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always satisfied [1, 27]. There is some evidence that atmospheric turbulence is not
necessarily normal, and while the observed departures from a normal amplitude
distribution are small, pilots seem to be quite sensitive to these effects. Actual atmospheric
turbulence possesses what is sometimes called a ‘patchy’ structure [1].
Taylor’s hypothesis seems to be valid as long as the aircraft’s velocity is large relative
to the encountered turbulence velocity. For this reason it is somewhat doubtful
that the hypothesis is fully valid when simulating the final approach and landing
of S/VTOL aircraft, i.e. aircraft which can fly at very low airspeeds, and which are
able to take off and land vertically (Vertical Take-Off and Landing) or using only a
very short runway (Short Take-Off and Landing). This includes the DeHavilland
DHC-2 Beaver aircraft, which served as the example vehicle for the aircraft model
from chapter 3.
The assumptions of homogeneity and isotropy appear not to be very valid in
the vicinity of the ground (i.e. within the Earth’s boundary layer). Near the ground,
there are fairly rapid changes in turbulence velocity with altitude, induced by vertical
windshear, which is closely related to the shape and roughness of the terrain. The
assumption of stationarity is satisfied only over short periods of time during which
the meteorological conditions remain reasonably constant, and is also affected by the
shape and roughness of the ground surface.
However, for many practical applications these six assumptions seem to be quite reasonable,
which is why these simplifications will all be maintained for the derivations
in the next sections. It is always possible to enhance these models in the future, if
48 Chapter 4. External atmospheric disturbances
Dryden
Von Kármán
A
S(W )
S(0)
WLg
1
.1
.01
Dryden
Von Kármán
B
S( W )
S(0)
WLg
1
.1
.01
.1 1 10 .1 1 10
Figure 4.3: Von Kármán and Dryden spectra (A: longitudinal, B: lateral/vertical)
a more accurate description of the atmospheric turbulence is required, e.g. for detailed
analysis of the final approach and landing. Alternatively, it is also possible to
insert actual measurements of atmospheric turbulence velocities into the simulation
models.
4.2.2 Power spectra of atmospheric turbulence
Due to the simplifications we have made in the previous section, it is possible to
distinguish between two fundamental correlation functions: the correlation between
velocities parallel to a connecting line between two points in space, and the correlation
between velocities normal to this connecting line. The first function is termed
‘longitudinal correlation’, the second ‘lateral correlation’.
Several authors have obtained analytical power spectral density functions for
the turbulence velocities, using measured data. The Von Kármán spectral density
functions seem to best fit the available theoretical and experimental data on atmospheric
turbulence, particularly at higher spatial frequencies [27]. Analogous to the
correlation functions, these spectra can also be divided in ‘longitudinal’ and ‘lateral’
functions. Applying those functions to the turbulence velocity components in the
aircraft’s body-axes yields:
Sugug (W) = 2su
2Lu
1
(1 + (1.339 LuW)2) 56
(4.8)
Svgvg (W) = sv
2Lv
1 + 8
3 (1.339LvW)2
(1 + (1.339 LvW)2) 11
6
(4.9)
Swgwg (W) = sw
2Lw
1 + 8
3 (1.339LwW)2
(1 + (1.339 LwW)2) 11
6
(4.10)
where Sugug represents the longitudinal Von Kármán spectrum, while Svgvg and Swgwg
are both instances of the lateral Von Kármán spectrum.1 The cross spectral density
1Some textbooks may use a scaled version of these spectra, due to the application of a different definition
for the Fourier transform (the partition of the constant 1
2p may differ). However, an agreement
on the correlation functions exists, because they originate from measured wind velocities.
4.2. Stochastic disturbances 49
functions are zero in isotropic turbulence at any point in space. Although this approximation
is not very valid at low altitudes, the cross covariances – and hence, the
cross power spectral densities – are usually neglected [19].
The von Kármán spectra yield an asymptotic behaviour of S(W) W−5/3 as W approaches
infinity. This condition is imposed by the rate at which the most energetic
eddies of turbulence loose their kinetic energy: this energy is not immediately lost to
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