Journal of Applied Mechanics Vol.11, pp.735-743
(August 2008)
JSCE
3-D Flood Flow Structures in a Doubly Meandering
under Dominant
Compound
Channel
Relative Depth
G.M.TarekulIslam*,Yoshihisa
Kawahara
**andNobuyuki
Tamai***
* Member
, PhD,VisitingResearcher,
DivisionofSocialandEnvrn.Eng.,Graduate
SchoolofEng.,Hiroshima
University
(1-4-1Kagamiyama,
Higashi-Hiroshima,
Hiroshima
739-8527)
**Member
, Dr.ofEng.,Professor,
Division
of SocialandEnvrn.Eng.,GraduateSchoolofEng.,Hiroshima
University
(1-4-1Kagamiyama,
Higashi-Hiroshima,
Hiroshima
739-8527)
***Member
, Dr.ofEng.,Professor,
GraduateSchool,Kanazawa
GakuinUniversity
(Sue-Machi
10,Kanazawa,
Ishikawa
920-1392)
A doubly meandering compound channel is one in which both lower and upper channel
meander. This paper examines the 3-D flood flow structures in terms of primary velocity
distribution,stream-wise velocity distribution and secondary currents in a doubly meandering
compound channel under dominantrelative depth. The relative depth is the ratio of the depth of
water overthe floodplainto the totaldepth of water whilethe dominantrelativedepth is the relative
depth at whichthe differencein dischargebetweenthe rising and falling stagestakes the maximum
value.The dominantrelativedepth of a doublymeanderingcompoundchannelis foundto be 0.17.
The primary velocity distributionunder the dominant relative depth appears to be essentiallythe
same for both rising and fallingstages.The exchangeof flow betweenmain channel and floodplain
results in the retardationand deviation of flow fields. The extent of retardationand the angle of
deviationof the stream-wisevelocityvaryin the rising and fallingstages.The maximumretardation
occurs in the cross-over section at the floodplain level for both rising and falling stages. The
maximum deviation occurs at the bend apex section for both rising and falling stages. The
magnitudeof the secondarycurrent under dominantrelative depth can be as high as 30% of the
bulk velocity.The evolution and decay processes of secondary currentsduring rising and falling
stagesare essentiallythe same but onlydiffer in strength.
Key Words: 3-Dstructure;food flow; doublymeandering; compoundchannel; relativedepth
1. Introduction
For a particular
that
A natural meander river is usually composed of a deep main
levee alignment,
it requires
works.
less maintenance
Here
lies
the
channel.
flow
at low flows, while the floodplainsgive storage for floods during
high flows. This is why a two-stagechannel or compound channel
a low floodplain
main channel and the floodplain
is proposed as a design cross-sectionto retain large parts of the
lateral momentum
natural environmentunchanged. When the main meander channel
is flanked by floodplains with meandering levee, it results in a
channel
and the floodplain12).
complex
for meandering
doubly meanderingcompound channel. The surrounding lands of
the floodplainmight compel one to constructmeanderinglevees.In
the three-dimensional
addition, the alignment of the meandering levees might be of
nature of flow.
―735―
even for straight compound
depth the large velocity
transfer
rivers,
and other
of
structure
a doubly
is complex
channel. It has been pointed
difference
The
compound
flow
between
mechanisms
channel
the
and the floodplain
of the flows
the main
are more
due to an increase
nature of the flow, the interaction
most
out that at
between
induces a strong shear layer and a
across the interface
the flows in the main channel
In natural
The
might be such
protection
significance
meandering
compound
for bank
engineering
channel and adjacent shallow floodplains. Such compound
channelsoffer advantagesmainlythat of ensuring reasonabledepth
specialinterest as far as maintenanceof the channel is concerned.
the flow structures
in
between
and the unsteady
are unsteady
especially
during
flood.
The
unsteadiness
velocity
distribution,
sediment
transport
to
understand
section
studied
unsteady
and
the velocity
obtained
straight
distribution
by
using
investigated
the
effects
rough
beds
effects
in flood
those
surface
unsteady
They
flow
found
the
property
of
the
the
by the effects
studied
the
compound
unsteady
and
maximum
first
conducted
experiments
compound
channel.
features,
i.e., the
unsteadiness,
hydraulic
flow
relations.
