Multiobjective optimization of a laminar transonic NLF airfoil for a swept wing
A Natural Laminar Flow (NLF) transonic airfoil for a swept wing has been designed applying the “Quasi–3D” approach and optimized using genetic algorithms. Several multipoint and multiobjective preliminary optimization cycles was used to setup the configuration of a final optimization environment.
Optimization procedure and numerical modelling
The airfoil geometry was generated by a linear combination of the initial geometry and some modification functions whose parameters represents the variables of design:
Physics modeling
The aerodynamic analysis is performed using the MSES code which solves the Euler flow field and the integral formulation of the boundary layer [1]. The transition location is estimated by the semiempirical e^{N} criterion. The wing section boundary layer is solved by a 2.5D boundary layer code based on the conical flow assumption (also known as Kaups–Cebeci approach) [2]. This approximation permits to obtain a tree–dimensional solution for swept tapered wing, in a form similar to two–dimensional boundary layer equations. The stability analysis is performed by a simplified method able to estimate both the Crossflow and Tollmien–Schlichting instabilities development [3]. The aerodynamic characteristics are computed referring to the so called “principle of cosine” [4]. This rule can be applied to finite wings with high aspect ratio and low angle of incidence and relates the 2D airfoil characteristics to the wing section according to the following relations:
Multiobjective optimization and Genetic Algorithms
Multiobjective optimizations are based on the “Pareto” dominance criterion. The dominance criterions is defines as follows:
If p and q are two candidate solutions for a minimization problem that involves n different objectives, we say that p dominates q if and only if
and …and and or …or |
Fig. 1. Pareto front |
The set of non–dominated solutions forms the “Pareto front” (red points of figure 1).
Genetic Algorithms (GAs) are search criteria that mimic the process of natural evolution [5]. The principle is to represent the combination of all possible design variables values as chromosomes. The genetic algorithm creates a population of solutions, applies genetic operators such as mutation and crossover to create new generations and drives the evolution toward the improvement of the species.
Fig. 2. Genetic algorithm operators
A Genetic Algorithm using the dominance criterion to drive the evolution of the population can directly face multiobjective optimization problems.
Procedure setup
The performances of candidates are evaluated by the steps included in the dotted frame named “Analysis” in the flow chart of figure 3. Three methods are integrated:
- a boundary layer code able to compute the three-dimensional velocity profiles of a swept wing section
- a stability analysis method which estimates the transition location in presence of Crossflow and Tollmien–Schlichting contamination
- a two–dimensional Euler/Thin Layer code that performs the aerodynamic analysis of the airfoil and provides the input pressure distribution for the BL code in an iterative process
Fig. 3. Flow chart of the procedure
The optimization cycle is closed by the search criterion which drives the evolution of generations. The population is passed to a geometric module that creates the airfoils coordinates to be analysed.
NLF transonic airfoil design
The described optimization procedure has been applied for the design of a NLF airfoil 12%c thick suitable for a transonic wing with 20 degree of sweep angle at the leading edge. Four design conditions have been selected: one for the climb condition, two at cruising Mach number (to guarantee acceptable off–design performance) and one at high speed cruise. The objective of the optimization is the minimization of the drag. Table 1 summarizes the four target design points.
Design point | Mach | Reynolds | C_{l} | C_{m} |
---|---|---|---|---|
Climb | 0.7 | 8 million | 0.63 | – |
Cruise 1 | 0.75 | 8 million | 0.55 | – 0.11 |
Cruise 2 | 0.75 | 8 million | 0.4 | – 0.11 |
High speed cruise | 0.78 | 8 million | 0.5 | – 0.13 |
Tab. 1. Airfoil design points
The starting reference geometry is the airfoil adopted within the ELFIN I and II European research project [6]. Several single and multiobjective (up to four cost functions) preliminary optimization cycles were performed. The information collected during this process gave the indication of the opportune configuration of the final optimization loop. The design targets were alternatively used as cost functions or as constraints (function penalties). The table 2 summarizes the drag performance improvement of the compromising geometry obtained with the optimization process in comparison to the starting reference airfoil at the four design conditions.
Design point | ΔC_{d} |
---|---|
Climb | – 5.1 dc |
Cruise 1 | + 0.6 dc |
Cruise 2 | – 11.8 dc |
High speed cruise | – 15.4 dc |
Tab. 2. Final design performance improvement
More than 60% of laminar flow was obtained at normal cruise and high speed cruising conditions on both sides of the airfoil. The performances were confirmed by an experimental test campaign conducted in the PT1 transonic wind tunnel at the Italian Aerospace Research Center (CIRA). Figure 4 shows the model after a test performed applying Acenaphthene to visualize the laminar flow.
Fig. 4. Wind tunnel model after a laminar test
Conclusions
A multiobjective optimization procedure, based on genetic algorithms, for the design of NLF transonic wing sections was setup and tested. Several numerical methods able to include the three dimensionality of swept wing flow was coupled. The airfoil aerodynamics was solved using a coupled Euler/Integral Boundary Layer solver. The “Quasi–3D” approach was used for the stability analysis. A semiempirical method was used to estimate the transition locations caused by Crossflow and Tollmien–Schlichting instabilities.
The procedure was applied for the design of a NLF transonic airfoil for which four design conditions were imposed. Several single and multiobjective optimization cycles produced a solution that improved significantly the performance of the airfoil taken as a starting design. An extensive experimental test campaign confirmed the performance of the design.
Bibliography
[1] Mark Drela, “A User’s Guide to MSES 3.05”, MIT Department of Aeronautics and Astronautics, July 2007.[2] Kaups, K. and Cebeci, T., “Compressible Laminar Boundary Layers with Suction on Swept and TaperedWings”, Journal of Aircraft, Vol. 14, No. 7 (1977), pp. 661–667.
[3] Donelli, R. S. and Casalis, G., “Database Approach – Stationary Mode Treatment”, Tech. Rep. ALTTA-CIRATR-AEP-01-039, CIRA and ONERA, April 2001.
[4] Losito, V., “Fondamenti di Aeronautica Generale”, Accademia Aeronautica, 1991.
[5] Quagliarella, D. and Vicini, A., “Coupling Genetic Algorithms and Gradient Based Optimization Techniques, Genetic Algorithms and Evolution Strategies in Engineering and Computer Science”, edited by D. Quagliarella, J. P´eriaux, C. Poloni, and G. Winter, JohnWiley & Sons Ltd., England, Nov. 1997, pp. 289–309.
[6] Schrauf, G., “Stability Analysis of the F100 Flight Experiment – A Second Look,”, Diamler–Benz Aerospace Airbus, ELFIN II Rept. 173, Bremen, Germany, Jan. 1996.