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

Dimensionless number — where it appears

A ratio of two effects, formed from the quantities a problem contains and having no units. Its value decides which terms may be neglected, which is why naming the regime and naming the number are the same act.

Named by 16 essays across 4 fields — each of them below, with the objects they name alongside it.

The bath is ten thousand times too small. The Rossby number — the ratio of the inertial term to the Coriolis term in the momentum equation — for eight flows at 45 degrees, on a logarithmic axis. Above one, rotation is a correction; below one, it is the physics. A draining bath sits at 3.2e+3 and a mid-latitude depression at 1.9e-1. The Coriolis term is not absent from the bath: it is present, computable, and four orders of magnitude smaller than the terms that decide what happens. Saying so is not the same as saying it is zero.

The bath does not know the hemisphere

The Coriolis term in a draining bath is not absent. It is present, computable, and about four hundred times smaller than what a hand left in the water an hour ago is still doing — which is a ratio rather than an opinion, and it is the same kind of argument as a Reynolds number.

misconceptions · Coriolis scale 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 Two terms, one with viscosity in it and one without. Ergun's two contributions to the pressure gradient, against the pore Reynolds number, on logarithmic axes. The viscous term rises with the first power of the velocity and the inertial one with the square, so on these axes they are straight lines of slope one and two and there is exactly one crossing. The second term contains no viscosity at all — it is the price of accelerating fluid into every pore and out again — which is why a linear resistance law has to fail eventually whatever the fluid is.

Where Darcy stops

A linear resistance law has to fail eventually, because pushing fluid into a pore and out again costs energy that has nothing to do with viscosity. Where it fails is one dimensionless number, and that number is 150/1.75 — read out of a correlation's own constants rather than measured.

applied · Porous Storage and loss moduli, for one relaxation time and for a spectrum. The two moduli of a Maxwell fluid and of a Rouse chain, against frequency in units of the longest relaxation time. One relaxation time makes the storage modulus overtake the loss modulus at λω = 1 and then leave it behind without limit. A spectrum of 1000 modes makes them rise together as the square root of frequency and stay a fixed ratio apart, so the material never becomes the solid the single time predicts.

A solid, if it is not given time

The Deborah number is the only group on this site with no fluid in it — two times and nothing else — and it says a material is a solid or a liquid depending on how long anybody watches. What it throws away is that no real material has one time, and the spectrum it replaces changes the answer in kind rather than in degree.

regimes · Deborah Three bluff bodies, whose Strouhal numbers agree once the wake's width is used. The measured Strouhal number of each body, and Roshko's universal number formed with the wake's width and the speed on the free streamline that bounds it. The raw numbers span a factor of 1.462; the collapsed ones span 1.0011, with a mean of 0.16281. The shedding was never body-dependent — the length in the number was.

The frequency a wake chooses

Bluff bodies shed at Strouhal numbers from 0.145 to 0.212, and the spread is not a fact about shedding — it is a fact about which length went into the number. Change the length to the wake's own width and three bodies agree to a tenth of a per cent. Then let the body move, and the number stops deciding anything at all.

regimes · Strouhal Drop deformation in simple shear, against the capillary number. The shape model's steady deformation for six viscosity ratios. Every curve is linear in the capillary number at small Ca — which is Taylor's result — and every one of them saturates, at 5/2(2λ+3), because the shear's own rotation turns the drop out of the stretching direction. A drop in simple shear cannot be deformed beyond that however hard it is sheared.

The number that cannot break a drop

The capillary number sets the stress that stretches a drop against the stress that holds it round, and it predicts the deformation beautifully. It cannot predict the breakup, because above a viscosity ratio of about four a drop in simple shear cannot be broken at any shear rate — and the theory that says the ratio hardly matters is the same theory that gets the deformation right.

regimes · Capillary Morison's two terms over one cycle, at KC = 10. The drag term, in phase with the velocity and going as its square; the inertia term, in phase with the acceleration and ninety degrees ahead of it; and their sum, which is what a load cell records. The peak of the total is not the peak of either, and its position in the cycle is the only thing in the record that says how the two are divided.

