Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
The collection

Every essay — page 41

Page 41 of 53, continuing through the fields in the same order.

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

Regimes and numbers

Reynolds, Mach, Froude, Strouhal. One number decides whether a flow creeps, separates or shocks, and the same shape behaves differently at each.

The exponent a pulsing tree wants is between two and three. The tree's reflection at its root, averaged over the pulse's ten harmonics with the pulse's own weights, against the branching exponent k in r_parent^k = 2 r_daughter^k — two is area-preserving, three is Murray's law. With a wave speed that does not change with radius the best exponent is 2.15, just above the inviscid match of two; with one that rises as smaller arteries stiffen, 2.58, just above 2.5. Viscosity in the smallest branches pushes the best exponent a little towards Murray's.

Murray's law is not the rule for a pulse

Murray's law sizes a branching vessel for the cheapest steady flow, and at every junction built to it a pressure pulse is partly reflected. The rule that makes a junction transparent to a pulse is a different exponent, set by how the wave speed changes with radius. A tree is not the sum of its junctions, either: a Murray tree six generations deep reflects a third of the pulse at the heart rate, three times what one of its junctions does, and viscosity in the smallest branches means no area rule can make it transparent at every frequency. The best a pulsing tree can do lies between area-preserving and Murray.

6 figures What a tilted pane holds is a difference of two cosines. The largest ridge a tilted plate holds, as a cross-section in square capillary lengths, against the tilt, for clean glass (advancing 30°, receding 10°), a plastic (90°, 70°) and a water-repellent coating (115°, 95°): the force balance (cos θᵣ − cos θₐ) ÷ sin α, and, as points, the areas of drops shot from Young–Laplace, which do not use it. Clean glass holds least, not because water sticks to it less but because on a surface it wets well the two cosines are nearly equal.

The force a contact line holds is a range

Capillary rise and the drop on a window are usually drawn with one contact angle, and a contact line with one angle makes a force that is a single number. A real contact line pins, and stops anywhere between a receding and an advancing angle. The force it holds is then a range, as static friction is, and its width is surface tension times the difference of two cosines. A tube holds its column at any height in the range, so which way the meniscus last moved matters more than how patchy the wall is — and a tilted pane holds a drop only as large as that difference allows.

6 figures Area-preserving in the aorta, Murray's cube in the small arteries. The branching exponent k, in r_parent^k = r₁^k + r₂^k, that makes one junction reflect least of the heart's pulse, against the parent artery's radius, for a symmetric split and for side branches a half and 0.15 of the continuing trunk. In the aorta, where the pulse is carried by inertia, the transparent rule is close to 2 — area preserved — whatever the asymmetry. In arteries under a millimetre, where viscosity carries it, it is 3: Murray's law, the rule for the cheapest steady flow, is exactly the rule that passes the pulse. Asymmetry moves the answer only in between.

Murray's law passes the pulse where the pulse is viscous

The rule that makes an arterial junction transparent to the heart's pulse was an exponent, and real arteries do not split evenly: the aorta sheds side branches a fraction of its size and carries on. For a lopsided junction the transparent rule is still an exponent — the same one, exactly, while viscosity is negligible. With viscosity it is a single function of the parent's Womersley number: close to area-preserving in the aorta, and exactly Murray's cube in arteries under a millimetre, where the pulse moves as the steady flow does. The asymmetry matters only in between, and a lopsided tree reflects far less than an even one.

6 figures Past its threshold a ridge slides at a speed its angles set. The steady speed of a ridge of liquid one square capillary length in cross-section, as a capillary number, against the plate's tilt, on three surfaces. Below each surface's threshold it is stuck. Past it the speed rises from zero, linearly at first, as far as the tilt allows. On clean glass the curve barely exists: between its threshold and the speed at which its uphill contact line fails there is a sliver of tilt, and a ridge pushed past that cannot slide steadily with a clean trailing edge.

A sliding drop is held harder the faster it goes

A ridge of liquid on a tilted plate starts to slide when its weight beats the difference between its two contact angles' cosines. Once it moves, the angles move too: the front steepens and the back flattens, by a law set in the viscous corners at each edge. So the resistance rises with speed from exactly the static value, and a sliding drop has no kinetic friction lower than its static one — it stops at the tilt it started at. The back edge's angle falls to nothing at a finite speed, and past that no drop slides with a clean back.

