3d-simulation of flow over submerged weirs with … · 3d-simulation of flow over submerged weirs...
TRANSCRIPT
3D-Simulation of Flow over Submerged Weirs with Free-Surface
R. Maghsoodi1, M.S. Roozgar2, H. Sarkardeh3
1 Ph.D Student, Department of Civil Engineering, Shahrood University of Technology,
Shahrood, Iran, email: [email protected] 2 M.Sc, Department of Civil Engineering, Shahrood University of Technology, Shahrood, Iran,
email: [email protected] 3 Civil Engineering Group, Department of Engineering, Tarbiat Moallem University of Sabzevar,
Sabzevar, Iran, email: [email protected]
Abstract
In the present study, by using the Computational Fluid Dynamics (CFD), free surface flow over
Submerged Weirs was simulated. A numerical model known as Fluent was used to numerical
modeling. The model solved the fully three-dimensional, Reynolds-averaged Navier–Stokes
(RANS) equation to predict flow near the structure where three dimensional flows are dominant.
To treat the complex free-surface flow, the volume of fluid (VOF) method with geometric
reconstruction scheme was applied and turbulence was simulated by using standard k−ε
equations. The computed results using numerical model on compressed mesh systems are found in
good agreement with measured experimental data.
Key words: Submerged Weir; Numerical Simulation; Fluent; k-ε Turbulence Model; Free Surface
Flow; VOF.
Introduction
Weirs and spillways build for passing water flow in critical conditions or for regulating the water
surface elevation. The most common types of weir crest in the practice are broad-crested weir,
sharp-crested weir, ogee crest weir and circular-crested weir. These structures which built for
measuring or regulating rate of flow in open channels usually consist of a converging transition
where subcritical flow of water is accelerating, a trapezoidal weir which it accelerates to
supercritical flow and a downstream transition where the flow velocity is reduced to an
acceptable subcritical velocity. That section of the approach channel where this water surface
elevation is measured is known as the “head measurement section” or “gauging station” (Figure
1).
Figure 1. Schematic view of a weir
Many experimental and numerical researches were carried out on the weir structures. Marc et al
(1986) studied numerically flow over weirs in a channel with thin weirs and various depths and
also for broad-crested weirs of infinite depth. They also computed discharge coefficient for a thin
weir, and presented a formula that applies when the height of the weir is large compared to the
height of the upstream free surface above the top of the weir. Dias et al (1988) simulated, flow
over rectangular weirs and calculated the corresponding discharge coefficient by using a 2D
model. Boiten (2002) derived head-discharge relations of weirs with a horizontal crest. Lin and
Liu (2005) developed an analytical solution for linear long-wave reflection by an obstacle of
general trapezoidal shape. Jia et al (2005) made a numerical simulation to study the helical
secondary current and the near-field flow distribution around one submerged weir. Göğüş et al
(2006) experimentally investigated effects of width of lower weir crest and step height of broad-
crested weirs of rectangular compound cross section on the values of the discharge coefficient,
the approach velocity coefficient and the modular limit. Xia and Jin (2007) used a multilayer
model for improvement depth-averaged model to obtain the velocity and pressure distributions in
the vertical direction. Honnorat et al (2008) presented a Lagrangian data assimilation experiment
in an open channel flow above a broad-crested weir. They observed trajectories of particles
transported by the flow and extracted from a video film, in addition to classical water level
measurements. Abdalla, Yang and Cook (2008) adopted Large-eddy simulation (LES) of
transitional separated-reattached flow over a surface mounted obstacle and a forward-facing step.
Castro-Orgaz, Giráldez and Ayuso (2008) developed a one-dimensional model based on the
critical flow in curvilinear motion. Sargison and Percy (2009) investigated flow of water over a
trapezoidal, broad-crested, or embankment weir with varying upstream and downstream slopes.
