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Title: Flow separation control in open-channel bends
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Flow separation control in open-channel bends
a

Omid Seyedashraf & Ali Akbar Akhtari

b

a

Department of Civil Engineering, Kermanshah University of Technology, 67178 Pardis St.,
Kermanshah, Iran
b

Department of Civil Engineering, Razi University, 67149 Bagh Abrisham St., Kermanshah,
Iran
Published online: 01 Sep 2015.

To cite this article: Omid Seyedashraf & Ali Akbar Akhtari (2015): Flow separation control in open-channel bends, Journal of
the Chinese Institute of Engineers, DOI: 10.1080/02533839.2015.1066942
To link to this article: http://dx.doi.org/10.1080/02533839.2015.1066942

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Journal of the Chinese Institute of Engineers, 2015
http://dx.doi.org/10.1080/02533839.2015.1066942

Flow separation control in open-channel bends
Omid Seyedashraf a

and Ali Akbar Akhtarib*

a

Department of Civil Engineering, Kermanshah University of Technology, 67178 Pardis St., Kermanshah, Iran; bDepartment of Civil
Engineering, Razi University, 67149 Bagh Abrisham St., Kermanshah, Iran

Downloaded by [University of Nebraska, Lincoln] at 22:35 01 September 2015

(Received 7 September 2012; accepted 2 March 2015)
The investigation of fluid flow in sharp open-channel bends is key to controlling undesired sedimentation in natural river
reaches. The difficulties are associated with controlling the flow separation in meanderings. Flow separation decreases
the width of the flow, and consequently, the conveyance capacity while increasing erosion and mixing. This study proposes a novel approach to reduce the flow separation at the inner banks of sharp open-channel bends. Three-dimensional
numerical experiments were conducted. To find the most reliable procedure, five turbulence models were examined. The
employed numerical approach is formulated within the framework of the finite volume method and the volume of fluid
(VOF) technique to solve the Navier–Stokes equations. Water levels and velocity profiles are obtained in different sections of the channel and are compared to experimental studies of a 90° sharp open-channel bend. A close agreement is
observed using the RSM (Reynolds stress model) turbulence model. Moreover, the evaluation of acquired velocity profiles demonstrates that in a regular bend, the lowest velocity occurs near the inner bank, where it has a flow separation
tendency. The same numerical procedure is employed to simulate water flow through a sharp converging open-channel
bend. The measurements of velocity profiles and velocity vectors in the curved sections support the idea that decreasing
the channel width considerably reduces the overall velocity variations in cross-sectional areas of the test case and is
effective to control flow separation.
Keywords: flow separation; open-channel bend; converging bend; turbulence modeling

1. Introduction
As noticed by Leopold and Gordon-Wolman (1960),
river channels do not remain straight for any appreciable
distance. It is very unusual to find a straight stream with
a length longer than 10-channel section widths. Meandering was defined by Yalin (1992) as “a self-induced plan
deformation of a stream, under ideal conditions, is periodic and anti-symmetrical with respect to an axis x, say.”
One of the significant characteristic attributes of flow
in an open-channel bend is its secondary flow and therefore, the helical motion that is the main reason of the
winding river morphology and the tendency to create a
succession of shoals and deeps along its way. The physical explanation of the phenomenon has been identified
as the attenuation of current velocities by secondary
flows, which are formed due to disequilibrium in pressure gradient and centrifugal force at an arbitrary section.
Sudden alterations in flow direction could induce flow
separation from boundaries as regularly found on the
inner banks of sharp river bends, and around hydraulic
structures such as bridge piers. Sudden alterations are
closely related to the pressure distribution on the openchannel side surfaces; its presence, together with vortex
bar formation, reduces the channel width and its conveyance capacity. The issue modifies patterns of bank
and bed erosion in river meanders and frequently induces
*Corresponding author. Email: akhtari@razi.ac.ir
© 2015 The Chinese Institute of Engineers

