It is observed in investigating of velocity component in bends that observed
helical flow at bend by adding pigments to water. This helical flow is formed
by secondary currents. It is stated that a secondary current occurs due to imbalance
between centrifugal forces and pressure gradient induced forces at surface.
In other words, close to the inner bank and also at channel bed, pressure gradient
exceeds centrifugal forces and conveys water in transverse direction toward
the inner bank. At free surface, centrifugal forces drive flow to the outer
bank. This kind of flow is called secondary current (Lien
et al., 1999).
Rozovskii (1957) worked on rectangular channels with
180° bend in completely smooth and rough beds. Channel width was 80 cm and
bend radius of curvature on the centerline was assumed to be 80 cm, so a ratio
of Rc/b = 1 was chosen where, Rc was the radius of curvature
and b was the width of channel. Discharge value of 12.3 L sec-1,
flow depth of 5.8 cm, Froude number of 0.35 and Reynolds number of 14000 were
selected. He also, performed an extensive research on trapezoidal channels.
He illustrated the changes of the tangential velocity distribution at transversal
direction and spelt out the cause of generation and intensification of the secondary
current through the bend. In these studies no researches were fulfilled on water
surface transversal profile at bend entry and bend exit in strongly-curved open
channels, also on its varieties through bend.
Engelund (1974) described the theory of helical flow
in circular bends for a wide rectangular open channel.
Leschziner and Rodi (1979) represented a numerical
model in conformity with Rozowski tests on a 90° strongly-curved bend with
a ratio of average radius to width equal to 1. In that investigation he didn't
reach to a linear surface slope at cross section but in both numerical investigation
and laboratory model, surface slope near inner bank was obtained greater than
De Vriend (1979) carried out experiments on a 180°
bend, which was one of his investigations on a bend at University of Delft,
Netherlands. This bend had rectangular cross section with a width of 1.7 m and
central radius of 4.25 m. The lengths of straight channel before and after bend
were 6 m. Flow discharge for this test was 190 L sec-1 and flow depth
for downstream channel was 0.18 with Froude number of 0.215. De
Vriend and Geoldof (1983) worked on another model with central angle of
90°, discharge of 0.61 m3 sec-1, width of 6 m, depth
of 0.25, 50 m radius of curvature and Froude number of 0.25. In these studies
the water surface transversal profile through the bend was linear but water
surface elevation was not calculated at upstream and downstream of the bend.
Steffler et al. (1985) studied a channel with
270° central angle. This channel had rectangular cross section with width
of 1.07 m and depth of 0.2 m. The central radius of curvature is considered
to be 3.66 m. Two conditions were considered, one with depth of 6.1 cm and the
other with depth of 8.5 cm and bed slope of 0.00083. He presented a new 2D numerical
model based on depth-averaged velocity, such that the secondary current having
3D nature was involved and evaluated validity of the model by comparing the
numerical and experimental results. After surveying the experimental and numerical
results, water surface transversal profile was calculated linear and he didnt
comment on it concerning its quantity at bend entry and bend exit.
Anwar (1986) worked on 31 and 180° bends with natural-topography
beds. Shimuzu et al. (1990) assumed a logarithmic
vertical distribution of the longitudinal velocity and developed a 3D hydrodynamic
model. Molls and Chaudhry (1995) developed a 2D depth-average
model to solve unsteady flow in open channel.
Ye and McCorquadale (1998) worked on two test cases,
one was a 270° bend with a one-sided trapezoidal cross section and Froude
number of 0.475 and the other was a meander bend consisted of two 90° bends
with a width of 2.34 m and central radius of 8.53 m. Length of straight part
between two bends was 4.27 m and length of straight approach channels to the
bend were 2.13 m. In this research water depth and approach velocity at entry
are assumed to be 0.115 m and 0.366 m sec-1, respectively. By comparison
of 3D hydrodynamic simulation based on k-ε model and investigation of the
experimental results, they found out that the secondary current and super-elevation
begin upstream the bend and gradually reach the bend. Although, they studied
on the water surface longitude profile in inner and outer bank and the axis
of bend, they didnt comment on the water surface transversal profile,
the beginning and end of super elevation position and effect of transversal
distribution of tangential velocities on super-elevation.
Blanckaert and De Vriend (2003) also studied a 120°
bend, 0.4 m width and central radius of 2 m. However the channel bed was fixed
in that research, but using sand with an average grain size of 2.1 mm enabled
testing with moveable bed.
