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.
What the ground actually gives a wing. Induced drag near the ground as a fraction of the same wing's in free air, against height in spans, at constant lift. The ground is a plane of symmetry, so the wake is joined by a mirrored wake of opposite circulation below it, and the upwash from that image is what takes the drag away. At a tenth of a span the wing keeps 0.516 of its induced drag — a saving of 48 per cent — and by a span and a half the effect is 1.3 per cent and going. The lift is held fixed all the way along this curve: what the ground gives here is not more lift, it is the same lift for less drag, and the two are different claims about a landing aeroplane.

Ground cushion

  1. 1 The cushion that is not there
  2. 2 The cushion that is there after all
  3. 3 A cushion that changes its physics
  4. 4 The borrowed mass the boundary decides
  5. 5 The borrowed mass that goes negative
  6. +4 more
9 essays · misconceptions Three times the wind, on a reach. The polar diagram: boat speed in every direction, as a multiple of the true wind speed, for drag angles of 14 and 6 degrees. The shaded wedge at the top is the no-go zone, whose half-angle is exactly the sum of the two drag angles. Everywhere outside about twice that angle the boat is faster than the wind, and the maximum is 2.92 times the wind at 110 degrees — which is 1/sin λ at 90° + λ, both checked.

Sailing

  1. 1 Faster than the wind that drives it
  2. 2 The fastest way is not the straight one
  3. 3 A breeze the boat cannot use
  4. 4 A keel flies wherever the course puts it
  5. 5 The rudder pays for the keel's wake
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9 essays · applied One spectrum, two fields. Two one-dimensional fields built from the same amplitudes and different phases. Their energy spectra are identical to the last bit, their two-point correlations agree to 2e-15, their second-order structure functions are the same function — and no measurement of either kind can tell them apart. Everything that distinguishes them is in the phases, which is where the cascade lives.

Spectrum

  1. 1 The moment a spectrum cannot hold
  2. 2 The range a real Reynolds number does not have
  3. 3 A record is as long as its integral scales
  4. 4 The best estimate of a scale assumes its shape
  5. 5 A snapshot counts areas, and a flow counts its edge
  6. +4 more
9 essays · turbulence 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
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9 essays · misconceptions The acceleration field of a steady flow. How hard the fluid is being accelerated at each point of a steady flow past a cylinder. The flow does not change with time anywhere in this picture, and yet almost nowhere in it is a parcel travelling at constant velocity — the pattern stands still while the fluid running through it is thrown about.

Acceleration

  1. 1 Steady does not mean nothing is happening
  2. 2 How long the fluid has been in there
  3. 3 A parcel goes straight while the streamlines curve
  4. 4 Drag decides whether a cloud folds, and the air decides at what
  5. 5 A lattice of vortices folds a cloud at the saddles' quarter
  6. +3 more
8 essays · kinematics d'Alembert's paradox, measured. Surface pressure round a cylinder in ideal flow, plotted against angle. The distribution is symmetric front to back, so every push on the front is matched by an equal push on the back, and the total force along the stream is exactly zero.

Dalembert

  1. 1 The exact theory says nothing has any drag
  2. 2 The force of getting going
  3. 3 The drag that is made of waves
  4. 4 The paradox is for the pair, not for each body
  5. 5 A pair set free in a stream collides or parts
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8 essays · inviscid 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
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8 essays · circulation The air a wing carries along, and how little of it there is. The Blasius profile, in the wing's frame, with the free stream at one. No slip says the air at the surface is at rest relative to the surface, which in the ground's frame means it is moving with the wing — but only exactly at the wall. The deficit, integrated across the layer, is the displacement thickness: at a Reynolds number of 1e+6 and a metre of chord it is 1.72 millimetres of air moving at flight speed, which is the whole of what is 'carried'. The step drawn on the axis is that same deficit as a solid slab. A wing does not drag a blanket of air with it; it leaves a boundary layer behind it, and the layer is made of air that keeps being replaced.

The no-slip condition

  1. 1 The air a wing does not carry
  2. 2 The drop a no-slip wall would never let spread
  3. 3 A length you can only measure by squeezing
  4. 4 The wall a melt really does slip on
  5. 5 A melt's slip length has an exponent it cannot choose
  6. +3 more
8 essays · misconceptions 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
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8 essays · misconceptions The cliff a rough ball reaches sooner. The drag coefficient of a sphere against Reynolds number, on log axes. The smooth curve is Morrison's correlation, which is a fit to measurements and is drawn in the colour this site reserves for a borrowed claim. The other is the same curve shifted along the Reynolds axis by a factor of 6 — a stated model of what roughness does, which is to trip the boundary layer early, and not a measurement of any real ball.

