Page 64 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 49

3.6 The Unique Form of the Fluid Equations
In the sequel we will analyze the conservative form of all the equations
associated with fluid flows — the equations of continuity, of momentum

torsor and of energy within a unique frame. Then we will show which
are the most appropriate forms for CFD. We notice, first, that all the
mentioned equations (even on axes projection if necessary) could be
framed in the same generic form

oU 1 OF 4 OG 4 OH
=J

Ot Ox Oy Oz
where U, F, G, H and J are column vectors given by

p
0
pvt phi
U= pv2 ; J= pfe
)
pv3 pfs
2
p(e+)
p(ui fi + vefe + vs fs) + pot
f p ,

pvy +p—o1n

F= ¢ pv2v) — O12
PU3V) — 013
2
p (c + +) V1 + PU — VITIL — V2012 — V3013

pv2

pU1V2 — 021
G= pus +p — 022
PU3Vv2 — 023
2
p (c + +) v2 + pv2 — V1021 — 2022 — 03023

PvU3
PUV1 U3 — 031
H = pPvU2U3 — 032
pU303 + p — 033
2
p (c + =) v3 + pv3 — V1031 — V1031 — 3033
.
where oj; are the components of the tensor [a], fj of the vector f and v;
of the vector v.
In the above equations the column vectors F, G and H are called
the flux terms while J is a “source” term (which will be zero if the
external forces are negligible). For an unsteady problem U is called

the solution vector because its elements are dependent variables which