The aspect ratio of an aerofoil has a most important bearing upon the lift/drag ratio and is defined as span2/aerofoil area. In the case of a wing the aspect ratio is given by
S b A
2
= (5-11)
When an aerofoil is rectangular the area is bc and the aspect ratio is, therefore, b/c. Any planform may be reduced to an equivalent rectangle having the same span and area. The chord of the equivalent rectangular aerofoil is called the mean chord, denoted c , and there is no point here in defining differences between the geometric mean chord, c , which we shall use, and the slightly different aerodynamic mean chord, c , both are the same for most practical purposes. Figure 5.11 shows the areas of the wing included in aspect ratio and aerodynamic calculations, together with the mean chord and approximate location of the centre of gravity, at 0.25c.
Fig. 5.11 (c) A crude but effective method after A. Spence and D. Lean (1962) enables the aerodynamic centre of an uncambered wing of low aspect ratio (A) to be found at low airspeed, using its centre of area (cut out planform from card and balance on a knife-edge to find xca).
The ac = NP lies approximately at
(
xac L) (
=2xca L)
−0.75 (5-11d)(Note: the author has found that the method works for a wide variety of low aspect ratio thin wing and wing-plus-tail model combinations, all of which fly when balanced at or close to xac.)
Geometrical construction of mean chord c
A simplified planform of one half of the wing is drawn as in Fig. 5.11(a), with an equivalent tip chord, ct,
constructed parallel with the plane of symmetry of the wing, as shown in the inset to (a). The leading and trailing edges of the wing are produced to intersect the plane of symmetry, thus forming chord cc. The chords cc and ct, are bisected and a line drawn joining their bisectors.
The tip chord is extended forward a distance cc, the centre-line chord rearwards a distance ct. The two
ends of the extended chord lines are then joined by a diagonal. The intersection of the diagonal and the line bisecting the centre-line and tip chords gives the distance of the mean chord outboard of the plane of symmetry. A line parallel to the plane of symmetry drawn through the lateral position and joining the leading and trailing edges gives the length of the mean chord, c .
When compound taper is used the same construction can be applied, but this time each separate portion of the wing is treated as a complete entity, as shown in Fig. 5.11(b). Mean chords c and 1 c are 2 determined for the inboard and outboard portions, respectively. Leading and trailing edges are drawn joining c1 and c2 to form a mean wing. The chords c and 1 c are then treated as c2 c and ct for the construction of a mean chord between them. The method can be applied ad infinitum for taper of more complicated compound forms.
Aspect ratio, span loading and lift-dependent drag
We have already noted that the strength of circulation varies inversely with aerofoil span. It follows, therefore, that the lower the aspect ratio of an aerofoil the more intense the circulation required to generate a given lift. The stronger downwash behind a low aspect ratio wing reduces the effective angle of attack compared with a wing of higher aspect ratio, so that the lower aspect ratio wing has to be flown at a larger angle of attack to generate the same lift. The stronger circulation around the low aspect ratio wing has the effect of inclining the
resultant force rearwards, as shown in Fig. 5.12.
Fig. 5.12 The effect of aspect ratio upon wings of equal area generating equal lift (note increased attitude of low aspect ratio wing to flight path).
Fig. 5.12 (c) Ground or surface effect alters the cross-section of the mass of air being worked on by a wing, increasing the effective span and aspect ratio, so reducing lift-dependent (induced) drag.
Fig. 5.12 (d) (Lift/drag) ratios of ram-wings, with sidewalls or wing-tip fences, as a function of cruise altitude above the surface over which they operate. Identities (5-ha and b) are derived from Eqn (5-11).
(After Dr-Ing. S. F. Hoerner (1975) Fluid Dynamic Lift.)
Note: for reasons of operational risk the author considers it likely that: S
1 . 0
h= (5-11c)
becomes a significant factor.
The relative airflow is no longer almost tangential to the flight path (it is only tangential in theory, when an aerofoil is infinitely long and the distant tips have moved the trailing vortices right out of the picture). As both lift and drag are resolved relative to the flight path, the drag component must therefore be increased by a
reduction of aspect ratio.
