Page 118 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 101

Fig. 2.34

Exercises

1 Continuity Equation for axisymmetricflow
(a) Consider an axisymmetric flow field expressed in terms of the cylindrical
coordinate system (r, 4, z) where all flow variables are independent of the azimuthal
angle 4. For example, the axial flow over a body of revolution. If the velocity
components (u, w) correspond to the coordinate directions (r, z) respectively, show
that the continuity equation is given by

du
u dw
-+-+-=o
dr r dz
(b) Show that the continuity equation can be automatically satisfied by a stream-
function 11, of a form such that

2 Continuity equation for two-dimensional flow in polar coordinates
(a) Consider a two-dimensional flow field expressed in terms of the cylindrical
coordinate system (r, 4, z) where all flow variables are independent of the azimuthal
angle 4. For example, the flow over a circular cylinder. If the velocity components
(u, v) correspond to the coordinate directions (r, 4) respectively, show that the
continuity equation is given by

du u ldv
-+-+--=o
dr r r&i5
(b) Show that the continuity equation can be automatically satisfied by a stream-
function $ of a form such that