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296 10. Inviscid Compressible Flow

challenge to the CFD community in the 1970's. The Transonic Small Distur-
bance (TSD) equation was first solved by Murman and Cole in a landmark
paper [2], in which they introduce a switch from elliptic to hyperbolic difference
operators. The theory is discussed in Section 10.4 and numerical results on a
non-lifting airfoil presented in Section 10.5. This type-differencing scheme was
extended to solve the full-potential equation by Jameson [3], and Section 10.6
will describe the added complexity of applying such operators when the flow is
not aligned with the streamwise coordinates.
All the previously described difficulties are imbedded in the compressible
continuity equation, which involves scalar algebra. Solving the equations for
conservation of mass and momentum simultaneously involves matrix algebra, as
discussed in Chapter 5. Appropriate boundary conditions must then be imposed,
especially at the far-field boundaries and a complete theory using the method
of Characteristics has been developed. These issues, presented in Section 10.7,
were tackled during the 1980's and the methods developed have lead directly to
the solution of the incompressible and compressible Navier-Stokes equations in
Chapters 11 and 12, respectively.
Stability analyses of the finite-difference operators discussed in Chapter 5
indicated that upwinding must be used to achieve a stable numerical algorithm
but upwinding reduces the accuracy of the discretized equations while being
unconditionally stable under some circumstances due to the truncation error
terms. To retain second-order accuracy, central difference operators must be
coupled to added dissipation operators to obtain converged flow solutions. This
dissipation can either be added explicitly (artificial dissipation), or implicitly
by the truncation error of the finite-difference operators (numerical viscosity),
which are discussed in Section 10.8
The explicit MacCormack scheme applied to the one-dimensional compress-
ible Euler equations is presented in Section 10.9. Applications of the scheme to
the unsteady 1-D Euler equations and the steady ID Euler equation with the
addition of a source terms on a nozzle flow computation are found in Sections
10.10 and 10.11, respectively.
The implicit method of Beam-Warming with explicit numerical dissipation
is presented in Section 10.12, with applications on the same model problems as
discussed above presented in Sections 10.13 and 10.14.

10.2 Shock Jump Relations

The compressible flow equations are typically used to solve flow situations that
contain discontinuous flow regimes such as a shock wave. Shock waves are pro-
duced when the gas undergoes a sudden compression and the second law of ther-
modynamics models the entropy jump associated with the compression wave.
As opposed to elliptic equations, the compressible flow phenomena are non-