What a mean profile cannot tell anybody

A mean velocity profile is the most-measured object in fluid mechanics. It is what a traverse produces, what a code is validated against, what a correlation is fitted to and what a textbook prints. It is also, on its own, very nearly uninformative about the flow it came from — and this essay is an attempt to say exactly how uninformative, with numbers.

The mean is not the flow makes the neighbouring point: averaging an unsteady flow produces an object that is not a solution of anything, and the averaged momentum equation has a term in it that the mean cannot supply. That is a statement about the equation. This one is about the inverse problem: given a mean profile, what can be said about the flow?

The answer is: less than almost anybody assumes.

Two flows, one profile #

The construction is deliberately as simple as it can be made. The first flow is a plain shear across a channel — velocity proportional to distance across, steady, nothing else happening. The second is the same shear carrying a disturbance of amplitude 0.4 whose two velocity components oscillate in time with a fixed phase of 30 degrees between them and vanish at both walls. The disturbance is forty per cent of the free stream and moves the mean profile by 3.6·10⁻¹⁷.

The disturbance has zero mean at every point, by construction, so the time-averaged velocity of the second flow is the first flow’s profile. Measured over sixty-one stations across the channel, the largest disagreement anywhere is 3.6·10⁻¹⁷, which is the arithmetic and not the flow.

Any instrument that reports a time-averaged velocity returns the same answer for both, and the computation says how far beyond an instrument’s precision that goes: the largest departure between the two mean profiles anywhere across the channel is 3.6·10⁻¹⁷. A pitot traverse, a laser system in mean mode, a pressure-drop measurement inverted through a friction correlation: identical to sixteen decimal places.

What one of them is carrying #

The two flows are not the same, and the first thing that differs is the momentum crossing the shear.

The disturbance’s two components are correlated. Where the fluid moves towards one wall it is also moving slightly faster or slower along the channel, and averaged over a cycle that correlation is a flux of streamwise momentum across the flow. It peaks at 0.069 of a dynamic head in the middle of the channel and vanishes at both walls.

This is the Reynolds stress, arrived at without any turbulence. It is a term in the mean momentum equation and it is not a function of the mean profile. The plain shear’s value is exactly zero everywhere; the disturbed flow’s peaks at 0.0693 in units of the free stream squared, from a disturbance of amplitude 0.4 with a phase of 30 degrees between its two components; and no measurement of the mean profile distinguishes them.

The consequence for anybody fitting a model is direct. A closure that predicts the stress from the mean profile — which is what every eddy-viscosity model is — is predicting a quantity from data that does not contain it. That does not make eddy viscosity useless: in a real flow the disturbance is generated by the mean shear, so the two are correlated in practice. It makes it an empirical correlation between two things rather than a derivation of one from the other, which is exactly what a guess with a constant in it says about the constants such models carry.

What one of them costs #

The second difference is energetic, and it is larger.

Dissipation depends on the mean square of the velocity gradient, and a mean square is not the square of a mean. The plain shear dissipates at a flat rate across the channel. The disturbed flow dissipates 1.40 times as much overall, and nearly twice as much in the middle where the disturbance is strongest.

So two flows with the same mean profile need different power to maintain: the mean dissipation is 1.401 against the plain shear’s 1, a difference of 40.1 per cent in the power required to hold a profile that the two flows share to sixteen decimal places. That is a statement with an immediate practical form: a pressure-drop measurement and a mean-velocity traverse in the same duct are not two measurements of the same thing, and reconciling them through the mean profile alone is possible only if the fluctuation happens to be negligible — which is a hypothesis, and one the mean profile cannot test.

The price of a gradient is this collection’s account of where the dissipation goes; the point here is that the gradient whose price is being paid is not the mean one.

Exactly quadratic in something invisible #

The size of everything that distinguishes the two flows scales in one particular way, and measuring the exponent is what turns the observation into a rule.

The transported momentum against disturbance amplitude has a local exponent of 2.000000000, over amplitudes spanning a factor of six, and across the whole sweep the mean profile does not move by more than 10⁻¹² anywhere.

That is the general shape of the problem and it is worth stating carefully. The mean is a linear functional of the flow; everything that distinguishes these two flows is quadratic in it. A linear functional of a zero-mean disturbance is zero. A quadratic one is not, and it is not small either — 0.069 of a dynamic head, from a disturbance of forty per cent, is a substantial fraction of what a turbulent channel carries at the same Reynolds number.