Data are presented comparing the effect of slopes of 2H:1V, 1H:1V and vertical in various
combinations on the upstream and downstream faces of the weir. Yazdi et al. (2010) simulated
flow around a spur dike with free-surface flow by using fully three-dimensional, Reynolds-
averaged Navier–Stokes equation. They also to model the free-surface flow, applied the volume
of fluid method with geometric reconstruction scheme. The turbulence model which they used
was standard k−ε equations.
In other researches which were done by other researchers, the Finite Volume Method (FVM) was
used to discretize the governing equations together with a staggered-grid system (Versteeg and
Malalasekera, 1995).
Experimental data collected
To ensure that the model is applicable, five experimental cases were selected. The selected test
cases included three fixed-bed cases. The three experimental tests were conducted in a flume
with dimensions that were 0.3 m wide, 0.38 m deep and about 7.1 m in length. Symmetrical
trapezoidal profile weirs of 150 mm high, crest length 100 mm, 150 mm and 400 mm
respectively and side slope 1V:2H were tested at different discharges. Three discharges 424.1
cm3/s/cm, 710.2 cm3/s/cm and 495.7 cm3/s/cm selected for crest length 100 mm, 150 mm and
400 mm respectively (Zerihun and Fenton, 2005) (Figure 2).
Figure 2. Constructed physical model of Zerihun and Fenton (2005)
Two other experimental test cases which were used in this study were performed by Kirkgoz et
al (2008) in a glass-walled, hydraulically smooth, horizontal laboratory channel. Their
experimental model had 0.2 m wide, 0.2 m deep and 2.4 m long. Two different test structures: a
rectangular-profile broad-crested weir and a triangular profile broad crested weir, with sharp
edges and smooth surfaces were in turn installed in the channel at a distance of 1 m from the
upstream end of the channel. Symmetrical rectangular profile weir of 88 mm, crest length 230
mm and Symmetrical triangular profile weir of 75 mm and upstream slope 1.00V:1.33H,
downstream slope 1.00V:4.47H were tested at two discharges 173.23 cm3/s/cm and 190.48
cm3/s/cm for rectangular weir and triangular weir respectively (Figure 3).
Figure 3. Constructed physical model of Kirkgoz et al (2008).
The geometric and flow parameters of these physical models were presented in Table 1.
Table 1. Model geometries and flow characteristics of the experimental cases
Weir Type Length
(m)
Width
(m)
Discharge
(cm3/s/cm)
Crest
Length
(mm)
Upstream Slope
V:H
Downstream Slope
V:H
Trapezoidal 7.1 0.3
424.1 100
1:2 1:2 710.2 150
495.7 400
Rectangular 2.4 0.2
173.23 230 - -
Triangular 190.48 - 1:1.33 1:4.47
Numerical modeling
In the present study, the fluent CFD code was employed. Numerical modeling involves the
solution of the Navier-Stokes equations in three-dimensional position, which are based on
principles of physics, i.e. mass conservation (for the continuity equation) and Newton’s Second
Law (for the momentum equations). The conservation of mass is:
( ) ( ) 0jj
Ut xρ ρ∂ ∂
+ =∂ ∂
(1)
where ρ and U are density and velocity, respectively.
For incompressible flows, the density is constant and therefore
( ) 0jj
Ux
ρ∂=
∂ (2)
The momentum equation is:
( ) ( ) jii i j i
j i i j i
UUPU U U g Ft x x x x xρ ρ μ ρ
⎡ ⎤⎛ ⎞∂∂∂ ∂ ∂ ∂+ = − + + + +⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦
r (3)
where P = pressure, gr is acceleration due to gravity. μ = 0μ + tμ , 0μ is viscosity of fluid, tμ =
turbulence viscosity and Fr
is the body force.
