 3-D acoustic Doppler velocity profiler. J. Atm. Oc. Techn., AMS , accepted for publication.
Lemmin, U. and Rolland, T. (1997). Acoustic velocity profiler for laboratory and field studies. J. Hydr. Eng. ,ASCE, 123(12), 1089-1098.
Odgaard, J. A. and Bergs, M. A. (1988). Flow processes in a curved alluvial channel. Wat. Resourc. Res. , AGU, 24(1), 45-56.
Rozovskii, I. L. (1957). Flow of Water in Bends of Open Channels . Ac. Sc. Ukr. SSR, Isr. Progr. Sc. Transl., Jerusalem, 1961.
Steffler, P. M. (1984). Turbulent flow in a curved rectangular channel. Ph.D., The University of Alberta, Alberta.
Yeh, K. C. and Kennedy, J. F. (1993). Moment model of nonuniform channel-bend flow. I: Fixed beds. J. Hydr. Eng. , ASCE, 119(7), 776-795.
Fig.4 Fig.5