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

Every essay — page 9

Page 9 of 61, continuing through the fields in the same order.

Flows and fields Ideal flow Circulation and lift Viscosity Regimes and numbers Compressible flow Transition and turbulence Fluids at work What is taught wrongly Series Concepts Regimes Refutations Search

Circulation and lift

Where lift actually comes from. The Kutta condition, the Joukowski aerofoil, and lift derived rather than assumed.

Every one of these is a solution, and they lift different amounts. Lift coefficient against the circulation the aerofoil was told to carry, with the Kutta condition's own answer marked. Each point is a complete solve: the sources were found for that circulation and the surface is a wall to within 1.3e-4 at every collocation point. Every member satisfies the equations of motion and the boundary condition, and the lift runs through them at exactly 2Γ/Uc — Kutta–Joukowski appearing as a property of the family rather than as a result about any member of it. Ideal flow round a closed body does not have a unique answer, and the Kutta condition is the extra sentence that picks one.

Nothing but the edge

Cut an aerofoil into panels, put a singularity on each, and require the surface to be a wall. The system that comes out has one more unknown than it has equations, and the row that is missing is not a bookkeeping slip — it is the fact that ideal flow round a closed body has no unique answer at all.

9 figures A slotted flap at 30°, in a flow with no viscosity anywhere. Streamlines through a main element and a flap, computed by a two-body panel solve. Each element carries its own circulation and its own Kutta condition, and the two interfere through their velocity fields and through nothing else — there is no boundary layer here, no wake, no mixing region and no high-energy air. The system's lift coefficient is 2.757 against 0.698 for the main element alone at the same incidence, and the main element itself is carrying 3.98 times the circulation it carries by itself.

A slot is not a nozzle

The gap between a wing and its flap is supposed to blow fast air into a tired boundary layer. A solver with no boundary layer in it at all — no viscosity, no wake, no mixing — produces most of the lift increment anyway, and produces it on the element nobody moved.

8 figures 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.

8 figures The rectangular wing stalls at the root; the tapered one stalls at the tip. Section lift coefficient across the half span for three planforms, each drawn at the wing incidence where its own worst section first reaches 1.5. A rectangular wing's peak is at the root, which is where a designer wants it: the stall starts inboard, ahead of the ailerons, and the pilot feels it. A tapered wing's peak has moved out to 0.62 of the semi-span — over the ailerons — because taper takes chord away from the tip faster than it takes circulation. The elliptic wing is the degenerate case: every section reaches the limit at once, which is elegant and is the worst possible stall behaviour.

Which part of a wing stalls first

A wing has one lift coefficient and its sections have a hundred, and no section is at the wing's. Which of them runs out first is decided by the planform, it decides whether the pilot keeps the ailerons, and on any wing that was close to elliptic the standard fix costs span efficiency for every degree of it applied.

8 figures Five quarters of the span, the same bending moment, 64/75 of the drag. The bell-loaded wing's induced drag and root bending moment against its span, both as fractions of an elliptic wing of unit span carrying the same lift. At equal span the bell is worse: its span efficiency is exactly three quarters. But its bending moment is exactly four fifths, and bending moment grows in proportion to span while drag falls as its square — so there is a span at which the bell has bought back the structure and is ahead on drag. It is at exactly five quarters, where the two curves are at 1 and 0.8533. Both numbers are rational and neither was put in by hand.

The loading nobody used

Elliptic loading is the least-drag answer to a question no aeroplane asks. Constrain the moment the lift makes about the wing root instead of the span, and a different curve comes out — five quarters of the span for sixty-four seventy-fifths of the drag, both exact — with an upwash over the outer wing and a yaw that turns the right way.

9 figures The wake, and the velocity it gives itself. The cross-section of the wake far behind a wing with turned-up tips, with the velocity the wake induces on itself drawn as arrows normal to the trace. Induced drag is the integral of the circulation against that velocity and nothing else — this picture contains the entire quantity. It also contains no information whatever about where the surfaces were: two wings a chord apart and two wings ten chords apart produce the same picture and therefore the same drag, which is Munk's stagger theorem stated as a fact about what the arithmetic can see.

A wing that leaves the plane

A winglet is not a fence and it does not block anything escaping round the tip. The induced drag of any system of lifting surfaces depends on one cross-section of its wake and on nothing else whatever, and a wake that reaches upwards is cheaper for the same reason a wake that reaches sideways is.

