A finite force from an infinite speed

Here is a contradiction that every course in aerodynamics sets up and most of them leave standing.

A flat plate at incidence in ideal flow carries a circulation, and Kutta–Joukowski says the force on it is $\rho U \Gamma$ perpendicular to the free stream. But the only thing acting on a flat plate is pressure, and pressure on a flat plate acts along the plate’s normal. Those two directions differ by the angle of attack.

If the force is normal to the plate it has a component along the stream, which is a drag — and an inviscid steady flow round a body has no drag. If it is perpendicular to the stream it has a component along the plate, and there is nothing on a flat plate for such a force to act on.

Where the missing force is #

It comes from the leading edge, and it comes from the one feature of the solution that is usually apologised for.

Thin-aerofoil theory represents the plate by a vortex sheet of strength $\gamma(\theta) = 2U\alpha(1+\cos\theta)/\sin\theta$ on $x = \tfrac{c}{2}(1-\cos\theta)$. At the trailing edge, $\theta = \pi$, it goes smoothly to zero: that is the Kutta condition, and it is the whole reason the circulation has the value it has. At the leading edge, $\theta \to 0$, it goes to infinity, as the inverse square root of the distance from the edge.

The surface perturbation velocity is half the sheet strength, so near the edge

$$u \;\to\; U\alpha\sqrt{c/x} \;=\; \frac{C}{\sqrt{2x}},\qquad C = U\alpha\sqrt{2c}.$$

By Bernoulli the pressure there is minus infinity. The area it acts over is zero. And the product is finite:

$$T = \tfrac12\pi\rho C^2 = \pi\rho U^2\alpha^2 c,$$

directed forwards along the plate, which is exactly $L\sin\alpha$ to leading order. The books balance.

Why this is not a conjuring trick #

A finite force extracted from a point where the model has stopped making sense deserves suspicion, and the way to settle it is to remove the singularity and see whether the force goes with it.

Round the leading edge into a parabola of nose radius $r$. The local problem has a closed-form solution — the map $z = \zeta^2/2$ takes a straight line to a parabola, and requiring that line to be a streamline while matching the outer $C/\sqrt{2x}$ gives the surface speed

$$q(\xi) = \frac{C}{\sqrt{\xi^2 + r}},\qquad x = \frac{\xi^2 - r}{2}.$$

Every part of that is finite. The peak speed is $C/\sqrt r$, at the nose; the width of the peak is of order $\sqrt r$; and the force, which is the pressure defect resolved along the axis, is

$$T = \int \tfrac12\rho\,\frac{C^2}{\xi^2+r}\,\sqrt r\;d\xi = \tfrac12\pi\rho C^2$$

— the same number for every $r$, including in the limit $r \to 0$ where the speed is infinite.

What it is worth on an aeroplane #

The suction is a small number and it decides a large one, because it is the difference between a section with no pressure drag and a section whose pressure drag is quadratic in lift.

Whether it is collected is not a question about Laplace’s equation. It is a question about whether the flow actually goes round the edge, which is a question about the boundary layer. Two answers:

A rounded, attached leading edge collects it. The suction peak is real, the pressure over the first few per cent of the chord is genuinely very low, and the resultant is perpendicular to the stream. The section has no pressure drag at all, and its measured drag is skin friction and nothing else.

A sharp edge at incidence does not. The flow separates at the edge, the suction peak never forms, the whole force is normal to the plate, and the drag is

$$C_D = C_L\tan\alpha \;\approx\; \frac{C_L^2}{2\pi}.$$

The size of it #

Numbers, because the argument so far is geometric and it is easy to leave with the impression that this is a large force.

As a coefficient the suction is $C_T = 2\pi\alpha^2$, against a lift coefficient of $2\pi\alpha$. At four degrees that is 0.031 against 0.44 — seven per cent of the lift — and at ten degrees 0.19 against 1.10, which is seventeen per cent. It is small compared with the lift and it is enormous compared with the drag it is competing with: a good section at four degrees has a friction drag coefficient of about 0.006, so losing the suction multiplies the section’s drag by six.

The quadratic growth is what makes it matter more as things get interesting. Lift is linear in incidence, suction is quadratic, so the fraction of the resultant that hangs on this one point rises steadily as the aeroplane is asked to do more. At the incidence a wing uses on approach it is not a correction.

