Page 374 - Computational Fluid Dynamics for Engineers

12.6 Model Problem: Sudden Expansion Laminar Duct Flow 365

and
- eRtj.^l + 2£)i?-. e /?= + 1 J i?* j (12.5.18b)

which can be solved by using the block elimination method, see subsection 4.4.3.
The region close to the boundaries are treated separately from the procedure
used for Eq. (12.5.5) and is discussed in Appendix 12B.

12.6 Model Problem: Sudden Expansion Laminar Duct Flow

To demonstrate the solution of the Navier-Stokes equation for a two-dimensional
compressible flow with the methods discussed in Sections 12.3 to 12.5, we again
use the model problem discussed in Section 11.5 for the INS method. Clearly
the boundary and initial conditions for the velocity profiles remain the same.
The initial and boundary conditions for temperature T and pressure p must
be specified. We use the following procedure for a given Prandtl number Pr,
Mach number M ^ and ratio of specific heats 7, with the dynamic viscosity \i
calculated from, n <i26i)
»=^ W%k -

where
= 198.6 r ^ = 540° (12.6.2)
C 2

12.6.1 Initial Conditions

We assume
£ = 1 . 0 , T = 1 . 0 (12.6.3)
and compute p and Et from

12.6.2 Boundary Conditions

Along the wall (y = 0), the wall pressure p w can be calculated from

| = 0 , (12.6.5)

and approximated by
Pw = Pi,i ~ (Pi,2 ~ Pz,i)/2 (12.6.6)
with the known interior values. Similarly, the wall temperature can be calculated
from the adiabatic relation
dT
^ = 0, (12.6.7)