Page 147 - Bird R.B. Transport phenomena

§4.3 Flow of Inviscid Fluids by Use of the Velocity Potential 131

Fig. 4.3-2. The streamlines for the
potential flow into a rectangular channel,
as predicted from potential flow theory in
Eqs. 4.3-30 and 31. A more realistic flow
pattern is shown in Fig. 4.3-5.
I „ - -

EXAMPLE 4.3-3 Figure 4.3-3 shows the potential flow in the neighborhood of two walls that meet at a cor-
ner at O. The flow in the neighborhood of this corner can be described by the complex
Flow Near a Corner 8 potential

w(z) = -cz a (4.3-38)
in which с is a constant. We can now consider two situations: (i) an "interior corner flow/ 7
with a > 1; and (ii) an "exterior corner flow/' with a < 1.
(a) Find the velocity components.
(b) Obtain the tangential velocity at both parts of the wall.
(c) Describe how to get the streamlines.
(d) How can this result be applied to the flow around a wedge?

SOLUTION (a) The velocity components are obtained from the complex velocity
dw = -caz ~ ] = - (4.3-39)
a
dz

Streamlines
ii)

(ii) Fig. 4.3-3. Potential flow near a corner. On
a
x the left portion of the wall, v,.= —cr ~\ and
a
on the right, v = +cr ~\ (i) Interior-corner
r
flow, with a > 1; and (ii) exterior-corner flow,
with a < 1.

• R. L. Panton, Compressible Flow, Wiley, New York, 2nd edition (1996).