Sport ball

  1. 1 The drag that falls as it speeds up
  2. 2 A ball that swings without spinning
  3. 3 The ball that never forgets its spin
  4. 4 The terminal speed sets the best launch angle
  5. 5 A spinning ball should leave low
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8 essays · applied 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. +3 more
8 essays · kinematics How hot seven films get. The peak temperature rise inside the film itself, on a logarithmic axis, for seven lubricated contacts with ordinary engineering numbers. A journal bearing runs tens of kelvin above its own housing, which is why lubrication systems are designed around heat rather than around load; a water-lubricated bush runs at a hundredth of a kelvin, because water conducts six hundred times better per unit of viscosity. The air bearing is marked: its gap is smaller than air's mean free path, so its film is not a continuum at all.

Viscous heating

  1. 1 The film that heats itself
  2. 2 A lost steady state still holds the film
  3. 3 A motor turns the runaway into a jump
  4. 4 A rotor's inertia slows the jump and cannot make it ring
  5. 5 A slow sensor makes the bearing cycle
  6. +3 more
8 essays · viscous The plateau that is the log law. y⁺ du⁺/dy⁺ across a channel, at five Reynolds numbers. Millikan's argument says this quantity must be constant wherever neither the viscous length nor the channel width may appear, and its value there is 1/κ. At Re_τ = 180 there is no flat part at all; at Re_τ = 100,000 it is flat over 2.16 decades and gives κ = 0.4120. The log law is a statement about a limit, and this is the picture of the flow approaching it.

Wall law

  1. 1 The layer with no length in it
  2. 2 What a code says to a wall
  3. 3 Three buffer layers, one friction
  4. 4 The wake is a tenth of the velocity and a third of the displacement
  5. 5 An adverse gradient grows the wake, not the log law
  6. +3 more
8 essays · turbulence 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. +3 more
8 essays · applied α = 13.0: a plug in the middle and everything happening at the wall. The velocity profile at eight phases of one cycle, at a Womersley number of 13.0 — the human aorta's at rest. The core moves almost as a solid plug, because viscosity cannot reach it within a cycle; all the shear is in a layer of thickness √(ν/ω) = 0.77 mm against a radius of 10.0 mm. At some phases the fluid near the wall is moving backwards while the core still moves forward, which is the reversal a Poiseuille profile can never show and which is routinely measured in arteries.

Womersley

  1. 1 Too fast for a profile
  2. 2 Where the parabola goes
  3. 3 The pulse that has to travel
  4. 4 The pulse that grows as it leaves the heart
  5. 5 Murray's law is not the rule for a pulse
  6. +3 more
8 essays · regimes The shock radius against time, from an equation that was given no exponent. The thin-shell energy balance integrated forward from a small initial radius, on logarithmic axes. The two-fifths power is not put in: the ordinary differential equation is Ṙ = √(E/AρR³), and the straight line is what it does. The fitted slope is 0.39983 and the fitted prefactor is 0.90721 against a closed form of 0.90702 — the small residuals being the integration's memory of where it started, which the similarity solution has no equivalent of.

Blast wave

  1. 1 A radius that gives the energy away
  2. 2 The fireball is hollow
  3. 3 A hypersonic body leaves a line explosion behind it
  4. 4 The gas beside a hypersonic body remembers its nose
  5. 5 The sheath outlasts the body
  6. +2 more
7 essays · compressible Two decay laws from two invariants, and nothing in the equations to choose. The energy of a decaying turbulence against time, integrated from dK/dt = −A K^(3/2)/l with the large scales conserving u² l³ in one case and u² l⁵ in the other. The exponents come out at 1.1997 and 1.4282 against the closed forms 6/5 and 10/7. Which invariant holds is decided by the shape of the spectrum at the very largest scales, at the moment the stirring stops.

Decay

  1. 1 What decay never forgets
  2. 2 Universal, and one of five
  3. 3 The constant that travels
  4. 4 The decay inside the four-fifths law
  5. 5 Forcing at the integral scale leaves the cascade alone
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7 essays · turbulence A control volume round an aerofoil. A rectangle drawn in the fluid around a lifting section. The arrows on the right-hand face show the downward velocity of the air leaving it, drawn to scale. Adding the momentum carried through all four faces to the pressure acting on them gives the force on whatever is inside, without the calculation ever going near the surface.

Momentum lift

  1. 1 Air must be pushed down, and the usual sum is wrong
  2. 2 A right total from a wrong picture
  3. 3 Weighing what is missing
  4. 4 The drag a wake keeps however it rolls up
  5. 5 Behind a finite wing the wake face keeps its share
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7 essays · misconceptions A 10° wedge at Mach 2.00 has two shocks that solve it. The wedge turns the flow through a fixed angle, and the θ–β–M relation offers two shock angles that achieve it. The weak solution, drawn steeply forward, leaves the flow supersonic and is what a wedge in a free stream produces. The strong solution leaves it subsonic and appears where downstream pressure forces it. Nothing local to the wedge chooses between them.

Oblique shock

  1. 1 A shock that leans
  2. 2 When the wedge is too blunt
  3. 3 A cone finishes its turn after the shock
  4. 4 A cone intake's shock sees the whole capture
  5. 5 The last shock should be the weakest
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7 essays · compressible 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 here: nothing above resolves 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
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7 essays · applied

All essays