Page 103 - Aerodynamics for Engineering Students

86 Aerodynamics for Engineering Students

The other direct strains are obtained in a similar way; thus the rates of direct strain
are given by

Thus we can introduce a rate of strain tensor analogous to the stress tensor (see
Section 2.6) and for which components in two-dimensional flow can be represented
in matrix form as follows:

(2.75)

where ( ‘ ) is used to denote a time derivative.

2.7.4 Vorticity
The instantaneous rate of rotation of a fluid element is given by (ci - ,8)/2 - see
above. This corresponds to a fundamental property of fluid flow called the vorticity
that, using Eqn (2.71), in two-dimensional flow is defined as
da dp av au
5 = - - - = - - - (2.76)
dt dt ax ay
In three-dimensional flow vorticity is a vector given by
aw av au aw av au
ay az’az wax ay
It can be seen that the three components of vorticity are twice the instantaneous
rates of rotation of the fluid element about the three coordinate axes. Mathematically
it is given by the following vector operation

Q=Vxv (2.78)
Vortex lines can be defined analogously to streamlines as lines that are tangential
to the vorticity vector at all points in the flow field. Similarly the concept of the
vortex tube is analogous to that of stream tube. Physically we can think of flow
structures like vortices as comprising bundles of vortex tubes. In many respects
vorticity and vortex lines are even more fundamental to understanding the flow
physics than are velocity and streamlines.

2.7.5 Vorticity in polar coordinates
Referring to Section 2.4.3 where polar coordinates were introduced, the correspond-
ing definition of vorticity in polar coordinates is

(2.79)
Note that consistent with its physical interpretation as rate of rotation, the units of
vorticity are radians per second.