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

Jet — where it appears

A stream of fluid issuing into surroundings at rest, which spreads by entraining them. Its momentum flux is conserved and its mass flux is not, so it slows as it grows and its noise depends on the eighth power of its speed.

Named by 12 essays across 7 fields — each of them below, with the objects they name alongside it.

The Ohnesorge diagram, with the boundaries where they belong. The classical map of jet break-up: the Ohnesorge number against the jet Reynolds number, with the five nozzles placed on it. The three sloping lines are Reitz's transitions in the gas Weber number, and their geometry is computed rather than sketched — a fixed We_g means Oh·Re is fixed, which is a straight line of slope exactly −1 in these coordinates, and a decade in Reynolds number is checked to move each line by exactly one decade. Where they sit is borrowed; that they are straight and parallel is not. A nozzle below and to the right of the last line atomises.

Where a jet stops being a jet

A tap makes drops a few centimetres down, a garden hose makes a stream that carries, a sprayer makes a mist and a diesel injector makes fog. Same liquid, same mechanism, four regimes — and the number that separates them is not the jet's inertia but the surrounding air's.

regimes · Atomisation The jet divides 75% to 25%. A jet striking a plate at 60 degrees. Both sheets leave at the jet's own speed, because their surfaces are at ambient pressure and Bernoulli allows nothing else, and the plate can exert no force along itself because the fluid has no viscosity. Momentum along the plate then fixes the split at (1 + cos β)/2 = 0.7500, and the normal force at ṁV sin β = 0.8660. Nothing about the plate's material, size or roughness enters either.

What a jet cannot push sideways

A jet striking a plate divides in two, and how it divides is fixed by a single sentence — an inviscid fluid exerts no force along a surface. That one statement, plus mass, gives the split exactly — and the same sentence turns a flat plate into a bucket worth twice as much.

applied · Jet Most power at exactly half the jet speed, found by search at 0.5000. The power a bucket takes from a jet, against how fast the bucket runs, for four deflection angles. Every curve is a parabola with roots at zero — where the force is greatest and the bucket is not moving — and at the jet speed, where the bucket is running away and there is no force at all. The peak is halfway between, at U = V/2, and it is there for every angle and every flow rate. A golden-section search that knows none of the algebra puts it at 0.500000.

Half the jet speed takes everything

A bucket standing still feels the largest force and does no work; a bucket running with the jet does no work either. Between them the power peaks at exactly half the jet speed, for every bucket shape and every flow rate — and at that speed a perfect bucket leaves the water motionless.

applied · Jet The sound is what fails to cancel. A compact quadrupole of kd = 0.01, followed outwards. The upper curve is what one of its four sources produces on its own at each radius — every one of them is as loud as a monopole — and the lower curve is what all four produce together. Inside the source they barely cancel at all; by one radian of wavelength the sum is 9.3e-5 of what a single source is doing, and it is falling as 1/R from there on because that is what radiation does. Between the two the field falls as the cube of the distance, which is a near field rather than a sound.

The sound is what does not cancel

A quadrupole is not a weak source. Every one of the four monopoles in it is as loud as a monopole of the same strength, and what makes the assembly quiet is that they very nearly cancel — one part in ten thousand survives at a hundredth of a wavelength. The eighth-power law is a statement about how nearly, not about how little.

compressible · Aeroacoustics Where the suction on a cylinder is actually made. The share of the surface's pressure deficit generated within a given distance of it. Twelve per cent comes from the first twentieth of a radius, half from within a third, and a tenth of it from beyond one and a half radii. There is no thin layer doing the work: the suction is made by the curvature of the whole outer field.

The effect that is real, and where it stops

The Coandă effect has a genuine mechanism, and it is the same equation that makes the suction on a wing. What separates the two is where the integral comes from: a wall jet generates all of its pressure deficit inside a layer two per cent of the radius thick, and a cylinder in a stream needs a third of a radius for half of it.

misconceptions · Coanda Most of the fluctuation is the switching. What a hot wire at the edge of a shear layer records, taken apart. The conditional intensity — the fluctuation inside the turbulent fluid — is flat at 0.16 by construction. The measured intensity peaks 20 per cent above it, and the excess is the third curve: the variance of a signal that keeps switching between two mean velocities, which peaks where the switching is most even and is not turbulence at all. A model calibrated against the measured curve is being fitted to an artefact of averaging.

Turbulent some of the time

At the edge of a jet or a wake a probe is inside turbulent fluid for part of the time and in perfectly smooth flow for the rest, and an ordinary time average mixes the two. Most of the fluctuation it records there is not turbulence at all — it is the switching between two states, and it peaks where the switching is most even rather than where the turbulence is strongest.

turbulence · Intermittency 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.

viscous · Free shear The water climbs the body, and the wetted width outruns the drawing. The wetted half-width against penetration for a wedge of 10° deadrise and a circular cylinder of unit radius: where the body crosses the undisturbed level, von Kármán's width, and where the risen water meets it, Wagner's. For the wedge Wagner's width is π/2 = 1.571 times the geometric one at every depth; for the circle it is √2 = 1.414 times, at small penetration.

Water climbs a falling wedge, and the load comes from the climbing

A hull that strikes the sea sets the water under it moving, and the force is the rate at which it does so. The obvious estimate measures the wetted width where the hull crosses the undisturbed surface. But the water does not wait: pushed aside, it rises up the hull and wets it sooner, a factor π/2 wider for a wedge and √2 for a round bottom. The force carries that factor twice, and the peak pressure, where a thin jet leaves the hull, carries it squared on a cotangent that grows without bound as the bottom flattens.

inviscid · Unsteady bernoulli 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.

viscous · Free shear 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.

viscous · Free shear 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.

viscous · Free shear 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.

viscous · Free shear

Named alongside it

The objects these essays reach for when they reach for this one.

EntrainmentMeasurementModel limitMomentum fluxSimilarity solutionBoundary layerConservationConserved quantityMomentum theoremThe no-slip conditionBernoulli's equationClosure

All concepts