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

Particle tracking — where it appears

Integrating the equation of motion of a small body through a computed velocity field, rather than assuming it follows a streamline. It is how collection efficiencies, spray penetration and preferential concentration are computed, and it needs a drag law and a relaxation time.

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

Particles at St = 1, against the flow that carries them. Particle paths and the streamlines they were released on, in this site's exact cylinder solution. At small Stokes number the two are indistinguishable and the body catches nothing; as the particles get heavier their paths straighten, cross the streamlines, and begin to strike. The paths are integrated with Stokes drag and nothing else — no gravity, no lift, no effect of the particles on the flow.

Whether the droplet turns

The air goes round the wing. Whether what is carried in it goes round too is decided by one number — and below a critical value of that number the body collects nothing at all, however many droplets are thrown at it, because the flow turns every one of them in time.

regimes · Particle How small is small enough. The error in Stokes' law for the drag on a sphere, against the Reynolds number, both logarithmic. Oseen's correction is the first term the neglected inertia puts back, and it says the error is 3Re/16: one per cent at Re = 16/297 = 0.054, five per cent at 0.28, and already sixteen per cent at Re = 1 — which is the value at which the two terms the Reynolds number compares are equal, and is where every textbook draws the boundary of creeping flow.

How small is small enough

Creeping flow drops the inertia terms and gets an exact answer for the drag on a sphere. The Reynolds number at which those terms are equal is one — and at one, Stokes' law is already sixteen per cent low. The honest boundary is 0.054, and the reason it is so far down is the same reason the theory needed repairing in the first place.

regimes · Reynolds The relaxation time a particle actually has. Two quantities against the particle-to-fluid density ratio. β = 3ρ_f/(2ρ_p + ρ_f) is three for a bubble, one for a neutrally buoyant particle and nearly zero for anything heavy; it is the factor by which the fluid's own acceleration is felt. The other curve is the true relaxation time over the usual formula's, which is one for a heavy droplet, exactly three halves for a neutrally buoyant tracer, and unbounded for a bubble — the usual formula gives a bubble a relaxation time of zero, and therefore no dynamics at all.

The tracer that is not one

Every Stokes number is built on a relaxation time that counts the particle's own inertia and nothing else. Adding the two terms it leaves out gives a bubble a relaxation time where the usual formula gives zero, makes a neutrally buoyant tracer half as slow again as advertised, and sends bubbles into vortex cores that droplets are flung out of.

regimes · Particle

Named alongside it

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

DimensionlessThresholdInertiaRelaxation timeStokes' dragStokes numberAdded massAsymptoticsBuoyancyCavitationCollection efficiencyCreeping flow

All concepts