Aerodynamics and Design for Ultra-Low Reynolds Number Flight

Chapter 3
the number of variables considered, but also simplifies the problem by removing
minimum thickness constraints.
The free design element is the camber-line. A wide range of variation is achieved by
modeling the camber-line with an Akima spline [23] anchored at the leading and trailing
edges. Akima splines provide a curve that has the benefits of a tensioned spline,
avoiding the undulations that can occur with simple cubic splines, but are generally
smoother across the control points and require no tuning to achieve a satisfactory
interpolation. Four interior control points, or knots, are used to define the camber-line.
These are evenly distributed and their chordwise locations are fixed. The knots move
perpendicular to the chord-line bounded by upper and lower camber limits.
The optimization study utilizes a constrained simplex optimizer, a modified Nelder-
Mead simplex [24], coupled with the INS2d code and a grid generator to maximize the
section lift to drag ratio. With only four design variables, this simple optimization
method is sufficient. In addition, each 2-D steady-state solution of the flow solver is
relatively inexpensive. The simplex method is simple to implement and does not require
(possibly noisy) gradient calculations. It is likely not the most efficient option, but the
small problem size and inexpensive flow calculations make it a good solution.
Function evaluations are also designed to increase the robustness of the method. A
coarse performance polar is generated for each function evaluation. Calculations begin
at a low angle of attack where convergence is likely, regardless of geometry. The angle
of attack is then increased until the L/D passes its maximum or the flow solver fails to
converge. In either case, an objective function value is generated for that geometry and
the process continues. This is computationally expensive and not particularly efficient,
but it is robust and effective.
Two airfoils have been developed using this approach for Re=6000 (R6) and Re=2000
(R2). Both sections are shown in Figure 3.18. The optimization runs were initialized
with a flat plate airfoil, but the converged solutions have been checked by restarting with
a geometry near the upper camber limits. Both airfoils exhibit similar features with a
44