Page 133 - Aerodynamics for Engineering Students

1 16 Aerodynamics for Engineering Students

Fig. 3.10

amount of fluid crossing any curve between 0 and P. For convenience take OTP
where T is given by (x, 0). Then

1c, = flow across OT + flow across TP
=-Vx+O

Note here that when going from 0 towards T the flow appears from the right and
disappears to the left and therefore is of negative sign, i.e.

+ = -vx (3.14)
The streamlines being lines of constant + are given by x = -+/V and are parallel to
Oy axis.
Again consider flow streaming past the Ox, Oy axes with velocity V parallel to the
Oy axis (Fig. 3.10). Again, taking the most convenient boundary as OTP where T is
given by (x, 0)

= flow along OT + flow along TP
=o+vy

Therefore
q!I = VY (3.15)

The lines of constant velocity potential, q!I (equipotentials), are given by
Vy = constant, which means, since Vis constant, lines of constant y, are lines parallel
to Ox axis.

Flow of constant velocity in any direction
Consider the flow streaming past the x, y axes at some velocity Q making angle 0 with
the Ox axis (Fig. 3.11). The velocity Q can be resolved into two components U and V
parallel to the Ox and Oy axes respectively where Q2 = U2 + V2 and tan0 = V/U.
Again the stream function 1c, at a point P in the flow is a measure of the amount of
fluid flowing past any line joining OP. Let the most convenient contour be OTP,
T being given by (x, 0). Therefore