Page 143 - Introduction to Computational Fluid Dynamics

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2D CONVECTION – CARTESIAN GRIDS
Special care is, however, needed for the mass residual. On staggered grids, the
mass residual R m is checked via Equation 5.30 [51]. That is,
0.5

2
R m = ( ˙ m P ) . (5.72)
all nodes
However,oncollocatedgrids,onecannotusethisequationdirectlybecause ˙ m i, j = 0
even at convergence. Therefore, Equation 5.72 is written as
⎡ ⎤ 0.5
2

R m = ⎣ AP p − A k p ⎦ , (5.73)
m,i, j m,k
all nodes k
where AP and A k are coefficients of the pressure-correction equation. It will
be recognized that this equation simply represents the discretised version of the
left-hand side of Equation 5.32 (or see Equation 5.28 with ˙ m R = 0). Thus, R m is
evaluated after p is recovered in step 4 of the calculation procedure. This is
m,i, j
an important departure from the staggered-grid practice that a casual reader may
overlook.

5.3.3 Underrelaxation

Global Relaxation
As mentioned in Chapter 2, in steady-state problems ( t →∞), underrelaxation
is effected by augmenting Su and Sp as
(1 − α)
l
Su i, j = Su i, j + B , Sp i, j = Sp i, j + B, B = (AP i, j + Sp i, j ),
i, j
α
(5.74)
where α is the underrelaxation factor and l is the iteration level. The value of α is
the same for all nodes but it may be different for different s. This is called global,
or constant, underrelaxation.

False Transient
In multidimensional problems, underrelaxation is often effected in another way.
Thus, consider a steady-state problem in which t =∞ and, therefore, the transient
term is zero. However, one can imagine that the steady state is achieved following
a transient and each time step is likened to a change in iteration level by one. In
this case, o may be viewed as l and the time step t as the false-transient
i, j i, j
step. Then, combining Equation 5.65 with Equation 5.74, we can deduce that the
resulting equation may be viewed as one in which

AP i, j + Sp i, j
α eff,i, j = , (5.75)
o
AP i, j + Sp i, j + (ρ V/ t) i, j