# 03Lect6BoundLayer by gegouzhen12

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```									      Wind loading and structural response
Lecture 6 Dr. J.D. Holmes

Atmospheric boundary layers and
turbulence I
Atmospheric boundary layers and turbulence

153 metres       64 metres        12 metres

35
30
Wind speed (m/s)

25
20
15
10
5
0
0         1      2           3    4               5
Time (minutes)

Wind speeds from 3 different levels recorded from a synoptic gale
Atmospheric boundary layers and turbulence

Features of the wind speed variation :

• Increase in mean (average) speed with height

• Turbulence (gustiness) at each height level

• Broad range of frequencies in the fluctuations

• Similarity in gust patterns at lower frequencies
Atmospheric boundary layers and turbulence

• Mean wind speed profiles :
• Logarithmic law
dU
is a function of (z, ρ a τ 0 )
dz
0 - surface shear stress        a - air density

dU             u
 constant . 
dz              z
u = friction velocity =  (0/a)
integrating w.r.t. z :
U  (1/ k ) . u log e z  constant
Atmospheric boundary layers and turbulence

• Logarithmic law

u
U(z)       log e (z/z0 )
k

• k = von Karman’s constant (constant for all surfaces)

• zo = roughness length (constant for a given ground surface)
logarithmic law - only valid for z >zo and z < about 100 m
Atmospheric boundary layers and turbulence

• Modified logarithmic law for very rough surfaces
(forests, urban)
u      z - zh 
 z 
U(z)  log e         
k           o 

• zh= zero-plane displacement

zh is about 0.75 times the average height of the roughness
Atmospheric boundary layers and turbulence

• logarithmic law applied to two different heights

U(z 1 ) log e z1/z o 

U(z 2 ) log e z 2 /z o 

• or with zero-plane displacement :

U(z 1 ) log e (z 1  z h )/z o 

U(z 2 ) log e (z 2  z h )/z o 
Atmospheric boundary layers and turbulence

• Surface drag coefficient :
Non-dimensional surface shear stress :
0     2
u
        2
U10 U10
2

u        10 
from logarithmic law :     U10       log e  
z 
k         o
2
              
              
               
k
        10  
 log e   
z 

        o  
Atmospheric boundary layers and turbulence

• Terrain types :

Terrain Type               Roughness       Surface Drag
Length (m)       Coefficient

Very flat terrain (snow, desert)      0.001 - 0.005   0.002 – 0.003

Open terrain (grassland, few trees)   0.01 – 0.05     0.003 – 0.006

Suburban terrain (buildings 3-5 m)      0.1 – 0.5     0.0075 – 0.02

Dense urban (buildings 10-30 m)           1–5          0.03 – 0.3
Atmospheric boundary layers and turbulence

• Power law

 z 
U ( z )  U10  
 10 

•  = changes with terrain roughness and height range

        1      
                
 log ( z / z ) 
    e   ref 0 

zref = reference height
Atmospheric boundary layers and turbulence

• Matching of power and logarithmic laws :

zo = 0.02 m                          = 0.128                 zref = 50 metres

Logarithmic law
Power law

100

80
Height, z (m)

60

40

20

0
0.0     0.5         1.0           1.5
Atmospheric boundary layers and turbulence

• Mean wind speed profiles over the ocean:
• Surface drag coefficient () and roughness length (zo) vary with mean
wind speed
2
au 2* aκU10
zo                         (Charnock, 1955)
g      g
g - gravitational constant       a - empirical constant

a lies between 0.01 and 0.02
2
              
                                                     2
a  kU10 
κ              
k
substituting :                                 zo                   
        10                  g  log e 10/zo  
 log e   
z 

        o  
Implicit relationship between zo and U10
Atmospheric boundary layers and turbulence

• Mean wind speed profiles over the ocean:
Assume g = 9.81 m/s2 ; a = 0.0144 (Garratt) ; k =0.41

U10 (m/s)    Roughness Length (mm)

10                   0.21

15                   0.59

20                   1.22

25                   2.17

30                   3.51

Applicable to non-hurricane conditions
Atmospheric boundary layers and turbulence

• Relationship between upper level and surface winds :
• Geostrophic drag coefficient          Cg 
u*
Ug
Ug
Rossby Number :        Ro 
fzo

balloon measurements : Cg = 0.16 Ro-0.09                 (Lettau, 1959)

Can be used to determine wind speed near ground level over different terrains :

Log law        Lettau       Lettau       Log law
U10, terrain 1  u*,terrain 1  Ug  u*,terrain 2  U10, terrain 2
Atmospheric boundary layers and turbulence

