Mechanics of Fluids

110 The principles governing fluids in motion
Fig. 3.10
Stagnation point
Stagnation pressure
Dynamic pressure
of the so-called discharge coefficient. This dimensionless quantity is further
discussed in Chapter 5.
3.7.1 The Pitot tube and the Pitot-static tube
A point in a fluid stream where the velocity is reduced to zero is known as a
stagnation point. Any non-rotating obstacle placed in the stream produces
a stagnation point next to its upstream surface. Consider the symmetrical
object illustrated in Fig. 3.10 as an example. On each side of the central
streamline OX the flow is deflected round the object. The divergence of the
streamlines indicates that the velocity along the central streamline decreases
as the point X is approached. The contour of the body itself, however, consists
of streamlines (since no fluid crosses it) and the fluid originally moving
along the streamline OX cannot turn both left and right on reaching X. The
velocity at X is therefore zero: X is a stagnation point.
By Bernoulli’s equation 3.9 the quantity p + 1
2ϱu2 + ϱgz is constant along a
streamline for the steady frictionless flow of a fluid of constant density. Consequently,
if the velocity u at a particular point is brought to zero the pressure
there is increased from p to p+ 1
2ϱu2 . For a constant-density fluid the quantity
p+ 1
2ϱu2 is therefore known as the stagnation pressure of that streamline.
That part of the stagnation pressure due to the motion, 1
2ϱu2 , is termed the
dynamic pressure. (If heads rather than pressures are used the term total head
is often preferred to stagnation head.) A manometer connected to the point
X would record the stagnation pressure, and if the static pressure p were
also known 1
2ϱu2 could be obtained by subtraction, and hence u calculated.
Henri Pitot (1695–1771) adopted this principle in 1732 for measuring
velocities in the River Seine, and Fig. 3.11 shows the sort of device he used.
A right-angled glass tube, large enough for capillary effects to be negligible,
has one end (A) facing the flow. When equilibrium is attained the fluid at
A is stationary and the pressure in the tube exceeds that of the surrounding
stream by 1
2ϱu2 . The liquid is forced up the vertical part of the tube to
a height
h = �p/ϱg = 1
2ϱu2 /ϱg = u 2 /2g
above the surrounding free surface. Measurement of h therefore enables
u to be calculated.
Such a tube is termed a Pitot tube and provides one of the most accurate
means of measuring the velocity of a fluid. For an open stream of liquid
only this single tube is necessary, since the difference between stagnation
and static pressures (or heads) is measured directly. (In practice, however,