The Reynolds number Re = ρVD/μ compares the inertia of a moving fluid with its viscosity. In a pipe, flow is smooth and layered (laminar) when Re is below about 2300 and chaotic (turbulent) above about 4000. Which regime you are in decides almost every other fluid calculation, starting with the friction loss.
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What the Reynolds number measures
A moving fluid carries momentum, which tends to keep disturbances growing, and viscosity, which damps them out. Re is the ratio of the two: Re = ρVD/μ = VD/ν, where ν = μ/ρ is the kinematic viscosity. It has no units, so the same Re means the same flow pattern whatever the fluid or pipe size.
For a pipe, the length scale D is the inside diameter and V is the mean velocity, Q divided by the flow area. Other shapes use their own length: the hydraulic diameter 4A/P for a duct, the chord for a wing, the diameter for a sphere. Their critical values differ from the pipe values.
Laminar, transitional and turbulent flow
Below Re ≈ 2300, pipe flow is laminar: fluid moves in smooth layers, the velocity profile is a parabola, and friction loss is proportional to velocity.
Above Re ≈ 4000, flow is turbulent: eddies mix the fluid across the pipe, the profile is much flatter, and friction loss grows roughly with velocity squared. Between the two, flow can switch back and forth, and results there are uncertain.
These limits are typical values for ordinary pipes. Very smooth, vibration-free laboratory pipes have stayed laminar far above 2300, so treat the numbers as engineering guides, not physical constants. The next lesson, the Darcy–Weisbach equation, turns the regime into a pressure drop.
Dynamic and kinematic viscosity
Dynamic viscosity μ is in Pa·s; the centipoise (cP) equals 1 mPa·s. Kinematic viscosity ν = μ/ρ is in m²/s; the centistokes (cSt) equals 1 mm²/s. Oil data sheets usually give ν in cSt at 40 °C and 100 °C. Use whichever form you have, but do not mix them in one formula.
Viscosity depends strongly on temperature. Liquids thin out as they warm, while gases get slightly more viscous, so always use properties at the operating temperature.
Key formulas
Formulas used on this page, with the textbook each is cited from
Formula
Expression
Where
Cited from
Reynolds number
Re = ρ·V·D / μ = V·D / ν
D = inside diameter, V = mean velocity
White, Fluid Mechanics
Mean velocity in a full pipe
V = Q / (π·D² / 4)
Q = volumetric flow rate
Munson et al., Fundamentals of Fluid Mechanics
Kinematic viscosity
ν = μ / ρ
1 cSt = 10⁻⁶ m²/s, 1 cP = 10⁻³ Pa·s
White, Fluid Mechanics
Hydraulic diameter
D_h = 4·A / P
A = flow area, P = wetted perimeter; D_h = D for a round pipe
Munson et al., Fundamentals of Fluid Mechanics
Worked examples
Example: Flow regime in a hydraulic return line
Given
Hydraulic oil flows at 6 L/min through a line with a 12 mm bore.
The oil is an ISO VG 46 grade at 40 °C. The grade number is its nominal kinematic viscosity, so ν = 46 mm²/s = 46 × 10⁻⁶ m²/s.
Find
The Reynolds number and the flow regime.
The flow rate at which Re would reach 2300.
Solution
Convert the flow rate to m³/s.Q = 6 L/min = 6 × 10⁻³ / 60 = 1.0 × 10⁻⁴ m³/s
Find the flow area and mean velocity.A = π × 0.012² / 4 = 1.131 × 10⁻⁴ m², V = 1.0 × 10⁻⁴ / 1.131 × 10⁻⁴ = 0.884 m/s
Use the kinematic form, since ν is given.Re = V·D / ν = 0.884 × 0.012 / 46 × 10⁻⁶ = 231
Re is proportional to Q for a fixed pipe and fluid, so scale the flow to reach 2300.Q = 6 L/min × 2300 / 231 = 59.8 L/min
Answer: Re ≈ 231, well inside the laminar range. The line would stay laminar up to about 60 L/min at this oil temperature.
Example: Air in a ventilation duct
Given
Air flows at 5 m/s in a round duct with a 200 mm inside diameter.
Take ρ = 1.2 kg/m³ and μ = 1.8 × 10⁻⁵ Pa·s, typical handbook values for air near room temperature.
Find
The Reynolds number and the flow regime.
The velocity below which the flow would be laminar.
Solution
Use the dynamic form, since ρ and μ are given.Re = ρ·V·D / μ = 1.2 × 5 × 0.2 / 1.8 × 10⁻⁵ = 66 700
Re is far above 4000, so the flow is turbulent.66 700 > 4000
Solve Re = 2300 for the velocity.V = 2300 × 1.8 × 10⁻⁵ / (1.2 × 0.2) = 0.173 m/s
Answer: Re ≈ 66 700, so the flow is turbulent. It would only be laminar below about 0.17 m/s, which is why nearly all duct and plant air flow is turbulent.
Common mistakes
Using the nominal pipe size for D. Use the actual inside diameter from the pipe schedule or tube wall thickness.
Mixing dynamic and kinematic viscosity. ρVD/μ needs μ in Pa·s; VD/ν needs ν in m²/s. Convert cP and cSt first.
Using viscosity at the wrong temperature. Oil can be several times thicker cold than at operating temperature. Use the temperature the flow actually runs at.
Applying the pipe limits to other shapes. 2300 and 4000 are for round pipes based on diameter. Ducts use the hydraulic diameter, and flows over objects have their own critical values.
Treating 2300 as a sharp switch. Between about 2300 and 4000, flow can be either. Check a design in that band under both assumptions.
In industry
Frequently asked questions
What Reynolds number is turbulent?
For flow in a round pipe, usually above about 4000. Below about 2300 it is laminar, and between the two it is transitional.
Does the Reynolds number have units?
No. ρVD/μ is dimensionless when consistent units are used, which is why it can compare flows of different fluids and sizes.
Why does a higher viscosity make flow laminar?
Viscosity damps the small disturbances that grow into turbulence. A larger μ lowers Re for the same velocity and pipe, so the damping wins.
How do I calculate it for a pipe quickly?
Use Coplain’s pipe pressure drop calculator, linked below. It reports Re and the regime along with the friction factor and pressure drop.