The organising axis

Every figure, by regime

A shape does not decide how a fluid behaves around it. One dimensionless number does, and the same cylinder creeps, sheds or goes turbulent as that number crosses its thresholds. This is every figure on the site sorted by the hypothesis it was drawn under.

Every figure here carries a strip along its foot naming two things: the model that produced it, and the regime it holds in. That is an invariant of this site rather than a courtesy — a result quoted without its Reynolds or Mach number is a result quoted without its hypothesis, and a figure that cannot state its regime throws and stops the build.

This page is built by rendering all 5899 figure placements and reading those strips back out of the finished SVG, rather than from the code that wrote them. The two can differ, and where they do it is the picture that is right. 165 of the 171 generators used in essays are called at more than one regime, which is the difference between a family of figures and one picture shown repeatedly.

The bands below are one page each. They were one page in total until the collection passed three hundred and seventy essays, at which point the combined listing went past the raw-byte budget this site holds every page to — so the index carries the counts and each hypothesis carries its own rows.

The figure families, by how far they are stretched

How many distinct regimes each family of figures has been asked to draw. One regime is a picture; several make a family.

Storage and loss moduli, for one relaxation time and for a spectrum 84 regimes
One signature, aged four times 81 regimes
The growth rate a discretised sheet has, at every wavelength it can carry 71 regimes
The roughness function, and the asymptote in which the viscosity has gone 67 regimes
The two induction factors across the blade, from the fixed point 55 regimes
The velocity field, arrows to scale 47 regimes
One spectrum, two fields 46 regimes
What a probe in a flame sees 43 regimes
A wave that dies within one wavelength — 100 Hz in air 41 regimes
20.45 MPa out of a film 25 µm thick 40 regimes
Ideal flow past a cylinder 40 regimes
Six profiles with nothing in common but their two ends 39 regimes
A wing at h/c = 0.4 38 regimes
The wing's lift coefficient is an average, and no section is at it 36 regimes
Four wakes carrying exactly the same drag 34 regimes
A normal shock at Mach 2.00, and what crosses it unchanged 34 regimes
The two depths a reach has, and the Froude number that separates them 33 regimes
A steady pressure field, from a flow with no steady part 32 regimes
One more condition, and the price of it 31 regimes
The section the story is about, with its two surfaces measured 31 regimes
Each cancellation costs two powers of the Mach number 29 regimes
The wake's memory, as a gain and a phase 29 regimes
A particle released in still fluid, with the history and without 28 regimes
α = 13.0: a plug in the middle and everything happening at the wall 28 regimes
One layer, and the four others inside and outside it 28 regimes
72.0 kPa at the top, and the atmosphere is doing the holding 27 regimes
One field, and the two parts the theorem splits it into 26 regimes
The density ratio and the nose pressure coefficient, against Mach number 26 regimes
The cross term is not small, it is zero 26 regimes
One flow, two observers, two pictures 26 regimes
The temperature behind a shock, which is not the jump condition's 26 regimes
One pair of strainings, two orders, two lengths 26 regimes
A material region, and the dye that stays inside it 25 regimes
The core that does not move, and where its edge is 25 regimes
One material loop, at five stages of being drawn out 25 regimes
Two lifetimes, crossing 25 regimes
Two forces on one cylinder, a quarter of a cycle apart 25 regimes
The fluid a moving cylinder carries with it 24 regimes
The Hugoniot of a gas that can burn, and the gap in the middle of it 24 regimes
