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

Grashof — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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 10 per cent low 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.

    part 1 · regimes
  2. Each collapsed form of the plate's heat transfer reaches its limit decades away from air. The exact local Nusselt number of a vertical isothermal plate in its two collapsed forms against the Prandtl number: over (Gr·Pr²)^¼, which tends to Le Fevre's 0.600 as Pr → 0, and over (Gr·Pr)^¼, which tends to 0.503 as Pr → ∞. At Pr = 10⁻⁵ the first reads 0.5994; at 10⁵ the second reads 0.5023. The two limit laws cross at Pr = 0.494, where the exact value is 0.736 of either; air, at 0.72, reads 0.421 and 0.387.

    Over a plate in mercury the warm layer flows as if inviscid

    The exact solution for a warm vertical plate has been tabulated since 1953 and is usually shot from the wall, which fails in a liquid metal unless the shot starts within a thousandth of the answer. Posed instead as one algebraic system, it solves from a Prandtl number of a hundred-thousandth to a hundred thousand — and shows that in mercury the layer's speed is set without viscosity, that the integral method runs low there rather than high, and that air sits where neither limiting law is within a fifth.

    part 2 · regimes
  3. A flat plate's layer is pushed along by the pressure its own warmth leaves, strongest at the wall. Across the layer of a horizontal plate warm side up, in air: the velocity scaled to its peak, the temperature excess, and the along-plate push −∂p′/∂x scaled to its wall value, against the similarity distance η = (y/x)·Gr^⅕. The push is the hydrostatic deficit under the warm fluid, (2/5)(ηθ − P), 0.694 at the wall in units of gβΔT·Gr^−⅕, falling to nothing at the layer's edge; the velocity peaks at η = 1.58 and the warm layer ends near 5.93.

    Laid flat, a warm plate is pushed by the pressure of its own layer

    A warm vertical plate drives its layer with buoyancy acting along it. Lay the plate flat and buoyancy acts across the layer instead, so the only push along it is the pressure deficit its own warm fluid leaves under the cold — deeper where the layer is thicker. That push gives a heat transfer growing as the fifth root of the Grashof number, not the fourth, works only on the plate's upper face, and survives a tilt of a degree for a third of a metre.

    part 3 · regimes
  4. Of the four common plumes, only the laminar one above a line speeds up as it rises. Centreline speed against height for four plumes, each scaled to its own speed at one reference height: laminar above a line source, rising as the fifth root of height; laminar above a point source and turbulent above a line, both constant; and turbulent above a point, falling as height to the minus one third. Each exponent follows from the heat flux being fixed and the width growing by diffusion, √(νx/u), when laminar, or in proportion to height when turbulent.

    Above a warm wire the plume speeds up as it rises

    A plume above a line of heat has no wall, no imposed speed and one conserved quantity, the heat it carries. That alone fixes a centreline speed growing as the fifth root of height — the only one of the four common plumes that accelerates — a temperature excess falling as the three-fifths power, and a warm core that is narrower than the moving sheet around it in any fluid more viscous than five ninths of its own diffusivity. At that Prandtl number the equations have a closed form, and the solver lands on it to three parts in a million.

    part 4 · regimes

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