Two averages of one flow

Averaging a flow throws information away, and this collection has already worked out what that costs: an averaged field is a new object with its own properties and is not a solution of anything, and the terms that survive the averaging cannot be closed by anything the averaging itself supplies.

Both of those assume there is one average. In a flow whose density varies there are two.

The two definitions #

Reynolds averaging is the ordinary one. Sit at a point, watch for long enough, and take the mean: $u = \bar{u} + u'$ with $\overline{u'} = 0$.

Favre averaging weights by mass:

$$\tilde{u} = \frac{\overline{\rho u}}{\bar{\rho}}, \qquad u = \tilde{u} + u''.$$

Both are averages of the same signal. Neither is an approximation to the other. And they are related exactly, with no modelling anywhere in the derivation:

$$\bar{u} - \tilde{u} = -\frac{\overline{\rho' u'}}{\bar{\rho}}.$$

The gap between them is a density-velocity correlation over the mean density. In an incompressible flow that correlation is zero and the two are one average; in a flame, a mixing layer between gases of different molecular weight, or any flow with heat release, it is not small.

The identity is worth deriving in one line, because it explains why the gap is a correlation rather than an arbitrary difference. Write $\rho = \bar{\rho} + \rho'$ and $u = \bar{u} + u'$, and average the product: $\overline{\rho u} = \bar{\rho}\bar{u} + \overline{\rho' u'}$. Divide by $\bar{\rho}$ and the left side is $\tilde{u}$ by definition. Rearranged, that is the relation above. There is no approximation in it and no assumption about the fluid, the flow or the size of anything.

It also says which way the gap goes. Where the fluid is fast it is hot, so it is light, so $\rho'u'$ is negative and the Favre mean sits below the Reynolds mean. That is the usual sign in a flame or a heated jet and it can be reversed — a cold fast stream in a hot slow one gives the opposite — but the direction is not a convention, it is the correlation’s sign.

The size of it #

The field above is the simplest one in which the two differ, and it is not a toy: it is what a probe in the outer part of a jet flame actually sees. Half the time hot light fluid at a hundred metres a second, half the time cold heavy fluid at twenty, with a density ratio of seven.

The Reynolds mean is sixty metres a second. The Favre mean is thirty. A factor of two, on the most basic quantity anybody reports about a flow, from a definition rather than from a model.

At a density ratio of one the gap is exactly zero and stays zero however violent the turbulence is. It opens as the ratio rises: eight metres a second at a ratio of one and a half, twenty at three, thirty at seven. The ratio across a hydrocarbon flame front is about seven, across a helium-air mixing layer about seven the other way, and across a hypersonic boundary layer larger still.

Those numbers are not obscure regimes. A ratio of seven is a stoichiometric hydrocarbon flame; a ratio of three or four is a heated jet a laboratory can make with a hot-air gun; a ratio of one and a half is the density difference between the two sides of a helium-air mixing layer, and even there the two means are thirteen per cent apart. Anything with combustion in it, anything with a large temperature difference, and anything mixing two gases of different molecular weight is in this territory.

There is one more feature of the sweep worth noticing. The gap does not saturate. At a density ratio of ten it is 33 metres a second against a Reynolds mean of 60 — more than half — and it goes on rising, because the mass weighting keeps concentrating on the heavy slow fluid. There is no ratio above which the effect stops growing and no regime in which it becomes a small correction.

The two-state field could be accused of being a special case chosen to be extreme. A continuous sinusoidal velocity with the density in antiphase — hot where the fluid is fast, which is what a heated jet does — gives a gap of fifteen metres a second against a Reynolds mean of sixty, and the identity holds on it to 1.5·10⁻¹³.

What each average is for #

The two are not interchangeable and they are not rivals. They answer different questions, and which one a quantity belongs to is decided by what the quantity is.

The mass flux belongs to Favre. $\overline{\rho u} = \bar{\rho}\tilde{u}$ is an identity — it is what the Favre average is defined to make true — and $\bar{\rho}\bar{u}$ is not the mass flux at all. On the field above it is out by a factor of two, which for a flow rate is not a correction.

The mean equations belong to Favre. Average the continuity equation with Reynolds weighting and a correlation $\overline{\rho' u'}$ appears in it, in an equation that had no closure problem to begin with. Average it with Favre weighting and it is $\partial\bar{\rho}/\partial t + \nabla\cdot(\bar{\rho}\tilde{\mathbf{u}}) = 0$, which is continuity again, exactly. The same happens to momentum. That is the whole reason compressible turbulence modelling is written this way.

A hot wire belongs to Reynolds. It sits at a point and samples time, so what it reports is the time-weighted mean, and no amount of care in the calibration changes which average it is.

