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

Series — page 3

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.
The laminar line does not end; the flow leaves it. Friction factor against Reynolds number in a pipe. The laminar law f = 64/Re is exact and is drawn continuing past the transitional Reynolds number, faintly, because it remains a solution there — the flow simply stops taking it. The turbulent branch is Blasius' correlation and begins where experiments find transition, not where any calculation puts it.

Transition

  1. 1 The solutions stop being chosen
  2. 2 The number that is not a number
  3. 3 Every mode decays and it grows anyway
  4. 4 A puff that does not know how old it is
4 essays · turbulence 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
4 essays · applied Two of opposite sign go somewhere. Two vortices of equal and opposite strength. Each is carried by the other's field, both are carried the same way, and the pair travels in a straight line at Γ/2πd forever, keeping its separation exactly. The speed is a consequence of one vortex's field evaluated at the other, and nothing else.

Vortex dynamics

  1. 1 Vortices move each other
  2. 2 Three is the most that can be predicted
  3. 3 What a point vortex is not
  4. 4 Reversible, and unusable
4 essays · inviscid α = 13.0: a plug in the middle and everything happening at the wall. The velocity profile at eight phases of one cycle, at a Womersley number of 13.0 — the human aorta's at rest. The core moves almost as a solid plug, because viscosity cannot reach it within a cycle; all the shear is in a layer of thickness √(ν/ω) = 0.77 mm against a radius of 10.0 mm. At some phases the fluid near the wall is moving backwards while the core still moves forward, which is the reversal a Poiseuille profile can never show and which is routinely measured in arteries.

Womersley

  1. 1 Too fast for a profile
  2. 2 Where the parabola goes
  3. 3 The pulse that has to travel
  4. 4 The pulse that grows as it leaves the heart
4 essays · regimes 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
3 essays · compressible A steady pressure field, from a flow with no steady part. The time-averaged pressure round a cylinder in a stream that oscillates as U₀cos ωt. The mean velocity is exactly zero at every point — the flow spends as long going one way as the other — and the mean pressure is not, because pressure depends on the square of the speed and a square has no sign. The mean coefficient reaches -2.00 at the shoulders and averages -1.00 over the surface, and its resultant is 6.6e-16: a real field with no force in it. The pale lines are the instantaneous streamlines, which reverse every half cycle.

Averaging

  1. 1 The mean is not the flow
  2. 2 Two averages of one flow
  3. 3 What a mean profile cannot tell anybody
3 essays · kinematics The stretch of surface that is boiling. The pressure coefficient along both surfaces of a section at 4 degrees, computed from the same potential-flow solution as the other ideal-flow aerofoil figures. The horizontal line is the vapour pressure at a cavitation number of 1: wherever the suction curve is above it, the liquid there has been pulled below its vapour pressure and is boiling at whatever temperature it happens to be. This section cavitates at any σ below 1.428.

Cavitation

  1. 1 When a body tears the water
  2. 2 The bubble that hammers
  3. 3 A threshold that is also a duration
3 essays · applied The wall is what cancels the waves it made. The characteristic net of a minimum-length nozzle designed for Mach 2.4, with 18 waves. The pale lines run from the sharp throat down to the axis, reflect there by symmetry, and run back up to the wall; the wall turns through exactly the angle needed to cancel each one as it arrives, so nothing reflects back into the flow and the exit is uniform at Mach 2.400 and parallel to 0.0 degrees. The area ratio decides the Mach number and this net decides the shape, and the two agree on the exit height to 0.51 per cent at this resolution.

Characteristics

  1. 1 The wall that cancels its own waves
  2. 2 Every compression becomes a shock in the end
  3. 3 Two numbers that do not change
3 essays · compressible Every curved streamline has a pressure gradient across it. Twelve points in the flow past a cylinder, with the arrow at each showing the pressure gradient across the streamline. It points away from the centre of curvature everywhere, and its size is ρq²κ: the fluid is being pushed round a bend, and something has to do the pushing. Both sides are computed here and they share no arithmetic — one is Bernoulli's pressure differenced across the flow, the other is the turning rate of the velocity direction along it — and they agree to 6.7e-5. This is the whole content of the effect usually named after Coandă, and it is happening on every curved streamline of every flow.

Coanda

  1. 1 The effect that explains nothing
  2. 2 The effect that is real, and where it stops
  3. 3 The teapot effect is a tension, not a pressure
3 essays · misconceptions Where the number says, and where it happens. Fourteen dimensionless groups on one logarithmic axis. The open circle on each row is the value at which the two terms the group compares are equal, which is one by the way the group is formed; the filled mark is the value at which the thing a reader cares about first changes by 1%. The bar between them is the distance the folklore phrase "of order one" hides, and it runs from nothing at all to a factor of 594.

Crossover

  1. 1 What "of order one" is worth
  2. 2 One formula for both ends
  3. 3 The three that never converge
3 essays · regimes d'Alembert's paradox, measured. Surface pressure round a cylinder in ideal flow, plotted against angle. The distribution is symmetric front to back, so every push on the front is matched by an equal push on the back, and the total force along the stream is exactly zero.

