Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: What is conserved — page 49

Page 49 of 49, continuing through the 436 essays this motif runs through.

436 essays carry this thread — page 49 of 49.

One formula draws a smooth layer, a row of cat's eyes, and in the limit a row of point vortices. Streamlines of ψ = ln(cosh y + A cos x) at A = 0.5, over two periods, in the frame of the eyes. The separatrix encloses eyes of half-height 1.32; the peak vorticity at their centres is 3 times the layer's, and on the braids 0.333. Ideal flow

A shear layer and a row of vortices are one formula, and not one flow

There is an exact steady solution of Euler's equations with one parameter in it, written in a line by Stuart in 1967. At one end it is the smooth shear layer every instability calculation starts from, and at the other it is a row of point vortices, with cat's-eye streamlines between. It looks exactly like the history of a layer rolling up, and textbooks draw it next to one. It cannot be. Each member has a different area of fluid at each vorticity and a different energy, and those are the two things an inviscid flow cannot change.

 A cart that leaks straight down keeps its speed, and the rule that says it gains one has lost track of what left. A frictionless cart losing three quarters of its mass at a steady rate, moving at speed 1. With its leak leaving backward at 0.5 of that speed relative to the cart it ends at 1.69. The rule F = d(mv)/dt, read as d(mv)/dt = 0, says the cart reaches 4 whatever the leak does. Flows and fields

Mass leaving a cart pushes nothing unless it leaves with a speed of its own

A cart leaking straight down keeps its speed, though its mass falls and every first course says force is the rate of change of mass times velocity. Apply that form to the leak and the cart speeds up fourfold, at a rate that depends on who is watching, which no force may. The transport theorem has the right rule in a line: only what crosses the boundary at a speed relative to it pushes. The rule then does four more things, from a cart that slows with a constant mass to a tank whose thrust is exactly twice the missing wall's.

 Every polymer leaves the Newtonian line at its own point, runs along its own branch, and stops at the same asymptote. The friction factor against Reynolds number. Newtonian Prandtl–Kármán is 0.0045 at 10⁵; Virk's asymptote 8.86·10⁻⁴. The three-layer profile with an elastic layer out to y⁺ = 30, 100, 300 gives 0.00239, 0.00137, 9.97·10⁻⁴ at that Reynolds number. A laminar pipe at the same Reynolds number would be at 1.6·10⁻⁴: the asymptote lies between the two and is still turbulent. Transition and turbulence

Polymers stop at an asymptote that no polymer chooses

A few parts per million of a long-chain polymer can cut a pipe's turbulent friction by four fifths. That is astonishing enough, but the stranger fact was measured by Virk around 1970. The reduction has a limit, and the limit is a single friction law, 19 log Re √f − 32.4, that does not depend on which polymer, what weight, which solvent or how much of it. Every polymer solution leaves the ordinary curve at its own point and arrives on the same line. An asymptote with no properties of the additive in it is a statement about turbulence, and half a century on, nobody can derive it.

 A bubble in water speeds up, slows as it flattens, and bottoms out at the speed of the slowest ripple. Rise speed against equivalent diameter in water. A clean surface reaches 0.298 m/s at 1.54 mm; a contaminated one has no such peak. Both then join the wave law, whose minimum is 0.236 m/s at 5.61 mm. Kelvin's slowest capillary–gravity wave is 0.231 m/s. Regimes and numbers

A bubble in water bottoms out at the speed of the slowest ripple

A bigger bubble should rise faster, and up to a couple of millimetres it does. Then it flattens, starts to wobble and slows down, and between 2 and 10 millimetres across it rises at nearly the same speed whatever its size, a little over 23 centimetres a second. That number is familiar from somewhere else. It is the speed of the slowest possible ripple on a water surface, the same capillary–gravity minimum that sets Kelvin's wind. Mendelson's wave analogy of 1967 makes the link a formula, and the two speeds agree to two per cent in every liquid.

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