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Introduction Governing Equations Numerical Schemes Test Cases Summary
Coupled Model of Shallow Water Equations
and Sediment Transport on Unstructured
Mesh
Xiaofeng Liu1 Marcelo H. Garca1
1Hydrosystems Lab, UIUC, USA
International Symposium on Environmental Hydraulics 2007
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Introduction Governing Equations Numerical Schemes Test Cases Summary
Outline
1 Introduction
2 Governing Equations
3 Numerical Schemes
4 Test Cases
Hydrodynamic Part
Coupled Model
5 Summary
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Introduction Governing Equations Numerical Schemes Test Cases Summary
Outline
1 Introduction
2 Governing Equations
3 Numerical Schemes
4 Test Cases
Hydrodynamic Part
Coupled Model
5 Summary
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Introduction Governing Equations Numerical Schemes Test Cases Summary
Introduction
Numerial Models for Scour Problem
Two-dimensional Models: Fast Evaluation, Large DomainThree-dimensional Models: Details, Small Domain,
Computational CostTwo-dimensional Models
Depth-averaged equationsHydrostatic usually assumed
Model Developed in This Paper
Shallow water equations are usedSediment Transport + Exner EquationHydrodynamcs and Morphadynamics are coupledUnstructured mesh
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Introduction Governing Equations Numerical Schemes Test Cases Summary
Outline
1 Introduction
2 Governing Equations
3 Numerical Schemes
4 Test Cases
Hydrodynamic Part
Coupled Model
5 Summary
I d i G i E i N i l S h T C S
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Introduction Governing Equations Numerical Schemes Test Cases Summary
Governing Equations
Shallow Water Equations
t+(uh)
x+(vh)
y=0
(uh)
t+(u2h)
x+(uvh)
y
(hux)
x+(huy)
y = wxbx
gh
x+ hfv
(vh)
t+(uvh)
x+(v2h)
y
(hvx)
x+(hvy)
y
=
wyby
gh
yhfu
Exner Equation
z
t+
1
10
qsx
x+qsy
y
=0
Sediment Transport Rate (Grass formulation):
qsx =Au
u2 + v2 m12
qsy =Av
u2 + v2 m12
I t d ti G i E ti N i l S h T t C S
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Introduction Governing Equations Numerical Schemes Test Cases Summary
Governing Equations
Shallow Water Equations
t+(uh)
x+(vh)
y=0
(uh)
t+(u2h)
x+(uvh)
y
(hux)
x+(huy)
y = wxbx
gh
x+ hfv
(vh)
t+(uvh)
x+(v2h)
y
(hvx)
x+(hvy)
y
=
wyby
gh
yhfu
Exner Equation
z
t+
1
10
qsx
x+qsy
y
=0
Sediment Transport Rate (Grass formulation):
qsx =Au
u2
+ v2 m12
qsy =Av
u2
+ v2 m12
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Introduction Governing Equations Numerical Schemes Test Cases Summary
Governing Equations
Splitting Purpose: obtain the hyperbolic formulation
Slope Term Splitting
gh
x=
1
2g(2 + 2hs)
x+ gSox (1)
gh
y
= 1
2
g(2 + 2hs)
y
+ gSoy (2)
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Introduction Governing Equations Numerical Schemes Test Cases Summary
Governing Equations
New Shallow Water Equations
(uh)
t +
(u2h+ 12
g(2 + 2hs))
x +
(uvh)
y (hux)
x +
(huy)
y
= wxbx
gSox+ hfv
(3)
(vh)
t +(uvh)
x +(v2h+ 1
2
g(2 + 2hs))
y (hv
x)
x +(hv
y)
y
= wyby
gSoyhfv
(4)
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Introduction Governing Equations Numerical Schemes Test Cases Summary
Governing Equations
Figure: Scheme of the Computational Domain
Equations System
t
Q d +
S
F ndS=
Hd
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Introduction Governing Equations Numerical Schemes Test Cases Summary
Governing Equations
Vectors
F= [f g]T
Q=
huhvhz
f=
uh
u2h+ gh2/2hu/xuvhhv/x
1/(10)qsx
g=
vh
uvhhu/yv2h+ gh2/2hv/y
1/(10)qsy
H= 0
ghSoxghSfx+ wx/ + hfvghSoyghSfy+ wy/hfu0
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Introduction Governing Equations Numerical Schemes Test Cases Summary
Governing Equations
Viscous and inviscid fluxes
F= [f g]T =fI gV =
fI fV
nx+
gI gV
ny
where
fI =
uh
u2h+ g(2 + 2hs)/2uvh1
10qsx
gI =
vhuvh
v2h+ g(2 + 2hs)/21
10qsy
fV =
0
huxhv
x0
gV = 0
huyhv
y
0
Introduction Governing Equations Numerical Schemes Test Cases Summary
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g q y
Governing Equations
