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

Series — page 2

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.
1.139 asks for a Francis. The specific-speed axis, with the four machine types on it and one duty marked: 3 m³/s at 60 m, on a shaft turning at 750 rev/min. The number is 1.1387, and the choice of runner follows from it before any blade has been drawn. What the number contains is a ratio of flow to head; what it does not contain is any size at all, which is why one axis serves a garden pump and a gigawatt turbine.

Specific speed

  1. 1 One number picks the machine
  2. 2 The duty that had no machine
  3. 3 The group with no head in it
  4. 4 The specific speed a pump spends its life at
  5. 5 One fast pump or two slow ones
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7 essays · applied A film with nothing useful to show for itself. The pressure under two discs squeezed together, as a fraction of the pressure at the centre. It is a parabola, and it holds an enormous load — but nothing is sliding, so there is no output at all and every joule put in becomes heat in the oil. The two routes to that heat share no arithmetic: one integrates the dissipation function over the film, the other multiplies the force by the approach speed.

Squeeze film

  1. 1 The last of the oil
  2. 2 A damper that turns into a spring
  3. 3 A ball that bounces in water and not in oil
  4. 4 A torn film still pulls
  5. 5 The heat of a squeeze leaves across the gap
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7 essays · viscous Two triangles, and the work is the difference between them. The velocity triangles at inlet and outlet of a rotor at constant blade speed and constant axial velocity. The horizontal arrow is the blade speed; the arrow from the origin is the absolute velocity of the fluid; the arrow closing the triangle is what the blade sees. Euler's equation says the work is the blade speed times the change in the swirl component alone — the horizontal distance between the two upper corners, times U — and nothing else in the picture appears in it.

Turbomachine

  1. 1 Work out of a change of swirl
  2. 2 What a turning frame keeps
  3. 3 The stress that picks the aerodynamics
  4. 4 Matched at one speed and at no other
  5. 5 The plenum decides whether a compressor surges
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7 essays · applied 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.

Vorticity

  1. 1 Spin is not the same as going round
  2. 2 Circulation is vorticity, added up
  3. 3 The spin that feeds itself
  4. 4 Where vorticity comes from
  5. 5 A wall puts in exactly its own speed
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7 essays · kinematics The tube widens because the air slows. The streamtube through an actuator disc at an induction factor of 0.333. The three radii are not drawn to taste: each is fixed by requiring the same mass to pass every station, and the slowest station is therefore the widest. The tube widening in front of a wind turbine is why some of the wind goes round it rather than through it, and it is the whole reason a disc cannot take everything.

Actuator disc

  1. 1 The most a disc can take
  2. 2 A big slow push
  3. 3 The wake that has to spin
  4. 4 Between hover and twice the hover inflow
  5. 5 A duct is worth the square root of two
  6. +1 more
6 essays · applied Each cancellation costs two powers of the Mach number. Radiated power against compactness for three source clusters: a single monopole, two of opposite sign, and four on a square with alternating signs. The fitted slopes are 0.00, 2.00, 4.00 — zero, two and four in (kd), measured by integrating the far field over a sphere rather than assumed. A turbulent eddy turns over in about the time sound crosses it, so kd is of order the Mach number, and those exponents become the fourth, sixth and eighth powers of speed. A flow with no moving surfaces has no monopole and no dipole available to it, which is Lighthill's whole argument, and the eighth power is what is left.

Aeroacoustics

  1. 1 The sound that only leaves
  2. 2 The sound is what does not cancel
  3. 3 The sound now is the source then
  4. 4 A wire sings with the force on it, not the flow round it
  5. 5 More blades hush a propeller only while its tips are slow
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6 essays · compressible A duct does the opposite thing above Mach one. The four cases of dA/A = (Ma² − 1) dV/V. Below the speed of sound a narrowing duct accelerates the flow, which is what continuity leads anyone to expect. Above it the sign of the bracket flips, and a narrowing duct decelerates: density is falling faster than the speed is rising, so the stream tube needs more room rather than less.

