318
10. Compressible Flow
the cell-center values (node P ) in place of values denoted by overbar; linear
extrapolation from W and P is also easily implemented on structured grids.
The mass fluxes constructed from these velocities do not, in general, satisfy the continuity equation and must therefore be corrected. Both the velocity and density need normally to be corrected, as described above. The
velocity correction is:
The convective (density) contribution to the mass flux correction would turn
out to be zero (since p; = (Cop1),, and p; = 0 since the pressure is prescribed);
however, although pressure is prescribed, the temperature is not fixed (it is
extrapolated from inside), so the density does need to be corrected. The
simplest approximation is the first-order upwind approximation, i.e. taking
p; = pb. The mass flux correction is then given by (10.13). Note, however,
that the density correction is not used to correct the density at the outflow
boundary - it is calculated always from the equation of state once the pressure
and temperature are calculated. The mass flux, on the other hand, has to
be corrected using the above expression, since only the correction used to
derive the pressure-correction equation does ensure mass conservation. Since,
at convergence, all corrections go to zero, the above treatment of density
correction is consistent with other approximations and does not affect the
accuracy of the solution, only the rate of convergence of the iterative scheme.
The coefficient for the boundary node in the pressure-correction equation
contains no contribution from the convective term (due to upwinding) - its
contribution goes to the central coefficient Ap. The pressure derivative in the
normal direction is usually approximated as:
where LP,E is the distance from cell center P to the outflow cell face E.
The coefficient Ap in the pressure correction equation for the control volume next to the boundary thus changes compared to those at inner CVs. Due
to the convective term in the pressure-correction equation and the Dirichlet
boundary condition where static pressure is specified, it usually converges
faster than for incompressible flow (where Neumann boundary conditions
are usually applied at all boundaries and the equation is fully elliptic).
Non-Reflecting and Free-Stream Boundaries. At some portions of the
boundary the exact conditions to be applied may not be known, but pressure
waves and/or shocks should be able to pass through the boundary without
reflection. Usually, one-dimensional theory is used to compute the velocity
at boundary, based on the prescribed free-stream pressure and temperature.
If the free-stream is supersonic, shocks may cross the boundary and one
10. Compressible Flow
the cell-center values (node P ) in place of values denoted by overbar; linear
extrapolation from W and P is also easily implemented on structured grids.
The mass fluxes constructed from these velocities do not, in general, satisfy the continuity equation and must therefore be corrected. Both the velocity and density need normally to be corrected, as described above. The
velocity correction is:
The convective (density) contribution to the mass flux correction would turn
out to be zero (since p; = (Cop1),, and p; = 0 since the pressure is prescribed);
however, although pressure is prescribed, the temperature is not fixed (it is
extrapolated from inside), so the density does need to be corrected. The
simplest approximation is the first-order upwind approximation, i.e. taking
p; = pb. The mass flux correction is then given by (10.13). Note, however,
that the density correction is not used to correct the density at the outflow
boundary - it is calculated always from the equation of state once the pressure
and temperature are calculated. The mass flux, on the other hand, has to
be corrected using the above expression, since only the correction used to
derive the pressure-correction equation does ensure mass conservation. Since,
at convergence, all corrections go to zero, the above treatment of density
correction is consistent with other approximations and does not affect the
accuracy of the solution, only the rate of convergence of the iterative scheme.
The coefficient for the boundary node in the pressure-correction equation
contains no contribution from the convective term (due to upwinding) - its
contribution goes to the central coefficient Ap. The pressure derivative in the
normal direction is usually approximated as:
where LP,E is the distance from cell center P to the outflow cell face E.
The coefficient Ap in the pressure correction equation for the control volume next to the boundary thus changes compared to those at inner CVs. Due
to the convective term in the pressure-correction equation and the Dirichlet
boundary condition where static pressure is specified, it usually converges
faster than for incompressible flow (where Neumann boundary conditions
are usually applied at all boundaries and the equation is fully elliptic).
Non-Reflecting and Free-Stream Boundaries. At some portions of the
boundary the exact conditions to be applied may not be known, but pressure
waves and/or shocks should be able to pass through the boundary without
reflection. Usually, one-dimensional theory is used to compute the velocity
at boundary, based on the prescribed free-stream pressure and temperature.
If the free-stream is supersonic, shocks may cross the boundary and one