Inverse Aerodynamic Shape Design Methodologies

The earliest approaches to aerodynamic shape design appear to be by inverse methods. This class offers a myriad of analytical and numerical techniques that can structurally and aerodynamically improve the flow performance associated with boundary surfaces. Such methods control physical quantities related to a target performance, i.e., a flow field, associated with a boundary geometry. The inverse methodologies provide potential means of achieving prescribed flow fields, where loss mechanisms can be avoided, but come with inherent difficulties associated with specifying a comprehensive target field. This drawback is often augmented by geometrical considerations, that can translate into non-physical boundaries (usually associated with constrained problems). Therefore, the feasibility of the solution may lead to a further reconsideration of the problem, increasing its time-cost properties. There are a number of methods in the literature that try to mitigate these drawbacks, or require as little designer’s expertise as possible, and are briefly presented in the following.

The idea of attaining a specified and desirable pressure distribution has come a long way since the mid 1940’s, when work was based on conformal mapping. The basic idea is that an airfoil is generated from a circle through a mapping function. Lighthill (1945a,b) applied conformal mapping in the case of two-dimensional incompressible flow to the design of cascade blades. A constraint was applied to ensure the profile/cascade is closed and a closed form solution is readily derived, as the solution φ is known for incompressible inviscid flow over a circle, therefore the analytical mapping is easily derived through the general relation of the speed over the profile

q= φ

h =qtarget. (3.1)

where φ is the velocity potential for flow past a circle and h is the modulus of the conformal mapping function between the closed profile and the circle. A two-dimensional compressible potential flow was modeled using this approach (McFadden (1979)) with a remapping function and extended to transonic regime, with artificial viscosity to mitigate the occurence of shock waves.

An inverse method for two and three dimensional potential flows with prescribed veloc- ities along the boundary surface was developed by Stanitz (1953, 1980). The governing equations of flow permitted a system of independent variables (i.e., the coordinates) related to streamlines to be formulated. This method considered a pair of stream func- tionsη(x, y, z), ψ(x, y, z) and the second independent coordinate was the the velocity

vector potential φ(x, y, z), implicitly satisfying mass conservation. For different choices of natural coordinate (i.e., the potential), further research (see Finnigan (1983), Keller (1998), Keller (1999)) was somewhat limited to axially symmetric flows. This drawback is overcome by the streamfunction-as-a-coordinate method, where the axial coordinate

x is used as an independent variable and the only remaining variables arey(x, η, ψ) and

z(x, η, ψ) (Giles and Drela (1987); An and Barron (2005)).

On a different basis, thesurface approach is an alternative method suggested in the lit- erature for inverse design (Campbell (1992)) . In this method the difference between the target and actual pressures is translated into surface changes in subsonic and supersonic flow. It has been successfully applied for 2D flows and Navier-Stokes equations, but it meets difficulties for 3D flows, where the surface curvature and slopes are calculated plane by plane. The grid points of the surface are iteratively loaded with ∆Cp (i.e., the difference between the actual and target distributions) until some form of convergence is met (i.e., a stable surface). The surface is governed by a second order partial differential equation, whose coefficientsβi are user specified entities, chosen based on the designer’s experience or flow parameters (Dulikravich and Dennis (2000)):

β0n+β1∂xn+β2∂yn+β3∂xxn+β4∂yyn= ∆Cp, (3.2)

where n=n(x,y) denotes a local normal surface displacement. Such approaches, based on residual correction, are also presented in the literature in monolithic schemes, where the shapes of boundaries are updated iteratively, and the governing flow parameters are solved under as artificial time parameter scheme (Varona (1999)). This last approach usually requires many iterations, is computationally complex and there are issues with the compatibility of target pressure distribution. An alternative to this limitation is pro- vided by Campbell (1998), using a weighted averaged of geometries, based on principal design requirements and desired gradients.

The formulation of inverse design problems also considers the constraints in boundary value problems. The inverse methodology can either be applied to the solid boundary where the pressure fields coincide and requires non-zero boundary velocity, so that the flow distorsion is possible (Leonard (1990)), or keeps the boundaries fixed and updates in geometry are linked to the computed residuals (Demeulenaere (1997)). The last author outlines that mechanical constraints are readily achieved only if a limited area of pressure surface is prescribed, and an additional degree of freedom is introduced to control the target flow angle.

Boundary problems can also be posed with respect to geometry updates, to enhance the feasibility of the solutions (Volpe and Melnik (1986) first addressed the issue of ill- posed inverse transonic design, by using constraints). If the flow field has a non-zero velocity on the boundary (i.e., Dirichlet boundary), the updates are determined by a

transpiration model, based on mass flux conservation between the old geometry and the current one (transonic wing conditions are studied by Henne (1981), Demeulenaere (1997)). A streamline model, based on flow tangency u_·n = 0, in the inviscid case (u is the tangential velocity described by target velocity and n is the surface normal and unknown in the problem), is studied by Meauze (1982), Dang et al. (2000).

An alternative for the design of transonic airfoils is offered by the hodograph method. Although it can guarantee the design of shock-free airfoils in transonic flows (Garabedian and Korn (1971), the hodograph transformation (i.e., spatial variables are functions of velocity components) lacks control over flow characteristics and geometry and also engineering applicability, as it cannot be applied to three-dimensional cases (Bauer et al. (1972)).