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

Invariant — where it appears

A quantity a flow's own dynamics cannot change, whatever else it does. Which invariants a problem has decides more of its behaviour than the equations look as though they should allow — a decaying turbulence's exponent is set by one, and a linked pair of vortex tubes cannot be unlinked because of another.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

Two of opposite sign go somewhere. Two vortices of equal and opposite strength. Each is carried by the other's field, both are carried the same way, and the pair travels in a straight line at Γ/2πd forever, keeping its separation exactly. The speed is a consequence of one vortex's field evaluated at the other, and nothing else.

Vortices move each other

A vortex alone in an infinite fluid sits exactly still, forever — its own field is antisymmetric about it and there is nothing at its centre to be carried by. Everything a vortex does, another vortex did, and two of them already exhaust what can be written down.

inviscid · Vortex dynamics Two decay laws from two invariants, and nothing in the equations to choose. The energy of a decaying turbulence against time, integrated from dK/dt = −A K^(3/2)/l with the large scales conserving u² l³ in one case and u² l⁵ in the other. The exponents come out at 1.1997 and 1.4282 against the closed forms 6/5 and 10/7. Which invariant holds is decided by the shape of the spectrum at the very largest scales, at the moment the stirring stops.

What decay never forgets

Stir a box of fluid and stop. The turbulence decays, at a rate with no viscosity in it — so the rate cannot come from the fluid. It comes from an invariant of the very largest scales, fixed at the moment the stirring stops, and never revisited.

turbulence · Decay 4 nodes and 2 saddles, and the difference is the shape. The skin-friction pattern on a body, drawn as the streaks a film of oil would leave, looking at the nose. Every critical point of the field is marked: 4 nodes, where the streaks converge or diverge, and 2 saddles, where two streaks cross. Their indices sum to 2, which is the Euler characteristic of a sphere, and nothing about the flow enters that number — the stream is at 0 and the crossflow at 0, and neither may change it. What the flow decides is where the points are and what they look like; what the body decides is how many there are of each.

The count computed on a body

The rule for a closed surface is usually quoted and seldom solved for. This solves for one, and what the computation adds is not confirmation — it is the discovery that the count survives two events in which the number of stagnation points falls, and that both of them have closed forms.

kinematics · Topology Four kinds of critical point, and the curve that separates them. The invariants of a trace-free velocity gradient, with the discriminant curve 27R²/4 + Q³ = 0 drawn through them. Inside the two upper lobes the cubic has one real root and a complex pair, which is a spiral being stretched along its own axis on the left and squeezed on the right; below the curve all three roots are real and the point is a node with two saddle directions. Of 820 random incompressible gradients, 509 land in the spiral region and 311 in the real one. A plane flow is the vertical line R = 0 and nothing else, which is why a plane has two kinds and space has four. The marked points are the cases the calculation checks that fall inside this window; the two vortex cases it also checks sit at Q = 3.25 and |R| = 4.25, off the top corners, because a window wide enough to hold them would flatten the curve the figure is about.

Two kinds is a plane flow's privilege

A plane incompressible flow has a saddle or a centre and nothing else, and the proof is one line about a trace. The same line in three dimensions constrains three numbers instead of two, which is far less, and what it leaves is four kinds of point separated by a curve — with the one a plane cannot have being the structure the whole of turbulence is made from.

kinematics · Topology A pair of points meets on R = 0, the one line an index can change on. The paths of the ABC flow's stagnation points across the (R, Q) diagram as C rises from 0.3 towards √2 with A = B = 1. The four points of index +1 share one path on the left and the four of index −1 its mirror image on the right. At C = 0.3 they sit at R = −0.088, Q = −1.045; at C = 1 they touch the discriminant curve at R = −0.707, Q = −1.500, where the strain has a repeated rate, and turn away from it without crossing; and as C approaches √2 they run in to R = -7.5e-3, Q = −2.000. Crossing into a lobe would have changed a node into a focus, which a Beltrami flow's stagnation point cannot be; reaching R = 0 is where each meets a partner of the other index.

The sign a stagnation point carries in space

In three dimensions a stagnation point's index is the sign of one determinant, and that determinant is minus the R of the invariant diagram — so the diagram's left and right halves are the two indices. An exact Euler flow in a periodic box has eight such points, four of each sign, never a spiral among them, and they can only disappear in pairs that meet on the one line where the sign is allowed to change.

kinematics · Topology Five spectra, five exponents, one linear equation. The energy of a decaying turbulence after the nonlinear term has stopped mattering, computed by integrating the exact modal solution E(k,0)exp(−2 nu k² t) at five different shapes of the spectrum at the origin. Each is a straight line on these axes and no two have the same slope: the exponent is (m+1)/2, where k^m is the spectrum's behaviour at wavenumbers smaller than any eddy. The 5/2 that is quoted as the final period's universal exponent is the m = 4 line and one of five.

Universal, and one of five

A turbulence that has run its Reynolds number down stops being turbulent, the equations go linear, and the decay picks up a new exponent. That exponent is quoted everywhere as 5/2 and as universal. It is neither: it is the same corner of the same spectrum deciding the answer a second time.

turbulence · Decay The coefficient that was a constant, against the number it is said not to depend on. The dissipation coefficient Cε = eps·l/u³ along two decays, plotted against the Taylor-scale Reynolds number they pass through. One is flat because it was put in flat; the other falls as the reciprocal of the Reynolds number, which is what is measured in the near field of a grid. Neither line is a derivation. What is exact is the relation between them, Cε = 15(ℓ/λ)/Reλ, which is a rearrangement of two definitions and holds along both curves to 5·10⁻¹⁶.

The constant that travels

Every decay law in the subject rests on the dissipation being some constant times u³ over a length. The constant is not one. Letting it move the way grid measurements say it moves changes the decay exponent by a quarter — and lands one of the answers five per cent from another that is entirely different physics.

turbulence · Decay Six that are symmetries and five that look like them. Each transformation applied to an exact solution, with the Navier-Stokes residual recomputed from the transformed field by finite differences — nothing differentiated by hand. The six symmetries leave the residual at the differencing floor, a few parts in 10^8. The five near-misses leave between 0.048 and 4.3, which is six to nine orders of magnitude larger. The gap is what makes this a test rather than an illustration: a transformation that is nearly a symmetry does not exist here, and every one of the five is something a reader might reasonably believe.

Why the list is this long

Every textbook list of exact solutions of the Navier–Stokes equations is about a dozen long, and the usual explanation is that the equations are hard. It is not the reason. A similarity reduction is a solution invariant under a subgroup of the equations' own symmetries, so the catalogue of possible reductions is the catalogue of subgroups — and that is a finite, countable object.

viscous · Exact layer

Named alongside it

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

Model limitCritical pointDecayIntegral scaleMeasurementPower lawReynolds numberSelf-similarityTopologyVorticityBifurcationDissipation

All concepts