Coupled model of SWE and sediment transport on unstructured mesh

8/14/2019 Coupled model of SWE and sediment transport on unstructured mesh

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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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Outline

1 Introduction

2 Governing Equations

3 Numerical Schemes

4 Test Cases

Hydrodynamic Part

Coupled Model

5 Summary


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Outline

1 Introduction


3 Numerical Schemes

4 Test Cases

Hydrodynamic Part

Coupled Model

5 Summary


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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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Outline

1 Introduction


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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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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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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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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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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Governing Equations

Figure: Scheme of the Computational Domain

Equations System

t

Q d +

S

F ndS=

Hd



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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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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



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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


3 Numerical Schemes

4 Test Cases

Hydrodynamic Part

Coupled Model

5 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


3 Numerical Schemes

4 Test Cases

Hydrodynamic Part

Coupled Model

5 Summary



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Hydrodynamic Part

Outline

1 Introduction


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


3 Numerical Schemes

4 Test Cases

Hydrodynamic Part

Coupled Model

5 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


Figure: Velocity Vectors



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Coupled Model


Figure: Water Surface Elevation



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Coupled Model


Scour and Deposition around the Spur Dike

Figure: Experimental Figure: Numerical



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Outline

1 Introduction


3 Numerical Schemes

4 Test Cases

Hydrodynamic Part

Coupled Model

5 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

Coupled model of SWE and sediment transport on unstructured mesh

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