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

Prandtl number — where it appears

The ratio of a fluid's momentum diffusivity to its thermal one, which fixes how a temperature layer's thickness compares with a velocity layer's. It is near one for gases, seven for water and a hundredth for liquid metals, spanning a factor of forty in the layer ratio.

Named by 14 essays across 3 fields — each of them below, with the objects they name alongside it.

One layer, and the four others inside and outside it. The velocity profile over a flat plate, and the temperature profile in the same layer at four Prandtl numbers: a liquid metal at 0.01, air at 0.71, water at 7 and a heavy oil at 100. The equations differ by one number and the profiles differ by a factor of twenty in thickness. At Pr = 1 the two are the same function — not similar, identical, to eight decimal places — because the equations and the conditions are then the same, which is what every statement called a Reynolds analogy rests on.

The other layer, and the one number that separates them

A wall in a stream carries two boundary conditions and grows two layers. Their thicknesses differ by a factor of twenty across ordinary fluids, and at exactly one Prandtl number the two profiles are not similar but identical.

regimes · Peclet What the skin settles at, before anything is done to it. The adiabatic wall temperature against Mach number, in air at 216.7 K, with the stagnation temperature above it. The gap between the two is the recovery factor, which is 0.8417 here and stays there at every Mach number — it is a property of the Prandtl number and not of the speed. At Mach 2 the skin sits at 363 K, at Mach 3 at 545 K, and at Mach 5 at 1128 K, which is past what aluminium will do. Nothing has been burnt and nothing has been rubbed: the air was brought to rest, and this is where its kinetic energy went.

The wall that heats itself

A surface told nothing about its temperature does not settle at the air's. It settles most of the way to the stagnation temperature, and the heat flux is driven from that invented temperature rather than from the free stream's — so a wall hotter than the air can be being heated by it.

compressible · Recovery The velocity through a shock at Mach two, as a function of position. The Becker profile, integrated outwards from its own inflection point. It runs from the upstream velocity to the downstream one — the two states the jump conditions give, which appear here as the equilibria of a first-order differential equation — and its steepest gradient matches the closed form to one part in ten thousand. The horizontal axis is in units of the thickness, which for this shock is 139 nanometres.

The discontinuity that has a thickness

The jump conditions do not contain the viscosity, which is why they are exact. The thickness is entirely viscosity — 289 nanometres at Mach 1.5, 35 at Mach 5, against a mean free path of 64. At Mach five the continuum equations have produced a structure thinner than the distance between collisions.

compressible · Shock Nu/√Re against the Prandtl number, over eight decades. The whole of the flat plate's heat transfer, as one curve. It is not a power law: it goes as Pr^½ at the bottom, where the thermal layer is far thicker than the viscous one, and as Pr^⅓ at the top, where it is buried inside it. The Pr^⅓ everybody quotes is the upper half. The two asymptotes are drawn beside it, and the low one is √(Pr/π) in closed form.

The number that is an answer

Almost every dimensionless group is a hypothesis: somebody sets the speed, the size and the fluid, and the number licenses a model. The Nusselt number is not. It is what the experiment produces, it sits on the left of the equals sign, and a regime diagram drawn on it is a category error.

regimes · Nusselt Seven shock structures, and the two states they all connect. The velocity through the shock for seven dissipation models — Prandtl numbers from a quarter to two, viscosities from constant to linear in temperature. Each curve is shifted so its midpoint sits at the origin. They start at the same speed, end at the same speed, and are nothing alike in between.

The jump does not ask what made it

Seven different dissipation mechanisms are made to smear the same shock. Their interiors are a factor of two and a third apart in thickness, their entropies overshoot the final value by between a quarter and a doubling, and the state they all arrive at agrees to seven parts in ten billion — because the end states are conservation and the interior is transport.

compressible · Shock 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.

regimes · Grashof A cooled wall at β = 1: the linear relation is 158 K out inside the layer. The static temperature across a laminar layer at Mach 5 with an edge temperature of 220 K, over a wall whose total enthalpy is 0.5 of the edge's, at a pressure gradient β = 1: exact (thick) and from the Crocco–Busemann linear relation (thin), at a Prandtl number of one with constant properties. The wall is at 660 K in both. The exact profile peaks at 684 K and the linear one at 759 K; the largest difference, −158.3 K, is at η = 0.81.

