The cost of going turbulent

The site has a laminar skin-friction law it derived: Blasius’ similarity solution, shot rather than tabulated, giving C_D = 1.328 Re^(−1/2) for a flat plate.

It has no turbulent one, because nothing on this site computes a turbulent flow. What it has instead is a correlation — a curve fitted to experiments, with an exponent that came out of the data rather than out of an equation — and the whole of this essay depends on keeping those two objects distinct while comparing them.

Two laws, and only one of them is a solution #

The laminar law comes from solving the boundary-layer equations under a similarity assumption that the equations themselves justify, integrating the wall shear along the plate, and dividing by the dynamic pressure. Every step is a derivation and the site performs the first of them itself.

$$C_D = 1.328\,Re_L^{-1/2}$$

The turbulent law comes from assuming a 1/7-power velocity profile — which is not a solution of anything, and which does not even satisfy the boundary condition at the wall, since its slope there is infinite — and calibrating the result against measurement.

$$C_D = 0.072\,Re_L^{-1/5}$$

Different authors quote 0.072, 0.074 or 0.0735 for the leading constant, and the spread between them is the honest measure of how much of it is fitted. Nobody quotes a range for 1.328.

The exponents are the interesting difference. The laminar layer’s friction falls as Re^(−1/2) and the turbulent layer’s as Re^(−1/5), so the two curves are not parallel and the penalty for transitioning is not a fixed multiple.

The ratio, which is what a design turns on #

Dividing one law by the other gives

$$\frac{C_{D,\text{turb}}}{C_{D,\text{lam}}} = 0.0542\,Re_L^{0.3}$$

At Re = 10⁵ that is a factor of about 3. At Re = 5·10⁶ it is about 11. At Re = 10⁸ it is about 27.

The design consequence is direct and is the reason natural-laminar-flow sections exist at all: the value of keeping a layer laminar rises with the size and speed of the surface. On a model aeroplane the saving is worth having; on a sailplane wing at Re of a few million it is worth designing the whole section around; on a large transport wing at Re of 5·10⁷ it would be worth a great deal, and it is also where holding laminar flow is hardest, because a long run of favourable gradient is difficult to arrange and easy to spoil.

Where the correlation stops being valid #

The two curves cross, and the crossing is a useful place to be careful.

The build bisects for it and finds Re ≈ 1.66·10⁴. Below that Reynolds number the turbulent correlation predicts less drag than the laminar solution, which is nonsense: a turbulent layer at Re = 10³ is not a thing that exists, and if it did it would not have less friction than a laminar one.

What the crossing marks is the bottom of the correlation’s domain of validity, and the fact that it is not marked in most textbooks is a small example of the general problem. A fitted curve carries no information about where it stops applying, and plotting it outside the range it was fitted on produces a confident wrong number with no warning attached.

The site’s practice is to state the range on the figure — the model note carries “Re 10⁵ to 10⁹” — which is the closest a picture can come to refusing to be read outside it.

The 1/7 profile, and why an indefensible assumption works #

It is worth taking the turbulent law’s foundation apart, because the gap between how bad the assumption is and how well the answer works is instructive rather than embarrassing.

The assumed profile is u/U = (y/δ)^(1/7). Its slope at the wall is infinite, so it predicts infinite wall shear and cannot be used to compute the very quantity it is being used to compute. The friction has to be imported from somewhere else — Blasius’ pipe correlation, in the original derivation — and the profile is used only for the momentum integral.

So the derivation is: an inadmissible profile, plus an imported friction law from a different geometry, integrated through the momentum-integral equation. It gets the exponent right and the constant within a few per cent over four decades.

The reason it works is that the momentum integral is forgiving. It needs the integral of the profile, and a profile can be substantially wrong in shape while having nearly the right area — the same forgiveness that lets thin-aerofoil theory get lift right while getting the leading-edge suction badly wrong.

The reason it should still be labelled is that the forgiveness has a boundary and nothing in the formula says where it is. The log-law profile gives a better-behaved friction law which is implicit rather than explicit, and which is what a careful estimate uses; the 1/7 law survives because it can be written on one line.

What the friction penalty buys #

A turbulent layer is not merely a more expensive laminar one. It behaves differently in the way that matters most, and the difference is worth more than the friction it costs on any body where separation is in question.

A turbulent profile is fuller — the mixing carries fast fluid down towards the wall — so it carries more momentum in the region that decides whether the layer can climb a pressure hill. Where a laminar layer separates at a Falkner–Skan parameter of −0.1988, a turbulent one survives gradients several times steeper.

