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

Streamfunction — where it appears

A scalar whose contours are the streamlines, defined so that its difference between two points is the volume flow between them. It exists only for two-dimensional or axisymmetric flow, and it turns a two-component velocity into one function.

Named by 21 essays across 5 fields — each of them below, with the objects they name alongside it.

Flow net — a stream past a cylinder. Two families of curves drawn over the same flow: the streamlines, along which the streamfunction is constant, and the equipotentials, along which the velocity potential is constant. They cross at right angles at every point, because they are the two parts of a single analytic function of position.

One function instead of two

A velocity field carries two numbers at every point and must satisfy two constraints. Both constraints can be solved once and for all by writing the whole flow as a single scalar function — and the two families of curves that function generates cross at right angles everywhere, for reasons that have nothing to do with fluids.

inviscid · Ideal flow The number on a streamline is a flow rate. Two streamlines and a crooked line drawn between them. The volume of fluid crossing that line every second, integrated from the velocity field, is the difference between the two streamfunction values at its ends — which is what makes a streamline a label rather than merely a curve.

The number on a streamline is a flow rate

Streamlines get drawn as decoration — curves the flow follows, spaced however the plotting looked best. Each one carries a number, the difference between two of those numbers is the volume of fluid passing between them every second, and it does not matter what route the measurement takes.

kinematics · Continuity The circle plane and the aerofoil plane. A circle with a polar net around it, and the same net after the Joukowski map. Curves that crossed at right angles still cross at right angles everywhere except at the single point where the map's derivative vanishes, and that point is the sharp trailing edge.

From a circle to a wing

The flow past a circular cylinder is known exactly and is of no interest to anybody who wants to fly. A change of variable turns that circle into a wing section — and, because the change of variable preserves angles, it carries the whole solution across with it. Nothing is solved twice.

inviscid · Ideal flow 1.5U at the equator, and no drag at all. The exact ideal flow past a sphere, in the meridional plane, with speed contoured behind the streamlines. The fastest fluid is at the equator at 1.5U — a cylinder's is at 2U — and the field is fore-and-aft symmetric, so the pressure integral over the surface gives a drag of -1.2e-16 against a dynamic scale of order one. The streamline spacing here does not measure speed the way it does in a plane flow: the flux between two meridional streamlines depends on the distance from the axis as well, which is why the speed is contoured rather than left to be inferred.

Three dimensions are kinder

The ideal flows usually drawn first are plane, and plane flow is the harsh case. Put the third dimension back and the fastest surface speed drops from twice the free stream to one and a half times, the disturbance dies as the cube of distance instead of the square, and the body cannot carry circulation at all.

inviscid · Axisymmetric One field, and the two parts the theorem splits it into. A velocity field made of a smooth source, a smooth vortex and a uniform stream, and the two fields the Helmholtz decomposition returns for it. The first carries the whole divergence and has no curl anywhere; the second carries the whole curl and has no divergence. They are computed by solving two Poisson problems on a grid, with the divergence and the vorticity differenced from the field rather than taken from the expressions that built it. Adding the two back together does not recover the field.

Every flow is two flows

Any velocity field splits into a part carrying all of the divergence and a part carrying all of the vorticity. The theorem says so and does not say which split — the two halves can be moved between each other by anything harmonic, and on a bounded region that is an infinite family.

kinematics · Helmholtz Two divergence-free fields with the same boundary conditions. On the left, the exact potential flow round a cylinder moving through fluid at rest. On the right, the same flow with a divergence-free eddy added — one that has no normal velocity on the body or on the outer circle, so it changes nothing about what crosses a boundary. Both fields conserve mass, both satisfy the wall condition, and only one is the flow. Nothing in the drawing says which.

The flow with the least energy in it

Draw a flow that conserves mass and does not go through the walls, and it will look exactly like a solution. There are infinitely many of them and one is the flow. What separates it from the others is not visible anywhere in the picture — it is a number, and the number is an energy.

inviscid · Least energy The hodograph plane, where the unknown boundary is the known one. The same flow drawn in the plane of its own velocity, ζ = (u − iv)/U. The plate, whose shape is known in the physical plane, becomes a segment of the imaginary axis; the axis of symmetry becomes a segment of the real one; and the free streamline — whose shape nobody knows — becomes an arc of the unit circle, because the speed on it is exactly the free stream. The unknown and the known have changed places, which is why the problem can be solved at all.

