The inverse problem has been formulated well before it could be reasonably solved by
numerical techniques. Goldstein’s thin airfoil theory (Goldstein, 1945) and
Theodorsen’s exact theory (Theodorsen, 1932) resulted so elaborate that airfoil
design could not be afforded.

Most of the methods falling under this category make use of some conformal mapping.
A pionering method for incompressible flow has been formulated by Lighthill, 1945. The
airfoil is mapped onto a circle in the complex plane. The surface speed distribution
is prescribed in the complex plane as function of the polar angle .

Eqs. 1 are a landmark in the history of aerodynamic design, and have been referred to
in all the literature concerned with the inverse problem. The methods of Woods (1955),
Arlinger (1970), Strand (1973) and Polito (1974) are improvements of Lighthill’s
formulation at various degrees.

Wood found that constraints similar to Eqs. 1 are required for the solution of the
mixed-inverse problem, and using a Karman-Tsien model gas approximation extended the
theory to compressible flows. Arlinger and Polito reformulated the problem in a way
such that the surface speed distribution could be prescribed as a function of the
airfoil arclength in the physical plane. In Arlinger’s method the prescribed
pressure is adjusted with the shape functions and their coefficients, that require to
be computed.

The solution also requires the inversion of a matrix, whose entries
depend on the shape functions. For a random choice of these functions the matrix may
be ill-conditioned, which turns out causing a severe correction to the pressure
distribution. More recently this method has been revisited by Drela 1988. The
prescribed pressure distribution is corrected with two shape functions inside the
target segment.

The non-linearity in the boundary conditions leads to a further problem: the one of
generating boundary-fitted coordinate system. Drela and Giles’ transonic design
method is based on a finite-volume formulation of the Euler equations (Drela-Giles, 1987),
discretized on an intrinsic grid in which one family of coordinate lines is made of
streamlines. The advantage of this formulation is at least twofold.

From the point
of view of the flow analysis the solution is simplified. In fact, the continuity and
energy equation are reduced to simple constant mass flux and stagnation enthalpy in
each streamtube, because there is no convection through the streamlines. Therefore
the number of unknowns per grid node is reduced to two. The inverse solution of the
equations comes from their formulation. as the streamlines evolve as part of the
solution.

The airfoil, which is thought to be made by two dividing streamlines can
evolve as well. In the direct problem boundary conditions are specified over the
airfoil contour. In the inverse problem the fixed-wall boundary condition is replaced
by the condition that the pressure at each wall grid node matches a prescribed
pressure distribution. This method is fairly flexible, and allows for the solution of
inverse and mixed-inverse boundary value problem for transonic flow around airfoils
and compressor cascades. In the latter case periodicity of boundary conditions is
enforced at the mean line of the cascade inlet. The Euler equations are coupled with
the integral boundary layer equations.

A Newton-Raphson scheme solves iteratively for all the flow and design variables. The
ability of this method to acheive good performances has been tested against a number
of cases, from low-Reynolds number transonic airfoils, to human-powered flight
(Drela, 1990).

Some methods based on the theory of small perturbations by-pass the problem of
generating a boundary-fitted grid by prescribing approximate boundary conditions on
the airfoil/wing mean plane. To see how this can be accomplished, consider first a
case of incompressible flow. From the thin airfoil theory the thickness and the
lifting problem can be separated. The correspondent velocity perturbations are
described by the following Cauchy integrals

Multi-point design is a requirement for many aerodynamic components. Only
airfoils have been solved satisfactorily (Selig, 1992, and later works), both
single- and double elements.

By using a Newton-Raphson method, Selig solves the problem through specification
of velocity distributions along segments, for given angle of attack,
by using a conformal mapping and three integral constraints (similar to Eqs. 1).