The ladder that never closes

The previous rung ended with a deficit: ten unknowns and four equations, with the six extra unknowns being the components of the Reynolds stress.

The obvious response is that the Reynolds stress is a physical quantity in a fluid governed by known equations, so it must obey an equation of its own, and that equation can be derived. This is correct. It can be derived, it is exact, and it does not help.

This essay is about why it does not help, and the argument is a count rather than a piece of physics.

Deriving the six equations #

The derivation is mechanical and worth outlining, because the terms that come out have names and the names are used everywhere.

Subtract the averaged momentum equation from the instantaneous one, and what is left is an equation for the fluctuation u′ᵢ. Multiply it by u′ⱼ. Write the same equation with the indices exchanged and multiply by u′ᵢ. Add the two and average.

What comes out is the Reynolds-stress transport equation, and its terms are:

Production. −⟨u′ᵢu′ₖ⟩ ∂⟨uⱼ⟩/∂xₖ − ⟨u′ⱼu′ₖ⟩ ∂⟨uᵢ⟩/∂xₖ. This is the rate at which the mean flow’s strain feeds the fluctuations, and it involves only quantities already in the system. It is closed.

Dissipation. 2ν⟨∂u′ᵢ/∂xₖ · ∂u′ⱼ/∂xₖ⟩. This is where the fluctuating motion loses energy to heat, and it involves the gradients of the fluctuations, which are not among the unknowns.

Pressure–strain. ⟨p′(∂u′ᵢ/∂xⱼ + ∂u′ⱼ/∂xᵢ)⟩. This redistributes energy between the components without changing the total, and it involves the fluctuating pressure, which is not among the unknowns either.

Turbulent transport. ∂⟨u′ᵢu′ⱼu′ₖ⟩/∂xₖ. This carries stress from one place to another, and it is the triple correlation.

Three of those four terms contain something new. The equation is exact and it is not a closure.

The count, one level up #

The triple correlation ⟨u′ᵢu′ⱼu′ₖ⟩ is a symmetric third-order tensor in three dimensions. Its number of independent components is the number of multisets of size 3 from an alphabet of 3, which is

$$\binom{3 + 3 - 1}{3} = \binom{5}{3} = 10$$

So deriving equations for the six second-order unknowns has introduced ten third-order ones. The system is now worse off than it was: it has gained six equations and ten unknowns, and the deficit has grown from six to ten.

Climb again. The transport equation for the triple correlation introduces the quadruple, which has

$$\binom{6}{4} = 15$$

components. Ten equations, fifteen new unknowns; deficit fifteen. Then twenty-one, then twenty-eight.

The general term is C(n+2, n) = (n+1)(n+2)/2, which grows quadratically, and the deficit at level n is the difference between consecutive terms, which is n+3 and grows without bound.

That is the whole argument. It is arithmetic on binomial coefficients and there is no fluid mechanics in it anywhere — which is exactly what makes it decisive, because nothing about a fluid can be adjusted to change a binomial coefficient.

The check that keeps this honest #

The build asserts the property rather than trusting the prose. At every rung of the ladder the number of new unknowns must strictly exceed the number of equations gained, and a ladder that ever closed would throw and stop the build.

That may look like ceremony for a statement about binomial coefficients, and it is not, for the reason this site keeps meeting: the figure is a bar chart, and a bar chart is equally convincing whichever way the bars go. Had the counting function been written with an off-by-one in the dimension — C(n+d−2, n) rather than C(n+d−1, n) — the bars would still have risen, the essay would still have read correctly, and the ladder would have closed at the fourth rung without anybody noticing.

Three unclosed terms, three different difficulties #

The four terms above are not equally hard, and the differences are what every modelling scheme is organised around.

Production is closed and is the easy one. It contains only the mean gradient and the stress itself, both already in the system. It is also the term that carries the physics an engineer most cares about, since it is where the mean flow’s energy goes, and the fact that it needs no model is the reason Reynolds-stress closures behave better than eddy-viscosity ones in strongly strained flows: they get the production exactly right by construction.

Dissipation is small-scale and nearly isotropic. At high Reynolds number the gradients of the fluctuations live at the far end of the spectrum, where the turbulence has largely forgotten the geometry that made it. That makes the dissipation tensor approximately (2/3)ε δᵢⱼ — one scalar instead of six components — which is the single most useful simplification in the whole subject, and it is a consequence of the cascade rather than of any modelling assumption.