Because separation is what produces a wide low-pressure wake, and because pressure drag on a bluff body is far larger than friction drag, postponing separation can reduce the total by much more than the friction increase costs.

The sphere is the standard case. Below the critical Reynolds number the laminar layer separates at about 80° from the front stagnation point, the wake is wide, and the drag coefficient is about 0.5. Above it the layer is turbulent before separating, separation moves back to about 120°, the wake narrows sharply, and the drag coefficient falls to about 0.1.

A fivefold reduction in total drag, from a change that increases the friction component. This is the drag crisis, and it is the single most counter-intuitive result in elementary aerodynamics.

Why a golf ball is dimpled #

The dimples move the crisis to a Reynolds number a golf ball actually reaches.

A smooth sphere’s drag crisis occurs near Re = 3·10⁵. A golf ball in flight is at Re of roughly 10⁵ — below it. Left smooth, the ball would fly in the high-drag regime for its whole trajectory.

The dimples trip the layer at a much lower Reynolds number, so the ball flies in the post-crisis regime throughout. The drag coefficient is roughly halved relative to a smooth ball at the same speed, and the carry is very roughly doubled.

Three things are worth noticing about that.

It is a case where roughness reduces drag, which is the opposite of the general rule and is worth having as a standing counter-example to it.

It works only in a Reynolds-number window. A dimpled ball well above the crisis Reynolds number would simply carry the extra friction with no benefit, and dimpling an aircraft wing would be straightforwardly harmful.

The dimples do a second job. They also increase the lift generated by backspin, which for a golf shot is comparable in value to the drag reduction, and the two effects are usually conflated.

The whole budget, and where this fits in it #

Skin friction is one line of an aircraft’s drag account, and the trade above is easier to weigh with the other lines beside it.

For a transport aircraft in cruise the account runs roughly: skin friction about half of the total, induced drag — the price of having ends — about a third, form drag and interference the remainder, and wave drag whatever the design has chosen to accept near its cruise Mach number.

Half the total being friction is what makes the laminar question worth the trouble. A section holding laminar flow over the first half of both surfaces removes perhaps a third of that half, which is a larger single saving than almost anything else available on a mature airframe — and it is why the subject keeps returning despite the fragility that has defeated it repeatedly in service.

The comparison also shows why the same argument reads differently for a bluff body. On a car, a lorry or a building, friction is a small fraction of the total and pressure drag is nearly all of it, so the entire calculus inverts: the question is never how to keep a layer laminar but how to keep the flow attached, and every device fitted to that end pays friction to buy pressure.

The third law, where the Reynolds number drops out #

Both curves above are for a smooth plate, and roughness does something to the turbulent one that it does not do to the laminar one at all — it removes the Reynolds number from the answer.

The mechanism is that a turbulent layer has a very thin viscous region right at the wall, of thickness of order $\nu/u_\tau$, and what matters is whether a surface irregularity fits inside it. The comparison is made by a roughness Reynolds number built on the friction velocity,

$$k^+ = \frac{k\,u_\tau}{\nu},$$

and it splits the problem into three. Below about $k^+ = 5$ the irregularities are buried in the viscous sublayer, the outer flow never meets them, and the surface is hydraulically smooth — the correlation above applies unchanged, and polishing further buys nothing. Above about 70 the elements protrude through the sublayer entirely, their own form drag is the whole of the wall stress, and the surface is fully rough: $c_f$ becomes a function of $L/k$ alone and stops falling with Reynolds number. In between is a transitional band that every practical surface manages to sit in.

The fully-rough regime is the one worth dwelling on, because it breaks the essay’s arithmetic. The whole comparison above rests on two curves that both descend, one at $-1/2$ and one at $-1/5$. A rough surface’s curve is horizontal, so its penalty against a laminar plate grows without any exponent difference being needed — and its drag rises as the square of the speed with no relief at all. That is why a fouled ship’s hull is a fuel problem rather than a nuisance: barnacles put the plating firmly into the fully-rough regime, where nothing about going faster makes the friction coefficient smaller.

Setting $k^+ = 5$ and using a typical friction velocity gives the number a manufacturer needs — an admissible roughness of about $100\nu/U$, below which a surface behaves as though it were polished. The values are uncomfortably small. A transport wing at cruise is allowed about six micrometres; a light aircraft at fifty metres a second about thirty; a ship’s hull about twenty. Insect debris, a lapped paint edge, a rivet head and a squashed sealant bead are all far larger than any of them.

