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

The Kolmogorov scale — where it appears

The length at which viscosity finally acts on a turbulent flow, formed from the viscosity and the dissipation rate. It falls as the Reynolds number to the three-quarter power, which is why resolving a turbulent flow costs so much.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The inertial range, and the slope read back off it. The model energy spectrum at Re = 1e+6, with production rolling off below the integral scale and dissipation cutting it off above the Kolmogorov scale. The straight middle is the inertial range, and the number printed beside it is the slope least-squares fitted to the drawn points over the middle of that range — not the −5/3 that went in.

Where the energy goes

Energy enters a turbulent flow at the largest scale and leaves it at the smallest, and in between there is nothing for it to depend on but the rate at which it is passing through. Two quantities and one dimensional argument fix the shape of the spectrum, and the exponent is −5/3.

turbulence · Cascade Grid points against Reynolds number, and where a wing sits. The number of grid points needed to resolve every scale of a turbulent flow, which is Re^(9/4) — the cube of the ratio between the largest scale and the Kolmogorov scale. The line is the arithmetic and the marks are flows a reader can picture. An airliner's wing needs about 10¹⁷ points, and the largest calculations ever run are around 10¹².

The grid nobody can build

Resolving every scale of a turbulent flow needs Re to the nine-quarters grid points and Re cubed point-updates. An airliner's wing comes to 2·10¹⁷ points against the 10¹² of the largest calculation ever run, and no amount of patience closes a gap of five orders of magnitude.

turbulence · Cascade 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 Model spectra at four Reynolds numbers, compensated. The spectrum multiplied by k^(5/3) and divided by eps^(2/3), so that a true inertial range is a horizontal line at the Kolmogorov constant. What a finite Reynolds number has instead is a single maximum: it reaches 1.4996 at the highest and 1.49 at the lowest, and the band over which it is flat to one per cent goes from a third of a decade to two.

The range a real Reynolds number does not have

Kolmogorov's minus five thirds is a statement about a band of scales that has forgotten the forcing and does not feel the viscosity. Both conditions are about separation, and separation is exactly what a finite Reynolds number does not have much of.

turbulence · Spectrum The scalar spectrum, with its two ranges. A model scalar spectrum at a Schmidt number of two thousand — dye in water. Below the Kolmogorov wavenumber it is Obukhov and Corrsin's five-thirds, inherited from the velocity; above it there is no turbulence left and the spectrum is Batchelor's minus one, which contains no velocity spectrum at all.

The scalar has its own cascade

Below the Kolmogorov scale there is no turbulence left, and a dye stirred into the flow goes on cascading anyway — on a spectrum whose exponent is minus one and whose amplitude contains no velocity spectrum at all. Resolving it costs the three-halves power of the Schmidt number, which for dye in water is a factor of ninety thousand.

turbulence · Mixing One curve from two to one, with four thirds somewhere in the middle. The ratio of the transverse second-order structure function to the longitudinal one, against separation, at three Reynolds numbers. Every value on every curve follows from the longitudinal function alone by a relation with no dynamics in it. It is exactly 2 where the field is smooth, exactly 1 beyond the correlation length, and it passes through four thirds on the way — but it passes through rather than resting there, and how nearly it rests is the whole of what a Reynolds number buys.

A relation with no turbulence in it

Isotropy and incompressibility alone fix the transverse structure function from the longitudinal one. Divide the relation through and it says the ratio of the two is one plus half the local slope — so the exponent everybody measures as 0.70 and the ratio everybody measures as 1.35 are one measurement, and a model spectrum with no intermittency in it produces both.

turbulence · Structure function

Named alongside it

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

DissipationInertial rangeSpectrumEnergy cascadeReynolds numberTurbulenceDimensional analysisIntermittencyMeasurementModel limitStrain rateAsymptotics

All concepts