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

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.
Mach number: one number, four different flows. Mach number is speed ÷ speed of sound. 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.

Mach

  1. 1 When air stops being incompressible
  2. 2 The pocket on top of the wing
  3. 3 Which speed goes in the number
3 essays · regimes Two flows with the same rate of strain, doing different things to a blob. A circle of fluid carried by two flows chosen to have exactly the same rate-of-strain magnitude, drawn at four times. Pure strain pulls it into an ellipse whose axes stay put; simple shear pulls it into an ellipse whose axes rotate as fast as they stretch. Both have zero divergence, so both preserve the area. The difference between them is not the strength of the straining but what the rotation does to the direction being stretched.

Material lines

  1. 1 Longer, with nothing pulling it
  2. 2 The stretching rate that is not one number
  3. 3 Two strainings, and the order they came in
3 essays · kinematics The same reading, and only one of them gives it back. A Venturi and an orifice plate at the same diameter ratio, with the pressure along the axis drawn beneath each. Both narrow the flow by the same amount, both read the same difference between the pipe and the narrowest section, and both infer the same flow rate from it. Downstream they part company: the Venturi's diffuser turns the throat's speed back into pressure, and the orifice's jet expands into the pipe and destroys 73% of the reading. The picture is a section rather than a solved field: nothing here computes the jet, and the recirculating corner is not drawn.

Metering

  1. 1 The price of knowing the flow rate
  2. 2 The instrument in the answer
  3. 3 The profile a meter cannot see
3 essays · applied 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
3 essays · compressible 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
3 essays · applied 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
3 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
3 essays · regimes A millionth, doubling every three-quarters of a second. The separation of two trajectories started a millionth apart, on log axes against time. It grows as a straight line until it saturates at the size of the attractor, and the slope of that line is the largest Lyapunov exponent — measured here on the system by renormalising a nearby pair, not quoted. Sensitive dependence is what the straightness of the line means.

Randomness

  1. 1 The randomness that is not in the equations
  2. 2 An hour for every tenfold
  3. 3 The same statistics, and a different load
3 essays · misconceptions 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 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
3 essays · compressible 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
3 essays · applied 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 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
3 essays · turbulence The one exact law, and where viscosity takes it back. The two terms of the Kármán–Howarth relation against separation, measured in Kolmogorov lengths. The four-fifths term rises linearly with r and is the whole of the law at large separation; the viscous term, 6ν dS₂/dr, falls as r^(−1/3) and takes over below 5.64η. Nothing in the calculation was told what η is: the crossing is at the same multiple of it at every viscosity tried, which is what makes the dissipation scale the lower end of the inertial range rather than a separate assumption.

Structure function

  1. 1 The one exact result
  2. 2 The fraction that is really four thirds
  3. 3 A relation with no turbulence in it
3 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
3 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
3 essays · misconceptions 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
3 essays · turbulence 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
2 essays · kinematics 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 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
2 essays · circulation

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