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![Page 1: Fritz R. Fiedler University of Idaho Department of Civil Engineering Simulation of Shallow Discontinuous Flow over Complex Infiltrating Terrain.](https://reader036.fdocuments.us/reader036/viewer/2022082211/56649e4a5503460f94b3ea8b/html5/thumbnails/1.jpg)
Fritz R. FiedlerUniversity of IdahoDepartment of Civil Engineering
0 1 0 0 2 0 0
x (cm )
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
y (c m )
0 1 0 0 2 0 0
x (cm )
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
y (c m )
2 0 m in u te s 4 0 m in u te s
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
0 5 10 15 20
time (min)
dis
char
ge
(mm
/hr)
1993
1994
Veg. Dist. 1
Veg. Dist. 2
h
t+
p
x+
q
y- q l
0
p
t+
x
p
h+
g h2
+y
p q
h- g h ( S - S ) +
p
hq = 0
2 2
o x f x l
q
t+
y
q
h+
g h2
+x
p q
h- g h ( S - S ) +
q
hq = 0
2 2
o y f y l
0
5
10
15
20
25
30
0 2 4 6 8 10 12 14 16 18 20 22 24 26
time(min)
disch
arge
(mm
/hr)
Cv=0Cv=0.2Cv=0.4Cv=0.6Cv=0.8Cv=1.0
Simulation of Shallow Discontinuous Flow over Complex Infiltrating Terrain
![Page 2: Fritz R. Fiedler University of Idaho Department of Civil Engineering Simulation of Shallow Discontinuous Flow over Complex Infiltrating Terrain.](https://reader036.fdocuments.us/reader036/viewer/2022082211/56649e4a5503460f94b3ea8b/html5/thumbnails/2.jpg)
What is shallow discontinuous flow?
Shallow: depth << wavelength vertical acceleration negligible depth-averaged NS equations
Discontinuous: both dry and wet areas shocks topographic control infiltration variability
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What is complex terrain?
Topography with characteristic length scales (amplitude and wavelength) similar to flow depth two-dimensional flow
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Examples
Flooding inundation mapping dam breaks
Overland Flow hydraulics hydrologic response
Wetlands and Estuaries, and Tidal Flats
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Physical Objectives
Determine how Dynamic Surface Interactions affect Hydrologic ResponseEvaluate the Effects of Grazing
– degenerates plant community • changes infiltration• changes microtopography
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Study Area Description
Central Plains Experimental RangelandLight- and heavy-grazed enclosures 1/2-hour, 100-year rain: ~100 mm/hr 1-hour, 100-year rain: ~75 mm/hrPatchy vegetation
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CPER
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Outline
Field MeasurementsMathematical ModelResults
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Infiltration Measurements
Disc infiltrometersLight- and heavy-grazed areasBare and vegetated
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Infiltration Variability
High K vegetated (locally high elevation)Low K bare (locally low elevation)
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Microtopography
The ground surface topography with approximately the same order amplitude and frequency as the overland flow depth in a given situation:
–related to rainfall intensity–related to infiltration characteristics–caused by vegetation growth
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Ground Microtopography
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Shaded Relief Map
0 1 0 0 2 0 0
x (c m )
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
y (c m )
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Mathematical Modeling
Infiltration spatial variability (G-A model)Microtopography (2-D dynamic equations)Uniform rainfallSimplified flow resistance
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Surface Water Equations
h
t+
p
x+
q
y- q l
0
p
t+
x
p
h+
g h2
+y
p q
h- g h ( S - S ) +
p
hq = 0
2 2
o x f x l
q
t+
y
q
h+
g h2
+x
p q
h- g h ( S - S ) +
q
hq = 0
2 2
o y f y l
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Numerical Challenges
