A stalled plate keeps a quarter of its lift slope and moves its centre aft

Drag in the theory that forbids it followed Helmholtz and Kirchhoff out of d’Alembert’s paradox. The paradox is a theorem about flows that close behind a body; let two streamlines leave the edges of a plate held across the stream and never come back, fill the space between them with still fluid at the stream’s own pressure, and the same equations — no viscosity, no vorticity — give a drag of 0.8798 of the dynamic pressure on the plate’s width. Where the unknown boundary is the known one explained the method that makes it soluble: drawn in the plane of its own velocity, the free streamline, whose shape is the answer, becomes an arc of the unit circle, known in advance.

The normal plate was chosen because it is symmetric. Tilt it to an incidence $\alpha$ and the construction describes something else as well: a flat plate whose flow leaves its leading edge as well as its trailing edge — a wing that has stalled from its leading edge, as a thin plate does at a few degrees. Rayleigh solved that problem in 1876, and his two formulas, for the force and for where it acts, are the content of this essay, computed here through the hodograph rather than quoted.

A stalled plate is not a plate without lift. Its force is a quarter of the attached plate’s near zero incidence, it acts at five sixteenths of the chord rather than a quarter, and it is perpendicular to the plate, so the plate’s lift-to-drag ratio is the cotangent of its incidence. What the theory cannot give is how much stronger a real stalled plate’s force is, because that is set by the pressure in its wake.

Three planes and two maps #

The construction has three planes. In the plane of the complex potential $w = \phi + i\psi$, with $\phi$ zero at the stagnation point, the plate’s two faces and both free streamlines are the single line $\psi = 0$, traced from the stagnation point outward along both sides of the positive $\phi$ axis; the flow fills the $w$-plane cut along it. The map $s = \sqrt{w/k}$ opens the cut onto the upper half of an $s$-plane: the windward face’s upper part and its free streamline onto positive $s$, the lower part and its free streamline onto negative $s$.

In the hodograph plane of $t = (\mathrm{d}w/\mathrm{d}z)/U$ — the velocity, scaled and conjugated — the free streamlines are arcs of the unit circle, because the wake is at the stream’s pressure and so the speed on them is the stream’s. The two parts of the face are segments of one diameter, at the plate’s own inclination, since the flow on them runs along the plate in opposite directions from the stagnation point at the centre. The flow is a half-disc. Rotated to lie flat, $T = te^{i\alpha}$, the half-disc is opened onto an upper half-plane by the Joukowski map $\tau = -(T + 1/T)/2$, which sends the semicircle onto $(-1, 1)$, the diameter onto $|\tau| > 1$, the stagnation point to infinity and the stream’s own direction, at infinity, to $\tau = -\cos\alpha$.

Two upper half-planes are joined by a Möbius map, and only one sends the stagnation point $s = 0$ to $\tau = \infty$ and infinity to $-\cos\alpha$: $\tau = -\cos\alpha - 1/s$. The plate’s edges are then at $s = 1/(1 - \cos\alpha)$ and $s = -1/(1 + \cos\alpha)$. Everything else is quadrature: the plate’s shape from $z = \int \mathrm{d}w/(Ut)$ along the faces, its length fixing the constant $k$, and the pressure on the windward face from Bernoulli, $\tfrac12\rho(U^2 - q^2)$ above the wake’s, with $q = U|t|$.

One numerical lesson cost a first attempt. Near the stagnation point $\tau$ is enormous and the small root of $T^2 + 2\tau T + 1 = 0$ is lost to cancellation if computed as a difference; the speed came out exactly zero and the plate’s length infinite. The large root has no cancellation, and the two multiply to one, so the small one is computed as the large one’s inverse.

The force, a quarter of the attached plate’s #

The first figure is the normal force over the dynamic pressure on the chord. The quadrature’s dots lie on Rayleigh’s curve, $C_N = 2\pi\sin\alpha/(4 + \pi\sin\alpha)$, to four parts in ten million at every incidence checked, and at 90° the curve is Kirchhoff’s 0.8798. At 5° the stalled plate carries 0.128; at 20°, 0.423.

The dashed line is the same plate with its flow attached — the thin-aerofoil plate of the lift curve, its flow round the leading edge and leaving the trailing edge smoothly: $2\pi\sin\alpha$, 0.548 at 5°. The ratio of the two is exactly $1/(4 + \pi\sin\alpha)$, a quarter at small incidence and less than a quarter beyond. Stalling from the leading edge takes three quarters of the lift slope away, and the theory says where it went: most of the attached plate’s force is suction over its lee face, concentrated behind the leading edge, where the flow races round the edge at the infinite speed a finite force from an infinite speed found there. A free streamline leaving the leading edge puts the whole lee face at the wake’s pressure, and the suction is gone; only the windward face’s overpressure is left.

