Reynolds Number Calculator

Re = ρ · v · D / μ = v · D / ν, with ρ in kg/m³, v in m/s, D in m, μ in Pa·s and ν in m²/s

ρ

Density of the fluid, in kg/m³ — the mass per unit volume that the flow has to drag along. It enters as a multiplier, so doubling the density doubles the Reynolds number: a denser fluid carries more momentum at the same speed, and more momentum means more inertia for the viscosity to fight. Water at 20 °C is 998.2 and dry air is 1.205, a factor of 800, and that is most of the reason the same duct at the same speed gives air a Reynolds number about fifteen times lower than water's rather than eight hundred times lower — the rest of the reason is that air's viscosity is correspondingly small. Density is also the quantity that process conditions move around: water's changes by about 4 % between 0 °C and 100 °C, which the water table below shows, while a gas's changes with pressure as well as temperature, so a compressed line is a different Reynolds number problem from the same line at atmospheric.

v

The average velocity of the flow along the pipe, in m/s — the volumetric flow rate divided by the cross-sectional area, not the speed of any particular parcel of fluid. Two warnings come with it. It is an average over everything in the pipe, and it is nowhere the same everywhere: in laminar pipe flow the fluid on the centre line travels twice as fast as this number, and in turbulent flow the profile is much flatter, so the velocity of a real pipe is only ever an average. And it is the quantity that changes when the pipe changes: the same flow rate squeezed into half the diameter gives four times the velocity and therefore twice the Reynolds number, which is the quickest way to turn a quiet laminar line into a noisy one.

D

The internal diameter of the pipe, in millimetres in the unit you choose. It is the bore the fluid actually sees, not the nominal size on the order: a DN50 steel pipe and a DN50 plastic pipe are the same size class and not the same hole, because the nominal figure describes a size rather than a measured passage, and the bore depends on the wall thickness. This is the length scale in the formula — a pipe is round, so its diameter is the obvious choice, and for a duct that is not round the same role is played by the hydraulic diameter, four times the cross-sectional area divided by the wetted perimeter. Using an outside measurement in this field overstates the Reynolds number by the ratio of the two diameters, which on a thick-walled line is easily ten percent.

μ

Dynamic (absolute) viscosity, in mPa·s — the fluid's internal resistance to being sheared, and the quantity that does the work of keeping a flow in line. It is what makes honey honey and water water: water at 20 °C is 1.002 mPa·s, air is 0.018, and a gear oil is in the hundreds. Because it sits in the denominator, a more viscous fluid has a lower Reynolds number at the same speed in the same pipe, which is the same statement as saying it is harder to stir. The unit is worth fixing in your head: 1 mPa·s = 0.001 Pa·s = 1 cP, so the centipoise of older chemical-engineering tables and the SI millipascal-second are the same number under two names.

ν

Kinematic viscosity, in cSt, printed as the second output because the Reynolds number only ever uses viscosity in this combination: ν = μ / ρ, the dynamic viscosity divided by the density. It is the quantity that answers how easily a fluid flows under its own weight, and its unit is defined rather than measured — 1 cSt = 1 mm²/s exactly — so the mm²/s that European and Chinese handbooks print for water at 20 °C (about 1.0) and the cSt this page prints are one number written two ways. The tables below show why it is worth having on its own. Water's ν falls by a factor of six from 0 °C to 100 °C, which is largely its viscosity falling, while air's ν rises with temperature because its density falls faster than its viscosity climbs — a hot liquid and a hot gas move in opposite directions along this axis, and the Reynolds number follows.

2300 / 4000

The two lines that turn a number into a name, and the one part of this page that is a convention rather than a measurement. Below 2300 a pipe flow is called laminar, above 4000 turbulent, and between them transitional — the range where the flow may sit still, may oscillate, or may flip because a fitting upstream disturbed it. The bands are half-open, so 2300 itself is already transitional and 4000 already turbulent, and the badge you see is read off the rounded value in the result panel rather than off the exact number behind it: an answer printed as 2300 is labelled transitional even if the unrounded value was 2299.6. That choice is deliberate — a panel reading 2300 under a laminar badge would be an error the reader could see and could not explain.