The coincidence that is not one #

$C_L^2/2\pi$ is the induced-drag formula $C_L^2/\pi A\!R$ with the aspect ratio replaced by two, and the resemblance is structural rather than numerical.

Induced drag is what is left of the streamwise force when the resultant on each section is tilted backwards by the downwash of the trailing vortices. Zero-suction drag is what is left when the resultant is tilted backwards by the loss of a leading-edge force. In both cases the section’s own lift is unchanged and the direction of the resultant is not, and in both cases the penalty goes as the square of the lift because the tilt angle is itself proportional to lift.

The practical consequence is the design of a whole class of aeroplane. A thin, sharp-edged wing at supersonic speeds cannot have a rounded leading edge — the wave drag of a blunt nose is prohibitive — so it flies with the suction lost, and its subsonic drag polar is the lower curve above. That is why a supersonic fighter’s take-off run is long and why its approach is flown at high power.

Getting it back #

There are three ways to recover a force worth this much, and all three are in service.

Round the edge and keep it attached, which works up to the incidence at which the peak gets too strong for the layer and the section stalls from the leading edge rather than the trailing one. A sharp stall with no warning is the price of a small leading-edge radius.

Blow or suck at the edge, which is what a slat does: it does not delay stall by “energising the boundary layer”, it changes the pressure distribution so that the peak the layer has to survive is smaller.

Or give up and use the separation. A slender delta with sharp edges separates on purpose, and the sheet it sheds rolls into a pair of vortices that sit over the upper surface and produce suction of their own. That is vortex lift, and its lift-curve climbs well past where an attached section would have stalled — with a drag polar that is the zero-suction one, because by construction the leading-edge suction was never collected.

The same point, used as a propeller #

There is a second thing the leading edge does with that force, and it is the reason anything flaps.

Take the plate and heave it up and down instead of holding it at incidence. The instantaneous incidence it sees is the heaving velocity over the flight speed, so the circulation oscillates, the lift oscillates, and — averaged over a cycle — the lift is nothing, because the motion is symmetric. What does not average to nothing is the leading-edge suction. It goes as the square of the instantaneous incidence, so it is positive on both halves of the stroke, and it points forwards along the chord on both.

A heaving aerofoil produces thrust, and in linear theory the thrust is the leading-edge suction and nothing else. Every other term in the force balance is oscillatory and averages away; the one term that is quadratic survives, and it is the one carried by the singular point. Garrick worked that out in 1936, using exactly the function the unsteady essay computes to supply the phase, and the result is the basis of every calculation of flapping propulsion since.

Which immediately explains a fact about animals. A flapping propulsor has to collect its suction, so its leading edge has to be round and its flow attached — and the leading edges of a tuna’s tail, a dolphin’s fluke and a bird’s wing are rounded, thickened and smooth, while their trailing edges are thin. That is the opposite of the shape a fixed sharp-edged plate would suggest, and it is exactly the shape the suction argument demands: the thrust is generated at the front, so the front is where the structure and the radius go.

It also explains the alternative, which is what small flapping fliers do instead. An insect’s wing is thin and sharp-edged and cannot collect a leading-edge suction in the attached sense; what it does is separate deliberately, roll the shed sheet into a leading-edge vortex that sits over the upper surface, and take the suction from the vortex. That is the same trade the delta wing makes, at a Reynolds number four orders of magnitude lower, and for the same reason — if the suction cannot be collected by an attached flow, it can be collected by a captive vortex, and the price is that the mechanism now depends on the vortex staying put.

The efficiency of the attached version turns on the same phase the previous section’s ladder is about. The suction is large when the instantaneous incidence is large, and the instantaneous incidence is set by how fast the foil is heaving relative to how fast it is travelling — which is a Strouhal number $fA/U$, with $A$ the stroke amplitude. Too small and the incidence never rises enough to produce useful suction; too large and the flow separates and the suction is lost outright. Measured across swimming and flying animals the value sits between about 0.2 and 0.4, over four orders of magnitude in size, from a hovering hummingbird to a whale — a band that is the compromise between those two failures, and one of the few genuinely universal numbers in animal locomotion.

What a plate has that a wing does not #

It is worth being clear that the singularity is a property of the sharp edge and not of thinness.

