Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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Series — page 4

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.
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.

Particle

  1. 1 Whether the droplet turns
  2. 2 The tracer that is not one
  3. 3 A particle is a low-pass filter
  4. 4 Inertia empties the vortices that would hold a settling particle, and then speeds its fall
  5. 5 Flickering vortices cannot hold a settling particle up, and still make it fall faster
5 essays · regimes One layer, and the four others inside and outside it. The velocity profile over a flat plate, and the temperature profile in the same layer at four Prandtl numbers: a liquid metal at 0.01, air at 0.71, water at 7 and a heavy oil at 100. The equations differ by one number and the profiles differ by a factor of twenty in thickness. At Pr = 1 the two are the same function — not similar, identical, to eight decimal places — because the equations and the conditions are then the same, which is what every statement called a Reynolds analogy rests on.

Peclet

  1. 1 The other layer, and the one number that separates them
  2. 2 How far before the heat arrives
  3. 3 The heat the eddies do not carry
  4. 4 The longest thermal entrance belongs to neither metal nor gas
  5. 5 A steel wall lengthens a sodium entrance by half
5 essays · regimes 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 Three waves out of one discontinuity. The x–t diagram of the burst diaphragm. A shock runs right at a speed of its own, a contact surface follows it more slowly, and an expansion fan spreads left as a family of rays that opens with time — the three wave families the Euler equations possess, produced at once by an initial condition with no waves in it at all. Every straight line here is a speed the solution computed, and the fan is drawn as the rays it actually consists of.

Shock tube

  1. 1 One diaphragm, every wave
  2. 2 A surface that remembers the diaphragm
  3. 3 The shock that passes without an echo
  4. 4 A tailored tube buys its test time with its driver
  5. 5 The gas a reflected shock refuses is inside the layer
5 essays · compressible A source at Mach 0.55, and the sound it has already made. Each circle is one pressure pulse, expanding at the speed of sound from the point the source was at when it left. Below the speed of sound every circle still contains the source, so the pressure disturbance reaches every point of the fluid before the source does. Above it the circles have an envelope, and outside that envelope nothing has been heard at all.

Speed of sound

  1. 1 What a signal travels at
  2. 2 When the warning cannot arrive
  3. 3 Slower than either of them
  4. 4 Above their resonance, bubbles make water faster
  5. 5 The more small bubbles a cloud holds, the later it turns fast
5 essays · compressible A cross at 30.0° for every wavelength it makes. A body oscillating at ω = 0.5N in a stratified fluid, and the four beams along which its energy leaves. The angle is arccos(ω/N) from the vertical — 30.0 degrees from the horizontal here — and it is the same for every wavelength the body excites, because the dispersion relation has no length in it. The short strokes are the crests, which lie along the beams rather than across them: the phase advances perpendicular to the energy, and the two are exactly at right angles. Raise the frequency and the cross closes towards the vertical; reach ω = N and it shuts entirely, because nothing above the buoyancy frequency propagates.

Stratification

  1. 1 The number that stops the mixing
  2. 2 Five numbers, one name
  3. 3 Sufficient, and not necessary
  4. 4 A thin interface keeps its waves past a quarter
  5. 5 What holds the shear decides whether the waves last
5 essays · turbulence 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.

Superposition

  1. 1 Flows add up
  2. 2 Bodies made out of nothing
  3. 3 The one thing that does not add up
  4. 4 Thickness carries no load of its own and still moves the load forward
  5. 5 A NACA section multiplies its lift by nearly one plus its thickness, and a nose correction cannot supply it
5 essays · inviscid 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.

Unsteady bernoulli

  1. 1 The pressure that depends on the past
  2. 2 Water climbs a falling wedge, and the load comes from the climbing
  3. 3 A light hull slams softly, and a round one does not
  4. 4 A hull bottom feels the slam it has time to answer
  5. 5 Plating that gives way slows the water that slams it, but only on a flat bottom
5 essays · inviscid There is a constriction, and it is a consequence rather than a cause. A cambered section at 5 degrees, with one streamtube traced above it and one below. The tube above narrows by 16 per cent at mid-chord and the tube below by -112 per cent, so the venturi story's premise is true: the flow over the top really is squeezed more than the flow underneath. The difficulty is that the tube's upper boundary is a streamline, not a wall. Nothing put it there but the solution of the whole flow — the same solution that already contains the lift — so the narrowing is a way of describing the answer rather than a reason for it.

