A coil's friction is one number, and the coil holds its flow laminar

A number that is only the shape of the hole wrote down the fully developed laminar flow in any straight duct, and in its list of what wasn’t claimed it stated: a curved duct develops secondary flow and a quite different resistance. In a plumbing system, nearly all pipes bend somewhere, and most heat exchangers are coils, so the exception is a good deal commoner than the rule.

A straight pipe’s Poiseuille flow is so easily found because it’s one-dimensional: the velocity is along the pipe, depends only on the radius, and the pressure has a constant gradient. Bend the axis into a circle of radius $R$, and consider what must change. Every fluid parcel now has a centripetal acceleration, $w^2/R$, towards the centre of the coil. It has to come from a pressure gradient across the section, with higher pressure on the outer wall. But $w$ isn’t uniform. It’s fastest at the centre of the pipe and zero at the walls, and a pressure gradient that’s right for the mean speed is too weak for the fast core and too strong for the slow fluid near the walls. So the core moves outward, the wall fluid moves inward along the top and bottom of the section, and a pair of counter-rotating cells appears.

Eustice saw it in 1910 and 1911, by injecting dye into coiled pipes and watching the threads follow helical paths. Dean, in 1927 and 1928, worked out the mathematics of a loosely coiled pipe, in the limit where $a/R$ is small.

One number #

Dean’s reduction has the same logic as many a dimensionless collapse. In the loose-coiling limit, the secondary velocity scales as $\sqrt{a/R}$ times the axial speed. Writing the axial momentum equation and the curl of the sectional equations, in polar coordinates $(r,\theta)$ with $\psi$ the secondary flow’s stream function, $\zeta$ its vorticity and $G$ the pressure gradient, every remaining appearance of the coil’s radius can be gathered into one group:

$$De = Re\sqrt{\frac{a}{R}} ,$$

with $Re$ on the mean speed and the pipe’s diameter. So the tightness of the coil and the speed are not two parameters. A slow flow in a tight coil and a fast one in a slack coil are the same. And a section’s axial and secondary fields depend on $De$ and nothing else. The steady equations are

$$\mathbf u\cdot\nabla w = \nabla^2 w + G,\qquad \mathbf u\cdot\nabla\zeta=\nabla^2\zeta-2\,\delta\,w\frac{\partial w}{\partial y},\qquad \nabla^2\psi=-\zeta .$$

The last term in the middle equation is what drives everything: it’s the curl of the centrifugal force $\delta w^2$ along the outward direction, and it’s zero if $w$ doesn’t vary across the coil’s plane. Marching them to steady state on a polar grid, with the velocity held symmetric about the plane of the coil, yields the friction ratio.

The result has a clear shape. Below about $De=12$ a coil is a straight pipe to within a per cent. White’s measurements of 1929 and Dean’s series agree on that, with a correction that starts at the fourth power of $De$. Beyond $De$ of about 20, it rises steadily, and at large values it approaches $\sqrt{De}$. So a pipe in a coil of a hundred radii at a Reynolds number of 1000 is at $De=100$, and it has about half as much friction again as a straight one.

Why nothing happens at first #

The slow start of the curve is exact, and its reason is worth having. Dean’s own series has the friction ratio departing from one as the fourth power of the Dean number, at $1+0.03\,(K/576)^2$ with $K=2De^2$: 0.7 per cent at $De=11.6$, which is what the march finds.

Why so high a power? Reversing the coil’s sense of curvature cannot change the friction, so no odd powers of the centrifugal term can appear; that rules out any term odd in the curvature. But the term in $De^2$ also vanishes, and that’s the interesting part. Poiseuille’s flow has a special property: among all flows through a pipe with a given flow rate, it has the least viscous dissipation. Its pressure drop is therefore at a minimum against any small change in the shape of the velocity field, and such a change affects the pressure drop only at second order in the change. The secondary flow distorts the axial field by an amount proportional to $De^2$. So the friction rises as the square of that, $De^4$.

So friction is blind to a coil’s small effects for the same reason that the period of a pendulum is blind to its small swing’s size. The profile itself is changing visibly: at $De=11.6$, the peak has moved about a tenth of the radius, a change anyone could measure with dye, while the friction has moved by under one per cent. It also says something for measurement. A friction measurement near $De=10$ can’t distinguish a coil from a straight pipe, while a velocity profile can, easily.

