Every essay — page 6
Flows and fields Ideal flow Circulation and lift Viscosity Regimes and numbers Compressible flow Transition and turbulence Fluids at work What is taught wrongly Series Concepts Regimes Refutations Search
Fluids at work
Turbines, pipes, weirs, balls, sails, arteries and blades. What the conservation laws say about machines, which is more than a designer expects and less than a brochure claims.
A duct beats Betz only on the area it chooses
Put a wind turbine inside a flaring duct and it can take more than 16/27 of the wind's power through its rotor, which is the claim. Measure the same power against the duct's exit, the area the device actually fills in the wind, and a plain diffuser takes less than a bare rotor of that size would. It crosses the limit only when the duct holds a suction behind its exit, and the suction it needs is exactly the pressure on the back face of Betz's own disc.
Air at a valve softens the hammer only in quantity
A vapour cavity at a closed valve holds the head at one value, and that is why the pressure after it collapses climbs a staircase of equal steps. Put a pocket of free air there instead and the valve becomes a spring. A millionth of the pipe's volume trims only the tallest, briefest pulses; a hundred-thousandth takes a third off them and moves them; a ten-thousandth can make the pulse taller than it was, even on a line that never cavitates. The air that reliably removes the hammer is a thousandth of the pipe, which is a deliberate air vessel rather than a little dissolved gas.
An ejector's nozzle exit is a condition, not a choice
A steam ejector has two areas a designer can pick, the mixing tube's and the nozzle exit's, and one of them looked like a second way round the trade between entrainment and compression. It is not. Both are largest with the exit matched to the pressure the jet meets, as a rocket's thrust is, and every other exit draws a curve inside the matched one. What moves the machine is loss, and the losses sort themselves: only the nozzle's reaches the entrainment, and the sharp edge of the characteristic belongs to the ideal machine alone.
A foil under the surface flies with a phantom
At foiling speed the water's surface cannot hold a pressure, and the image that condition requires is not the reversed one a wall makes. For a horizontal foil it is an identical wing, equally loaded, above the surface — a biplane partner that takes lift and gives nothing back, doubling the induced drag as the foil rises. For the board that pierces the surface it is a reversed copy, which makes the board pay two and a half times what a keel under a hull pays. It is the board, not the foil, that decides how deep a foiling boat rides.
A foil flies level through a short sea and follows a long one
A foiling boat in waves has two ways to fly. It can hold its height and let the surface rise and fall over its foil, or it can follow the surface and heave with every wave. Flying level costs a deeper ride and a few per cent of drag whatever the wavelength; following costs the square of the heave, which grows with the frequency the boat meets the waves at. Into a head sea the two cost the same at a wavelength of about sixteen metres, running with the waves at about six, and the wave's height hardly moves either.
The air a cavity releases is capped by the cavity
When a shut valve pulls the pressure behind it down, the air dissolved in the water starts to come out, and a little gas was already known to soften the hammer. But air leaves the water only while its pressure in the cavity is below the pressure the water was saturated at, so a cavity can fill with at most its own volume of air at that pressure — a seven-thousandth of the pipe here, however much water gives up its air and however fast. That is less than a third of what removes the hammer. Released quickly, the air does not cushion the collapse at all: it turns the vapour cavity into an air cavity, and the pulse becomes the column-separation pulse with the margin measured to the atmosphere — taller, on this main.
A throttle can hold the peak only in a small plenum
A compressor makes its most pressure at the peak of its characteristic, and at the peak it either surges or stalls. A throttle moved by feedback can fight both. Driven by the plenum's pressure it damps surge, with a gain that grows as the square of Greitzer's B until it meets a ceiling B does not move; driven by the flow, it cannot. Driven by the stall cell's amplitude it turns the sudden jump into deep stall into a gradual one and closes the hysteresis loop — but only while the plenum is small. Above B of about a third the same feedback turns the jump into a cycle.
A foil in a random sea follows it up to its peak
In a regular wave a foiling boat chooses between holding its height and following the surface. A real sea is a spectrum, and a wand that filters its signal can do both at once: follow the long waves and fly level through the short. The best place to divide them is close to the frequency of the sea's own peak, the divided control beats either pure strategy, and in a rough sea it is the only one of the three that keeps the foil a safe distance under the surface and the drag near its calm-water value.
The terminal speed sets the best launch angle
In a vacuum every ball goes furthest at forty-five degrees. In air the best angle is lower, and it is lower by an amount that depends on one number only: how many times its own terminal speed the ball is launched at. A shot put, a baseball and a shuttlecock are three points on one curve, which falls from 45° to 41° at once and then needs a thousandfold more speed to reach 20°.
A spinning ball should leave low
Drag alone takes a ball's best launch angle from 45° to the high thirties at the speeds sport is played at. Backspin takes it into the teens, because a ball with lift is a glider and a glider buys its height with lift rather than with angle. Every tenth of lift-to-drag ratio takes a fixed slice off the angle, and above a line that depends on the launch speed the best flight leaves the ground level.
The cheapest turbulent pipe follows two and a half, not three
Murray's cube law comes from Poiseuille's resistance, and it fails the moment the flow in a vessel turns turbulent. The same cost minimised with turbulent friction gives a flow that grows as the radius to the power 27/11 in a smooth pipe and 7/3 in a rough one. Murray's invariant, a wall shear the same in every vessel, goes with it, and so does the local rule that made the law credible as biology: a vessel that holds its shear at a set point is twice too wide.
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.