Mechanics of Fluids

106 The principles governing fluids in motion
Fig. 3.7
Thus H is the total head for the streamline on which the point 1 lies.
If no energy is dissipated by friction the total head H is constant along
that streamline and may therefore be represented by a line parallel to the
datum plane.
At a point 2 in the pipe the pressure is indicated by the rise p2/ϱg of the
liquid in the piezometer tube. (For reasons that will become apparent in
Section 3.6, there should be no appreciable curvature of the streamlines at
positions 2 and 3.) The amount by which the sum of p2/ϱg and z2 falls short
of the total head corresponds to the velocity head u2 2 /2g for the streamline
considered. There is a similar state of affairs at point 3, although here the
cross-section of the pipe is smaller and so the mean velocity is greater than
at 2 by virtue of the continuity equation Au = constant.
In practice, friction leads to a loss of mechanical energy, so the total head
line (sometimes known as the total energy line) does not remain horizontal,
but drops. The height of any point on this line above the datum plane always
represents the total head (p/ϱg) + (u2 /2g) + z of the fluid at the point in
question. Another line that may be drawn is that representing the sum of
the pressure head and elevation only: (p/ϱg) + z. This line, which would
pass through the surface levels in the piezometer tubes of Fig. 3.7, is known
as the pressure line or hydraulic grade line and is always a distance u2 /2g
vertically below the total head line. The geometrical representation that these
lines afford is frequently useful, and it is therefore important to distinguish
clearly between them.
Strictly speaking, each streamline has it own total head and pressure lines.
When one-dimensional flow is assumed, however, it is usual to consider
only the streamline in the centre of the pipe, so that the z measurements
are taken to the centre line and the static head p/ϱg is measured upwards
from there. The mean total head line is then a distance αu2 /2g vertically
above the pressure line. Other conventional assumptions about these lines
are mentioned is Section 7.7.
The one-dimensional continuity relation shows that, for a fluid of constant
density, a reduction in the cross-sectional area causes an increase in
the mean velocity, and the energy equation (3.16) indicates that unless additional
energy is given to the fluid an increase of velocity is accompanied by