Reynolds Number Calculator
Water at 20 °C moving 1.5 m/s through a 50 mm pipe reaches a Reynolds number of 74,715.6. That sits far above the 4,000 mark, so the flow is turbulent. The calculator also reports the speed at which the same pipe stays laminar and accepts metric or US units.
Enter density, velocity, a characteristic length and either dynamic or kinematic viscosity. Pick pipe flow for the 2,300 / 4,000 thresholds, or external surface for boundary-layer flow.
Water at 20 C needs mu = 1.002 mPa s. Air at sea level on a standard day needs about 0.0173 mPa s.
Kinematic viscosity is mu divided by rho. Water at 20 C is about 1.0038 cSt.
| Reynolds number | |
| Same value in scientific form | |
| Kinematic viscosity used | |
| Flow regime | |
| Speeds that set the regime | |
Re = rho x v x L / mu = v x L / nu. For pipes use the inside diameter as L and the 2,300 / 4,000 thresholds; for plates and wings use the length from the leading edge. All calculations run in your browser, and nothing you type is uploaded or stored.
Enter density, velocity, a characteristic length and either dynamic or kinematic viscosity to get the Reynolds number, the kinematic viscosity and the flow regime, with the speeds that mark the regime change shown alongside.
Key takeaways
- Re = rho x v x L / mu, which is the same as v x L / nu, and the result carries no units
- Water at 20 °C gives mu = 1.002 mPa s and nu = 1.0038e-6 m2/s (NIST reference data)
- Standard-day sea-level air gives rho = 1.229 kg/m3 and mu = 0.0000173 Pa s (NASA Glenn)
- Pipe flow is laminar below 2,300, transitional from 2,300 to 4,000, and turbulent above 4,000
- Use the inside diameter for pipes, the chord for wings, and the distance from the leading edge for flat plates
On this page
What the Reynolds number measures
The Reynolds number compares two forces inside a moving fluid: the inertial force that keeps the fluid moving, and the viscous force that resists shearing between layers. NASA's Glenn Research Center defines it in exactly those terms, as the ratio of inertial to viscous forces, and shows how the same ratio lets a wind tunnel test stand in for a full-size aircraft. A large Reynolds number means inertia dominates and the flow tends to break into eddies. A small one means viscosity smooths disturbances out and the flow stays in layers.
The number is dimensionless, so it survives any unit system as long as the units are consistent. That is the reason it is useful: a 1:10 model at ten times the speed can match the Reynolds number of the full object, and the flow pattern around it will look the same.
Engineers use it to pick a drag correlation, to decide whether a pipe will mix or stratify, to size heat exchangers, and to predict when a boundary layer will separate. For a single pipe or a single wing, it is usually the first number calculated after the geometry is fixed.
The formula and what each term means
The standard form is Re = rho x v x L / mu, where rho is density, v is velocity, L is a characteristic length, and mu is the dynamic viscosity. Dividing mu by rho gives the kinematic viscosity nu, which turns the formula into Re = v x L / nu. The NASA Glenn similarity parameters page writes it as rho V L / mu and states that L is a characteristic length of the problem, which is the part most people get wrong on the first try.
| Symbol | Meaning | SI unit | Typical value |
|---|---|---|---|
| rho | Fluid density | kg/m3 | Water at 20 °C: 998.2 |
| v | Flow velocity | m/s | 0.02 in a small tube, 40 over a wing |
| L | Characteristic length | m | Pipe diameter, plate length, wing chord |
| mu | Dynamic viscosity | Pa s | Water at 20 °C: 0.001002 |
| nu | Kinematic viscosity, mu / rho | m2/s | Water at 20 °C: 1.0038e-6 |
A quick unit check shows why the result is a pure number: (kg/m3) x (m/s) x (m) / (Pa s) leaves nothing behind once a pascal is written as a kilogram per metre per second squared. If your answer has a unit attached, a conversion has gone wrong somewhere.
The two viscosity forms are interchangeable as long as you know the density. Water at 20 °C has nu = 1.0038e-6 m2/s, or 1.0038 cSt, because 1 cSt equals 1e-6 m2/s. Air on a standard day is about 1.4076e-5 m2/s, fourteen times the kinematic viscosity of water, so an air flow needs a larger speed times length to reach the same Reynolds number.