Inside a drop, at a viscosity ratio of 0.5 23 regimes
Three instruments, three derivatives, one field 23 regimes
Two camber lines, one lift and one moment 23 regimes
Three times the wind, on a reach 23 regimes
12 bar from stopping one metre per second 23 regimes
Where four insects have to turn round 23 regimes
The head at a valve that was shut in no time at all 22 regimes
A logarithm against a power, over a hundred and twenty-eight decades 22 regimes
A Joukowski aerofoil at 6° 21 regimes
Two velocities, 2.50 apart, and only one of them is anybody's 21 regimes
The mean lift is not the lift at the mean angle 21 regimes
The cliff a rough ball reaches sooner 21 regimes
What a barometer on the wing would read at 60 m/s 20 regimes
Two centres, two saddles, and a sum of nothing 20 regimes
A plane jet, slowing and widening at once 20 regimes
Six bodies at one circulation, and one lift 20 regimes
The temperature a film makes for itself 20 regimes
Two triangles, and the work is the difference between them 20 regimes
The plateau that is the log law 19 regimes
The entropy of a weak shock is third order in its strength 18 regimes
A pump curve out of the momentum theorem 17 regimes
1.139 asks for a Francis 17 regimes
Twice as fast is not twice as thick 16 regimes
The flow is solenoidal by construction, which is what the law needs 16 regimes
The lift curve that made flight look impossible 16 regimes
Two bills, and the radius that settles them 15 regimes
Four drops, drawn to scale 15 regimes
Particles at St = 1, against the flow that carries them 15 regimes
The bath is ten thousand times too small 15 regimes
The same number, by two integrals that share no arithmetic 14 regimes
Three equations, and the set they never leave 14 regimes
A nozzle at pb/p₀ = 0.70: shock in the divergent section 14 regimes
A sphere does not go at the speed of the flow it is in 14 regimes
A cross at 30.0° for every wavelength it makes 14 regimes
A million times faster, and the constant is 48.0 14 regimes
Where the flow over the section first reaches Mach one 14 regimes
Prandtl's interference factor, computed — 0.484 at a fifth of a span 14 regimes
There is a constriction, and it is a consequence rather than a cause 14 regimes
Every curved streamline has a pressure gradient across it 14 regimes
1.5U at the equator, and no drag at all 13 regimes
A flat plate at 6° and Mach 2.00: a shock below, a fan above 13 regimes
The only candidate the far field allows, and the wall it slips past 13 regimes
A double integral that comes out an integer 12 regimes
Ballistic first, diffusive later 12 regimes
The Kelvin wake, and the angle it comes out at 12 regimes
The stretch of surface that is boiling 11 regimes
A closed wake, and the loading of least drag on it 11 regimes
A control volume round an aerofoil 11 regimes
Three thresholds, and none of them is one 11 regimes
Flow net — a stream past a cylinder 11 regimes
80 per cent of the energy goes to the larger scale 11 regimes
Two divergence-free fields with the same boundary conditions 11 regimes
Wound up, and worth exactly what it started with 11 regimes
How wrong the ordinary answer already is 11 regimes
A source at Mach 1.60, and the sound it has already made 11 regimes
The laminar boundary-layer profile 10 regimes
The Ohnesorge diagram, with the boundaries where they belong 10 regimes
Three waves out of one discontinuity 10 regimes
The speed depends on the core; the slope does not 10 regimes
Seven fluids in one pipe at one pressure gradient 10 regimes
45° of corner, and an infinite number of eddies in it 9 regimes
The core spreads and the outside never notices 9 regimes
The Kirchhoff flow past a flat plate 8 regimes
The vortex sheet on a flat plate, and where it goes to infinity 8 regimes
The left wall is not a speed, it is an angle drawn in speed coordinates 8 regimes
Two depths, and a gap the model will not describe 8 regimes
