Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: A smooth picture proves nothing — page 30

Page 30 of 31, continuing through the 272 essays this motif runs through.

272 essays carry this thread — page 30 of 31.

Moving the saddles never lowers the quarter; fast large wobbles raise it. The smallest Stokes number at which a cloud of particles in the oscillating lattice folds within sixty time units, against the oscillation's amplitude ε, at frequencies of 1, 3 and 10 times the saddles' strain rate. With ε = 0 it is the steady lattice's 0.251. Moving the lattice never takes it below a quarter; at ε = 1 it is 0.283, 0.311 and 0.324 at the three frequencies. Flows and fields

Wobbling saddles keep the quarter, and raise it

Heavy particles in a steady lattice of vortices fold onto the cell walls only above a Stokes number of a quarter, the converging saddles' own threshold. Random flows fold at every Stokes number, so it was natural to suspect the quarter of belonging to steady saddles. Set the lattice wobbling and the particles cross from cell to cell, seventy per cent of them in forty time units, and still none folds below a quarter. The threshold belongs to the strongest strain the flow has; moving the saddles only moves particles out of reach of it, and raises the threshold.

 Viscosity slows every ripple, and the short ones most. The growth rate of a varicose ripple on a liquid jet against its wavenumber times the jet's radius, at Ohnesorge numbers of 0, 0.1, 1 and 10, in units of the capillary time. Every ripple longer than the circumference grows. Viscosity damps each one in proportion to the square of its wavenumber, so the short ones lose most: the fastest moves from ka = 0.697 inviscid to 0.344 at Oh = 1 and 0.123 at Oh = 10, and its rate falls from 0.343 to 0.0114. Regimes and numbers

Viscosity lets a jet break, later and into bigger drops

A thread of honey falls for metres before it breaks, and a thread of water for centimetres, which suggests that viscosity holds a jet together. It does not: it cannot stop any ripple longer than the jet's circumference from growing. It slows them, the short ones most, so the ripple that wins is longer, it takes the viscous time rather than the capillary one to win, and each drop it makes is bigger. The wavelength grows as the square root of the Ohnesorge number and the drop as its sixth root.

 Above the logarithm the profile lifts away: the wake. The mean velocity in wall units against the distance from the wall, for a flat-plate layer at Reθ = 10⁴ (δ⁺ = 3484), from Spalding's inner law alone and with Coles's wake of strength Π = 0.3, 0.55 and 1. Below about a fifth of the layer the curves coincide on the logarithm; above it the wake lifts the profile by up to 2Π/κ, 2.68 wall units for a flat plate — a tenth of the edge velocity, and the part of the profile a log law cannot describe. Transition and turbulence

The wake is a tenth of the velocity and a third of the displacement

The logarithmic law describes a band in the middle of a turbulent boundary layer, and above it the profile lifts away by an amount Coles called the wake. It is a tenth of the edge velocity, so it looks like a correction. It is not: it carries a third of the layer's displacement, it is what turns the log law into a friction law for a boundary layer, and with it the friction comes out within a few per cent of a measured correlation that contains no logarithm at all.

 The waves a wing makes on the sea cost it almost nothing. The wave drag of a wing skimming deep water, as a fraction of its lift, against speed, at heights of half a metre, one and two. Each curve peaks at the speed √(2gh) — 4.43 m/s at one metre, whatever the chord — and there the wave drag is 4.4·10⁻⁴ of the lift, the density ratio of air to water times Cₗ c/4eh. At a cruise of 50 m/s it is 9.31·10⁻⁶ of the lift. What is taught wrongly

A wing over the sea barely touches it

A wing skimming the water presses its whole weight onto the surface beneath it, spread over a width about its own height. The sea is a deformable ground, and one might expect it to give way: a trough under the wing, waves behind, a drag to pay. It barely notices. Air is eight hundred times lighter than water, and that one ratio sets the dent at millimetres, the waves at a few millimetres high, and the wave drag at less than a two-thousandth of the lift even at the worst speed.

