The wake is a tenth of the velocity and a third of the displacement

The layer with no length in it derived the logarithm in a turbulent wall profile from a single argument: in a band of heights too far from the wall for viscosity and too near it for the layer’s own thickness, no length can enter, and the velocity gradient must be the friction velocity over the distance. It was careful about where that band is — from about thirty wall units to a tenth or a fifth of the layer’s thickness, under two decades and under a fifth of the layer — and it named what lies above: a systematic departure Coles called the wake, strong in a boundary layer, weaker in a pipe, dominant near separation. Three buffer layers, one friction then integrated three wall laws across a pipe and attributed what they missed to “the missing wake”. This essay puts it in and asks what it carries.

The profile above the logarithm #

Coles’s 1956 proposal was an additive function: the velocity is the law of the wall, which knows only the friction and the wall distance, plus a function of the outer variable $y/\delta$ that knows only the layer’s thickness,

$$u^+ = u^+_{\rm wall}(y^+) + \frac{2\Pi}{\kappa}\,\sin^2\!\frac{\pi y}{2\delta}.$$

The inner part here is Spalding’s single formula, which joins the viscous sublayer, the buffer layer and the logarithm in one expression. The outer part is zero at the wall, rises smoothly and reaches $2\Pi/\kappa$ at the edge; $\Pi$ is the wake’s strength, measured to be about 0.55 for a layer on a flat plate at moderate Reynolds numbers. The figure shows what it does at a momentum-thickness Reynolds number of $10^4$, a layer of 3,484 wall units: below a fifth of the layer the curves coincide on the logarithm, and above it the wake lifts the profile by up to 2.68 wall units. The edge velocity is 27.6 wall units, so the wake is a tenth of it.

In wall units the wake is the same size at every Reynolds number — $2\Pi/\kappa$ does not depend on it — while the logarithm keeps climbing with the layer’s thickness, so the wake’s share of the edge velocity falls slowly: twelve per cent at a momentum-thickness Reynolds number of a thousand, seven at a million. A tenth is the size that invites calling something a correction. The rest of the essay is about why it is not one.

Why the outer part is a separate function #

The additive form is not an arbitrary fit, and its justification is the same argument the logarithm came from, run from the other side. Far from the wall the eddies that carry momentum are as large as the layer, and the velocity there cannot know the viscosity or the wall distance in wall units; it can know only how far it is through the layer and how strongly the wall is pulling. So the outer profile, measured as a defect below the edge velocity in units of the friction velocity, must be a single function of $y/\delta$ for every layer with the same history. Clauser showed in 1956 that measured profiles collapse on such a defect law. Near the wall the profile is a function of $y^+$ alone. A profile that is both must, in the band where both hold, be the logarithm — which is the first essay’s result — and Coles’s wake is simply the outer function with the logarithm taken out of it. What remains after the logarithm is removed is not small: it is whatever the large eddies do that the overlap’s dimensional argument has no access to.

That also says why the wake is the part that depends on the layer’s history. The inner law is universal because the wall region forgets its past within a few eddy turnovers; the outer region’s eddies are large and slow, and they carry the effect of whatever pressure gradient, roughness change or free-stream disturbance the layer met upstream for a long way downstream. The wake strength is the single number in which that memory is kept.

Where the wake lives #

The velocity defect — how far below the edge velocity the profile is at each height — shows the wake’s geography. Without the wake the defect falls only logarithmically with height, so the profile is still well short of the edge velocity close to the edge and jumps to it there with a kink. With the wake the defect is carried smoothly to zero across the outer four-fifths of the layer. That region is most of the layer’s thickness, so whatever happens there is weighted by most of the thickness in every integral of the profile — and the integrals are what the drag and the layer’s growth are made of.

The friction law the wake supplies #

The edge velocity in wall units is the friction coefficient in disguise: $c_f = 2/U_e^{+2}$, because $U_e^+$ is the edge velocity divided by the friction velocity. So the profile’s value at the edge — the logarithm evaluated at the layer’s thickness plus the whole wake — is a friction law. Without the wake it would be the log law’s value at $\delta^+$, two and a half wall units too low at this Reynolds number, and the friction coefficient would come out twenty per cent too high. The wake is the term that makes the log law a law for a boundary layer’s drag rather than for a pipe’s, because a pipe’s centreline and a boundary layer’s edge are different places with different outer flows.

