03 Lect 6 Bound Layer

Atmospheric boundary layers and turbulence I

Wind loading and structural responseLecture 6 Dr. J.D. Holmes

Atmospheric boundary layers and turbulence

0

5

10

15

20

25

30

35

0 1 2 3 4 5

Time (minutes)

Win

d sp

eed

(m/s

)

153 metres 64 metres 12 metres

Wind speeds from 3 different levels recorded from a synoptic gale


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


• Mean wind speed profiles :

• Logarithmic law

0 - surface shear stress a - air density

)τρ(z, offunction a is dzUd

0a

zu .constant

dzUd

constantlog . )/1(U zuk e

integrating w.r.t. z :

u = friction velocity = (0/a)


• Logarithmic law

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

)(z/zlogku(z)U 0e

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

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


• Modified logarithmic law for very rough surfaces (forests, urban)

• zh= zero-plane displacement

o

he z

z-zlogku(z)U

zh is about 0.75 times the average height of the roughness


• 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


• Surface drag coefficient :

Non-dimensional surface shear stress :

from logarithmic law :

210

2

210

0

Uu

U

oe10 z

10logkuU

2

10log

oe z

k


• Terrain types :

Terrain Type Roughness Length (m)

Surface Drag 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


• Power law

= changes with terrain roughness and height range

10

)( 10zUzU

)/(log1

0zzrefe

zref = reference height


• Matching of power and logarithmic laws :

0

20

40

60

80

100

0.0 0.5 1.0 1.5

Hei

ght,

z (m

)

Logarithmic law

Power law

zo = 0.02 m = 0.128 zref = 50 metres


• Mean wind speed profiles over the ocean: • Surface drag coefficient () and roughness length (zo) vary with mean wind

speed

g - gravitational constant a - empirical constant

substituting :

a lies between 0.01 and 0.02

gUaκ

gauz

210*

2

o (Charnock, 1955)

2

oe z

10log

kκ

2

oe

10o 10/zlog

Ukgaz

Implicit relationship between zo and U10


• 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


• Relationship between upper level and surface winds : • Geostrophic drag coefficient

Rossby Number :

balloon measurements : Cg = 0.16 Ro-0.09

g

*

UuC g

o

g

fzU

Ro

(Lettau, 1959)

U10, terrain 1 u*,terrain 1 Ug u*,terrain 2 U10, terrain 2

Log law Lettau Lettau Log law

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


• 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


• Mean wind profiles in hurricanes :

• Northern coastline of Western Australia

Exmouth

EXMOUTH

GULF

North West Cape US Navy

antennas

100 km

• Profiles from 390 m mast in late nineteen-seventies


• 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)

Hei

ght z

, (m

)

)3.0/10(log)3.0/(logUU 10z

e

e z for z < 100 m

Uz =U100 for z 100 m


• 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/εz/zr/R2

eee12r

λRU2

• Horizontal wind profile shows peak at 50-100 m


• 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)

Heig

ht (m

)

r/R = 1.121


• 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)

Hei

ght (

m)

Uconst = 35 m/s


0

5

10

15

20

25

30

35

0 1 2 3 4 5

Time (minutes)

Win

d sp

eed

(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


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)


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)



v 2.2u*

Iu = u /U

from logarithmic law

0e0e z/zlog1

z/zlog/0.4u2.5u

0ev z/zlog

0.88I

w 1.37u* 0ew z/zlog

0.55I

near the ground, u 2.5u*



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


Probability density :

for u(t) :

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

2

uuu σ

Uu21exp

2πσ1uf

2

vvv σ

v21exp

2πσ1vffor v(t) :

for w(t) :

2

www σ

w21exp

2πσ1wf

End of Lecture 6

John Holmes225-405-3789 [email protected]

Documents

03 Lect 6 Bound Layer