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105
Rodney J. Sobey
Lateral structure of uniform flow
Journal of Hydroinformatics
06.2
2004
with cubic spline interpolation providing smooth and
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.
tion will not be improved by the spline interpolation.
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 error-correcting, adaptive
involved in the definition of Equation (22b, c). Evaluation
step size (mixed fourth- and fifth-order 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
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 first-order 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
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
error-correcting adaptive step size RungeKutta code as
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