Rodney J. Sobey
Lateral structure of uniform flow
Journal of Hydroinformatics
(ii) no longitudinal variation in flow,
f 2P e
(iii) constant channel resistance parameters f9 and e.
The place of f9 and e in the depth-integrated description is
Equation (7) together with no flow boundary bound-
taken by f alone in the cross-section-integrated 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 cross-section-integrated
The residual terms in Equation (8) become
and to depth-integrated descriptions of the channel
∂2qx 8 zqxzqx
e 2 f
gdnS0 f 2 0
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 second-order 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.
Suitable boundary conditions on Equation (14) are no
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
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 depth-integrated flow
is achieved by computing A and P from the ordinary
Q *qx y dy
that identifies the cross-section-integrated discharge Q. Q
is a given parameter.
The relationship between the cross-section-integrated
friction factor f, the depth-integrated 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,