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The thread: The exact theory is wrong — page 34

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335 essays carry this thread — page 34 of 38.

Along the carry's ridge the run hardly moves the best spin, because every drive on it lands alike. The total distance of a 70 m/s drive, carry and run, at each spin rate launched at the angle that carries furthest for that spin, on a firm, a medium and a soft fairway, with the carry alone. The carry peaks at 234.6 m at 2273 rpm; the total along the ridge peaks at 1728 rpm on the firm turf, 274.3 m, 2052 rpm on the medium turf, 249.3 m, 2375 rpm on the soft turf, 239.9 m. Every drive on the ridge lands between 45.2° and 49.2°.

The run moves the best drive below the ridge that carries furthest

A drive's carry is greatest along a ridge of launches on which more spin trades for a lower angle at almost no cost. Every drive on that ridge lands at about the same steep angle, so the run after landing hardly chooses between them. What the run rewards is leaving the ridge: less spin and a launch several degrees lower, a lower flight that lands flatter, gives up thirteen metres of carry and gets back twenty in the run. On an ordinary fairway the best drive in all lands near thirty-five degrees, at every ball speed; on a firm one the model's ground stops being a turf anybody has measured before the run stops paying.

The paint pushes a scraper off the wall, and roughly as one over the square of the angle it is held at. The force the fluid exerts on a still blade at angle α to a wall sliding into the corner, per unit span over μU ln(R/ε): its component pushing the blade away from the wall and the size of its component dragging it along with the wall, towards the apex. At 10° they are 196 and 22.9; at 45°, 8.56 and 4.88; at 90°, 1.36 and 2.14; at 150°, 0.0757 and 0.924.

Viscosity

A scraper held flat is pushed off the wall it scrapes

A corner stirred from far away fills with an endless sequence of eddies. Drive the corner at the corner instead — let one of its walls slide under a still blade, as paint slides under a scraper — and the eddies vanish: every streamline turns once, and the stress rises into the corner without limit. G. I. Taylor solved it in 1962. The paint pushes the blade away from the wall with a force that grows as one over the square of the blade's angle, so a blade held flat rides up on what it was meant to remove, and the force is finite only because something too small to appear in the problem ends the corner.

$Pressed into their grooves, the menisci cost a striped surface at most a quarter of its slip before it floods. The slip length along the stripes as a share of the flat meniscus's, against the liquid's pressure over the gas's as aΔp/γ, for gas fractions of 0.5, 0.8 and 0.95; the groove floods when the arc meets a vertical wall at its advancing angle — 110°, 120° and 150° at pressures 0.342, 0.5 and 0.866. 0.5: 0.93, 0.903 and 0.84 kept at those floods; 0.8: 0.913, 0.879 and 0.804 kept at those floods; 0.95: 0.888, 0.845 and 0.752 kept at those floods.$ Flows and fields

A bowed meniscus keeps most of its slip until the groove floods

The slip lengths of a striped superhydrophobic surface are computed with flat gas surfaces. Press the liquid against the texture and every meniscus bows into its groove as an arc pinned at the solid's edges, and the slip falls. It falls by less than the obvious worry says: at the pressure where a groove wall advancing at 120° lets the liquid in, a surface four-fifths gas still keeps 88 per cent of its slip, and even at a 150° wall's flood three-quarters remains. A striped surface does not lose its slip gradually under pressure. It keeps most of it, and then loses all of it when the groove floods.

$At the cheapest fork each daughter runs at the parent's velocity along it, so the split loses only the sideways part. The power lost at a symmetric split, as a multiple of the flow times the parent's dynamic pressure, against the daughters' radius as a fraction of the parent's, at each of three cheapest angles. For laminar, k = 3 the angle is 37.5°, the curve's minimum is at a ratio of 0.7937 — the cheapest ratio — and its value is sin²θ = 0.37; for smooth turbulent, 27/11 the angle is 28.4°, the curve's minimum is at a ratio of 0.754 — the cheapest ratio — and its value is sin²θ = 0.226; for rough, 7/3 the angle is 25.1°, the curve's minimum is at a ratio of 0.743 — the cheapest ratio — and its value is sin²θ = 0.18.$ Fluids at work

The cheapest fork loses only its sideways speed, and pricing it closes the fork

A turbulent network built at least cost forks at 28.4°, with daughters three quarters of the parent's radius, and those numbers priced the pipes and never the split. Price the split as Borda priced a sudden widening and something exact appears: at the cheapest fork each daughter already runs at the parent's velocity component along it, so the junction loses nothing but the sideways part of the parent's speed, and its loss cannot be reduced by resizing the daughters. It can be reduced by narrowing the fork. A split costing under two per cent of the bill moves the cheapest angle by seven degrees, because the pipes' bill is flat at its best angle and the split's loss is not.

A symmetric panel with no spring stiffness washes out exactly as each lift coefficient wants. The outer panel's washout against the wing's lift coefficient over the trainer's sortie, with a spring stiffness of 0 cruise dynamic pressures: each lift coefficient's own best, 5.31° per unit of it; the best fixed washout, 2.31°; a symmetric panel at its best hinge and preload, 1.34% above an elliptic wing; and a panel with a cambered trainer section's moment of −0.05, 9.31%. The fixed twist spends 2.65% and the floor is 1.34%.

