Page 154 - Aerodynamics for Engineering Students

Potential flow 137

Equating (3.53) and (3.54):

(3.55)
The rate of change of total pressure His

and substituting for Eqn (3.55):

aff - 4: dqt
dr
- P- r + P4t - pqt
=
Now for this system (l/r)(aq,/%) is zero since the streamlines are circular and
therefore the vorticity is (qt/r) + (dq,/dr) from Eqn (2.79), giving

(3.56)

3.4 Axisymmetric flows (inviscid and
incompressible flows)

Consider now axisymmetric potential flows, i.e. the flows around bodies such as
cones aligned to the flow and spheres. In order to analyse, and for that matter to
define, axisymmetric flows it is necessary to introduce cylindrical and spherical
coordinate systems. Unlike the Cartesian coordinate system these coordinate systems
can exploit the underlying symmetry of the flows.

3.4.1 Cylindrical coordinate system
The cylindrical coordinate system is illustrated in Fig. 3.27. The three coordinate
surfaces are the planes z = constant and 0 = constant and the surface of the cylinder
having radius r. In contrast, for the Cartesian system all three coordinate surfaces are

X
Fig. 3.27 Cylindrical coordinates