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

Prandtl–Glauert correction — where it appears

The transformation that turns a compressible subsonic flow into an incompressible one about a stretched shape. It shows that compressibility is a change of geometry rather than of physics, and in three dimensions the shape that changes is the aspect ratio.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Where the flow over the section first reaches Mach one. Two curves running towards each other. The falling one is the section's peak suction, solved exactly in incompressible flow and then deepened by the Prandtl–Glauert factor as the free-stream Mach number rises. The rising one is the pressure coefficient at which the local flow would be exactly sonic. Where they cross is the critical Mach number, and above it there is a pocket of supersonic flow on a subsonic aeroplane.

The pocket on top of the wing

An airliner cruising at Mach 0.85 has subsonic flow almost everywhere and a patch of supersonic flow over its wing. Closing that patch takes a shock, the shock separates the layer beneath it, and the correction that predicts all of this also predicts, in the plainest terms, where it stops being true.

regimes · Mach The correction belongs to the wing, not to the air. Lift-curve slope against Mach number for three aspect ratios, each computed by solving the lifting line for the wing the transformation actually implies — one of aspect ratio βAR — and dividing by β. The dashed curve is the two-dimensional rule, which is what an aerofoil section gets. At Mach 0.7 the aspect ratio of 20 has gained 35 per cent of slope and the aspect ratio of 4 has gained 24, against the 40 per cent the section rule promises both. The β in the finite-wing term cancels the β in front of it, so the shorter the wing the less compressibility does to it.

The wing the equation is really solving

The Prandtl–Glauert rule is usually quoted as a factor on the answer. It is a change of shape — and in three dimensions the shape it changes is the aspect ratio, so a short wing is far less affected by compressibility than a long one.

regimes · Similarity Which speed the number is formed on. The fractional change in air density at three places on a body, against the free-stream Mach number. At a stagnation point the density rises, and it reaches five per cent at M = 0.314 — which is where the familiar 0.3 comes from, and it is a five per cent tolerance rather than a physical boundary. At the suction peak the density falls instead, and how fast depends on the body: a lightly loaded section is milder than its own nose, and one working at cp₀ = −2 reaches five per cent at M = 0.22 and is at Mach 0.55 over its shoulder while the free stream is at 0.3.

Which speed goes in the number

The most quoted threshold in the subject — air is incompressible below Mach 0.3 — is a five per cent tolerance on the density at a stagnation point wearing a physical boundary's clothes. A wing working for its living is at Mach 0.55 over its shoulder while the free stream is still at 0.3.

regimes · Mach The coefficient of the equation's second derivative, along a chord. The bracket multiplying the streamwise second derivative in the transonic small-disturbance equation, along a chord at Mach 0.85. Where it is positive the equation is elliptic and the flow is subsonic; where it is negative the equation is hyperbolic and the flow is supersonic. Which it is at a given point depends on the perturbation velocity there, which is the thing being solved for. Forty-two per cent of this chord is hyperbolic, and no amount of inspecting the problem beforehand could have said so.

The equation that changes type inside its own answer

Near Mach one the coefficient of the streamwise second derivative depends on the perturbation velocity, which is what is being solved for. Two solutions of the linear equation no longer add — the leftover is three times the term the linear theory keeps — and the critical Mach number approaches one as the two-thirds power of thickness.

compressible · Transonic

Named alongside it

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

Critical machMach numberCompressibilityPressure coefficientSimilarityTransonicArea ruleAspect ratioBoundary conditionBoundary layerCharacteristicsDimensionless

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