A Study of Flow in Open Orthogonal and Trapezoidal Channels with
Gabion Walls
D. KASITEROPOULOU
Department of Environmental Studies,
University of Thessaly, Larissa,
GREECE
Abstract: - The turbulent characteristics of the flow in open channels with gabion walls are studied numerically.
Two trapezoidal and one orthogonal channel are used along with four different heights: 100mm, 150mm,
200mm, and 250mm of sand roughness in the gabion walls. Calculations of velocity, pressure, turbulence
kinetic energy and eddy viscosity show clearly that the presence of the roughness affect considerably the fluid
motion. As expected, the rough elements affect the mean velocity inside the channel and near the walls. Near
the solid walls, the velocity profile is significantly affected and sharply lower velocities are observed very close
to the walls.
Key-Words: - Open channel flow, Gabions, Computational Fluid Dynamics, Turbulent flow, orthogonal
channel, trapezoidal channel, computer simulation.
Received: October 25, 2022. Revised: August 13, 2023. Accepted: September 24, 2023. Published: October 6, 2023.
1 Introduction
The type and the roughness of the solid walls can
affect considerably the turbulent characteristics of
the flow in open channels. Experimentally there is a
lot of interest for aquatic flows in channels with
vegetation and gabion walls. These studies present
the influence of the wall construction on the
velocity, pressure, turbulence kinetic energy, and
eddy viscosity distribution near the bottom and up to
the channel.
Gabions are used to reduce energy loss, stabilize
the hydraulic jump, [1], [2], [3], [4], and increase
discharge coefficients, [5], [6]. Many investigators
used gabion walls in road construction as retaining
structures in areas with hills, [7], and in open
channels for protection against bed and bank
erosion, [8], [9], [10].
Many studies are performed in open channels
whose bed is covered with vegetation. The study,
[11], investigated mass and momentum transfer
across the Sediment-Water and found that there is a
variation of the mean flow and turbulence across the
SWI as a function of a dimensionless permeability.
The flow in open channels with vegetated solid
walls was also investigated, [12], [13], [14], and
found that the velocity distribution depends on the
vegetation height.
The last, compared the experimental results with
the results of a CFX-based numerical model and
found that there is a good agreement between the
two of them. In all cases examined the vegetation
height can reduce the velocity values near the walls.
The higher vegetation leads to lower velocities.
The study, [15], investigated the vegetation
geometry and the drag resistance as a function of the
flow depth. In this theory, they added the turbulent
kinetic energy equation for the layers model. The
study, [16], studied the vegetation influence on
velocity distribution by using a model that solves
the momentum and continuity equations for each
element and uses the k-ε model for the turbulence
modeling. The study, [17], investigated flows in
gabion channels and observed that the zero-place
displacement parameter decreases with the increase
in the area density of roughness elements.
In this work, we numerically investigate the
flow in open channels with gabion walls. Velocity,
pressure, turbulence kinetic energy, and eddy
viscosity patterns are presented according to the
roughness height and the flow rate in two
trapezoidal and one orthogonal open channel.
2 Mathematical Model – Numerical
Simulation
2.1 Mathematical Model
The mathematical model of the turbulent flow in
this work consists of the Reynolds-Averaged
Navier-Stokes (RANS) equations coupled with the
k-ε turbulence model. Each primitive flow variable
is decomposed into an averaged-in-time part and a
fluctuation term. For example, the velocity vector at
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DOI: 10.37394/232013.2023.18.7
D. Kasiteropoulou
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a point in the flow field is given as the sum of the
time-averaged velocity
U
and a time-dependent
velocity fluctuation
u
, i.e., we write
uUU
(1)
The time-averaged velocity vector is defined as
dtU
t
Utf
f
1
(2)
where T is a time interval much longer than the
characteristic periods of the turbulence fluctuations.
The use of mean values (in time) in the conservation
equations leads to the Reynolds-Averaged Navier-
Stokes (RANS) equations:
0
U
t
(3)
M
SuuUU
t
U
(4)
In Equation (4),
uu
are the Reynolds stresses
and τ denotes the stress tensor due to molecular
viscosity.
After introducing the concept of an effective
viscosity, μeff, the conservation of mass equation is
unchanged and the conservation of momentum
equation is written as
T
effeff UpUUU
t
U
(5)
where
is the total body force per unit mass, μeff is
the effective viscosity, and p′ is the modified
pressure defined as
)
3
2
(
3
2
eff
Ukpp
(6)
In Equation (6), ζ is the fluid bulk viscosity, ρ is the
fluid density and k denotes the turbulent kinetic
energy.
