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The thread: One number decides the regime — page 33

Page 33 of 34, continuing through the 305 essays this motif runs through.

305 essays carry this thread — page 33 of 34.

Six nozzles, six pump curves, and where each one is best. The head ratio a water jet pump delivers against the flow ratio it entrains, for six area ratios from a narrow nozzle to one filling four-fifths of the throat. A wide nozzle makes a tall, steep curve that is finished at a small flow; a narrow one makes a low, long curve. The dots are each curve's best-efficiency point. Nozzle, suction, throat-friction and diffuser losses are included at borrowed representative values, and the mixing loss is computed.

The nozzle that is best at one thing

Put the four losses back into a jet pump and three questions get three answers. The most head comes from a nozzle four-fifths of its throat, in closed form; the best efficiency from one a quarter of it; and the most flow from whichever nozzle is smallest, because flow has no optimum at all.

The induced velocity, after a step in thrust. A rotor's induced velocity following a thirty per cent increase in thrust applied at fifty milliseconds. It does not jump: the air the disc has to accelerate has an apparent mass, and the response is a first-order climb to the new momentum-theory value.

Circulation and lift

The inflow that takes time to arrive

Momentum theory gives a rotor's induced velocity from its thrust, instantly. It does not arrive instantly: the air the disc has to accelerate has a mass, and the response is a first-order climb with a time constant of 33 milliseconds — a twentieth of the time the wake itself takes to convect a radius.

One curve from two to one, with four thirds somewhere in the middle. The ratio of the transverse second-order structure function to the longitudinal one, against separation, at three Reynolds numbers. Every value on every curve follows from the longitudinal function alone by a relation with no dynamics in it. It is exactly 2 where the field is smooth, exactly 1 beyond the correlation length, and it passes through four thirds on the way — but it passes through rather than resting there, and how nearly it rests is the whole of what a Reynolds number buys.

Transition and turbulence

A relation with no turbulence in it

Isotropy and incompressibility alone fix the transverse structure function from the longitudinal one. Divide the relation through and it says the ratio of the two is one plus half the local slope — so the exponent everybody measures as 0.70 and the ratio everybody measures as 1.35 are one measurement, and a model spectrum with no intermittency in it produces both.

Six that are symmetries and five that look like them. Each transformation applied to an exact solution, with the Navier-Stokes residual recomputed from the transformed field by finite differences — nothing differentiated by hand. The six symmetries leave the residual at the differencing floor, a few parts in 10^8. The five near-misses leave between 0.048 and 4.3, which is six to nine orders of magnitude larger. The gap is what makes this a test rather than an illustration: a transformation that is nearly a symmetry does not exist here, and every one of the five is something a reader might reasonably believe.

Viscosity

Why the list is this long

Every textbook list of exact solutions of the Navier–Stokes equations is about a dozen long, and the usual explanation is that the equations are hard. It is not the reason. A similarity reduction is a solution invariant under a subgroup of the equations' own symmetries, so the catalogue of possible reductions is the catalogue of subgroups — and that is a finite, countable object.

The characteristic gets a wall, and the wall does not care about the discharge. One jet pump's characteristic, at an area ratio of 0.275, with the flow ratio at which its throat entry reaches vapour pressure drawn for three values of the cavitation parameter σ = (Pₛ − pᵥ)/(Pₘ − Pₛ). Left of a wall the machine runs on its curve. At the wall no lower discharge pressure raises the flow: the head ratio can fall to zero along the vertical and the flow ratio stays where it is. The wall's position contains the nozzle and suction losses and nothing downstream of the throat entry.

Fluids at work

The wall the suction puts in the curve

A liquid jet pump's lowest pressure is where the entrained stream enters the throat, and when that reaches vapour pressure the pump curve stops being a curve. The flow ratio freezes at a value no lower discharge pressure can move — and raising the motive pressure, the obvious cure, brings the wall closer.

The kept transport spirals into nothing as the sea deepens. The net Lagrangian transport as a vector, scaled on the Stokes transport, traced as the water depth increases from a quarter of an Ekman depth to eight, for an 8-second swell with an eddy viscosity of 0.01 m²/s. Shallow water keeps the whole transport pointing with the waves, at the right-hand end. As the sea deepens the vector shortens and swings to the right, crosses the across-wave axis near two Ekman depths, and winds into the origin, which is the open ocean's exact cancellation.

Flows and fields

The floor that gives the drift back

In the open ocean the Coriolis force drives a current that cancels a swell's Stokes transport exactly. Over a continental shelf the sea floor holds a stress, and whatever it holds is transport the rotation does not take back. How much survives depends almost only on the depth in Ekman depths; which way it points depends on the wave.

What is taught wrongly

The spot a local theory cannot see

Newtonian theory gives every panel of a hypersonic vehicle a pressure set by its own angle to the stream, and no panel more than the stagnation pressure behind a normal shock. Let a shock from one part cross the bow shock of another and a supersonic jet forms that reaches the surface through weaker shocks. At Mach 8 it stagnates at 8.6 times the ceiling — and the worst amplification at every Mach number is close to the Mach number itself.

One number decides which pulse grows. The pressure pulse and the flow pulse at the far end of the tube, each as a multiple of its value at the entrance, against the load's reflection coefficient. A load that reflects pressure with the same sign — a stiffer or narrower continuation — amplifies the pressure pulse and damps the flow pulse. One that reflects it inverted — a wider continuation, or many branches — does the opposite. With no reflection both fall slightly, by the wave's own attenuation. The two curves cross near Γ = 0 and pull apart on either side.

Regimes and numbers

The pulse that grows as it leaves the heart

The pressure pulse measured at the wrist is larger than the pulse in the aorta that drives it, and the flow pulse is smaller. Nothing downstream is pumping. A wave reflected from the end of an elastic tube arrives back in step with the outgoing wave near the end and out of step near the start, and a single number — the reflection coefficient — decides whether it is the pressure or the flow that grows.

What the rays say the edge is: nearly as loud, and then nothing. The overpressure across the carpet relative to the value under the track, from ray-tube spreading alone, at three Mach numbers from 15 km. It falls gently and then stops: at Mach 1.8 it is 0.92 ten kilometres out, 0.77 at twenty-three and 0.65 on the last ray, 36 km out, with silence beyond. Geometrical acoustics says the carpet ends at a cliff, and the boom at the edge of a real carpet fades as a rumble. The cliff is what the model says; it is also where the model stops.

Compressible flow

The edge is a rumble, not a quieter bang

The rays that reach the outer half of a sonic-boom carpet arrive nearly horizontally, having travelled almost three times as far as the one under the track. Ray theory says they still carry two-thirds of the overpressure, right up to a line beyond which there is nothing. Neither half of that is what is heard — which is the useful result, because it says the edge's loudness is not a ray quantity at all.

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