Page 69 - Computational Fluid Dynamics for Engineers

54 2. Conservation Equations

t = r
x = x(f,T7,r) (2.2.44)
y = y(£,v,T)
so that £, r/ and r are now the independent variables, then the metric coefficients
in Eqs. (2.2.38) can be obtained from the relations given by Eq. (2.2.43).
At this point, we notice that Eqs. (2.2.38) are in a weak conservation form.
That is, even though none of the flow variables (or more appropriately, func-
tions of the flow variables) occur as coefficients in the differential equations, the
metrics do. However, as discussed by Pulliam [6], the expressions

d_
(J d£\J M5).
dr dr)\J.
k 0 (T]x
-( ) J I + dn\J J'
'
and

dZ\Jj dr]\J<
are denned as invariants of the transformation and are analytically equal to
zero. Eqs. (2.2.38) can then be expressed in the strong conservation form and
written as
d
® + ^ + &i. - _L dE v dF v (2.2.45)
dr d£ drj ~ Re
where

gU
Q QV
QU QUU + £ xp QUV + T] xp
1
Q = J~ E = J~ F = J~
QV gvll + £ yp QVV + TJyP
U{E t+p)~itP + P) - T)tP
E t V(E t
(2.2.46a)
with the contravariant velocities U and V defined by
U = & + ixu + £yv, V = Tit + r] xu + r\ yv (2.2.46b)

The viscous flux terms are
= l + £ yF v), = + r] yF v) (2.2.46c)
E v J~ {£ xE v F v J-\r) xE v
with E v and F v given by Eq. (2.2.32b). The stress terms, such as a xx, a yy, etc.
are also transformed in terms of the £ and r\ derivatives where

&XX = -o[4(£xU£ + T] XU V) - 2(t yV£ + r) yV v)}
a 2 (2.2.47)
yy = 3 - (Cx^ + VxU v) + 4 ( ^ ^ + T] yv v)]
[
°xy = V{£yU£ + T)yU v + £ xVt + T} xVr,)