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236 Chapter 12 Compressible Flow Analysis

12.2.2 Computational Procedure

The finite volume equations are discretized by applying
the forward difference approximation to the integral term associat-
ed with time,

  U dA = U m 1  U m A
A t  t 
th
where superscript m refers to the m step and t is the time step.
For the integral term associated with the flux across the cell edge,
we replace it by the numerical flux,


 FdS =   FdS
n
n
S S
So that the finite volume equations become,

m
U m 1  U =  t    FdS
n
A S
~
The numerical flux F from the left cell L to the right cell R with
n
the common edge of length  is determined using the Roe’s
S
averaging method,
~ 1 1
F = ( n F )F   A ( U )U 
2 nL nR 2 L R
where F and F are the fluxes of the left cell L and the right
nL
nR
cell R, respectively. The determinant A is computed from the
Jaco-bian matrix which will be shown later. The quantities U and
L
U represent the conservation variables of the left cell L and the
R
right cell R, respectively. The final form of finite volume equations
becomes,

U m 1  U =  m t     F  m m A m ( F  m U m )U  
2A S  nL nR L R 
m
The computational procedure starts from using U at
time step m to determine U m 1 at time step m+1. The procedure is
performed for all the cells in the flow domain for transient analysis.
For steady-state analysis, the computation is terminated when the