The thread: A smooth picture proves nothing — page 28
272 essays carry this thread — page 28 of 31.
Flows and fields
A post holds liquid back by its radius, a stripe by its length
A surface of no-slip posts standing in a gas-filled texture lets liquid slip over it, and the slip length grows as one over the square root of the solid fraction, where a surface of stripes grows only as its logarithm. The reason is the oldest contrast in slow flow: a disc dragged through a viscous liquid has a drag proportional to its radius, which is Stokes' law, and a line has no finite drag at all, which is Stokes' paradox. But sparse posts beat stripes only while they are sparse.
What is taught wrongly
A prior the pictures can refute is the one worth having
A few views of a flow with no symmetry leave most of the field unseen, and a reconstruction has to fill that part in by assuming something. Three common assumptions were tried on the same field and the same views. Non-negativity saves three views of ten, and five on a field of puffs; smoothness saves one; a dictionary of known shapes saves none unless it is exactly right, and when it is nearly right it can be wrong by a factor of a hundred. The prior that helps most is also the one the data can catch being wrong.
Regimes and numbers
A tree transparent at every junction is not the quietest
The branching rule that lets a heart's pulse through one junction without reflection is set by that junction's Womersley number: area-preserving in the aorta, Murray's in the small arteries. Build a whole arterial tree that way, every junction at its own transparent rule, and it reflects more of the pulse than the best tree built to a single exponent. The single exponent wins by leaving its junctions slightly mismatched, in the way that cancels the echo from the tree's own ends — the trick an antireflection coating plays on light.
Fluids at work
The air a cavity releases is capped by the cavity
When a shut valve pulls the pressure behind it down, the air dissolved in the water starts to come out, and a little gas was already known to soften the hammer. But air leaves the water only while its pressure in the cavity is below the pressure the water was saturated at, so a cavity can fill with at most its own volume of air at that pressure — a seven-thousandth of the pipe here, however much water gives up its air and however fast. That is less than a third of what removes the hammer. Released quickly, the air does not cushion the collapse at all: it turns the vapour cavity into an air cavity, and the pulse becomes the column-separation pulse with the margin measured to the atmosphere — taller, on this main.
Compressible flow
The sheath outlasts the body
A blunt hypersonic body wraps itself in a sheath of hot, thin gas from its nose's shock, and its boundary layer grows by eating that gas from the inside. The usual picture has the boundary layer through the sheath within a few nose diameters. It is not: a laminar boundary layer takes about a third of a Reynolds number of diameters to swallow the sheath at Mach 15, which is thousands to millions of diameters, and a turbulent one hundreds. On any body of ordinary length the boundary layer never sees the cooler gas outside, and it is heated by the sheath's gas all the way down.
Transition and turbulence
A thin interface keeps its waves past a quarter
Miles' quarter rules a shear layer whose density changes over the same depth as its velocity. Make the density interface three times thinner and the stationary billow dies early, but a pair of travelling waves takes its place and is still growing at five times the quarter. No theorem is broken: at the edges of the shear, where the waves draw their energy, the local Richardson number has fallen to nothing.
Flows and fields
A lattice of vortices folds a cloud at the saddles' quarter
A cloud of heavy particles folds in still air above a Stokes number of one, and in converging air above a quarter. Air that turns as well as converges might have settled between them. In a lattice of vortices it settles exactly on the quarter: the vortices fling the particles to the cell walls, but only the saddles where the walls meet can make them cross. Below the quarter nothing folds, and the cloud is gathered onto the walls without limit instead.
Ideal flow
A trailing wake hurries a pair together
Two cylinders free in a stream and damped by their drag always end up meeting near side by side, and the obvious candidate for what holds real pairs apart is their wakes: a body beside another's wake is pushed towards the faster fluid, away from it. Give each cylinder a wake that carries its drag and the push is there — but it points the wrong way. It throws a following cylinder out of the leader's wake and round towards side by side, where neither is in the other's wake and nothing opposes the stream's pull. Near tandem the wakes make the pair meet seven to sixty times sooner, and the only balance they create is a saddle.
What is taught wrongly
A soft wall is read as slip only at second order
A drainage force smaller than Taylor's has been read as the liquid slipping at the wall. A wall that gives under the drainage pressure lowers the force too, and it can be mistaken for slip. Solved together, the flow and the elastic wall say how: the softness first adds a spring to the force, in phase with the motion, and only at second order takes anything from the damping a slip length is read from. When it does, the slip it imitates grows as the square of the frequency and falls as the square of the gap.