The k-ε model is used in this work for the
calculation of the turbulent viscosity at each point of
the flow field. The k-ε model is a two differential
equation model where the effective viscosity is
calculated as the sum of turbulent viscosity (μt) and
molecular viscosity (μ) i.e.,
teff
(7)
The turbulent viscosity is computed at each point of
the flow field in terms of the turbulence kinetic
energy, k, and the turbulence kinetic energy
dissipation rate, ε, by the relation
2
k
C
t
(8)
where
09,0
C
The required values of k and ε are computed at each
point of the turbulent flow field by concurrently
solving the following two partial differential
equations, [18]:
k
k
tPkkU
t
k])[()(
)(
(9)
)(])[()(
)( 21
eke
tCPC
k
U
t
(10)
where
1,45
1
e
C
,
1,90
2
e
C
,
1,00
k
,
30,1
and
k
P
is the rate of production of
turbulence kinetic energy calculated by
)3(
3
2
)( kUUUUUP t
T
tk
(11)
The ANSYS-CFX, [19], computer package is used
in this work under the assumption of incompressible
flow with constant properties (ρ= constant, μ=
constant) and bulk viscosity ζ=0.
2.2 Channel Geometry and Modeling
We use a 3D model to study the flow inside: 1)
case1: a trapezoidal channel of bottom width 2m,
top width 6m and flow depth 1m, 2) case 2: a
trapezoidal channel of bottom width 2m, top width
4m and flow depth 1m and 3) case3: an orthogonal
channel of bottom width 3m and flow depth 1m.
The gabion channel walls were simulated as gravel
bed walls with four different gravel: 100mm,
150mm, 200mm, and 250mm. To minimize the
computational burden we simulated flow by using
periodic boundary conditions in the main flow
direction. Consequently, our computational domain
has a length of 1m. A 3-D (isometric) view of the
computational domain is shown in Figure 1. Five
values of discharge were used: Q1=0.52m3/s,
Q2=0.79 m3/s, Q3=1.58 m3/s, Q4=15.41 m3/s,
Q5=41.96 m3/s.
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Fig. 1: A 3-D view of the channel as modeled in the
ANSYS-CFX environment
2.3 Boundary Conditions and Mesh Design
To complete the mathematical model, the free slip
boundary condition was specified at the free surface.
In solid weir flows walls were assumed smooth
without slip velocity (no-slip boundary condition)
whereas at the gabion channels, four different gravel
diameters were used to simulate sand-grained
roughness. A 3-D view of the mesh generated using
the ANSYS-CFX preprocessor is shown in Figure 2.
Fig. 2: A 3-D view of the mesh. Case 2
The computational domain has been discretized
using tetrahedral elements. Three to six mesh
designs were evaluated to choose the
optimal/suboptimal mesh and ensure that the
solution is independent of the mesh used. Sensitivity
to global quantities, such as mass conservation
helped judge the approximate convergence of the
solutions. The mesh finally chosen provides a good
balance between the stability of the solution and the
flow field resolution. The grid for the three mesh
designs used for all cases is shown in Table 1.
Table 1. Mesh Parameters
3 Results and Discussion
Various aspects of the computed 3-D velocity field
are presented in Figure 3, Figure 4, Figure 5, and
Figure 6. The calculated dimensionless mean total
velocity for constant discharge is shown in Figure 3
and Figure 4 for the coarse grid (Mesh 3). The
dimensionless velocity is useful when comparing
the computational results obtained with channel
flows of different gravel bed roughness diameters
and different geometries.
Fig. 3: Computed mean dimensional total velocity
value. Q=0,52m3/s
We observe that the total velocity value depends
on the roughness size, the geometry shape, and the
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flow rate. In particular, for constant discharge equal
to Q=0,52m3/s as the gravel diameter increases for
the same geometry channel shape the mean velocity
decreases. Moreover, for the same gravel diameter,
we observe that case 1 reveals larger values than
cases 2 and 3 where values are close enough.
Regarding now larger discharge (Figure 4) we
observe that as the gravel diameter increases the
dimensional velocity decreases but the values for all
channel shapes are close enough.
Fig. 4: Computed mean dimensional total velocity
value. Q=41,96m3/s
Fig. 5: Streamwise velocity value near the bottom
(point 1: 3, 0.05, 0.5) and near the top of the channel
(point 2: 3, 0.95, 0.5). Volume flow rate Q = 0.79
m3/s.
The streamwise velocity for constant discharge
near the bottom and the top of the channel is
presented in Figure 5. The velocity at these points is
useful to investigate how the gravel bed can affect
the flow pattern. Again, we observe that the gravel
effect is important both near the bottom and near the
top of the channel. However, when comparing
smooth and rough solid walls, we note that the
velocity values are significantly affected by the
roughness because sharply lower velocities are
observed very close to the rough walls: for example,
in case 2 the velocity near the smooth wall is
0.13m/s while near the rough walls is 0.08m/s (the
last can also be observed in Figure 6).
We also observe that the channel shape along
with the gravel diameter can affect considerably the
velocity values at the top of the channel. Velocity
values near the top of the channel in case 2 is higher
than in case 1 and 3. In all cases, the gravel effect is
important, and sharply lower values are detected
when the gravel diameter increases.
