Training course: boundary layer II Similarity theory: Outline Goals, Buckingham Pi Theorem and examples Surface layer (Monin Obukhov) similarity Asymptotic.

training course: boundary layer II

Similarity theory: Outline

• Goals, Buckingham Pi Theorem and examples

• Surface layer (Monin Obukhov) similarity

• Asymptotic behaviour and free convection scaling

• The outer layer






• The outer layer


Similarity theory

Motivation: •Closure problem requires empirical expressions for turbulent diffusion coefficients (which include dependency on flow characteristics).•Number of independent parameters has to be limited.

Procedure:•Select relevant parameters and plot dimensionless functions.•Use constraints from asymptotic cases.•Apply empirical functions as turbulence closure.

Similarity theory is an intelligent way of organizing data e.g. from field experiments or large eddy simulations.

Note: turbulence closure is based on observations not theory.


Buckinham Pi dimensional analysis (Stull, 1988): example

U

2. Count number of fundamental dimensions.

kgsm ,,

2

3

velocity ( / )

dimension ( )

mass ( )

acceleration of gravity ( / )

air density ( / )

U m s

m

W kg

g m s

kg m

1. Define relevant variables and their dimensions.


Buckinham Pi dimensional analysis: example

3. Form n dimensionless groupswhere n is the number of variables minus the number of fundamental dimensions.

235 nn ,..1

221 U

Wg

gravitational force

lift force 32 W

mass of airplane

mass of displaced air

4. Measure as a function of 1 2

5. Further simplification; assume: 3

2 1~ constant constantW i.e.

63/2222 ~~~ UWWUUW


Weight as a function of cruising speed ( “The simple science of flight” by Tennekes, 1997, MIT press)

106

10-6

Wei

ght (

New

tons

)

Flying objects range from small insects to Boeing 747

W~U6

Speed (m/s)

The great flight diagram


Dimensional analysis example: windmill/anemometer

1. Define relevant variables and their dimensions.

)(

)(

)(

)(

)/(

3

1

22

mkgdensityair

sspeedangular

smkgmNtorqueT

mdiameterD

smvelocityU

DU

kgsm ,,2. Count number of fundamental dimensions.


Dimensional analysis example: windmill/anemometer

3. Form n dimensionless groupswhere n is the number of variables minus the number of fundamental dimensions.

235 nn ,..1

231 DU

T

produced power

wind power U

D2

speed of rotor tip

wind speed

5. For an anemometer:

DU

torqueno

/

constant)(0

2

21

4. Measure as a function of 1 2 by changing load.






• The outer layer


h

.surf

Flux profile

layersurface0 o

For z/h << 1 flux is approximately equal to surface flux.

Relevant parameters:

)/()''(

)/()''()''(/||

)(

32

222/122

smwg

smwvwu

msizeeddyorheightz

ovv

ooo

Say we are interested in wind shear:

z

U

)( 1s

Surface layer similarity (Monin Obukhov similarity)

Considerations about the nature of the process:• z/zo >> 1• distance to surface determines turbulence length scale• shear scales with surface friction rather than with zo


MO similarity

Four variables and two basic units result in two dimensionless numbers, e.g.:

2/1)/|(| o

z

z

U

2/3)/|(|

)''(

o

ov

v

zwgand

The standard way of formulating this is by defining:1/ 2

*

3*

(| | / )

( ' ')

o

v

v o

u friction velocity

uL Obukhov length

g w

Resulting in:

*u

z

z

Um

dimensionless shear

Stability parameter

L

z

(von Karman constant) is defined such that 1 for / 0m z L


Observations of as a function of z/L, with

m4.0

Empirical gradient functions to describe these observations:

0//51

0/)/161( 4/1

LzforLz

LzforLz

m

m

Note that eddy diffusion coefficients and gradient functions are related:

z

UKwu m

''

m

m

zu

K

1

*

then

ifstableunstable

MO gradient functions


MO-similarity applied to other quantities

)/(*

Lzz

q

q

zh

z

q

**

)''(

u

qwq o

Quantity

z

scaling parameter

**

)''(

u

w o

dimensionlessfunction

)/(*

Lzz

zh

u *u )/(*

Lzfu uu

* )/(

*

Lzf


Integral profile functions

Dimensionless wind gradient (shear) or temperature gradient functions can be integrated to profile functions:

)/()ln(** Lzz

zuU

z

u

z

Um

omm

with:

omz integration constant (roughness length for momentum)

m wind profile function, related to gradient function:

L

zwithm

,1

Profile functions for temperature and moisture can be obtained in similar way.


MO wind profile functions applied to observations

Stable Unstable

Limit of stable layer






• The outer layer


Asymptotic behaviour

Limiting cases can help to constrain functions e.g.:

/ 0 : 1 (defines von Karman constant, )mfor z L

*/ : becomes irrelevantfor z L u

Therefore e.g.:1/3*

*

1/3*

( / ) ( / )( ' ')

( / )( ' ')

h

o

o

zuz L u z L

z w

zuz L

z w

free convection scaling


Example of free convection scaling






• The outer layer


Above the surface layer (z/h>0.1)

•Fluxes are not constant but decrease monotonically with height•Boundary layer height h and Coriolis parameter f are additional scales.

Neutral PBL; velocity defect law:

hofinsteadscaledepthBLisfuwhere

f

zuf

u

VV

f

zuf

u

UUV

GU

G

/

),(),(

*

*

*

*

*


Mixed layerConvective boundary layer (mixed layer scaling):

•Effects of friction can often be neglected.•Profiles well mixed, so gradient functions become less important• w* is important turbulence velocity scale

3/1* })''({ hw

gw ov

v


Stable boundary layer

Local scaling extension of surface layer scaling, where surface fluxes are replaced by local fluxes, in other words:Surface layer closure applies to outer layer as well

3/ 2(| | / )

( ' ')v

v

local Obukhov lengthg w

)/()/|(| 2/1

zz

z

Um

Z-less scalingfar away from the surface, z should drop:

z

zz

z

Um ~)/(

)/|(| 2/1


Local scaling

Z-less regime


scaling regions for the unstable BL

Holstlag and Nieuwstadt, 1986: BLM, 36, 201-209.


scaling regions for the stable BL

Holstlag and Nieuwstadt, 1986: BLM, 36, 201-209.


Geostrophic drag law

Match surface layer and outer layer to obtain relation between surface drag and geostrophic wind.

B

u

V

A

zf

u

u

U

G

om

G

*

*

*

)ln(1

drag

ageostrophic angle

Plot A and B as a function of stability parameter.

Wangara data

Training course: boundary layer II Similarity theory: Outline Goals, Buckingham Pi Theorem and examples Surface layer (Monin Obukhov) similarity Asymptotic.

Documents

boundary layer

training course

outer layer slide

number of variables

similarity theory motivation

mo similarity

z o slide

dimensionless functions