Turbulent some of the time

A hot wire held at the edge of a turbulent jet records something that does not look like turbulence. For a while the trace is violent and full of structure; then it goes quiet, almost perfectly smooth, for as long again; then the violence returns. The probe is not moving and the jet is not changing. What is happening is that the boundary between turbulent fluid and the still air outside is itself convoluted, and it sweeps back and forth across the probe.

Everything measured at that station is an average over the two states, and the average is not a description of either.

The interface is sharp, and it is not flat #

The most important experimental fact about a free shear layer is that its edge is a surface. On one side the fluid carries vorticity and is turbulent; on the other it carries none and is smooth, irrotational and in perfectly ordinary potential motion. The transition between them happens across a few Kolmogorov lengths — a fraction of a millimetre in a laboratory jet — and it is far thinner than the eddies on either side of it.

That surface is deeply folded, and it moves. A probe at a fixed height therefore spends a fraction $\gamma$ of its time inside and $1-\gamma$ outside, and $\gamma$ falls from one in the core to zero in the free stream over a distance comparable to the layer’s own width.

The irrotational side is not still. The turbulent region pushes the outside fluid about, and the outside fluid responds as potential flow does: it moves, and it fluctuates, with no vorticity in it anywhere. A probe outside the interface therefore measures a genuine unsteadiness that has nothing to do with turbulence, and distinguishing the two needs a measurement of vorticity rather than of velocity — which is why intermittency was measured properly only once hot wires could be arranged to give a velocity gradient.

What the average actually contains #

Take the simplest possible model of the situation: the signal is $U_t$ plus a fluctuation of intensity $q$ for a fraction $\gamma$ of the time, and $U_n$ with no fluctuation for the rest. Then

$$\bar{U} = \gamma U_t + (1-\gamma)U_n,$$

$$\langle u'^2\rangle = \gamma q^2 + \gamma(1-\gamma)(U_t - U_n)^2.$$

The first term is the turbulence, weighted by how often it is there. The second term is not turbulence at all. It is the variance a two-valued signal has because it is two-valued, it depends on the difference between the two states rather than on either of them, and it peaks at $\gamma = 1/2$ where the switching is most even.

In the model drawn here — with a conditional intensity that is flat by construction, so that nothing about the turbulence varies across the edge — the measured intensity still has a peak. It sits at $\gamma = 0.64$, and it is twenty per cent above the conditional value. A feature with no counterpart in the fluid.

That is the essay’s whole point, and it generalises past this model: any quantity averaged across an intermittent region acquires a contribution from the intermittency itself, and the contribution is largest where the two states are most evenly mixed.

The size of the artefact is worth pricing, because it depends on a ratio that varies between flows. The switching term is $\gamma(1-\gamma)(\Delta U)^2$ and the turbulent term is $\gamma q^2$, so at $\gamma = 1/2$ their ratio is $(\Delta U/q)^2/2$ — the square of the velocity difference across the edge divided by the local turbulence intensity. In a jet issuing into still air that ratio is large, because $\Delta U$ is most of the jet’s own speed and $q$ is a tenth of it; in a wake behind a streamlined body the two velocities differ by only the deficit, and the artefact nearly disappears. The same statistic is a serious contamination in one flow and a footnote in another, and which it is can be worked out from the mean profile alone before any conditional sampling is attempted.

This also settles which measurements need the repair. An intensity profile across a jet’s edge needs it badly; a mean velocity profile needs it hardly at all, because the mean is linear in the signal and a linear average over two states is an honest average of them. Nonlinearity is the whole mechanism — as it is wherever an average meets a curve — and the order of the statistic decides how much of it there is.

Why the intermittency profile has the shape it has #

The $\gamma$ curve in the figures is an error function, and it is fitted rather than derived — but there is a reason it takes that form, and having it makes the model less arbitrary than a fit.

$\gamma(y)$ is the probability that the interface lies above the height $y$, which is to say it is the cumulative distribution of the interface’s own position. Measurements of that position, taken at a fixed streamwise station over a long record, come out close to Gaussian: the interface is displaced by the superposition of many large eddies, and a sum of many contributions is normally distributed for the usual reason. The cumulative of a Gaussian is an error function, which is exactly the profile that is fitted.

