Fluid dynamics · rotating shear

Taylor–Couette Flow

Why can fluid swirling between two cylinders break up into stacked rolls? Because the angular momentum per unit mass can fall off with radius. When its square decreases outward, a fluid ring nudged outward carries more angular momentum than the fluid it moves into, so the centrifugal push flings it farther out. By Rayleigh's criterion the flow is unstable, and secondary radial and axial circulation can grow into stacked Taylor cells.

Orbit the three-dimensional coaxial-cylinder rig and compare solid-body rotation, flow driven by the outer cylinder and counter-rotation. Tracers move with the exact velocity, and a Rayleigh profile shows where the inviscid flow is prone to centrifugal instability. The optional rolls only illustrate the structure.

Physical time $0.0000\,\mathrm{s}$

Inner Reynolds

\mathrm{Re}_i

$31.42$

Outer Reynolds

\mathrm{Re}_o

$0.00$

Minimum Rayleigh

\Phi

$-31.58\,\mathrm{s^{-2}}$

Rayleigh-negative gap $100.0\%$

Inner base torque on fluid $1.092\,\mathrm{mN\,m}$

Base-flow power $3.429\,\mathrm{mW}$

Shown modelExact base flow

Coaxial-cylinder rig

Inspect rotating shear

Integrated tracersTrails follow physical azimuthal velocity and any prescribed secondary flow.Exact velocity profileBoth wall velocities set the signed circular profile.Inviscid Rayleigh profileAmber marks the negative-discriminant radial band, not a viscous onset prediction.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}}

Prescribed roll strength

0.00

Roll strength is chosen by hand for illustration; it is not a measured instability amplitude. The rolls stay inside the band where the Rayleigh criterion is negative and are suppressed when there is no such band.

The Lab computes the exact base flow and the inviscid, axisymmetric criterion. It does not solve for viscous onset, bifurcation or turbulence. The optional rolls only illustrate the structure. The periodic axial window $H=6(r_o-r_i)$ has no endwalls. Base torque and power use the reference density $\rho=1000\,\mathrm{kg\,m^{-3}}$ and leave out dissipation from the prescribed rolls.

Physics tutorial

Taylor–Couette flow: separate base state, criterion, and rolls

BackgroundTwo coaxial cylinders turn independently, and viscous shear drags the fluid in the gap along with them. The exact circular flow in the gap is a steady solution of the equations even when small disturbances could upset it.

Why it mattersThis rig lets you compare absolute rotation, the speed difference between the cylinders and viscosity separately, without mistaking a pretty roll pattern for a computed bifurcation.

Start with the essentials

Focus question: Why can two flows with the same speed difference have different centrifugal stability?
One-sentence intuition: The inviscid, axisymmetric Rayleigh criterion looks at how the square of the specific angular momentum changes with radius, not at the angular momentum itself. Where $\Phi=r^{-3}\frac{d(r v_\theta)^2}{dr}=4A\Omega(r)$ is negative, the criterion marks the flow as prone to centrifugal instability; it does not give the critical speed for a fluid with finite viscosity.

Core mathematical model

Circular Couette base flow

v_\theta=Ar+\frac{B}{r},\qquad A=\frac{\Omega_o r_o^2-\Omega_i r_i^2}{r_o^2-r_i^2},\qquad B=\frac{r_i^2r_o^2(\Omega_i-\Omega_o)}{r_o^2-r_i^2}

The fluid sticks to both walls, so the two wall speeds fix the exact profile. The base flow has no radial or axial velocity.

A Reynolds number for each wall

d=r_o-r_i,\qquad \mathrm{Re}_i=\frac{|\Omega_i|r_i d}{\nu},\qquad \mathrm{Re}_o=\frac{|\Omega_o|r_o d}{\nu}

Solid-body rotation has no shear, yet both wall Reynolds numbers are nonzero. When the inner wall is at rest, its Reynolds number is always zero.

Inviscid centrifugal criterion

\Phi(r)=\frac{1}{r^3}\frac{d(r^2\Omega)^2}{dr}=4A\left(A+\frac{B}{r^2}\right)

Reversing both cylinders leaves the criterion unchanged. With counter-rotation, the negative band may cover only part of the gap. The criterion is not the threshold at which a viscous flow becomes unstable.

Base torque and power

T_i=4\pi\rho\nu BH,\qquad T_o=-T_i,\qquad P_{\mathrm{base}}=T_i(\Omega_i-\Omega_o)\geq0

The torques are the ones the cylinders exert on the fluid. The power equals the viscous dissipation of the exact base flow. The reference density and the periodic axial length are given in the model notice, and dissipation from the prescribed rolls is left out.

Prescribed secondary circulation

\Psi=C\sin^2(\pi s)\sin(kz),\qquad u_r=-\frac{1}{r}\partial_z\Psi,\qquad u_z=\frac{1}{r}\partial_r\Psi

The normalized radius spans only the band where the criterion is negative. Building the velocity from a streamfunction keeps the flow axisymmetric and incompressible, and both secondary velocity components vanish at the edges of the band. The strength is chosen by hand, not solved from the Navier–Stokes equations.

Common difficulties

A negative criterion does not mean rolls have formed

Typical misconceptionOnce the Rayleigh value is negative, I can read off the real critical speed and the roll amplitude.

Better mental modelThe criterion only covers inviscid, axisymmetric disturbances. Onset with finite viscosity, endwalls, non-axisymmetric modes and turbulence all need a fuller model. The optional rolls here are only an illustration.

Solid-body rotation moves but does not shear

Typical misconceptionIf the two cylinders turn at the same rate, the fluid is at rest and the Reynolds numbers are zero.

Better mental modelThe tracers still circle the axis, and both wall Reynolds numbers are nonzero. What vanishes is the base torque and the dissipated power.

Run the experiment

01
Step through solid-body rotation
Choose Solid-body rotation, then use Single step or Advance one second.
What to observe: Each tracer keeps its radius and height; only its angle advances, at the physical rotation rate. The base torque and power are zero.
02
Keep the shear, change the overall rotation
Compare Prescribed rolls with Outer cylinder faster, then adjust both speeds yourself.
What to observe: The size of the speed difference alone does not set the Rayleigh profile. The Outer cylinder faster preset is stable by this criterion, so the rolls you selected are suppressed.
03
Find a negative band in part of the gap
Choose Counter-rotation, show the Rayleigh profile and let the tracers run.
What to observe: The illustrative rolls fill only the radial band where the criterion is negative. Fluid farther out can keep moving in pure circles.