Reynold's number

Osborne Reynolds (1842-1912) worked out how to predict, in advance, whether the flow of a fluid through a cylindrical pipe will be streamlined or turbulent, by combining the relevant physical quantities into a single dimensionless number now called REYNOLD'S NUMBER: $R_c=\dfrac{\rho v D}{\eta}$, where $\rho$ is the density of the fluid, v is its flow velocity, D is the diameter of the pipe carrying the flow, and $\eta$ is the fluid's coefficient of viscosity. Because $R_c$ is dimensionless, its numerical value comes out the SAME regardless of which consistent system of units (SI, CGS, or any other) is used to compute it. Reynolds established, from experiment, threshold values of $R_c$ that separate the three flow regimes: when $R_c<1000$ the flow is streamline; when $1000<R_c<2000$ the flow is unsteady (a transitional regime between the two); and when $R_c>2000$ the flow is fully turbulent. A further important result is the LAW OF SIMILARITY: the critical value of $R_c$ at which turbulence sets in is the SAME for any two geometrically similar flows, even if the two fluids involved have completely different densities and viscosities -- for instance, oil and water flowing through pipes of identical shape and size both become turbulent at essentially the same value of $R_c$. This law of similarity is exploited heavily in engi …