Page 126 - Aerodynamics for Engineering Students

Potential flow 109

But, since 4 is a function of two independent variables;

and (3.3)

Again, in general, the velocity q in any direction s is found by differentiating the
velocity potential q5 partially with respect to the direction s of q:
ad
q=-
dS

3.2 Laplace's equation

As a focus of the new ideas met so far that are to be used in this chapter, the main
fundamentals are summarized, using Cartesian coordinates for convenience, as
follows:
(1) The equation of continuity in two dimensions (incompressible flow)
au av
-+-=o
ax ay

(2) The equation of vorticity
av du
--_ =5 (ii)
ax ay
(3) The stream function (incompressible flow) .IC, describes a continuous flow in two
dimensions where the velocity at any point is given by
(iii)

(4) The velocity potential C#J describes an irrotational flow in two dimensions where
the velocity at any point is given by

Substituting (iii) in (i) gives the identity

g$J @$J =o
axay axay

which demonstrates the validity of (iii), while substituting (iv) in (ii) gives the identity
824 824 =o
axay axay