Transition along a plate is a count of spots, and one number gives its length

A puff that does not know how old it is described what transition is in a pipe: isolated turbulent patches, each with a lifetime and a chance of splitting, that the mean flow carries along and nobody can draw. Its statement was about statistics. The patches could be counted without being computed. Turbulent some of the time defined the intermittency, the fraction of time a place is turbulent, and in one sentence mentioned that a boundary layer’s transition can be written the same way. This essay does the arithmetic behind that sentence, since it turns out to be very nearly the whole of what’s known in a form an engineer can use, and it’s simple enough to check directly.

In 1951 Howard Emmons was at Harvard running a sheet of water over a sloping glass table. A water table’s shallow flow is an analogue of a compressible flow, and he was looking at something else. He noticed, at a low speed and on the sheet’s surface, small triangular patches of disturbed flow that appeared at apparently random places and times, and swept down the table, growing as they went. Nobody had seen one before. Schubauer and Klebanoff in 1955 created single spots in a real boundary layer in a wind tunnel with an electrical spark, and measured them: an arrowhead, with its leading point at 0.88 of the free-stream speed and its trailing edge at about half of it, spreading sideways at about eleven degrees.

So a transition zone isn’t a place where the flow gradually changes. Every station in it is alternately laminar and turbulent, with clean edges between. Mean quantities change smoothly because of how often each state is present, and the fraction is what needs a theory.

Coverage #

The theory is a piece of pure geometry. A point in the plate at distance $x$ from where spots begin, at a moment $t$, is turbulent if and only if some spot’s region has reached it. Which spots could have reached it? Those born in the cone of dependence: a stated range of birth positions and times. If spots are born independently at a constant rate, then the number born in that region is a Poisson variable, and the point is laminar only if that number is zero. Therefore

$$1-\gamma = \exp\bigl(-n\,V(x)\bigr),$$

where $V$ is the dependence volume. This requires one further physical fact: that spots pass through one another. They don’t interact in a way that alters the sum of their areas. If they just merge, a point covered by two is only as turbulent as one covered by one, and the coverage’s formula is exactly right.

Emmons wrote that in 1951. For a spot with a straight-sided arrowhead shape and speeds $U_L$ and $U_T$ at its edges, the dependence region at distance $x$ for a source on a line has a height that opens with time from nothing at the leading edge’s arrival to $x\tan\alpha$ at the trailing edge’s. So the volume is a pyramid,

$$V(x)=\frac{x^2}{U}\tan\alpha\left(\frac{U}{U_T}-\frac{U}{U_L}\right)=\frac{\sigma x^2}{U},$$

with Emmons’s spot-shape constant $\sigma$, which is 0.168 for measured spots. The dimensional content is that the region’s streamwise and spanwise sizes both grow as $x$, so its area grows as $x^2$. Therefore the intermittency is

$$\gamma=1-\exp\left(-\frac{n\sigma (x-x_t)^2}{U}\right).$$

Concentrated breakdown #

The source’s shape is what made this a working theory. Emmons had spots born over the whole plate at a rate depending on $x$, and this adds an unknown function. Narasimha showed in 1957, by plotting measurements a particular way, that spots in natural transition are in fact born in a narrow band: there’s a line at $x_t$ upstream of which there are essentially no spots, and downstream of which virtually no more begin. It’s now called concentrated breakdown, and it reduces the whole of that unknown function to two numbers, $x_t$ and $n$, with the second being a spot production rate per unit span and time on that line. The figure’s source is such a line.

The Monte Carlo draws each spot’s birth at random on a long line in span and time, and tests each station against every spot’s arrowhead. It knows nothing of the exponential, and it comes out on the formula. The reason it’s worth doing is that this is a statement about the geometry of coverage rather than about the fluid, so it confirms the arithmetic and identifies what must be measured. With a real fluid in place, the same plot can be made. Dhawan and Narasimha made it in 1958 on many experiments, and the axis they found put nearly every one of them on one line.

On that axis, every transition ramp becomes a straight line, whose intercept gives $x_t$ and slope gives $\sqrt{n\sigma/U}$. It’s rather like plotting a logarithmic friction law on semi-log paper, where a whole family of measurements falls on one line and what remains is one constant. Once it’s a straight line, what remains is a number, and in a zone of length $\lambda$, between 25 and 75 per cent intermittency, the whole of the rest is

$$\lambda=\left(\sqrt{\ln 4}-\sqrt{\ln\tfrac43}\right)\sqrt{\frac{U}{n\sigma}}.$$

The zone’s shape is universal, and its length is a statement about one number.

