A second length at the wall

The layer with no length in it is the whole argument for the logarithm, and it is worth restating because everything here depends on it.

Near a wall a turbulent flow has two lengths: the viscous length $\nu/u_\tau$ and the layer thickness $\delta$. Between them there is a region too far from the wall for the first to matter and too close to the edge for the second to, so a velocity gradient there has nothing to depend on but the distance to the wall itself. Dimensionally that forces

$$\frac{du^+}{dy^+} = \frac{1}{\kappa y^+},$$

and integrating gives the logarithm. The constant $\kappa$ has never been derived from anything; it is the coefficient of an argument that says a certain dependence is forbidden.

Now roughen the wall. There is a third length: the roughness height $k$, which belongs to neither the fluid nor the flow.

The slope cannot change, and the reason is the argument itself #

The first thing to say is what does not happen, because it is the thing a reader expects.

$\kappa$ does not change. It cannot, because the argument that produces the logarithm is an argument about which lengths are forbidden in the overlap region, and $k$ is forbidden there for exactly the same reason $\nu/u_\tau$ is: the overlap is far from the wall compared with both.

Roughness therefore acts entirely through the boundary condition at the bottom of the overlap, and the only thing a boundary condition can do to a logarithm whose slope is fixed is move it up or down. The profile becomes

$$u^+ = \frac{1}{\kappa}\ln y^+ + B - \Delta U^+(k^+),$$

with $k^+ = ku_\tau/\nu$, and $\Delta U^+$ is called the roughness function. It is the whole of the effect, and every measurement ever made on a rough wall is consistent with that.

What the argument does and does not license #

It is worth being careful here, because the claim is stronger than the usual one and a reader who over-reads it will get into trouble.

What the overlap argument licenses is that within the overlap region the gradient depends only on $y$, so the slope is $1/\kappa y^+$ whatever the wall is made of. It says nothing whatever about the region below the overlap, where the roughness elements are, and where the flow is a complicated three-dimensional business of separations and wakes behind individual bumps.

That region is where the intercept is set, and it is where all the physics of a particular surface lives. The theory’s achievement is to compress it into one number — and its limitation is that it cannot compute that number from a surface.

The same division runs through a guess with a constant in it, where Prandtl’s mixing length produces the whole structure of a wall profile from one line of assumption and cannot produce $\kappa$; and through what a code says to a wall, where a wall function is an asymptotic result applied at one grid point on the assumption that the point lies in a region the calculation has not checked exists.

Where the viscosity leaves the answer #

The interesting limit is $\nu \to 0$ at fixed $k$, and it is the one this essay is built on.

At large $k^+$ the roughness function becomes $(1/\kappa)\ln k^+ + B - 8.5$. Substituting it back into the profile, the $\ln k^+$ cancels the $\ln$ of the viscous length inside $y^+$, and what is left is

$$u^+ = \frac{1}{\kappa}\ln\frac{y}{k} + 8.5.$$

There is no $\nu$ in that at all. The viscosity has left the answer entirely, and what decides the profile is the wall’s own length.

The check that this is really what the machinery does, rather than an algebraic rearrangement, is to evaluate $u^+ - (1/\kappa)\ln(y/k)$ on computed profiles at $k^+ = 200$, $2{,}000$ and $20{,}000$. It comes out at 8.755, 8.803 and 8.814 — a spread of 0.06 wall units across a hundredfold change in the viscosity.

The interpolation used for $\Delta U^+$ is not fitted to that. It is $(1/\kappa)\ln(1 + ck^+)$ with $c$ fixed by requiring the large-$k^+$ limit to be the fully rough asymptote, which gives $c = e^{\kappa(B - 8.5)} = 0.2381$ — the number usually rounded to 0.3.

The three regimes, which are three regimes of $k^+$ and not of the wall #

$k^+$ is the roughness measured in viscous lengths, so the same surface moves between regimes as the flow speeds up.

Below $k^+ \approx 5$ the roughness is buried inside the viscous sublayer, $\Delta U^+$ is negligible, and the wall is hydraulically smooth. Above $k^+ \approx 70$ the sublayer is gone and the flow is fully rough. Between them is a transitional band where both matter.

That is why the roughness a wall cannot feel can say that a rough pipe and a polished one carry the same flow for the same pressure over three decades of Reynolds number and then suddenly do not. Nothing about the pipe changed. The viscous length shrank past the roughness.

What the friction does, which is the practical statement #

The consequence a designer meets is the shape of the friction curve.

