Lesson 7 · 30 min

Twist, Dihedral and Wing Position

The planform is drawn. Four smaller choices finish the wing: how much to twist it, at what angle to set it on the fuselage, how far to tilt it up from the root, and where to attach it, high, low or in the middle. Each is a few degrees or a few lines on a drawing, and each affects how the aircraft flies near the stall and in a sideslip.

Learning objectives

Explain geometric and aerodynamic twist, and why most wings have washout.
Find the wing incidence that levels the fuselage in cruise.
Explain the dihedral effect and choose dihedral for a wing position and sweep.
Compare high, mid and low wings, and describe what wingtips and winglets do.

Twist

A wing is twisted when its sections are not all set at the same angle. Geometric twist changes the angle of the chord along the span; aerodynamic twist changes the airfoil, so that the zero-lift angle varies. Most wings have washout: the tip is set at a lower angle than the root.

Washout reduces the load near the tips. That moves the start of the stall inboard, away from the ailerons, which keep working, and away from the swept tips that would pitch the nose up (Lesson 5). On a wing with little taper, or a swept one, whose tips carry too much load, it also brings the loading closer to elliptic. A strongly tapered wing is the opposite case: its tips already carry less than the elliptic share, even though their small chords make their lift coefficients high, so washout fixes its stall at a small cost in induced drag. Washout is also right for one lift coefficient only. The twist adds a load of its own, the same at every lift coefficient, so at low lift coefficients the loading departs further from elliptic, and the induced drag is higher than it need be (though small anyway). Typical washout is 0 to 5 degrees, with about 3 degrees common on light aircraft.

Figure 7.1 shows these effects for an unswept wing of aspect ratio 8, computed with Prandtl's lifting-line theory (Lesson 4). Every section is given the same maximum lift coefficient, so the first section to reach it marks where the stall begins.

Root and tip sections, seen along the span (washout drawn twice its size)

Figure 7.1 Washout on an unswept wing of aspect ratio 8, from Prandtl's lifting-line theory, with linear twist from root to tip. Top: the tip section (blue) is set nose-down from the root section (gray) by the washout. Left: the span loading with washout (blue), without twist (gray, dashed) and the elliptic ideal (green, dashed). Right: the local lift coefficient at the chosen wing \(C_L\), against an airfoil \(c_{l,\max}\) of 1.6 (orange, dashed). The highest point stalls first. Add washout to a tapered wing: the peak moves inboard. Then lower \(C_L\): the twisted wing's loading departs further from elliptic, and its span efficiency \(e\) falls. Compare a rectangular wing, where washout also improves \(e\).

Incidence

The wing incidence is the angle between the root chord and the fuselage axis. It is chosen so that the fuselage, which has its least drag when it meets the air head-on, is level in cruise: the wing must then reach the cruise lift coefficient by itself.

Incidence for a level fuselage in cruise

\[ i_w = \alpha_{L0} + \frac{C_{L,\text{cruise}}}{\CLa} \]

with \(\CLa\) in per radian (Lesson 4) and the angle converted to degrees. Typical values are about \(2^\circ\) for light aircraft, \(1^\circ\) for transports and nearly \(0^\circ\) for fighters, which must work over a wide range of lift coefficients.

Example 7.1 — The business jet's incidence

The business jet cruises at \(C_L = 0.30\) with \(\CLa = 5.98\) per radian (Lesson 4), and its airfoil has a zero-lift angle of about \(-2.1^\circ\) (a 2% camber, Lesson 2). Find the wing incidence.

Show solution

\[ \frac{C_L}{\CLa} = \frac{0.30}{5.98} = 0.0502\ \text{rad} = 2.87^\circ \] \[ i_w = -2.1^\circ + 2.87^\circ = 0.8^\circ \]

About one degree, typical of a transport. At the start of a long cruise the aircraft is heavier and flies slightly nose-up; at the end, slightly nose-down. Lesson 3's cruise-climb keeps \(C_L\), and so the attitude, nearly constant.

The incidence is fixed when the wing is built, but the lift coefficient is not. At any other \(C_L\), the wing still needs the angle of attack \(\alpha_{L0} + C_L/\CLa\), so the whole aircraft pitches until the wing gets it, and the fuselage meets the air at an angle (Figure 7.2).