The force is normal to the plate because nothing in the wake can push along it: the lee face is at the wake’s uniform pressure, and the windward face’s pressure acts perpendicular to the plate. So the stalled plate’s lift and drag are $C_N\cos\alpha$ and $C_N\sin\alpha$, and its lift-to-drag ratio is $\cot\alpha$ exactly — 2.75 at 20°. The attached inviscid plate has a lift-to-drag ratio of infinity, because the leading-edge suction cancels the normal force’s drag component; the stalled plate has lost the suction, and with it the cancellation.

The centre moves aft #

The second figure is where the force acts. The attached plate’s centre of pressure is at the quarter chord at every incidence. The stalled plate’s is behind it: 0.318 of the chord from the leading edge at 2°, approaching $1/2 - 3/16 = 5/16$ as the incidence falls to zero, 0.361 at 20° and the middle of the chord at 90°, where the flow is symmetric. Rayleigh’s formula puts it $(3/4)\cos\alpha/(4 + \pi\sin\alpha)$ chords ahead of mid-chord, and the quadrature agrees to $3\cdot10^{-7}$ chords.

That aft movement is the pitch-down that accompanies a leading-edge stall. A wing trimmed about a point near its quarter chord, with its flow attached, has its lift acting there; when the flow leaves the leading edge the lift falls to a quarter of its slope and moves back by at least a sixteenth of a chord, and the moment about the old point becomes nose-down. For a thin section that is a stabilising surprise as well as a loss: the stall itself pitches the wing towards smaller incidence. Which part of a wing stalls first is the question of where along the span that happens first; this is what happens at a section once it has.

The stagnation point, also drawn, stays at the leading edge to within a few thousandths of a chord up to about 20°, then moves back, reaching the middle only at 90°. The windward face is loaded from its leading edge back.

The windward face carries everything #

The third figure draws the windward face’s pressure. At the dial’s first setting, 20°, it peaks at the stagnation point, a few thousandths of a chord from the leading edge, falls steeply and then slowly back to the stream’s pressure at the trailing edge, where the flow leaves along the free streamline at the stream’s speed. The lee face is entirely wake. Integrated, the windward face gives the normal force of 0.423; its first moment gives the centre at 0.361.

The dial shows the pressure’s shape changing with incidence. At 5° the windward face’s pressure is low everywhere except a sliver at the leading edge; at 45° the stagnation point has moved seven hundredths of the chord back and the loading has filled out; at 90° it is Kirchhoff’s symmetric distribution. A pressure tapping row on a stalled flat plate measures this face directly, which is one of the few places a free-streamline prediction can be compared with an experiment point by point.

The wake’s edges #

The fourth figure draws the free streamlines themselves, by integrating $\mathrm{d}z = \mathrm{d}w/(Ut)$ outward from each edge, where $t$ lies on the unit circle. Each leaves its edge tangent to the plate. The trailing edge’s runs downstream on the windward side; the leading edge’s curves away over the lee face, slowly at a small incidence — at 10° it is still only about a tenth of a chord from the plate a third of the way along — and faster at a large one. Neither ever returns: the wake grows without limit downstream, as the normal plate’s did, because a wake held at the stream’s pressure cannot close.

The small-incidence picture is the one that matters for a wing. A free streamline lying close over the lee face is a thin separated region, a bubble with no closure; a real flow in that position reattaches, forming the short or long leading-edge bubble of a thin section near stall, and the free-streamline theory describes the state that bubble bursts into rather than the bubble itself.

The one number missing #

The fifth figure confronts the theory with the one thing it is known to get wrong. A real flat plate held across a stream is measured at a normal force of about 2, not 0.88, because its wake is not at the stream’s pressure: the turbulent wake behind a bluff plate is at an underpressure of about 1.2 dynamic pressures. The free-streamline theory accepts that number as an input, through the cavitation number of the earlier essay — the wake’s underpressure $\sigma$ over the dynamic pressure — and to first order the force on a face of fixed shape scales with $1 + \sigma$. With $\sigma = 1.2$ the normal plate’s 0.88 becomes 1.94, the measurement.

At incidence the same rule scales every point of the curve, which is the figure. It is a rule, not a solution, and its weakness at small incidence is plain: a thin stalled plate’s wake is narrow and its underpressure smaller than a bluff plate’s, so the right $\sigma$ falls with the incidence. The inviscid theory says how the force depends on the geometry once the wake’s pressure is given; the wake’s pressure itself is set by the turbulent mixing at its edges, which is the subject of when the flow lets go and nothing in Laplace’s equation.