Where the unknown boundary is the known one

A free surface is the hardest kind of boundary — its shape is part of the answer, so the region the problem is posed in is not known until the problem is solved. Draw the same flow in the plane of its own velocity and the shape becomes an arc of a circle, known in advance and exactly.

inviscid · Free-streamline A flow that is incompressible and carries a density that varies three to one. Ideal flow past a cylinder, shaded by a density that is constant along each streamline and runs from one to three across the field. Every parcel keeps the density it started with, so the divergence is zero — measured at 10⁻¹⁰, which is the differencing — and the flow is incompressible in the only sense the word has. The density is not uniform anywhere. Incompressible is a statement about what the flow does to a parcel's volume, not about what the fluid is made of.

Incompressible is not a property of the fluid

A flow whose density varies three to one across it, with a divergence of 10⁻¹⁰ everywhere. And a flow of air at Mach 0.1, whose divergence is three per cent of U/a and which every textbook calls incompressible. The word is about what the flow does to a parcel's volume, and about nothing else.

kinematics · Dilatation Ideal flow past a sphere, drawn in a meridional plane. The Stokes stream function of ideal flow past a sphere, contoured at equal intervals. Its relations to the velocity carry factors of r sin θ that the plane stream function does not have, and differencing it reproduces the closed-form velocity to 10⁻¹¹. The surface speed at the equator is exactly one and a half times the free stream, against twice for a circular cylinder: a three-dimensional body lets the flow past in two directions rather than one.

The one number that runs out at three dimensions

A stream function is one function where the velocity is two, and four essays here are built on it. It exists because the divergence vanishes, it is single-valued only if nothing inside is making fluid, and in three dimensions it is not one function at all.

kinematics · Stream function A cylinder in a uniform shear, K = 0.4. A stream whose velocity increases with height, meeting a circular cylinder. The oncoming profile is drawn at the left. The flow carries uniform vorticity −K, so it is a solution of Euler's equations and not of Laplace's, the pattern is no longer symmetric top to bottom, and the body feels a lift towards the fast side with no circulation anywhere.

Inviscid does not mean irrotational

Dropping viscosity gives Euler's equations. Assuming nothing is spinning gives Laplace's — one scalar, linear, unique. The second step is a separate hypothesis about the flow's history, and a flow that fails it is still an inviscid flow with exact solutions of its own.

inviscid · Euler rotational Hill's spherical vortex. A sphere of rotating fluid travelling steadily through fluid at rest, drawn in the frame that moves with it. Outside the sphere the flow is the ordinary potential flow past a sphere; inside, the vorticity is proportional to the distance from the axis and the fluid recirculates. The two solutions match in value and in slope across the surface, and there is no body anywhere — the boundary is a streamline and nothing else.

The one rotational solution anybody can write down

A sphere of spinning fluid travelling steadily through fluid at rest, with no body anywhere in it — the boundary is a streamline and nothing else. It is exact, it is two lines long, and the reason it is the famous one turns out to be the reason it is the only one a real fluid can settle into.

inviscid · Euler rotational Two exact solutions of the same problem. Two steady Euler flows in the same square cell with the same boundary condition, differing only in the function relating vorticity to streamfunction. On the left the vorticity is proportional to the streamfunction, which is the textbook cellular flow; on the right it is uniform. Both satisfy the equations exactly. Nothing in the ideal theory prefers either, and at the same peak streamfunction their kinetic energies differ by thirty-one per cent.

The vorticity nothing decides

A streamline that comes from upstream carries its vorticity with it. A closed one comes from nowhere, so nothing determines what it carries — the ambiguity is not one number per body but a whole function. What closes it is a limit, and setting the viscosity to zero gives a different answer from letting it go to zero.

inviscid · Euler rotational The image system of a wedge of pi/3. The vortex and its images, with the two walls. Reflection in one wall and then the other generates a dihedral group, and the group is finite exactly when the angle is pi over a whole number — here 2n vortices, alternating in sign round the circle, with both walls streamlines to a part in 10¹⁶.

