Concept

Production — where it appears

The rate at which mean-flow energy is fed into turbulence, which is the Reynolds stress working against the mean strain. It is the source term in the turbulent kinetic energy budget, and it is balanced by dissipation only when the flow has had time to settle.

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

A normal stress the closure makes negative. The Boussinesq closure's first normal stress in a plane strain, against the strain measured in units of the turbulence's own time scale. It crosses zero at S k/eps = 1/(3 C_mu) = 3.704 — eleven per cent above the value the constant was calibrated at — and goes on falling. A variance below zero is not a small error; it is a statement that cannot be true.

The constant that makes a variance negative

Every engineering turbulence calculation in the world rests on one number, C-mu equals 0.09. It is not a property of turbulence. It is the assertion that a particular ratio is ten thirds, which is true in one flow — and eleven per cent above that flow the same closure reports a mean square below zero.

turbulence · Closure

The production jumps and the dissipation does not. Production and dissipation against time, through a step change in the strain rate. The production follows the strain immediately — it is the strain squared times an eddy viscosity — and the dissipation moves by less than one per cent at the instant of the step, because it is set by a cascade that has not been told yet.

A dissipation that lags its production

Change the strain rate on a patch of turbulence and the production of energy follows instantly — it is the strain squared. The dissipation moves by less than one per cent, because it is the far end of a cascade that has not been told yet, and the two are out of balance by a factor of six for the next turnover.

turbulence · Dissipation

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.

Dissipation Cascade Closure Eddy viscosity Boussinesq Calibration Decay Direct numerical simulation Equilibrium Four-fifths law Inertial range Measurement

All concepts