Airfoil Design and Optimization - IAG

Lutz, Th.
Airfoil Design and Optimization
The aerodynamic efficiency of mildly swept wings is mainly influenced by the characteristics of the airfoil sections.
The specific design of airfoils is therefore one of the classical tasks of aerodynamics. Since the airfoil characteristics
are directly dependent on the inviscid pressure distribution the application of inverse calculation methods is obvious.
The direct numerical airfoil optimization offers an alternative to the manual design and attracts increasing interest.
1. Inverse Subsonic Airfoil Design
In order to find a proper airfoil shape for a specific aerodynamic application different approaches may be used. In
former times, when no theoretical methods were available, relevant geometric parameters were varied systematically
and the effect on the aerodynamic characteristics was measured in a series of wind-tunnel tests. The first extensive
experimental investigations in this sense were performed at the AVA after completion of the Göttinger low-speed
wind-tunnel in 1917/1918.
After L. Prandtl developed the boundary-layer theory it became obvious that the outer-flow velocity distribution
along the airfoil surface determines the airfoil performance. This gave reason to develop inverse methods which
enable the calculation of the airfoil contour from a given velocity distribution. First methods for the approximate
inverse solution of the Laplace equation were developed in the late twenties and the thirties. Exact solutions based
on a conformal mapping procedure were proposed by Mangler (1938) and Lighthill (1945).
When inverse methods were available, knowledge was established how to shape the velocity distribution in order to
obtain favourable airfoil characteristics, e. g. with respect to minimum drag or maximum lift. A milestone represent
the well-known NACA laminar flow airfoils designed in the late thirties. As further examples the investigations of
Eppler, Stratford and Wortmann should be mentioned.
A significant progress in airfoil design was achieved in the fifties by coupling the potential-flow methods with integral
boundary-layer procedures to consider viscous effects. Since then, a large number of subsonic airfoil design tools
were developed, for example the Eppler/Somers code or the methods of Drela. In the hands of experienced users
such methods enable a carefully directed design and the adaption of the airfoil characteristics to specific applications.
A good example in this respect is documented in the progress achieved in the aerodynamic performance of modern
sailplanes. A significant portion of the improvements can be attributed to the sucessful design of low-drag laminar
flow airfoils (e. g. DU, E, FX or HQ airfoils).
Another significant amount of investigations on inverse methods for the regime of transonic flows were performed in
the seventies after the discovery of the supercritical airfoils. Due to the availability of supercomputers it is nowadays
possible to handle semi-inverse solutions of the Euler equation, the coupled Euler boundary-layer equations or even
for the Reynolds-averaged Navier-Stokes (RANS) equations.
2. Direct Numerical Airfoil Optimization
An alternative to the inverse or semi-inverse procedure is offered by direct numerical optimization. With this
approach an automated search for an optimal solution with respect to a user-specified objective function, e. g.
minimzation of drag, is performed. This is done by means of an iterative variation of the chosen design variables.
To parameterize the airfoil contour mostly geometric shape functions are applied with the respective coefficients
representing the design variables. The choice of a proper optimization algorithm strongly depends on the topology
of the considered objective function, the airfoil parameterization, the number of design variables and finally on the
efficiency of the aerodynamic model. In general, gradient methods converge fast for simple topologies of the objective
function but one may be trapped in a local optimum if a multimodal objective function is considered. Stochastic
algorithms offer a greater chance to avoid this problem and can also cope with complex topologies but usually require
much more iterations. The combination of stochastic optimizers with costly high-accuracy flow-solvers, e. g. based
on the solution of the RANS equations, is therefore still limited to a relatively small number of design variables.

Numerical optimizations of subsonic airfoils featuring a detailed representation of the complete contour are hardly
known. To reduce the computational effort, potential-flow methods coupled with integral boundary-layer procedures
are preferred for the aerodynamic analysis, the number of optimization cycles is limited and a moderate number of
design variables (usually below 10 ∼ 15) is considered. Actually, most of the numerically optimized subsonic airfoils
published so far show a great similarity to the initial shape.
3. Example on Numerical Shape Optimization of Subsonic NLF Airfoils
The objective of numerical optimizations performed by the present author
was to design natural laminar flow (NLF) airfoils which show minimized
average drag for a user-specified design lift region [1], [2]. In order
to enable a detailed airfoil representation a large number of 34 design
variables was considered. Contrary to the usual approach the airfoil was
not parameterized by geometric shape functions. Instead, an inverse
conformal mapping procedure according to Eppler was applied to generate
the airfoil contour. The input parameters of this method directly
control the local outer-flow velocity gradient and finally the boundarylayer
development. A spline representation of the critical leading-edge
region is avoided with this approach.
The potential-flow method was coupled with an integral boundary-layer
procedure utilizing closure relations according to Eppler for laminar flow.
The method proposed by Drela with a new shape-factor relation is used
to calculate turbulent boundary-layers. To predict the laminar to turbulent
transition location, an e n database method based on spatial stability
analysis for Falkner-Skan self-similar profiles was implemented. The effect
of ’short’ transitional separation bubbles is considered by means of a
new efficient bubble model. The complete aerodynamic model was coupled
with a commercial hybrid optimizer which consists of a combination
of genetic algorithm, downhill simplex and a gradient method.
Figure 1: Inviscid velocity distribution
for the optimized airfoil
One optimization result is depicted in Figs. 1
and 2. The objective was to minimize the
average drag coefficient for angles of attack
α design =[2 ◦ , 3 ◦ , ..., 8 ◦ ] relative to the
zero-lift line. Two Reynolds numbers were
considered, namely Re design =3· 10 6 and
9 · 10 6 . In order to prevent a breakdown
in lift at off-design conditions and to enhance
the stall characteristics, the curvature
of the lift curve was limited for α =
[2 ◦ , 3 ◦ , ..., 15 ◦ ]. The resulting airfoil shows
features which are well-known from manual
airfoil design such as smooth transition
ramps or a Stratford-like turbulent pressure
recovery on the lower side (see Fig. 1).
1.5
Present method, n
c crit.
= 11.13
l
Eppler code
Experiment IAG
1.0
Re=3x10 6
0.5
0.0
0.000 0.005 0.010
c d
0.0 0.5 x tra /c
1.0
Figure 2: Predicted and measured drag polar for the optimized airfoil
Wind-tunnel tests for the optimized airfoil showed very low drag coefficients inside the laminar bucket which exactly
coincides with the design lift region, see Fig. 2. The optimization method has furthermore been applied to design of
airfoils which show minimized trailing-edge noise or minimal derivation to a user-specified drag polar.
4. References
1 Lutz, Th.: Berechnung und Optimierung subsonisch umströmter Profile und Rotationskörper. VDI Fortschritt–Berichte,
Reihe 7: Strömungstechnik, Nr. 378, ISBN 3-18-337807-8.
2 Lutz, Th. and Wagner, S.: Numerical Shape Optimization of Subsonic Airfoil Sections. Proc. ECCOMAS 2000: European
Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, September 11–14, 2000.
Addresses: Thorsten Lutz, Institute for Aerodynamics and Gas Dynamics, University of Stuttgart (IAG),
Pfaffenwaldring 21, D-70550 Stuttgart, email: lutz@iag.uni-stuttgart.de