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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0
50
100
150
200
250
300
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0
0.151
0.235
0.245
0.210
0.145
0.068
0.318
0.363
0.456
0.538
0.565
0.560
0.508
0.382
0.229
0.0825
0.662
0.666
0.681
0.690
0.688
0.670
0.620
0.506
0.307
0.123
I
II III
H [ m ]
Vw
9.15
swg [ - ]
Figure 4.5: Standard deviation of vertical turbulence velocity as a function of altitude, given
for three different temperature lapse rates. Curve I: dT
dH = 0.003 C m−1, curve II:
dT
dH = 0.0065 C m−1, curve III: dT
dH 0.01 C m−1.
This representation of the Dryden filters is very suitable for implementation in a
simulation package like SIMULINK. If the white noise is approximated by a sequence
of Gaussian distributed random numbers, it is easy to determine the time-trajectories
of the turbulence velocity components. It is important to ensure that these random
sequences are completely independent, which may not be obvious if the simulation
software uses some initial starting value or ‘seed’ for its random-generator.
Better approximations of the Von Kármán turbulence spectra can be obtained by
using a series of additional lead-lag networks, adding differentiating and integrating
terms to the basic first-order Dryden filters. Although the resulting equations will be
more complicated than the Dryden equations, the resulting functions will still remain
rational. A detailed discussion of this technique can be found in ref.[1].
4.2.4 Turbulence intensity and scale
In the power spectral density functions from the previous sections, the turbulence
scale was represented by the scale-lengths Lu, Lv, and Lw, while the turbulence inten52
Chapter 4. External atmospheric disturbances
0 50 100 150 200 250 300 350
0
50
100
150
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48
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33
26
9
160
158
156
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149
145
139
108
69
18
300
301
305
310
315
326
329
284
161
36
H [ m ]
L w [ m ]
I
II III
Figure 4.6: Scale of (longitudinal) turbulence as a function of altitude, given for three
different temperature lapse rates. Curve I: dT
dH 0.005 C m−1, curve II:
0.005 < dT
dH < 0.01 C m−1, curve III: dT
dH 0.01 C m−1.
sity was represented by the standard deviations su, sv, and sw. If the turbulence field
is assumed to be isotropic, the scale lengths can all be replaced by a single length Lg,
and the standard deviation factors can be replaced by a single factor sg, but this is
not necessarily true for lower altitudes.
The standard deviation of the vertical gust velocity sw was experimentally found
to be a function of height, atmospheric stability (as expressed by the temperature
lapse rate), wind velocity, and a terrain factor. Since the latter influence appears to
be relatively small and the available data show a large scatter, the wind velocity as
a function of the altitude, and the atmospheric stability appear to be the primary
factors in determining turbulence intensity. With wind profiles for the boundary
layer of the atmosphere experimentally determined at different atmospheric stabilities
(see e.g. figure 4.1 for the standard temperature lapse rate l = −0.0065 K m−1),
the standard deviation of atmospheric turbulence turns out to be dependent on the
windspeed at a certain reference height (usually 9.15m, or 30 ft), and the temperature
lapse rate l.
Figure 4.5 presents the standard deviation sw in relation to the windspeed at
9.15m, as a function of the height above the ground. The standard deviations for
4.2. Stochastic disturbances 53
the other directions are equal to sw if the turbulence is isotropic. For lower altitudes,
ref.[19] presents the following relations for all stability conditions of the atmosphere,
based on the analysis of data from measurements:
su
sw
= sv
sw
= 2.5 0m h < 15m
su
sw
= sv
sw
= 1.25 − 0.001h 15m h < 250m (4.25)
su
sw
= sv
sw
= 1 h > 250m
The scale of the turbulence is represented by the scale length Lg, which also depends
on the height above the ground and the temperature lapse rate. This has been shown
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FDC 1.4 – A SIMULINK Toolbox for Flight Dynamics and Contro(30)