Fukuoka
structures
meandering
transfer
and
of
rising
They
found
during
vertical
falling
direction
stage.
structure
in
a
doubly
dominant
relative
depth.
the
This
and
paper
meandering
wave
the
convective
compound
front,
of the
shift
of
channel
a positive
doubly
of
on
channel.
et al.10) studied
geometric
in a compound
parameters
momentum
3-D
stage
measuring
probes
flood
flow
the
on
flume
transverse
under
2. Experimental Setup and Measurement
desired
position.
were
was
constructedwith vinyl chloride,was mounted on 2 m wide steel
channel structures. Fig. 1 shows the general layout of the
direction
experimentalchannel. The longitudinalgradient of the channel is
adjustablewith a jack and hinge and can be varied up to 5%. The
The
channel consists of a main channel flankedby meandering levees
to measure
on both sides of the main channel. The outer levee which has the
same sinuosity as that of main meander channel has a phase
discharge
difference of about 300 ahead of main channel. The meanders are
convertor.
expressed as combinationsof arcs and straightreaches. The flume
consists of nine consecutive meandering waves with straight
personal
approachchannelsat the beginningand end of the meanderingpart
of the channel. The bed of the main channel and floodplains is
considered
smooth.The width and depth of the main channel are 300 mm and
50mm, respectively.The angle of arc is 600with a sinuosityof 1.05.
of about
for
direction
parallel
to the cross
electromagnetic
and
flow
The
A/D
the
main
were
for automatic
channel
to be the same
developed.
Thus
15m
fully
developed
board
is
at positions
flow
I- and
was used
the L-type
probe
is
in
the
is in the direction
was
outflow
direction.
5
pipe
sampling
Hz.
was
An
used
frequency
electromagnetic
of
current
to an A/D
(Analog/Digital)
was
connected
then
regular,
in phase
reach
the beginning
viz.
probe
two
to
a
data acquisition.
meander
the measuring
from
The
by
to the
by
x-velocity
The
connected
convertor
meters
meter
in the
1 Hz.
measured
is in the vertical
discharge.
was
meter
computer
As
installed
the instantaneous
performed
y-velocity
current
in the
the probe
while
The
the
along
carriage
The I-type
and z-velocity
the
traveled
current
section,
of
meter
geometric
velocities,
were
components
cross
measurement
meters
ensure
flow
different
and
of the
velocities
component.
section
of the
of
which
to move
Japan).
y-velocity
frequency
300 ahead
automatically
electromagnetic
to each
sampling
is
point
z-velocity
normal
levels
movement
Company,
x- and
used
of this
different
on a carriage
The
3-D
(Kenek
for measuring
water
mounted
can be programmed
The
it is said
1.
of
vertical
meander
shift
sketch
of
of the
then
phase
is about
definition
of
the
depending
of the
channel,
The
that
to the wave
or negative
versa.
magnitudes
system.
which
probes
width and depth are 30m, 1.5m and 1m, respectively.The flume,
―736―
The
of two-component
L-type
The experimentswere conductedin a tiltingflume whose length,
and
motor
the
levee
If the apex
channel
shows
in Table
a rail
electric
sets
2
and
as
precisely,
the peak
flood
of the main
and vice
measurement
rising
channel
shift
compound
Fig.
are shown
the
levee.
apex
bend
between
positive
channel
is defined
More
of the meander
meander
parameters.