6 figures Stratification makes a hill easier to block. The strength at which a current over a Gaussian hill first stops and traps fluid, δc = (h₀/H)/Ro at onset, against the stratification B = NH/fa — the depth over the height fa/N to which a rotating stratified flow feels the hill. Homogeneous, it is the 3.134 of the unstratified layer. As B grows the hill's anticyclone gathers at the bottom, where it is stronger, and the threshold falls; in deep water it falls as 2.243/B, which is a Froude number: the current is blocked when N h₀/U exceeds 2.243, and the rotation has dropped out.

A stratified sea blocks a current sooner and traps less

A slow current in a rotating layer stops over a hill once the hill's height over the depth, divided by the Rossby number, passes 3.134, and a column of water over the hill is trapped from floor to surface. In a stratified sea the hill's anticyclone gathers near the floor and fades upward over a height of fa/N. It is stronger there, so the current stops sooner; and it is confined there, so what it traps is a cap, not a column. In deep water the threshold loses the rotation altogether and becomes a Froude number: a current is blocked when N h₀/U exceeds 2.24.

6 figures The integral's hull is fine-ended slow and full-ended fast. The waterline half-breadth, bow to the right, of the hull of least wave resistance with the Wigley hull's length, draught, depth profile and displacement, designed for Froude numbers of 0.25, 0.30, 0.40 and 0.50, beside the Wigley hull's parabola. At low speed the optimum pulls volume into its middle and leaves long, fine ends; at high speed it pushes volume out towards the ends. Every one is symmetric fore and aft.

The hull Michell's integral prefers

Michell's integral gives a thin ship's wave resistance as a quadratic in the hull's offsets, so the hull of least resistance at a given speed and displacement is a quadratic minimisation. Allowed only to reshape its waterline, the integral rediscovers the naval architect's oldest rule: fine ends for a slow ship, full ends for a fast one. Allowed to reshape its depth too, it drains the waterline and piles volume at the keel until the hull is not a ship. And it cannot grow a bulb at the bow, because it cannot tell the bow from the stern.

6 figures With a wake, the bulb belongs at the bow. The Wigley hull's wave resistance with a spherical bulb, as a fraction of the bare hull's, against Froude number, with a wake fraction of 0.25: the bulb just ahead of the bow, and the same bulb just behind the stern. Without a wake the two curves are one. With it they separate: the stern's waves are made by slowed water and are weaker, so the bulb's cancelling wave has less to cancel there, and at the design speed of 0.30 the bow bulb leaves 0.443 of the bare resistance and the stern bulb 0.624.

The stern's wake puts the bulb at the bow

Michell's thin-ship integral cannot tell a ship's bow from its stern: reverse any hull and its wave resistance is unchanged, so the least-resistance hull is symmetric and a bulb is worth as much at the stern as at the bow. Real ships are fuller aft and put their bulbs forward. Let the stern's waves be made by water the hull's own boundary layer has slowed, and both follow: the optimum hull leans aft, and a bulb at the bow cuts the waves by far more than the same bulb at the stern.

6 figures Local transparency is the best tree only when the ends absorb. The root's reflection at the heart rate against the reflection at the tree's leaves, for the map tree, for the single exponent that is best at that leaf reflection, and for Murray's. With leaves that reflect nothing the map tree is the best of all, 0.035 against 0.0419: making every junction transparent is then the whole job. As the leaves start to reflect, the best single exponent pulls ahead, because it leaves its junctions slightly mismatched in the way that cancels the echo coming back from the ends; at a leaf reflection of 0.5 it reflects 0.0735 to the map tree's 0.13.

A tree transparent at every junction is not the quietest

The branching rule that lets a heart's pulse through one junction without reflection is set by that junction's Womersley number: area-preserving in the aorta, Murray's in the small arteries. Build a whole arterial tree that way, every junction at its own transparent rule, and it reflects more of the pulse than the best tree built to a single exponent. The single exponent wins by leaving its junctions slightly mismatched, in the way that cancels the echo from the tree's own ends — the trick an antireflection coating plays on light.

