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

Series — page 7

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 sphere does not go at the speed of the flow it is in. How far a force-free sphere on the axis of a round tube lags the fluid at its own centre, against its size. The lag is two-thirds of the square of the size ratio and comes from the reciprocal theorem in one line: the sphere moves at the average of the ambient flow over its own surface, and a parabolic profile is slower everywhere on that surface than at the middle. At a fifth of the tube's radius the lag is 2.7 per cent, which is the difference between a tracer and a thing being measured.

Reciprocity

  1. 1 A force without the flow that makes it
  2. 2 Two spheres settling at a slant keep their slant for ever, and three do not
  3. 3 A larger sphere overtakes its neighbour and leaves on the line it came in on
3 essays · viscous 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
  3. 3 Sand and a commercial pipe part company at six wall units
3 essays · turbulence Reynolds number: one number, four different flows. Reynolds number is inertia ÷ viscosity. It is not a property of the fluid or of the shape but of the combination, and crossing a threshold changes the physics rather than the magnitude.

Similarity

  1. 1 The model that cannot be matched
  2. 2 The wing the equation is really solving
  3. 3 An exponent dimensions cannot give
3 essays · regimes The left wall is not a speed, it is an angle drawn in speed coordinates. The V–n diagram: every combination of speed and load factor the aircraft can reach. The curved left boundary is the wing at its stalling angle — n = ½ρV²S C_Lmax /W, a parabola — and it is the same limit at every point along it. The flat top and bottom are the structure. Where the two meet is the corner speed, 56.6 metres per second here: the slowest speed at which the aircraft can reach its limit load factor, and therefore the speed at which it turns hardest. Above it the wing can pull more than the structure allows — at the never-exceed speed it could reach 9.6g before stalling — and the pilot's limit stops being the air.

Stall speed

  1. 1 An angle, not a speed
  2. 2 A roll that feeds itself
  3. 3 A stall that is a place
3 essays · misconceptions A swimmer that dissipates the same everywhere. Taylor's waving sheet, with the wave drawn along the bottom and the dissipation drawn against height above it. The dissipation function works out at 4μb²c²k⁶y²e^{−2ky} — with no x in it at all, so the sheet is destroying energy at the same rate under every part of the wave and at every instant of the cycle. It peaks one radian of wavelength above the sheet and is gone within about three.

Swimming

  1. 1 A swimmer that cannot go backwards
  2. 2 A stroke is worth the area it encloses
  3. 3 Two drags, or nothing swims
3 essays · viscous 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
  3. 3 A load carried to the trailing edge needs a hook
3 essays · circulation The speed depends on the core; the slope does not. A ring's self-induced speed against the logarithm of the cutoff used to compute it. The line is a quadrature of the reduced integral; the dots are an independent sum of cross products over a hundred and twenty thousand straight segments. Both diverge as the cutoff is thinned — a filament of zero thickness would move infinitely fast — so no answer here is a ring's speed until a core model is chosen. The slope, Γ/4πR, is the same whatever is chosen, and is what the check measures.

Vortex ring

  1. 1 A ring moves because it is bent
  2. 2 Two rings leapfrog only if they start alike
  3. 3 Leapfrogging rings end by merging, not by parting
3 essays · circulation Four corners, and what the flow does in each. The local flow in corners of four different interior angles, drawn from the exact local solution ψ = r^(π/α) sin(πθ/α). The exponent of the speed is π/α − 1 and depends on nothing else: at a right angle the corner is stagnant, at a flat wall nothing happens, and at any angle greater than a straight line the speed has no bound at the corner. The last panel, at 360 degrees, is the flow round the edge of a plate.

Wedge flow

  1. 1 Nothing turns a sharp corner
  2. 2 A step's face is pushed at its foot and pulled at its lip, equally
  3. 3 A block's front face is pulled upstream by its own lip
3 essays · inviscid Same disc, same solidity, different number of blades, different answer. Thrust and power coefficients for six blade counts at a fixed solidity. They span a factor of 1.101 in thrust, and every one of those rotors is the same actuator disc: same area, same blade area, same tip-speed ratio. The disc theory cannot distinguish them because the blade count is not one of its variables.

Blade-element

  1. 1 A disc that knows no blades
  2. 2 The inflow that takes time to arrive
2 essays · circulation One calculation is about the animal; the other is about how fast it happens to be going. The mean lift coefficient required of each animal's wings, computed two ways, on a log scale. Treating the wings as fixed and flying them at the animal's forward speed gives answers spanning a factor of 29 — from 0.88 to 26 — because the number is governed by a speed that has nothing to do with how the animal makes its lift. Doing the flapping arithmetic gives answers spanning a factor of 1.90. And in a hover the fixed-wing calculation has no answer at all: there is no dynamic pressure, and no coefficient however large will do. That is the version of the famous claim that is actually true, and it is a statement about the calculation rather than about the bee.

