Question 3 of 7: Airfoils, High-Lift Devices and Compressibility Effects

Question 3: Airfoils, High-Lift Devices and Compressibility Effects (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Part (a) — the laminar-flow airfoil. A laminar-flow airfoil is a section shaped deliberately so that the boundary layer stays laminar over a much larger fraction of the chord than on a conventional section, because laminar skin friction is several times lower than turbulent. Transition from laminar to turbulent flow is triggered chiefly by an adverse pressure gradient, so the design lever is the chordwise pressure distribution: the point of maximum thickness is moved well aft, typically to 40–50 per cent of chord instead of the 25–30 per cent of a classical section, which keeps the flow accelerating — a favourable gradient — over most of the forward surface. The NACA 6-series sections are the archetype. The reward is a “drag bucket”: a narrow band of lift coefficients over which the profile drag is markedly lower than a conventional section of the same thickness. The penalties are real and explain why the benefit is rarely fully realised. The bucket is narrow, so off-design lift coefficients lose the advantage entirely; the boundary layer is very sensitive to surface waviness, rivet heads, insect debris, rain and ice, any of which trips it turbulent; and because a long favourable gradient must be followed by a steep recovery near the trailing edge, stall can be abrupt. In practice, sailplanes and some general-aviation and business aircraft with smooth composite or bonded surfaces obtain much of the benefit; large transports historically obtained little, though modern natural- and hybrid-laminar-flow wings and nacelles are changing that.

Part (b) — what happens when an airfoil stalls. As the angle of attack is increased, the suction peak near the leading edge grows and the pressure recovery that the boundary layer must negotiate on the rear upper surface becomes steeper. The boundary layer is slowed by both friction and this adverse pressure gradient; at some angle it no longer has the momentum to reach the trailing edge against the gradient and it separates, leaving a region of recirculating, largely stagnant air over the upper surface. Once that happens the upper-surface suction collapses, so the lift coefficient stops rising, reaches its maximum $C_{L,max}$ and then falls with further increase in angle of attack, while the pressure drag increases sharply. The wake is unsteady, so the aircraft buffets, and because the separated flow washes over the tail and ailerons, control effectiveness degrades and pitch and roll behaviour can become uncommanded. Three stall types are distinguished by where separation begins: trailing-edge stall, typical of thick sections, in which the separation point creeps forward gradually and gives a rounded, benign $C_L$ peak; leading-edge stall, typical of moderately thin sections, in which a short laminar separation bubble near the nose bursts and the whole upper surface separates almost at once, giving a sharp and abrupt loss of lift; and thin-airfoil stall, in which a bubble forms at the nose at low angle and lengthens progressively down the chord. The critical point for the pilot is that stall is a function of angle of attack alone, not of speed: an aircraft can be stalled at any airspeed and any attitude, which is why the angle-of-attack limit rather than the placarded stalling speed is the true boundary.

Part (c) — high-lift devices. A high-lift device is any deployable feature of the wing whose purpose is to raise $C_{L,max}$ temporarily, so that the aircraft can fly safely at a much lower speed for take-off and landing than the cruise wing would otherwise allow. Since $V_{min} = \sqrt{2W/(\rho S C_{L,max})}$, raising $C_{L,max}$ from about 1.4 clean to 2.5–3.2 in the landing configuration cuts the stalling speed by a quarter to a third, and with it the landing distance, which scales with the square of that speed. The family divides into trailing-edge devices — plain, split, single-slotted, double- and triple-slotted, and Fowler flaps — which work by adding camber and, in the Fowler case, wing area; and leading-edge devices — fixed slots, movable slats, Krueger flaps and drooped leading edges — which work by delaying leading-edge separation to a higher angle of attack. They are retracted in cruise because the drag and structural weight they carry would be unacceptable there.

Part (d) — why slotted flaps are used. A plain or split flap raises camber, but the sharply increased adverse gradient over the deflected surface separates the boundary layer at quite modest deflections, so $C_{L,max}$ saturates near about 30 degrees of flap and the extra drag is largely wasted. A slotted flap leaves a carefully shaped, converging gap between the flap leading edge and the wing trailing edge. Air from the high-pressure lower surface is ducted through this gap and emerges as a fast, energetic sheet tangential to the flap upper surface, where it re-energises the boundary layer and lets it negotiate the steep recovery without separating. The consequence is that far larger deflections — 40 degrees and beyond, and more still with double- and triple-slotted arrangements — remain effective, so the achievable $C_{L,max}$ is much higher for a given drag penalty. Most slotted flaps also translate rearwards as they deflect (Fowler motion), adding wing area as well as camber, which raises the lift at a given angle of attack still further.

