Page 45 - Mechanical Engineers Reference Book

1/34 Mechanical engineering principles
Stream functions may be superposed so that if Tangential velocity vg = -
a*
= fn(x,y) describes flow pattern A, ar (1.126)
and
1.5.9.2 The velocity potential
$2 = fn(x,y) describes flow pattern B,
In a gravitational field there is a property the change in which
then = $1 + $2 describes the flow pattern produced by the is independent of the path of the change: potential energy. In a
combination of A and B. continuous, irrotational flow field there is also a property the
The x and y components of the flow velocity v are given by change in which is independent of the path of the change. This
a* a* property is the velocity potential (4). It can be shown that
v, = - -; and v = - (1.124)
ay ax
(1.127)
or
a* or
v, = -, and vy = - -
a4
aY ax a4 and v, = - - (1.128)
depending on sign convention. In polar coordinates the com- "a=--' ra0' ar
ponents are Lines of constant 4 are known as velocity potential lines with
-a* an equation
Radial velocity v, = - (1.125)
ra0 v,dx + vydy = 0 (1.129)

-
-
V
-
Parallel flow
J, = Vy = Vr cos 0
@ = Vx=Vrsin0

42

44-

I
$6 46
Source +=-d Vortex J, = mr
2n 2n
m
@ =- -lnr -+2
271
Doublet in parallel flow

Figure 1.48 Simple flow patterns