Two ways for a leading edge to let go #

The free streamline is one answer to the question every sharp leading edge poses: what the flow does when it cannot go round. The sharp edge decides at a trailing edge by insisting that the flow leave smoothly, and the attached plate is the result of allowing the leading edge to be gone round instead, at an infinite speed. Kirchhoff’s flow makes the opposite choice at the leading edge as well: the flow leaves both edges, and the lee face is surrendered to a wake.

A slender delta wing makes a third choice, which lift out of a failure described. Its leading edges are swept, the sheets they shed roll up into a pair of vortices that stay above the wing and reattach behind them, and the suction under those vortices is a lift of its own — Polhamus’s analogy turns the lost leading-edge suction back into a normal force. The two-dimensional plate cannot do that: a sheet shed from a straight leading edge has no spanwise flow to carry its vorticity away, cannot settle into a standing vortex, and either reattaches as a bubble or bursts into the open wake drawn here. The difference between losing three quarters of the lift and gaining lift with incidence is the difference between a vortex that is held and a free streamline that is not.

That makes the stalled flat plate the lower bound of a family. Every section that separates from its leading edge without a standing vortex and without a reattaching bubble keeps at least Rayleigh’s quarter, plus whatever its wake’s underpressure adds; a delta’s vortex and a thin section’s bubble each recover part of the difference, by holding the separated vorticity close to the surface.

The convention: chord, incidence and the wake’s pressure #

Forces are per unit span over $\tfrac12\rho U^2$ times the chord; the normal force is perpendicular to the plate, lift and drag perpendicular and parallel to the stream. Incidence is the angle between the plate and the stream. Distances along the plate are from the leading edge, in chords. “Attached” means the inviscid flat plate with the Kutta condition at its trailing edge and the flow round its leading edge; “stalled” means Kirchhoff’s flow with free streamlines from both edges and a wake at the stream’s pressure, unless an underpressure is stated. Incidences are in degrees in the figures and in radians inside the formulas, where the small-angle forms read most simply.

What the theory leaves out #

Where the flow leaves. Both edges are sharp, so the separation points are not in doubt; on a rounded leading edge the flow can leave anywhere, and finding where is the whole difficulty of stall.

The wake’s closure. The wake is infinite, as in every Kirchhoff flow; Riabouchinsky’s and Roshko’s closure models give a finite wake with a stated underpressure, and change the force by the factor the fifth figure shows.

Unsteadiness. A stalled plate sheds vortices from both edges alternately; the steady free-streamline flow is a model of their mean, and at large incidence the fluctuating force is a large fraction of the mean.

The bubble. At small incidence a real thin section first forms a reattaching bubble behind its leading edge, which keeps much of the lift; the free-streamline flow is the state after the bubble has burst.

Who solved it #

Helmholtz introduced free streamlines in 1868 and Kirchhoff solved the normal plate in 1869. Rayleigh treated the inclined plate in 1876, finding the normal force and the centre of pressure as above, in the paper “On the resistance of fluids”; Levi-Civita’s general method of 1907 made the construction routine for curved bodies. The three-plane construction here is the classical one; the quadrature and its agreement with Rayleigh’s formulas are this calculation’s.

How it was checked #

The normal force by quadrature matches Rayleigh’s formula to four parts in ten million at 5°, 15°, 30°, 60° and 90°; the centre of pressure’s distance from mid-chord matches his to three parts in ten million of a chord; the normal plate is Kirchhoff’s $2\pi/(4 + \pi)$ to the same precision. At 1° the stalled force is 0.247 of the attached one, approaching the quarter the formulas give in the limit. The construction’s map was checked indirectly by the plate’s own length, which the quadrature reproduces as one chord by construction, and by the free streamlines leaving each edge tangent to the plate.

Still open: a stalled plate that is also cambered #

Every number here is for a flat plate. A cambered plate — a sail, a thin cambered wing, a bird’s flight feather — stalls with its flow leaving the leading edge along a surface that curves, and the windward face’s curvature changes both the force and where it acts. The hodograph is then no longer a half-disc bounded by a straight diameter, and Levi-Civita’s method is needed: the next calculation would solve Kirchhoff’s flow past a circular-arc plate at incidence, and ask whether camber, which raises an attached plate’s lift at every incidence, raises a stalled one’s by the same amount, and whether the stalled centre of pressure still sits behind the quarter chord, which decides whether a stalled cambered wing pitches down as a flat one does.