2d finite element modeling of bed elevation change in a curved channel

Environmental Hydrodynamics Lab.Yonsei University, KOREA RCEM 2005

2D finite element modeling 2D finite element modeling of bed elevation change of bed elevation change

in a curved channelin a curved channel

S.-U. Choi, T.B. Kim, & K.D. MinYonsei UniversitySeoul, KOREA

Environmental Hydrodynamics Lab.Yonsei University, KOREA

RCEM 2005

Introduction

• Most natural streams are sinuous and meandering.

• In a curved channel, the centrifugal force makes the flow structure and sediment transport mechanism extremely complicated.

• To simulate the flow and morphological change in a curved channel, the secondary currents and the gravity effect due to morphological change should be properly considered (Kassem & Chaudhry, 2002).


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Why 2D Model?

• 1D Model– Impossible to account for sediment transport in the transverse

direction

• 3D Model– Still Expensive– Not readily applicable to many engineering problems

• Turbulence Closure

• Sediment Transport Model

• Boundary Conditions


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Previous Study

• Only applicable to the steady flow condition, constant channel width and constant radius curvature– Koch & Flokstra (1981), Struiksma et al. (1985), Shimizu & Itakura

(1989), Yen & Ho (1990) and so on.

• The coordinate transformed, unsteady FDM & FVM– Kassem & Chaudhry (2002), Duc et al. (2004), Wu (2004)

• The finite element model for bed elevation change in a curved channel has never been proposed!!– FEM provides greater flexibility in handling spatial domain.


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Purpose

• To Develop a 2D FEM model– capable of predicting time-dependent morphological change in a c

urved channel.

• For flow analysis, the shallow water equations are solved by the SU/PG scheme.

• To assess the be elevation change, Exner’s equation is solved by BG scheme.

• For validation, we applied the model to two laboratory experiments.


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Limitations

• Decoupled modeling approach• Flow equations and Exner’s equations are solved separately.

• Uniform sediment• Neglecting armoring or grain sorting effects.


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Flow Equations

• 2D shallow water equations with the effective stress terms

• Eddy viscosity model

2 2 2

1/ 22 27 /3

2 2

0

2 02

22

bt t

t t

h p q

t x y

zp p gh pq p p q gngh p p q

t x h y h x x y y x x h

q pq q gh p q q

t x h y h x y x y y

2

1/ 22 27 /3

0bz gngh q p q

y h

*6t U h


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Bed Sediment Conservation

• Exner’s equation

• Total Sediment Load

1 0tyb txqz q

pt x y

cos

sintx t

ty t

q q

q q

1/ 2 3/ 2

2 00.05/ 1t s

s s

dq V

g d

Engelund & Hansen’s formula


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Finite Element Method (1)

• Flow Equations

• Weighted Residual Equations

• 2D SU/PG Method (Ghanem, 1995)* i ii i y

N NN N x y

x y

xW W

2 2 2 2, y

x

A BW W

A B A B

t x y x y

yx

DDU U UA B F 0

0i ii

N NN x y d

x y t x y x y

yxx y

DDU U UW W A B F


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Finite Element Method (2)

• Exner’s Equation

• Weighted Residual Equation

• BG Method*i iN N

1 0tyb txqz q

pt x y

* 1 ' 0tyb txi

qz qN p d

t x y


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Boundary Conditions

• Upstream & Downstream BCs

• Sidewall BC

tan

k

k

NdA

yNdA

x

(Akanbi, 1986)


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Flow Characteristics of a Curved Channel

(a) Under a flat (& fixed) bed condition- The centrifugal force makes higher flow depth, but lower mean velocity, at the outer bank. This generates the secondary flows satisfying the continuity. - Observed in Experiments and Numerical simulations.

