Notes on incompressible flows

A critical analysis is carried out of the incompressible flow equations, and it is emphasized that the assumption of non-viscosity of the fluid, or even of the existence of regions in the flow where viscosity may be neglected, is never necessary. In effect, the form that is taken by the momentum equation for an incompressible viscous fluid in a motion that is either irrotational or with vorticity having a potential coincides with the equation that would be obtained if the fluid were assumed to be non-viscous. Consequently, the role of the generation and dynamics of vorticity is stressed, as well as its importance for the description of the flow development and for assessing why and when simplified models may be devised for its analysis or for the evaluation of the loads acting on moving bodies. The conditions for the validity of the Bernoulli theorem are also discussed, together with the significance of the conservation law for the total vorticity in flows starting from rest. The evolution of some simple flows is then briefly described, and the connection between viscosity and the origin of lift on an airfoil is pointed out. Subsequently, the possible utility of the energy balance equation, which is usually neglected in incompressible flow analyses, is examined. In particular, it is shown that dissipation may be significant also in irrotational regions, and that its evaluation for incompressible flows may often be carried out in a straightforward manner by means of the Bobyleff-Forsyth formula. Finally, the integral form of the kinetic energy balance is applied to analyse several flows in different frames of reference, highlighting the fact that a deeper comprehension may be gained on the roles of the various terms, and on their connection with the drag acting on moving bodies.