🌀 Taylor-Couette Flow Stability & Vortex Cells

Compute Taylor number (Ta), critical onset speed (N1c), toroidal Taylor vortex pair wavelength (λ ≈ 2d), and viscous rotor torque in concentric rotating cylinders.

⚡ Fortran 90 Engine Double Precision (IEEE 754) ✓ ISO / ASME Validated

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👁️ Consultations 22

⚡ Calculs faits 19

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📅 Mise en service Jun 2026

⏱️ Latence < 1 ms

🌀 Concentric Rotating Cylinders & Toroidal Taylor Vortex Cells

Real-time visual simulation of centrifugal instability vortex pairs forming between inner rotor & outer stator

📝 Configuration & Presets

🧪 Water Rheometer (150 RPM) 🛢️ Oil Journal Bearing 🧫 Vortex Bioreactor ⚡ Rotating Electrode (RCE)

📐 Cylinder Dimensions & Rotation

Inner Cylinder Radius R₁ [mm]:

Outer Cylinder Radius R₂ [mm]:

Cylinder Length / Height H [mm]:

Inner Rotor Speed N₁ [RPM]:

💧 Fluid Physical Properties

Kinematic Viscosity ν [cSt]: Water: 1.0 cSt, Light Oil: 10–32 cSt

Fluid Density ρ [kg/m³]:

Taylor-Couette Stability Formulations:
• Taylor Number: Ta = [ 2 Ω₁² R₁ d³ ] / [ ν² (1 + η) ]
• Critical Stability: Ta_c ≈ 1708 / (1 − 0.652 d/R₁)
• Critical Speed: N_1,c = (60/2π) · √[ Ta_c ν² (1+η) / (2 R₁ d³) ]
• Vortex Pair Wavelength: λ ≈ 2 d [mm]

📊 Hydrodynamic Stability Results

📊 Output Summary
💾 Fortran Source

Taylor Number (Ta)

Ta = 13,458,551 (Ta / Ta_c = 6852.20)

Regime: ⚡ Turbulent Taylor-Couette Flow

N_1,c = 1.8 RPM

Critical Onset Speed (N_1,c) 1.8 RPM Operating: 150.0 RPM

Toroidal Vortex Wavelength (λ) 20.0 mm 10.0 vortex pairs

Viscous Rotor Torque 0.0043 N·m Power: 0.07 W

Annular Gap Ratio (η = R₁/R₂) 0.833 Gap d = 10.0 mm

📈 Taylor Number (Ta) vs Rotor Speed (RPM)

📉 Viscous Torque (N·m) vs Kinematic Viscosity (cSt)

=================================================================
 THERMOFLUIDCALC — TAYLOR-COUETTE HYDRODYNAMIC REPORT
=================================================================
Case Title                 : Laboratory Cylindrical Couette Water Apparatus
Cylinder Radii (R1 x R2)   : 50.00 mm x 60.00 mm (Gap d = 10.00 mm, eta = 0.833)
Cylinder Height (H)        : 200.0 mm
Inner Rotor Speed (N1)     : 150.0 RPM
Fluid Kinematic Viscosity  : 1.00 cSt (Density = 998.0 kg/m3)
-----------------------------------------------------------------
TAYLOR NUMBER (Ta)         : 13,458,551.5
Critical Taylor (Ta_c)     : 1,964.1
CRITICAL ONSET SPEED (N1c) : 1.81 RPM
Hydrodynamic Regime        : ⚡ Turbulent Taylor-Couette Flow
Taylor Vortex Wavelength   : 20.00 mm (10.0 cell pairs)
Viscous Torque & Power     : 0.00432 N.m / 0.068 Watts
=================================================================

📘 Calculation Methodology & Taylor Stability Standards

Centrifugal Instability Mechanism

When the inner cylinder rotates faster than the outer wall, centrifugal forces destabilize the fluid element, giving birth to toroidal Taylor vortices when $Ta > Ta_c \approx 1708$:

                    Ta = [ 2 Ω₁² R₁ d³ ] / [ ν² (1 + η) ]
                

Toroidal Vortex Cells Structure

Each pair of toroidal vortex cells occupies an axial wavelength $\lambda \approx 2d$, rotating in alternating counter-clockwise and clockwise directions.

Key Engineering Assumptions

Isothermal Newtonian fluid with constant kinematic viscosity.
Narrow to medium annular gap ($0.5 \le \eta \le 0.98$).
Negligible end-wall Ekman layer perturbations.

ThermoFluidCalc Engineering Report