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

The thread: The exact theory is wrong — page 22

Page 22 of 38, continuing through the 335 essays this motif runs through.

335 essays carry this thread — page 22 of 38.

Three answers to one question: what happens after a body is jerked into motion. The force following a step change in a body's velocity, for three models. The ideal one is a spike at the instant and nothing afterwards. The viscous one falls as the inverse square root of time and never reaches zero. The compressible one holds while the signal is still crossing the body and then settles. Ideal flow

The theory with no memory in it

Laplace's equation has no time in it, so an ideal flow's response to a body being jerked into motion is instantaneous and complete. Its indicial kernel is a spike and nothing afterwards. Beside it sit the two kernels that are not, and the comparison says which ingredient every memory in this collection came in through.

 Where the eddies outconduct the molecules. The ratio of turbulent to molecular heat diffusivity across the pipe at Reτ = 2000, for five fluids, on logarithmic axes. Wherever it is above one the eddies carry more heat than conduction does. For air it passes one inside the buffer layer and reaches 135; for water, earlier and higher. For liquid sodium it never reaches one anywhere: at its peak, halfway to the axis, the eddies carry just over half what conduction carries, and the temperature profile is set by conduction across the whole pipe. Regimes and numbers

The heat the eddies do not carry

In a laminar layer the temperature and the velocity have different thicknesses in every fluid but one. In a turbulent pipe the eddies carry both, and the difference nearly vanishes — for air, water and oil alike. It does not vanish for a liquid metal, whose molecules conduct heat faster than the eddies can, and the boundary between the two behaviours is a Péclet number of about four hundred at every Reynolds number.

 The plateau everybody looks for is a summit, and a low one. −Dₗₗₗ/((4/5)εr) against separation in decaying turbulence at five Taylor-scale Reynolds numbers. None has a plateau at one. Each rises through the viscous range and turns over, peaking at 0.49, 0.63, 0.75, 0.85, 0.90 for Reλ = 50, 100, 200, 500, 1000. A measurement of ε that takes the largest value of this curve as four-fifths reads each of those shortfalls as a smaller dissipation. Transition and turbulence

The decay inside the four-fifths law

The four-fifths law gives the dissipation of turbulence from one measured moment with no constant in it, which makes it the obvious way to measure how the dissipation coefficient travels during a decay. But the law is exact only in a limit, and a decaying flow is not in it. The decay itself takes a quarter off the moment at the Reynolds numbers grids reach — and the bias moves as the flow decays, by a sixth, in the direction opposite to the effect being looked for.

 Three lobes, then a filament. An ellipse of aspect ratio 4 with a three-lobed bump of three thousandths, as contour dynamics carries it, drawn in the frame turning with the undisturbed ellipse at t = 0, 30 and 42. By t = 30 the bump has grown to a visible three-fold asymmetry — one end fattened, the other thinned — and by t = 42, about a turn and a tenth of the ellipse, the thinned end is being drawn out into a filament. The march is stopped there, while the area is still conserved to a few parts in a thousand; resolving the filament needs a contour that adds nodes, which this one does not. Ideal flow

Past three, an ellipse is a shear layer

Kirchhoff's elliptical vortex turns for ever without changing shape, and Love showed in 1893 that it stops being stable at an aspect ratio of exactly three. Computed, that threshold turns out to be the first of a sequence — a new way of coming apart every one and a half aspect ratios — and the sequence ends somewhere recognisable. A long enough ellipse is a strip of vorticity, and it comes apart the way a shear layer does, at a rate Rayleigh found for the strip.

 Below a Stokes number of ten a ball does not come back. The restitution of an impact through a liquid film, as a fraction of the same impact's dry restitution, against Stokes number. Nothing rebounds below the threshold and the recovery above it is a hyperbola: the restitution, as a fraction of its dry value, is one minus the critical Stokes number over the Stokes number, with that critical value 10.18 computed from the film and the dry restitution alone. The measured curve, drawn beside it, is the same expression with ten in place of that number — and ten is what four decades of viscosity and four materials all give. Viscosity

A ball that bounces in water and not in oil

A squeeze film cannot be closed with any finite energy, so nothing should ever touch anything. A sphere dropped into a tank nevertheless rebounds, and whether it does is decided by a number near ten that four materials and four decades of viscosity all agree on.

 Three different flows, drawn on top of one another. The surface pressure of three parabolic arcs of different thickness, each at the Mach number that gives them the same transonic similarity parameter, divided by the scale the similarity rule prescribes. They lie on one curve. The three solutions were computed separately from an equation with different coefficients in each case, and nothing in the solver knows what the parameter is — the scaled peaks differ by 0.01 per cent. Compressible flow

One curve for every thickness

Near Mach one there is no linear theory, so there is no superposition, no scaling of a solution by the thing that caused it, and no shortcut of any kind. There is exactly one thing left — a change of variables in which the thickness and the Mach number vanish and only their combination survives — and it is worth more than the theory that was lost.

 The line that decides how much a thickener can pass. The batch settling flux against concentration, the same curve the closed column was solved on, with the operating line of a continuous thickener drawn across it. The line runs from the underflow concentration on the axis to the flux the plant can take, and it is placed by one condition: it must touch the flux curve rather than cut it. Where it touches is the concentration at which the thickener is working hardest. Flows and fields

The tangent that sizes a plant

A closed cylinder of settling suspension is a kinematic wave problem with a flux curve. Open the cylinder — feed it at the top and draw thickened slurry from the floor — and the capacity of the whole plant is a straight line laid against that same curve, touching it rather than cutting it.

 Three strokes, and only the flat one is a theorem. Three cycles drawn in the swimmer's shape space: a square, a circle of the same width, and an out-and-back along the diagonal. The first two enclose area and carry the swimmer forward; the third encloses none and carries it exactly nowhere, which is the scallop theorem with no symmetry argument in it. What the third lacks is not a broken symmetry but an interior. Viscosity

A stroke is worth the area it encloses

The usual account of swimming without inertia is a symmetry argument about reciprocal strokes, which says what cannot work and nothing about what does. Draw the stroke in the space of the swimmer's own shapes and the displacement is a line integral — so it is an area, it does not depend on how fast the stroke is played, and the scallop theorem is Stokes' theorem.

 A drag that is nothing until there is something to be discontinuous about. The scaled wave drag of a parabolic arc against the similarity parameter. Above about 1.4 it is not small but zero to five decimal places — d'Alembert's paradox, which the small-disturbance equation inherits. Below it the drag rises by four orders of magnitude over a range of the parameter that corresponds to a few hundredths in Mach number, and the only thing that changed is that the flow now contains a shock. Compressible flow

A drag that needs a discontinuity

A closed body in a steady inviscid flow has no drag, and the transonic equation obeys that as faithfully as any other — until a shock appears inside its own answer. Then the same pressure integral that was zero to five decimal places becomes four decades larger, with no viscosity added and nothing about the body changed.

All themes