Page 91 - Aerodynamics for Engineering Students

74 Aerodynamics for Engineering Students

Fig. 2.14

the bank can be represented by the Ox axis, and the line joining you to your friend at
A the Oy axis in the two-coordinate system. Now if the stream speed is 2ms-' the
amount of water passing between you and your friend is 40 x 1 x 2 = 80 m3 s-l and
this is the amount of water flowing past any point anywhere along the river which
could be measured at a weir downstream. Suppose you now throw a buoyant rope to
your friend who catches the end but allows the slack to fall in the river and float into
a curve as shown. The amount of water flowing under the line is still 80m3 s-' no
matter what shape the rope takes, and is unaffected by the configuration of the rope.
Suppose your friend moves along to a point €3 somewhere downstream, still
holding his end of the line but with sufficient rope paid out as he goes. The volume
of water passing under the rope is still only 80m3 s-l providing he has not stepped
over a tributary stream or an irrigation drain in the bank. It follows that, if no water
can enter or leave the stream, the quantity flowing past the line will be the same as
before and furthermore will be unaffected by the shape of the line between 0 and €3.
The amount or quantity of fluid passing such a line per second is called the stream
function or current function and it is denoted by +.
Consider now a pair of coordinate axes set in a two-dimensional air stream that is
moving generally from left to right (Fig. 2.15). The axes are arbitrary space references
and in no way interrupt the fluid streaming past. Similarly the line joining 0 to a point
P in the flow in no way interrupts the flow since it is as imaginary as the reference axes
Ox and Oy. An algebraic expression can be found for the line in x and y.

X / --.

Fig. 2.15