10. Compressible Flow
10.1 Introduction
Compressible flows are important in aerodynamics and turbomachinery
among other applications. In high speed flows around aircraft, the Reynolds
numbers are extremely high and turbulence effects are confined to thin boundary layers. The drag consists of two components, frictional drag due to the
boundary layer and pressure or form drag which is essentially inviscid in nature; there may also be wave drag due to shocks which may be computed
from the inviscid equations provided that care is taken to assure that the
second law of thermodynamics is obeyed. If frictional drag is ignored, these
flows may be computed using the inviscid momentum Euler equations.
Due t o the importance of compressible flow in civilian and military applications, many methods of solving the equations of compressible flow have
been developed. Among these are special methods for the Euler equations
such as the method of characteristics and numerous methods that may be
capable of extension to viscous flows. Most of these methods are specifically
designed for compressible flows and become very inefficient when applied to
incompressible flows. A number of variations on the reason for this can be
given. One is that, in compressible flows, the continuity equation contains a
time derivative which drops out in the incompressible limit. As a result, the
equations become extremely stiff in the limit of weak compressibility, necessitating the use of very small time steps or implicit methods. Another version of
the argument is that the compressible equations support sound waves which
have a definite speed associated with them. As some information propagates
at the flow velocity, the larger of the two velocities determines the allowable
time step in an explicit method. In the low speed limit, one is forced to take
a time step inversely proportional to the sound speed for any fluid velocity; this step size may be much smaller than the one a method designed for
incompressible flows might allow.
Discretization and solution of the compressible flow equations can be carried out with methods already described. For example, to solve the timedependent equations, one can use any of the time-advance methods discussed
in Chap. 6. As the effect of diffusion is usually small in compressible flows
because the Reynolds numbers are high, there may be discontinuities e.g.
shocks, in the flow. Special methods for producing smooth solutions near
10.1 Introduction
Compressible flows are important in aerodynamics and turbomachinery
among other applications. In high speed flows around aircraft, the Reynolds
numbers are extremely high and turbulence effects are confined to thin boundary layers. The drag consists of two components, frictional drag due to the
boundary layer and pressure or form drag which is essentially inviscid in nature; there may also be wave drag due to shocks which may be computed
from the inviscid equations provided that care is taken to assure that the
second law of thermodynamics is obeyed. If frictional drag is ignored, these
flows may be computed using the inviscid momentum Euler equations.
Due t o the importance of compressible flow in civilian and military applications, many methods of solving the equations of compressible flow have
been developed. Among these are special methods for the Euler equations
such as the method of characteristics and numerous methods that may be
capable of extension to viscous flows. Most of these methods are specifically
designed for compressible flows and become very inefficient when applied to
incompressible flows. A number of variations on the reason for this can be
given. One is that, in compressible flows, the continuity equation contains a
time derivative which drops out in the incompressible limit. As a result, the
equations become extremely stiff in the limit of weak compressibility, necessitating the use of very small time steps or implicit methods. Another version of
the argument is that the compressible equations support sound waves which
have a definite speed associated with them. As some information propagates
at the flow velocity, the larger of the two velocities determines the allowable
time step in an explicit method. In the low speed limit, one is forced to take
a time step inversely proportional to the sound speed for any fluid velocity; this step size may be much smaller than the one a method designed for
incompressible flows might allow.
Discretization and solution of the compressible flow equations can be carried out with methods already described. For example, to solve the timedependent equations, one can use any of the time-advance methods discussed
in Chap. 6. As the effect of diffusion is usually small in compressible flows
because the Reynolds numbers are high, there may be discontinuities e.g.
shocks, in the flow. Special methods for producing smooth solutions near