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

Upstream influence — where it appears

The transmission of information against the flow direction in a boundary layer, which the classical parabolic equations forbid and every experiment shows. It is restored when the pressure is allowed to respond to the layer's own displacement, over a region whose length is a new power of the Reynolds number.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

The backwater curve behind a weir. The depth at the control is 1.6 times the normal depth; integrating upstream the profile relaxes onto normal depth over about seven hundred metres and stays there. That relaxation is the reach forgetting the weir, and nothing about the weir survives past it.

The section that decides the river

A reach of open channel has a normal depth and a critical depth, and the water has neither. What it has is a profile obeying a first-order equation, which needs exactly one condition — and whether that condition belongs at the upstream end or the downstream end is not a choice, because the equation is stable in one direction and unstable in the other.

applied · Open-channel The two long-wave speeds, and the swirl at which one of them stops. For uniform axial velocity the wave speeds follow from the criticality condition by a Galilean boost: c = W(1 ± 2S/j), with j the first zero of J1. The upstream-running root crosses zero exactly at S = j/2 = 1.9159, and above that swirl no disturbance can travel upstream — which is what subcritical and supercritical mean here and in an open channel.

The swirl that holds a wave still

A swirling flow down a pipe carries waves, and above a certain swirl one of them stops moving. Below it, a disturbance downstream can send information upstream; above it, the flow has outrun its own waves. The words are open-channel flow's words, and they are the same words for the same reason.

inviscid · Swirl The eighths nobody chose. Four physical statements — the inner layer sits in the classical one's shear, its inertia balances its own viscous stress, the pressure is of the order of that inertia, and the displacement it makes produces that pressure — are a linear system in four exponents. Solving it gives three eighths, five eighths, one eighth and a quarter, exactly.

The length the limit invents

Prandtl's equations are parabolic, so nothing at one station can depend on anything downstream of it. Every experiment shows the pressure rising ahead of a shock or a step. The resolution is a region three eighths of a power of the Reynolds number long, which the limit that produced the equations was supposed to have removed.

inviscid · Interaction

Named alongside it

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

EigenvalueFroude numberHydraulic jumpModel limitAiry functionAsymptotic matchingBoundary conditionBoundary layerBragg hawthorneCirculationCriticalityDisplacement thickness

All concepts