Engineering & Science
Taylor-Couette Rotating Annulus: Taylor Number & Toroidal Vortex Onset
Calculate the root Taylor Number Ta_1/2 = ((Omega * R_i * d) / nu) * SQRT(d / R_i) for concentric journal bearings and chemical Taylor-vortex reactors against G. I. Taylor’s 1923 critical instability threshold (Ta_1/2 >= 41.3).
Predict exact toroidal Taylor-vortex roll formation inside journal bearings & Couette reactors!
G. I. Taylor’s landmark 1923 Philosophical Transactions paper provided the first triumphant quantitative confirmation of linear hydrodynamic stability theory: when an inner cylinder of radius R_i (m) rotates at angular velocity Omega (rad/s) inside a stationary outer cylinder across annular gap d = R_o - R_i (m) filled with fluid of kinematic viscosity nu (m^2/s), purely azimuthal Couette shear becomes centrifugally unstable and snaps into stacked counter-rotating toroidal Taylor Vortices as soon as the root Taylor Number Ta_1/2 = Re_i * SQRT(d / R_i) = ((Omega * R_i * d) / nu) * SQRT(d / R_i) exceeds 41.3 (equivalent to squared Ta = Ta_1/2^2 >= 1,708)!
In emulsion-polymerization and protein-crystallization Taylor-Couette reactors, engineers deliberately operate in laminar Taylor-vortex flow (41.3 <= Ta_1/2 < ~400) for uniform plug-like shear without turbulent blade damage—whereas high-speed turbine journal bearings monitor Ta_1/2 crossing 41.3 because vortex onset spikes bearing friction torque! For R_i = 0.040 m, gap d = 0.004 m, nu = 2.0e-6 m^2/s: at slow Omega = 1.0 rad/s, Ta_1/2 = 25.3 (< 41.3 -> Laminar Circular Couette); spinning up to Omega = 5.0 rad/s yields Ta_1/2 = 126.5 (>= 41.3 -> Axisymmetric Taylor Vortex Flow)!
| Inner Cylinder Angular Speed (Ri = 40 mm, Gap d = 4 mm, nu = 2 mm^2/s) | Inner Couette Re_i | Root Taylor Number Ta_1/2 | Annular Flow Regime |
|---|---|---|---|
| Slow Rotation (Omega = 1.0 rad/s) | 80.0 | 25.3 (< 41.3) | Stable Laminar Circular Couette |
| Critical Threshold (Omega ~ 1.63 rad/s) | 130.6 | 41.3 (Ta = 1,708) | G. I. Taylor Toroidal Vortex Onset |
| Moderate Reactor Spin (Omega = 5.0 rad/s) | 400.0 | 126.5 (>= 41.3) | Laminar / Wavy Taylor Vortex Rolls |
Dimensionless Root Taylor Number Ta_1/2 = ((Omega * R_i * d) / nu) * SQRT(d / R_i)
Computes Re_i * SQRT(d / R_i) from angular speed Omega (rad/s, B2), inner radius R_i (m, B3), gap d (m, B4), and kinematic viscosity nu (m^2/s, B5).
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Couette Inputs", ROUND(((B2 * B3 * B4) / B5) * SQRT(B4 / B3), 1))Taylor-Couette Hydrodynamic Stability Regime Classification
Compares Ta_1/2 against G. I. Taylor’s narrow-gap critical onset 41.3 and turbulent vortex regime > 400.
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Couette Inputs", IF((((B2 * B3 * B4) / B5) * SQRT(B4 / B3)) >= 400, "Turbulent Taylor Vortices (Ta_1/2 >= 400)", IF((((B2 * B3 * B4) / B5) * SQRT(B4 / B3)) >= 41.3, "Taylor Vortex Flow Active (41.3 <= Ta_1/2 < 400)", "Stable Circular Couette Flow (Ta_1/2 < 41.3)")))Try the example
Use the original values or change them to test your understanding.