Dalembert

  1. 1 The exact theory says nothing has any drag
  2. 2 The force of getting going
  3. 3 The drag that is made of waves
3 essays · inviscid The core spreads and the outside never notices. The swirl velocity at four times a factor of four apart, with the free vortex Γ/2πr drawn behind them. Every curve leaves the free-vortex line at its own core radius and turns over into solid-body rotation inside it; outside the core all four are the same curve, to the precision of the plot. Viscosity has rounded off the singularity and changed nothing else. The peak swirl falls from 52.4 to 6.6 m/s across the four, and the circulation is identical for all of them.

Diffusion

  1. 1 What viscosity cannot take away
  2. 2 The energy a vortex cannot have
  3. 3 The drag that integrates a whole history
3 essays · viscous A wing's two drags, and the lift at which they are equal. Friction drag and induced drag plotted against lift coefficient, with their sum above them. Friction is flat, because a surface costs the same whatever the wing is doing; induced drag rises as the square of the lift. The total is least where the two are equal, and that is also the point of best glide.

Drag budget

  1. 1 The two drags a wing pays
  2. 2 The cheapest way to stay up
  3. 3 The cost of going turbulent
3 essays · viscous A cylinder in a uniform shear, K = 0.4. A stream whose velocity increases with height, meeting a circular cylinder. The oncoming profile is drawn at the left. The flow carries uniform vorticity −K, so it is a solution of Euler's equations and not of Laplace's, the pattern is no longer symmetric top to bottom, and the body feels a lift towards the fast side with no circulation anywhere.

Euler rotational

  1. 1 Inviscid does not mean irrotational
  2. 2 The one rotational solution anybody can write down
  3. 3 The vorticity nothing decides
3 essays · inviscid Circulation across the span, for three planforms. How much circulation each part of the wing carries, plotted across the span. It has to reach zero at both tips, because a wing cannot carry circulation off its end, and the rate at which it falls is what determines the vorticity shed into the wake.

Finite wing

  1. 1 The price of having ends
  2. 2 The span is the whole story
  3. 3 Where the line stops being a line
3 essays · circulation Streamlines and pathlines are not the same curve. In an unsteady flow the line tangent to the velocity everywhere at one instant, and the track a single particle actually follows, are different curves. They coincide only when the flow is steady, which is the hypothesis most figures forget to state.

Flow curves

  1. 1 Streamlines are not the paths particles take
  2. 2 The line the dye actually draws
  3. 3 The curve that measures a gradient
3 essays · kinematics The momentum of the fluid, against the shape of the region it is added up over. Momentum of the fluid around a cylinder moving through it, divided by the body's hydrodynamic impulse, against the aspect ratio of the rectangle the integral was taken over. Every rectangle has the same area and contains the same body. A tall region gives minus the impulse, a long one gives plus it, a square gives exactly zero, and the limit of a large region is whichever of those the region was shaped like. The momentum of an unbounded ideal flow is not a number.

Impulse

  1. 1 The momentum with no value
  2. 2 The mass a body has to borrow
  3. 3 Everything about the start, except one vector
3 essays · inviscid Most of the fluctuation is the switching. What a hot wire at the edge of a shear layer records, taken apart. The conditional intensity — the fluctuation inside the turbulent fluid — is flat at 0.16 by construction. The measured intensity peaks 20 per cent above it, and the excess is the third curve: the variance of a signal that keeps switching between two mean velocities, which peaks where the switching is most even and is not turbulence at all. A model calibrated against the measured curve is being fitted to an artefact of averaging.

Intermittency

  1. 1 Turbulent some of the time
  2. 2 The exponents that stop being thirds
  3. 3 A dissipation correlated across every scale
3 essays · turbulence The chart, with one exact line on it. The friction factor of a pipe against Reynolds number, for five relative roughnesses. Every curve here except one is Colebrook's correlation, solved by iteration rather than read off a chart. The exception is the short straight line at the left: f = 64/Re is the laminar solution and it is exact. The curves flatten to the right because once the roughness pokes out of the viscous layer the Reynolds number has nothing left to change.

Internal flow

  1. 1 The roughness a wall cannot feel
  2. 2 A loss with no viscosity in it
  3. 3 A number that is only the shape of the hole
3 essays · applied The jet is 61.1% of the hole. Flow out of a slot in a plane wall, solved by Kirchhoff's free-streamline method. The outer curve is not a wall and not a guess: it is the streamline on which the pressure is ambient, and where it goes is part of the solution. It leaves the edge of the slot travelling straight down the wall and turns through ninety degrees, settling to a jet whose width is π/(π+2) = 0.6110 of the opening. Every streamline drawn is a level set of the streamfunction the conformal map supplies.

Jet

  1. 1 The hole that halves the flow
  2. 2 What a jet cannot push sideways
  3. 3 Half the jet speed takes everything
3 essays · applied

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