Page 93 - Computational Fluid Dynamics for Engineers

78 2. Conservation Equations

then the values of the second derivative u xx, u xy and u yy cannot be determined
at the points of T.

2-16. Show that in Eq. (2.2.32a) pressure is related to the conservative flow
variables, Q, by the equation of state g.

2
P= ( 7 - 1) E t--Q(u 2 + v ) (P2.16.1)

2-17. In the numerical solution of the incompressible Navier-Stokes equations
by the pseudocompressibility method [13] a time derivative of pressure is added
to the continuity equation so that the two-dimensional version of Eq. (2.2.1)
becomes + +
i '(S £)=° ™

where /3 is known as the pseudocompressibility constant. Here t represents
pseudo-time and is not related to physical time.
Show that with the continuity equation given by Eq. (P2.17.1), the incom-
pressible Navier-Stokes equations in Cartesian coordinates for two-dimensional
flows can be written in the following form

§ + gjW-*.> + £<F-f.)-o (P2.17.2)

where
V 0u (5v
D u , E = u 2 +p , F = uv (P2.17.3)
V uv 1 v 2 +p
0 0

E v = ®xx 1 F v = \ &xy (P2.17.4)
®xy G yy

2-18. Using the transformation given by Eq. (2.2.34), show that the Laplace
equation expressed in the physical plane (x, y)
2
2
d u d u
0
2
2
dx dy
can be written in the form
2
d u
a? <&+$ drf 2 (Vl + Vy) + 2 d^dri \Vx£,x H~ Vy<,y
(P2.18.1)
du,^ ^ x du
+ o>. \£xx + €yy) + o xHxx + Vyy) — 0
in the computational plane (£,77).