The thread: What is conserved — page 38
384 essays carry this thread — page 38 of 43.
Regimes and numbers
Murray's law is not the rule for a pulse
Murray's law sizes a branching vessel for the cheapest steady flow, and at every junction built to it a pressure pulse is partly reflected. The rule that makes a junction transparent to a pulse is a different exponent, set by how the wave speed changes with radius. A tree is not the sum of its junctions, either: a Murray tree six generations deep reflects a third of the pulse at the heart rate, three times what one of its junctions does, and viscosity in the smallest branches means no area rule can make it transparent at every frequency. The best a pulsing tree can do lies between area-preserving and Murray.
![The number of passes grows in proportion to Reynolds number. Passes the two rings make before their cores are close enough to merge, against the vortex Reynolds number, marched with the merging threshold at 0.24 of the core separation and at 0.30, and the closed form [(0.24 dₘᵢₙ)² − σ₀²] Re ÷ 4Tₚₐₛₛ, with the closest approach dₘᵢₙ and the time between passes Tₚₐₛₛ read from the inviscid pair. Both thresholds give a count in proportion to Re: one or two passes at a few thousand, ten or thirty at thirty thousand.](/f/the-number-of-passes-grows-in-proportion-to-reynolds-number-671303d0.svg)
Circulation and lift
Leapfrogging rings end by merging, not by parting
Two smoke rings that leapfrog in an ideal fluid do it for ever, because their energy binds them. A real fluid drains the energy, and the obvious guess is that the rings drift out of the bound state and part. They do not: diffusion drains the pair's energy and the energy of every possible pair of free rings together, and the margin that binds them never goes negative. What viscosity does instead is fatten the cores until, as one ring threads the other, the two are close enough to merge. That takes a number of passes in proportion to the Reynolds number — one to three at the few thousand of a laboratory smoke ring.
Transition and turbulence
The streamfunction says which relaxed state
Decaying two-dimensional turbulence ends in a large pair of vortices, and three theories say what that pair should look like: a sinh relation between vorticity and streamfunction, a tanh, or a straight line. Their scatter plots differ only in curvature, and a real flow's scatter hides curvature. Solve the three states in the same periodic box, at the same energy and enstrophy, and a statistic that separates them turns out to be one nobody looks at: the flatness of the streamfunction, which sits above the straight line's value for every sinh state and below it for every tanh state, and does not move when unrelaxed small eddies are added.
Fluids at work
The plenum decides whether a compressor surges
Throttle a compressor past the peak of its characteristic and it does one of two things. It settles into rotating stall, a steady state with a cell of dead flow running round the annulus, or it surges, the whole flow through the machine collapsing, reversing and recovering over and over. Which one is not decided by the blades. It is decided by the volume the compressor discharges into, against the inertia of the air in its duct — one number, Greitzer's B, which grows with the blade speed, so the same machine stalls at part speed and surges at full speed.
Compressible flow
A hypersonic body leaves a line explosion behind it
A blunt body at hypersonic speed does work on the air at a rate equal to its drag, and each slice of air it passes through is struck once and left to expand. Seen from the ground, that is a line explosion, and Sedov's cylindrical blast wave gives the bow shock's width and the pressure on the afterbody without a Mach number in either. The analogy's classical constants come straight out of the blast solution. So do its limits: it holds only while its own shock stays strong, over a length that grows as the square of the flight Mach number, and it puts the body inside a core hotter than anything the flow can reach.
Regimes and numbers
The force a contact line holds is a range
Capillary rise and the drop on a window are usually drawn with one contact angle, and a contact line with one angle makes a force that is a single number. A real contact line pins, and stops anywhere between a receding and an advancing angle. The force it holds is then a range, as static friction is, and its width is surface tension times the difference of two cosines. A tube holds its column at any height in the range, so which way the meniscus last moved matters more than how patchy the wall is — and a tilted pane holds a drop only as large as that difference allows.
Circulation and lift
The follower beats in time with the wake, not the bird
A gliding bird can sit in its neighbour's upwash and stay there. A flapping bird's wake is a wave left in the air — the path its wingtip traced, rising and falling with every beat — and a bird behind gains only if its own wing is where that wave is when it arrives. The best timing is a rule with no aerodynamics in it: lag the bird ahead by the time the wake took to come, so that each wingtip retraces the path of the one before. Directly behind, the rule flips by half a beat, and it buys a smaller loss rather than a gain.
Ideal flow
A pair set free in a stream collides or parts
Two cylinders held in an ideal stream pull together side by side and push apart in tandem. Let them go and the forces become a motion, and the motion has a law of its own: the stream's force on each is exactly the slope of how large the pair looks from far away, so the pair moves to look bigger. Side by side that means closing, and the fluid squeezed out of the gap costs so little that nothing stops them: released four radii apart they collide in six and a half radii of stream. In tandem it means parting, for good.
Flows and fields
The pressure a parcel remembers keeps it finite
The restricted Euler equation follows a parcel's velocity gradient with the pressure's shape thrown away, and every gradient it follows blows up. Give the parcel back a pressure that remembers how its neighbourhood was deformed over the last Kolmogorov time — nothing more — and no gradient blows up at all. The ensemble settles into the teardrop measured in turbulence, its vorticity lines up with the middle strain axis, and its intermittency grows as the memory shortens. What the memory cannot do is keep the one identity homogeneity demands, and that miss says where the rest of the pressure lives.