Page 148 - Bird R.B. Transport phenomena

132 Chapter 4 Velocity Distributions with More Than One Independent Variable

Fig. 4.3-4. Potential flow along a wedge. On
a
the upper surface of the wedge, v x = cx ~ l =
P/(2-p)^ j^e quantities a and j3 are related by
cx
P = (2/a){a - 1).
!тг

Streamlines

Hence from Eq. 4.3-12 we get

v x = +car ~' 1 cos (a? - 1)6 (4.3-40)
a
a
v = -car ~ ] sin (a - 1)0 (4.3-41)
y
(b) The tangential velocity at the walls is
a
a
at 6 = 0: v = v = car ~ l = cax ~ ] (4.3-42)
x r
at 6 = тт/a: v r = v x cos 6 + v y sin 6
a
a
l
= +car ~ l cos (a - 1)6 cos 6 - car ~ sin (a - 1)6 sin 6
a
]
= car ~ cos a6
a ]
= -car ' (4.3-43)
Hence, in Case (i), the incoming fluid at the wall decelerates as it approaches the junction, and
the departing fluid accelerates as it moves away from the junction. In Case (ii) the velocity
components become infinite at the corner as a — 1 is then negative.
(c) The complex potential can be decomposed into its real and imaginary parts
w = ф + 1ф= -cr (cos a6 + i sin a6) (4.3-44)
a
Hence the stream function is
ф= -cr a sin a6 (4.3-45)
To get the streamlines, one selects various values for the stream function—say, ф и ф ъ ф • • •
3
—and then for each value one plots r as a function of 6.
(d) Since for ideal flow any streamline may be replaced by a wall, and vice versa, the results
found here for a > 0 describe the inviscid flow over a wedge (see Fig. 4.3-4). We make use of
this in Example 4.4-3.

A few words of warning are in order concerning the applicability of potential-flow
theory to real systems:
a. For the flow around a cylinder, the streamlines shown in Fig. 4.3-1 do not con-
form to any of the flow regimes sketched in Fig. 3.7-2.
b. For the flow into a channel, the predicted flow pattern of Fig. 4.3-2 is unrealistic
inside the channel and just upstream from the channel entrance. A much better
approximation to the actual behavior is shown in Fig. 4.3-5.

Both of these failures of the elementary potential theory result from the phenomenon of
separation: the departure of streamlines from a boundary surface.
Separation tends to occur at sharp corners of solid boundaries, as in channel flow,
and on the downstream sides of bluff objects, as in the flow around a cylinder. Gener-
ally, separation is likely to occur in regions where the pressure increases in the direction