Worked example: water through a steel pipe
Take water at 20 °C flowing at 1.5 m/s through a 50 mm inside-diameter pipe. The NIST Chemistry WebBook fluid data gives a density of 998.2 kg/m3 and a dynamic viscosity of about 1.002e-3 Pa s at that condition. Put the numbers in SI: rho = 998.2, v = 1.5, L = 0.05, mu = 0.001002. Then Re = 998.2 x 1.5 x 0.05 / 0.001002 = 74,715.6.
The same answer comes out of the kinematic form, since nu is 0.001002 / 998.2 = 0.0000010038 m2/s and 1.5 x 0.05 / 0.0000010038 gives the same figure. At 74,715.6 the flow is well past the 4,000 line, so the pipe runs turbulent and a laminar friction factor would be the wrong choice.
The reverse question is often more useful: how slow must this pipe run to stay laminar? Setting Re to 2,300 gives a velocity of 2,300 x 1.0038068e-6 / 0.05, which is 0.0462 m/s. The upper edge of the transitional band, Re = 4,000, arrives at 0.0803 m/s. Below 0.05 m/s this pipe is a calm, layered flow, and by 0.1 m/s it is mixing.
Worked example: air over a wing
Air is a different story because its density is roughly 800 times lower than water while its viscosity is only about 60 times lower. NASA Glenn lists standard-day sea-level values of 1.229 kg/m3 for density and 0.0000173 Pa s for dynamic viscosity, at 15 °C and 101.3 kPa. For a light aircraft wing with a 1.5 m chord flying at 40 m/s, Re = 1.229 x 40 x 1.5 / 0.0000173 = 4,262,427.7.
That is 57 times the Reynolds number of the pipe example. Speed and length alone would suggest a factor near 800, but the higher kinematic viscosity of air pulls the number back down to 57. Density and viscosity both matter, which is why model testing in a water tunnel can match the Reynolds number of a full-size wing at a much lower speed.
For comparison, water creeping at 0.02 m/s through a 10 mm tube sits at Re = 998.2 x 0.02 x 0.01 / 0.001002 = 199.2. That flow is laminar with a wide margin, and the same tube would need about 0.231 m/s before the number reached 2,300.
Reading the number: laminar, transitional, turbulent
The thresholds are not universal constants, they are conventions tied to a geometry. For flow inside a circular pipe, the long-standing convention is that the flow is laminar below Re = 2,300, transitional between 2,300 and 4,000, and turbulent above 4,000. Course notes from Purdue University use the same 2,300 line for the laminar friction factor and describe the 2,300 to 4,000 range as transitional with a friction factor that cannot be pinned down.
| Regime | Pipe range | What the flow looks like |
|---|---|---|
| Laminar | Below 2,300 | Smooth layers, dye stays in a thread, friction factor 64 / Re |
| Transitional | 2,300 to 4,000 | Intermittent bursts, unstable pressure drop |
| Turbulent | Above 4,000 | Mixing eddies, flatter velocity profile, higher friction |
External flow uses a different scale. A smooth flat plate keeps a laminar boundary layer up to roughly 500,000 based on distance from the leading edge, with transition usually complete by about 1,000,000. Princeton's notes on transition put a conservative laminar limit near 200,000 for boundary layers that see small disturbances, and note that roughness, vibration and noise can trip transition earlier. That spread is why the calculator reports a band rather than a single verdict.
Osborne Reynolds found the pipe threshold in 1883 with dye injected into a glass tube, and the number still carries his name. The 2,300 value has survived because it reproduces well, not because it is exact: an unusually smooth pipe with a quiet inlet can stay laminar past 4,000, and a rough one can break down well before 2,300.
Choosing the characteristic length and the units
The length in the formula is whatever dimension sets the scale of the flow, and picking the wrong one shifts the answer by the ratio of the two lengths. The table lists the usual choices.
| Geometry | Length to use | Notes |
|---|---|---|
| Circular pipe or tube | Inside diameter | Not the outside diameter, not the radius |
| Non-circular duct | Hydraulic diameter, 4 x area / wetted perimeter | Gives an equivalent diameter for the same friction |
| Flat plate | Distance from the leading edge | Local Re grows along the plate |
| Wing or blade | Chord | Mean chord for a tapered wing |
| Sphere or cylinder in cross flow | Diameter | Used for drag coefficient tables |
Unit discipline matters as much as geometry. The calculator converts on the way in, using 1 ft/s = 0.3048 m/s, 1 mph = 0.44704 m/s, 1 km/h = 0.2777778 m/s, 1 in = 0.0254 m, 1 lb/ft3 = 16.018463 kg/m3, 1 cP = 0.001 Pa s and 1 cSt = 1e-6 m2/s. Typing a viscosity in cP into a field that expects Pa s makes the result 1,000 times too large, which is the single most common error in hand calculations.
