373
Now, integrating the vertical component of the momentum equation. the hydrostatic
form of the pressure is obtained:
(20)
For the sake of simplicity, let us suppose that Pa is constant. It can be supposed then
that Pa = 0 without loss of generality. Using (20) in equation (12), together with the
boundary conditions on the surface and the bottom, the expression
is obtained, where
U = (Ub U2) = u- u
represents velocity fluctuations from the mean. In order to dose the equations, the
effects of these fluctuations may be neglected or parameterized. Here, variations from
the mean are supposed small enough so that terms involving quadratic expressions ofu
can be neglected. The non-linear expression arising from the bottom drag can also be
evaluated in u with second-order error. In this case, the following equation is obtained
8( hu l)
8(huD
8( hu I U 2)
h~ _ fh- ~ I~I_ gUl~
"'t + ' "
+ ' "
+ 9 '" - U2 + /,Vl V
C 2 '
U
uXI
UX2
UXI
P
Vertical integration of (13) gives an analogous equation.
In order to write these equations in conservative form, the flux
Q = hu
is introduced and the shallow water or St.Venant equations are then written under
the form,
where Ii is the unit tensor, and
0,
in D
f(Q2, -Ql)
/'liJli!
glftl_
(fiu,
Finally, for certain reasons related to the numerical methods, it is more convenient
to rewrite these equations under the form:
0,
in D
(21)
~
1
g'1VH+F+-(Tw-Tj),
inD.(22)
Po