Page 244 - Finite Element Analysis with ANSYS Workbench

12.2 Finite Volume Method 235

 U
 dA   ( I x F n y )E n  dS = 0
I
A t  S
where n and n are directions cosines of the unit vector normal to
x
y
the cell edge.
The integrand in the second integral term represents the
flux F normal to the cell edge,
n
F = E I n  F I n
y
x
n
So that the finite volume equations reduce to,
  U dA =   F dS
n
A t  S
The fluxes normal to the cell edge for the four Navier-Stokes
equations are,
 u  n x  v  n y    V n 
  2  p  u  n  uv n    uV  np 

F
 =  x y  =  n x 




n
  n x   uv 2  p  v  n y    vV n  np y 
    u n   pu v  pv    n    
 x y   V n  pV n 
In the above equation, V is the velocity normal to the cell edge.
n
As an example of a triangular cell in the figure, the normal velocity
to the cell edge is,
V = nu  n v y
x
n
while the tangential velocity to the cell edge is,
V =  u n  n v x
y
t

3 n y n ˆ 3 v V n
V t
S 1 n x S u
S 2 S 2 1
2 2
y S 3 y S 3
1 1

x x