A wall that is not quite there

A wall made of sintered metal, or of sandstone, or of a catalyst pellet bed, has structure at every scale from the pore upwards. Resolving that structure in a calculation of the flow along it is out of the question — a centimetre of medium with ten-micron pores has a million pores across it — and the usual approach is to ignore it and apply no slip at a smooth surface drawn where the medium starts.

That is wrong by a computable amount, and the computable amount has a name.

Two equations, joined at a plane #

Above the interface the fluid is ordinary and the flow satisfies Stokes’ equation, which for a channel is $\mu\,u'' = dp/dx$. Below it the medium resists, and the standard description is Brinkman’s — Darcy’s law with a viscous term restored so that the equation is of second order and can meet a condition at the interface:

$$\mu\frac{d^2u}{dy^2} - \frac{\mu}{K}u = \frac{dp}{dx}.$$

$K$ is the permeability, an area, and $\sqrt{K}$ is the pore scale. Deep inside the medium the second derivative is negligible and the equation reduces to Darcy’s, giving the seepage velocity $u_D = -K(dp/dx)/\mu$ — the velocity nobody has, since it is a flux per unit total area rather than the speed of anything.

The two regions are joined by requiring the velocity and the shear stress to be continuous. Both solutions are elementary — a parabola above, an exponential plus a constant below — and matching them gives the interfacial velocity in closed form:

$$u(0) = \underbrace{\frac{K}{\mu}\left(-\frac{dp}{dx}\right)}_{\text{Darcy}} \;+\; \underbrace{\sqrt{K}\,\frac{du}{dy}\bigg|_{0}}_{\text{slip}}.$$

The second term is a Navier slip condition with a slip length of exactly $\sqrt{K}$. The medium has been replaced by one number, and the number is the pore scale.

Checked twice, because a matched pair of guesses is not a solution #

The closed form above is two assumed functional forms with their constants fitted to the matching conditions, which is precisely the sort of solution that can satisfy everything it was fitted to and none of the equations. So the coupled problem is also solved on a grid — one tridiagonal system spanning both regions, with the Brinkman term switched on below the interface and nothing assumed about the shape of anything.

The two agree to 0.008 per cent in the slip length at the resolutions used, and the agreement improves as the square of the spacing.

The check is not ceremony. A closed form of this kind is exactly what a plausible wrong field looks like in algebraic dress: it has the right limits, it satisfies the conditions it was fitted to, and being wrong in the interior would show up nowhere. Two details of that grid solve are worth recording because both cost time to find. The interface node’s control volume is half in each medium, so it takes half the Brinkman term; giving it none puts the interface half a cell into the porous block and the slip length comes out too large by one part in twice the number of points per pore — a fixed error that no refinement removes, because refining moves the cell and the interface together. And the shear has to be read from the free side only: a central difference across the interface straddles a jump in curvature and is first-order accurate at the one place the answer is being taken.

What the one number buys #

The comparison that matters is not the slip length but the flow rate, because that is what a calculation is for. Three channels, all with the same pressure gradient: the exact two-region solution, a channel with a plain no-slip wall, and a channel with the Navier condition and nothing else.

Read the last row: with pores a thousandth of the channel — sintered metal in a millimetre gap, or sandstone in a borehole — a plain wall is wrong by three parts in a thousand and the averaged condition by six parts in a million. Neither error is large in that case, and the point is that the ratio between them is five hundred, and that it goes on growing. Read the first row instead and both are substantial: a medium whose pores are three per cent of the channel is not really a wall at all, and the two-region problem has to be solved.

The orders are the whole story. The plain wall’s error is first order in the pore scale and the averaged condition’s is second, so the advantage of the second grows without limit as the structure being averaged over gets finer. That is the general property that makes an averaged boundary condition worth having: it improves as the thing it is standing in for becomes less resolvable.

And the error the averaged condition does make is identifiable rather than mysterious. It is exactly the Darcy term in the matching above — the flow the wall itself carries — and a wall told to slip has not been told that anything goes through it. Fitting the exponent of the discrepancy against $\sqrt{K}$ gives 1.009, which is the confirmation that the missing piece is that term and not something else.