For
difference
compound
shift
channel
direction.
can be either
phase
meandering
main
at the
downstream
of the main
phase
of the main
and that
phase
lies ahead
to have
and
the effects
during
the
ratio
bend
of the bend
stages
explores
is the
of the
three
meandering
channel
meandering
The
longitudinal
levee
on
on
compound
to the
the alignment
importance.
peak
in the
The
of
by floodplain
the main
flow
is
channel
the approach
in a doubly
between
between
shift
flow
is of great
floodplain
channel
enter
and
channel
main
meandering
of the outgoing
on the position
form
active
the relationship
levee
length.
et al.8)
of the
and
main
can
approach
approach
is flanked
Similarly
exit
Two
the end of the channel.
study
phase
a
the flow
in a meandering
falling
is more
Lai
its
so that
channel,
the
The
the
the
at the beginning
having
connects
disturbance.
through
placed
which
channel
From
of pipeline
to the experimental
sump.
channel.
channel
smooth
it flows
been
compound
based
Morishita
that
provides
the
meandering
depth.
overbank
without
the difference
is
et al.7)
reaches
et al.9) studied
flow.
channel
a
waves
cross-sectional
flood
during
channel.
in the
flood
shape,
water
smoothly
channel
For the
the
have
one.
by means
underground
of 0.30m•~0.05m
channel
reservoir
peak
the
compound
This
is a closed
is transported
Consequently
to
(length=2.75m)
sides.
end
of
double
returns
straight
both
in the channel
the water
reservoir.
and
cross-section
Tominaga
Jayaratne
propagation
studied
hydrograph
planform
main
of
on flood-wave
characteristics
the
velocity
and
al.5)
system
reservoir
of the meandering
a
over
increases
its
that
the
end
floodplains.
and
characteristics
by discharge
They
in
et
characteristics
arrangement.
found
followed
stage-velocity
flood
flow
channel
stress.
floodplains
velocity
of vegetation
and
significantly
vegetated
velocity
Graf4)
channels
the unsteadiness
differ
shear
with
in the
channel
depth-averaged
changed
profiles
the hydraulic
vegetation
main
New
that
model
shear-stress
theory.
supply
to the upstream
single
flow
the
velocity
law
bed
channels
and
water
downstream
conducted
Tu
as
found
and
investigated
that
of
and
velocity
in compound
unsteadiness
than
flow
on
in the log-wake
on the friction
et al.6) experimentally
wave
unsteadiness
on the ƒÎ-parameter
from
the
kinematic
of
well
load
straight
open-channel
as
The
the
bed
been
in
channel.
in unsteady
velocity
the
have
characteristics
compound
friction
influences
transport,
investigations
flow
distribution
the
waves
sediment
etc.3). Many
the
channel
they
of the flood
suspended
has
been
the
selected
of the meander
condition.
The
flow
after the flow
velocity
field
is
is fully
at a distance
channel
so as to
in five sections
has been measured in the meander reach covering half of the
meander wave (Fig. 3). Sections 1 and 5 are bend apex sections
the centerline of the section. The transverse grid spacing in the
while section 3 is the cross-over section.The measuringgrid in the
The grid spacing in the vertical direction is 1cm from the main
transverse direction was kept fine at the interface and coarse far
from the interface.The transverse grid spacing in the main channel
view of the measuring grid is shown in Fig. 4.
floodplains is 3cm from the interface which is followed by 8cm.
channel bed up to 3cm and then followed by 2cm. The sectional
is 1cm from the interfacewhich is followedby 2cm and 4cm until
Fig. 1 General layout of the experimental
setup
Fig. 4 Sectional(section 1) view of measuringgrid
Table
1 Geometric
parameters
of the compound
meandering
channel
Fig.2 Definition
sketch of the geometric
parameters
3. Unsteadiness
It
is
necessary
characterizes
surface
(1993)
to
the effects
flows.
expressed
Fig. 3 Measuring
Parameter
The
by
an
defined
define
on velocity
overall
a parameter,
flood
―737―
water
parameter,
surface
to
profiles
the
s/Uc
and turbulence
Nezu
that
in flood
and
can
be
Nakagawa
unsteadiness:
ƒ¿
a is the ratio
convection
parameter
of a hydrograph
parameter.
a to express
=V
unsteadiness
unsteadiness
characteristics
unsteadiness
reach and sections
The
an
(1)
of the rising
velocity
speed
Uc of
Vs of
turbulent
eddies.
and
Graf
and Suszka11) proposed
express the overall characteristics
an unsteadiness
parameter
of a hydrograph.
