Reynolds Number Formula and Flow Regimes

The Reynolds number, $Re$ , compares the inertia of a flow with its viscosity. It is the main way to predict whether a flow will be laminar or turbulent. Laminar flow moves in smooth layers. Turbulent flow mixes chaotically.

It has no units. For the same shape, the same value means the same behaviour whether the fluid is air, water or oil, and whatever the size.

The formula

Re = \frac{\rho V L}{\mu} = \frac{V L}{\nu}

Symbol	Meaning	Unit
ρ	Fluid density	kg/m³
V	Flow speed, usually the mean or free-stream speed	m/s
L	Characteristic length	m
μ	Dynamic viscosity	Pa·s
ν	Kinematic viscosity, μ divided by ρ	m²/s

The length depends on the flow, and each threshold only holds for its own length.

Pipes use the inside diameter. A non-circular duct uses the hydraulic diameter, four times the flow area over the wetted perimeter.
Flat plates use the distance from the leading edge, so the Reynolds number grows along the plate.
Wings and blades use the chord.
Spheres and cylinders use the diameter.

Laminar or turbulent

Flow	Laminar	Turbulent
Pipe	below about 2,300	above about 4,000
Flat plate	below about 500,000	beyond transition

Pipe flow between 2,300 and 4,000 is transitional. On a flat plate, transition can come anywhere from 300,000 to 3,000,000, depending on how disturbed the incoming flow is.

A worked example

A 25 mm pipe carries fluid at 1.5 m/s and 20 °C.

Water gives $1.5 \times 0.025 / 10^{-6} =$ 37,500. Fully turbulent.
Air gives $1.5 \times 0.025 / (1.5\times10^{-5}) =$ 2,500. Transitional, so small changes can tip it either way.

Same pipe, same speed. Air's kinematic viscosity is 15 times water's, so the Reynolds number is 15 times lower.

Where it matters

Choosing the CFD model. Laminar flow can be solved directly. Turbulent flow needs a turbulence model.
Meshing near walls. A higher Reynolds number means a thinner boundary layer, so the first cell sits closer to the wall.
Scale models. A wind-tunnel model behaves like the real part only if its Reynolds number matches.
Heat transfer. Convection formulas are written in the Reynolds number. See heat transfer.

Common mistakes

Using the wrong length, such as a pipe's length instead of its diameter.
Mixing up the two viscosities. Dynamic viscosity goes with density. Kinematic viscosity goes alone.
Taking viscosity at the wrong temperature. Water's kinematic viscosity roughly halves between 20 °C and 50 °C.
Borrowing the 2,300 threshold for anything that isn't a pipe.

Reynolds number using Navier AI

Describe a flow in Prism and the agent chooses the solver, sets up the case and runs it on your machine. Prism shows every step as it happens, and you can turn on approval before each one.

Our heat exchanger duct study runs a turbulent k–ω SST case to find why an elbow starves part of a tube face. Our tank mixing study finds that mixing depends on inlet momentum, not flow rate. Download Prism to run your own.

References

O. Reynolds, "An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels," Philosophical Transactions of the Royal Society, 174, 1883.
F. M. White, Fluid Mechanics, McGraw-Hill.