Area mach

  1. 1 The duct that works backwards
  2. 2 The throat that stops listening
  3. 3 One area, two answers
  4. 4 The jet a cone sprays sideways
  5. 5 Friction moves the sonic point past the throat
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6 essays · compressible One more condition, and the price of it. The model problem ε u″ + u′ = 0 with a condition at each end, at three values of ε. The outer solution is the flat line at one — that is the whole of the answer when ε is zero, and it is a first-order equation that can meet one condition, so it meets the one at the far end and misses the one at the wall by the whole range. Restoring ε restores the second condition and pays for it with a layer of thickness ε, inside which the gradient is of order 1/ε. The product of those two — which is what a stress is — does not depend on ε at all: it is 1.0000 at every value tried, to nine decimal places. Drag does not vanish as viscosity does. It converges.

Boundary conditions

  1. 1 How many things a flow must be told
  2. 2 A wall that is not quite there
  3. 3 The vorticity a clean surface cannot refuse
  4. 4 Twice as slippery along as across
  5. 5 A post holds liquid back by its radius, a stripe by its length
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6 essays · kinematics Every disturbance has its own threshold; one of them is lowest. The Rayleigh number at which a disturbance of horizontal wavenumber a becomes neutral, Ra = (π² + a²)³/a². Every wavenumber has a threshold and the layer goes unstable at the lowest of them, which a golden-section search on this curve puts at a = 2.221441 and Ra = 657.5114 — the exact π/√2 and 27π⁴/4 to fourteen digits. Below the curve the layer conducts and nothing moves.

Convection

  1. 1 A threshold with a closed form
  2. 2 Three numbers left of a fluid
  3. 3 A millionth is enough
  4. 4 The threshold the walls decide
  5. 5 The truncation that cannot carry three times the heat
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6 essays · turbulence A flow that is incompressible and carries a density that varies three to one. Ideal flow past a cylinder, shaded by a density that is constant along each streamline and runs from one to three across the field. Every parcel keeps the density it started with, so the divergence is zero — measured at 10⁻¹⁰, which is the differencing — and the flow is incompressible in the only sense the word has. The density is not uniform anywhere. Incompressible is a statement about what the flow does to a parcel's volume, not about what the fluid is made of.

Dilatation

  1. 1 Incompressible is not a property of the fluid
  2. 2 The air flows in against the heat
  3. 3 A flat flame is unstable at every size
  4. 4 A wrinkled flame has one cusp and a speed limit
  5. 5 A flame's speed limit holds only in silence
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6 essays · kinematics 5 quantities, 3 rows, 2 left over. The dimension matrix for the drag on a sphere: one column per quantity, one row per base dimension, and every entry an exponent. Buckingham's theorem is a statement about this matrix and nothing else — the number of independent dimensionless groups is the number of columns minus the rank, computed here by elimination. Nothing about fluids enters until somebody decides which columns to write down.

Dimensional

  1. 1 Counting what matters
  2. 2 A limit nothing reaches
  3. 3 Where a pure number comes from
  4. 4 Exact in the total, free in the profile
  5. 5 The groups are not the only groups
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6 essays · regimes Two parcels released together do not arrive together. The most repeated explanation of lift says that air parting at the leading edge must meet again at the trailing edge, so the longer upper path forces a higher speed. Released into the solved field, the upper parcel arrives long before the lower one — the premise is simply false, and the real speed difference is larger than it would require.

Equal transit

  1. 1 The story about air meeting up again
  2. 2 The wing that is flat, and flies
  3. 3 The upper parcel leads by the circulation
  4. 4 Inside the layer the upper parcel leads by more
  5. 5 The side that turns turbulent first decides who arrives first near the wall
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6 essays · misconceptions A wave that dies within one wavelength — 100 Hz in air. The velocity profile above an oscillating wall, at eight phases of one cycle, with depth in units of δ = √(2ν/ω). The motion is a wave travelling into the fluid, and its amplitude falls by 1/e in the same distance it turns by one radian — so it is dead within about one wavelength, and the fluid three δ up hardly knows the wall is moving at all. This is one of the very few exact solutions the Navier–Stokes equations have. The dashed line is one fixed distance above the wall in millimetres: in these units it climbs as the square root of the frequency, which is the whole of how far the motion reaches.

Exact layer

  1. 1 The wall that shakes
  2. 2 The layer that stops growing
  3. 3 The wall the fluid is listening to
  4. 4 The solution that keeps its nonlinear term
  5. 5 One channel, one flux, two flows
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6 essays · viscous Tip to tip, two wings cost exactly half of what they cost apart. The induced-drag saving of a pair of wings against the gap between their tips, as a fraction of what the two pay flying alone. With the tips touching the pair is one wing of twice the span carrying twice the lift, and the arithmetic of that is exact: the drag halves, and the computation returns 0.499257 of the separate figure. Pull them apart and the saving falls away with the square of the distance. Birds fly in a V because the tips are the part worth overlapping, and the spacing that pays is a small fraction of a span rather than any distance a formation could hold by eye.