Use it before you trust a flow meter, because most meters are calibrated in turbulent flow and read low in laminar flow, and the Reynolds number is the cheapest way to know which one you are in. Use it when sizing a pipe or a pump, because the regime decides which pressure-drop correlation applies — laminar flow is a tidy 64/Re and turbulent flow needs the roughness of the pipe as well. Use it when heat transfer matters, since turbulent flow mixes and carries heat away several times faster than laminar flow at the same speed, which is why a heat exchanger is designed to be turbulent and why a hot pipe run that is barely trickling behaves like an insulated one. And use it when something has to stay mixed, suspended or settled: whether the sand in a slurry stays in the pipe is a Reynolds number question, and so is the speed at which a wind tunnel model has to be tested to say anything true about a full-size aircraft. It is a question about the character of a flow — it says nothing by itself about how much pressure that flow costs you.

1. First screen: water at 20 °C in a DN50 pipe, 2 m/s

   1. Re = ρ · v · D / μ = 998.2 · 2 · 0.050 / 0.001002
   2. 998.2 · 2 = 1996.4 kg/(m²·s)
   3. 1996.4 · 0.050 = 99.82 kg/s per metre
   4. 99.82 / 0.001002 = 99 621
   5. ν = μ / ρ = 0.001002 / 998.2 = 1.004 × 10⁻⁶ m²/s = 1.004 cSt

   Ninety-nine thousand, comfortably turbulent, which is what you expect from a water main at two metres per second: pipes that carry water for a living are almost always turbulent, and the interesting cases are the ones at the quiet end of the table. Both numbers printed here are in the water table below — 998.2 and 1.002 are its 20 °C row, and the 1.004 cSt in the second line of the results is the same row's last column recomputed — so you can check the page against a handbook without leaving it. Note that the panel prints no unit next to the Reynolds number: it is a ratio of two forces and has none.

2. The same pipe, slowed down: 0.02 m/s

   1. Re = 998.2 · 0.02 · 0.050 / 0.001002
   2. 998.2 · 0.02 = 19.964 kg/(m²·s)
   3. 19.964 · 0.050 = 0.9982 kg/s per metre
   4. 0.9982 / 0.001002 = 996
   5. Same fluid, same pipe, so ν is unchanged at 1.004 cSt

   Exactly one hundredth of the previous velocity and exactly one hundredth of the Reynolds number: 996 instead of 99 621, and the badge flips from turbulent to laminar. The linearity is the whole point — Re scales with velocity and nothing else here changes — and it also tells you how little it takes. Twenty millimetres per second is a slow trickle, the sort of flow you get from a partly closed valve or a gravity drain, and it is already a different fluid mechanically. This is the pair to remember when a meter that reads correctly at full flow starts lying at low flow.

3. A gas instead of a liquid: air in a 300 mm duct at 10 m/s

   1. Re = 1.205 · 10 · 0.300 / 0.00001813
   2. 1.205 · 10 = 12.05 kg/(m²·s)
   3. 12.05 · 0.300 = 3.615 kg/s per metre
   4. 3.615 / 0.00001813 = 199 393
   5. ν = μ / ρ = 0.00001813 / 1.205 = 15.046 × 10⁻⁶ m²/s = 15.046 cSt

   Air in a ventilation duct at a brisk ten metres per second, and still turbulent — but look at how it gets there. Air is eight hundred times lighter than water, which alone would make its Reynolds number eight hundred times smaller; its viscosity is fifty-five times smaller as well, which pushes back the other way. The net result is a ν of 15.046 cSt against water's 1.004, so for the same diameter and the same speed a gas gives roughly a fifteenth of the Reynolds number a liquid does. Every value used here is the 20 °C row of the air table below, which is there so this calculation can be redone by hand from the page.

