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7. Solution of the Navier-Stokes Equations
been solved for, the velocities are updated using Eqs. (7.36) and (7.37). This
is known as the SIMPLE algorithm (Caretto et al., 1972), an acronym whose
origin will not be detailed. We shall discuss its properties below.
A more gentle way of treating the last term in the pressure-correction
equation (7.39) is to approximate it rather than neglecting it. One could
approximate the velocity correction u: at any node by a weighted mean of
the neighbor values, for example,
u:,~ %
E L A,"' 4,l
El Al"' '
This allows us to approximate ii:,p from Eq. (7.38) as
which, when inserted in Eq. (7.37), leads to the following approximate relation
between u: and p':
With this approximation the coefficient A; in Eq. (7.39) is replaced by
A;4' + El A;' and the last term disappears. This is known as the SIMPLEC
algorithm (van Doormal and Raithby, 1984).
Still another method of this general type is derived by neglecting iii in
the first correction step as in the SIMPLE method but following the correction with another corrector step. The second correction to the velocity u" is
defined by (see Eq. (7.37)):
where ii!, is calculated from Eq. (7.38) after u: has been calculated from Eq.
(7.37) with iii neglected. Application of the discretized continuity equation
(7.33) to corrected velocities leads to the second pressure-correction equation:
Note that the coefficients on the left hand side are the same as in Eq. (7.39),
which can be exploited (a factorization of the matrix may be stored and
reused). Still further corrector steps can be constructed in the same way,
but this is seldom done. This procedure is essentially an iterative method for
solving Eq. (7.39) with the last term treated explicitly; it is known as the
PIS0 algorithm (Issa, 1986).
7. Solution of the Navier-Stokes Equations
been solved for, the velocities are updated using Eqs. (7.36) and (7.37). This
is known as the SIMPLE algorithm (Caretto et al., 1972), an acronym whose
origin will not be detailed. We shall discuss its properties below.
A more gentle way of treating the last term in the pressure-correction
equation (7.39) is to approximate it rather than neglecting it. One could
approximate the velocity correction u: at any node by a weighted mean of
the neighbor values, for example,
u:,~ %
E L A,"' 4,l
El Al"' '
This allows us to approximate ii:,p from Eq. (7.38) as
which, when inserted in Eq. (7.37), leads to the following approximate relation
between u: and p':
With this approximation the coefficient A; in Eq. (7.39) is replaced by
A;4' + El A;' and the last term disappears. This is known as the SIMPLEC
algorithm (van Doormal and Raithby, 1984).
Still another method of this general type is derived by neglecting iii in
the first correction step as in the SIMPLE method but following the correction with another corrector step. The second correction to the velocity u" is
defined by (see Eq. (7.37)):
where ii!, is calculated from Eq. (7.38) after u: has been calculated from Eq.
(7.37) with iii neglected. Application of the discretized continuity equation
(7.33) to corrected velocities leads to the second pressure-correction equation:
Note that the coefficients on the left hand side are the same as in Eq. (7.39),
which can be exploited (a factorization of the matrix may be stored and
reused). Still further corrector steps can be constructed in the same way,
but this is seldom done. This procedure is essentially an iterative method for
solving Eq. (7.39) with the last term treated explicitly; it is known as the
PIS0 algorithm (Issa, 1986).