Page 25 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

10 BASICS OF FLUID MECHANICS AND INTRODUCTION TO CFD

For a fixed time t, by a vortex (vorticity, rotation) line (surface) we

understand those curves (surfaces) whose tangents, at every point of
them, are directed along the local vorticity (curl, rotation) vector.
Of course the particles distributed along a vortex line rotate about
the tangents to the vortex line at their respective positions.
A vortex (vorticity, rotation) tube is a vortex surface generated by
vortex lines drawn through each point of an arbitrary simple closed curve

(there is a diffeomorphism between the continuum surface enclosed by
this simple curve and the circular disk).
If the vortex tube has a very small (infinitesimal) sectional area it is
known as a vortex filament.

1.2.4 Circulation

The circulation along an arc AB is the scalar T(AB) = f v-dr. The
AB
following result is a direct consequence of the Stokes theorem [110):
“The circulation about two closed contours on a vortex tube at a given
instant ¢, — closed contours which lie on the vortex tube and encircle it
once, in the same sense — are the same” (this result of pure kinematic
nature is known as the “first theorem of Helmholtz’).
The invariance of the circulation vis-a-vis the contour C which encir-

cles once the vortex tube supports the introduction of the concept of the
strength of the vortex tube. More precisely, this strength would be the
circulation along the closed simple contour (C) which encircles once, in
a direct sense, the tube.
The constancy of this circulation, which is equal to the rotation flux

through the tube section bounded by the contour (C), leads to the fact
that, within a continuum, both vortex and filament lines cannot “end”
(the vanishing of the area bounded by (C) or of the vortex would imply,
respectively, the unboundedness of the vorticity or the mentioned area,
both cases being contradictions).
That is why the vortex lines and filaments either form rings in our
continuum or extend to infinity or are attached to a solid boundary.

(The smoke rings from a cigarette make such an example).°

>The circulation of a vector u, from a continuous derivable field, along the simple closed
contour (L), is equal to the flux of rotu through a surface ( © ) bounded by (ZL), ive.
f u-dr = ff rotu-ndo , provided that the reference frame (system), made by the positively
(LE) (2)
oriented tangent at a point P € (L), the outward normal n to ( © ) at a point M and the
vector MP, for any points M and P, is a right-handed system.
°For a line vortex (which is distinct from a vortex line and which is a mathematical ideal-
ization of a vortex filament assumed to converge onto its axis, i.e. a vortices locus) the same
assertion, often made, is false (rot v could have zeros within the continuum in motion!)