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.
The same patch, 6 periods later, in two flows. A round patch of 848 marked particles, advanced 6 periods by the blinking flow and by a steady flow of the same strength. The steady flow has drawn the patch into a smooth ribbon along a streamline and every particle in it is still on the streamline it started on; the blinking flow has folded the patch through itself repeatedly and its particles are spread across the whole region. Neither flow has any diffusion in it and neither has lost a particle. The difference between them is that one depends on time.

Advection

  1. 1 No randomness, and it mixes anyway
  2. 2 Steady, three-dimensional, and mixing anyway
  3. 3 The area that must not move
  4. 4 A scalar is a record of where its fluid was
4 essays · kinematics The hypotheses Bernoulli's equation needs. The equation is correct and its hypotheses are strict. Most misuse is not a wrong formula but a right formula carried across a streamline, through a machine, or into a region where viscosity dominates.

Bernoulli's equation

  1. 1 Where Bernoulli's equation applies
  2. 2 Energy instead of pressure
  3. 3 Four Bernoullis and one name
  4. 3 Two totals, one of which a shock cannot touch
4 essays · misconceptions 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
4 essays · kinematics A streamtube narrows and the flow speeds up. Two neighbouring streamlines bound a tube that no fluid crosses. Where the tube pinches, the same mass has to pass through a smaller gap every second, so it must move faster — which is mass conservation with no equations in sight.

Continuity

  1. 1 Mass has nowhere to go
  2. 2 The number on a streamline is a flow rate
  3. 3 A wave on the wall is a pump
  4. 4 The pump that is better the more it squeezes
4 essays · kinematics Two decay laws from two invariants, and nothing in the equations to choose. The energy of a decaying turbulence against time, integrated from dK/dt = −A K^(3/2)/l with the large scales conserving u² l³ in one case and u² l⁵ in the other. The exponents come out at 1.1997 and 1.4282 against the closed forms 6/5 and 10/7. Which invariant holds is decided by the shape of the spectrum at the very largest scales, at the moment the stirring stops.

Decay

  1. 1 What decay never forgets
  2. 2 Universal, and one of five
  3. 3 The constant that travels
  4. 4 The decay inside the four-fifths law
4 essays · turbulence A pump curve out of the momentum theorem. The pressure an ejector delivers, against how much it is entraining, at a fixed nozzle. It has the shape of every pump characteristic ever measured — a shut-off pressure with no flow, falling to no pressure at free delivery — and it was obtained from a momentum balance on a tube with nothing in it. The shut-off value here is 36.0 kPa and the machine at its best power runs at 4.56 times its own motive flow.

Ejector

  1. 1 Mixing is a pump
  2. 2 The nozzle that is best at one thing
  3. 3 The wall the suction puts in the curve
  4. 4 A choke that belongs to two streams
4 essays · applied Two roots walking towards each other, and one of them crosses. The roots of the characteristic quartic in the complex plane as the airspeed is raised from nothing to 105 metres per second — growth rate across, frequency up. At rest the two sit on the imaginary axis at the uncoupled frequencies. As the speed rises, the aerodynamic coupling drags them towards each other in frequency while pushing one left and the other right, and at 80.8 metres per second the right-hand one crosses the axis. Everything about the failure is in this picture: the coalescence, the crossing, and the fact that the flutter frequency is neither of the two the structure started with.

Flutter

  1. 1 The shake that is not resonance
  2. 2 The control that works backwards
  3. 3 The speed where the damping is exactly zero
  4. 4 The lag that makes flutter possible
4 essays · circulation What the ground actually gives a wing. Induced drag near the ground as a fraction of the same wing's in free air, against height in spans, at constant lift. The ground is a plane of symmetry, so the wake is joined by a mirrored wake of opposite circulation below it, and the upwash from that image is what takes the drag away. At a tenth of a span the wing keeps 0.516 of its induced drag — a saving of 48 per cent — and by a span and a half the effect is 1.3 per cent and going. The lift is held fixed all the way along this curve: what the ground gives here is not more lift, it is the same lift for less drag, and the two are different claims about a landing aeroplane.

Ground cushion

  1. 1 The cushion that is not there
  2. 2 The cushion that is there after all
  3. 3 A cushion that changes its physics
  4. 4 The borrowed mass the boundary decides
4 essays · misconceptions Flow net — a free vortex. Two families of curves drawn over the same flow: the streamlines, along which the streamfunction is constant, and the equipotentials, along which the velocity potential is constant. They cross at right angles at every point, because they are the two parts of a single analytic function of position.

