Convergence — where it appears
Named by 23 essays across 8 fields — each of them below, with the objects they name alongside it.
Nothing but the edge
Cut an aerofoil into panels, put a singularity on each, and require the surface to be a wall. The system that comes out has one more unknown than it has equations, and the row that is missing is not a bookkeeping slip — it is the fact that ideal flow round a closed body has no unique answer at all.
An hour for every tenfold
Turbulence is deterministic and unpredictable, and the exchange rate between those two is exact: measuring the initial state ten times better buys the same extra forecast time every time, for ever. A constant, and it belongs to the flow rather than to the instrument.
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.
The momentum with no value
A cylinder is pushed from rest to a steady speed. Work was done, energy went into the fluid, something was pushed. How much momentum does the fluid carry? The integral converges, the answer is finite, and it is a different finite number for every shape of region it is summed over.
The threshold the walls decide
A layer heated from below convects at a Rayleigh number of 1707.762, and nothing whatever happens at one. The free–free case has a closed form and the two that do not differ from it by a factor of 2.6 — produced by nothing but what the top and bottom surfaces are permitted to do.
How far before the heat arrives
A duct's thermal entry length is quoted everywhere as x/(D·Re·Pr) = 0.05, with no tolerance attached. Working out what it delivers gives a Nusselt number 1.45 per cent above the developed value — and the developed value itself is not a term ratio at all but an eigenvalue, 3.6568, which is also the rate at which the duct forgets its inlet.
A number that is only the shape of the hole
The friction factor times the Reynolds number for laminar flow in a duct is a pure number that depends on the cross-section's shape and on nothing else — not the fluid, not the size, not the flow rate. It is exactly 64 for a circle and exactly 96 for a slot, and the correlation that treats them as the same is wrong by a third.
The mesh that makes its own mass
The transport theorem holds for a region moving at any velocity, which is what makes a moving-mesh calculation possible. Discretised carelessly it is not an identity but an approximation, and a fluid at rest with a uniform density then gains density from the motion of a grid — smoothly, plausibly, and looking exactly like a physical transient.
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.
The part of the flow inside the body
A potential flow outside a body is an analytic function, and an analytic function does not stop at the boundary it was defined on. It continues inward until it meets a singularity — and every body in this collection has at least one inside it, in a place that decides how the flow behaves outside.
One formula for both ends
Two limits, each with its own description, neither valid everywhere. The composite is the sum less the part they agree about, and it is uniformly good — but the overlap region that justifies the construction does not exist at ε = 0.01, and the composite is still accurate to two per cent there.
Where the line stops being a line
Prandtl's lifting line replaces a wing with a single bound vortex and its trailing sheet, and the formula that comes out is the most quoted in low-speed aerodynamics. Solved numerically at aspect ratio one it returns its own closed form to sixteen decimals — and the answer is forty-one per cent too high.
The gap that carries the most
A bearing's most consequential dimension is decided by a number with no oil in it, no speed and no size — the root of one transcendental equation, at 2.1887048. And the maximum it sits on is flat enough that the value printed in every handbook is not it.
The three that never converge
Oseen's drag coefficient, the overlap layer's power-law exponent and the enstrophy range's slope are three separate disappointments in three different fields. They are one piece of arithmetic, and the arithmetic says why none of them will ever be reached.
Where a pure number comes from
Sixty-four for a round pipe, sixteen twenty-sevenths for a wind turbine, 0.332 for Blasius. Counting dimensions produces none of them — it produces the list of arguments, and the function has to be solved. Which of the four ways it was solved decides how many digits are worth printing.
The corners that can be done with mirrors
The method of images works for a wall and for a circle, and for a corner it works only when the angle is pi over a whole number. At every other angle the reflections never come back, the image set is infinite and dense, and the flow exists anyway — which says the method is a statement about symmetry rather than about fluids.
The optimum that does not matter
Elliptic loading gives the least induced drag, exactly, and the proof is one line: the penalty is a sum of squares. Which is also why the optimum is flat enough that a quarter of the design space sits inside a tenth of a per cent of it.
The speed where the damping is exactly zero
Flutter is an algebraic condition on a quartic: one root crosses the imaginary axis, at one speed, exactly. And the quantity that locates it is so nearly flat there that the standard way of finding it from flight test overshoots by a quarter.
A wake that keeps the drag and forgets the body
Two very different wakes with the same momentum deficit converge to the same profile, because the deficit is conserved and everything else diffuses away. The convergence is a power law rather than an exponential, so it takes two hundred widths for twenty per cent agreement and nine hundred for five.
Twice as slippery along as across
A surface of alternating gas and solid stripes lets a liquid slip, and a flow far above it sees one number in place of the pattern — but the number depends on which way the flow goes. Along the stripes it is Philip's logarithm; across them it is exactly half, for a reason that takes one substitution to show. And the logarithm means that the slip is bought by the pattern's period rather than by how much of it is gas.
The truncation that cannot carry three times the heat
Lorenz's three modes give a convecting layer's heat flux in one line — one plus twice (r − 1) over r — and it can never reach three times what conduction carries. The same rolls computed with forty-two modes agree with that line exactly at onset, carry 7.1 per cent more heat at twice the critical Rayleigh number, and twice as much at thirty times it. The ceiling is not in the convection; it is in having one sine to draw the temperature with.
The drift was the instrument
Kelvin's theorem was checked on this site by carrying a loop and watching its circulation move by eight parts in a hundred thousand. That drift is not the flow forgetting. The exact number is a count of what is inside the loop, it does not move by anything at all, and the drift belongs entirely to the two instruments used to measure it.
A duct that forgets everything but one number
Whatever is fed into a pipe, what survives a little way down it is one shape. The disturbance's higher modes decay as the square of their mode number, so the sixth is gone in thirteen centimetres where the first survives four and a half metres — and the entrance length is that one mode's decay rate and an arbitrary threshold.
Named alongside it
The objects these essays reach for when they reach for this one.
MeasurementModel limitModel validityToleranceBoundary conditionDimensionlessDiscretisationEigenvalueRegimeThresholdConformal mapLinear stability