concentrated bank erosion. According to Blanckaert and
de Vriend (2003), this considerably varies depending on
the structure’s section form and morphodynamics in
sharp bends.
Numerous meandering rivers have curvatures quite
sharp for flow separation. Predicting the morphological
properties of flow separation could be significant
information to stop bank erosion and maintain shipping
depth. To the authors’ knowledge, despite the significance of the subject in describing the morphology and
erosion shapes of rivers, few studies have been carried
out in this regard by previous researchers. However,
numerous experimental and numerical investigations
have been carried out to analyze other characteristics of
flows in open-channel bends. Thomson (1876) published one of the first reports dealing with flow patterns
in open-channel bends. He described the development
of a helicoidal flow approach and described the phenomenon as being the result of the variation in velocity
of fluid segments along the water depth. Rozovskii
(1957) conducted a series of experiments on a tight
180° bend of rectangular cross section with straight
inlet and outlet reaches. He measured velocity profiles
near the walls and noticed that the utmost velocities
occur below the water surface. Moreover, as a recent
experimental investigation, Akhtari, Abrishami, and

Downloaded by [University of Nebraska, Lincoln] at 22:35 01 September 2015

2

O. Seyedashraf and A.A. Akhtari

Sharifi (2009); Akhtari (2010) has carried out studies
on 30°, 60°, and 90° strongly curved open-channel
bends with central radius of 60 cm. With a 1.5 ratio of
curvature radius to channel width considering five different discharge values, he collected extensive data like
velocity and water depth profiles. In his reports, he has
mentioned that in a distance equal to channel width
from the bend entry and bend exit, water surface was
not affected by the curvature.
The following researchers have also contributed to
the implementation of the numerical methods to predict
flow behaviors in open-channel bends. De Vriend
(1976) applied a three-dimensional mathematical model
to simulate the flow features in open-channel bends.
However, the model seems not to behave well in sharp
curvatures. Kuipers and Vreugdenhil (1973) developed
a model to deal with axial current features. As the
model was based on depth-averaged procedure, it could
not simulate secondary flows and therefore, velocity
distribution along the bend was not correctly simulated.
In order to work out this difficulty, Ghamry (1999)
developed a system of two-dimensional vertically averaged governing equations to account for the respective
problems and employed an implicit Petrov–Galerkin
finite element scheme to solve them. Moreover, Lee
(1989) showed that the one-dimensional equations of
motion in conjunction with theories, which connect the
strength of the secondary flow components to flow
depth, channel plan form curvature, and depth-averaged
mean velocity, can be also used to numerically model
the flow characteristics of meandering rivers. Overall,
excluding the research published by DamaskinidouGeorgiadou and Smith (1986), few comprehensive
investigations on converging bends can be found in the
literature. Damaskinidou-Georgiadou and Smith investigated the flow pattern in a converging bend with both
experimental and numerical approaches. His research
involved three-dimensional velocity measurements and a
flow visualization technique for the surface and bottom
currents, a branching curved converging open channel
and a simplified finite difference method to simulate the
phenomenon.
In this investigation, Akhtari’s 90° strongly curved
bend data are utilized to verify a computational fluid
dynamics (CFD) model, which is employed here to
investigate the flow behaviors in open-channel bends
and optimize the bend’s efficiency. The flow was
numerically modeled and the flow separation tendency
was captured fairly well. As a result of the work, a
novel solution was proposed to reduce the separation
tendency, which was concluded by analyzing the data
obtained from another model that was afterwards simulated to evaluate the convergence effects on the flow
separation zone.

