A flapping plate has no best Strouhal number until it pays for drag

The frequency a wake chooses was about a body at rest in a stream and the frequency it sheds at, near a Strouhal number of 0.2. A wake told what to do took a body forced to move and showed how its wake gets captured by the forcing. There’s a third case, and it’s the most striking: a plate moved strongly enough that the wake reverses and the plate’s flapping propels it. Now it’s the Strouhal number $St=fA/U$, with $A$ the full width of the stroke, that sets the behaviour.

The striking part is what it measures in nature. Triantafyllou and his colleagues pointed out in 1991 that fish cruise with $St$ between about 0.25 and 0.35. Taylor, Nudds and Thomas found in 2003 that the same holds for a bird, a bat and a good many flying insects: birds and bats from a few grams to several kilograms, and a dolphin, have values of 0.2 to 0.4. Few numbers in biology hold across such a range. There’s an appealing explanation for it. A flapping foil’s wake is a reverse Kármán street, with its own instability, and its most amplified frequency sits near that band, so animals would be tuning to their own wake. The result is widely cited and has real experimental support.

The purpose of this essay isn’t to dispute what wakes do. It’s to ask what the exact theory of a flapping plate says, in the regime where that can be done in closed form, and to find that it has a different thing to say: a plate has no best Strouhal number at all, until it’s asked to do what an animal has to do.

Garrick’s calculation #

The lift that arrives late introduced Theodorsen’s function, the factor $C(k)=F+iG$ between what a moving plate’s quasi-steady circulation would be and what the shed wake allows, as a function of the reduced frequency $k=\omega b/U$, with $b$ the half-chord. Garrick used it in 1936 to answer a question Knoller and Betz had raised in 1909 and 1912: why a flapping wing gets thrust from a plain up-and-down motion at all. Lift is perpendicular to the local flow, and a plunging plate’s local flow tilts, so the lift has a component forward. The other part is the suction at the sharp leading edge, the same finite force from an infinite speed that comes from the attached flow around a thin edge.

For a plate plunging with amplitude $h_0$ at angular frequency $\omega$, in attached, inviscid flow, his answer is, per unit span,

$$\bar T=\pi\rho b\,\omega^2h_0^2\,(F^2+G^2), \qquad \bar P=\pi\rho bU\,\omega^2h_0^2\,F .$$

The first is the mean thrust; the second, the power the plate’s actuator supplies. Dividing, the efficiency is

$$\eta=\frac{\bar TU}{\bar P}=\frac{F^2+G^2}{F}.$$

It depends only on $k$ and contains no amplitude: Theodorsen’s linear theory scales everything with the square of the motion. When $k\to0$, $F\to1$ and $G\to0$, so $\eta\to1$. It’s perfectly efficient: an infinitely slow, gentle flap is the most efficient of all. When $k\to\infty$, $F\to\frac12$ and $G\to0$, and $\eta\to\frac12$. Half the power goes into the wake. Between the two there’s no maximum. The efficiency simply falls.

In terms of the Strouhal number, $St=2h_0f/U$, the thrust coefficient on the chord is

$$C_T=\pi^3St^2\,(F^2+G^2).$$

So inviscid theory has only one recommendation, which is to flap as little as possible with as large a surface as possible. A stream-tube argument agrees: thrust from a large area of small acceleration wastes least. The disc says exactly the same about a propeller. Why then does nothing cruise at $St=0.05$?

The drag term #

Because cruising isn’t free. An animal in steady flight must produce thrust equal to its drag, and that’s a given. There’s a skin friction on the fin itself, and there’s a body to push. Let $C_D$ be the total drag expressed on the fin’s own area. The useful output is what’s left over:

$$\eta_{\rm net}=\frac{(C_T-C_D)\,U}{P},$$

where the thrust’s own rate of work is only partly useful. If the plate merely flaps enough to overcome its own friction, it delivers nothing, so $\eta_{\rm net}=0$. Further up, $C_T\gg C_D$, the drag doesn’t matter, and $\eta_{\rm net}$ falls back to Garrick’s curve. So between the two there must be a peak.

There it is. For a drag of 0.1 at a plunge of one chord, the best Strouhal number is 0.24. There’s no wake instability in the calculation and no animal. The band arises from the competition between two curves, one rising as $St^2$ against a fixed cost and one falling slowly from Garrick’s, and the position of the peak depends directly on the drag that has to be paid.

