A guess with a constant in it

Two rungs of this ladder have established that the mean-flow equations are short of six pieces of information and that no derivation supplies them. This rung takes the oldest and simplest supply, follows it exactly, and asks what the result is worth.

The supply is Prandtl’s, from 1925, and it is one sentence: an eddy near a wall cannot be larger than its distance from the wall.

The closure, written out #

Prandtl’s argument is a transport analogy. In the kinetic theory of gases, a molecule travels one mean free path before colliding and exchanging momentum, and the viscosity that results is proportional to the mean free path times a characteristic speed. Prandtl proposed that a lump of fluid does the same over a distance ℓ — the mixing length — carrying its streamwise momentum with it before mixing.

If a lump displaced by ℓ arrives carrying a velocity deficit of order ℓ du/dy, and if the vertical fluctuation is of the same order, the momentum flux is

$$-\langle u'v'\rangle = \ell^2 \left(\frac{\mathrm{d}u}{\mathrm{d}y}\right)^2$$

with the sign arranged so that momentum flows down the gradient. Near a wall, ℓ = κy.

That is the entire model. It has one adjustable number in it, κ, and one structural assumption, that the length scale is the distance from the wall.

Both parts deserve to be looked at squarely. The structural assumption has a plausible physical motivation — an eddy of diameter larger than y would have to pass through the wall, which is the same geometric argument that makes the layer thin in the first place — and it is not a derivation. The constant κ is measured, and its measured value has moved: it was long taken as 0.40, then 0.41, and careful recent experiments in high-Reynolds-number pipes and boundary layers put it between 0.384 and 0.421 depending on the flow, which is itself evidence that it is not a universal constant of nature.

Integrating it, which is the part usually skipped #

Very close to a wall, the total shear stress is nearly the wall stress, so

$$\nu \frac{\mathrm{d}u}{\mathrm{d}y} + \ell^2\left(\frac{\mathrm{d}u}{\mathrm{d}y}\right)^2 = u_\tau^2$$

In wall units — y⁺ = yu_τ/ν and u⁺ = u/u_τ — this is a quadratic in the velocity gradient with the solution

$$\frac{\mathrm{d}u^+}{\mathrm{d}y^+} = \frac{2}{1 + \sqrt{1 + 4\ell^{+2}}}$$

which the build integrates numerically on a logarithmically spaced grid, because the sublayer is four decades thinner than the log region and a linear grid resolves one of them.

Two limits come out of it without being put in.

Near the wall the mixing length vanishes, the quadratic term with it, and the gradient tends to

1. Integrating gives u⁺ = y⁺: the viscous sublayer, produced rather than pasted on.

Far from the wall the viscous term is negligible, the balance is ℓ⁺² (du⁺/dy⁺)² = 1, and with ℓ⁺ = κy⁺ that integrates to

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

which is the logarithmic law of the wall.

The check, and why it is a check #

The essential discipline here is that the constants are fitted back out of the integrated curve rather than quoted. A least-squares fit of u⁺ against ln y⁺ over 50 < y⁺ < 500 returns a slope of 2.454, so 1/κ = 2.454 and κ = 0.4076 against the 0.41 that went in, with B = 5.209.

The build asserts that agreement and refuses a profile whose fitted κ differs from the input by more than three per cent. That is not decoration. If the integration were wrong — a factor in the quadratic, a mis-scaled grid, a damping function applied to the wrong quantity — the profile would still look exactly like a log law with a sublayer under it, because that is what a curve on semi-logarithmic axes with a bend in it looks like. The eye cannot check a slope on log axes and the fit can.

The published B for a smooth wall is between 5.0 and 5.2 depending on the experiment, and the 5.209 here is a consequence of the van Driest damping constant A⁺ = 26 rather than an independent confirmation. Changing A⁺ moves B and leaves κ alone, which is the correct behaviour and is worth knowing when reading a table of these constants: κ is a property of the log region and B is a property of the sublayer, and they are usually presented as a pair as though they came from the same place.

What wall units are, and why they are the right variables #

The collapse this model achieves happens in a particular pair of variables, and they are worth defining carefully because a great deal of the apparent universality is in the choice.

The wall shear stress τ_w has dimensions of pressure. Divided by the density and square-rooted it gives a velocity, u_τ = √(τ_w/ρ), called the friction velocity — the only velocity scale that can be built out of the wall stress and the fluid. Combined with the viscosity it gives a length, ν/u_τ, the only length scale that can be built out of the wall stress, the fluid and nothing else.

