Question 2 of 7: High-Speed and Low-Speed Limits, Critical Mach Number and Area Ruling

Question 2: High-Speed and Low-Speed Limits, Critical Mach Number and Area Ruling (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.

(a) Highest speed at 8000 m for a limiting Mach number of 0.89

Given. The aircraft is Mach-limited at $M_{\max} = 0.89$ and is flying at $h = 8000\ \text{m}$ in the standard atmosphere.

Find. The corresponding true airspeed.

Approach. A Mach limit is a limit on the ratio of true airspeed to the local speed of sound, and the local speed of sound depends only on temperature. So the whole calculation is the ISA temperature at 8000 m.

1. ISA temperature at 8000 m. The point lies in the troposphere ($h < 11\,000\ \text{m}$), so $$T = 288.15 - 0.0065(8000) = 236.15\ \text{K}\quad (-37.0\ ^\circ\text{C}).$$
2. Local speed of sound. $$a = \sqrt{\gamma R T} = \sqrt{1.4 \times 287.05 \times 236.15} = 308.1\ \text{m}/\text{s}.$$
3. Maximum true airspeed. $$V_{\max} = M_{\max}\,a = 0.89 \times 308.1 = \boxed{274.2\ \text{m}/\text{s} = 987\ \text{km}/\text{h}}$$

It is worth noting how much this differs from the sea-level answer: at $T_0 = 288.15\ \text{K}$ the speed of sound is 340.3 m/s and the same Mach number would correspond to 302.8 m/s (1090 km/h). A Mach-limited aeroplane therefore flies slower in true airspeed as it climbs through the troposphere, which is one reason cruise altitude selection is a compromise rather than "as high as possible".

(b) Minimum flight speed with and without high-lift devices

Given. $m = 40\,000\ \text{kg}$, $S = 120\ \text{m}^2$, sea-level ISA density $\rho_0 = 1.225\ \text{kg}/\text{m}^3$, $C_{L\max} = 1.3$ clean and $C_{L\max} = 2.3$ with all high-lift devices deployed.

Find. The stalling (minimum) speed in each configuration at sea level.

Approach. The minimum speed in level flight is the speed at which the wing must work at its maximum lift coefficient to carry the weight; invert the lift equation with $C_L = C_{L\max}$.

1. Weight. $W = mg = 40\,000(9.81) = 392\,400\ \text{N}$, so the wing loading is $W/S = 3270\ \text{N}/\text{m}^2$.
2. Minimum-speed relation. Setting $L = W$ with $C_L = C_{L\max}$, $$V_{\min} = \sqrt{\frac{2W}{\rho_0 S\,C_{L\max}}}.$$
3. Clean configuration ($C_{L\max} = 1.3$). $$V_{\min} = \sqrt{\frac{2(392\,400)}{1.225 \times 120 \times 1.3}} = \sqrt{4107} = \boxed{64.1\ \text{m}/\text{s} = 231\ \text{km}/\text{h}}$$
4. All high-lift devices deployed ($C_{L\max} = 2.3$). $$V_{\min} = \sqrt{\frac{2(392\,400)}{1.225 \times 120 \times 2.3}} = \sqrt{2321} = \boxed{48.2\ \text{m}/\text{s} = 173\ \text{km}/\text{h}}$$

The high-lift system reduces the minimum speed by 25 %, exactly the factor $\sqrt{1.3/2.3} = 0.752$ predicted by the inverse square-root dependence. Because landing distance scales with the square of the approach speed, that 25 % speed reduction is worth roughly a 44 % reduction in the kinetic energy that the brakes and reversers must absorb — which is the entire commercial justification for the mechanical complexity of slats and multi-element flaps.

(c) Critical Mach number and the transonic drag rise

The critical Mach number $M_{crit}$ is the free-stream Mach number at which the flow first reaches sonic velocity somewhere on the aircraft — in practice at the point of minimum pressure on the upper surface of the wing, where the local flow has been accelerated well above the free-stream speed. Below $M_{crit}$ every point on the aeroplane is subsonic; at $M_{crit}$ one point is exactly sonic; above it a pocket of supersonic flow grows on the surface.

Nothing dramatic happens to the drag at $M_{crit}$ itself. What matters is what happens a little above it. As the supersonic pocket grows it must be terminated by a shock wave in order to return the flow to subsonic conditions, and that shock does two damaging things: it dissipates energy directly (wave drag), and the steep adverse pressure gradient it imposes on the boundary layer provokes shock-induced separation, which sharply increases the pressure drag as well. The Mach number at which the resulting rise becomes commercially unacceptable — conventionally where $dC_D/dM = 0.10$, or where $C_D$ has risen by 0.0020 above its low-speed value — is called the drag-divergence Mach number $M_{dd}$, and it lies typically 0.05 to 0.10 above $M_{crit}$.

The full transonic behaviour is therefore as sketched in Figure 2.1. The drag coefficient is nearly constant up to $M_{crit}$; between $M_{crit}$ and $M_{dd}$ it creeps upward; from $M_{dd}$ it rises very steeply, by a factor of three to five for an unswept aerofoil, reaching a peak just above $M = 1$ where the bow shock has formed but is still detached; thereafter it decreases slowly with increasing supersonic Mach number as the shock system becomes attached and more oblique. The steep part of this curve is the "sound barrier" of the 1940s, and delaying it is the reason for wing sweep, thin sections, and supercritical aerofoil design.

(d) Compressibility drag and the area rule

Compressibility drag is the increment in drag that arises purely because the air can no longer be treated as being of constant density — that is, everything in the curve of Figure 2.1 above the flat low-speed value. It has two physically distinct parts. Wave drag is the momentum loss associated with the shock waves themselves; the entropy rise across a shock corresponds directly to a drag force, and it exists even in an inviscid flow. Shock-induced separation drag is a viscous effect: the pressure jump across the shock is imposed on a boundary layer that frequently cannot negotiate it, so the layer separates behind the shock and the pressure recovery on the aft surface is lost. On a transport wing the two together can double the total aircraft drag over a Mach number increment of 0.05, and the associated separation also produces buffet, control heaviness and the pitch-down known historically as "Mach tuck".

The area rule, due to Whitcomb (1952), is the design principle that near $M = 1$ the wave drag of a complete aircraft depends primarily on the distribution of its total cross-sectional area along the longitudinal axis, and only secondarily on how that area is divided among wing, fuselage and nacelles. It follows that the cross-sectional area distribution should be made as smooth as possible — no local bumps and no abrupt changes of slope — and ideally should approximate a Sears–Haack body, which is the shape of minimum wave drag for a given length and volume. In practice this means the fuselage must be waisted where the wing and tailplane are attached, so that the area added by the wing is offset by area removed from the body: the classic "coke-bottle" fuselage of the F-102A, whose transonic drag was reduced enough by the modification to let it pass through $M = 1$ in level flight when the original design could not. The same idea, applied more subtly, shapes the wing-body fairings and nacelle placement of every modern transport aircraft.