Why the friction rises #

The obvious story is that the secondary flow dissipates energy. That’s true, but the numbers don’t support it as the main explanation. With a secondary speed of ten per cent of the mean, the secondary kinetic energy is a per cent of the mean’s, and friction has risen by 50 per cent. The bulk of the extra loss comes from what the secondary flow does to the axial flow.

The fast core has moved towards the outer wall. A parabolic profile had its maximum at the centre, half a radius from anywhere, in a smooth distribution. A profile with its peak at 0.7 radii has a very thin layer between that peak and the outer wall, with a much steeper gradient, and it pays for that in wall stress.

So nearly all the increase is on the outer wall, where the shear reaches more than three times a straight pipe’s, and the inner wall sees a smaller shear than it would have if the pipe were straight. Most of the fluid passes the outer wall at high speed in a thin layer. This is an important point for design. It’s the outer wall that erodes in a real bend, and that has the highest heat transfer, in an exchanger, and the thin wall layer’s thickness is set by the secondary flow’s sweeping action.

There’s a clean way of viewing the large-Dean limit. The core is then nearly inviscid, with uniform vorticity across its centre, and a thin viscous layer runs round the wall from the outer point towards the inner, where the two layers meet and discharge inward. That layer has a thickness that goes as $De^{-1/2}$ in units of the radius, so the wall shear and the friction ratio go as $De^{1/2}$. Barua and Ito worked that out in the 1950s and 1960s, and it’s the origin of the square root: 0.1064 $\sqrt{De}$ in this form. Below De of about 500 the layer isn’t thin, and the curve hasn’t reached it, as the figure shows, and the calculation’s grid doesn’t go far enough to meet it.

A secondary flow in other clothes #

The same balance turns up in a surprising number of other places. A stirred cup of tea has a rotating core whose centrifugal force is held by the radial pressure gradient, and a slow bottom layer on which that gradient acts unopposed: it pushes the fluid inward, which is why tea leaves gather at the centre. The river has the same one in a bend: James Thomson worked it out in 1876 for meanders, and he recognised it as the reason bed material moves to the inside and banks wear on the outside. Einstein offered the tea-cup and river argument in 1926.

So Dean’s flow is the same statement with one difference: here the fluid in the core has been dragged along by a pressure gradient rather than by rotation, and the secondary flow is an unavoidable product of a pipe’s profile. Prandtl called all of these secondary flow of the first kind, the ones produced by a mean flow’s curvature, as distinct from those produced by turbulent stresses in a straight square duct, which are his second kind. In all of them, the principle is that a pressure field is set by the bulk of the flow and then acts on the boundary layer, which has less speed, and it’s a surprising force in that thin layer. The layer that stops at a depth has its own version with rotation.

And there’s a practical connection that runs opposite to the friction. A straight pipe is an excellent way of spreading a pulse of dye along its length: the fast core carries it ahead and the slow layer holds it back, and Taylor’s dispersion has an effective diffusivity proportional to the square of the speed. A coil’s cross-flow carries particles between the fast and slow regions more often, and that reduces the axial dispersion. A helical tube in a chromatograph or a chemical reactor narrows the residence-time distribution at a modest price in pressure. It’s the same secondary flow that makes the friction worse, seen as a mixing mechanism across the section.

The surprise at transition #

The most surprising result is what happens when the flow speeds up. Taylor showed in 1929, in a short paper, with a dye filament in a helical tube, that a coiled pipe stays laminar at a much greater Reynolds number than a straight one. Many later experiments confirm it, Ito’s among them.

So the very thing that would be expected to destabilise a flow, a persistent disturbance to its velocity field, instead makes it more stable. A straight pipe at Reynolds number 5000 is fully turbulent, whereas a coil with a tightness of a twentieth is laminar there. By White’s fit, its friction is 3.7 times the laminar straight-pipe value, a Fanning factor of 0.0118, and that is only a quarter above the 0.0094 that a turbulent straight pipe has at that Reynolds number. The penalty for a coil very nearly disappears in that range, since the pipe it’s being compared with has gone turbulent and it hasn’t.

Why that happens is less settled than the fact of it. A straight pipe’s instability is a subcritical one, needing a finite disturbance that grows through the non-normal mechanism, and the streaks and vortices of the self-sustaining process have to fit in a section. The secondary flow in a coil sweeps across the section and it removes the near-wall structures by convection, and the base flow’s profile is no longer the same. Recent measurements by Kühnen and others have shown that the transition in a coil happens at a relaxed, supercritical instability for tight coils, of a travelling wave, rather than through puffs. So at high curvature, a coil’s transition differs in kind from a puff in a straight pipe. Dean’s equations can’t reach it, since the loose-coiling limit’s solution has no critical number in it.

