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The thread: What is conserved — page 10

Page 10 of 33, continuing through the 292 essays this motif runs through.

292 essays carry this thread — page 10 of 33.

The current at the surface is 45° from the wind, and nothing sets that angle. The Ekman spiral drawn as a hodograph: each point is the velocity at one depth, and depth runs along the curve. At the surface the flow is at exactly 45 degrees to the wind that drives it — not approximately, exactly, and independently of the wind, the viscosity and the latitude. By one Ekman depth the flow has turned another radian and lost 1/e of its speed; by three it is a hundredth of the surface value and pointing back the way it came. The angle is a property of the equation having two terms in it, and nothing else.

The layer that stops at a depth

Every other boundary layer grows. This one does not — rotation supplies a frequency, the balance against diffusion supplies a length, and the transport that comes out contains the stress on the surface and not the viscosity underneath it.

What the skin settles at, before anything is done to it. The adiabatic wall temperature against Mach number, in air at 216.7 K, with the stagnation temperature above it. The gap between the two is the recovery factor, which is 0.8417 here and stays there at every Mach number — it is a property of the Prandtl number and not of the speed. At Mach 2 the skin sits at 363 K, at Mach 3 at 545 K, and at Mach 5 at 1128 K, which is past what aluminium will do. Nothing has been burnt and nothing has been rubbed: the air was brought to rest, and this is where its kinetic energy went.

Compressible flow

The wall that heats itself

A surface told nothing about its temperature does not settle at the air's. It settles most of the way to the stagnation temperature, and the heat flux is driven from that invented temperature rather than from the free stream's — so a wall hotter than the air can be being heated by it.

The jet divides 75% to 25%. A jet striking a plate at 60 degrees. Both sheets leave at the jet's own speed, because their surfaces are at ambient pressure and Bernoulli allows nothing else, and the plate can exert no force along itself because the fluid has no viscosity. Momentum along the plate then fixes the split at (1 + cos β)/2 = 0.7500, and the normal force at ṁV sin β = 0.8660. Nothing about the plate's material, size or roughness enters either.

Fluids at work

What a jet cannot push sideways

A jet striking a plate divides in two, and how it divides is fixed by a single sentence — an inviscid fluid exerts no force along a surface. That one statement, plus mass, gives the split exactly — and the same sentence turns a flat plate into a bucket worth twice as much.

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.

Flows and fields

The picture belongs to whoever is watching

Photograph the flow past a cylinder from the tunnel and it has two stagnation points. Photograph the same flow from a frame moving with the air and it has none at all, and its surface speed is exactly the free stream at every angle. Both pictures are correct and no measurement distinguishes them.

Fluids at work

Half the jet speed takes everything

A bucket standing still feels the largest force and does no work; a bucket running with the jet does no work either. Between them the power peaks at exactly half the jet speed, for every bucket shape and every flow rate — and at that speed a perfect bucket leaves the water motionless.

Each cancellation costs two powers of the Mach number. Radiated power against compactness for three source clusters: a single monopole, two of opposite sign, and four on a square with alternating signs. The fitted slopes are 0.00, 2.00, 4.00 — zero, two and four in (kd), measured by integrating the far field over a sphere rather than assumed. A turbulent eddy turns over in about the time sound crosses it, so kd is of order the Mach number, and those exponents become the fourth, sixth and eighth powers of speed. A flow with no moving surfaces has no monopole and no dipole available to it, which is Lighthill's whole argument, and the eighth power is what is left.

Compressible flow

The sound that only leaves

A flow is a catastrophically bad radiator, and the reason is that it has no monopole and no dipole available to it. What is left is the eighth power of speed — and the equation is equally happy with sound converging on a jet, which is ruled out by a condition imposed at infinity.

$Two profiles, and they are the same profile. The chordwise and spanwise velocity profiles in the boundary layer of a yawed flat plate, each as a fraction of its own edge velocity. They are computed by different code — the chordwise one by shooting a third-order nonlinear equation, the spanwise one by a single pass through a linear second-order one — and they agree to 2.1e-8 over the whole layer. They are the same function of η, because the two equations reduce to the same equation. A swept flat plate has no crossflow at any sweep angle, and that is the independence principle in the only form that has no wriggle room in it.$ Circulation and lift

The wind a swept wing feels

Sweeping a wing back is usually justified by saying it meets a slower wind. It does not meet a slower wind. The equations split exactly in two, and the flow along the span is a passenger that exerts no force and changes nothing — until a pressure gradient breaks the split, and then it becomes the reason a swept wing is a different problem rather than a harder one.

The two bodies the far field cannot tell apart. A circular cylinder and the Rankine oval that has the same doublet strength: 1.17 radii long against the circle's one, and 0.94 tall against its one, with a source and a sink 1.2 apart inside it. On the pale ring, one and a half radii out, the two flows differ by 19 per cent of the disturbance; at six radii by one per cent; at infinity not at all. What a far field records of a body is three numbers — its circulation, its net outflow and its doublet — and nothing else survives the journey.

Ideal flow

What the far field remembers

Three numbers survive the journey to infinity — a circulation, a net outflow and a dipole — and nothing else about a body does. Two shapes with nothing in common can therefore make the same flow a few radii away, and the difference between them dies two orders faster than the disturbance either one makes.

One field, and the two parts the theorem splits it into. A velocity field made of a smooth source, a smooth vortex and a uniform stream, and the two fields the Helmholtz decomposition returns for it. The first carries the whole divergence and has no curl anywhere; the second carries the whole curl and has no divergence. They are computed by solving two Poisson problems on a grid, with the divergence and the vorticity differenced from the field rather than taken from the expressions that built it. Adding the two back together does not recover the field.

Flows and fields

Every flow is two flows

Any velocity field splits into a part carrying all of the divergence and a part carrying all of the vorticity. The theorem says so and does not say which split — the two halves can be moved between each other by anything harmonic, and on a bounded region that is an infinite family.

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