Lect8 Similarity

7/30/2019 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

Lect8 Similarity

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