10.3 Methods Designed for Compressible Flow
325
mentioned several times in this book that there are methods specifically designed for the solution of compressible flows. In particular, these methods can
be used in conjunction with the artificial compressibility methods described
in Chap. 7. In this section, we briefly describe some of these methods. The
purpose is to give enough information about these methods to allow comparison with the methods described above. We shall not present them in sufficient
detail to allow the reader to develop codes based on them. The latter task requires a separate volume; readers interested in such a treatment are referred
to the texts by Hirsch (1991) and Anderson et al. (1984).
Historically, the development of methods for the computation of compressible flows proceeded in stages. Initially (until about 1970), only the
equations for linearized potential flow were solved. Later, as computer capacity increased, interest moved progressively to the non-linear potential flow
equations and, in the 1980's, to the Euler equations. Methods for the viscous
or Navier-Stokes equations (more properly, the RANS equations, because the
high Reynolds numbers assure that the flows are turbulent) are the subject
of current research and are proving very difficult to develop. So, we see that,
in contrast to the situation for low speed flows, most solvers for high speed
flows are designed t o deal only with the inviscid case.
If there is a major theme running through these methods, it is explicit
recognition that the equations are hyperbolic and thus have real characteristics along which information about the solution travels at finite speeds. The
other essential issue (which arises from the existence of characteristics) is
that the compressible flow equations support shock waves and other kinds
of discontinuities in the solutions; the discontinuities are sharp in inviscid
flows but have finite width when the viscosity is non-zero. Respecting these
properties is important so it is explicitly taken into account in most methods.
These methods are mainly applied to the aerodynamics of aircraft, rockets, and turbine blades. In nearly all cases, the flow is steady. Since the speeds
are high, explicit methods would need to use very small time steps and would
be very inefficient. Consequently, implicit methods would be useful and have
been developed. However, the nature of the equations makes it difficult to
construct efficient implicit methods and, as we shall see below, many of the
methods used are explicit.
The need to treat discontinuities raises another set of issues. We have
seen that, in the attempt to capture any kind of rapid change in a solution,
discretization methods are likely to produce results that contain oscillations
or 'wiggles'. This is especially so when non-dissipative discretizations (which
includes essentially all central-difference schemes) are used. A shock (or any
other discontinuity) represents the extreme of a rapidly varying solution and
therefore presents the ultimate challenge to discretization methods. It can be
shown that no discretization method of order higher than first can guarantee
a monotonic solution when the solution contains discontinuities. Since accuracy is best obtained through the use of central-difference methods (or their
325
mentioned several times in this book that there are methods specifically designed for the solution of compressible flows. In particular, these methods can
be used in conjunction with the artificial compressibility methods described
in Chap. 7. In this section, we briefly describe some of these methods. The
purpose is to give enough information about these methods to allow comparison with the methods described above. We shall not present them in sufficient
detail to allow the reader to develop codes based on them. The latter task requires a separate volume; readers interested in such a treatment are referred
to the texts by Hirsch (1991) and Anderson et al. (1984).
Historically, the development of methods for the computation of compressible flows proceeded in stages. Initially (until about 1970), only the
equations for linearized potential flow were solved. Later, as computer capacity increased, interest moved progressively to the non-linear potential flow
equations and, in the 1980's, to the Euler equations. Methods for the viscous
or Navier-Stokes equations (more properly, the RANS equations, because the
high Reynolds numbers assure that the flows are turbulent) are the subject
of current research and are proving very difficult to develop. So, we see that,
in contrast to the situation for low speed flows, most solvers for high speed
flows are designed t o deal only with the inviscid case.
If there is a major theme running through these methods, it is explicit
recognition that the equations are hyperbolic and thus have real characteristics along which information about the solution travels at finite speeds. The
other essential issue (which arises from the existence of characteristics) is
that the compressible flow equations support shock waves and other kinds
of discontinuities in the solutions; the discontinuities are sharp in inviscid
flows but have finite width when the viscosity is non-zero. Respecting these
properties is important so it is explicitly taken into account in most methods.
These methods are mainly applied to the aerodynamics of aircraft, rockets, and turbine blades. In nearly all cases, the flow is steady. Since the speeds
are high, explicit methods would need to use very small time steps and would
be very inefficient. Consequently, implicit methods would be useful and have
been developed. However, the nature of the equations makes it difficult to
construct efficient implicit methods and, as we shall see below, many of the
methods used are explicit.
The need to treat discontinuities raises another set of issues. We have
seen that, in the attempt to capture any kind of rapid change in a solution,
discretization methods are likely to produce results that contain oscillations
or 'wiggles'. This is especially so when non-dissipative discretizations (which
includes essentially all central-difference schemes) are used. A shock (or any
other discontinuity) represents the extreme of a rapidly varying solution and
therefore presents the ultimate challenge to discretization methods. It can be
shown that no discretization method of order higher than first can guarantee
a monotonic solution when the solution contains discontinuities. Since accuracy is best obtained through the use of central-difference methods (or their