Work-Energy Equation for the Streamline

The equation of energy for shear-free flow may be generalized through incorporation of the Saint-Venant terms for the work done by tangential as well as normal stresses. Although the resulting equation is conservative, a dissipative term may be introduced by substituting for the work done conservatively the difference between the total work and that done dissipatively. In the case of laminar flow, the usual Bernoulli terms for the streamline become augmented by two: one involving the energy transferred to (or work done upon) the neighboring fluid by viscous shear; and the other the viscous generation of heat. For the mean pattern of turbulent flow, the Reynolds stresses can be used to form two comparable terms; however, the dissipation term now represents the production of turbulence rather than the immediate generation of heat. Application of the equation is illustrated for three representative cases of fluid motion: two-dimensional Poiseuille flow; the boundary-layer wake of a flat plate; and the hydraulic jump.