Measuring velocity in units of u_τ and distance in units of ν/u_τ is therefore not a convenience: it is the only non-dimensionalisation available in a region where the outer geometry has not yet entered. The claim that profiles from different flows collapse in these variables is the claim that near a wall the flow does not know what shape it is on, and that claim is what the collapse tests.

It also explains why the region is so thin in ordinary units and so wide in wall units. On an aircraft wing at cruise the viscous length ν/u_τ is a few microns, so y⁺ = 1000 — deep in the log region by the standards of the figure — is about five millimetres. The four decades the figure shows are four decades of a layer whose whole thickness is a centimetre or two.

The other route to the same logarithm #

There is a derivation of the log law that does not use the mixing length at all, and it is worth having because it makes clear which part of the result is robust.

Consider a wall-bounded flow with two length scales: the viscous scale ν/u_τ, small, and the outer scale δ, large. Very near the wall the profile can depend only on the viscous scale. Very far from it, in the outer region, the profile can depend only on δ. If the Reynolds number is high enough there is an overlap region in which both descriptions must hold at once, and therefore in which the profile can depend on neither scale.

A function whose derivative depends on neither of two scales, in a region parameterised by their ratio, must have du⁺/dy⁺ ∝ 1/y⁺ — and integrating that gives the logarithm.

This is Millikan’s argument, from 1938, and it is stronger than Prandtl’s in one respect and equally assumption-laden in another. It does not require any picture of eddies carrying momentum, so it survives the observation that lumps of fluid do not in fact travel a fixed distance and then mix. But it does require the assumption that an overlap region exists and that the profile in it is independent of both scales, and that assumption is exactly as underived as Prandtl’s was.

The agreement between two routes with different assumptions is genuine evidence, and it is evidence that the logarithm is robust rather than evidence that either derivation is a derivation from the Navier–Stokes equations. Neither is.

A third route, from the same argument in reverse #

There is a well-known objection to Millikan’s derivation that is worth meeting rather than skirting, because it recurs in the literature under the name of the power-law alternative.

The overlap argument shows that if a region exists where the profile depends on neither the inner nor the outer scale, the profile there is logarithmic. It does not show that such a region exists. Barenblatt and others have argued that the correct statement involves an incomplete similarity, in which the overlap profile retains a weak dependence on the Reynolds number, and that the resulting profile is a power law with Reynolds-number-dependent exponents rather than a logarithm.

The two forms are extremely hard to tell apart with real data over the decade or two of Reynolds number a single facility spans, which is why the argument ran for two decades. The current position, from the highest-Reynolds-number experiments, is that the logarithm survives, with κ close to 0.39 in pipes, and that the region over which it holds begins further from the wall than the traditional y⁺ = 30.

None of this changes the point of the present essay: whichever functional form is right, it follows from an assumption about what the profile may depend on, and not from the equations of motion. The dispute is between two closures, not between a closure and a derivation.

What the model gets right, and it is a lot #

It would be a mistake to leave the impression that a fitted model is a bad model. This one is remarkably good, and the reasons it is good are instructive.

The profile shape is right over four decades. Measurements in pipes, channels and boundary layers collapse onto the curve above, with the same κ and B, at Reynolds numbers spanning several orders of magnitude and in geometries that differ substantially. That is a strong universality and it is real.

It produces a friction law. Integrating the profile across the layer and matching to the free stream gives the skin-friction relation from which most practical drag estimates descend, including the 1/7-power correlation used elsewhere on this site, and with it the resistance to separation that makes tripping a layer worth its friction penalty.

It transfers to heat and mass. The same argument with temperature in place of velocity gives a thermal law of the wall, and the ratio between the two — the turbulent Prandtl number — is close enough to one in most flows for the analogy to be useful.

Its failures are legible. Where the mixing length is wrong, it is wrong in identifiable places rather than everywhere, and the places are known: near separation, in strong adverse gradients, in strongly curved or rotating flows, and anywhere the turbulence has not had time to reach equilibrium with the local mean shear.

How much of a boundary layer this describes #

The collapse in the figures is genuine and it is worth asking how much of a layer it covers, because the answer is a small fraction of it and the fraction that is left over is where the useful integrals live.

The log region runs from about $y^+ = 30$ at the bottom — below that the viscous term is not negligible — to roughly fifteen per cent of the layer’s thickness at the top, above which the outer flow has entered and the profile stops being independent of $\delta$. So the law of the wall describes the inner sixth of a boundary layer, and the remaining five sixths depart from it, upwards.

