The solutions stop being chosen

There is a picture of transition that almost everybody carries, and it is wrong in a specific and interesting way. In that picture the laminar solution is valid up to some Reynolds number, breaks down, and is replaced by turbulence — the way a material yields, or an approximation runs out of accuracy.

Pipe flow refuses that account more cleanly than anything else in the subject, because pipe flow has an exact solution and it is easy to check whether the solution is still a solution.

The solution, and what makes it exact #

Take a long straight pipe, a fluid of constant density and viscosity, and a flow that does not change along the pipe or with time. Every term in the Navier–Stokes equations that involves acceleration vanishes, because nothing accelerates: a parcel entering the pipe at a given radius leaves at the same radius and the same speed. What is left is a balance between the pressure gradient pushing the fluid along and the viscous stress resisting it, and that balance has one solution:

$$u(r) = 2\bar{u}\left(1 - \frac{r^2}{R^2}\right)$$

This is not an approximation, a similarity solution or the leading term of an expansion. It is the answer, and it is the answer at every Reynolds number, because the Reynolds number measures the size of the term that has already been shown to be zero here.

That is worth dwelling on, because it is exactly where the intuition goes wrong. The Reynolds number is the ratio of inertia to viscosity, and in fully developed pipe flow the inertial term is not small — it is absent. A parcel in this flow has no acceleration at all. So there is nothing for a large Reynolds number to make large.

Two routes to the flow rate, and how far in it applies #

The build integrates the profile over the cross-section numerically — twenty thousand annular strips, each of area 2πr dr — and compares the result against the closed form ūπR².

They agree to better than one part in a million, and the check is in the site’s gate rather than in this paragraph, so an edit that broke the profile would stop the build rather than quietly changing the picture. This is the site’s standing habit: two routes to the same number, sharing no line of code.

The agreement is not surprising and it is not meant to be. What it establishes is the thing the rest of the essay depends on: this profile really is the solution, at the Reynolds numbers being discussed, and any claim that it stops being one has to explain how.

One honest qualification before the transition argument, because it is the qualification that gets skipped and it matters for reading any experiment.

The parabola is the fully developed profile. Fluid entering a pipe from a reservoir arrives with a nearly flat profile and a thin boundary layer on the wall; the layer grows inward, and only when it has met itself on the axis is the profile the one written above. The distance that takes is the entrance length, and for laminar flow it is roughly 0.06 Re diameters.

At Re = 2000 that is 120 diameters — six metres of a 50 mm pipe before the exact solution is the right description of anything. Reynolds’ own apparatus had a bell-mouthed inlet precisely to keep the entering flow smooth, and the care taken over the inlet is the single largest reason different experimenters find different transitional Reynolds numbers.

The same statement in reverse is the useful one: most of the pipe in most real installations is not carrying the profile this essay calls exact, because most real pipes have bends, fittings and valves closer together than the entrance length. The exact solution is a statement about an idealised configuration, and the idealisation is stated here rather than assumed.

What actually happens at 2300 #

Osborne Reynolds ran the experiment in 1883: a glass tube, a reservoir, and a thread of dye. Below a certain flow speed the dye ran the length of the tube as a line. Above it, at some point along the tube, the line burst into a cloud.

The friction factor tells the same story in numbers. Below transition it follows 64/Re exactly, which is the laminar solution’s own prediction. Above it the pressure drop jumps to a different and much larger value, and follows a different law.

Two things about that jump are worth being precise about.

It is a jump, not a bend. The two branches do not meet smoothly. Somewhere in the transitional band a pipe carrying a given flow rate demands substantially more pressure than it did a moment before, and the amount depends on things a Reynolds number does not name.

The number is not a constant. Reynolds himself found values from about 2000 to over 13,000 depending on how carefully the inlet was made and how quiet the laboratory was. Later workers, taking extreme care, have kept pipe flow laminar past 100,000. The figure’s 2300 is a conventional value for an ordinary pipe, and the next essay in this ladder is about why the number is not a number.

What the jump costs, in units somebody pays #

The friction factor is dimensionless and therefore easy to discuss without noticing what it means. The ratio between the two branches is not.

