The mean is not the flow

Almost everything published about a turbulent flow is a mean. The velocity profile in a pipe, the pressure distribution on a wing, the wake behind a cylinder — each is an average over time or over realisations, and each is presented as a picture of the flow. It is worth being exact about what an average of a flow actually is, and the cleanest way to be exact is to average something whose fluctuation is known in closed form.

Take a cylinder in an inviscid stream that oscillates: $U(t) = U_0\cos\omega t$, with the flow at every instant the exact potential solution for that instantaneous speed. Everything about it can be written down. Its mean velocity is exactly zero at every point. And its mean is not a flow.

Zero mean velocity, and a pressure field anyway #

The instantaneous flow is $U(t)\,\mathbf{q}(\mathbf{x})$, with $\mathbf{q}$ the unit-speed potential solution. Averaging over a cycle:

$$\langle \mathbf{u}\rangle = \langle U\rangle\,\mathbf{q} = 0$$

at every point, exactly and by symmetry. There is no mean flow. Nothing is going anywhere.

The pressure is a different matter. Bernoulli’s equation for this flow gives $p = -\tfrac{1}{2}\rho U^2|\mathbf{q}|^2$ plus an unsteady term that averages to nothing, and $\langle U^2\rangle = U_0^2/2$, so

$$\langle p \rangle = -\tfrac{1}{4}\rho U_0^2\,|\mathbf{q}|^2.$$

Half the peak suction, everywhere, permanently. In coefficient form the mean reaches $-2.00$ at the shoulders where the instantaneous value reaches $-3$, and it averages $-1.00$ over the whole surface where the instantaneous distribution averages zero. A body in this flow is held in a steady field of suction by a flow that delivers no net momentum at all.

That the mean pressure has no positive part is not an artefact of the geometry. Every point on the surface is a stagnation point twice per cycle and a shoulder never, or a shoulder twice and a stagnation point never; the average of $1 - q^2$ over the cycle is $1 - q^2$ evaluated with $\langle U^2 \rangle$, and $q$ exceeds one over most of the body.

The resultant is still zero. Integrating the mean pressure round the body gives a force of $7\times10^{-16}$, which is d’Alembert’s paradox arriving in an unexpected place: a real, steady, strongly varying pressure field with no force in it.

The term the mean equation cannot supply #

Write the momentum equation, average it, and one term does not survive intact. The advective term is quadratic, so its average is not the advective term of the average:

$$\langle (\mathbf{u}\cdot\nabla)\mathbf{u}\rangle \;\ne\; (\langle\mathbf{u}\rangle\cdot\nabla)\langle\mathbf{u}\rangle.$$

For this flow the left-hand side is exactly $\tfrac{1}{2}U_0^2\,(\mathbf{q}\cdot\nabla)\mathbf{q}$ and the right-hand side is exactly zero. The mean equation is short of a term whose size is $U_0^2/2$, and the missing term is the divergence of the correlation $\langle u_iu_j\rangle$ — which is, by definition, a Reynolds stress.

That is the closure problem, with everything known. In turbulence the correlation is unknown and has to be modelled, and the whole subject of turbulence modelling exists to guess it. Here it is computed, and its value is instructive: it is largest where the flow is fastest, it points towards the body along the shoulders, and it is precisely what holds the mean pressure field in place.

The example matters because it removes every excuse. There is no randomness here, nothing chaotic, no sensitivity to initial conditions, no unresolved scales. The closure problem is not caused by turbulence. It is caused by averaging a nonlinear equation, and turbulence merely makes the missing term impossible to compute.

What an average has to be, for any of this to work #

The decomposition above went through cleanly, and it is worth noticing that it did not have to. An average is an operator, and the tidiness of splitting a field into a mean and a fluctuation with zero mean depends on three properties that Reynolds stated and that not every average has.

It must be linear, so that the average of a sum is the sum of the averages. It must commute with differentiation, so that averaging the equations of motion is a legitimate operation at all. And it must satisfy $\langle \langle a\rangle b\rangle = \langle a\rangle\langle b\rangle$ — an already-averaged quantity passes through untouched. That third rule is the load-bearing one: set $b = 1$ and it says the average is idempotent, $\langle\langle a\rangle\rangle = \langle a\rangle$, from which $\langle u'\rangle = 0$ follows immediately, and from which the cross terms in $\langle (\langle u\rangle + u')(\langle u\rangle + u')\rangle$ vanish and leave exactly one correlation to worry about.

