The thread: One number decides the regime — page 54
509 essays carry this thread — page 54 of 57.
Ideal flow
Plating that gives way slows the water that slams it, but only on a flat bottom
The slamming pressure on a hull's bottom is a ridge that rides at the edge of the wetted width, and how fast that edge moves depends on how far the water has to climb. A panel of plating that bends inward under the load gives the water further to climb, and while it is moving away it slows the climb itself. Fed back into Wagner's condition, that takes a seventh off the peak stress of a panel on a four-degree bottom loaded to the edge of what aluminium can bear, and almost nothing on a deep V — because what matters is the plating's deflection against the bottom's own rise across the panel.
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.
Compressible flow
A wire sings with the force on it, not the flow round it
A free flow can only radiate as a quadrupole, four sources nearly cancelling, which is why a jet's noise grows as the eighth power of its speed and so little of its power becomes sound. Put a body in the flow and the cancellation is broken: the fluctuating force the body exerts is a dipole, its sound grows as the sixth power, and at the speed of any wind it outshouts the turbulence that made it by a factor of tens of thousands. The whistle of a wire is the force of its own wake, radiated.
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.
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.
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.
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.
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.
Regimes and numbers
Below separation a cylinder's drag is a Stokes layer, and KC stops deciding the force
Morison's equation splits the force on a cylinder in an oscillating flow into inertia and drag, and the Keulegan–Carpenter number says which dominates — once the flow separates and leaves a wake. Below that, nothing separates, and the drag term describes the thin oscillating boundary layer on the cylinder's surface. Its coefficient then grows without limit as the motion shrinks, as one over the Keulegan–Carpenter number, and that exactly cancels the number's power to decide anything: the drag's share of the force is fixed by the frequency parameter alone, a few per cent at most, until Honji's instability ends the regime.