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that the wind is blowing from the north. This notation does not take into account
vertical wind components, because such vertical windspeeds are often short-lived,
whereas the horizontal wind pattern is usually much more steady.
46 Chapter 4. External atmospheric disturbances
If Vwhor is the horizontal wind velocity and yw is the wind direction, the horizontal
wind velocity components in the Earth axes are equal to:
uEw
= Vwhor · cos(yw − p)
vEw
= Vwhor · sin(yw − p) (4.4)
For sake of completeness, we will also introduce the ‘vertical wind direction’ angle
gw, for cases where the wind also has a vertical component; this angle is positive
when the wind is blowing upwards. Using that angle, the horizontal wind velocity
is found to be:
Vwhor = Vw · cos gw (4.5)
In this generalized situation, the vertical wind velocity component can be non-zero:
wEw
= −Vw · sin gw (4.6)
where the minus sign reflects the fact that the ZE points downwards.
Equations (4.3) to (4.6) can be used to model a steady wind pattern, while equation
(4.1) can be used to describe the changing wind velocity in the Earth’s boundary
layer. Notice however, that the horizontal wind direction yw will normally also vary
with height. The above given representation of the vertical wind component may be
practicable for modelling microburst wind patterns like the one shown in figure 4.2,
but for most other purposes the vertical windspeed can simply be neglected (i.e. gw
can be kept identical zero).
4.2 Stochastic disturbances
Atmospheric turbulence is often regarded to be a ‘random’ process. This is actually a
bit misleading, because the evolution of turbulent flows are governed by the general
Navier-Stokes equations (a set of deterministic, nonlinear, coupled partial differential
equations), even when the creation of an eddy out of an instability somewhere in
the flow field is a matter of ‘chance’ [27]. However, the theory of stochastic processes
does provide a convenient means to describe the atmospheric turbulence velocities
for the types of simulation problems considered in this report (e.g. the assessment of
handling characteristics of the airplane or the performance of automatic flight control
systems).
4.2.1 Stochastic properties of atmospheric turbulence
Auto power density spectra form the basic elements of the stochastic turbulence model.
In the literature, several sets of these spectra can be found. They all require the
selection of intensity levels and scale lengths before they can be applied in simulations.
In order to simplify the statistical equations as far as practicable, the stochastical
processes describing atmospheric turbulence are usually assumed to have the
following six restrictive characteristics [1, 27]:
1. Normality, which means that the probability density function of each turbulence
velocity component is Gaussian. As a consequence of this assumption,
the covariance matrix alone provides sufficient information for a complete statistical
description of the atmospheric turbulence.
4.2. Stochastic disturbances 47
2. Stationarity, which deals with temporal properties of turbulence. If the statistical
properties of a process are not affected by a shift in the time origin, this
process is called stationary.
3. Taylor’s hypothesis of a ‘frozen atmosphere’, which implies that gust velocities
are functions of the position in the atmosphere only. This hypothesis allows
spatial correlation functions and frequencies to be related to correlation
functions and frequencies in the time-domain.
4. Homogeneity, which deals with the spatial properties of turbulence. Turbulence
may be called homogeneous if its statistical properties are not affected by
a spatial translation of the reference frame.
5. Ergodicity, which means that time averages in the process are equal to corresponding
ensemble averages. This condition follows from the previous assumptions
of the turbulence being stationary and homogeneous. As a consequence,
all required statistical properties related to a given set of atmospheric
conditions can be determined from a single time history of sufficient length.
6. Isotropy, which means that the statistical properties are not changed by a rotation
or a deflection of the frame of reference. Complete isotropy implies homogeneity.
Because of isotropy, the three mean-square velocity components and
their scale lengths are equal:
su
2 = sv
2 = sw
2  s2
Lu = Lv = Lw  Lg
(4.7)
Experimental data on atmospheric turbulence shows that these assumptions are not
　
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