fritz r. fiedler university of idaho department of civil engineering simulation of shallow...
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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
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
What is complex terrain?
Topography with characteristic length scales (amplitude and wavelength) similar to flow depth two-dimensional flow
Examples
Flooding inundation mapping dam breaks
Overland Flow hydraulics hydrologic response
Wetlands and Estuaries, and Tidal Flats
Physical Objectives
Determine how Dynamic Surface Interactions affect Hydrologic ResponseEvaluate the Effects of Grazing
– degenerates plant community • changes infiltration• changes microtopography
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
CPER
Outline
Field MeasurementsMathematical ModelResults
Infiltration Measurements
Disc infiltrometersLight- and heavy-grazed areasBare and vegetated
Infiltration Variability
High K vegetated (locally high elevation)Low K bare (locally low elevation)
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
Ground Microtopography
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 )
Mathematical Modeling
Infiltration spatial variability (G-A model)Microtopography (2-D dynamic equations)Uniform rainfallSimplified flow resistance
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
Numerical Challenges
Non-linear hyperbolic systemStrong source terms (sometimes “stiff”)Small depths / dry areas (discontinuous)Large gradients in dependent variables
Vector Form
)()()(
USUHUGU
= y
+ x
+ t
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
[
]
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:
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 - =
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 - =
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
Lateral Inflow
l j,k j,kaveq = r - f
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,
not ponded:
x
q +
x
p +r = i
non
non
kj,a
i = f kj,aave
kj,
High-performance computing
Fortran Loop optimizations
most dependencies eliminated unrolling, fusion single-stride memory access
Shared-memory parallel processing PC environment
Comparative Numerical Examples
Steady state kinematic wave solution (analytical)Dam break problem (analytical)Published results
Iwagaki, 1955 (experimental)Woolhiser et al., 1996 (characteristics- based)
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
Microtopographic Surface
Overland Flow Depths
Flow Depths and Velocity
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 )
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
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
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
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
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
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 )
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
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)
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
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
Effect of Microtopographic Amplitude
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
Conclusions
Plane approximation gross distortionVegetation controls responseAverage/effective K not applicableInteractive infiltration importantReynolds No. - Friction FactorK-W assumption
Watch Your Step!
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