And it puts the golf ball at the other end of the same axis. A dimple is a quarter of a millimetre deep, which is forty times the admissible roughness for a body of that size and speed — not marginally rough but emphatically so, and deliberately. The whole of this essay’s trade sits between those two numbers: six micrometres of unwanted roughness is a defect on a wing, and two hundred and fifty micrometres of wanted roughness is the design of a ball.

Where the model stops #

Three limits, all of which the figures are inside.

The flat plate is not a wing. Both laws are for zero pressure gradient. A real surface has one, and the boundary layer on it is thicker or thinner accordingly. A practical drag estimate applies a form factor to the flat-plate result, and the form factor is itself a correlation.

Transition is a placement. Neither curve says where transition occurs; a real plate has a laminar run followed by a turbulent one and a total drag between the two curves, with the split decided by a transition Reynolds number that is not a property of the fluid.

The site cannot compute either the turbulent friction or the drag crisis. Both are stated here from measurement, and the reason is arithmetic rather than diffidence: a grid that resolved the turbulent layer on a sphere at the crisis Reynolds number would need something over 10¹² points, and this site has eleven thousand.

The separation angles are quoted, not solved. The 80° and 120° above come from experiment. What the site’s own stepper computes is the separation angle at Reynolds numbers in the hundreds, where the layer is laminar throughout and the crisis is nowhere near — so the mechanism is visible in the figures and the numbers attached to the crisis are not the figures’ own. What the site does compute is the laminar half, the separation criterion behind the crisis, and the split between friction and pressure drag on its own solved fields at Reynolds numbers in the hundreds.

What a laminar section actually looks like #

It is worth being concrete about what the design change amounts to, because “keep the flow accelerating” is easy to say and constrains a shape severely.

A conventional section has its maximum thickness at about 30% of chord, which puts the pressure minimum near there and gives an adverse gradient over the aft 70%. A laminar section moves the maximum thickness back — to 40%, 50%, occasionally further — so that the flow is still accelerating over the front half.

Three costs come with that.

The pressure recovery is steeper. Whatever pressure rise is required has to be accomplished over a shorter distance, so the aft gradient is more adverse than a conventional section’s, and the layer arrives at it with less margin. That is a design nearer to separation, not further from it.

The low-drag range is narrow. The favourable gradient exists only over a band of lift coefficients — the drag polar shows a characteristic bucket — and outside the bucket the section is worse than a conventional one rather than merely no better.

The surface tolerance is severe. Waviness and steps that would be irrelevant on a conventional section trip the layer here, and the requirement is measured in tenths of a millimetre over a chord of metres. That is a manufacturing and maintenance specification, and it is the reason so many laminar designs have worked on a prototype and not in service.

None of this is an argument against the approach; sailplanes have exploited it successfully for decades, and modern composite construction makes the surface requirement attainable. It is an argument for reading a quoted laminar drag figure together with the conditions it assumed, which is the same caution the transition Reynolds number needed.

Who found it, and when #

Blasius solved the laminar plate in 1908.

Prandtl and von Kármán obtained the 1/7-power turbulent law in the early 1920s, on Blasius’ own pipe-friction correlation transferred to a plate, and both were clear that it was a correlation. The better-behaved logarithmic friction laws — Schoenherr’s from 1932, and the Kármán–Schoenherr form derived from the log-law profile — followed, and are what a serious estimate uses.

Eiffel measured the drag crisis on spheres in 1912 and was disbelieved, since the drag coefficient appeared to fall discontinuously with speed. Prandtl settled it in 1914 with the tripwire experiment: a wire round a smooth sphere ahead of the separation point reproduced the low-drag state at a Reynolds number below the crisis, which demonstrated that the cause was the state of the boundary layer rather than anything about the free stream.

Dimpled golf balls predate the explanation by a generation. Players had noticed by the 1890s that scuffed gutta-percha balls flew further than new ones, and manufacturers were moulding patterns before anybody could say why.

Where the ladder goes next #

This anchor’s earlier rungs establish the two components of a wing’s drag and the speed at which their sum is smallest. This rung supplies the third quantity the budget needs, which is what changes when the layer’s state changes.

From here the turbulence field continues with the structure the arithmetic forbids drawing: the street this site cannot draw is the wake behind a cylinder, built as an ideal-flow model and named as one.