Page 132 - Aerodynamics for Engineering Students

Potential flow 1 15

Fig. 3.9

OTP is taken where T is on the Ox axis x along from 0, i.e. point T is given by (x, 0).
Then

$ = flow across line OTP
= flow across line OT plus flow across line TP
= O+ U x length TP
=o+uy

Therefor e
$= UY (3.12)
The streamlines (lines of constant $) are given by drawing the curves
@ = constant = Uy
Now the velocity is constant, therefore

1cI
y = - = constant on streamlines
U
The lines $ = constant are all straight lines parallel to Ox.
By definition the velocity potential at a point P(x, y) in the flow is given by the line
integral of the tangential velocity component along any curve from 0 to P. For
convenience take OTP where T has ordinates (x, 0). Then

#I = flow along contour OTP
= flow along OT + flow along TP
= ux+o
Therefore
#I = ux (3.13)
The lines of constant #I, the equipotentials, are given by Ux = constant, and since the
velocity is constant the equipotentials must be lines of constant x, or lines parallel to
Oy that are everywhere normal to the streamlines.

Flow of constant velocity parallel to 0 y axis
Consider flow streaming past the Ox, Oy axes at velocity Vparallel to Oy (Fig. 3.10).
Again by definition the stream function $ at a point P(x, y) in the flow is given by the