2. Governing equations
The governing equations used in the numerical model
are based on conservation of mass, momentum, and
energy. A state of the art CFD package, fluent 6.3, which
uses the finite volume method (FVM), was employed to
carry out the numerical computations (Fluent 2006).
FVM involves discretization and integration of the governing equations over the control volumes. The basic
equations for steady-state laminar currents are conservation of mass and momentum. Here, as heat transfer or
compressibility is not involved, the energy equation is
omitted from the equation system. The governing equations, Reynolds-averaged Navier–Stokes (RANS) and
continuity equations, in generalized Cartesian coordinates
(x, y, z) are written in their conservative forms:
@ ðqÞ @ ðqui Þ
þ
¼ Sm ;
@t
@xi




@ ðqui Þ @ qui uj
@p
@
@ui @uj
þ
¼
þ
l
þ
@t
@x i @xj @xj @xi
@xj
þ

@ qu0i u0j
@xj

;

(1)

(2)

where t is the time, ui is the ith component of the
Reynolds-averaged velocity, xi the ith axis (with the
axis-x3 vertical and oriented upward), ρ is the water
density, p is the Reynolds-averaged pressure, g is the
acceleration due to the gravity, μ is the viscosity, which
is equal to zero in this study, and Sm is the mass
exchange between the two phases (water and air). It
should be noted that the unsteady solver, which will be
used to get the velocities and other solution variables,
now represents time-averaged values instead of instantaneous values.
The term ( qu0i u0j ) is called Reynolds stress and has
to be modeled in order to close the momentum equation.
To do this, the Boussinesq hypothesis has been used,
which relates Reynolds stresses to the mean rate of
deformation:


@ui @uj
0
0
qui uj ¼ lt
þ
;
(3)
@xj @xi
where μt is the turbulent viscosity.
3. Turbulence modeling
Five different turbulence models were adopted in conjunction with the non-equilibrium wall function treatment
to perform the numerical simulations presented in this
work. The standard k–ε model, realizable k-ε model,
renormalization group (RNG) k–ε, k–ω, and the
Reynolds stress model (RSM) are the turbulence models
that were used in the present study and included in the
equations system to approximately solve the governing

Downloaded by [University of Nebraska, Lincoln] at 22:35 01 September 2015

Journal of the Chinese Institute of Engineers
equations. Only the essential features of the turbulence
models will be discussed here since the reader may
refer to the earlier paper published by the authors
(Seyedashraf and Akhtari 2013).
The standard k–ε model is a model on the basis of
model transport equations for the turbulence kinetic
energy (k) and its dissipation rate (ε). The equation for k
contains additional turbulent fluctuation terms, which are
unknown. Again, using the Boussinesq assumption, these
fluctuation terms can be linked to the mean flow.
Simplified model equation for k:


@ðqkÞ
l
þ divðqkUÞ ¼ div t grad k þ 2lt Eij :Eij qe;
@t
rk
(4)
Simplified model equation for ε:


@ðqeÞ
l
e
þ divðqeUÞ ¼ div t grad e þ C1e 2lt Eij :Eij
@t
k
re
e2
C2e q ;
k
(5)
where U is the velocity vector, Eij is the mean rate of
deformation tensor, C1ε and C2ε are constants and

3

typically values 1.44 and 1.92 are used, respectively. The
Prandtl number σk, connects the diffusivity of k to the
eddy viscosity and normally a value of 1.0 is used, while
the Prandtl number σε connects the diffusivity of ε to the
eddy viscosity and generally, a value of 1.30 is used.
RNG k–ε equations are derived from application of a
precise statistical technique to the instantaneous Navier–
Stokes equations. They are similar in form to the standard k–ε equations, but with an additional term in the ε
equation for dealings between turbulence dissipation and
mean shear. Its noting of the effect of swirl on turbulence makes the method appropriate to model rather
swirling and secondary flows. Below are the equations
written for steady incompressible flow neglecting the
body forces.
Turbulent kinetic energy:


@k
@
@k
qUi
¼ lt S 2 þ
ak leff
qe;
(6)
@xi
@xi
@xi
Dissipation rate:


2
e
@e
@
@e
e
2
¼ C1e
ae leff
qUi
lS þ
C2e q
@xi
k t
@xi
@xi
k
R;
(7)

Figure 1. Schematic representation of the open channel: (a) section view; (b) and the plan view; and (c) of the mesh form used to
perform the computations.

