High reynolds number simulation dating, other resources
For a semi-circular channel, it is quarter the radius. Laminar flow tends to dominate in the fast-moving center of the pipe while slower-moving turbulent flow dominates near the wall. Both approximations alter the values of the critical Reynolds number.
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.
Sphere in a fluid For a sphere in a fluid, the characteristic length-scale is the diameter of the sphere and the characteristic velocity is that of the sphere relative to the fluid some distance away from the sphere, such that the motion of the sphere does not disturb that reference parcel of fluid.
For grains in which measurement of each axis is impractical, sieve diameters are used instead as the characteristic particle length-scale. Spheres are allowed to fall through the fluid and they reach the terminal velocity quickly, from which the viscosity can be determined.
The Reynolds number is very small and Stokes' Law can be used to measure the viscosity of the fluid. Do is the inside diameter of the outer pipe, Di is the outside diameter of the inner pipe.
OpenFOAM v6 User Guide: 2 Turbulence models
Some texts then use a characteristic dimension that is four times the hydraulic radius, chosen because it gives the same value of Re for the onset of turbulence as in pipe flow, while others use the hydraulic radius as the characteristic length-scale with consequently different values of Re for transition and turbulent flow.
Flow in a wide duct For a fluid moving between two plane parallel surfaces—where the width is much greater than the space between the plates—then the characteristic dimension is equal to the distance between the plates.
It characterizes the nature of the surrounding flow and its fall velocity.