Airfoil Optimization By Evolution Strategies - Auburn University
2013 IEEE Congress on Evolutionary ComputationJune 20-23, Cancún, MéxicoAirfoil Optimization by Evolution StrategiesDrew A. CurristonAlice E. SmithDepartment of Aerospace EngineeringDepartment of Industrial and Systems EngineeringAuburn UniversityAuburn, ALAbstract— This paper addresses subsonic airfoil optimizationusing Evolution Strategies (ES) and devises a means of definingthe airfoil geometry that reduces unnecessary restrictions in thesearch space. The solution encoding uses Bezier Control Pointsto define the geometry of the airfoil, but does not restrict themovement of the control points as was common in previousairfoil optimization algorithms. The ES move operator combinedwith this improved solution encoding expands the search space toinclude superior solutions while also enabling a more efficientsearch to reduce computational cost.Keywords—Evolution Strategies; airfoil optimization; subsonic;Bezier; multi-objective optimization, aerodynamic optimization.I.INTRODUCTIONAirfoil optimization has been attempted in a variety ofways for a wide range of objectives. Typically, an airfoiloptimization problem tries to maximize the performance of anairfoil with respect to a specific set of performance parametersat a specified flight regime. However, the airfoil must alsoperform well across a wide variety of flight regimes, as wouldbe experienced in actual flight. Adaptive optimizers have beenused often in recent years to optimize airfoils, and many papershave been published with impressive results, but often thesearch space is restricted significantly to simplify the problem.This paper discusses the development of a subsonic airfoiloptimizer that will minimize a specific set of objectivefunctions while imposing the least amount of artificialrestrictions on the problem.II.the use of the same flow solver, XFoil, in their objectivefunction evaluation as used in this paper. By using the sameflow solver, the two optimization algorithms produce the sameresult for the same solution, therefore improving the basis forcomparison.B. Benchmarks for ComparisonThe first paper is “Designing Airfoils using a Reference Pointbased Evolutionary Many-objective Particle SwarmOptimization Algorithm,” written by Wickramasinghe [6].Wickramasinghe’s paper attempted to optimize the geometryof the NLF0416 airfoil used on a UAV with respect to certaincriteria. Six different objectives were evaluated and a ParticleSwarm Optimizer was used for his algorithm. The objectivesare listed below:1. Minimize Coefficient of Drag(at CL 0.5, Re 4.0*10 6, and Mach 0.3)2. Minimize CD/CL(3/2)(at floating α, Re 4.0*10 6, and Mach 0.3)3. Minimize Cm02(at CL 0, Re 4.0*10 6, and Mach 0.3)4. Minimize 1/CLmax2(at floating α, Re 4.0*10 6, and Mach 0.3)5. Minimize 1/CL2(at α 5o, Re 2.0*10 6, and Mach 0.15)6. Minimize 1-xtr(at α 5o, Re 2.0*10 6, and Mach 0.15)OPTIMIZATION PROBLEMA. Airfoil OptimizationIn general, airfoil optimization is either conducted by direct(specify airfoil geometry and calculate performance) or indirect(specify pressure curves for desired performance and calculatecorresponding geometry) methods. Indirect methods aregenerally solved in a deterministic way while direct methodshave become more popular in recent years through the use ofmeta-heuristics. This paper will present an algorithm to findthe optimal airfoil geometry using a direct method. In order toevaluate the performance of the algorithms, two previouslypublished papers on airfoil optimization were selected asbenchmarks for comparison. Two separate optimizationproblems were applied to the algorithm, each selected from onethe papers discussed below. These papers were selected due to978-1-4799-0454-9/13/ 31.00 2013 IEEEThe second paper selected for comparison is “MultiObjective Evolutionary Optimization of Subsonic Airfoils byMeta-Modeling and Evolution Control,” written by D’Angelo[1]. In this paper a multi-objective optimization is performedusing a Pareto Front to minimize the coefficient of drag andmaximize the coefficient of lift. This paper also includes workin meta-modeling, artificial neural networks, and evolutioncontrol, but the comparison will focus on solution encodingand results.III.METHODS OF AIRFOIL PARAMETERIZATIONThe solution encoding for this problem must provide thegeometry of the airfoil, and this encoding of the geometry mustalways consist of smooth surfaces in order to evaluate them1451
Fig. 1. The 11 variables used in PARSEC representation of an airfoil [6]properly in the flow solver. Obviously, encoding the solutionas a series of x and y coordinates is not feasible because it leadsto surfaces that are not smooth and not easy to manipulate witha move operator. Two common representations used in thepast are PARSEC and Bezier Curve representations.A. PARSEC EncodingPARSEC encoding consists of 11 variables,including angles and distances, that define the shape of theairfoil. These variables are depicted in Figure 1. While thislends to a smooth surface that is easily manipulated by a moveoperator, it also comes with some problems. The biggestdisadvantage is the fact that PARSEC encoding cannotrepresent all geometric shapes, especially certain desirablefeatures around the trailing edge of the airfoil. This limits thesearch space and may prevent an optimal or near-optimalsolution from being found. The PARSEC encoding is theencoding used by Wickramasinghe in his paper.B. Bezier Control PointsAnother common representation is using BezierCurves. These use control points and a series of BernsteinPolynomials to define the shape of the airfoil. This lends to asmooth shape and the control points can be easilymanipulated, but often the range of movement on the controlpoints is severely restricted. Figure 2 shows a Bezier Curverepresentation of an airfoil from Feinekos [2]. In thisrepresentation the abscissae are fixed along the chord lengthand the vertical range of the control point movement islimited. This will also limit the possible airfoil geometriesfound by the algorithm. D’Angelo’s paper [1] used BezierControl Points to encode solutions, and similarly restricted themovement of the control points to ranges defined by the user.In 2000, Naujoks et al. conducted a multi-pointairfoil optimization using Bezier control points and notedseveral deficiencies with the parameterization. First, theairfoil surface had a tendency to oscillate, which created awavy surface on the top and bottom of the airfoil. Thisreduced the performance of the airfoil and produced far fromideal pressure curves. The article found that using threecontrol points rather than six or 12 provided a sub-parsolution. As stated in the article, “the use of only three Bezierpoints on the lower and upper surface might not be theFig. 2. Bezier Curve Representation. The Bezier control points used in todefine an airfoil. The “X” marks represent control point location and the“I” marks represent the range of movement for each control point.[2]ultimate choice but it provides ”non-oscillating” shapes [5].”An example of this oscillatory behavior from [5] is shown inFigure 3. The second issue found was the weak representationof the leading edge of the airfoil, where the highest fidelity inthe flow is required because it is the location of the greatestturn in the flow. One of the two major conclusions from [5] isthe need for a parameterization that resolves these two issues.Naujoks asserted the need for future work in improving theairfoil parameterization, specifically to “involve x-coordinatein design optimization [5].”The ES algorithm devised for this paper uses BezierControl Points to encode the solution, however greatly reducesthe restrictions on the movements allowed for the controlpoints by allowing control point movment in both the x and ydirections. This expands the search space to include solutionsthat cannot be represented by other means, while still ensuringsolutions produce a smooth surface that can most likely beevaluated by the flow solver. This allowed movement of thecontrol points in the x direction significantly reduces theoscillations in the airfoil surface, similar to the reductions inoscillations in polynomial interpolation through the use ofunequally spaced Chebyshev nodes. The allowed movement inthe x direction also allows the control points to beconcentrated in certain regions, allowing more points andgreater fidelity at the leading edge of the airfoil where it isneeded. By making these changes, better solutions may befound compared to other algorithms and the solutions are notrestricted to what the user is expecting the solution to look1452TABLE I.ControlPoint123456SOLUTION ENCODINGUpperSurfaceX ValueUpperSurface YValueLowerSurface XValueLowerSurface 4-0.00521-0.09111-0.0911
Fig. 4. Flow Chart.IV.Fig. 3. An example of oscillatory behavior from [5]like. It is entirely possible that the optimum solution isdifferent than what the user is expecting, and imposing theserestrictions may prohibit those solutions from beingconsidered.The only necessary restrictions placed on themovements of the control points were to ensure the airfoil kepta chord length of 1 and the surface of the leading edge wasperpendicular to the flow to allow the flow solver XFoil to beused. The leading and trailing control points will remain atrespective x coordinates of 0 and 1 to ensure a chord length of1, however the vertical movement of the trailing point isunrestricted. The 2nd point from the leading edge on both theupper and lower surface will remain at an x value of 0 as wellto ensure the tangent line to the leading edge of the airfoil isperpendicular to the flow. Without this restriction the flowsolver could fail due to the sharp angle on the leading edge.These 2nd points on the upper and lower surfaces may movevertically in any amount as long as their x location stays at 0,so restrictions on possible airfoil geometries areinconsequential. If a CFD Euler or Reynold’s AveragedNavier Stokes (RANS) solver were used instead of Xfoil,these restrictions could also be removed. All other points maymove freely in any direction, which will allow virtually anyairfoil shape to be represented by this method. An examplesolution encoding for an airfoil is listed below in Table 1,which simply gives the (x,y) coordinates of each BezierControl Point. Six Bezier Control Points were used for eachsolution’s upper and lower surfaces because this achieved abalance between sufficiently representing the airfoil andsearch efficiency. Up to 12 control points per surface