That reading also supplies the width. The standard deviation of the interface’s position is comparable with the layer’s own thickness — because the eddies displacing it are the layer’s largest eddies — so the intermittent region is as wide as the flow rather than being a thin skin on it. A jet’s edge is not a boundary with a little unsteadiness at it; a substantial fraction of the whole jet is sometimes outside it.

And the contamination is worse in higher moments #

The switching term in the variance is the leading case of a general result, and the general result decides which measurements survive an intermittent region and which do not.

Take the same two-state signal and compute the third central moment. Along with the turbulence’s own skewness it acquires

$$\gamma(1-\gamma)(1-2\gamma)\,(\Delta U)^3,$$

which is a pure artefact of the switching, is odd about $\gamma = 1/2$, and changes sign there. So a measured skewness profile across an intermittent edge has a positive lobe on one side and a negative lobe on the other, crossing zero where the probe is inside half the time — a striking, antisymmetric feature that is entirely a property of the interface’s position statistics and contains no information about the turbulence whatever.

The pattern continues upwards. Each moment picks up a switching contribution carrying a higher power of the velocity difference, so relative to the turbulence’s own contribution the artefact grows as $(\Delta U/q)^n$. With $\Delta U/q$ of order ten at a jet’s edge, the second moment is contaminated by tens of per cent and the fourth by a factor of thousands. A flatness or a kurtosis measured across an intermittent region is a measurement of the interface, and the numbers reported for it in the older literature — values of ten and twenty where isotropic turbulence gives three — are very largely that.

Which puts a boundary on where the site’s other essays about high moments may be read. The inertial-range results that depend on third and fourth moments — the one exact law of turbulence among them — are statements about fully turbulent fluid, and applying them to a record taken at an edge measures the interface’s wanderings raised to a power. The repair is the same conditional sampling as before, and the need for it grows with the order of the statistic being asked for.

What was concluded from the peak before it was understood #

Measured intensity profiles across jets, wakes and boundary-layer edges all show a shoulder or a secondary peak near the outer edge. Before conditional sampling was available, that feature was interpreted physically: as evidence of a distinct outer region with its own dynamics, of a production mechanism at the interface, of large-scale structures with a preferred location.

Some of that turned out to be real. The interface does have its own dynamics, entrainment does happen there, and the large-scale structures are a genuine feature. But the peak in the intensity profile is largely arithmetic, and separating the two required measuring $\gamma$ and taking the average over the turbulent state alone.

The wake, where the same statistic decides an engineering number #

A wake behind a bluff body is intermittent over most of its width, and there the arithmetic has a consequence that is not merely a matter of interpretation. The drag of the body can be inferred from the momentum deficit in its wake — that is the standard tunnel technique — and the deficit is computed from a mean profile which, across the intermittent region, is the two-state average.

This collection has already met the difficulty from the other side: the momentum-deficit route to a bluff body’s drag fails on a coarse grid because the pressure has not recovered, and it fails in a real measurement partly because the outer edge of the profile is intermittent and the mean there is not the mean of anything. Both failures are recorded rather than smoothed over, because a wake survey that returns a number is more persuasive than one that returns a range.

Conditional averaging, and what it costs #

The repair is to average separately: take the turbulent-state average $\langle\;\rangle_t$ and the non-turbulent-state one, and report both along with $\gamma$. The three together contain everything the conventional average does, and the conventional average can be recovered from them, so nothing is lost.

What it costs is a criterion — a rule for deciding, at each instant, which state the probe is in. Since the physical distinction is the presence of vorticity, the natural criterion is a threshold on a velocity derivative or on a fluctuation’s local variance, and every criterion has a threshold in it. Too high and brief turbulent excursions are missed; too low and the potential fluctuations outside are counted as turbulence. Published values of $\gamma$ at the same station in the same flow differ by several per cent between laboratories, and almost all of that spread is the threshold.

So the honest version of the measurement carries a stated criterion, and the conclusions that depend on $\gamma$ carry a sensitivity to it. This is the same discipline as stating a model and a regime, applied to a statistic instead of to a figure.

Where else the same term appears #

In a boundary layer’s outer edge, where the same interface separates the layer from the free stream, and where the intermittency profile is one of the reasons a layer’s “thickness” is a quantity with several definitions and no single value.