to
The parameter
then
(or discharge)
total
is:
decreasing
the
where
U*b = friction
of
the
base
flow
(before
the
in
the
flow
hydrograph);
depth
duration
the
of the
In this
AD
and
= difference
base
flow
between
depth;
the
and •¢T
the following
parameter
has
maximum
= total
been
used
it was gradually
flow
was
difference
to express
(3)
the curvature
flow
consists
of
of the hydrograph
aspects
point
In
of a hydrograph
the Figure
duration
are schematically
are:
H = water
hydrograph;
the hydrograph;
duration
depth
Hpeak= water
limb,
Different
in Fig.5.
duration
The
of
consecutive
depth
corresponding
branch
before
Schematic
in
time
branch
of the
passage
to the peak
H = Hpeak-Hbase..
Fig.5
of the
varies
the hydrograph
steady
flow
value
to reach
for
was
flow
was
10 minutes.
in 30 minutes.
its base
Then
for the selected
flow
in 62
for
study
1.14x10-4
and
base
logging
while
the
generating
unsteady
other
a software
software
is
unsteady
flow
from
The
which
used
unsteady
One
to
separate
instantaneous
for
flow
flow.
is how
the
is the
4.33•~10-2.
by using
data
are as
m3/s2, where •¢Q
flow; ƒÐ=
generated
is
hydrograph
of the
the
velocity,
this
to
were
the
study,
obtain
moving
the
arranged
hydrograph.
time
in time
Then
5 second-period
average
and
dependent
method
flow
has
been
fields,
the
so as to be t = 0 at the rising
the
the
time
velocities
was
were
divided
into
averaged
in
Relative
Depth
shows the discharge and water depth (section 1)
hydrographs.It is seen that the peak of dischargeis followedby the
notations
of the rising
flow
third
period.
Fig.6
and
= total
of the falling
of the base
peak
components
in a hydrograph; •¢T
time
= time
Hbase = water
rising
flow.
shown
depth
of the hydrograph; •¢Tr=
of the hydrograph; •¢Tf
flow,
used
to
order
signals
5. Dominant
of Hydrograph
base
data
In
+ u'(t).
4. Selection
by base
of
u(t)
each
of
One
component, u(t)
expressedby Eq. (3) has been used to expressthe unsteadiness.
consists
was
parts.
mean-velocity
velocity
is followed
one
depth
near
the parameteris that it relates verticalaccelerationof water level to
the gravitationalacceleration.In this study,the above parameteras
limb which
flow
hydrograph
two
software
difficult
longitudinalbed slope of the channel.The physicalsignificanceof
hydrograph
gradually
water
parameter
beginning,
as base
to a peak
parameters
peak
logging
generating
applied.
A
at the
= 1/3; •¢Q/•¢Tr=
unsteady
the origin of time, g is the gravitational acceleration, i is the
falling
at about
in mind,
discharge
decreased
between
automatic
represents
that
The
unsteadiness
these
increased
The different
follows: •¢Tr/•¢T
time
the unsteadiness:
where•@
a way
value
The
Keeping
branch.
passage
hydrograph.
study,
its maximum
by a constant
minutes.
of
such
maintained
Then
in the falling
of the hydrograph.
1•~10-2-6•~10-2.
selected
(2)
reaches
duration
from
slowly
of
peak of water depth. This is attributedto the fact that dischargedata
corresponds to the one as measured by the electromagneticflow
meter installedin the deliverypipe. It is also seenthat the rate of rise
of water level is fasterin the initialstagethan that of dischargewhile
the rate of fall of water level is slower than that of discharge.This
can be attributedto the cross-sectionalshape of the channel.
flow;
•¢
representation
Fig.6 Hydrographsof discharge(in the deliverypipe) and water
of a hydrograph
depth (section 1)
Islam12) studied natural hydrographs
hydrographs
and found that the natural
and found that a hydrograph
shape with water level first increasing
The relative
usually take a skewed
rapidly
in the rising branch
depth, Dr is defined
water over the floodplain
― 738―
as the ratio of the depth
to the total depth of water. Fig.7
of
shows
the variation
of discharge
shown in Fig.7
transverse
and
direction
relative
characterized
difference
depth
relationship
gradients over that region. In the cross-over section, the maximum
velocity field continues to move further towards the left bank.
between the rising and falling stage takes the
Before reaching the next bend apex, both maximum and minimum
depth
at a relative
the maximum
depth
can be defined
in discharge
The
velocity contours lie near the water surface: the maximum being at
of 0.17 and then gradually
that
the left bank while the minimum near the middle of the
in the rising and falling stages
cross-section. At section 5, the shift of maximum velocity field
of 0.17.
