Geoscience Reference
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in which here
w
u
)
ν
∂
u
τ
zx
=
ρ
z
−
(
(5.29)
∂
is the shear stress and in which the subscript
z
denotes the direction normal to the plane
on which the stress acts and the subscript
x
denotes the direction of this stress itself.
Equation (5.28) is readily integrated with the boundary condition that the shear stress at
the water surface is zero. If
τ
0
is the shear stress at the bottom wall where
z
=
0, this
integration yields
τ
0
γ
S
f
=
(5.30)
h
u
2
or, in terms of the friction velocity
S
f
=
gh
. Unfortunately, an expression for
S
f
in terms of the shear stress at the wall is not of much help at this point. Instead, to
be able to solve the shallow water equations an expression is required in terms of the
main dependent variables, namely
h
and
V
. Hence, to repeat briefly, first a relationship is
obtained relating the slope
S
0
with the flow variables
h
and
V
(or the analogous variables,
such as
A
c
and
Q
) under uniform steady conditions. In accordance with Equation (5.27),
this relationship obtained for
S
0
is then used in the shallow water momentum equation
to parameterize
S
f
in terms of the same flow variables. In the following two sections,
relationships are presented for laminar and turbulent flow.
∗
/
5.3.1
Laminar flow
The case of two-dimensional steady uniform flow, that is plan-parallel flow down a plane
surface, can be solved exactly for laminar conditions, when both
u
and
are zero. For
such conditions all terms in Equation (5.14) (or (5.15)) are zero, except the first and third
on the right-hand side. Integrating these remaining two terms twice (with the conditions
that
w
∂
u
/∂
z
=
0at
z
=
h
and that
u
=
0at
z
=
0) and making use of Equation (5.27) or
−
sin
θ
=
S
f
, one obtains the velocity profile
gS
f
ν
z
2
u
=
(
hz
−
/
2)
(5.31)
After normalization with the maximal velocity
u
h
at
z
=
h
, this velocity profile can be
written as
u
h
)
2
, which is illustrated in Figure 5.3. Integrating (5.31)
over
z
, according to (5.8), one obtains the average velocity,
/
u
h
=
2(
z
/
h
)
−
(
z
/
gS
f
h
2
3
V
=
(5.32)
ν
In the absence of lateral inflow by precipitation the applicability of Equation (5.32)
depends mainly on the Reynolds number Re
); as Re increases the flow will
become turbulent, but the transition may also depend on the smoothness of the surface,
the uniformity and stationarity of the flow, and possibly other factors. Experimentally,
Equation (5.32) has been found to fail for Re values as low as 300, and for flows over
smooth surfaces without lateral inflow it has also been observed to be valid up to Re
≡
(
Vh
/ν
=
1000 (see, for example, Chow, 1959; Woo and Brater, 1961). An upper limit of
