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10.4 The Transonic Small Disturbance (TSD) Equation 301

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and the solution is not function of the interior discretization, but only a function
of the boundary conditions as required by Eq. (10.3.4). Schemes having this
property are referred to as conservative and preserve the correct shock jump
relations when discontinuities are present.

10.4 The Transonic Small Disturbance (TSD) Equation

The numerical solution of the transonic small disturbance equation has histori-
cal significance and Murman and Cole [2] were the first to numerically compute
steady transonic flows using this equation. Their breakthrough was to use cen-
tral difference operators in subsonic flow regions, and upwind difference opera-
tors in the supersonic regions. This seemingly simple idea gave rise to extensive
research in the field during the 1970's and eventually led to significant advances
in CFD. The conservative TSD equation, Eq. (10.2.2), is derived by making
an asymptotic expansion of the perturbation velocity potential. Eq. (10.2.2) is
rewritten as
(1 - Ml)<p x - I^M^ + <fiyy = 0 (10.4.1)

where the perturbation velocity potential has been scaled by [Too- The non-
conservative form of Eq. (10.4.1) is

[1 - Ml - (7 + 1)M^ X}^ XX + p yy = 0 (10.4.2)

The non-conservative form is used in the following description since its type
can be readily determined by evaluating the coefficient of the (p xx term. If pos-
itive, Eq. (10.4.2) is elliptic whereas the equation is hyperbolic if negative. The
limiting case where the coefficient is zero leads to Eq. (10.4.2) being parabolic.
Clearly, central difference operators can be used for the elliptic case, while up-
wind difference operators must be used on the streamwise derivative for the
hyperbolic case to retain the proper domain of dependence (see Section 2.6).
Because the body-surface boundary conditions of the TSD equation can
be shifted onto a mean plane, with the requirement that the flow direction be
tangential to the surface normal, a Cartesian based mesh is enough to discretize
the computational domain. One advantage of Cartesian meshes is that they lend
themselves to automatic mesh generation, a much more difficult task on body
conforming meshes (see Chapter 9). It is also natural to cluster points around
the leading and trailing edges of the airfoil, and around the shock region, if any,
to increase the flow resolution. If this is the case, appropriate terms need to be
added to the discrete form of Eq. (10.4.2) in the computational space.