Geoscience Reference
In-Depth Information
5
WATER ON THE LAND SURFACE:
FLUID MECHANICS OF FREE
SURFACE FLOW
Owing to the irregular topography of the Earth's continents, surface runoff, that is the
flow of water over land, takes place in many different ways. When for some reason,
such as rainfall, snowmelt, the overtopping of small depressions, or the emergence of
groundwater at a source, surface flow is initiated, it may at first proceed as a thin sheet
flow; however, as a result of local irregularities, the flow soon gathers in small gullies and
rills, which in turn join to form rivulets in the fashion of a tree-like network. Eventually
these merge with others to become larger rivers, which finally end up in some lake or in
the ocean. Thus the flow system consists of an intricate combination of many different
types of flow regimes, in channels of different geometries and sizes. For purposes of
analysis, to describe the basic hydraulic elements of landsurface runoff, it is convenient
and useful to distinguish between two major types of free surface flow; these are first,
sheet flow or overland flow, which is most likely to occur under conditions of heavy
precipitation in source areas where runoff is being generated which feeds into streams;
and second, the flow that occurs in larger permanent open channels. Both types of flow
are usually unsteady and spatially varied. In this chapter a general description is given of
free surface flow. The general principles are then applied to overland flow and to channel
flow and streamflow routing in the next two chapters.
5.1
FREE SURFACE FLOW
The flow of water on a solid surface is governed by the usual conservation equations
of fluid mechanics, namely the continuity equation for mass and the Navier-Stokes
equations for momentum. One important condition on the boundaries can be formulated
by observing that once a fluid particle is on an impermeable surface, it stays on it (see
Lamb, 1945, p. 7). In other words, it moves with the surface, and its velocity relative to
the surface is either purely tangential or zero (in the case of no slip), for otherwise a finite
flow of fluid would take place across the surface. Thus, if the surface is described by a
function
F
0, then any displacement occurring with the fluid particles
should leave that function unchanged, i.e.
=
F
(
x
,
y
,
z
,
t
)
=
DF
Dt
=
0
(5.1)
The operator
D
Dt
, already defined in Equation (1.3), is the time derivative
following the
motion
, also called the
fluid mechanical
time derivative, the
substantial
time derivative,
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