solution of the st venant equations / shallow-water equations of open channel flow
DESCRIPTION
Solution of the St Venant Equations / Shallow-Water equations of open channel flow. Dr Andrew Sleigh School of Civil Engineering University of Leeds, UK www.efm.leeds.ac.uk/CIVE/UChile. Shock Capturing Methods. Ability to examine extreme flows Changes between sub / super critical - PowerPoint PPT PresentationTRANSCRIPT
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Solution of the St Venant Equations / Shallow-Water equations of open channel flow
Dr Andrew SleighSchool of Civil EngineeringUniversity of Leeds, UK
www.efm.leeds.ac.uk/CIVE/UChile
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Shock Capturing Methods
Ability to examine extreme flows Changes between sub / super critical
Other techniques have trouble with trans-critical Steep wave front Front speed
Complex Wave interactions Alternative – shock fitting Good, but not as flexible
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More recent
Developed from work on Euler equations in the aero-space where shock capturing is very important (and funding available)
1990s onwards Euler equations / Numerical schemes:
Roe, Osher, van Leer, LeVeque, Harten, Toro Shallow water equations
Toro, Garcia-Navarro, Alcrudo, Zhao
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Books
E.F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer Verlag (2nd Ed.) 1999.
E.F. Toro. Shock-Capturing Methods for Free-Surface Flows. Wiley (2001)
E.F. Toro. Riemann Problems and the WAF Method for Solving Two-Dimensional Shallow Water Equations. Philosophical Transactions of the Royal Society of London. A338:43-68 1992.
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Dam break problem
The dam break problem can be solved
It is in fact integral to the solution technique
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Conservative Equations
As earlier, but use U for vector
USUFU xt
Q
AU
1
2
gIA
UF
fo SSgAgI
US2
0
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I1 and I2
Trapezoidal channel Base width B, Side slope SL= Y/Z
Rectangular, SL = 0
Source term
322
1LS
hB
hI
dx
dSh
dx
dBhI L
32
122
B
ABhI
22
22
1 dx
dB
B
AI
2
2
2 2
)3/4(2
2
RA
nQQS f
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Rectangular Prismatic
Easily extendible to 2-d
Integrate over control volume V
hu
hU
22
2
1ghhu
huUF
fo SSgh
US0
dUUSdtUFdxUVV
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2-dimensions
In 2-d have extra term:
friction
USUGUFU yxt
hv
hu
h
U
huv
ghhu
hu
UF 22
2
1
22
2
1ghhv
huv
hu
UG
yy
xx
fo
fo
SSgh
SSghUS
0
22)3/1(
2
vuuh
nS
xf
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Normal Form
Consider the control volume V, border Ω
dUUSdUHdVUt VV
n x
V
n x
n x
V
UGnUFnUGUFUH 21sincos
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Rotation matrix
H(U) can be expressed
UFUGnUFnUH TT 121
cossin0
sincos0
001
T
cossin0
sincos0
0011T
vh
uh
h
U
ˆ
ˆT
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Finite volume formulation
Consider the homogeneous form i.e. without source terms
And the rectangular control volume in x-t space
0V
dtUFdxU
xi-1/2 xi+1/2
tn+1
i
Fi-1/2Fi+1/2
x
t
xi-1/2 xi+1/2
tn+1
i
Fi-1/2Fi+1/2
x
t
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Finite Volume Formulation
The volume is bounded by xi+1/2 and xi-1/2 tn+1 and tn
The integral becomes
dttxUFdttxUFdxtxUdxtxU
n
n
n
n
i
i
i
i
t
t i
t
t i
x
x n
x
x n
112/1
2/1
2/1
2/1
,,,, 2/12/11
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Finite Volume Formulation
We define the integral averages
And the finite volume formulation becomes
dxtxUx
Ui
i
x
x nni
2/1
2/1
,1 dxtxU
xU
i
i
x
x nni
2/1
2/11
1 ,1
dttxUFt
Fn
n
t
t ii
1
,1
2/12/1 dttxUFt
Fn
n
t
t ii
1
,1
2/12/1
2/12/11
iini
ni FF
x
tUU
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Finite Volume Formulation
Up to now there has been no approximation
The solution now depends on how we interpret the integral averages
In particular the inter-cell fluxes Fi+1/2 and F1-1/2
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Finite Volume in 2-D
The 2-d integral equation is
H(u) is a function normal to the volume
dUUSdUHdVUt VV
dUGnUFndUHFs
s
s
ss
A
A
A
An
11
21
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Finite Volume in 2-D
Using the integral average
Where |V| is the volume (area) of the volume then
dVUV
UV
i 1
N
ss
ni
ni Fn
V
tUU
1
1FnFn
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Finite Volume in 2-D
If the nodes and sides are labelled as :
Where Fns1 is normal flux for side 1 etc.
