Potential Flows

Abstract

As the discussions in Sects 4.1 and 4.3 have already shown, solid walls and discontinuities in the tangential velocity represent surfaces from which angular velocity ($\vec\omega={\rm curl}\,\vec u/2$) diffuses into the flow field. Since the widths of the developing regions (boundary layers) tend to zero in the limit $Re\to\infty$, the flow can be treated within the framework of potential theory. Because of the kinematic restriction of irrotationality, only the kinematic boundary condition, but not the no slip condition, can be satisfied. Therefore potential flows, although they are exact solutions of the Navier-Stokes solutions in the incompressible case, can in general only describe the flow field of an inviscid fluid (with exceptions, like the potential vortex for the flow around a rotating cylinder). However, the results of a calculation for inviscid fluid can be carried over to real flows as long as the flow does not separate. If separation does occur, the boundaries of the separation region are generally not known. In cases where these boundaries are known or can be reasonably estimated, a theory based on inviscid flow can also be useful.