4. On the line itself: Re = 2300 exactly

   1. Re = 1000 · 2.3 · 0.001 / 0.001
   2. 1000 · 2.3 = 2300 kg/(m²·s)
   3. 2300 · 0.001 = 2.3 kg/s per metre
   4. 2.3 / 0.001 = 2300
   5. ν = 0.001 / 1000 = 1 × 10⁻⁶ m²/s = 1 cSt

   One thousand kilograms per cubic metre, one millipascal-second, one millimetre: the numbers are chosen so the arithmetic collapses to Re = 1000 × v, and at 2.3 m/s the answer lands exactly on the lower band line. The badge says transitional, because the bands are half-open and 2300 already belongs to the range above. Push the same setup to 4.0 m/s and you get exactly 4000 and the turbulent badge, so the two lines can be explored from a single field. In a real pipe it would be unwise to trust either label right at the line: this is the range where an upstream valve, a weld bead or a vibration decides the answer.

The 2300 and 4000 lines are a convention, not a law of nature, and the sources do not fully agree on them: the textbook referenced below gives about 2000 for the lower line and about 3000 for the upper one, and calls the band between them unstable rather than transitional. What the transition actually does depends on things this page does not ask for — how the flow entered the pipe (a bell mouth and a sharp-edged entry differ), the roughness of the wall relative to the diameter, vibration, and how far downstream of the last fitting you are. A laboratory pipe with a smooth inlet can hold a laminar flow well past 2300, and the same pipe after a partly closed valve can be turbulent below it. The lines are also specific to a circular pipe: the critical Reynolds number is a property of the geometry as well as of the fluid, and a boundary layer along a flat plate stays laminar to values two orders of magnitude higher, while flow around a cylinder or a sphere has its own behaviour entirely. For a duct that is not round, the diameter must be replaced by the hydraulic diameter, four times the area over the wetted perimeter, which for a square duct is the side length and for a wide thin slot is twice the gap. The diameter must be the true internal bore at the working temperature, not the nominal size: DN is a size class, and the bore of a DN50 pipe depends on whether it is steel, copper or plastic and on its wall thickness. The fluid must be Newtonian — blood, paint, slurries and polymer melts have a viscosity that depends on how fast they are being sheared, so a single Reynolds number does not describe them and the bands above carry no meaning for them. Nothing here says whether the flow is steady, whether it separates from the wall, or what it costs in pressure: the Reynolds number is the input to those correlations, not an answer to them, and for a gas moving fast enough to compress, the Mach number is needed alongside it, just as the Froude number is needed for a free surface in an open channel. Finally, the property values below are for pure water and dry air at one atmosphere between 0 °C and 100 °C; the water's kinematic viscosity changes six-fold across that range, so a number computed with room-temperature properties is not the number for a hot line.

What is the Reynolds number formula?

Re = ρ · v · D / μ — density times velocity times diameter, divided by the dynamic viscosity — which is also written Re = v · D / ν, with the kinematic viscosity ν = μ / ρ doing the work of two quantities at once. Every length and unit cancels, which is why Re is dimensionless: 99 621 is 99 621 whether the pipe is measured in millimetres or miles. The formula is a ratio in disguise. The top line is inertia — mass in motion, the tendency of a fast, dense, wide flow to keep going and to start curling — and the bottom line is viscosity, the tendency of the fluid to be dragged into line by its neighbours. Whichever wins tells you which regime you are in.

What are the laminar, transitional and turbulent ranges?

For flow in a pipe, the engineering convention is laminar below 2300, turbulent above 4000, and transitional in between — a band where the flow may hold still, may oscillate between the two, or may be pushed one way by an upstream fitting. Be aware that textbooks differ: the reference below uses about 2000 and about 3000 for a tube and calls the middle band unstable, and those numbers are as defensible as these. Both sets describe the same physics and neither is exact, because the transition depends on the entry conditions and the roughness of the wall as much as on the number. This page uses 2300 / 4000, applies the bands half-open (2300 itself is transitional), and reads the badge off the rounded value the panel prints.

What does the Reynolds number actually mean?

