Page 344 - Computational Fluid Dynamics for Engineers

334 11. Incompressible Navier-Stokes Equations

«E*H
Ay

—» a J) — »
|(i-l/2,j) ,0+1/2, jb
<N

- 1 ^ -
— • Fig. 11.1. Difference stencil for derivatives
L\X in the x-direction.

<Pi+l/2,j ~ AE t+l/2,j ~ AE i+l/2,j (11.4.17a)
(11.4.17b)
&J+1/2 = ^ ^ ' + 1 / 2 " ^ 7 i + l / 2
where AE ± and AF ± are the flux difFerences across the positive or negative
traveling waves. As discussed in [9], they are given by

(11.4.18a)

[11.4.18b)
Here

D
A+i/2,j = ^ A + i j + M") ^ A + i / 2 , j = A+i,j - A , j (11.4.19a)
(
A j + i / 2 = « ( A j + i + A j ) ^ A j + i / 2 = A j + i - A j (11.4.19b)

t
11.4.3 Discretization of he Viscous Fluxes
The discretization of the viscous fluxes is much more simple than the discretiza-
tion of the convective fluxes. Viscous diffusion occurs in all direction, and the
discretization of the viscous terms is always performed with central formulas
[11]. Here we approximate the viscous fluxes in Eqs. (P2.17.2) with second-
order accuracy on a compact stencil

E
E
6E V _ ( v)i+l/2,j - ( v)i-l/2,j
(11.4.20a)
dx Ax
F
F
_ ( v)jj+l/2 - ( v)i,j-l/2
dF v
(11.4.20b)
dy Ay