At Re = 4000 the laminar law gives f = 0.016 and Blasius’ correlation gives f = 0.040. The pressure gradient needed to drive a given flow rate is proportional to f, so the pipe on the turbulent branch costs two and a half times the pumping power for the same delivery.

At Re = 10⁵ the comparison is more dramatic and more academic — the laminar branch is not attainable there in any ordinary pipe — but the arithmetic is instructive: f_laminar = 0.00064 against f_turbulent = 0.0178, a factor of twenty-eight. That factor is why laminar flow is worth engineering for wherever it can be had, and why so much of what is called drag reduction is really transition delay.

Linearly stable, and unstable anyway #

Here is the part that took a century.

The standard way to find out whether a solution is the one a flow will take is linear stability analysis: perturb the solution by something infinitesimal, linearise, and ask whether the perturbation grows. If some disturbance grows, the solution is not observable and the flow will find something else.

Applied to pipe flow, this analysis returns the answer stable — at every Reynolds number, for every disturbance, without exception. The result has been checked with increasing rigour since the 1970s and is not in doubt. The parabolic profile is linearly stable at Re = 10⁶.

So a pipe carrying laminar flow at Re = 4000 is sitting in a solution which is exact, which is stable to every infinitesimal disturbance, and which it will nevertheless leave the moment anybody taps the pipe.

The resolution took until the 1990s and 2000s and turns on two facts that sound technical and are not. The first is that the linearised operator is non-normal: its eigenfunctions are not orthogonal, so a disturbance can grow enormously — by factors of thousands — for a while, and then decay, without any eigenvalue ever having a positive real part. The second is that if the transient growth is large enough for long enough, the nonlinear terms are no longer negligible, and the flow can be carried to an entirely different state before the linear decay has time to happen.

The consequence is that transition here is subcritical: it requires a disturbance of finite size, and the size required falls as the Reynolds number rises. That is why the transitional Reynolds number depends on the inlet, the vibration and the roughness — those are what set the size of the disturbances available.

Puffs, slugs, and a transition that is not a point #

The subcritical picture has an observable consequence that the friction chart hides, and it is worth stating because it is what a transitional pipe actually contains.

Just above threshold, turbulence in a pipe is not everywhere. It appears in localised patches — puffs, some twenty diameters long, that travel with the flow, keep their length, and are surrounded by laminar fluid on both sides. A puff can decay back to laminar flow spontaneously, and whether it does is a matter of probability rather than of Reynolds number: the mean lifetime of a puff rises steeply with Reynolds number but stays finite.

Higher up, puffs begin to split faster than they decay, and the balance between splitting and decaying is what sets the Reynolds number above which turbulence persists indefinitely. Measured carefully by Avila and co-workers in 2011, that number is 2040, and it is the closest thing pipe transition has to a genuine critical value — a point at which two competing rates cross rather than a point at which a solution ceases to exist.

Above it the patches grow and merge into slugs that fill the pipe, and the flow is turbulent in the ordinary sense.

None of that is drawn on this site, and it cannot be: a puff is an unsteady three-dimensional structure twenty diameters long, and this site’s solver is a two-dimensional stepper on a coarse grid. The description above is reported, not computed, and is marked as such here for the same reason the turbulent profile is drawn in the colour reserved for borrowed claims.

Where the model stops, and it stops early #

Everything above is about what the equations permit. Nothing above is a calculation of the turbulent state, and there is none anywhere on this site.

The turbulent profile in the first figure is a fitted correlation — the 1/7-power law — and it is drawn in the colour this site reserves for a claim it has not derived. The turbulent friction branch is Blasius’ 1913 fit to experimental data. Neither came out of a solver, and the reason is arithmetic: resolving the flow in a pipe at Re = 10⁵ needs roughly 10¹¹ grid points, which is a supercomputer calculation rather than a build step.

So this essay can say with confidence what the laminar solution does and cannot say what replaces it. That asymmetry is a fact about the subject rather than a limitation of this site, and it runs through the whole of this field.

The door swings both ways #

There is a consequence of the essay’s thesis that is easy to miss and is the strongest evidence for it. If the laminar solution never stopped being a solution — and never stopped being a stable one — then a flow that has left it can be put back.