Figure 7.2 The business jet in level flight, seen from the side, with its angles drawn three times their size. The wing root chord (blue) is set at the incidence \(i_w\) to the fuselage axis (gray, dashed). The wing must meet the oncoming flow at \(\alpha_{L0} + C_L/\CLa\), with \(\CLa = 5.98\) per radian and \(\alpha_{L0} = -2.1^\circ\) (Example 7.1), so the fuselage flies at whatever angle is left over, \(\alpha_f\). Raise \(C_L\), as at the heavy start of a cruise: the nose comes up. Then set the incidence that levels the fuselage again.

Dihedral

Dihedral \(\Gamma\) is the upward angle of the wing from the root, seen from the front; a downward angle is anhedral. In a sideslip, the wing toward the slip meets the air at a higher angle of attack and the other at a lower one, so the aircraft rolls away from the slip, back toward level flight. This dihedral effect gives lateral stability. Too little and the aircraft slowly diverges into a spiral; too much and it wallows from side to side in the Dutch roll, and rolls sluggishly.

Geometric dihedral is not the only source. Aft sweep adds a dihedral effect, roughly \(1^\circ\) for every \(10^\circ\) of sweep, and a high wing adds one through the flow around the fuselage, roughly \(1^\circ\) to \(2^\circ\), while a low wing subtracts it. So a high, swept wing may need anhedral, and a low, straight wing several degrees of dihedral:

Dihedral guidelines in degrees (source: Raymer, Table 4.2; check the values against the edition you use)
Wing	Low wing	Mid wing	High wing

Figure 7.3 The front view of a wing on a fuselage, with its dihedral (exaggerated scale on the angle). Choose the wing position and type: the shaded band is the guideline range, and the readout says whether the chosen dihedral falls inside it. High swept wings droop; low straight ones tilt up.

High, mid or low wing

High wing. The fuselage hangs low, close to the ground, so cargo aircraft load through a rear ramp without a high step. Engines and propellers clear the ground easily, and the view downward is clear. The wing box passes over the cabin without cutting it. The landing gear usually goes into pods on the fuselage. Military transports and many turboprops are high-winged.
Low wing. The wing box passes under the cabin floor, the main gear retracts into the wing, and the wing gives a floor to stand on and absorbs energy in a ditching or belly landing. Nearly all airliners are low-winged.
Mid wing. The least interference drag where wing meets fuselage, preferred on fighters and aerobatic aircraft; but the wing box must pass through the middle of the fuselage, where the cabin, the engine or the weapons would be.

The business jet has a low wing, like nearly all business jets: the spar passes under the cabin floor and the gear retracts into the wing. With \(25^\circ\) of sweep the guideline is \(3^\circ\) to \(7^\circ\) of dihedral; about \(3^\circ\) is a reasonable start, to be checked against the lateral stability analysis of Module 11.

Wingtips and winglets

The tip vortex begins where the wing ends, so the shape of the tip affects the induced drag a little. A rounded tip lets the pressure leak around early and acts slightly shorter than its span; a sharp-edged, cut-off tip holds the flow and acts slightly longer. Raked tips, swept back more than the wing, give a little more effective span for their weight.

Winglets, small near-vertical surfaces at the tips, reduce the induced drag by spreading the tip vortex, much as more span would. Airliners use them to cut cruise fuel burn by a few percent where the airport gates limit the span. They add weight and bending at the root, so for a new wing without a span limit, a little more span usually does the same job more cheaply.

Check your understanding

Key takeaways

Washout (tip set lower) moves the stall inboard and keeps the ailerons working; 0 to 5 degrees is typical. It suits one lift coefficient best, and on a strongly tapered wing it costs a little induced drag.
Incidence \(i_w = \alpha_{L0} + C_{L,\text{cruise}}/\CLa\) levels the fuselage in cruise: about \(1^\circ\) to \(2^\circ\).
Dihedral gives the dihedral effect; aft sweep and a high wing add to it, so swept high wings may need anhedral.
High wings suit cargo and ground clearance, low wings airliners, mid wings fighters.
Next, Lesson 8 chooses the tail's arrangement and its first size.