4.7 Examples
89
than the original Eq. (4.40). Equation (4.42) is called the modified equation
for this problem. By transforming this equation into coordinates parallel and
perpendicular to the flow, one can show that the effective diffusivity in the
normal direction is:
r,fi = U sin 6 cos AX cos 6 + Ay sin 6) ,
(4.43)
where U is the magnitude of the velocity and 6 is the angle of the flow with
respect to the x-direction. A similar and widely quoted result was derived by
de Vahl Davis and Mallinson (1972).
To sum up our findings, we have shown:
0 High-order schemes oscillate on coarse grids but converge to an accurate
solution more rapidly than low order schemes as the grid is refined.
0 First-order UDS is inaccurate and should not be used. This scheme is
mentioned because it is still used in some commercial codes. Users should
be aware that high accuracy cannot be obtained on affordable grids with
this method, especially in 3D. It introduces a large diffusive error in both
the streamwise and normal directions.
0 CDS is the simplest scheme of second-order accuracy and offers a good
compromise among accuracy, simplicity and efficiency.