(b) Under a mobile bed condition- Secondary flows induces sediment erosion & deposition at the outer & inner banks, respectively.- The flow depth and mean velocity at the inner bank is lower. - Observed in natural meandering rivers and Experiment by Yen (1967 & 70)


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Direction of sediment transport

• Gravity effect on a slope

• Angle of bed shear stress due to the secondary flow effect

*

*

1sin

tan1

cos

b

s

b

s

z

f yz

f x

(Struiksma et al., 1985)

12 1/ 6

2tan , 1

s

n gFh F

R h

(Rozovskii, 1957)

1tanv

u

2 23

1 1

s

v v u uu uv uv v

R V x y x y


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Applications1. 180º Curved Channel Experiment

Lab. of Fluid Mech. (LFM) in Delft Univ. of Tech. (Sutmuller & Glerum, 1980)

2. 140º Curved Channel ExperimentDelft Hydraulics Lab. (DHL) (Struiksma, 1983)


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LFM 180º Curved Channel

Q(m3/s)

B(m)

h(m)

u(m/s)

S0

(×10-3)C

(m1/2/s)d50

(mm)Rs

(m)L

(m)

0.17 1.7 0.2 0.5 1.8 26.4 0.78 4.25 13.35

x (m)

y(m

)

0 5 10 15 20

0

5

10

Flow

• 1400 elements, 1551 nodes

• Porosity = 0.4

• 10 times extension of width

• Fr = 0.36

• Fixed bed B.C. for upstream & downstream boundaries

• Experimental Conditions


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x (m)

y(m

)

16 18 20 22

0

5

10

x (m)y

(m)

16 18 20 22

0

5

10

10 min. 150 min.

Direction of Sediment Transport (LFM)

• At the initial stage, the particles are heading for the inner bank. This induces sediment deposition & erosion at the inner and outer banks.

• After for a while, the gravity effect due to changed bed reduces the secondary flow effect.


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x (m)

y(m

)

16 18 20 22

0

5

10

0.220.210.20.190.180.170.160.150.14

x (m)y

(m)

16 18 20 22

0

5

10

0.40.350.30.250.20.150.10.05

10 min. 150 min.

Flow Depth (LFM)

• At the initial stage, the flow depth near the outer bank is higher than that near inner bank.

• A similar pattern at 150 min. But, considering deposition & erosion, the water surface elevation across the width is nearly uniform.


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x (m)y

(m)

16 18 20 22

0

5

10

0.650.60.550.50.450.40.350.30.250.20.15

x (m)

y(m

)

16 18 20 22

0

5

10

0.650.60.550.50.450.40.350.30.250.20.15

10 min. 150 min.

Velocity Distribution (LFM)

• At the initial stage, the mean velocity near the inner bank is slightly higher.

• Later, we have an opposite situation after bed deformation.


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Evolution of Depth-Averaged Velocity


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Bed Elevation Change (LFM)

0.50.250

-0.25-0.5-0.75-1

Measured data bySutmuller & Glerum (1980)

Simulated Result

• In the numerical simulation, the bed elevation change became negligible after 150 min.

• A good agreement. But the location of max deposition is slightly different. This may be …


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Longitudinal Bed Profile (LFM)

• Overall trend is the same.

• Near the inner bank, the amount of sediment deposition is over-predicted.

•Near the outer bank, the amount of sediment erosion is under-predicted.


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DHL 140º Curved Channel

• Experimental Conditions

(Struiksma, 1983)

Q(m3/s)

B(m)

h(m)

u(m/s)

S0

(×10-3)

0.062 1.5 0.1 0.41 2.03

• 945 elements, 804 nodes• Porosity = 0.4• 10 & 15 times extension of width for US & DS, respectively• Fixed bed B.C. for both US & DS

x (m)

y(m

)

0 5 10 15 20 25 30

0

10

20

30

Flow

C(m1/2/s)

d50(mm)

Rs

(m)L

(m)

28.8 0.45 12.0 29.35


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Longitudinal Bed Profile (DHL)

• Simulated result is after 10 hr.

• Overall trend is the same.

• Especially good agreement in max deposition & erosion.

• The simulated results fluctuate with distance while...


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Variation with Time (DHL)

• Spatial fluctuation increases with time.

• This is due to BG scheme applied to Exner’s eq.


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Conclusions

• Development of 2D FEM model for bed elevation change– SU/PG method for shallow water eqs.

– BG method for Exner’s eq.

– Secondary flow effect and gravity effect on sloping bed

• Applications to 2 curved channel experiments– The model predicts the flow and bed morphology well.

• Specially, the time-evolution of changing bed morphology from the flat bed. • Sediment deposition & erosion at the inner & outer banks.

• Necessity of introducing the upwind scheme to Exner’s eq.– Spatial fluctuations in the simulated bed profiles increase with time.

– Weighting is required in the upwind direction along the trajectory of sediment particles.

2d finite element modeling of bed elevation change in a curved channel

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