A1
A3
A2
A4
V
L4L3
L2
L1
A1
A3
A2
A4
V
L4L3
L2
L1
443322111 LFnLFnLFnLFn
V
tUU ssss
ni
ni
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FV 2-D Rectangular Grid
For this grid
Solution reduces to
Fi-1/2Fi+1/2
x
y
Gj-1/2
Gj+1/2
i+1/2,j-1/2
i-1/2,j+1/2
i-1/2,j-1/2
i+1/2,j+1/2
i, jFi-1/2
Fi+1/2
x
y
Gj-1/2
Gj+1/2
i+1/2,j-1/2
i-1/2,j+1/2
i-1/2,j-1/2
i+1/2,j+1/2
i, j
2/1,2/1,,2/1,2/11
jijijijini
ni GG
y
tFF
x
tUU
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Flux Calculation
We need now to define the flux Many flux definitions could be used to that
satisfy the FV formulation
We will use Godunov flux (or Upwind flux)
Uses information from the wave structure of equations.
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Godunov method Assume piecewise linear data states
Means that the flux calc is solution of local Riemann problem
n
n+1
i-1 i i-1
Fi+1/2Fi-1/2Cells
Data states
n+1
n
U(0)i+1/2U(0)i-1/2
n
n+1
i-1 i i-1
Fi+1/2Fi-1/2Cells
Data states
n+1
n
U(0)i+1/2U(0)i-1/2
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Riemann Problem
The Riemann problem is a initial value problem defined by
Solve this to get the flux (at xi+1/2)
0 xt UFU
2/11
2/1,i
ni
ini
nxxifU
xxifUtxU
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FV solution
We have now defined the integral averages of the FV formulation
The solution is fully defined First order in space and time
2/12/11
iini
ni FF
x
tUU
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Dam Break Problem
The Riemann problem we have defined is a generalisation of the Dam Break Problem
Dam wall
Deep water at rest
Shallow water at rest
Dam wall
Deep water at rest
Shallow water at rest
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Dam Break Solution Evolution of solution
Wave structure
x
x
x
Water levels at time t=t*
t*t
Velocity at time t=t*
v
h
Shock
Rarefaction
xx
xx
xx
Water levels at time t=t*
t*t
Velocity at time t=t*
v
h
Shock
Rarefaction
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Exact Solution
Toro (1992) demonstrated an exact solution
Considering all possible wave structures a single non-linear algebraic equation gives solution.
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Exact Solution
Consider the local Riemann problem
Wave structure
0 xt UFU
0
00,
xifU
xifUxU
R
L
x
t
Right wave, (u+c)Left wave (u-c)
Star region, h*, u*
(u)
vL
tShear wave
vR
hL, uL, vL hR, uR, vR
0xx
t
Right wave, (u+c)Left wave (u-c)
Star region, h*, u*
(u)
vL
tShear wave
vR
hL, uL, vL hR, uR, vR
0
hv
hu
h
U
huv
ghhu
hu
UF 22
2
1
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PossibleWave structures
x
tRight Shock
Left Rarefaction Shear wave
x
t
Left Shock Right RarefactionShear wave
x
t
Right RarefacationLeft Rarefaction
Shear wave
x
t
Right ShockLeft Shock
Shear wave
xx
tRight Shock
Left Rarefaction Shear wave
xx
t
Left Shock Right RarefactionShear wave
xx
t
Right RarefacationLeft Rarefaction
Shear wave
xx
t
Right ShockLeft Shock
Shear wave
Across left and right wave h, u change v is constant
Across shear wave v changes, h, u constant
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Determine which wave
Which wave is present is determined by the change in data states thus:
h* > hL left wave is a shock h* ≤ hL left wave is a rarefaction
h* > hR right wave is a shock h* ≤ hR right wave is a rarefaction
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Solution Procedure
Construct this equation
And solve iteratively for h (=h*). The functions may change each iteration
uhhfhhfhf LLLL ,,
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f(h) The function f(h) is defined
And u*
uhhfhhfhf LLLL ,,
)(2
1
)(2
shockhhifhh
hhghh
nrarefactiohhifghgh
fL
L
LL
LL
L
)(2
1
)(2
shockhhifhh
hhghh
nrarefactiohhifghgh
fR
R
RR
RR
R
LR uuu
LLRRRL hhfhhfuuu ,,2
1
2
1 ***
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Iterative solution
The function is well behaved and solution by Newton-Raphson is fast (2 or 3 iterations)
One problem – if negative depth calculated! This is a dry-bed problem. Check with depth positivity condition:
RLLR ccuuu 2
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Dry–Bed solution
Dry bed on one side of the Riemann problem
Dry bed evolves Wave structure is
different.
