Reynolds Number Calculator

Reynolds number: Re = ρ·v·D/μ

The Reynolds number is a dimensionless ratio between inertial and viscous forces: Re = ρ·v·D/μ, where ρ is density (kg/m³), v is velocity (m/s), D is a characteristic length (e.g., pipe diameter, m), and μ is dynamic viscosity (Pa·s). For pipe flow: Re < 2,300 is laminar (smooth, ordered), 2,300–4,000 is the transition zone, and Re > 4,000 is turbulent (chaotic, with eddies). Example: water (ρ ≈ 1,000, μ ≈ 10⁻³) at 1 m/s in a 5 cm pipe gives Re = 50,000 — fully turbulent. For flow around an object, the picture differs: a baseball at Re ~ 10⁵ has turbulent wakes. Stokes's law (Re ≪ 1) describes particles falling through viscous fluids. The Navier–Stokes equations describing viscous flow are so complex that their general solution remains one of the Clay Mathematics Millennium Problems (US$ 1 M prize).

Applications

Household plumbing (Re ~ 10⁴ → turbulent), aviation (CFD computes Re around wings to estimate drag and lift), bioengineering (blood in capillaries at Re ~ 0.01 is laminar; in the aorta Re ~ 3,000 is transitional), naval architecture (hull design) and hemodialysis (membrane flow control).

FAQ

Why is Re dimensionless? Because the units of ρ·v·D and μ exactly cancel: (kg/m³)·(m/s)·(m) ÷ (Pa·s) = 1. This makes Re a universal scaling number — wind-tunnel scale models work because Re is preserved.

What is the critical Re threshold? In a circular pipe, ~2,300 is the classic transition. For flow around a sphere, transition happens near Re ~ 3·10⁵. The threshold depends on the geometry.

Why is turbulence so hard to model? Turbulent flow involves coupled eddies across many scales. Navier–Stokes describes them but lacks a closed analytic solution — that's why we rely on numerical CFD and empirical models like k-ε.