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

with
8
><
Fi ˆ
8
><
Q ˆ
>:
0
ÿ f 1
ÿ f 2
u i
u1ui ‡ p 1i >=
u2ui ‡ p 2i
u3ui ‡ p 3i >;
Hu i
8
><
Gi ˆ
><
Gx ˆ
9
0
ÿ 1i
ÿ 2i
ÿ 3i
8
><
or Fx ˆ
ÿ… ijuj†ÿk @T
>:
>;
8
ÿ f3 >:
>;
ÿ fiui ÿ qH @x i
0
9
>=
ÿ xx
ÿ yx
ÿ zx
>:
or
u
u 2 ‡ p >=
uv ; etc: …1:25b†
uw
Hu
9
ÿ… xxu ‡ xyv ‡ xzw†ÿk @T
>:
>;
@x
9
8
0
>=
ij ˆ
><
or Q ˆ
@ui ‡
@xj @uj @xi The governing equations of ¯uid dynamics 9
9
>;
>=
; etc: …1:25c†
ÿ f x
ÿ f y
9
ÿ fz >:
>;
ÿ … fxu ‡ fyv ‡ fzw†ÿqH 2 @uk ÿ ij
3 @xk >=
; etc:
…1:25d†
The complete set of (1.24) is known as the Navier±Stokes equation. A particular
case when viscosity is assumed to be zero and no heat conduction exists is known
as the `Euler equation' … ij ˆ k ˆ 0†.
The above equations are the basis from which all ¯uid mechanics studies start and
it is not surprising that many alternative forms are given in the literature obtained
by combinations of the various equations. 2 The above set is, however, convenient
and physically meaningful, de®ning the conservation of important quantities. It
should be noted that only equations written in conservation form will yield the
correct, physically meaningful, results in problems where shock discontinuities are
present. In Appendix A, we show a particular set of non-conservative equations
which are frequently used. There we shall indicate by an example the possibility
of obtaining incorrect solutions when a shock exists. The reader is therefore