Download - Lect8 Similarity
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Similarity theory
1. Buckingham Pi Theorem and examples
Turbulent closure problem requires empirical expressions for
determining turbulent eddy diffusion coefficients. The development
of turbulence closure is based on observations not theory.
We need to find an intelligent wayof organizing observational data.
Similarity theory is a method to find relationships among variables
based on observations.
U
We want to find the relationship between
the cruising speed and the weight of airplane.
2
3
velocity ( / )
dimension ( )
mass ( )
acceleration of gravity ( / )
air density ( / )
U m s
m
W kg
g m s
kg m
a. Define relevant variables and
their dimensions.
b. Count number of fundamental
dimensions.kgsm ,,
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c. Form n dimensionless groups
where n is the number of variables
minus the number of fundamental
dimensions.
235 n
n ,..1
221U
Wg
gravitational force
lift force 32
W
mass of airplane
mass of
displaced air
d. Measure as a function of1 2
e. Further simplification; assume:3
2 1~ constant constantW
i.e.63/2222
~~~ UWWUUW
)(f 21
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Weight as a function ofcruising speed ( The
simple science of flight by
Tennekes, 1997, MIT press)
Flying objects range
from small insects toBoeing 747
W~U6
Speed (m/s)
The great flight diagram
Weigh
t(Newtons)
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Procedure of Buckingham Pi Analysis
Step 1, Hypothesize which variables could be important to the flow.
e.g., stress, density, viscosity, velocity, ..
Step 2, Find the dimensions of each of the variables in terms of the
fundamental dimensions. Fundamental dimensions are:
L=length
M=mass
T=time
K=temperature
Dimensions of any other variables can be represented by these
fundamental dimensions.
1-1-
0
2-1-
1-
3-
TMLtCoefficienViscosity
LheightLayerBoundaryH
Lroughnessz
TMLstresswindLTtyvelociU
MLdensity
Example
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H;,,;zU,,H;U,, 0
Step 3, Count the number of fundamental dimensions in the problem
there are 3 dimensions in this example: L, M, T
Step 4, Pick up a subset of original variables to become key variables,
subject to the following restrictions:
The number of key variables must equal the number of fundamental dimensions.All fundamental dimensions must be represented in terms of key variables.
No dimensionless group is allowed from any combination of key variables.
e.g. Pick up 3 variables:
Invalid set: U;,,;zH,, 0
Step 5, Form dimensionless equations of the remaining variables in terms of
the key variables.
e.g.
ihg0
fed
cba
UHz
UH
UH
Step 6, Solve for the unknowns a, b, c, d, e, f, g, h, i
e.g.
2c0,b1,a
)LT(L)()ML(TML
UH
c1-ba3-2-1-
cba
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Step 7, Form dimensionless (PI) groups.
e.g.,,,
H
z3UH2U
10
2
Step 8, Form other PI groups if you want as long as the total number is the same.
e.g.,,
33
2 154
,,,00
2 zH
5Uz4U1
Which PI groups are right?
They are all right, but some groups are more commonly used and follow
Conventions.
Next, find relations betweenPIs through experiments.
roughnessrelative,number;Reynolds,H
z3
UH1 0
2
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Surface layer similarity (Monin Obukhov similarity)
Surface layer: turbulent fluxes are nearly constant. 20-30 m
Relevant parameters:
)sm()''w(
velocityfrictional,)sm()wv()wu(/||u
)m(z
32g
222/1222*
ov
ooo
v
sizeeddyorheight
Say we are interested in wind shear: zu
Four variables and two basic units result in two dimensionless numbers, e.g.:
3*
0
* u
z)w(g
zu
uz and v
v
The standard way of formulating this is by defining:
0
3*
)w(
u
gL vv
Monin-Oubkhov length
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constantKarman-Von
0.35(0.4),
),()(Lz
zu
uz
*
mm
PI relation
stableunstable
Empirical gradient functions to
describe these observations:
0510)161(
4/1
forfor
m
m
Note that eddy diffusion coefficients
and gradient functions are related:
zuwu mk
,0wvAssuming
*zu1
m
m
k
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Now we are interested in the vertical gradient of virtual potential temperature.