(a)
(b)
(c)
Fig. 6: Contour plots of speed velocity at depth y =
0.250 m, for the channel with gravel 250mm
(R250). (a) case1, (b) case2, (c) case3. Volume
flow rate Q = 0.79 m3/s.
At the bottom of the channel, a nearly linear
velocity profile is exhibited in the turbulent case and
is completely dominated by viscous effects. It
should be noted that to accurately resolve the
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boundary layers, an extremely fine grid must be
used and, even if the resolution is adequate, the
mean (in time) turbulent velocity profile is not
modeled adequately when wall functions are used in
the implementation of the k-ε model. For a
discussion on k-ε model modification, to resolve the
boundary layer up to the solid wall, [18].
Regarding the computed total mean dimensional
pressure along the channel presented in Figure 7,
Figure 8 and Figure 9 we observe that as the
volume flow rate increases the effect of the gravel
diameter becomes more important. For the same
volume flow rate the total mean pressure decreases
as the grain diameter increases whereas, for the
same roughness the total pressure increases as the
volume flow rate also increases. It is important to
notice that in case 1 the reduction is lower than in
case 2 and 3 when the gravel diameter increases.
Cases 2 and 3 reveal lower velocities than in case 1
for the same discharge and the same gravel
diameter, meaning that gravel effects are more
important in bigger shapes for low discharges.
Fig. 7: Computed mean dimensional total pressure
value. Q=0,79m3/s
Fig. 8: Computed mean dimensional total pressure
value. Q=1,58m3/s
Examining the distribution of the turbulence
kinetic energy (Figure 9) and the turbulence kinetic
energy value near the bottom (point 3, 0.05, 0.5) and
near the top of the channel (point 3, 0.95, 0.5)
(Figure 10) we observe that lower values are
detected as we move closer to the top of the channel
for smooth and rough channels. Near the bottom of
the channel higher values are detected in the rough
channels for all channel shapes and the thickness of
the maximum turbulence kinetic energy layer
depends on the roughness height and the shape of
the channel.
(a)
(b)
(c)
Fig. 9: Turbulence Kinetic energy distribution on
midplane z=0.5 m for the channel with gravel
250mm (R250). (a) case1, (b) case2, (c) case3.
Volume flow rate Q = 0.79 m3/s.
We also observe that in case 1 as the roughness
increases the turbulence kinetic energy decreases
near the bottom whereas in cases 2 and 3 higher
velocity values are detected at the bottom of the
channel as the gravel diameter increases. According
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to the top of the channel, we observe that the
turbulence kinetic energy increases as the roughness
height increases for all channel shapes.
Fig. 10: Turbulence Kinetic energy value near the
bottom (point 1: 3, 0.05, 0.5) and near the top of the
channel (point 2: 3, 0.95, 0.5). Volume flow rate Q
= 0.79 m3/s.
In Figure 11, the eddy viscosity distribution for
the channel with gabions of gravel diameter 250mm
is presented for all channel shapes. We observe that
low eddy viscosity values are detected at the solid
side walls near the surface for all cases and high
eddy viscosity values are observed in the central
part of the channel throughout the depth of the
channel. In Figure 10, we observe that both near the
surface and the bottom of the channel the eddy
viscosity increases as the roughness increases for
constant flow rate. Therefore, it is concluded, that
the gabion roughness increases eddy viscosity
values in a much greater volume of the channel for
both orthogonal and trapezoidal channels.
(a)
(b)
(c)
Fig. 11: Eddy viscosity distribution on midplane
z=0.5 m for the channel with gravel 250mm (R250).
(a) case1, (b) case2, (c) case3. Volume flow rate Q
= 0.79 m3/s.
Fig. 12: Eddy viscosity value near the bottom (point
1: 3, 0.05, 0.5) and near the top of the channel (point
2: 3, 0.95, 0.5). Volume flow rate Q = 0.79 m3/s.
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4 Conclusion
In this paper, we have presented the simulations of
turbulent flow in two trapezoidal and one
orthogonal channel whose walls are simulated as
gabion walls. The gabion walls are modeled by sand
roughness of four different heights: 100mm,
150mm, 200mm, and 250mm. Calculations of
velocity, pressure, turbulence kinetic energy, and
eddy viscosity (Figure 12) show clearly that the
presence of the roughness affects considerably the
fluid motion for all channel shapes examined. As
expected, the rough elements affect the mean
velocity inside the channel and near the walls and
this effect becomes more important depending on
the channel shape. Near the solid walls, the velocity
profile is significantly affected and sharply lower
velocities are observed very close to the walls. The
last behavior becomes more important for bigger
channel shapes at low discharges.
For the same volume flow rate the total mean
pressure decreases as the grain diameter increases.
Moreover, at the top of the channel, the turbulence
kinetic energy and the eddy viscosity increase as the
roughness increases for a constant flow rate for all
channel shapes.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
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Creative Commons Attribution License 4.0
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