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

Closure problem — where it appears

The fact that averaging the equations of motion produces more unknowns than equations, six of them, and that no exact route closes the gap. Every practical calculation of a turbulent flow guesses at the missing terms, and the guesses carry measured constants.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Ten unknowns, four equations. What is left after the Navier–Stokes equations are averaged. The mean velocities and mean pressure were there before; the six Reynolds stresses are new, and they arrived from the one term that does not average away. Nothing in the count is an approximation — the averaged equations are exact — and that is what makes the gap uncomfortable.

What averaging costs

Split the velocity into a mean and a fluctuation, average the equations, and the result is exact. It is also short of six equations, because the one nonlinear term does not average away and leaves six new unknowns behind that nothing determines.

turbulence · Closure Every rung of the hierarchy is worse than the last. Writing a transport equation for the Reynolds stress does not close the system: that equation contains the triple correlation, whose equation contains the quadruple. The bars are the number of independent components at each order and the number of new unknowns its own equations introduce. The second is always larger, and the gap widens.

The ladder that never closes

Being six equations short is a problem with an obvious remedy — derive six more. The remedy works, produces an exact equation for the Reynolds stress, and leaves ten new unknowns behind. The gap does not narrow at any level, and the counting says why.

turbulence · Closure u⁺ = (1/κ) ln y⁺ + B, integrated rather than asserted. The velocity profile in wall units, produced by integrating the mixing-length closure outward from the wall. The straight portion is the log law and the constants beside it were least-squares fitted to the integrated curve over 50 < y⁺ < 500 — so the 1/κ is a measurement on the drawing rather than the number that was fed in. The viscous sublayer u⁺ = y⁺ comes out rather than being pasted on.

A guess with a constant in it

Prandtl's mixing length is one line — an eddy near a wall can only be as big as its distance from the wall. Integrate it and the whole structure of a turbulent wall profile falls out, sublayer and log region and all. That is a fact about the assumption, and the essay is careful about which.

turbulence · Closure 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 What a probe in a flame sees. Half the time hot light fluid at a hundred metres a second, half the time cold heavy fluid at twenty. That is what intermittency in a jet flame looks like at a point, and it is the simplest field in which the two averages of the velocity are different numbers.

Two averages of one flow

In a flow whose density varies there are two mean velocities, they are both correct, and across a flame they differ by a factor of two. One of them is what a hot wire returns; the other is what every compressible turbulence model is written in; and the mass flux is the single product they agree on.

kinematics · Averaging

Named alongside it

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

TurbulenceReynolds stressAveragingMomentum fluxNonlinearityBoundary layerCompressibilityCorrelationDensityDimensional analysisDirect numerical simulationDissipation

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