The drag increment that varies with aspect ratio is called the vortex drag, sometimes it is still referred to as induced drag. The vortex drag is a measure of how much the trailing vortices are intruding to affect the total airflow and pressure field around an aeroplane. The vortex drag is a part of the lift-dependent drag of the whole aeroplane and this depends upon other factors, besides aspect ratio, which influence the whole pressure field surrounding the aircraft.
Once the planform of a wing is fixed, however, in terms of area and aspect ratio, the lift-dependent drag varies inversely with the span loading: defined as the lift carried per unit span, and equal to the weight/wing span (W/b). Examination of Eqn (5-4). shows that
c p y l =
Δ = lift/unit span for the strip of aerofoil considered
For a given wing, in which the chord distribution is already fixed, the span loading is, therefore, dependent only upon the value of p , the average pressure differential between the upper and lower surfaces. From Eqn (5-5) we see that at a given speed and height (q = constant) the pressure differential is a measure of the lift
coefficient of the wing, in other words the angle of attack of the wing to the air. We may argue that, for a given set of ambient conditions and fixed wing geometry, the lower the span loading the lower the lift-dependent drag. For simplicity we may say vortex drag instead of lift-dependent drag, as long as we remember that we are neglecting certain other aspects that alter with, for example, angle of attack at high Mach numbers.
The lower the span loading, the lower the wing loading of a given wing. It follows from the foregoing arguments that, all else being equal, the longer the span of the wing and the lighter the weight of the aeroplane the smaller will be the lift-dependent drag.
The reduced effective angle of attack of very low aspect ratio wings delays the stall considerably. Some delta wings have no measurable stalling angle up to 400 or more inclination to the flight path. The drag is so high that the flight path is usually inclined downwards at a steep angle to the horizontal, with the aircraft descending rapidly. Apart from a rapid rate of descent and possible loss of stability and control, such aircraft may have a shallow attitude to the horizon that can be deceptive to a casual observer. The condition is called, picturesquely, the superstall or deep stall, although the wing may be far from a true stall and still be generating appreciable lift. Super stalling is a characteristic of the geometrically ‘slender’ aircraft (see Table 12-3).
Taper
Rectangular chord wings are heavy and uneconomical, the ease of manufacture no longer offsetting the structural weight penalties when aeroplanes are larger than a certain size. Taper is therefore employed to shift the spanwise loading inboard, which reduces the bending moment at the root. Furthermore, taper enables a deeper root to be built, so that a lighter structure can be used in that region to resist the stresses set up by bending and torsion.
Taper has an aerodynamic disadvantage, however. Each slice, or section of an aerofoil may be thought of as generating a circulation that is modified by the adjacent sections. If the span of an aerofoil is sliced into sections of equal width, those inboard, having broader chords and greater thicknesses than those outboard, generate more powerful circulations. The tip vortices do not originate at the tip — vortices are shed across the whole of the trailing edge — but roll into a vortex-skein behind the tip, rather like the strands of a rope. The strong inboard vortices cause a powerful upwash outboard that cancels the downwash inboard of the weaker outboard vortices shed from the trailing edge. Their effect is, therefore, the opposite of that illustrated in Fig. 5.12(b), in which the effective angle of attack is decreased by the downwash. With a tapered aerofoil the effective angle of attack outboard is increased by the upwash effect outboard of the stronger vortices, so that the effective angle of attack near the tips is increased. The tips work at higher lift coefficients and tend, therefore, to stall first. Tip stalling is undesirable, for it leads to asymmetric wing dropping and the danger of a spin. We often find that wings are twisted nose down towards the tips, i.e. they are washed-out, by having a smaller angle set at the tip than at the root. In this way tip stalling may be averted.
The ideal planform for minimum vortex drag is an ellipse, because the downwash is then constant across the span. The spanwise lift distribution is also elliptical. When an aerofoil is joined to a non-lifting body there is a loss of lift at the junction, and a trough occurs in the spanwise lift distribution. Fairings and fillets are therefore fitted to smooth out the troughs, for decreased lift means lost circulation — vortices shed in the wake without doing useful work first — increased drag and reduced performance.
Fairings and fillets are never too large, however, because they increase wetted area and skin friction drag. Where a junction is right-angled one finds either small fillets or none at all. Fillets are most commonly
employed for very high or very low wing—fuselage combinations, for then the angle between curved wing and fuselage is acute and generates the most interference.