Concept

Direct numerical simulation — where it appears

A computation of turbulence that solves the full Navier–Stokes equations on a grid fine enough to resolve every eddy, with no turbulence model. The grid must reach the Kolmogorov length, so the cost grows roughly as the cube of the Reynolds number, which limits it to modest Reynolds numbers.

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

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

Forced at the integral scale, the third moment nearly reaches four-fifths. The Kármán–Howarth balance at a Taylor-scale Reynolds number of 200, on the same model spectrum, for a flow forced in a band at the spectrum's peak and for one decaying. Each term is divided by (4/5)εr and plotted against separation in Kolmogorov lengths. The viscous term is the same for both. The forcing term is negligible until the separation approaches the integral scale, and −Dₗₗₗ/((4/5)εr) for the forced flow peaks at 0.965 at 90 Kolmogorov lengths, where the decaying flow's peaks at 0.747 at 32.

Forcing at the integral scale leaves the cascade alone

A decaying flow's third moment falls a quarter short of the four-fifths law at the Reynolds numbers a grid reaches. A flow forced at its largest scales, on the same spectrum at the same Reynolds number, falls short by three and a half per cent. The shortfall is not what a finite Reynolds number does to every flow; it is what the source does, and how fast it closes is set by how far the source reaches into the inertial range.

turbulence · Decay

Named alongside it

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

Reynolds number Cascade Closure problem Decay Dimensional analysis Dissipation Four-fifths law Inertial range The Kolmogorov scale Large-eddy simulation Production Resolution

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