Free Reynolds Number Calculator
Calculate the Reynolds number (Re) from fluid density, velocity, characteristic length, and dynamic viscosity, and instantly classify the flow as laminar, transitional, or turbulent.
Enter the fluid and flow values — Re = ρ·v·L/μ is calculated live.
Re = ρ·v·L/μ (dimensionless). Pipe-flow guide: laminar < 2300, transitional 2300–4000, turbulent > 4000. Viscosity μ must be in Pa·s (1 cP = 0.001 Pa·s) and cannot be zero.
Quick answer
The Reynolds number is Re = ρ·v·L/μ, where ρ is density (kg/m³), v is velocity (m/s), L is the characteristic length (m), and μ is dynamic viscosity (Pa·s). For water at ρ=1000, v=1 m/s, L=0.1 m, and μ=0.001 Pa·s, Re = 1000·1·0.1/0.001 = 100,000, which is turbulent. As a rule of thumb for internal pipe flow, Re below 2300 is laminar, 2300–4000 is transitional, and above 4000 is turbulent.
Formula & method
Reynolds number
Re = ρ · v · L / μ
- Re — Reynolds number (dimensionless)
- ρ — Fluid density (kg/m³); water ≈ 1000
- v — Flow velocity (m/s)
- L — Characteristic length (m), e.g. pipe diameter
- μ — Dynamic viscosity (Pa·s); water ≈ 0.001
Equivalent form using kinematic viscosity ν = μ/ρ: Re = v·L/ν. The characteristic length L is the pipe inner diameter for internal flow, or the chord/length of a body for external flow.
Examples
- Input
- ρ = 1000 kg/m³, v = 1 m/s, L = 0.1 m, μ = 0.001 Pa·s
- Result
- Re = 100,000 (turbulent)
- Why
- Re = ρ·v·L/μ = 1000 × 1 × 0.1 / 0.001 = 100 / 0.001 = 100,000. Since 100,000 > 4000, the flow is fully turbulent.
- Input
- ρ = 1260 kg/m³, v = 0.5 m/s, L = 0.1 m, μ = 1.49 Pa·s
- Result
- Re ≈ 42.3 (laminar)
- Why
- Re = 1260 × 0.5 × 0.1 / 1.49 = 63 / 1.49 ≈ 42.28. Glycerin's very high viscosity keeps Re far below 2300, so the flow is smooth and laminar.
- Input
- ρ = 1000 kg/m³, v = 0.06 m/s, L = 0.05 m, μ = 0.001 Pa·s
- Result
- Re = 3,000 (transitional)
- Why
- Re = 1000 × 0.06 × 0.05 / 0.001 = 3 / 0.001 = 3000. This sits between 2300 and 4000, so the flow is in the unstable transitional range.
- Input
- ρ = 1.225 kg/m³, v = 50 m/s, L = 1.5 m, μ = 1.81×10⁻⁵ Pa·s
- Result
- Re ≈ 5,075,970 (turbulent)
- Why
- Re = 1.225 × 50 × 1.5 / 0.0000181 = 91.875 / 0.0000181 ≈ 5,075,967. High velocity and low air viscosity give a Reynolds number in the millions, typical of aerodynamic flows.
When to use this tool
- Deciding whether pipe or channel flow will be laminar or turbulent before sizing pumps, heat exchangers, or pressure-drop calculations.
- Aerodynamics and hydrodynamics scaling — matching the Reynolds number between a model and a full-scale body in wind-tunnel or towing-tank tests.
- Choosing friction-factor correlations: laminar flow uses f = 64/Re, while turbulent flow needs the Moody chart or Colebrook equation.
- Checking microfluidic, blood-flow, or lubrication problems where high viscosity or tiny length scales usually force flow into the laminar regime.
Common mistakes
- Confusing dynamic viscosity μ (Pa·s) with kinematic viscosity ν (m²/s). If you only have ν, use Re = v·L/ν instead of dividing by μ — do not mix the two.
- Using the wrong units. The formula gives a dimensionless Re only when ρ is in kg/m³, v in m/s, L in m, and μ in Pa·s. Centipoise (cP) must be converted to Pa·s (1 cP = 0.001 Pa·s) first.
- Picking the wrong characteristic length L. For pipe flow L is the inner diameter, not the radius or the pipe length; for flow over a flat plate or wing it is the chord/streamwise length.
- Treating the 2300 and 4000 thresholds as exact. They are conventional guidelines for smooth circular pipes; transition depends on geometry, surface roughness, and inlet disturbances.
Frequently asked questions
What is the Reynolds number formula?
Re = ρ·v·L/μ, where ρ is fluid density (kg/m³), v is flow velocity (m/s), L is the characteristic length (m), and μ is dynamic viscosity (Pa·s). It can also be written Re = v·L/ν using kinematic viscosity ν = μ/ρ. The result is dimensionless.
What Reynolds number is laminar vs turbulent?
For flow inside a circular pipe, Re below about 2300 is laminar, 2300–4000 is transitional, and above 4000 is turbulent. These are conventional thresholds; the exact transition shifts with pipe roughness, vibration, and inlet conditions, and is different for external flows like wings or flat plates.
Is the Reynolds number dimensionless?
Yes. Reynolds number is a ratio of inertial forces (ρ·v·L) to viscous forces (μ), so all units cancel and Re has no units. This is why it works as a universal similarity parameter for comparing flows of different fluids, speeds, and sizes.
What characteristic length L should I use?
It depends on the geometry. For full flow in a round pipe, L is the inner diameter. For non-circular ducts, use the hydraulic diameter (4 × area / wetted perimeter). For external flow over a plate, sphere, or airfoil, L is the streamwise length, sphere diameter, or wing chord, respectively.
How do I convert viscosity units for this calculator?
Dynamic viscosity must be in pascal-seconds (Pa·s). 1 Pa·s = 1000 centipoise (cP), so water at 0.001 Pa·s equals 1 cP. To convert cP to Pa·s, multiply by 0.001. If you have kinematic viscosity in centistokes (cSt), 1 cSt = 1×10⁻⁶ m²/s.
Why does a higher Reynolds number mean turbulence?
A high Re means inertial forces dominate viscous forces, so small disturbances are no longer damped out and grow into chaotic eddies. A low Re means viscosity dominates and smooths disturbances, keeping the flow in orderly layers (laminar). The Reynolds number is the balance point between these two effects.
Sources & references
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