Page 373 - Computational Fluid Dynamics for Engineers

364 12. Compressible Navier-Stokes Equations

The terms on the right all have a similar form: for example

d S S i 2 i 1 i( i
z+l/2j = ^[ i +i/2j(Q^^J~Q^~ i+l/2j(Q + d~^ + d~^' ^ d~Qi-lj)]
(12.5.12)
where h v is the area of the cell. For subsonic flows, only the 4 t h order dissipation
coefficient is needed while for supersonic flows, a 2 n d order coefficient is added
to reduce dispersion effects around shock waves. Define

» • • - « ± ^ - ^ . + f t - W . . (12.5.13a)
\Pi+ij\ + 2\Pij\ + \Pi-ij\

which activates when large pressure gradient occur. Then
e w{2)
i+i/2j = ™*x("i+ij,Vij), e£ 1/2J = max(0, w^ - s^ 1/2J)
(12.5.13b)
2
where typical values of the constants x&( ' and w^> are
w(2) = w(4) (12513c)
i = 2k --

Next, a residual smoother is introduced not only to add additional implicit
characters, similar to the MacCormack method, but also to smooth the high
frequency variations of the residual. The high frequency residual smoothing
concept is used as a smoother in the multi-grid method and can speed-up the
calculations. Here, we discuss explicit smoothing and implicit smoothing. The
explicit smoothing residual R is defined by

R
Rij = Rij + £{S X + Sy) ij
= e(Ri-i,j + Ri+ij + Rij-i + Ri,j+i) + (1 - *e)Ri,j (12.5.14)

The implicit average is defined by

[l-e(6l + Sl)]Rij = Rij (12.5.15)
which can be approximated using ADI factorization

(1 - e6l)(l - eS^Rij = Rij. (12.5.16)

The solution of Eq. (12.5.16) can be obtained by solving the following two
equations
(1 - e6l)R?j = Rij (12.5.17a)
and
(l-eS^RTjRlj. (12.5.17b)
These equations form the following block tridiagonal systems

- eR*_ hj + (1 + 2s)R*j - sR* +1J = Rij (12.5.18a)