Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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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.
A circle of fluid, 0.90 of a gradient time later. A material circle in a uniform velocity gradient, carried by the exact matrix exponential of that gradient. It becomes an ellipse — always an ellipse, for every gradient — and the axes it stretches along are the eigenvectors of the symmetric half. In an incompressible flow the area is unchanged however extreme the distortion, which is the statement that stretching in one direction is squashing in the other.

Deformation

  1. 1 What a parcel does in the first instant
  2. 2 A gradient left to itself
  3. 3 The pressure a parcel remembers keeps it finite
3 essays · kinematics 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 million times faster, and the constant is 48.0. The effective diffusivity along a pipe, divided by the molecular one, against the Péclet number — both logarithmic. Below Pe ≈ 7 the tracer simply diffuses and the curve is flat at one. Above it the dispersion is all Taylor's, rising as the square of the Péclet number, so a thousandfold Péclet number is a millionfold enhancement. The constant in D(1 + Pe²/48) is not quoted here: it is recovered from a numerical solution of the cell problem across the section, giving 48.0000 for a tube and 52.5 for a plane channel, which is Aris' 2/105.

Dispersion

  1. 1 Two slow things make a fast one
  2. 2 The best a tube can do is its own radius
  3. 3 Transport with nothing transported
3 essays · regimes 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 One flow, two observers, two pictures. The same ideal flow past a circular cylinder, drawn in the frame of the tunnel and in the frame of the undisturbed air. The two are related by subtracting one constant velocity. On the left the flow arrives from infinity, divides at a stagnation point on the nose and closes at another on the tail. On the right the air is at rest far away, the body pushes through it, the streamlines are closed loops, and there is no stagnation point anywhere in the field. Every force, every pressure and every measurement either observer can make is identical.

Frames

  1. 1 The picture belongs to whoever is watching
  2. 2 Every snapshot says vortex, and every particle leaves
  3. 3 A vortex pair carries eddies no snapshot can see
3 essays · kinematics The Kirchhoff flow past a flat plate. A uniform stream meeting a flat plate held across it, with two streamlines leaving the edges and never returning. Between them is a wake of fluid at rest at a constant pressure. The equations solved are the same equations that give d'Alembert's paradox for a closed body, and this flow has a drag coefficient of 0.8798.

Free-streamline

  1. 1 Drag in the theory that forbids it
  2. 2 Where the unknown boundary is the known one
  3. 3 A stalled plate keeps a quarter of its lift slope and moves its centre aft
3 essays · inviscid The error in the balance is the number itself. The fractional error in the geostrophic wind, against the Rossby number, on logarithmic axes. It is a straight line of slope one through the origin, and that is not an approximation: keeping the centripetal term gives V_g/V = 1 ± Ro exactly, so the error and the number are the same quantity. The geostrophic wind is one per cent right at Ro = 0.01 and a hundred per cent wrong at Ro = 1, which is the value the number is named for and is quoted as the boundary of the approximation.

Geostrophy

  1. 1 The balance that is its own error
  2. 2 The hill a slow current will not climb
  3. 3 A stratified sea blocks a current sooner and traps less
3 essays · regimes 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 Morison's two terms over one cycle, at KC = 10. The drag term, in phase with the velocity and going as its square; the inertia term, in phase with the acceleration and ninety degrees ahead of it; and their sum, which is what a load cell records. The peak of the total is not the peak of either, and its position in the cycle is the only thing in the record that says how the two are divided.

Keulegan–Carpenter number

  1. 1 Long enough to make a wake
  2. 2 Two forces, and only one of them remembers
  3. 3 Below separation a cylinder's drag is a Stokes layer, and KC stops deciding the force
3 essays · regimes How wrong the ordinary answer already is. The error in a no-slip continuum calculation of the flow through a channel, against the Knudsen number, both logarithmic. The rule at Kn = 1 is where a molecule crosses the whole channel between collisions — the value the number is named for. The error is one per cent at Kn = 1/594, five per cent at 1/114 and ten at 1/54, so the continuum regime of the usual classification, which runs to Kn = 0.01, is a region in which the continuum answer is already six per cent out at its far end.

Knudsen

  1. 1 Where a fluid stops being one
  2. 2 Slip is a memory of one mean free path
  3. 3 A thinning gas flows least at one free path across, and then more without end
3 essays · regimes 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 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

All essays