# Theoretical calculation of the power spectra of the rolling and yawing moments on a wing in random turbulence Page: 3 of 20

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ROLLING AND YAWING MOMENTS ON A WING IN RANDOM TURBULENCE

The implication of these assumptions and the limitations

they impose on the results of the analysis are discussed in a

subsequent section of the report.

, Coordinate system and gust components.-The system of

axes and the local velocity field relative to the lifting surface

are shown in figure 1 (a). The velocity at each point in the

field is resolved into components lying in the three planes of

an orthogonal set of axes, the X-axis of which is tangent at

every point to the flight path. Throughout this report these

three components are designated as follows: The component

alined with the X-axis is referred to as the horizontal gust

u,; the component alined with the Y-axis is referred to as

the side gust v,; and the component alined with the Z-axis

is referred to as the vertical gust w,.

As the wing moves through the local velocity field, the

random variations in the horizontal and vertical gust com-

ponents are defined both in the flight-path direction and in

the spanwise direction at every position along the flight path.

Random variations of these gust components across the

chord are taken into account by indicial-response-functions

and, hence, need not be considered separately.

The side gust component of the gust velocity field is

treated in only a limited manner. Neither the chordwise

nor the spanwise variations of v, are considered along the

flight path; rather, v, is assumed to act on the wing as a

point velocity with a variation only along the flight-path

direction. Contemporary aircraft exhibit such wide varia-

tions in distribution of dihedral across the span that it is

doubtful that a generalized analysis could be utilized. The

point or centroid analysis should be fairly accurate when the

dihedral distribution is predominant over only a small section

of the span near the fuselage. Such a distribution is ex-

hibited by an unswept wing with zero geometric dihedral

mounted very high or low on a fuselage. For a wing with

zero aerodynamic dihedral, this component could be neglected

completely.

Definition of gust correlation functions.--In order to define

random variations of the gust velocities both along the

flight path and across the span of the wing as it moves

through the turbulence, it is necessary to define the correla-

tion between any two velocities in the gust field through

which the wing passes. The space correlation function of a

velocity u is defined in terms of the distance r as1 5 x r

()=lim 1 u(r )(r z+r)dr

X- m l-(1)

Von Krmln and Howarth (ref. 8) have shown that, in

homogeneous isotropic turbulence, the correlation between

two velocity vectors a distance r apart can be defined in

terms of two scalar functions f(r) and g(r) and that this

relationship is invariant with respect to rotation and reflec-

tion of the coordinate axes. These one-dimensional corre-

lation functions relate the paired velocity components

obtained by resolving the velocity vector at any two points

a distance r apart into two parts: The pair lying along the

straight-line path between the points are known as the

longitudinal components and the pair normal to the straight-

line path are known as the lateral components. These twopairs of components are pictorially shown in figure 1(b).

Such velocity components may be measured in wind tunnels

downstream of a grid mesh. (See ref. 9.)

In reference 8, it is further shown that these correlation

functions are interrelated by the differential equationr df(r) +f(r)=g(r)

2 dr )=gr)(2)

By defining the variable r in the coordinate system of this

report and using the correlation tensor of reference 8, a two-

dimensional analysis of the turbulence as it affects the wing

may be made in terms of f(r) and g(r). The variable in the

correlation functions of the horizontal and vertical gust com-

ponents in the two-dimensional XY-plane of the wing is

given simply byr= (Ax)'+(Ay)= (Ur)2+ (Ay) 2

(3)

The correlation function of the horizontal gust components,

as derived from the correlation tensor of reference 8, is

defined in terms of x- and y-components of the present

analysis by the formula

t(Ax,A=y) { t f (Ax)2+(

=,(Azy)= V (Az)2'S(Ay)2 f (x)+(y)VA)(Ay)2 (A2(A

(4)

The relationship between the components is shown schemati-

cally in figure 2 (a).

In a like manner, the correlation function of the vertical

gust components affecting the wing, given in terms of the

mean-square value of the vertical gust velocity Vf may be,A

(a)

- Longitudinal components, f (r)

Sr --(b)

Lateral components, g (r)

(a) Wing passing through three-dimensional turbulence.

(b) Components of turbulence as a function of distance r.

FIGURE 1.--Sign convention and stability axes of a wing passing

through a turbulent velocity field. Arrows denote positive direction,

where applicable.953

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Eggleston, John M. & Diederich, Franklin W. Theoretical calculation of the power spectra of the rolling and yawing moments on a wing in random turbulence, report, September 6, 1956; (https://digital.library.unt.edu/ark:/67531/metadc60721/m1/3/: accessed April 21, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.