Similarity theory: Outline
•
•
•
•
Goals, Buckingham Pi Theorem and examples
Surface layer (Monin Obukhov) similarity
Asymptotic behaviour and free convection scaling
The outer layer
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
training course: boundary layer II
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.
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.
Procedure:
•Select relevant parameters and plot dimensionless
functions.
•Use constraints from asymptotic cases.
•Apply empirical functions as turbulence closure.
training course: boundary layer II
Buckinham Pi dimensional analysis (Stull, 1988): example

U
U velocity (m / s)
1. Define relevant variables and
dimension (m)
their dimensions.
W mass (kg )
g acceleration of gravity (m / s 2 )
 air density (kg / m3 )
2. Count number of fundamental
dimensions.
m, s, kg
training course: boundary layer II
Buckinham Pi dimensional analysis: example
3. Form n dimensionless groups 1,..n
where n is the number of variables
minus the number of fundamental
dimensions.
1 
Wg
U 2  2
gravitational force
lift force
2 
n 53 2
W
3
4. Measure 1 as a function of  2
5. Further simplification; assume:
W~
3
 2  constant  1  constant
i.e.
W ~ U 22 ~ U 2W 2 / 3  W ~ U 6
training course: boundary layer II
mass of airplane
mass of
displaced air
The great flight diagram
106
Weight as a function of
cruising speed ( “The
simple science of
flight” by Tennekes,
1997, MIT press)
W~U6
Flying objects range
from small insects to
Boeing 747
Speed (m/s)
10-6
training course: boundary layer II
Dimensional analysis example: windmill/anemometer
U
D
1. Define relevant variables and
their dimensions.
U velocity (m / s )
D diam eter (m)
T
torque ( N m  kg m 2 s  2 )
 angularspeed ( s 1 )
 air density (kg m  3 )
2. Count number of fundamental
dimensions.
m, s, kg
training course: boundary layer II
Dimensional analysis example: windmill/anemometer
3. Form n dimensionless groups 1,..n
where n is the number of variables
minus the number of fundamental
dimensions.
1 
T
U 3 D 2
produced power
wind power
D
2 
U
n 53 2
speed of rotor tip
wind speed
4. Measure 1 as a function of  2 by changing load.
5. For an anemometer:
1  0 (no torque)   2  constant
   2 U / D
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
training course: boundary layer II
Surface layer similarity (Monin Obukhov similarity)
h
For z/h << 1 flux is approximately equal
to surface flux.
surface layer
Relevant parameters:
0
z height or eddy size

|  o | /   (u ' w')  (v' w')
g
v
2
o
Flux profile

2 1/ 2
o
( m)
2
2
(m / s )
( w' v ')o
(m 2 / s 3 )
Say we are interested in wind shear:
U
z
( s 1 )
training course: boundary layer II
o
surf .
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.:
U
z
z (|  o | /  )1 / 2
and
g ( w' v ')o z
 v (|  o | /  )3 / 2
The standard way of formulating this is by defining:
u*  (|  o | /  )1/ 2 friction velocity
v
u*3
L
Obukhov length
g ( w ' v ') o
Resulting in:
U z
m 
z u*
z
L
dimensionless shear
Stability parameter
 (von Karman constant) is defined such that m  1 for z / L  0
training course: boundary layer II
MO gradient functions
Observations of m as a
function of z/L, with   0.4
Empirical gradient functions to
describe these observations:
m  (1  16z / L)1 / 4 for z / L  0
m  1  5 z / L
for z / L  0
Note that eddy diffusion
coefficients and gradient
functions are related:
if
then
U
 u ' w'  K m
z
unstable
Km
1

zu* m
training course: boundary layer II
stable
MO-similarity applied to other quantities
Quantity

z
q
z
u

scaling
parameter
dimensionless
function
 ( w' ')o
* 
u*
z 
 h ( z / L )
* z
q* 
 ( w' q')o
u*
z q
q* z
u
u*
u*
 h ( z / L )
 f u ( z / L)

 f ( z / L)
*
*
training course: boundary layer II
Integral profile functions
Dimensionless wind gradient (shear) or temperature gradient
functions can be integrated to profile functions:
U
u*
 m
z
z


u* 
z
U  ln( )  m ( z / L)
  zom

with:
zom
integration constant (roughness length for momentum)
m wind profile function, related to gradient function:

z
m  1  
, with  

L
Profile functions for temperature and moisture can be obtained
in similar way.
training course: boundary layer II
MO wind profile functions applied to observations
Unstable
Stable
Limit of
stable layer
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
training course: boundary layer II
Asymptotic behaviour
Limiting cases can help to constrain functions e.g.:
for z / L  0 : m  1 (defines von Karman constant,  )
for z / L   : u* becomes irrelevant
Therefore e.g.:
  zu*
 h ( z / L) u*  ( z / L) 1/ 3
z ( w ' ')o
  zu*

z ( w ' ')o
( z / L)1/ 3
free convection scaling
training course: boundary layer II
Example of free convection scaling
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
training course: boundary layer II
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:
U  UG
zu*
V  VG
zu*
 fU ( ),
 fV ( ),
u*
f
u*
f
where u* / f is BL  depth scale instead of h
training course: boundary layer II
Mixed layer
Convective boundary layer (mixed layer scaling):
•Effects of friction can often be neglected.
•Profiles well mixed, so gradient functions become less
important
g
1/ 3
w

{
(
w
'

'
)
h
}
• w* is important turbulence velocity scale *
v o
v
training course: boundary layer II
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
v (|  | /  )3/ 2

local Obukhov length
g (w 'v ')
U
z
 m ( z /  )
1/ 2
z (|  | /  )
Z-less scaling
far away from the surface, z should drop:
U
z
z
 m ( z /  ) ~
1/ 2
z (|  | /  )

training course: boundary layer II
Local scaling
Z-less regime
training course: boundary layer II
scaling regions for the unstable BL
training course: boundary layer II
Holstlag and Nieuwstadt, 1986:
BLM, 36, 201-209.
scaling regions for the stable BL
training course: boundary layer II
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.
UG 1
u*
A
 ln(
)
u* 
f zom

VG B

u* 
drag
Wangara data
ageostrophic angle
Plot A and B as a
function of stability
parameter.
training course: boundary layer II