Free surface flow

We can develop a theory of flood propagation from the Saint Venant equations (Abbott, 1979; Cunge et al., 1980) for gradually varying flow in open channels:

Conservation of momentum:

(1)

Conservation of mass:

(2)

The Saint-Venant equations consist of a number of terms, especially thedynamic equation. This equation can be simplified in certain circumstancesthat are particular to floods in a river. To do this we make a scale analysis on the terms in the dynamic equation.

Suppose we have a (small) river that has dimensions:

where B = Top width of the channel, n = Manning’s roughness coefficient, S0 = channel bed slope, g = acceleration due to gravity, A = cross-sectional area, Q = discharge, y = depth of flow q = lateral inflow and t = time. The superposition of a ‘bar’ above a variable denotes the scale for thatvariable . With this data compare the magnitudes of the terms in the Saint Venant equations:

We can conclude, in this case, that

• the momentum of the flow in the river is governed primarily by the bottom and friction slopes, and is modified by the water surfaceslope, which is defined relative to the bottom slope of the channel

• the acceleration and convection of momentum terms can be ignored • the contribution to the momentum in the main channel from

tributaries and lateral inflow can also be ignored • the lateral inflow from small tributaries and direct runoff can be

significant under snow-melt conditions, but in general its effect issmall

• the length scale of a flood wave is considerably greater than the lengths of most rivers.

If we ignore the inertial terms (the local acceleration and convection ofmomentum) then we obtain

(3)

We now have a advection-diffusion equation. The equation has the remarkable property of having only Q as the primary dependent variable within the differentials. It is still, however, a function of the dependent variable h (or A), and therefore the equation is incomplete without the massbalance equation (Eq 2). Note that the advection speed and diffusioncoefficients are given by

(4)

(5)

Four boundary conditions are needed to solve Eq (2) and (3): Initial condition Q(x,0)=Q0(x) Initial condition h(x,0)=h0(x) Upstream condition Q(0,t)=Qu(t) Downstream condition Q(x,t)=f (h(x,t)) say The non-inertial equation clearly indicates that there is a translation/advection process occurring in terms of the disturbance ofthe flow, and there is a corresponding diffusion associated with it. These processes underlie the basic equations. The pair of equations can be solved numerically using, say, thePriessman 4 point finite difference scheme.

References

Cunge, J.A., Holly, F.M., and Verwey, A. (1980) Practical aspects ofcomputational river hydraulics. Iowa State Univ. Press.

Abbott, M.B. (1979). Computational hydraulics; elements of thetheory of free surface flows, Pitman, London