Three buffer layers, one friction

The law of the wall has two ends that nobody argues about.

Next to the wall the flow is a viscous shear layer with a linear profile, $u^+ = y^+$, because the turbulent stress has to vanish at a surface where the fluid does not move. Far enough out it is a logarithm, $(1/\kappa)\ln y^+ + B$, because there is no length in the problem there and a logarithm is what a scale-free region gives.

Between them, from about $y^+ = 5$ to $y^+ = 30$, is the buffer layer. The turbulence production peaks in it. Every photograph of near-wall turbulence is of it. And no theory says what the profile there is.

Four guesses at the gap #

Every wall model in use is a different guess. Van Driest damps a mixing length towards the wall with an exponential; Reichardt fits an expression to measurements; Spalding writes the relation the wrong way round, $y^+$ as a function of $u^+$, so that one formula holds everywhere at once.

The fourth here is a control that nobody would defend: no buffer layer at all, the viscous sublayer joined straight to the logarithm where the two cross, at $y^+ = 11.635$.

At $y^+ = 10$ the control is 19.6 per cent above the three fitted models, which agree with each other to 1.5 per cent. That agreement is what one should expect and is not evidence of anything: all three were fitted to the same measurements. What is worth pricing is the departure of the control.

It is worth saying what each model is actually asserting, because the three are not three fits to the same curve. Van Driest’s is a mechanism: the mixing length is $\kappa y$ far from the wall and is suppressed near it by a factor that represents the viscous damping of the eddies, and the profile follows from integrating an eddy viscosity built out of it. Reichardt’s is a fit: an expression with three terms chosen to have the right limits at both ends and to pass through the measurements in between. Spalding’s is an inversion: an exact series for $y^+$ in terms of $u^+$ whose truncation reproduces both limits, with no claim about mechanism at all.

Three quite different kinds of statement, and they agree with each other in the buffer layer to a per cent and a half. That is the first thing the comparison establishes, and it is worth having before the second: the disagreement that follows is not a disagreement about the buffer layer, because there barely is one.

What the friction actually does #

Turning a profile into a friction is an integral. The bulk velocity in a pipe is

$$U_b^+ = \frac{2}{R^{+2}}\int_0^{R^+} u^+(y^+)\,(R^+ - y^+)\,dy^+,$$

and the friction factor is $8/U_b^{+2}$ with a Reynolds number of $2R^+U_b^+$. Nothing is fitted: given a profile and a pipe radius in wall units, both numbers follow.

The expectation was that the control would be the outlier and the three fitted models would agree. The measurement says the opposite. The three respectable models spread by between four and eleven per cent, and the control with no buffer layer at all is out by one to two and a half.

That is the wrong way round, and it means the buffer layer is not what the friction is sensitive to.

The immediate reaction to that measurement is that something must be wrong with the control, and it is worth ruling that out. Its profile is exactly $u^+ = y^+$ below the crossing and exactly the log law above it, so it has the correct viscous sublayer, the correct logarithmic region and the correct constants by construction. What it lacks is any smoothing between the two, which shows as a corner in the profile and a discontinuity in the gradient.

That corner is a serious defect for anything that reads a derivative — a production, an eddy viscosity, a turbulent stress — and is almost no defect at all for an integral of the profile itself, because the profile is continuous and the corner occupies no area. So the control is exactly the right instrument for this question: it has one large, localised error and no others.

The constant nobody was looking at #

Measured in the logarithmic region rather than quoted from the papers, the four models’ additive constants are 5.28, 5.63, 5.00 and 5.00. That is a spread of six tenths in a quantity everybody writes as “about five”.

An additive constant shifts the whole outer profile, and the outer profile is nearly all of the pipe. So six tenths of $u^+$ across ninety-eight per cent of the radius is worth far more to an integral than a nineteen per cent disagreement across two per cent of it.

Shifting the four models to a common additive constant collapses the spread. At a friction Reynolds number of two thousand it goes from 6.6 per cent to 0.66; at ten thousand from 5.2 to 0.10; at fifty thousand from 4.4 to 0.034, a factor of a hundred and thirty. And the control joins them.

So the constraint that decides the friction is the log law’s own constants, and the freedom left in the buffer layer costs less than the third digit of $B$.

Why the buffer layer cannot matter much #

The reason is arithmetic and worth having explicitly.

At a friction Reynolds number of 180 — a direct numerical simulation of a channel, the smallest turbulent flow anybody computes — everything below $y^+ = 30$ is a sixth of the radius and carries a fifth of the bulk velocity. There the buffer layer is a large part of the flow and the models genuinely disagree about the friction by eleven per cent.

