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

Every essay — page 31

Page 31 of 39, continuing through the fields in the same order.

Flows and fields Ideal flow Circulation and lift Viscosity Regimes and numbers Compressible flow Transition and turbulence Fluids at work What is taught wrongly Series Concepts Regimes Refutations Search

Transition and turbulence

Where the laminar solutions stop being the ones the flow takes, and what can honestly be said about what follows — which is less than the textbooks imply and more than nothing.

Two drag laws, one derived and one fitted. Flat-plate drag coefficient against Reynolds number, laminar and turbulent, on log axes. The laminar curve is Blasius' similarity solution, solved by shooting; the turbulent one is the 1/7-power correlation, which was fitted to experiment and is drawn dashed-in-kind to keep the difference visible. Their slopes differ — −1/2 against −1/5 — so the gap between them widens rather than staying put.

The number that is not a number

Transition Reynolds numbers are quoted to three figures and vary by two decades. That is not sloppiness in the measurement — it is the honest report of a quantity that depends on the laboratory as much as on the fluid, and knowing which part is which decides what may be designed on it.

6 figures A shear layer, and the point of inflection in it. The velocity profile U = tanh y across a layer of finite thickness, with the inflection point located by searching for a sign change in the second derivative rather than by reading it off the algebra. Rayleigh's theorem says an inviscid parallel flow can only be unstable if such a point exists — a necessary condition, not a sufficient one.

A layer with a kink in it

Rayleigh proved in 1880 that an inviscid shear flow cannot be unstable unless its velocity profile bends the other way somewhere. It is one line of algebra, it is necessary and not sufficient, and it ties instability to separation through the sign of a single derivative at the wall.

6 figures Every wavelength grows, and the shortest grows fastest. Growth rate against wavenumber for a vortex sheet of zero thickness and for a layer of thickness δ. The sheet's curve rises without limit, which is the model announcing that it has no shortest scale in it; the layer's turns over and dies at kδ = 1, the neutral wavenumber computed from Rayleigh's equation rather than fitted.

Every wavelength at once

A vortex sheet of zero thickness is unstable at every wavelength, and the shorter the wavelength the faster it grows. The answer has no smallest scale in it, which is not a fact about fluids — it is the model reporting that it left something out.

6 figures 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.

6 figures 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.

6 figures 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.

6 figures 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.

6 figures 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.

7 figures Every disturbance has its own threshold; one of them is lowest. The Rayleigh number at which a disturbance of horizontal wavenumber a becomes neutral, Ra = (π² + a²)³/a². Every wavenumber has a threshold and the layer goes unstable at the lowest of them, which a golden-section search on this curve puts at a = 2.221441 and Ra = 657.5114 — the exact π/√2 and 27π⁴/4 to fourteen digits. Below the curve the layer conducts and nothing moves.

A threshold with a closed form

A layer of fluid heated from below sits still until buoyancy overcomes both diffusions at once, and then it convects. Unlike every other threshold in this field, that one is an eigenvalue with an exact answer, and the answer is 27π⁴/4.

5 figures Three equations, and the set they never leave. The Lorenz trajectory at r = 28, projected on x and z, after the transient has been discarded. It never repeats, never leaves, and never crosses itself in three dimensions. The Lyapunov exponent printed beside it is measured on this system by separating a nearby pair, so the claim of sensitive dependence is a computation.

Three numbers left of a fluid

Saltzman truncated convection to three Fourier modes and Lorenz studied what was left. The result changed science, and it stopped being a description of a fluid at about a fifth of the way to the parameter everybody quotes it at.

6 figures A millionth, doubling every three-quarters of a second. The separation of two trajectories started a millionth apart, on log axes against time. It grows as a straight line until it saturates at the size of the attractor, and the slope of that line is the largest Lyapunov exponent — measured here on the system by renormalising a nearby pair, not quoted. Sensitive dependence is what the straightness of the line means.

A millionth is enough

Two trajectories a millionth apart separate by a factor of e every three-quarters of a second, so a millionfold improvement in the measurement buys about ten seconds of extra prediction. That exchange rate, and not the size of the error, is what limits forecasting.

6 figures The street, as two rows of point vortices. The exact velocity field of a staggered double row of point vortices at the stable spacing ratio, with the row spacing and the circulation of one core printed from a line integral of the field rather than from the number that built it. This is a model of a wake. It contains no body, no viscosity and no mechanism that would shed anything, and the viscous stepper used here does not produce a street at any Reynolds number.

The street this site cannot draw

The alternating wake behind a cylinder is the most photographed structure in fluid mechanics, and this site's solver does not produce one. What can honestly be drawn instead is a model of it — and the model settles one thing exactly, which is the spacing.

7 figures