1d numerical approach to model the flow over a piano key weir (pkw).pdf
TRANSCRIPT
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7/27/2019 1D NUMERICAL APPROACH TO MODEL THE FLOW OVER A PIANO KEY WEIR (PKW).pdf
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1D NUMERICAL APPROACH TO MODEL THE FLOW OVER A
PIANO KEY WEIR (PKW)
S. Erpicum, O.Machiels, P. Archambeau, B. Dewals*, M. Pirotton
Research unit HACH, Department ArGEnCo, University of Liege (Belgium)*F.R.S.-FNRS Belgian National Fund for Scientific Research
email: [email protected]
4th June 2010
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7/27/2019 1D NUMERICAL APPROACH TO MODEL THE FLOW OVER A PIANO KEY WEIR (PKW).pdf
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Introduction Modeling principles Math/num model ConclusionsApplications
Beside physical modeling(EDF, EPFL, Ulg...) and 3D numerical modeling
(EDF), attend to developed a simplified model
Piano Key Weir (PKW) = a new type of free weir first time devised by Lemprire
(2001) to improve the design of a labyrinth weir
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Introduction Modeling principles Math/num model ConclusionsApplications
Main goals of the simplified model:
- to help in identify the most relevant geometrical parameters regarding release capacity
- to assess their pertinent range of variation
Modeling of the inlet and the outlet as 1D channels
- possibly interacting by exchange of mass and momentum along the lateral crest,
- linked by an upstream reservoir.
Inlet bottom
Outlet bottom
Lateral crestPossible exchange of
mass and momentum
Downstream
steep slope
channels
Usptreamreservoir
B
P
c d
Flow
Plane view
Elevation
x
xoutlet
xinlet
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Introduction Modeling principles Math/num model ConclusionsApplications
Cross-section averaged equations of mass and momentum conservation
Assumption : velocities normal to the main flow direction
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Introduction Modeling principles Math/num model ConclusionsApplications
Bottom slope term discretized in agreement with the FVS (water at rest)
Bottom friction with Mannings formula and modified hydraulic radius
Exchange discharge on the basis of a simple water depth difference over the crest
level and a discharge coefficient
1 1cos cos2
i b bi i ib
i
z zzg gx x
2
b
4 3
xgn u u
R
min( , )s bR
L h z z
, ,max 0, max 0,b in in s b out out sH z h z z h z
32 sgnlq g H H
,l in l q q
,l out l q q
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Introduction Modeling principles Math/num model ConclusionsApplications
Reservoir = two special twin 1D finite volumes, with distinct discharges but a single cross
section value
Upstream global discharge = only boundary condition (Fr>1 downstreamno BC)
Explicit RK time integration scheme with CFL number condition
Reservoir 1 2 N
Outlet
Inlet
Lateral crestQUp QR,out
Finite volume surface
Finite volume node
x x
QR,in
RQout,1
Qin,1
out,1
in,1
Solver written in Visual Basic (Excel VBA environment)
Convergence criteria regarding discharge value (transient flow computation)
Typical time step of 5 mm, for standard PKW scale model 50-cm long and 10-cm wide less
than 2 min of computation time on a standard desktop
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Introduction Modeling principles Math/num model Conclusions
Numerical model: 5mm space step
=.385 (thick crest)
n=.011s/m1/3 (PVC)
=1 (full exchange of mass and momentum)
Outlet axis inclination=49.7
q=.055m/s to .55m/s with step of .001m/s
Applications
Comparison of the numerical results with experimental data from scale model studies
Scale model (Machiels et al., 2009): P=.525m, B=.63m
c=d=.18m
a=b=.18m
q=.013m/s to .47m/s
32w
T
QC
W gH
Comparison of non dimensional release efficiency
curve (Cw-H/P)
2T
a bwith W n
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Introduction Modeling principles Math/num model ConclusionsApplications
Comparison of the numerical results with experimental data from scale model studies
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
0 0.1 0.2 0.3 0.4 0.5 0.6
Cw
H/P
Experimental results
Numerical results
Numerical results +/-10%
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Introduction Modeling principles Math/num model ConclusionsApplications
Comparison of the numerical results with experimental data from scale model studies
0
2
4
6
8
10
12
14
16
18
20
0.000
0.005
0.010
0.015
0.020
0 0.2 0.4 0.6 0.8 1
Fr[-]
Q[m/s
]
x [m]
Q - Inlet
Q - Outlet
Fr - Inlet
Fr - Outlet
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0 0.2 0.4 0.6 0.8 1
Elevation[m]
x [m]
Channels bottom
Free surface level in the inlet
Free surface level in the outlet
0
2
4
6
8
10
12
14
16
18
20
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0 0.2 0.4 0.6 0.8 1
Fr[-]
Q[m/s]
x [m]
Q - Inlet
Q - Outlet
Fr - Inlet
Fr - Outlet
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0 0.2 0.4 0.6 0.8 1
Elevation[m]
x [m]
Channels bottom
Free surface level in the inlet
Free surface level in the outlet
q=.50m/s
q=.11m/s
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Introduction
Comparison with experimental data shows a 10% accuracy to predict the release capacity of
a PKW geometry
Development of a simplified numerical model of the flow over a PKW
using a separated 1D modeling of the inlet and the outlet
with a common upstream reservoir
and possible interaction along the lateral crest (exchange of mass and momentum)
Physical modeling results
Numerical models ValidationLinking strategy ConclusionsApplications
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Introduction
Comparison with other
experimental data (EDF,
Univ Biskra)
Modification of the reservoir flow model (energy conservation instead of momentum)
Modification of the outlet space step (exchange with the inlet in terms of free surface level
difference)
=0 in the outlet (no gain in momentum),=1 in the inlet (full lost of momentum)
Additional developments (to be published soon)
Numerical models ValidationLinking strategy ConclusionsApplications
0.5
1.0
1.5
2.0
2.5
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
C
w
Exp 1 (ULg-HACH, 2009) Exp 2 (EDF-LNHE, 2003)
Exp 3 (Univ Biskra, 2006) Exp 4 (ULg-HACH, 2008)
Num 1 Num 2
Num 3 Num 4
Confirmation of the
solver potential