4

O. Seyedashraf and A.A. Akhtari

Table 1. Experimental characteristics.
Radius of curvature (m)

Depth of flow (m)

Channel width (m)

Mean velocity (m/s)

Reynolds number

0.12

0.403

0.394

36,765

0.6

Table 2. Root-mean-square errors of numerical velocity results of different turbulence models.

o

Downloaded by [University of Nebraska, Lincoln] at 22:35 01 September 2015

0
45o
90°
40 (cm)
80 (cm)

Standard k-ε

Realizable k-ε

RNG k-ε

k-ω

RSM

5.80
6.56
9.45
4.40
5.60

4.80
6.05
9.25
6.30
6.10

7.30
5.36
11.45
6.10
9.30

5.00
6.87
10.08
9.94
7.04

2.80
5.59
9.96
7.00
4.80

where R is an additional term related to mean strain and
turbulence quantities, which is the main difference
between the RNG and Standard k–ε model. S could be
calculated from the velocity gradients, and αε, αk, C1ε,
and C2ε are derived through RNG theory (Gaudio,
Malizia, and Lupelli 2011).
Realizable k–ε is another improved turbulence model,
which shares identical turbulent kinetic energy equation
with the Standard k–ε model with an enhanced equation
for the ε parameter. Its ‘Realizability’ stems from the
modifications that allow certain mathematical limitations
to be followed, which ultimately improve the performance of the model. It resembles the RNG model, but is
possibly more accurate and easier to converge. Improved
performance for flows involving planar and round jets,
boundary layers under strong adverse pressure gradients
or separation, rotation, recirculation, and strong streamline curvature are the most significant properties of the
model. Its simplified equation is as follows:


De
@
l @e
e2
pffiffiffiffi
q
¼
lþ t
þ qc1 Se qc2
Dt @xj
re @xj
k þ me
e
(8)
þ c1e c3e Gb ;
k
where Gb is the generation of turbulent kinetic energy
due to buoyancy.
K–ω is another two equation model, which offers largely the same benefits as RNG. In this model, ω is an
inverse timescale that is associated with the turbulence.
The model solves two additional partial differential equations, a modified version of the k equation used in the
k–ε model in addition to a transport equation for ω. Its
numerical behavior is similar to that of the k–ε
models and suffers from some of the same impediments,
such as the assumption that μt is isotropic. The turbulent
viscosity is calculated as follows:
lt ¼ q

k
;
x

(9)

Figure 2. Comparison of water levels in different sections: (a)
0°; (b) 45°; and (c) 90°. Lines: standard k–ε method; dots:
experimental measurements (Akhtari 2010; Akhtari, Abrishami,
and Sharifi 2009).

Downloaded by [University of Nebraska, Lincoln] at 22:35 01 September 2015

Journal of the Chinese Institute of Engineers

5

Figure 3. Numerical velocity profiles in different sections of the channel compared with the experiments of Akhtari, Abrishami, and
Sharifi (2009) and Akhtari (2010), for different turbulence models: (a) The beginning of the bend; (b) at 45° of the bend; (c) At 90°
of the bend; (d) 40 (cm) after the bend; and (e) 80 (cm) after the bend.

RSM closes the RANS equations by solving extra transport equations for the six independent Reynolds stresses.
Its transport equations resulting from Reynolds averaging
are the outcomes of the momentum equations with a
fluctuating property and one equation for turbulent dissipation. Here, the isotropic eddy viscosity assumption is
avoided, and the equations contain terms that need to be
modeled. The model is suitable for accurately predicting
complex flows like streamline curvature, swirl, rotation,
and high strain rate currents, and is recommended to
model open-channel flow problems (Lu et al. 2003).
The exact equation for the transport of the Reynolds
stress Rij is as follows.
DRij
¼ Pij þ Dij eij þ Pij þ Xij ;
Dt

