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