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

Viscous diffusion — where it appears

The spreading of vorticity or momentum by viscosity, at a rate that makes a layer's thickness grow as the square root of time. It is what limits how concentrated a vortex can stay and what sets every boundary-layer thickness in the subject.

Named by 5 essays across 4 fields — each of them below, with the objects they name alongside it.

The one place the stretching argument closes. Burgers' vortex: an axisymmetric strain carrying vorticity inwards at exactly the rate viscosity spreads it outwards. The vorticity profile is a Gaussian of radius √(4ν/α), the swirl velocity peaks at 1.12 core radii rather than at the core radius itself, and the circulation reaches its full value by about two. The steady vorticity equation is evaluated on this profile by differencing it, not by re-deriving it.

The spin that feeds itself

Stretch a vortex tube and its spin rises in exact proportion, because the circulation round it cannot change and its area has fallen. Nothing in that argument sets a limit — and the one flow where the limit can be written down exactly puts it at a length of √(4ν/α).

kinematics · Vorticity A wave that dies within one wavelength — 100 Hz in air. The velocity profile above an oscillating wall, at eight phases of one cycle, with depth in units of δ = √(2ν/ω). The motion is a wave travelling into the fluid, and its amplitude falls by 1/e in the same distance it turns by one radian — so it is dead within about one wavelength, and the fluid three δ up hardly knows the wall is moving at all. This is one of the very few exact solutions the Navier–Stokes equations have. The dashed line is one fixed distance above the wall in millimetres: in these units it climbs as the square root of the frequency, which is the whole of how far the motion reaches.

The wall that shakes

Slide a wall back and forth in its own plane and the fluid above it does not follow — a wave travels upwards into the fluid and dies within one wavelength. The depth it reaches is √(2ν/ω), it contains no length from the geometry at all, and the whole thing is one of the very few exact solutions the Navier–Stokes equations have.

viscous · Exact layer Five spectra, five exponents, one linear equation. The energy of a decaying turbulence after the nonlinear term has stopped mattering, computed by integrating the exact modal solution E(k,0)exp(−2 nu k² t) at five different shapes of the spectrum at the origin. Each is a straight line on these axes and no two have the same slope: the exponent is (m+1)/2, where k^m is the spectrum's behaviour at wavenumbers smaller than any eddy. The 5/2 that is quoted as the final period's universal exponent is the m = 4 line and one of five.

Universal, and one of five

A turbulence that has run its Reynolds number down stops being turbulent, the equations go linear, and the decay picks up a new exponent. That exponent is quoted everywhere as 5/2 and as universal. It is neither: it is the same corner of the same spectrum deciding the answer a second time.

turbulence · Decay Three profiles that do not depend on the radius. The radial, azimuthal and axial velocities of the flow above a rotating disc, as functions of one similarity variable. The radial one is a jet: fluid thrown outward by the swirl it has picked up, peaking at 0.181 of the local disc speed a fifth of the way through the layer. The azimuthal one falls from the disc's own speed to nothing. And the axial one is the surprise — it does not vanish far from the disc but tends to a constant, so the disc draws fluid down onto itself at 0.8845 times the square root of the viscosity times the rotation rate, at every radius and for ever.

The solution that keeps its nonlinear term

Every exact solution before this one has been exact because the nonlinear term vanished. A rotating disc's does not vanish — at the wall it is the whole of the balance — and the reduction is exact anyway, because the radius divides out of all three momentum equations at once.

viscous · Exact layer The number of passes grows in proportion to Reynolds number. Passes the two rings make before their cores are close enough to merge, against the vortex Reynolds number, marched with the merging threshold at 0.24 of the core separation and at 0.30, and the closed form [(0.24 dₘᵢₙ)² − σ₀²] Re ÷ 4Tₚₐₛₛ, with the closest approach dₘᵢₙ and the time between passes Tₚₐₛₛ read from the inviscid pair. Both thresholds give a count in proportion to Re: one or two passes at a few thousand, ten or thirty at thirty thousand.

Leapfrogging rings end by merging, not by parting

Two smoke rings that leapfrog in an ideal fluid do it for ever, because their energy binds them. A real fluid drains the energy, and the obvious guess is that the rings drift out of the bound state and part. They do not: diffusion drains the pair's energy and the energy of every possible pair of free rings together, and the margin that binds them never goes negative. What viscosity does instead is fatten the cores until, as one ring threads the other, the two are close enough to merge. That takes a number of passes in proportion to the Reynolds number — one to three at the few thousand of a laboratory smoke ring.

circulation · Vortex ring

Named alongside it

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

Model limitNavier–Stokes equationsBoundary layerExact solutionPenetration depthReynolds numberBurgers vortexCirculationDecayEnergy cascadeFinal periodHamiltonian

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