The cycle average used in this essay has all three exactly. So does an ensemble average over realisations, and so does a time average over an infinite record. A running average over a finite window has none of them exactly, and the third not even approximately.

That is not a pedantic point, because a finite window is what every large-eddy simulation uses. Filter the velocity with a kernel of width $\Delta$ and the filtered field is smooth but not idempotent: filtering it again changes it, so the residual $u - \bar{u}$ has a filtered value that is not zero. Work through the same expansion and the stress appearing in the filtered momentum equation is no longer one correlation but three — the cross terms between resolved and residual fields survive, and there is a further piece, $\overline{\bar{u}_i\bar{u}_j} - \bar{u}_i\bar{u}_j$, built entirely out of quantities the simulation already has.

That last piece is not a modelling problem. It can be evaluated exactly, from the resolved field, at every time step. It exists because the filter is not a projection, and it would be there in a flow that was perfectly smooth and entirely laminar — which is the same lesson this essay’s zero-mean cylinder teaches, arriving from the other direction. The closure term is a property of the operator, not of the physics it was applied to.

The commutation rule fails too, and it fails where it is least convenient. A filter of constant width commutes with $\partial/\partial x$; a filter whose width varies with position does not, and every practical grid stretches towards a wall. The difference between filtering a derivative and differentiating a filtered field is then a term of its own, with no physical content whatsoever — it is the mesh, appearing in the equations of motion as though it were a stress.

So the honest statement of what an averaged equation is has a clause in it that is usually left out. It is not the equation for the mean flow; it is the equation for the flow under a stated operator, and changing the operator changes the equation. A cycle average, an ensemble average, a running mean over ten seconds and a box filter one grid cell wide produce four different sets of extra terms from the same instantaneous field. The mean is not the flow, and it is not even a single object — which average was taken is part of the statement, and it belongs beside the numbers rather than in the methods section.

There is a practical corollary that follows immediately and is worth stating before the list. Any paper reporting a mean field owes its reader the operator alongside it — the window, the ensemble, the filter width — and a great many report only the field. The number that would let somebody else reproduce the extra terms is the one that goes missing, and it goes missing because it is thought of as a detail of processing rather than as half of what was measured.

Three more things the mean does not carry #

A mean streamline is not a path. The mean field’s streamlines here are undefined, because the mean field is zero; more generally, in an unsteady flow the streamlines of the average and the average of the paths are different curves, and neither is what a marked parcel does. This collection makes that distinction in its first field, and averaging sharpens it: a time-averaged velocity field is a legitimate object whose integral curves are not trajectories of anything.

A mean acceleration is not the acceleration of the mean. The parcel acceleration $D\mathbf{u}/Dt$ contains the same quadratic term, so its average has the same extra piece. That is why the mean flow in a duct with a fluctuating supply is not the flow the mean supply would produce.

A mean vorticity is not the vorticity of the mean, once viscosity is admitted. In this inviscid example the vorticity is zero at every instant so nothing is lost, but in a real flow the mean vorticity equation carries its own correlation term, and the vorticity a wall makes — which is the pressure gradient along it — is a statement about the instantaneous field that has to be averaged before it can be compared with a measured mean.

And a mean energy is not the energy of the mean. The kinetic energy of this flow averages $\tfrac{1}{4}\rho U_0^2\int|\mathbf{q}|^2$ over the field, while the energy of the mean field is zero — so all of the energy in the problem is in the fluctuation, and none of it appears in a description that carries only the mean.

What an average does keep #

The essay so far is a list of things the mean does not have, and it would be a misreading to conclude that mean fields are not worth computing. What survives an average is exactly what is linear, and a good deal of fluid mechanics is.

Mass conservation is linear in the velocity, so the mean field is divergence-free if the instantaneous one is — which is why a measured mean velocity profile can be integrated for a flow rate and the answer is right. Circulation is a linear functional of the velocity, so the mean circulation round a loop is the circulation of the mean field. A pressure gradient enters the momentum equation linearly, so the mean pressure gradient is the gradient of the mean pressure.

Every difficulty in this essay comes from exactly one term, the advective one, and every other term in the equations of motion averages honestly. That is worth knowing because it says where to look: any statement about a mean flow that uses only linear operations is safe, and any statement that multiplies two fluctuating quantities together needs the correlation.

A pile in the surf, with the numbers #

The practical version of this essay is the load on a structure in a wave, and it is worth doing because the answer is not small and the arithmetic is the same.

