🌀 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)
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🌀 Concentric Rotating Cylinders & Toroidal Taylor Vortex Cells
Real-time visual simulation of centrifugal instability vortex pairs forming between inner rotor & outer stator📝 Configuration & Presets
Taylor-Couette Stability Formulations:
• Taylor Number: Ta = [ 2 Ω₁² R₁ d³ ] / [ ν² (1 + η) ]
• Critical Stability: Tac ≈ 1708 / (1 − 0.652 d/R₁)
• Critical Speed: N1,c = (60/2π) · √[ Tac ν² (1+η) / (2 R₁ d³) ]
• Vortex Pair Wavelength: λ ≈ 2 d [mm]
• Taylor Number: Ta = [ 2 Ω₁² R₁ d³ ] / [ ν² (1 + η) ]
• Critical Stability: Tac ≈ 1708 / (1 − 0.652 d/R₁)
• Critical Speed: N1,c = (60/2π) · √[ Tac ν² (1+η) / (2 R₁ d³) ]
• Vortex Pair Wavelength: λ ≈ 2 d [mm]
📊 Hydrodynamic Stability Results
📊 Output Summary
Taylor Number (Ta)
Ta = 31,595 (Ta / Tac = 17.90)
Regime: 〰️ Modulated Wavy Vortex
N1,c = 709.2 RPM
Critical Onset Speed (N1,c)
709.2 RPM
Operating: 3000.0 RPM
Toroidal Vortex Wavelength (λ)
4.0 mm
20.0 vortex pairs
Viscous Rotor Torque
0.2446 N·m
Power: 76.85 W
Annular Gap Ratio (η = R₁/R₂)
0.952
Gap d = 2.0 mm
📈 Taylor Number (Ta) vs Rotor Speed (RPM)
📉 Viscous Torque (N·m) vs Kinematic Viscosity (cSt)
================================================================= THERMOFLUIDCALC — TAYLOR-COUETTE HYDRODYNAMIC REPORT ================================================================= Case Title : High-Speed Lubrication Journal Bearing Sleeve Cylinder Radii (R1 x R2) : 40.00 mm x 42.00 mm (Gap d = 2.00 mm, eta = 0.952) Cylinder Height (H) : 80.0 mm Inner Rotor Speed (N1) : 3000.0 RPM Fluid Kinematic Viscosity : 32.00 cSt (Density = 870.0 kg/m3) ----------------------------------------------------------------- TAYLOR NUMBER (Ta) : 31,594.8 Critical Taylor (Ta_c) : 1,765.6 CRITICAL ONSET SPEED (N1c) : 709.18 RPM Hydrodynamic Regime : 〰️ Modulated Wavy Vortex Taylor Vortex Wavelength : 4.00 mm (20.0 cell pairs) Viscous Torque & Power : 0.24462 N.m / 76.848 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.