It is the ratio of inertial forces to viscous forces in a flow, and it is the reason two flows of different fluids in different pipes can behave identically. At low values viscosity wins and the fluid slides along in orderly layers — disturb it and the disturbance is damped out. At high values inertia wins and the flow breaks into eddies, mixes itself, and the disturbance grows instead of dying. The number is not a velocity and not a flow rate: it is a property of the whole arrangement of fluid, speed and geometry, and doubling the pipe diameter has the same effect on it as doubling the speed.

Is the length in the formula the pipe's diameter or its radius?

It is the diameter here — specifically the internal diameter, the hole the fluid passes through. The radius form exists too and carries a factor of two: a textbook may write the tube form as 2ρvr/η, which is the same number as ρvD/η because D = 2r. Pick one and stay with it, since mixing the two gives an answer that is wrong by a factor of two and looks entirely reasonable. Two related traps: a DN50 pipe is a size class rather than a bore, so the actual internal diameter depends on the material and wall thickness, and a duct that is not circular needs its hydraulic diameter — four times the cross-sectional area divided by the wetted perimeter — in place of this field.

Which viscosity do I use, dynamic or kinematic?

Either one, as long as it is paired with the right formula: dynamic viscosity μ with the density, or kinematic viscosity ν alone, since ν = μ / ρ. This page takes the dynamic one in mPa·s and prints the kinematic one in cSt for you, because that is usually the number the next calculation needs. The units are worth fixing: 1 mPa·s = 1 cP, and 1 cSt = 1 mm²/s by definition, so the mm²/s printed in European and Chinese tables and the cSt used in older American literature are the same unit. Water at 20 °C is 1.002 mPa·s and 1.004 cSt, air is 0.01813 mPa·s and 15.046 cSt, and the two tables below give both columns for six temperatures.

Why does the answer change when the water gets hot?

Because kinematic viscosity falls as a liquid is heated — water goes from 1.787 cSt at 0 °C to 0.294 cSt at 100 °C, a factor of six — so for the same pipe and the same speed the Reynolds number rises six-fold and a flow that was laminar when cold can be turbulent when hot. It works the other way for gases: air's kinematic viscosity climbs from 13.3 to 23.0 cSt over the same range, because its density falls faster than its viscosity rises, so heating a gas lowers its Reynolds number. The two property tables below exist for exactly this reason: a Reynolds number is only as good as the pair of properties behind it, and both change with temperature far more than most people expect.

Does a high Reynolds number mean the flow is loud or rough?

Not necessarily. The Reynolds number predicts the regime, not the noise: a turbulent flow in a well-lagged pipe can be silent, and a laminar flow past a sharp obstruction can hiss. What the number does tell you is that mixing, heat transfer and pressure drop have changed character, and that a correlation built for the other regime will not apply. It also says nothing about whether the flow stays attached to the wall, whether it is steady over time, or — for a fast-moving gas — whether compressibility matters. Those are separate questions with separate numbers, which is why the Reynolds number is best thought of as the first thing to compute about a flow rather than the last.

• 12.5 The Onset of Turbulence — the Reynolds number for flow in a tube, the definitions of its terms, and the threshold values this page treats as a convention (about 2000 and about 3000, with the band between them described as unstable) — OpenStax (Rice University)
• Reynolds Number — the definition as the ratio of inertial to viscous forces, the derivation down to Re = ρVL/μ, the dimensionless reading (about 10 million for essentially inviscid flow, a few hundred where viscosity cannot be ignored) and the kinematic form Re = VL/ν — NASA Glenn Research Center
• Water — Dynamic and Kinematic Viscosity at Various Temperatures and Pressures — the viscosity column of the water table below — The Engineering ToolBox
• Water — Density, Specific Weight and Thermal Expansion Coefficients — the density column of the water table below — The Engineering ToolBox
• Air — Dynamic and Kinematic Viscosity at Various Temperatures and Pressures — the viscosity column of the air table below — The Engineering ToolBox
• Air — Density, Specific Weight and Thermal Expansion Coefficients at Varying Temperatures and Pressures — the density column of the air table below — The Engineering ToolBox