And a laser-Doppler measurement is somewhere between, because it reports a velocity every time a seeding particle passes, and particles arrive at a rate proportional to the mass flux. A seeded measurement in a variable-density flow is closer to a Favre average than to a Reynolds one, and how much closer depends on how the particles were mixed in.

There is a fifth entry in that list which is easy to miss, and it is the one that causes the most trouble in practice. A Reynolds-averaged momentum equation is not wrong; it is inconvenient. It can be written down, and when it is, it acquires terms containing $\overline{\rho' u'}$ and $\overline{\rho' u' u'}$ that nobody has a model for. The choice of Favre weighting is not a claim that the Reynolds equation is false — it is a choice of variables in which the unclosed terms are the ones that have been modelled for eighty years.

That framing matters because it decides what to do when the two conventions meet. The right move is never to convert one average into the other by an approximation; it is to compute the correlation that separates them, which is a well-defined statistic that a simulation has and an experiment usually does not. Where the correlation is unavailable, the honest statement is that the two numbers are not comparable, and the size of the incomparability is what this page measures.

Two Reynolds stresses #

The fluctuation is measured from a mean, so there are two fluctuations, so there are two Reynolds stresses.

They differ by fifty-six per cent on this field. And only one of them does the job the Reynolds stress exists to do: the mean momentum flux is exactly $\bar{\rho}\tilde{u}\tilde{u} + \bar{\rho}\widetilde{u''u''}$, reconstructed here to twelve figures, while the Reynolds-weighted assembly of the same pieces is out by two hundred and twenty-five per cent.

That is worth being precise about, because it is the difference between a naming convention and an error. Both stresses are well-defined statistics of the same signal. Only the Favre one appears in the conservation law, so only the Favre one can be modelled by a closure whose job is to make that law close. A closure calibrated on Reynolds-weighted stresses and inserted into a Favre-weighted equation is not slightly miscalibrated; it is supplying a different quantity.

The same doubling reaches every derived statistic. A turbulence intensity is a root-mean-square fluctuation over a mean, and there are two of each: sixty-seven per cent by one convention and eighty-eight by the other, on one flow. Papers compare those numbers between experiment and computation routinely, and the convention is often not stated.

The kinetic energy has the same problem twice over. It is a mean of a square, so it carries the weighting; and it is often reported normalised by the square of a mean velocity, which carries the weighting again. Two conventions applied to a ratio of two quantities give four possible numbers for one flow, and they span more than a factor of two here.

None of this is a subtlety about high-order statistics. It is present in the first moment, it is present in the second, and it is present in every ratio built from them — which is why a convention that is not stated is not a small omission.

Where it goes away #

At constant density the gap is exactly zero, the two stresses are the same stress, and every statistic on this page collapses to one number. That is why incompressible turbulence has one average and one closure problem, and it is why the complication here is a complication about density rather than about turbulence.

It also means the distinction cannot be avoided by making the turbulence weaker. The gap is the density-velocity correlation, so it scales with the density variation, and a nearly laminar flame still has it. Incompressible is a statement about the flow rather than about the fluid; whether the two averages differ is decided by the same statement.

The closure problem, once more, with the density in it #

The other half of what averaging costs is that the averaged equations do not close, and it is worth seeing what the density does to that.

In an incompressible flow, averaging the momentum equation produces exactly one unclosed term, the Reynolds stress, and the ladder of equations for it never terminates. In a variable-density flow the same averaging produces the Favre stress plus a turbulent mass flux $\overline{\rho' u'}$ — the very correlation that separates the two means — and that quantity is unclosed too.

So the two halves of this essay are the same problem seen twice. The gap between the averages is a correlation; the extra unclosed term in the equations is that same correlation; and a model that does not represent it is both mis-predicting the mean velocity and leaving a term out of the momentum balance. It is not two difficulties, it is one, and it has one measurement.

That is also why the distinction cannot be finessed by defining it away. Somebody could declare that the mean velocity means the Favre mean and be internally consistent; the correlation would still appear in the equations, and it would still be what a probe measures the difference of.

The general shape of it #

There is a pattern here that this collection keeps arriving at from different directions, and it is worth naming.

An exact constraint fixes one combination of the field and leaves the rest open. The mass flux is the constraint: both averages return it, one directly and one after being multiplied by the wrong density. The mean velocity is not a constraint; it is a second functional of the same signal, and the two averaging weights reach two different values of it.

So this is the same statement as the one about wakes and contours, in a different guise. There, four profiles carrying one drag disagreed about everything else. Here, one profile under two weightings gives one mass flux and two mean velocities. In both, what is determined is what the conservation law is about, and the quantity everybody quotes is not it.

The practical form is a question to ask of any reported mean in a variable-density flow: weighted how? If the answer is “by time”, it is comparable with a probe and not with a model. If it is “by mass”, it is comparable with a model and not with a probe. If there is no answer, the number is somewhere between two values that can differ by a factor of two.

