[PPT] - A model for bed-load transport and morphological evolution in PowerPoint Presentation

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A model for bed-load transport and morphological evolution in rivers: description and pertinence

André Paquier Kamal El kadi Abderrezzak

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During floods, bed of rivers moves up and down

What is the peak water surface elevation during a flood when depositions/erosions? What is the minimum bed elevation reached during flood (stability of banks and structures)?

Solution requires:

Estimate of bed load transport Calculation method for water surface elevation when a moving bed

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!

"#$%#&' " "

−

= ∂ ∂

=

+ ∂ ∂ ∂ ∂

+

− = +

+
∂

∂ ∂ ∂ ∂ ∂ ( ) ( ) ( )

ρ ρ ρ ρ ρ ρ − − − =

(

)

=

+ − ∂ ∂ ∂ ∂

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" ( %$)*(+,-'(./(0(∆ ∆ ∆ ∆1./(2/(α α α α3./(/ % /(-/4 /(+/'

1.82 1.92 2.02 2.12 2.22 2.32 2.42 5 10 15 20 25 x (m) z (m)

Initial profile Without space lag With space lag Measurements Water surface Bottom

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5 67 8 7

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#! ! !

=

=

#

σ

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91

7 7 7 7

=

= + = =

⊕
+

+ + +

σ

σ σ σ σ

$

$ σ

"
$

$ σ

"

⊕ σ $ $

"

:1;

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"

$

$ σ

"
∅
σ

$ $

"
$

$ σ

"
×

= × =

− × − × ∆ − − × − × ∆

σ

σ σ σ

σ

σ σ

:

×

= × =

− × − × ∆ − × − × ∆ −

σ

σ σ σ

σ

σ σ

Exponential laws with one or two calibration parameters

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17

)( ! .(σ

σ σ σ .(0

< ./(σ

σ σ σ .2

&58 = (77 <

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" 5

5∆zj related to (τj- τc,j)m

» ∆zj=0 si τc,j ≥ τj

&

) *+,- ) ∆+.- '/-" ) ∆+. / τ$.0 ζ×τ." %∆zj=0 if ζ× ζ× ζ× ζ×τj≥ τc,j) ) ∆+. / τ."

=

=

τ

τ ς ς

$

∆

∆ = =

!

! τ τ ς ς

$

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:

>
ρ

τ =

5 ?5

@ 9 9

%9:'%1/--$(('

A

# # # @ A @ # @ # @ # " @ (# @ (#(A @ "

"

"

ρ τ =

#" #" "

$ %

=

Water surface

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@ %B )(+-'(8

0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 MPC G-R-Directes G-R-Vitesses G-R-Pressions

Bank Bottom section Bank

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5

7.4 7.7 8 8.3 8.6 8.9 9.2 2 4 6 8 10 12 14 16 18

y (m) z (m)

Initial bed Horizontal Layers Uniform

( )

$
$
τ

τ τ τ ς ς ς ς τ τ τ τ −

( )

$
$
τ

τ τ τ ς ς ς ς τ τ τ τ −

( )

$
τ

τ τ τ

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5

7 8 9 10 11 12 13 14 15 16 17 18 19 20 5 10 15 20 25 30 35 40 45 50 55 y (m) z (m)

Initial bed Uniform Varied

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First step: water fluxes at cross sections by 2nd
rder MUSCL type explicit scheme

Second step: sediment mass balance and

geometry changes

Third step: water variables calculated

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9" 9"

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9"

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Crosssection

Crosssection Crosssection Crosssection

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"

Mam Ac Mav Bs

,-.

,#""/ , 0 /AC)Bs ,"/ 1 +,"/ 2,#""/

Dam, σ

σ σ σam

Dav, σ

σ σ σav

Dc, σ

σ σ σc

Ds, σ

σ σ σs

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"

Space step

Discretization effect Space lag limits this effect

Time step

No effect between limits ensuring stability Classical: Courant number 0.5 to 1

On field cases, influence can be neglected face

to geometry evolution questions

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<

7.75 8 8.25 8.5 8.75 9 9.25 100 200 300 400 500 600 700 800 900 1000

x (m) z (m)

Dx=10 m Dx=25 m Dx=50 m

Bed at equilibrium Bed 10 days

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&C%1'

At 0.2 s

9.88 9.9 9.92 9.94 9.96 9.98 10 10.02 10.04 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Distance along the channel (m) Elevation (m) Zf0.1 Z0.1 t=0 t=0

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"7

9.88 9.93 9.98 10.03 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Distance along the channel (m) Elevation (m) z0.005s z0.001s z0.0001s Z0.5 Z0.1

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!

9.88 9.93 9.98 10.03 10.08 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4

Distance along the channel (m) Elevation (m) zf0.005S zf0.001s zf0.0001s Zf0.5 Zf0.1 Zf1

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D

68 "7 >. 1

& %τ

τ τ τ(# ζ ζ ζ ζ×τ ×τ ×τ ×τ'(2

15 m 10 m 10 m

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27 6.5 7 7.5 8 8.5 9 9.5 100 200 300 400 500 600 700 800 900 1000 x (m) z (m)

Initial