


104
Rodney J. Sobey
Lateral structure of uniform flow
Journal of Hydroinformatics
06.2
2004
y
*Sh
D
∂qx
qy
yR
8 Q2
2
8
qx
dqx
∂
0, and
(ii) no longitudinal variation in flow,
yR
gAS0
f 2P e
f
(17)
dy.
∂x
∂x
h
A
dy
L
yL
The place of f9 and e in the depthintegrated description is
Equation (7) together with no flow boundary bound
taken by f alone in the crosssectionintegrated descrip
ary conditions at the channel sides gives qy[0. Equation
tion. Equation (17) is, in fact, two equations, relating the
(9) gives ∂h/∂y = 0, so that h = h(x), a function of x only.
constant water surface slope to crosssectionintegrated
The residual terms in Equation (8) become
and to depthintegrated descriptions of the channel
∂2qx 8 zqxzqx
∂h
gh h
e 2 f
(13)
0
h h2
∂x
∂y
or
NUMERICAL ALGORITHM
q2
d2qx
Stage 1
x
gdnS0 f 2 0
(14)
e
2
dy
8dn
For a natural channel, the depth varies across the channel,
but the water surface elevation remains horizontal across
in which both dn = (h + h) and qx are functions of y. Given
the channel. The flow cross section at uniform flow can be
dn(y), Equation (14) is a secondorder ordinary differential
characterized by the elevation Dh of the water surface (or
equation for the lateral velocity profile qx(y). It is in the
equivalently by the flow cross section). Equation (17a) (or
familiar form of a boundary layer equation, requiring no
(5)) becomes the implicit algebraic equation
slip at the banks and describing the lateral diffusion of
boundary shear from the banks.
Q2
Suitable boundary conditions on Equation (14) are no
f
gAS0
f Dh
(18)
P
8 A2
flow at yL, the left hand channel bank and at yR, the right
hand channel bank:
with A and P defined as in Equation (2) but with zb =  h.
A number of numerical algorithms (bisection, Newton
qx(yL) = 0 qx(yR) = 0
(15)
Raphson, secant method, etc) are suitable for implicit
algebraic equations in a single unknown.
together with the integral condition
Numerical precision that is consistent with the bal
ance of the subsequent discussion of depthintegrated flow
is achieved by computing A and P from the ordinary
yR
differential equations
Q *qx y dy
(16)
yL
dY1
h Dh
dy
that identifies the crosssectionintegrated discharge Q. Q
D
2
is a given parameter.
dh
dY2
OE1 S
(19)
The relationship between the crosssectionintegrated
dy
dy
friction factor f, the depthintegrated friction factor f9
and the horizontal eddy viscosity e is established from
in which dY1/dy is dA/dy and dY2/dy is dP/dy. Both h
Equations (5) and (14). Integrating (14) over the cross
and dh/dy are required as continuous functions of y.
section and comparing terms gives
Bathymetry specified as discrete (yi,hi) pairs is anticipated,