The three-dimensional governing equations of turbulent kinetic energy, turbulent energy
dissipation rate, using the standard k–ε model are as follows (Launder and Spalding, 1974):
Turbulent kinetic energy equation
( ) ( ) ti k b M
i j k i
kk kU G G Yt x x x
μρ ρ μ ρεσ
⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂+ = + + + − −⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦
(4)
Dissipation rate of turbulent kinetic energy
( ) ( ) ( )2
1 3 2t
i k bi i i
U C G C G Ct x x x k kε ε ε
ε
μ ε ε ερε ρε μ ρσ
⎡ ⎤⎛ ⎞∂ ∂ ∂ ∂+ = + + + −⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦
(5)
where µt is: 2
tkC μμ ρε
= (6)
In Equations 4 and 5, kG and bG are the generation rates of the turbulent kinetic energy due to
the mean velocity gradients and buoyancy, respectively. MY represents the contribution of the
fluctuating dilatation in compressible turbulence to the overall dissipation rate. 1C ε , 2C ε , and
C μ are constants and equal to 1.44, 1.92, and 0.09, respectively. kσ and εσ are the turbulent
Prandtl Numbers for k and ε equal to 1.0, 1.3, respectively.
The Volume of Fluid (VOF) method appears to be a powerful computational tool for the analysis
of free-surface flows (Hirt and Nichols 1981). The tracking of the interface(s) between the
phases is accomplished by the solution of a continuity equation for the volume fraction of one (or
more) of the phases. For the qth phase, this equation has the following form:
( ). . 0qqt
αν α
∂+∇ =
∂ (7)
where qα is the volume fraction of qth phase. In each control volume, the volume fractions of all
phases sum to unity. The following three conditions are possible for each cell:
- 0qα = : the cell is empty (of the qth fluid).
- 1qα = : the cell is full (of the qth fluid).
- 0 1qα< < : the cell contains the interface between the qth fluid and one or more other
fluids.
Thus, it can be assumed the free surface is on the volume fraction of 0.5. The properties
appearing in the transport equations are determined by the presence of the component phases in
each control volume. For example, density and dynamic viscosity in each cell of two phases are:
( )1 1 1 21μ α μ α μ= + − , ( )1 1 1 21ρ α ρ α ρ= + − , respectively. The phases are represented by the
subscripts 1 and 2.
In the present work, the geometric reconstruction method of Young (1982) was employed. The
geometric reconstruction scheme represents the interface between fluids using a piecewise-linear
approach. It assumes that the interface between two fluids has a linear slope within each cell, and
uses this linear shape for calculation of the advection of fluid through the cell faces. Moreover,
both structured and unstructured mesh were used. The area around the weir used a finer mesh
than the other region (Figure 4). The number of mesh in vertical view was increased near the free
surface to tracing more accurate the free surface. Also In boundary layers a more dense mesh is
used to consider the viscous flow in sublayers. The first grid surface off the solid boundaries was
at 0.0005yΔ = , which ensures that the first grid surface off the wall is located almost
everywhere at ( )*1.0y y u y ν+ += = Δ and that at least two grid surfaces are located within the
laminar s
shear stre
Boundary
inlets we
These in
sublayer (y+
ess, and ν is
Figure 4.
y conditions
ere needed to
lets were de
< 5.0), whe
kinetic visc
Computation
s which wer
o define the
efined as str
ere Δy is the
osity
nal grid in th
re employed
water flow
ream-wise ve
e distance of
he vicinity o
d in this inv
(Inlet I) and
elocity inlet
f first grid fr
of weir: 3D v
vestigation a
d air flow (In
ts that requir
rom the solid
view and pla
are (Figure 5
nlet II) in th
re the value
d wall, *u is
n view
5): Two diff
he model dom
s of velocity
s wall
ferent
main.
y. To
estimate the effect of walls on the flow, empirical wall functions known as standard wall
functions (Launder and Spalding 1974) were used. The k−ε turbulence model was used with
standard-wall functions. The upper boundary above the air phase was specified as a symmetry
condition, which enforces a zero normal velocity and a zero shear stress.