The same idea, four times on this site #

Roughness. A wall covered in grains is replaced by one length, and the additive constant in the law of the wall absorbs it. The condition for that substitution to work is that the grains be small compared with the layer — and it stops working, in the other direction, when they are smaller than the viscous sublayer, at which point the wall cannot feel them at all.

A wall function. A calculation that cannot resolve the sublayer replaces it with a relation between the velocity at one point and the stress at the wall. That is the same species of object and it is conspicuously worse behaved, because it has no small parameter: its accuracy depends on where a grid point happens to lie, and there is no expansion in which its error is one order smaller than the effect.

A lubrication film. The whole of a bearing’s behaviour is replaced by a relation between the gap and the pressure, with the film’s own profile never computed in the outer problem.

A free surface, where the condition applied is no shear rather than no slip, and where the same question — what the interface between two fluids does to the one being computed — is answered by one relation instead of by resolving the other fluid. The difference from the porous case is that there the substitution is exact rather than asymptotic, because a free surface really does carry no stress.

And suction through a surface, which is the other way to modify what a wall does to the fluid next to it — and which gives a layer of constant thickness rather than a slip.

Why the flow rate rises at all #

It is worth pausing on the direction of the effect, because a porous wall sounds like a leak and a leak sounds like a loss. The flow rate through the channel rises: 3.03 per cent at a pore scale of one per cent of the channel, against a plain wall at the same pressure gradient.

The reason is that a boundary condition is a constraint, and a weaker constraint costs less. A plain wall holds the fluid at rest and the whole of the profile is pulled down to meet it; a wall that permits a slip holds it less firmly, and the parabola sits higher. Nothing has been gained by the fluid inside the medium — the seepage there is a second-order trickle — and everything is gained by the condition at the top of it.

That is the same accounting as a drag-reducing surface. A riblet, a superhydrophobic texture and a lubricated groove all work by presenting an effective slip length to the flow, and the figure of merit is precisely $\sqrt{K}/h$ in this notation — a slip length compared with the scale of the flow being reduced. A surface that offers a slip length of a micron is worth nothing to a pipe and a great deal to a microchannel, and the reason is the ratio rather than either length.

Why a slip length is the wrong design variable for a turbulent surface #

The section above ends by saying that a structured surface’s slip is a tensor rather than a scalar, and it is worth following that further, because the tensor’s two components enter a turbulent flow in a way that reverses the conclusion of this whole essay.

In the laminar channel computed here, slip is simply a weaker constraint and any amount of it helps. Slip along the flow raises the profile; slip across the flow is irrelevant, since there is no flow across. So the design goal would be to maximise the slip length in every direction, and an isotropic porous surface with the largest available $\sqrt{K}$ would be the best surface obtainable.

In a turbulent boundary layer it is not, and it is not by an argument that is exact at leading order. The near-wall region is organised by streamwise vortices lying just above the surface, and what decides the drag is where those vortices think the wall is. A streamwise slip moves the apparent wall away from the flow for the mean profile; a spanwise slip moves it away for the vortices, letting them sit closer and mix harder. The two effects have opposite signs, and Luchini’s analysis makes that quantitative: the shift in the logarithmic profile — which is what a drag reduction is — depends on the difference between the streamwise and spanwise virtual origins,

$$\Delta U^+ = h_\parallel^+ - h_\perp^+ ,$$

and on nothing else at first order.

The consequence is blunt. An isotropic slip surface gives no turbulent drag reduction at all, however large its slip length, because the two origins move together and their difference stays zero. The sintered wall this essay computes, which raises a laminar channel’s flow rate by three per cent, would do nothing whatever to a turbulent one — and a naive reading of the slip length as the figure of merit predicts the opposite.

What works is anisotropy, deliberately arranged. A riblet is a set of streamwise grooves that offer a long path along the flow and a short one across it, so $h_\parallel$ exceeds $h_\perp$ by roughly a tenth of the groove spacing, and the drag reduction follows that difference rather than either origin. The optimum spacing is a few tens of viscous lengths — a physical size that shrinks as the Reynolds number rises, which is why riblet film for an airliner has grooves tens of microns apart and why the same film does nothing on a slow-moving hull. A superhydrophobic texture works the same way and for the same reason, with gas pockets supplying a much larger $h_\parallel$ than any permeability could, and its performance is decided by how much of the spanwise slip it manages not to give away.