Formation

  1. 1 The lift beside a wing
  2. 2 The bird at the point pays for the V
  3. 3 A fair V is a curved V
  4. 4 The follower beats in time with the wake, not the bird
  5. 5 A flapping follower can drift fore and aft, but not sideways
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6 essays · circulation The jet has to be fed from the sides. The velocity field of the plane jet with streamlines integrated through it. The seven central streamlines run down the jet and spread; the ten started at the top and bottom edges bend inwards and join it, which is entrainment and is a consequence of the solution rather than an addition to it. The dashed lines are the half-speed edges, widening as x^{2/3}. The transverse velocity far from the axis is 5.70e-3 m/s at this station, inward on both sides — a jet is a sink as seen from a distance, which is why two parallel jets pull together.

Free shear

  1. 1 What a jet keeps, and what it collects
  2. 2 Exactly similar, and one number short
  3. 3 A point force makes a jet only when it is strong
  4. 4 A jet along a wall keeps the flux of its momentum flux
  5. 5 A jet striking a plate leaves a stronger wall jet the further it travelled
  6. +1 more
6 essays · viscous The Kelvin wake, and the angle it comes out at. The wake behind something moving over deep water. Each curve joins everything radiated at one moment, wherever it has since travelled to. All of it stays inside a wedge whose half-angle is the inverse sine of one third, about nineteen and a half degrees, and that number does not depend on the speed, on gravity, or on the size of the vessel.

Froude

  1. 1 The angle that does not care
  2. 2 The number that really is one
  3. 3 Hull speed is not the hump
  4. 4 The hull Michell's integral prefers
  5. 5 The stern's wake puts the bulb at the bow
  6. +1 more
6 essays · regimes The mean lift is not the lift at the mean angle. A finite wing's lift curve, with a gust distribution of standard deviation 3° about a mean angle of 10° drawn along the foot. The wing spends its time spread across that distribution, so what it averages is the average of the curve: 0.7951 against the 0.8368 the mean angle promises, a deficit of 5.0 per cent. Nothing has stalled, and no gust has taken the wing past the stall angle: the deficit comes entirely from the curve bending over, and it is there at every angle where the curve is not straight.

Gust

  1. 1 The lift at the mean angle
  2. 2 The aeroplane rises out of its own gust
  3. 3 The span takes away the infinity, not the gust
  4. 4 A wing averages a gust through its own loading
  5. 5 One wing root feels the rolling gust its lift cannot
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6 essays · circulation A shear layer, and the point of inflection in it. The velocity profile U = tanh y across a layer of finite thickness, with the inflection point located by searching for a sign change in the second derivative rather than by reading it off the algebra. Rayleigh's theorem says an inviscid parallel flow can only be unstable if such a point exists — a necessary condition, not a sufficient one.

Instability

  1. 1 A layer with a kink in it
  2. 2 Every wavelength at once
  3. 3 Every unstable wave is inside one circle
  4. 4 The profile Rayleigh cleared and viscosity did not
  5. 5 How far apart a ceiling drips
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6 essays · turbulence 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
  3. 3 The further apart, the faster they part
  4. 4 A pair's memory shapes the cloud it spreads into
  5. 5 Bursts and memory pull a cloud both ways
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6 essays · turbulence What the skin settles at, before anything is done to it. The adiabatic wall temperature against Mach number, in air at 216.7 K, with the stagnation temperature above it. The gap between the two is the recovery factor, which is 0.8417 here and stays there at every Mach number — it is a property of the Prandtl number and not of the speed. At Mach 2 the skin sits at 363 K, at Mach 3 at 545 K, and at Mach 5 at 1128 K, which is past what aluminium will do. Nothing has been burnt and nothing has been rubbed: the air was brought to rest, and this is where its kinetic energy went.

Recovery

  1. 1 The wall that heats itself
  2. 2 The skin that lags the flight
  3. 3 The thermometer that heats itself
  4. 4 Three readings, and the one each answer leans on
  5. 5 The gradient the heat never hears
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6 essays · compressible

All essays