Mechanics of Fluids

the junction, the curvature of the streamlines and the acceleration of the fluid
cause the pressure at the annular face to vary in an unknown way. However,
immediately downstream of the junction a vena contracta is formed, after
which the stream widens again to fill the pipe. A region of separated flow is
formed between the vena contracta and the wall of the pipe, and this causes
practically all the dissipation of energy. Between the vena contracta plane
and the downstream section 2 – where the velocity has again become sensibly
uniform – the flow pattern is similar to that after an abrupt enlargement, and
so the loss of head is assumed to be given by eqn 7.15:
h l = u2 2
2g
� A2
Ac
− 1
� 2
= u2 2
2g
� 1
Cc
− 1
� 2
(7.16)
where Ac represents the cross-sectional area of the vena contracta, and the
coefficient of contraction Cc = Ac/A2.
Although the area A1 is not explicitly involved in eqn 7.16, the value of
Cc depends on the ratio A2/A1. For coaxial circular pipes and fairly high
Reynolds numbers Table 7.1 gives representative values of the coefficient k
in the formula
h l = ku2 2
2g
(7.17)
Other losses in pipes 263
Fig. 7.10
As A1 → ∞ the value of k in eqn 7.17 tends to 0.5, and this limiting Entry loss
case corresponds to the flow from a large reservoir into a sharp-edged pipe,
provided that the end of the pipe does not protrude in to the reservoir (see
Fig. 7.11a). A protruding pipe, as in Fig. 7.11b, causes a greater loss of head.
For a non-protruding, sharp-edged pipe the loss 0.5u2 2 /2g is known as the
entry loss. If the inlet to the pipe is well rounded, as in Fig. 7.11c, the fluid
can follow the boundary without separating from it, and the entry loss is
Table 7.1 Loss coefficient k for abrupt
contraction
d 2/d 1 0 0.2 0.4 0.6 0.8 1.0
k 0.5 0.45 0.38 0.28 0.14 0