How to simulate quickly and efficiently a flow over a sharp-crested weir ?

L. Goffin, S. Erpicum, B. J. Dewals, M. Pirotton, P. Archambeau

How to simulate quickly and efficiently a flow over a spillway ?

L. Goffin, S. Erpicum, B. J. Dewals, M. Pirotton, P. Archambeau

Context

Due to climate changes, many hydraulic structures need to be checked and maybe modified

Need for efficient numerical methods

Context

A spillway flow is irrotational (proven

experimentally by Escande, 1937) 0

A x b

Free surface ? BC ?

Boundary conditions

Irregular vs. regular mesh

Free surface

Test cases

Spillway flow

Table of contents

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

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

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

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

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

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

BC – Free surface

First approach based on analytical velocity profiles fitted on perpendicular velocities on linear sections

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”

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

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

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

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

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

Boundary conditions

Irregular vs. regular mesh

Free surface

Test cases

Spillway flow

Table of contents

Test cases – Bumps

[Certificate by Doug Cavendish from The Noun Project]

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

Boundary conditions

Irregular vs. regular mesh

Free surface

Test cases

Spillway flow

Table of contents

Spillway flow

WES standard profile with Hd = 15cm

H = 1.01Hd – q = 0.1275 m³/s/m

Cd = 0.506 (exp. : 0.505)

Spillway flow

WES standard profile with Hd = 15cm

H = 4.97Hd – q = 1.7632 m³/s/m

Cd = 0.618 (exp. : 0.615)

Spillway flow

Horizontal velocities

Vertical section

At crest

Horizontal component ONLY

Spillway flow

Pressures

Along spillway

Crest = 2 m

exp = pressure probes

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]

ww

w.h

ece.

ulg

.ac.

be

Contact information Louis Goffin

L.Goffin@ulg.ac.be http://www.hece.ulg.ac.be

(Question by Martin Delin from The Noun Project)