🌀 Taylor-Couette Flow — Rotating Cylinder Vortices

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.