Hence
as the relative
between
value.
the
ceases and starts to move in the right bank. It is observed that the
'dominant
depth at which
the
primary velocity contour in the bend apex section is more or less
uniformly distributed which implies that the velocity gradients are
the rising and falling stages takes
times needed
low. Further downstream at the cross-over region, the uniformity of
to attain the dominant
velocity distribution is reduced which implies that the velocity
depth for both rising and falling stages are 160 and 5420
respectively.
It can be concluded
depth of a doubly meandering
flow condition
Morishita
compound
that the dominant
channel
is 0.17. This finding is different
et al.11) which involved
with straight floodway.
high velocity
The
loop
down to zero at the peak water depth. This suggests
depth'
representing
and
is distinctive
seconds,
highly
are rather densely-spaced
formation.
value at a relative
difference
section
non-linear
of flow characteristics
relative
is
clock-wise
the difference
relative
cross-section whereas the contours in the remaining part of the
water depth in the
1. As can be seen that the discharge
familiar
in discharge
maximum
narrows
in section
by the
bank. The evenly-spaced contours spread over the two-third of the
with relative depth. The relative depth as
has been based on the average
meandering
gradients across the section is higher than that in the previous
section. This is due to the exchange of flow between main channel
and floodplain. The flow from right floodplain impinges in the
from the study of
channel
main channel which in turn causes the primary velocity vectors to
in discharge
retard. This retardation is maximum at the cross-over section as can
compound
They found that the difference
in the rising and falling stage takes the maximum
relative
under the given
be seen in the contour plots. The primary velocity distribution and
value at a relative
the extent of retardation due to momentum exchange appear to be
depth of 0.30.
essentially the same for both rising and falling stages under
dominant relative depth.
(a) Rising stage
Fig.7 Variation
of discharge with relative depth
(b) Falling stage
6. Primary
Velocity
Fig.8 Primaryvelocity(u/Ub1)distributionat section 1
Distribution
Figs.8 to 12 show the primary velocity distribution for rising
and falling stages under dominant relativedepth at sections 1 to 5,
respectively.The primaryvelocity(u) has been made dimensionless
by dividingit with the bulk velocityat section 1 (Ub1).The vertical
(a) Rising stage
(z) and lateral distances (y) have been made dimensionless by
dividing with the height of floodplain (h). It is seen that the
magnitudes of primary velocity fields are higher in case of rising
stagethan falling stage at all the sections.At section 1,it is seen that
the maximumvelocityfield lies near water surfaceof the right bank
while the minimum velocityfield lies near bed of the left bank for
(b) Falling stage
Fig.9 Primaryvelocity(u/Ub1)distributionat section2
both rising and falling stages. The velocity contours are almost
evenlyspaced except near the right bank where the contours are a
bit closely-spaced.A core of high velocityfield appears to evolve
near the bed at the right bank and becomes fully developed at the
cross-overregion section.At section 2, the closely-spacedcontours
in the right bank extend in the lateral direction. The maximum
velocityfield still lies near the surface but shifts towards the left
―739―
(a) Rising stage
Fig.10 Primaryvelocity(u/Ub1)distributionat section3
occurs near the left bank for both rising and falling stages. The
stream-wise velocity fields in the cross-over section follow the
alignment of the main channel. This means that the flow fields are
perpendicular to the section at the point of inflection where the
curvature effect is non-existent. Following the cross-over region at
(b) Falling stage
Fig.10 Primaryvelocity(u/Ub1)distributionat section3
section 4, it can be seen that the maximum velocity occurs near the
(Continued)
left bank while the minimum velocity occurs near the right bank for
both rising and falling stages.
