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103
Rodney J. Sobey
Lateral structure of uniform flow
Journal of Hydroinformatics
06.2
2004
SF
GD S D
characterize uniform flow by the cross-section area at
8 zqzqy
∂qx
∂qy
qy
∂
∂
2
f
∂
(9)
e
e
uniform flow, the normal area An. An might be estimated
h h2
∂x
∂y
∂x
∂y
∂y
implicitly from Equation (5). But again there is the
additional assumption that the lateral water surface pro-
from which both Coriolis accelerations and surface wind
file is horizontal in the identification of the local elevation
stresses have been omitted. The friction model becomes
of the water surface and also in the identification of the
lateral locations of both the left and right banks.
8 zqzqa
t0a f r
, a x,y
(10)
The lateral structure of the water surface and the
h h2
lateral location of the left and right banks are issues that
are directly addressed in the following consideration of a
locally, in which the DarcyWeisbach friction factor f ′ for
depth-integrated description of uniform flow. The cross-
this depth-integrated description is consistent (but not
section-integrated description precludes any prediction of
identical; see Equation (17)) with the f for the cross-
the lateral (cross-stream) structure of h, Q and dn or An.
section-integrated model (Equation (3)).
Lateral momentum transfer has been modelled as
SF
GD
h
∂qb qa
∂
t ab ru au b dz
*
∂ re
∂
(11)
∂xb
∂xb
∂xa ∂xb
DEPTH-INTEGRATED DESCRIPTION
h
The lateral flow structure of nearly horizontal flow in
by analogy with the general form of Newton's law of
natural channels is retained by including the lateral pos-
viscosity; tab and ru9 u9 are the local viscous and
ition y along with the longitudinal position x and time t
a b
Reynolds stresses in the horizontal plane and e is the
as the independent variables. The dependent variables
horizontal eddy viscosity. In the cross-section-integrated
become water surface elevation h(x,y,t) and the depth-
description, both vertical and horizontal momentum
integrated flows
transfer are scaled by a constant f. In the depth-integrated
description, vertical momentum transfer is scaled by f9 and
h
h
horizontal momentum transfer is scaled by e. Consistent
*uxdz,
*uydz
qx x, y, t
qy x, y, t
(6)
with the practice in cross-section-integrated descriptions,
h
h
both f9 and e are assumed constant in the depth-integrated
description.
in which the bed elevation is at zb = - h(x,y) in common
The integral parameter Q is
practice, and (ux,uy) are the local velocity components.
The depth-integrated mass and momentum conservation
Q *qxdy
(12)
are
A
qx qy
∂h
∂
∂
0
where A was defined in Equation (2), except for the
(7)
∂t ∂x ∂y
sign change convention in the representation of the bed
elevation.
S D S D
S D
q2
∂qx
qxqy
∂qx
∂h
∂
x
A definition of uniform flow for depth-integrated
∂
gh h
∂
2
e
∂x h h
∂y h h
∂t
∂x
∂x
∂x
descriptions of channel flow that is consistent with
S F GD
8 zqzqx
∂qx qy
Equation (5) for a cross-section-integrated description
∂ e
f
∂
(8)
h h2
∂y
∂y ∂x
would have
S D S D
2
qxqy
qy
∂qy
∂h
qx
qy
∂h
∂
gh h
∂
∂ e ∂
0
(i) steady flow,
∂x h h
∂y h h
∂t
∂y
∂t ∂t
∂t
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