Reynolds Number Calculator

The Reynolds number compares the inertia of a flow with the viscosity holding it back. Small, and viscosity damps every disturbance, so the fluid slides in smooth layers (laminar). Large, and disturbances grow into eddies that mix the flow (turbulent). Pressure drop, heat transfer and mixing all change with the regime, so it is the first number to know about a flow.

Re = ρ V L ÷ μ = V L ÷ ν

V is the mean velocity, ν = μ ÷ ρ the kinematic viscosity, and Lthe length that sets the flow's scale: the inside diameter of a pipe running full; for any other duct the hydraulic diameter D_h = 4A ÷ P, four times the flow area over the wetted perimeter (2ab ÷ (a + b) for a rectangle, D_o − D_i for an annulus); and along a flat plate, the distance x from the leading edge. From a flow rate, the velocity is Q ÷ Aover the duct's real area, not an area worked back from D_h.

The regime lines are conventions, not sharp limits. In a pipe, flow is usually taken as laminar below about 2,300, turbulent above about 4,000, and transitional between (Çengel and Cimbala, Section 8-2); some texts put the lower line at 2,000 or 2,100. Laminar flow has been held far higher in very smooth, quiet pipes, and a rough or disturbed inlet trips it sooner. Ducts use the same lines on D_h, an approximation. On a flat plate the boundary layer is commonly taken to turn turbulent at Re_x ≈ 5 × 10⁵, a representative value: depending on surface roughness and how turbulent the oncoming flow is, the real transition falls anywhere from about 10⁵ to 3 × 10⁶ (Incropera et al., Section 6.3).

Fluid data.Water, 0.01 to 99.97 °C at 1 atm: the NIST Chemistry WebBook (IAPWS-95 density, IAPWS 2008 viscosity) at 1 °C steps, interpolated between them. Dry air, −40 to 200 °C: the U.S. Standard Atmosphere, 1976 (Sutherland's law for viscosity, the ideal gas for density, so density follows the absolute pressure you enter). Mineral oils: Machinery's Handbook, 27th ed., p. 2333, with density carried to 100 °F by its correction on p. 2334. 50% ethylene glycol: the MEGlobal Ethylene Glycol Product Guide (2013), Figs. 7 and 17. All of these are typical values for the stated temperature; for a design, use your own fluid's data sheet.

Units.Every conversion is exact, from the defined foot (0.3048 m), pound (0.45359237 kg), standard gravity (9.80665 m/s²) and US gallon (231 in³), Machinery's Handbook pp. 2545 to 2577: 1 cP = 0.001 Pa·s, 1 cSt = 10⁻⁶ m²/s (Table 55b, p. 2586), 1 lbf·s/ft² = 47.880 Pa·s, 1 ft²/s = 0.09290304 m²/s. A density in lb/ft³ is pound-mass; the calculator converts it, so there is no g_c to forget.

Worked example, SI. Water at 20 °C (998.207 kg/m³, 1.0016 mPa·s), 30 L/min through a 25 mm bore. The area is 4.9087 × 10⁻⁴ m², so V = 5.0 × 10⁻⁴ ÷ 4.9087 × 10⁻⁴ = 1.0186 m/s, and Re = 998.207 × 1.0186 × 0.025 ÷ 0.0010016 ≈ 25,380: turbulent.

Worked example, US. Light mineral oil, 32 cSt (3.444 × 10⁻⁴ ft²/s), 5 gal/min through a 1/2 in bore. 5 × 231 ÷ 60 = 19.25 in³/s over 0.1963 in² is 8.170 ft/s, and Re = 8.170 × (0.5 ÷ 12) ÷ 3.444 × 10⁻⁴ = 988: laminar. Load either with the button under the inputs, in its unit system.

Not covered: non-Newtonian fluids (slurries, greases, polymer melts), pipes running part full, gas flows fast enough to compress (past about a third of the speed of sound), and two-phase flow. The regime is for fully developed flow; near an entrance, a bend or a valve the flow is still settling.

References: Çengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, Section 8-2; Incropera, DeWitt, Bergman and Lavine, Fundamentals of Heat and Mass Transfer, Section 6.3; NIST Chemistry WebBook, SRD 69; U.S. Standard Atmosphere, 1976; Machinery's Handbook, 27th ed., pp. 2333 to 2334 and 2545 to 2586; MEGlobal Ethylene Glycol Product Guide (2013), pp. 18 and 28. Reviewer: pending.