Page 346 - Computational Fluid Dynamics for Engineers

336 11. Incompressible Navier-Stokes Equations

f)R n i f)

t
"lj
(11.4.25)
v -\B--,, 2 + B. . . / 2 - B. . 1 / 2 - S.
/
2 Ay M+V2 ^ M-i/2 M + I / 2 "ij-i/2>
dR?j l , , ^ ^ a
Z (A +lj + A
^ 0^77 * 2^ * ^ii/aj i+i/2j) ^ ^
n
BTR 1 iQ

where I m is a modified identity matrix given by

TO o 0]
0
0]
ro
1
0
I m = 0 0 1 0 (11.4.26)
.0 0 lj
The solution of Eq. (11.4.24) is obtained with the ADI method discussed in
subsection 4.5.2. Thus, we first write Eq. (11.4.24) for a given time n+1. Along
the x-direction, this gives
B[X,Y,Z]AD = - R - ADj- ±V - ADj+iW (11.4.27)
which is solved with the block elimination method discussed in subsection 4.4.3
[see Eqs. (4.4.32) and (4.4.34)]. Similarly, Eq. (4.2.24) is written along the y-
direction,

B[V, y, W]AD = - R - XADi- lt - ZAD i+h (11.4.28)

which is again solved with the block elimination method. Equations (11.4.27)
and (11.4.28) are solved iteratively until convergence of the iterative process due
to the calculation of the inexact Jacobian. Then the time step is incremented
and the above procedure is repeated for the next time step until convergence of
the steady-state system (11.4.5).

11.5 Model Problem: Sudden Expansion Laminar Duct Flow

To demonstrate the solution of the incompressible Navier-Stokes equations with
the numerical method of Section 11.4, we consider a model problem correspond-
ing to the symmetrical half of a laminar flow downstream of a sudden expansion
in a duct of length L and height H as shown in Fig. 11.2.