The corners that can be done with mirrors

The method of images works for a wall and for a circle, and for a corner it works only when the angle is pi over a whole number. At every other angle the reflections never come back, the image set is infinite and dense, and the flow exists anyway — which says the method is a statement about symmetry rather than about fluids.

inviscid · Images In the frame of the wave the walls stand still, and a bolus rides between them. Streamlines of a peristaltic channel of amplitude ratio 0.7 over two wavelengths, drawn in the frame moving with the wave, where the flow is steady and the walls are themselves streamlines. The time-mean flow is Θ = 0.5904 of the wave speed times the mean half-width, so the flow rate between centreline and wall in this frame is q = −0.4096 and the pressure rise per wavelength is 0.000 in units of μcλ/a². The centreline velocity changes sign at 0.106π and 0.894π, and the streamline through those points closes round a bolus holding 30.5 per cent of the fluid in each wavelength, which travels with the wave.

A wave on the wall is a pump

A channel whose wall only moves in and out, in a wave travelling along it, delivers a steady net flow with no part of the wall moving along the channel. In the frame of the wave the walls stand still and are streamlines, so continuity alone fixes how the laboratory flow rate follows the wall shape — and the momentum equation is needed only for one number, which also decides whether fluid rides along with the wave or leaks back against it.

kinematics · Continuity In the pair's own frame, two eddies with no vorticity. The flow round two equal co-rotating vortices (orange dots), seen from the frame that turns with them, where it is steady; shading is the stream function. Round each vortex a lobe of fluid circulates; a band round both is bounded by a figure-eight through the saddle at the centre; and above and below, centred on the two points that make equilateral triangles with the vortices (blue dots), are two large eddies of fluid that circulate in this frame and carry no vorticity at all, bounded by the streamline through the two outer saddles at √5 half-separations.

A vortex pair carries eddies no snapshot can see

Two equal vortices circling each other are, by every snapshot of the velocity gradient, two small vortices in a straining flow: outside their cores the flow is irrotational, and the gradient there is pure strain. Seen from the frame that turns with them, the same flow contains two large eddies, one on each side, centred where a third point would complete an equilateral triangle with the pair. Their fluid goes round with the pair for ever, and it carries no vorticity at all. They hold seventy times the area of the cores, and the only way to see them is to follow the fluid or to turn with it.

kinematics · Frames Three relaxed states with the same energy and enstrophy. The vorticity along the diagonal of the periodic square, through the centres of both vortices of the dipole, scaled by its rms value, for the three relations at the same ratio of enstrophy to energy, Z/E = 1.1 — except the linear state, which exists only at Z/E = 1. The sinh state concentrates its vorticity into sharp cores; the tanh state spreads it into flat-topped patches with steep edges; the linear state is a sine. All three carry the same two quadratic invariants in proportion.

The streamfunction says which relaxed state

Decaying two-dimensional turbulence ends in a large pair of vortices, and three theories say what that pair should look like: a sinh relation between vorticity and streamfunction, a tanh, or a straight line. Their scatter plots differ only in curvature, and a real flow's scatter hides curvature. Solve the three states in the same periodic box, at the same energy and enstrophy, and a statistic that separates them turns out to be one nobody looks at: the flatness of the streamfunction, which sits above the straight line's value for every sinh state and below it for every tanh state, and does not move when unrelaxed small eddies are added.

turbulence · Two-dimensional A flow's window mean converges a power faster. The variance of the mean over an L × L window, as a fraction of the point variance, against the window's side in integral lengths, on logarithmic axes. The scalar field's falls as the inverse area. The plane cut through a three-dimensional flow falls the same way at half the level. The velocity of a two-dimensional incompressible flow falls as the inverse cube of the side, because its mean over any window is a streamfunction difference around the window's edge. Lines are closed forms; dots are Monte Carlo means over fifty generated fields.

A snapshot counts areas, and a flow counts its edge

A record at one point holds one independent value for every two integral scales it lasts. A snapshot of a field holds one for every integral area it covers, so at the same number of samples it holds far fewer, and sampling it more finely adds nothing. Except for a velocity in a two-dimensional incompressible flow, whose mean over a window is fixed by the window's edge alone: it converges a whole power faster than its area allows, and a large enough snapshot of it beats the record.

turbulence · Spectrum A frozen run counts the area it sweeps. Independent values in a run of frames while a frozen pattern is swept 100 integral lengths past a window 12.8 integral lengths square, against the distance the pattern moves between frames. For a scalar the run holds 353 values with a frame every half integral length, 362 with one every four and 388 with one every window width: the swept area in integral areas, whatever the frame rate. Only past a window width, when gaps open between frames, does the count fall. Treating frames an integral scale or two apart as independent, as a record would allow, counts 2075 at two integral lengths — 5.8 times too many.