(10)

where Pij is the rate of production, Dij is the transport by
diffusion, εij is the rate of dissipation, Pij is the transport

due to turbulent pressure–strain interactions, and Ωij is
the transport due to rotation.
This equation describes six partial differential equations; one for the transport of each of the six independent Reynolds stresses. As noticed by the authors, RSM
is the most time-consuming and accurate model. It
should be noted that in cases, where there is no possibility to determine Reynolds stress parameters explicitly, it
is more convenient to use the turbulent kinetic energy
parameters instead.
4. Geometry and grid form
The three-dimensional flow domain was divided into
numbers of non-overlapping unstructured mesh with a
total of 386,400 segments and 409,683 nodes. Out of
different possible meshing schemes, the chosen form is
suitable for both accuracy and the time duration of the
convergence, and was obtained from a grid-independence

6

O. Seyedashraf and A.A. Akhtari
curvature of the duct was discretized by diminutive cells.
One hundred and seventy-five longitudinal, 50 latitudinal, and 28 altitudinal segments were created in the
specified computational domain. To simulate the fully
developed flow, the experimental channel was modeled
in duplicate. Figure 1(a)–(c) shows the geometric layout
and the mesh form in the section and plan view.
The inlet discharge flow was 19.1 (m3/s) while the
water level in the entrance channel was 12(cm).

Downloaded by [University of Nebraska, Lincoln] at 22:35 01 September 2015

5. Numerical calculations

Figure 4. Distribution of mean velocities along channel bend
for the numerical simulation of a regular sharp bend.

study (Akhtari 2010). To obtain sound data of the
separation, flow depth, and the secondary currents, the

The governing Equations (1) and (2) are a set of convection equations with velocity and pressure coupling based
on the control volume method. A general-purpose CFD
code was used for all the numerical simulations presented
in this research. The code employs the FVM in conjunction with a coupling technique, which simultaneously
solves all the transport equations in the whole domain
through a false time-step algorithm. Convection terms are
discretized using the third-order Monotone UpstreamCentered Scheme for Conservation Law. The linearized
system of equations is preconditioned in order to reduce
all the eigenvalues to the same order of magnitude. The
Pressure-Implicit with Splitting of Operators (PISO)
method was employed to deal with the problem of velocity and pressure coupling. PISO methods incorporate

Figure 5. Development of secondary flow along channel bend at typical cross section: (a) 45° of the bend; (b) 90° of the bend;
(c) 40 (cm) after the bend; and (d) 80 (cm) after the bend (height-to-width ratio is 2:1).

Downloaded by [University of Nebraska, Lincoln] at 22:35 01 September 2015

Journal of the Chinese Institute of Engineers

7

Figure 6. Velocity profiles in a numerically simulated converging bend: (a) The beginning of the bend; (b) 45° of the bend; (c) 90°
of the bend; (d) 40 (cm) after the bend; and (e) 80 (cm) after the bend.

pressure effect through momentum equations into
continuity equation to obtain correction equations.
The volume of fluid (VOF) method, a free surface
tracking technique, was employed to simulate the
air–water interaction. Hirt and Nichols (1981) have
developed the model, of which the formulation relies on
the fact that two or more phases are not interpenetrating.
For each additional phase added to the system, a variable
was introduced in the volume fraction of the phase in
the computational cell. In each control volume, the
volume fractions of both air and water phases sum to
unity.
For a full convergence, about 3500 iterations were
conducted for each simulation. The structured curvilinear-grid form used to discretize the computational
domain, has significantly enhanced the acceleration of
convergence.
The hydraulic parameters for the fluid flow are
shown in Table 1.

