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

Unsteady flow — where it appears

A flow whose properties at a fixed point change with time. It separates streamlines from pathlines, adds an added-mass force to every acceleration, and makes the mean of the flow a different object with different properties.

Named by 14 essays across 7 fields — each of them below, with the objects they name alongside it.

The fluid a moving cylinder carries with it. Kinetic energy density around a cylinder moving through fluid that is at rest far away. The fluid is not dragged along in a lump: it is pushed aside in front and closes in behind, and the energy in that motion is what has to be supplied to change the body's speed.

The force of getting going

The exact theory says a body moving steadily through an ideal fluid feels no force at all. It does not say the fluid is free. Accelerating the body has to accelerate the fluid too, and the bill for that is exactly the mass of fluid the body displaces.

inviscid · Dalembert A wave that dies within one wavelength — 100 Hz in air. The velocity profile above an oscillating wall, at eight phases of one cycle, with depth in units of δ = √(2ν/ω). The motion is a wave travelling into the fluid, and its amplitude falls by 1/e in the same distance it turns by one radian — so it is dead within about one wavelength, and the fluid three δ up hardly knows the wall is moving at all. This is one of the very few exact solutions the Navier–Stokes equations have. The dashed line is one fixed distance above the wall in millimetres: in these units it climbs as the square root of the frequency, which is the whole of how far the motion reaches.

The wall that shakes

Slide a wall back and forth in its own plane and the fluid above it does not follow — a wave travels upwards into the fluid and dies within one wavelength. The depth it reaches is √(2ν/ω), it contains no length from the geometry at all, and the whole thing is one of the very few exact solutions the Navier–Stokes equations have.

viscous · Exact layer One sheared stream, and the total pressure across it. A parallel shear flow is an exact steady solution of the Euler equations, and the momentum equation requires its static pressure to be uniform. So the total pressure is entirely the dynamic pressure, which varies with the speed — by three and a fifth dynamic heads across this layer, on a flow where the static pressure does not vary at all.

Four Bernoullis and one name

"Bernoulli's equation" names at least four statements with four different constants, three domains of validity and one shared reputation for being misapplied. A single sheared stream separates the first three: its total pressure is constant along every streamline, varies by three dynamic heads across them, and its static pressure never moves at all.

misconceptions · Bernoulli's equation 12 bar from stopping one metre per second. The head at the valve after it shuts, computed by the method of characteristics on a 600 m pipe. The rise is 122.4 m of water, which is ρaΔV/ρg to 1.4e-14 m — and the scheme was told neither ρaΔV nor anything else about the answer. The wave then runs to the reservoir and back every 2.000 s, and with no friction in the model it never decays: a real pipe damps this out in a few tens of cycles.

Stopping water costs more than moving it

Shut a valve on water running at one metre per second and the pressure that appears is twelve bar — not because the water was pushing hard, but because the only way to stop a column of fluid is to send a message back along it, and the message travels at the speed of sound in the pipe.

applied · Water hammer A tube opened at the foot of a reservoir. Speed in a two-metre tube fed by a one-metre head, from the instant it is opened. The quasi-steady answer — Torricelli's √(2gh), which is what dropping the time derivative from Bernoulli's equation gives — is reached at the instant of opening, before any fluid has moved. The true answer is a hyperbolic tangent with a time constant of 2L/√(2gh), and it takes 2.4 seconds to come within one per cent. The whole of that transient is one term.

The pressure that depends on the past

Bernoulli's equation for an unsteady flow has a term nobody writes down and a right-hand side that is a function of time rather than a constant. The term is exactly zero once a flow has started and is the whole of the flow while it is starting, which is why it never appears in an answer and is never negligible in getting to one.

inviscid · Unsteady bernoulli Morison's two terms over one cycle, at KC = 10. The drag term, in phase with the velocity and going as its square; the inertia term, in phase with the acceleration and ninety degrees ahead of it; and their sum, which is what a load cell records. The peak of the total is not the peak of either, and its position in the cycle is the only thing in the record that says how the two are divided.

Long enough to make a wake

A Reynolds number cannot ask whether an oscillating flow gets round a body before it turns and comes back, because it has no time in it. The Keulegan–Carpenter number can, and it decides which of Morison's two terms is the force. What it discards is the phase — and a peak force measurement cannot recover it.

regimes · Keulegan–Carpenter number Stokes' drag stops being the answer almost at once. The damping on a millimetre sphere oscillating in air, divided by Stokes' steady drag on the same sphere, against frequency. The extra term is the sphere's radius over the layer thickness, so it takes over as soon as the layer is thinner than the body — which for a millimetre sphere in air is below one hertz. At a kilohertz the damping is fifteen times the steady value, and the exponent is a half rather than zero.

