Theoretical, Numerical and Experimental

Study of the Laminar Macroscopic Velocity

Profile near Permeable Interfaces

Uri Shavit

Civil and Environmental Engineering, Technion, Haifa, Israel

Kyiv, May 8th, 2004

The laminar flow field at the vicinity of permeable surfaces

• Rainfall events

• Fractures

• Wetlands

• Industrial processes

Free flow

Porousregion

z

u(z)

?

The Beavers and Joseph studyAnd the Brinkman Eq.

00

dB

BJ

z

uukz

u

z Beavers and Joseph

(Beavers and Joseph ,1967)

B u

d u

02

2*

z

uu

kx

P

The Brinkman Eq. (1947)

The Taylor Brush The Cantor-Taylor Brush

(G.I. Taylor, 1971)(Vignes-Adler et al., 1987)

(Shavit et al., 2002, WRR)

z x

y

Spatially averaged N-S equation (x-comp)

SSS

dSz

undS

y

undS

x

un

z

u

y

u

x

u

x

P

wuwuz

vuvuy

uuuux

)(1

)(1

)(1

12

2

2

2

2

2

"uuu

2

2

2

2

2

21

z

u

y

u

x

u

x

P

z

uw

y

uv

x

uu

yx

<u>

u

u"

SSS

dSz

undS

y

undS

x

un

z

u

y

u

x

u

x

P

wuwuz

vuvuy

uuuux

)(1

)(1

)(1

12

2

2

2

2

2

v=w=0

0x

0

y

Spatial averaging for the parallel grooves configuration

REV

REV

REV

Hrev

0)(2

2

ff

uz

u

x

P

The result of the spatial averaging

2

222

112

1

Hrevzn

Hrevz

Hrevnz

Hrev

n

Hrevz

– Local porosity

n – The structure porosity (n = 5/9)

u

uf

k

n

0

012

2

11

0

2

2

2

2

2

2

ff

fff

f

uk

n

z

un

x

P

uk

n

z

u

Hrev

n

z

unz

Hrev

n

x

P

z

u

x

P

The Modified Brinkman Equation (MBE)

The MBE solution as a function of Hrev

Z (

cm)

A numerical solution of the microscale field

Y (cm)

Z (

cm)

The Modified Brinkman Equation (MBE)

The Cantor-Taylor brush

z

0

012

2

11

0

2

2

2

2

2

2

ff

fff

f

uk

n

z

un

x

P

uk

n

z

u

Hrev

n

z

unz

Hrev

n

x

P

z

u

x

P

The Modified Brinkman Equation (MBE)

kaeknH bnrev

),(52.8a

29.2b (Shavit et al., 2004)

MBE’s analytical solution

)c(Hrev

zn

k

x

peCu

)b(Hrev

zHrev

n

k

x

p)z(SC)z(SCu

)a(Hrev

zhCzhx

pz

x

pu

kz

f

f

f

2 1

22 1302

24

1

2

1 2

And C1, C2, C3, C4 are constants.

m

mm zazS

0

0)(0 1

0

1)(1

m

mm zazS

Where:

Experimental

30 x 5 = 150 sets

150 wide columns 1200 narrow columns

Sierpinski Carpet

n = 0.79

36m

m

12m

m

4 mm

L = 108 cm, B = 20.4 cm

Nd:YAG Lasers

PIV

Camera

Opt

ics

Laser sheet

Flow Direction

Z = -5 mm

h = 10 mm

Q = 150 cc/s

PIV Results

The Velocity Vertical Profile (Q = 150 cm3 s-1)

The RMS Velocity Profile

(Q = 150 cm3 s-1)

Numerical

CFD (Fluent)

Contours of u(x,y)

CFD (Fluent)

Z = -2 mm

Flow direction

Contours of u(x,y)

Numerical

Solution

of the

Laminar

Flow

versus

the MBE

Turbulent Numerical Solution versus PIV

(Q = 150 cm3 s-1)

• Ravid Rosenzweig

• Shmuel Assouline

• Mordechai Amir

• Amir Polak

Acknowledgments:

• The Israel Science Foundation

• Grand Water Research Institute

• Technion support

• Joseph & Edith Fischer Career

Development Chair