Another form of the Colebrook-White equation exists for free surfaces. Such a condition may exist in a pipe that is flowing partially full of fluid. For free surface flow:
The above equation is valid only for turbulent flow. Another approach for estimating f in free surface flows, which is valid under all the flow regimes (laminar, transition and turbulent) is the following:
where a is:
and b is:
where Re is Reynolds number where h is the characteristic hydraulic length (hydraulic radius for 1D flows or water depth for 2D flows) and R is the hydraulic radius (for 1D flows) or the water depth (for 2D flows). The Lambert W function can be calculated as follows: