Page 93 - Aerodynamics for Engineering Students

76 Aerodynamics for Engineering Students

Fig. 2.17

All other points Pz, P3, etc. which have a stream function equal in value to that of P
have, by definition, the same flow across any lines joining them to 0, so by the same
argument the velocity of the flow in the region of PI, Pz, P3, etc. must be along PP1,
Pz, P3, etc., and no fluid crosses the line PP1, P2,. . .,P,. Since $pl = h2 =
1clp3 = +p = constant, the line PP1, Pz, . . . P,, etc. is a line of constant $ and is called
a streamline. It follows further that since no flow can cross the line PP, the velocity
along the line must always be in the direction tangential to it. This leads to the two
common definitions of a streamline, each of which indirectly has the other’s meaning.
They are:

A streamline is a line of constant $

and/or
A streamline is a line of fluid particles, the velocity of each particle being
tangential to the line (see also Section 2.1.2).
It should be noted that the velocity can change in magnitude along a streamline but
by definition the direction is always that of the tangent to the line.
2.5.3 Velocity components in terms of w

(a) In Cartesian coordinates Let point P(x, y) be on the streamline AB in Fig. 2.18a
of constant $ and point Q(x + Sx, y + Sy) be on the streamline CD of constant
$ + S$. Then from the definition of stream function, the amount of fluid flowing
across any path between P and Q = S$, the change of stream function between
P and Q.
The most convenient path along which to integrate in this case is PRQ, point R
being given by the coordinates (x + Sx, y). Then the flow across PR = -vSx (since
the flow is from right to left and thus by convention negative), and that across
RQ = uSy. Therefore, total flow across the line PRQ is

S$ = uSy - VSX (2.55)
Now $ is a function of two independent variables x and y in steady motion, and thus
a$ a$
S$ = -Sx + -Sy (2.56)
ax ay