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)

Atmospheric boundary layers and turbulence 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’ T=10 min.

Atmospheric boundary layers and turbulence 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

Atmospheric boundary layers and turbulence 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. : Sometimes u * 2 or  U 2 is used in the denominator

Atmospheric boundary layers and turbulence 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.S u (n)  n -2/3, or S u (n)  n -5/3 U,  u, u must be specified to numerically determine S u (n)

Atmospheric boundary layers and turbulence Wind spectra : von Karman spectrum : at high frequencies, n.S u (n)  n -2/3, or S u (n)  n -5/3 at zero frequencies, S u (0)  4  u 2 u /  U The latter is a property of turbulence in a frequency range known as the inertial sub-range

Atmospheric boundary layers and turbulence Wind spectra : zero frequency limit : (von Karman spectrum satisfies this) From Lecture 5 : since auto-correlation is a symmetrical function of  :  u (-  ) =  u (  ) setting n = 0 T 1 is time scale (Lecture 5)

Atmospheric boundary layers and turbulence von Karman spectrum : Maximum value of occurs at n. u /U of 0.146

Atmospheric boundary layers and turbulence Busch and Panofsky spectrum for vertical component w(t): Length scale in this case is height above ground, z Maximum value of occurs at n.z/U of 0.30

Atmospheric boundary layers and turbulence 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 Disadvantages : 1) goes to 1 as n  0, even for very large  z 2) does not allow negative values

Atmospheric boundary layers and turbulence 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 [-C  z 1 - z 2  ] As separation distance  z 1 - z 2  increases, correlation coefficient  uu decreases

Atmospheric boundary layers and turbulence Effects of topography : Shallow topography : no separation of flow (follows contours) Predictable from computer models, wind-tunnel models shallow escarpment shallow hill or ridge

Atmospheric boundary layers and turbulence 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

Atmospheric boundary layers and turbulence Effects of topography : Effective upwind slope : about 0.3 (17 degrees) Upper limit on speed up effect as upwind slope increases effective slope  0.3

Atmospheric boundary layers and turbulence Topographic multiplier : denoted by M t : : Can be greater or less than 1. Codes only give values > 1 for mean wind speeds for peak gust wind speeds : ASCE-7 : K z,t = (1 + K 1 K 2 K 3 ) 2 M t = 1 + K 1 K 2 K 3

Atmospheric boundary layers and turbulence Shallow hills :  is the upwind slope = H/2L u k is a constant for a given type of topography s is a position factor LuLu H /2 crest  = 1.0 at crest <1 upwind and downwind, and with increasing height

Atmospheric boundary layers and turbulence 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

Atmospheric boundary layers and turbulence Shallow topography : Gust multiplier : Assume that standard deviation of longitudinal turbulence,  u, is unchanged as the wind flow passes over the hill

Atmospheric boundary layers and turbulence Steep topography : Can be treated approximately by taking an effective slope,  ' = 0.3 then same formulae are used, i.e. : 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

Atmospheric boundary layers and turbulence 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 x i (z) roughness length z o1 roughness length z o2 x Within the inner boundary layer, the logarthmic law with the roughness length, z 02, applies, but the wind speeds must match at the edge of the inner boundary layer

Atmospheric boundary layers and turbulence Change of terrain : For flow from smooth to rougher terrain (z 02 > z 01 ) : (Deaves, 1981) For flow from rough to smoother terrain (z 02 < z 01 ) :

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

End of Lecture 7 John Holmes