The grid nobody can build

Every claim this site makes about turbulence is downstream of one calculation, and it is short enough to do in a paragraph.

The largest motions in a turbulent flow are the size of whatever is stirring it, L. The smallest are the Kolmogorov scale η, and the ratio between them is Re^(3/4). A grid that resolves both, in three dimensions, needs

$$N = \left(\frac{L}{\eta}\right)^3 = Re^{9/4}$$

points. Advancing it needs a timestep short enough for the smallest eddies, of which there are about Re^(3/4) in one turnover of the largest, so the work is about Re³ point-updates for one large-eddy turnover.

Those two exponents govern the whole field.

Four flows, and what each would cost #

The arithmetic is more persuasive with numbers in it than as exponents, so the figures put four flows on the axis.

A bee’s wing, Re = 10³. About 5.6 million grid points. This is entirely feasible and has been done many times; insect flight is one of the few genuinely aerodynamic problems where direct simulation is the standard tool rather than an aspiration.

A model in a wind tunnel, Re = 10⁵. About 5.6·10¹¹ points. This is at the edge of what the very largest machines have done, for the simplest possible geometry, on a research allocation.

A sailplane wing, Re = 10⁷. About 5.6·10¹⁵ points. Out of reach by three to four orders of magnitude.

An airliner’s wing, Re = 5·10⁷. About 2.1·10¹⁷ points. Out of reach by five.

Those are memory figures and they are the optimistic half. The work rises as Re³, so the airliner is about eleven orders of magnitude more expensive to advance than the tunnel model, not five.

In hours, which is where the argument ends #

Converting to time requires two assumptions, and both are chosen to flatter the calculation.

A hundred floating-point operations per point-update. A real high-order scheme uses several times that, and none of the associated memory traffic is counted here at all — which on a modern machine is usually the binding constraint rather than the arithmetic.

10¹⁷ operations per second. That is the sustained rate of the largest machines in existence, available to one user, with perfect scaling.

Under those assumptions the tunnel model at Re = 10⁵ is a matter of minutes and the airliner wing is several thousand years. And that is for one large-eddy turnover — a single flow-through time, not long enough to gather statistics, at one angle of attack, at one Mach number, in one configuration.

A design process needs thousands of such cases. Multiplying a millennium by a thousand is not a statement about computing.

The memory figure is the friendlier one #

There is a detail in the two exponents that is easy to miss and that changes which resource binds first.

Memory goes as Re^(9/4) = Re^2.25 and work as Re³. The ratio between them is Re^(3/4), which is the number of timesteps — so for every doubling of Reynolds number the work grows about 1.7 times faster than the storage.

The practical consequence is that a calculation which fits in memory may still be entirely unaffordable in time, and that the gap between the two widens as the problem gets harder. Published direct simulations are almost always limited by time rather than by storage, and the storage figures quoted in the previous section are therefore the optimistic description of the difficulty.

A second detail, in the same direction: the arithmetic assumes the calculation is perfectly parallel. Splitting a grid across a hundred thousand processors requires communicating the boundaries between subdomains at every timestep, and the pressure solve in an incompressible formulation is global — every point depends on every other, immediately. That is the single hardest thing to scale in the whole calculation, and it is why an incompressible direct simulation does not simply get faster in proportion to the machine.

Why faster machines do not fix it #

The response that faster computers will eventually arrive deserves an answer with arithmetic in it rather than a shrug.

The work goes as Re³. A machine a thousand times faster therefore buys a factor of ten in Reynolds number — one decade — and the gap between a tunnel model and an airliner is two and a half decades in Reynolds number, which is seven and a half orders of magnitude in work.

At the historical doubling rate of the past few decades, seven and a half orders of magnitude is about fifty doublings. Even taking the optimistic historical rate at face value, that is somewhere over a century for the first flow-through time of one case.

The point is not that it will take a long time. It is that the exponent, rather than the constant, is what stands in the way, and an exponent is not something an engineering improvement addresses. This is the same structural fact as the transition Reynolds number’s exponential sensitivity, one level up: wherever a fluid problem is governed by an exponent, changes in the input arrive as very large changes in the outcome.

What the field does instead, and what each choice costs #

There are three routes past the wall, and they differ in what they resolve.