Venturi

  1. 1 Not half a venturi
  2. 2 The venturi that stops listening downstream
  3. 3 A choked throat buys time, not silence
  4. 4 Carburettor ice is made by the fuel, not the venturi
  5. 5 The throttle plate is a wet bulb for petrol
5 essays · misconceptions Lift and wave drag on a flat plate at Mach 2, exactly and to first order. The lift and wave-drag coefficients of a supersonic section against incidence, computed face by face from shocks and fans, with Ackeret's linear result dashed over them. The two agree to three decimal places at small angles and part company slowly — which is what a first-order theory is supposed to do, and evidence rather than tautology, since the two routes share no algebra.

Wave drag

  1. 1 Drag with nothing to rub
  2. 2 The least drag a volume can have
  3. 3 Busemann's biplane has to be flown past its design point
  4. 4 A flap that closes the mouth starts Busemann's biplane below its design point
  5. 5 Busemann's biplane lifts only on its outer faces, and its channel turns the lift into a quarter-chord load
5 essays · compressible The same patch, 6 periods later, in two flows. A round patch of 848 marked particles, advanced 6 periods by the blinking flow and by a steady flow of the same strength. The steady flow has drawn the patch into a smooth ribbon along a streamline and every particle in it is still on the streamline it started on; the blinking flow has folded the patch through itself repeatedly and its particles are spread across the whole region. Neither flow has any diffusion in it and neither has lost a particle. The difference between them is that one depends on time.

Advection

  1. 1 No randomness, and it mixes anyway
  2. 2 Steady, three-dimensional, and mixing anyway
  3. 3 The area that must not move
  4. 4 A scalar is a record of where its fluid was
4 essays · kinematics Five quarters of the span, the same bending moment, 64/75 of the drag. The bell-loaded wing's induced drag and root bending moment against its span, both as fractions of an elliptic wing of unit span carrying the same lift. At equal span the bell is worse: its span efficiency is exactly three quarters. But its bending moment is exactly four fifths, and bending moment grows in proportion to span while drag falls as its square — so there is a span at which the bell has bought back the structure and is ahead on drag. It is at exactly five quarters, where the two curves are at 1 and 0.8533. Both numbers are rational and neither was put in by hand.

Bell shape

  1. 1 The loading nobody used
  2. 2 A bending moment never chooses a span
  3. 3 A spar that must lift itself stops the span at a quarter of the weight
  4. 4 A stiffness floor keeps the span and gives the tips their load back
4 essays · circulation The hypotheses Bernoulli's equation needs. The equation is correct and its hypotheses are strict. Most misuse is not a wrong formula but a right formula carried across a streamline, through a machine, or into a region where viscosity dominates.

Bernoulli's equation

  1. 1 Where Bernoulli's equation applies
  2. 2 Energy instead of pressure
  3. 3 Four Bernoullis and one name
  4. 3 Two totals, one of which a shock cannot touch
4 essays · misconceptions A closed wake, and the loading of least drag on it. The wake of a box wing in the plane that decides its drag, with the circulation of least induced drag drawn as a thickness along it. The horizontal members carry a loading close to elliptic and the vertical ones carry a share that lifts nothing — they contribute no lift, since lift is Γ dy and dy is zero on a vertical, and they change the drag by changing where the wake's vorticity is. At a gap of 20 per cent of span this system costs 67.1 per cent of what a single wing of the same span and lift would.