The next step is to ask what drag coefficient a real animal presents, on the area of its fin. A trout’s caudal fin has an area of a fifth or less of its total wetted area. A skin friction coefficient of about 0.005 on the body’s wetted area, a form-drag factor of 1.2 or so and the skin friction of the fin together give something around 0.05 to 0.1, on the fin’s area, more if the fish undulates; and a flying bird, with its body and legs, has a drag coefficient on its wing area of order 0.03 to 0.1 in cruising flight, depending on its aspect ratio. Those are only estimates, but they land in the part of the curve at $St$ around 0.15 to 0.3. So a large fraction of the observation seems to require nothing beyond the drag and the plate. That doesn’t falsify the wake argument. It says the two have a common range and can’t readily be separated by observing where animals cruise.

A broad peak, and what that means #

The peak is not sharp, and that matters more than its position. At a drag of 0.1 and a plunge of a chord, the net efficiency is 0.39 at a Strouhal number of 0.1, then 0.55 at 0.15, 0.58 at 0.2, 0.59 at 0.24, 0.585 at 0.3, 0.57 at 0.4 and 0.56 at 0.5. So below the peak the curve falls off a cliff, and above it hardly anything happens. An animal choosing its stroke therefore pays heavily for going too slowly, where its thrust is hardly covering its drag, and very little for going a good way too fast.

That asymmetry predicts where animals should sit. Anything with an uncertain drag, a fish in a current or a bird in gusty air, would do better to err on the high side, and the band then ought to extend further above the peak than below it. It’s a broad target, which is much the same statement as the observed range of 0.2 to 0.4: a factor of two in Strouhal number is about a five per cent change in efficiency. It also limits how much weight a “tuned” explanation can bear, since nothing in such a flat optimum needs tuning at all. Comparing a dolphin’s stroke of 0.6 metres at two beats a second and six metres a second gives 0.2. A cruising pigeon, at about fifteen metres a second with six or seven beats a second and a wingtip stroke of about 0.4 metres, has about 0.17. Each is a plain measurement a reader could make with a stopwatch and a video, and each lands on the plateau’s left-hand shoulder.

The steep left side also has a simple reading. Below the peak, the animal’s thrust barely pays the drag, and the drag term’s share of thrust is what costs it. So what makes a stroke’s Strouhal number low is low drag: any smooth, streamlined body can afford a lower Strouhal number. The calculation therefore predicts that the better-streamlined an animal, the nearer the bottom of the band it should cruise, which a survey could test. A wake told what to do wouldn’t by itself give that ordering.

The amplitude #

There’s a second variable in the problem that the Strouhal number doesn’t contain. At a fixed $St$, a plate can have a small amplitude at a high frequency, or a large amplitude at a low one. Those are at different $k$: $k=\pi St/(2h_0/c)$. So at a fixed $St$ a larger stroke means a lower $k$ and thus a higher efficiency in the inviscid limit. A plate gains from having a large amplitude.

That’s visible in the figure: a best-efficiency curve at two chords of plunge sits at a lower Strouhal number for each drag and has a higher peak than the curve at half a chord. It also explains the shape of a swimming animal’s motion, since a tuna’s tail moves about a chord side to side, or more, and most animals flap their wings through a stroke comparable to the wing’s length. In this theory, a small amplitude and a large frequency never make sense. A plate with a low amplitude of a quarter of a chord doesn’t even have an interior peak beyond a drag of about 0.05, because by the time its thrust has paid the drag, it’s already at high $k$ and Garrick’s efficiency has reached its floor of a half.

So the theory’s two-parameter map gives a clear statement: animals must have large stroke amplitudes, to be near the favourable end of the efficiency curve, and then a Strouhal number that’s set by their drag. That’s a different kind of argument for why the numbers should hold across species. Since skin friction and geometry vary rather little between fliers of similar shape, the Strouhal number doesn’t either.

Where the plate would stall #

The theory has a limit, and it is a severe one.

At $St=0.3$, a plate that only plunges would see 43 degrees. No attached flow survives that. So Garrick’s calculation, taken literally, would fail well before its own best point at any drag that gives an interesting Strouhal number. Real flappers pitch as they plunge, feathering the foil to follow the path, so the effective angle at mid-stroke is kept to around 15–25°, at a cost in thrust for a given Strouhal number. The later work by Anderson, Triantafyllou and others in 1998 showed in a water tunnel that a foil with appropriate pitch reaches efficiencies above 80 per cent at $St$ of 0.3–0.4, which has the sign of this calculation and a larger peak value. It also showed a role for a leading-edge vortex.

