What a code says to a wall

Every calculation of a turbulent flow over a surface faces the same arithmetic. The viscous sublayer is a few microns thick on an aeroplane wing; the wing is tens of metres long; and resolving the first with cells small enough to see it, over the whole of the second, costs more cells than the calculation can carry. Something has to be assumed about what happens between the wall and the first grid point, and what is assumed is almost always a wall function.

A wall function is the law of the wall, used backwards. It takes the velocity at one point, assumes that point lies in the logarithmic region, and solves for the friction velocity that would put it there. This essay measures what that assumption costs, and the answer has a shape nobody would choose: it is a valley.

What the calculation actually knows #

A finite-volume code storing velocity at cell centres knows $u_1$, the velocity at height $y_1$, and needs $\tau_w$, the stress the wall exerts. The two are related by the profile between them, and the profile between them is the thing the calculation cannot afford to compute.

The log-law route closes the gap in one line:

$$u_1 = u_\tau\left[\frac{1}{\kappa}\ln\frac{y_1 u_\tau}{\nu} + B\right].$$

Everything in it is known except $u_\tau$, which appears on both sides and inside a logarithm, so it is found by iteration — half a dozen bisections, once per wall cell, once per time step. That is the whole of a standard wall function, and its cost is nothing at all. What it buys is the ability to put the first cell four decades further from the wall than a resolved calculation needs, which is a saving measured in orders of magnitude.

The assumption is not that the flow is turbulent. It is that $y_1$ lies in the overlap layer — the band where neither the viscous length nor the layer thickness is felt. That band exists, has a measurable width, and moves with the Reynolds number, none of which the calculation checks.

The measurement, with the closure on both sides #

The comparison made here is deliberately narrow. The velocity fed to each wall treatment is the one the mixing-length closure gives at that height, and the friction the treatment infers is compared against the friction that same closure has. So the closure’s own shortcomings — and they are substantial — appear on both sides of the comparison and cancel. What is left is the error of the boundary condition, on its own.

Three treatments are measured. The log-law function above; the sublayer treatment, which assumes instead that $y_1$ is inside the viscous region where $u^+ = y^+$, which is what a resolved calculation does; and the blended treatment that most production codes ship, which joins the two with a weight that is a function of $y^+$.

The log-law column is the one to read first. It is accurate to a few tenths of a per cent from $y^+ = 30$ up to about half the channel width, and it degrades catastrophically below that: at $y^+ = 1$, which is where a careful analyst puts the first cell of a resolved calculation, the inferred friction is a third of the true value.

That column runs the wrong way. Every other numerical error in a calculation falls as the grid is refined; this one grows. A practitioner who has been told that finer is safer, and who halves the near-wall spacing to be certain, has made the answer worse by a factor that no grid-convergence study will diagnose, because the sequence of grids does not converge to anything.

The band that is quoted, and the band that exists #

Textbooks and code manuals give the guidance as $30 < y^+ < 300$, and the lower bound is right. The upper one is not a property of wall functions at all: it is a property of the flows those manuals were written about.

Measured against the closure at $\mathrm{Re}_\tau = 20{,}000$, the log-law function stays within two per cent up to $y^+ \approx 10{,}000$ — half the channel. At $\mathrm{Re}_\tau = 2{,}000$ the same treatment is out by two per cent by $y^+ \approx 1{,}000$, because the overlap has ended by then. The upper limit is not a number; it is a fraction of the layer thickness, and expressing it in wall units disguises the fact that it moves.

The practical difficulty is that $y^+$ is not known when the grid is made. It contains $u_\tau$, which is an output of the calculation; a grid is built on an estimate, and the estimate is often wrong by a factor of two, which moves the first cell across half the valley. On a body where the friction varies by an order of magnitude between the leading edge and the rear — which is every body — no single grid puts every wall cell in the band, and the parts of the surface that fall outside it are the parts where the flow is doing something interesting.