Spots in a wind tunnel, counted #

The size of these numbers is worth working out, since a rate per unit span per unit time doesn’t mean much until someone tries it. Take a flat plate in a wind tunnel at twenty metres a second in air, with one per cent free-stream turbulence. Mayle’s correlation gives $\hat n\sigma=1.5\times10^{-11}$, which in dimensional terms is about 3200 spots per metre of span every second. That is a lot of spots. Half the layer is turbulent 16 centimetres past the breakdown line, and there a spot is about 3 centimetres wide on either side and takes about 7 milliseconds to pass. Multiply the rate by the dependence volume at that place and the answer is about 0.7 spots in the dependence region at any moment, which is what a half must mean: $1-e^{-0.69}$. So a transition zone is not dense with spots. A probe sees them passing singly, with gaps, and that’s why the square wave in the first figure looks the way it does.

That’s also part of why the independence assumption holds so much better than it deserves to. Each spot’s own region of influence is a fraction of the zone, and through most of the ramp spots overlap rarely, up to the final stage, where any interaction hardly matters since they’d occupy the same region anyway. The formula’s most doubtful assumption is thus tested only in the part of the ramp where it isn’t needed.

A blade in a turbine at 200 metres a second, hot enough to have a kinematic viscosity of $3\times10^{-5}$, at five per cent turbulence, is a very different place. The rate rises by a factor of more than a thousand, and the zone falls to about 6 millimetres. Its zone has become a small part of the blade’s chord, and it’s where the blade’s cooling must suit the largest heat load on the surface, which comes at the end of the zone. So the same formula that states a zone as long as a wing’s laminar run also states a zone that’s only just resolved by the instruments. It covers the whole range without a change of form.

The opposite limit is the quietest flow. Extrapolating the correlation to a twentieth of a per cent, and it’s a long way beyond where it was fitted, gives a zone Reynolds number above two million, longer than the laminar run itself on many surfaces. That would be a very long transition indeed, and it’s in the range where the non-normal growth of disturbances, rather than the turbulence level, sets the source of spots. Quiet flight is a case for which the correlation shouldn’t be believed.

Kolmogorov’s crystals #

The same formula has been independently written at least four times, and that’s the surprising connection here. Kolmogorov worked it out in 1937, for crystals growing in a melt. Avrami, Johnson and Mehl did so around 1939. A melt held below its freezing point nucleates crystals at random, each one grows until it runs into a neighbour, and the fraction crystallised rises as

$$1-\exp\left(-k\,t^{m}\right),$$

with $m$ depending on the dimension and the nucleation history. The argument is exactly Emmons’s: a point stays uncrystallised only if no nucleus has appeared in its dependence region, and crystals’ overlap is accounted for automatically by working with the uncovered fraction. A transition zone with concentrated breakdown has $m=2$ in distance: a one-dimensional line of nuclei, each growing in two dimensions. Change the line to a volume distribution of nuclei and the exponent changes, as it does in crystals. It’s the same statistics behind the coverage of random discs, the fraction of a forest burned from random lightning strikes, and the percentage of a plate rusted by random pit formation.

It also makes sense of why Dhawan and Narasimha’s plot looks so clean: it’s the same straight line by which metallurgists read off nucleation rates from an Avrami plot, and it has the same weaknesses, in that a straight line doesn’t prove the mechanism.

The friction, read off #

The skin friction during transition is exactly what a mean of a square wave would give, as a weighting of the two laminar and turbulent states.

The result’s overshoot of the fully turbulent curve at the end of the zone isn’t a surprise, though it has often been treated as one. The turbulent layer just past the breakdown is thin, since it started at $x_t$ and not at the leading edge, and a thin turbulent layer has a high friction. That is a simple consequence of what a virtual origin is, and it’s why a measured maximum in heat transfer in a turbine blade lies near the end of the zone. A computed heat-transfer profile on a turbine blade depends on the ramp a good deal, because a heat-transfer coefficient can rise several-fold across it. That’s what made Mayle’s 1991 survey so important to the gas-turbine industry, and it’s why they, far more than aircraft designers, care about the zone’s length.

How long the zone is #

So the zone’s length rests on a single measured quantity, the spot production rate. Its value is what Narasimha’s non-dimensional parameter $\hat n\sigma=n\sigma\nu^2/U^3$ expresses, and it has no known derivation from first principles. In a quiet flow it depends on free-stream turbulence, and Mayle’s survey gives

$$\hat n\sigma\approx 1.5\times10^{-11}\,\mathrm{Tu}^{7/4}$$

for a turbulence level $\mathrm{Tu}$ in per cent.

