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

5
Free surfaces, buoyancy and
turbulent incompressible ¯ows
5.1 Introduction
In the previous chapter we have introduced the reader to general methods of solving
incompressible ¯ow problems and have illustrated these with many examples of
newtonian and non-newtonian ¯ows. In the present chapter, we shall address three
separate topics of incompressible ¯ow which were not dealt with in the previous
chapters. This chapter is thus divided into three parts. In the ®rst two parts the
common theme is that of the action of the body force due to gravity. We start with
a section addressed to problems of free surfaces and continue with the second section
which deals with buoyancy e€ects caused by temperature di€erences in various parts
of the domain. The third part discusses the important topic of turbulence and we shall
introduce the reader here to some general models currently used in such studies. This
last section will inevitably be brief and we will simply illustrate the possibility of
dealing with time averaged viscosities and Reynolds stresses. We shall have occasion
later to use such concepts when dealing with compressible ¯ows in Chapter 6.
However the ®rst two topics of incompressible ¯ow are of considerable importance
and here we shall discuss matters in some detail.
The ®rst part of this chapter, Sec. 5.2, will deal with problems in which a free
surface of ¯ow occurs when gravity forces are acting throughout the domain. Typical
examples here would be for instance given by the disturbance of the free surface of
water and the creation of waves by moving ships or submarines. Of course other
problems of similar kinds arise in practice. Indeed in Chapter 7, where we deal
with shallow water ¯ows, a free surface is an essential condition but other assumptions
and simpli®cations have to be introduced. Here we deal with the full problem
and include either complete viscous e€ects or simply deal with an inviscid ¯uid
without further physical assumptions. There are other topics of free surfaces which
occur in practice. One of these for instance is that of mould ®lling which is frequently
encountered in manufacturing where a particular ¯uid or polymer is poured into a
mould and solidi®es. We shall brie¯y refer to such examples. Space does not permit
us to deal with this important problem in detail but we give some references to the
current literature.
In Sec. 5.3, we invoke problems of buoyancy and here we can deal with pure
(natural) convection when the only force causing the ¯ow is that of the di€erence