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

6
Compressible high-speed gas ¯ow
6.1 Introduction
Problems posed by high-speed gas ¯ow are of obvious practical importance. Applications
range from the exterior ¯ows associated with ¯ight to interior ¯ows typical of
turbomachinery. As the cost of physical experiments is high, the possibilities of
computations were explored early and the development concentrated on the use
of ®nite di€erence and associated ®nite volume methods. It was only in the 1980s
that the potential o€ered by the ®nite element forms were realized and the ®eld is
expanding rapidly.
One of the main advantages in the use of the ®nite element approximation here is its
capability of ®tting complex forms and permitting local re®nement where required.
However, the improved approximation is also of substantial importance as practical
problems will often involve three-dimensional discretization with the number of
degrees of freedom much larger than those encountered in typical structural problems
(10 5 ±10 7 DOF are here quite typical).
For such large problems direct solution methods are obviously not practicable and
iterative methods based generally on transient computation forms are invariably
used. Here of course we follow and accept much that has been established by the
®nite di€erence applications but generally will lose some computational e ciency
associated with structured meshes typically used here. However, the reduction of
the problem size which, as we shall see, can be obtained by local re®nement and
adaptivity will more than compensate for this loss (though of course structured
meshes are included in the ®nite element forms).
In Chapters 1 and 3 we have introduced the basic equations governing the ¯ow of
compressible gases as well as of incompressible ¯uids. Indeed in the latter, as in
Chapter 4, we can introduce small amounts of compressibility into the procedures
developed there speci®cally for incompressible ¯ow. Here we shall deal with highspeed
¯ows with Mach numbers generally in excess of 0.5. Such ¯ows will usually
involve the formation of shocks with characteristic discontinuities. For this reason
we shall concentrate on the use of low-order elements and of explicit methods,
such as those introduced in Chapters 2 and 3.
Here the pioneering work of the ®rst author's colleagues Morgan, LoÈ hner and
Peraire must be acknowledged. 1ÿ38 It was this work that opened the doors to practical