Where the assumption is not merely inaccurate but false #

The valley above is measured in an equilibrium channel, which is the flow the law of the wall was derived for, and which is the least demanding case a calculation ever meets. Three common situations violate its assumptions outright, and in each the error is not a number on a curve but a missing effect.

A separating flow has no friction velocity to scale with. At the point of separation the wall stress is zero, so $u_\tau = 0$, the wall unit is infinite, and every cell is at $y^+ = 0$. The logarithm is not slightly wrong there; it is undefined. Codes patch this with a velocity scale built from the turbulent kinetic energy instead, which is a different assumption with its own range of validity and no derivation of the kind the log law has.

A layer under a strong pressure gradient is not in equilibrium. The overlap argument assumes the stress is nearly constant across the matched region. Under a rising pressure it is not — the total stress varies by tens of per cent across the very band where the wall function is being applied — and the profile acquires a term the log law has no room for.

A transitional layer is not turbulent at all. A calculation using wall functions has assumed turbulence everywhere, so it has assumed that transition happened upstream of everything, which on a wing at flight Reynolds number is wrong for the first several per cent of the chord and matters because that is where the friction is highest. Whether a layer has become turbulent by a given station is a question this collection treats as the hard one it is, and no wall treatment asks it.

The blended treatment, and what it costs to be never wrong #

The third curve in the hero figure is the compromise every commercial code now ships: a blend that tends to the sublayer solution at small $y^+$, to the log law at large $y^+$, and to something in between in the buffer layer. Its error stays within a few per cent everywhere across four decades of placement.

That is genuinely valuable — it removes the trap of a grid that is accidentally refined into the buffer layer — and it should be read for what it is. The blend is exact nowhere. In the sublayer the resolved treatment is exact and the blend is a few per cent out; in the log layer the wall function is exact to a tenth of a per cent and the blend is half a per cent out. It converts a large error that occurs sometimes into a small error that occurs always, which is the right trade for industrial work and the wrong one for a calculation intended to establish a number.

The honest summary of what a wall function is #

It is a modelled boundary condition — the same species of object as the effective slip length that stands in for a porous wall, or the surface roughness that stands in for a distribution of grains. In each case a structure too fine or too expensive to resolve is replaced by a relation between quantities the calculation does carry, and in each case the replacement is exact in a limit and approximate outside it.

Roughness is the third member of the family and the most familiar: a wall covered in grains is replaced by one number, and the additive constant $B$ in the log law absorbs it. That substitution has the same structure and the same limitation — it is exact once the grains are much smaller than the layer, and the wall stops being able to feel them once they are smaller than the sublayer, which is a condition on the flow rather than on the surface.

The difference is that the porous-wall condition has a small parameter and an error estimate, and the wall function has neither. Its accuracy depends on where a grid point happens to be relative to a layer whose thickness is an output, in a flow whose equilibrium is assumed rather than checked. That is not a criticism of anybody’s code — the alternative is not computing the flow at all — but it is the reason a friction coefficient from a wall-function calculation carries an uncertainty that no residual plot displays. On an aircraft that uncertainty is not academic: friction is roughly half of what a transport aeroplane spends its fuel on, and the split between the two drags is decided by exactly this number.

The velocity is not the only thing the wall is told #

The measurement above is of one quantity — the friction inferred from one velocity — and a calculation needs more than that at a wall. A two-equation model carries a turbulent kinetic energy and a dissipation rate as well, and both of those need boundary conditions at the same first cell, supplied by the same treatment and resting on the same assumption.

They are set by local equilibrium: assume that turbulence production equals dissipation in the first cell, which in the overlap layer is very nearly true, and the two conditions follow — $k = u_\tau^2/\sqrt{C_\mu}$ and a dissipation set by $u_\tau^3/\kappa y_1$. Both inherit whatever error the friction has, and both then inherit a second one of their own, because local equilibrium is a stronger assumption than the log law is. A layer can be logarithmic in the mean while its turbulence is transporting energy across the region rather than dissipating it where it is made, and under a pressure gradient it routinely is. So the boundary condition on the mean velocity and the boundary condition on the turbulence fail at different rates, and the second is not measured by any figure in this essay.

