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

Symmetry — where it appears

A transformation that leaves a problem unchanged, which fixes part of the answer before any calculation. A wall can be replaced by a mirror image, a front-to-back symmetric pressure field can exert no drag, and each is a result obtained by looking rather than solving.

Named by 12 essays across 7 fields — each of them below, with the objects they name alongside it.

Flow net — a free vortex. Two families of curves drawn over the same flow: the streamlines, along which the streamfunction is constant, and the equipotentials, along which the velocity potential is constant. They cross at right angles at every point, because they are the two parts of a single analytic function of position.

A wall made by reflection

Imposing a boundary condition on a plane is work. Putting a mirrored copy of everything on the far side of where the plane would be is not, and the plane then appears on its own — as a consequence of the symmetry rather than as a condition anybody enforced.

inviscid · Images A wing at h/c = 0.4. A wing close to the ground, with the mirrored vortex row that makes the ground a streamline drawn faintly beneath it. The air between wing and ground is squeezed through a narrowing gap, and the wing carries more lift than it would in free air.

The wall that pushes back

A wing near the ground carries more lift than the same wing in free air, and the usual explanation — a cushion of compressed air underneath — is not what the equations say. The ground is a mirror, and what changes is not the pressure under the wing but the circulation the wing is forced to carry.

circulation · Images Flow net — a stream past a cylinder. Two families of curves drawn over the same flow: the streamlines, along which the streamfunction is constant, and the equipotentials, along which the velocity potential is constant. They cross at right angles at every point, because they are the two parts of a single analytic function of position.

The mirror that is a circle

A flat wall is made by reflecting everything in it. A round one is made the same way, except that the mirror is an inversion — the image of a point at distance d sits at a²/d, and a vortex acquires a second image at the centre that nothing about the wall requires.

inviscid · Images Four sections, and what the shape story says about each. A flat plate, a symmetric section twelve per cent thick, a cambered one, and a section with a wavy upper skin. The first two have upper and lower surfaces of exactly equal length; the last has an upper surface two and a half per cent longer than its lower one, which is twice the cambered section's excess.

The wing that is flat, and flies

Refuting the equal-transit story by computing the parcels leaves its premise standing, and the premise is the part most readers believe: that the shape is what makes the lift. It is a claim about shapes, so it is tested with shapes — a flat plate, a symmetric section, and a cambered one flown upside down.

misconceptions · Equal transit Ideal flow past a sphere, drawn in a meridional plane. The Stokes stream function of ideal flow past a sphere, contoured at equal intervals. Its relations to the velocity carry factors of r sin θ that the plane stream function does not have, and differencing it reproduces the closed-form velocity to 10⁻¹¹. The surface speed at the equator is exactly one and a half times the free stream, against twice for a circular cylinder: a three-dimensional body lets the flow past in two directions rather than one.

The one number that runs out at three dimensions

A stream function is one function where the velocity is two, and four essays here are built on it. It exists because the divergence vanishes, it is single-valued only if nothing inside is making fluid, and in three dimensions it is not one function at all.

kinematics · Stream function The loading over the disc in forward flight, at μ = 0.4. Section lift per unit span over the rotor disc, with the flight direction upwards, the advancing side to the right and the reversed-flow region outlined. The loading is not axisymmetric and cannot be: at this advance ratio the advancing blade meets 1.40 times the tip speed and the retreating one 0.60. Every quantity a hover calculation reports as a function of radius is here a function of two variables.

The side that cannot keep up

A hovering rotor is axisymmetric, so one radial distribution of circulation describes the whole disc. Move it forward and the advancing blade meets one and a half times the tip speed while the retreating one meets a half — and lift goes as the square of that. The rotor does not roll over, because the pitch is made a function of azimuth.

circulation · Rotor The image system of a wedge of pi/3. The vortex and its images, with the two walls. Reflection in one wall and then the other generates a dihedral group, and the group is finite exactly when the angle is pi over a whole number — here 2n vortices, alternating in sign round the circle, with both walls streamlines to a part in 10¹⁶.

