Hydraulically smooth boundary turbulent flow where the velocity distribution, mean velocity
(1)
and resistance to flow are independent of the boundary roughness of the bed but depend
on fluid kinematic viscosity. Then with
δ′
11.6υ
δ′ =
we find y′ =
(2.76)
107
τo
ρ
(2)
Hydraulically rough boundary turbulent flow where velocity distribution, mean velocity and
resistance to flow are independent of viscosity and depend entirely on the boundary
roughness. For this case, y′ = ks/30.2 where ks is the height of the roughness element.
Transition where the velocity distribution, mean velocity and resistance to flow depend on
(3)
both fluid viscosity and boundary roughness. Then
δ′
k
< y′ < s
(2.77)
107
30.2
The boundary roughness effects can be merged into one equation by using y′ = ks/(30.2X) where
X is determined from Figure 2.10. As a result, the velocity distribution v, mean velocity V, and
resistance to flow equations can be written in the following dimensionless form which is related to
the above flow types by Figure 2.10.
Xy
Xy
v
= 5.75 log 30.2
= 2.5 In
30.2
(2.78)
ks
ks
V*
and
Xy
Xy o
V
C
= 5.75 log 12.27 o = 2.5 In 12.27
=
(2.79)
ks
ks
V*
g
Note that any system of units can be used as long as yo and ks (and V, v and V*) have the same
dimensions. The symbols of Equations 2.78 and 2.79 denote:
X
=
Coefficient given in Figure 2.10
=
Height of the roughness elements, for sand channels
ks
v
=
Local mean velocity at depth y
=
Depth of flow
yo
V
=
Depth-averaged velocity
=
V*
Shear velocity τ o / ρ which for steady uniform flow is gRS f
=
Shear stress at the boundary and for steady uniform flow the average is
τo
τ o = γRS f
(2.80)
R
=
Hydraulic radius, equal to the cross-sectional area A divided by the
wetted perimeter P
2.19
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