Theme

The thread: A smooth picture proves nothing — page 6

Page 6 of 20, continuing through the 172 essays this motif runs through.

172 essays carry this thread — page 6 of 20.

Three accounts of lift, one of them tuned to be exactly right at five degrees. Thin-aerofoil theory, which is the answer; Newtonian impact theory with its constant tuned so that it passes exactly through the truth at five degrees; and the equal-transit story, which has no free constant and sits along the bottom. At the tuning point the tuned model and the truth are indistinguishable, and no measurement there separates them.

A right total from a wrong picture

Newtonian impact theory has a free constant in it. Tuned at five degrees it reproduces a NACA 2412's lift exactly, and no measurement at five degrees can tell it from the truth. What separates them is the derivative — a lift-curve slope 2.8 times too steep — which is a second constraint rather than a better one.

Where the first grid point may go. The error in the friction a wall treatment infers, against the height of the first grid point, at Re_τ = 20,000. The velocity fed to each treatment is the closure's own, so what is plotted is the modelling of the boundary condition with nothing else in it. The log-law function is exact between y⁺ 30 and 10261 and is 64 per cent wrong at y⁺ = 1; the sublayer treatment is exact below y⁺ = 5 and hopeless above it; and the blend that most codes ship is within a few per cent everywhere and exact nowhere.

Transition and turbulence

What a code says to a wall

A calculation that cannot afford to resolve the viscous sublayer has to tell the wall something else instead, and what it tells it is the law of the wall — an asymptotic result, applied at one grid point, on the assumption that the point lies in a region the calculation has not checked exists. Where it does, the answer is exact. Where it does not, the friction is out by tens of per cent, and refining the grid makes it worse.

One spectrum, two fields. Two one-dimensional fields built from the same amplitudes and different phases. Their energy spectra are identical to the last bit, their two-point correlations agree to 2e-15, their second-order structure functions are the same function — and no measurement of either kind can tell them apart. Everything that distinguishes them is in the phases, which is where the cascade lives.

Transition and turbulence

The moment a spectrum cannot hold

A spectrum and a two-point correlation are the same object, transformed, so everything one of them records the other records too. Give a field the exact Kolmogorov amplitudes and independent phases and every second-order measurement comes out right — while the cascade, which lives in the phases, is not there at all.

How big before gravity shows. A drop's height over its width against the Bond number, which is the ratio of its weight to the force its own skin can supply. The number is one where those two are equal, and by then the drop is a bun: it is one per cent from a ball at Bo = 0.0079, five per cent at 0.054 and ten at 0.13. Every one of those is below one, and the first is below it by a factor of a hundred and twenty-six.

Regimes and numbers

The size a drop is allowed

The Bond number sets a drop's weight against the force its own skin can supply, and it is one when they are equal. By then the drop is a bun — it is a per cent from being a ball at Bond number 0.0079, which is a water drop half a millimetre across.

The rectangular wing stalls at the root; the tapered one stalls at the tip. Section lift coefficient across the half span for three planforms, each drawn at the wing incidence where its own worst section first reaches 1.5. A rectangular wing's peak is at the root, which is where a designer wants it: the stall starts inboard, ahead of the ailerons, and the pilot feels it. A tapered wing's peak has moved out to 0.62 of the semi-span — over the ailerons — because taper takes chord away from the tip faster than it takes circulation. The elliptic wing is the degenerate case: every section reaches the limit at once, which is elegant and is the worst possible stall behaviour.

Circulation and lift

Which part of a wing stalls first

A wing has one lift coefficient and its sections have a hundred, and no section is at the wing's. Which of them runs out first is decided by the planform, it decides whether the pilot keeps the ailerons, and the standard fix costs span efficiency in exact proportion to how much of it is applied.

Two divergence-free fields with the same boundary conditions. On the left, the exact potential flow round a cylinder moving through fluid at rest. On the right, the same flow with a divergence-free eddy added — one that has no normal velocity on the body or on the outer circle, so it changes nothing about what crosses a boundary. Both fields conserve mass, both satisfy the wall condition, and only one is the flow. Nothing in the drawing says which.

Ideal flow

The flow with the least energy in it

Draw a flow that conserves mass and does not go through the walls, and it will look exactly like a solution. There are infinitely many of them and one is the flow. What separates it from the others is not visible anywhere in the picture — it is a number, and the number is an energy.

A material region, and the dye that stays inside it. The same fluid at four times, carried by an unsteady straining flow whose strain rate oscillates. The outline is a circle of the fluid at the first instant, tracked by integrating the velocity field; the shading is a blob of passive dye. The region is stretched to nearly seven to one and its area is unchanged to fifteen decimal places, because the flow is incompressible. The amount of dye inside it is unchanged to thirteen, because the dye is carried by the same fluid.

Flows and fields

A rate of change that will not hold still

Three boxes drawn in one flow at one instant give three different answers to how fast the dye inside them is changing — one falling, one falling twice as fast, one rising. All three reconcile with a single material rate, and that rate is zero.

What is taught wrongly

The effect that is real, and where it stops

The Coandă effect has a genuine mechanism, and it is the same equation that makes the suction on a wing. What separates the two is where the integral comes from: a wall jet generates all of its pressure deficit inside a layer two per cent of the radius thick, and a cylinder in a stream needs a third of a radius for half of it.

The tunnel makes the stream 7.4% faster. The flow past a cylinder between two walls, solved from the closed form of an infinite image row rather than from a truncated sum. The walls are streamlines exactly — that is what the images are for, and the normal velocity on them is zero to the last bit — and the flow beside the body is squeezed between the body and the wall, which is the whole of the blockage effect. The stream at the model is 7.40% faster than the speed the tunnel's own instruments report far upstream.

Fluids at work

The instrument in the answer

A model in a wind tunnel is not a model in the sky. The walls are supplied by an infinite row of reflections, the stream at the model is faster than the tunnel's own instruments report, and the correction is not an empirical fudge — it is a series with a closed form.

All themes