Figure 5. Domain of solutions and boundaries for a weir
To complete the description of the CFD simulation, the PRESTO pressure discretization scheme
was applied because this scheme was showed the best convergence in this simulation. The PISO
pressure-velocity coupling algorithm was used purely because it is designed specifically for
transient simulations. The unsteady, free-surface calculations required fine grid spacing and
small initial time steps. The grid spacing used was adequate for solution convergence and
showed good agreement with the experimental results. A time step equal to 0.001 was selected.
During the 3D simulation runs, solution convergence and the water-surface profiles were
monitored. Convergence was reached when the normalized residual of each variable was on the
order of 1000. The free surface was defined by a value of VOF = 0.5, which is a common
practice for volume fraction results (Fluent Manual 2005, Dargahi 2006). After the convergence
of the numerical solution, in order to obtain more accurate results, again mesh was refined
according to gradients of two phases and velocities and the model was run. The final number of
mesh in various conditions changed in the ranges 242000–450000 cells. A sensitivity analysis
was used and number of mesh increased two times that showed the results of model were valid.
Verification
Inlet II
Inlet I
Symmetry
WeirWall
Before employing the numerical model to study the flow pattern around the weir, it was
necessary to ensure about the accuracy of the numerical model. For this purpose, experimental
cases which mentioned in previous section were employed. To evaluate the free surface, the first
case was selected regarding the available flume data.
To assess results of flume and simulation of water-surface profiles, one longitudinal section (in
the mid-width) was selected. The compared results are shown in Figure 6.
Case 1
Case 2
0
50
100
150
200
250
-500 0 500 1000 1500
y (m
m)
x (mm)
Numerical Experimental Bed
0
50
100
150
200
250
300
-500 0 500 1000 1500
y (m
m)
x (mm)
Numerical Experimental Bed
Case 3
Case 4
Case 5
Figure 6. Surface profiles of flow over broad crested weirs
0
50
100
150
200
250
300
-1000 -500 0 500 1000 1500 2000
y (m
m)
x (mm)
Numerical Experimental Bed
80
90
100
110
120
130
875 925 975 1025 1075
y (m
m)
x (mm)
Bed Experimental Numerical
60
70
80
90
100
110
910 940 970 1000 1030 1060 1090 1120
y (m
m)
x (mm)
Bed Experimental Numerical
Flow velocity was evaluated by the forth and fifth cases, regarding the available data. In the forth
case and for the convenience of comparison, two directions are selected: direction 1, x = 0.905
m, z = 0.1 m and direction 2, x = 0.995 m, z = 0.1 m, at zone behind the weir. Figure 7, shows
good agreements between measured and computed velocities. Maximum error (in terms of
depth) was observed in direction 2, which have small depths. For such small depths, two
turbulence model equations cannot predict turbulence precisely due to the existence of a shear
layer between the recirculation zones and flow in the downstream direction.
x=0.905 m
x=0.995 m
Figure 7. Comparisons of the computed and measured horizontal velocity profiles for
rectangular weir flow (Case 4)
For fifth case, the results of simulation were compared with the experimental data and are shown
in Figure 8. Two directions are selected: Direction 1, x = 0.960 m, z = 0.1 m and Direction 2, x =
0
20
40
60
80
100
120
140
0 20 40 60 80 100 120
y (m
m)
u (mm/s)
Numerical Experimental
0
20
40
60
80
100
120
140
-50 50 150 250 350
y (m
m)
u (mm/s)
Numerical Experimental
1.066 m, z = 0.1 m, at zone behind the weir. It can be seen that agreements between the measured
and computed velocities are satisfactory.
x=0.960 m
x=1.066 m
Figure 8. Comparisons of the computed and measured horizontal velocity profiles for triangular
weir flow (Case 5)
In Figures 9 and 10, experimental and numerical velocity field vectors and the corresponding
flow patterns represented by the streamlines for the flows upstream of rectangular and triangular
weirs, are given. The streamlines which produced from the velocity field vectors are so
constructed that the flow discharge per unit width of the flow section is approximately evenly
distributed among all the stream tubes.
