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

Equation of state
p ˆ RT
c p
ˆ RT ˆ … ÿ 1† T …3:15†
In the above equation R ˆ c p ÿ c v is used. The following forms of non-dimensional
equations are useful to relate the speed of sound, temperature, pressure, energy, etc.
E ˆ T ‡ 1
2 u i u i
c 2 ˆ… ÿ 1†T …3:16†
p ˆ… ÿ 1† E ÿ 1 Ui Ui 2
The above non-dimensional equations are convenient when coding the CBS
algorithm. However, the dimensional form will be retained in this and other chapters
for clarity.
3.2 Characteristic-based split (CBS) algorithm
3.2.1 The split ± general remarks
The split follows the process initially introduced by Chorin 1;2 for incompressible ¯ow
problems in the ®nite di€erence context. A similar extension of the split to ®nite
element formulation for di€erent applications of incompressible ¯ows have been
carried out by many authors. 3ÿ27 However, in this chapter we extend the split to
solve the ¯uid dynamics equations of both compressible and incompressible forms
using the characteristic±Galerkin procedure. 28ÿ46 The algorithm in its full form
was ®rst introduced in 1995 by Zienkiewicz and Codina 28;29 and followed several
years of preliminary research. 47ÿ51
Although the original Chorin split 1;2 could never be used in a fully explicit code, the
new form is applicable for fully compressible ¯ows in both explicit and semi-implicit
forms. The split provides a fully explicit algorithm even in the incompressible case for
steady-state problems now using an `arti®cial' compressibility which does not a€ect
the steady-state solution. When real compressibility exists, such as in gas ¯ows, the
computational advantages of the explicit form compare well with other currently
used schemes and the additional cost due to splitting the operator is insigni®cant.
Generally for an identical cost, results are considerably improved throughout a
large range of aerodynamical problems. However, a further advantage is that both
subsonic and supersonic problems can be solved by the same code.
3.2.2 The split ± temporal discretization
Characteristic-based split (CBS) algorithm 67
We can discretize Eq. (3.4) in time using the characteristic±Galerkin process. Except
for the pressure term this equation is similar to the convection±di€usion equation