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

146 Free surfaces, buoyancy and turbulent incompressible ¯ows
full Navier±Stokes equations or, neglecting viscosity e€ects, a pure potential or Euler
approximation. Both assumptions have been discussed in the previous chapter but it
is interesting to remark here that the resistance caused by the waves may be four or
®ve times greater than that due to viscous drag. Clearly surface e€ects are of great
importance.
Historically many solutions that ignore viscosity totally have been used in the ship
industry with good e€ect by involving so-called boundary solution procedures or
panel methods. 1ÿ11 Early ®nite element studies on the ®eld of ship hydrodynamics
have also used potential ¯ow equations. 12 A full description of these is given in
many papers. However complete solutions with viscous e€ects and full non-linearity
are di cult to deal with. In the procedures that we present in this section, the door is
opened to obtain a full solution without any extraneous assumptions and indeed such
solutions could include turbulence e€ects, etc. We need not mention in any detail the
question of the equations which are to be solved. These are simply those we have
already discussed in Sec. 4.1 of the previous chapter and indeed the same CBS
procedure will be used in the solution. However, considerable di culties arise on
the free surface, despite the fact that on such a surface both tractions are known
(or zero). The di culties are caused by the fact that at all times we need to ensure
that this surface is a material one and contains the particles of the ¯uid.
Let us de®ne the position of the surface by its elevation relative to some
previously known surface which we shall refer to as the reference surface (see
Fig. 5.2). This surface may be horizontal and may indeed be the undisturbed water
surface or may simply be a previously calculated surface. If is measured in the
direction of the vertical coordinate which we shall call x 3, we can write
…t; x 1; x 2†ˆx 3 ÿ x 3ref
Noting that is the position of the particle on the surface, we observe that
dx 1
dt ˆ u 1;
and from Eq. (5.1,) we have ®nally
where
d
dt ˆ u3 ˆ @
@t ‡ u1 u ˆ‰u 1; u 2Š T
dx 2
dt ˆ u 2;
@
@x 1
@
‡ u2 @x 2
dx 3
dt
r ˆ @
;
@x1 @
@x2 ˆ d
dt ˆ u 3
…5:1†
…5:2†
ˆ @
@t ‡ uT r …5:3†
T
…5:4†
We immediately observe that obeys a pure convection equation (see Chapter 2) in
terms of the variables t; u 1; u 2 and u 3 in which u 3 is a source term. At this stage it is
worthwhile remarking that this surface equation has been known for a very long
time and was dealt with previously by upwind di€erences, in particular those introduced
on a regular grid by Dawson. 2 However in Chapter 2, we have already
discussed other perfectly stable, ®nite element methods, any of which can be used
for dealing with this equation. In particular the characteristic±Galerkin procedure
can be applied most e€ectively.