The general lesson is the one this collection keeps arriving at about averaged conditions. An effective boundary condition is a summary, and which feature of the summary matters depends on what the outer flow is doing with it. For the laminar channel here the relevant feature is the magnitude of the slip length, and one number suffices. For a turbulent layer it is the anisotropy of the slip tensor, and the magnitude is nearly irrelevant. The same surface, honestly characterised by the same $\sqrt{K}$, is a three-per-cent improvement in one flow and worth nothing in the other — and no amount of care in measuring the slip length would have revealed that, because the quantity the second flow cares about is a difference that an isotropic measurement reports as zero without ever printing it.

Where the number came from, historically #

Beavers and Joseph measured the effect in 1967 by running a channel over a block of foam metal and finding the flow rate several per cent above what a solid wall gives. They wrote the condition as

$$\frac{du}{dy}\bigg|_0 = \frac{\alpha}{\sqrt{K}}\left[u(0) - u_D\right],$$

with $\alpha$ an empirical constant of order one depending on the surface’s structure — so their slip length is $\sqrt{K}/\alpha$. Saffman derived essentially the same condition from a statistical average of the medium in 1971 and showed that the $u_D$ term is a higher order and may be dropped, which is the result the calculation above reproduces.

Brinkman’s equation, from 1949, is the other half of the story and has a subtlety worth noting: its viscous term carries an effective viscosity that need not equal the fluid’s, and taking the two to be equal — as is done here — is a modelling choice rather than a derivation. Taking a different effective viscosity changes $\alpha$ and not the structure of the result.

So the honest statement is that the slip length is $\sqrt{K}$ times a number of order one, with the number depending on how the interface is built and measurable rather than derivable. This collection computes the case where that number is exactly one and says so.

What the picture cannot show #

The medium is homogeneous and isotropic, and its interface is flat. A real porous surface has a transition region a few pores deep in which the porosity varies, and that region is where the effective interface actually is — so the plane at which the condition is applied has an uncertainty of about one pore scale, which is the same size as the slip length. Every measurement of $\alpha$ carries that ambiguity inside it.

The flow is slow. The interface condition is derived in creeping flow, where reversibility and linearity make the matching exact. Brinkman’s equation is a Stokes-flow description of the medium, valid while Darcy’s law still holds, which is up to a pore Reynolds number of about ten. Faster than that, inertia inside the medium adds a quadratic resistance and the whole matching has to be redone.

The channel is laminar. Over a turbulent flow the effective condition interacts with the layer whose own length is set by the friction velocity, and the comparison that matters is between the slip length and the viscous length rather than the channel height — which is exactly the comparison a roughness has to pass, and it puts most porous surfaces firmly in the “the flow cannot feel it” category at high Reynolds number.

And the same substitution is worse in three dimensions. A slip length is a scalar here because the geometry is one-dimensional; over an anisotropic medium the effective condition is a tensor, and over a patterned surface — grooves, riblets, a superhydrophobic texture — it has different values along and across, which is precisely the property those surfaces are designed to exploit.

Who found it, and when #

Darcy’s law is from 1856 and a measurement rather than a derivation; Brinkman added the viscous term in 1949 to allow a boundary condition to be applied; Beavers and Joseph measured the slip in 1967 and Saffman derived it in 1971. The subject has been active ever since because the effective condition over a structured surface — grooves, posts, a lattice of gas pockets — is the design variable for drag reduction, and its computation is a genuinely open problem for anything but the simplest geometries.

The surprising connection is with the counting argument this ladder starts from. The number of conditions a flow may be told is set by the order of its equation, and the rung below shows that no slip is what viscosity’s extra order buys. This rung says something the counting argument does not: the condition need not be a statement about the fluid’s velocity at all. It can be a statement about a relation between the velocity and its gradient, with a coefficient that encodes a structure nobody resolves — and the counting still works, because a Navier condition is one condition exactly as no slip is.

Where the ladder goes next #

Beside this rung sits the wall treatment a turbulence calculation uses, which is the same idea with no small parameter and no error estimate. Above it lies the structured surface — riblets, superhydrophobic textures, patterned porosity — where the effective condition is anisotropic and is being designed rather than measured, and where the whole question is how much slip a surface can be given before it stops being a surface.