Reynolds Number Calculator
Reynolds number calculator. Re = ρ·v·D / μ, with the circular-pipe regime shown under this calculator's bands and two textbook conventions. Includes preset fluids (water, air, oil, honey).
Reynolds Number
| Kinematic viscosity ν (μ/ρ) | — |
| Density used | — |
| Viscosity used | — |
Circular-pipe regime, by whose bands
The bands are conventions, not physical constants, and sources draw them in different places. This calculator's 2,300 is the figure Rott (Annual Review of Fluid Mechanics, 1990) says most textbooks quote for pipes; its 4,000 is the upper bound it shares with its friction-factor code, which we have not traced to a primary source. OpenStax's physics texts call the middle band "unstable"; the US Department of Energy's handbook says it is "sometimes referred to as transitional". They apply to flow in a circular pipe only.
How to use the Reynolds number calculator
Pick fluid + enter velocity
Velocity in m/s is the bulk speed of the fluid through the pipe; it cannot be negative. A fluid preset writes its fixed reference values into the density and viscosity boxes and locks them (water at 20 °C: ρ = 1000 kg/m³, μ = 0.001 Pa·s; air at sea level: 1.225 kg/m³, 1.81 × 10⁻⁵ Pa·s). Choose Custom to type your own properties, for water at another temperature or a fluid not listed.
Enter characteristic length D
For a circular pipe, the inner diameter in metres. For a non-circular duct, the hydraulic diameter 4A/P (four times the flow area over the wetted perimeter). The regime bands shown are circular-pipe bands; for flow around an object the length is the object's size, and those bands do not apply.
Read Re + regime
The headline uses this calculator's bands: below 2,300 laminar, 2,300 up to 4,000 transitional, 4,000 and above turbulent. Underneath, the same Re is labelled under the bounds of OpenStax's physics texts (2,000 / 3,000) and the US Department of Energy's fluid-flow handbook (2,000 / 3,500), so you can see where the conventions disagree. A blank box, a zero density, viscosity or length, or a negative speed shows a message instead of a number.
Compare the conventions
Near the bands, the label depends on whose bands you use: a Re of 2,100 is laminar here, "unstable" under OpenStax and "transitional" under the DOE handbook. The calculator reports all three rather than choosing.
Cross-check with friction factor
For pressure drop, Re selects the friction-factor formula: 64/Re for laminar flow, the Colebrook-White equation (or a fit to it) for turbulent flow. The head loss is then f·(L/D)·v²/2g. The pipe flow calculator does that step with the same 2,300 boundary.
Reynolds number: inertia against viscosity in one ratio
The Reynolds number is Re = ρvD/μ, where ρ is fluid density, v is bulk velocity, D is a characteristic length and μ is dynamic viscosity. It compares inertial with viscous forces, and it is dimensionless, so a pipe of water and a pipe of oil at the same Re behave alike. Osborne Reynolds showed in 1883, with dye streaks and pressure-drop measurements in pipes, that this combination decides whether the flow is "direct or sinuous". Stokes had noted the similarity in 1850, and the name came later: Arnold Sommerfeld called it "the Reynolds number" in 1908, according to Rott's 1990 history of it.
Three bands, and whose they are
This calculator labels pipe flow laminar below 2,300, transitional from 2,300 up to 4,000, and turbulent from 4,000. In laminar flow the friction factor is 64/Re; in turbulent flow it depends on Re and the pipe's roughness. The bands are a convention. OpenStax's physics texts put laminar flow below about 2,000 and turbulent flow above about 3,000, with the band between called "unstable"; the US Department of Energy's 1992 fluid-flow handbook uses less than 2,000 and greater than 3,500. Rott's history reports 2,300 as the figure most textbooks quote for pipes without a streamlined inlet.
None of these is where the physics changes sharply for every pipe. Reynolds could hold flow laminar to about 13,000 by reducing disturbances at the inlet, and later experiments held it laminar up to 100,000 (Pfenniger, 1961), as Avila, Barkley and Hof summarise in their 2023 review. In the other direction, short-lived puffs of turbulence can be detected down to about 1,500.
Beyond pipes
For an object moving through a fluid, the length in Re is the object's size and the numbers change completely. OpenStax gives flow around an object as laminar below about 1, either laminar or turbulent between 10 and 10⁶, and entirely turbulent above about 10⁶. Pipe bands do not transfer to other geometries, which is why this calculator labels the regime for circular pipes only.