Mixing systems is the other trap. Pounds per cubic foot with metres per second and pascal seconds cannot be combined directly, so convert everything to one system first and convert the answer only if you need it in another form. The Reynolds number itself needs no conversion, since it has no units.
Where the number stops being enough
Several conditions break the simple formula, and each has a workaround. Air above about 0.3 times the speed of sound compresses as it flows, so density stops being constant and the Mach number takes over as the leading parameter; the Reynolds number still matters, but not on its own. Non-Newtonian fluids such as paint, blood and ketchup have a viscosity that changes with shear rate, so a single mu is only a local approximation and a generalized Reynolds number is used instead.
Transitional flow is genuinely unpredictable. Pressure drop can jump between laminar and turbulent values at the same flow rate, which is why designers avoid the 2,300 to 4,000 band when a stable pressure matters. Two-phase flow, packed beds and porous media all need modified definitions with a different length or an effective viscosity.
Used within its limits, the number answers practical questions quickly: whether a pipe will clog, whether a model test is representative, whether a flow meter will read reliably, and how much pump power friction will consume. The NASA Glenn page on air properties lists the standard-day values used above, and its page on similarity parameters covers how Reynolds and Mach numbers work together. For the pipe side, the Purdue pipe flow notes and Princeton's transition notes give the thresholds and their caveats. All calculations run in your browser, and nothing you type is uploaded or stored.
Frequently asked questions
What is the Reynolds number formula?
Re = rho x v x L / mu, or the same expression with kinematic viscosity, Re = v x L / nu. For water at 20 °C in a 50 mm pipe at 1.5 m/s, Re = 998.2 x 1.5 x 0.05 / 0.001002 = 74,715.6.
Is the Reynolds number dimensionless?
Yes. Density, velocity, length and viscosity cancel out, so the number has no units and can be compared across any unit system.
What Reynolds number means laminar flow in a pipe?
Below about 2,300 the flow inside a circular pipe is treated as laminar. The 2,300 figure comes from Osborne Reynolds and is still the standard convention.
What is transitional flow?
Between 2,300 and 4,000 in a pipe, the flow shifts between laminar and turbulent. Pressure drop is unstable in this band, which is why design work avoids it.
What Reynolds number is turbulent?
Above 4,000 in a pipe, the flow is classed as turbulent. In that regime the friction factor depends on both the Reynolds number and the pipe roughness.
Which length should I use for L?
Use the inside diameter for a circular pipe, the hydraulic diameter for a non-circular duct, the chord for a wing, the diameter for a sphere, and the distance from the leading edge for a flat plate.
Can I use kinematic viscosity instead of dynamic viscosity?
Yes. If you know nu, use Re = v x L / nu. Water at 20 °C has nu = 1.0038e-6 m2/s, which is 1.0038 cSt.
What is the viscosity of water at 20 °C?
NIST reference data give about 1.002 mPa s for dynamic viscosity and 998.2 kg/m3 for density, so the kinematic viscosity is 1.0038e-6 m2/s.
What is the viscosity of air at sea level?
NASA Glenn lists 0.0000173 Pa s for a standard day at sea level, with a density of 1.229 kg/m3 at 15 °C and 101.3 kPa.
Does temperature change the Reynolds number?
Yes, because viscosity depends strongly on temperature. Water viscosity falls from about 1.002 mPa s at 20 °C to about 0.466 mPa s at 60 °C, so the same flow roughly doubles its Reynolds number.
Why is my answer 1000 times too large?
The usual cause is a viscosity in cP entered as if it were Pa s. One cP equals 0.001 Pa s, so 1.002 cP must be entered as 0.001002.
What Reynolds number does a wing see?
A light aircraft with a 1.5 m chord at 40 m/s in standard-day air sits near 4,262,427.7, while a model at the same speed and a 0.15 m chord would be ten times lower.
Is a higher Reynolds number more turbulent?
In general yes. Inertia grows relative to viscosity as the number rises, so disturbances grow instead of being damped.
Who was Osborne Reynolds?
A British engineer who showed in 1883 that the transition to turbulence in a pipe depends on a single combined ratio of density, velocity, diameter and viscosity.
Spotted a wrong number or a broken result?
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