The gradient the heat never hears

On a flat plate at a Prandtl number of one, a boundary layer's total enthalpy is a straight-line function of its velocity, whatever the wall's temperature. Put the same layer in a pressure gradient and the straight line fails everywhere except on an insulated wall, because the gradient enters the velocity's equation and not the enthalpy's — and a favourable gradient can leave a band of gas colder than the free stream above a wall three times hotter than it.

compressible · Recovery Where the eddies outconduct the molecules. The ratio of turbulent to molecular heat diffusivity across the pipe at Reτ = 2000, for five fluids, on logarithmic axes. Wherever it is above one the eddies carry more heat than conduction does. For air it passes one inside the buffer layer and reaches 135; for water, earlier and higher. For liquid sodium it never reaches one anywhere: at its peak, halfway to the axis, the eddies carry just over half what conduction carries, and the temperature profile is set by conduction across the whole pipe.

The heat the eddies do not carry

In a laminar layer the temperature and the velocity have different thicknesses in every fluid but one. In a turbulent pipe the eddies carry both, and the difference nearly vanishes — for air, water and oil alike. It does not vanish for a liquid metal, whose molecules conduct heat faster than the eddies can, and the boundary between the two behaviours is a Péclet number of about four hundred at every Reynolds number.

regimes · Peclet The entrance grows with the Péclet number, peaks, and then shrinks. The distance from a wall-temperature step at which the local Nusselt number has come within 5 per cent of its developed value, in diameters, against the Péclet number, for turbulent pipes at Reτ = 500, 1000, 2000 and 5000, with the slug's laminar line L/D = 0.029·Pe. The liquid metals sit on the line; the entrance peaks at 11.6 diameters at Pe ≈ 4700 for Reτ = 1000 and falls to about three and a half diameters for water.

The longest thermal entrance belongs to neither metal nor gas

A liquid metal carries its heat across a turbulent pipe by conduction, as a laminar flow would, and so its thermal entrance should be long. It is long only when its Péclet number is large. Below a few hundred the entrance grows in proportion to the Péclet number with the constant of a fluid moving as a solid block, and it leaves that scaling close to the threshold at which the eddies first match conduction. The longest entrance in a turbulent pipe belongs to the fluids between the metals and the gases, and how long it is depends on a number the measurements have never pinned down.

regimes · Peclet A resistive wall lengthens a slug's thermal entrance by half, and no more. The 5 per cent thermal entrance length of a slug flow, as a multiple of its length with the wall held at a fixed temperature, against the wall Biot number hw·D/kf. A wall that conducts well changes nothing; as the wall's resistance grows the entrance lengthens to a peak of 1.501 times at a Biot number of 2.29, and then shortens slightly to 1.457 times, the fixed-flux value, for a wall that conducts hardly at all.

A steel wall lengthens a sodium entrance by half

A liquid metal's thermal entrance is long because its heat crosses the pipe by conduction, and a steel tube conducts a tenth as well as sodium, so the wall is part of the path. Put it in series with the fluid and the entrance does grow — but by half at the very most, at a wall Biot number near two, which is where sodium in a steel tube happens to sit. A wall that resists more changes the boundary condition from a fixed temperature to a fixed flux, not the length.

regimes · Peclet A free interface's thickness ratio heads to the root of its Prandtl number. The ratio of the shear layer's thickness to the density interface's against time, for an interface starting four times thinner than its shear, heat and salt in water. A free shear layer thickens by viscosity faster than the interface by conduction, so the ratio heads to √Pr: 2.64 for heat, 25.8 for salt. Held at its thickness, the shear lets the ratio fall through 2 — at about 50 s for heat and 4725 s for salt.

What holds the shear decides whether the waves last

Holmboe's travelling waves need a density interface about half as thick as the shear around it, and diffusion thickens both. If the shear layer is held at its thickness, the interface catches up — in under a minute for heat in a laboratory tank, over an hour for salt — and the waves get two e-folds or a hundred and seventy. If the shear layer is free it thickens faster than the interface, the ratio heads to the root of the Prandtl number, and the waves last until the Richardson number ends them instead.

turbulence · Stratification 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.

regimes · Grashof 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.

regimes · Grashof 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.

regimes · Grashof

Named alongside it

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

Heat transferModel limitBoundary layerSimilarity solutionBuoyancyNusselt numberPeclet numberBoundary conditionCorrelationDimensionless numberDissipationMeasurement

All concepts