1 – Wind phenomenon
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1 Wind phenomenon
The study of wind behavior and the determination of its intensity is extremely
complicated mainly because it is impossible to represent physically the forces field.
Whereas for the representation of almost every load acting on the structure is possible to
use vectors or distributed load, for wind it is necessary to use a statistical approach.
Even the regulation code requests a precise description of wind strength considering its
dependence on a large amount of parameters and statistically analyzing the collected
data.
The horizontal load of wind on the vertical side structure will be considered; it is
difficult to establish the response of a structure to this load due to the trouble in defining
univocally the force for it is unknown in every time null .
The modal analysis and the dynamic’s study of building requests the knowledge of the
force and of its behavior in time. Those are the deterministic forces (periodic or non-
periodic ones). In Appendix A there is a statistical approach to the study of this kind of
deterministic forces.
1.1 The study of the wind
The main characteristics of real turbulent wind is the subject of this paragraph. The
complex wind behaviour requires a statistical explanation, as it was underlined above.
Wind is caused by a difference in atmospheric pressure. At great heights the air
movement is not influenced by earth surface, whereas below a certain height, called
delta gradient height (or boundary layer), surface’s friction modifies wind flow, creating
turbulence. The boundary layer changes according to the ground underneath.
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Approximately the suggestion is considering its value 300 m on the ocean and 600 m
above towns (some studies suggest to put the level even higher, up to 1000 m according
to Jonh Holmes, for skyscrapers are growing higher and higher).
Time story of longitudinal velocity is:
null null null null = null
null
+ null null null null
Equation 1
Figure 1 - Longitudinal velocity
Figure 2 - Mean velocity wind profile and atmospheric turbulence
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The first term (null
null
) is the velocity mean value on period that goes from 10 minutes to 1
hour according to the reference codes; the nullnullnullnull is the dynamic waving part, which
causes the turbulence.
Figure 3 - Composition of the measured gust wind velocities
Such a distinction allows separating our study in two: the boundary layer is
characterized by the mean velocity in a way, and by a turbulence part, which can be
added.
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Furthermore real wind has a vertical and transversal component, but their mean value is
none. A precise mathematical description, which uses a Cartesian system, requires the
three directions together (longitudinal, transversal and vertical one):
null null null ,null ,null ,null null = null
null
null null null + null null null ,null ,null ,null null
null null null ,null ,null ,null null = null null null ,null ,null ,null null Equation 2
null null null ,null ,null ,null null = null null null ,null ,null ,null null
Figure 4 - Generic components of wind speed
It is useful therefore splitting wind behaviour into mean wind velocity and wind
turbulence.
1.1.1 Mean wind velocity
The average value of the wind’s velocity depends on the height (null ) from where the
velocity is measured, as it is shown in the picture below (Figure 4 and Figure 5) that
reports values for different type of ground: the results granted include also the turbulent
part and, still, the dependence on the height remains.
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Figure 5 - Examples of wind velocity measurements at different heights
The decisive magnitudes of the limit layer are the ground’s roughness, above all in the
lower part, and height in the part closer to the free regime of flow. Mathematically
speaking the more accurate expression is the logarithmic law that grants proper results
up to 300 m for velocity greater than 20 m/s. Codes prefer to adopt the power law which
is the result of empiric studies and it’s also easier to use. Eurocode 1 makes use of the
logarithmic law up to height of 200 m. It won’t be dwelt upon the calculation of
roughness parameters of the ground used by the codes. It will be just illustrated the
more suitable theoretic formulation of the logarithmic trend of the wind profile
(obtained by integration):
null null null ,null ,null ,null null = null
null
null null null =
null
∗ null
ln
null
null
null
Equation 3
null
∗ =
null
null
null
null
Equation 4
null
null
stands for the tangential stress at the surface level, null stands for the air density, null
null
is
the parameter related to roughness (roughness’ height) and null is the Von Karman’s
constant. The following picture (Figure 6) simply shows the dependence on roughness:
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Figure 6 - Dependence on the roughness
null
∗ is the friction velocity that has the same dimensions as velocity, but, physically
speaking , it is not exactly a velocity.