The same point, asked for a heat flux #

The harder version of all of this is thermal, and it is the version most industrial calculations actually want, because a heat exchanger, a turbine blade and an electronics enclosure are all asking for a wall heat flux rather than a wall stress.

A thermal wall function has the same shape as the momentum one —

$$T^+ = \frac{Pr_t}{\kappa}\ln y^+ + P(Pr)$$

— and the difference between the two is entirely in that additive term. For a gas, where the Prandtl number is near one, the conductive and viscous sublayers have the same thickness, $P$ is small, and the thermal function is the momentum function with a constant of the same size: this is the Reynolds analogy holding, and heat transfer in air is about as reliable as friction is.

For a liquid it is not. The conductive sublayer is thinner than the viscous one by roughly $Pr^{-1/3}$, so in water it is half the thickness and in an oil a fifth, and the additive term that accounts for the extra resistance grows steeply with the Prandtl number — from nearly nothing for air to tens of wall units for oils. That term is a correlation. It has no derivation of the kind the log law has, it was fitted to measurements in the 1960s, and every liquid-side heat-transfer coefficient a wall-function calculation produces has it as a multiplier.

The practical position is worth stating in one sentence, because it is not the one a residual plot suggests. A calculation whose friction is good to two per cent can be out by tens of per cent in its wall heat flux, in the same cells, at the same grid spacing — because the two treatments have different sublayers, different sensitivities to placement and different provenance, and only one of them is asymptotic. And the turbulent Prandtl number in that formula, quoted as 0.85 in the middle of a layer, is not 0.85 next to a wall: it rises towards the surface in every measurement that has resolved it, in a way the models represent with another correlation.

None of which shows up in the valley plot. A wall treatment is a set of modelled conditions rather than one, they fail independently, and the one this essay can measure is the one with the best derivation behind it.

What the picture cannot show #

The comparison is against a closure and not against a flow. Both the wall function and the profile it is tested on came from Prandtl’s mixing length, and the agreement in the middle of the valley is therefore a statement about internal consistency. What it establishes is the shape of the error against placement, which is the thing being argued about; it does not establish that the absolute friction from either is right, and a direct simulation would give a different valley floor.

The channel is in equilibrium and the surface is smooth. Both are assumptions the essay has already said a real application violates, and no figure here shows the size of the resulting error, because computing it needs the flow that is not available. The same limitation applies to the laminar comparison: where a real layer separates, the friction goes to zero and the whole scaling collapses, and the error of a wall treatment there is not a percentage but a category error.

And no error bar accompanies a wall-function result in practice. Everything in the table above required knowing the answer in advance. In a calculation of a new configuration the same numbers are unavailable by construction, which is the position every practitioner is actually in.

Who found it, and when #

Wall functions in this form date from Launder and Spalding’s work of the early 1970s, arriving with the two-equation turbulence models they were built to serve, and the $30 < y^+ < 300$ guidance is about as old. Blended treatments came later: Kader’s composite profile of 1981 is the ancestor of most of them, and the automatic wall treatments in current codes descend from work in the 1990s that made the switch continuous rather than a user’s choice.

The surprising connection is with the singular-perturbation model problem that this collection uses to introduce boundary layers. There, an equation of higher order takes an extra condition and pays for it with a thin layer, and the outer solution is recovered by matching rather than by resolving. A wall function is that procedure carried out numerically: the outer problem is solved on a coarse grid, the inner problem is not solved at all, and a matching condition is applied at one point. The whole of the technique is an asymptotic method implemented as a boundary condition — and its failure mode is the classical one, which is that a matching applied where the layers do not overlap gives an answer with no error estimate and no warning.

Where the ladder goes next #

Beside this rung sits the question of what else a wall may be told when its structure is not resolved: an averaged wall condition, where the small parameter is explicit and the error can be established. Above it lies the same problem in a flow that is not in equilibrium, which needs a description of the layer’s history and is the frontier of the subject rather than a settled rung.