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

Images — where it appears

Singularities placed outside the fluid so that a wall's boundary condition is met by symmetry rather than imposed. The construction is exact and costs no extra unknowns, and an infinite row of them is what a pair of tunnel walls requires.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

What the ground actually gives a wing. Induced drag near the ground as a fraction of the same wing's in free air, against height in spans, at constant lift. The ground is a plane of symmetry, so the wake is joined by a mirrored wake of opposite circulation below it, and the upwash from that image is what takes the drag away. At a tenth of a span the wing keeps 0.516 of its induced drag — a saving of 48 per cent — and by a span and a half the effect is 1.3 per cent and going. The lift is held fixed all the way along this curve: what the ground gives here is not more lift, it is the same lift for less drag, and the two are different claims about a landing aeroplane.

The cushion that is not there

Ground effect is usually explained as air trapped under the wing and compressed into a cushion. At the heights an aeroplane actually flies at, the air under the wing is moving at four fifths of flight speed and most of what the ground gives is a drag saving the story never mentions.

misconceptions · Ground cushion 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 The wall is a boundary condition, solved for rather than reflected. A model's trailing vortices in a closed working section, with the sheet of sources that makes the wall a wall drawn as a displacement of the wall itself. The interference is an upwash — the model looks better than it is, and the correction factor comes to 0.1242 against the exact value 0.1250 that the image at the inverse point gives for a circle. The arrows are the interference velocity alone, with the model's own downwash removed, drawn to a common scale set by the longest of them.

The walls are in the answer

A wind tunnel measures a wing in a box the aeroplane will never fly in, and the box is worth a fifteenth of the induced drag. The correction is exactly an eighth for a closed circular section and exactly minus an eighth for an open jet, and the sign is the whole argument.

circulation · Tunnel interference A body of no volume with a finite added mass. Thin an ellipse towards a plate and the fluid it displaces goes to nothing while its broadside added mass does not move at all — it stays at πρa² to the last digit. Whatever added mass measures, it is not how much fluid a body carries with it: the ratio of the two diverges as the reciprocal of the thickness, reaching a million at a thickness ratio of a millionth.

The mass a body has to borrow

Accelerate a sphere through water and it resists as though it were half again as heavy. The half is exact, it is a rational number rather than a measurement, and almost everything a reader infers from it about carried fluid is false.

inviscid · Impulse One coefficient, two errors, two places they vanish. The two wall-interference errors against the slot parameter, in units of the tunnel's half-height. Both are positive for an open jet at the left and negative for a solid wall at the right, so each passes through zero — the blockage at 1.184 and the streamline curvature at 1.590. A wall has one coefficient and the two zeros are a third of a tunnel height apart.

One coefficient, two errors

A closed tunnel's walls push a measurement one way and an open jet's push it the other, so a wall somewhere between them should push it neither. There is such a wall, its coefficient is a length, and it nulls the blockage at 1.184 tunnel half-heights — and the streamline curvature at 1.590.

circulation · Tunnel interference The water a submerged circle carries depends on how fast it is shaken. The added mass of a horizontal circular cylinder heaving under deep water, as a multiple of ρπa², against the wavenumber of the waves its frequency makes, Ka = ω²a/g, for five depths of its centre, with the depth set drawn dark: 1.1 radii, least added mass -0.182. Slowly shaken, every depth borrows more than its free-space mass; quickly shaken, less. Between, each curve rises to a peak and then falls through a trough, and for the shallowest circle the trough goes below zero — to -0.182 at Ka = 0.76.

The borrowed mass that goes negative

A body's added mass is taught as the water it carries with it, fixed by its shape. Put the body under a surface that can make waves and the amount depends on how fast it is shaken, runs from a wall's value to a free boundary's and past both, and for a circle close enough to the surface falls below zero.

misconceptions · Ground cushion

Named alongside it

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

Potential flowBoundary conditionCirculationMeasurementAdded massBlockageCorrectionDownwashInduced dragKinetic energyLift coefficientMisconception

All concepts