Polymers stop at an asymptote that no polymer chooses

Sand and a commercial pipe part company took the log law, the line on which every turbulent wall flow lies, and asked what roughness does to it: it drags the line down, leaving the slope alone, by an amount that’s a function of the roughness in wall units. The roughness a wall cannot feel established the log law’s own terms. Roughness is the wall’s way of adding friction, and it moves the law one way.

Something else moves it the other way, by a larger amount, with no change to the wall at all. Dissolve a very small amount of a very long polymer in the fluid, a few parts per million by weight, and a turbulent pipe flow at the same pressure gradient carries a great deal more liquid. Toms presented that in 1948, at the first international congress on rheology, from an experiment on polymethyl methacrylate in monochlorobenzene. It seemed to him a curiosity about a dilute solution. It’s since become the largest drag reduction available in any flow by any means, and the crude oil in the Alaska pipeline has flowed with a dose of it since 1979.

The viscosity of a solution so dilute barely changes: it’s a per cent higher, perhaps. That’s why it was a surprise. A polymer solution in a laminar pipe is nearly indistinguishable from the solvent. Only in a turbulent flow do the chains seem to do anything, and when they do, their effect is out of all proportion to their share.

Three layers #

The whole of the empirical picture is a mean velocity profile on the same wall-unit axes as the log law. It has three parts, as Virk and his colleagues laid out at MIT around 1970. Nearest the wall is the same viscous layer as ever, $u^+=y^+$. Farthest from it is the same log law as ever, with the same von Kármán slope of 2.5, but displaced upward. Between them is a new layer, the elastic sublayer, with a much steeper slope. It lies on a law of its own,

$$u^+=11.7\ln y^+-17.0,$$

and the elastic layer grows from the wall towards the pipe’s centre as the polymer’s effect increases. What it does to the log law above is carry it up, by $S^+$, as the figure shows. At the largest effect, the elastic layer fills the pipe, and the whole profile is this one law.

There’s an easily overlooked curiosity in this law. It meets the viscous line almost exactly as a tangent: at $y^+=11.7$ its slope is exactly one, and its height there is $0.08$ above $u^+=y^+$. So the join is smooth and it doesn’t need a blending function. It’s hard to say whether that’s a deep fact or a coincidence. It comes from the numerical values of two fitted constants. Nobody has shown that it means anything.

Integrating such a profile over the pipe’s section gives the mean speed, and a mean speed gives a friction factor. So the whole family of friction curves comes from a single parameter, the edge of the elastic layer.

The two limits are a real test of the arithmetic, because the integrated three-layer profile isn’t the same thing as the well-known friction formulae. With no elastic layer, the integrated profile gives 0.00448, against Prandtl and Kármán’s 0.00450; with the elastic law throughout, it gives 0.000899, against the published asymptote 0.000886. Those are two independent checks of the same quadrature, within one and a half per cent, and both of the friction laws being compared to were fitted to experiment.

What is strange about it #

In Prandtl–Kármán coordinates, Virk’s asymptote is the straight line

$$\frac{1}{\sqrt f}=19\log_{10}\left(Re\sqrt f\right)-32.4 .$$

The strange thing is what’s not in it. It has no polymer concentration, no molecular weight, no relaxation time, no solvent viscosity other than through the Reynolds number. Every other number in the problem has been dropped. The data come from dozens of polymers in water and in organic solvents, and from pipes of a few millimetres to tens of millimetres, and they all reach this one line and stop. Add more polymer and nothing further happens. A higher concentration eventually raises the viscosity, which raises $f$ for a mundane reason, but it doesn’t take the flow past the line. Virk, Mickley and Smith’s 1970 paper noted this, and Virk’s review of 1975 is the definitive statement.

The way to appreciate that is by comparison with what polymers do before the asymptote. On the way there, every polymer behaves differently. Its onset point depends on the molecular weight and the pipe’s size, and the slope of its own branch depends on its concentration. Each solution has its own curve. Then all of them arrive on one line, the way a family of flows arrive on the same log law near a wall. The asymptote is therefore a law of turbulence, or of something about turbulence and any elastic fluid. It’s not a law of polymers.

It’s also not a statement that the flow has become laminar, though at high concentration that’s often the casual assumption. The laminar line at a Reynolds number of $10^5$ is 0.00016, and the asymptote 0.00089 is more than five times higher. So there’s a sustained turbulence on the asymptote, and it’s a very particular one. Its Reynolds shear stress is small, its streaks are widely spaced and its turbulent energy is mostly in the streamwise component, whereas Newtonian turbulence has a much larger stress. An elastic fluid at the asymptote seems to maintain just enough of a self-sustained process to stay turbulent, and hardly any more. Exactly why that level is independent of everything else is the open question.

