44
2 Water at Rest and in Motion
if the surface is rough. For Rex < 10 5 , the laminar layer is stable; but at Rex
near 2 x 10 5 , it is difficult to prevent transition.
After transition, the main part of the flow in the layer is turbulent, except
immediately at the boundary where the turbulent fluctuations are suppressed
by the presence of the wall. Therefore, the flow in vicinity of the solid surface
can be divided into three sublayers: the laminar sublayer, the turbulent boundary layer, and the free stream. There are no sharp boundaries between each
layer, so the description of the flow in each particular layer should be treated
to some extent as qualitative only. We have to note that when the surface is
rough, the laminar sublayer may be destroyed by the surface roughness.
Within a laminar sublayer (see Fig. 2.17), the velocity is controlled by the
viscous stress, T, as was given by Eq. (1.2), i.e. T = f.1 (du/dz) = pwl/ (du/dz).
For a very thin laminar sublayer, stress, T, is approximately constant and equal
to its value at the wall TO. Then, an integration of Eq. (1.2) gives:
u 2
u(z) = ~z,
1/
in which:
_ (TO) 1/2
U. -
Pw
(2.51 )
(2.52)
Equation (2.51) shows that in the laminar sublayer (a term 'viscous sublayer'
is sometimes used), velocity u is a linear function of the distance z from the
boundary. The quantity u. is called the friction velocity because it has
dimension (m/s), although it is not actually a flow velocity. Rather, it is
a measure of the turbulent fluctuations in the velocity near the wall. The
magnitude of u. can only be estimated in an indirect way, and a more detailed
discussion of velocity u. will be given later. For now we only note that when, for
example, wind blows over a rough sea surface, the ratio of the friction velocity
to the mean wind velocity at a standard height of 10 m above sea level is of
the order of 3 to 5% (Massel, 1996a). On the other hand, for water flow over
the smooth bottom of a basin 10 m deep, the u. velocity is about 3% of the
mean flow velocity, while for a rougher bottom it varies from about 5% to 15%
of the mean velocity (Vogel, 1994).
The laminar sub layer is at most only a few millimeters thick. However, its
influence on the flow in the boundary layer is quite considerable. Whether
the laminar sublayer exists or not depends on the free stream velocity and the
bottom elements size (Fig. 2.17). When the bottom element diameter is less
than one-third of the thickness of the laminar sublayer, the sublayer remains
intact (Fig. 2.17a). The main turbulent flow above is not affected at all by
the presence of the bottom elements. When bottom elements are greater than
about one-third of the laminar sublayer thickness, the elements begin to disrupt
the flow in the sublayer and once these elements reach about seven times the
2 Water at Rest and in Motion
if the surface is rough. For Rex < 10 5 , the laminar layer is stable; but at Rex
near 2 x 10 5 , it is difficult to prevent transition.
After transition, the main part of the flow in the layer is turbulent, except
immediately at the boundary where the turbulent fluctuations are suppressed
by the presence of the wall. Therefore, the flow in vicinity of the solid surface
can be divided into three sublayers: the laminar sublayer, the turbulent boundary layer, and the free stream. There are no sharp boundaries between each
layer, so the description of the flow in each particular layer should be treated
to some extent as qualitative only. We have to note that when the surface is
rough, the laminar sublayer may be destroyed by the surface roughness.
Within a laminar sublayer (see Fig. 2.17), the velocity is controlled by the
viscous stress, T, as was given by Eq. (1.2), i.e. T = f.1 (du/dz) = pwl/ (du/dz).
For a very thin laminar sublayer, stress, T, is approximately constant and equal
to its value at the wall TO. Then, an integration of Eq. (1.2) gives:
u 2
u(z) = ~z,
1/
in which:
_ (TO) 1/2
U. -
Pw
(2.51 )
(2.52)
Equation (2.51) shows that in the laminar sublayer (a term 'viscous sublayer'
is sometimes used), velocity u is a linear function of the distance z from the
boundary. The quantity u. is called the friction velocity because it has
dimension (m/s), although it is not actually a flow velocity. Rather, it is
a measure of the turbulent fluctuations in the velocity near the wall. The
magnitude of u. can only be estimated in an indirect way, and a more detailed
discussion of velocity u. will be given later. For now we only note that when, for
example, wind blows over a rough sea surface, the ratio of the friction velocity
to the mean wind velocity at a standard height of 10 m above sea level is of
the order of 3 to 5% (Massel, 1996a). On the other hand, for water flow over
the smooth bottom of a basin 10 m deep, the u. velocity is about 3% of the
mean flow velocity, while for a rougher bottom it varies from about 5% to 15%
of the mean velocity (Vogel, 1994).
The laminar sub layer is at most only a few millimeters thick. However, its
influence on the flow in the boundary layer is quite considerable. Whether
the laminar sublayer exists or not depends on the free stream velocity and the
bottom elements size (Fig. 2.17). When the bottom element diameter is less
than one-third of the thickness of the laminar sublayer, the sublayer remains
intact (Fig. 2.17a). The main turbulent flow above is not affected at all by
the presence of the bottom elements. When bottom elements are greater than
about one-third of the laminar sublayer thickness, the elements begin to disrupt
the flow in the sublayer and once these elements reach about seven times the