The corners that can be done with mirrors

The method of images works for a wall and for a circle, and for a corner it works only when the angle is pi over a whole number. At every other angle the reflections never come back, the image set is infinite and dense, and the flow exists anyway — which says the method is a statement about symmetry rather than about fluids.

inviscid · Images The same wavelength drawn as rolls and as hexagons. Plan views of a convecting layer with one critical wavelength, drawn from the amplitude equations' two stable states. On the left, rolls: a single set of parallel bands, rising fluid along one set of lines and sinking along the next. On the right, hexagons: three sets of rolls at 120° to each other with equal amplitudes, whose sum has its maxima on a triangular lattice with spacing 2/√3 of the wavelength, each maximum at the centre of a hexagonal cell. At ε = 0.0250, inside the window, both are stable: rolls with amplitude 0.158 and hexagons with 0.081 in each of their three rolls. A hexagon is not a different kind of cell; it is three roll patterns that the quadratic term lets reinforce one another.

Hexagons remember how the heat was turned up

A layer heated from below convects in rolls, unless its top and bottom are not mirror images of each other. Then three sets of rolls at 120° can feed one another through a term the symmetry used to forbid, hexagonal cells appear before the layer is formally unstable, and there is a range of heating in which rolls and hexagons are both stable — so the pattern a layer shows depends on whether the heat was turned up or down to get there.

turbulence · Convection With a wake, the bulb belongs at the bow. The Wigley hull's wave resistance with a spherical bulb, as a fraction of the bare hull's, against Froude number, with a wake fraction of 0.25: the bulb just ahead of the bow, and the same bulb just behind the stern. Without a wake the two curves are one. With it they separate: the stern's waves are made by slowed water and are weaker, so the bulb's cancelling wave has less to cancel there, and at the design speed of 0.30 the bow bulb leaves 0.443 of the bare resistance and the stern bulb 0.624.

The stern's wake puts the bulb at the bow

Michell's thin-ship integral cannot tell a ship's bow from its stern: reverse any hull and its wave resistance is unchanged, so the least-resistance hull is symmetric and a bulb is worth as much at the stern as at the bow. Real ships are fuller aft and put their bulbs forward. Let the stern's waves be made by water the hull's own boundary layer has slowed, and both follow: the optimum hull leans aft, and a bulb at the bow cuts the waves by far more than the same bulb at the stern.

regimes · Froude 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.

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.

regimes · Froude Two spheres settle faster together than apart, faster in line than side by side, and still faster at thirty radii than either alone. The settling speed of a pair of equal spheres over a single sphere's, against the distance between their centres in radii: along the line of centres, exactly (the bispherical solution) and by Batchelor's far-field series; across it, by the series; dashed, each sphere's Stokeslet alone, 1 + 3/(2ρ) and 1 + 3/(4ρ). Touching in line the pair falls 1.55 times as fast; at 5 radii, 1.286 in line and 1.154 side by side; at 30, 1.05 and 1.025.

Two spheres settling at a slant keep their slant for ever, and three do not

Two equal spheres settling through a viscous liquid fall faster together than either alone — half as fast again if they touch in line, and still five per cent faster thirty radii apart. Slanted, the pair drifts sideways as it falls, by up to three and a half degrees, and keeps the slant, the separation and the drift for as long as it falls: the two spheres always move with exactly the same velocity. Release a third and the shape is lost at once, the middle sphere drawing ahead and the outer two closing in behind it. What a pair keeps and a trio does not is a statement about symmetry, and reciprocity is half of it.

viscous · Reciprocity A sphere two per cent larger overtakes its neighbour round a path that is its own mirror image, and leaves at the offset it came in at. The centre of a sphere two per cent larger than its neighbour, relative to the smaller sphere, as it overtakes it in a viscous liquid, from four horizontal offsets; the circle is the distance at which they would touch. Each path bulges outward as it passes and is symmetric about the smaller sphere's level: from 0.85, 1.5, 2.5, 4 radii the larger sphere leaves at 0.8499, 1.5, 2.5, 4, after closest approaches of 2.27, 2.83, 3.74, 5.15 radii.

A larger sphere overtakes its neighbour and leaves on the line it came in on

Two equal spheres settling together never change their shape. Make one of them two per cent larger and the pair becomes an encounter: the larger sphere sinks past the smaller round a path that bulges out and comes back, slowed sixteenfold at contact but never stopped, and it leaves at exactly the offset it arrived with. The smaller sphere swings seven radii sideways while the larger is above it and gives every radius back while the larger is below. In Stokes flow, meeting a neighbour displaces nothing — unless the spheres touch, and a hundredth of a radius of roughness is enough to make every touching path leave on the same line.

viscous · Reciprocity

Named alongside it

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

Model limitBoundary conditionCirculationLift coefficientMethod of imagesSourceWall interferenceBoundary layerCreeping flowDisplacement hullFroude numberGround effect

All concepts