The optimiser used here employs an objective function to determine the optimum and so the selection pressure is towards a small area in the design space. However, another way of finding the optimum is to find the designs that provide the best compromise between all the performance parameters. This is known as the Pareto front. The advantage of using a Pareto-front-type optimiser (PFO) is that it provides you with a range of design points that represent the best combination of performance measure parameters. In essence, the PFO method provides the user with the designs that give the best performance for each parameter and subsequently leaves the weighting of these performance parameters to them at the end of the optimisation process. The advantage of using an objective-function-type optimiser (OFO) is that it allows designs that do not necessarily fall on the Pareto front to be included in the genetic algorithm’s selection based on the end-point required. So, rather than spreading out the performance along a front, it clusters it at a specific objective.
For example in Figure 3.22, if CM has less value in the objective of the design, point 2 is still a good design even though it may not fall on the Pareto front. However, having the Pareto front is a good indicator that there may be a slightly better design, possibly at point 4.
CM CD 3 1 2 4
Chapter 4
Aerofoil Optimisation
To demonstrate the optimisation framework on aerofoils, the RAE 2822 transonic aerofoil was used. The reason for using this aerofoil is that it is designed with a very specific objective in mind, which is to have good performance in terms of lift and drag at high Mach numbers, especially in the transonic to supersonic flow regime141. This makes it a good candidate for optimisation since its objectives are clear, and there is potential for further improvement within constraints such as lift-to-drag ratio and moments. The added advantage of using this aerofoil is that the experimental data on this aerofoil at these conditions are readily available. Also, it does not have a flexible parameterisation method and so this provides an opportunity to develop a generic method usable for shape optimisation. This parameterisation technique uses Chebyshev polynomials and is described in Section 3.2.1. Its application to the upper surface of the RAE 2822 aerofoil is also shown in Figure 3.2.
4.1
Parameterisation Technique
First the method was applied to a variety of aerofoils (NACA 0012, RAE 2822, NACA 23009 and ONERA OA213), and their Cp distribution compared in order to find what error convergence values were needed to obtain a shape curve that was accurate enough to produce the same results. For simple aerofoils like the NACA 0012, the parameterisation technique works very well with almost zero error in the shape using only a few coefficients. However for more complex aerofoils like the NACA 23009, the OA213 and the RAE 2822, more coefficients are required. Figure 4.1 is a Cp plot obtained from XFOIL for the NACA 23009 aerofoil and that of the reconstructed one. The error of the shape was 0.013728 with 3 coefficients and 0.005775 with 5. The moment values tend to be the most sensitive to the aerofoil shape. For both the actual and the reconstructed aerofoil, the values are within about 0.2%. The maximum error was for drag, approximately a 1% difference. For the ONERA OA 213 aerofoil which is used on wind turbines, the error was 0.068819 with 3 coefficients and 0.027594 with 5 coefficients. Figure 4.2 is the corresponding Cp plot. The error in moment coefficient is over 16% which is high suggesting that this error conver- gence value for the geometry is too large.
Therefore an error convergence less than approximately 0.01 is selected. However, it is also good practice to look at the recreated curve, even if the error convergence falls within this value. This is because the error is the sum of the absolute difference along the curve, therefore the same value can be obtained whether the error is a small but well distributed difference in the curves or a very accurate curve that has a large localised error, in which case the former would most likely produce better results.
For the RAE 2822 aerofoil, six coefficients were required to parameterise the upper surface with an error convergence of 0.00741. Figure 4.3 shows the Cpcurve for the original RAE 2822 aerofoil and its parameterised version. The difference is minimal resulting in a difference in Cl, Cd and Cmof negligible values as shown in Table 4.1.
Parameterised NACA 23009 Original NACA 23009
Figure 4.1: Cpplots predicted by XFOILfor the parameterised aerofoil NACA 23009 (a) and the original
NACA 23009 aerofoil (b) at Mach 0.2 and Re = 1×106.
Parameterised OA213 Original OA213
Figure 4.2: Cp plots predicted by XFOIL for the parameterised ONERA aerofoil OA 213 (a) and the original ONERA aerofoil OA 213 (b) at Mach 0.2 and Re = 1×106. The error convergence was about 0.03
resulting in large differences in moments especially. This suggest a smaller error is required and hence a value of 0.01 was chosen.
4.2
Objective Function
The objective was to improve the lift for a reduced amount of drag. Therefore, the lift-to-drag ratio (Cl/Cd) was the objective used. However, a constraint was placed on the drag coefficient (Cd) so that it never exceeded the original aerofoils drag value. This also ensured that the optimised aerofoil is one that will have the same or higher lifting capability than the original design. As this is a transonic aerofoil, an important constraint is to ensure that the drag divergence Mach number does not fall below the value for the original design. In addition, its moments must be maintained to within a small margin of the original value. The latter is quantified using the moment coefficient (Cm) about the quarter chord point of the aerofoil. The former is obtained analytically using Korn’s method with a technology factor of 0.95142. The program for this can be found in Appendix B.12.