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

Particle — where it appears

A small discrete body carried by a flow, either a real one being transported or a tracer added to make the flow visible. It follows the fluid only up to a frequency set by its own inertia, so it is a low-pass filter with a corner rather than a perfect marker.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

A sphere does not go at the speed of the flow it is in. How far a force-free sphere on the axis of a round tube lags the fluid at its own centre, against its size. The lag is two-thirds of the square of the size ratio and comes from the reciprocal theorem in one line: the sphere moves at the average of the ambient flow over its own surface, and a parabolic profile is slower everywhere on that surface than at the middle. At a fifth of the tube's radius the lag is 2.7 per cent, which is the difference between a tracer and a thing being measured.

A force without the flow that makes it

A sphere carried along by a flow does not travel at the speed of the fluid at its centre. It travels at the average of the flow over its own surface — and getting that result needs no solution of the flow around the sphere at all, only the answer to a completely different problem that everybody already knows.

viscous · Reciprocity The one curve of the four that measures a gradient. A row of particles released along a line at one instant, seen at 4 later times, in a layer profile. Each line has been displaced by the local velocity times the elapsed time and by nothing else, so dividing the displacement back out returns the profile exactly — checked here to machine precision at every one of two hundred heights. A streamline, a pathline and a streakline each report where fluid went; this one reports how fast neighbouring fluid was going relative to its neighbour, which is the quantity a boundary layer is made of and the one the other three never show.

The curve that measures a gradient

Three of the four curves drawn through a flow answer the same question — where did the fluid go. The fourth answers a different one. A line of particles released together is displaced by the local velocity and by nothing else, so its shape is the velocity profile, and dividing the elapsed time back out returns that profile exactly rather than approximately.

kinematics · Flow curves A particle released in still fluid, with the history and without. The velocity of a hundred-micron particle released at one metre a second in air, computed with the unsteady Stokes force and with the quasi-steady drag alone. After five particle time constants the one that remembers is still moving nearly four times as fast.

The drag that integrates a whole history

A particle released in still air does not stop exponentially. The unsteady drag on it carries a term that is an integral over everything the particle has already done, weighted by the inverse square root of how long ago — so there is no time constant, and five particle time constants later it is still moving four times faster than the quasi-steady answer allows.

viscous · Diffusion 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

Named alongside it

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

MeasurementConvolutionInstrumentMemory kernelModel validityRegimeRelaxation timeStokes flowAdded massBoundary conditionBoundary layerCreeping flow

All concepts