314
10. Compressible Flow
If Eq. (10.10) is used to express pb in terms of pi, and the approximation
(10.13) of the mass flux correction is substituted into this equation, we arrive
at an algebraic system of equations for the pressure correction:
The coefficients in this equation depend on the approximations used for the
gradients and cell face values of the pressure correction. The part which stems
from the velocity correction is identical to that for the incompressible case,
see Eqs. (8.59) and (10.13). The second part depends on the approximation
used for the 'convective' term; it corresponds to the convective contribution
to the conservation equations, see Chap. 4 and 7 for examples.
Despite the similarity in appearance to the pressure-correction equation
for incompressible flows, there are important differences. The incompressible
equation is a discretized Poisson equation, i.e. the coefficients represent an
approximation to the Laplacian operator. In the compressible case, there are
contributions that represent the fact that the equation for the pressure in a
compressible flow contains convective and unsteady terms, i.e. it is actually
a convected wave equation. For an incompressible flow, if the mass flux is
prescribed at the boundary, the pressure may be indeterminate to within
an additive constant. The presence of convective terms in the compressible
pressure equation makes the solution unique.
The relative importance of the two terms in the mass flux correction depends on the type of flow. The diffusive term is of order 1/Ma2 relative to the
convective term so the Mach number is the determining factor. At low Mach
numbers, the Laplacian term dominates and we recover the Poisson equation.
On the other hand, at high Mach number (highly compressible flow), the convective term dominates, reflecting the hyperbolic nature of the flow. Solving
the pressure-correction equation is then equivalent to solving the continuity
equation for density. Thus the pressure-correction method automatically adjusts to the local nature of the flow and the same method can be applied to
the entire flow region.
For the approximation of the Laplacian, central difference approximations are always applied. On the other hand, for the approximation of convective terms a variety of approximations may be used, just as is the case
for the convective terms in the momentum equations. If higher-order approximations are used, the 'deferred correction' method may be used. On the
left-hand side of the equation, the matrix is constructed on the basis of the
first-order upwind approximation while the right-hand side contains the difference between the higher-order approximation and the first-order upwind
approximation, assuring that the method converges to the solution belonging
to the higher-order approximation; see Sect. 5.6 for details. Also, if the grid
is severely non-orthogonal, deferred correction can be used to simplify the
pressure-correction equation as described in Sect. 8.8.
10. Compressible Flow
If Eq. (10.10) is used to express pb in terms of pi, and the approximation
(10.13) of the mass flux correction is substituted into this equation, we arrive
at an algebraic system of equations for the pressure correction:
The coefficients in this equation depend on the approximations used for the
gradients and cell face values of the pressure correction. The part which stems
from the velocity correction is identical to that for the incompressible case,
see Eqs. (8.59) and (10.13). The second part depends on the approximation
used for the 'convective' term; it corresponds to the convective contribution
to the conservation equations, see Chap. 4 and 7 for examples.
Despite the similarity in appearance to the pressure-correction equation
for incompressible flows, there are important differences. The incompressible
equation is a discretized Poisson equation, i.e. the coefficients represent an
approximation to the Laplacian operator. In the compressible case, there are
contributions that represent the fact that the equation for the pressure in a
compressible flow contains convective and unsteady terms, i.e. it is actually
a convected wave equation. For an incompressible flow, if the mass flux is
prescribed at the boundary, the pressure may be indeterminate to within
an additive constant. The presence of convective terms in the compressible
pressure equation makes the solution unique.
The relative importance of the two terms in the mass flux correction depends on the type of flow. The diffusive term is of order 1/Ma2 relative to the
convective term so the Mach number is the determining factor. At low Mach
numbers, the Laplacian term dominates and we recover the Poisson equation.
On the other hand, at high Mach number (highly compressible flow), the convective term dominates, reflecting the hyperbolic nature of the flow. Solving
the pressure-correction equation is then equivalent to solving the continuity
equation for density. Thus the pressure-correction method automatically adjusts to the local nature of the flow and the same method can be applied to
the entire flow region.
For the approximation of the Laplacian, central difference approximations are always applied. On the other hand, for the approximation of convective terms a variety of approximations may be used, just as is the case
for the convective terms in the momentum equations. If higher-order approximations are used, the 'deferred correction' method may be used. On the
left-hand side of the equation, the matrix is constructed on the basis of the
first-order upwind approximation while the right-hand side contains the difference between the higher-order approximation and the first-order upwind
approximation, assuring that the method converges to the solution belonging
to the higher-order approximation; see Sect. 5.6 for details. Also, if the grid
is severely non-orthogonal, deferred correction can be used to simplify the
pressure-correction equation as described in Sect. 8.8.