Between 25 °C and 35 °C, the Reynolds number of water in the same pipe at the same speed rises 23.4%, by our calculation from NIST data. That is more than the 15% between 2,000 and 2,300, the two figures most often given for where laminar flow, smooth and layered, gives way to eddying turbulence. The Reynolds number, Re = ρVD/μ, divides a flow's inertia by its viscosity. OpenStax's physics texts and the US Department of Energy's 1992 fluid-flow handbook put the laminar limit at 2,000; a 1990 history of the number reports 2,300 as the value most textbooks quote. Dry air goes the other way, and by less: its Re falls 5.7%.
Dye and pressure readings forecast the onset of turbulence at two different points
Osborne Reynolds' 1883 pipe experiments produced two critical values from two kinds of test. In one, a streak of dye ran down a tube until the flow broke it up; a 1995 paper by Kanda lists results for three tubes between 12,600 and 13,300. In the other, Reynolds measured how pressure drop changed with speed, in two different pipes, and found a much lower point.
Reynolds stated these results through a parameter, B, never as a Reynolds number. Rott's 1990 history of the number converts them, using water's viscosity at 0 °C: 12,830 for the dye tests and a "lower critical" value of 2,020 for the pressure tests. Re-running his arithmetic, we get 12,837 and 2,022. Rott adds that Reynolds first quoted the number itself in 1895, giving a value between 1,900 and 2,000. The name came later still: Sommerfeld proposed "the Reynolds number" in 1908, and Stokes had noted the similarity behind it in 1850.
Disturbance-free inlets calm pipe flow as far as 100,000
The upper value was never fixed. Reynolds himself "could shift the onset of turbulence from Re ≈ 2,000 to 13,000 by reducing disturbances at the pipe inlet", according to a 2023 review by Avila, Barkley and Hof. About a quarter-century later V. Walfrid Ekman travelled to Manchester to use Reynolds' own apparatus, smoothed its wooden trumpet-shaped inlet and reached critical values of up to 44,000, as Rott reports; Kanda's paper puts Ekman's figure at 51,000. The same review cites Pfenniger in 1961 holding flow laminar up to Re 100,000.
The review gives the reason no ceiling appears: laminar flow in a straight round pipe is stable to infinitesimal disturbances up to very high, and possibly infinite, Reynolds numbers. Turbulence needs a disturbance of finite size.
Textbook conventions cloud the lower critical value
The sources we could open or trace:
| Source | Laminar below | Turbulent above | Middle band called |
|---|---|---|---|
| OpenStax, University Physics Vol. 1 and College Physics 2e (same passage) | about 2,000 | about 3,000 | "unstable" |
| US DOE handbook DOE-HDBK-1012/3-92 | 2,000 | 3,500 | "sometimes referred to as transitional" |
| Rott 1990, on "most contemporary textbooks" | 2,300 | none given | none given |
| Schiller, as recorded by Kanda | 2,320 | none given | none given |
Neither of the teaching and standards texts we opened, OpenStax or the DOE handbook, uses 2,300. That figure rests on Rott's description of what textbooks quote for pipes without a streamlined inlet. Rott also notes that a 2,000 extrapolated by Rotta in 1956 "happens to be the value first proposed by Reynolds."
The upper bounds run from 3,000 to the 4,000 common on calculator pages, a 33% gap that exceeds water's 23.4%, so the comparison in the opening holds for the lower bound only. A course's own text decides which figure its exams expect; we opened no syllabus.
Turbulent puffs blow over below about 2,040
Near the lower bound, turbulence arrives as puffs: patches of eddies travelling inside otherwise smooth flow. A puff can die away or split into two. In experiments and simulations by Avila and colleagues, published in Science in 2011 and summarised in their 2023 review, the mean times for those two events cross at Re 2,040 ± 10, and turbulence is "first sustained indefinitely" above that point.
Solving the review's two fitted curves for their crossing, we get 2,036.7. The edge is steep: at 1,900 on the same fits, a puff's mean decay time is shorter than its splitting time by a factor of about fifty million, an extrapolation beyond the measured range that shows only the steepness. Below 2,040 is no promise of laminar flow either: the review cites puffs detected down to about 1,500 by Hof and colleagues in 2005.
Air and water weather a ten-degree swing in opposite directions
Temperature enters through viscosity. Dynamic viscosity, μ, is a fluid's resistance to shearing, about 1.0 mPa·s for water at 20 °C; kinematic viscosity, ν = μ/ρ, in m²/s, divides it by density. Since Re = VD/ν, a lower ν means a higher Re at the same speed V and diameter D. Our water figures divide the NIST Chemistry WebBook values at one atmosphere; for air we used Lemmon and Jacobsen's 2004 equation at sea-level pressure.
| Temperature | Water ν (10⁻⁶ m²/s) | Dry air ν (10⁻⁵ m²/s) |
|---|---|---|
| 20 °C | 1.0034 | 1.5123 |
| 25 °C | 0.8927 | 1.5586 |
| 30 °C | 0.8007 | 1.6054 |
| 35 °C | 0.7234 | 1.6527 |
As an illustration, take water in a 15 mm bore at about 0.113 m/s, the speed that gives Re 1,900 at 25 °C. At 30 °C the same flow gives 2,118. At 35 °C it gives 2,344, inside OpenStax's unstable band and the DOE's transitional one. That places a number against each convention and predicts nothing about a particular pipe. Reynolds checked temperature himself: Rott records critical velocities at 5 °C and 22 °C "found to have the ratio of about 1.4". Humid air is slightly less dense than the dry air tabled, an effect our density altitude guide measures at Changi.
Until 15 October 2026 our Reynolds number calculator labelled a zero diameter TURBULENT, called a negative speed laminar, and credited 2,300 to Reynolds' original experiments in its FAQ. The revision made that day refuses those inputs and shows the same Re under three sets of bands: its own 2,300 and 4,000, OpenStax's and the DOE's. Its water preset stays fixed at 20 °C, which by our NIST figures puts Re about 20% low for water at 30 °C.
Objects moving through still fluid make their own weather
Every figure above belongs to flow inside a round pipe, measured on its diameter. For an object moving through a fluid, OpenStax defines the Reynolds number on the object's size and describes the flow as laminar when it is "less than about 1". From there up to about 10, the flow is in transition, "depending on surface roughness". Between 10 and 10⁶ it "may be either laminar or turbulent", and above about 10⁶ it is "entirely turbulent".
Those bands span a factor of a million, while every pipe convention sits between 2,000 and 4,000. A pipe's threshold of 2,000 or 2,300 means nothing for a falling raindrop. The characteristic length in the equation has changed, and so has the physical meaning of the result.
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