For instance, it is convenient quoting the Eurocode 1 that refers to such a mathematical
expression for the average profile of the wind (although it also gives reference values
for each zone trough tabs).
On the other hand, the following picture (Figure 7) represents the power law that is
applied by the American and the Canadian codes:
Figure 7 - Different functions to define wind profile
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The variation of the average value of velocity can be expressed as a function of height
by an exponential law such as:
null
null
null null null = null
null
null null null null
null
null
null
null
null
null
Equation 5
null
null
null null null null
is the average velocity defined at the height’s gradient null
null
=null : the
coefficient null depends on the type of ground and it stays within a range of 0.16 (for areas
at the open sea) and 0.40 (for areas inside big cities).
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1.1.2 Turbulence of the wind
Figure 8 - Mean wind profile with turbulence intensity clearly dependent on different roughness
The picture above (Figure 8) clearly shows the turbulent nature of wind. Furthermore it
shows a measurement close to the earth’s surface with a modest velocity. It is clear that
the velocity is not stationary. Quantities such as mean velocity and RMS may be
calculated for the random variable, i.e. the turbulence. In the matter of motion’s
equations of the fluid, they change according with the turbulence component. They
must be taken into account in a study more precise.
The waving part, from which depends the turbulence in the boundary layer, is less
sensible to the variation of height. It is also a random variable both in time and space.
The main statistical values which describe the vein’s fluctuations are: turbulence
intensity null
null
, spectral power density null
null
null null null , cross-correlation between measured
velocities in different point of space null
nullnull
null null null and the probability distribution of the
velocity. To be more precise, the turbulent part includes three components null , null and null .
Therefore the calculation of the standard deviation for each component.
The experimental results of Davenport (1967), Harris (1970) and Armitt (1976) show
that the standard deviations fall in a small account according with heights up to ordinary
building’s one. Armitt even claims that they are constant up to half the boundary layer.
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Until the height of 100 - 200 m upon the homogeneous surface, the standard deviation
of the three components are approximately:
null
null
= null null
∗
Equation 6 null
null
≈ 0 . 75null
null
null
null
≈ 0 . 5null
null
Where null is a constant value of 2.5 if null
null
=0.05 and null=1.8 if null
null
=0.3 .
The turbulence’s intensity null
null
is defined as follow:
null
null
=
null
null
null
null
Equation 7
Where null
null
is the standard deviation of the vein’s velocity. This definition is true in all
the three dimensions; i.e. longitudinal, lateral and vertical one. Obviously null
null
has mean
equal to zero, for it refers to the oscillation close to the mean velocity.
null
null
usually depends from the drag coefficient of ground null (and another formula is):
null
null
= null null null
Equation 8
The value of null goes from 0.005 in blue water, to 0.05 in the city centre.
The beta coefficient in the horizontal component is usually equal to 2.45. Therefore null
null
goes from 5% to 25%.
null
null
is possible to define the turbulence intensity’s pattern as follow:
null
null
=
null
null
null
null
=
2. 5null
∗ null
∗ 0 . 4
ln
null
null
null
=
1
ln
null
null
null
Equation 9
In this way, it is related only to the ground’s roughness. The same process can be
extended also to the other directions, even though it is less interesting.
This elaboration is very useful for the first part of the chapter about the wind tunnel test.
One of the main topic is indeed to build the wind profile as close as possible to the real
behaviour of it. Hence each parameter is meaningful and necessary.
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The wind tunnel which will be used for the test, is able to produce the boundary layer
and its characteristics; the one which will be used at the University of Miami for the
PIV simulation, isn’t able.