A smooth wall’s friction coefficient falls by a factor of 3.75 over four decades of Reynolds number, because the viscous length keeps shrinking and the log law keeps gaining decades. A wall with a roughness one per cent of the layer thickness stops falling: over the last four decades it moves by 3 per cent.

That flattening is the whole of Reynolds-number independence, and it is not an approximation. In the fully rough state the friction is a function of $\delta/k$ alone, and $\delta/k$ does not depend on the Reynolds number. At $Re = 10^8$ the rough wall’s friction is 3.76 times the smooth wall’s, from an intercept shift and nothing else.

The practical reading of that picture is a rule of thumb worth carrying. The cost of going turbulent prices the transition from laminar to turbulent friction on a wing; this is the further price of the surface not being smooth in the flow’s own terms, and on a large fast vehicle it is the larger of the two.

What the outer layer is supposed not to notice #

There is a second claim usually made alongside all this, and it needs to be labelled carefully because the computation here cannot test it.

Townsend’s hypothesis is that outside a few roughness heights the turbulence does not know what the wall is made of: the velocity defect $(U_e - u)/u_\tau$ is the same function of $y/\delta$ for a rough wall as for a smooth one. Only the intercept moves, so only the region where the intercept lives is affected.

In the composite profile used here the collapse is exact by construction: the roughness function enters as a constant and cancels identically in the defect, to $3.6\times10^{-15}$. A check that “found” the collapse would be reporting its own input, and it is worth saying so rather than presenting the figure as evidence.

What the figure is good for is the size of the thing being cancelled. The four profiles differ by 11.1 wall units in $u^+$ across the whole layer and by a factor of nearly four in friction, and the hypothesis is that the outer turbulence is unaffected by all of it. That is a strong claim, not a weak one, and the evidence for it is measured profiles rather than anything computed here.

The length that is fitted, which is the weak link #

The whole apparatus rests on a single number $k$, and $k$ is not a measurement of a surface.

It is an equivalent sand-grain roughness: the height of Nikuradse’s uniformly packed sand that would produce the same friction. Two surfaces with the same root-mean-square height can differ several fold in $k$, because the roughness function depends on the shape of the elements, their density, their arrangement and how sharp they are — a forest of ribs behaves nothing like a field of dimples of the same height.

So the theory is exact in its structure and calibrated in its input. That is a common position in this subject and it is worth being precise about which half is which. The statement “roughness moves the intercept and nothing else” is a consequence of the dimensional argument and is as solid as $\kappa$. The statement “this surface has $k = 0.2$ millimetres” is a fit.

Nothing here derives $k$ from a surface, and nothing in the literature does either.

Why this is a limit essay and not a correlation essay #

The reason this belongs beside its neighbours is the shape of the limit rather than the engineering.

Take $\nu \to 0$ on a smooth wall and nothing survives: the viscous length goes to zero, the sublayer disappears, the friction coefficient goes to zero logarithmically, and the wall stops resisting. That is the ordinary expectation for an inviscid limit and it is why the limit that is not the value is such a surprise when the dissipation refuses to follow it.

Take $\nu \to 0$ on a rough wall and the friction goes to a finite constant. The fluid’s own length has left the problem, and the surface’s length has taken its place. What survives the limit is not a residual of the viscosity; it is a completely different quantity that was there all along and was invisible while the viscous length was larger.

That is the rule these essays run on, in an unusual form: the limit does not leave a residue of the thing it removed, it promotes something else.

Two ways of quoting the same wall, and the trap in it #

There is a bookkeeping hazard worth naming because it produces confident wrong numbers.

The roughness function is a function of $k^+$, which contains the friction velocity, which is what is being solved for. So computing the friction of a given surface at a given Reynolds number is a fixed point: guess $u_\tau$, get $k^+$, get $\Delta U^+$, get the profile, get a new $u_\tau$, iterate. That is what the friction figures here are doing, and it converges in a few dozen passes.

The hazard is that a roughness quoted as a fraction of the layer thickness and a roughness quoted in wall units are different statements about the same wall, and they move in opposite directions as the Reynolds number rises. At fixed $k/\delta$, raising the Reynolds number raises $k^+$ and makes the wall rougher in the sense that matters. At fixed $k^+$ — which is what a computation holds if it is careless — the physical roughness is shrinking.

Every figure in this essay states which is held, and the regime map is the place where the two are shown together.

The mechanism, in one sentence about drag #

It is worth saying what the fully rough wall’s force actually is, because the word “friction” is misleading there.

