2.7 Laminar and Turbulent Flow in Ducts
73
in which w is the width of the plates. Thus, average and maximum velocities
are:
_ b,.pD2
u = - -
12J..tl '
(2.109)
Now, the maximum velocity is only one and a half times the average. There
are quite a few biological situations which involve closely spaced and parallel
flat plates such as, for example, flow through the gill of fish. This and other
internal flows in marine organisms are discussed in more detail in Part III of
this book.
At the end of this section we note that the above relationships are valid for
laminar flow when the Reynolds number is below 2000 (see Sect. 2.4) or so,
depending on the smoothness of the pipe or plate walls and entry upstream.
The threshold Reynolds number Re ~ 2000 is about 0.4 m/s. Within organisms
these limits are rarely exceeded and the internal flow is mostly laminar.
2.7.3 Turbulent Flow in Ducts
Although laminar flow dominates within marine organisms, there are many
situations when organisms find themselves in a turbulent environment. As the
flow becomes turbulent, the analytical determination of the velocity distribution is not possible. Within a pipe flow, momentum is transported across the
pipe resulting in a velocity along the axis which is less than twice the average
speed of laminar flow. In contrast to laminar flow, the roughness of the inside
wall of a pipe plays a decisive role in the determination of the pipe's resistance.
To derive the velocity distribution in a pipe, we assume that velocity, u,
at some distance from the pipe wall depends on the tangential stress, TO, at
the wall, viscosity, J..t, and density, p, of the fluid. Dimensional analysis as
well as a detailed examination of experimental data and logical arguments
given in Sect. 2.5.4 suggest that the velocity profile obeys the logarithmic law
(see Fig. 2.33). An advantage of such laws, as compared with the l/nthpower law (see Eq. 2.56) consists of its being an asymptotic expression for
very large Reynolds numbers. Experiments conducted by Nikuradse yields the
following form of the velocity distribution for very large Reynolds numbers and
for smooth pipes (Schlichting, 1960):
u(z) = 2.51n (u*z) + 5.5 for
u'
v
or:
u(Z)
(u*Z)
-
= 5.7510g 10 -
+ 5.5
u*
v
u*Z
->70,
v
for
U.Z
->70.
v
(2.110)
(2.111)
73
in which w is the width of the plates. Thus, average and maximum velocities
are:
_ b,.pD2
u = - -
12J..tl '
(2.109)
Now, the maximum velocity is only one and a half times the average. There
are quite a few biological situations which involve closely spaced and parallel
flat plates such as, for example, flow through the gill of fish. This and other
internal flows in marine organisms are discussed in more detail in Part III of
this book.
At the end of this section we note that the above relationships are valid for
laminar flow when the Reynolds number is below 2000 (see Sect. 2.4) or so,
depending on the smoothness of the pipe or plate walls and entry upstream.
The threshold Reynolds number Re ~ 2000 is about 0.4 m/s. Within organisms
these limits are rarely exceeded and the internal flow is mostly laminar.
2.7.3 Turbulent Flow in Ducts
Although laminar flow dominates within marine organisms, there are many
situations when organisms find themselves in a turbulent environment. As the
flow becomes turbulent, the analytical determination of the velocity distribution is not possible. Within a pipe flow, momentum is transported across the
pipe resulting in a velocity along the axis which is less than twice the average
speed of laminar flow. In contrast to laminar flow, the roughness of the inside
wall of a pipe plays a decisive role in the determination of the pipe's resistance.
To derive the velocity distribution in a pipe, we assume that velocity, u,
at some distance from the pipe wall depends on the tangential stress, TO, at
the wall, viscosity, J..t, and density, p, of the fluid. Dimensional analysis as
well as a detailed examination of experimental data and logical arguments
given in Sect. 2.5.4 suggest that the velocity profile obeys the logarithmic law
(see Fig. 2.33). An advantage of such laws, as compared with the l/nthpower law (see Eq. 2.56) consists of its being an asymptotic expression for
very large Reynolds numbers. Experiments conducted by Nikuradse yields the
following form of the velocity distribution for very large Reynolds numbers and
for smooth pipes (Schlichting, 1960):
u(z) = 2.51n (u*z) + 5.5 for
u'
v
or:
u(Z)
(u*Z)
-
= 5.7510g 10 -
+ 5.5
u*
v
u*Z
->70,
v
for
U.Z
->70.
v
(2.110)
(2.111)