1.1.3 Density of probability
As it was showed in the opening picture (Figure 8), the wind variations in the boundary
layer are random and never repeat themselves in time. The probability density is defined
and our measurements showed that the velocity components in the boundary layer
follow the normal distribution of Gauss. Hence knowing the mean value and the
standard deviation is enough in order to calculate the probability of each wind velocity.
null
null
null null null =
1
null
null
null 2null
nullnullnull null − 1
2
null
null − null
null
null
null
null
null
null Equation 10
This study though does not give any information about the variation of the wind
phenomenon in time. In order to describe the turbulence distribution using frequency, is
used the spectral power density or spectrum.
In many codes, among which the American code, for the wind loads, is used the peak
gust wind speed, that is the peak velocity due to the wind gusts. Considering the
conditions of random variable, it is possible to estimate approximately the value,
considering the normal distribution of Gauss.
null
null
= null
null
+ null null
null
Equation 11
Where null is the peak factor equal to 3,5. Instead more interesting and used is the gust
factor, null , that is the ratio between the maximum gust velocity in a specified period and
the mean wind velocity.
null =
null
null
null
null
Equation 12
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To assign a fixed value to such coefficient is not easy; different experimental studies
have been made and depending on the zone obviously are been found different values.
Deacon for a 10 m height, based on the mean velocity of 10 minutes, has found values
of 1.45 for grounds in open country and 1.96 in town.
Obviously in case of hurricanes and tropical storms such value is destined to increase.
Many authors using different formulations have associated such result to the intensity of
the turbulence. Especially as regards the study of the wind loads the American code
refers to some pressure coefficients, as it will be clear in the following part, not pure,
but considering such peak effect, in the project phase. It will not be the null
null
as for the
Eurocode, but rather the nullnull
null
.
1.1.4 Turbulence spectrum
The energy associated to the fluctuations of the vein is arranged on a wide field of
frequencies: such distribution depending on the frequency is described through the
function “spectral power density” null
null
null null null that is correlated to the variance through the
relation:
null null
null
null null null nullnull
null null
null
= null
null
Equation 13
Often this function is represented in non dimensional form, as the picture shows, where:
null is the frequency in Hz and null
null
is the variance, derivable from the turbulence index
through the superscript formula (or rather from the kind of ground),
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Figure 9 - Power spectrum density of the wind
The ratio between frequency and mean velocity is defined as the inverse of the wave
length, associated to the dimensions of the atmospheric vortexes. Such length can be
compared to a characteristic dimension, called integral scale null , defined as the wave
length of the vortexes in correspondence to the spectrum peak (such size is also defined
as the barycentre of the auto-correlation): usually null is of the order of hundred m.
In this way it is possible to define a non dimensional frequency (called also reduced
frequency):
null
null
=
nullnull
null
null
Equation 14
Obviously this relation can be specified for every direction, as was already done before:
null null
null
null null null nullnull
null null
null
= σ null
null
Equation 15 null null
null
null null null nullnull
null null
null
= σ null
null
null null
null
null null null nullnull
null null
null
= σ null
null
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The function of spectral power density null
null
null null null =null
null
nullnull
null
null is obtained, through wind
measure, with the methodologies defined in the previous paragraph and it is possible to
interpolate this function through the formula of Von Karman:
null null null
null
null = 4null null
null
null
null
null
null 2+ null
null null
null
null
null
Equation 16
Where null
null
is the adimensional frequency, null
null
is the mean value of velocity (m/s) and null is
the integral scale (m).
In literature some variations of the formula of Von Karman with different coefficients
are reported, but with the same form; avoiding to enter into details of the theoretical
treatment, this is because of an asymptotic behavior of the spectral density. Harris has
reported to the case of wind the studies of Von Karman. Further Davenport, Busch and
Panofsky, Kaimal, Simiu and Scanlan and others have analyzed and tested this problem:
null null
null
σ null
null
=
4
null null
null
null
null
null 1+70 . 78null
null null
null
null
null
null
null
null
null
null
Equation 17
Only the spectrum of the longitudinal velocity it is reported, because the others are less
interesting as regards the problem of this thesis. For example for the horizontal
structures, as the bridges, the vertical component performs an important role.
The following picture shows different solutions reported in literature as regards the
spectrum of the frequency density about the turbulent component of the windy action.
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