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

Series

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
A Joukowski aerofoil at 6°. A cambered aerofoil in a uniform stream. The circulation is not chosen: it is whatever value makes the flow leave the sharp trailing edge smoothly, and that single condition fixes the lift.

Lift

  1. 1 What actually holds a wing up
  2. 2 The sharp edge decides
  3. 3 The lift curve, and why it is a straight line
  4. 4 Lift with no wing at all
  5. 5 The vortex a wing leaves behind
  6. +3 more
8 essays · circulation Four ways of photographing one flow, and what each of them records. The same solved flow, rendered as four different laboratory techniques would record it. Smoke from a port gives a streakline; tufts give direction with no speed in it at all; an oil film gives the direction of the friction on the surface rather than the flow above it; pressure taps give a scalar with no direction in it. None of the four is the velocity field, and only the first happens to coincide with a streamline, because this flow is steady.

Visualisation

  1. 1 What a photograph of a flow shows
  2. 2 The shutter is part of the answer
  3. 3 An instrument that takes a derivative
  4. 4 The paint that measures the wrong field
  5. 5 The window every vector is averaged over
  6. +2 more
7 essays · misconceptions Every disturbance has its own threshold; one of them is lowest. The Rayleigh number at which a disturbance of horizontal wavenumber a becomes neutral, Ra = (π² + a²)³/a². Every wavenumber has a threshold and the layer goes unstable at the lowest of them, which a golden-section search on this curve puts at a = 2.221441 and Ra = 657.5114 — the exact π/√2 and 27π⁴/4 to fourteen digits. Below the curve the layer conducts and nothing moves.

Convection

  1. 1 A threshold with a closed form
  2. 2 Three numbers left of a fluid
  3. 3 A millionth is enough
  4. 4 The threshold the walls decide
  5. 5 The truncation that cannot carry three times the heat
  6. +1 more
6 essays · turbulence 5 quantities, 3 rows, 2 left over. The dimension matrix for the drag on a sphere: one column per quantity, one row per base dimension, and every entry an exponent. Buckingham's theorem is a statement about this matrix and nothing else — the number of independent dimensionless groups is the number of columns minus the rank, computed here by elimination. Nothing about fluids enters until somebody decides which columns to write down.

Dimensional

  1. 1 Counting what matters
  2. 2 A limit nothing reaches
  3. 3 Where a pure number comes from
  4. 4 Exact in the total, free in the profile
  5. 5 The groups are not the only groups
  6. +1 more
6 essays · regimes 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.

Exact layer

  1. 1 The wall that shakes
  2. 2 The layer that stops growing
  3. 3 The wall the fluid is listening to
  4. 4 The solution that keeps its nonlinear term
  5. 5 One channel, one flux, two flows
  6. +1 more
6 essays · viscous Four turns of a wave, and the parcel is not back. Two parcels traced through four periods of a linear deep-water wave of steepness 0.1, by integrating the exact velocity field. Each orbit is very nearly a closed circle and misses closing by a little, every time, in the same direction — that miss is the whole of the Stokes drift. The near-surface parcel advances 0.0357 of a wavelength over the four cycles and the one a tenth of a wavelength down advances 0.0122, a third as far — because the drift falls off twice as fast with depth as the orbit's own size does.

Stokes drift

  1. 1 The drift in a wave that has none
  2. 2 The drift a closed box will not allow
  3. 3 A drift made of two things that average to zero
  4. 4 The drift a rotating planet takes back
  5. 5 The drift that turns a current into rolls
  6. +1 more
6 essays · kinematics 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.

Water hammer

  1. 1 Stopping water costs more than moving it
  2. 2 A pump with no engine
  3. 3 The part of the closure a pipe cannot see
  4. 4 A tank that turns a hammer into a swing
  5. 5 The better tunnel needs the bigger tank
  6. +1 more
6 essays · applied A duct does the opposite thing above Mach one. The four cases of dA/A = (Ma² − 1) dV/V. Below the speed of sound a narrowing duct accelerates the flow, which is what continuity leads anyone to expect. Above it the sign of the bracket flips, and a narrowing duct decelerates: density is falling faster than the speed is rising, so the stream tube needs more room rather than less.

Area mach

  1. 1 The duct that works backwards
  2. 2 The throat that stops listening
  3. 3 One area, two answers
  4. 4 The jet a cone sprays sideways
  5. 5 Friction moves the sonic point past the throat
5 essays · compressible Flow past a cylinder at Re 40. A real fluid past a circular cylinder. At low Reynolds number the flow closes up behind the body much as the ideal theory says; as it rises the flow separates and a region of reversed flow appears behind, which is where drag comes from.

Boundary layer

  1. 1 Everything happens in a layer you cannot see
  2. 2 How thick is thin
  3. 3 The singularity a layer makes for itself
  4. 4 Four profiles, one drag
  5. 5 A layer that is an integral of everything upstream
5 essays · viscous The inertial range, and the slope read back off it. The model energy spectrum at Re = 1e+6, with production rolling off below the integral scale and dissipation cutting it off above the Kolmogorov scale. The straight middle is the inertial range, and the number printed beside it is the slope least-squares fitted to the drawn points over the middle of that range — not the −5/3 that went in.

Cascade

  1. 1 Where the energy goes
  2. 1 A row is not a set of aerofoils
  3. 2 The grid nobody can build
  4. 2 A row that meets the row before it
  5. 3 A flux that runs both ways
5 essays · turbulence Ten unknowns, four equations. What is left after the Navier–Stokes equations are averaged. The mean velocities and mean pressure were there before; the six Reynolds stresses are new, and they arrived from the one term that does not average away. Nothing in the count is an approximation — the averaged equations are exact — and that is what makes the gap uncomfortable.

