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

Frequency response — where it appears

The magnitude and phase with which a system follows a forcing, as a function of how fast the forcing is. A quasi-steady model asserts that the magnitude is one and the phase is zero, and the phase is usually the half that decides whether something is stable.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

The stress a closure predicts, and the stress there is. The Reynolds-stress anisotropy through a step change in the strain rate, against what an eddy viscosity gives — which is the equilibrium value at every instant. The real stress takes about a turnover to get there, and during that turnover the closure is wrong by up to sixty per cent.

A closure with no memory at all

An eddy viscosity says the Reynolds stress is the mean strain rate times a number, now. The stress it is standing in for takes a turnover to arrive, so the closure is the zero-frequency limit of a response that has a lag in it — and the curve it is the limit of is the same shape as an aerofoil's lift deficiency.

turbulence · Closure A particle is a low-pass filter. The fraction of a fluctuating flow's velocity that a particle follows, and the phase by which it lags, against the Stokes number. At one the particle follows 71 per cent of the motion and lags by 45 degrees — which is the corner frequency of a first-order filter, arrived at from mechanics rather than from electronics.

A particle is a low-pass filter

A seeding particle does not report the flow; it reports the flow through its own transfer function. At a Stokes number of one it follows 71 per cent of the motion and lags it by 45 degrees, and a fifty-micron droplet at a kilohertz is following two per cent of what it is supposed to be measuring.

regimes · Particle Eight numbers, one construction. The dimensionless groups this collection has produced, placed on a logarithmic axis at a representative value. Each is a memory time divided by a process time, each was named separately in a different field, and each decides the same question: whether the past is still present.

Every memory number is one time over another

These essays produced eight dimensionless groups in eight different fields, named after eight different people, spanning a factor of four hundred in value. Written out, all eight are a memory time divided by a process time, and all eight govern the same curve.

regimes · Dimensional A gust spectrum at a point, across a span, and as a load. The spectrum of vertical gust velocity at a single point — von Kármán's, falling as the minus five-thirds power — and the same gust averaged across a wing whose span is a tenth of the turbulence scale and across one as wide as the scale. Averaging removes the wavenumbers shorter than the span, one power steeper. Through Sears' filter for a chord of a five-hundredth of the scale, the tenth-span load falls as the minus eleven-thirds power, steep enough to converge.

The span takes away the infinity, not the gust

A two-dimensional wing in turbulence reverses its load infinitely often, because the gust spectrum never falls fast enough. A real wing averages the gust across its span, and the average keeps ninety-five per cent of the gust's strength and removes the infinity. The load's crossing rate settles at about two per turbulence scale flown and stops depending on where anyone cut the spectrum off.

circulation · Gust An elliptic wing filters the gust like a round aperture. How much of a gust component with spanwise wavenumber k₂ survives the span average, against κ = k₂b/2, for three weights. A uniform strip is a slit and its filter is sinc², falling as κ⁻² between its zeros. An elliptic loading is a circular aperture and its filter is the Airy pattern (2J₁(κ)/κ)², whose first zero is at 3.83 and whose envelope falls as κ⁻³, but which stays near one to larger κ: the ellipse's weight is narrower, so its filter is wider. A rectangular wing of aspect ratio nine weights the span by its own loading and lies between, with the ellipse's κ⁻³ fall because its loading too goes to zero at the tips as a square root.

A wing averages a gust through its own loading

A finite span averages a turbulent gust and so tames the load it causes, but it does not average uniformly. The reverse-flow theorem says exactly how it weights the span: by the loading the wing makes at a uniform incidence. An elliptic wing therefore filters the gust the way a round aperture diffracts light, passes eight per cent more of its short scales than a uniform strip, and crosses its mean load up to four per cent more often.

circulation · Gust Heading into the sea, follow the long waves and fly level through the short. The dinghy's mean drag against the crossover encounter frequency below which it follows the surface, in units of the sea's peak frequency, heading into seas of significant height 0.2, 0.4 and 0.6 m. Following everything — the right-hand end — costs the heave's lift, and flying level through everything — the left — costs the foil's swinging depth and, in the roughest sea, breaches. Between them each sea has a best crossover: for 0.4 m at 2.92 times the peak frequency, 67.6 N against 67.9 flying level and 199 following.

A foil in a random sea follows it up to its peak

In a regular wave a foiling boat chooses between holding its height and following the surface. A real sea is a spectrum, and a wand that filters its signal can do both at once: follow the long waves and fly level through the short. The best place to divide them is close to the frequency of the sea's own peak, the divided control beats either pure strategy, and in a rough sea it is the only one of the three that keeps the foil a safe distance under the surface and the drag near its calm-water value.

applied · Sailing A root's bending moment weighs the gust by the loading of a wing twisted like its moment arm. Across the span of an elliptic wing of aspect ratio 9, each normalised to unit area: the lift's weight, the loading at uniform incidence; the two roots' bending moment's, the loading of a wing washed in as |y|, with a hole at the root; and the right root's alone, the loading of a wing washed in as y on the right half and not at all on the left. The one-root weight reaches across to the left wing through the induced flow, and it is not symmetric.

One wing root feels the rolling gust its lift cannot

The reverse-flow theorem says the gust weight of any load is the loading the wing would carry at an incidence shaped like that load's arm. For the lift that is the familiar uniform-incidence loading. For the bending moment at one root it is the loading of a wing washed in towards one tip only — a weight that is not symmetric, and so passes the rolling half of a gust that the lift and the two roots together cancel. That half is a sliver of the variance and adds up to a fifth to how often the root's moment crosses its mean.

circulation · Gust A soft filter flies through most of the waves it is meant to follow, and follows them late. Against the encounter frequency over the crossover, for the ideal cut and the first- and second-order complementary pairs: left, the share of a wave's height the boat flies through rather than follows; right, how far the followed part lags the surface, in degrees. At the crossover itself the first-order pair flies through 0.707 of each wave and follows it 45° late; the second-order pair 1.22 and 90°.

A wand that lags its sea makes flying level the better bet

A foiling dinghy whose height control divides the sea sharply follows the long waves, flies level through the short and beats both pure strategies. Give the division the lag a real wand or filter has and the gain disappears in a moderate sea; in a rough one no crossover both keeps the foil under and spares it from pulling down. A second-order filter, which cuts more sharply, does worse rather than better, because it flies through more of the sea near its crossover than the sea contains.

applied · Sailing Sweep slows every load's crossings by about a tenth at 35°, and leaves one root's excess over the lift standing. Against the sweep, for an elliptic wing of aspect ratio nine whose span is eight hundredths of the turbulence's integral scale: left, the lift's and one root's crossing rates as fractions of the straight wing's; right, one root's rate over the lift's. At 35° the lift crosses its mean 0.899 times as often as on the straight wing and the root 0.887; the root's excess over the lift goes from 1.19 to 1.17, and at 45° to 1.16.

Sweep slows a wing root's gust cycles and leaves their excess over the lift

On a straight wing every station meets a gust at the same instant, and one wing root's bending moment crosses its mean nearly a fifth more often than the lift, because it reads the rolling half of the gust. Sweep the wing back and the outer stations meet the gust later, by their distance aft. That delay averages the gust along the flight path as a second, softer chord would: at 35° every load crosses its mean about a tenth less often, the root slightly more than the lift since its weight sits further out, and the variance hardly changes at all. The rolling excess survives almost whole.

circulation · Gust

Named alongside it

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

Model limitSpectrumMeasurementTurbulenceGustConvolutionFatigueMemory kernelModel validityReciprocityRegimeRelaxation time

All concepts