Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: What is conserved — page 21

Page 21 of 33, continuing through the 292 essays this motif runs through.

292 essays carry this thread — page 21 of 33.

The marginal Taylor number against the axial wavenumber. The smallest Taylor number at which a disturbance of a given axial wavenumber is neutral. Its minimum is 1707.757 at a wavenumber of 3.1158, which are the critical Rayleigh number and critical wavenumber of a layer of fluid heated between two rigid walls — the same numbers, because in the narrow-gap limit the two problems are the same sixth-order eigenvalue problem. This curve is computed by that essay's own solver. Regimes and numbers

A transition that needs a second number

Fluid between rotating cylinders goes unstable at a Taylor number of 1707.762 — which is the same number, to every digit, as a layer of fluid heated between two rigid walls. It is not an analogy. And two things the number cannot carry decide whether the transition happens at all and what it looks like when it does.

 The growth rate a discretised sheet has, at every wavelength it can carry. Kelvin–Helmholtz gives a growth rate proportional to the wavenumber and without bound. A sheet represented by N point vortices has pi m (1 − m/N) instead — the same rate at long waves and half of it at the shortest wave the grid carries, with the fastest-growing mode at the grid scale itself. Smoothing the kernel over a length delta moves that mode back to a wavelength the physics chose. Ideal flow

A sheet that cannot stay a sheet

Let a shear layer's thickness go to zero and it becomes a surface across which the velocity jumps. The model is used everywhere in this subject, it is unstable at every wavelength, and the thing it does next is worse: it develops a singularity in its own shape, at a finite time, from a smooth start.

 Every unstable mode of every profile, inside one circle. The complex phase speeds of the unstable modes, scaled so that each profile's own semicircle is the unit one. Howard's theorem says every one of them must lie inside — the centre and the radius are the mean and half-range of the velocity profile and nothing else — and every one of them does, with the closest approach at 0.915 of the radius. Transition and turbulence

Every unstable wave is inside one circle

Before solving anything, you know where the answer is. Howard's theorem says the complex phase speed of any growing disturbance in a shear flow lies inside a circle fixed by the fastest and slowest parts of the profile — and by nothing else about it at all.

 Seven shock structures, and the two states they all connect. The velocity through the shock for seven dissipation models — Prandtl numbers from a quarter to two, viscosities from constant to linear in temperature. Each curve is shifted so its midpoint sits at the origin. They start at the same speed, end at the same speed, and are nothing alike in between. Compressible flow

The jump does not ask what made it

Seven different dissipation mechanisms are made to smear the same shock. Their interiors are a factor of two and a third apart in thickness, their entropies overshoot the final value by between a quarter and a doubling, and the state they all arrive at agrees to seven parts in ten billion — because the end states are conservation and the interior is transport.

 Five points, and where each one's fluid came from. Back-trajectories through six units of time in an unsteady double gyre. Each curve ends at the place the fluid now at the marked point started; nothing about that place can be read off the velocity at the marker. Flows and fields

A scalar is a record of where its fluid was

A conserved scalar has no value of its own. Its value at a point is whatever it was at the place that point's fluid started from, which makes a dye field a photograph of the past — and makes the map from now to then the only thing in the flow that carries the past at all.

 How much the peak cares about the valve at each instant. A small bump is added to the valve's opening at one moment and the peak is recomputed; the difference, divided by the bump, is plotted against the moment. It is flat and zero for the first 0.45 seconds — that part of the closure is invisible to the peak — rises through the reflection arrivals, and falls to nothing after the peak has happened, which is the one part of the shape that needs no fluid mechanics. Fluids at work

The part of the closure a pipe cannot see

A surge specification says how long the valve takes to shut. The line does not integrate the duration, it integrates the shape — and it is measurably blind to the first half-second of an eight-second closure, while three closures of identical length give peaks 97 per cent apart.

 The one term in ground effect that really is about carried air. The added mass of a plate approaching a plane, as a multiple of its free-air value. In free air a plate borrows ρπc²/4 per unit span — exactly 0.79 for a unit chord in unit density, which is the mass of the circle its chord spans. Near a wall the fluid in the gap must leave sideways through a narrowing passage, so it moves faster than it would in the open and carries more energy: the borrowed mass grows as the inverse of the gap and diverges as the gap closes. At half a chord it is 1.21 times, at a tenth 2.06, at a twentieth 3.12. This is a cushion in the ordinary sense — fluid that has to be got out of the way and resists being — and it is the term the steady argument has none of. What is taught wrongly

The cushion that is there after all

A wing near the ground makes more lift for reasons that have no cushion in them, and that account cannot reach two cases — a wing descending, and a rotor in the hover, where there is no steady flight for the image-vortex account to be about. Those cases have a term that really is about carried air, and it grows without limit as the gap closes.

 Deviation: how far the flow leaves from the blade angle, against solidity. The angle between the outlet flow and the blade, for a row of flat plates at 30° stagger meeting a flow at 45°. An open row barely turns the flow at all and the deviation is nearly the whole of the intended turning; a tight row guides it to within a thousandth of a degree of the blade angle. Nothing about the blade changed between the two ends of this curve. Circulation and lift

A row is not a set of aerofoils

An isolated aerofoil's incidence is measured from the free stream. A compressor blade's cannot be, because an infinite row of identical blades above and below it has a circulation that is part of its own free stream — and the velocity the theorem uses is a vector mean that exists nowhere in the machine.

 Kirchhoff's rotation rate, which a point vortex does not have. A patch of uniform vorticity bounded by an ellipse turns rigidly at omega a b/(a+b)², a rate that depends on the shape alone. It is largest for a circle, where it is unobservable, and falls away as the patch is drawn out. A point vortex has no shape and therefore no entry on this axis at all. Ideal flow

The shape a vortex keeps

Outside a circular patch of uniform vorticity the flow is exactly the point vortex's — not nearly, exactly — so replacing one by the other looks free. It is not. The patch has a shape, the shape has a rotation rate of its own, and there is a strain above which no shape exists at all.

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