What a fluid takes out of a swing

The damping a body feels from the air around it is not Stokes' drag, and stops being it far earlier than anybody expects — a millimetre sphere in air is already forty-six per cent above the steady answer at one hertz. Past that the damping rises as the square root of the frequency, and the fluid it is fighting is a shell a fraction of its own size.

viscous · Damping The line the dye actually draws. A streakline in an oscillating uniform stream: everything released from the origin over the last 3.4 units of time, drawn where it has got to at one instant. The parameter along it is release time, not distance and not time of flight, which is why it is a record rather than a curve of the field. The mean speed of 1 exceeds the amplitude of 0.6, so the fluid never reverses and the filament is single-valued in x.

The line the dye actually draws

The streakline is the curve most photographs really show, and of the three curves it is the hardest to compute, because it needs the whole history of the flow. This draws it, in a flow where all four curves have closed forms, and the third curve turns out to be a different kind of object from the other two rather than a third example of the same one.

kinematics · Flow curves A 204 m hammer traded for a 8.57 m swing over 299 seconds. The water level in a 10 m surge tank at the end of a 2 km tunnel 3 m across, carrying 2 m/s, after the turbine is shut off at once, with the level measured from the reservoir's. Without friction it rises to V₀√(L Aₜ/g Aₛ) = 8.57 m and swings with a period 2π√(L Aₛ/g Aₜ) = 299.1 s, the integration agreeing with both closed forms. With the tunnel's 5 m of friction the level starts 5 m below the reservoir, peaks at 5.61 m after 98 s, falls to −3.70 m, and decays. The same tunnel shut at its end with no tank would take the Joukowsky rise of 204 m. The tank does not remove the column's momentum; it gives it a free surface to push against, slowly.

A tank that turns a hammer into a swing

Shut a turbine at the end of a two-kilometre tunnel in two seconds and the valve takes a rise of 256 metres of head. Put a shaft open to the air beside it and the rise is 51, the tunnel never carries the closure as a wave at all, and its water slows instead against a level that climbs for a minute and a half — to a height that is a closed form with the tank's area under a square root.

applied · Water hammer 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 Two forces on one cylinder, a quarter of a cycle apart. The inertia and drag terms of Morison's equation over one wave period, at a Keulegan-Carpenter number of ten. The inertia term follows the acceleration and peaks where the velocity is zero; the drag term follows the velocity and peaks where the acceleration is. They are a quarter of a cycle apart and they are different kinds of quantity.

Two forces, and only one of them remembers

Morison's equation adds an inertia term to a drag term and is usually presented as an empirical patch. It is not: the two terms are the two kinds of memory this collection has been separating, one a function of the present acceleration and one a function of the wake left by the previous half cycle.

regimes · Keulegan–Carpenter number Six ways of reaching one speed. Six velocity histories, all starting from rest and all reaching exactly one at the same moment. Two are ramps, two are eased, one overshoots and comes back, and one goes backwards before it goes forwards.

Everything about the start, except one vector

Six ways of accelerating a body from rest to the same speed produce six force histories with nothing in common — peaks spanning a factor of thirty-nine, two of them negative for part of the journey. The impulse left in the fluid is the same ten-figure number in every case, and so is the energy.

inviscid · Impulse The induced velocity, after a step in thrust. A rotor's induced velocity following a thirty per cent increase in thrust applied at fifty milliseconds. It does not jump: the air the disc has to accelerate has an apparent mass, and the response is a first-order climb to the new momentum-theory value.

The inflow that takes time to arrive

Momentum theory gives a rotor's induced velocity from its thrust, instantly. It does not arrive instantly: the air the disc has to accelerate has a mass, and the response is a first-order climb with a time constant of 33 milliseconds — a twentieth of the time the wake itself takes to convect a radius.

circulation · Blade-element Three answers to one question: what happens after a body is jerked into motion. The force following a step change in a body's velocity, for three models. The ideal one is a spike at the instant and nothing afterwards. The viscous one falls as the inverse square root of time and never reaches zero. The compressible one holds while the signal is still crossing the body and then settles.

The theory with no memory in it

Laplace's equation has no time in it, so an ideal flow's response to a body being jerked into motion is instantaneous and complete. Its indicial kernel is a spike and nothing afterwards. Beside it sit the two kernels that are not, and the comparison says which ingredient every memory in this collection came in through.

inviscid · Ideal flow

Named alongside it

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

Added massMeasurementRegimeMemory kernelModel validityCompressibilityOscillationPotential flowThe Stokes layerBernoulli's equationBoundary layerColumn separation

All concepts