Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: The exact theory is wrong — page 30

Page 30 of 31, continuing through the 276 essays this motif runs through.

276 essays carry this thread — page 30 of 31.

A wake held at the stern moves the volume aft a third as far. The least-resistance hull's centre of volume, in per cent of the half-length from midships, negative aft, against the design Froude number, holding the Wigley hull's length, draught, depth profile and displacement. With the linear wake it sits 2.8 per cent aft at Fr 0.30; with the shaped wake of the same propeller-disc fraction, 0.86 if uniform in depth, 1.5 if deepest at the waterline, and 0.11 if deepest at the keel, where the sources make the fewest waves. Regimes and numbers

A wake held at the stern keeps the bulb and loses the lean

A wake that slows the water steadily from bow to stern makes Michell's least-resistance hull fuller aft and puts its bulb at the bow. A real ship's wake is nearly nothing along most of the hull and strong only in its last few metres, deepest at the keel. Held there, with the same wake at the propeller, it moves the hull's volume aft by a twentieth to a half as much, and it keeps a third to two-thirds of the bulb's preference for the bow. The lean was a property of the water along the run, and the bulb a property of the water at the stern.

 The drag follows the surfactant load, and a surface pressure of μU pays nearly all of it. How far the drag has climbed from the clean bubble's to the rigid sphere's, against the mean surfactant load over the whole bubble as a surface pressure in units of μU. A load of 0.2 μU makes a 60° cap and a third of the climb; 0.56 a 90° cap and 70 per cent of it; 1 a 120° cap and 94 per cent. The dashed line is the cap's share of the surface for the same caps: the drag runs ahead of the area covered. Viscosity

A thousandth of a monolayer holds a bubble still

A clean bubble rising slowly through water feels two-thirds of a rigid sphere's drag, and real bubbles almost never do, because surfactant swept to the rear holds the surface still over a cap there. Solving the flow with the cap in it shows how little that takes. The drag runs ahead of the area covered — half-way to rigid with a third of the surface held — and the surfactant needed is set by the viscous stress, not by the surface tension. For a bubble a tenth of a millimetre across, a thousandth of a monolayer, spread as a cap, makes it rise within a few per cent of a solid ball.

 The ring is drawn off while the core keeps its shape. The edges of a vortex with a core of twice the ring's vorticity out to six-tenths of its radius, at four times, in a strain raised to 0.245 — just past the 0.227 that strips its ring. The strain stretches along the horizontal. Both edges lean into the ellipses of a strained vortex; then the ring's edge is pulled out at its two tips into arms, while the core inside stays close to an ellipse. Ideal flow

A strained vortex loses its ring at the limit of what it holds

A uniform patch of vorticity survives a strain up to 0.150 of its vorticity and no further. A real vortex is not uniform: its vorticity falls off outward, and in a strain its weak outer layers are drawn off while its core survives. The obvious way to predict how much survives is to apply the uniform limit layer by layer, each layer to its own vorticity. It is wrong by up to a factor of three. Built from a core inside a ring and followed by contour dynamics, the ring is stripped not at the limit of its own vorticity but at the limit of all the vorticity inside its edge, and a little after it.

 Below the bubbles' resonance the mixture is slow; above its stop band it is faster than water. The phase speed of sound in water carrying millimetre air bubbles, against frequency, at void fractions of 10⁻⁴, 10⁻³ and 10⁻². At low frequency each is Wood's mixture speed with the gas isothermal — 312 m/s at 10⁻³, not the 366 an adiabatic gas would give. Approaching the bubbles' resonance near 3.2 kHz the speed falls further, then jumps through a stop band in which the wave hardly propagates, and above it the phase speed exceeds water's — at 10⁻³ 1640 m/s at 30 kHz — before returning to 1481 m/s from above. Compressible flow

Above their resonance, bubbles make water faster

Wood's formula says a pinch of air makes water's sound slower than air's, and it is right at low frequency. Each bubble, though, is a spring with water for its mass, and the wave drives it. Below the bubbles' resonance the mixture is slow; through a band above it no sound propagates at all; and above that band the same bubbly water carries sound faster than pure water does. Even the slow end is not quite Wood's, because a slowly squeezed bubble keeps its heat.

 Every ball's best angle is on one curve. The launch angle that carries a ball furthest over level ground, against β, the square of its launch speed measured in terminal speeds. At small β the air does nothing and the answer is 45°. It falls slowly as β grows: 41° at β = 1, 32.6° at 10 and 20.3° at 1000. A shot put sits at β = 0.009 and 44.9°, a golf ball struck without spin at β = 2.39 and 38.1°, a shuttlecock hit at 30 m/s at β = 19.5 and 30.2°. Fluids at work

The terminal speed sets the best launch angle

In a vacuum every ball goes furthest at forty-five degrees. In air the best angle is lower, and it is lower by an amount that depends on one number only: how many times its own terminal speed the ball is launched at. A shot put, a baseball and a shuttlecock are three points on one curve, which falls from 45° to 41° at once and then needs a thousandfold more speed to reach 20°.

 The water climbs the body, and the wetted width outruns the drawing. The wetted half-width against penetration for a wedge of 10° deadrise and a circular cylinder of unit radius: where the body crosses the undisturbed level, von Kármán's width, and where the risen water meets it, Wagner's. For the wedge Wagner's width is π/2 = 1.571 times the geometric one at every depth; for the circle it is √2 = 1.414 times, at small penetration. Ideal flow

Water climbs a falling wedge, and the load comes from the climbing

A hull that strikes the sea sets the water under it moving, and the force is the rate at which it does so. The obvious estimate measures the wetted width where the hull crosses the undisturbed surface. But the water does not wait: pushed aside, it rises up the hull and wets it sooner, a factor π/2 wider for a wedge and √2 for a round bottom. The force carries that factor twice, and the peak pressure, where a thin jet leaves the hull, carries it squared on a cotangent that grows without bound as the bottom flattens.

 A weak push stirs the whole fluid; a strong one makes a jet. Streamlines of the exact solution in a plane through the force, for a weak force (jet Reynolds number 1, left) and a strong one (100, right), the force pointing right from the origin. The dashed line is the cone inside which the fluid moves outwards: 89.4° from the axis for the weak force, nearly a hemisphere, and 22.8° for the strong one. Outside it the fluid is drawn back towards the origin from every direction and turned into the jet. Viscosity

A point force makes a jet only when it is strong

Push on a fluid at one point and there is an exact solution of the full Navier–Stokes equations for what follows, at any strength. Weak, it is Stokes's point force: fluid pushed forward over a whole hemisphere and drawn in behind. Strong, it is Schlichting's slender jet, with fluid drawn in from every direction outside a narrowing cone. One constant joins them, and it says how strong a push must be before the boundary-layer jet everyone uses is right — at the axis by a jet Reynolds number of a hundred, at the edges only by a thousand.

 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.

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