Question 3 of 7: Compressibility drag, critical Mach number, induced drag, high-lift devices and static stability

Question 3: Compressibility drag, critical Mach number, induced drag, high-lift devices and static stability (20 marks — 4 marks each)

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.

(a) Compressibility drag. Compressibility drag is the increment of drag that appears once the flow over some part of the aircraft becomes locally supersonic, and which has no counterpart at low speed. At subsonic Mach numbers the air behaves almost as a constant-density fluid: the drag is the sum of skin friction and the pressure (form) drag associated with the boundary layer, and the drag coefficient at a fixed lift coefficient is essentially independent of Mach number. As the free-stream Mach number rises, the flow accelerating over the curved upper surface of the wing eventually reaches sonic speed at one point. Beyond that condition a pocket of supersonic flow grows on the surface, and because the flow must return to subsonic conditions before leaving the aerofoil, that pocket is closed by a nearly normal shock wave. Three separate mechanisms then add drag. First, the shock itself is an irreversible process: the entropy rise across it appears to the aircraft as wave drag, a pressure force with no viscous origin at all. Second, the steep adverse pressure gradient imposed by the shock on the boundary layer beneath it thickens that layer and, at sufficient strength, separates it — shock-induced separation both increases the form drag directly and can destroy lift over the rear of the aerofoil, forcing a higher angle of attack and hence more induced drag. Third, the separated and thickened wake alters the effective shape of the aerofoil, shifting the centre of pressure rearwards and changing the trim drag. The combined effect is the drag-divergence curve sketched below: the drag coefficient stays flat, begins to creep up beyond the critical Mach number, and then rises very steeply beyond the drag-divergence Mach number. Modern transonic designs push that rise to higher Mach numbers by sweeping the wing (which reduces the velocity component normal to the leading edge), by using supercritical aerofoil sections with flatter upper surfaces that terminate the supersonic pocket in a weaker shock, and by applying the area rule to the fuselage.

(b) Critical Mach number. The critical Mach number $M_{\text{crit}}$ of a body is the free-stream Mach number at which the maximum local Mach number anywhere on the surface first reaches unity. It is therefore a property of the body and of its attitude, not of the air: a thicker aerofoil, or the same aerofoil at a higher lift coefficient, accelerates the flow more strongly over its upper surface and so reaches the sonic condition at a lower free-stream Mach number. Sweeping the wing raises it, because only the component of the free-stream velocity normal to the leading edge does the accelerating. The critical Mach number is not itself the point at which drag rises sharply — a small supersonic pocket with a weak terminating shock costs very little. The practically important speed is the drag-divergence Mach number $M_{dd}$, conventionally defined as the Mach number at which $\mathrm{d}C_{D}/\mathrm{d}M=0.10$ or at which $C_{D}$ has risen a stated amount above its subsonic value; it lies a little above $M_{\text{crit}}$, and a transport is cruised just below it.

(c) Induced drag. Induced drag, also called drag due to lift or vortex drag, is the streamwise force that a finite wing must pay in order to generate lift. A wing of finite span carries a higher pressure below than above, and at the tips the air spills from the lower surface to the upper one. The result is a pair of trailing vortices that leave the wing and persist far downstream. Those vortices induce a downward velocity — the downwash — at the wing itself, so the air meeting each section arrives at a slightly smaller effective angle of attack and, more importantly, the local lift vector, which is by definition perpendicular to the local relative wind, is tilted rearwards through the induced angle. Its component along the flight path is the induced drag. For an untwisted wing with a parabolic polar the coefficient is $$\begin{aligned} C_{D,i} &= \frac{C_{L}^{2}}{\pi\,e\,A\!R}, \\ A\!R &= \frac{b^{2}}{S} \end{aligned}$$ where $e$ is the span (Oswald) efficiency factor and $A\!R$ the aspect ratio. Four influences follow directly. The lift coefficient enters squared, so induced drag grows as weight increases, as load factor increases in a turn or pull-up, and as speed falls — at the stall it is the dominant term. Aspect ratio enters inversely, which is why gliders and long-range transports have slender wings and why winglets, which raise the effective aspect ratio by displacing the tip vortex outboard and upward, pay for themselves in cruise. The span efficiency factor depends on the spanwise lift distribution, being unity for the elliptical loading and typically $0.75$ to $0.90$ for practical planforms with taper, twist and fuselage interference. Finally, altitude enters through density: at a given weight and true airspeed, thinner air demands a higher $C_{L}$ and hence more induced drag, while proximity to the ground has the opposite effect, since the ground plane inhibits the downwash and produces the well-known reduction in drag during the landing flare.

(d) Why high-lift devices are fitted. The wing of a transport aircraft is sized by its cruise condition, where the aeroplane is fast, light on lift coefficient and required to be efficient. The same wing at approach speed would stall far too fast, because the minimum speed $V_{\min}=\sqrt{2W/(\rho SC_{L,\max})}$ depends on the largest lift coefficient the clean wing can reach, and a wing chosen for efficient cruise has a modest $C_{L,\max}$. High-lift devices resolve that conflict by giving the wing a second, temporary identity for the few minutes of take-off and landing. Trailing-edge flaps increase camber and, in the Fowler arrangement, translate rearwards to increase area as well; leading-edge slats and slotted flaps duct high-energy air from the lower surface onto the upper one, re-energising the boundary layer so that it survives a much steeper adverse pressure gradient before separating. The practical benefits are several. Approach and landing speeds fall, and since the energy that must be dissipated in the landing run scales with the square of the touchdown speed, the runway length required falls sharply — the $1.6$ to $2.8$ increase in Question 2 cut the stalling speed by a quarter and the ground run by more than $40\%$. Take-off performance improves for the same reason, allowing a heavier payload from a given field. The approach can be flown at a steeper angle and a lower deck angle, which improves the pilot's view of the runway and reduces the noise footprint, and the additional drag of a fully deployed flap is itself useful, because it lets the engines be kept spooled up on approach while still descending. The costs — weight, mechanical complexity, a large nose-down pitching moment that must be trimmed, and drag that would be unacceptable in cruise — are all accepted precisely because the devices are retracted for the cruise.

(e) Inherent static stability. An aircraft is inherently statically stable if, when a disturbance displaces it from a trimmed equilibrium and the controls are then left alone, the aerodynamic forces and moments generated by the displacement act to return it towards that equilibrium. The word static restricts the statement to the initial tendency: it says the restoring moment has the correct sign, and says nothing about whether the ensuing motion is well damped, which is the separate question of dynamic stability. The word inherent stresses that the restoring tendency comes from the airframe's own shape and mass distribution, without any contribution from the pilot or from a stability-augmentation system. Longitudinally the criterion is that the pitching moment coefficient about the centre of gravity must fall as angle of attack rises, $\mathrm{d}C_{m}/\mathrm{d}\alpha \lt 0$, which is achieved when the centre of gravity lies ahead of the neutral point; the distance between them, expressed as a fraction of the mean aerodynamic chord, is the static margin, and the tailplane is what supplies the moment. Directionally, a sideslip must generate a yawing moment that turns the nose back into the relative wind, which is the function of the fin and rear fuselage. Laterally, a sideslip must generate a rolling moment that raises the down-going wing, which is supplied mainly by wing dihedral, by sweep and by a high wing position. Stability is not free: a large static margin makes an aircraft sluggish in manoeuvre and increases the download the tail must carry and hence the trim drag, so transport aircraft are designed with a modest positive margin, while combat aircraft are deliberately made statically unstable in pitch and flown through a full-authority digital flight-control system that supplies the missing restoring moment artificially.