7.3 Calculation of the Pressure
169
where 6 / 6 x represents a discretized spatial derivative (which could represent
a different approximation in each term) and Hi is shorthand notation for the
advective and viscous terms whose treatment is of no importance here.
For simplicity, assume that we wish to solve Eq. (7.17) with the explicit
Euler method for time advancement. We then have:
To apply this method, the velocity at time step n is used to compute H r
and, if the pressure is available, 6pn/6xi may also be computed. This gives
an estimate of pui a t the new time step n + 1. In general, this velocity field
does not satisfy the continuity equation:
We have stated an interest in incompressible flows, but these include flows
with variable density; this is emphasized by including the density. To see how
continuity may be enforced, let us take the numerical divergence (using the
numerical operators used to approximate the continuity equation) of Eq.
(7.18). The result is:
The first term is the divergence of the new velocity field, which we want to
be zero. The second term is zero if continuity was enforced at time step n;
we shall assume that this is the case but, if it is not, this term should be left
in the equation. Retaining this term is necessary when an iterative method is
used to solve the Poisson equation for the pressure and the iterative process is
not converged completely. Similarly, the divergence of the viscous component
of Hi should be zero for constant p, but a non-zero value is easily accounted
for. Taking all this into account, the result is the discrete Poisson equation
for the pressure pn:
Note that the operator 6/6xi outside the parentheses is the divergence operator inherited from the continuity equation, while 6p/6xi is the pressure
gradient from the momentum equations. If the pressure pn satisfies this discrete Poisson equation, the velocity field at time step n + 1 will be divergence
free (in terms of the discrete divergence operator). Note that the time step
169
where 6 / 6 x represents a discretized spatial derivative (which could represent
a different approximation in each term) and Hi is shorthand notation for the
advective and viscous terms whose treatment is of no importance here.
For simplicity, assume that we wish to solve Eq. (7.17) with the explicit
Euler method for time advancement. We then have:
To apply this method, the velocity at time step n is used to compute H r
and, if the pressure is available, 6pn/6xi may also be computed. This gives
an estimate of pui a t the new time step n + 1. In general, this velocity field
does not satisfy the continuity equation:
We have stated an interest in incompressible flows, but these include flows
with variable density; this is emphasized by including the density. To see how
continuity may be enforced, let us take the numerical divergence (using the
numerical operators used to approximate the continuity equation) of Eq.
(7.18). The result is:
The first term is the divergence of the new velocity field, which we want to
be zero. The second term is zero if continuity was enforced at time step n;
we shall assume that this is the case but, if it is not, this term should be left
in the equation. Retaining this term is necessary when an iterative method is
used to solve the Poisson equation for the pressure and the iterative process is
not converged completely. Similarly, the divergence of the viscous component
of Hi should be zero for constant p, but a non-zero value is easily accounted
for. Taking all this into account, the result is the discrete Poisson equation
for the pressure pn:
Note that the operator 6/6xi outside the parentheses is the divergence operator inherited from the continuity equation, while 6p/6xi is the pressure
gradient from the momentum equations. If the pressure pn satisfies this discrete Poisson equation, the velocity field at time step n + 1 will be divergence
free (in terms of the discrete divergence operator). Note that the time step