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

Series — page 6

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.
Ballistic first, diffusive later. The mean square displacement of a parcel carried by a fluctuating velocity, from Taylor's 1921 integral with an exponential correlation of time scale 1. Below the correlation time it follows the straight-line law u²t² exactly — the parcel has not yet changed its mind — and above it the curve joins the diffusion law 2u²T·t, offset by the head start the ballistic phase gave it. The eddy diffusivity is u²T = 1.000, and it is a property of the correlation rather than of any equation of motion.

Mixing

  1. 1 How far a parcel gets
  2. 2 The scalar has its own cascade
2 essays · turbulence 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
2 essays · circulation Five flows past one cylinder, every one of them a solution. Surface pressure round a cylinder in a stream, at five circulations. Each satisfies Laplace's equation, the tangency condition on the body and the condition at infinity, and each has a different lift. Nothing in the problem chooses between them: the domain has a hole in it, so the potential is many-valued and the circulation is a free constant.

Multiply-connected

  1. 1 The constant a hole leaves behind
  2. 2 Nothing in the present picks the flow
2 essays · inviscid 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
2 essays · misconceptions γ for air, against temperature. The ratio of specific heats for air as a mixture of nitrogen and oxygen, with the vibrational mode filling according to the Einstein function. It is 1.400 at room temperature, where only translation and rotation are available; 1.337 at a thousand kelvin; and 1.288 at six thousand. Every compressible result on this site has used 1.4, and that is the value for a gas that is not hot — which, behind any shock worth drawing, it is not.

Real gas

  1. 1 When gamma stops being a number
  2. 2 A gas that has not finished being shocked
2 essays · compressible Quasi-steady stops being true a long way before one. The magnitude of Theodorsen's function, which is the factor a quasi-steady lift calculation is wrong by, and the phase the lift lags the motion. Quasi-steady means C = 1, and the amplitude is already one per cent low at k = 0.0061 and fifteen per cent low at k = 0.1 — a reduced frequency at which nobody hesitates to call a flow quasi-steady. The lag is worse: it reaches a degree at k = 0.003, and a flutter calculation is decided by phase rather than by amplitude.

Reduced frequency

  1. 1 Slow enough to be steady
  2. 2 The gusts that cancel along the chord
2 essays · regimes The current at the surface is 45° from the wind, and nothing sets that angle. The Ekman spiral drawn as a hodograph: each point is the velocity at one depth, and depth runs along the curve. At the surface the flow is at exactly 45 degrees to the wind that drives it — not approximately, exactly, and independently of the wind, the viscosity and the latitude. By one Ekman depth the flow has turned another radian and lost 1/e of its speed; by three it is a hundredth of the surface value and pointing back the way it came. The angle is a property of the equation having two terms in it, and nothing else.

Rotating

  1. 1 The layer that stops at a depth
  2. 2 How long a fluid takes to forget it was not rotating
2 essays · viscous The loading over the disc in forward flight, at μ = 0.4. Section lift per unit span over the rotor disc, with the flight direction upwards, the advancing side to the right and the reversed-flow region outlined. The loading is not axisymmetric and cannot be: at this advance ratio the advancing blade meets 1.40 times the tip speed and the retreating one 0.60. Every quantity a hover calculation reports as a function of radius is here a function of two variables.

Rotor

  1. 1 The side that cannot keep up
  2. 2 A blade that flies through what it shed
2 essays · circulation The roughness function, and the asymptote in which the viscosity has gone. The whole effect of a rough wall on a turbulent boundary layer is one number: the downward shift of the logarithmic profile. It vanishes on a smooth wall, rises through a transitional band, and becomes (1/kappa)ln(k+) + B − 8.5 — at which point substituting it back leaves u+ = (1/kappa)ln(y/k) + 8.5, with the fluid's own length gone from the answer entirely.

