MathBench

Reynolds Number Calculator

Enter velocity, pipe diameter, and fluid, or your own viscosity, and get the Reynolds number with a clear laminar, transitional, or turbulent verdict.

Result

—Reynolds number
—flow regime
  • Laminar < 2,300
  • Transitional 2,300–4,000
  • Turbulent > 4,000

How this calculator works

The calculator converts your velocity and diameter to SI units, looks up the kinematic viscosity for the fluid preset you chose (or takes yours in centistokes), and evaluates the definition directly. The verdict applies the standard pipe-flow thresholds: laminar below about 2,300, turbulent above about 4,000, and an unpredictable transitional band between.

The scale under the result places your Reynolds number on a logarithmic axis spanning four decades, so you can see which regime you are in and how much margin you have before the flow changes character. A value sitting just past 4,000 is a very different design proposition from one at 200,000, though both read "turbulent". Beneath it, the pipe drawing shows what that regime looks like: the velocity profile across the pipe and the streamlines along it, redrawn as your inputs move between laminar, transitional, and turbulent.

Presets are handbook values at the stated temperatures. If your fluid or temperature isn't listed, find the kinematic viscosity on its datasheet (usually quoted in cSt at 40 °C and 100 °C for oils) and use the custom field.

The formula

Re = V × D ÷ ν

V = mean flow velocity (m/s)
D = pipe inside diameter (m)
ν = kinematic viscosity (m²/s); 1 cSt = 10⁻⁶ m²/s

Velocity profiles drawn in the graph (r = distance from centre, R = pipe radius):
  laminar    u/umax = 1 − (r/R)²        exact (Hagen-Poiseuille)
  turbulent  u/umax = (1 − r/R)^(1/7)   Prandtl's one-seventh-power law

The two profile shapes in the graph are the standard results for each regime: the parabola is exact for fully developed laminar pipe flow, while the blunt turbulent profile uses Prandtl's one-seventh-power law, the usual textbook approximation at moderate Reynolds numbers (see White, Fluid Mechanics). Both are drawn to show shape, not to compute a number.

Equivalently Re = ρVD/μ with dynamic viscosity μ and density ρ: kinematic viscosity is just μ/ρ, so one number per fluid suffices here. The regime thresholds (2,300 / 4,000) are the conventional circular-pipe values used with the Moody chart, per standard references such as White, Fluid Mechanics.

Worked example

Say water at 20 °C flows at 1.5 m/s through a 25 mm pipe:

  1. Convert: D = 25 mm = 0.025 m; ν = 1.004 cSt = 1.004 × 10⁻⁶ m²/s
  2. Re = 1.5 × 0.025 ÷ 1.004 × 10⁻⁶ = 37,351
  3. 37,351 > 4,000 → turbulent, comfortably

To make that same pipe laminar you'd have to slow the water below about 0.09 m/s, a trickle. That's the general lesson: ordinary water systems are turbulent, and laminar water flow takes deliberate effort or very small tubes.

Assumptions & tips

  • Compute velocity from flow rate. V = Q ÷ A: divide the volumetric flow by the pipe's cross-sectional area. For quick US-unit work, V(ft/s) = 0.4085 × GPM ÷ d², with d in inches.
  • Use the operating temperature's viscosity. A glycol loop sized at summer temperature can drop into the transitional band in winter as viscosity climbs. Check both ends of the operating range.
  • The transitional band is a place to avoid. Between 2,300 and 4,000, friction and heat transfer are unpredictable and can oscillate. Nudge the design to one side, usually by changing pipe size.
  • Entrance effects linger. Downstream of a bend or valve, the flow needs roughly 10 diameters (turbulent) to several dozen (laminar) to settle. Plan measurement points accordingly.
  • Pair with the right friction model. Laminar pipe friction is exactly f = 64/Re, and in turbulent flow it follows the Colebrook-White correlation. The pipe pressure drop calculator applies both, and reports which one it used.

Frequently asked questions

What does the Reynolds number tell you?

It is the ratio of inertial forces to viscous forces in a flow. When viscosity dominates (low Re), the fluid moves in smooth, orderly layers: laminar flow. When inertia dominates (high Re), the flow tumbles into chaotic mixing: turbulence. Because it is dimensionless, the same number describes water in a pipe, air over a wing, or syrup in a straw.

Why are the thresholds 2,300 and 4,000?

Those are the conventional limits for flow in circular pipes, established by experiment and used with the Moody chart: below about 2,300 pipe flow is reliably laminar, above about 4,000 it is reliably turbulent, and between them it flickers unpredictably between the two. They apply to pipes specifically: a flat plate transitions near Re 500,000, and a sphere's drag crisis happens around 300,000.

Why does temperature change the answer so much?

Viscosity depends strongly on temperature. Water's kinematic viscosity halves between 20 °C and 55 °C, so the same flow can double its Reynolds number just by warming up. Oils are more extreme. An SAE 30 oil is roughly ten times less viscous at 100 °C than at 40 °C. Always use the viscosity at the operating temperature, not the catalog value.

What about ducts that are not circular?

Use the hydraulic diameter: four times the cross-sectional area divided by the wetted perimeter. For a full square duct of side a, that works out to exactly a. For a wide flat channel of gap h, it approaches 2h. Feed the hydraulic diameter into this calculator as the diameter and the usual thresholds apply approximately.

Sources

  1. 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. Osborne Reynolds, Philosophical Transactions of the Royal Society of London 174, 935–982, 1883. archive.orgThe original dye-streak experiments that established the dimensionless group this page computes and the existence of a critical value separating direct from sinuous motion.
  2. Fluid Mechanics. Frank M. White, McGraw-Hill. mheducation.comThe reference for the 2,300 and 4,000 pipe thresholds, Prandtl's one-seventh-power turbulent velocity profile drawn in the graph, the exact parabolic laminar profile, and the hydraulic-diameter rule 4A/P used for non-circular ducts.
  3. Friction Factors for Pipe Flow. Lewis F. Moody, Transactions of the ASME 66(8), 671–678, 1944. DOI 10.1115/1.4018140. The chart the laminar, transitional and turbulent bands reported here are conventionally read against.
  4. Release on the IAPWS Formulation 2008 for the Viscosity of Ordinary Water Substance (R12-08). International Association for the Properties of Water and Steam, 2008. iapws.orgThe international reference formulation behind the water presets: 1.004 cSt at 20 °C and 0.475 cSt at 60 °C, combined with the corresponding densities.
  5. Technical Paper No. 410: Flow of Fluids Through Valves, Fittings and Pipe. Crane Co. tp410.comThe customary-unit shortcut in the tips, V(ft/s) = 0.4085 × GPM ÷ d² with d in inches, and the standard treatment of velocity from volumetric flow.