0
20
40
60
80
100
120
0 20 40 60 80 100 120 140
y (m
m)
u (mm/s)
Numerical Experimental
0
10
20
30
40
50
60
0 50 100 150 200 250 300
y (m
m)
u (mm/s)
Numerical Experimental
Exp
N
Expe
perimental vel
Numerical veloc
erimental veloc
locity field vect
city field vecto
city field stream
tors (Case 4)
ors (Case 4)
mlines (Case 4)
)
Figure 9
9. Compariso
Num
ons of the co
Exp
merical velocity
omputed and
perimental vel
y streamlines fi
d measured v
4)
locity field vect
fields (Case 4).
velocity field
tors (Case 5)
d vectors flo
w patterns (Case
N
Expe
Num
Numerical veloc
erimental veloc
merical velocity
city field vecto
city field stream
ty field streaml
ors (Case 5)
mlines (Case 5)
lines (Case 5)
)
Figure 10. Comparisons of the computed and measured velocity vector fields flow patterns
(Case 5)
Results of this simulation show the accuracy of the numerical model in calculating the flow field over a
weir.
Analysis of computational results
The analysis of the shear-stress field at the channel bed presents a particular interest for studying
the sediment transport over a weir. The criterion of initial motion of sediment at the streambed is
generally estimated using a critical-shear stress threshold (Ouillon and Le Guennec 1996).
Potential depositional and zones erosion are estimated from the bed shear-stress values.
Downstream from a weir in the recirculation zone, shear-stress decreases and deposition occurs.
Downstream of the weir, velocity and shear stress increase and bed erodes. The effect of slope
and length of the weir and discharge on the bed-shear stress are shown in Figures 11 and 12.
Figure 11. Bed-shear stresses of rectangular broad-crested weir (N/m2)
Figure 12. Bed-shear stresses of triangular broad-crested weir (N/m2)
Figures 13 and 14 shows the respective free-surface flow profiles from the VOF analyses, using
the k–ε turbulence model, when rectangular and triangular broad-crested weirs are placed in the
channel.
Figure 13. Computed profile changes of flow over a rectangular broad-crested weir
Figure 14. Computed profile changes of flow over a triangular broad-crested weir
Conclusions
In the present numerical study, flow over weirs was simulated by using a three dimensional code
(Fluent Software). The k−ε turbulence model with the VOF method was employed to simulate
fully 3D flow. The numerical simulating solves the Navier–Stokes equations within the flow
domain upstream and downstream of a weir that used experimental flume data obtained by other
researchers. By comparing the 3D simulation results with the flume data, the simulation was
found to produce flow over a weir with sufficient accuracy and have good agreement with
experimental data.
References
Cenling Xia, Yee-Chung Jin, 2007, "Multilayer depth-averaged flow model with implicit
interfaces", Journal of Hydraulic Engineering, 133 (10), 10.1061/(ASCE), 1145-1154.
Dargahi, B., 2006. "Experimental study and 3D numerical simulations for a free-overflow
spillway". Journal of Hydraulic Engineering, 132 (9), 899, 10.1061/(ASCE), 0733–9429.
Fluent Manual, 2005. "Manual and user guide of Fluent Software". Fluent Inc.
Frédéric Dias, Joseph B. Keller, Jean-Marc Vanden-Broeck, 1988, "Flows over rectangular
weirs", Phys. Fluids, 31(8), 2071-2076.
Hirt, C.W. and Nichols, B.D., 1981, "Volume of fluid methods for the dynamics of free
boundaries". Journal of Computational Physics, 39, 201–225.
Ibrahim Elrayah Abdalla, Zhiyin Yang, Malcolm Cook, 2008, "Computational analysis and flow
structure of a transitional separated-reattached flow over a surface mounted obstacle and a
forward-facing step", International Journal of Computational Fluid Dynamics, Vol. 23, No. 1,
25–57.