At fifty thousand, which is a pipe a person could stand in, everything below $y^+ = 30$ is six parts in ten thousand of the radius and carries four parts in ten thousand of the bulk velocity. A disagreement about a region that small cannot move an integral across the whole pipe, whatever the disagreement is.

So the insensitivity is not a property of the models. It is a property of the weight the integral puts on the region they disagree about, and that weight shrinks like the reciprocal of the Reynolds number. This is the same statement four assumed boundary-layer profiles make about a drag: an integral with a smooth weight averages out shape, and how much it averages out depends on where the shape variation is.

A second consequence follows from the same arithmetic, and it is the one that decides how a computation should be set up.

The buffer layer’s weight falls with Reynolds number, so the accuracy a wall model needs falls with Reynolds number too. At a friction Reynolds number of 180 a wall model has to be right, because a fifth of the flow is in the region it is modelling; at fifty thousand it can be crude, because four parts in ten thousand are. That is exactly backwards from the usual intuition, which is that high Reynolds numbers are harder.

What is harder at high Reynolds number is resolving the wall region rather than modelling it. The first cell of a resolved calculation has to sit at $y^+$ of order one, so the number of cells needed across the layer grows with the Reynolds number even though the layer’s importance to the answer shrinks. That combination — expensive to resolve, unimportant to the answer — is the entire justification for wall modelling, and this page is the quantitative form of it.

Which is why the constants are argued about #

The result explains a feature of this subject that can look like pedantry from outside.

The von Kármán constant and the additive constant have been measured, re-measured and disputed for ninety years, over differences in the second decimal place: $\kappa$ between 0.38 and 0.42, $B$ between 4.9 and 5.5. Meanwhile the shape of the buffer layer — the part that is photographed, the part that has coherent structures in it, the part every near-wall model is written about — is not the subject of the same scrutiny.

That looks backwards and is not. The friction is an integral over the whole layer, the constants sit in the integrand across all of it, and the buffer layer’s shape sits in a shrinking sliver. Six tenths in $B$ is worth 6.6 per cent in the friction at a modest Reynolds number and more at a high one; the whole buffer layer is worth 2.6.

What a wall model is being asked to do #

There is a practical version of this that is worth stating, because it changes what a wall model should be validated against.

A wall-modelled computation replaces the near-wall region by a relation between the velocity at the first grid point and the wall stress. Every model on this page can supply that relation, and the question is what error it introduces. The answer this page gives is: almost none from the buffer layer, and all of it from the constants, provided the first grid point is in the logarithmic region.

That last proviso is the whole of it. If the first point is at $y^+ = 100$, the model is being asked only to reproduce the logarithm, and the constants are the entire model. If it is at $y^+ = 15$, the model is being asked for the buffer layer directly, and the differences between the three become the model’s error rather than an integrated nuisance.

So the sensitivity to the buffer layer is a sensitivity to where the grid is put, not to the flow. A common failure mode in practice is a mesh whose first point drifts into the buffer layer somewhere along a body — near a stagnation point, or where the boundary layer thins — so that part of the surface is being modelled in a regime where the model matters and part is not. Nothing in the output distinguishes the two.

What the buffer layer is for, then #

It is not for the friction, and saying what it is for makes the division clean.

The production of turbulent kinetic energy peaks between $y^+ = 11$ and 13 in the three fitted models, and nowhere at all in the control, whose profile has a corner instead. That peak is where the turbulence is made — the streaks, the ejections and sweeps, the vortices that every near-wall visualisation shows — and it is what sets the whole flow’s turbulence intensity.

So the buffer layer decides the turbulence and the log law decides the drag. A model built to reproduce near-wall structure and one built to reproduce skin friction are answering different questions, and doing well at one is not evidence about the other.

The gap to a real pipe #

The profiles land within about a tenth of Blasius’ empirical correlation for smooth pipes rather than on it, and the gap is not a defect of the models.

A law of the wall integrated to the centreline is missing the wake component — the outer-layer excess above the logarithm that Coles measured, which is worth a few per cent of the bulk velocity and therefore several per cent of the friction. Quoting the agreement as exact would be quoting a fit; quoting the size of what is left out is the honest form, and it is comparable with the spread between the models.

That also gives the right reading of the whole comparison. All four models are missing the same thing, so shifting them to a common additive constant makes them agree with each other and does not make any of them right. What has been isolated is where their disagreement lives, not where their error lives.

Reading it as a constraint and a freedom #

The result has the same shape as several others in this collection and it is worth putting in those terms, because the shape is what makes it transferable.

The constraint is the pair of asymptotic conditions: $u^+ = y^+$ at the wall, and a logarithm with particular constants outside. Every model satisfies both, so every model is admissible.