Long enough to make a wake

A Reynolds number cannot ask whether an oscillating flow gets round a body before it turns and comes back, because it has no time in it. The Keulegan–Carpenter number can, and it decides which of Morison's two terms is the force. What it discards is the phase — and a peak force measurement cannot recover it.

regimes · Keulegan–Carpenter number 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 The marginal Taylor number against the axial wavenumber. The smallest Taylor number at which a disturbance of a given axial wavenumber is neutral. Its minimum is 1707.757 at a wavenumber of 3.1158, which are the critical Rayleigh number and critical wavenumber of a layer of fluid heated between two rigid walls — the same numbers, because in the narrow-gap limit the two problems are the same sixth-order eigenvalue problem. This curve is computed by that essay's own solver.

A transition that needs a second number

Fluid between rotating cylinders goes unstable at a Taylor number of 1707.762 — which is the same number, to every digit, as a layer of fluid heated between two rigid walls. It is not an analogy. And two things the number cannot carry decide whether the transition happens at all and what it looks like when it does.

regimes · Taylor 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 Three exponents for one dimensionless group. The local slope of each error, measured over one decade at a time. The duct's is exactly 1, the long wave's is exactly 2, and the slender body's drifts from 1.900 to 1.733 across the range and never reaches either. The same geometric ratio, in three problems that look alike, and the third one has no exponent at all.

One group, three exponents

Lubrication theory, shallow water and slender-body theory are taught in three places and are one expansion in one group — a ratio of two lengths, with no speed, no viscosity and no fluid in it at all. The error is supposed to be second order. In three problems that look alike it is first order, second order, and an exponent that does not exist.

regimes · Slenderness The exact solution and its three approximations, at ε = 0.02. The outer solution is excellent everywhere except in a layer of width ε at the left, where it is wrong by a whole unit. The inner solution is excellent inside that layer and wrong everywhere else. The composite is their sum less the part they agree about, and it is within order ε of the exact solution across the whole interval — which is the entire content of matched asymptotics, drawn.

One formula for both ends

Two limits, each with its own description, neither valid everywhere. The composite is the sum less the part they agree about, and it is uniformly good — but the overlap region that justifies the construction does not exist at ε = 0.01, and the composite is still accurate to two per cent there.

regimes · Crossover The same data, in a basis the theorem allows just as much. The identical two hundred and forty points, plotted as F/(mu U d) — which is the drag coefficient times the Reynolds number, a perfectly legitimate pi group forming a complete pair with the abscissa. The result is a straight line of slope 0.9985 through five decades with an r-squared of 0.9985, and it contains no physics: the ordinate contains the abscissa.

The groups are not the only groups

Buckingham's theorem fixes how many dimensionless groups an answer can depend on and says nothing about which. Two of the infinitely many legitimate choices are used here on the same data: one manufactures a straight line through five decades out of a constant, and the other erases Stokes' law completely.

regimes · Dimensional A wave that travels and a wave that spreads. A harmonic pressure wave's amplitude and its instantaneous value along a tube, over two wavelengths of the inviscid wave, at four Womersley numbers. At α = 15 the wave marches on, a little weaker each wavelength. At α = 5 it is visibly damped. At α = 2 it is nearly gone within a wavelength. At α = 0.5 there is no wave to speak of: the disturbance falls away within a small fraction of the inviscid wavelength, as heat does into a wall.

The pulse that has to travel

In a rigid tube the Womersley number decides the shape of an oscillating flow. Make the wall elastic and the pressure pulse has to travel, a second number appears — the tube's length in wavelengths — and the first number turns out to decide something more basic than the profile: whether the tube carries a wave at all, or only a disturbance that spreads like heat.

regimes · Womersley One number decides which pulse grows. The pressure pulse and the flow pulse at the far end of the tube, each as a multiple of its value at the entrance, against the load's reflection coefficient. A load that reflects pressure with the same sign — a stiffer or narrower continuation — amplifies the pressure pulse and damps the flow pulse. One that reflects it inverted — a wider continuation, or many branches — does the opposite. With no reflection both fall slightly, by the wave's own attenuation. The two curves cross near Γ = 0 and pull apart on either side.

The pulse that grows as it leaves the heart

The pressure pulse measured at the wrist is larger than the pulse in the aorta that drives it, and the flow pulse is smaller. Nothing downstream is pumping. A wave reflected from the end of an elastic tube arrives back in step with the outgoing wave near the end and out of step near the start, and a single number — the reflection coefficient — decides whether it is the pressure or the flow that grows.

regimes · Womersley 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

Named alongside it

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

Model limitMeasurementRegimeScalingThresholdBoundary layerCorrelationHeat transferPrandtl numberReynolds numberAdded massAngular momentum

All concepts