That length is a sizeable fraction of a boundary layer’s whole Reynolds number range on many surfaces, and it depends only weakly on the turbulence level. On an aeroplane’s wing in quiet air, it’s about the size of the laminar run before it. On a turbine blade with five per cent free-stream turbulence, it’s short, and in a fan’s wake the spots come in bursts tied to the upstream blade’s passing, making a periodic transition with time-varying spot production that requires the same formula in time rather than steady.

The production rate’s dependence is therefore doing a job similar to that of the number that is not a number for a critical Reynolds number. The onset of transition, $x_t$, depends on the free stream’s disturbances, as that essay said. Mack’s and Abu-Ghannam and Shaw’s correlations express it. In the same way the zone’s length depends on the same disturbances, and the two are correlated with one another. A real design takes the free stream’s level as a measured input, and it’s frank about the rest.

What the picture cannot show #

Spots that are not independent. Coverage assumes each spot’s birth is uncorrelated with the others. In a real flow, a passing spot leaves behind a calmed region, a stretch of the layer stabilised by the spot’s own wake, at the speed of about half the free stream and in which new spots can’t form. Schubauer and Klebanoff first saw it, and Gostelow and others have shown it’s a major part of what’s seen behind a spot on a turbine. Real spots also interact when they pass closely, and they may merge or induce new ones. The Monte Carlo has none of this in it.

The spot’s shape. The straight arrowhead is a simplification of a complicated, overhanging, three-dimensional object that has a tongue at its leading edge and a tail of streaks behind. The shape’s numbers come from visual and hot-wire measurements in a zero-gradient layer. An adverse pressure gradient widens spots and raises their angle, and shortens the zone. With one constant $\sigma$, the formula has no way of expressing that, and a varying $\sigma$ along a real blade gives up the closed form.

Everything inside a spot. This is a theory with the turbulent regions treated as occupied and that’s all. It’s silent on what goes on inside. The grid nobody can build has the size of the computation needed to look inside one, and even then, the spot-production rate isn’t something that can be got from a statistically small sample.

What creates a spot. Coverage needs the rate as an input, as the transition’s first statement would lead anyone to expect. It’s nowhere predicted, and that’s the hardest question in the subject. On an aeroplane’s wing the first spots grow from the breakdown of Tollmien–Schlichting waves, on a turbine blade they come from free-stream eddies, and on a rough plate from the roughness itself, with no shared mechanism, and a rate that depends on each. The profile Rayleigh cleared and viscosity did not is about the first of these, and it gets as far as the amplification.

The convention #

Lengths and times are in arbitrary units with $U=1$, since the formula’s form is what the Monte Carlo tests; the rate $n$ is in spots per unit span and time, and the coefficient $\sigma$ is dimensionless. The spot shape is a straight-edged arrowhead with its apex at the leading edge, whose width at a given station grows linearly with its age. Intermittency is the fraction of a time window of 4000 units, sampled at two thousand instants, on a span of 400 with the span treated as periodic. Its scatter is about 0.02 at mid-ramp. The skin-friction figure uses Blasius’s law for the laminar part and $0.0592\,Re^{-1/5}$ from a virtual origin for the turbulent part.

How it was checked #

The principal test is the Monte Carlo itself, run with no reference to the formula it’s compared to. Treating a spot’s width at a station as its full value from its first arrival, rather than as opening in proportion to its age, makes the exponent nearly twice too large, which is a useful reminder of what the pyramid means. The model refuses a rate of zero and a span of zero.

Who found it, and when #

Emmons’s water table and coverage formula are from 1951, Schubauer and Klebanoff’s spark-generated spots from 1955, and Narasimha’s concentrated breakdown from 1957. Dhawan and Narasimha’s universal plot is from 1958; Abu-Ghannam and Shaw’s correlation from 1980; and Mayle’s survey from 1991. Kolmogorov’s nucleation statistics are from 1937, with Johnson and Mehl’s and Avrami’s from 1939.

Still open: what the calmed region does to the coverage #

The coverage formula assumed that a spot leaves nothing behind. The next calculation would add to the Monte Carlo an exclusion zone behind each spot, of a stated length and at a stated speed, inside which no new spot may start, so that births aren’t Poisson. It would ask how large an exclusion moves the intermittency ramp from a clean straight line on Narasimha’s axis, at what upstream rate the effect is detectable, and whether the small but persistent curvature often seen at high intermittency on that plot could be read as a measurement of the calmed region’s length.