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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 8 th , 2004. The laminar flow field at the vicinity of permeable surfaces. Rainfall events - PowerPoint PPT Presentation
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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