Part (e) — subsonic, transonic and supersonic flow. In subsonic flow the Mach number is below unity everywhere in the flow field, including at the point of maximum local velocity on the body; disturbances propagate ahead of the body, so the flow adjusts smoothly and no shock waves exist. In transonic flow the field contains both subsonic and supersonic regions simultaneously — typically a pocket of supersonic flow over the upper surface of a wing terminated by a shock wave — and this occurs over a free-stream range of roughly $0.8 < M_\infty < 1.2$ for a typical aircraft, beginning at the critical Mach number. In supersonic flow the Mach number exceeds unity everywhere of consequence in the field; disturbances cannot propagate upstream, the body announces itself only through attached or detached shock waves, and the governing equations change character from elliptic to hyperbolic.

Part (f) — why the drag coefficient rises above the critical Mach number. The critical Mach number $M_{crit}$ is the free-stream Mach number at which the flow first reaches $M = 1$ somewhere on the body, normally at the point of maximum thickness on the upper surface. Just above $M_{crit}$ the drag hardly changes, because the supersonic pocket is small and terminated by a very weak shock. As the free-stream Mach number rises further the pocket grows and the terminating shock strengthens, and two mechanisms then act together. First, a shock wave is an irreversible, entropy-producing process; the total-pressure loss across it appears in the momentum balance as a drag force, the wave drag. Second, and usually larger at transonic speeds, the shock imposes an abrupt and severe adverse pressure gradient on the boundary layer beneath it, which commonly separates the flow from that point to the trailing edge. The separated wake destroys the rear-surface pressure recovery, so form drag climbs sharply, and the same separation produces buffet and a rearward shift of the centre of pressure. The Mach number at which the drag coefficient begins its steep climb, conventionally where $dC_D/dM$ reaches a defined value or $C_D$ has risen by 0.002 above its subsonic level, is the drag-divergence Mach number $M_{DD}$, and it lies a little above $M_{crit}$. Beyond $M = 1$ the coefficient peaks and then falls again as the bow shock attaches and the flow becomes fully supersonic.

Part (g) — why swept-back wings are used. The dominant reason is to raise the critical and drag-divergence Mach numbers, so that a jet can cruise faster before the transonic drag rise sets in. To a first approximation, the pressure distribution over a wing of infinite span swept at angle $\Lambda$ is governed only by the velocity component normal to the leading edge, $V_\infty \cos\Lambda$; the spanwise component simply sweeps along the section and does no compressive work. The effective Mach number that the section experiences is therefore $M_\infty \cos\Lambda$, so a wing swept 35 degrees does not feel its critical condition until the free-stream Mach number is about $1/\cos 35^\circ = 1.22$ times what an unswept wing of the same section could tolerate. Sweep also thickens the effective section in the streamwise direction for a given structural depth, and gives the designer a lever on the position of the aerodynamic centre relative to the centre of gravity. The price is paid at low speed: sweep reduces $C_{L,max}$, promotes a spanwise boundary-layer drift towards the tips that provokes tip stall and pitch-up, and produces a heavier, more flexible structure — which is why swept wings carry elaborate high-lift systems, vortilons, fences or leading-edge extensions.

Part (h) — the area rule and its influence on commercial jet design. The area rule, established by Whitcomb in 1952, states that at transonic speeds the wave drag of a complete aircraft is governed, to first order, not by the shape of its individual components but by the distribution of its total cross-sectional area along the longitudinal axis. Two configurations with the same area distribution have nearly the same wave drag, and the drag is minimised when that distribution is smooth and free of bumps or kinks — ideally approaching the Sears–Haack body for the given length and volume. The practical consequence is that wherever the wing, tailplane, nacelles or canopy add cross- sectional area, an equal area must be removed elsewhere, most conveniently from the fuselage, giving the waisted or “coke-bottle” fuselage of early transonic fighters such as the F-102, whose transonic drag fell dramatically when it was rebuilt to the rule.

On modern commercial jets the influence is real but subtle, because a constant-section cabin is worth far more commercially than a waisted one. Designers therefore satisfy the rule by other means: carefully shaped wing-body fairings that fill in the area distribution ahead of and behind the wing carry-through; nacelles staggered forward and below the wing so that their area peaks where the wing area is still growing rather than at the same station; wing-root fillets and rear-fuselage upsweep contoured to smooth the distribution; and, on some aircraft, local antishock bodies. The result is that the fuselage looks almost cylindrical, yet the cross-sectional area plot is smooth, which is what actually matters. This is why the characteristic pinched fuselage is rare on airliners while the underlying principle governs their fairing and nacelle geometry throughout.