Reynolds Number Calculator

The Reynolds number tells you which kind of flow you have. In the 1880s Osborne Reynolds ran water through a glass tube, added a thread of dye, and watched what happened as he raised the speed. At low speed the dye stayed in a thin straight line. At higher speed it broke up and smeared across the tube. The number that predicts this switch is now named after him.

It compares two effects. Inertia is the tendency of moving fluid to keep going and carry disturbances along. Viscosity is internal friction that damps those disturbances. When viscosity wins, the Reynolds number is low and the flow is laminar, with smooth layers sliding past each other. When inertia wins, the number is high and the flow is turbulent, with swirling eddies and rapid mixing. Because it is a ratio, the number has no units. A small model in a lab and a full-size pipe have similar flow when their Reynolds numbers match.

The number matters in plumbing, HVAC ducts, chemical process lines, oil pipelines, heat exchangers, aircraft wings and even blood flow. It decides how much pressure a pump must supply, how fast a fluid mixes, and how well heat moves from a pipe wall into the fluid.

The usual formula for a pipe is Re = rho x v x D / mu. Here rho is the fluid density, v is the mean velocity, D is the inside diameter and mu is the dynamic viscosity. Dividing viscosity by density gives the kinematic viscosity nu, which lets you write Re = v x D / nu. Handbooks that list viscosity in centistokes are giving you nu.

If you know the flow rate Q instead of the velocity, use v = Q / A with A = pi x D squared / 4, which gives Re = 4 x rho x Q / (pi x mu x D). For mass flow m it is Re = 4 x m / (pi x mu x D). That is the same formula, rearranged. The formula for turbulent flow is not a different equation. Turbulent flow is simply the case where this number is high.

Rearranged, the same relation gives velocity, v = Re x mu / (rho x D), and diameter, D = Re x mu / (rho x v). That is how the velocity and diameter modes work. In the first example, water at 1.5 m/s in a 50 mm pipe gives Re = 74,746. For a stirred tank in chemistry, the length is the impeller diameter and the speed term is N x D, so Re = rho x N x D squared / mu. The Wikipedia article on the Reynolds number lists these forms and others.

For full pipe flow the usual guideline is laminar below Re 2,300, transitional from 2,300 to 4,000, and turbulent above 4,000. The Wikipedia article on the Reynolds number notes that experiments find laminar flow in a fully developed pipe below about 2,300 and fully turbulent flow above about 2,900, with intermittent bursts of turbulence between. The wider 2,300 to 4,000 band used in many textbooks is the safer engineering guideline, so this page uses it.

These limits depend on the setup. A smooth pipe with a calm inlet can stay laminar well past 2,300. A rough pipe, a bend or a valve just upstream will trip turbulence earlier. The critical value is also different for every geometry, which is why other shapes and external flows use other thresholds. Treat the labels on this page as a guide to the likely regime, not as a precise boundary.

What changes across the bands is practical. Laminar flow has low friction, slow mixing and a parabolic velocity profile. Turbulent flow has a flatter profile, higher friction, rapid mixing and better heat transfer. If the Reynolds number lands in the transitional band, designers usually avoid it or use conservative assumptions, because the behavior can swing between the two extremes.

Non-round passages need a single length to feed into the formula. That length is the hydraulic diameter, Dh = 4A / P, where A is the cross-sectional area of the flow and P is the wetted perimeter, the part of the boundary that touches the fluid. For a round pipe Dh equals the diameter. For a rectangle a x b it is 2ab / (a + b). For an annulus it reduces to Do - Di. The rectangular duct and annulus modes above calculate it for you.

The substitution is a good approximation for ducts that are not too flat, roughly when the aspect ratio stays between 1:4 and 4:1. For very flat ducts or unusual shapes, treat the answer with extra caution. The table of equal-area rectangles on this page shows the effect: a 245 × 245 mm square and a 1,000 × 60 mm slot have the same area, but the slot has a much smaller hydraulic diameter and a much lower Reynolds number at the same air speed.

The wetted perimeter counts only the walls the fluid touches. A half-full pipe has a different perimeter and area than a full one, so this calculator assumes a full conduit. For open channels and partly full pipes use a method written for them.

Density and viscosity decide the result, and both depend on temperature. Water is the biggest surprise: its viscosity falls by about two thirds between 5 °C and 60 °C, so the Reynolds number for the same pipe and speed more than triples. The calculator includes water at 5, 20 and 60 °C, air at 25 °C, olive oil and glycerol, each with the density and viscosity from the sources listed below. The table of fluid presets shows the values.

Gases follow the same logic. Air viscosity depends mostly on temperature and barely on pressure, while density changes with both pressure and temperature. At altitude, air is thinner, so for the same duct and speed the Reynolds number falls roughly in proportion to density. If you need air at another temperature or altitude, or a fluid not in the list, pick Custom fluid and enter the density in kg/m3 and the dynamic viscosity in mPa·s (the same as cP). Interpolating between two catalogue values works well; the linear interpolation calculator linked below does it quickly.

Be careful with units. Centipoise and mPa·s are the same. Centistokes are kinematic and equal mm2/s. Pascal-seconds are 1,000 times larger than mPa·s. Mixing these up is the most common source of a Reynolds number that is wrong by a factor of 1,000.

Pick what you want to find. For a pipe with a known speed, use the round pipe mode: choose the fluid, enter the diameter and velocity, and read the Reynolds number and the regime label. If you know the flow instead, use the flow rate mode, which accepts L/s, L/min, m3/h, m3/s, gpm, cfm and ft3/s and converts to velocity first. For ducts and annular gaps use the rectangular or annulus modes.

To work backward, the velocity mode takes a target Reynolds number and a diameter and gives the speed that produces it. The diameter mode takes a target Re and a speed and gives the pipe size. A common use is to find the largest pipe or highest speed that keeps a viscous fluid laminar. In the examples, water at 20 °C in a 25 mm pipe stays at Re 2,300 only at 0.0923 m/s, while olive oil at 1 m/s stays laminar in a pipe up to 212 mm.

The results panel also shows velocity in m/s and ft/s, hydraulic diameter in mm and inches, flow in L/s and gpm, and kinematic viscosity in centistokes, so you can cross-check against pump data sheets or pipe schedules.

It is a regime indicator, not a design. It does not give pressure drop, which also needs pipe length, roughness, fittings and the friction factor. It does not account for entrance effects, where flow takes many diameters to become fully developed. It assumes a Newtonian fluid with constant properties and a full conduit. Slurries, polymers, paints and blood can have viscosity that changes with shear, and then a single viscosity value is only an approximation.

The numbers here are estimates for checking the flow regime. Follow the applicable standards, the pump and pipe manufacturer data and local code when you design a system, and have a qualified engineer check anything that matters for safety, cost or compliance. The bands of 2,300 and 4,000 are widely used guidelines, and real flow can differ.

If you want to go further, a few related tools on this site may help: the linear interpolation calculator for properties between two table values, the temperature converter for °F and °C, and the percentage calculator for percent changes in Re when speed or temperature changes.