The wall is what cancels the waves it made 8 regimes
Where the number says, and where it happens 8 regimes
A cylinder in a uniform shear, K = 0.4 8 regimes
The circle plane and the aerofoil plane 7 regimes
Four corners, and what the flow does in each 7 regimes
Circulation across the span, for three planforms 7 regimes
Two drag laws, one derived and one fitted 7 regimes
Every wavelength grows, and the shortest grows fastest 7 regimes
Two depths for the same energy, and one for the least 7 regimes
The downwash approaches twice the value at the wing — from above 7 regimes
Four camber lines, and the angle at which each stops lifting 7 regimes
The element that did not move is where the lift went 7 regimes
Quasi-steady stops being true a long way before one 7 regimes
A wing's two drags, and the lift at which they are equal 6 regimes
A supersonic corner: Mach 2.00 turning 10° away from itself 6 regimes
The jet is 61.1% of the hole 6 regimes
Three contours, one force, three different accounts of it 6 regimes
Two of opposite sign go somewhere 6 regimes
Reynolds number: one number, four different flows 6 regimes
The ask, drawn against the circle rather than the chord 6 regimes
Two profiles, and they are the same profile 6 regimes
The lift does not arrive when the incidence does 6 regimes
Elliptic loading rolls up to πb/4, and it is π that puts it there 6 regimes
The tube widens because the air slows 5 regimes
The hypotheses Bernoulli's equation needs 5 regimes
The inertial range, and the slope read back off it 5 regimes
Creeping flow, and the same body with inertia 5 regimes
A lift curve that keeps climbing to 49 degrees 5 regimes
Two different physics, one destination 5 regimes
One calculation is about the animal; the other is about how fast it happens to be going 5 regimes
Two roots walking towards each other, and one of them crosses 5 regimes
64 panels, and the pressure is right nearly everywhere 5 regimes
A circle of fluid, 0.90 of a gradient time later 5 regimes
The laminar line does not end; the flow leaves it 5 regimes
Nothing can deliver more than h/H, and here that is 10.0% 5 regimes
The same patch, 6 periods later, in two flows 5 regimes
A boundary layer with no x in it 5 regimes
The tunnel makes the stream 7.4% faster 5 regimes
Flow past a cylinder at Re 40 5 regimes
Ninety microseconds, and most of it spent barely moving 4 regimes
The flap moves the curve and leaves its slope alone 4 regimes
The jet divides 75% to 25% 4 regimes
The current at the surface is 45° from the wind, and nothing sets that angle 4 regimes
The same reading, and only one of them gives it back 4 regimes
The chart, with one exact line on it 4 regimes
5 quantities, 3 rows, 2 left over 4 regimes
The dynamic pressure is not ½ρU², and by Mach 0.85 it is out by a fifth 4 regimes
The pressure, relaxed rather than quoted 4 regimes
Pressure recovery along the upper surface at 6° 4 regimes
Stable in every mode, and 100 times larger first 4 regimes
A tube stretched to 3.0 times its length 4 regimes
The wake, and the velocity it gives itself 4 regimes
Most power at exactly half the jet speed, found by search at 0.5000 4 regimes
What an isentropic acceleration does to pressure, density and temperature 3 regimes
Stokes' theorem on a solved wake at Re 40 3 regimes
Two sides of one ball, at different pressures 2 regimes
Every disturbance has its own threshold; one of them is lowest 2 regimes
Grid points against Reynolds number, and where a wing sits 2 regimes
The number a duct settles at is an eigenvalue 2 regimes
The starting vortex, and the circulation it pays for 2 regimes
Ten unknowns, four equations 1 regime
Two parcels released together do not arrive together 1 regime
The box, and the one thing assumed about it 1 regime
The street, as two rows of point vortices 1 regime
Finding the aerodynamic centre by sweeping the chord 1 regime
u⁺ = (1/κ) ln y⁺ + B, integrated rather than assumed 1 regime