Closure

  1. 1 What averaging costs
  2. 2 The ladder that never closes
  3. 3 A guess with a constant in it
  4. 4 The constant that makes a variance negative
  5. 5 A closure with no memory at all
5 essays · turbulence The same number, by two integrals that share no arithmetic. Three flows whose dissipation is in closed form both ways. The volume route integrates the dissipation function over the fluid; the boundary route multiplies a force or a torque by the speed of whatever is applying it. Neither calculation contains the other, and the residual column is what is left when they are subtracted.

Dissipation

  1. 1 The price of a gradient
  2. 1 The limit that is not the value
  3. 2 Where the heat of a drag is made
  4. 2 Equal on average, and nothing else
  5. 3 A dissipation that lags its production
5 essays · viscous Ideal flow past a cylinder. A uniform stream past a circular cylinder in a fluid with no viscosity. The solution is exact and closed-form: streamlines part at a stagnation point, run round the surface and close up perfectly behind, and the pressure recovers to exactly what it was in front.

Ideal flow

  1. 1 The theory that solves everything
  2. 2 One function instead of two
  3. 3 From a circle to a wing
  4. 4 The lowest pressure is on the body
  5. 5 The theory with no memory in it
5 essays · inviscid Two velocities, 2.50 apart, and only one of them is anybody's. The velocity in Darcy's law is the flow rate divided by the whole cross-section, solid included: a speed no fluid particle ever has, since the fluid occupies only the fraction ε of that area. The speed the fluid actually averages is larger by exactly 1/ε — 2.50 times here — and it is the one that belongs in a residence time, in a pore Reynolds number and in any statement about when a tracer arrives. The grains are drawn to say that the pore-scale flow is not computed anywhere: no figure on this site claims to resolve it.

Porous

  1. 1 A velocity nobody has
  2. 2 Where Darcy stops
  3. 3 The bed that weighs itself
  4. 4 A permeability that is only the geometry
  5. 5 The outlet is the inlet, a while ago
5 essays · applied What the skin settles at, before anything is done to it. The adiabatic wall temperature against Mach number, in air at 216.7 K, with the stagnation temperature above it. The gap between the two is the recovery factor, which is 0.8417 here and stays there at every Mach number — it is a property of the Prandtl number and not of the speed. At Mach 2 the skin sits at 363 K, at Mach 3 at 545 K, and at Mach 5 at 1128 K, which is past what aluminium will do. Nothing has been burnt and nothing has been rubbed: the air was brought to rest, and this is where its kinetic energy went.

Recovery

  1. 1 The wall that heats itself
  2. 2 The skin that lags the flight
  3. 3 The thermometer that heats itself
  4. 4 Three readings, and the one each answer leans on
  5. 5 The gradient the heat never hears
5 essays · compressible A normal shock at Mach 2.00, and what crosses it unchanged. The state in front of the shock and the state behind it. Every ratio was computed from the standard jump relations and then substituted back into mass, momentum and energy, which is an independent route — a mistyped exponent in the total-pressure expression cannot survive a momentum balance it never appeared in. The residuals are printed below because a check nobody can see is a check nobody can audit.

Shock

  1. 1 The jump the equations allow
  2. 2 The only law that forbids it
  3. 3 What a shock costs
  4. 4 The discontinuity that has a thickness
  5. 5 The jump does not ask what made it
5 essays · compressible Two heights, and only one of them is in the answer. A siphon, with the two heights that get confused. The drop from the source surface to the outlet is what drives the flow: the exit speed is √(2gΔz) = 4.43 m/s and nothing else enters it. The rise to the crown decides the pressure at the top — 72.0 kPa absolute here, against an atmosphere of 101.3 — and therefore whether the column holds together at all. A siphon over a high wall and one over a kerb, draining to the same place, flow at exactly the same rate.

Siphon

  1. 1 The siphon that does not need the air
  2. 2 The one place the atmosphere pushes
  3. 3 The siphon that does not break
  4. 4 The margin friction lends a siphon
  5. 5 The air that breaks a siphon nothing else can
5 essays · misconceptions One signature, aged four times. The pressure signature of a slender body at four distances, computed by the exact Lax formula for the nonlinear propagation. Each point of the waveform moves forward in proportion to its own overpressure, so the compression at the front catches the undisturbed air and a shock forms there, while the expansion at the rear falls behind and forms a second one. What is left is an N-wave: two discontinuities and a straight line between them, spreading and weakening.

Sonic boom

  1. 1 The signature that forgets the shape
  2. 2 The boom that turns back before the ground
  3. 3 The carpet an accelerating aeroplane folds
  4. 4 The edge is a rumble, not a quieter bang
  5. 5 A boom is aged in the thin air it starts in
5 essays · compressible The velocity field, arrows to scale. The same flow drawn as arrows. Scaled to the local speed the picture is honest and crowded; drawn all the same length it is legible and hides the very variation the figure is about.

Vorticity

  1. 1 Spin is not the same as going round
  2. 2 Circulation is vorticity, added up
  3. 3 The spin that feeds itself
  4. 4 Where vorticity comes from
  5. 5 A wall puts in exactly its own speed
5 essays · kinematics The tube widens because the air slows. The streamtube through an actuator disc at an induction factor of 0.333. The three radii are not drawn to taste: each is fixed by requiring the same mass to pass every station, and the slowest station is therefore the widest. The tube widening in front of a wind turbine is why some of the wind goes round it rather than through it, and it is the whole reason a disc cannot take everything.

Actuator disc

  1. 1 The most a disc can take
  2. 2 A big slow push
  3. 3 The wake that has to spin
  4. 4 Between hover and twice the hover inflow
4 essays · applied

All essays