Frames that share their eddies count as one

A particle-image run is a sequence of snapshots, and snapshots closer together than an integral time photograph the same eddies. When a mean flow carries a frozen pattern past the window, a run holds exactly the area it sweeps, counted in integral areas, and a faster camera adds nothing until frames stop overlapping — a threshold set by the window, not by the turbulence. In a flat flow one component escapes the rule: its run mean is fixed by the pattern at the two ends.

turbulence · Spectrum Four of six decayed states sit on the sinh branch; the other two lie above it. The streamfunction's flatness against the enstrophy-to-energy ratio Z/E: the sinh and tanh relaxed dipoles as curves, from the linear dipole at Z/E = 1 outwards, and the six decayed states at t = 600 as points. The tanh branch falls below 9/4 and stays near Z/E of one; the sinh branch rises. The two runs begun as two-level patches lie on the sinh branch at their own Z/E, 0.0049 to 0.055 from it, and so do the two begun on a k⁻³ spectrum, 0.04 to 0.072. The two random-phase runs lie above it, by 0.3 to 0.31: on the sinh side of 9/4, but not sinh dipoles.

Decaying flows end on the sinh side

Three theories predict the state a decaying two-dimensional flow relaxes to, and the streamfunction's flatness tells them apart: above 9/4 for the point-vortex theory's sinh, below for the two-level theory's tanh. Run the decay from three kinds of start and every run ends above 9/4 — including the one begun as two-level patches, the case the tanh theory was built for. Four of six land on the sinh branch itself; the other two are sinh-like without being sinh.

turbulence · Two-dimensional Followed along their streamlines, deep parcels at 4° arrive nearly together, and at 0° the upper leads by more than outside. The arrival lag at 95% chord, lower parcel minus upper, as a multiple of the lag outside both layers, against the streamline both parcels follow, as a share of the upper layer's mass flux at the trailing edge. At 4° it is 0.46 for the deepest streamline drawn, −0.16 at a twentieth, 0.47 at a fifth and 1 at the layer's edge; at 0°, 1.47, 1.68, 1.33 and 1.

Followed rather than held, the parcels near a wing nearly tie

Parcels held at a fixed fraction of a boundary layer's thickness made the side that turns turbulent first decide which arrives first near the wall, and at a wing's ordinary incidence they gave the upper parcel a lead ten times the one outside the layer. Real smoke is not held at a depth: it follows a streamline. Near the attachment point the layer is thinner than any filament's height, so every filament starts outside it and is swallowed later, and on the upper surface, whose turbulent layer grows faster, it is swallowed deeper. Followed this way, the deep parcels at 4° arrive nearly together, and at a Reynolds number of a million the lower one can come first.

misconceptions · Equal transit Freed by its inertia, a settling particle falls through a lattice of vortices several times faster than through still air. The mean settling speed of a cloud released over one cell, over its speed in still air, against the Stokes number, for still-air speeds of 0.1, 0.3 and 0.6 of the vortices' speed. 0.1: 4.82 at St = 0.02, 4.39 at 1 and 1.86 at 10; 0.3: 2.36 at St = 0.02, 1.22 at 1 and 0.605 at 10; 0.6: 1.43 at St = 0.02, 0.984 at 1 and 0.959 at 10. Without inertia most of the cloud never falls at all.

Inertia empties the vortices that would hold a settling particle, and then speeds its fall

A particle that settles slower than a lattice of vortices turns can be carried round a cell for ever: without inertia, its path is a closed curve, and Stommel showed in 1949 that half a cell can be such a trap. Give the particle the slightest inertia and every trap leaks, in a time that grows as one over its Stokes number; the freed particle is flung to the cells' edges, rides their downdraughts, and falls on average several times faster than it would through still air. More inertia undoes some of that, and a narrow band of it can hold a slow particle up indefinitely on an orbit of its own.

regimes · Particle

Named alongside it

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

Model limitBoundary conditionPotential flowVorticityDivergenceIrrotationalLaplace's equationVelocity potentialConvergenceExact solutionMass conservationStreamline

All concepts