atmospheric boundary layers and turbulence ii wind loading and structural response lecture 7 dr....

Atmospheric boundary layers and turbulence II

Wind loading and structural response

Lecture 7 Dr. J.D. Holmes

Atmospheric boundary layers and turbulence

• Topics :

• Turbulence (Section 3.3 in book)

- gust factors, spectra, correlations

• Effect of topography (Section 3.4 in book)

• Change of terrain (Section 3.5 in book)


Gust speeds and gust factors :

The ‘expected value’ or average peak can be written as :

• The peak wind speed in a given time period (say 10 minutes) is a random variable

where g is a peak factor, in this case equal to 3.5

From the response time of anemometers (Dines, cup) used for long-term wind measurements, measured peak gusts are often quoted as a ‘3 second gusts’

ugσUU u3.5σU

T=10 min.

U

U

u3.5σ


Gust speeds and gust factors :

• Gust factor, G, is the ratio of the maximum gust speed to the mean wind speed :

At 10 metres height in open country, G 1.45 ( higher latitude gales)

In hurricanes, G 1.55 to 1.66

U

UG


Wind spectra :

As discussed in Lecture 5, the spectral density function provides a description of the frequency content of wind velocity fluctuations

Empirical forms based on full scale measurements have been proposed for all 3 velocity components

These are usually expressed in a non-dimensional form, e.g. :

2u

u

σ

(n)n.S

Sometimes u*2 orU2 is used in the denominator


Wind spectra :

The most important turbulence component is the longitudinal component u(t). The most commonly used spectrum for u is known as the von Karman spectrum :

u is the integral scale of turbulence, which can also be obtained from the auto-correlation function (Lecture 5).

Note that at high frequencies, n.Su(n) n-2/3, or Su(n) n-5/3

6/522

8.701

4)(.

U

n

U

nnSn

u

u

u

u

U, u , u must be specified to numerically determine Su(n)


Wind spectra :

von Karman spectrum :

at high frequencies, n.Su(n) n-2/3, or Su(n) n-5/3

6/522

8.701

4)(.

U

n

U

nnSn

u

u

u

u

at zero frequencies, Su(0) 4 u2 u /U

The latter is a property of turbulence in a frequency range known as the inertial sub-range


Wind spectra :

zero frequency limit :

(von Karman spectrum satisfies this)

From Lecture 5 :

-

n2uu dτ)e(2(n)S i

0

n2uu dτ)e(4(n)S i

since auto-correlation is a symmetrical function of : u(-) = u()

0

n2u

2u dτ)e(4(n)S i

u R

0 u

2u )dτ(4(0)S Ru

setting n = 0

12

uu T4σ(0)S T1 is time scale (Lecture 5)

U/4σ(0)S u2

uu


von Karman spectrum :

0.0

0.1

0.2

0.3

0.01 0.10 1.00 10.00

von Karman

n.u / U

n.Su(n)/u2

Maximum value of 0.271 occurs at n.u/U of 0.146


Busch and Panofsky spectrum for vertical component w(t):

Length scale in this case is height above ground, z

Maximum value of 0.258 occurs at n.z/U of 0.30

5/32

w

w

Unz

11.161

Unz

2.15

σ

(n)n.S

0.0

0.1

0.2

0.3

0.01 0.10 1.00 10.00

Busch & Panofsky

n.Sw(n)/w2

n.z / U


Co-spectrum of longitudinal velocity component :

As discussed in Lecture 5, the normalized co-spectrum represents a frequency-dependent correlation coefficient :

It is important use is to determine the strength of wind forces at the natural frequency of a structure, and hence the resonant responseExponential decay function :

As separation distance z increases, or frequency, n, decreases, co-spectrum (z,n) decreases

U

zk.n.expn)ρ(Δz,

Disadvantages : 1) goes to 1 as n0, even for very large z

2) does not allow negative values


Correlation of longitudinal velocity component :

Covariance and cross-correlation coefficient were discussed in Lecture 5

The correlation properties for the longitudinal velocity components, at points with vertical or horizontal separation are important for wind loads on tall towers, buildings, transmission lines etc.

Exponential decay function : uu exp [-Cz1 - z2]

As separation distance z1 - z2 increases, correlation coefficient uu

decreases


Effects of topography :

Shallow topography : no separation of flow (follows contours)

Predictable from computer models, wind-tunnel models

shallow escarpment

shallow hill or ridge



Steep topography : separation of flow occurs

Less predictable from computer models, wind-tunnel models OK at large enough scale

steep escarpment

separation

steep escarpment

steep hill or ridge

separation separation



Effective upwind slope : about 0.3 (17 degrees)

Upper limit on speed up effect as upwind slope increases

effective slope 0.3


Topographic multiplier :

denoted by Mt : :

Can be greater or less than 1. Codes only give values > 1

upwindgroundflattheabovezheightatspeedWind

featuretheabovezheightatspeedWindMultipliercTopographi

,,

,,

tM for mean wind speeds

tM for peak gust wind speeds :

ASCE-7 : Kz,t = (1 + K1K2K3)2 Mt = 1 + K1K2K3


Shallow hills :

is the upwind slope = H/2Lu

k is a constant for a given type of topography

s is a position factor

ksM t 1

Lu

H /2

crest

= 1.0 at crest <1 upwind and downwind, and with increasing height


Shallow topography :

k is a constant for a given type of topography (ridge, escarpment, hill)

4.0 for two-dimensional ridges

1.6 for two-dimensional escarpments

3.2 for three-dimensional (axisymmetric) hills


Shallow topography : Gust multiplier :

Assume that standard deviation of longitudinal turbulence, u, is unchanged as the wind flow passes over the hill

sk1M t

u

utt gσU

gσM.UM

u

u

gσU

gσ)ks.(1U

u

uu

gσUU

gσ1

ks1).gσU(

Ugσ1

ks1

u

Ugσ1

kk in which,

u


Steep topography :

Can be treated approximately by taking an effective slope, ' = 0.3

then same formulae are used, i.e. :φks1M t

φsk1M t

However these formulae are less accurate than those for shallow hills and do not account for separations at crest of escarpment or on lee side of a hill or ridge


Change of terrain :

At a change of terrain roughness, adjustment takes place within an ‘inner boundary layer

Full adjustment of the magnitudes of the mean wind speeds does not occur until the inner boundary layer fills the entire atmospheric boundary layer - this could take as much as 50 km (30 miles) of the new terrain

inner boundary layer

z

xi(z)

roughness length zo1roughness length zo2

x

Within the inner boundary layer, the logarthmic law with the roughness length, z02, applies, but the wind speeds must match at the edge of the inner boundary layer


Change of terrain :

For flow from smooth to rougher terrain (z02 > z01) :

3/4

22 36.0

)(

ooi z

zzzx (Deaves, 1981)

For flow from rough to smoother terrain (z02 < z01) :

2/1

2

114)(

o

oi z

zzzx


Change of terrain :

Turbulence and gust wind speeds adjust faster than mean speeds to a change of terrain

(Melbourne, 1992)

For gust speed at 10 metres, an exponential adjustment can be assumed :

2000

exp1)ˆˆ(ˆˆ121,2

xUUUU x

where and are the asymptotic gust velocities over fully-developed terrain of type 1 (upstream) and 2 (downstream).

1U 2U

End of Lecture 7

John Holmes225-405-3789 [email protected]

atmospheric boundary layers and turbulence ii wind loading and structural response lecture 7 dr....

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