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National Advisory Committee for Aeronautics, Report - Theoretical Calculation of the Power Spectra of the Rolling a Yawing Moments on a Wing in Random Turbulence

The correlation functions and power spectra of the rolling
and yawing moments on an airplane wing due to the three
components of continuous random turbulence are calculated.
The rolling moments due to the longitudinal (horizontal) and
normal (vertical) components depend on the spanwise distribu-
tions of instantaneous gust intensity, which are taken into
account by using the inherent properties of symmetry of iso-
tropic turbulence. The results consist of expressions for
correlation functions or spectra of the rolling moment in terms
of the point correlation functions of the two components of
turbulence.

Specific numerical calculations are made for a pair of correla-
tion functions given by simple analytic expressions which fit
available experimental data quite well. Calculations are made
for four lift distributions. Comparison is made with the
results of previous analyses which assumed random turbulence
along the flight path and linear variations of gust velocity across
the span.

The rolling moment due to lateral (side) gusts, which is small,
is expressed in terms of the instantaneous value of the gust near
the center line of the fuselage, so that the efi'ect of spanwise
variation in gust intensity is ignored. The gaming moments
are considered to be proportional to the rolling moments with the
constants of proportionality given by simple aerodynamic
relations.

The gust velocities acting on an airplane flying through
turbulent air are functions of position or time known only
in a statistical sense. Consequently, aerodynamic forces and
moments produced by the lifting surfaces of the airplane
can be known only in a statistical sense. If the statistical
characteristics of the turbulence are assumed to be invariant
with position along the flight path, flight through turbulent
air may be considered to be a stationary random process
and the mathematical techniques developed for such pro-
cesses (see ref. 1, for instance) may then be used, in this
problem.