 The camber lines that carry them. The camber lines that carry the four loads at their ideal angles, heights in chords at a design lift coefficient of one. The uniform load's line is symmetric about mid-chord, 5.52% high; tapering the load over the last fifth moves the peak to 0.515 of the chord and raises it to 6.79%; tapering it over the whole chord, a = 0, puts the peak at 0.323. Circulation and lift

A load carried to the trailing edge needs a hook

Thin-aerofoil theory runs backwards: ask for a chordwise load and it returns the camber line that carries it. Ask for the simplest load of all, the same everywhere, and the line comes back with a vertical tangent at both ends. The one at the nose is the price of a clean entry; the one at the tail is the Kutta condition being broken, and the hook it makes is what a Gurney flap is. Every NACA a-series line is a way of not paying it.

 Zero wave drag at one Mach number, inside a loop it cannot enter from below. The wave drag of the pair, per unit of one element's chord, against Mach number, beside a single diamond of the pair's combined thickness. The biplane's drag is zero at its design Mach number, 1.944, and rises either side. Shaded: below 1.6 the channel cannot pass the flow at all; between 1.6 and 2.15 it can, but only if it was started above the band. Accelerating from subsonic, the biplane stays choked until Mach 2.15, past its design point. Compressible flow

Busemann's biplane has to be flown past its design point

Two half-diamonds facing across a channel cancel each other's waves at one Mach number and carry their thickness at no wave drag. The cancellation needs the channel narrowed at its middle, and a narrowed supersonic channel will not swallow the shock that forms in it on the way up. A biplane 3 per cent thick per element, designed for Mach 1.94, stays choked until Mach 2.15, and only one that is thinner than about 2.6 per cent can start by itself at any speed.

 Long waves grow, short ones are held flat, and one in between grows fastest. The growth rate of a ripple on the interface where air displaces oil in a Hele-Shaw cell, against its wavenumber, at displacement speeds of 0.5, 1 and 2 mm/s. The viscosity contrast drives every wavelength at a rate proportional to its wavenumber; surface tension holds back short ones as the cube. At 1 mm/s the cut-off is 0.245 per mm and the fastest wavenumber 0.141 per mm, a wavelength of 44.4 mm, growing at 0.0942 per second. Ideal flow

The fastest finger is set by the gap and one number

Push air into oil between two glass plates and the interface breaks into fingers. The averaged law of the cell makes the instability exact in its linear stage: every wavelength is driven in proportion to its wavenumber, surface tension holds back the short ones as the cube, and the fastest finger is π gap widths divided by the square root of a capillary number. A narrow channel stays flat; a wide one chooses how many fingers to make; and a growing bubble of air makes more of them the larger it gets.

 The source points along the span; the flow does not. Round a swept circular cylinder, the share of the wall's vorticity source that points along the local direction of the flow outside the layer, against the angle from the attachment line, at sweeps of 10°, 20°, 35° and 50°. The source lies along the span everywhere. At the attachment line the flow does too and the share is one; at the suction peak, 90°, it is 0.0878, 0.179, 0.33 and 0.512. What is not along the flow is the ordinary across-the-flow vorticity a two-dimensional layer has. Flows and fields

A swept wall makes vorticity along its isobars, not across its flow

A still wall in a pressure gradient puts vorticity into the fluid at a rate set by the gradient, and the source is a vector: it lies in the wall, along the isobars. On an unswept body the isobars run across the flow and so does the vorticity, which is the ordinary boundary layer. On a swept one the isobars run along the span and the flow does not, so part of every new vortex line points along the flow — half of it, over the accelerating front of a cylinder swept 35°. That part is where a swept wing's crossflow comes from.

 The parcel over the top arrives first, and never waits for the other. Two parcels released together far upstream, a whisker above and below the streamline that divides at the nose of a Joukowski section 11.8% thick at 4°. Both crawl past the nose; then the upper one is carried over the top faster and reaches the far line 1.13 time units ahead — 0.279 chord-transit times — against the 1.59 that the circulation divided by the speed squared predicts. Behind the section they travel on side by side at the stream's speed and the gap between them never closes. What is taught wrongly

The upper parcel leads by the circulation

Transit time looked like a quantity that could not be measured: a parcel released near the dividing streamline crawls past the nose for as long as one likes. The difference between two parcels' transit times, one either side of that streamline, does not crawl. It converges, and on a thin section it is the circulation divided by the square of the speed — the lift, measured in seconds. The upper parcel arrives first by that much, and behind the wing the two never close the gap.

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