x
t
x
t
x
t
Wet bed
Dry bed
Wet bed
Wet bed
Wet bed
Dry bed
Dry bed
xx
t
xx
t
xx
t
Wet bed
Dry bed
Wet bed
Wet bed
Wet bed
Dry bed
Dry bed
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Dry-Bed Solution
Solutions are explicit Need to identify which applies – (simple to do)
Dry bed to right
Dry bed to left
Dry bed evolves h* = 0 and u* = 0 Fails depth positivity test
LL cuc 23
1* LL cuu 2
3
1*
RR cuc 23
1* RR cuu 2
3
1*
gch /2**
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Shear wave
The solution for the shear wave is straight forward. If vL > 0 v* = vL
Else v* = vR
Can now calculate inter-cell flux from h*, u* and v* For any initial conditions
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Complete Solution
The h*, u* and v* are sufficient for the Flux But can use solution further to develop exact
solution at any time. i.e. Can provide a set of benchmark solution
Useful for testing numerical solutions.
Choose some difficult problems and test your numerical code again exact solution
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Complete Solution
x
x
x
Water levels at time t=t*
t*t
Velocity at time t=t*
v
h
Shock
Rarefaction
xx
xx
xx
Water levels at time t=t*
t*t
Velocity at time t=t*
v
h
Shock
Rarefaction
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Difficult Test Problems
Toro suggested 5 testsTest No. hL (m) uL (m/s) hR (m) uR (m/s)
1 1.0 2.5 0.1 0.0
2 1.0 -5.0 1.0 5.0
3 1.0 0.0 0.0 0.0
4 0.0 0.0 1.0 0.0
5 0.1 -3.0 0.1 3.0
Test 1: Left critical rarefaction and right shockTest 2: Two rarefactions and nearly dry bedTest 3: Right dry bed problemTest 4: Left dry bed problemTest 5: Evolution of a dry bed
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Exact Solution
Consider the local Riemann problem
Wave structure
0 xt UFU
0
00,
xifU
xifUxU
R
L
x
t
Right wave, (u+c)Left wave (u-c)
Star region, h*, u*
(u)
vL
tShear wave
vR
hL, uL, vL hR, uR, vR
0xx
t
Right wave, (u+c)Left wave (u-c)
Star region, h*, u*
(u)
vL
tShear wave
vR
hL, uL, vL hR, uR, vR
0
hv
hu
h
U
huv
ghhu
hu
UF 22
2
1
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Returning to the Exact Solution We will see some other Riemann solvers that
use the wave speeds necessary for the exact solution.
