Page 155 - Aerodynamics for Engineering Students

138 Aerodynamics for Engineering Students

planes. As a consequence for the Cartesian system the directions (x, y, z) of the
velocity components, say, are fixed throughout the flow field. For the cylindrical
coordinate system, though, only one of the directions (z) is fixed throughout the flow
field; the other two (r and 8) vary throughout the flow field depending on the value of
the angular coordinate 8. In this respect there is a certain similarity to the polar
coordinates introduced earlier in the chapter. The velocity component qr is always
locally perpendicular to the cylindrical coordinate surface and qg is always tangential
to that surface. Once this elementary fact is properly understood cylindrical coord-
inates become as easy to use as the Cartesian system.
In a similar way as the relationships between velocity potential and velocity
components are derived for polar coordinates (see Section 3.1.3 above), the following
relationships are obtained for cylindrical coordinates
& 1 a4
&
qr = dr 4s = -- 42 = & (3.57)
rd8'
An axisymmetric flow is defined as one for which the flow variables, i.e. velocity
and pressure, do not vary with the angular coordinate 8. This would be so, for
example, for a body of revolution about the z axis with the oncoming flow directed
along the z axis. For such an axisymmetric flow a stream function can be defined.
The continuity equation for axisymmetric flow in cylindrical coordinates can be
derived in a similar manner as it is for two-dimensional flow in polar coordinates
(see Section 2.4.3); it takes the form

(3.58)

The relationship between stream function and velocity component must be such as to
satisfy Eqn (3.58); hence it can be seen that

(3.59)

3.4.2 Spherical coordinates

For analysing certain two-dimensional flows, for example the flow over a circular
cylinder with and without circulation, it is convenient to work with polar coord-
inates. The axisymmetric equivalents of polar coordinates are spherical coordinates,
for example those used for analysing the flow around spheres. Spherical coordinates
are illustrated in Fig. 3.28. In this case none of the coordinate surfaces are plane
and the directions of all three velocity components vary over the flow field, depending
on the values of the angular coordinates 0 and p. In this case the relationships
between the velocity components and potential are given by

(3.60)

For axisymmetric flows the variables are independent of 8 and in this case the
continuity equation takes the form

(3.61)