Bound Layer

8/12/2019 Bound Layer

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Atmospheric boundary layers and

turbulence I

Wind loading and structural response

Lecture 6 Dr. J.D. Holmes


2/27

Atmospheric boundary layers and turbulence

0

5

10

15

20

25

30

35

0 1 2 3 4 5

Time (minutes)

Windspeed(m/s)

153 metres 64 metres 12 metres

Wind speeds from 3 different levels recorded from a synoptic gale


3/27


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


4/27


Mean wind speed profiles :

Logarithmic law

0 - surface shear stress a - air density

)(z,offunctionaisdz

Ud0a

z

u .constantdz

Ud

constantlog.)/1(U zuk e

integrating w.r.t. z :

u = friction velocity = (0/a)


5/27


Logarithmic law

k = von Karmans constant (constant for all surfaces)

)(z/zlogk

u(z)U 0e

zo= roughness length (constant for a given ground surface)

logarithmic law - only valid for z >zo and z < about 100 m


6/27


Modified logarithmic law for very rough surfaces

(forests, urban)

zh= zero-plane displacement

o

he

z

z-zlog

k

u(z)U

zhis about 0.75 times the average height of the roughness


7/27


logarithmic law applied to two different heights

or with zero-plane displacement :

o2e

o1e

2

1

/zzlog/zzlog

)(zU)(zU

oh2e

oh1e

2

1

)/zz(zlog

)/zz(zlog

)(zU

)(zU


8/27


Surface drag coefficient :

Non-dimensional surface shear stress :

from logarithmic law :

2

10

2

2

10

0

U

u

U

o

e10z

10log

k

uU

2

10log

o

e

z

k


9/27


Terrain types :

Terrain Type Roughness

Length (m)Surface Drag

Coefficient

Very flat terrain (snow, desert) 0.001 - 0.005 0.0020.003

Open terrain (grassland, few trees) 0.010.05 0.0030.006

Suburban terrain (buildings 3-5 m) 0.10.5 0.00750.02

Dense urban (buildings 10-30 m) 15 0.030.3


10/27


Power law

= changes with terrain roughness and height range

10

)( 10z

UzU

)/(log1

0zzrefe

zref= reference height


11/27


Matching of power and logarithmic laws :

0

20

40

60

80

100

0.0 0.5 1.0 1.5

Height,z

(m)

Logarithmic law

Power law

zo= 0.02 m = 0.128 zref= 50 metres


12/27


Mean wind speed profiles over the ocean:

Surface drag coefficient () and roughness length (zo) vary with meanwind speed

g - gravitational constant a - empirical constant

substituting :

a lies between 0.01 and 0.02

g

Ua

g

auz

2

10*2

o (Charnock, 1955)

2

o

ez

10log

k

2

oe

10o

10/zlog

Uk

g

az

Implicit relationship between zoand U10


13/27


Mean wind speed profiles over the ocean:

Assume g = 9.81 m/s2 ; a = 0.0144 (Garratt) ; k =0.41

Applicable to non-hurricane conditions

U10(m/s) Roughness Length (mm)

10 0.21

15 0.59

20 1.22

25 2.17

30 3.51


14/27


Relationship between upper level and surface winds : Geostrophic drag coefficient

Rossby Number :

balloon measurements : Cg= 0.16 Ro-0.09

g

*

U

uC g

o

g

fz

URo

(Lettau, 1959)

U10, terrain 1 u*,terrain 1 Ug u*,terrain 2 U10, terrain 2Log law Lettau Lettau Log law

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


15/27


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

Sonic radar (SODAR) measurements in Okinawa

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

Tower measurements not enough

usually in outer radius of hurricane and/or higher latitudes


16/27


Mean wind profiles in hurricanes :

Northern coastline of Western Australia

Exmouth

EXMOUTH

GULF

North

West CapeUS Navy

antennas

100 km

Profiles from 390 m mast in late nineteen-seventies


17/27


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

10

100

1000

0.0 1.0 2.0

U(z)/U(10)

Heig

htz,

(m)

)3.0/10(log

)3.0/(logUU 10z

e

e z for z < 100 m

Uz=U100 for z 100 m


18/27


Mean wind profiles in thunderstorms (downbursts) : Doppler radar

Model of Oseguera and Bowles (stationary downburst):

Some tower measurements (not enough)

r - radial coordinate

R - characteristic radius

z*- characteristic height out of the boundary layer

- characteristic height in the boundary layer

- scaling factor

z/z/zr/R2

eee12r

RU

2

Horizontal wind profile shows peak at 50-100 m


19/27


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

R = 1000 m

r/R = 1.121

z*= 200 metres

= 30 metres

= 0.25 (1/sec) 0

200

400

600

0 20 40 60

Wind speed (m/s)

Height(m)

r/R = 1.121


20/27


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

R = 1000 m

r/R = 1.121

z*= 60 metres

= 50 metres

= 1.3 (1/sec)0

200

400

600

0 20 40 60 80 100

Wind speed (m/s)

Height(m)

Uconst= 35 m/s


21/27


0

5

10

15

20

25

30

35

0 1 2 3 4 5

Time (minutes)

Windspe

ed(m/s)

153 metres 64 metres 12 metres

Turbulence represents the fluctuations (gusts) in the wind speed

It can usually be represented as a stationary random process


22/27


Components of turbulence :

u(t) - longitudinal - parallel to mean wind direction

- parallel to ground (usually horizontal)

ground

U+u(t)

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

w(t)

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

v(t)


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Turbulence intensities :

standard deviation of u(t) :

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

21

2

0

})(1

{ dtUtUT

T

u

Iv= v /U (lateral turbulence intensity)

Iw= w /U (vertical turbulence intensity)


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v 2.2u*

Iu= u /U

from logarithmic law

0e0e z/zlog

1

z/zlog/0.4u

2.5u

0ev

z/zlog

0.88I

w 1.37u* 0ew

z/zlog

0.55I

near the ground, u 2.5u*


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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


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Probability density :

for u(t) :

The components of turbulence (constantU) can generally berepresented quite well by the Gaussian, or normal, p.d.f. :

2

uu

u

Uu

2

1exp

2

1uf

2

vv

v

v

2

1exp

2

1vffor v(t) :

for w(t) :

2

ww

w

w

2

1exp

2

1wf


27/27

End of Lecture 6

John Holmes225-405-3789 [email protected]

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