The layer with no length in it

A turbulent boundary layer has two lengths in it and they are separated by four orders of magnitude. The first is the viscous length ν/u_τ, built from the fluid’s viscosity and the friction velocity at the surface, and on an aeroplane wing it is a few microns. The second is the thickness of the layer itself, which is a few centimetres. Between them lies a region that is very much larger than the first and very much smaller than the second, and the whole of the law of the wall is a statement about what can happen there.

The answer is a logarithm, and the reason is that nothing else is allowed.

Two descriptions, each complete on its own side #

Close to the wall, the only quantities available are the distance $y$, the viscosity $\nu$ and the shear stress the wall is carrying. The last two combine into a velocity $u_\tau = \sqrt{\tau_w/\rho}$ and a length $\nu/u_\tau$, and dimensional analysis then says the profile must be some universal function of the ratio:

$$u^+ = f(y^+), \qquad u^+ = \frac{u}{u_\tau}, \quad y^+ = \frac{y\,u_\tau}{\nu}.$$

That is the inner description. It knows nothing about how thick the layer is, and it is exact in the limit of a layer infinitely thick compared with the viscous length.

Far from the wall, viscosity is not what is carrying the stress — the turbulence is — so $\nu$ ought not to appear at all. What is available instead is the layer thickness $\delta$ and, still, the friction velocity. The natural statement is not about the velocity but about the defect from the free stream:

$$\frac{U_\infty - u}{u_\tau} = g\!\left(\frac{y}{\delta}\right).$$

That is the outer description, and it is exact in the limit of vanishing viscosity.

Neither is a description of the whole layer. What makes the pair useful is that when the two lengths are widely separated there is a band of $y$ that is simultaneously very large in wall units and very small in layer thicknesses — and in that band both descriptions must hold at once.

The one function that can match #

Take the overlap seriously. In it, $u^+ = f(y^+)$ and $U^+_\infty - u^+ = g(\eta)$ with $\eta = y/\delta$, and both hold at the same physical point. Differentiate each with respect to $y$ and multiply by $y$, which removes the scaling constants:

$$y^+ \frac{df}{dy^+} = -\eta\frac{dg}{d\eta}.$$

The left-hand side depends only on $y^+$. The right-hand side depends only on $\eta$. The two variables are independent — a given $y^+$ can be reached at any $\eta$ by changing the Reynolds number — so both sides must equal the same pure number, and that number is written $1/\kappa$.

Integrating the left-hand side gives the law of the wall,

$$u^+ = \frac{1}{\kappa}\ln y^+ + B,$$

and integrating the right-hand side gives the velocity-defect law with the same $\kappa$ in it. No model of turbulence has been used anywhere in this. The only assumptions are that the two scalings exist, that they overlap, and that the overlap is wide.

The mixing length is a way of getting the same result, not the reason for it. Prandtl’s hypothesis — that an eddy near a wall can be no larger than its distance from the wall — gives $\ell = \kappa y$ and hence, in a constant-stress layer, exactly the profile above. It is a good argument and this collection makes it in full. But the logarithm survives every closure that has replaced the mixing length in the ninety years since, and it survives the abandonment of closures altogether in a direct simulation, because it was never a consequence of any of them.

What the plateau actually looks like #

The argument above is about an asymptotic limit, and a real flow is at a finite Reynolds number. Whether there is an overlap at all, and how wide it is, is a question with a numerical answer, and the way to get it is to plot the quantity the argument says must be constant.

$\Xi(y^+) = y^+\,du^+/dy^+$ is that quantity — the diagnostic function — and the hero figure is it, computed by integrating the mixing-length closure across a channel with the total stress falling linearly to zero at the centre, so that the Reynolds number is in the problem rather than assumed away. The results are worth stating as numbers:

Two things follow, and both of them matter more than the law itself.

At the Reynolds numbers of most laboratory experiments there is barely a log layer. A channel at $\mathrm{Re}_\tau = 180$ — for many years the standard direct numerical simulation, and still a common one — has no flat region whatever. Fitting a logarithm to it produces a number, because fitting a logarithm to anything produces a number, and that number is not $\kappa$.

The constant approaches its value from above. A fit made before the limit is reached overestimates $1/\kappa$, and the error is a quarter at $\mathrm{Re}_\tau = 180$. The published values of $\kappa$ range from about 0.384 to about 0.421 depending on the flow and the fitting range, and the spread is not carelessness: it is what happens when a constant defined in a limit is measured at a finite Reynolds number, in flows whose overlaps are of different widths.

