wall free shear flow turbulent flows free of solid boundaries jet two-dimensional image of an...
TRANSCRIPT
WALL FREE SHEAR FLOW
Turbulent flows free of solid boundaries
JET
Two-dimensional image of an axisymmetric water jet, obtained by the laser-induced fluorescence technique. (From R. R. Prasad and K. R. Sreenivasan, Measurement and interpretation of fractal dimension of the scalar interface in turbulent flows, Phys. Fluids A, 2:792–807, 1990)
x
y
Irrotational
Turbulent
WAKE
http://www.ifh.uni-karlsruhe.de/science/envflu/
SHEAR LAYER
Laser-Induced Fluorescence (LIF) VISUALIZATION OF AN AXISYMMETRIC TURBULENT JET, C. Fukushima and J. Westerweel, Technical University of Delft, The Netherlands
x
x
Laser-Induced Fluorescence (LIF) VISUALIZATION OF AN AXISYMMETRIC TURBULENT JET, C. Fukushima and J. Westerweel, Technical University of Delft, The Netherlands
x
Turbulent Kinetic Energy (q2) Balance in a Jet
2
222
2wvu
q
x
y
q2
<v2 > <u2 >
<w2 >
-<uv >
y
m2 /
s2
2
222
2wvu
q
Laser-Induced Fluorescence (LIF) VISUALIZATION OF AN AXISYMMETRIC TURBULENT JET, C. Fukushima and J. Westerweel, Technical University of Delft, The Netherlands
x
Turbulent Kinetic Energy (q2) Balance in a Jet
02
22
0 vpvqyy
uuv
y
qv
x
qu
No local accelerationsNo viscous transportPart of the shear production = 0No buoyancy production
wg
x
uuueuuqup
xdt
dq
oj
i
jiijijjoj
21 2
2
x
y
02
22
0 vpvqyy
uuv
y
qv
x
qu
ym2 /
s3
Gai
nL
oss
x
qu
2
y
qv
2
y
uuv
02 vpvq
y
http://www.symscape.com/node/447
U0
WALL-BOUNDED SHEAR FLOW
Nominal limit of boundary layer
0.99U0
Viscous sublayer
For fully developed, bounded turbulent flow (not changing in x):
zx
p
0 uwz
u
zx
p
0Function of x only Function of z only
CONSTANTS!
uwz
u
z
centerline or surface
zz
uw
x
uu
0
1in boundary layer:
uwz
u
z
z
edge of boundary layer
stress is a function of x and z
Near the wall – Different Layers
zuu ,,, 0
http://furtech.typepad.com/
z
ū (x)
u(x,z)
0,Only involve mass dimension
Should appear together in nondimensional groups
0
* u Friction Velocity
zuuu ,,*
zuuu ,,*
This relates 4 variables involving the dimensions of length and time
According to the PI THEOREM, this relationship has 4 variables and 2 dimensions
Then, only two (4 – 2) non-dimensional groups can result:
zf
zuf
u
u
*
*
Law of the Wall
Inner part of the wall layer, right next to the wall, is called the viscous sublayer – dominated by viscous effects
*u
u
*u
u
z z
z (m
) =
z+ν/
u*
viscous sublayer
z
u
0
zu 0
zu
u
*
buffer layer
logarithmic layer
*u
u
z
viscous sublayer
buffer layer
logarithmic layer
zf
zuf
u
u
*
*
outer layer
F
zF
u
Uu
*
Velocity defect law
Law of the wall
dz
dfu
z
u
2*
d
dFu
z
u *
z
fz
d
dF
1
Karman constant = 0.41
Equating and multiplying times z/u*
z
fz
d
dF
1
Karman constant = 0.41
Integrating: Azzf ln1
BF
ln
1
From experiments: 5ln1 *
*
zu
u
u1ln
1
*
z
u
Uu
*u
u
z
Velocity distributions for theOverlap layer,Inertial sublayer,Logarithmic layer
Logarithmic velocity distribution near a boundary can also be derived from dimensional analysis
z
u
can only depend on z, and the only relevant velocity scale is u*
z
u
z
u *1
Czu
u ln*
*zu
*u
u
*
5
u
*
30
u
*
300
u
0@0 zzu 0
* lnz
zuu
0
* lnz
zuu
m005.0
sm04.0
0
*
z
u
0
* u
Pa20
Data from Ponce de Leon Inlet
FloridaIntracoastal Waterway
Florida