Taylor–Couette flow

A simple Taylor–Couette flow is a steady flow created between two rotating infinitely long coaxial cylinders.[3] Since the cylinder lengths are infinitely long, the flow is essentially unidirectional in steady state. If the inner cylinder with radius ${\displaystyle R_{1}}$ is rotating at constant angular velocity ${\displaystyle \Omega _{1}}$ and the outer cylinder with radius ${\displaystyle R_{2}}$ is rotating at constant angular velocity ${\displaystyle \Omega _{2}}$ as shown in figure, then the azimuthal velocity component is given by[4]

${\displaystyle v_{\theta }=Ar+{\frac {B}{r}},\quad A=\Omega _{1}{\frac {\mu -\eta ^{2}}{1-\eta ^{2}}},\quad B=\Omega _{1}R_{1}^{2}{\frac {1-\mu }{1-\eta ^{2}}}}$

where

${\displaystyle \mu ={\frac {\Omega _{2}}{\Omega _{1}}},\quad \eta ={\frac {R_{1}}{R_{2}}}}$.

Lord Rayleigh[6][7] studied the stability of the problem with inviscid assumption i.e., perturbing Euler equations. The criterion states that in the absence of viscosity the necessary and sufficient condition for distribution of azimuthal velocity ${\displaystyle v_{\theta }(r)}$ to be stable is

${\displaystyle {\frac {d}{dr}}(rv_{\theta })^{2}\geq 0}$

everywhere in the interval; and, further, that the distribution is unstable if ${\displaystyle (rv_{\theta })^{2}}$ should decrease anywhere in the interval. Since ${\displaystyle |rv_{\theta }|}$ represents angular momentum per unit mass, of a fluid element about the axis of rotation, an alternative way of stating the criterion is: a stratification of angular momentum about an axis is stable if and if only it increases monotonically outward.

Streamlines showing Taylor–Couette vortices in the radial-vertical plane, at Re = 950

Taylor vortices (also named after Sir Geoffrey Ingram Taylor) are vortices formed in rotating Taylor–Couette flow when the Taylor number (${\displaystyle \mathrm {Ta} }$) of the flow exceeds a critical value ${\displaystyle \mathrm {Ta_{c}} }$.

For flow in which

${\displaystyle \mathrm {Ta} <\mathrm {Ta_{c}} ,}$

instabilities in the flow are not present, i.e. perturbations to the flow are damped out by viscous forces, and the flow is steady. But, as the ${\displaystyle \mathrm {Ta} }$ exceeds ${\displaystyle \mathrm {Ta_{c}} }$, axisymmetric instabilities appear. The nature of these instabilities is that of an exchange of stabilities (rather than an overstability), and the result is not turbulence but rather a stable secondary flow pattern that emerges in which large toroidal vortices form in flow, stacked one on top of the other. These are the Taylor vortices. While the fluid mechanics of the original flow are unsteady when ${\displaystyle \mathrm {Ta} >\mathrm {Ta_{c}} }$, the new flow, called Taylor–Couette flow, with the Taylor vortices present, is actually steady until the flow reaches a large Reynolds number, at which point the flow transitions to unsteady "wavy vortex" flow, presumably indicating the presence of non-axisymmetric instabilities.

The idealized mathematical problem is posed by choosing a particular value of ${\displaystyle \mu }$, ${\displaystyle \eta }$, and ${\displaystyle \mathrm {Ta} }$. As ${\displaystyle \eta \rightarrow 1}$ and ${\displaystyle \mu \rightarrow 1}$ from below, the critical Taylor number is ${\displaystyle \mathrm {Ta_{c}} \simeq 1708}$[4][8][9][10][11]⁠⁠

In 1975, J. P. Gollub and H. L. Swinney published a paper on the onset of turbulence in rotating fluid. In a Taylor–Couette flow system, they observed that, as the rotation rate increases, the fluid stratifies into a pile of "fluid donuts". With further increases in the rotation rate, the donuts oscillate and twist and finally become turbulent.[12] Their study helped establish the Ruelle–Takens scenario in turbulence,[13] which is an important contribution by Floris Takens and David Ruelle towards understanding how hydrodynamic systems transition from stable flow patterns into turbulent. While the principal, governing factor for this transition is the Reynolds number, there are other important influencing factors: if the flow is open (meaning there is a lateral up- and downstream) or closed (flow is laterally bound; e.g. rotating), and bounded (influenced by wall effects) or unbounded (not influenced by wall effects). According to this classification the Taylor–Couette flow is an example of a flow pattern forming in a closed, bounded flow system.

Wikimedia Commons has media related to Taylor–Couette flow.

• Chossat, P.; Iooss, G. (1992). The Couette–Taylor Problem. Applied Mathematical Sciences. 102. Springer. doi:10.1007/978-1-4612-4300-7. ISBN 978-0387941547.
• Koschmieder, E. L. (1993). Bénard Cells and Taylor Vortices. Cambridge University Press. ISBN 978-0-521-40204-0.