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The thread: One number decides the regime — page 18

Page 18 of 34, continuing through the 305 essays this motif runs through.

305 essays carry this thread — page 18 of 34.

Drop deformation in simple shear, against the capillary number. The shape model's steady deformation for six viscosity ratios. Every curve is linear in the capillary number at small Ca — which is Taylor's result — and every one of them saturates, at 5/2(2λ+3), because the shear's own rotation turns the drop out of the stretching direction. A drop in simple shear cannot be deformed beyond that however hard it is sheared.

The number that cannot break a drop

The capillary number sets the stress that stretches a drop against the stress that holds it round, and it predicts the deformation beautifully. It cannot predict the breakup, because above a viscosity ratio of about four a drop in simple shear cannot be broken at any shear rate — and the theory that says the ratio hardly matters is the same theory that gets the deformation right.

The mean flux, flat across the inertial shells and equal to the dissipation. The time-averaged transfer out of the first n shells, computed as the rate at which the nonlinearity changes their energy rather than from a remembered formula. It is constant to a tenth across the middle of the ladder and equal to the dissipation, which is the cascade — and it is an average.

Transition and turbulence

A flux that runs both ways

The cascade is a statement about a mean. Kolmogorov's four-fifths law fixes an average and the constant flux through the inertial range is an average, and neither says anything about what the transfer is doing at any instant — which turns out to be running backwards a substantial part of the time.

Two different moments of one distribution. The permeability and the specific surface of a log-normal bundle, against the width of its pore-size distribution at a fixed median. The permeability rises by five orders because it is a fourth moment and the widest tubes dominate it; the surface falls because it is a first moment and the narrowest tubes dominate that.

Fluids at work

A permeability that is only the geometry

Kozeny–Carman says that a porous medium's permeability follows from its porosity and its specific surface. Both are real, both are exactly measurable, and they do not determine the answer: forty-nine tube bundles built with identical values of each span a factor of eighty-two in permeability.

Identical lift at every altitude, and the skin load nearly doubles. The same wing at the same dynamic pressure at seven altitudes. The lift is identical at every one of them — the flat line, computed from the absolute pressures rather than assumed, because a net force cannot depend on where the pressure datum is set. The load on a vented panel is identical too, at 4.86 kPa, because both sides of it moved together. The load on a sealed panel with 75.26 kPa inside is not: it is 28.27 kPa at sea level and 60.84 kPa at twelve kilometres. The datum cancelled in the force and it is one of the two numbers in the stress.

What is taught wrongly

A force forgets the datum, a stress cannot

Nothing sucks, because the pressure datum cancels — the normals of a closed body sum to nothing, so the lift is the same in gauge or absolute pressure. That identity is about a resultant, and it is routinely carried one step too far. The load on a panel of skin has the ambient in it as one of two numbers, not as a datum.

Morison's two terms over one cycle, at KC = 10. The drag term, in phase with the velocity and going as its square; the inertia term, in phase with the acceleration and ninety degrees ahead of it; and their sum, which is what a load cell records. The peak of the total is not the peak of either, and its position in the cycle is the only thing in the record that says how the two are divided.

Regimes and numbers

Long enough to make a wake

A Reynolds number cannot ask whether an oscillating flow gets round a body before it turns and comes back, because it has no time in it. The Keulegan–Carpenter number can, and it decides which of Morison's two terms is the force. What it discards is the phase — and a peak force measurement cannot recover it.

The condensate, and the limit of no friction which does not exist. In a steady state the friction must remove everything the forcing puts in, so the energy is eps/(2 alpha) and the coherent velocity is sqrt(eps/alpha) — exactly a minus one half power, checked to 10⁻¹². As the friction is weakened the condensate grows without bound: the limit alpha to zero is not a flow with a weak condensate, it is a flow with no steady state at all.

Transition and turbulence

Where the inverse cascade stops

Two-dimensional turbulence sends its energy upward in scale, and the upward direction has an end: the box. Without something to remove the energy before it arrives, it accumulates there in a pair of vortices filling the domain, and the limit of no friction has no steady state at all.

So splitting a turn into N ramps costs one over N squared. A twelve-degree compression at Mach 3, done in one ramp and in up to sixty-four. The entropy is N times a cube of one Nth, so it falls as exactly the inverse square of the number of ramps — the measured exponent is −2.00 — and sixty-four ramps cost a two-hundred-and-fifty-sixth of what one costs.

Compressible flow

A compression that costs nothing in the end

Turning a supersonic stream away from itself is free and turning it into itself is not. But the price of a compression is the cube of its strength, so splitting one turn into N turns costs one over N squared — and in the limit the compression is free too.

The junction the minimisation is over. A parent vessel entering from the left and two daughters leaving to fixed points. The radii are settled by Murray's law; what is left free is where the branch point sits, and the cost of the junction depends on it. The point drawn is the one the minimisation finds.

Fluids at work

The angle a junction chooses

Murray's law fixes the radii at a branching vessel and is where every account of it stops. The same minimisation fixes the angles completely — 74.93 degrees for a symmetric bifurcation, a right angle for a vanishing side branch — and it does so as a triangle of forces, with tensions proportional to the squares of the radii.

Four bodies, four drag coefficients, and no flow was solved. Four bodies with their Newtonian drag coefficients, each computed as a quadrature over its own surface with no flow solution anywhere. The cone's answer is exactly 2sin²δ, checked against the closed form to nine decimal places; the flat disc's is exactly 2, since every element of it faces the stream; and the sphere's is 1 against the classical Newtonian value of 1. Every one of those is an integral of one expression over a shape, and none of them required knowing what the air was doing anywhere.

What is taught wrongly

The only theory simple enough to optimise

Whether Newton's sine-squared law is right has two answers — hopeless at the speeds he argued about, nearly exact behind a strong shock. This asks a different question about the same formula. Its pressure depends only on the local surface angle, so a shape's drag is a quadrature rather than a solution, and the best shape can be found by calculus.

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