Return to this to see where there come from
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PossibleWave structures
x
tRight Shock
Left Rarefaction Shear wave
x
t
Left Shock Right RarefactionShear wave
x
t
Right RarefacationLeft Rarefaction
Shear wave
x
t
Right ShockLeft Shock
Shear wave
xx
tRight Shock
Left Rarefaction Shear wave
xx
t
Left Shock Right RarefactionShear wave
xx
t
Right RarefacationLeft Rarefaction
Shear wave
xx
t
Right ShockLeft Shock
Shear wave
Across left and right wave h, u change v is constant
Across shear wave v changes, h, u constant
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Conditions across each wave
Left Rarefaction wave
Smooth change as move in x-direction Bounded by two (backward) characteristics Discontinuity at edges
Left bounding characteristic
x
t
hL, uL h*, u*
Right bounding characteristic
t
LL cudtdx /**/ cudtdx
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Crossing the rarefaction
We cross on a forward characteristic
States are linked by:
or
constant2 cu
** 22 cucu LL
** 2 ccuu LL
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Solution inside the left rarefaction The backward characteristic equation is For any line in the direction of the rarefaction
Crossing this the following applies:
Solving gives
On the t axis dx/dt = 0
cucu LL 22
cudt
dx
dt
dxcuc LL 2
3
1
dt
dxcuu LL 2
3
1
LL cuc 23
1 LL cuu 2
3
1
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Right rarefaction
Bounded by forward characteristics Cross it on a backward characteristic
In rarefaction
If Rarefaction crosses axis
RR cuc 23
1
** 22 cucu RR RR ccuu ** 2
dt
dxcuc RR 2
3
1
dt
dxcuu RR 22
3
1
cudt
dx
RR cuu 23
1
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Shock waves
Two constant data states are separated by a discontinuity or jump
Shock moving at speed Si Using Conservative flux for left shock
LL
LL uh
hU
**
** uh
hU
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Conditions across shock
Rankine-Hugoniot condition
Entropy condition
λ1,2 are equivalent to characteristics. They tend towards being parallel at shock
LiL UUSUFUF **
*USU iiLi
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Shock analysis
Change frame of reference, add Si
Rankine-Hugoniot gives
LLL Suu ˆLSuu **ˆ
LL
LL uh
hU
ˆˆ
**
** ˆ
ˆuh
hU
222*
2**
**
2
1ˆ
2
1ˆ
ˆˆ
LLL
LL
ghuhghuh
uhuh
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Shock analysis
Mass flux conserved
From eqn 1
Using
also
LLL uhuhM ˆˆ**
L
LL uu
hhgM
ˆˆ2
1
*
22*
** /ˆ hMu L LLL hMu /ˆ
LLL hhhhgM **2
1
LL uuuu ** ˆˆ
L
LL uu
hhgM
*
22*
2
1
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Left Shock Equation
Equating gives
Also
LLL hhfuu ,**
L
LLLL hh
hhghhhhf
*
*** 2
1,
LLLL qauS
2
**
2
1
L
LL h
hhhq
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Right Shock Equation
Similar analysis gives
Also
RRR hhfuu ,**
R
RRRR hh
hhghhhhf
*
*** 2
1,
RRRR qauS
2
**
2
1
R
RR h
hhhq
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Complete equation
Equating the left and right equations for u*
Which is the iterative of the function of Toro
LLL hhfuu ,** RRR hhfuu ,**
0,, LRLLLR hhfhhfuu
0,, *** uhhfhhfhf LRLL
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Approximate Riemann Solvers No need to use exact solution Expensive
Iterative When other equations, exact may not exist
Many solvers Some more popular than others
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Toro Two Rarefaction Solver
Assume two rarefactions Take the left and right equations
Solving gives
For critical rarefaction use solution earlier
** 2 ccuu LL RR ccuu ** 2
24*RLRL ccuu
c
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Toro Primitive Variable Solver Writing the equations in primitive variables
Non conservative
Approximate A(W) by a constant matrix
Gives solution
0 xt WWAW
u
h
g
u
u
hWAW
LR WWW 2
1
RL
RLLRRL
cc
hhuuhhh
42*
RL
RLLRRL
hh
cchhuuu
42*
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Toro Two Shock Solver
Assuming the two waves are shocks
Use two rarefaction solver to give h0
RL
RLRRLL
uuhqhqh
*
LLRRRL qhhqhhuuu *** 2
1
2
1
Lo
LoL hh
hhgq
2
Ro
RoR hh
hhgq
2
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Roe’s Solver
Originally developed for Euler equations Approximate governing equations with:
Where is obtained by Roe averaging
xtxtxt UAUAUUUFU~
A~
LL
RRLL
hh
huhuu
~RLhhh
~ 22
2
1~RL ccc
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Roe’s Solver
