10.2 Pressure-Correction Methods for Arbitrary Mach Number
319
makes then a distinction between the parallel velocity component, which is
simply extrapolated to the boundary, and the normal component, which is
computed from theory. The latter condition depends on whether compression
or expansion (Prandtl-Meyer) waves hit the boundary. The pressure is usually
extrapolated from the interior to the boundary, while the normal velocity
component is computed using the extrapolated pressure and the prescribed
free-stream Mach number.
There are many schemes designed to produce non-reflecting and freestream boundaries. Their derivation relies on the outgoing characteristics
computed via one-dimensional theory; the implementation depends on the
discretization and the solution method. A detailed discussion of these (numerical) boundary conditions can be found in Hirsch (1991).
Supersonic Outflow. If the flow a t the outflow is supersonic, all of the
variables a t the boundary must be obtained by extrapolation from the interior, i.e. no boundary information needs to be prescribed. The treatment of
the pressure correction equation is similar to that in the case in which the
static pressure is prescribed. However, since the pressure at the boundary is
not prescribed but is extrapolated, the pressure correction also needs to be
extrapolated - it is not zero as in the above case. Since pk is expressed as a
linear combination of pb and p t , (if the pressure gradient can be neglected,
one may also set pb = pb), the node E does not occur in the algebraic equation, so AE = 0. The coefficients of nodes appearing in the approximation
of the mass-flux correction through the boundary are different from those in
the interior region.
Some examples of application of the pressure-correction scheme to solving
compressible flow problems are presented below. More examples can be found
in DemirdiiC et al. (1993) and in Lilek (1995).
10.2.3 Examples
We present below the results of the solution of Euler equations for a flow
over a circular arc bump. Figure 10.2 shows the geometry and the predicted
isolines of Mach number for the subsonic, transonic and supersonic conditions. The thickness-to-chord ratio of the circular arc is 10% for subsonic
and transonic cases and 4% for the supersonic case. Uniform inlet flow a t
Mach numbers Ma = 0.5 (subsonic), 0.675 (transonic) and 1.65 (supersonic)
is specified. Since Euler equations are solved, viscosity is set to zero and slip
conditions are prescribed a t walls (flow tangency, as for symmetry surfaces).
These problems were the test cases in a workshop in 1981 (se Rizzi and Viviand, 1981) and are often used to assess the accuracy of numerical schemes.
For subsonic flow, since the geometry is symmetric and the flow is inviscid,
the flow is also symmetric. The total pressure should be constant throughout
the solution domain, which is useful in assessing numerical error. In the transonic case, one shock is obtained on the lower wall. When the oncoming flow
319
makes then a distinction between the parallel velocity component, which is
simply extrapolated to the boundary, and the normal component, which is
computed from theory. The latter condition depends on whether compression
or expansion (Prandtl-Meyer) waves hit the boundary. The pressure is usually
extrapolated from the interior to the boundary, while the normal velocity
component is computed using the extrapolated pressure and the prescribed
free-stream Mach number.
There are many schemes designed to produce non-reflecting and freestream boundaries. Their derivation relies on the outgoing characteristics
computed via one-dimensional theory; the implementation depends on the
discretization and the solution method. A detailed discussion of these (numerical) boundary conditions can be found in Hirsch (1991).
Supersonic Outflow. If the flow a t the outflow is supersonic, all of the
variables a t the boundary must be obtained by extrapolation from the interior, i.e. no boundary information needs to be prescribed. The treatment of
the pressure correction equation is similar to that in the case in which the
static pressure is prescribed. However, since the pressure at the boundary is
not prescribed but is extrapolated, the pressure correction also needs to be
extrapolated - it is not zero as in the above case. Since pk is expressed as a
linear combination of pb and p t , (if the pressure gradient can be neglected,
one may also set pb = pb), the node E does not occur in the algebraic equation, so AE = 0. The coefficients of nodes appearing in the approximation
of the mass-flux correction through the boundary are different from those in
the interior region.
Some examples of application of the pressure-correction scheme to solving
compressible flow problems are presented below. More examples can be found
in DemirdiiC et al. (1993) and in Lilek (1995).
10.2.3 Examples
We present below the results of the solution of Euler equations for a flow
over a circular arc bump. Figure 10.2 shows the geometry and the predicted
isolines of Mach number for the subsonic, transonic and supersonic conditions. The thickness-to-chord ratio of the circular arc is 10% for subsonic
and transonic cases and 4% for the supersonic case. Uniform inlet flow a t
Mach numbers Ma = 0.5 (subsonic), 0.675 (transonic) and 1.65 (supersonic)
is specified. Since Euler equations are solved, viscosity is set to zero and slip
conditions are prescribed a t walls (flow tangency, as for symmetry surfaces).
These problems were the test cases in a workshop in 1981 (se Rizzi and Viviand, 1981) and are often used to assess the accuracy of numerical schemes.
For subsonic flow, since the geometry is symmetric and the flow is inviscid,
the flow is also symmetric. The total pressure should be constant throughout
the solution domain, which is useful in assessing numerical error. In the transonic case, one shock is obtained on the lower wall. When the oncoming flow