Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: The exact theory is wrong — page 3

Page 3 of 22, continuing through the 193 essays this motif runs through.

193 essays carry this thread — page 3 of 22.

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. Viscosity

The two drags a wing pays

A wing pays for having a surface, and it pays for making lift with a finite span. One of those bills falls as it flies faster and the other rises, so there is a speed at which the total is least — and the condition for it turns out to be that the two are equal.

 The starting vortex, and the circulation it pays for. A wing that has just begun to lift, and the vortex it shed as it started. The circulation round the wing and the circulation round the shed vortex are equal and opposite, so a circuit large enough to contain both has no circulation at all — which is what Kelvin's theorem requires of a circuit that began at rest. Circulation and lift

The vortex a wing leaves behind

A wing at rest has no circulation. A wing in flight has a great deal. Circulation round a circuit of fluid particles cannot change, so the difference had to come from somewhere — and it did, as an equal and opposite vortex dropped on the runway.

 Ten unknowns, four equations. What is left after the Navier–Stokes equations are averaged. The mean velocities and mean pressure were there before; the six Reynolds stresses are new, and they arrived from the one term that does not average away. Nothing in the count is an approximation — the averaged equations are exact — and that is what makes the gap uncomfortable. Transition and turbulence

What averaging costs

Split the velocity into a mean and a fluctuation, average the equations, and the result is exact. It is also short of six equations, because the one nonlinear term does not average away and leaves six new unknowns behind that nothing determines.

 Flow net — a stream past a cylinder. 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. Ideal flow

Bodies made out of nothing

Put a source in a stream and a solid nose appears in front of it. Put a sink downstream of the source and the nose closes into a finite body. Nothing was placed in the fluid — the surface is a level set of a function, and it behaves like a wall because nothing crosses it.

 The wall slope against pressure gradient, and where it runs out. How steeply the flow leaves the wall, plotted against the pressure gradient the layer is running into. A favourable gradient presses the profile against the surface and steepens it; an adverse one hollows it out. The curve reaches zero at a definite value, and beyond that there is no attached solution at all. Viscosity

How much uphill a layer can take

A boundary layer running into rising pressure is climbing a hill on the last of its momentum. There is a definite steepness at which it can no longer do it, and the number is not a rule of thumb — it is where a family of solutions stops existing.

 One nozzle, five back pressures, five different flows. The duct above, and the static pressure along it below, computed station by station from the local area. Where a shock stands inside the divergent section its position was solved for rather than placed: the shock spends total pressure, which sets the subsonic Mach number at the exit, which has to match the imposed back pressure. Compressible flow

One area, two answers

The area of a duct at a station fixes the Mach number there — twice over. A ratio of 2.5 is satisfied at Mach 0.24 and again at Mach 2.44, nothing local chooses between them, and where the choice cannot be made consistently a shock appears inside the duct to join the two.

 Every rung of the hierarchy is worse than the last. Writing a transport equation for the Reynolds stress does not close the system: that equation contains the triple correlation, whose equation contains the quadruple. The bars are the number of independent components at each order and the number of new unknowns its own equations introduce. The second is always larger, and the gap widens. Transition and turbulence

The ladder that never closes

Being six equations short is a problem with an obvious remedy — derive six more. The remedy works, produces an exact equation for the Reynolds stress, and leaves ten new unknowns behind. The gap does not narrow at any level, and the counting says why.

 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. What is taught wrongly

The theory that forbade flight

Newton treated air as a hail of particles that give up their normal momentum on impact, and got a lift coefficient of 2sin²α cos α. At five degrees that is a thirty-sixth of what a wing actually makes, and the quadratic is why powered flight looked arithmetically impossible for two centuries. The same formula is exact at Mach twenty.

 Creeping flow, and the same body with inertia. The exact creeping-flow solution beside a solved field at a Reynolds number where inertia matters. The creeping flow is a mirror image of itself front to back — a photograph of it run backwards is a photograph of it — and the field with inertia has a wake, which is what a direction of time looks like. Regimes and numbers

The world with no inertia

Drop the viscosity and the equations become exactly solvable and wrong about drag. Drop the inertia instead and they become exactly solvable again — and for a cylinder in an unbounded fluid there is no solution at all, which took fifty years to notice and longer to fix.

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