Atmospheric Boundary Layers and Turbulence - Wind Engineering - Lecture Slides, Slides of Environmental Law and Policy

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Atmospheric boundary layers and

turbulence II

• 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 :

• 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.

U

Uˆ

G 

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. :

2 u

u

n.S (n)

Sometimes u*^2 orU^2 is used in the denominator

Wind spectra :

von Karman spectrum :

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

2 2 5 / 6 1 70. 8

4

. ( )



 

 

 

  

  

 

 

  

 

U

n

U

n nS n

u

u

u

u l

l



at zero frequencies, Su(0)  4 u^2 lu /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 : 

 

2 n

Su (n) 2 u( )e dτ

  i ^ 



 (^) 

Su (n)  4 0  u( )e i^2 n^ dτ

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



 (^)  Su (n)  4  (^) u^20 R u( )e i^2 n^ dτ



 Su (0)  4  (^) u^20 R u( )dτ^ setting n = 0

1

2

Su (0) 4 σu T T 1 is time scale (Lecture 5)

S u (0) 4 σu^2 lu/U

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.



 

 

 

  

  

 

 

  

 2 ^ 5/ w

w

U

1 11.16 nz

U

2.15 nz σ

n.S (n)

0.01 0.10 1.00 10.

Busch & Panofsky

n.Sw(n)/ w^2

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 response

Exponential decay function :

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

 

  

  

  

    U

k.n. z ρ(Δz,n) exp

Disadvantages : 1) goes to 1 as n0, even for very large z

2. does not allow negative values

Effects of topography :

Shallow topography : no separation of flow (follows contours)

Predictable from computer models, wind-tunnel models

shallow escarpment

shallow hill or ridge

Effects of topography :

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

Topographic multiplier :

denoted by Mt : (^) :

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

Wind speed at height z abovethe flat groundupwind

Wind speed at height z abovethe feature Topographic Multiplier , ,

, , 

Mt for mean wind speeds

Mˆ t for peak gust wind speeds (^) :

ASCE-7 : Kz,t = (1 + K 1 K 2 K 3 )^2 Mt = 1 + K 1 K 2 K 3

Shallow hills :

 is the upwind slope = H/2Lu

k is a constant for a given type of topography

s is a position factor

Mt  1  ks 

Lu

H /

crest

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

Shallow topography : Gust multiplier :

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

Mˆ^ t  1 ks

u

t u t (^) U gσ

U.M gσ Mˆ 

u

u U gσ

U.(1 ks ) gσ 



u

u

u

U gσ

U

1 gσ

ks (U gσ ). 1



U

1 gσ

ks 1 u



U

1 gσ

k in which, k  u

Steep topography :

Can be treated approximately by taking an effective slope, ' = 0.

then same formulae are used, i.e. :

Mt  1 ksυ

Mˆ^ t  1 ksυ

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