Bodnar and Prihoda (2006) presented numerical simulation
of turbulent free-surface flow accordance with finite-volume method by SST k-ω
turbulence model and analyzed strongly-curved bend with 90° angle. In that
study, he calculated the nonlinear slope of water surface in the bend.
According to investigations of Rozovskii (1957), Leschziner
and Rodi (1979), if the ratio of bend radius to the channel width was less
than 3, bend was considered strongly curved, otherwise it was mild. Since, the
aim of this study was to investigate flow pattern in strongly-curved bends,
the ratio of central bend radius to the channel width is chosen to be 1.5. Respecting
to survey of other studies, it was destined that in most studies water surface
transversal profile, situation of the variation of water surface at upstream
and downstream due to existence the bend hasnt been surveyed and the effect
of transversal distribution of tangential velocities on super-elevation hasnt
been calculated. In this research, laboratorial study has been done to evaluate
variation of super elevation of the flow at upstream, entry, cross sections,
exit and down stream the bend in a strongly curved open channel with different
curve angles and different discharges. In addition, the effect of transversal
tangential velocity distribution on super elevation was investigated.
Flume characteristics: A laboratorial flume was designed, as illustrated
in Fig. 1. The flume has a square section with 403x403 mm2
dimensions. Flume side walls and bed were made of Plexiglas, in order to provide
the possibility of changing bend direction and observation of the flow characteristics
as well. At channel entry, a storage reservoir made of galvanized sheets with
dimensions of 150x100x50 cm, was constructed. The function of this reservoir
was to convey pumped water from main reservoir to the channel entry. At the
conjunction of pumping pipe to the reservoir pressure-reducing baffles were
erected perpendicular to the connecting pipes, same as in the USBR stilling
basin type VI (Mays, 1999).
Five plastic bar screens perforated to an aluminum bar screen were used to
reduce velocity head and create an evenly distributed homogeneous flow in the
entry reservoir. To improve developed flow conditions in channel, having no
influence of entry conditions on bend and better adaptation of flow lines with
the channel, an elliptical transition was used at the connection of channel
to reservoir, according to the following equation (USBR, 1985).
where, x and y values are in centimeters. The larger half-diameter of ellipse is 40 cm, equal to channel width and the smaller half-diameter is equal to two third (2/3) of channel width.
After installation of entrance transition according to Fig. 2,
a 3.6 m straight channel made of Plexiglas was added to bend entry to provide
sufficient expanding-length before the bend.
||Plan of experimental flume with 60° bend
||Channel photo at upstream of bend
Three bends with central angles of 30, 60 and 90° were connected to the
channel separately. After completion of experiments on each bend, the new bend
Right after the bend, there was a 1.8 m length straight channel; at the end of this channel a rectangular sharp crested weir was constructed in a way that water surface and velocity variations at weir wouldnt affect the bend. When water passes the weir, pours into a collecting reservoir with dimensions of 1x1x1.25 m. At reservoir entry a 1x1 m mesh with 1 mm openings was installed. This resulted in energy loss and water diffusion at the end of flume.
To calculate discharge value, a triangular weir with central angle of 90°,
was utilized. This weir was placed in main reservoir with dimensions of 1x1x4
m; 0.80 m of which allocated to weir and the remaining 3.2 m was allocated to
storage reservoir (which should have a volume, greater than volume of circulating
||Values of water discharge and water depth upstream triangular
Acentrifugal pump was used for pumping water from storage reservoir into the
flume entry. Flow discharge was regulated by a gate valve with diameter of 150
mm which was installed right after the pump.
To measure velocity and depth values a Verniet ruler was utilized and enabled depth measurement with accuracy of 1 mm in transverse direction and 0.1 mm downward to depth direction. Velocity was measured by a propeller velocity-meter in a range of 2.5 to 150 cm sec-1. Accuracy of this velocity meters based on manufacturers catalog was as follow:
Discharge and water depth range: To study flow in strongly curved open channel in different conditions, flow depth and the discharge must be constant at downstream of the channel. In this study, variations in gate valve opening has changed water elevation upstream of the triangular weir and a fixed depth yields a constant discharge during experiment. Water depth of downstream straight channel is also adjusted by changing height of sharp-crested weir installed at downstream of straight channel. In all cases, depth is measured within a distance of 80 cm downstream of the bend and is constant with measurement error, less than 0.5 mm. Values of water depths in straight channel downstream of the bend and depths upstream triangular weir and applied discharges during the experiments are shown in Fig. 3.