Images

  1. 1 A wall made by reflection
  2. 2 The wall that pushes back
  3. 3 The mirror that is a circle
  4. 4 The corners that can be done with mirrors
4 essays · inviscid 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
4 essays · turbulence A control volume round an aerofoil. A rectangle drawn in the fluid around a lifting section. The arrows on the right-hand face show the downward velocity of the air leaving it, drawn to scale. Adding the momentum carried through all four faces to the pressure acting on them gives the force on whatever is inside, without the calculation ever going near the surface.

Momentum lift

  1. 1 Air must be pushed down, and the usual sum is wrong
  2. 2 A right total from a wrong picture
  3. 3 Weighing what is missing
  4. 4 The drag a wake keeps however it rolls up
4 essays · misconceptions The lift curve that made flight look impossible. Lift coefficient against incidence, by Newton's impact theory and by thin-aerofoil theory. One is quadratic in the angle and the other linear, so at small incidence — which is where aircraft fly — they differ by more than an order of magnitude. Newton's version says a wing large enough to carry a man would need an engine nobody could build, and for a century that arithmetic was taken as settling the question.

Newtonian

  1. 1 The theory that forbade flight
  2. 2 The only theory simple enough to optimise
  3. 3 The face Newton left in shadow
  4. 4 The spot a local theory cannot see
4 essays · misconceptions Seven fluids in one pipe at one pressure gradient. Velocity profiles of power-law fluids in a round pipe, all at the same pressure gradient and the same consistency, scaled to the fastest. A shear-thinning fluid (n below one) is flatter in the middle and steeper at the wall; a shear-thickening one is the reverse. The flattening is often called plug-like, which invites the reading that the fluid is moving more freely — what has actually happened is that all the shear has been pushed into a thin annulus at the wall, which is the expensive place to put it.

Non-newtonian

  1. 1 A viscosity that depends on the question
  2. 2 The core that does not move
  3. 3 The stress a pipe knows
  4. 4 The fluid that has not finished its last deformation
4 essays · viscous 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.

Pressure

  1. 1 Fast means low pressure
  2. 2 The number that does not depend on the tunnel
  3. 3 What the airspeed indicator believes
  4. 4 Pressure has no speed
4 essays · inviscid 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.

Reynolds

  1. 1 One number decides which physics applies
  2. 2 The Reynolds number, and the length in it
  3. 3 The world with no inertia
  4. 4 How small is small enough
4 essays · regimes Three times the wind, on a reach. The polar diagram: boat speed in every direction, as a multiple of the true wind speed, for drag angles of 14 and 6 degrees. The shaded wedge at the top is the no-go zone, whose half-angle is exactly the sum of the two drag angles. Everywhere outside about twice that angle the boat is faster than the wind, and the maximum is 2.92 times the wind at 110 degrees — which is 1/sin λ at 90° + λ, both checked.

Sailing

  1. 1 Faster than the wind that drives it
  2. 2 The fastest way is not the straight one
  3. 3 A breeze the boat cannot use
  4. 4 A keel flies wherever the course puts it
4 essays · applied Flow past a cylinder at Re 100. A real fluid past a circular cylinder. At low Reynolds number the flow closes up behind the body much as the ideal theory says; as it rises the flow separates and a region of reversed flow appears behind, which is where drag comes from.

Separation

  1. 1 When the flow lets go
  2. 2 How much uphill a layer can take
  3. 3 Where the straight line stops
  4. 4 The gradient that does both
4 essays · viscous 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
4 essays · applied What a barometer on the wing would read at 60 m/s. The absolute pressure along an aerofoil's surface, in kilopascals, with the atmosphere put back. Nothing on the wing is near zero: the deepest point of the so-called suction peak is a few per cent below a hundred kilopascals, and the air there is still pushing on the wing hard enough to hold up a column of itself eight kilometres tall. The lift is the difference between two large pushes, not a pull.

Suction

  1. 1 Nothing sucks
  2. 2 A force forgets the datum, a stress cannot
  3. 3 Where a liquid does pull
  4. 4 A breaking strength that is the size of a flaw
4 essays · misconceptions Two centres, two saddles, and a sum of nothing. A separation pattern: a uniform stream with two counter-rotating cored vortices in it, which reproduces the arrangement of critical points behind a body at a Reynolds number of a few tens. There are exactly four — a saddle where the flow divides, a centre in each recirculating cell, and a saddle where it closes — and their indices sum to 0. The winding number of a loop enclosing all of them is 0, which is what a uniform stream far away requires. A bubble costs nothing in this bookkeeping, which is why one is free to appear.

Topology

  1. 1 The count a pattern cannot break
  2. 2 The count computed on a body
  3. 3 Two kinds is a plane flow's privilege
  4. 4 The sign a stagnation point carries in space
4 essays · kinematics

All essays