74
2 Water at Rest and in Motion
Note the different logarithms used in the above relationships. The equations
above for turbulent flow are valid for situations where the laminar shearing
stresses can be neglected in comparison with the turbulent stresses. Very close
to the pipe wall, a laminar sublayer exists in which laminar stresses dominate
and, as was shown for a flat plate (Eq. 2.51), velocity changes linearly with
distance from the wall, i.e.:
u(z)
for
v
u.z
-<5.
v
(2.112)
In an intermediate range 5 < u.z/v < 70, both contributions, laminar and
turbulent, are of the same order of magnitude.
Assuming a velocity distribution as in Eq. (2.111), and integrating over a
pipe cross-section, an average velocity, ii, results as follows:
ii
(u.D)
-=5.7510g lO -
+1.75.
u.
2v
(2.113)
Most pipes and ducts in marine organisms as well as those made by humans
can not be regarded as smooth, at least at higher Reynolds numbers. Surface
roughness induces resistance to flow and the effect of the roughness elements
depends on the relationship between the height of elements and the laminar
sublayer. When the sublayer is so thick that it covers the roughness elements,
the surface can be considered as hydrodynamically smooth. However, if the
size of the roughness elements are large compared with the laminar sublayer,
the surface is completely rough and the effect of viscosity can be neglected.
To introduce the roughness effect into the formula for velocity distribution in
a pipe, we present Eq. (2.111) in more general form:
u(z)
(z )
(U.k s )
- - = 5.7510g 10 -
+ B - - ,
u.
ks
v
(2.114)
where ks is the roughness size; factor B depends on the so called 'shear Reynolds
number', u.ks/v. A notation ks was introduced by Nikuradse who used closely
packed sand grain roughness elements in his study of flow resistance in pipes.
In general, it is necessary to consider three regions for the factor B (Schlichting, 1960):
1. hydraulically smooth regime when 0 ~ u.ks/v ~ 5; the size of the
roughness elements is very small, covered totally by the laminar sublayer
(Fig. 2.17). The factor B = 5.5 + 2.51n (u.ks/v) (Fig. 2.35);
2. transition regime when 5 ~ u.ks/v ~ 70. Some of the roughness elements
extend outside the laminar sublayer and contribute some resistance. The
value of B in the transition region is shown in Fig. 2.35;
2 Water at Rest and in Motion
Note the different logarithms used in the above relationships. The equations
above for turbulent flow are valid for situations where the laminar shearing
stresses can be neglected in comparison with the turbulent stresses. Very close
to the pipe wall, a laminar sublayer exists in which laminar stresses dominate
and, as was shown for a flat plate (Eq. 2.51), velocity changes linearly with
distance from the wall, i.e.:
u(z)
for
v
u.z
-<5.
v
(2.112)
In an intermediate range 5 < u.z/v < 70, both contributions, laminar and
turbulent, are of the same order of magnitude.
Assuming a velocity distribution as in Eq. (2.111), and integrating over a
pipe cross-section, an average velocity, ii, results as follows:
ii
(u.D)
-=5.7510g lO -
+1.75.
u.
2v
(2.113)
Most pipes and ducts in marine organisms as well as those made by humans
can not be regarded as smooth, at least at higher Reynolds numbers. Surface
roughness induces resistance to flow and the effect of the roughness elements
depends on the relationship between the height of elements and the laminar
sublayer. When the sublayer is so thick that it covers the roughness elements,
the surface can be considered as hydrodynamically smooth. However, if the
size of the roughness elements are large compared with the laminar sublayer,
the surface is completely rough and the effect of viscosity can be neglected.
To introduce the roughness effect into the formula for velocity distribution in
a pipe, we present Eq. (2.111) in more general form:
u(z)
(z )
(U.k s )
- - = 5.7510g 10 -
+ B - - ,
u.
ks
v
(2.114)
where ks is the roughness size; factor B depends on the so called 'shear Reynolds
number', u.ks/v. A notation ks was introduced by Nikuradse who used closely
packed sand grain roughness elements in his study of flow resistance in pipes.
In general, it is necessary to consider three regions for the factor B (Schlichting, 1960):
1. hydraulically smooth regime when 0 ~ u.ks/v ~ 5; the size of the
roughness elements is very small, covered totally by the laminar sublayer
(Fig. 2.17). The factor B = 5.5 + 2.51n (u.ks/v) (Fig. 2.35);
2. transition regime when 5 ~ u.ks/v ~ 70. Some of the roughness elements
extend outside the laminar sublayer and contribute some resistance. The
value of B in the transition region is shown in Fig. 2.35;