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

Buoyancy — where it appears

The force on a body or a parcel arising from a density difference in a gravitational field. It drives convection, it decides whether a grain is lifted from a bed, and stratification is the case where it opposes a parcel's displacement rather than assisting it.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Every disturbance has its own threshold; one of them is lowest. The Rayleigh number at which a disturbance of horizontal wavenumber a becomes neutral, Ra = (π² + a²)³/a². Every wavenumber has a threshold and the layer goes unstable at the lowest of them, which a golden-section search on this curve puts at a = 2.221441 and Ra = 657.5114 — the exact π/√2 and 27π⁴/4 to fourteen digits. Below the curve the layer conducts and nothing moves.

A threshold with a closed form

A layer of fluid heated from below sits still until buoyancy overcomes both diffusions at once, and then it convects. Unlike every other threshold in this field, that one is an eigenvalue with an exact answer, and the answer is 27π⁴/4.

turbulence · Convection 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 The pressure drop stops rising at 0.213 m/s. The pressure drop across a bed of 500 µm sand, against the velocity through it, in units of the fluidisation velocity. The rising branch is Ergun's resistance and the flat one is the bed's buoyant weight, which the flow cannot exceed however hard it is pushed: past the corner the bed expands rather than resisting more. That flat line is the reason fluidisation is unmistakable in practice — the corner is a crossing of two curves rather than a gradual departure, and it can be read off a gauge.

The bed that weighs itself

Blow hard enough through a pile of sand and the pressure drop stops rising. It cannot rise — a control volume round the bed says the drop can never exceed the buoyant weight of the solid in it, and at the velocity where the two meet the bed stops being a structure and starts being a fluid.

applied · Porous Nu/Gr^¼ against the Prandtl number, with the exact solution's points on it. The closed form 0.508 Pr^½(20/21 + Pr)^−¼, over eight decades, with Ostrach's exact similarity values marked. The integral method is two to eight per cent high from Pr = 0.7 upwards and 27 per cent high at Pr = 0.01 — which is where the thermal layer is ten times the momentum layer and giving them one thickness stops being an approximation to anything.

A speed nobody imposed

Every regime number in this collection contains a velocity somebody chose. Natural convection has none: a warm plate makes its own flow, and the Grashof number is what is left when the speed is taken out. The Reynolds number of the result — six thousand, on an ordinary radiator — is an output of the solution rather than a setting on an apparatus.

regimes · Grashof Instability up to a quarter, and none past it. The fastest growth rate of a stratified shear layer against its Richardson number, on a profile whose gradient Richardson number is the same at every height. It falls smoothly towards zero and reaches it at a quarter: at Ri = 0.2499 the fastest mode still grows at 0.00106, and at 0.26 the solver finds no unstable mode at all.

Sufficient, and not necessary

A stratified shear layer whose Richardson number exceeds a quarter everywhere cannot go unstable. That is a theorem with an exact number in it. What it does not say — and what it is constantly read as saying — is that a layer below a quarter will.

turbulence · Stratification The same wavelength drawn as rolls and as hexagons. Plan views of a convecting layer with one critical wavelength, drawn from the amplitude equations' two stable states. On the left, rolls: a single set of parallel bands, rising fluid along one set of lines and sinking along the next. On the right, hexagons: three sets of rolls at 120° to each other with equal amplitudes, whose sum has its maxima on a triangular lattice with spacing 2/√3 of the wavelength, each maximum at the centre of a hexagonal cell. At ε = 0.0250, inside the window, both are stable: rolls with amplitude 0.158 and hexagons with 0.081 in each of their three rolls. A hexagon is not a different kind of cell; it is three roll patterns that the quadratic term lets reinforce one another.

Hexagons remember how the heat was turned up

A layer heated from below convects in rolls, unless its top and bottom are not mirror images of each other. Then three sets of rolls at 120° can feed one another through a term the symmetry used to forbid, hexagonal cells appear before the layer is formally unstable, and there is a range of heating in which rolls and hexagons are both stable — so the pattern a layer shows depends on whether the heat was turned up or down to get there.

turbulence · Convection

Named alongside it

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

Linear stabilityMeasurementRegimeThresholdDimensionless numberEigenvalueModel limitRayleigh–Bénard convectionAdded massBifurcationBoundary layerCavitation

All concepts