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

Free shear — the series

6 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. The jet has to be fed from the sides. The velocity field of the plane jet with streamlines integrated through it. The seven central streamlines run down the jet and spread; the ten started at the top and bottom edges bend inwards and join it, which is entrainment and is a consequence of the solution rather than an addition to it. The dashed lines are the half-speed edges, widening as x^{2/3}. The transverse velocity far from the axis is 5.70e-3 m/s at this station, inward on both sides — a jet is a sink as seen from a distance, which is why two parallel jets pull together.

    What a jet keeps, and what it collects

    A jet leaving a nozzle into still fluid has no boundary anywhere and one conserved quantity. Its momentum flux is exactly the same at every station downstream; its mass flux is not conserved at all and grows without limit, because a jet is a machine for acquiring fluid it did not start with.

    part 1 · viscous
  2. One shape, at every station and for every jet. The axial velocity of Schlichting's round jet at four distances, each scaled on its own centreline speed and its own half-width. The four curves are one curve: the shape function (1 + eta²/4)⁻² has no parameter in it at all, and every station of every jet — laminar or modelled, weak or strong — collapses onto it exactly.

    Exactly similar, and one number short

    The round jet has an exact solution of the Navier–Stokes equations, and a turbulent jet is modelled by the same formula with the viscosity replaced by a fitted constant. The shape is identical. Nothing a measurement of the profile can do will tell the two apart.

    part 2 · viscous
  3. A weak push stirs the whole fluid; a strong one makes a jet. Streamlines of the exact solution in a plane through the force, for a weak force (jet Reynolds number 1, left) and a strong one (100, right), the force pointing right from the origin. The dashed line is the cone inside which the fluid moves outwards: 89.4° from the axis for the weak force, nearly a hemisphere, and 22.8° for the strong one. Outside it the fluid is drawn back towards the origin from every direction and turned into the jet.

    A point force makes a jet only when it is strong

    Push on a fluid at one point and there is an exact solution of the full Navier–Stokes equations for what follows, at any strength. Weak, it is Stokes's point force: fluid pushed forward over a whole hemisphere and drawn in behind. Strong, it is Schlichting's slender jet, with fluid drawn in from every direction outside a narrowing cone. One constant joins them, and it says how strong a push must be before the boundary-layer jet everyone uses is right — at the axis by a jet Reynolds number of a hundred, at the edges only by a thousand.

    part 3 · viscous
  4. 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.

    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.

    part 4 · viscous
  5. Hold the nozzle further from the plate and the jet arrives slower but leaves a stronger wall jet. For a laminar jet from a 2 mm slot at 5 cm/s in water, against the nozzle's distance from the plate, each over its value at 10 cm: the arriving centreline speed, which falls as h^−⅓; the inherited invariant F = 3KQ/32, which grows as h^⅓ because the jet has gathered fluid; and the wall jet's peak speed a metre along the plate, which grows as h^⅙. At 2 cm the arriving speed is 1.71 of its value at 10 cm and the far wall jet 0.765; at 2 m, 0.368 and 1.65.

    A jet striking a plate leaves a stronger wall jet the further it travelled

    A plane jet that strikes a wall splits into two wall jets, and each keeps not the jet's momentum but Glauert's flux of momentum flux. Turned without loss at the stagnation point, each half arrives laid along the wall with its axis on the wall, and its invariant is the momentum it brought times the fluid it had gathered on the way: F = 3KQ/32. Hold the nozzle further away and the jet arrives slower but leaves a stronger wall jet, by the cube root of the distance — and the wall jet takes a distance of about the Reynolds number in jet widths to forget it was ever free.

    part 5 · viscous
  6. A round jet held further from the plate arrives slower in proportion and leaves a radial wall jet stronger in proportion. For the round water jet from a 2 mm nozzle, against the nozzle's distance from the plate, each over its value at 10 cm: the arriving centreline speed, the radial invariant F = Kνh/(2π) it hands on, and the radial wall jet's peak speed 5 cm from the stagnation point; dashed, a plane jet's speed and invariant for comparison. At 40 cm the round jet arrives at 0.25 of its speed at 10 cm and hands on 4 times the invariant, and its wall jet runs 2 times as fast; a plane jet's invariant would have grown 1.59 times.

    A round jet hands its wall jet a strength proportional to its throw

    A round jet striking a plate spreads into a radial wall jet, and like the plane jet before it, the wall jet keeps not the jet's momentum but Glauert's flux of momentum flux — the momentum the jet brought times the fluid it gathered on the way. A round jet gathers fluid at a rate set by the viscosity alone, eight π ν per metre of travel, whatever its strength. So the invariant it hands on grows in exact proportion to the distance it was thrown, where a plane jet's grew as the cube root: hold the nozzle four times further away and the wall jet is four times as strong and runs twice as fast.

    part 6 · viscous

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