SECOND ORDER MODELLING OF COMPOUND

OPEN CHANNEL-FLOWS

école nationale d'ingénieurs de Tunis

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Laboratoire de Modélisation en Hydraulique et Environnement

Prepared by :

Olfa DABOUSSI

Presened by

Zouhaïer HAFSIA

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Plan

Introduction.

Secondary currents in compound open channel flow.

Turbulence model.

Numerical results.

Conclusions.

Experimental results.

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INTRODUCTION

In laboratory, compound channel are represented by the main channel and one floodplain with rectangular sections.

After strong rain.

The compound channel is composed from many stages. Mean channel

The turbulence model : second order model Rij.

We use the CFD code PHOENICS for numerical simulations.

It is interesting to study the compound channel flow to understand main channel – floodplain interaction.

Numerical results are compared to experimental data of Tominaga and al. (1989).

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THE RECTANGULAR OPEN COMPOUND CHANNEL FLOW

I – Rectangular compound channel

Symmetric

Asymmetric

B

b

HhH

Free surface

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Three cases of β values are simulated (Mesures of Tominaga and al., 1989).

β = 0.5

β = 0.343

β = 0.242

λ = 2.07

II – Tested cases

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Cas B (m) b (m) λ = B/b h (m) H (m) β = (H-h)/HWmoy (m

s-1)

1 0.195 0.094 2.07 0.0501 0.1001 0.500 0.315

2 0.195 0.094 2.07 0.0501 0.0661 0.242 0.32

3 0.195 0.094 2.07 0.0501 0.0763 0.343 0.273

Tominaga and al. (1989) data

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NUMERICAL SIMULATIONS

I – Governing Equations

In incompressible Newtonian fluid and parabolic flow through the z direction, the

Reynolds stress turbulence model of Launder and al. (1975) is written as :

U V0 (1)

x y

- Continuity :

- Momentum :

U U ² U ² U u ' ²U V (2)

x y x² y² x

V V 1 P ² V ² V v ' ²U V g (3)

x y y x² y² y

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W W ² W ² W u 'w ' v 'w 'U V gI (4)

x y x² y² x y

2 2 2t tt

k k

k k k k U V W WU V ( ) ( ) (( ) ( ) ( ) ) (5)

x y x x y y y x x y

- Kinetic equation :

22 2 2t t

1 2

U V W WU V ( ) ( ) C (( ) ( ) ( ) ) C (6)

x y x x y y k y x x y k

- ε equation :

i j i j i j i jijs k k ijl

l l ll

ij k1 i j ij 2 ij

' ' ' ' ' ' ' 'k 2u u u u u u u u( )C u ' u ' PU

t x 3x x x

2 2( ' ' k) ( ) (7)C u u C P P

k 3 3

- Reynolds stress :

i, j, k = 1, 2, 3

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)xU'u'u

xU'u'u(P

l

ilj

l

jliij

xU'u'uP

k

lklk

C1, C2 and Cs are constants.

- Boundary conditions :

1ln AyU

- Smooth wall logarithmic law :

yU

y * 0.41

- Near walls, k , ε and Rij are :

3*

w

w

Uy

*

UU

U

2*

wU

kC

- The free surface is considered as a symmetric plane :

(U, W,k, )0

y

u 'v ' 0 v 'w ' 0 V 0

- On the vertical symmetric plane : (V, W,k, )0

x

u 'v ' 0 u 'w ' 0 U 0

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I – In PHOENICS

There are four derivations of the Rij model : IPM, IPY, QIM and SSG.

• IPM is the Isotropisation of production model.

• IPY is the IPM model of Younis (1984).

• QIM is the quasi-isotropic model

• SSG is the model of Speziale, Sarkar and Gatski

Where :

PHOENICS use the finite volume numerical method.

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PHOENICS take the z axis as the main flow direction for the parabolic ones. The cell along z is a slab.

j

j j

( U ) S (8)t x x

The general form of the transport equations is :

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k-ε model :

Using the CFD code PHOENICS, we have tested two cases for β = 0.5.

The k- can not

reproduce the

isovelocity bulging

shown experimentally

RESULTS OF SIMULATIONS

Results with coarse grid :

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The k-ε model is isotropic.

The second order model Rij take account of the turbulence anisotropy.

k- do not

reproduce the

isovelocity

bulging

Results with thin grid :

Experience shows a strong anisotropy

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Results with the Rij model:

The four derivations of Rij give the same results.

• Secondary currents :

Main channel vortex

Free surface vortex

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Tranversal velocity profils

on X = 0.087 m : = 0.5

on X = 0.101 m : = 0.5

on X = 0.055 m : = 0.5

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Longitudinal velocity variations :

Wall law at X = 0.02 m : β = 0.5 Vertical averaging of the velocity : β = 0.5

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Comparaison of numerical isovelocity with Tominaga et al. (1989) data : β = 0.5

The Rij model reproduce well the isovelocity.

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Wall shear stress variation :

The shear stress on the floodplain raises near the main channel- floodplain junction.

Momentum transfer from the main channel to the floodplain.

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z2

(SW) (SWW) 1 (S P) ² (SW) (S w"w" ) (S w 'w ' )M

t z z z z z

The adimensional dispersion coefficient appears after integration of the momentum equation through the transversal section.

Needs a closure law

Two approches :

Gradient closure :zz

Ww"w" D

z

Correlation based on the momentum distribution

W W

W W

THE ADIMENSIONAL DISPERSION COEFFICIENT

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Whith λ = 2.07, α varies as second degrees polynomial in function of as follow :

α variation in function of β :

21.05 0.3 0.66

Calculations show that α is different from 1. It depends of : from 1.04 to 1.35

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CONCLUSIONS

Numerical computation show that the dispersion coefficient α is expressed as a polynomial function of β

Secondary currents modify longitudinal iso-velocity.

The first order k-ε model do not reproduce the isovelocity bludging

The second order turbulence model can reproduce the interaction between the main channel and flood-plain (momentum transfer)