What is Reynolds Number?
The Reynolds number (Re) is a dimensionless quantity that describes the ratio of inertial forces to viscous forces in a flowing fluid. It predicts whether flow will be laminar or turbulent — the single most important parameter in fluid mechanics.
Where:
- ρ = fluid density (kg/m³)
- v = flow velocity (m/s)
- D = pipe inner diameter (m)
- μ = dynamic viscosity (Pa·s or kg/m·s)
- ν = kinematic viscosity, ν = μ/ρ (m²/s)
Flow Regimes
| Regime | Reynolds Number (Pipe Flow) | Characteristics |
|---|---|---|
| Laminar | Re < 2,300 | Smooth, layered flow; parabolic velocity profile; predictable; mixing only by molecular diffusion |
| Critical/transition | 2,300 < Re < 4,000 | Unstable; intermittent turbulence; design should avoid this range |
| Turbulent | Re > 4,000 | Chaotic eddies; flat velocity profile; good mixing; predictable statistically |
| Fully turbulent rough | Re > 10,000 with ε/D significant | Friction factor depends only on roughness |
Laminar Flow (Re < 2,300)
In laminar flow, fluid moves in smooth parallel layers (laminae). There is no cross-flow mixing between layers.
Key characteristics:
- Velocity profile: Perfect parabola, with vmax = 2 × vaverage at centerline
- Friction factor: f = 64/Re (exact, no iteration needed)
- Pressure drop: Directly proportional to velocity (not v²!) — Hagen-Poiseuille equation
- Entrance length: ~0.05 × Re × D to reach fully developed profile
Applications where laminar flow matters:
- Very viscous fluids (heavy oils, polymers, honey)
- Microfluidic channels
- Flow around small particles (settling)
- Lubrication films in bearings
Turbulent Flow (Re > 4,000)
In turbulent flow, chaotic eddies and vortices cause rapid mixing across the pipe cross-section.
Key characteristics:
- Velocity profile: Flat (power-law profile, ~1/7 power), vmax ≈ 1.2 × vaverage
- Friction factor: Depends on both Re and relative roughness (ε/D) — use Colebrook-White or Moody chart
- Pressure drop: Proportional to v² (Darcy-Weisbach)
- Entrance length: ~30-50 × D (shorter than laminar, mixing speeds development)
Velocity Profiles Compared
| Feature | Laminar | Turbulent |
|---|---|---|
| Shape | Parabolic | Flat (log law) |
| vmax/vavg | 2.0 | ~1.2 |
| Mixing | None (molecular only) | Vigorous (turbulent eddies) |
| Wall shear | 8μv/D | ½ρv² × f/4 |
Reynolds Number Calculator
Worked Example
Water at 20°C flows through a 100mm ID pipe at 2 m/s.
- ρ = 1000 kg/m³, μ = 0.001 Pa·s, D = 0.1m, v = 2 m/s
- Re = 1000 × 2 × 0.1 / 0.001 = 200,000 → turbulent regime.
Heavy fuel oil at 20°C in same pipe at same velocity:
- μ = 1.0 Pa·s (1000× more viscous)
- Re = 1000 × 2 × 0.1 / 1.0 = 200 → laminar!
How to Interpret Reynolds Number for Design
| Re Range | Design Implication |
|---|---|
| < 500 | Very viscous; laminar; pressure drop linear with flow; may need PD pumps |
| 500-2,300 | Laminar; ensure Re-based friction calculations |
| 2,300-10,000 | Transition zone; unstable; avoid designing in this range |
| 10,000-100,000 | Fully turbulent; friction depends on Re and roughness |
| > 100,000 | High turbulence; friction mostly depends on pipe roughness (fully rough) |
Reynolds Number in Other Flow Geometries
| Geometry | Characteristic Length | Critical Re |
|---|---|---|
| Circular pipe | Diameter | 2,300 |
| Flow over flat plate | Distance from leading edge | 500,000 |
| Flow around sphere | Sphere diameter | ~2,000 (upper 300,000) |
| Flow around cylinder | Cylinder diameter | ~2,000 |
| Open channel | Hydraulic radius (A/P) | 500 |
| Annulus | Douter − Dinner | ~2,000 |
| stirred tank | Impeller diameter | 10,000 (for mixing) |
Practical Design Implications
1. Pumping Power
In turbulent flow, power ∝ Q³ (double flow → 8× power). In laminar flow, power ∝ Q² (double flow → 4× power). Increasing pipe diameter reduces power much more in turbulent systems.
2. Heat Transfer
Turbulent flow gives much higher heat transfer coefficients (5-10× laminar). For heat exchangers, turbulent flow is desirable — design for Re > 10,000 on both sides when possible.
3. Flow Measurement
Orifice plates, venturis, and most DP meters require turbulent flow (Re > 10,000) for rated accuracy. At low Re, discharge coefficient changes unpredictably.
4. Mixing
In stirred tanks, Reynolds number based on impeller diameter determines mixing regime:
- Re < 10: Laminar blending
- 10 < Re < 10,000: Transitional
- Re > 10,000: Turbulent mixing
5. Drag Reduction
Adding small amounts of polymers can reduce turbulent drag by 50-70% by suppressing turbulent eddies. This is used in pipelines (Trans-Alaska pipeline uses this).
Reynolds Number and Friction Factor (Moody Chart)
The Moody chart plots f vs Re for various ε/D:
- Laminar region: f = 64/Re (straight line on log-log, independent of roughness)
- Transition (2,300-4,000): Not well-defined
- Turbulent smooth: f depends on Re only (Blasius: f = 0.316/Re0.25 for Re < 100,000)
- Transition (turbulent): f depends on both Re and ε/D (Colebrook-White)
- Fully rough: f depends on ε/D only (horizontal lines on Moody chart)
Summary
Reynolds number determines whether flow is laminar or turbulent. For pipe flow, Re < 2,300 = laminar (f = 64/Re, parabolic profile, linear pressure drop), Re > 4,000 = turbulent (f depends on Re and roughness, flat profile, v² pressure drop). Most industrial flows are turbulent. Higher Reynolds number increases mixing, heat transfer, and pressure drop. Calculate Re early in any pipe flow problem — it tells you which equations and correlations apply.