Page 62 - Computational Fluid Dynamics for Engineers

2.2 Navier-Stokes Equations 47

dg dg du dg dv dg dw . rt ^ .
{)
u
v
-£ + ir + ejr + 7^ + ejr + w ^ + eir = ' 2 2 17
- -
ot ox dx dy dy dz dz
it is in nonconservative form.
The momentum equations in conservation form are
dgu d , 2 o x 9 , ^ 9
-{gu + p-a xx) + ^-(QUV - <r xy) + -^-{guw - a xz) = gf x (2.2.18)

uv
a
~KT + ~^7.(@ ~ yx) + ^ ~ ( ^ 2 + P ~ ^j/s/) + -^-(0 VW ~ °yz) = Qfy (2.2.19)

dgw d , . d , x ^ / 2 \ r /^ * ™\
- ^ + ^(QUW-G ZX) +—(gvw-a zy) +-—(gw + p - a zz) = gf z (2.2.20)

where the components of the viscous stress tensor follow from Eq. (2.2.13),

2 (~du dv dw\

2 (^dv du dw\ / ^ ^ ^ N
(2 2 21)
*«> = H%-te- *;) --
2 f 0dw du dv
° zz = 3 M V a ! " ~d~x ~ ~0y

°xy = H | — &yx
f dw du \
1
0"x* = /i | = O'zx
{dx dz J
( dv dw \
cr^ = /i | = &zy
{dz dy J
Similarly, the energy equation can be written in conservation form as

dQ
dE t
Q{fxU + f yv + f zw)
dt dt
9 ,T,
E
+ ~^~\ tu + pu- ua xx - va xy - wa xz + q x)
0x
(2.2.22)
9 /rt . x
E V
+ ~fi-\ t + P V - U(J Xy ~ VGyy ~ WGy Z + q y)
d ,
+ Tj-{E tw +pw - UG XZ - va yz - wa zz + q z) = 0