Coles gave the standard description of the departure in 1956: add to the log law a wake component, a fixed function of $y/\delta$ scaled by a parameter $\Pi$. The name is apt — the outer part of a boundary layer looks like a wake sitting on top of a wall layer — and the addition is substantial. For an ordinary flat-plate layer $\Pi \approx 0.55$, and the wake adds a couple of wall units to the velocity at the edge, which is a large fraction of the defect between the edge and the log line.

And $\Pi$ is not universal. It is near zero in a pipe or a channel, where the geometry constrains the outer flow and there is barely a wake at all. It is about 0.55 for a boundary layer in zero pressure gradient. And it climbs steeply in an adverse gradient, reaching values of several as separation is approached — so the parameter that measures the non-universal part of the profile is precisely the one that responds to the pressure gradient the layer is in.

Which qualifies the essay’s own result. The shape of the profile near the wall is universal, and the quantities a designer wants — the displacement and momentum thicknesses, and the shape factor built from them — are integrals dominated by the outer five sixths. So the celebrated universality belongs to a thin region, and the number that warns of separation is a property of the part that is not universal at all.

Where the model stops, and it stops in a specific place #

The mixing length’s central defect is that it is local and instantaneous. It says the Reynolds stress at a point is determined by the mean velocity gradient at that point, through a length that depends only on the distance to the wall.

Turbulence is neither local nor instantaneous. The eddies carrying momentum through a plane were created somewhere upstream and have a lifetime comparable with the time they take to cross the layer. In a flow where conditions change along the surface faster than the turbulence can adjust — which is any flow approaching separation, and any flow just downstream of a change in surface condition — the stress at a point reflects history rather than the local gradient. It is the same non-locality that makes the pressure–strain term resist modelling, arriving at the simplest possible level of description.

Three consequences follow, and all three are seen in practice.

The model predicts zero stress where the gradient is zero. In a pipe the mean velocity gradient vanishes on the axis, so the mixing-length model predicts no turbulent momentum flux there. The measured flux is not zero. The same failure appears at the maximum of an asymmetric channel’s profile, displaced from where the model puts it.

It cannot represent counter-gradient transport. In several flows the measured momentum flux runs up the mean gradient over part of the domain — in a wake behind a body, and in some buoyancy-affected layers. A model whose flux is a positive multiple of the gradient cannot produce that at all.

It requires a length scale for each geometry. The κy prescription is for the near-wall region of an attached layer, and it is silent about the free shear layer that a separated flow leaves behind. A jet, a wake and a mixing layer each need their own ℓ, fitted to their own measurements, and there is no formula inside the model that produces them.

The response to all three — carrying transport equations for turbulence quantities rather than an algebraic length — is what one- and two-equation models do, and they buy generality with more constants rather than with fewer assumptions.

Who found it, and when #

Boussinesq proposed the eddy-viscosity idea in 1877, before the Reynolds decomposition existed to give it a subject.

Prandtl’s mixing length is from 1925 and Kármán’s similarity hypothesis, which reaches the same logarithm by a different route, from 1930. Nikuradse’s measurements in smooth and artificially roughened pipes, published in 1932 and 1933, are what established the constants and are still the data most often plotted against the curve.

Millikan’s overlap argument is from 1938. Van Driest’s damping function, which is what makes the sublayer come out of the integration rather than being appended to it, is from 1956.

The value of κ has been re-measured with increasing care since the 2000s — the Princeton Superpipe and the Melbourne wind tunnel are the two best-known campaigns — and the result is that it is not quite the same in a pipe and in a boundary layer. That is a small discrepancy and it is exactly the kind of small discrepancy that would be impossible if the log law were a theorem.

Where the ladder goes next #

This anchor has gone as far as it usefully can: the gap was counted, the hierarchy was climbed and found to diverge, and the oldest closure was integrated and its constants recovered.

It is also the rung where this ladder makes contact with what a reader can check. Every turbulent boundary-layer number quoted elsewhere on this site — the fuller profile, the shape factor of 1.35, the turbulent plate friction — descends from this closure or from a measurement, never from a solved field.

The next anchor asks a different question — not what the mean flow does, but where the energy goes after the mean flow gives it up. Where the energy goes is the cascade, and it contains the one part of this subject where a statement about high-order structure is derived rather than fitted.