Page 94 - Computational Fluid Dynamics for Engineers

Problems 79

2-19. Using the transformation

(P2.19.1)
r/ = r/(r, 0)

Show that the Laplace equation in polar coordinates
2
2
9 u Idu 1 <9 t
2
Q r2 r g r r2 QQ2
can be expressed in the following form

dl? \^ r*) drj* \' r ' r*) ' d^d V ^ l r r* J ( p 2 - 1 9 - 3 )
\
,du( ,ir,iee\,du( rj r r]ee\ n

in the computational plane.

2-20. Determine the metric coefficients in Eq. (P2.19.3) for £ and rj defined by

1 (P2.20.1)
rj= -
r
and show that the Laplace equation, Eq. (P2.19.3) can be written as

2
2
2d u + + d u (P2202)
du
"V ^ **=° - -
2-21. For a perfect gas, show that

p=(T-l)g\Et-YJ (P2.21.1)

Hint: Note that
- R
Cy — — , C — C v 1
7 - 1

2-22. Show that, with mass transfer (QV W), the momentum integral integration,
Eq. (2.4.36), can be written as

±{ul9) + u ev w + 6*u ep- = ^ (P2.22.1)
ax ax Q