Page 214 - Soil and water contamination, 2nd edition

Substance transport 201

Box 11.IV Physics of surface water flow

The governing equations that describe surface water flow are based on the conservation
laws of mass (continuity) and momentum (Newton’s second law of motion). Working out
these basic equations for one-dimensional, unsteady (dynamic) flow in an open channel
and neglecting lateral inflow (from precipitation or tributary rivers), wind shear, and
energy losses due to eddies yields the so-called Saint-Venant equations:

Continuity equation
Q A
0 (11.IVa)
x t
Momentum equation
1 Q 1 Q 2 H
g g S S 0 (11.IVb)
A t A x A x 0 f
Local Convective Pressure Gravity Friction
acceleration acceleration force force force
term term term term term
Kinematic wave
Diffusion wave
Dynamic wave

3
-1
2
where Q = the water discharge [L T ], A = the cross-sectional area of the channel [L ],
-2
-2
H = the water depth [L], g = the gravitational acceleration constant (= 9.8 m s ) [L T ],
S = the bed slope [-], and S = the friction slope [-]. If the water level or flow velocity (=
0 f
Q/A) changes at a given location in a channel, the effects of this change propagate back
upstream. These backwater effects are represented by the first three terms of Equation
(IVb). These terms may be neglected for overland flow and flow through channels with a
more or less uniform cross-section along their length. The remaining terms comprise the
kinematic wave, which assumes S = S : the friction and gravity forces balance each other.
0 f
The relation between the friction slope and the flow rate is described by either Chezy’s
equation or Manning’s equation. Chezy’s equation reads:
u C R S (11.IVc)
f
-1
-1
0.5
where u = cross-sectional averaged flow velocity [L T ], C = Chezy’s coefficient [L T ],
and R = hydraulic radius [L]. The hydraulic radius equals the wetted perimeter divided
by the cross- sectional area. For most rivers that are much wider than they are deep, the
hydraulic radius is approximately equal to the water depth. Typical values for C for rivers
0.5 -1
are in the order of 30–50 m s . Manning’s equation reads:
R 3 / 2 S f 2 / 1
u (11.IVd)
n
-1
where n = Manning roughness coefficient, with u expressed in m s and R expressed in
m. Typical values for n range between 0.012 for smooth, concrete channels, via 0.040 for
meandering streams, to 0.100 for channels in densely vegetated floodplains. Chezy’s and
Manning’s equations are both only applicable for turbulent flow, which is almost always

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