The reduction is a little different from the two laws’ constancy. As a fraction of the Newtonian friction, the limit rises slowly with Reynolds number, from two thirds at the bottom of the turbulent range to almost nine tenths at the highest pipe flows. It’s a result of the asymptote’s slope being steeper than Prandtl’s, so it’s the shape of the line and not a separate fact.

Which chains act, and when #

The onset, by contrast, has a clear mechanism. A polymer chain in a dilute solution is a loose coil that relaxes in a time $\tau$, typically fractions of a millisecond for a million-weight chain, and scaling as the three-halves power of its weight by the Zimm theory. It’s a passive tracer until the flow can stretch it faster than it can relax. Lumley proposed in 1969 that that’s the condition for an effect: the Weissenberg number on the wall’s time scale, $\tau u_\tau^2/\nu$, must reach about one.

So a heavier chain acts sooner, at a lower wall stress. Virk’s onset is expressed in almost those terms. For a fire-hose’s flow the numbers come out comfortably, and a fire brigade dosing its hoses, as New York’s did in the 1970s with a product named Rapid Water, gets a substantially longer throw at the same pump pressure. And the polymer’s fate has the mechanism in it too: the same stretching that drives the effect breaks long chains, so a solution being recirculated through a pump degrades and loses its effect after a few passes, the heavy chains first. That’s why a pipeline’s additive has to be injected continuously, every few pumping stations.

The Lumley idea explains where the chain acts. How it works once stretched is much less clear. Stretched chains exert a stress along their length, and direct simulations with dumbbell models, since Sureshkumar and his colleagues’ in 1997, reproduce the drag reduction: the chains store energy near the wall, damp the streamwise vortices that lift low-speed streaks, and return energy at a slower rate. The picture agrees about the near-wall layer. Reaching the asymptote in simulation depends on a model’s details in a way that the measurements don’t, however. De Gennes offered a different scale-by-scale argument in 1986, that the chains act through elasticity on a range of eddies, truncating the cascade at a scale where elastic stress matches the turbulent. Neither version derives 11.7 or 19.

The most striking recent work, by Samanta and others in 2013, has shown that a polymer solution at modest Reynolds numbers can reach what looks like the asymptote from below, by a separate, purely elastic-inertial form of turbulence that has no Newtonian equivalent. So perhaps the asymptote isn’t the end of a sequence of damped Newtonian turbulence but the beginning of a different kind. It’s unsettled. Xi and Graham’s calculation of 2010 finds “hibernating” intervals, in which the Newtonian turbulence nearly shuts off and the flow rests on a state close to Virk’s asymptote, which suggests a state the flow can always fall back on. Nobody can yet say what gives it that value.

An eddy with a fifth of its reach #

The numbers in the log law’s constants can be turned into a statement about mixing. Prandtl’s guess with a constant in it gives a log law whose slope is one over $\kappa$, a constant saying that an eddy’s reach is $\kappa$ times its distance from the wall. For Newtonian flow, $\kappa=0.4$. For the elastic sublayer, the same reading says $\kappa=1/11.7=0.085$. So the eddies in that layer mix momentum as if their reach were a fifth of what it is in a Newtonian fluid at the same distance. Yet the fluid’s viscosity is effectively unchanged.

That reading has its limits, since the elastic stress itself carries some of the momentum, which the mixing-length picture doesn’t count. A chain being stretched by the flow is carrying part of the total stress in the layer, and measurements near the asymptote show the Reynolds stress falling far below the total, with the difference, a “stress deficit”, in the polymer. It can be a large fraction of the whole. Yet the reading makes the comparison clear. It says that polymers change the mean profile by reaching into the same layer that’s responsible for most of Newtonian friction: the buffer layer, where turbulence production peaks and where a wall’s viscous layer hands over to its eddies. No other method alters that layer by anything like that amount.

The comparison with other approaches underlines that. Riblets, the small streamwise grooves on a racing yacht or a swimmer’s suit, get a reduction of eight or ten per cent at best, by holding the near-wall vortices off the wall. Wall-normal suction and spanwise oscillation of the wall reach a few tens of per cent, at a power cost. A polymer gets to eighty per cent, with no moving parts and at a low dose. That’s because a chain acts on the small scales from inside the flow, as elasticity competing with the cascade itself, rather than from the wall.