Pitching changes the formula’s terms, but the structure stays. A plate that pitches and plunges in attached flow still has a falling inviscid efficiency against $k$, with the phase of the pitch shifting it, and it still gets its peak in the net efficiency from the drag it has to pay. A leading-edge vortex, once it forms, can provide extra thrust at high angles, and insects seem to depend on it, as the calculation with no memory in it showed for bumblebees. In that regime this calculation is not the right one, and small animals, whose flows are viscous and separated, are more likely to be there.

What it means for the wake #

Is the wake then irrelevant? Not at all. A flapping plate does generate a reverse Kármán street, at the same frequency as the flap, and the street’s own stability has a most-amplified frequency. Triantafyllou’s argument is that this is where the wake’s energy capture is best, and that real foils respond there. Moored and his colleagues in later work showed a resonance of that kind for a flexible fin, with a real peak. There’s also experimental evidence for it at the same range of Strouhal numbers, where flow and body can’t be separated.

The two arguments aren’t rivals so much as two statements on one curve. A real swimmer’s body, with its drag and its stroke, sets the Strouhal number at which it can afford to cruise. If that happens to coincide with a band in which its wake is also receptive, all the better, and evolution would then have had two reasons rather than one. What this calculation removes is the implication that a flapping foil must have a best Strouhal number by virtue of being one. The formation of animals that use one another’s wakes is a further complication that neither argument has in it.

What the picture cannot show #

A viscous wake. The wake in the calculation is a sheet at the plate’s level that carries circulation away, with no roll-up. Real wakes roll up into discrete vortices at the amplitudes that count, and their sideways spacing changes the induced flow at the plate. A finite-amplitude panel method shows modest shifts of these curves.

The body’s coupling. A swimming body’s drag isn’t a constant. Lighthill showed in 1971 and others have measured that an undulating body’s drag rises by a large factor, the “Bone–Lighthill boundary-layer thinning”, so that the drag a fish’s fin has to pay depends on how hard it’s flapping. That effect is absent here, and it would move each animal’s peak somewhat to the right.

Three dimensions. A real wing or fin has a finite aspect ratio, and its tip vortices take part of the power: a bird’s wing is closer to a lifting line than to a two-dimensional plate. Efficiencies would be lower across the board, and the shape of the curve is preserved.

Lift. Cruising birds carry weight on the same wing that provides thrust, so a flapping bird isn’t doing what this plate does. Weight support adds its own induced power, and the bird’s flight is only partly a thrust problem. It has less bearing on a fish, whose body is neutrally buoyant.

The convention #

The Strouhal number is $St=2h_0f/U$, on the full width of the stroke, $2h_0$, so it isn’t the usual one on a body’s width. The reduced frequency is $k=\omega b/U$, on the half-chord. Plunge amplitude is written $h_0/c$, with $c$ the chord. Thrust, drag and power coefficients are per unit span on the chord, $\frac12\rho U^2c$ and $\frac12\rho U^3c$. Net efficiency uses the total thrust’s power as the denominator and thrust minus the stated drag as the output, so that a value of zero is a flapper which only pays its own way. Leading-edge suction is taken as fully realised, as it is in Garrick’s formula, which would be over-optimistic for a thick or blunt edge.

How it was checked #

The two limits of Theodorsen’s function and the quasi-steady thrust are independent checks, since they depend on the Bessel function evaluation over the range of reduced frequency used. The function itself is the one that an earlier unsteady-lift calculation here had verified against its published values. The model refuses a zero Strouhal number, a zero amplitude and a negative drag.

Who found it, and when #

Knoller’s and Betz’s explanation of flapping thrust dates from 1909 and 1912, and Theodorsen’s function from 1935. Garrick’s propulsion paper is from 1936, Lighthill’s elongated-body theory from 1970 and 1971. Triantafyllou, Triantafyllou and Grosenbaugh’s observations on fish and foils date from 1991 and 1993, Anderson and colleagues’ efficiency measurements from 1998, and Taylor, Nudds and Thomas’s survey of fliers from 2003.

Still open: where the pitching foil’s peak goes #

A plain plunge at the band’s Strouhal number is far beyond stall, so the plate above isn’t the object that animals actually use. The next calculation would add pitching in phase with the plunge to Garrick’s formula, which he gave in full, with the pitch set to hold the largest effective angle at a stated value, and find the best Strouhal number at each drag and amplitude under that constraint. It would ask whether keeping the angle below stall moves the net-efficiency peak into the measured band at realistic drags, or moves it away, and so how much of the observed band can be got with no separation at all, before a leading-edge vortex needs to be called on.