The other scaling, and why both are needed #

The outer description is the half that is usually skipped, and skipping it hides what the log law is for. Plotted in wall units, the five channel profiles agree beautifully near the surface and separate completely in the middle. Plotted as a defect against $y/\delta$, they do the opposite.

Neither picture is the profile. The profile is the object that both pictures are approximations to, and the logarithm is the region where the two approximations agree. That is why the argument is called matched asymptotics: the log law is the matching condition, and $\kappa$ is the number the two limits have to share.

There is a consequence worth drawing out, because it is where the whole apparatus pays for itself. Since the outer flow is described by a defect law with $u_\tau$ in it, and the inner flow by a wall law with $u_\tau$ in it, evaluating both at the same point in the overlap gives a relation between the free-stream velocity and the friction velocity — which is a friction law, obtained without solving anything. It comes out implicit and logarithmic, and it is the reason the friction factor of a smooth pipe is given by an implicit logarithmic formula rather than by a power.

Colebrook’s formula, which every pipe in every building is sized with, is that friction law with a roughness term added, and its logarithm is this one. So is the reason a wall cannot feel roughness smaller than its own sublayer: the sublayer is measured in wall units, the roughness is measured in millimetres, and which of them is larger is a question whose answer changes with speed. The overlap argument is not a piece of theory that stayed in the seminar room. It is in the sizing of pipework, and it got there by being the only available route from a wall stress to a bulk velocity that does not require solving the flow.

Everything a surface does arrives as one number #

The remark that $B$ rather than $\kappa$ carries the effect of roughness, suction and curvature deserves more than a clause, because it is the reason the law of the wall is usable in engineering at all.

The overlap argument fixes the slope of the logarithm from a symmetry, and it says nothing whatever about the constant of integration. That constant is what the wall region contributes, and for a smooth impermeable wall it comes out near 5.0 — a number measured, like $\kappa$, and with a similar spread for similar reasons.

Change something about the surface and the profile in the overlap does not change shape. It shifts down, by an amount written $\Delta u^+$, and the whole effect of the change is that one number. Roughness gives the largest and most-used case: in the fully rough regime the shift grows as $(1/\kappa)\ln k_s^+$, with $k_s^+$ the roughness height in wall units, so a surface’s entire aerodynamic character is compressed into an equivalent sand-grain height and thereafter into a single downward displacement of the log line. Suction shifts it up; blowing shifts it down; a favourable pressure gradient shifts it up.

That is an extraordinarily strong statement and it is why correlations exist. A surface treatment does not have to be understood to be used — it has to be measured once, converted into a $\Delta u^+$, and it can then be applied to any Reynolds number and any geometry whose overlap is wide enough, because the slope it is displacing is universal. Every roughness table, every riblet performance figure and every entry in a hull-fouling penalty chart is a $\Delta u^+$ wearing other units.

And how little of a real layer is logarithmic #

There is a second qualification that a reader of the profiles above should carry, and it moves the law from something that describes a boundary layer to something that describes a part of one.

The overlap runs from about $y^+ = 30$ at its lower end — below which viscosity is felt — to about a tenth or a fifth of the layer thickness at its upper end, above which the layer’s own size is felt. At $\mathrm{Re}_\tau = 10^4$ that is from 30 to roughly 1,500: under two decades of $y^+$, and under a fifth of the physical thickness of the layer.

The other four fifths is not logarithmic and is not a small correction to a logarithm. In a boundary layer developing under no pressure gradient it departs upward by a systematic amount that Coles described in 1956 as a wake component — an additive function of $y/\delta$ whose strength is roughly a fifth of the whole profile’s range, and which is the outer flow asserting the layer’s own scale. In a pipe or a channel the same component is present and weaker; under an adverse pressure gradient it grows until it dominates, and a layer approaching separation is nearly all wake.

So a figure of $u^+$ against $\log y^+$ that looks like a straight line over most of its span is misleading twice over. The straightness is partly the semilogarithmic axis flattering a slowly varying function, as the earlier figure warns, and the genuinely logarithmic part is a band in the middle with a viscous region below it and a wake above. The law of the wall is a law about an interval, and the interval is narrower than the pictures suggest.

The awkward fact about power laws #

A logarithm is the $n \to 0$ member of the family $u^+ = C(y^+)^n$, and that family has an inconvenient property: at any Reynolds number a laboratory can reach, a power law fits the overlap about as well as the logarithm does.