It is seen that the stream-wise velocity pattern near main channel
bed is the same for both rising and falling stages. The stream-wise
velocity fields near main channel bed follow the main channel
(a) Rising stage
alignment. The transverse profiles of the flow fields are regular near
the channel bed. At higher depths, the transverse profiles lose its
regularity and their irregularity becomes more pronounced for both
rising and falling stages. This suggests that the effect of the
exchange of momentum between main channel and floodplain is
(b) Falling stage
less near the channel bottom for both rising and falling stages. The
Fig.11 Primaryvelocity(u/Ub1)distributionat section4
difference in magnitude
of the stream-wise velocities for a
particular section near main channel bed is high at the bend apex
section and low at the cross-over section for both rising and falling
stages. At the floodplain height, the difference in magnitudes of the
stream-wise velocities is high at the cross-over section and low at
(a) Rising stage
the bend apex as opposed to the case near the main channel bed.
(a) Falling stage
Fig.12 Primaryvelocity(u/Ub1)distributionat section5
(a) z = 1cm
40cm/s (Rising stage)
•¨ 40
Fig.13 Flow visualization
Velocity
(Falling
stage)
showing how flow from floodplain
impinges
7. Stream-wise
cm/s
main channel
Distribution
Fig.14 shows the stream-wise velocity distribution at the
dominantrelativedepths in the rising and fallingstages,respectively.
The stream-wisevelocity is the resultant of the x-velocity (u) and
the y-velocity(v). It is seen that the magnitudeof velocityfields is
higherin case of rising stage than that of falling stage.At section 1,
the maximum velocity occurs near the right bank while the
minimum velocity occurs near the left bank for both rising and
falling stages. Following the bend apex section at section 2, the
stream-wisevelocityfields show that the maximum velocityoccurs
near right bank of the main channel while the minimum velocity
― 740―
(b) z = 3cm
40 cm/s (Rising stage)
40 cm/s (Falling stage)
Fig.14 Transverse
distribution
of steam-wise
velocity
line perpendicular to the measuring section. It can be seen that the
angle of deviation of the stream-wise velocity in the bend apex
section is towards the left bank and is higher in the falling stage than
that of the rising stage. The angle of deviation of the stream-wise
velocity in the cross-over section is towards the right bank near
channel bed for both rising and falling stages while at the floodplain
height, it is towards the left bank for the rising stage and towards the
right bank for the falling stage. It is also seen that the deviation is
higher in the rising stage than that of the falling stage near channel
bed while it is higher in the falling stage than that of the rising stage
(c) z = 5cm
at floodplain height. The maximum deviation occurs at the bend
apex section for both rising and falling stages.
40 cm/s (Rising stage)
40 cm/s (Falling stage)
8. Secondary
Currents
Fig.14 Transversedistributionof streamwise velocity(Continued)
Turbulent flow in a compound channel is characterizedby a
The exchange of momentum between main channel and
shear layergeneratedby the velocitybetween the main channeland
floodplain causes the stream-wise velocity to deviate from
the floodplainflow. In this region,there are not only vortices with
vertical axis but also those with longitudinal axis. The latter is
floodplaintowards main channel and from main channel towards
floodplain.It is seen that the deviation of the stream-wisevelocity
increasesat higher depth and becomes maximum at the floodplain
level which is subject to severe interactionbetween main channel
and floodplain.The exchange of flow between main channel and
floodplainresults in the retardationof flow fields. The flow from
right floodplain impinges in the main channel (Fig.13) which in
turn causes the velocity vectors to retard and deviate. As the
retardationof stream-wise velocity arises due to the exchange of
flow between main channel and floodplain, it is estimated by
comparing to the streamwise velocity near channel bed. The
maximum retardation occurs in the cross-over section at the
floodplainlevel for both rising and falling stages.In order to have a
closer look, the retardation and deviation of the steam-wise
velocityat right bank for sections 1 and 3 are analyzed. Table 2
shows the magnitudeof the stream-wisevelocity and the angle of
deviationat right bank for dominantrelativedepth during risingand
falling stages and for the bend apex (section 1) and cross-over
(section3) sectionsat depths of 5 cm and 1 cm which correspondto
floodplainlevel and near channelbed, respectively.