A slender pile in an oscillating flow experiences a drag force proportional to $u|u|$ — the absolute value is there because the force reverses with the flow — and an inertia force proportional to $du/dt$. Average over a cycle and the inertia term gives nothing, since it is linear in a zero-mean quantity. The drag term gives nothing either, because $u|u|$ is odd.

So the mean force is zero and the structure still fails. What matters is the mean of the square: the root-mean-square load is $\sqrt{\langle F^2\rangle}$, the fatigue damage accumulates as a high power of the load range, and the design case is the extreme rather than the mean. The whole of offshore structural design lives in the second moment, and a description carrying only means would say a pile in the surf is unloaded.

The same is true, less dramatically, of a building in a gusty wind, of a heat exchanger tube in a cross-flow, and of an aeroplane in rough air. In each case the mean is honest, uninformative and insufficient, and the quantity that matters is the one this essay’s arithmetic produces.

Why the effect is not small #

It would be reasonable to expect a mean-square effect to be a correction. It is not, and the reason is worth stating: the mean of the square is not a correction to the square of the mean when the mean is zero. It is the entire quantity.

Where there is a mean, the second-order term is a correction and is usually small: a wing in smooth air at ten degrees has fluctuations of a few per cent, and the mean lift and the lift at the mean differ by a fraction of a per cent, which is why nobody thinks about it. The correction becomes the whole answer only when the first-order term vanishes, and that is the situation in every oscillatory flow with no mean — a sound field, a wave, a shaking building, a heart valve during diastole — and in each the steady loads, the steady drift and the steady mixing are second-order quantities with no first-order counterpart to be corrections to. A structure in the surf is loaded by a flow whose mean velocity is nothing at all — and the same arithmetic, applied to a fluctuating speed rather than a fluctuating direction, is why the dynamic pressure a gusty wind delivers exceeds what its mean speed accounts for by the square of the gust intensity.

The three essays this one points at are the three ways that plays out. In a viscous layer the second-order term drives an actual steady flow, which is Rayleigh’s streaming. In a wave the parcels advance even though the field at each point averages to zero, which is Stokes’ drift. And in a wing flying through gusts the mean lift is not the lift at the mean angle, which is Jensen’s inequality with aerodynamics attached.

The middle one of the three can be measured here rather than quoted, because the wave field is known exactly and the parcels can simply be integrated through it.

The viscous member of the same family is the one that produces a current rather than a displacement, and it is the sharpest of the three because the fluid it moves has nowhere to store the momentum.

What the picture cannot show #

The flow is quasi-steady, and that is an assumption with a number in it. Treating the field at each instant as the steady potential solution requires the oscillation to be slow compared with the time for the flow to establish itself round the body — a frequency parameter $\omega a/U_0$ much less than one. Where it is not, the added-mass term matters and the instantaneous field is not the steady one; the mean pressure then acquires a further contribution, which this figure does not have.

The fluid is inviscid, so nothing here separates. A real cylinder in an oscillating flow sheds vortices, and at moderate amplitudes the shedding organises into patterns that depend on the ratio of stroke length to diameter. That is a large effect and the mean pressure of the real flow differs substantially from this one. What survives is the arithmetic — that the mean of the square is not the square of the mean — which is what the essay is about.

And the average is over a cycle, not over an ensemble. For a periodic flow the two agree; for a turbulent one they need not, and the difference is what the intermittency essay is about in one particular case.

Who found it, and when #

Reynolds’ 1895 paper introduced the decomposition and the stresses that bear his name, and it did so for a turbulent flow — which is why the stresses are usually explained as a turbulence phenomenon. The observation that any averaging of a nonlinear system produces the same structure is older than fluid mechanics and appears wherever a mean-field description is attempted; in fluid mechanics the cleanest early statement of the oscillatory case is Rayleigh’s work on acoustic streaming in 1884, which starts from exactly this cancellation.

The surprising connection is with the site’s own account of what a flow is. The first field of this collection insists that a flow is a field — a velocity at every point at every instant — and that pictures which summarise it are not it. An average is the most respectable summary available, and it is still a summary: it obeys different equations, carries different energy, and has properties the flow does not have. The mean pressure field above is a real thing that a manometer would measure, and it belongs to no flow at all.

Where the ladder goes next #

The rung above is the same statement carried into a viscous layer, where the residual term does not merely exist but drives a measurable steady current with a coefficient of $-3/4$. Beside it lies the kinematic version — a parcel’s mean displacement in a wave whose mean velocity is zero — and beyond both lies the ordinary closure problem, where the correlation is real and nobody can compute it.