# Inverse Methods for Airfoils

#### Summary

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 .

### Existence of Solutions

Lighthill proved that a solution to the problem does not exist unless three integral constraints are satisfied : the airfoil to be closed at one end, and the free stream to be equal to one at the prescribed angle of attack

In order to comply with the above conditions must be corrected with three free parameters, and is generally given in the form . The airfoil is then constructed by using conjugate Fourier series from the mapping modulus. The method can be applied to the design of a cascade of airfoils, by arranging periodically (with period ) the airfoils in the complex plane.

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.

### Newton-Raphson Methods

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

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).

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