Mechanics of Fluids

To ensure that the pressure measured at each section is the average,
connections to the manometer are made via a number of holes into an annular
ring. The holes are situated where the walls are parallel so that there
is no variation of piezometric pressure across the flow. To discourage the
formation of eddies, sharp corners at the joins between the conical and the
parallel-sided sections are avoided.
Rapidly converging flow, as between the inlet and throat of a
venturi-meter, causes the velocity to become more uniform over the crosssection.
Over the short length involved the loss of head to friction, hf, is negligible in comparison with (p1 − p2)/ϱg. For a single streamline
Bernoulli’s equation then gives
u 2 2 = u2 �
p∗ 1 − p
1 + 2g
∗ �
2
ϱg
where p∗ represents the piezometric pressure p+ϱgz. On a nearby streamline,
slightly different values u1+δu1 and u2+δu2 may be found instead of u1 and
u2. If the streamlines are straight and parallel at these sections, however, no
difference of piezometric pressure across the flow can be sustained and so all
streamlines have the same values of p∗ 1 and of p∗ 2 . Subtracting the Bernoulli
equations for the two streamlines therefore gives
(u2 + δu2) 2 − u 2 2 = (u1 + δu1) 2 − u 2 1
Hence 2u2δu2
neglected.
= 2u1δu1, higher orders of the small quantities being
∴ δu2
u2
� �2 u1 δu1
=
u2 u1
Since u1 < u2,
|δu2|
<
u2
|δu1|
u1
in other words, the proportionate variation of velocity is less after the contraction
than before it. This is why a rapid contraction is placed before the
working section of a wind tunnel or water tunnel, where a uniform velocity
is especially important.
In a few extreme cases, where friction between inlet and throat of a venturimeter
is very small, the uniformity of velocity may be so much improved at
the throat as to outweigh the effect of friction and give a value of Cd slightly
greater than unity. (A Cd greater than unity may also result from faulty
manometer connections.) For accurate and predictable results, it is desirable
that the venturi-meter be approached by a sufficient length of straight pipe
for the flow to be reasonably uniform and to be free from large eddies and
similar disturbances, caused by fittings upstream of the flow-meter.
The function of the diverging part of the meter is to reduce the velocity
gradually, with the aim of restoring the pressure as nearly as possible to its
original value. In a rapidly diverging tube complete pressure recovery is not
possible as the flow tends to separate from the walls, eddies are formed and
Simple applications of Bernoulli’s equation 121