Page 188 - Computational Fluid Dynamics for Engineers

Problems 175

5-6. For two-dimensional inviscid flows, Eq. (2.2.30) can be written as
3Q 9 | 9 ^
(P5.6.1)
dt dx dy

where Q, E and F are given by Eq. (2.2.32a). It can also be written in the form
dQ dQ dQ t

with the Jacobian matrices A and B of the flux vectors E and F defined by

(P5.6.3)
dcy dQ
Show that for a perfect gas, A and B can be written as

0 1 0 0
2
^ M + ^ (3 — 7)M (1 — j)v 7 —1
A
—uv v u 0
3
2
2
2
2
*f + (7 - 1M« + v ) f + V( « + v ) (! - 7)«v 7«
(P5.6.4)
0 0 1 0
—uv v u 0
B
(1 — 7)7/ (3 — 7)1; 7 — 1
2
2
+ (7 - l)v(u 2 + i; ) (1 - j)uv ^ + ^ ( 3 i ; 2 + u ) *yv
(P5.6.5)
5-7. Apply the von Neumann stability analysis to show that the Euler explicit
method, Eq. (5.7.17), is unstable.

5-8. Show that the implicit method

,.n+l
^ + \Kti-^ u 0
At i-i
for Eq. (5.1.1) is unconditionally stable.

5-9. Show that the explicit upwind method, Eq. (5.5.1) is stable 0 < a < 1.

5-10. Show that
u(x,t) = f(x-ct) (P5.10.1)

satisfies the linear convection Eq. (5.1.1) and the initial condition
u(x,0) = f(x) (P5.10.2)