z
v
We can form a new variable*
* u
)'w( o
Again, four variables and two basic units result in two dimensionless numbers,
Lz
zz and*
v
),()(Lz
zz
*
hhv
PI relation
Similarly, we have
),()(Lz
z
q
qz
* qq
Normally, ),()( qm
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Surface wind profile
1. Neutral condition 0Lz 1)0( m
1zu
uz
*
)ln(u0
*
zzu
)exp(
zz
*uu0
disappear.windsreheight whetheisz0)ln(
uu
0zz*
Aerodynamic roughness length
i
N
1i iti
N
1i it0 whL
25.0
shS
25.0
z
elementiofwidth:wlength;total:L
elementiofarea:selement;iofheight:harea;total:S
it
iit
Over land
Over water
Kondo and Yamazawa
(1986)
0.016,guz
2
*0
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If you have observations at three levels,
you may determine displacement as,
)ln(
u
)ln(
u
)ln(
uu
0
3
0
2
0
1
z
dz3
z
dz2
z
dz1
*
0z
d
Displacement distance
)ln(u0
*
zd-zu
)ln()ln(uu
uudz
dz
dz
dz
13
12
1
2
1
3
2. Non-neutral condition 0Lz
051
0)161( 4/1
for
for
m
m
051
0)161( 2/1
for
for
hq
hq
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)]()[ln(u0
*
zzu
m
Integral form of wind and temperature profiles in the surface layer
0,5)(
0,)161(,tan2)ln()ln(2)( 4/12
12
12
1 2
for
forxx
m
xxm
*/uu
)ln(0zz
0,L
0,0L
0,0L
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)()ln(t*
0
zz)(
h
vv
Integral form of wind and temperature profiles in the surface layer
0,5)(
0,)161(),ln(2)( 2/121
for
fory
h
yh
t0 zzat vv
0t zzNormally, Similarly,
)()ln(q*
0
zz
q
)qq(
q
)()( hq
q0 zzatqq
0t zzNormally,
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Bulk transfer relationsHow to estimate surface fluxes using conventional surface observations,
surface winds (10m), surface temperature (2m),?
).qq(u)qw(
),(u)w(
,uu
00
v0v0v
22*
Q
H
D
C
C
C
:,,QHD
CCC Drag coefficient of momentum, heat, and moisture.
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,)(2
0
2*
)]()z/z[ln(
2u
u
mDC
,)(
20
2*
)]z/z[ln(
2u
u DNC
,)]()z/z)][ln(()z/z[ln( t0
2
hmHC
,)]z/z)][ln(z/z[ln( t0
2HNC
,)]()z/z)][ln(()z/z[ln( q0
2
qmQC
,)]z/z)][ln(z/z[ln( q0
2QNC
0-0.5 0.5
1.0
1.5
DN
D
C
C
Lz
2
z
z 100
5zz 100
0-0.5 0.5
1.0
1.5
HN
H
C
C
Lz
2zz 100
5zz 100
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Flux footprint
General concept of the flux footprint. The darker the red color,
the more contribution that is coming from the surface area certaindistance away for the instrument.
Relative contribution of the land surface area to the flux for two
different measurement heights at near-neutral stability.
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Relative contribution of the land surface area to the flux for two different surface
roughnesses at near-neutral stability.
Relative contribution of the land surface area to the flux for two different casesof thermal stability.
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Bulk transfer relationsHow to estimate surface fluxes using conventional surface observations,
surface winds (10m), surface temperature (2m),?
).qq(u)qw(
),(u)w(,uu
00
v0v0v
22*
Q
H
D
C
CC
:,, QHD CCC Drag coefficient of momentum, heat, and moisture.
numberStanton:HC numberDalton:QC
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,)(2
0
2*
)]()z/z[ln(
2u
u
mDC
,)(
20
2*
)]z/z[ln(
2u
u DNC
,)]()z/z)][ln(()z/z[ln( t0
2
hmHC
,)]z/z)][ln(z/z[ln( t0
2HNC
,)]()z/z)][ln(()z/z[ln( q0
2
qmQC
,)]z/z)][ln(z/z[ln( q0
2
QNC
0-0.5 0.5
1.0
1.5
DN
D
C
C
Lz
2
z
z 100
5zz 100
0-0.5 0.5
1.0
1.5
HN
H
C
C
Lz
2zz 100
5zz 100
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Problem: Assuming we have wind observations but no temperature
observations at two levels, say, 5 m and 10 m, in the surface layer,
can we estimate surface roughness and stability?
),()ln( Lz
z
z
u
u 10
0
10
*
10
),()ln( Lz
z
z
u
u 5
0
5
*
5
),()()ln(L
z
L
z
z
z
u
)uu( 510
5
10
*
510
0)ln(:Neutral5
10
*
510
z
z
u
)uu(
)()ln(
,0)ln(:Stable
L
z
L
z
z
z
u
)uu(
z
z
u
)uu(
510
5
10
*
510
5
10
*
510
4/1Lz
51
101
x1
x1
)x1(
)x1(
z
z
u
)uu(
z
z
u
)uu(
)16(1x
)}xtanx(tan2]ln[]{ln[)ln(
,0)ln(:Unstable
25
210
25
210
5
10
*
510
5
10
*
510