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The Reynolds number compares inertial and viscous effects in a flow. It is dimensionless; its interpretation depends on geometry, characteristic length and boundary conditions.
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Formula
Re dimensionless; ρ in kg/m3; v in m/s; L in m; η in Pa·s; ν in m2/s
| Symbol | Meaning | Unit |
|---|---|---|
| Reynolds number of the flow considered | dimensionless | |
| Fluid density | kg/m3 | |
| Characteristic flow speed | m/s | |
| Characteristic length suited to the geometry | m | |
| Dynamic viscosity | Pa·s | |
| Kinematic viscosity, ν=η/ρ | m2/s |
\mathrm{Re} = \frac{\rho v L}{\eta} = \frac{vL}{\nu}Re arises as the ratio of inertial to viscous effects in the Navier–Stokes equations. Small values mean viscosity acts relatively strongly; at large values, inertia tends to dominate. For a circular pipe, the inside diameter is often used as L. OpenStax writes its pipe form as Re=2ρvr/η with radius r, equivalent to ρvD/η for D=2r. Transition values are not universal; they depend on flow type, geometry and disturbances.
The dimensionless number needs a length and speed suited to the flow geometry; transition thresholds are not universal.
Scaling inertial forces against viscous forces leaves the ratio Re=ρvL/η.
Kinematic viscosity
ν is in m2/s; the length choice must match the definition of Re.
Dynamic viscosity
η is in Pa·s.
Air: ρ=1.2 kg/m3, v=5 m/s, L=0.10 m, η=1.8·10−5 Pa·s. Compute Re.
Re=1.2·5·0.10/(1.8·10−5)≈3.3·104.
Whether flow is turbulent also depends on geometry and disturbances.
Why does Re not change when ν replaces η/ρ in the formula?
Because ν=η/ρ: ρvL/η=vL/(η/ρ)=vL/ν.
Using radius for pipe flow when the chosen formula expects diameter.
Choose the length consistently; OpenStax uses 2r, which is D.
Assuming one fixed Reynolds threshold for every shape and flow.
Transitions depend on geometry, inlet, roughness and disturbances.
Formula cluster
The number becomes physically interpretable only with geometry and boundary conditions.
Water with ρ=1000 kg/m3 and η=1.0·10−3 Pa·s flows at v=0.20 m/s through a pipe with D=0.010 m. Re=1000·0.20·0.010/0.001=2000.
Compare dynamically similar flows, scale model experiments and interpret regime boundaries for a specific geometry.
Here is how to work through a typical Reynolds number (Re = ρvL/η = vL/ν) task step by step:
Task
Air: ρ=1.2 kg/m3, v=5 m/s, L=0.10 m, η=1.8·10−5 Pa·s. Compute Re.
Solution path
Re=1.2·5·0.10/(1.8·10−5)≈3.3·104.
Whether flow is turbulent also depends on geometry and disturbances.
Task
Why does Re not change when ν replaces η/ρ in the formula?
Solution path
Because ν=η/ρ: ρvL/η=vL/(η/ρ)=vL/ν.
The Reynolds number compares the scales of inertial and viscous effects in a flow. At low Re, viscosity strongly damps velocity differences; at high Re, inertia matters more and instabilities can arise more readily. This is guidance, not a complete diagnosis of the flow state. The number is dimensionless, allowing similar flows in differently sized models to be compared. A meaningful calculation requires speed and characteristic length suited to the geometry, as well as the correct viscosity. A specific laminar-or-turbulent judgment also needs details about the experimental setup.
The characteristic length depends on the flow problem. For fully developed flow in a round pipe, the inside diameter D is usually appropriate. OpenStax equivalently writes Re=2ρvr/η using pipe radius r. For a plate, the distance along the flow may be relevant; for an immersed body, a body dimension is often used. Choosing radius rather than diameter changes the value by a factor of two. Always state Re together with geometry and convention. Record the chosen length alongside Re so comparisons between problems remain meaningful.
Dynamic viscosity η measures resistance to shear and has units Pa·s. Kinematic viscosity is ν=η/ρ and has units m2/s. Substituting this into Re=ρvL/η gives Re=vL/ν. The forms are mathematically identical when values and units match. A common error is using η in the ν form or inserting density again there. Check dimensions: the complete Reynolds expression must be dimensionless. The two viscosity quantities describe the same fluid behavior with different reference quantities. Since ν depends on density, it also changes with fluid temperature.
There is no universal threshold for every flow. For a smooth straight pipe, a transition range of a few thousand is often discussed; OpenStax gives below about 2000 as laminar and above about 3000 as turbulent for its pipe example. Intermediate values depend on disturbances. External flows, channels, rough pipes and inlet conditions transition differently. Use a threshold only with the geometry and experiment it describes, not as a general definition of turbulence. The intermediate range can be sensitive to roughness, inlet effects and small disturbances.
For similar flow patterns, relevant dimensionless numbers should be comparable in model and prototype. Reynolds number helps assess whether inertia and viscosity have similar relative importance. If a model is scaled down, L and often attainable speed change; keeping Re constant may require a different fluid or operating condition. Re alone does not guarantee full similarity: geometry, boundary conditions, surface roughness and other dimensionless groups may also matter. It is a tool for experiment design and scaling, not an automatic proof that results transfer. If other forces matter, matching Re alone is not sufficient.
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Curated exam set for "Reynolds number":
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Which formula describes Reynolds number?
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How do you rearrange Reynolds number for Kinematic viscosity?
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Question (front)
Which common mistake happens with Reynolds number?
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