• Mean wind profiles in hurricanes :
• Aircraft flights down to 200 metres

• Drop-sonde (probe dropped from aircraft - tracked by satellite) : recently started

• Sonic radar (SODAR) measurements in Okinawa

•   Tower measurements
•   not enough
•   usually in outer radius of hurricane and/or higher latitudes
Atmospheric boundary layers and turbulence

• Mean wind profiles in hurricanes :
North
US Navy
West Cape            antennas

Exmouth

EXMOUTH
GULF

100 km

• Northern coastline of Western Australia
• Profiles from 390 m mast in late nineteen-seventies
Atmospheric boundary layers and turbulence

• Mean wind profiles in hurricanes :
• In region of maximum winds : steep logarithmic profile to 60-200 m

• Nearly constant mean wind speed at greater heights

1000

log e ( z / 0.3)
U z  U10
Height z, (m)

log e (10 / 0.3)    for z < 100 m
100

Uz =U100 for z  100 m

10
0.0      1.0       2.0

U(z)/U(10)
Atmospheric boundary layers and turbulence

• Mean wind profiles in thunderstorms (downbursts) :
• Doppler radar

• Some tower measurements (not enough)

• Horizontal wind profile shows peak at 50-100 m

• Model of Oseguera and Bowles (stationary downburst):
 λR 2 
U
 2r   1 e r/R 2
e 
 z/z 
 e z/ε   
      
r - radial coordinate
R - characteristic radius
z* - characteristic height out of the boundary layer
 - characteristic height in the boundary layer
 - scaling factor
Atmospheric boundary layers and turbulence

• Mean wind profiles in thunderstorms (downbursts) :
Model of Oseguera and Bowles (stationary downburst) :

r/R = 1.121
R = 1000 m                             600

r/R = 1.121
400
z* = 200 metres           Height (m)

 = 30 metres                          200

 = 0.25 (1/sec)                        0
0     20         40      60
Wind speed (m/s)
Atmospheric boundary layers and turbulence

• Mean wind profiles in thunderstorms (downbursts) :
Add component constant with height (moving downburst) :

R = 1000 m                              600

r/R = 1.121
400

Height (m)
z* = 60 metres
 = 50 metres                           200

 = 1.3 (1/sec)                          0
0   20   40   60   80   100
Wind speed (m/s)

Uconst = 35 m/s
Atmospheric boundary layers and turbulence

153 metres       64 metres        12 metres

35
30
Wind speed (m/s)

25
20
15
10
5
0
0         1      2           3    4               5
Time (minutes)

Turbulence represents the fluctuations (gusts) in the wind speed

It can usually be represented as a stationary random process
Atmospheric boundary layers and turbulence

Components of turbulence :

• u(t) - longitudinal - parallel to mean wind direction
- parallel to ground (usually horizontal)

• v(t) - parallel to ground - right angles to u(t)

• w(t) - right angles to ground (usually vertical)

w(t)     v(t)
U+u(t)

ground
Atmospheric boundary layers and turbulence

Turbulence intensities :
2

 u  {  U (t )  U  dt} 2
T
1                   1
• standard deviation of u(t) :
T 0

Iu = u /U (longitudinal turbulence intensity) (non dimensional)

Iv = v /U (lateral turbulence intensity)

Iw = w /U (vertical turbulence intensity)
Atmospheric boundary layers and turbulence

Turbulence intensities :

near the ground, u  2.5u*
2.5u                   1
Iu = u /U                              
u  /0.4 log e z/z 0  log e z/z 0 
from logarithmic law

0.88
v  2.2u*         Iv 
log e z/z 0 

0.55
w  1.37u*        Iw 
log e z/z 0 
Atmospheric boundary layers and turbulence

Turbulence intensities :

rural terrain, zo = 0.04 m :

Height, z (m)    Iu
2          0.26

5          0.21

10         0.18

20         0.16

50         0.14

100         0.13
Atmospheric boundary layers and turbulence

Probability density :

• The components of turbulence (constantU) can generally be
represented quite well by the Gaussian, or normal, p.d.f. :

 1  u  U 2 
f u u  
1
for u(t) :                            exp  σ   
σ u 2π     2 u  
              
 1  v 2 
f v v  
1
for v(t) :                             exp   
 
σ v 2π     2  σv  
          
 1  w 2 
f w w  
1
for w(t) :                             exp   
 
σ w 2π     2  σw  
          
End of Lecture 6

John Holmes
225-405-3789 JHolmes@lsu.edu

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