Mechanics of Fluids

124 The principles governing fluids in motion
Fig. 3.22
number of the flow, and the location of the wall pressure tappings, but, as
it does not depend on what happens beyond the throat, it is little different
from that for a venturi-meter.
A still simpler and cheaper arrangement is a sharp-edged orifice fitted
concentrically in the pipe (Fig. 3.22). Application of Bernoulli’s equation
between a point 1 upstream of the orifice and the vena contracta (2) gives,
for an inviscid fluid and uniform velocity distribution:
u 2 2
2g = p∗ 1
ϱg − p∗ 2
ϱg + u2 1
2g = �h + u2 1
2g
For a real fluid we introduce a coefficient of velocity:
u2 = Cv{2g(�h + u 2 1 /2g)}1/2
(3.24)
We now put u1 = Q/A1 and u2 = Q/Ac = Q/CcAo where Q represents
the discharge, Ac the cross-sectional area of the vena contracta, Ao the area
of the orifice itself and Cc the coefficient of contraction. Then
Q =
CvCcAoA1
(A 2 1 − C2 v C2 c A2 o )1/2 (2g�h)1/2 =
C dAo(2g�h) 1/2
{1 − C 2
d (Ao/A1) 2 } 1/2
since CvCc = Cd (eqn 3.19).
The discharge coefficient C may be introduced by writing
�
1 − (Ao/A1)
C = Cd 2
1 − C2 �1/2 (Ao/A1) 2
d
so that finally
Q =
CAo(2g�h) 1/2
{1 − (Ao/A1) 2 } 1/2