Non-linear hyperbolic systemStrong source terms (sometimes “stiff”)Small depths / dry areas (discontinuous)Large gradients in dependent variables
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Vector Form
)()()(
USUHUGU
= y
+ x
+ t
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Vector Form
U = h , p , qT[ ]
G U( ) = p ,p
h+
g h
2,
p q
h
2 2T[ ]
H U( ) = q ,q
h+
g h
2,
q p
h
2 2T[ ]
S U( ) 1 = q , - g hz
x-
K p
8 h-
p
hq (
p
x+
p
y) ,
- g hz
y-
K q
8 h-
q
hq (
q
x+
q
y)
l
o
2 l
2
2
2
2
o
2 l 1
2
2
2
2
T
[
]
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Basic MacCormack Scheme
j , kn + 1
x y y x j , kn = L ( t / 2 ) L ( t / 2 ) L ( t / 2 ) L ( t / 2 )U U2 2 1 1
j , k*
j , kn
j , kn
j-1 , kn
x ; j , kn = -
t
2 x ( - ) +
t
2 U U G G S
j , k**
j, kn
j, k*
j+ 1 , k*
j, k*
x ; j , k* = 0 .5 - -
t
2 x ( - ) +
t
2 U U U G G S
Lx1 Operator:
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Friction Slope: Point-Implicit Treatment
)p(Opp
SSS 2
n
fxnfx
1nfx
pSx
fxt - 1
1 = D
SGGUU nkj, x;x
nk1,-j
nkj,x
nkj,
*kj,
2
tD + ) - (
x2
tD - =
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Convective Acceleration Upwinding
h
p -
h
pn
kj,
2n kj,
nk1,j+
2n k1,j+
SGGUU nkj, x;
nk1,-j
nkj,
nkj,
*kj,
2
tD + ) - (
x2
tD - =
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Smoothing Function
h + h2 + h
|h + h2 - h|
t
x =
**k1,-j
**kj,
**k1,j+
**k1,-j
**kj,
**k1,j+
kj,
) , (max = kj,k1,j+2k1/2,j+
)h-h( - )h-h( + h = h **k1,-j
**kj,k1/2,-j
**kj,
**k1,j+k1/2,j+
**kj,
***kj,
) , (max = kj,k1,j2k1/2,j
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Lateral Inflow
l j,k j,kaveq = r - f
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ponded:
y
q +
x
p +
th
+r = inon
non
nkj,
kj,a
tF - F
= fn
kj,1+n
kj,avekj,
tK = + F
+ Fln - F - F kj,
kj,kj,n
kj,
kj,kj,1+n
kj,kj,kj,
nkj,
1+nkj,
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not ponded:
x
q +
x
p +r = i
non
non
kj,a
i = f kj,aave
kj,
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High-performance computing
Fortran Loop optimizations
most dependencies eliminated unrolling, fusion single-stride memory access
Shared-memory parallel processing PC environment
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Comparative Numerical Examples
Steady state kinematic wave solution (analytical)Dam break problem (analytical)Published results
Iwagaki, 1955 (experimental)Woolhiser et al., 1996 (characteristics- based)
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Dam Break Problem
500
600
700
800
900
1000
wate
r su
rface e
lev
ati
on
(cm
)
400 600 800 1000 1200 1400 1600 x (m)
model results analytical solution
dam
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Microtopographic Surface
![Page 30: Fritz R. Fiedler University of Idaho Department of Civil Engineering Simulation of Shallow Discontinuous Flow over Complex Infiltrating Terrain.](https://reader036.fdocuments.us/reader036/viewer/2022082211/56649e4a5503460f94b3ea8b/html5/thumbnails/30.jpg)
Overland Flow Depths
![Page 31: Fritz R. Fiedler University of Idaho Department of Civil Engineering Simulation of Shallow Discontinuous Flow over Complex Infiltrating Terrain.](https://reader036.fdocuments.us/reader036/viewer/2022082211/56649e4a5503460f94b3ea8b/html5/thumbnails/31.jpg)
Flow Depths and Velocity
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Spatial Distribution ofInfiltration Parameters
0 1 0 0 2 0 0
x (cm )
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
y (cm )
![Page 33: Fritz R. Fiedler University of Idaho Department of Civil Engineering Simulation of Shallow Discontinuous Flow over Complex Infiltrating Terrain.](https://reader036.fdocuments.us/reader036/viewer/2022082211/56649e4a5503460f94b3ea8b/html5/thumbnails/33.jpg)
Flow Channels
0 1 0 0 2 0 0
x (cm )
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
y (c m )
0 1 0 0 2 0 0
x (cm )
