Dispersion within Emergent Vegetation Using PIV and Concentration Measurements Uri Shavit Technion, Haifa, Israel
Mar 28, 2015
Dispersion within Emergent Vegetation
Using PIV and Concentration Measurements
Uri Shavit
Technion, Haifa, Israel
x
y
Flow
The advective dispersive equation
y
cv
x
cu
t
c
The local (micro-scale) transport equation
x
C
A
QCD
t
C
Q
A
- Flow rate
- Cross – section area
1. Fickian dispersion (Concentration only)
2. Decomposition and averaging (Euler) (Simultaneous concentration & velocity)
3. Ensemble of path-lines (Lagrange) (Velocity only)
x
y
Flow
We examine the PIV ability to measure dispersion,applying the following three methods:
Experimentalsection
LevelcontrollerInjector
Straightener
Flow
-mete
r
Pres
sure
regu
lator
y
x
z
x
Laser sheetCamera
The Experimental Setup
The experimental setup:
Visualization
The experimental challenge is to measure
simultaneously concentration & velocity.
Image Pair (1)(Visualization and conc.
measurements)
Image Pair (2)(Velocimetry)
Experimental ConditionsArray
DensityFlume
DischargeMeasured Averaged Velocity
ReFlume Re cylinder
% L/min cm/s - -
0 19.5 1.31 26200 40.5 2.90 58000 61 4.22 84400 63.6 4.61 9220
1.7 19.5 1.03 2060 521.7 40.5 2.07 4140 1041.7 61 3.16 6320 1581.7 69 3.53 7060 1773.5 19.8 1.42 2840 713.5 40.5 1.92 3840 963.5 58.5 2.69 5380 1353.5 66 2.93 5860 147
)4
)(exp(
)(/4)(),(
2
Dx
AQy
AQDxAQ
MyxC
x
y
Flow
][ 12 scmgrMwhere is the injection discharge
1. Fickian Dispersion
02
x
C
A
QCD
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
35 40 45 50
2
4
6
8
10
12
14
16
18
cm
cm
d = 3.5%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1.0 2.0 3.0 4.0 5.0
Q/A [cm/s]
Time-averaged normalized concentration (following an intensive calibration).
Q/A=4.58cm/s, d= 3.5%.
Fickian Dispersion
D [
cm2 /
s]
2. Decomposition and double averaging of the
convective equation (Eulerian)
Requires simultaneousmeasurements of velocity
and concentration
Decomposition
ccc uuu vvv
vvv
vvv uuu
uuu
y
cv
x
cu
t
c
x
y
Flow
Considering the commutativity rules:
The averaging end result:
0
y
Cv
x
Cu
y
Cv
x
Cu
y
Cv
x
Cu
t
C
x
C
A
Q
0The dispersion term
CD 2
Q=66 min-1, Array Density = 3.5%50mm Lens
2
4
6
8
10
12
14
16
18
20
2 4 6 8 10 12 14 16 18
Y(cm)
X(cm)
200mm Lens
Y(c
m)
X(cm)
2
2
2
2
yc
xc
yc
vxc
uyc
vxc
uD
Spatial variations
Longitudinal Lateral
Temporal fluctuations
The calculated dispersion coefficient
x
y
Flow
-0.3
-0.2
-0.1
0
0.1
0.2
35
40
45
50
4
6
8
10
12
14
16
-0.5
0
0.5
2
2
2
2
4321
yC
xC
DDDDD
xCuD
1
-1
-0.5
0
0.5
1
1.5
35
40
45
50
4
6
8
10
12
14
16
-2
0
2
2
2
2
2
4321
yC
xC
DDDDD
yCvD
2
3. An Ensemble of Path-lines
(a Lagrangian approach)
The location of a particle released
at (x0, y0) at time t0 is,
dttyxuttyxXttyxX iii ),,(),,,(),,,( 11000000
dttyxvttyxYttyxY iii ),,(),,,(),,,( 11000000
0dt
2.0/ ufdS
Kundu, 1990, p. 324 or Williamson (1996)
Hzf 17.157.0
The Strouhal number:
)(2
1YY
tDyy
0000000 ),,,(),,,( yttyxYttyxY ii
Lateral dispersion is then calculated using the mean square of the lateral variations,
Where Y’ is:
Q=66 min-1, Array Density = 3.5%50mm Lens,
2
4
6
8
10
12
14
16
18
20
2 4 6 8 10 12 14 16 18
Y(cm)
X(cm)
The Evolution of Pathlines
36 38 40 42 44 46 48 50 52
2
4
6
8
10
12
14
16
18
x (cm)
y (
cm
)
The Results of the Lagrangian Approach:
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20 25 30 35 40
t0 (s)
D(x
0,y 0
) (c
m2 /s
)
0
0.2
0.4
0.6
0.8
1
1.0 2.0 3.0 4.0 5.0Q/A [cm/s]
D
[cm2/s]
Eulerian Fickian Lagrangian
Nepf 97 Nepf 99
The dispersion coefficient d = 3.5%
4 cm
4 cm
A Moving Frame of Reference:
Q = 23 min-1, Array Density = 3.5%
Acknowledgments:
• The Israel Science Foundation (ISF)
• Grand Water Research Institute
• Joseph & Edith Fischer Career Development Chair
• Tuval Brandon
• Mordechai Amir
• Ravid Rosenzweig