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

The thread: One number decides the regime — page 35

Page 35 of 37, continuing through the 325 essays this motif runs through.

325 essays carry this thread — page 35 of 37.

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.

 Where the inverse cube stops being an inverse cube. The force resisting a sphere's approach to a plane, against the gap, for four slip lengths. With no slip the force goes as one over the gap and has no limit. Any slip at all bends the curve over: below the slip length the force grows only as a logarithm. The curves separate where the gap falls through the slip length, which is exactly where a measurement has to be made and is why the apparatus has to close to nanometres rather than microns. What is taught wrongly

A length you can only measure by squeezing

No slip is a boundary condition rather than a law, and nothing in the equations requires it. The evidence that it is right for an ordinary liquid to within a nanometre comes from the one experiment whose small length is chosen rather than given — a squeeze film closed to tens of nanometres.

 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.

 The spreading per metre has a best speed; the spreading per second has none. The plate height — how much variance a slug picks up per unit length travelled — against the velocity, for four retention factors. Each curve falls as one over the velocity, because molecular diffusion along the tube matters when the flow is slow, and rises with it, because that is Taylor's dispersion. The minimum is what a chromatograph is run at, and Taylor's effective diffusivity has no such point anywhere. Regimes and numbers

The best a tube can do is its own radius

Taylor's dispersion coefficient rises without limit with velocity, so there is no best speed to run a tube at. Ask instead how much a slug spreads per metre travelled and an optimum appears at once — and the spreading there is the tube's radius over the square root of three, with no diffusivity and no velocity left in it.

 The barrier is pinched to nothing in one direction. Where the inverse cascade is stopped, drawn in the plane of wavenumbers. The curve is the scale at which a Rossby wave oscillates as fast as an eddy turns over, and inside it the cascade cannot proceed. It is not a circle: it closes to a point on the axis of modes with no variation in longitude, so the cascade runs on unimpeded towards the largest scales in exactly one direction — and what it makes there is a band. Transition and turbulence

A cascade that arrives as stripes

Two-dimensional turbulence sends its energy upward in scale and, on a plane, piles it into a pair of vortices filling the box. On a rotating planet one extra term in the vorticity equation blocks that in every direction but one, and what arrives at the large scales is not a vortex at all but a set of bands.

 The current that runs across the river. The transverse velocity through the depth in a bend, beside the downstream velocity that causes it. The surface water is thrown outward and the bed water is driven inward, the two cross at about half the depth, and the whole profile carries no net discharge — which it cannot, because the cross-section has to pass what it is given. What it does carry is the bed load, inward. Fluids at work

The current that runs across the river

The standard channel calculation is one-dimensional, and a river is not. Put a bend in it and the surface tilts by a few centimetres, which is invisible — and a helix forms, a few per cent of the main flow, which decides where the river will be in a hundred years.

 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