Roughness

  1. 1 A second length at the wall
  2. 2 How far downwind a surface is remembered
2 essays · turbulence 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
2 essays · compressible The singularity a thin aerofoil drives onto its own nose. The Joukowski map has a critical point that maps to a place inside the body, a distance 4 mu²/(1 + 2 mu) from the leading edge. Against thickness that distance is a clean square: a twelve per cent section is analytic only within eight thousandths of a chord of its own nose, and the thin-aerofoil limit is the limit in which the singularity arrives on the surface.

Singularities

  1. 1 The part of the flow inside the body
  2. 2 The inside a flow does not decide
2 essays · inviscid 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
2 essays · turbulence Three bluff bodies, whose Strouhal numbers agree once the wake's width is used. The measured Strouhal number of each body, and Roshko's universal number formed with the wake's width and the speed on the free streamline that bounds it. The raw numbers span a factor of 1.462; the collapsed ones span 1.0011, with a mean of 0.16281. The shedding was never body-dependent — the length in the number was.

Strouhal

  1. 1 The frequency a wake chooses
  2. 2 A wake told what to do
2 essays · regimes Two profiles, and they are the same profile. The chordwise and spanwise velocity profiles in the boundary layer of a yawed flat plate, each as a fraction of its own edge velocity. They are computed by different code — the chordwise one by shooting a third-order nonlinear equation, the spanwise one by a single pass through a linear second-order one — and they agree to 2.1e-8 over the whole layer. They are the same function of η, because the two equations reduce to the same equation. A swept flat plate has no crossflow at any sweep angle, and that is the independence principle in the only form that has no wriggle room in it.

Sweep

  1. 1 The wind a swept wing feels
  2. 2 The sweep a root does not have
2 essays · circulation 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
2 essays · circulation The marginal Taylor number against the axial wavenumber. The smallest Taylor number at which a disturbance of a given axial wavenumber is neutral. Its minimum is 1707.757 at a wavenumber of 3.1158, which are the critical Rayleigh number and critical wavenumber of a layer of fluid heated between two rigid walls — the same numbers, because in the narrow-gap limit the two problems are the same sixth-order eigenvalue problem. This curve is computed by that essay's own solver.

Taylor

  1. 1 A transition that needs a second number
  2. 2 The state a machine was started into
2 essays · regimes Four camber lines, and the angle at which each stops lifting. Four mean lines on the same chord: symmetric, a circular arc, a four-digit line with its crest at forty per cent, and a reflexed line whose tail turns up. The zero-lift angle beside each is computed from that line's own slope by quadrature and is a property of the shape alone — no incidence, no speed, no thickness enters it. The symmetric line's is exactly zero, the arc's is −2m to ten decimal places, and the reflexed line's is positive: it needs to be pointed up before it stops lifting.

Thin-aerofoil

  1. 1 Where lift starts
  2. 2 Three numbers out of a camber line
2 essays · circulation Elliptic loading rolls up to πb/4, and it is π that puts it there. Three loadings on the same span carrying the same lift, with the rolled-up core positions marked below each. The core sits at the centroid of the shed vorticity, which is a quadrature over the loading and needs nothing about the roll-up itself. Elliptic loading gives πb/8 from the centreline, so the pair ends up 0.7854 of the span apart — the number every wake-separation rule is written against, and one of the few places in this subject where π turns up in an answer an engineer uses directly. The bell rolls up to 0.586 and a nearly rectangular loading to 0.978, because it sheds at the tips.

Tip vortex

  1. 1 Where the wake ends up
  2. 2 A wake that says what made it
2 essays · circulation A material region, and the dye that stays inside it. The same fluid at four times, carried by an unsteady straining flow whose strain rate oscillates. The outline is a circle of the fluid at the first instant, tracked by integrating the velocity field; the shading is a blob of passive dye. The region is stretched to nearly seven to one and its area is unchanged to fifteen decimal places, because the flow is incompressible. The amount of dye inside it is unchanged to thirteen, because the dye is carried by the same fluid.

Transport theorem

  1. 1 A rate of change that will not hold still
  2. 2 The mesh that makes its own mass
2 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
2 essays · turbulence

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