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

Momentum conservation
@U i
@t
@
ˆÿ …u
@x
jUi†‡ j
@ ij
ÿ
@xj @p
ÿ gi @xi In the above we de®ne the mass ¯ow ¯uxes as
Energy conservation
@… E†
@t
@
ˆÿ …uj E†‡
@xj @
@xi …3:4†
U i ˆ u i …3:5†
k @T
@x i
ÿ @
…u
@x
jp†‡
j
@
…
@x
ijuj† …3:6†
i
In all of the above u i are the velocity components; is the density, E is the speci®c
energy, p is the pressure, T is the absolute temperature, g i represents body forces
and other source terms, k is the thermal conductivity, and ij are the deviatoric
stress components given by (Eq. 1.12b)
ij ˆ
@ui ‡
@xj @uj ÿ
@xi 2
3 ij
@uk @xk Introduction 65
…3:7†
where ij is the Kroneker delta ˆ 1, if i ˆ j and ˆ 0ifi 6ˆ j. In general, in the above
equation is a function of temperature, …T†, and appropriate relations will be used.
The equations are completed by the universal gas law when the ¯ow is coupled and
compressible:
p ˆ RT …3:8†
where R is the universal gas constant.
The reader will observe that the major di€erence in the momentum-conservation
equations (3.4) and the corresponding ones describing the behaviour of solids (see
Volume 1) is the presence of a convective acceleration term. This does not lend
itself to the optimal Galerkin approximation as the equations are now non-selfadjoint
in nature. However, it will be observed that if a certain operator split is
made, the characteristic±Galerkin procedure valid only for scalar variables can be
applied to the part of the system which is not self-adjoint but has an identical form
to the convection±di€usion equation. We have shown in the previous chapter that
the characteristic±Galerkin procedure is optimal for such equations.
It is important to state again here that the equations given above are of the
conservation forms. As it is possible for non-conservative equations to yield multiple
and/or inaccurate solutions (Appendix A), this fact is very important.
We believe that the algorithm introduced in this chapter is currently the most
general one available for ¯uids, as it can be directly applied to almost all physical
situations. We shall show such applications ranging from low Mach number viscous
or indeed inviscid ¯ow to the solution of hypersonic ¯ows. In all applications the
algorithm proves to be at least as good as other procedures developed and we see
no reason to spend much time describing alternatives. We shall note however that
the direct use of the Taylor±Galerkin procedures which we have described in the
previous chapter (Sec. 2.10) have proved quite e€ective in compressible gas ¯ows
and indeed some of the examples presented will be based on such methods. Further,