It can, and it happens by itself in a situation every wind tunnel exploits. Accelerate a turbulent boundary layer hard enough and it relaminarises: the turbulence weakens, the fluctuations die, and the layer reverts to a laminar profile. The criterion is an acceleration parameter

$$K = \frac{\nu}{U^2}\frac{dU}{dx},$$

and reversion sets in above about $3\times10^{-6}$ — a small number that a strong contraction comfortably exceeds. That is one of the reasons a wind tunnel’s contraction is shaped as it is: it does not merely accelerate the flow, it relaminarises whatever the settling chamber left in it.

The same reversion is produced by anything that suppresses the near-wall production: suction through a porous wall, heating a gas at the wall or cooling a liquid there — both of which raise the viscosity where the turbulence lives — and strong stratification or rotation, which give a displaced parcel a restoring force and stop it from carrying momentum across the layer. All of them are used, and none of them is a repair of a broken solution: each is a way of driving the flow back into a state that was available throughout.

The deliberate version is the most direct test the argument has had. Turbulence in a pipe can be destroyed on purpose — by injecting a disturbance chosen to flatten the near-wall profile, which removes the shear the turbulence lives on — and the flow then stays laminar downstream, at a Reynolds number where it had been turbulent a metre earlier. The control costs a small fraction of the pumping power the laminar branch saves, which is the factor of two and a half computed above.

Nothing about that would be possible if transition were a breakdown. A material that has yielded does not un-yield when the load is removed the right way; an approximation that has run out of accuracy does not become accurate again. What the pipe does instead is exactly what a system with two available states does: it can be moved from one to the other in either direction, and the direction that is easy is the one for which a large enough disturbance is lying around.

Which sharpens the essay’s central claim into something with an experimental consequence. The asymmetry between the two directions is not in the equations, which admit both states at every Reynolds number. It is in the basins: above about two thousand the turbulent state is easy to reach because ordinary disturbances suffice, and the laminar one is hard to reach because returning to it requires a disturbance shaped rather than merely large. Transition looks like a one-way door because the world supplies the push in only one direction, and not because the door has a hinge.

The same shape, elsewhere #

Pipe flow is the clean case, not the only one.

Plane Couette flow — fluid dragged between two sliding plates — is also linearly stable at every Reynolds number, and also goes turbulent, at around Re = 350.

A flat-plate boundary layer is the opposite case and is genuinely unstable to infinitesimal disturbances above Re ≈ 91 based on displacement thickness, through the Tollmien–Schlichting waves. It transitions at Re_x of roughly 10⁶ anyway, three or four decades later than the instability sets in, because the growth rate is small and the amplification takes distance.

Rayleigh–Bénard convection is the case where the linear threshold is exactly right, and it is the one this site can compute in closed form: below a critical Rayleigh number nothing moves and above it convection begins, at the value linear theory says.

Three problems, three different relationships between the linear threshold and what is observed. The lesson is not that linear stability is useless; it is that a linear threshold is a statement about infinitesimal disturbances, and whether that is the relevant statement depends on the problem.

Who found it, and when #

Hagen measured the pressure drop in 1839 and Poiseuille in 1840, both empirically, both before the Navier–Stokes equations were generally trusted. The derivation came later and the law kept both names.

Reynolds published the transition experiment in 1883 and named the ratio in the same paper. He was explicit that the transition was sensitive to disturbance and that the number was not a property of the fluid alone — a caution that was widely dropped over the following century.

The linear stability of pipe flow was established piecemeal and settled by Salwen, Cotton and Grosch in 1980. The transient-growth explanation is due to Trefethen, Trefethen, Reddy and Driscoll in 1993, and the identification of the exact travelling-wave solutions that structure the transitional state came in the 2000s, from Faisst and Eckhardt and from Wedin and Kerswell.

That is 110 years between the observation and the explanation, on the simplest geometry in the subject, with an exact solution in hand the whole time.

Where the ladder goes next #

The next rung takes the transitional Reynolds number seriously as a quantity and finds that it is not one: the number that is not a number is about what a transition Reynolds number actually reports, which is as much about the laboratory as about the fluid.

A different route from here goes to the mechanism rather than the threshold. A layer with a kink in it is Rayleigh’s criterion — the one general statement about which velocity profiles can be unstable at all — and it connects transition to separation through the sign of a single derivative.