The figure library · What is refuted here · All essays

Kinematics only — an identity about the field, whatever moves it

Any Reynolds number — viscosity is absent

An exact Beltrami solution of Euler — steady, inviscid, three-dimensional

Re → 0 — creeping flow, inertia absent

Set by the reduced Reynolds number Re·(h/L) — not by Re

Set by the film's aspect ratio — one length small enough to delete inertia

Re below 1 — creeping, no wake at all

Re 1 to 47 — attached, or a steady standing pair

Re 47 to 10³ — the wake sheds

Re 10³ to 2·10⁵ — laminar separation, wide wake

Re above 2·10⁵ — turbulent boundary layer

Swept in Reynolds number — the figure is about the variation, not one value

Swept in pipe Reynolds number — the figure is about the variation

Re below 2300 in a pipe — the exact solution, and the flow takes it

Re above 4000 in a pipe — turbulent, and the loss is a correlation

Set by the channel Reynolds number — internal flow between walls

Set by the pipe Reynolds number — internal flow, not a wake

Set by the wedge angle and the flux — a purely radial similarity flow

Set by the suction coefficient — a layer held at a fixed thickness

Set by the vortex Reynolds number Γ/ν — circulation against viscosity

Set by the jet's own Reynolds number — its width and its centreline speed

Swept in pore Reynolds number — the figure is about where the linear law stops

Re_p below 10 in a bed — Darcy's law, and the linear term carries it

Re_p 10 to 86 in a bed — both of Ergun's terms matter

Set by the pore Reynolds number — a bed, not a body in a stream

Set by the position along a duct — x⁺ = x/(D·Re·Pr), axial conduction dropped

Set by the Womersley number — the radius against a cycle's diffusion depth

An averaged wall condition — one length standing in for a structure

Set by the Péclet number — advection against molecular diffusion

Set by the tip-speed and advance ratios — a rotor with blades rather than a disc

Boundary-layer theory — Re large enough that the layer is thin, and the number cancels

Set by the reduced frequency — linear unsteady aerofoil theory, planar wake

Set by the reduced frequency — how much of its own wake the section is still in

Trimmed level flight — the hypothesis is a force and a moment balance

Set by the Keulegan–Carpenter number — whether the flow has time to make a wake

Set by a stated gust distribution — the average of a curve, not of a flow

A self-excited oscillator — a caricature of a wake, with no fluid in it

The limit of vanishing viscosity — where the answer does not follow the parameter

Second order in the amplitude — a mean flow driven by an oscillation with none

Set by being local — an exponent the geometry fixes, at any scale

A truncation — a model system standing in for a flow, and named as one

Stated in the wall's own variables — a friction velocity, a roughness, a kappa

Set by a stated closure — a model rather than the equations

Set by the class of fields compared — same boundary values, same flux

Set by the Brinkman number — the heat a shear makes against the heat conducted away

Set by the Rayleigh number — buoyancy against diffusion

Vorticity as an object — a sheet, a patch or a filament, inviscid throughout

The narrow-gap limit — a Taylor number that is also a Rayleigh number

A topological count — linking and writhe, of curves rather than of a field

Set by the domain's symmetry — images where they exist, a map where they do not

Linear free-surface theory — an inviscid flow that radiates, and therefore drags

An ensemble statistic — a distribution over material elements, not a field

Set by the cavitation number — the pressure margin above vapour

Set by the fluid's transport coefficients — a stated gas at a stated state

Stated by the turbulence's own statistics — a dissipation, a scale, a spectrum

Set by the Ekman number — viscosity against the planet's rotation

A closure evaluated in a strain — a model's own algebra, with no flow round it

A reduced set of equations — Prandtl's, Stokes' or the triple deck's, with what was dropped named

Set by the Rossby number — inertia against the planet's rotation

A comparison class at fixed boundary data — admissible fields, with no equation being solved

A translating body's dipole — added mass from the disturbance's own energy, inviscid

Set by the pressure's own Poisson equation — its source is Q, and the sign of Q decides

A resultant on a closed contour — the shape entering only through the circulation

A loading in harmonics — where the induced drag is a sum of squares of the departure

An aeroelastic boundary — a root of the characteristic quartic, quasi-steady

A yield stress — a fluid with a threshold below which it does not shear at all

A converging film — Reynolds' equation, whose load separates into a bracket in the taper alone

A free jet's similarity solution — momentum conserved, the scale fitted

A field built from a stated spectrum — a statistic, not a solution

Linear stability of a parallel flow — an eigenvalue problem with no Reynolds number in it

A constructed solenoidal field — exact derivatives, and not a solution of anything

A chaotic advection map — deterministic, area-preserving, and exponentially unpredictable

A shock read by its strength — the expansion of the Hugoniot about a sound wave

Set by the intermittency factor — turbulent some of the time

The method of characteristics — one-dimensional unsteady flow, with two invariants

A wave's second-order transport — a constraint on an integral, and not on a profile

A logarithmic correction — exact in a limit, and approached at a constant rate per decade

A shell model — a nonlinear cascade with no space in it

A number's provenance — how it was located, rather than what flow it belongs to

Fully developed duct flow — the Reynolds number scaled out, leaving only the cross-section