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

Relaxation time — where it appears

How long a particle takes to give up its velocity difference from the surrounding fluid. The usual formula counts the particle's own inertia alone; adding the added mass and the undisturbed flow's pressure gradient changes it by half for a tracer and by everything for a bubble.

Named by 19 essays across 6 fields — each of them below, with the objects they name alongside it.

Particles at St = 1, against the flow that carries them. Particle paths and the streamlines they were released on, in this site's exact cylinder solution. At small Stokes number the two are indistinguishable and the body catches nothing; as the particles get heavier their paths straighten, cross the streamlines, and begin to strike. The paths are integrated with Stokes drag and nothing else — no gravity, no lift, no effect of the particles on the flow.

Whether the droplet turns

The air goes round the wing. Whether what is carried in it goes round too is decided by one number — and below a critical value of that number the body collects nothing at all, however many droplets are thrown at it, because the flow turns every one of them in time.

regimes · Particle The relaxation time a particle actually has. Two quantities against the particle-to-fluid density ratio. β = 3ρ_f/(2ρ_p + ρ_f) is three for a bubble, one for a neutrally buoyant particle and nearly zero for anything heavy; it is the factor by which the fluid's own acceleration is felt. The other curve is the true relaxation time over the usual formula's, which is one for a heavy droplet, exactly three halves for a neutrally buoyant tracer, and unbounded for a bubble — the usual formula gives a bubble a relaxation time of zero, and therefore no dynamics at all.

The tracer that is not one

Every Stokes number is built on a relaxation time that counts the particle's own inertia and nothing else. Adding the two terms it leaves out gives a bubble a relaxation time where the usual formula gives zero, makes a neutrally buoyant tracer half as slow again as advertised, and sends bubbles into vortex cores that droplets are flung out of.

regimes · Particle Storage and loss moduli, for one relaxation time and for a spectrum. The two moduli of a Maxwell fluid and of a Rouse chain, against frequency in units of the longest relaxation time. One relaxation time makes the storage modulus overtake the loss modulus at λω = 1 and then leave it behind without limit. A spectrum of 1000 modes makes them rise together as the square root of frequency and stay a fixed ratio apart, so the material never becomes the solid the single time predicts.

A solid, if it is not given time

The Deborah number is the only group on this site with no fluid in it — two times and nothing else — and it says a material is a solid or a liquid depending on how long anybody watches. What it throws away is that no real material has one time, and the spectrum it replaces changes the answer in kind rather than in degree.

regimes · Deborah The backwater curve behind a weir. The depth at the control is 1.6 times the normal depth; integrating upstream the profile relaxes onto normal depth over about seven hundred metres and stays there. That relaxation is the reach forgetting the weir, and nothing about the weir survives past it.

The section that decides the river

A reach of open channel has a normal depth and a critical depth, and the water has neither. What it has is a profile obeying a first-order equation, which needs exactly one condition — and whether that condition belongs at the upstream end or the downstream end is not a choice, because the equation is stable in one direction and unstable in the other.

applied · Open-channel The temperature behind a shock, which is not the jump condition's. The static temperature along the flow behind a Mach 6 normal shock, with the frozen value the jump conditions give and the equilibrium value they give a long way behind. The gas arrives at 2382 K and settles at 2059, over about four tenths of a millimetre.

A gas that has not finished being shocked

The jump conditions give the state a long way behind a shock. Immediately behind it the molecules have not started vibrating yet, so the temperature is 2,382 K where the equilibrium answer is 2,059 — and the gas takes four tenths of a millimetre to get from one to the other.

compressible · Real gas Two lifetimes, crossing. The mean time for a turbulent puff to decay and the mean time for it to split into two, against Reynolds number, on a logarithmic axis spanning thirty orders of magnitude. Below the crossing puffs die faster than they multiply; above it they multiply faster than they die, and the flow stays turbulent.

A puff that does not know how old it is

A patch of turbulence in a pipe below the critical Reynolds number dies at random, and its chance of dying in the next second does not depend on how long it has already lasted. The flow that contains it has a memory anyway, because the patches multiply — and where multiplying overtakes dying is a Reynolds number.

turbulence · Transition The structure the sensitivity produces. A detonation front, schematically: a leading shock, an induction zone in which nothing measurable happens, and a reaction zone behind it. The induction zone's length is set by the shock's own strength, and because that dependence is exponential the front is unstable and breaks into cells.

A gas that has not decided to react yet

Behind a detonation's leading shock there is a zone in which nothing measurable happens. Its length is set by the temperature the shock produced, exponentially — a one per cent change in the shock shortens it by fifteen per cent — and that sensitivity is why a detonation front cannot stay flat.

compressible · Detonation The production jumps and the dissipation does not. Production and dissipation against time, through a step change in the strain rate. The production follows the strain immediately — it is the strain squared times an eddy viscosity — and the dissipation moves by less than one per cent at the instant of the step, because it is set by a cascade that has not been told yet.