Bumblebee

  1. 1 The bee that cannot fly
  2. 2 A calculation with no memory in it
2 essays · misconceptions Twice as fast is not twice as thick. The film a plate carries out of water, against the speed it is withdrawn at, both logarithmic. The slope is exactly two-thirds, so doubling the speed thickens the film by 58.7 per cent and never by more. The open marks are outside the range the derivation holds in — above a capillary number of about a hundredth the film is no longer thin against the capillary length, and the measured thickness leaves this line.

Coating

  1. 1 What a plate takes with it
  2. 2 A thin fibre coats by its own radius, and beads by it
2 essays · viscous 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.

Continuum

  1. 1 What a flow is
  2. 2 A gas's viscosity ignores its pressure until the free path reaches the walls
2 essays · kinematics Storage and loss moduli, for one relaxation time and for a spectrum. The two moduli of a Maxwell fluid and of a Rouse chain, against frequency in units of the longest relaxation time. One relaxation time makes the storage modulus overtake the loss modulus at λω = 1 and then leave it behind without limit. A spectrum of 1000 modes makes them rise together as the square root of frequency and stay a fixed ratio apart, so the material never becomes the solid the single time predicts.

Deborah

  1. 1 A solid, if it is not given time
  2. 2 A liquid with one relaxation time rings in a tube, and a spectrum of times silences it
2 essays · regimes The Hugoniot of a gas that can burn, and the gap in the middle of it. Pressure against specific volume, both scaled on the unburnt gas. The lower curve is the ordinary shock Hugoniot, which passes through the initial state because a jump of zero strength satisfies mass, momentum and energy. Adding a heat release lifts it away, and the initial state now sits in a region no wave can reach: between the two branches a Rayleigh line would need a positive slope, and its slope is minus the square of the mass flux. A burning gas has no weak waves available to it at all.

Detonation

  1. 1 The other branch of the same curve
  2. 2 A gas that has not decided to react yet
2 essays · compressible A falling drop is a bun, not a tear. The shape of a drop pressed on by the air it is falling through, computed by matching the Legendre component of a sphere's own potential-flow pressure distribution against the change in curvature it produces. The result is oblate — flattened along the direction of travel — because the pressure is high at the poles and low round the equator. The teardrop of every illustration has the wrong symmetry entirely: a drop in free fall has no up, and one at terminal speed is being pressed on from in front.

Drop shape

  1. 1 The drop that is not a tear
  2. 2 A drop rings like a bell
2 essays · regimes The wall's condition, on its way to the middle. Five profiles across the half-channel, from just inside the entrance to fully developed, each drawn at the station where it occurs. The march starts from a slab of uniform flow and never assumes a shape: what arrives at the far end is a parabola, with a centre-line speed of 1.4979 times the mean against the exact 3/2 and a momentum flux of 1.1995 against 6/5. Notice what the middle does while the edges are being slowed: it speeds up, because the flow rate is held, and that acceleration is what the entrance's extra pressure drop pays for.

Entrance

  1. 1 How far before a duct forgets what was fed into it
  2. 2 A duct that forgets everything but one number
2 essays · viscous The same 10° turn, taken both ways. A supersonic stream turned away from itself expands through a fan of Mach waves and keeps every bit of its total pressure. Turned into itself through the same angle it shocks, and pays. Nothing in the equations distinguishes the two cases except the sign of the angle: compression waves converge and steepen into a front, expansion waves diverge and spread.

Expansion fan

  1. 1 Turning the other way is free
  2. 2 A compression that costs nothing in the end
2 essays · compressible Three contours, one force, three different accounts of it. The same vortex, and the same total force on it, computed by a momentum balance over three contours of the same area. All three give ρUΓ to eight figures. What differs is the bookkeeping: the tall box gets 16 per cent of it from pressure and the rest from momentum flux, the wide box gets 84 per cent from pressure, and the circle gets exactly half. Neither part converges on its own as the contour is enlarged — each falls off like 1/r while the contour grows like r — so the split is a property of the shape of the limit rather than of the flow.

Far field

  1. 1 Where the reaction to a wing's lift is
  2. 2 What the far field remembers
2 essays · inviscid The wall the outer flow is really solving for. A flat plate, the edge of its boundary layer, and the line the outer flow behaves as though the plate were on. The displacement thickness is the mass deficit divided by ρU — checked here against the profile's own integral rather than quoted — and moving the wall out by that much reproduces exactly the flow rate the viscous layer lets past. It is a third of the visible thickness of the layer and it is the only part of the layer the outer problem knows about.

Interaction

  1. 1 The body the outer flow actually sees
  2. 2 The length the limit invents
2 essays · inviscid Why a flat plate has no drag, drawn as a triangle. A flat plate at incidence with the three forces that must balance. Pressure can only act along the plate's normal, so the pressure force is the arrow perpendicular to the plate. Kutta–Joukowski says the resultant is perpendicular to the free stream. The difference between the two directions is the suction force at the leading edge, which acts forwards along the plate and is exactly L sin α. Without it the plate would have a drag of L sin α, and an inviscid fluid does not permit one.

Leading edge suction

  1. 1 A finite force from an infinite speed
  2. 2 A wing that cannot hold its suction gains little from its span
2 essays · circulation

All essays