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

Critical mach — where it appears

The free-stream Mach number at which the flow somewhere on a body first reaches the speed of sound. Above it a supersonic pocket forms and usually ends in a shock, so it marks where the drag begins to rise steeply.

Named by 7 essays across 3 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 Two profiles, and they are the same profile. The chordwise and spanwise velocity profiles in the boundary layer of a yawed flat plate, each as a fraction of its own edge velocity. They are computed by different code — the chordwise one by shooting a third-order nonlinear equation, the spanwise one by a single pass through a linear second-order one — and they agree to 2.1e-8 over the whole layer. They are the same function of η, because the two equations reduce to the same equation. A swept flat plate has no crossflow at any sweep angle, and that is the independence principle in the only form that has no wriggle room in it.

The wind a swept wing feels

Sweeping a wing back is usually justified by saying it meets a slower wind. It does not meet a slower wind. The equations split exactly in two, and the flow along the span is a passenger that exerts no force and changes nothing — until a pressure gradient breaks the split, and then it becomes the reason a swept wing is a different problem rather than a harder one.

circulation · Sweep 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 Three answers for the lift-curve slope, with three different signs. Thin-aerofoil theory has no thickness term at all and returns 2π for every section. The exact potential solution rises: 2π(1 + 0.766 t/c), measured off the Joukowski map. Real sections do the opposite, because the boundary layer thickens towards the trailing edge and decambers the section. Two of these curves are computed here; the third is what measurement says.

The half that carries nothing

Thin-aerofoil theory splits a section into a camber line that carries all the lift and a thickness distribution that carries none — at any incidence, exactly none. That is very nearly true, and what it discards decides the peak suction, the critical Mach number and where the boundary layer gives up.

circulation · Thickness The local sweep of the isobars, across the span. The sweep of the half-load line at each spanwise station, for four geometric sweeps. Over the middle of the span it is the wing's own sweep, which is the simple theory being right. At the root it collapses — by twenty-one degrees at a geometric thirty-five — and at the tip it falls again. That root region is where the shock forms first on every swept wing ever built, and it is why they have waisted fuselages.

The sweep a root does not have

Simple sweep theory is one of the cleanest arguments in aerodynamics: an infinite yawed wing cannot know about the velocity along its own span, so only the normal component matters. A real wing has a root and two tips, and at the root of a thirty-five-degree wing the isobars are swept fourteen.

circulation · Sweep

Named alongside it

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

Model limitPrandtl–Glauert correctionBoundary layerMach numberTransonicCompressibilityLift curve slopeLifting linePressure coefficientPressure distributionSeparationSimilarity

All concepts