Axisymmetric simulation of Taylor-Couette flow. A viscous fluid fills the gap between a rotating inner cylinder and a stationary outer cylinder. Below the critical Taylor number the flow is smooth azimuthal Couette flow; above it the flow destabilises into a vertical stack of counter-rotating toroidal Taylor vortices, visualised here with streamlines and tracer particles.
This is an axisymmetric model of the flow of a viscous fluid trapped in the gap between two concentric cylinders. The inner cylinder rotates while the outer one is held still. At low speed the fluid slides smoothly in circles (laminar Couette flow). Once the rotation passes a critical threshold the flow buckles into a neat stack of doughnut-shaped, counter-rotating vortices — the famous Taylor vortices.
Ta ≈ (Ω² · R · d³) / ν²
Ta_c ≈ 1708 (narrow gap)
v_θ(r) = A·r + B/r (Couette base flow)
Sir Geoffrey Ingram Taylor's 1923 study of this flow is celebrated as one of the most precise agreements ever achieved between fluid-dynamics theory and experiment — his predicted onset of vortices matched the lab to within a percent, a rarity in turbulence research.
Taylor-Couette flow is the motion of a viscous fluid confined in the gap between two concentric cylinders, one or both of which rotate. It is a cornerstone experiment in fluid dynamics for studying the transition from smooth laminar flow to ordered vortices and ultimately turbulence.
Taylor vortices are stacked, doughnut-shaped (toroidal) rolls of fluid that fill the gap between the cylinders. Adjacent vortices counter-rotate, so the fluid alternately moves outward and inward along the gap, producing a regular striped pattern along the cylinder axis.
The Taylor number Ta is a dimensionless group that compares destabilising centrifugal forces to stabilising viscous forces. It scales as Ta ≈ (Ω² R d³)/ν², where Ω is the inner cylinder angular speed, R the inner radius, d the gap width and ν the kinematic viscosity. Vortices appear once Ta exceeds a critical value.
For a narrow gap with a stationary outer cylinder, the critical Taylor number is approximately Ta_c ≈ 1708. Below this value viscosity damps disturbances and the flow stays purely azimuthal (Couette flow); above it the flow becomes unstable and Taylor vortices form.
Rayleigh's centrifugal criterion says that flow is unstable when angular momentum decreases outward. When the inner cylinder spins fast enough, fluid near it carries more angular momentum than the slower fluid further out, so it tends to fling outward. Viscosity resists this, but above the critical Taylor number the centrifugal drive wins and overturning vortices form.
Conservation of mass forces the rolls to alternate. Where one vortex pushes fluid outward toward the outer wall, the neighbouring roll must carry it back inward toward the inner wall. This continuity of flow links the vortices into counter-rotating pairs stacked along the axis.
Near onset each Taylor vortex is roughly square in cross-section, so its axial height is close to the gap width d. The most unstable disturbance has an axial wavelength of about 2d, meaning a counter-rotating pair occupies roughly two gap widths along the cylinder.
As speed increases further the steady Taylor vortices give way to wavy vortices, then modulated and chaotic states, and eventually fully turbulent Taylor-Couette flow. This well-defined sequence makes the system a classic laboratory model for the route to turbulence.
It models lubrication in rotating bearings and seals, mixing and reaction in Taylor-Couette chemical reactors, journal-bearing stability, and the rotating shear flows found in accretion disks around stars. It is also a benchmark for validating computational fluid dynamics codes.
No. It is an axisymmetric, qualitative model. It computes the azimuthal Couette base flow analytically and switches on an idealised stack of counter-rotating Taylor vortices once the Taylor number exceeds its critical value. The streamline pattern, vortex count and onset behaviour are realistic, but it does not solve the full Navier-Stokes equations.
This simulation models Taylor-Couette flow: a viscous fluid trapped in the gap between a rotating inner cylinder and a fixed outer cylinder. Each frame it computes the dimensionless Taylor number Ta ≈ (Ω²⋅R⋅d³)/ν² from the inner cylinder's angular speed Ω, the gap width d and the fluid's kinematic viscosity ν (a measure of internal friction), then compares it to the critical value Ta_c ≈ 1708. Below that threshold the fluid slides in smooth concentric circles (laminar Couette flow); above it, the code switches on an analytic stack of counter-rotating toroidal (ring-shaped) Taylor vortices, drawn as nested streamlines with drifting tracer particles.
An axisymmetric cross-section of the annular gap between two concentric cylinders. Below Ta_c the canvas shows plain vertical shear bands; above it, alternating clockwise/counter-clockwise elliptical rolls appear, their count set by how many gap-widths fit along the column, always rounded to an even number so the counter-rotating pairs stack cleanly.
Drag the Inner cylinder speed slider (0-400 rpm) to spin the inner wall faster or slower, the Fluid viscosity slider (0.4x-3.0x) to make the fluid thicker or thinner, and the Gap width slider (0.10-0.45 of the inner radius) to widen or narrow the annulus — each one changes Ta live in the status panel. Pause/Play freezes the tracers, Reset restores the defaults (120 rpm, 1.0x viscosity, 0.20 gap), and the "?" button opens a popup with the full equations.
Geoffrey Ingram Taylor's 1923 experiments on this exact flow are still cited as one of the most precise matches ever obtained between fluid-dynamics theory and laboratory measurement — his predicted onset speed for the vortices agreed with experiment to within about one percent.
The Taylor number Ta ≈ (Ω²⋅R⋅d³)/ν² compares the destabilising centrifugal effect of the spinning inner cylinder (which grows with the square of its angular speed Ω and the cube of the gap width d) against the stabilising effect of viscosity ν. The simulation recomputes Ta every frame from the current slider values and displays it live in the status panel next to the fixed critical value Ta_c.
Taylor-Couette flow undergoes a genuine bifurcation: for Ta below the critical value Ta_c ≈ 1708, viscosity damps out any small disturbance and the fluid stays in smooth laminar Couette flow. The instant Ta crosses Ta_c, the code switches on the idealised vortex stack, so in the simulation (as in the real experiment) the transition is a threshold effect rather than a smooth ramp.
Because Ta scales with d cubed, even a modest change in gap width produces a large change in the Taylor number. Widening the gap from 0.10 to 0.45 of the inner radius can push Ta from well below Ta_c to far above it at the same rotation speed, which is why that slider can flip the flow between laminar and vortex regimes almost by itself.
Mass conservation forces the rolls to alternate: where one vortex pushes fluid outward toward the fixed outer wall, its neighbour must carry fluid back inward toward the rotating inner wall. The simulation renders this with an orange arc for one rotation sense and a blue arc for the other, alternating row by row up the column.
At very high Ta (the simulation flags this once Ta exceeds about eight times Ta_c) the status panel's regime label switches from "Taylor vortices" to "Wavy / turbulent," reflecting the real physical sequence in which steady vortices give way to wavy, then chaotic, and eventually fully turbulent Taylor-Couette flow as rotation speed keeps rising.