Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

7
Shallow-water problems
7.1 Introduction
The ¯ow of water in shallow layers such as occur in coastal estuaries, oceans, rivers,
etc., is of obvious practical importance. The prediction of tidal currents and elevations
is vital for navigation and for the determination of pollutant dispersal which,
unfortunately, is still frequently deposited there. The transport of sediments
associated wth such ¯ows is yet another ®eld of interest.
In free surface ¯ow in relatively thin layers the horizontal velocities are of primary
importance and the problem can be reasonably approximated in two dimensions.
Here we ®nd that the resulting equations, which include in addition to the horizontal
velocities the free surface elevation, can once again be written in the same conservation
form as the Euler equations studied in previous chapters:
@
@t ‡ @Fi ‡
@xi @Gi ‡ Q ˆ 0 for i ˆ 1; 2 …7:1†
@xi Indeed, the detailed form of these equations bears a striking similarity to those of
compressible gas ¯ow ± despite the fact that now a purely incompressible ¯uid (water)
is considered. It follows therefore that:
1. The methods developed in the previous chapters are in general applicable.
2. The type of phenomena (e.g. shocks, etc.) which we have encountered in compressible
gas ¯ows will occur again.
It will of course be found that practical interest focuses on di€erent aspects. The
objective of this chapter is therefore to introduce the basis of the derivation of the
equation and to illustrate the numerical approximation techniques by a series of
examples.
The approximations made in the formulation of the ¯ow in shallow-water bodies
are similar in essence to those describing the ¯ow of air in the earth's environment
and hence are widely used in meteorology. Here the vital subject of weather prediction
involves their daily solution and a very large amount of computation. The interested
reader will ®nd much of the background in standard texts dealing with the subject,
e.g. references 1 and 2.