


105
Rodney J. Sobey
Lateral structure of uniform flow
Journal of Hydroinformatics
06.2
2004
yR
0 *qx y dy Q
z,e,f
continuous estimates for h and dh/dy as required.
2
Bathymetric resolution must be adequate to follow the
yL
*Sh y
D
yR
significant detail of the cross section. Inadequate resolu
8
qx y
0 gAS0
Z2 yR
f
z
f3 z,e,f
(22)
dy.
Dh
yL
Initial conditions are Y1 = Y2 = 0 at y = yL. Integration
from yL to yR gives A = Y1(yR) and P = Y2(yR). Excellent
This system is nonlinear through Equation (21b), which is
precision is achieved with an errorcorrecting, adaptive
involved in the definition of Equation (22b, c). Evaluation
step size (mixed fourth and fifthorder RungeKutta)
code for numerical integration.
Equations (21) with initial conditions Z1 = 0 and Z2 = z at
Estimation of yL and yR is formulated as the implicit
yL; z is an unknown. As the integrals in Equations (22b, c)
algebraic equation
must also be evaluated numerically, consistent numerics is
assured by redefining the ODE system as
fL,R(y) = 0 = h(y) + Dh.
(20)
Given Dh and the bathymetry h(y), there are two solutions
dZ1
2
Z
to Equation (20), respectively yL at the left bank and yR at
dy
e
the right bank. Equations (20) may be solved by the same
2
Z1
dZ2
g h h S0
f
numerical algorithm adopted for Equation (18).
8 h Dh 2
dy
This Stage 1 algorithm requires knowledge of the
dZ3
Z1
channel geometry, together with the assigned Q, S0 and f.
dy
A successful numerical solution provides Dh, yL, yR, A
2
dZ4
1
Z
and P.
(23)
h h2
dy
where
Stage 2
The lateral boundary layer Equation (14) is equivalent to
y
y
q2
the simultaneous firstorder ODE system:
x
Z3 *qxdy Q
Z4 *
(24)
and
dy
h Dh 2
yL
yL
dZ1
2
Z
dy
e
with initial conditions Z1 = 0, Z2 = z, Z3 =  Q and Z4 = 0
2
8Z1
dZ2
f 2
gdnS0
at yL. The simultaneous implicit algebraic equations
(21)
dy
dn
become
where Z1 is qx and Z2 is edqx/dy. But numerical integration
0 Z1 yR
f1 z,e,f
of Equations (21) requires initial conditions at a known y
0 Z3 yR
f2 z,e,f
on both Z1 and Z2. Z1(yL) = 0, but Z2(yL) = z is unknown.
8
The second boundary condition is Z1(yR) = 0. Locally,
0 gAS0
Z2 yR
f Z4 yR .
z
z,e,f
(25)
1
dn(y) = h(y) + Dh, in which Dh is known from Stage 1.
The problem is formulated as the simultaneous
implicit algebraic equations
Newton's method (Press et al. 1992) is a suitable choice
for the numerical solution of Equations (25), with a
0 qx yR
f
f1 z,e,f
errorcorrecting adaptive step size RungeKutta code as