Page 144 - Introduction to Computational Fluid Dynamics

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5.3 METHOD OF SOLUTION
where the suffix eff is added for two reasons. Firstly, note that this equation arises 12:28 123
out of comparison with Equation 5.74; secondly, α eff is not a global constant but
will vary for each node (i, j). In fact, this variation also proves to be most appropri-
ate. This can be understood as follows. When AP i, j + Sp i, j is small, the change in
from iteration level l to l + 1 will be large (see Equation 5.65). It is precisely this
large change that is to be controlled by underrelaxation. Equation 5.75 shows that
α eff is indeed small when AP i, j + Sp i, j is small. Conversely, when AP i, j + Sp i, j
is large, the implied change in is small; therefore, we can afford a larger value
of α. Thus, underrelaxation through the false-transient method is proportionate to
the requirement. Of course, the smaller the value of the false t, the smaller is the
value of the estimated α eff .
Although in most nonlinear problems use of constant α suffices, the false-
transient method needs to be invoked when couplings between equations for dif-
ferent s are strong or when the source terms for a given vary greatly over a
domain or when the initial guess of different variables is very poor. Most practi-
tioners invoke the false-transient method when the global underrelaxation method
fails.

5.3.4 Boundary Conditions for Φ
In fluid flow and convective transport, five types of boundaries are encountered:
inflow, outflow or exit, symmetry, wall, and periodic. At all these boundaries, mainly
three types of conditions are encountered:

1. b specified,
2. ∂ /∂n| b specified, and
2
2
3. ∂ /∂n | b specified,
where n is normal to the boundary. We shall discuss each boundary type separately.

Inflow Boundary
At the inflow boundary, values of all variables are specified and are therefore
8
known. Thus, at a west boundary (see Figure 5.4), for example, we can write
Su 2, j = Su 2, j + AW 2, j 1, j , Sp 2, j = Sp 2, j + AW 2,J , AW 2, j = 0.

(5.76)

8
Care is needed in specifying inflow conditions for turbulence variables e and . Typically, e in =
2
(Tu u in ) , where Tu is the prescribed turbulence intensity. Now, the dissipation is specified through
2
the definition of turbulent viscosity. Thus, in = C µ ρ e /(µ VISR), where the ratio VISR = µ t /µ
is assumed (typically, of the order of 20 to 40). In practical applications, Tu and VISR are rarely
known and, therefore, the analyst must assume their magnitudes.