Pressure–strain is the hard one. The fluctuating pressure satisfies a Poisson equation whose source is the velocity field everywhere, so it is non-local: the pressure at a point depends on the motion throughout the flow, including at the walls. A term that is non-local cannot be modelled by any expression in local quantities, and every scheme that does so is making an approximation whose error is not bounded by anything.

Turbulent transport is the one that redistributes rather than creates. Its integral over a closed domain vanishes, so it moves stress about without changing the total — which is why a gradient-diffusion model for it is comparatively benign even though it has no more justification than the others.

Why nonlinearity is the culprit, and which nonlinearity #

It is worth being precise about which feature of the equations causes this, because the answer identifies exactly what would have to be different for the problem to go away.

The hierarchy arises from quadratic nonlinearity in the evolution equation. If the equation for u were linear, the equation for ⟨u⟩ would involve only ⟨u⟩ and there would be no hierarchy at all. Because the convective term is quadratic, the equation for the nth moment involves the (n+1)th, and the chain has no end.

Two consequences follow that are usually left implicit.

It is not about randomness. Nothing in the derivation assumed the flow was random, chaotic, or turbulent. The hierarchy exists for any nonlinear system whose state has fluctuations about a mean, including entirely deterministic ones. A laminar oscillating flow has the same structure; it simply happens that its fluctuations can be computed directly, so nobody needs the hierarchy.

It is not special to fluids. The same structure is the BBGKY hierarchy in kinetic theory, the Schwinger–Dyson equations in field theory, and the moment-closure problem in population dynamics and epidemiology. Fluid turbulence is where the hierarchy has the most engineering consequence, not where it is mathematically worst.

What a closure actually is #

Given that the ladder cannot be climbed to the top, every practical method stops somewhere and supplies the missing quantity from outside. The stopping point is the model’s order, and the supply is its closure.

Zero-equation models stop at the mean flow and give the Reynolds stress directly in terms of the mean velocity gradient. Prandtl’s mixing length is one; the algebraic models used in early panel codes are others. They contain a length scale that must be specified for each geometry.

One- and two-equation models carry transport equations for one or two scalar properties of the turbulence — usually the kinetic energy k and either the dissipation ε or a frequency ω — and build the stress from those. The k–ε and k–ω families are here, and they are the workhorses of industrial computation. They contain between four and six constants, all calibrated.

Reynolds-stress models carry all six components and close at the third order, modelling the pressure–strain and transport terms. They are more faithful in flows with strong streamline curvature or rotation, and they contain more constants and are harder to converge.

Detached-eddy and hybrid methods switch between a Reynolds-averaged treatment near a wall and a resolved treatment away from it, on the argument that the near-wall region is where models are best calibrated and the separated region is where they are worst. They inherit both sets of difficulties and a new one at the interface between the two treatments, and they are nevertheless what most industrial computations of separated flow now use.

Large-eddy simulation does something different in kind: it stops at a scale rather than at a moment, resolving the large motions and modelling only what is smaller than the grid. It is defensible in a way the others are not, because the modelled part is the part that is closest to universal — but its cost still rises steeply with Reynolds number, and near a wall it approaches the cost of resolving everything.

In every case the missing information came from outside the equations. That is not a criticism; it is the definition of a closure, and the honest form of a modelling paper says which experiment supplied it.

The one place the hierarchy nearly pays #

There is a partial exception worth stating, because it is the reason the hierarchy is studied rather than merely lamented, and it belongs to the next anchor.

At the smallest scales of a high-Reynolds-number flow, the turbulence is thought to forget the geometry that produced it. If so, the statistics there depend only on the rate at which energy is arriving and on the viscosity, and those statistics might be universal — the same in a jet, a pipe and the atmosphere.

That hypothesis is Kolmogorov’s, it is what the cascade argument is about, and its consequences at second order are checked routinely and hold well. It does not close the hierarchy — the closure it would supply is at the far end of the spectrum from where an engineer needs one — but it is the one part of the subject where a genuinely derived statement about high-order statistics exists.

The consequence for the boundary-layer arguments elsewhere on this site is worth stating: every statement made there about a turbulent profile — that it is fuller, that it resists an adverse gradient, that its shape factor is 1.35 — rests on a closure or on a measurement, never on a solution. The laminar statements beside them are solved. Two neighbouring sentences in the same paragraph can therefore have entirely different epistemic status, which is why this site’s figures carry a model note.

Averaging removed the information; the cascade hypothesis is a claim that at small enough scales the removed information did not matter, because the small scales are statistically the same whatever produced them. If true, it is the one part of the hierarchy that can be filled from theory rather than from an experiment on the particular flow in hand.

