Fig.D-14
Flow development:
It is an important - and often forgotten - fact that the formulas as shown in Fig.D-12 for the friction loss coefficient are valid only for fully developed flow. The flow development requires sufficiently long pipe of equal cross section located upstrem from the investigated pipeline element. Computation of the development is a rather difficult two- or three-dimensional problem (requiring computer solution), which cannot be treated in the present text. Fig.D-15 below presents the value of the drag coefficient as a function of relative length of the pipe, if the entrance velocity profile is a rectangular one. Two typical velocity profiles (results of numerical computations) in the development region are shown in Fig.D-16. Immediately downstream from the entrance, flowner the pipe wall has a character of a boundary layer (cf. Fig.I-12). Thickness of boundary layer increases with increasing distance from the entrance. Note that velocity profiles near the entrance (Fig.D-16) exhibit larger slope .
Fig.D-15 Results
of numerical solutions
of laminar (low Re)
flow development:
Dependence of the
loss (drag) coef-
ficient on the rela-
tve pipe length.
Readings from this
curve may be used
for estimates
of hydraulic loss
in short pipes.
It follows from the Newton law in this region the tangential (shear) stress on the wall is higher. As a consequence, drag coefficient in Fig.D-15 increases there faster with downstream than the increase which corresponds to the Hagen-Poiseuille law in the fully developed flow.

Fig.D-16
It is a typical property of turbulent flow that because of more intensive mixing it develops quite fast. It requires for full developement a distance downstream from a disturbance (e.g. an orifice, Fig.D-18) equal to about 1O . Laminar flow requires a much longer development distance. As a good estimate it is possible to use for the development length the Boussinesq formula:

Note that there is the quadratic loss caused by the development (hatched vertically in Fig.D-15) which remains as a consequence of the intitial development region even after the flow has fully developed. This quadratic component is to be added to the frictional loss - calculations with, of course, it follow the pattern for local losses.


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This is page Nr. D08 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS"
Any comments and suggestions concerning this text may be mailed to the author to his address tesar@fsid.cvut.cz

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