The exponent that comes out, about an eighth, is not a coincidence. It is where the famous one-seventh power law comes from. Nikuradse’s pipe data of the 1930s were fitted by $u \propto y^{1/7}$, Blasius’ friction correlation follows from it, and both are still in use; the essay on what turbulence costs a flat plate uses that correlation and says so. The power law is not a rival theory that lost. It is what the logarithm looks like over a decade and a half.

Barenblatt made the point sharply in the 1990s by arguing that the exponent is not a constant at all but a slowly falling function of Reynolds number — which is exactly what the fitted exponents in the figure above do. The dispute is genuinely open in the sense that no measurement settles it, and genuinely closed in the sense that the overlap argument predicts the logarithm from a symmetry and the power law predicts nothing at all. That asymmetry is the reason to prefer the logarithm, and it is a different kind of reason from “the data fit better”.

What the picture cannot show #

Everything here is a consequence of a closure, and the closure is Prandtl’s. The profiles were produced by integrating $\ell^+ = \kappa y^+(1 - e^{-y^+/A^+})$ through a channel with a linearly falling total stress. What that establishes is that the diagnostic function of such a profile behaves as the overlap argument says — it cannot establish that a real channel does, because no figure on this site solves the Navier–Stokes equations at any of these Reynolds numbers, and the arithmetic of why not is elsewhere in this field.

The constant is measured, not derived. Ninety years of work have not produced $\kappa$ from anything more fundamental, and the honest statement of the law of the wall is that its form comes from a symmetry argument and its number comes from experiment. There are arguments that $\kappa$ should be related to other constants, and none of them is settled. This is the usual position in this field rather than an embarrassment peculiar to it: the averaged equations are short of six equations and every route to closing them ends in a constant somebody measured, so a law whose shape is forced and whose single constant is empirical is close to the best outcome available.

The wall is smooth, flat, impermeable and not moving. Every one of those can be relaxed and each changes the additive constant $B$ while leaving $\kappa$ alone — which is itself a strong statement, and the reason $B$ rather than $\kappa$ carries the effect of roughness, suction and curvature. A layer with suction through the surface is the sharpest case: it has a perfectly good logarithmic region and an entirely different constant.

And the argument assumes the overlap is wide. Where it is not — a low-Reynolds-number channel, a strongly accelerated layer, a boundary layer with separation approaching — the two descriptions do not agree anywhere, there is no logarithm, and quoting one is quoting a limit the flow is not in. That is not a small class of cases: it includes most of what happens near the trailing edge of a wing.

The same argument, three fields away #

The shape of the reasoning here — two scalings, an overlap, and a functional form forced by the overlap rather than by any mechanism — is not particular to walls. It is the same argument that fixes the inertial range of the energy spectrum, where the two scalings are the energy-containing range and the dissipation range, the overlap is the band that knows neither, and the form forced by the overlap is a power law rather than a logarithm because the quantity being scaled is different.

Both are cases of a general habit worth carrying: when a region of a problem is told it may not depend on either of the available scales, the functional form is decided before any physics is put in. The mechanism decides the constant; the symmetry decides the shape. Where a subject is hard enough that no mechanism can be computed, the symmetry may be all there is — and it is a great deal more than nothing.

Who found it, and when #

Prandtl and von Kármán arrived at the logarithm around 1930 from mixing-length arguments, and von Kármán’s name is on the constant. Millikan gave the overlap argument in 1938, in three pages, and it is the version that has lasted, because it explains why every subsequent closure keeps reproducing the same law. The measurement of $\kappa$ has been refined continuously since — Nikuradse’s pipes, Coles’ compilation of boundary layers in 1968, the Princeton superpipe from 1998 — and the value has moved, by more than the quoted uncertainties, in a way that the width of each experiment’s overlap largely accounts for.

The surprising connection is with the exact theory that predicts no drag at all. Ideal flow fails at a wall because it cannot be told about no slip; the law of the wall exists because a wall can be told, and because the length that condition creates is so much smaller than the flow that a region opens up between them where neither length is felt. The log layer is the gap between two failures of scale, and it is the most useful thing in wall turbulence precisely because nothing local happens in it.

Where the ladder goes next #

The rung above is what a computation does when it cannot resolve the wall at all, and has to impose the boundary condition somewhere in the middle of this region instead: what a code says to a wall. The overlap argument turns from a piece of theory into an engineering decision there, and the decision is made worse by refining the grid — which is a sentence that ought to sound alarming.