### 2d_fem

```Environmental Hydrodynamics Lab.
Yonsei University, KOREA
RCEM 2005
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.
• 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
curved 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
h p q
 
0
t x y
1/ 2
zb gn 2
p   p 2 gh 2    pq   
p     p q  
2
2
 


2





gh

p
p

q
0


 

t 

 t

t x  h
2  y  h  x 
x  y   y x  
x h 7 / 3
1/ 2
zb gn 2
q   pq    q 2 gh 2     p q    
q 
2
2
 






2


gh

q
p

q
0





 t

t 

t x  h  y  h
2  x   y x   y 
y 
y h7 / 3




• Eddy viscosity model
t 

6
U *h
Environmental Hydrodynamics Lab.
Yonsei University, KOREA
RCEM 2005
Bed Sediment Conservation
• Exner’s equation
1  p
qtx  cos   qt
zb qtx qty


0
t
x
y
qty  sin   qt
1/ 2


d
qt  0.05 sV 2 

 g  s /   1 
Environmental Hydrodynamics Lab.
Yonsei University, KOREA
 0



  s    d 
3/ 2
Engelund & Hansen’s formula
RCEM 2005
Finite Element Method (1)
• Flow Equations
U
U
U Dx Dy
A
B


F  0
t
x
y
x
y
• Weighted Residual Equations


  U
Ni
Ni
U
U Dx Dy
N



x
W



y
W

A

B



F
 d  0
x
y 
  i
x
y

t

x

y

x

y


• 2D SU/PG Method (Ghanem, 1995)
Ni
N
Wx  y i Wy
x
y
A
B
Wx 
, Wy 
A2  B2
A2  B2
Ni*  Ni  x
Environmental Hydrodynamics Lab.
Yonsei University, KOREA
RCEM 2005
Finite Element Method (2)
• Exner’s Equation
1  p
zb qtx qty


0
t
x
y
• Weighted Residual Equation
zb qtx qty 
*
N
1

p
'




 d  0
 i 
t
x
y 
• BG Method
Ni*  Ni
Environmental Hydrodynamics Lab.
Yonsei University, KOREA
RCEM 2005
Boundary Conditions
• Upstream & Downstream BCs
• Sidewall BC
N k
 y dA
tan  
N k
 x dA
Environmental Hydrodynamics Lab.
Yonsei University, KOREA
(Akanbi, 1986)
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
1 zb
f s* y
tan  
1 zb
cos  
f s* x
sin  
(Struiksma et al., 1985)
• Angle of bed shear stress due to the secondary flow effect
v
 
  tan 1    
u
F 
2  n g
  tan 1  h  , F  2 1  1/ 6 
   h 
 Rs 
(Rozovskii, 1957)
1
1  v
v   u
u  
 3  u 2  uv    uv  v 2  
Rs V  x
y   x
y  
Environmental Hydrodynamics Lab.
Yonsei University, KOREA
RCEM 2005
Applications
1. 180º Curved Channel Experiment
Lab. of Fluid Mech. (LFM) in Delft Univ. of Tech.
(Sutmuller & Glerum, 1980)
2. 140º Curved Channel Experiment
Delft Hydraulics Lab. (DHL)
(Struiksma, 1983)
Environmental Hydrodynamics Lab.
Yonsei University, KOREA
RCEM 2005
LFM 180º Curved Channel
• Experimental Conditions
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
• 1400 elements, 1551 nodes
Flow
• Porosity = 0.4
10
y (m)
• 10 times extension of width
• Fr = 0.36
5
• Fixed bed B.C. for upstream
& downstream boundaries
0
0
5
10
15
20
x (m)
Environmental Hydrodynamics Lab.
Yonsei University, KOREA
RCEM 2005
Direction of Sediment Transport (LFM)
5
10
y (m)
y (m)
10
10 min.
0
16
5
150 min.
0
18
20
x (m)
22
16
18
20
22
x (m)
• 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
Flow Depth (LFM)
0.22
0.21
0.2
0.19
0.18
0.17
0.16
0.15
0.14
5
10 min.
0
16
10
y (m)
y (m)
10
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
5
150 min.
0
18
20
x (m)
22
16
18
20
22
x (m)
• 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
Velocity Distribution (LFM)
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
5
10 min.
0
16
10
y (m)
y (m)
10
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
5
150 min.
0
18
20
22
16
18
x (m)
20
22
x (m)
• 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)
0.5
0.25
0
-0.25
-0.5
-0.75
-1
Measured data by
Sutmuller & 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)
B
(m)
h
(m)
u
(m/s)
S0
(×10-3)
0.062
1.5
0.1
0.41
2.03
C
(m1/2/s)
d50
(mm)
Rs
(m)
L
(m)
28.8
0.45
12.0
29.35
• 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
30
y (m)
Q
(m3/s)
20
10
Flow
0
0
5
10
15
20
25
30
x (m)
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.
Environmental Hydrodynamics Lab.
Yonsei University, KOREA
RCEM 2005
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