Reynolds Number — Internal (Pipe) Flow

Dimensionless ratio of inertial to viscous forces in internal pipe flow. Determines whether flow is laminar, transitional, or turbulent.

Fluid Mechanics I core concept

Common fluids — click to auto-fill density and viscosity

Inputs

Density ρ (kg/m³)

Mean velocity V (m/s)

Internal pipe diameter D (mm)

Dynamic viscosity μ (cP)

1 cP = 1 mPa·s = 0.001 Pa·s

Test Case — Water at 20°C in a 50 mm pipe

Fluid	Water at 20°C
Density ρ	998 kg/m³
Dynamic viscosity μ	0.001002 Pa·s (1.002 cP)
Internal diameter D	50 mm = 0.05 m
Mean velocity V	1 m/s

Re = (998 × 1 × 0.05) / 0.001002 = 49,800 → Turbulent

To verify: click Water at 20°C in the Common fluids panel, set D = 50 mm and V = 1 m/s, then click Calculate.

Where to go next

→

Viscosity Conversion

Fluid not in the presets? Convert μ and ν across all unit systems (Pa·s, cP, cSt …) and cross-convert using density.

→

Reynolds Number — External Flow

Flow over flat plates, cylinders, and spheres — same formula, different characteristic length and regime thresholds.

→

Pipe Head Loss

Once you know the flow regime, calculate friction head loss using the Darcy-Weisbach equation.

Theory

Main Equation:

Re = (ρ × V × D) / μ = V × D / ν

Where:

ρ (rho) = fluid density [kg/m³]
V = mean flow velocity [m/s]
D = internal pipe diameter [m]
μ (mu) = dynamic viscosity [Pa·s]
ν (nu) = kinematic viscosity [m²/s] = μ / ρ

Flow Regime Classification (approximate — smooth, straight pipe):

Re < 2,300 — Laminar (viscous forces dominate)
2,300 ≤ Re < 4,000 — Transitional (unstable)
Re ≥ 4,000 — Turbulent (inertial forces dominate)

These thresholds are approximations for smooth, straight circular pipes with fully-developed, low-disturbance flow. In practice, transition can start near Re ≈ 2,000 with rough walls or high-disturbance inlets, and laminar flow can persist to Re ~ 10,000 under carefully controlled lab conditions.