Navier-Stokes Equations: Vorticity Transport Theorems

Navier-Stokes Equations
Vorticity Transport Theorems: Introduction

The vorticity z º Ñ x v is discussed in nearly every introductory fluid mechanics course and even in many courses on engineering fluid mechanics. As is pointed out in these courses, the physical interpretation of the vorticity is that it is a measure of the angular velocity or angular momentum of the fluid particles; click on the highlighted text to see a somewhat more complete discussion of the standard interpretation. To complete the presentation, I'll provide the standard discussion of the ways in which vorticity is diffused, distorted, and generated by the fluid flow.

Motivation
The motivation for describing the dynamics of vorticity is two-fold. The first is that most introductory treatments of fluid mechanics place great emphasis on irrotational flows, i.e., those for which the vorticity is zero over the bulk of the flow. For example, most of hydrodynamics, aerodynamics, and water wave theory is normally presented in the context of irrotational flows. An inspection of any discussion of the unsteady form of the Bernoulli equation will reveal that the condition of irrotational flow is critical to the derivation of that form of the Bernoulli equation. Thus, it is important to understand the conditions under which the vorticity vanishes. The second motivation for presenting the vorticity transport theorems is that vorticity dynamics can play an important role in the theory of turbulence, meterology, and oceanography.

Outline
In these notes, I'll follow the standard approach by first describing how vorticity diffuses through the flow. As is usually done, I'll restrict the discussion of diffusion to incompressible flows. Again, I will follow the standard approach in my discussion of the creation of vorticity; this will be carried out using the inviscid flow approximation. Finally, I'll discuss the mechanisms by which vorticity is distorted by the flow. In this section, I'll deviate somewhat from more elementary treatments by first presenting the results of Cauchy's integral. The concepts of stretching and distortion will then be discussed in the context of this wonderful little result. In the section entitled Shock Waves, the jump in vorticity across a stationary shock wave is presented. The value of the vorticity generated by the shock serves as an initial value for the Vorticity Transport Equation.