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For flow in a pipe of diameter D, experimental observations show that for "fully developed" flow, [n 2] laminar flow occurs when Re D < 2300 and turbulent flow occurs when Re D > 2900. [ 13 ] [ 14 ] At the lower end of this range, a continuous turbulent-flow will form, but only at a very long distance from the inlet of the pipe.
The Reynolds and Womersley Numbers are also used to calculate the thicknesses of the boundary layers that can form from the fluid flow’s viscous effects. The Reynolds number is used to calculate the convective inertial boundary layer thickness that can form, and the Womersley number is used to calculate the transient inertial boundary thickness that can form.
Dimensionless numbers (or characteristic numbers) have an important role in analyzing the behavior of fluids and their flow as well as in other transport phenomena. [1] They include the Reynolds and the Mach numbers, which describe as ratios the relative magnitude of fluid and physical system characteristics, such as density, viscosity, speed of sound, and flow speed.
For Reynolds number greater than 4000, the flow is turbulent; the resistance to flow follows the Darcy–Weisbach equation: it is proportional to the square of the mean flow velocity. Over a domain of many orders of magnitude of Re ( 4000 < Re < 10 8 ), the friction factor varies less than one order of magnitude ( 0.006 < f D < 0.06 ).
The behavior of pipe flow is governed mainly by the effects of viscosity and gravity relative to the inertial forces of the flow. Depending on the effect of viscosity relative to inertia, as represented by the Reynolds number, the flow can be either laminar or turbulent.
A turbulent flow in a fluid is defined by the critical Reynolds number, for a closed pipe this works out to approximately R e c ≈ 2000. {\displaystyle \mathrm {Re} _{\text{c}}\approx 2000.} In terms of the critical Reynolds number, the critical velocity is represented as
As an example for pipe flows, with the Reynolds number based on the pipe diameter: =. Here l is the turbulence or eddy length scale, given below, and c μ is a k – ε model parameter whose value is typically given as 0.09;
In fluid dynamics, the entrance length is the distance a flow travels after entering a pipe before the flow becomes fully developed. [1] Entrance length refers to the length of the entry region, the area following the pipe entrance where effects originating from the interior wall of the pipe propagate into the flow as an expanding boundary layer.