The calculation of mechanical energy loss for incompressible steady pipe flow of homogeneous fluid

The calculation of the mechanical energy loss is one of the fundamental problems in the field of Hydraulics and Engineering Fluid Mechanics. However, for a non-uniform flow the relation between the mechanical energy loss in a volume of fluid and the kinematical and dynamical characteristics of the flow field is not clearly established. In this paper a new mechanical energy equation for the incompressible steady non-uniform pipe flow of homogeneous fluid is derived, which includes the variation of the mean turbulent kinetic energy, and the formula for the calculation of the mechanical energy transformation loss for the non-uniform flow between two cross sections is obtained based on this equation. This formula can be simplified to the Darcy-Weisbach formula for the uniform flow as widely used in Hydraulics. Furthermore, the contributions of the mechanical energy loss relative to the time averaged velocity gradient and the dissipation of the turbulent kinetic energy in the turbulent uniform pipe flow are discussed, and the contributions of the mechanical energy loss in the viscous sublayer, the buffer layer and the region above the buffer layer for the turbulent uniform flow are also analyzed.