Osborne Reynolds found that if the free- stream velocity of a fluid increases when it moves relative to a solid body, a point is reached where the steady flow becomes turbulent. From experiments, he found that the transition from steady to turbulent flow depends on the value of the quantity \(\frac{v_0d}{η/p}\), where v₀ is the free-stream velocity, d is some characteristic dimension of the system, ρ the density of the fluid and η its coefficient of viscosity. For a sphere in a fluid stream, d is its diameter; for water in a pipe, d is the pipe diameter.

This dimensionless number, defined as Re = \(\frac{v_0dp}η\) is called the Reynolds number. In a system of particular geometry, transition from a steady to turbulent flow is given by a certain value of the Reynolds number called the critical Reynolds number. The free-stream velocity for this critical Reynolds number is called the critical velocity, V_critical = \(\frac{nR_e}{pd}\). For a given system geometry, the free stream velocity beyond which a streamline flow becomes turbulent is called critical velocity.

Steady flow takes place for Re up to about 1000. For 1000 < Re < 2000, there is a transition region in which the flow is extremely sensitive to all sorts of small disturbances. For Re > 2000, the flow is completely turbulent.

[Notes : (1) for “free-stream velocity”. (2) The dimensionless number is named after Osborne Reynolds (1842-1912), British physicist.]