Geoscience Reference
In-Depth Information
may become plane again. Although the bed is plane, sediment particles are still
moving on the bed.
(e) Further increase in flow strength will induce anti-dunes. In the anti-dune stage,
the flow Froude number usually is larger than 1, and the sediment movement
is strongly influenced by the free surface flow. While water and sediment move
downstream, the bed and water surface waves actually propagate upstream in
phase. They may break like sea surfs or subside as standing waves.
(f) Chutes and pools occur at relatively large slopes, with high flow velocities and
sediment concentrations. Sediment particles move intensively in this stage.
The stationary flat bed, ripples, and dunes are usually called the lower flow regime,
while the moving plane bed, anti-dunes, and chutes/pools are called the upper flow
regime. Anti-dunes and chutes/pools are mostly observed in laboratory flumes but
rarely found in natural rivers.
In addition, other large-scale bed forms, such as point bars, alternate bars, and
islands, often exist in natural rivers. They are usually generated by channel meandering,
expansion, and contraction as well as tributary confluence. Their dimensions are thus
related to channel width, depth, curvature, etc.
3.3.2 Division of grain and form resistances
For a channel bed with sand grains and bed forms (such as sand ripples and dunes),
the bed shear stress,
τ
b
, may be divided into the grain (skin or frictional) shear stress,
τ
b
, and the form shear stress,
τ
b
:
τ
b
=
τ
b
+
τ
b
(3.48)
The bed shear stress is usually calculated by
τ
b
=
γ
R
b
S
f
(3.49)
where
R
b
is the hydraulic radius of the channel bed.
Einstein (1942) suggested the division of the hydraulic radius
R
b
into two parts
R
b
and
R
b
, corresponding to the grain and form roughnesses, and determined the grain
and form shear stresses as
τ
b
=
γ
R
b
S
f
,
τ
b
R
b
S
f
=
γ
(3.50)
R
2
/
3
b
S
1
/
2
f
R
2
/
3
b
S
1
/
2
f
n
, and
U
The assumption of equal velocity:
U
=
/
n
,
U
=
/
=
R
2
/
3
b
S
1
/
2
2
. Here,
U
is the average
flow velocity,
n
is the Manning roughness coefficient of channel bed, and
n
and
n
are the Manning coefficients corresponding to the grain and form roughnesses, respec-
tively. Therefore, from these two relations and Eqs. (3.49) and (3.50), the following
relations for the grain and form shear stresses are obtained:
n
yields
R
b
=
n
/
3
/
2
and
R
b
=
n
/
3
/
f
/
R
b
(
n
)
R
b
(
n
)
n
n
3
/
2
n
n
3
/
2
τ
b
=
τ
b
τ
b
,
=
τ
b
(3.51)