Jane E. Sargison, Aaron Percy, 2009, "Hydraulics of broad-crested weirs with varying side
slopes", Journal of Irrigation and Drainage Engineering, 135 (1), 10.1061/(ASCE), 115-118.
Jean-Marc Vanden-Broeck, Joseph B. Keller, 1987, "Weir flows", Journal of Fluid Mechanics,
176, 283-293.
Launder, B.E. and Spalding, D.B., 1974. "The numerical computation of turbulent flows".
Journal of Computer Methods in Applied Mechanics and Engineering, 3, 269–289.
M. Göğüş, Z. Defne, V. Özkandemir, 2006, "Broad-crested weirs with rectangular compound
cross sections", Journal of Irrigation and Drainage Engineering, 132 (3), 10.1061/(ASCE),
272-280.
M. Salih Kirkgoz, M. Sami Akoz, A. Alper Oner, 2008. "Experimental and theoretical analyses
of two-dimensional flows upstream of broad-crested weirs". Can. J. Civ. Eng. 35: 975–986.
Marc Honnorat, Jérôme Monnier , Nicolas Rivière, Étienne Huot and François-Xavier Le Dimet,
2008, "Identification of equivalent topography in an open channel flow using Lagrangian data
assimilation", Comput Visual Sci, 13:111–119.
O. Castro-Orgaz, J. V. Giráldez, J. L. Ayuso, 2008, "Critical flow over circular crested weirs",
Journal of Hydraulic Engineering, 134 (11), 10.1061/(ASCE), 1661-1664.
Ouillon, S. and LeGuennec,B., 1996. "Modeling non cohesive suspended sediment transport in
2D vertical free surface flows" (in French). Journal of Hydraulic Research, 34 (2), 219–236.
Pengzhi Lin, Huan-Wen Liu, 2005, "Analytical study of linear long-wave reflection by a two-
dimensional obstacle of general trapezoidal shape", Journal of Engineering Mechanics, 131
(8), 10.1061/(ASCE), 822-830.
Versteeg, H.K. and Malalasekera, W., 1995. "An introduction to computational fluid dynamics:
the finite volume method". Harlow: Longman Publications.
Wubbo Boiten, 2002, "Flow measurement structures", Flow Measurement and Instrumentation,
13: 203-207.
Yafei Jia, Steve Scott, Yichun Xu, Suiliang Huang, Sam S. Y. Wang, (2005), "Three-
dimensional numerical simulation and analysis of flows around a submerged weir in a channel
bendway", Journal of Hydraulic Engineering, 131(8), 10.1061/(ASCE), 682-693.
Yazdi, J., Sarkardeh, H., Azamathulla, H.MD. and AB. Ghani, A., (2010). "3D simulation of
flow around a single spur dike with free-surface flow". Intl. J. River Basin Management Vol.
8, No. 1, pp. 55–62.
Young, D.L., 1982. "Time-dependent multi-material flow with large fluid distortion". In: K.W.
Morton and M.J. Baines, eds. Numerical methods for fluid dynamics. New York: Academic
Press, 273–285.
Y.T. Zerihun, J. D. Fenton, 2006. "One-dimensional simulation model for steady transcritical
free surface flows at short length transitions". Advances in Water Resources 29: 1598-1607.
Notations
t time
F body force
U velocity
P pressure
kG generation rate of the turbulent kinetic energy due to the mean velocity gradients
bG generation rate of the turbulent kinetic energy due to buoyancy
g gravity acceleration
MY dilatation dissipation
ρ density of fluid
ν kinematics viscosity of water
μ viscosity of fluid
tμ turbulent viscosity
k turbulent kinetic energy
kσ turbulent Prandtl Numbers for k
εσ turbulent Prandtl Numbers for ε
ε dissipation rate of turbulent kinetic energy
1 2 3, ,C C Cε ε ε constants
qα volume fraction of qth phase
yΔ the distance of first grid from the solid wall
*u wall shear stress