Temperature and the presets
The presets are fixed reference values, and temperature matters mainly through viscosity. From the NIST Chemistry WebBook's water data, kinematic viscosity falls by about a fifth between 20 °C and 30 °C, so the same pipe at the same speed has a Reynolds number about a quarter higher at 30 °C. Air goes the other way and by less: warmer air is less dense and slightly more viscous, so its Re falls by about 6% over the same 10 °C (our calculation from Lemmon and Jacobsen's 2004 air-viscosity equation, dry air at sea-level pressure). Near a band, use Custom with properties at the actual temperature.
Where the pipe bands came from, and how far real flows stray from them
Re = ρvD/μ, or vD/ν with kinematic viscosity ν = μ/ρ. Named by Sommerfeld in 1908 after Reynolds' 1883 pipe experiments.
Reynolds stated his 1883 results through a parameter, not as Re. Converted by Rott, his lower critical value is about 2,020; in 1895 Reynolds himself quoted a value between 1,900 and 2,000.
His upper value, from dye in the tube, is about 12,830 on Rott's conversion. Ekman later held flow laminar far higher on Reynolds' own apparatus.
The 2,300 most textbooks quote traces to later work: Schiller reported 2,320, as Kanda records in a 1995 paper on Reynolds' experiments.
In long, straight pipes, turbulence first lasts indefinitely above about 2,040: the point where puffs split faster than they decay, from Avila and colleagues' 2011 experiments and simulations as summarised in their 2023 review.
Laminar-flow ceilings depend on the inlet: about 13,000 in Reynolds' own work, and up to 100,000 in Pfenniger's 1961 experiments.
Lower bounds in use: 2,000 (OpenStax, US DOE), 2,300 (this calculator). Upper bounds: 3,000, 3,500 and 4,000.
Laminar friction factor: f = 64/Re. Turbulent: the Colebrook-White equation or the Moody chart drawn from it.
Dimensionless: two geometrically similar flows at the same Re are dynamically similar, whatever their size. Wind-tunnel testing relies on it.
Around an object, OpenStax gives laminar flow below a Re of about 1 and entirely turbulent flow above about 10⁶, with the object's size as the length.
Frequently asked questions
For a circular pipe, D is the inner diameter. For other cross-sections the usual length is the hydraulic diameter D_h = 4A/P. Open-channel texts may use the hydraulic radius A/P instead, which is a quarter of D_h, so the same flow has a Re four times smaller on that definition. Check which length a published threshold uses before comparing. This calculator's regime bands are circular-pipe bands and are not open-channel thresholds.
Which equations describe the flow. In a pipe, Re selects the friction-factor formula (64/Re when laminar, Colebrook-White when turbulent), and through it the pressure drop. Because Re is dimensionless, it also says when two flows of different size or fluid will behave alike.
This calculator assumes one fixed viscosity. Fluids such as paint or polymer melts have a viscosity that changes with shear rate, so a single μ, and the Re built from it, does not describe them, and the pipe bands here do not apply.
It is a convention, not Reynolds' figure. Reynolds' own lower critical value was about 2,000: Rott's conversion of his 1883 parameter is about 2,020, and in 1895 Reynolds quoted between 1,900 and 2,000. The 2,300 traces to later work (Schiller reported 2,320), and Rott says it is the value most contemporary textbooks quote; OpenStax and the US DOE handbook use 2,000. Avila and colleagues later put the point where turbulence first sustains itself in a long pipe at about 2,040. The upper limit is less settled still: with a smooth inlet, flow has been held laminar up to 100,000.
Mostly through viscosity. For water, dynamic viscosity is 1.79 mPa·s at 0 °C, 1.00 at 20 °C, 0.80 at 30 °C and 0.47 at 60 °C (NIST Chemistry WebBook): it falls by almost half between 0 °C and 20 °C and by more than half again between 20 °C and 60 °C. So warmer water has a higher Re at the same speed. Reynolds himself compared runs at 5 °C and 22 °C. The presets are fixed at their reference temperature; use Custom for a hot or cold fluid.
Because a named preset supplies them. While Water, Air, Light oil or Honey is selected, the calculator writes that preset's reference values into the two boxes and locks them, so what you see is what is used. Switch the preset to Custom and the boxes unlock for your own values. The "used" rows under the result repeat the pair that went in.
Two geometrically similar flows at the same Re behave alike in dimensionless form, whatever the fluid or size, which is what lets a wind-tunnel model stand in for a full-size body. It has limits when a second number matters too. A ship model in the same water would have to go 50 times faster than the ship to match Re at 1:50 scale, but about 0.14 times as fast to match the Froude number that governs its waves, so it cannot match both, and ship models are not run at the ship's Re.