8 figures Four aeroplanes, one number. Four biplanes with the same gap and the same loadings, staggered by nothing, by four tenths, by nine tenths and by one and six tenths of a chord. Their induced drags agree to the last bit of the arithmetic — the calculation cannot even express the stagger, because the Trefftz plane is a cross-section and everything drawn here projects onto the same one. That is Munk's stagger theorem, and stating it as the drag is unchanged understates it: there is no place in the computation where the stagger could be entered.

Two wings and it does not matter where

Move one wing of a biplane a chord forward and the induced drag does not change. Not approximately, not to a good approximation — the calculation that gives the induced drag has nowhere to put the stagger, because everything projects onto the same cross-section of the wake.

8 figures Tip to tip, two wings cost exactly half of what they cost apart. The induced-drag saving of a pair of wings against the gap between their tips, as a fraction of what the two pay flying alone. With the tips touching the pair is one wing of twice the span carrying twice the lift, and the arithmetic of that is exact: the drag halves, and the computation returns 0.499257 of the separate figure. Pull them apart and the saving falls away with the square of the distance. Birds fly in a V because the tips are the part worth overlapping, and the spacing that pays is a small fraction of a span rather than any distance a formation could hold by eye.

The lift beside a wing

Fly two aeroplanes with their wingtips touching and the pair costs exactly half what the two cost apart. Not approximately half — the arithmetic is a closed form, because two wings tip to tip are one wing of twice the span, and induced drag goes as the square of it.

8 figures Elliptic loading rolls up to πb/4, and it is π that puts it there. Three loadings on the same span carrying the same lift, with the rolled-up core positions marked below each. The core sits at the centroid of the shed vorticity, which is a quadrature over the loading and needs nothing about the roll-up itself. Elliptic loading gives πb/8 from the centreline, so the pair ends up 0.7854 of the span apart — the number every wake-separation rule is written against, and one of the few places in this subject where π turns up in an answer an engineer uses directly. The bell rolls up to 0.586 and a nearly rectangular loading to 0.978, because it sheds at the tips.

Where the wake ends up

The sheet a wing sheds rolls up into two cores within a few spans, and nobody can compute the roll-up cheaply. Nobody has to: what the cores conserve is fixed before they form, and for an elliptically loaded wing the answer contains π and comes out at 78.5 per cent of the span.

9 figures Two roots walking towards each other, and one of them crosses. The roots of the characteristic quartic in the complex plane as the airspeed is raised from nothing to 105 metres per second — growth rate across, frequency up. At rest the two sit on the imaginary axis at the uncoupled frequencies. As the speed rises, the aerodynamic coupling drags them towards each other in frequency while pushing one left and the other right, and at 80.8 metres per second the right-hand one crosses the axis. Everything about the failure is in this picture: the coalescence, the crossing, and the fact that the flutter frequency is neither of the two the structure started with.

The shake that is not resonance

A steady airstream contains no oscillation at any frequency, so nothing is driving anything. What happens instead is that the aerodynamic forces couple two structural motions that were independent, drag their frequencies together, and turn one damping negative — and the wing takes the energy out of the air itself.

8 figures The downwash approaches twice the value at the wing — from above. The downwash behind an elliptically loaded wing of aspect ratio 8, as a multiple of the induced angle at the wing itself, against distance in spans. Every account of tail sizing quotes a factor of two here. Two is the value at infinity: the trailing legs of the horseshoe system contribute a factor (1 + x/√(x² + a²)) which is one at the lifting line and two far downstream. Close behind, the bound vortex dominates and the field is much larger, and the curve comes down to its limit. A tailplane sits two or three chords behind, which on this wing is 0.31 of a span — where the factor is 2.46, a quarter above the number in the formula.

The surface in the wake

A wing has no opinion about its own incidence, which is why it needs a second surface behind it. How much that surface is worth depends on how much of the wing's downwash it is sitting in, and the factor of two everybody quotes for that is the value at infinity — where no tailplane has ever been put.

8 figures A closed wake, and the loading of least drag on it. The wake of a box wing in the plane that decides its drag, with the circulation of least induced drag drawn as a thickness along it. The horizontal members carry a loading close to elliptic and the vertical ones carry a share that lifts nothing — they contribute no lift, since lift is Γ dy and dy is zero on a vertical, and they change the drag by changing where the wake's vorticity is. At a gap of 20 per cent of span this system costs 67.1 per cent of what a single wing of the same span and lift would.

A wake that closes on itself

Prandtl's best wing system is a rectangle, not a wing. Solving for the loading on a closed wake turns up a circulation that costs nothing and does nothing — a gauge freedom in the middle of an optimisation — and a drag that keeps falling with no floor under it.

6 figures