Reynolds-averaged modelling resolves nothing of the turbulence. It solves the averaged equations with a closure supplying the Reynolds stress, and its cost is nearly independent of Reynolds number — a full aircraft at flight Reynolds number is an overnight job. It is what almost all industrial aerodynamics uses, and its accuracy is good for attached flow and poor for separated flow, which is exactly where the interesting design questions are.

Large-eddy simulation resolves the large motions and models only what is smaller than the grid. The argument for it is the cascade: the small scales are the ones most likely to be universal, so they are the ones it is safest to model. Away from walls its cost scales far more gently than Re³. Near a wall the eddies that matter shrink in proportion to the viscous length, and the cost approaches direct simulation again — which is why wall-modelled large-eddy simulation, with its own closure at the wall, is where most of the current effort is.

Direct numerical simulation resolves everything and is what the arithmetic above prices. It is a research instrument, used to produce the data that the other two are calibrated against, at Reynolds numbers far below anything anyone flies at.

The reason to be clear about the three is that all three are called “CFD”, and a result carries completely different weight depending on which produced it.

What this site does, stated plainly #

This site’s own solver is a two-dimensional vorticity–streamfunction stepper on a grid of roughly 150 by 76 points, marched for a few hundred steps.

By the arithmetic above, that resolves a fully turbulent flow at a Reynolds number of about 20.

The site therefore does not compute turbulent flow anywhere, at any Reynolds number, in any figure, and it never will. What it does compute is:

Laminar solutions, including the boundary layer by shooting, the Falkner–Skan family and the steady and unsteady wakes its stepper can hold at Reynolds numbers in the hundreds.

Exact results about instability, which are eigenvalue problems rather than flows: the neutral wavenumber of a shear layer, the critical Rayleigh number, the vortex street’s spacing ratio.

Consequences of stated closures, drawn with the closure named on the figure.

Counts and dimensional arguments, of which this essay is one.

Every figure carries a model note saying which of those it is, and the regime index reads those notes back out of the rendered SVG rather than out of the source — so a figure that misdescribed itself would show up in the wrong band rather than merely being wrong.

Where the model stops, which here is a question about the estimate #

The Re^(9/4) count is an estimate and deserves its own qualifications.

It assumes the smallest scale must be resolved. For some questions it need not be: a computation interested only in the mean drag may tolerate under-resolving the dissipation range, at the price of an error nobody can bound in advance.

It assumes a uniform grid. Real computations stretch the grid, and near a wall the anisotropy of the eddies means the streamwise spacing can be far coarser than the wall-normal one. That buys back perhaps an order of magnitude, and it does not change the exponent.

It uses L for the whole domain. A real aircraft has a domain far larger than a chord, and the region requiring the finest resolution is a small fraction of it. That is a constant factor in the favourable direction and again not an exponent.

Re^(3/4) timesteps is one turnover. Useful statistics need tens of turnovers, which is another factor that the essay has not charged.

And it prices the wrong thing for some questions. The estimate is for resolving all scales everywhere and for all time. A great deal of useful aerodynamics needs only the mean forces on a body, which are integrals of a pressure distribution, and integrals are far more forgiving of under-resolution than the fields they are integrals of. That is the reason the averaged methods work at all, and it is a genuine qualification rather than a technicality.

Taken together the qualifications move the numbers by perhaps two orders of magnitude in each direction and leave the conclusion where it was. That insensitivity is the sign of a good estimate: if the answer had depended on the assumptions, the argument would be worth less than the effort of making it.

What is bought by giving up on resolving the walls #

There is one more asymmetry worth naming, because it explains why so much recent effort is aimed at one particular part of the problem.

Away from a wall, the eddies that carry the energy are the size of the flow — a fraction of the boundary-layer thickness, or of the jet’s width — and their size does not shrink as the Reynolds number rises. A large-eddy simulation of a free shear flow therefore has a cost that depends on Reynolds number only through the smallest resolved scale, which the modeller chooses, and it grows roughly linearly rather than as Re³.

Near a wall this fails. The eddies that carry the momentum flux there scale with the viscous length ν/u_τ, which shrinks as the Reynolds number rises, so resolving them costs almost as much as resolving everything. Estimates put wall-resolved large-eddy simulation at about Re^1.8 — better than Re³ and not by enough to matter for an aircraft.