Circulation and lift

A pivoted wingtip follows the lift only with no spring stiffness and no camber

The washout a trainer's sortie wants grows in proportion to the lift coefficient, and the twist a wing gets for free follows the dynamic pressure instead. A tip panel free to pivot on a hinge ahead of its own aerodynamic centre can do better, and on one condition it does exactly what is wanted: held by a constant torque and carrying no camber moment, its own lift coefficient stays in proportion to the wing's, and every segment of the sortie flies at its own best twist. Every unit of spring stiffness gives part of that back, a cambered section's moment turns the panel into the wrong law altogether, and the free panel sheds a quarter of a gust's lift for nothing.

Bluntness delays a cone's transition by a factor that peaks, and the correlation's Mach exponent sets how high. The distance at which a blunted 8° cone's layer transitions, over the sharp cone's 0.3 m, against the nose Reynolds number, at Mach 6 and 2.6·10⁷ per metre, for three correlations of the form Reθ/Mₑⁿ = C. Reθ alone: a largest delay of 4.96 at a nose of 43.9 mm, nose Reynolds number 1.14·10⁶; Reθ / √Mₑ: a largest delay of 2.57 at a nose of 20.1 mm, nose Reynolds number 5.23·10⁵; Reθ / Mₑ: a largest delay of 1.41 at a nose of 7.29 mm, nose Reynolds number 1.9·10⁵. Past a nose Reynolds number of 8.81·10⁵ the Mach-ratio criterion transitions within ten nose radii of the tip, where the model's edge is least trustworthy.

Compressible flow

A blunted nose delays its cone's transition only so far, and the correlation decides how far

A blunted tip leaves a sheath of hot, slow gas along a hypersonic cone, and the boundary layer growing inside it sees a lower Reynolds number and a lower Mach number than a sharp cone's. That is why bluntness is said to push transition back. Read the layer's edge through the sheath and the push has a limit: the delay grows with the nose, peaks, and falls again once the transition point lies inside the sheath. How large the peak is depends on whether the transition correlation divides by the edge Mach number — five times the sharp distance at Mach 6 for the momentum-thickness Reynolds number alone, 1.4 for its ratio to the Mach number — and every peak sits forty to sixty-five nose radii from the tip.

The lagged layer arrives at the release with the worse shape and leaves it with the better one. The shape factor of the layer from its flat-plate run through an outer flow decelerating from x = 1 m and released at 1.6 m into a flat one: Green's lagged layer and the layer whose entrainment follows its own shape. At the release the lagged layer's shape factor is 1.66 against 1.57; 11.8 thicknesses later it is below the shape-only layer's, and at the end, 4.5 m, it is 1.282 against 1.301.

Transition and turbulence

A layer released from a gradient overshoots the shape it left

A turbulent layer that meets an adverse gradient suddenly entrains only half what the gradient asks, because its large eddies were made upstream, and it separates sooner than a layer that answers at once. Release it — let the outer flow stop decelerating, as at a diffuser's exit or a wing's trailing edge — and the same lag runs the other way. The eddies made in the gradient keep entraining after it has gone. The layer arrives at the release with the worse shape and, twelve thicknesses later, has the better one, its shape factor falling past the flat plate's and its friction rising past the shape-only layer's.

Free in heave, the craft is an oscillator whose spring is its own cushion: it follows long swells and flies level over short ones. The craft's heave over the swell's amplitude against the encounter frequency over its own natural frequency, 2.07 rad/s with 0.69 of critical damping, for swells 30 to 800 m long; the curve is a linear oscillator driven through its spring with that damping. 30 m: 0.042 at 5.76; 50 m: 0.0972 at 3.58; 200 m: 0.723 at 1.03; 800 m: 1 at 0.324.

What is taught wrongly

Free to rise and fall, a ground-effect craft rides its own cushion over the swell

A wing held still over a swell sees its gap change and its lift swing with it. A real ground-effect craft is not held: its weight sits on that lift, and ground effect, whose lift rises as the gap closes, gives it a height stiffness — a spring made of its own cushion — while its vertical speed changes its angle of attack and damps it. Free in heave it is an oscillator driven through that spring. Over long swells it follows the sea and keeps twice the held wing's clearance; over short ones it flies level and rides a dozen centimetres higher, because the cushion's swing is convex; and it comes closest to the crests over the swells it meets near its own natural frequency, never as close as the held wing does.

Stalled from its leading edge, a flat plate keeps a quarter of its attached lift slope. The normal force on a flat plate over the dynamic pressure on its chord against incidence: Kirchhoff's stalled flow, by quadrature through its hodograph, with Rayleigh's closed form 2π sin α/(4 + π sin α); and the attached flat plate's 2π sin α. At 5° the stalled plate carries 0.128 against the attached 0.548; at 20°, 0.423; at 90°, 0.8798, Kirchhoff's drag.

Ideal flow

A stalled plate keeps a quarter of its lift slope and moves its centre aft

Kirchhoff's free streamlines gave a plate held across a stream a drag with no viscosity in it. Tilt the plate, let the flow leave both its edges, and the same construction is a wing that has stalled from its leading edge. Rayleigh solved it in 1876, and the hodograph confirms him to seven figures: near zero incidence the stalled plate lifts at a quarter of the attached plate's rate, its centre of pressure sits at five sixteenths of the chord instead of a quarter, and its force is normal to it, so its lift-to-drag ratio is the cotangent of its incidence. A real stalled wing lifts more, by a factor the wake's pressure decides.

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