Restricted and Vortical Flows

Abstract

The fluid dynamical equations of motion are a formidable set of nonlinear PDEs (partial differential equations). It seems hopeless to look for solutions of these equations in any general case, if specific boundary and initial conditions are given. We can, however, learn a lot about the physical properties of flows, i.e., solutions of the equations by defining auxiliary functions and deriving theorems about flows that are valid under special circumstances. That is the subject of this chapter. We shall introduce and use here the vorticity field $\boldsymbol{\omega }= \nabla \times \mathbf{u}$, whose importance in FD is paramount and it will accompany the discussions along most of the book. It is very useful to consider flows occurring under various restricted conditions, e.g., steady, inviscid, incompressible, barotropic, and irrotational or a combination of a small number thereof. We include in this category flows with simplistic geometries and boundary conditions, including those with manageable initial conditions. In this chapter we choose a number of flows with explicitly defined restricted conditions and we go on to examine what can be learned analytically about such flows. It is important, in our view, to know the physical properties of relevant special flows and understand them before attempting to numerically solve more complicated general flows. We shall also introduce here some mathematical approximation methods to help in the derivations and understanding of approximate analytical solutions. Naturally, analytical approximations are best suited to equations that are simplified by special conditions.