Page 174 - Introduction to Computational Fluid Dynamics

P1: IWV
CB908/Date
0521853265c05
EXERCISES

B 0 521 85326 5 May 20, 2005 12:28 153
C
S R

T
h

OUTFLOW
P Q
M
H F
Y
g G E INFLOW
A
D
X
Z
Figure 5.23. Long chamber of Exercise 11.

5. Starting with Equation 5.40, derive Equation 5.47.

6. Show the validity of Equations 5.55 and 5.56.
7. Identify the differences and similarities between Equations 5.57 for collocated
grids and Equation 5.32 for staggered grids.
8. Confirm that on collocated grids AP u 1 = AP .
u 2
9. It is of interest to derive a total pressure-correction equation for compressible
flows in which p = ρR g T . To do this, start with Equation 5.57 and write

p m (p − p )
sm
l
l
l

ρ l+1 = ρ + ρ = ρ + = ρ + .
m
R g T R g T
Withthissubstitutionshowthatthe p -equationtakestheformofageneraltrans-

port equation for any with appearance of convection–diffusion-like terms.
$
Also, V sound = γ R g T . Hence, show the Mach number dependence in the
equation. If CDS is used, can the coefficients in the discretised equation (5.60)
turn negative? If yes, suggest a remedy.
10. Explain the need for evaluating the mass residual via Equation 5.73 when
computing on collocated grids.
11. Consider the chamber shown in Figure 5.23. The chamber is long in the
z-direction so that the flow and heat transfer can be considered 2D. Assume
that all relevant dimensions are given. The flow enters the chamber with