Properties of matrix Eigen values
Right eigen vectors
Wave strengths
Flux given by
cu ~~~1 cu ~~~
1
cu
R ~~1~ )1(
cu
R ~~1~ )2(
u
c
hh ~
~
2
1~1
u
c
hh ~
~
2
1~2
LR hhh
2
112/1
~~~2
1
2
1
j
jjj
ni
nii FFF R
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HLL Solver
Harten, Lax, van Leer Assume wave speed Construct volume Integrate round
Alternative gives flux
x
t
URUL
U*
FRFL F*
t
xL xR
x
t
URUL
U*
FRFL F*
t
xL xR
0* LRRLRRLL tUtUUxxUxUx
t
xS L
L
t
xS R
R
LRRLLLRR SSFFUSUSU /*
LRLRRLRLLR SSUUSSFSFSF /*
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HLL Solver
What wave speeds to use? Free to choose One option:
For dry bed (right)
Simple, but robust
TRTRLLL cucuS ,min TRTRRRR cucuS ,min
LLL cuS RRR cuS 2
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Higher Order in Space
Construct Riemann problem using cells further away
Danger of oscillations Piecewise
reconstruction
Limiters
i-1 i i+1 i+2
i-1 i i+1 i+2
L
R
LR
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Limiters
Obtain a gradient for variable in cell i, Δi
Gradient obtained from Limiter functions Provide gradients at cell face
Limiter Δi =G(a,b)
iiL xUU 2
111 2
1 iiR xUU
ii
iii xx
uua
2/1
12/1
1
12/1
ii
iii xx
uub
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Limiters
A general limiter
β=1 give MINMOD β=2 give SUPERBEE
van Leer
Other SUPERBEE expression s = sign
0,max,,max,0min
0,min,,min,0max,
aforbaba
aforbababaG
ba
bababaG
,
asbasbsbaG 2,min,,2min,0max,
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Higher order in time Needs to advance half time step MUSCL-Hancock
Primitive variable Limit variable Evolve the cell face values 1/2t:
Update as normal solving the Riemann problem using evolved WL, WR
0 xt WWAW
Ri
Li
ni
RLi
RLi x
tWWWAWW
2
1,,
2/12/11
iini
ni FF
x
tUU
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Alternative time advance
Alternatively, evolve the cell values using
Solve as normal
Procedure is 2nd order in space and time
Li
Ri
ni
ni FF
x
tUU WW
2
12/1
2/12/11
iini
ni FF
x
tUU
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Boundary Conditions
Set flux on boundary Directly Ghost cell
Wall u, v = 0. Ghost cell un+1=-un
Transmissive Ghost cell hn+1 = hn
un+1 = un
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Wet / Dry Fonts
Wet / Dry fronts are difficult Source of error Source of instability
Common near tidal boundaries Flooding - inundation
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Dry front speed
We examined earlier dry bed problem
Front is fast Faster than characteristic. Can cause problem with time-step / Courant
LLL
LL
L
cuS
c
cucu
cuS
2
0
22
*
*
xx
t
Wet bed Dry bed
SL
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Solutions
The most popular way is to artificially wet bed
Give a small depth, zero velocity Loose a bit of mass and/or momentum
Can drastically affect front speed E.g. a 1.0 dambeak with 1cm gives 38% error 1mm gives 25% error – try it!
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Conservation Errors
The conserved variable are h and hu
Often require u
Need to be very careful about divide by zero
Artificial dry-bed depths could cause this
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Source Terms
“Lumped” in to one term and integrated Attempts at “upwinding source” Current time-step
Could use the half step value E.g.
x
B
BK
uu
x
zghS
2
02
2
n
RK
3/2
2/1CRK
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Main Problem is Slope Term
Flat still water over uneven bed starts to move.
Problem with discretisation ofx
zgh
x
zzgh lr
i
i-1 i i+ 1
z, hx
d a tu m le v e l
b e d le v e l
w a te r su r fa c e
iz1izlz1iz
rh
rz
1ihihlh1ih
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Discretisation
Discretised momentum eqn
For flat, still water
Require
lri
rl
rlini zzgh
x
tghghhuhu
x
thuhu
22
22221
022
221
ri
rli
lni zh
hzh
h
x
tghu
rir
lil zh
hzh
h
22
22
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A solution
Assume a “datum” depth, measure down
Momentum eqn: Flat surface
x
hg
x
hhg
x
zhg ii
ii
2
22
2
1
x
zzhzzhg liirii
22
2
1
2222'
2 liiriirli zzhzzhhhx
tghu
liil zzhh riir zzhh
and
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Some example solutions
Weighted mesh gives more detail for same number of cells
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Dam Break - CADAM Channel with 90° bend
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Secondary Shocks