Table
2 Magnitude
of stream-wise
velocity,
V (m/s)
and angle
of
known as the secondary currents. Secondarycurrents are defined
as currentswhich occur in the plane normal to the local axis of the
primary velocity. It is one of the physical mechanisms by which
linear momentumis transferredperpendicularto the directionof the
flow. They are important in that they distort the distribution of
primaryvelocitiesand boundaryshear stressfrom those expectedin
simple sectionflow and thus affectthe processesof flow resistance,
sediment transport, bed and bank erosion and influence the
development of channel morphology. The secondary flow cell in
single meandering channel occurs mainly due to the action of
centrifugalforces.However,when there is flow over the floodplain
in a compound meandering channel the interchange of flow
between main channel and floodplaingives use to a differenttype
of secondaryflow.
Figs.15 to 19 show the secondaryflow structuresin sections 1
to 5, respectively during rising and falling stage under dominant
relativedepth. The vertical (z) and lateral distances(y) have been
made dimensionlessby dividingwith the height of floodplain(h).
At section 1,the overalldirectionof secondaryflow is from rightto
left floodplainfor both rising and falling stages.The magnitude of
the secondarycurrent section 1 is in the orderof more than 30%
deviation,Į(degree)
of the bulk velocity. The secondary flow from right floodplain
enters the main channel and from main channel to left floodplain.
This means that the momentumis transferredfrom right floodplain
to main channel and from main channel to left floodplain. The
transfer of momentum along with the centrifugal force drives a
clockwisecell to develop near water surface in the left bank of the
main channel. Two small anticlockwisecells appear to develop at
both the corners of the main channel for both rising and falling
It is found that the extent of retardation
of stream-wise
velocity
at section 1 is higher in the rising stage than that in the falling stage
while at section 3 it is higher
in the falling stage than that in the
rising stage. The angle of deviation
is measured
with respect
to a
stages.The mechanismof the developmentof secondarycell during
falling stagesis similarto that of the rising stage. The differenceis
that the magnitudesof the flow components are larger during rising
stage than that during falling stage. As the flow approaches from
― 741―
section 1 to section 2, it is seen that the overall direction of
secondary flow is still from right to left. The clockwise cell
developed near water surface in section 1 continues to grow. The
anticlockwisecells in the corners grow in size for both rising and
falling stages. As the flow goes down from section 2 to section 3
(b) Falling stage
(cross-over), the secondary flow is dominated by the vertical
components.The clockwisecell near water surfacedisappears.The
anticlockwisecells in the two sides of the main channel develop
Fig.17 Secondary
currents at section 3
further for both rising and falling stages. At section 4, the
anticlockwisecell in the main channel near bed of the left bank
completely disappears. The other anticlockwisecell in the main
channelnear bed of the right bank grows and the extentof the cell is
(a) Rising stage
from channelbed to the water surface.The orientationof secondary
cell during falling stages is similar to that of the rising stage. The
differenceis that the magnitudes of the flow components are larger
during rising stage than that during falling stage.At section 5, the
(b) Falling stage
secondaryflow structuresand the secondaryflow cells during rising
Fig.18 Secondary
currents at section 4
and falling stages are same as of section 1 but in the opposite
directions.This is because section 1 is the bend apex of the main
channel while section 5 is bend apex of the main channel in
opposite phase. It can be concluded that the evolutionand decay
(a) Rising stage
processesof secondarycurrentsduring rising and falling stagesare
the same but only differin strength.
(b) Falling stage
(a) Rising stage
Fig.19 Secondary
currents at section 5
9. Conclusions
This paper examines the 3-D flood flow structures in terms of
primary velocity distribution, stream-wise velocity distribution and
secondary currents in a doubly meandering compound channel
(b) Falling stage
Fig.15 Secondary
currents at section 1
under dominant relative depth. The dominant relative depth of a
doubly meandering compound
channel is found be 0.17. The
primary velocity distribution under the dominant relative depth
appears to be essentially the same for both rising and falling stages.