The qualification, which the next anchor takes seriously, is that the higher moments do not obey the 1941 theory, and the departures are the phenomenon called intermittency.

What a constant is worth, and what it costs #

The constants in a closure deserve a sentence of their own, because their number is often quoted as though it were the measure of a model’s honesty and it is not.

The k–ε model has five. That is not many for something asked to describe every turbulent flow in engineering, and it sounds like a triumph of economy. The difficulty is not their number but their status: they were calibrated on a small set of canonical flows — decaying grid turbulence, a homogeneous shear, a log-layer — and their values are the ones that make those flows come out right. A flow outside that set is being extrapolated to, and nothing inside the model reports how far.

The comparison that makes this concrete is with the viscosity in the unaveraged equations. There is one constant there too. It is measured in a viscometer, it is a property of the substance, and it transfers to every flow of that substance without qualification. That is what a constant looks like when the equation it appears in is derived.

The five constants of a closure look superficially similar and are a different kind of object: they are the coordinates of a fitted surface, and their transferability is exactly as good as the similarity between the new flow and the calibration set. This is the same distinction the previous rung drew about the word stress, and it keeps recurring because the notation invites it.

What the hierarchy does supply, free #

The counting says no closure can be derived. It does not say nothing at all is available without a calibration, and the thing that is available is worth having, because it is a set of tests a model can fail.

The Reynolds stress is a matrix of averaged products of a quantity with itself — a covariance. That forces three properties on it before any fluid mechanics is consulted. Each diagonal entry is a mean square and cannot be negative. Each off-diagonal entry is bounded by the geometric mean of the two diagonals it sits between, by the Cauchy–Schwarz inequality. And the matrix as a whole must be positive semi-definite, since a variance measured along any direction is still a variance.

Those are inequalities rather than equations, so they close nothing. What they do is reject, and they reject models that are in wide use. The standard eddy-viscosity form writes each normal stress as a share of the kinetic energy minus a term proportional to the local strain rate — and where the strain rate is large enough, that expression goes negative. A model has then predicted that the mean square of a velocity fluctuation is less than zero. It is not slightly wrong; it is describing something that cannot exist, and it happens in exactly the places engineers care about: stagnation regions, strongly curved flows, the front of any bluff body.

The repair is to cap the eddy viscosity so the inequalities hold, and that is the whole content of the word realizable in the names of several standard models.

The habit is the site’s own, arriving from the other direction. A constraint that has never rejected anything proves nothing — and here is a family of constraints that costs no physics, is derivable from the definition alone, and throws out models that had been calibrated, published and used for years.

Where the model stops #

Nothing in this essay is a flow. Both figures are counts, both are drawn as counts, and the model notes on them say arithmetic rather than naming a solver.

That is the right presentation of the material and it is unusual for this site, where the standing rule is that every field is solved before it is drawn. The counting arguments are the exception in form and the strongest thing in the field in content: a count cannot be a plausible smooth wrong answer in the way a velocity field can.

The limit worth naming is the one the counting does not reach. It says a closure cannot be derived from the hierarchy. It says nothing about whether a given closure is accurate, which is an empirical question about a particular class of flows, and it says nothing about whether some other formulation of the problem — a functional equation, a renormalisation-group treatment, a statement about weak solutions — might succeed where the moment hierarchy does not. Those are open, and the counting is not an argument against them.

Who found it, and when #

Keller and Friedmann set out the moment hierarchy and its non-closure in 1924, before most of the models that route around it were proposed.

Boussinesq’s eddy viscosity is from 1877, Prandtl’s mixing length from 1925, Kármán’s similarity hypothesis from 1930. Rotta wrote the first Reynolds-stress closure in 1951 and the pressure–strain model that most later ones descend from.

Launder and Spalding’s k–ε model appeared in 1974 and became the industrial default for two decades; Wilcox’s k–ω in 1988 and Menter’s blend of the two in 1994 are what most current codes actually run. Smagorinsky proposed the subgrid model that large-eddy simulation is built on in 1963, for weather prediction rather than for engineering.

None of those is a derivation and none of their authors claimed otherwise. The claim that turbulence modelling is a solved problem waiting on computer power is a later invention, and the counting above is the shortest refutation of it.

Where the ladder goes next #

The next rung takes the most transparent closure in the subject and follows it all the way: a guess with a constant in it integrates Prandtl’s mixing length, recovers the law of the wall exactly, and is careful about what that does and does not establish.

The other direction is the one place where a genuinely derived statement exists. Where the energy goes is the cascade, the −5/3 spectrum, and the dimensional argument that produces it from two quantities and nothing else.