Mechanics of Fluids

7.9 CONDITIONS NEAR THE PIPE ENTRY
The formulae for the head loss to friction considered in the preceding
sections are all applicable, strictly, only to the fully developed flow found
some distance downstream from the entry to the pipe or from other causes
of disturbance to the flow. Near the entry to the pipe the variation of
velocity across the section differs from the fully developed pattern, and
gradually changes until the final form is achieved.
Suppose that fluid from a large reservoir steadily enters a circular pipe
through a smooth, bell-mouthed entry as indicated in Fig.7.23. At first all the
particles – expect those in contact with the wall – flow with the same velocity.
That is, the velocity profile is practically uniform across the diameter as
shown at the left of the diagram. The effect of friction at the wall, however,
is to slow down more and more of the fluid near the wall, so forming the
boundary layer which increases in thickness until, ultimately, it extends to
the axis of the pipe. Since the total flow rate through any section of the pipe is
the same, the velocity of the fluid near the axis must increase to compensate
for the retardation of fluid near the walls. The shape of the velocity profile
thus changes until its final fully developed form – for laminar or turbulent
flow according to the Reynolds number – is achieved.
Theoretically, an infinite distance is required for the final profile to be
attained, but it is usual to regard the flow as fully developed when the velocity
on the axis of the pipe is within 1% of its ultimate value. Figures given
for the entry length required to establish fully developed laminar flow vary
somewhat, but a simple expression derived by H. L. Langhaar is 0.057(Re)d,
where Re = ud/ν and u = mean velocity Q/A. So for Re = 2000, the highest
value at which laminar flow can be counted on, the entry length is about 114
times the diameter. For turbulent flow the final state is reached sooner, the
entry length is less dependent on Reynolds number and a value of about 50
times the diameter is common for smooth pipes. If, however, the entry is
Conditions near the pipe entry 283
Fig. 7.23 Growth of
boundary layer in a pipe
(not to scale). (a) Laminar
flow. (b) Turbulent flow.