Critical Mach Number and Mach Tuck

The critical Mach number (Mcrit) marks the start of transonic flight. It is the free-stream Mach number at which the air accelerating over some part of the aircraft, usually the wing upper surface near its thickest point, first reaches Mach 1.0. A jet transport usually cruises a little above it: in normal cruise, weak shocks already stand on its wings. What matters to the pilot is what happens as the Mach number rises further: the drag rise, shock-induced separation and buffet, and a nose-down pitching tendency, strongest on swept-wing jets, called Mach tuck, which feeds on itself if left alone.

The principles are introduced in high-speed flight; this article adds worked figures, the certification and training rules, type examples of Mach trim and the lesson of Air France 447.

Local Mach number and critical Mach

The local Mach number is the flow speed at one point on the aircraft divided by the local speed of sound. Over a lifting wing the upper-surface flow is faster than the free stream; the FAA handbook notes that at extreme angles of attack, in some aircraft, it can reach double the aircraft's speed. The wing therefore goes sonic before the airplane does, and Mcrit is the highest free-stream Mach number with no supersonic flow anywhere.

The FAA handbook illustrates it with one airfoil. At M 0.50 the highest local speed is still subsonic. At M 0.72 one point on the upper surface reaches sonic speed: that airfoil's Mcrit. At M 0.77 a pocket of supersonic flow has formed and ends in a normal shock, with possible separation behind it.

Mcrit is not fixed; it depends on how much the flow accelerates over the wing:

Mach number itself is true airspeed divided by the local speed of sound, which depends on temperature alone: about 38.95 × √T kt, with T in kelvin. At the ISA tropopause (−56.5 °C, 216.65 K) the speed of sound is about 573 kt, so M 0.78 is about 447 kt TAS. On a day 10 °C warmer it is about 586 kt, and the same M 0.78 gives about 457 kt. Temperature changes the true airspeed at a given Mach number, not the Mach number at which shocks appear.

Effects above critical Mach

Just above Mcrit, a small supersonic region forms on the upper surface. Supersonic flow cannot slow down gradually to meet the subsonic flow behind it, so the region ends in a shock wave at its rear boundary. As the Mach number rises the region grows, and the shock strengthens and moves aft. The FAA handbook's sequence shows a normal shock on the upper surface at M 0.82, larger supersonic regions with shocks nearer the trailing edge at M 0.95, and a bow wave ahead of the wing at M 1.05.

Drag divergence

The drag divergence Mach number (MDD) is the Mach number at which drag begins to rise rapidly. It is always above Mcrit, and the two are different events: Mcrit is the first sonic point, MDD the point where shocks and separation have grown enough to dominate the drag. AC 61-107B warns, "Do not confuse drag divergence with Mcr": beyond it, further speed demands substantial increases in thrust. The FAA handbook adds that the drag of the separated flow is much greater than the energy lost in the shock itself.

Because the rise is gradual at first, manufacturers fix MDD by their own criterion, such as Boeing's wave-drag increase of 20 drag counts. High-speed cruise schedules sit above Mcrit, with weak shocks on the wing, but close to and not beyond the drag rise.

Shock-induced separation and buffet

The pressure jump across a shock imposes a steep adverse pressure gradient on the boundary layer. Once the shock is strong enough, the boundary layer separates immediately behind it: shock stall. The separated wake strikes the tail surfaces, and the airframe shakes: high-speed buffet, or Mach buffet. It is a boundary to stay clear of, not one to explore.

Weight, altitude, load factor and a forward center of gravity all raise the angle of attack at a given Mach number, so all of them bring buffet onset to a lower Mach number, while the low-speed buffet boundary rises. AC 61-107B puts figures on the squeeze. A typical turbojet airplane at 51,000 ft and 1.0 g may meet Mach buffet just above its MMO of M 0.82 and low-speed buffet at M 0.60. At only 1.4 g, the load factor of a level turn at about 44° of bank, buffet can begin at its optimum speed of M 0.73, and any change of speed, bank or gust load may remove what protection is left. In the same AC, a typical jet's buffet-free speed range at 1.0 g is 135 kt at FL350 but only 26 kt at FL450. Vortex generators, which energize the boundary layer behind the shock, delay the separation; some straight-wing jets use them to obtain a higher MMO.

Buffet at altitude is not always Mach buffet. On Air France 447 in 2009, cruising at FL350 near M 0.8, the margin between the flight angle of attack and the stall warning threshold was of the order of 1.5°, and each degree was worth about 25 kt. After the measured airspeeds became inconsistent, nose-up inputs made the airplane climb, and it stalled. The BEA noted that the crew's training material associated buffet with both the stall and overspeed, whereas on the A330 buffet occurs only on the approach to the stall, and listed possible confusion with overspeed buffet among the reasons the stall was never identified.

Mach tuck

Mach tuck, also called tuck under, is the nose-down pitching moment that develops above Mcrit. AC 61-107B gives its two causes:

• Flow separation. Shock-induced separation, which normally begins near the wing root, reduces the downwash over the tailplane. The tailplane's download falls, and the nose drops.
• Center of pressure movement. The center of pressure moves aft, upsetting the balance about the center of gravity that held in slower flight, and the nose-down moment grows.

A swept wing with a thicker root section makes both worse. The root has the lowest Mcrit, so it separates first; lift is lost inboard, which on a swept wing is also forward, and the center of pressure moves outboard and aft. This is why sweepback, for all that it raises Mcrit, increases the severity of Mach tuck.

The danger is that Mach tuck makes the airplane unstable with speed. A small increase in Mach number pitches the nose down, the airplane accelerates, and the tuck strengthens. Without compensation the stick force gradient reverses: holding the nose up takes a growing pull as the speed rises, the reverse of normal speed stability. AC 61-107B describes how intentional excursions beyond VMO/MMO, including on corporate jets whose airspeed and Mach warnings had been disabled, can bring flutter, aileron buzz or snatch and Mach tuck together, "leading to catastrophic loss of the airplane".

Overspeed recovery follows from the mechanism: reduce thrust, extend the speedbrakes as the manufacturer allows and raise the nose gently, never abruptly, since a sharp pull at high Mach raises the load factor and can trigger buffet or overstress the structure (see upset prevention and recovery).

Mach trim systems

A Mach trim system senses Mach number and adds nose-up trim as it rises, at high subsonic speeds only. Depending on the type it deflects the elevator up, reduces the incidence of the trimmable tailplane, or moves the center of gravity aft by transferring fuel to a tail tank. ATPL texts note that the correction is made larger than needed simply to cancel the trim change, so that the stick force still increases with speed, and describe the system as independent of the autopilot: armed at all times, active only at high subsonic speed.

View full schematic

AC 61-107B adds that if a jet's Mach compensating device (it calls it a stick puller) must be disabled after a malfunction, the airplane must be flown well below MMO as the flight manual prescribes.

US Part 121 training (Appendix E) includes tuck and Mach buffet training. The FAA's Flight Standardization Board report for the Boeing 737 states that the 737 family does not exhibit any Mach tuck tendency, so that training is not required, and accepts a demonstration of the overspeed protection capabilities instead. The 787 report, and draft revisions of the A220 and A320 reports, accept the same substitute.

MMO and the coffin corner

The maximum operating limit Mach number (MMO) may not be deliberately exceeded in any phase of flight, except where a higher speed is authorized for flight test or pilot training (CS 25.1505 and 14 CFR 25.1505). It is set from flight trials with a margin below the Mach number at which shock effects become unacceptable, and it lies above Mcrit. An aural warning must sound whenever the speed exceeds VMO + 6 kt or MMO + 0.01 (CS and 14 CFR 25.1303(c)(1)); the 737 sounds its clacker once VMO/MMO is exceeded. On jet transports the Machmeter's position error is biased so that it over-reads, and any error moves the pilot away from MMO.

MMO becomes the limit above the crossover altitude, because a fixed Mach number corresponds to a falling calibrated airspeed as the airplane climbs. The FAA handbook's example is an early jet with a VMO of 306 KCAS: at about FL310, M 0.82 equals 306 KCAS, and at FL380 it equals only 261 KCAS. Climbing at M 0.82 from sea level to FL380, the calibrated airspeed would fall from 543 to 261 kt. Meanwhile a jet that stalls at 152 KCAS stalls at M 0.23 at sea level and M 0.50 at FL380. For the A320 family, VMO 350 kt and MMO M 0.82 cross at about FL245 in ISA.

Climb high enough at a given weight and the limits meet, though sources word it differently. ATPL texts define the coffin corner as the altitude where the low-speed and high-speed buffet boundaries meet. The FAA handbook describes the stall speed in Mach terms reaching MMO. AC 61-107B calls the meeting point the aerodynamic ceiling, above the flight manual's altitude limit, where the airplane can neither speed up without the stick puller acting nor slow down without the stick shaker or pusher.

Airplanes are kept well below it by a buffet margin, conventionally 0.3 g, or 1.3 g of maneuver capability, equal to a level turn at about 40° of bank. On the 737 NG with flaps up, the bottom of the amber maximum maneuver speed bar on the speed tape gives that 1.3 g to high-speed buffet. Because Mcrit moves and MMO does not, a heavy airplane, high, turning or in turbulence, can meet buffet below MMO (see buffet and buffet margin).

Library articles are written for study and exam preparation. They do not replace your aircraft's approved documentation, your operator's procedures or the regulations themselves.