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The thread: A smooth picture proves nothing — page 36

Page 36 of 36, continuing through the 320 essays this motif runs through.

320 essays carry this thread — page 36 of 36.

$One rate of production and one shape give the entire transition ramp. The fraction of time a station is turbulent, against distance past the breakdown line, from a direct Monte Carlo of spots at a rate of 0.002 per unit span and time, against Narasimha's 1 − exp(−nσx²/U). The two agree within the Monte Carlo's scatter: at x = 40, 0.403 against 0.416. Half the layer is turbulent at x = 45.4.$ Transition and turbulence

Transition along a plate is a count of spots, and one number gives its length

A boundary layer does not turn turbulent everywhere at some station. Arrowhead patches of turbulence appear at random and sweep downstream, spreading as they go, and a probe at a fixed place reads a square wave. Treat those patches as independent, add one rate of production and one spot shape, and the entire transition zone is a line of arithmetic: the same formula Kolmogorov wrote down in 1937 for crystals growing in a cooling melt. It has only one free number in it, and that one number states the zone's length.

A drop on a plate too hot to touch lives a hundred times longer than one on a merely hot plate. The lifetime of a drop of 2.5 mm radius against the plate's temperature, in contact (nucleate boiling) and on its film. At 130 °C, in contact, 11 s; on its film at 210 °C 1062 s, and at 500 °C 229 s. Where the two branches meet depends on the plate: Leidenfrost's point on polished metal is about 200 °C for water, and on a rough plate can be well above it.

Viscosity

A Leidenfrost drop floats on a bearing it feeds itself

A bead of water on a hot pan vanishes in seconds, unless the pan is much hotter, when it skates about for minutes. Between the drop and the plate is a film of its own steam a tenth of a millimetre thick, and that film is a lubricated bearing with no shaft and no pump: its supply is the drop's own evaporation, which keeps it thin where it is thick and thick where it is thin. Its thickness obeys a fourth root of everything. On a plate too hot to touch, the drop's lifetime is a hundred times what it is on a plate that is merely hot.

One formula draws a smooth layer, a row of cat's eyes, and in the limit a row of point vortices. Streamlines of ψ = ln(cosh y + A cos x) at A = 0.5, over two periods, in the frame of the eyes. The separatrix encloses eyes of half-height 1.32; the peak vorticity at their centres is 3 times the layer's, and on the braids 0.333.

Ideal flow

A shear layer and a row of vortices are one formula, and not one flow

There is an exact steady solution of Euler's equations with one parameter in it, written in a line by Stuart in 1967. At one end it is the smooth shear layer every instability calculation starts from, and at the other it is a row of point vortices, with cat's-eye streamlines between. It looks exactly like the history of a layer rolling up, and textbooks draw it next to one. It cannot be. Each member has a different area of fluid at each vorticity and a different energy, and those are the two things an inviscid flow cannot change.

A cart that leaks straight down keeps its speed, and the rule that says it gains one has lost track of what left. A frictionless cart losing three quarters of its mass at a steady rate, moving at speed 1. With its leak leaving backward at 0.5 of that speed relative to the cart it ends at 1.69. The rule F = d(mv)/dt, read as d(mv)/dt = 0, says the cart reaches 4 whatever the leak does.

Flows and fields

Mass leaving a cart pushes nothing unless it leaves with a speed of its own

A cart leaking straight down keeps its speed, though its mass falls and every first course says force is the rate of change of mass times velocity. Apply that form to the leak and the cart speeds up fourfold, at a rate that depends on who is watching, which no force may. The transport theorem has the right rule in a line: only what crosses the boundary at a speed relative to it pushes. The rule then does four more things, from a cart that slows with a constant mass to a tank whose thrust is exactly twice the missing wall's.

Two equal balloons below the peak even out, past the peak split into a small one and a large one, and at great stretch even out again. Where two balloons that start one hundredth apart in stretch finish, against that starting stretch. From 1.2 they settle at 1.2 and 1.2; from 2, at 1.15 and 2.44; from 3, at 1.11 and 3.75; from 5.5, at 5.5 and 5.5. Two balloons of different size can then sit at one pressure, and no flow in the tube changes that.

What is taught wrongly

Two connected balloons part over a range of stretch, and even out on either side

Air flows from high pressure to low until the pressures match, and everyone concludes that two connected balloons end up the same size. Soap bubbles famously do not. Rubber does something stranger than either: a pair evens out at low stretch, parts into one large and one small balloon over a wide middle range, and evens out again at high stretch. Equal pressure is all the air ever asks for, and one rubber curve lets the same pressure be held at three different sizes. What the pair does also depends on how it was filled.

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