Fluid dynamics · rotating shear

Taylor–Couette Flow

Orbit a three-dimensional coaxial rig and compare solid-body, outer-driven, and counter-rotating flow. Exact velocity drives tracer paths; a Rayleigh profile identifies inviscid centrifugal vulnerability. Optional rolls illustrate structure only.

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 a chosen illustration, not a measured instability amplitude. Rolls are confined to the Rayleigh-negative band and suppressed when that band is absent.

Exact base flow and the inviscid axisymmetric criterion; no viscous onset, bifurcation, or turbulence solver. Optional rolls illustrate structure only. The periodic axial window $H=6(r_o-r_i)$ has no endwalls. Base torque and power use reference density $\rho=1000\,\mathrm{kg\,m^{-3}}$ and exclude prescribed-roll dissipation.

Start with the essentials

Focus question: Why does equal differential speed not imply equal centrifugal stability?
One-sentence intuition: The inviscid axisymmetric Rayleigh criterion examines squared specific angular momentum, not angular momentum itself. Negative $\Phi=r^{-3}\frac{d(r v_\theta)^2}{dr}=4A\Omega(r)$ indicates centrifugal vulnerability under this criterion, not the finite-viscosity critical speed.

Core mathematical model

Circular Couette base

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 two no-slip wall velocities fix the exact profile. The base has no radial or axial velocity.

Individual wall Reynolds numbers

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 but nonzero wall Reynolds numbers. A stationary inner wall always has zero inner Reynolds number.

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. Counter-rotation may have a negative band in only part of the gap. This is not a viscous onset threshold.

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

Torques are exerted by the cylinders on the fluid. Power equals exact-base viscous dissipation. Reference density and periodic axial length are specified in the model notice; prescribed-roll dissipation is excluded.

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

Normalized radius lies within the negative-discriminant band. The streamfunction enforces axisymmetric incompressibility and both secondary velocities vanish at the band boundaries. Strength is chosen, not solved from Navier–Stokes.

Common difficulties

A negative criterion is not computed roll onset

Typical misconceptionA negative Rayleigh value gives the actual critical speed and roll amplitude.

Better mental modelThe criterion concerns inviscid axisymmetric perturbations. Finite-viscosity onset, endwalls, non-axisymmetric modes, and turbulence require a fuller model. Optional rolls here are illustrative.

Solid-body rotation moves without shearing

Typical misconceptionZero differential speed means stationary fluid and zero Reynolds numbers.

Better mental modelTracers still orbit and both wall Reynolds numbers are nonzero. Base torque and dissipative power are zero.

Run the experiment

01
Measure solid-body rotation while paused
Choose Solid-body rotation, then single-step or advance one second.
What to observe: Radius and axial position stay fixed while the angle advances at the physical rate. Base torque and power vanish.
02
Keep shear, change absolute rotation
Compare Prescribed rolls with Outer cylinder faster, then adjust both speeds yourself.
What to observe: Differential-speed magnitude alone does not determine the Rayleigh profile. The stable outer-faster preset suppresses the selected rolls.
03
Inspect a local negative band
Choose Counter-rotation, display the Rayleigh profile, and advance tracer paths.
What to observe: Illustrative rolls occupy only the negative radial band. Fluid farther out can remain purely circular.