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The thread: The exact theory is wrong — page 18

Page 18 of 22, continuing through the 193 essays this motif runs through.

193 essays carry this thread — page 18 of 22.

The same plate borrows more near a wall and less near a free surface. The added mass of a plate closing broadside on a boundary, as a multiple of its free-air value, against the gap in chords on a logarithmic axis, for a solid wall and for a boundary held at constant pressure — a free surface struck quickly, or the edge of an open jet. At a tenth of a chord the wall gives 1.966 and the free boundary 0.677; at 0.035 chords 3.97 and 0.584. The wall's value grows without limit as the gap closes, because the fluid in the gap has to be squeezed out. The free boundary's falls towards exactly one half, because a plate lying on a free surface sets in motion only the half-space below it. Same plate, same fluid, same speed — the boundary decides the sign, through the one thing it is allowed to tell the flow.

The borrowed mass the boundary decides

A body accelerating near a solid wall has to squeeze out the fluid between them, and borrows more mass than it would in the open. The same body accelerating near a free surface, or inside an open-jet wind tunnel, borrows less. The fluid, the body and the speed are identical, and what reverses the answer is the one thing each boundary is allowed to tell the flow.

A 204 m hammer traded for a 8.57 m swing over 299 seconds. The water level in a 10 m surge tank at the end of a 2 km tunnel 3 m across, carrying 2 m/s, after the turbine is shut off at once, with the level measured from the reservoir's. Without friction it rises to V₀√(L Aₜ/g Aₛ) = 8.57 m and swings with a period 2π√(L Aₛ/g Aₜ) = 299.1 s, the integration agreeing with both closed forms. With the tunnel's 5 m of friction the level starts 5 m below the reservoir, peaks at 5.61 m after 98 s, falls to −3.70 m, and decays. The same tunnel shut at its end with no tank would take the Joukowsky rise of 204 m. The tank does not remove the column's momentum; it gives it a free surface to push against, slowly.

Fluids at work

A tank that turns a hammer into a swing

Shut a turbine at the end of a two-kilometre tunnel in two seconds and the valve takes a rise of 256 metres of head. Put a shaft open to the air beside it and the rise is 51, the tunnel never carries the closure as a wave at all, and its water slows instead against a level that climbs for a minute and a half — to a height that is a closed form with the tank's area under a square root.

The excess energy is the energy of the difference, exactly. Add any admissible perturbation to the potential flow and its kinetic energy rises by precisely the energy of the perturbation itself — not approximately, and not to leading order. The measured excess and the perturbation's own energy lie on one another to two parts in 10¹¹ across a sixty-fold range of amplitude.

Ideal flow

How much more than the least

Of all the flows that conserve mass and stay inside the walls, the ideal one carries the least energy. That is a theorem, and the useful half of it is the part nobody quotes: it says by exactly how much every other flow misses, and the answer is the square of how wrong it looks.

In clean water a bubble rises nearly three times as fast as the same bubble in tap water. The terminal rise speed of an air bubble in water at 20 °C against its radius, from buoyancy balanced against drag: with a clean, shear-free surface using Moore's law, and with a surface immobilised by contamination using the rigid-sphere correlation. At 0.3 mm the clean bubble rises at 13.0 cm/s against 6.7; at 0.5 mm at 31.0 against 11.2, a factor of 2.76. Beyond a radius of 0.47 mm the clean bubble's Weber number passes one, its shape flattens, and a spherical calculation stops describing it; that region is shaded. Nothing about the bubble's size, gas or liquid changes between the two curves — only whether its surface can move.

Flows and fields

The vorticity a clean surface cannot refuse

A clean bubble's surface cannot hold a shear stress, and it is easy to conclude that it makes no vorticity. On a curved surface it must carry exactly 2κu — three times the speed over the radius at a sphere's equator, whatever the Reynolds number. That is so much weaker than a rigid wall's that the flow stays irrotational to leading order, and the bubble's drag is the dissipation of that irrotational flow: 48/Re, three to ten times below a rigid sphere's.