A dissipation that lags its production

Change the strain rate on a patch of turbulence and the production of energy follows instantly — it is the strain squared. The dissipation moves by less than one per cent, because it is the far end of a cascade that has not been told yet, and the two are out of balance by a factor of six for the next turnover.

turbulence · Dissipation The stress a closure predicts, and the stress there is. The Reynolds-stress anisotropy through a step change in the strain rate, against what an eddy viscosity gives — which is the equilibrium value at every instant. The real stress takes about a turnover to get there, and during that turnover the closure is wrong by up to sixty per cent.

A closure with no memory at all

An eddy viscosity says the Reynolds stress is the mean strain rate times a number, now. The stress it is standing in for takes a turnover to arrive, so the closure is the zero-frequency limit of a response that has a lag in it — and the curve it is the limit of is the same shape as an aerofoil's lift deficiency.

turbulence · Closure 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 What the fluid at one height is listening to. The weight the fluid two millimetres above a moving wall gives to the wall's velocity a given delay earlier, in water. It peaks at two thirds of a second and has a tail that falls as the delay to the power minus three halves — so the fluid is responding to a broad stretch of the wall's past rather than to a moment of it.

The wall the fluid is listening to

Water two millimetres above a moving wall is responding to what the wall did two thirds of a second ago — most likely. Half of its response is older than four and a half seconds, a tenth is older than two minutes, and the average age of what it is responding to does not exist at all.

viscous · Exact layer The flow stops and the stress does not. Shear stress against time for a fluid sheared at a constant rate for two seconds and then left alone. Nothing is moving after the second mark and the fluid is still stressed: 37 per cent of the peak one relaxation time later, and 0.7 per cent after five.

The fluid that has not finished its last deformation

Shear a polymer solution for two seconds and stop. Nothing is moving and the fluid is still stressed — 37 per cent of the peak one relaxation time later, and still measurably stressed after five. Two histories imposing exactly the same total strain leave it in states differing by a factor of 3.7.

viscous · Non-newtonian Captured, and not captured. The amplitude of the wake's oscillation over time, for a forcing inside the capture band and one outside it. Inside, the amplitude settles; outside, it beats at the difference between the two frequencies, because the wake is keeping its own time and the forcing is keeping its.

A wake told what to do

Force a shedding wake near its own frequency and it abandons its own and adopts the forcing's. The band over which it will do so is proportional to how hard it is pushed — and near the edge of the band it takes six times as long to make up its mind.

regimes · Strouhal One incidence, two lifts. Lift coefficient against incidence for a wing pitched sinusoidally through the stall, with the static curve for comparison. The loop is traversed anticlockwise: at twelve degrees the wing carries 0.22 more lift going up than coming down.

Two lifts at one incidence

A wing pitched up and down through the stall does not retrace its own lift curve. At twelve degrees it carries 0.22 more lift going up than coming down, and the loop that opens between the two is the work the airstream does on it — which is where the energy for a stall flutter comes from.

circulation · Vortex lift How long a fluid takes to forget it was not rotating. The fraction of solid-body rotation a container's interior has reached, against time, by the two available routes. The Ekman layers on the end walls pump fluid radially and carry angular momentum inwards in a hundred seconds; diffusion alone would need ten thousand.

How long a fluid takes to forget it was not rotating

Spin a container of water and the fluid inside reaches solid-body rotation in a hundred seconds rather than the three hours diffusion would need. The shortcut is the thin layers on the end walls, and the advantage they give is exactly the reciprocal of the square root of the Ekman number.

viscous · Rotating 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 How fast a duct forgets each part of its inlet. The decay rate along the duct for each mode of a disturbance to the developed profile. It goes as the square of the mode number, so the third mode is forgotten nine times faster than the first and the tenth a hundred times faster.

A duct that forgets everything but one number

Whatever is fed into a pipe, what survives a little way down it is one shape. The disturbance's higher modes decay as the square of their mode number, so the sixth is gone in thirteen centimetres where the first survives four and a half metres — and the entrance length is that one mode's decay rate and an arbitrary threshold.

viscous · Entrance Eight numbers, one construction. The dimensionless groups this collection has produced, placed on a logarithmic axis at a representative value. Each is a memory time divided by a process time, each was named separately in a different field, and each decides the same question: whether the past is still present.

Every memory number is one time over another

These essays produced eight dimensionless groups in eight different fields, named after eight different people, spanning a factor of four hundred in value. Written out, all eight are a memory time divided by a process time, and all eight govern the same curve.

regimes · Dimensional 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

Named alongside it

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

MeasurementModel validityMemory kernelRegimeConvolutionDiffusionStokes numberAdded massBoundary conditionFrequency responseStabilityThreshold

All concepts