Study of water surface elevations in 30, 60 and 90°
Super-elevation upstream and downstream of the bend: In this investigation
water surface elevation was studied 40 cm upstream and downstream of the bend.
||Water surface slopes dispersion for different depths at 40
cm upstream and downstream of the bend
Water surface slopes at these cross sections for three bends with 30, 60 and 90° angles and for different values of discharges were measured and results are shown in Fig. 4.
It is obvious that in a distance equal to channel width upstream and downstream of the bend, with any central angle of 90° or less, water surface slopes are very closed to zero. Small positive and negative values for water surface slopes, illustrates that water surface is not a function of the discharge and the bend angle.
As shown in Table 1, laboratory data and their statistical distributions with upper limit of values of water surface slopes and their probabilities were calculated. Based on different statistical distributions, for a 99.5% probability, water surface slope at upstream of bend are less than 0.002. For the downstream of bent with 99.5% probability, water surface slope are less than 0.0037.
As shown in Fig. 4 and Table 1, water surface slope, dispersion in both sections is almost zero, so these sections are not affected by bend and different discharges.
Water surface variation at bend entry and bend exit: In this study laboratory
investigations on water surface transversal profile at bend entry and bend exit
have been surveyed. Water surface profile at bend entry and bend exit were studied
for bends of 30, 60 and 90° with central radius of 60 cm. The difference
between water levels at the outer and inner bank of the channel is called super-elevation
or banking. Measured super-elevation values at bend entry and bend exit were
normalized by dividing by downstream average depth in distance of 80 cm after
the bend. Analyses on these dimensionless parameters with different downstream
water depths and channel discharges, revealed no correlation between water depth
and channel discharge and dimensionless super-elevation values at bend entry
and bend exit.
||Water surface slope at 40 cm upstream and downstream of the
|Fig. 5(a, b):
||Channel depth and discharge Influence on dimensionless super-elevation
at bend entry
|Fig. 6(a, b):
||Channel depth and discharge Influence on dimensionless super-elevation
values at bend exit
||Influence of water depth in channel on super-elevation values
at the entry of 90° bend
As shown in Fig. 5a,b and 6a,b,
for all angles at bend entry, dimensionless super-elevations are independent
from discharge and downstream depths.
Mean values of dimensionless super-elevation at bend entry and bend exit were 0.0462 and 0.055, respectively. In this phase laboratory data for 30, 60 and 90° bends were analyzed separately and again it was found that the super-elevation is not a function of discharge and downstream channel depth but it is a function of channel curve angle and decreases by reduction in bend curvature angle. For example changes of dimensionless super-elevation with flow depths are schemed in Fig. 7 for 90° bend.
Poor correlation between dimensionless super-elevation values and flow depths
revealed no relationship between these quantities.
||Dimensionless values of super-elevation at bend entry
||Dimensionless values of super-elevation at bend exit
In Table 2 and 3 mean values of dimensionless
super-elevation for each curvature angle at bend entry and bend exit are given.
For each series of tests with aforementioned angles, normal, log-normal, 3-parameter
normal, Pearson type III and log-Pearson type III, probability distributions
were fitted. Normal distribution was the best fitted model. Predicted values
for 90, 95and 99.5% probability are given in Table 2 and 3.
Water surface at bend entry and bend exit increased as curvature angle changes
from 30 to 90°. Average values of dimensionless super-elevation in the bend
entry and bend exit change from 0.034 to 0.052 and from 0.0499 to 0.061, respectively.
Variation of dimensionless super-elevation at the bend entry and bend exit,
as a function curvature angle is given in Fig. 8.
The fitted parabolic equations of Fig. 8 for bend entry and bend exit are as Eq. 3 and 4:
where, θ is the curvature angle (in degrees), Δy is the difference
between water levels at inner and outer bank and is
the average depth at downstream of the channel. In Eq. 3 and
4, correlation of determination of 0.99 and 0.94 were determined
for entry and exit of bend.
Non-linear transversal water surface profile in bends: As previously
discussed in introduction, most of researchers assumed a linear equation for
water surface. In this study, according to the well known Eq.
