Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Scaling law — where it appears

A relation in which one quantity varies as a power of another, so that it plots as a straight line on logarithmic axes. Its exponent is read from the slope, and over a finite range that slope is a local measurement that can drift.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

Area-preserving in the aorta, Murray's cube in the small arteries. The branching exponent k, in r_parent^k = r₁^k + r₂^k, that makes one junction reflect least of the heart's pulse, against the parent artery's radius, for a symmetric split and for side branches a half and 0.15 of the continuing trunk. In the aorta, where the pulse is carried by inertia, the transparent rule is close to 2 — area preserved — whatever the asymmetry. In arteries under a millimetre, where viscosity carries it, it is 3: Murray's law, the rule for the cheapest steady flow, is exactly the rule that passes the pulse. Asymmetry moves the answer only in between.

Murray's law passes the pulse where the pulse is viscous

The rule that makes an arterial junction transparent to the heart's pulse was an exponent, and real arteries do not split evenly: the aorta sheds side branches a fraction of its size and carries on. For a lopsided junction the transparent rule is still an exponent — the same one, exactly, while viscosity is negligible. With viscosity it is a single function of the parent's Womersley number: close to area-preserving in the aorta, and exactly Murray's cube in arteries under a millimetre, where the pulse moves as the steady flow does. The asymmetry matters only in between, and a lopsided tree reflects far less than an even one.

regimes · Womersley Out to a hundred neutral wavelengths the noisy flame is still speeding up. The front's speed in excess of a flat flame's against its width, on logarithmic axes, at noise strengths of a thousandth and a millionth, with the quiet pole front's and a power fitted from 20 to 100. At a thousandth the speed climbs from 1.05 at 12.5 to 2.71 at 100; at a millionth from 0.668 to 1.71. Both follow the 0.42 and 0.43 power of the width to within 4 per cent, and neither bends over; the quiet front stays at a half.

A noisy flame is still speeding up a hundred wavelengths wide

A quiet wrinkled flame gains a fixed, bounded amount of speed over a flat one, however wide it is. A noisy flame keeps getting faster as it widens, and the open question was whether that ever stops. Out to a hundred neutral wavelengths it does not. The speed grows as about the 0.43 power of the width at two noise strengths a thousandfold apart. The noise sets only the prefactor, and the front's slope grows with the speed until it leaves the range the equation was derived for — which is where the growth stops being the equation's to decide.

kinematics · Dilatation

Named alongside it

The objects these essays reach for when they reach for this one.

Model limitBlood flowBranchingCombustionFlame frontFlame speedImpedanceInstabilityMichelson sivashinskyMurray's lawNoisePole solution

All concepts