Fluid dynamics · rotating-shear instability

Taylor–Couette Flow

A transparent outer cylinder surrounds an independently driven inner rotor. Change both rotation rates, gap, and viscosity to compare the analytic Couette profile with the axial Taylor cells created by centrifugal instability.

Taylor

\mathrm{Ta}

$1974$

Supercriticality

\mathrm{Ta}/\mathrm{Ta}_c

$1.16$

Inner Reynolds

\mathrm{Re}_i

$31$

Cell amplitude $0.20$

Angular-momentum gradient $-0.25\,\mathrm{m\,s^{-1}}$

Flow regimeTaylor vortices

Coaxial-cylinder rig

Tune rotating shear

Luminous tracersFollow secondary circulation in the annular gap.Couette profileCompare the exact laminar azimuthal profile across the gap.Stability discriminantRead whether specific angular momentum rises or falls outward.Inner-cylinder speed

\Omega_i

+30\,\mathrm{rpm}

Outer-cylinder speed

\Omega_o

0\,\mathrm{rpm}

Inner radius

r_i

0.040\,\mathrm{m}

Outer radius

r_o

0.060\,\mathrm{m}

Kinematic viscosity

\nu

8.0\times10^{-5}\,\mathrm{m^2\,s^{-1}}

Counter-rotation changes both shear and the outward angular-momentum gradient; the same speed difference need not have the same stability.

The laminar azimuthal speed obeys $v_\theta(r)=Ar+B/r$ . The instability layer uses a small-gap weakly nonlinear approximation and excludes endwalls, non-axisymmetric waves, and turbulence.

Start with the essentials

Focus question: Why does smooth circular motion suddenly grow a stack of toroidal secondary-flow cells?
One-sentence intuition: The analytic base state $v_\theta=Ar+B/r$ destabilizes when centrifugal drive defeats viscous diffusion, closing radial and axial velocity into Taylor cells.

Core mathematical model

Circular Couette base state

v_\theta(r)=Ar+\frac{B}{r}

With no radial or axial motion, the two wall speeds uniquely determine the laminar azimuthal profile.

Inner-cylinder Reynolds number

\mathrm{Re}_i=\frac{|\Omega_i-\Omega_o|r_i d}{\nu}

Uses gap width and differential rotation to compare shear with viscous diffusion.

Small-gap Taylor number

\mathrm{Ta}=\frac{4(\Omega_i-\Omega_o)^2r_i d^3}{\nu^2}

The teaching model uses it to organize circular flow, Taylor cells, and a higher-amplitude wavy state.

Common difficulties

Cells are not the primary circular flow

Typical misconceptionThe visible loops simply trace the inner cylinder rotation.

Better mental modelPrimary flow travels azimuthally around the axis; Taylor cells are additional radial–axial secondary circulation.

Speed difference is not the whole stability problem

Typical misconceptionEqual cylinder-speed difference guarantees equal stability.

Better mental modelAbsolute rotation and the outward angular-momentum gradient also matter, so co- and counter-rotation can share shear but not stability.

Run the experiment

01
Read the analytic base state
Choose Circular Couette and inspect the velocity profile.
What to observe: Tracers move primarily azimuthally while the profile joins the two wall speeds.
02
Create Taylor cells
Switch to Taylor cells.
What to observe: Paired radial–axial circulation appears, stacking along the cylinder axis with alternating rotation.
03
Compare counter-rotation
Choose Counter-rotation and inspect the stability discriminant.
What to observe: Shear, Taylor number, and angular-momentum gradient change together as the model enters a stronger wavy-cell state.