There are two important special cases of flows, namely laminar and turbulent flows, respectively. The flow typically is turbulent, if the Reynolds number exceeds a critical value. In general, for turbulent flow, the flow law cannot be derived from first principles. Normally it is measured. The flow can be expressed in terms of the appropriate flow characteristic.

The turbulent flow law is often written in the form

with a friction factor f. ∆p_R is the turbulent pressure loss. D is the diameter of the pipe, v is the flow speed, and rho is the density of the fluid. For smooth pipes, the friction factor is found to be

Re is the Reynolds number. This leads to the following flow law for smooth pipes:

eta: viscosity; r: radius of pipe; l: length of pipe; rho: density; IV: volume flux.

A simple relation for turbulent flow that is quite useful and accurate for many practical cases is the following

k is called the turbulent flow factor.

turbulent flow, turbulent flow characteristic, viscosity, laminar flow laws,

turbulente Strömungsgesetze