Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Dynamic pressure — where it appears

One half the density times the square of the speed, being the pressure a flow would produce if brought to rest. Because it is a square, its average over a fluctuating speed exceeds the value at the mean speed by the variance.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

The dynamic pressure is not ½ρU², and by Mach 0.85 it is out by a fifth. The pressure difference a pitot tube measures, divided by the incompressible dynamic pressure ½ρU², against Mach number. The dashed curve is the two-term series 1 + Ma²/4 + Ma⁴/40 that the rule of thumb comes from. An airspeed inferred from ½ρU² alone reads high, and the error is entirely predictable — which is why it is corrected rather than tolerated.

What the airspeed indicator believes

A pitot tube measures the difference between two pressures, correctly, at every speed. Everything wrong with an airspeed reading is in the arithmetic applied to that difference — and the error is 2.3 per cent at Mach 0.3 and 19.4 per cent at Mach 0.85.

misconceptions · Pressure The left wall is not a speed, it is an angle drawn in speed coordinates. The V–n diagram: every combination of speed and load factor the aircraft can reach. The curved left boundary is the wing at its stalling angle — n = ½ρV²S C_Lmax /W, a parabola — and it is the same limit at every point along it. The flat top and bottom are the structure. Where the two meet is the corner speed, 56.6 metres per second here: the slowest speed at which the aircraft can reach its limit load factor, and therefore the speed at which it turns hardest. Above it the wing can pull more than the structure allows — at the never-exceed speed it could reach 9.6g before stalling — and the pilot's limit stops being the air.

An angle, not a speed

The number is printed in the handbook, marked in white on the airspeed indicator and used in every briefing, and the wing has no way of knowing it. A wing stalls at an angle. The speed at which an aeroplane reaches that angle is an answer with four other variables in it, and every one of them moves.

misconceptions · Stall speed One sheared stream, and the total pressure across it. A parallel shear flow is an exact steady solution of the Euler equations, and the momentum equation requires its static pressure to be uniform. So the total pressure is entirely the dynamic pressure, which varies with the speed — by three and a fifth dynamic heads across this layer, on a flow where the static pressure does not vary at all.

Four Bernoullis and one name

"Bernoulli's equation" names at least four statements with four different constants, three domains of validity and one shared reputation for being misapplied. A single sheared stream separates the first three: its total pressure is constant along every streamline, varies by three dynamic heads across them, and its static pressure never moves at all.

misconceptions · Bernoulli's equation The mean lift is not the lift at the mean angle. A finite wing's lift curve, with a gust distribution of standard deviation 3° about a mean angle of 10° drawn along the foot. The wing spends its time spread across that distribution, so what it averages is the average of the curve: 0.7951 against the 0.8368 the mean angle promises, a deficit of 5.0 per cent. Nothing has stalled, and no gust has taken the wing past the stall angle: the deficit comes entirely from the curve bending over, and it is there at every angle where the curve is not straight.

The lift at the mean angle

A wing in rough air flies at every angle in turn, so what it averages is the average of its lift curve rather than the lift at its average angle. Where the curve bends over near the stall the two differ by five per cent at three degrees of gust and by fourteen at five, and the drag goes the other way.

circulation · Gust Identical lift at every altitude, and the skin load nearly doubles. The same wing at the same dynamic pressure at seven altitudes. The lift is identical at every one of them — the flat line, computed from the absolute pressures rather than assumed, because a net force cannot depend on where the pressure datum is set. The load on a vented panel is identical too, at 4.86 kPa, because both sides of it moved together. The load on a sealed panel with 75.26 kPa inside is not: it is 28.27 kPa at sea level and 60.84 kPa at twelve kilometres. The datum cancelled in the force and it is one of the two numbers in the stress.

A force forgets the datum, a stress cannot

Nothing sucks, because the pressure datum cancels — the normals of a closed body sum to nothing, so the lift is the same in gauge or absolute pressure. That identity is about a resultant, and it is routinely carried one step too far. The load on a panel of skin has the ambient in it as one of two numbers, not as a datum.

misconceptions · Suction Every output of the air-data reduction, and how hard it leans on each reading. The logarithmic sensitivity of each derived quantity to each of the three readings — the total pressure, the static pressure and the indicated total temperature — at Mach 0.3 and Mach 0.85, at 11 km with a probe recovering 0.98: a one per cent error in a reading times the number is the per cent error it puts into the output. The Mach number leans on the two pressures by 8.08 and −8.08 at Mach 0.3 and by 1.13 and −1.13 at Mach 0.85, and not at all on the temperature. The static temperature leans on its probe by exactly one and on the pressures by −0.280 at Mach 0.3 and −0.281 at Mach 0.85 — almost the same. The true airspeed leans on the probe by exactly one half and the density by exactly minus one, at every speed.

Three readings, and the one each answer leans on

An aircraft works out the air it flies through from three readings — a static pressure, a total pressure and a probe's temperature — and every derived number inherits their errors through one small table of sensitivities. Written out, the table says the airspeed is afraid of the pressure sensors almost to Mach one at cruise altitude and only to Mach 0.52 at sea level, and that the density belongs to the thermometer at every speed.

compressible · Recovery

Named alongside it

The objects these essays reach for when they reach for this one.

MeasurementBernoulli's equationCompressibilityFlight envelopeInstrumentLoad factorMach numberMisconceptionModel limitStagnation pressureStallTotal pressure

All concepts