6 figures The entrance grows with the Péclet number, peaks, and then shrinks. The distance from a wall-temperature step at which the local Nusselt number has come within 5 per cent of its developed value, in diameters, against the Péclet number, for turbulent pipes at Reτ = 500, 1000, 2000 and 5000, with the slug's laminar line L/D = 0.029·Pe. The liquid metals sit on the line; the entrance peaks at 11.6 diameters at Pe ≈ 4700 for Reτ = 1000 and falls to about three and a half diameters for water.

The longest thermal entrance belongs to neither metal nor gas

A liquid metal carries its heat across a turbulent pipe by conduction, as a laminar flow would, and so its thermal entrance should be long. It is long only when its Péclet number is large. Below a few hundred the entrance grows in proportion to the Péclet number with the constant of a fluid moving as a solid block, and it leaves that scaling close to the threshold at which the eddies first match conduction. The longest entrance in a turbulent pipe belongs to the fluids between the metals and the gases, and how long it is depends on a number the measurements have never pinned down.

7 figures A wake held at the stern moves the volume aft a third as far. The least-resistance hull's centre of volume, in per cent of the half-length from midships, negative aft, against the design Froude number, holding the Wigley hull's length, draught, depth profile and displacement. With the linear wake it sits 2.8 per cent aft at Fr 0.30; with the shaped wake of the same propeller-disc fraction, 0.86 if uniform in depth, 1.5 if deepest at the waterline, and 0.11 if deepest at the keel, where the sources make the fewest waves.

A wake held at the stern keeps the bulb and loses the lean

A wake that slows the water steadily from bow to stern makes Michell's least-resistance hull fuller aft and puts its bulb at the bow. A real ship's wake is nearly nothing along most of the hull and strong only in its last few metres, deepest at the keel. Held there, with the same wake at the propeller, it moves the hull's volume aft by a twentieth to a half as much, and it keeps a third to two-thirds of the bulb's preference for the bow. The lean was a property of the water along the run, and the bulb a property of the water at the stern.

7 figures Viscosity slows every ripple, and the short ones most. The growth rate of a varicose ripple on a liquid jet against its wavenumber times the jet's radius, at Ohnesorge numbers of 0, 0.1, 1 and 10, in units of the capillary time. Every ripple longer than the circumference grows. Viscosity damps each one in proportion to the square of its wavenumber, so the short ones lose most: the fastest moves from ka = 0.697 inviscid to 0.344 at Oh = 1 and 0.123 at Oh = 10, and its rate falls from 0.343 to 0.0114.

Viscosity lets a jet break, later and into bigger drops

A thread of honey falls for metres before it breaks, and a thread of water for centimetres, which suggests that viscosity holds a jet together. It does not: it cannot stop any ripple longer than the jet's circumference from growing. It slows them, the short ones most, so the ripple that wins is longer, it takes the viscous time rather than the capillary one to win, and each drop it makes is bigger. The wavelength grows as the square root of the Ohnesorge number and the drop as its sixth root.

7 figures The air pulls on the crests, and short ripples start to grow. The growth rate of a ripple on an inviscid jet against its wavenumber times the radius, at gas Weber numbers of 0, 0.4, 2, 6 and 13 on the diameter. The air flowing over a rippled jet is faster over the crests and its pressure lower there, which pulls them further out. Without it nothing shorter than the circumference grows; at a gas Weber number of 2 ripples up to ka = 1.45 grow, at 13 up to 6.19, and the fastest moves with them.

The air shortens a jet's fastest ripple, and the drops follow

A jet in a vacuum breaks into drops nearly twice its own width, whatever its speed. A jet in air does not, and the reason is a pressure the air puts on its surface: flowing over a rippled jet it is faster over the crests and its pressure lower there, which pulls them further out. That pull lets ripples shorter than the jet's circumference grow, shortens the fastest one, shrinks the drops, and puts a ceiling on how long a fast jet can be — at the gas Weber number where the measured break-up regimes change.

6 figures