The position and shift of maximum and minimum primary velocity
(a) Rising stage
fields in the downstream direction appear to be the same for both
rising and falling stages.
The stream-wise velocity pattern is the same for both rising and
falling stages. The effect of the exchange of momentum between
main channel and floodplain is less near the channel bottom and
high at the floodplain level for both rising and falling stages. The
(b) Falling stage
difference
Fig.16
Secondary
currents at section 2
in magnitude
of the
maximum
and minimum
stream-wise velocities for a particular section near main channel
bed is high at the bend apex section and low at the cross-over
section for both rising and falling stages. At the floodplain height,
the difference in magnitudes of the maximum and minimum the
stream-wise velocities is high at the cross-over section and low at
(a) Rising stage
the bend apex as opposed to the case near the main channel bed.
― 742―
The exchange
results
channel
4) Tu, H., and Graf, W.H., Velocity Distribution in unsteady
and floodplain
open-channel over gravel beds, Journal of Hydroscience and
HydraulicEngineering,JSCE, Vol.10, No.1, pp.11-25, 1992.
level for both rising and falling stages. It is found that the
5) Nezu, I., Nakagawa,H., Ishida,Y., and Fujimoto,H., Effects of
retardation
retardation
floodplain
main
at the
in the
maximum
of flow between
extent of retardation
and
occurs
deviation
in the
at section
of
flow
cross-over
1 is higher
fields.
section
The
unsteadiness on velocity profiles over rough beds in flood
in the rising stage than
that in the falling stage while at section 3 it is higher
surface flows, 25thIAHR Congress, Tokyo, A1, pp.153-160,
1993.
in the falling
stage than that in the rising stage. It can be seen that the angle of
deviation
than
of the stream-wise
that
of the
stream-wise
rising
velocity is higher
stage.
The
velocity in the cross-over
angle
of
section
6) Tominaga,A., Shibata,K., Mio,N., and Nagao,M., Hydraulic
characteristics
of unsteadyflow in compoundchannelswith
wooded zone on floodplains,Annual Journal of Hydraulic
Engineering,
JSCE,Vol.40,pp.693-698,1996.
7) Jayaratne,B.L., Kawahara,Y. and Kan, K., Unsteadyflow
characteristics
in a compoundchannel,27thCongressof IAHR,
SanFrancisco,
USA,pp.740-745,1997.
8) Lai, C., Liu, C., and Lin., Y., Experimentson flood-wave
in the falling stage
deviation
higher
of the
in the rising
stage than that of the falling stage near channel bed while it is higher
in the falling stage than that of the rising stage at floodplain
The maximum
deviation
occurs at the bend apex section
height.
for both
rising and falling stages.
The magnitude
of the secondary
current under dominant
relative
depth can be as high as 30% of the bulk velocity. In the bend apex
section, the secondary
component
flow is dominated
while that in the cross-over
vertical velocity component.
secondary
currents
by the horizontal
section is dominated
The evolution
propagationin compoundchannel, J.HydraulicEngineering,
ASCE,Vol.126,No.7, pp.492-501,2000.
velocity
9) Fukuoka,S., Watanabe, A., Okabe, H and Seki, K., Effects of
by the
and decay processes
unsteadiness,planformand cross-sectionalform of channels on
of
hydraulic characteristics of flood flow, Annual Journal of
HydraulicEngineering,JSCE, Vol.44, pp.867-872, 2000.
during rising and falling stages are essentially
the same but differ in strength.
10)Morishita,Y, Uchida T, KawaharaY, WatanabeA., Flood flow
structure during rising and falling stage in a meandering
compound channel, International Conference on Civil and
Acknowledgement
This
in-Aid
work
for JSPS
was
supported
by
KAKENHI(18•E06395),
Grant-
EnvironmentalEngineering,Hiroshima,pp. 287-292,2006.
11) Graf, W.H. and Suszka, L., Unsteady flow and its effect on
Fellows.
sediment transport, XXI Congress of IAHR, International
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―743―