Taylor Number (rotating-flow instability)

Take two concentric cylinders with fluid between them and turn the inner one. At low speed the fluid goes round in smooth concentric layers — circular Couette flow, the rotational analogue of flow between two flat plates. Raise the speed and at some point the flow reorganizes itself, abruptly and beautifully, into a stack of counter-rotating doughnut-shaped cells filling the gap. G. I. Taylor predicted where that would happen in 1923, measured it, and got agreement so close that the paper is still cited as one of the finest confirmations of hydrodynamic stability theory ever produced.

The mechanism is centrifugal. A ring of fluid near the inner cylinder is going fast and wants to fly outward; a ring near the outer cylinder is going slowly. Swap them and the fast ring finds itself where the required centripetal force is smaller, so it keeps going outward — the disturbance grows. Rayleigh had already written down the inviscid criterion: rotation is unstable when angular momentum decreases outward. Viscosity opposes the exchange, and the Taylor number is the ratio of the centrifugal driving to the viscous damping:

\[\mathrm{Ta} = \frac{\Omega^2 r\, d^3}{\nu^2}\]

with a critical value near 1708 in the narrow-gap limit with a stationary outer cylinder. That number is not a coincidence: it is also the critical Rayleigh number for Bénard convection between heated plates, because the linear stability problems reduce to the same equations. Buoyancy driving a lighter fluid up past viscosity, and centrifugal force driving a faster ring out past viscosity, are the same mathematics wearing different clothes.

The definition is not universal and you must check which one a threshold belongs to. The Taylor number appears in the literature in at least four normalizations, differing by factors of 2, 4 and $\pi^2$, and each carries its own critical value — 1708, 3390 and $41.3^2$ are all "the critical Taylor number" in somebody's notation. The form on this page is $\Omega^2 r d^3/\nu^2$, and 1708 belongs to it and to nothing else. It is worth knowing that this form equals $\mathrm{Re}^2 (d/r)$ with $\mathrm{Re} = \Omega r d/\nu$, so the critical value is the same statement as the old bearing-shop rule that vortices begin near $\mathrm{Re} = 41.3\sqrt{r/d}$.

Which brings us to why a journal bearing has a speed limit. A plain bearing is a Taylor–Couette apparatus that happens to carry a load: a shaft turning inside a shell with a thin oil film between. Petroff's law, the whole basis of hydrodynamic bearing friction, assumes that film is in laminar shear. Above the critical Taylor number it is not. Taylor vortices appear in the film, the torque rises above the laminar prediction, and the extra heat thins the oil, which raises the Taylor number further — a feedback that has to be modelled rather than ignored in large high-speed machines.

Look at where the sensitivity is. The clearance is cubed. A bearing worn to twice its design clearance reaches the instability at eight times lower Taylor number, which is 2.8 times less shaft speed. Viscosity enters squared, so a bearing running hot on thin oil is far closer to the limit than the same bearing cold. Radius enters only linearly, which makes it the weakest term and is why scaling a machine up does not, by itself, cause trouble — it is the clearance that has to grow with the shaft, and it is that growth, cubed, that carries the instability into large machines. Turbine and turbocharger bearings genuinely operate above the first transition, and their thermal models account for it.

The geometry is not symmetric, and it matters which member turns. Rayleigh's criterion says angular momentum increasing outward is stable, so rotating the outer cylinder while holding the inner one still is stable to far higher speeds. That asymmetry is one of the elegant results of the subject and it is easy to forget when reading a critical value out of a table.

Beyond first onset, this apparatus became one of the great testbeds for the route to chaos. Taylor vortices become wavy, then modulated, then chaotic, then turbulent, through a well-documented sequence of transitions that Gollub and Swinney used in 1975 to test dynamical-systems theory against a real fluid. And the geometry is used deliberately: a Taylor–Couette reactor exploits the vortices for excellent radial mixing with very narrow residence-time distribution, which is why the arrangement turns up in polymerization, in crystallization, and in blood-oxygenator design.