Below Thoma's 6.07 m² the governed tank's swing grows; above it, it dies. The tank level after the turbine's power demand drops by two per cent, with a governor holding the power constant, for tanks of 0.7 and 1.3 times Thoma's area of 6.07 m² — a tank 2.78 m across. The smaller tank's swing grows by a factor of 1.47 every 74 s cycle and has reached −12.20 m by 427 s; the larger one's keeps 0.75 of itself every 100 s and is barely visible. Carried on, the smaller tank's run is refused at 794 s, where the head at the turbine has fallen below a quarter of its design value and the governor would be asking for a flow no turbine passes. The instability has nothing to do with the tank's height: it is the governor drawing more water as the level falls, which feeds the swing, against the tunnel's friction, which is the only thing damping it.

Fluids at work

The better tunnel needs the bigger tank

A turbine governed to hold its power opens further when the head at it falls, and draws the tank down harder. That makes it a negative resistance, the tunnel's friction is the only thing damping the swing against it, and so the smallest stable tank grows as the friction shrinks — 2.78 metres across for five metres of friction, 6.09 for one.

A body of no volume with a finite added mass. Thin an ellipse towards a plate and the fluid it displaces goes to nothing while its broadside added mass does not move at all — it stays at πρa² to the last digit. Whatever added mass measures, it is not how much fluid a body carries with it: the ratio of the two diverges as the reciprocal of the thickness, reaching a million at a thickness ratio of a millionth.

Ideal flow

The mass a body has to borrow

Accelerate a sphere through water and it resists as though it were half again as heavy. The half is exact, it is a rational number rather than a measurement, and almost everything a reader infers from it about carried fluid is false.

A cone sends 1.7 per cent of its jet's momentum sideways at 15°. A conical divergent section of 15° half-angle and exit area ratio 25, drawn to scale from its throat to its lip, with its virtual apex to the left. The gas leaves as a source flow: straight streamlines from the apex, each at its own angle, and a Mach number uniform on spheres centred there. On the spherical cap through the lip the flow is normal to the surface and uniform, at Mach 3.925 for a gas with γ = 1.2; its axial momentum is ρV² times the cap's projection onto the exit disc, while the mass crossing it is ρV times the cap itself. The ratio of the two areas is (1 + cos α)/2 = 0.9830, and it is the only thing the cone's shape does to the momentum. On the flat exit plane the flow is not uniform: its edge is further from the apex than its centre, and the Mach number there is higher.

Compressible flow

The jet a cone sprays sideways

The one-dimensional nozzle sends all its gas straight out along the axis. A real divergent section is a cone, and the gas leaves it as a spray of straight lines from the cone's apex. Only the axial part of that momentum pushes, and the share that does is (1 + cos α)/2 — 98.3 per cent at fifteen degrees, 93.3 at thirty — whatever the gas, the Mach number or the area ratio, and it touches the momentum and never the pressure.

$A force ceiling ends the similarity: a breeze makes the boat slower as a share of the wind. Speed made good to windward as a fraction of the true wind, on the best course for each wind, for a rig that is flattened once the righting moment binds and for one that is reefed, with the hull's drag angle held at 6°. Below 3.70 m/s the two are the same boat, and the fraction does not depend on the wind — 1.122 at every speed, which is the similarity the two-angle polar rests on. Above it the flattened rig's fraction falls: 0.918 at 6 m/s, 0.572 at 10 m/s, 0.321 at 15 m/s and 0.169 at 20 m/s. The reefed rig's stays at 1.122, because a reefed sail keeps its least drag angle and the hull's angle is held. Once a force is limited, the triangle is no longer the same shape in every wind.$ Fluids at work

A breeze the boat cannot use

The two-angle polar makes a boat's speed a fixed fraction of the wind, in any wind. A righting moment ends that at 3.7 metres a second on the beat. Past it the crew must spill force, a flattened sail's drag angle climbs, and the best course to windward moves closer to the wind rather than away from it — which 45° + λ/2 cannot say, because λ now depends on the course.

Four solved flows, and the minimum is on the surface in every one. Sampling the whole exterior of each body on a grid and comparing the lowest pressure found there with the lowest found on the surface. The surface wins by a margin that is not marginal — between 0.18 and 0.56 in pressure coefficient — and it wins for a reason rather than by luck: the pressure of an irrotational flow is superharmonic, and a superharmonic function has its minimum on a boundary.

Ideal flow

The lowest pressure is on the body

In an ideal flow the minimum pressure is always on a surface — not usually, not for the shapes people draw, always. The proof is an identity about the velocity gradient, and the identity says exactly which flows are exempt.

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