The freedom is the buffer layer, which is a whole function joining the two.

And the wanted quantity is an integral of the profile across the pipe, weighted by $(R^+ - y^+)$ — a smooth weight, spread over the whole radius. That functional lies almost entirely in the span of the constraints, so the freedom hardly reaches it, and how much it reaches is a number rather than an impression: 2.6 per cent for the whole buffer layer against 6.6 for six tenths of $B$.

Read that way the surprise disappears. The additive constant is not a small detail of the constraint; it is the constraint, over most of the domain, and moving it moves the answer. The buffer layer is the freedom, it is confined to a sliver, and the answer is an average.

The same reading predicts what would be sensitive to the buffer layer without any further computation: anything local in it. A near-wall heat transfer, a particle deposition rate, a concentration of a reacting species at the wall — all of those are dominated by the region the friction ignores, and the three models that agree about the friction to a per cent would not be expected to agree about them.

Where the same integral appears elsewhere #

The pipe’s friction is one instance of a pattern that runs through this part of the subject, and seeing three more makes the sensitivity easier to predict without computing it.

Roughness enters the law of the wall as a shift in the additive constant and nothing else — a second length at the wall moves $B$ downwards by an amount that depends on the roughness in wall units. Since a shift in $B$ is exactly what this page finds the friction to be sensitive to, roughness is a first-order effect on drag, which it is. And since a wall cannot feel a roughness smaller than its own sublayer, the shift is zero until the roughness reaches into the buffer layer — which is the hydraulically smooth regime, stated in these terms.

The cross-section’s shape enters a laminar duct’s friction as a pure number that depends on the shape and nothing else, and that number is likewise an integral over the whole section rather than a property of any part of it.

And the drag of a turbulent boundary layer on a plate is the same integral with a different outer condition, which is why going turbulent costs what it costs — the log law is steeper at the wall than a laminar profile of the same thickness, and the integral registers it.

In all four, the thing to ask is where the weight is. An integral over a domain with a smooth weight is sensitive to whatever occupies most of the domain and insensitive to whatever occupies a sliver of it, however violently the sliver is disagreed about.

What the measurement suggests about calibrating a model #

The result has a consequence for how a wall model should be fitted, and it is not the obvious one.

The natural approach is to fit a model to near-wall velocity data, since that is what the model is a model of, and to check the friction afterwards. What the sensitivity says is that this gets the weighting backwards: the friction depends almost entirely on the logarithmic region’s constants and almost not at all on the buffer layer’s shape, so a fit that matches the buffer layer beautifully and the constants approximately will predict friction badly.

The reverse approach — fit the constants first, to friction data, and let the buffer layer be whatever a smooth interpolation gives — produces better friction and a worse-looking profile. Which of the two is preferable depends entirely on what the model is for, and the point is that it is a choice rather than a matter of accuracy.

There is also a warning about validating on both at once. A model tuned to reproduce a measured profile and a measured friction has two targets whose sensitivities are wildly different, so the optimiser will spend nearly all of its freedom on the constant and none on the buffer layer, and the resulting profile agreement will be whatever the functional form happens to give. That is worth knowing before reading a table of such agreements as evidence.

And a note on why the control was worth including. Three models fitted to the same measurements agreeing with one another is not evidence about anything; it is a statement that they were fitted to the same measurements. The whole informational content of this comparison came from adding a fourth member that was not fitted at all, which is what a control is for — and it is what turned an expected result into an unexpected one. A comparison of models is only as informative as its worst member.

And a note on the von Kármán constant. It has been reported between 0.38 and 0.42 over the last two decades, which is a five per cent spread in a multiplicative constant rather than an additive one, so its effect on the friction is larger still than the six tenths in B measured here. That is why the argument over it has been conducted as carefully as it has, and this page is a quantitative reason rather than a defence of the arguing.

What is not claimed #

The three models were fitted to the same data, so their agreement in the buffer layer is not independent evidence. That is why the control is in the comparison at all: it is the only member of the set that was not fitted, and its departure is the measurement.

The friction here is a wall-law integration, not a pipe. No wake component, no core, no roughness, no entrance. The numbers are what four profiles give under one integration, and they land near a real correlation rather than on it for a reason this page states.

Nothing here settles what $\kappa$ and $B$ are. What is measured is what a spread of six tenths in $B$ costs, which is a different question from which value is right, and the answer to the second is not settled by the same kind of argument.

And a wall model’s job in a computation is not only to supply a friction. It supplies a boundary condition for the whole outer solution, and a model that gets the friction right by a compensating error elsewhere in the profile has not supplied a good boundary condition. The measurement here is about one output.