On a smooth wall the wall force is viscous shear: $\mu\,du/dy$ at the surface, integrated. On a fully rough wall it is almost entirely pressure drag on the roughness elements — each bump has a high pressure on its front and a low one behind, and the sum of those is the force. Viscous shear on the element surfaces contributes a few per cent.

That is why the answer stops depending on the viscosity. Pressure drag on a bluff element at high Reynolds number is $\tfrac12\rho u^2$ times a coefficient of order one, and there is no viscosity in it — the same statement the two drags a wing pays makes about the split between friction and form drag, applied to objects a millimetre high.

The logarithm survives because the overlap argument survives. What changes underneath it is the mechanism by which the wall exerts a force at all.

What a smooth wall is, in this light #

The definition falls out and it is a satisfying one.

A wall is hydraulically smooth when its roughness is buried inside the viscous sublayer, which is about five viscous lengths thick. So “smooth” means $k < 5\nu/u_\tau$ — a statement about the flow and the surface together, with no absolute standard of polish in it.

At a Reynolds number of $10^4$ that is a fairly generous tolerance. At $10^8$ it is not: the viscous length on a ship’s hull at speed is a few microns, and no paint is that smooth. Which is why Reynolds-number independence is the normal state for large vehicles and the exceptional state for small ones, and why extrapolating a model test to full scale needs the roughness state of both to be known.

What this collection has said about wall layers, in order #

It is worth laying the four essays side by side, because together they are a complete account of a turbulent wall and each supplies exactly one thing.

The layer with no length in it supplies the logarithm, from a dimensional argument with no model in it, and measures $\kappa$ coming out of a computed profile at 0.412 after the limit is reached and 0.511 before.

A guess with a constant in it supplies a closure — Prandtl’s mixing length — that reproduces the same profile from an assumption, and is careful about what the agreement establishes.

What a code says to a wall supplies the practical consequence: a calculation that cannot afford the sublayer tells the wall the log law instead, exactly between $y^+ = 30$ and $10{,}000$ and sixty per cent wrong at $y^+ = 1$.

And this one supplies the third length. Nothing in the first three changes; a term is added to the intercept, the viscosity leaves the fully rough answer, and the friction stops depending on the Reynolds number.

Four essays, one profile, and the only thing that ever moves is where the logarithm sits.

One number, and where it is worth knowing #

The whole of a rough wall’s effect is $\Delta U^+$, so a reader who remembers one thing should remember what it is worth in ordinary terms.

Eleven wall units of shift is a large number. On a smooth wall at $Re_\tau = 5{,}000$ the edge velocity is about 27 wall units, so a shift of 11 is forty per cent of it — which is why the friction changes by a factor of nearly four rather than by a few per cent.

That is the correct order of magnitude for the effect of ordinary roughness on an ordinary vehicle, and it explains why hull fouling and paint quality are treated as seriously as they are. Nothing about the shape of the vessel has changed; the logarithm has moved down by a third of its own range.

Limits recorded rather than smoothed over #

The profile is a composite, not a solution. Van Driest’s sublayer, the log law and Coles’ wake are three pieces glued together with constants that were fitted to measurements. Everything computed on it inherits that: what the figures show is what the standard description of a wall layer implies, not what a Navier–Stokes solution would give.

The roughness function is an interpolation. Its two asymptotes are argued for; its shape in between is a smooth curve with one constant, chosen to meet them. Real transitional roughness functions are not all the same shape — Nikuradse’s sand and Colebrook’s commercial pipes disagree in exactly that band — and neither shape is derived from anything.

The grid matters more than it looks. The first version of this computation used a uniform grid in $y^+$, which at $y^+ = 10^6$ puts the first point at three thousand and integrates $du^+/dy^+ = 2$ across the entire sublayer, returning a velocity of three thousand. The failure was silent: the profile was smooth, monotone and enormous, and it appeared as a roughness function that had moved by eighty-eight wall units. The grid is linear through the sublayer and logarithmic above it now.

And nothing here is three-dimensional or unsteady. Riblets, which reduce drag by being rough in one direction and not the other, are outside this framework entirely.

The residue #

$\nu \to 0$ at fixed $k$. Everything viscous leaves the answer: the sublayer, the viscous length, the Reynolds number itself.

What is left is a profile set by the surface, a friction that is a property of the pipe rather than of the flow through it, and a force that is pressure drag on obstacles rather than shear on a wall. None of it is a correction to the smooth-wall answer, and none of it goes away as the limit is taken further.

The wall brought a length, and in the limit it is the only one left.