Time Integration
(VQ)
tn
i =
Si
Fn
i ndS
+Vn
i Hn
i
Qn+1i
=Qni + t
Vni
(VQ)
t
n+1i
+ (1) (VQ)
t
ni
t min Ri2maxj(
u2 + v2 + c)ij
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g q y
Outline
1 Introduction
2 Governing Equations
3 Numerical Schemes
4 Test Cases
Hydrodynamic Part
Coupled Model
5 Summary
Introduction Governing Equations Numerical Schemes Test Cases Summary
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Numerical Schemes
Standard Finite Volume Method
t
Q d +
S
F ndS=
Hd
FIi,j= 1
2FI(Q+i,j) + FI(Qi,j) |A| (Q+i,j Qi,j)
|A|= R|| L
whereQ+i,j
andQ+i,j
are reconstructed Riemann state
variables on the right and left sides, respectively. Ais theflux Jacobian matrix defined by
A= F n
Q
Fluxes evaluations:
Viscous fluxes: Roes approximate Riemann solver
Inviscid fluxes: Average value on the interfaces
Figure: Control Volume
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Numerical Schemes
Flux Jacobian Matrix
A= Axnx+ Byny
where
Ax =
0 1 0 0
u2
+ gh 2u 0 ghuv v u 0
3ku
u2+v2
h k
3u2+v2
h 2kuv
h 0
and
By =
0 0 1 0uv v u 0
v2 + gh 0 2v gh
3kv
u2+v2
h 2kuv
h k
u2+3v2
h 0
|A|= R|| L
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Numerical Schemes
Eigendecomposition ofA1 Properties ofA: asymmetric, real 4 4 matrix2 Eigenvalue/vectors
Numerical: Most based on iterative method, slow
Analytical: Asymptotic approximation, more efficient
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Outline
1 Introduction
2 Governing Equations
3 Numerical Schemes
4 Test Cases
Hydrodynamic Part
Coupled Model
5 Summary
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Hydrodynamic Part
Outline
1 Introduction
2 Governing Equations
3 Numerical Schemes
4 Test Cases
Hydrodynamic Part
Coupled Model
5 Summary
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Hydrodynamic Part
Test Case: Hydrodynamic Part
1D steady flow over a bump (Goutal and Maurel, 1997)
1 Bump definition:
zb(x) =
0.2 0.05(x 10)2 if8
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Hydrodynamic Part
Subcritical flow
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Hydrodynamic Part
Transcritical flow with no shock
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Hydrodynamic Part
Transcritical flow with shock
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Hydrodynamic Part
Test Case: Hydrodynamic Part
2D unsteady flow over a bump (LeVeque, 1998)
1 Bump definition:
hs(x, y) =1 12
exp
50
x 12
2+
y 12
2 for0 < x, y
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Hydrodynamic Part
2D unsteady flow over a bump
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Coupled Model
Outline
1 Introduction
2 Governing Equations
3 Numerical Schemes
4 Test Cases
Hydrodynamic Part
Coupled Model
5 Summary
Introduction Governing Equations Numerical Schemes Test Cases Summary
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Coupled Model
Test Case: Coupled Model
Scour Around Spur Dike During a Surge Pass
(Mioduszewski&Maeno, 2003)
Figure: Computational Domain
Figure: Unstructured Mesh
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Coupled Model
Test Case: Coupled Model
Figure: Velocity Vectors
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Coupled Model
Test Case: Coupled Model
Figure: Water Surface Elevation
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Coupled Model
Test Case: Coupled Model
Scour and Deposition around the Spur Dike
Figure: Experimental Figure: Numerical
Introduction Governing Equations Numerical Schemes Test Cases Summary
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Outline
1 Introduction
2 Governing Equations
3 Numerical Schemes
4 Test Cases
Hydrodynamic Part
Coupled Model
5 Summary
Introduction Governing Equations Numerical Schemes Test Cases Summary
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Summary
1 2D coupled model for scour: SWE + Exner equation
2
Numerical evaluation of wave speeds (Jacobian matrix A)3 Analytical evaluation of wave speeds can be used (future
development)
4 Unstructured mesh, complex domain
5
Coupled approach vs. quasi-steady approach