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![Page 1: How to simulate quickly and efficiently a flow over a ... · How to simulate quickly and efficiently a flow over a sharp-crested weir ? L. Goffin, S. Erpicum, B. J. Dewals, M. Pirotton,](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b93417a09d3f2d9098cf0b2/html5/thumbnails/1.jpg)
How to simulate quickly and efficiently a flow over a sharp-crested weir ?
L. Goffin, S. Erpicum, B. J. Dewals, M. Pirotton, P. Archambeau
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How to simulate quickly and efficiently a flow over a spillway ?
L. Goffin, S. Erpicum, B. J. Dewals, M. Pirotton, P. Archambeau
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Context
Due to climate changes, many hydraulic structures need to be checked and maybe modified
Need for efficient numerical methods
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Context
A spillway flow is irrotational (proven
experimentally by Escande, 1937) 0
A x b
Free surface ? BC ?
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Boundary conditions
Irregular vs. regular mesh
Free surface
Test cases
Spillway flow
Table of contents
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BC – Irregular vs. regular mesh
How to fit well free surface and boundaries ?
1. Refine all the mesh
2. Refine the mesh locally
3. Use irregular boundary conditions
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BC – Irregular vs. regular mesh
How to fit well free surface and boundaries ?
1. Refine all the mesh
2. Refine the mesh locally
3. Use irregular boundary conditions
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BC – Irregular vs. regular mesh
Evaluation of derivatives on boundary nodes
Based on Green-Gauss theorem
1
1
1
1 1
·N
i i
i i
i i i
m N
i i i i
i
y y
x xu
x y x y
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BC – Free surface
Free surface = imposed pressure on a boundary
Main idea:
1. Use a first approximate solution
2. Defined as impervious
3. Solve
4. Compute pressures on FS
0
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BC – Free surface
Free surface = imposed pressure on a boundary
Main idea:
1. Use a first approximate solution
2. Defined as impervious
3. Solve
4. Compute pressures on FS
5. Move Free Surface
6. Go to 2.
0
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BC – Free surface
How to move the free surface ?
For H ↑, Epot ↑ and Ecin↓ but, |ΔEpot|> or < |ΔEcin|
According to flow regime, ≠ behaviors
1. Fr < 1 : |ΔEpot|>|ΔEcin|
2. Fr > 1 : |ΔEpot|<|ΔEcin|
We need to identify a critical section
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BC – Free surface
First approach based on analytical velocity profiles fitted on perpendicular velocities on linear sections
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BC – Free surface
First approach based on analytical velocity profiles fitted on perpendicular velocities on linear sections
1(1 ) ln
(1 )1
Uq
H
H
Approach based on curvilinear coordinates. See Stilmant et al., “Depth-averaged flow modeling in curvilinear coordinates”
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BC – Free surface
A velocity profile in order to …
... Predict the pressure evolution
2( ) ( )
co s2
r
p UE z
g g
co s
s
s s
pd
g U dU
dH g dH
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BC – Free surface
co s
s
s s
pd
g U dU
dH g dH
From a free surface to another
1
s
i i
pd
gp H
dHp
1i i
s
pH
p
g
d
p
d
H
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BC – Free surface
3 kinds of zone
0
sp
dg
dH
1i i
s
pH
p
g
d
p
d
H
0
sp
dg
dH
0
sp
dg
dH
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BC – Free surface
Key role of the critical section:
Minimizes the required energy level for a given discharge
The critical section rules the energy level of the flow
1i i
s
pH
p
g
d
p
d
H
0o n F
( ,
S :
)cr
cr
E x y E
p p
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BC – Free surface
In theory, we have enough information to move the free surface…
… BUT some instabilities might appear
Solution: FS smoothing
1i i
s
pH
p
g
d
p
d
H
0o n F
( ,
S :
)cr
cr
E x y E
p p
Solution: FS smoothing
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Boundary conditions
Irregular vs. regular mesh
Free surface
Test cases
Spillway flow
Table of contents
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Test cases – Bumps
[Certificate by Doug Cavendish from The Noun Project]
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Test cases – Transcritical Bump
0
0.05
0.1
0.15
0 0.05 0.1 0.15 0.2
z [m
]
p [m]
computed pressure
hydrostatic pressure
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Boundary conditions
Irregular vs. regular mesh
Free surface
Test cases
Spillway flow
Table of contents
![Page 23: How to simulate quickly and efficiently a flow over a ... · How to simulate quickly and efficiently a flow over a sharp-crested weir ? L. Goffin, S. Erpicum, B. J. Dewals, M. Pirotton,](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b93417a09d3f2d9098cf0b2/html5/thumbnails/23.jpg)
Spillway flow
WES standard profile with Hd = 15cm
H = 1.01Hd – q = 0.1275 m³/s/m
Cd = 0.506 (exp. : 0.505)
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Spillway flow
WES standard profile with Hd = 15cm
H = 4.97Hd – q = 1.7632 m³/s/m
Cd = 0.618 (exp. : 0.615)
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Spillway flow
Horizontal velocities
Vertical section
At crest
Horizontal component ONLY
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Spillway flow
Pressures
Along spillway
Crest = 2 m
exp = pressure probes
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Conclusion
Good agreement between theory/experiments and numerical results
Improve treatment near critical section
Release smoothing parameter
Investigate possibility to use curved sections
… compute a sharp-crested weir !
[Thumbs Up by Milky - Digital innovation from The Noun Project]
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ww
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ulg
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Contact information Louis Goffin
[email protected] http://www.hece.ulg.ac.be
(Question by Martin Delin from The Noun Project)