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

Appendix A
Non-conservative form of
Navier±Stokes equations
To derive the Navier±Stokes equations in their non-conservative form, we start with
the conservative form.
Conservation of mass:
Conservation of momentum:
Conservation of energy:
@… E†
@t ‡ @…uj E†
@
@t ‡ @… ui† ˆ
@xi @
@t ‡ @ui @xi @… ui† @t ‡ @…uj ui† @x j
ÿ @
@x i
@x j
k @T
@x i
@
‡ ui Rewriting the momentum equation with terms di€erentiated as
@u i
@t ‡ u i
@
@t ‡ @u j
@x j
@
‡ uj @x j
@x i
ˆ 0 …A:1†
ÿ @ ij
‡
@xj @p
ˆ 0 …A:2†
@xi ‡ @…ujp† ÿ
@xj @… ijuj† ˆ 0 …A:3†
@xj ‡ u j
@ui @xj ÿ @ ij
‡
@xj @p
ˆ 0 …A:4†
@xi and substituting the equation of mass conservation (Eq. A.1) into the above equation
gives the reduced momentum equation
@u i
@t ‡ u j
@ui @xj ÿ 1 @ ij
‡
@xj 1 @p
ˆ 0 …A:5†
@xi Similarly as above, the energy equation (Eq. A.3) can be written with di€erentiated
terms as
E @
@t ‡ @u j
@x j
@
‡ uj @x j
‡ @E
@t ‡ u @E
j
@x j
ÿ @
@x i
k @T
@x i
‡ @…ui p†
ÿ
@xi @… ijuj† ˆ 0 …A:6†
@xi