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

Series — page 3

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.
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
  6. +1 more
6 essays · compressible The rectangular wing stalls at the root; the tapered one stalls at the tip. Section lift coefficient across the half span for three planforms, each drawn at the wing incidence where its own worst section first reaches 1.5. A rectangular wing's peak is at the root, which is where a designer wants it: the stall starts inboard, ahead of the ailerons, and the pilot feels it. A tapered wing's peak has moved out to 0.62 of the semi-span — over the ailerons — because taper takes chord away from the tip faster than it takes circulation. The elliptic wing is the degenerate case: every section reaches the limit at once, which is elegant and is the worst possible stall behaviour.

Taper

  1. 1 Which part of a wing stalls first
  2. 2 The optimum that does not matter
  3. 3 Washout is right at one lift coefficient
  4. 4 One twist is chosen for the speed flown per metre
  5. 5 A wing that twists with its speed washes out the wrong way
  6. +1 more
6 essays · circulation Two centres, two saddles, and a sum of nothing. A separation pattern: a uniform stream with two counter-rotating cored vortices in it, which reproduces the arrangement of critical points behind a body at a Reynolds number of a few tens. There are exactly four — a saddle where the flow divides, a centre in each recirculating cell, and a saddle where it closes — and their indices sum to 0. The winding number of a loop enclosing all of them is 0, which is what a uniform stream far away requires. A bubble costs nothing in this bookkeeping, which is why one is free to appear.

Topology

  1. 1 The count a pattern cannot break
  2. 2 The count computed on a body
  3. 3 Two kinds is a plane flow's privilege
  4. 4 The sign a stagnation point carries in space
  5. 5 A stagnation point can leave the flow through the wall, and the count is kept by halves
  6. +1 more
6 essays · kinematics 80 per cent of the energy goes to the larger scale. One unit of energy is taken out of the middle wavenumber and shared between its two neighbours. Two conservation laws decide the split completely: the energy must add up, and so must the enstrophy, which weights each wavenumber by k². For (1, 2, 4) the answer is that 80.0 per cent of the energy goes up in scale and 80.0 per cent of the enstrophy goes down. There is no model of turbulence anywhere in that: it is two linear equations in two unknowns, and it is why a two-dimensional flow organises itself into large vortices while its gradients get finer.

Two-dimensional

  1. 1 The cascade that runs backwards
  2. 2 Where the inverse cascade stops
  3. 3 A cascade that arrives as stripes
  4. 4 An equilibrium a three-dimensional flow cannot have
  5. 5 The streamfunction says which relaxed state
  6. +1 more
6 essays · turbulence 1.5U at the equator, and no drag at all. The exact ideal flow past a sphere, in the meridional plane, with speed contoured behind the streamlines. The fastest fluid is at the equator at 1.5U — a cylinder's is at 2U — and the field is fore-and-aft symmetric, so the pressure integral over the surface gives a drag of -1.2e-16 against a dynamic scale of order one. The streamline spacing here does not measure speed the way it does in a plane flow: the flux between two meridional streamlines depends on the distance from the axis as well, which is why the speed is contoured rather than left to be inferred.

Axisymmetric

  1. 1 Three dimensions are kinder
  2. 2 Thickness costs a fuselage far less than a wing
  3. 3 A fuselage lowers its wing root's critical Mach number
  4. 4 A waist moves the overspeed and cannot remove it
  5. 5 A free waist needs half the depth, and the sweep costs it nothing
5 essays · inviscid How big before gravity shows. A drop's height over its width against the Bond number, which is the ratio of its weight to the force its own skin can supply. The number is one where those two are equal, and by then the drop is a bun: it is one per cent from a ball at Bo = 0.0079, five per cent at 0.054 and ten at 0.13. Every one of those is below one, and the first is below it by a factor of a hundred and twenty-six.

Bond

  1. 1 The size a drop is allowed
  2. 2 The drop falls at a fold
  3. 3 A law that is exact as an average
  4. 4 The force a contact line holds is a range
  5. 5 A sliding drop is held harder the faster it goes
5 essays · regimes 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 Two bills, and the radius that settles them. The cost of a vessel against its radius: the pumping power, which falls as the inverse fourth power, and the price of owning the fluid and the wall, which rises as the square. Their sum has a minimum, found here by golden-section search and agreeing with the closed form to eight figures. At that radius the pumping bill is exactly a third of the total — for every set of constants, because it follows from the two exponents alone.