Rott, "Note on the history of the Reynolds number", Annual Review of Fluid Mechanics 22 (1990), doi:10.1146/annurev.fl.22.010190.000245. Avila, Barkley and Hof, "Transition to turbulence in pipe flow", Annual Review of Fluid Mechanics 55 (2023), doi:10.1146/annurev-fluid-120720-025957. OpenStax College Physics 2e, sections 12.5 and 12.6. Reynolds' own paper is in Philosophical Transactions of the Royal Society 174 (1883), pp. 935–982.
Method & sources
How it computes
Re = ρ·v·D/μ (= v·D/ν, ν = μ/ρ). Circular-pipe bands: laminar below 2300, transitional to 4000, turbulent from 4000 — this calculator's convention (2300 per Rott, Annu Rev Fluid Mech 1990), shown beside OpenStax (2000 / 3000) and US DOE-HDBK-1012/3-92 (2000 / 3500). Blank, zero ρ/μ/D or negative v is refused. Presets fill and lock fixed reference properties: water 1000 kg/m³, 0.001 Pa·s; air 1.225, 1.81×10⁻⁵; oil 870, 0.04; honey 1420, 10.
What this tool implements
- Circular-pipe regime bands are labelled as this calculator's convention (laminar below 2300, transitional 2300 up to 4000, turbulent from 4000 — the constants its shared friction-factor code also uses), and the same Re is shown under OpenStax's bounds (below about 2000 laminar, above about 3000 turbulent, "unstable" between) and the US DOE handbook's (less than 2000, greater than 3500). The bands are stated for circular pipes only.
- Inputs are refused rather than defaulted: a blank or non-numeric box, a zero or negative density, viscosity or length, or a negative speed shows a message and no regime. Speed 0 gives Re 0 and no regime.
- Preset properties are fixed reference values, not a temperature model; a preset writes its values into the density and viscosity boxes and locks them. Water viscosities quoted on the page (1.79 / 1.00 / 0.80 / 0.47 mPa·s at 0 / 20 / 30 / 60 °C) are from the NIST Chemistry WebBook.
- Characteristic length D is the inner diameter for a circular pipe and the hydraulic diameter 4A/P for other ducts; the page notes that open-channel thresholds may use the hydraulic radius A/P, a quarter of D_h.
Sources
- Reynolds O. 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 of London 1883;174:935-982. https://doi.org/10.1098/rstl.1883.0029
- Rott N. Note on the history of the Reynolds number. Annual Review of Fluid Mechanics 1990;22:1-11 — Reynolds' lower critical value converts to about 2020 and his dye value to 12,830; in 1895 he quoted 1900-2000; "the critical value quoted in most contemporary textbooks is 2300"; named by Sommerfeld in 1908. https://doi.org/10.1146/annurev.fl.22.010190.000245
- Kanda H. Radial pressure distribution for flow in a circular pipe. RIMS Kôkyûroku 921 (1995) 123-130 — "Schiller showed experimentally that it is 2320". https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/09…
- Avila M, Barkley D, Hof B. Transition to turbulence in pipe flow. Annual Review of Fluid Mechanics 2023;55:575-602 — laminar up to Re 100,000 (Pfenniger 1961); turbulence "first sustained indefinitely above the critical point Re ≈ 2,040" (Avila et al., Science 2011). https://doi.org/10.1146/annurev-fluid-120720-025957
- OpenStax. College Physics 2e, §12.5-12.6 — "For N_R below about 2000, flow is laminar. For N_R above about 3000, flow is turbulent." https://openstax.org/books/college-physics-2e/pages/12-5-the-onse…
- US Department of Energy. DOE Fundamentals Handbook: Thermodynamics, Heat Transfer, and Fluid Flow, Vol. 3 (DOE-HDBK-1012/3-92), 1992 — laminar if Re "is less than 2000", turbulent if "greater than 3500".
- NIST Chemistry WebBook, SRD 69, Thermophysical Properties of Fluid Systems — water density and viscosity at 0.101325 MPa. https://webbook.nist.gov/chemistry/fluid/
- Lemmon EW, Jacobsen RT. Viscosity and thermal conductivity equations for nitrogen, oxygen, argon, and air. International Journal of Thermophysics 2004;25(1):21-69 — air viscosity behind the page's warm-air figure.
What can make this go out of date
- None at runtime — preset properties are fixed reference values.
Abridged — the full review record for this tool runs longer than the list above.
Pick up where you left off
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