Saint-Venant equations

Most channel routing performed by computer modeling is based on some simplification of the Saint-Venant equations. These equations provide a very simple model of very complex processes. These equations are:

where:
A = cross-sectional flow area
V = average velocity of water
x = distance along channel
B = water surface width
y = depth of water
t = time
q = lateral inflow per unit length of channel
S_f = friction slope
S_o = channel bed slope
g = gravitational acceleration

The solutions to the momentum and continuity equations concurrently define the propagation of a floodwave with respect to distance along the channel and time. Assumptions for these equations include:

• The momentum and continuity equations are shown for one-dimensional flow in the downstream direction. The natural variation in velocity with respect to depth is ignored. In addition, these equations do not directly address lateral or vertical stream flows that would require a more complex equation.
• Flow is gradually varied so that hydrostatic pressure prevails, and vertical accelerations can be ignored.
• The effects of boundary friction and turbulence can be treated with resistance laws, as they are in steady flow.
• Fluid is incompressible and has a constant density.