Page 66 - Computational Fluid Dynamics for Engineers

2.2 Navier-Stokes Equations 51

la c
Re = (2.2.31)
Moo
and the Q, E, F, E v, F v vectors by

gu
Q gv
QU gu 2 + p guv
Q E = (2.2.32a)
QV guv gv 2 + p
(E t+p)u
E t (E t+p)v
0 0
®XX u xy
— = (2.2.32b)
E v F v
®xy a yy
Px Py
with 7 denoting the ratio of specific heats and a the speed of sound, which for
ideal gases is given by a 2 — jp/g. The viscous stresses are
2 / du dv
axx M 2
= 3 \ Yx ~ d~y
2 (dv du
a (2.2.33)
™ = r {% ~ d~ x
f du dv

and we also write
v d , 2
Px ua xx + va xy + p r ( 7 _ 1 } dx («')

M d
Py = Ua Xy + VCJyy +
P r ( 7 - 1 ) 0 ^

2.2.4 Navier-Stokes Equations: Transformed Form
The Navier-Stokes equations discussed in the previous subsections and ex-
pressed for a Cartesian coordinate system are valid for any coordinate sys-
tem. In many problems it is more convenient to write the equations in general
curvilinear coordinates by using a coordinate transformation from the rectan-
gular Cartesian form. To illustrate the procedure, consider a two-dimensional
unsteady flow and introduce the generic transformation

r = t
(2.2.34)

For an actual application, the transformation in Eq. (2.2.34) must be given in
some analytical or numerical form. Often the transformation is chosen so that