Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: What is conserved — page 44

Page 44 of 44, continuing through the 394 essays this motif runs through.

394 essays carry this thread — page 44 of 44.

Each oblique shock added before the normal one buys less than the last. The best total-pressure recovery of a train of oblique shocks ending in a normal shock, against the number of oblique shocks, at Mach 2, 2.5, 3 and 4. At Mach 3 one oblique shock recovers 58.1%, two 74.4%, four 88.9% and eight 96.4%; a normal shock alone keeps 32.8%. Compressible flow

The last shock should be the weakest

An intake slows a supersonic stream through a train of oblique shocks and one normal shock, and the rule for laying out the train is that its shocks should be equally strong. Solved exactly rather than quoted, the rule is right for the oblique shocks to four figures and wrong for the last one: the terminal normal shock should be about five per cent weaker than the others, and the turns that give equal strengths are not equal turns but grow along the train.

 The face behind a finite wing carries the share its own shape gives it. The share of the lift leaving through the face of the box 500 spans behind the wing as downward momentum, and the share carried as pressure on the faces above and below, against the box's height over its width, for boxes of thirty-six square spans, with the far-wake share (2/π)·arctan(H/W). A box sixteen times as tall as it is wide sends 0.96 of the weight through the face behind, a square one 0.501, and one sixteen times as wide 0.04. What is taught wrongly

Behind a finite wing the wake face keeps its share

Draw a box round a two-dimensional section and the face behind it carries almost none of the lift, so "the air is thrown down" keeps the wrong term. A finite wing is different. Its trailing vortices carry downward momentum through the face behind it however far back the face is. The share is set by the face's own shape — all of the weight through a tall face, half through a square one, none through a wide one — and the only things a box cannot change are the total and the induced drag.

 A slender nose at incidence makes most of its vorticity pointing along the flow. The share of the wall's vorticity source that points along the local surface flow, integrated over the region where the flow accelerates, against incidence, for prolate spheroids of fineness 1 (a sphere), 2, 4 and 8. A sphere makes none at any incidence. At 10° a body of fineness 2 makes 20% of its source streamwise, fineness 4 55% and fineness 8 81%, with no sweep anywhere. Flows and fields

A nose at incidence makes its vorticity along the flow

A wall makes vorticity along its isobars, and on a swept wing that sends part of it along the flow. A fuselage at a few degrees of incidence has no sweep anywhere, and still most of the vorticity its wall makes points along the flow — a fifth for a body twice as long as it is wide, four-fifths for one eight times — because at incidence its pressure varies round the body faster than along it. A sphere, at any incidence, makes none. The lee vortices begin at the wall.

 Below a specific speed of a half, a pump's losses are its discs and its narrow passages. Each loss as a share of the shaft power against the specific speed, for pumps of 40 m head at 1450 rpm, so that every one has the same impeller diameter and only the flow changes. At Ωₛ = 1.0 the fixed share is most of the loss and the efficiency is 0.902; at 0.3, disc friction is 7.65% and passage friction 5.32%, and the efficiency 0.777; at 0.15 the two together take 37.9%. Fluids at work

Two small pumps carry two big discs

Split a pumping duty between two pumps of the same head and speed and each one keeps the big pump's impeller diameter, because the head fixes the tip speed. So the pair spins two full-size discs in their casings for the flow one disc used to carry, and the efficiency penalty that a station comparison assumed to be two points is four. That moves the lift at which splitting saves energy from 44 per cent to 65, and below a specific speed of about a half it never does.

 A bending-moment wing has no best span until its tips are forbidden to push down. The least induced drag for the elliptic wing's lift and bending moment, against span, for a spar sized by its root moment and one sized by the moment along its whole span. Free, both fall without limit: at 1.7 times the span the root-sized optimum is 0.815 of the elliptic drag and the other 0.79, with the tips carrying downloads. With no section allowed to push down both stop where the free loading's tip load first reaches zero: at 1.225 and 0.8889 for the whole-span measure, and near 1.33 and 0.84375 for the root moment. Longer spans then leave their tips unloaded and gain nothing. Circulation and lift

A bending moment never chooses a span

Hold a wing's lift and its bending moment and ask for the least induced drag, and the answer does not stop: the span grows for ever and the tips push down. Both celebrated answers — Prandtl's bell at √(3/2) of the span for 8/9 of the drag, and the root-moment wing at 4/3 for 27/32 — are the span at which the least-drag loading's tips stop lifting. The structure sets how much is saved; a condition nobody states sets where the saving ends.

 A free jet keeps its momentum; a wall jet keeps the flux of its momentum flux. Each quantity against distance, relative to its value at 2 cm, on logarithmic axes. The free jet's momentum flux is constant and its volume flux grows as x^(1/3). The wall jet's momentum flux falls as x^(−1/4) — the wall's shear takes it — while its volume flux grows as x^(1/4), and the one thing it conserves is Glauert's ∫u(∫u²)dy, the flux, through each height, of the momentum flux carried above it. Viscosity

A jet along a wall keeps the flux of its momentum flux

A jet in open fluid keeps its momentum and gathers fluid as it goes. Blow the same jet along a wall and the wall takes its momentum away as drag, so it keeps neither. What it keeps is stranger: the flux, through each height, of the momentum flux carried above that height. That one invariant fixes everything else — the jet thickens as the three-quarter power of distance, slows as the inverse square root, and gathers fluid as the fourth root, more slowly than the free jet's cube root.

 At Mach 3 the cowl caps external compression at 88 per cent. Best recovery at Mach 3 against the number of oblique shocks, free and with the total turn capped at the free stream's detachment angle, 34.07°, which is as far as the cowl's outer surface can turn. Free, six shocks reach 94.2%; capped, 86.7%, and no number of shocks, nor a continuous isentropic ramp turning the same angle, can pass 88.5%. Compressible flow

The cowl caps what the ramps can buy

Every ramp in front of a supersonic intake turns the flow towards the cowl, and the cowl's outer surface has to turn the free stream at least as far. Past the free stream's own detachment angle it cannot, and that one angle caps every external intake: at Mach 3 no number of ramps, nor a smooth ramp, recovers more than 88 per cent, and the shock that is left at the throat is the strongest in the train. Above Mach 3 the cap falls fast, which is why fast intakes compress inside.

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