Branching

  1. 1 The radius that costs least
  2. 2 The angle a junction chooses
  3. 3 The cheapest turbulent pipe follows two and a half, not three
  4. 4 The cheapest fork loses only its sideways speed, and pricing it closes the fork
  5. 5 The layer every airway restarts bends Murray's cube towards the turbulent pipe's
5 essays · applied 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 The stretch of surface that is boiling. The pressure coefficient along both surfaces of a section at 4 degrees, computed from the same potential-flow solution as the other ideal-flow aerofoil figures. The horizontal line is the vapour pressure at a cavitation number of 1: wherever the suction curve is above it, the liquid there has been pulled below its vapour pressure and is boiling at whatever temperature it happens to be. This section cavitates at any σ below 1.428.

Cavitation

  1. 1 When a body tears the water
  2. 2 The bubble that hammers
  3. 3 A threshold that is also a duration
  4. 4 A cavity that cools the water it came from
  5. 5 A bubble is throttled by its age, and a sheet by its length
5 essays · applied 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 Every curved streamline has a pressure gradient across it. Twelve points in the flow past a cylinder, with the arrow at each showing the pressure gradient across the streamline. It points away from the centre of curvature everywhere, and its size is ρq²κ: the fluid is being pushed round a bend, and something has to do the pushing. Both sides are computed here and they share no arithmetic — one is Bernoulli's pressure differenced across the flow, the other is the turning rate of the velocity direction along it — and they agree to 6.7e-5. This is the whole content of the effect usually named after Coandă, and it is happening on every curved streamline of every flow.

Coanda

  1. 1 The effect that explains nothing
  2. 2 The effect that is real, and where it stops
  3. 3 The teapot effect is a tension, not a pressure
  4. 4 An edge holds any angle it is given
  5. 5 A rounded edge spills before its line goes round
5 essays · misconceptions A streamtube narrows and the flow speeds up. Two neighbouring streamlines bound a tube that no fluid crosses. Where the tube pinches, the same mass has to pass through a smaller gap every second, so it must move faster — which is mass conservation with no equations in sight.

Continuity

  1. 1 Mass has nowhere to go
  2. 2 The number on a streamline is a flow rate
  3. 3 A wave on the wall is a pump
  4. 4 The pump that is better the more it squeezes
  5. 5 A squeezed tube is held open by its own fluid
5 essays · kinematics 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 A pump curve out of the momentum theorem. The pressure an ejector delivers, against how much it is entraining, at a fixed nozzle. It has the shape of every pump characteristic ever measured — a shut-off pressure with no flow, falling to no pressure at free delivery — and it was obtained from a momentum balance on a tube with nothing in it. The shut-off value here is 36.0 kPa and the machine at its best power runs at 4.56 times its own motive flow.

Ejector

  1. 1 Mixing is a pump
  2. 2 The nozzle that is best at one thing
  3. 3 The wall the suction puts in the curve
  4. 4 A choke that belongs to two streams
  5. 5 An ejector's nozzle exit is a condition, not a choice
5 essays · applied 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 Flow net — a free vortex. 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.

Images

  1. 1 A wall made by reflection
  2. 2 The wall that pushes back
  3. 3 The mirror that is a circle
  4. 4 The corners that can be done with mirrors
  5. 5 Two eddies can stand behind a cylinder, but not for long
5 essays · inviscid The chord and the tangent, which are the two speeds. The flux of a conserved quantity against its own density, for a wide river and for traffic. At any point the slope of the chord from the origin is the speed the material moves at, and the slope of the tangent is the speed a disturbance moves at. They are the same number only if the curve is a straight line through the origin. For the river the tangent is five-thirds of the chord at every depth; for traffic the tangent turns negative above half the jam density while the chord never does.

Kinematic waves

  1. 1 A wave nothing in it travels with
  2. 2 The column the chord rule cannot settle
  3. 3 The tangent that sizes a plant
  4. 4 When the flux outruns the pressure
  5. 5 Roll waves stop merging at a spacing their fronts choose
5 essays · kinematics Finding the aerodynamic centre by sweeping the chord. How fast the pitching moment changes with incidence, plotted against where along the chord the moment is taken. The curve crosses zero once, and that crossing is the aerodynamic centre — the single point about which pitching the wing does not change the moment.

Moment

  1. 1 Where the lift acts
  2. 2 The surface in the wake
  3. 3 A canard pays for its stability in induced drag
  4. 4 A third surface is worth a square
  5. 5 The tail a canard needs costs more than the canard saves
5 essays · circulation Two depths, and a gap the model will not describe. The surface either side of a hydraulic jump at an arriving Froude number of 5.05. Both depths are exact consequences of the momentum balance. The distance between them is not: the shallow-water model has no length scale in it and cannot say how far the transition takes, so the region between the two levels is left blank and the six-depth rule of thumb beside it is somebody's measurement rather than this site's result.

Open-channel

  1. 1 The shock in a river
  2. 2 The depth that costs least
  3. 3 The section that decides the river
  4. 4 The current that runs across the river
  5. 5 The depth at the edge is not the critical one
5 essays · applied

All essays