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234 Chapter 12 Compressible Flow Analysis

2  u v  2  v u 
 = μ 2   ;  =  2   ;
x
3  x y  y 3  y x 
 u v 
 =    
xy
 y x 
The heat fluxes q and q vary with the temperature T according to
x
y
the Fourier’s law,
 T  T
q =  k x  ; q =  k y 
y
x
The fluid thermal conductivity k is determined from,
k = c  Pr
p
where Pr is the Prandtl number and  is the dynamic viscosity that
can be determined from the Sutherland’s law.

12.2 Finite Volume Method

12.2.1 Finite Volume Equations
For simplicity in understanding the derivation of the
finite volume equations, we will concentrate on the inviscid flow
analysis. A typical equation representing any one of the four
Navier-Stokes equations can be written in the form,
 U   E I   F I = 0
t  x  y 

If we consider the mass equation, then U   ; E  u  ; F  v  .
I
I
Similarly, if we consider the x-momentum equation, then U  u  ;
E   u  p ; F   uv. To derive the finite volume equations,
2
I
I
the method of weighted residuals is employed with unit weighting
function to yield,
 U dA      E I   F  I   dA = 0
t  x  y   
A A
The Gauss’s theorem is applied to introduce the boundary integral
term so that the equations become,