What a probe cannot do about it #

It is fair to ask whether the experimenter can simply measure the other average. In principle yes, in practice rarely, and the reason is instructive.

Getting a Favre average from a point measurement requires the density and the velocity at the same instant at the same place, because what is wanted is the mean of their product rather than the product of their means. That is a simultaneous two-quantity measurement in a hostile environment, and it is exactly the measurement that is hardest to make in the flows where it matters most.

A hot wire in a variable-density flow does not even return a clean Reynolds average: its heat transfer depends on the mass flux past it rather than on the velocity, so what it responds to is already partly $\rho u$. Untangling that requires knowing the density, which is the same problem again.

So the practical position is that the two averages differ by a quantity nobody in the experiment has and everybody in the computation does. The asymmetry is worth stating plainly, because it decides where the burden of conversion falls: a computation can report either average, and an experiment usually cannot.

Where this bites hardest #

Three places, in increasing order of how easily the mistake survives.

Comparing a measurement with a computation. A hot wire returns one average and a compressible code computes the other, and the discrepancy is a physical quantity — the density-velocity correlation — rather than an error in either. Correcting for it requires knowing that correlation, which is not measured by the hot wire and not usually output by the code.

Building a profile from a traverse. Integrating a measured mean velocity profile across a jet to get a flow rate gives $\int\bar{\rho}\bar{u}\,dA$ when the flow rate is $\int\bar{\rho}\tilde{u}\,dA$, and the two differ by the integral of the correlation. In a flame that is not a small error, and it appears as a mass balance that does not close.

And reporting a spreading rate or a decay exponent. Those are fitted to a profile of the mean velocity, so they inherit whichever average was used — and a self-similar collapse in one weighting is not a self-similar collapse in the other, because the density profile changes shape as the flow develops.

Where the same weighting appears elsewhere #

Mass weighting is not peculiar to turbulence, and seeing it in two other places makes it look less like a modelling convention and more like what it is.

A tracer particle in a flow is a mass-weighted sampler of exactly this kind: it reports where it is, and it is where it is because it was carried, so its statistics are weighted by the transport rather than by time. That is why a particle-based estimate of a residence time and a probe-based one are different numbers even in a constant-density flow with a non-uniform velocity.

A flow visualisation is the same thing again. Smoke marks fluid rather than space, so a photograph is a mass-weighted picture, and the mean of what it shows is not the mean of the field at a point.

And the distinction is the reason a spatial average and a flux average of a pipe’s velocity are different numbers — the second is the one a flowmeter’s calibration is built on, and the gap between them is what a profile correction is a correction for.

In all four the pattern is the same. There is a quantity being averaged and a weight deciding how much of the average each sample contributes, the weight is set by whatever is doing the sampling, and two instruments with different weights return different means of the same field.

What to do when both are needed #

The practical question this leaves is what to do in a study that has both a measurement and a computation, and there is a short answer.

Report the correlation. A simulation has $\overline{\rho' u'}$ available at every point and almost never outputs it. Reporting it alongside the mean velocity makes the two averages interconvertible, turns an incomparability into a comparison, and costs one extra field.

Where the correlation is not available, the honest position is to say which average is being quoted and to leave the conversion undone rather than approximating it. An approximate conversion needs a model of the density-velocity correlation, which is exactly the unclosed term of the problem, so it is a modelling assumption wearing the clothes of a data reduction.

And normalise carefully. A great many reported quantities are ratios — an intensity, a spreading rate, a decay exponent — and a ratio built from a numerator in one convention and a denominator in the other is a third quantity with no name. That is easier to do than it sounds when a fluctuation comes from an experiment and a mean from a computation.

None of this is difficult. It is bookkeeping, and the reason it is worth spelling out is that the two averages are equal in every incompressible flow anybody learned the subject on, so the habit of saying which one is meant has nowhere to form.

What is not claimed #

Neither average is more physical than the other. They are two weightings of the same signal. Favre’s is the one the conservation laws close in and is therefore the one models are written in; that makes it convenient rather than true, and a Favre mean is not what a stationary observer sees.

The fields here are constructed. A two-state signal with an intermittency and a sinusoid with a prescribed phase are chosen because their averages are exact — a sum of two terms and a quadrature respectively — so that nothing in the comparison depends on a sample size. No turbulent flow was solved.

The relation between the two averages is exact and the size of the gap is not general. A factor of two follows from a density ratio of seven at an intermittency of one half, and both of those are chosen. In a flow with a density ratio of 1.2 the gap is a few per cent and can be ignored, which is why most of this subject never meets it.

And nothing here is about compressibility in the acoustic sense. The density varies because the fluid is hot, not because it is moving fast, and the two are different reasons for a density to change. A low-speed flame has this problem in full and has no compressibility effects at all.