The check is against something that does not know about logarithms. Ludwieg and Tillmann fitted measured flat-plate and pressure-gradient layers in 1949 to a power-law formula in the momentum-thickness Reynolds number and the shape factor, $c_f = 0.246\cdot10^{-0.678H}\,\mathrm{Re}_\theta^{-0.268}$. Evaluated at the shape factor the composite profile gives, it agrees with the composite friction within 1.4 per cent at a momentum-thickness Reynolds number of 3,000, 0.26 per cent at $10^4$ and 4.9 per cent at $3\cdot10^4$, and the two part beyond that range — the correlation’s power law was fitted to layers between those numbers, and a power law cannot follow a logarithm far. Within the range both were built for, two independent descriptions of a layer, one from a dimensional argument plus a wake and one from fitting measurements, give the same drag.

A mixing length that stops growing #

The wake has a mechanical reading in the oldest turbulence model there is. Prandtl’s mixing length says an eddy near a wall is as large as its distance from the wall, $\ell = \kappa y$, and integrating that gives the logarithm everywhere — including all the way to the edge, where it is wrong. The repair every practical mixing-length calculation makes is to stop the length growing: Escudier’s cap of about nine hundredths of the layer’s thickness is the usual one. With the cap, the outer eddies are smaller than the logarithm assumes, they carry momentum less effectively, and the velocity rises faster towards the edge than the logarithm would let it. That faster rise is the wake. So the wake is, in the model’s terms, the statement that the largest eddies in a boundary layer are limited by the layer and not by the wall — the outer length entering, exactly where the overlap argument said it would.

Where the strength comes from #

The strength 0.55 is a measurement and it is not constant everywhere. Close to transition, below a momentum-thickness Reynolds number of a few hundred, measured wakes are weaker — Coles found the strength falling towards zero as the layer gets younger — and that is why the composite friction and Ludwieg and Tillmann’s part company at $10^3$: the correlation was fitted partly to such young layers, the composite profile holds the mature wake. A turbulent free stream weakens the wake too, by stirring the outer layer from above until its large eddies look less like a wake and more like the free stream; a layer under free-stream turbulence of a few per cent has a measurably smaller wake and a higher friction. In each case the wake strength is the single number that records the outer flow’s history, and like a transition number, it depends on the laboratory as well as the layer.

A layer that fills out slowly #

The shape factor, the ratio of the displacement to the momentum thickness, says how full the profile is. A laminar flat-plate layer has 2.59; the turbulent one is far fuller, 1.52 at a momentum-thickness Reynolds number of $10^3$, 1.33 at $10^4$, 1.21 at $10^6$. It falls towards one — a uniform profile — but only as fast as the inverse of the logarithm of the Reynolds number, because the velocity defect is a fixed number of wall units while the edge velocity grows logarithmically. A turbulent layer at any Reynolds number reached in practice is a long way from uniform, and a fixed shape factor of 1.3 or 1.4, which the simplest integral methods assume, is right only over a decade of Reynolds number.

A tenth of the velocity, a third of the displacement #

The plainest measure of what the wake carries is the displacement thickness — the distance by which the layer pushes the outer flow away — computed with the wake and without it at the same thickness. The wake carries 33 per cent of it at $\mathrm{Re}_\theta = 10^3$ and 35 per cent at $10^5$, roughly a third at every Reynolds number. A contribution that is a tenth of the velocity becomes a third of the displacement because it is spread over the outer four-fifths of the layer, where the deficit it adds is multiplied by the most thickness.

That is the sense in which the log law describes the layer’s friction and not its thickness. The friction is set close to the wall; the displacement, the momentum deficit and with them the layer’s growth and its effect on the outer flow are set mostly in the wake. A calculation that resolved only the log law would get the drag of a flat plate nearly right and the layer’s displacement a third wrong — and what the outer flow sees of a body is the displacement.

The drag of a whole plate #

The friction law at a point becomes the drag of a plate by the momentum integral: the momentum thickness grows at half the local friction coefficient per unit length, and the plate’s total drag is the momentum thickness it ends with. Marched from a momentum-thickness Reynolds number of 500, five centimetres behind the leading edge, along a ten-metre plate in a twenty-metre-a-second stream of air — a plate Reynolds number of $1.33\cdot10^7$ — the composite friction law gives a total drag coefficient of 0.002806. The Prandtl–Schlichting formula, $0.455/(\log_{10}\mathrm{Re}_L)^{2.58}$, a fit to measured plate drags, gives 0.002870. The two differ by 2.2 per cent, and most of that difference is in the first metre, where the wake strength is lower than 0.55 in reality and the composite law’s friction is therefore slightly low.