6. Validation tests
In order to use the same numerical procedure for subsequent simulations, the model should be validated. To this
end, comparison of velocity data from experiments and
direct numerical simulations are also presented. Accordingly, the root-mean-square errors (RMSE) of the
numerical velocity results are computed and listed in
Table 2.
Here, the test sections are located at 0°, 45°, 90°
upstream into the bend and 40 (cm) and 80 (cm) after
the outlet of the bend in the middle depths.
It can be seen from Table 2 that the RSM following
the standard k–ε model trends toward better results;
however, in this study the standard k–ε model has been
chosen as the preferred turbulence model because of its
fast convergence. The model demonstrates the separation
and secondary flows reasonably accurately. A numerical
and experimental analogy of the water levels in different

Downloaded by [University of Nebraska, Lincoln] at 22:35 01 September 2015

8

O. Seyedashraf and A.A. Akhtari

sections of the channel bend, which is called super
elevation, is shown in Figure 2(a)–(c).
The comparisons between the experimental and the
numerically predicted flow velocities; using five different
turbulence models are shown in Figure 3(a)–(e).
In order to have a better insight into the velocity discrepancies and the flow separation, Figure 4 depicts the
distribution of the numerically obtained mean tangential
velocities at 12 cross sections along the regular sharp
open-channel bend.
Figure 5(a)–(d) exhibits the growth process of transverse circulation of water – the previously mentioned
helical motion – through and after the bend. The
depicted vectors are the measured quantity of transverse
velocity vector fields.
As can be seen from the figures, the helical motion
has a center of rotation, which gently shifts inwards and
downwards through the bend.
In Figures 3–5, it is obvious that the distribution of
the velocity contours is non-symmetric, which is due to
the existence of secondary currents and pressure variations. The average velocity of flow is 39.4 (cm/s) and the
value approaches zero in the vicinity of the side walls differing from the utmost velocity below the water surface
and near the outer bank, which is due to the boundary
layer formation and the no-slip wall condition. In Figure 3,
there is a noticeable flow separation tendency near the
inner bank of the bend and the 90° section, which is captured reasonably accurately. Concerning Table 2 and the
obtained RMSE of water levels with 1.6655 in the section
of 0°, 2.1903 in the 45° section, and 1.0995 in the 90°
section, it can be acknowledged that the numerical model
is capable of reproducing the flow behaviors in sharp
open-channel bends.
7. Converging bend
The main objective of the present work is to examine
channel convergence effects on the separation zone
length, as can be ascertained from Figure 3(d) and (e).
This is, to the authors’ knowledge, the first such attempt
at converging the channel bend for suppression of flow
separation. A three-dimensional numerical model of a
converging bend was simulated. The geometry is identical, as illustrated before, nevertheless, the channel width
converges along the meander, changing from 40.3 (cm)
in the 0° section to 20.15 (cm) in the outlet section of
the bend. The meshing outline and the solution procedure are the same as used earlier in Sections 4 and 5.
Performing the numerical simulation, it can be seen
that the rapid changes in velocity distributions of the
same cross-sectional area of the bend are significantly
reduced (Figure 6).
Moreover, the flow tendency to separate from the
inner bank of the bend is totally removed; however, it is

Figure 7. Distribution of mean velocities along channel bend
for the numerical simulation of a converging sharp bend.

obvious that the situation has led to an increase in the
super elevation through the bend.
The distribution of the mean tangential velocities
along the channel is depicted in Figure 7.
According to Figures 6 and 7, the velocity distribution
in the converging bend is not symmetrical and is shifted
toward its utmost value at the outer bank of the curvature
at the section placed 80 (cm) after the bend. Additionally,
these Figures indicate that the flow separation tendency
was avoided in the vicinity of the inner bank. This is
attributed to a lower mean velocity; and therefore, a higher
average super elevation in the channel curvature.
8. Conclusions
Comparing the results obtained from five different turbulent models, it has been concluded that the RSM following the Standard k–ε model in conjunction with the VOF
free surface model have the capability of capturing specific flow behaviors in an open-channel bend reasonably
accurately. The location of the minimum velocities in an
ordinary sharp open-channel bend was evaluated; it is
notable that the least velocity occurs near the inner bank
of the bend and inside the zone that has a separation tendency along the bend. However, converging the openchannel bend with the similar channel specifications, the
flow separation tendency (tendency of the boundary layers to separate from the inward wall of the bend) in the
vicinity of the inner bank was totally avoided, while the
minimum and maximum flow velocity situations did not


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