Box wing

  1. 1 A wake that closes on itself
  2. 2 A lighter spar turns a box wing into a biplane
  3. 3 A box wing's fins earn their keep in the spar
  4. 4 A strut inside a box wing is invisible to the wake, and halves the spar
4 essays · circulation The bath is ten thousand times too small. The Rossby number — the ratio of the inertial term to the Coriolis term in the momentum equation — for eight flows at 45 degrees, on a logarithmic axis. Above one, rotation is a correction; below one, it is the physics. A draining bath sits at 3.2e+3 and a mid-latitude depression at 1.9e-1. The Coriolis term is not absent from the bath: it is present, computable, and four orders of magnitude smaller than the terms that decide what happens. Saying so is not the same as saying it is zero.

Coriolis scale

  1. 1 The bath does not know the hemisphere
  2. 2 The bath that was only ever a wait
  3. 3 The Earth bends every river a little
  4. 4 A river drifts right, and only a reach can show it
4 essays · misconceptions Two different physics, one destination. The Fanno and Rayleigh lines, each with temperature up the page and entropy across, each referred to its own sonic state. Neither is a curve anybody drew: the entropy was swept over Mach number and its maximum located, and it sits at Mach 1 on both to within the resolution of the sweep. The subsonic branch runs up to the nose from the left and the supersonic branch runs up from below, so whatever is being done to the flow, it can only be moved towards that point.

Fanno rayleigh

  1. 1 Two ways to choke
  2. 2 A duct that cannot be run backwards
  3. 3 A pipe cannot hold its gas at the wall's temperature
  4. 4 A broken line empties at the pace of its friction
4 essays · compressible Two roots walking towards each other, and one of them crosses. The roots of the characteristic quartic in the complex plane as the airspeed is raised from nothing to 105 metres per second — growth rate across, frequency up. At rest the two sit on the imaginary axis at the uncoupled frequencies. As the speed rises, the aerodynamic coupling drags them towards each other in frequency while pushing one left and the other right, and at 80.8 metres per second the right-hand one crosses the axis. Everything about the failure is in this picture: the coalescence, the crossing, and the fact that the flutter frequency is neither of the two the structure started with.

Flutter

  1. 1 The shake that is not resonance
  2. 2 The control that works backwards
  3. 3 The speed where the damping is exactly zero
  4. 4 The lag that makes flutter possible
4 essays · circulation Nu/Gr^¼ against the Prandtl number, with the exact solution's points on it. The closed form 0.508 Pr^½(20/21 + Pr)^−¼, over eight decades, with Ostrach's exact similarity values marked. The integral method is two to eight per cent high from Pr = 0.7 upwards and 10 per cent low at Pr = 0.01 — which is where the thermal layer is ten times the momentum layer and giving them one thickness stops being an approximation to anything.

Grashof

  1. 1 A speed nobody imposed
  2. 2 Over a plate in mercury the warm layer flows as if inviscid
  3. 3 Laid flat, a warm plate is pushed by the pressure of its own layer
  4. 4 Above a warm wire the plume speeds up as it rises
4 essays · regimes The same flow, from two different physics. Solid lines: the depth-averaged flow between two plates a small distance apart, with an obstacle standing between them. Dashed: the ideal-flow solution for the same obstacle. They are the same field to a part in ten billion, because averaging Stokes flow across a narrow gap gives a velocity that is the gradient of a harmonic potential. The cell has no inertia at all, which is the one hypothesis the ideal theory cannot do without.

Hele-shaw

  1. 1 The exact theory, drawn by viscosity
  2. 2 The cell draws a larger cylinder
  3. 3 The fastest finger is set by the gap and one number
  4. 4 The meniscus slows the finger and hardly moves it
4 essays · inviscid Wound up, and worth exactly what it started with. A material loop in a steady cellular flow — an exact solution of Euler's equations — drawn at four times. Each streamline in the cell has its own period, so the loop is stretched steadily into a spiral: by the last frame its perimeter is 5.7 times what it started as. The circulation round it is 0.903741 at the start and 0.903666 at the end. Nothing about the curve survives except the number.

Kelvin

  1. 1 What survives being wound up
  2. 2 The drift was the instrument
  3. 3 At thirty degrees the Earth keeps time with the sea breeze
  4. 4 The sea-breeze front runs till evening only near the equator
4 essays · inviscid

All essays