2.11 Vorticity

Vorticity describes the tendency for a fluid to rotate locally, defined as

! = r u ; \relax \special {t4ht=
where the eqn operator is the curl derivative. The curl of a vector eqn is evaluated in Cartesian co-ordinates by
 r u = @uz- @uy; @ux @uz; @uy- @ux- : @y @z @z @x @x @y \relax \special {t4ht=
The local rotation which vorticity measures is not due to the flow following a curved path. In fact, vorticity is often associated with shear when streamlines may be straight and parallel, i.e. not curved at all.

Vorticity is difficult to picture under shear because the deformation masks the local rotation. By separating the deformation, as in the figure in Sec. 2.10 , the local rotation is revealed.

PICT\relax \special {t4ht=

Vorticity eqn is often demonstrated by a vortex ring produced by an air “cannon” with smoke to visualise the flow. It reveals flow circulation around sections of the torus, from front to back. The vorticity vectors are normal to the planes of circulation, along the axis of the torus.

PICT\relax \special {t4ht=

Stokes’s theorem8 relates circulation — the integral of eqn around a closed curved line, eqn — to the integral of vorticity eqn over a section of surface eqn bounded by the curve, according to:

Z Z Z [dS (r u)] = (dS !) = (dL u): S S L \relax \special {t4ht=
In Eq. (2.39 ), eqn is a vector representing a segment of the line eqn. As you stand on eqn looking in the direction eqn, with your head in the direction eqn, eqn is oriented to your left.

Vorticity is related to the spin tensor eqn in Eq. (2.33 ) by eqn, where eqn is the Hodge dual operator which extracts components of a vector from a tensor eqn as shown below:

T = (Tyz; Txz;Txy) : \relax \special {t4ht=

The term “spin tensor” emphasises local rotation as opposed to the general path in a similar way that the spin on a ball is clearly distinguished from its trajectory, which curves due to gravity.
8posed by George Stokes as a examination question for the 1854 Smith’s Prize for progress in mathematics and natural philosophy at the University of Cambridge, awarded to James Clerk Maxwell and Edward Routh.

Notes on CFD: General Principles - 2.11 Vorticity