Page 349 - Computational Fluid Dynamics for Engineers

11.5 Model Problem: Sudden Expansion Laminar Duct Flow 339

The values of Uij for 0 < i < I and 0 < j < J are obtained by linear inter-
polation. From the continuity equation and the definition of Uij, the normal
velocity Vij can be written as

y<H-h
v(x,y)
y> H-h
2
3tf
(11.5.11)
The pressure pij is assumed to be zero.

11.5.3 Solution Procedure and Sample Calculations

With the boundary and initial conditions specified, the solution of Eq. (11.4.24)
can be obtained by the line iterative (ADI) method discussed in the previous
section. The computer program given in Appendix B has nine subroutines and
a MAIN program. Here we present sample calculations for this problem and
discuss the behavior of the solutions as a function of the pseudo-time step At
and the parameter /?.
J
The input data, given in MAIN, comprises a uniform grid (J, ), the geo-
:
metrical parameters (L,H, h), the Reynolds number RL = ^rS the parameter
j3. In addition, the time marching algorithm is provided with an upper limit
number of time iterations NLIMIN and a possibility for nonuniform time steps.
These at first increase linearly with initial Ato up to a specified Atf and remain
constant thereafter. The initial conditions (including boundary conditions) are
also specified in MAIN. Convergence is based on the solutions of Eq. (11.4.24)
obtained by solving Eqs. (11.4.27) and (11.4.28) iteratively. The banded matrix
B in Eq. (11.4.27) is computed in the subroutines FLUX_DX, PHLIH. Its right
hand side is computed in the subroutine RESIDUALX, the banded matrix B
in Eq. (11.4.28) in the subroutines FLUX.DF, PHLJH, and its right hand side
computed in the subroutine RESIDUALY. The unknowns AD are obtained by
the ADI method in the subroutine SOLVER.
Once a solution to Eq. (11.4.24) is obtained, the boundary conditions are up-
dated in MAIN and, since the boundary conditions at x = L are approximated,
it is also necessary to verify the conservation of mass determined at x = 0 by
integrating Eq. (11.5.1a) across y = 0 and y = H at x = L.
Figure 11.3 shows the effect of (3 on the velocity profiles U/UQ across the
duct at several x locations (x = 0.9, 1.8, 9, 15). These calculations were made
by taking the following parameters: / = 200, J = 100, H = 1, h = 1/2, R^ =
25, with f3 varying from 100, 50, 1, (the range as recommended in [8]) Ato —
- 5
0.1, Atf = 1, and for a convergence criteria \AD\ < 10 . As can be seen, the
velocity profiles are essentially independent of /3.