A section with a rounded leading edge has no singularity anywhere. Its surface speed is bounded, its pressure is bounded, and the suction force is not concentrated at a point but distributed over the nose region — the first per cent or two of the chord — with the same total. The singular calculation is the limit of that as the nose radius shrinks, and its value is that the total does not depend on the radius, so a quantity that would otherwise have to be computed shape by shape is universal.

That is the practical use of a singular solution and it is worth stating as a rule: a singularity is useful when the quantity it carries is independent of how it is regularised. Where the answer depends on the cut-off, the cut-off is physics and has to be modelled. Here it is not.

What the picture cannot show #

The pressure at the edge is not minus infinity in any fluid. Bernoulli gives a pressure that falls without bound as the speed rises, and a liquid boils and a gas becomes compressible long before that. The singular solution is an outer limit, and the inner region where it stops being true is the region the rounded-nose calculation above is a model of.

The peak is drawn as a speed and it acts as a pressure. The force is the integral of $\tfrac12\rho q^2$ against the surface’s slope, so it weights the peak twice over — once for the square and once for the fact that the surface near a nose is nearly perpendicular to the chord. A figure of $q$ against $x$ makes the peak look like a detail of the leading five per cent, and the force it carries is not a detail.

And nothing here separates. The whole calculation is inviscid, which means the difference between the two polars is put in by hand as an assumption about whether the flow is attached. What decides it is the layer’s ability to climb the pressure rise behind the peak, and that is a viscous calculation with a Reynolds number in it that appears nowhere above.

The general shape of the result #

It is worth extracting the pattern, because it recurs wherever a theory develops a singularity.

A singular solution is not automatically useless. What matters is whether the quantity being asked for is an integral against a measure that kills the singularity, and here it is. The rounded-nose calculation shows the shape of it exactly: the force integrand is a Lorentzian whose height goes as $1/r$ and whose width goes as $\sqrt{r}$, so the two dependences cancel in the area and the answer is independent of $r$.

The same structure appears at a crack tip in elasticity, where the stress is singular as $1/\sqrt r$ and the energy release rate — an integral — is finite and is the quantity fracture mechanics is built on. It appears in the added mass of a sharp-edged plate, and in the lift of a slender delta. The rule is to ask what the singular quantity is being integrated against before deciding whether the model has failed.

Who found it, and when #

The suction force is in Munk’s and Glauert’s work of the early 1920s, and the cleanest statement is usually attributed to von Kármán and Burgers in 1935. It arrived as an accounting device — the term needed to make thin-aerofoil theory consistent with d’Alembert — and became a design quantity in the 1950s, when Polhamus turned “the suction that a sharp edge fails to collect” into an analogy that predicts the vortex lift of a delta wing from the suction the attached flow would have had. That is one of the odder pieces of engineering reasoning in the subject: a quantity is computed for a flow that does not happen, and used to predict the lift of a completely different flow that does.

The surprising connection is with a number that is not aerodynamic at all. The suction force is $\tfrac12\pi\rho C^2$ where $C$ measures the strength of the inverse-square-root singularity, and the energy release rate at the tip of a crack is $K^2/E'$ where $K$ measures the strength of an inverse-square-root stress singularity. The two are the same formula, because both are the finite part of an integral of a squared field against a shrinking region. Griffith’s crack criterion of 1921 and the leading-edge suction of 1923 were arrived at independently, two years apart, in fields that did not read each other.

One more consequence of the same accounting, and it is the one that decides how a wing is measured. A wind-tunnel balance reads the resultant force and resolves it into two components. If the section is collecting its suction, the streamwise component is friction alone and the balance is measuring a boundary layer. If it is not, the streamwise component is dominated by a term that has nothing to do with viscosity at all, and a test that was intended to measure skin friction is measuring the failure of a singular point to stay attached. The two are separated by a pressure tapping at five per cent chord, and by nothing else the balance can see.

Where the ladder goes next #

Below this rung are the lift curve whose slope this force is a consequence of, and the paradox it exists to protect.

Beside it is lift out of a failure, which is what an aeroplane does when it decides not to collect the suction at all, and the price of having ends, whose polar has the same shape for a different reason.

And above it, the local analysis this rests on: what a flow does at a corner, where the exponent is set by the angle and nothing else.