TABLE OF CONTENTS Vortex motions are vital elements of fluid flows. In fact, most dynamical aspects of fluid motions are featured by vortex motions. Already we considered some motions of vortices: vortex sheet and shear layer [Sec. 4.6.2(b)], vortex line and point vortex (Secs. 5.6.5 and 5.8.5), etc.
Analytical study of vortex motions is mostly based on vorticity which is defined as twice the angular velocity of local rotation (Sec. 1.4.3). The vorticity equation is derived in Sec. 4.1 from the equation of motion. It is to be remarked that the no-slip condition and the boundary layer are identified as where vorticity is generated (Sec. 4.5). In Chapter 12, it will be shown that vorticity is in fact a gauge field associated with rotational symmetry of the flow field, on the basis of the gauge theory of modern theoretical physics.
Equation of the vorticity ? = curl v was given for compressible flows in Sec. 3.4 in an inviscid fluid ( ? = 0) as
For viscous incompressible flows, two equivalent equations given in Sec. 4.1 are reproduced here:
Let us consider a solenoidal velocity field v( x) (defined by div v = 0, i.e. incompressible) induced by a compact vorticity field ?( x). That is, the vorticity vanishes out of a bounded open domain D:
It is to be noted in this case that the following space...