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
y (c m )
2 0 m in u te s 4 0 m in u te s
![Page 34: Fritz R. Fiedler University of Idaho Department of Civil Engineering Simulation of Shallow Discontinuous Flow over Complex Infiltrating Terrain.](https://reader036.fdocuments.us/reader036/viewer/2022082211/56649e4a5503460f94b3ea8b/html5/thumbnails/34.jpg)
Overland Flow Depths
0 1 0 0 2 0 0
x (cm )
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
y (c m )
0 1 0 0 2 0 0
x (cm )
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
y (c m )
0 .0 c m
0 .2 c m
0 .6 c m
1 .0 c m
1 .4 c m
1 .8 c m
2 .2 c m
2 .6 c m
3 .0 c m
2 0 m in u te s 4 0 m in u te s
![Page 35: Fritz R. Fiedler University of Idaho Department of Civil Engineering Simulation of Shallow Discontinuous Flow over Complex Infiltrating Terrain.](https://reader036.fdocuments.us/reader036/viewer/2022082211/56649e4a5503460f94b3ea8b/html5/thumbnails/35.jpg)
Cumulative Infiltration
0 1 0 0 2 0 0
x (cm )
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
y (c m )
0 1 0 0 2 0 0
x (cm )
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
y (c m )
1 .4 c m
2 .0 c m
2 .6 c m
3 .2 c m
3 .8 c m
4 .4 c m
5 .0 c m
5 .6 c m
6 .2 c m
2 0 m in u te s 4 0 m in u te s
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0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
0 5 10 15 20
time (min)
dis
char
ge
(mm
/hr)
1993
1994
Veg. Dist. 1
Veg. Dist. 2
Simulated vs. Measured
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Simulated Grazing Effects
0
5
10
15
20
25
30
0 5 10 15 20 25
time (min)
dis
char
ge
(mm
/hr)
Heavy-grazedLight-grazed
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0 1 0 0 2 0 0
x (cm )
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
y (c m )
2 0
1 6 0
3 0 0
4 4 0
5 8 0
7 2 0
8 6 0
1 0 0 0
0 1 0 0 2 0 0
x (cm )
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
7 0 0
8 0 0
y (c m )
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5
3 .0
3 .5
4 .0
4 .5
Spatial Distribution of ReynoldsNumber and log(f )
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Re
0.1 1 10 100
f
1e+1
1e+2
1e+3
1e+4
1e+5
1e+6
1e+7
1e+8
20 minutes
f = 11713 Re-1.51
R2 = 0.67
Cross-Sectional Mean ReynoldsNumber vs. Friction Factor
![Page 40: Fritz R. Fiedler University of Idaho Department of Civil Engineering Simulation of Shallow Discontinuous Flow over Complex Infiltrating Terrain.](https://reader036.fdocuments.us/reader036/viewer/2022082211/56649e4a5503460f94b3ea8b/html5/thumbnails/40.jpg)
Distribution of log(KS)
0.00 100.00 200.000.00
100.00
200.00
300.00
400.00
500.00
600.00
700.00
800.00
-11.00
-10.50
-10.00
-9.50
-9.00
-8.50
-8.00
-7.50
-7.00
-6.50
-6.00
-5.50
-5.00
-4.50
-4.00
-3.50
x (cm )
y (c
m)
0.00 100.00 200.000.00
100.00
200.00
300.00
400.00
500.00
600.00
700.00
800.00
x (cm )
y (c
m)
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Plane Slope, Variable Ks
0
5
10
15
20
25
30
0 2 4 6 8 10 12 14 16 18 20 22 24 26
time(min)
disch
arge
(mm
/hr)
Cv=0Cv=0.2Cv=0.4Cv=0.6Cv=0.8Cv=1.0
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1.E-06
1.E-05
1.E-04
1.E-03
1.E-02 1.E-01 1.E+00
Mean Depth (cm)
Un
it D
isc
ha
rge
(c
m/s
)
CV=1.0
CV=0.8
CV=0.6
CV=0.4
CV=0.2
CV=0.0
Mean Depth vs DischargeVariable KS
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Effect of Microtopographic Amplitude
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1.E-06
1.E-05
1.E-04
1.E-03
1.E-02 1.E-01 1.E+00
Mean Depth (cm)
Uni
t D
isch
arge
(cm
/s)
20% reduced40% reduced60% reducedactual20% increasedplane surface
Mean Depth vs DischargeVariable Microtopography
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Conclusions
Plane approximation gross distortionVegetation controls responseAverage/effective K not applicableInteractive infiltration importantReynolds No. - Friction FactorK-W assumption
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Watch Your Step!