Nor is any of this a case of a viscosity that depends on the question in the ordinary sense. The solution’s shear viscosity, measured in a rheometer, barely registers the polymer, and a shear-thinning fluid would give a tiny effect. A solution’s extensional viscosity, on the other hand, can be a thousand times its shear viscosity, because a fully stretched chain resists further stretching strongly. That’s the property a turbulent eddy feels and a rheometer doesn’t, and it’s why such a dilute solution matters only in a flow that stretches it.

What it is worth #

That’s the largest number here. A line like the Trans-Alaska pipeline runs at about that Reynolds number, and a long pipeline’s capacity is limited by what its pumps can deliver in pressure over the distance between stations. Commercial additives since 1979 have been used for extra throughput and to avoid building extra pumping stations. They have run well short of the asymptote, at reductions of a half or so, for reasons of cost and degradation, and they’ve been worth a great deal. In the Alaskan case, Burger, Munk and Wahl reported a capacity increase of tens of per cent, with a dose in the parts per million.

What it hasn’t done is help outside pipes. The same effect on a ship’s hull would be very attractive, but a ship would have to carry the polymer and continuously inject it into the boundary layer over the whole hull, at a rate proportional to its wetted area. Trials have shown that it works, and that the logistics don’t. Polymer near a surface also has a way of being washed outwards and diluted, unlike a pipe, which is closed. So this is a story of pipelines.

What the picture cannot show #

A polymer. The three-layer picture is an empirical summary. There’s no chain in it, and the elastic layer’s edge is a fitted parameter rather than a derived one. Going from a named polymer to that edge needs a further correlation, of the kind Virk gave in his “slope increment” and fluid-specific parameter.

A rough wall. The chart is for smooth pipes. Polymer effects on rough pipes are large, as Virk also showed, and often largest in a range where the chains act on the layer between roughness elements, which changes the roughness function too. In a real line the two effects interact.

Non-circular ducts and a nonuniform wall stress. The law is written for the pipe’s section. A square duct’s secondary flows, or a curved pipe’s, would have a stress-dependent onset across their walls.

The flow near the pipe’s centre. A log law is imposed up to the centre, as is usual and is about right for the friction. It’s known to be off in wake components, for Newtonian flow, by a small correction, which the check above absorbs. In the elastic limit a real profile deviates more than that as the core becomes increasingly uniform. A profile law also implies a stress profile that the measurements need not obey.

The convention #

The friction factor is Fanning’s, $f=\tau_w/(\frac12\rho U^2)$, a quarter of Darcy’s, the usual form in this literature, and $Re$ is on the bulk speed and the pipe’s diameter. In wall units, $u^+=u/u_\tau$ and $y^+=yu_\tau/\nu$. The log law here is $2.5\ln y^++5.5$, the pipe flow’s traditional constants. $R^+$ is the pipe’s radius in wall units, the Kármán number. The zero-shear solvent viscosity is used throughout. The bars for the oil line take a kinematic viscosity of $8\times10^{-6}$ and a density of 860, and the friction factor from the smooth-pipe law.

How it was checked #

Each end of the family recovers a separately stated law, by a quadrature that used only the three profile segments. The difference between the integrated elastic profile and Virk’s published friction formula, one and a half per cent, is within his stated scatter. The model refuses a zero distance from the wall, a pipe of zero radius and a negative Reynolds number.

Who found it, and when #

Toms’s report is from 1948, and Mysels’s patent for gasoline with aluminium soaps, on the same effect in a military context, from 1949. Virk’s three-layer model and the asymptote date from 1967 to 1975, and Lumley’s time criterion from 1969. De Gennes’s elastic scale argument is from 1986. The Trans-Alaska pipeline’s use of additive began in 1979. Sureshkumar’s direct simulations date from 1997, Xi and Graham’s hibernating turbulence from 2010, and Samanta’s elasto-inertial transition from 2013.

Still open: where 19 comes from #

The asymptote’s two constants are measured. Any theory of it must produce the elastic layer’s slope of 11.7 from something, and the simplest possibility is that it’s whatever self-sustained state a shear flow can barely hold. The next calculation would take a low-order model of the near-wall cycle of streaks and vortices, ask for the mean profile at which that cycle is marginally self-sustaining at each wall distance, as a function only of the local mean shear, and see whether that profile has a logarithmic form and with what slope. A slope close to 11.7 would be evidence that the asymptote is the marginal state of Newtonian turbulence, with the polymer merely what removes everything above that margin, and a very different value would say that the elasticity must be in it.