First introduced in the early 1880's by Osborne Reynolds to characterize the transition between laminar and turbulent flow [Reynolds (1883)] the dimensionless term Reynolds number, Re, is now universally employed in the correlation of experimental data on frictional pressure drop and heat and mass transfer in convective flow. It is the basis of much physical modeling.
In general terms
where u is fluid velocity, ρ fluid density, η fluid viscosity and L the characteristic length. In the case of flow in a circular pipe this becomes
Reynolds number represents the ratio of force associated with momentum (ρu2) to force associated with viscous shear (ρu/L). Alternatively it may be regarded as a measure of the ratio of turbulent energy production per unit volume (ρu3/L) to the corresponding rate of viscous dissipation (ηu2/L2) [Lighthill (1970)].
Below a lower critical value of Reynolds number flow is laminar, or "streamline"; above a higher critical value flow is turbulent, or "sinuous" in Reynolds terminology. Between these values the flow is in what is called "transition". The higher critical value is strongly dependent on upstream conditions; Reynolds observed values of between 11,800 and 14,300 for water in bell mouth tubes of a few centimeters diameter. The lower critical value is less sensitive and is usually quoted simply as the "critical Reynolds number". Its value for smooth circular pipes and tubes is approximately 2,000.
Reynolds, O. (1883) On the experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and the law of resistance in parallel channels, Phil. Trans. Roy. Soc., 174, 935.
Lighthill M. J. (1970) Turbulence, Ch. 2. Osborne Reynolds and Engineering Science Today, Manchester University Press Barnes and Noble Inc. New York.
Hewitt, G. F., Shires, G. L., and Bott, T. R. (1994) Process Heat Transfer, CRC Press.