ABSTRACT
The basic equations for the one-dimensional analysis of unsteady flow in open channels are the continuity equation and the equation for the conservation of linear momentum. The continuity equation can be written as:
∂Q
∂x + ∂A
∂t = r(x, t) (8.1)
where Q is the discharge, A the area of flow, and r(x, t) the rate of lateral inflow. The above equation is a linear one and consequently poses no difficulties for us in this regard. The second equation used in the onedimensional analysis of unsteady free-surface flow is that based on the conservation of linear momentum, which reads
∂y
∂x + u
g
∂u
∂x + 1
g
∂u
∂t = S0 − Sf − ugy r(x, t) (8.2)
where y is the depth of flow, u is the mean velocity, S0 is the bottom slope and Sf is the friction slope. This dynamic equation is highly nonlinear. Consequently, it is not possible to obtain closed-form solutions for problems governed by equations (8.1) and (8.2). The extent of the
non-linearity can be appreciated if we examine the special case of discharge in an infinitely wide channel with Chezy friction, in which case the continuity equation takes the form
∂q
∂x + ∂y
∂t = r(x, t) (8.3)
where q= uy is the discharge per unit width; and the momentum equation takes the form
∂y
∂x + u
g
∂u
∂x + 1
g
∂u
∂t = S0 − u
C2y − u
gy r(x, t) (8.4a)
which appears to be non-linear in only three of its six terms. However, if we multiply through by gy, five of the six terms of the equation are seen to be non-linear. If, in addition, we express u in terms of q and y, which are the dependent variables in the linear continuity equation, we obtain
(gy3 − q2)∂y ∂x
+ 2qy ∂q ∂x
+ y2 ∂q ∂t
= S0gy3 − g C2
q2 (8.4b)
in which every term is seen to be highly non-linear (see Appendix D). On the basis of the above equations we would expect such processes
as flood routing, which is a case of unsteady flow with a free surface, toUnsteady flow with a free surface be characterised by highly non-linear behaviour. However, practically all
the classical methods of flood routing commonly used in applied hydrology are linear methods. In contrast most of the methods used in applied hydrology to analyse overland flow (which is another case of unsteady free surface flow) are non-linear in character.
