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

220 Shallow-water problems
reduced to
1 @p
‡ g ˆ 0 …7:3†
@x3 where g 3 ˆÿg is the gravity acceleration. After integration this yields
p ˆ g… ÿ x 3†‡p a …7:4†
as, when x 3 ˆ , the pressure is atmospheric ( p a) (which may on occasion not be
constant over the body of the water and can thus in¯uence its motion).
On the free surface the vertical velocity u 3 can of course be related to the total time
derivative of the surface elevation, i.e. (see Sec. 5.3 of Chapter 5)
Similarly, at the bottom,
u s 3 ˆ D
dt
u b 3 ˆ DH
dt
@
@
@t ‡ us1 @x1 @
u b 1
@x1 ‡ u s @
2
‡ u b 2
@
@x 2
@x 2
…7:5a†
…7:5b†
assuming that the total depth H does not vary with time. Further, if we assume that
for viscous ¯ow no slip occurs then
and also by continuity
u b 1 ˆ u b 2 ˆ 0 …7:6†
u b 3 ˆ 0
Now a further approximation will be made. In this the governing equations will be
integrated with the depth coordinate x3 and depth-averaged governing equations
derived. We shall start with the continuity equation (7.2a) and integrate this in the
x3 direction, writing
…
…
…
@u3 @u1 @u2 dx3 ‡ dx3 ‡ dx3 ˆ 0
ÿH @x3 ÿH @x1 ÿH @x2 …7:7†
As the velocities u1 and u2 are unknown and are not uniform, as shown in Fig. 7.1(b),
it is convenient at this stage to introduce the notion of average velocities de®ned so
that
…
ui dx3 ˆ Ui…H ‡ † Uih …7:8†
ÿH
with i ˆ 1; 2. We shall now recall the Leibnitz rule of integrals stating that for any
function F…r; s† we can write
…b …b @
@
F…r; s† dr F…r; s† dr ÿ F…b; s†
ÿa @s @s ÿa
@b @a
‡ F…a; s† …7:9†
@s @s
With the above we can rewrite the last two terms of Eq. (7.7) and introducing Eq. (7.6)
we obtain
…
@ui dx3 ˆ
ÿH @xi @
…U
@x
ih†ÿu
i
s @
i
…7:10†
@xi