This is the same arithmetic as a drift made of two things that average to zero, where a displacement and a gradient each average to nothing and their correlation is the Stokes drift, and as the mean pressure in the mean is not the flow, which reaches minus two dynamic pressures in a flow whose mean velocity is exactly zero. Three different quantities, one structure: the first-order description is blind by construction and the second-order one is where the physics is.

The number the mean cannot possibly contain #

There is one further variable, and it is the sharpest demonstration available because it changes the answer by everything while changing the data by nothing.

Hold the mean profile fixed. Hold the disturbance’s amplitude fixed. Change only the phase between its two velocity components, and sweep it through a quarter turn.

The transported momentum runs from its full value down to exactly zero. At ninety degrees the two components are in quadrature, the correlation between them integrates to nothing, and the flow carries no momentum across the shear at all — while fluctuating exactly as hard as before, and while presenting precisely the same mean profile and the same fluctuation intensity to every instrument.

So a mean profile and a turbulence intensity together are still not enough. Two flows can agree on both and differ by everything in what they transport, because transport is a correlation and a correlation needs a phase. That is why a measurement programme that reports means and root-mean-squares is reporting half of a covariance, and why the stress is measured directly with two-component instruments when it matters.

How large the family of flows with one profile is #

The two flows above are a pair, and a pair proves the profile does not determine the flow. It is worth asking how much bigger the family is, because the answer decides whether this is a curiosity or a structural limit.

It is very large. The construction adds to a given mean profile any disturbance field with zero time average, and the space of such fields is a whole function space rather than a handful of shapes: any amplitude, any cross-channel shape vanishing at the walls, any frequency, any phase, and any sum of those. Every member of it presents the same mean profile.

Within that family, the transported momentum ranges from zero — take the phase to a quarter turn, or take the two components to be uncorrelated in any other way — up to a bound set only by the amplitude, and the dissipation ranges from the plain shear’s value upwards without limit. So the mean profile constrains the transport not at all in one direction and only through the fluctuation amplitude in the other.

This is the same shape of statement as exact in the total, free in the profile, and it is worth putting them side by side because they are complementary. There, an exact integral constraint is imposed and the freedom left in some other quantity is computed. Here, the constraint is that a whole function — the mean profile — is fixed, which is far more than one number, and the freedom left in the second-order quantities is still complete. Constraining a function is not the same as constraining the flow, because the flow is not determined by any set of first moments.

What a measurement programme should do about it #

The practical consequence is a short list, and it is not a counsel of despair.

Measure the correlation, not the intensities. A two-component instrument at one point gives the stress directly. Two single-component instruments give two intensities and no phase, and the phase is where the answer lives — which is why the stress has been measured with crossed wires and two-component laser systems since it became possible to, rather than inferred.

Report what was averaged over. A mean is an average over a set, and the set is part of the answer: a mean over an interval short compared with the largest structure is not the same object as a mean over a long one, and neither is a mean over an ensemble. A profile quoted without its averaging window is a profile with a free parameter in it.

And validate on a second-order quantity. A code that reproduces a measured mean profile has agreed in the quantity least able to discriminate between models, for the reason this essay is about. Agreeing on the stress, the dissipation or a spectrum is a real test; agreeing on the mean is the entry requirement.

What the solver computed, and how it was checked #

The construction is analytic and the averages are quadratures, so there is nothing here that could be wrong through discretisation, and the checks are aimed instead at the claims themselves.

That the two profiles genuinely agree: the largest difference across sixty-one stations is required to be below 10⁻³ and reads 3.6·10⁻¹⁷. A construction that quietly moved the mean would make the whole essay a comparison of two different flows, which is not a result.

That the mean profile stays put across the amplitude sweep: the same difference is required to stay below 10⁻¹² at every amplitude, so the invisibility is not something that holds at one setting.

That the exponent is exactly two rather than approximately two: the check refuses a departure above 10⁻⁶ and reads 2.000000000. An exponent that came out near two would be consistent with a great many mechanisms; one that comes out at two to nine figures is the algebra, and it is what makes the statement about linear functionals rather than about this particular disturbance.

