WINGS OF FINITE SPAN IN INCOMPRESSIBLE FLOW

3- 1 INTRODUCTION

For an airfoil of infinite span, the flow field is equal in all sections normal to the airfoil lateral axis. This two-dimensional flow has been treated in detail by profile theory in Chap. 2. For an airfoil of finite span as in Fig. 3-1, however, the flow is three-dimensional. As in Chap. 2, incompressible flow is presupposed.

2- 1-1 Wing Geometry

The wing of an aircraft can be described as a flat body of which one dimension (thickness) is very small in relation to the other dimensions (span and chord). In general, the wing has a plane of symmetry that coincides with the plane of symmetry of the aircraft. The, geometric form of the wing is essentially determined by the wing planform (taper and sweepback), the wing profile (thickness and camber), the twist, and the inclination or dihedral of the left and right halves of the wing with respect to each other (V form) (see Fig. 3-1). In what follows, the geometric parameters that are significant in connection with the aerodynamic characteristics of a lifting wing will be discussed.

For the description of wing geometry, a coordinate system in accordance with Fig. 3-1 that is fixed in the wing will be established with axes as follows:

x axis, wing longitudinal axis, positive to the rear

у axis, wing lateral axis, positive to the right when viewed in flight direction, and perpendicular to the plane of symmetry of the wing z axis, wing vertical axis, positive in the upward direction, perpendicular to the xy plane

У

Figure 3-1 Illustration of wing geometry. (a) Planform, xy plane, (b) Dihedral (V form), yz plane, (c) Profile, twist, xz plane.

It is expedient to select the position of the origin of the coordinates as suitable for each case. Frequently it is advisable to place the origin at the intersection of the leading edge with the inner or root section of the wing (Fig. 3-1), or at the geometric neutral point [Eq. (3-7)]. The wing planform is given in the xy plane; the twist, as well as the profile, in the xz plane; and the dihedral in the yz plane.

The largest dimension in the direction of the lateral axis (y axis) is called the span, which will be designated by b = 7s, where s represents the half-span. Frequently the coordinates will be made dimensionless by reference to the half-span s, and abbreviated notations

f = T	(3-U)
	(3-lb)
f = T	(3-lc)

are here introduced.

The dimension in the direction of the longitudinal axis (x axis) will be designated as the wing chord c(y), dependent on the lateral coordinate y. The wing chord of the root or inner section of the wing (y = 0) will be designated by cr, and the corresponding dimension for the tip or outer section by ct. In Fig. 3-2, the geometric dimensions are illustrated for a trapezoidal, a triangular, and an elliptic planform.

For a wing of trapezoidal planform (Fig. 3-2a), an important geometric parameter is the wing taper, which is given by the ratio of the tip chord to the root chord:
(3-2)

A special case of the trapezoidal wing is the triangular wing with a straight trailing edge, also designated as a delta wing (Fig. 3-2b).

The wing area A (reference area) is understood to be the projection of the wing on the xy plane. For a variable wing chord, the area is obtained by integration of the wing chord distribution ciy) over the span b = 2s; that is,

(3-3)

Figure 3-2 Geometric designations of wings of various planforms. (a) Swept-back wing. (b) Delta wing, (c) Elliptic wing.

From the wing span b and the wing area A, there is obtained, as a measure for the wing fineness (slenderness) in span direction, the aspect ratio

II X|<s	(34a)
II if I»	(34b)
As mean chord and reference wing chord, especially for the introduction of dimensionless aerodynamic coefficients, the quantities
A cm=J	(3-5a)
s CjU = i fc2(y)dy	(3-5 b)
4

are used, where the ratio > 1. For the trapezoidal planform, it may be easily

demonstrated that the reference chord cM is equal to the local chord at the position of the center of gravity of the half-wing; that is, = c(yc) (Fig. 3-2a and b). The sweepback of a wing is understood to be the displacement of individual wing cross sections in the longitudinal direction (x direction). Representing the position of a wing planform reference line by x(y), the local sweepback angle of this line is

tan cp{y) = ^^~ (3-6)

If x(y) represents the connecting line of points of equal percentage rearward position, measured from the leading edge at the у section under consideration, then this fact is designated by giving the percentage number as an index of the value x. Accordingly, the position of the quarter-chord line is designated by x2s(y)- For the sake of simplicity, the index will be omitted in the case of the sweepback angle of the quarter-chord-point line. For aerodynamic considerations, furthermore, the geometric neutral point plays a special role. Its coordinates are given by

S

= j f c(y) Z2 5 {y)dy 5 = 0 (3 -7)

– S

For a symmetric wing planform, the geometric neutral point may be demonstrated to be the center of gravity of the entire wing area, whose quarter-chord-point line is overlaid by a weight distribution that is proportional to the local wing chord. The rearward distance of the geometric neutral point of a wing with a swept straight quarter-chord-point line is equal to the rearward distance of the quarter-chord point of the wing section at the planform center of gravity of the half-wing. Since, for a trapezoidal wing, the wing chord at the center of gravity of the half-wing is equal to the reference chord cM, the geometric neutral point for this wing lies at the cM/4 point (see Fig. 3-2a and b).

Of particular importance is the delta wing, a triangular wing with a straight

trailing edge (Fig. 3-2b). For the geometric magnitudes of this wing, especially simple formulas are obtained:

b _ 2 b _ 3 = 4 x _ Cy

cm cr tan cm 3 ■XiV25 2

For a wing of elliptic planform as in Fig. 3-2c, the geometric quantities become

A further geometric magnitude related to the wing planform is the flap (control-surface) chord сДу). The flap-chord ratio is defined as the ratio of flap chord (control-surface chord) to wing chord:

c(y)

For the description of the whole wing, data on the relative positions of the profile sections are required at various stations in span direction. They are required in addition to the knowledge of wing planforms and wing profiles. The relative displacement in longitudinal direction is specified by the sweepback, the displace­ment in the direction of the vertical axis by the dihedral, and the rotation of the profiles against each other by the twist.

In what follows, the geometric twist e(y) is defined as the angle of the profile chord with the wing-fixed xy plane (Fig. 3-3).[11] For aerodynamic reasons, in most cases the twist angle is larger on the outside than on the inside. The dihedral determines the inclination of the left and the right wing-halves with respect to the

Figure 3-3 Illustration of geometric twist.

xy plane. Let z^sx, у) be the coordinates of the wing skeleton surface. Then the local V form at station x, у is given by

tan у {x, у) = – (3-11)

The partial differentiation is done by holding x constant. If the wing is twisted, it must be specified in addition at which station xp(y) the angle v is to be measured. According to Multhopp [61], the aerodynamically effective dihedral has to be taken approximately at the three-quarter point xp =x75.