Mechanics of Fluids

250 Flow and losses in pipes and fittings
Fig. 7.3
in the form shown in Fig. 7.3. One of the first to present results in this way
was Sir Thomas E. Stanton (1865–1931) who, with J. R. Pannell, conducted
experiments on a number of pipes of various diameters and materials, and
with various fluids. Figure 7.3 is based on results obtained by the German
engineer Johann Nikuradse using circular pipes that had been artificially
roughened using sand grains.
At the left-hand side of the diagram is a single line corresponding to
Reynolds numbers ud/v less than 2000. In this range the flow is laminar and
is governed by Poiseuille’s equation (6.7). A rearrangement of that equation
gives
h f = �p∗
ϱg
8Qlµ
=
πR4 8ulµ
=
ϱg R2ϱg since the mean velocity u = Q/πR 2 . But a rearrangement of eqn 7.1 gives
f = (2gd/4lu 2 )h f and so, for laminar flow,
f = 2gd
4lu2 8ulµ
(d/2) 2 16µ 16
= =
ϱg udϱ Re
(7.4)
With the logarithmic scales used in Fig. 7.3, eqn 7.4 is represented by a
straight line. Experimental results confirm the equation and the fact that
laminar flow is independent of the roughness of the pipe walls (unless the
roughness is so great that the irregularities constitute an appreciable change
of diameter).
For Reynolds numbers greater than about 2000, that is, for turbulent flow,
the flow does depend on the roughness of the pipe, and different curves are
obtained for different values of the relative roughness. In Nikuradse’s experiments,
grains of sand of uniform size were glued to the walls of pipes of
various diameters, which were initially very smooth. Thus a value of the
relative roughness was readily deduced since k could be said to correspond
to the diameter of the sand grains. Such uniform artificial roughness is, of