So the wake, measured once on a few layers and carried through a dimensional profile and a momentum balance, predicts the drag of a ship’s hull or an aircraft’s fuselage skin to a couple of per cent. Without it — with the bare log law’s edge velocity — the friction would be about a fifth too high everywhere and the plate’s drag with it.

What a rough wall changes, and what it does not #

A rough wall changes the inner law and leaves the wake alone. A second length at the wall is the roughness height, and its effect is to shift the logarithm down by an amount that depends on the roughness in wall units; the outer flow, whose eddies are far larger than any roughness element, does not notice, and measured rough-wall profiles keep the smooth wall’s defect law and its wake. That separation is what makes the composite form useful for rough surfaces: the friction changes through the inner shift, the displacement keeps the third the wake gives it, and a ship’s fouled hull pays its extra drag at the wall and not in the outer layer.

How it was checked #

Spalding’s law, inverted by Newton’s method, returns $u^+ = y^+$ at half a wall unit and the logarithm at $10^5$ to $8\cdot10^{-4}$ wall units. With no wake the composite profile’s edge velocity is Spalding’s at the layer’s thickness, to the last digit. The thicknesses are integrated on a grid crowded towards the wall, and quadrupling it changes the shape factor by a part in ten million. The friction’s agreement with Ludwieg and Tillmann at $\mathrm{Re}_\theta = 10^4$ is 1.0026 — the independent check, and the one the others exist to make credible. The plate-drag comparison above is the second independent check, against a different correlation, built on the same friction law integrated over a plate’s length rather than read at one station.

The convention: Coles’s sin², κ = 0.41 and B = 5.0 #

The wake function is Coles’s later algebraic choice, $\sin^2(\pi y/2\delta)$, rather than his original tabulated one; the two differ by under a few per cent of the wake. The logarithm’s constants are $\kappa = 0.41$ and $B = 5.0$, inside Spalding’s formula, and the wake strength for a flat plate is 0.55. Other choices of the constants move the wake strength needed to reproduce a given friction by a tenth or two and leave the shares and shapes above nearly unchanged. Reynolds numbers are on the momentum thickness; $\delta$ is the thickness at which the composite profile reaches the edge velocity. The flat plate has no pressure gradient; a layer in one has a different wake strength, which is where the next essay begins, and the shares and friction above are for the zero-gradient layer only.

What the picture cannot show #

The composite profile is an interpolation between two limits, not a solution. Its wake is a fitted shape, and its strength 0.55 is a measurement that itself drifts at low Reynolds numbers, falling towards zero below a momentum-thickness Reynolds number of a few hundred — which is part of why the comparison with Ludwieg and Tillmann parts at $10^3$. The sin² function gives the profile a non-zero slope at the edge, where the real one meets the free stream smoothly; corrections that fix it change the thicknesses by a per cent or two. And the profile is the mean: the wake region is where turbulent bulges from the layer alternate with irrotational free-stream fluid, turbulent some of the time, and its mean is an average over two kinds of flow.

Who found it, and when #

Prandtl and von Kármán gave the logarithmic law in the early 1930s, and Millikan’s 1938 overlap argument is the one the first essay follows. Clauser in 1956 showed the outer part of the profile collapses on a defect law, and Coles in the same year proposed the wake function as the additive outer correction and compiled the measurements that fix its strength, a compilation he extended in 1968. Spalding’s single-formula inner law is from 1961. Ludwieg and Tillmann’s friction correlation of 1949 is older than all of it, which is what makes it a fair test.

Still open: a wake that remembers its pressure gradient #

The flat plate’s wake has one strength. A layer that has been through a pressure gradient has another, and a layer in a gradient that is held in a fixed relation to its own friction — Clauser’s equilibrium layers — has a wake whose strength is set by the gradient alone. The next calculation takes the same composite profile into adverse gradients, with the wake strength tied to Clauser’s pressure-gradient parameter by measurement, and asks what happens to the shape factor and the friction as the gradient steepens, how steep a rising pressure an equilibrium layer can climb, and what the layer looks like at the separation it approaches.