312
10. Compressible Flow
Here Qc-' represents the source term minus the contribution of the pressure
term; in what follows, the discretization methods applied to this term and
the pressure term are not important. Also, we consider only two time level
schemes and 2D geometry.
The velocities obtained by solving linearized momentum equations and
using 'old' pressure and density, Eq. (10.5), do not satisfy the mass conservation equations; that's why they carry an asterisk. When the mass fluxes
computed from these velocities and the 'old' density (denoted here by m * ;
see Eq. (8.12)) are inserted into the discretized continuity equation:
there results an imbalance Qh that must be eliminated by a correction
method. For incompressible flows, the mass flux and the velocity are essentially equivalent and the imbalance is corrected by correcting the velocity.
Since velocity correction is proportional to the gradient of the pressure correction, as was shown in Sect. 7.3.4, an equation for the pressure correction can
be derived and solved. This procedure is not applicable in the compressible
case.
In compressible flows the mass flux depends on both the velocity component normal to the cell face, v,, and the (variable) density, p. To correct the
mass flux imbalance, both the density and the velocity must be corrected.
The corrected mass flux on the 'e' face of a CV can be expressed as:
where p' and vk represent the density and velocity corrections, respectively.
The mass flux correction is thus:
The underscored term is usually neglected as it is of second order in the
corrections and thus becomes zero more rapidly than the other two terms.
Near convergence, this approximation is certainly permissible; one hopes that
it does not affect the rate of convergence of the method when the solution is far from converged. This term can be taken into account using a
predictor-corrector approach, as described in Sect. 8.8 for the treatment of
non-orthogonality in the pressure-correction equation.
The first of the two remaining terms in the mass flux correction is identical
to the one obtained for incompressible flows. In Sect. 8.8 it was shown that,
for the colocated variable arrangement, this term can be approximated in the
SIMPLE method as (see Eq. (8.59)):
10. Compressible Flow
Here Qc-' represents the source term minus the contribution of the pressure
term; in what follows, the discretization methods applied to this term and
the pressure term are not important. Also, we consider only two time level
schemes and 2D geometry.
The velocities obtained by solving linearized momentum equations and
using 'old' pressure and density, Eq. (10.5), do not satisfy the mass conservation equations; that's why they carry an asterisk. When the mass fluxes
computed from these velocities and the 'old' density (denoted here by m * ;
see Eq. (8.12)) are inserted into the discretized continuity equation:
there results an imbalance Qh that must be eliminated by a correction
method. For incompressible flows, the mass flux and the velocity are essentially equivalent and the imbalance is corrected by correcting the velocity.
Since velocity correction is proportional to the gradient of the pressure correction, as was shown in Sect. 7.3.4, an equation for the pressure correction can
be derived and solved. This procedure is not applicable in the compressible
case.
In compressible flows the mass flux depends on both the velocity component normal to the cell face, v,, and the (variable) density, p. To correct the
mass flux imbalance, both the density and the velocity must be corrected.
The corrected mass flux on the 'e' face of a CV can be expressed as:
where p' and vk represent the density and velocity corrections, respectively.
The mass flux correction is thus:
The underscored term is usually neglected as it is of second order in the
corrections and thus becomes zero more rapidly than the other two terms.
Near convergence, this approximation is certainly permissible; one hopes that
it does not affect the rate of convergence of the method when the solution is far from converged. This term can be taken into account using a
predictor-corrector approach, as described in Sect. 8.8 for the treatment of
non-orthogonality in the pressure-correction equation.
The first of the two remaining terms in the mass flux correction is identical
to the one obtained for incompressible flows. In Sect. 8.8 it was shown that,
for the colocated variable arrangement, this term can be approximated in the
SIMPLE method as (see Eq. (8.59)):