194
7. Solution of the Navier-Stokes Equations
where ii,' is shorthand notation for
Analogously, v,!, can be expressed as:
The velocities u* and v* need to be corrected to enforce mass conservation.
This is done, as outlined in Sect. 7.3.4, by correcting the pressure. The corrected velocities - the final values for the mth outer iteration - urn = u* + u'
and vm = v* + v' are required to satisfy the linearized momentum equations,
which is possible only if the pressure is corrected. So we write:
and
where pm = pm-' + p' is the new pressure. The indices refer t o the mass
CV. The relation between velocity and pressure corrections is obtained by
subtracting Eq. (7.102) from Eq. (7.105):
where 6' is defined as (see Eq. (7.103)):
Analogously, one obtains:
The corrected velocities are required to satisfy the continuity equation,
so we substitute urn and urn into the expressions for mass fluxes, Eq. (7.83),
and use Eq. (7.101):
7. Solution of the Navier-Stokes Equations
where ii,' is shorthand notation for
Analogously, v,!, can be expressed as:
The velocities u* and v* need to be corrected to enforce mass conservation.
This is done, as outlined in Sect. 7.3.4, by correcting the pressure. The corrected velocities - the final values for the mth outer iteration - urn = u* + u'
and vm = v* + v' are required to satisfy the linearized momentum equations,
which is possible only if the pressure is corrected. So we write:
and
where pm = pm-' + p' is the new pressure. The indices refer t o the mass
CV. The relation between velocity and pressure corrections is obtained by
subtracting Eq. (7.102) from Eq. (7.105):
where 6' is defined as (see Eq. (7.103)):
Analogously, one obtains:
The corrected velocities are required to satisfy the continuity equation,
so we substitute urn and urn into the expressions for mass fluxes, Eq. (7.83),
and use Eq. (7.101):