That is why the wall is modelled. A wall-modelled large-eddy simulation resolves the outer flow and supplies the wall stress from a closure of exactly the kind this field’s other anchor examines, and its cost then grows roughly linearly with Reynolds number. The price is that the least reliable component sits in the most influential place, which is a fair description of the current state of computational aerodynamics.

The place where the arithmetic is not close #

The four flows above run out at five orders of magnitude, which is enough to settle the question for aerodynamics. It is worth running the same estimate on a flow nobody thinks of as a fluid-mechanics problem, because there the gap is not five orders of magnitude and no future machine is relevant to it.

Take the atmospheric boundary layer: a kilometre deep, ten metres a second, in air. The Reynolds number is about $7\times10^8$, so the Kolmogorov scale is around a quarter of a millimetre, and the grid that resolved it in one kilometre-square box would need something near $10^{20}$ points. That is for one box. A global model has half a billion square kilometres to cover.

A weather model actually runs at a grid spacing of a few kilometres and a climate model at tens. So between the smallest thing such a model resolves and the smallest thing the flow contains lie about eight decades of scale, every one of which is represented by a parameterisation rather than computed.

Which reframes what those models are. An aerodynamicist can at least point at direct simulations of simplified flows and calibrate against them; there is no Reynolds number at which a resolved atmosphere has ever been computed or ever will be, so the sub-grid models cannot be validated by the route this essay’s first three sections describe. They are validated against observations of the atmosphere itself, which is one realisation of a chaotic system rather than a controlled experiment.

The consequence is a familiar one stated at a larger scale. The single largest source of spread among climate projections is not the radiative physics, which is well understood, nor the arithmetic, which is sound — it is the treatment of clouds, which are sub-grid by a wide margin and whose net effect depends on processes at scales the models cannot see. That is exactly the position this essay’s wall-modelled large-eddy simulation is in: the least reliable component sits in the most influential place, for reasons of arithmetic rather than of effort.

So the honest summary of the estimate is that it does not describe a frontier being approached. In aerodynamics it describes a wall five orders of magnitude away, and in geophysics it describes a wall so far off that the entire discipline is built on the far side of it, and knows that it is.

Who found it, and when #

The scale ratio Re^(3/4) is Kolmogorov’s, from 1941, though he had no interest in grids.

The explicit computational statement is usually credited to Emmons in 1970, who put the count in terms of the memory of the machines of the day and concluded that direct simulation of engineering flows would remain out of reach. Orszag and Patterson performed the first direct simulation of isotropic turbulence in 1972, on a 32³ grid, at a Reynolds number based on the Taylor microscale of about 35.

The largest published direct simulations have reached grids of order 10¹² points. The intervening half-century has bought roughly three decades of Reynolds number, which is exactly what a cubic scaling predicts from the growth in machine capability over the same period — a rare case of a projection made in 1970 being confirmed in its detail rather than its spirit.

What follows for reading anybody else’s figures #

The practical use of this arithmetic is as a reading aid, and it is worth stating as one because it transfers off this site entirely.

Given a picture captioned as a turbulent flow, three questions settle what it can be. What Reynolds number? What grid? Resolved or modelled? The arithmetic above turns the first two into an answer to the third, without any need to trust the caption: a figure at Re = 10⁶ on a grid of 10⁷ points has resolved about a thousandth of the scales the flow contains, whatever it says of itself.

That is not an accusation of dishonesty. Modelled computations are the right tool for most purposes and their pictures are useful. But a modelled field and a resolved one are different kinds of claim, and the visual difference between them is close to nil — a modelled field is smoother, which reads as cleaner rather than as coarser, and is therefore more convincing rather than less.

It is the same problem this site was built around, one level up. A smooth streamline picture proves nothing about whether the field satisfies the equations; a smooth turbulent picture proves nothing about whether the scales it does not show were computed or assumed. In both cases the remedy is the same and it is not aesthetic: say what produced the field, and give the reader the arithmetic to check whether that was possible.

Where the ladder goes next #

This anchor has taken the cascade from its dimensional argument to its computational consequence. What remains in this field is the structure that the arithmetic forbids drawing.

The street this site cannot draw takes the most famous picture in the subject — the alternating wake behind a cylinder — and builds the one honest substitute: an ideal-flow model, named as a model, with an exact stability result inside it and no body anywhere in it.