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2D finite element modeling of bed elevation change in a curved channel. S.-U. Choi, T.B. Kim, & K.D. Min Yonsei University Seoul, KOREA. Introduction. Most natural streams are sinuous and meandering. - PowerPoint PPT Presentation
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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).
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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.
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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.
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
Limitations
• Decoupled modeling approach• Flow equations and Exner’s equations are solved separately.
• Uniform sediment• Neglecting armoring or grain sorting effects.
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
Boundary Conditions
• Upstream & Downstream BCs
• Sidewall BC
tan
k
k
NdA
yNdA
x
(Akanbi, 1986)
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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)
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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)
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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.
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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.
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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.
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
Evolution of Depth-Averaged Velocity
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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 …
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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.
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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...
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
Variation with Time (DHL)
• Spatial fluctuation increases with time.
• This is due to BG scheme applied to Exner’s eq.
Environmental Hydrodynamics Lab.Yonsei University, KOREA
RCEM 2005
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.
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