Whatever the mechanism, it means a coil’s friction chart looks unlike a straight pipe’s: there’s no sudden jump where the laminar curve meets the turbulent one, since by then the two are nearly equal.

What it would mean in a real coil #

Take an ordinary domestic heat exchanger, a coil of 12 millimetres’ bore tube wound at a 150 millimetre radius, with water at a tenth of a metre a second. The Reynolds number is 1200, and $a/R$ is 0.04, so the Dean number is 240. The friction is about twice that of a straight tube, and the secondary flow’s top speed is about 0.7 $\sqrt{0.04}$ of the mean, or 1.4 centimetres a second. It sounds trivial, and a parcel makes a trip across the section of the tube, 12 millimetres, in about a second. Compared with thermal diffusion across the tube, which takes the square of the radius over the diffusivity, about 250 seconds for water, that’s a very large improvement, and it’s why a coiled tube has several times the heat transfer of a straight one at the same flow. That result is worth the doubled friction.

What the picture cannot show #

Tight coils. Everything has been in the loose-coiling limit, in which curvature enters only through the centrifugal term. For $a/R$ larger than about 0.1, geometric terms that were dropped begin to matter, and the Dean number alone no longer decides. A very tight bend isn’t governed by this at all.

Entry length and helical pitch. The flow is treated as fully developed, in a coil with no pitch. A real coil’s flow takes a certain distance to develop, longer than a straight pipe’s, and it oscillates along the way as the secondary flow overshoots; how far before a duct forgets has the straight version. A pitch adds a torsion that’s small unless the coil is steep. An isolated bend is entirely developing flow, and this isn’t applicable to it.

Extra cells and other solutions. At high Dean numbers, a second solution with four cells exists: Dennis and Ng in 1982 and others found it. For symmetric solutions on this grid, it isn’t reached. Its presence shows that the solution being marched to isn’t the only one, and it’s why experiments can be sensitive to how a flow is started.

Grid. The convection is first-order upwinded, so the computed friction is too high by an error proportional to the grid spacing, about 5 per cent at $De=100$ with the 36-point grid used in the figures, and more at larger $De$. That’s why the computed curve sits above White’s fit near the top of its range. The De = 100 check below records that error’s size, and the extrapolated value is about 4 per cent below White’s fit, which is itself a fit to measurements with a comparable scatter.

The convention #

The Reynolds number is on the mean speed and the diameter. The Dean number here is $Re\sqrt{a/R}$; others use $Re\sqrt{2a/R}$ or Dean’s $K=2De^2$ in a different velocity scale, so a quoted Dean number needs its definition checking. Friction ratio is at the same flow rate, the coil’s over the straight pipe’s. “Critical” is Ito’s, from the departure of the friction from the laminar curve. White’s fit is $f/f_0=1/[1-(1-(11.6/De)^{0.45})^{2.2}]$, and Ito’s is $2\times10^4(a/R)^{0.32}$.

How it was checked #

With curvature off the solver has to return Poiseuille’s flow, and it does so to the 0.08 per cent its treatment of the centre allows. The change with the grid is cleanly first-order: successive differences stand in the ratio of the reciprocal grid sizes. The model refuses a negative pressure gradient, a negative curvature and a negative Dean number. A check of the secondary flow’s symmetry is built in: the plane of the coil is a boundary of the grid.

Who found it, and when #

Eustice’s dye measurements are from 1910 and 1911; Dean’s analysis from 1927 and 1928. White’s friction measurements and Taylor’s observation of delayed transition are both from 1929. Thomson’s account of rivers is from 1876, Einstein’s of tea leaves from 1926. Barua’s boundary-layer treatment dates from 1963, Ito’s correlations from 1959 and 1969, and the four-cell solutions from Dennis and Ng in 1982.

Still open: how much heat the coil is really worth #

The heat-transfer estimate above was an order of magnitude. The same marching equations hold an energy equation with one extra parameter, the Prandtl number, and the calculation would march temperature on the same secondary flow, with a fixed wall temperature, at each Dean number. It would ask how the Nusselt number rises with the Dean number at Prandtl numbers from 1 to 100, whether at high Prandtl number the thermal layer is thin enough that only the outer wall’s share counts, and so at which Dean number and Prandtl number a coil’s heat transfer outweighs what its friction costs, with both given as powers of the same one number.