{"id":1858,"date":"2026-09-13T16:59:24","date_gmt":"2026-09-13T16:59:24","guid":{"rendered":"https:\/\/physicsfundamentalsinfo.com\/blog\/?p=1858"},"modified":"2026-09-13T16:59:25","modified_gmt":"2026-09-13T16:59:25","slug":"reynolds-number","status":"publish","type":"post","link":"https:\/\/physicsfundamentalsinfo.com\/blog\/fluids\/reynolds-number\/","title":{"rendered":"Reynolds Number: Formula and Meaning"},"content":{"rendered":"\n
Definition<\/div>

\nThe Reynolds number is the dimensionless ratio of inertial forces to viscous forces in a moving fluid, Re = \u03c1vD\/\u03bc, where \u03c1 is the density, v the mean velocity, D the pipe diameter and \u03bc the dynamic viscosity. It predicts the flow regime: in a circular pipe, flow is laminar below about 2,300 and turbulent above about 4,000.\n<\/p><\/div>\n\n

Open a kitchen tap a fraction and the water falls in a smooth, glassy column you could almost mistake for a glass rod. Open it further and the column roughens, hisses and splatters. Nothing about the water has changed except its speed, yet the flow has switched from one kind of motion to another.<\/p>\n\n

That switch has a number attached to it. Engineers sizing a water main, cardiologists listening for a heart murmur and aerodynamicists testing a wing model all reach for the same dimensionless ratio, and it tells them, before any experiment, whether the fluid will glide in layers or churn.<\/p>\n\n

What Is the Reynolds Number?<\/h2>\n\n

The Reynolds number is a dimensionless quantity that compares the inertial forces in a flow, which keep each parcel of fluid moving the way it was already moving, with the viscous forces that smooth out any disturbance. A large value means inertia wins and small wobbles grow into eddies. A small value means viscosity wins and the same wobbles are damped away before they can spread.<\/p>\n\n

Think of it as a tug of war between momentum and stickiness. Honey creeping from a spoon has plenty of stickiness and almost no momentum, so its Reynolds number is tiny and its motion is perfectly orderly. Water leaving a fire hose has the opposite balance, and the jet is a churning mess of eddies from the moment it leaves the nozzle.<\/p>\n\n

The number takes its name from Osborne Reynolds, who in 1883 injected a thread of dye into water flowing along a glass pipe and reported how the thread stayed straight at low speed, wavered as the speed rose, and finally shredded into eddies. He showed that one combination of speed, pipe size and fluid properties, rather than any of them alone, fixed the point of change. The Reynolds number<\/a> entry on Wikipedia gives a fuller account of that history and of the many other flows the ratio has since been applied to.<\/p>\n\n

Why it has no units<\/h3>\n\n

Multiply a density in kg\/m\u00b3 by a velocity in m\/s and a length in m and you get kg\/(m\u00b7s). A dynamic viscosity in Pa\u00b7s is also kg\/(m\u00b7s), so dividing one by the other leaves a bare number. That is the whole point: a Reynolds number of 24,905 describes the same balance of forces in water, air or oil, at any scale, in any consistent unit system.<\/p>\n\n

The Reynolds Number Formula<\/h2>\n\n

For flow through a pipe the Reynolds number formula is:<\/p>\n\n

Re = \u03c1 v D \/ \u03bc<\/div>\n\n

An equivalent form uses the kinematic viscosity \u03bd, the dynamic viscosity divided by the density:<\/p>\n\n

Re = v D \/ \u03bd, with \u03bd = \u03bc \/ \u03c1<\/div>\n\n

Both forms give the same number, and which you use depends on which viscosity your data table quotes. To check a case in seconds, or to find the velocity, diameter or viscosity that produces a target value, use the Reynolds Number Calculator<\/a>, which shows the substitution with units, names the regime and adds the smooth-pipe friction factor.<\/p>\n\n

Variables and SI units<\/h3>\n\n
    \n
  • Re<\/strong>: Reynolds number, dimensionless.<\/li>\n
  • \u03c1<\/strong> (rho): fluid density, in kg\/m\u00b3 (water at 20 \u00b0C 998.2 kg\/m\u00b3, air 1.204 kg\/m\u00b3).<\/li>\n
  • v<\/strong>: mean velocity of the flow, in m\/s; the volume flow rate Q divided by the cross-sectional area, v = 4Q \/ (\u03c0D\u00b2) for a circular pipe.<\/li>\n
  • D<\/strong>: characteristic length, in m; for a full circular pipe the inside diameter.<\/li>\n
  • \u03bc<\/strong> (mu): dynamic viscosity, in Pa\u00b7s (water at 20 \u00b0C 1.002 \u00d7 10-3<\/sup> Pa\u00b7s, which is 1.002 mPa\u00b7s or 1.002 cP).<\/li>\n
  • \u03bd<\/strong> (nu): kinematic viscosity, in m\u00b2\/s (water at 20 \u00b0C 1.004 \u00d7 10-6<\/sup> m\u00b2\/s; 1 cSt = 10-6<\/sup> m\u00b2\/s).<\/li>\n<\/ul>\n\n

    Two of those inputs cause most of the wrong answers. The velocity must be the mean over the cross-section, not the faster centreline value, and the viscosity must be in Pa\u00b7s, not mPa\u00b7s: type 1.002 where 0.001002 belongs and the domestic supply pipe of the first worked problem comes out with a Reynolds number a thousand times too small.<\/p>\n\n

    The length D is whatever dimension sets the scale of the flow. Inside a pipe it is the bore; for a duct that is not circular engineers use the hydraulic diameter, four times the flow area divided by the wetted perimeter; for a sphere it is the sphere’s diameter, and for a wing it is the chord. The thresholds quoted below belong to circular pipes only, and each geometry has its own.<\/p>\n\n

    How the Reynolds Number Works<\/h2>\n\n

    Why should one ratio decide so much? Start with a small disturbance, a wobble in the flow a pipe-width across. Viscosity spreads momentum sideways by diffusion, and it takes a time of order D\u00b2\/\u03bd to smooth a wobble of size D out of existence. Meanwhile the flow carries the wobble downstream, and in a time of order D\/v it has travelled its own size.<\/p>\n\n

    Divide the smoothing time by the travel time and you get (D\u00b2\/\u03bd)\/(D\/v) = vD\/\u03bd, which is the Reynolds number. When it is small the wobble is smoothed away long before it gets anywhere. When it is large the wobble travels, tumbles and feeds on the mean flow’s energy before viscosity can act, and the flow becomes turbulent.<\/p>\n\n

    Reynolds saw exactly this in his glass pipe. At low speed his dye thread ran the length of the tube as a single straight filament: laminar flow, every layer sliding past its neighbour with no mixing across the pipe. Raising the speed made the thread waver, and a little higher still it burst into eddies that tinted the whole cross-section within a few diameters of the injector.<\/p>\n\n

    The change is not a switch thrown at one exact number. Turbulence needs a seed, some disturbance from a rough inlet, a vibration or a bend, and a very still, smooth entry can keep pipe flow laminar well above 10,000. Below about 2,000 turbulence dies out along the pipe however the flow was disturbed; 2,300 is the conventional design limit, chosen with a margin, and between the two a disturbed flow can carry intermittent turbulent slugs.<\/p>\n\n

    <\/span>Reynolds Number Lab<\/span><\/div>