And that the dissipation ratio is what it claims: it is computed from the mean square of the gradient rather than from the square of the mean, at every station, and reads 1.40.

The refusals are exercised. The amplitude check is asked to certify the exponent to no tolerance and refuses; the profile check is asked to accept a dissipation ratio of a hundred and refuses.

Why this is not an argument against averaging #

The conclusion is easy to overstate, so it is worth putting the other side plainly.

Averaging is not a mistake. It is the only way to make a statement about a flow nobody can resolve, and the mean profile is genuinely the right object for a great many purposes: it gives the flow rate, it gives the wall shear stress through its gradient at the wall, and it is the quantity most engineering answers are integrals of.

What it does not do is determine the flow. The distinction matters most where a mean profile is being used as evidence for a mechanism rather than as a number in an integral — a claim that a flow is relaminarising, or that a control device has changed the turbulence, or that two facilities agree. Those are claims about the second moments, and the number that does not depend on the tunnel is the collection’s account of how much work it takes to make even a first-moment comparison between facilities mean anything. The mean is a projection, projections lose information, and the information a mean loses is exactly the information that the second-order quantities carry. The practical rule that follows is modest and useful: a quantity that is linear in the velocity can be predicted from the mean, and a quantity that is quadratic cannot.

Drag on a body in a steady stream is linear in the momentum flux and quadratic in the velocity, which is why the second class is larger than it first looks. Heat transfer, mixing, particle dispersion, acoustic radiation, structural loading and every closure term are all in it.

The same statement, with a different weighting #

There is a second way a mean can mislead, and it is worth putting beside this one because the two are often confused.

Two averages of one flow is about a flow with varying density, in which the plain time average and the mass-weighted average are both correct, both different, and differ by a factor of two across a flame. That is a question about which average, and both of the averages there are computable from the same data.

This essay is about a single average which is correct and incomplete. No reweighting fixes it, because the missing information is not in the first moment under any weighting: it is in the second.

What the picture cannot show #

The disturbance drawn into these figures is a single mode with one frequency and one shape. A real fluctuation is a continuum of scales, and every pair of them contributes to the transport with its own phase — so the total stress is a sum of correlations rather than one, and can be built from contributions of both signs.

Nothing in the figures shows that, and one consequence of it is worth carrying: because the stress is a signed sum, a flow can have a large fluctuation intensity and a small stress, or a modest intensity and a large stress. The two quantities are not ordered.

Who found it, and when #

Reynolds’ 1895 paper is the origin of the decomposition and states the difficulty in its first pages: the averaged equations contain a term the averaging cannot supply, and the term is a correlation. What that paper does not say, and what took another half century of measurement to establish, is how weakly the mean constrains the correlation — a fact that arrived experimentally, through flows with identical mean profiles and different stresses, rather than through theory.

The modern form of the observation is the one that matters for anybody building a model: the mean profile is a necessary validation and not close to a sufficient one, and a code reproducing a measured mean profile has demonstrated agreement in the one quantity least able to discriminate.

Limits recorded rather than smoothed over #

The disturbance is prescribed, not solved. It is a construction chosen to have zero mean and a non-zero correlation, and it is not a solution of the Navier-Stokes equations. That is deliberate: the argument is about what a mean profile determines, and a prescribed disturbance is the cleanest way to hold everything else fixed.

In a real flow the two are not independent. A disturbance in a shear flow is generated by that shear, so the profile and the stress are correlated in practice. The essay’s claim is about determination rather than about correlation, and the practical success of eddy-viscosity models rests on the correlation this construction deliberately breaks.

The dissipation here is a kinematic quantity. It is computed as the mean square velocity gradient, without a viscosity multiplying it, so the 1.40 is a ratio and not a power in watts. Multiplying by a constant viscosity leaves the ratio unchanged, which is why the comparison is made this way.

The phase sweep holds the shape fixed. Only the relative phase of the two components is moved, so the two intensities are constant along that axis by construction. A real disturbance changing its phase would generally change its shape too, and separating the two effects in measured data is harder than separating them here.

And the channel is a channel in name only. There is no wall law, no pressure gradient and no Reynolds number here — the profile is a linear shear because that is the simplest carrier for the argument, and nothing in the conclusion depends on it.