NUMERICAL ALGORITHM

|

104

Rodney J. Sobey

Lateral structure of uniform flow

Journal of Hydroinformatics

06.2

2004

∂qx

dqx

(ii) no longitudinal variation in flow,

dy.

(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

resistance, respectively.

∂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 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.

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 depth-integrated 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 cross-section-integrated discharge Q. Q

D

2

is a given parameter.

dh

dY2

OE1 S

(19)

The relationship between the cross-section-integrated

dy

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,