In a jet’s entrainment, where the fluid drawn in across the interface is what makes the jet grow, and where the momentum a jet keeps is conserved while its mass flux is not. In a turbulent spot’s own arithmetic, where the same factor decides the friction. And in transition, where a flat plate’s boundary layer becomes turbulent through the growth and merging of isolated turbulent spots. There, $\gamma$ is a function of distance along the plate rather than of height, it runs from zero to one across the transition region, and the mean friction in that region is a $\gamma$-weighted average of the laminar and turbulent values. Every transition correlation used in design is written that way, and the intermittency is doing the work — including the one this collection uses when it prices what a turbulent layer costs a flat plate, where the transition Reynolds number is really the midpoint of a γ profile rather than a place where anything switches.

And in combustion, in clouds, and in every reacting flow, where a rate that depends nonlinearly on concentration is being averaged over a fluid that is sometimes one thing and sometimes another. The mean of the rate is not the rate at the mean, which is the same inequality a wing meets in a gust — and where the nonlinearity is an exponential, as in a chemical rate, the difference is not twenty per cent but orders of magnitude.

What the picture cannot show #

The model is stated, not solved. The γ profile is an error function fitted to what measurement finds, the two states have flat properties by construction, and the interface has no dynamics in it whatever. Nothing here computes an intermittency factor from the equations of motion — that requires a turbulent field with a resolvable interface, which is beyond what this collection’s solver can do.

A two-state model is a caricature. The turbulent state’s own intensity varies across the layer in reality, the non-turbulent state is not quiescent, and the interface has a finite thickness with partially rotational fluid inside it. Each of those adds a term; none of them removes the switching term, which is the one the essay is about.

The intensity is measured in one component. A hot wire gives the streamwise fluctuation, and the two-state term involves the difference in mean velocity between the states, so the artefact is largest in the component the instrument measures best. A cross-wire measuring the shear stress sees a smaller version of the same term, because the non-turbulent state contributes no correlated transverse fluctuation — which is one reason stress profiles look better behaved at an edge than intensity profiles do.

And the entrainment mechanism is missing entirely. How irrotational fluid becomes turbulent as it crosses the interface — whether by small-scale nibbling or by large-scale engulfment — is an active question, and this collection’s figures say nothing about it because nothing here resolves the interface.

What it means for a model #

A turbulence model solves for the conventional averages: it transports $\overline{u'^2}$, or a turbulent kinetic energy, or a stress. At an intermittent edge, part of what it is transporting is the switching term, which is not turbulence and does not obey the equations the model is written from.

That is a genuine defect and its size is measurable, and it is of the same species as the difficulty a wall function meets at the other edge of a layer: a boundary condition or a calibration applied where its assumptions do not hold, with no error estimate attached and no residual that goes up when it fails. In a jet’s outer edge a substantial fraction of the measured kinetic energy is switching rather than turbulence, and a model calibrated to reproduce the measured profile has been calibrated to reproduce an artefact of averaging.

The best models handle it by not pretending otherwise: they are fitted to the conventional averages because those are what a calculation produces, they are known to be wrong in detail at the edges, and the errors are tolerated because the edges carry little of the momentum. That is a reasonable engineering position and a poor scientific one, and the difference between them is what this essay’s arithmetic makes visible.

Who found it, and when #

Townsend measured intermittency in a wake in the late 1940s and gave the two-state decomposition; Corrsin and Kistler’s 1955 report established the sharpness of the interface and named it the viscous superlayer, and their argument that its thickness is set by the Kolmogorov scale is still the standard one. Conditional sampling as a routine technique arrived with digital data acquisition in the 1970s.

The surprising connection is with what averaging costs at the beginning of this field. There the difficulty was that averaging a nonlinear equation leaves a term the mean cannot supply. Here the difficulty is one step earlier and worse: the average is over an ensemble of two different flows, so even the quantities the model transports are not properties of either. A closure problem is at least a problem about a well-defined quantity; an intermittent average is a quantity that has to be defined before it can be modelled.

Where the ladder goes next #

Beside this rung sits the small-scale intermittency of the dissipation field, which is a different phenomenon with the same name: there the quantity that is patchy is $\varepsilon$ itself, and the consequence is that the exact law’s neighbours stop obeying the dimensional argument. Above it lies the interface’s own dynamics — entrainment — which this collection has not yet earned the machinery to write about.