Question 3 of 7: Transonic Aerodynamics and High-Speed Wing Design

Question 3: Transonic Aerodynamics and High-Speed Wing Design (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.

This question is entirely descriptive; each part is answered in turn.

(a) Drag coefficient through the transonic range, and the critical Mach number. Below about $M_\infty = 0.5$ the drag coefficient of a subsonic aircraft is essentially independent of Mach number: the flow is everywhere subsonic, compressibility merely thickens the pressure distribution slightly, and $C_D$ is fixed by friction, form and induced contributions. As the free-stream Mach number rises further, the flow accelerating over the upper surface of the wing reaches progressively higher local Mach numbers. The critical Mach number $M_{cr}$ is defined as the free-stream Mach number at which the maximum local Mach number on the surface first reaches exactly unity. At $M_{cr}$ itself nothing dramatic happens to the drag — a single sonic point produces no loss. As $M_\infty$ is increased slightly beyond $M_{cr}$, however, a region of supersonic flow grows on the surface and is closed by a shock wave. The entropy rise across that shock, together with the strong adverse pressure gradient it imposes on the boundary layer (which thickens and often separates behind it), produces wave drag. The drag coefficient then rises steeply, typically by a factor of three to ten, beginning at the drag-divergence Mach number $M_{DD}$, which lies a little above $M_{cr}$ and is defined in practice by a specified slope $\mathrm{d}C_D/\mathrm{d}M$ or a fixed drag increment. The curve peaks slightly above $M_\infty = 1$, where shocks stand on both surfaces and at the trailing edge, and then falls again as the shock system moves to the trailing edge and the flow becomes fully supersonic. This peak is the physical content of the "sound barrier": an aircraft with insufficient installed thrust simply cannot push through it, whereas one with enough thrust finds conditions easier again beyond $M = 1.2$ or so. Design for high subsonic cruise therefore aims to raise $M_{cr}$ and $M_{DD}$ as far as possible, which is exactly what parts (b) and (c) address.

(b) Why swept-back wings are used. Sweep raises the critical Mach number. For an infinite wing swept back at angle $\Lambda$, the aerodynamics of the section is governed by the component of the free-stream velocity normal to the leading edge, $V_n = V_\infty \cos\Lambda$; the spanwise component simply slides along the wing and does no pressure-producing work. The section therefore behaves as though it were flying at the lower Mach number $M_\infty \cos\Lambda$, and to a first approximation $M_{cr,\text{swept}} \approx M_{cr,\text{unswept}}/\cos\Lambda$. A 35° sweep buys roughly a 0.08–0.10 increase in usable cruise Mach number, deferring the drag rise and allowing a thicker, lighter, higher-volume wing at a given cruise speed. Sweep is not free: it reduces the lift-curve slope and $C_{L,max}$, encourages spanwise boundary-layer drift towards the tips and hence tip stall (with its associated pitch-up), increases structural weight through the bending-torsion coupling, and complicates the high-lift system. Forward sweep achieves the same Mach relief with better stall behaviour but is prone to aeroelastic divergence.

(c) The area rule. The area rule, due to Whitcomb, states that near $M = 1$ the wave drag of a complete aircraft depends primarily on the longitudinal distribution of its total cross-sectional area, taken normal to the flight direction, and only secondarily on how that area is apportioned between fuselage, wing, nacelles and tail. The transonic wave drag of a slender body is minimised when this cross-sectional area distribution is smooth and closely approximates the Sears-Haack body, which grows and decays gradually with no discontinuity in slope or curvature. Simply adding a wing to a smooth fuselage produces a bump in the area distribution where the wing root is, and that bump costs wave drag. The remedy is to waist the fuselage where the wing joins it — the so-called Coke-bottle shape — so that the total area still varies smoothly; equivalently, volume may be added fore and aft (bulged fairings, "Whitcomb bumps" on the wing trailing edge) to fill the distribution out. Applying the rule to the YF-102 turned an aircraft that could not exceed $M = 1$ into one that comfortably did, on the same engine. The supersonic extension of the idea, the supersonic area rule, uses areas cut by oblique Mach planes rather than normal planes and depends on the design Mach number.

(d) The laminar-flow aerofoil. A laminar-flow aerofoil is a section shaped so that the point of transition from a laminar to a turbulent boundary layer is deliberately moved far aft, thereby reducing skin-friction drag over a substantial fraction of the chord. Transition is triggered by adverse pressure gradients, so these sections are designed with the maximum thickness well aft — typically 40 to 50 per cent of chord rather than the 25 to 30 per cent of a classical section — which keeps the flow accelerating (favourable gradient) over the forward half or more of the surface. The NACA 6-series sections are the canonical family; their drag polars exhibit a characteristic "drag bucket", a range of lift coefficients (set by the design $C_L$ and the extent of the favourable gradient) within which the profile drag is markedly lower than that of a conventional section, with the low-drag benefit lost abruptly outside the bucket. The practical caveats are important: the benefit requires a very smooth, wave-free, clean surface, because roughness, manufacturing waviness, insect debris, rain or ice will trip the boundary layer and forfeit the gain; the sections tend to have lower $C_{L,max}$ and sharper stall; and at high Reynolds number the achievable run of laminar flow shortens. The design remains central to sailplanes and general-aviation aircraft, and in modern form (natural laminar flow and hybrid laminar flow control) to transport wings and nacelles.

(e) Why slotted flaps are used. A plain or split flap increases camber and therefore increases the lift coefficient at a given angle of attack, but it also steepens the adverse pressure gradient on the flap upper surface, so the flow separates earlier and the gain in $C_{L,max}$ is limited (typically to about 0.9 for a plain flap). A slotted flap leaves a carefully shaped convergent gap between the trailing edge of the main element and the leading edge of the flap. High-energy air from the lower surface, which is at high pressure, is ducted through this slot and accelerated over the flap upper surface. Three effects follow: the fresh high-momentum flow re-energises the boundary layer on the flap and delays its separation; the circulation of the flap element modifies the pressure distribution on the main element, allowing it to carry more lift without stalling (the "dumping" and "circulation" effects); and each element operates at a lower peak suction than a single element producing the same total lift. The result is a much larger usable lift increment — of order 1.3 for a single-slotted flap and 1.6 or more for double- and triple-slotted arrangements, especially when combined with Fowler motion that also increases wing area. The penalty is mechanical complexity, weight, cost and (with the slot open in cruise, if sealing is poor) parasite drag, so slots are opened only for take-off and landing. The payoff is the one calculated in Question 2(b): a lower approach and lift-off speed, hence shorter field lengths and lower landing energy.