5, for water surface gradient, mentioned in most open channel flow reference
books like Henderson (1966), Subramanya
(1982) and Hosseini and Abrishami (1999) a nonlinear
transversal water surface profile was derived:
where, h denotes water height relative to a reference point, like channel bed
and v denotes average velocity for any vertical section, like AB shown in Fig.
9a and b, r denotes the curvature radius in plan of bed
and n is the vertical axis outward the bend.
||Effect of curvature angle dimensionless super-elevation
||Flow in plan and cross section of a curved channel (Henderson,
If the bend is a part of a circle
as in this work, all vertical vectors of n cross the center of curvature.
In Eq. 5, with large values of curvature radius and small value of velocity, water surface gradient (dh/dn) would be very small and negligible. However in channels with small radius of curvature as in strongly-curved bends, non-uniform transversal distribution of tangential velocity, water surface gradient (dh/dn) wouldn't be constant. Therefore, a nonlinear water surface profile is expected and will be derived later.
Centrifugal forces incline water surface and at half outer-side of the bend cross section water surface rises, which causes velocity decrease in the sub critical flow. In half inner-side of the bend cross section a depression in water surface occurs and hence velocity increase.
||Plan view of bend coordinates used in Eq. 9
Investigations on 30, 60 and 90° bends with constant downstream depths of 4.5, 6, 9, 12 and 15 cm, showed that from bend entry up to bend exit, where secondary current intensify, velocity distribution in all layers parallel to channel bed very linearly as shown in Fig. 10. Considering Linear distribution of velocity in channel width, velocity decreases close to outer bank and increases near inner bank. Since, curvature radius of outer bank is greater than the radius of inner bank so water surface gradient changes in transversal direction. So, water surface gradient will be larger near the inner bank because of large average tangential velocity and small curvature radius compared to the gradient near the outer bank.
In this research, tangential velocity and water depth profiles were measured for 30° bend at the entry , middle point and end of the bend, while for 60° bend at cross section with central angles of 10, 20, 30, 40, 50 and 60° and for 90° bend cross section with central angles of 0, 22.5, 45, 67.5 and 90°. Tangential velocity were measured, at different layers parallel to the channel bed, in all depths of 4.5, 6, 9, 12 and 15 cm thickness of layers were in the range of 1.5 to 3 cm. Depth average velocity profile was derived by calculating mean velocities in depth. The velocity profile coincides with the linear distribution formula given in Eq. 6.
where, a and V0 are the regression coefficients of depth average velocities.
Considering linear distribution of velocity with curvature radius and using the following variable transformation as shown in Fig. 10, integration of Eq. 5 will result in the Eq. 9.
||Calculated super-elevation versus experimental super-elevation
in strongly curved channel
where, x denotes distance of each point in transverse direction relative to channel axis. Rc is the central radius of curvature and b is the channel width which is equal to 60 and 40 cm, respectively.
V0 and a regression coefficients with R2 greater than 0.9, determined from experimental results of transversal distribution of tangential velocities were substituted in Eq. 9. The difference between water elevations at outer and inner bank were measured for each cross section and each discharge. Calculated super-elevations from Eq. 9 are plotted against experimental super-elevations in Fig. 11.
As shown in Fig. 11, linear regression line and coefficient of determination are as follows:
The slope of regression line is very close to 1 which shows that experimental results have closely been simulated by Eq. 9.
Based on laboratory investigations on 30, 60 and 90° bends in a strongly curve open channel, following results were obtained:
||Within a distance equal to the channel width at upstream and
downstream of the bend, water surface is not affected by bend and follows
conditions of upstream and downstream of straight channel
||At bend entry and bend exit, dimensionless values of super-elevation
are not a function of flow depth and discharge and are only a function of
curvature radius. For a determinate central angle, the aforementioned quantity
can be computed from Eq. 3 and 4
||In strongly-curved bends, water surface can not be assumed
linear due to non-uniform velocity distribution and influence of curvature
radius, for prediction of super-elevation in strongly curve channel presented
a new equation
The authors wish to express their gratitude to Dr. Mohamad Reza Jafarzadeh for his comments and suggestions in constructing the laboratorial model.
The following symbols are used in this study:
||Regression coefficient for tangential velocity
||Width of channel
||Water surface gradient
||Local depth of flow
||Radius of curve
||tangential velocity, velocity
||Regression coefficient for tangential velocity
||Depth of flow
||Downstream average depth in distance of 80 cm after bend
||Difference of water depth between internal and outer banks