wereattempted, but generally increasing the number of controlpoints above six did not improve results but greatly increasedcomputation time. However, it may be beneficial to conduct arough optimization using six control points to narrow in on aregion of interest, then conduct a follow on optimization usingmore control points for an intensive search.EVOLUTION STRATEGIES ALGORITHMEvolution Strategies (ES) was selected for the algorithmapplied to this problem for a variety of reasons. First, becausethis is a continuous problem an optimizer suited forcontinuous problems was selected. A population basedoptimizer was also desired to produce multiple feasiblesolutions at the end of each run rather than just a singlesolution. The ES move operator is a much better fit to theBezier Control Point representation than other algorithms dueto the high mutation rate and ability to search outwardly moreefficiently than Genetic Algorithms when the initial solutionsare seeded airfoils. Because of the increased search space dueto the solution encoding, pairing the ES move operator withthe Bezier solution encoding was crucial in keepingreasonable computation times. Evolution Strategies also aremore conducive to a micro-population, which is often used inairfoil optimization due to the high computational costs of theflow solvers. Past work has even shown that a population ofone member may provide good results using ES algorithms[5]. For these reasons, Evolution Strategies was viewed as thebest paradigm for this problem. The ES algorithm used veryclosely follows the canonical ES algorithm, which is keptsimple to focus on the benefits of the airfoil parameterizationused rather than complexities in the algorithm. Below isdiscussion on some of the major details of the algorithm.Figure 4 shows a flow chart of the algorithm’s operation.A. Initial SolutionsThe initial solutions used for the first objective problemwere seeded NLF0416 airfoils, matching the initial solutionsused in [6]. For the second objective problem, two differentsets of initial solutions were tried. The first were randomlygenerated Bezier Control Points, which produced randomshapes that may or may not resemble an airfoil. Randomlygenerated initial solutions were used in D’Angelo’s paper,however the method to generating the random initial solutionswas not elaborated on. Random initial solutions producedgood results, but needed extra computational time to movefrom the random shape to a generic airfoil shape. To reducethis unnecessary computation time, eight different subsonicairfoils were seeded for the initial solutions. The seeded1453
LS(1)-0417, 64-210, 63-015), a NLF0416 airfoil, and aRAE100 airfoil. By seeding the initial solutions thecomputational time was reduced, but the quality of the finalresults had no noticeable difference from using randominitial solutions.A micro-population of eight airfoils was used for eachobjective problem. Micro-populations have been shown toprovide comparable results with greatly reducedcomputational time when the function evaluations areparticularly costly. Using a micro-population allowed thealgorithm to perform well while keeping computation timesreasonable.Best Airfoil Objective Values010Obj Function Value-110-210ES AirfoilReferencePSO Airfoil-3101234Objective Number56Fig. 5a. Airfoil Comparison. The above graph compares the ES produced,PSO produced, and reference airfoils with respect to each objectivefunction. The values for each objective are sought to be minimized.Best Airfoil eBezier CPsBezier CPs0.2y/c0.10-0.1-0.2B. Move OperatorThe move operator for this algorithm is the canonicalmove operator for Evolution Strategies. The algorithmselects a parent through uniform probability and producesoffspring by applying a mutation. The mutation consists ofmoving each of the Bezier Control points a random distancein the x and y directions subject to a normal distribution anda standard deviation σ. The leading two control points andthe trailing edge control point are only mutated in the ydirection for reasons stated in earlier in the paper. The onefifth rule was applied to the standard deviation after every50 objective function evaluations. “Sigma Restart” was alsoimplemented, which reset the standard deviation to theinitial value of 0.03 if sigma was reduced to below 0.0002.This would diversify the search if the standard deviationbecame so small that moves were ineffective, but would notlose superior solutions due to the elitism in the method ofreplacement.-0.3-0.4-0.5C. Replacement and cement was accomplished through (µ λ) selection,where both the parents and offspring competed for selectionin the next generation. This method provided an elitistaspect to the algorithm that allowed superior airfoils toremain in the population until they are replaced by improvedoffspring. Recombination was not used in the algorithm.Fig. 5b. The ES produced airfoil geometry.Objective Function Results for All Runs0Obj Func tion V alue10ReferenceRun 1Run 2Run 3Run 4Run 5Run 6Run 7Run 8-110-210-3101234Objective Number56Fig. 5c. Results comparison between runs.airfoils were six NACA series airfoils (0012, 2412, 4415,1454D. Stopping CriteriaThe stopping criteria was completing the maximumnumber of generations. Because evaluating the performanceof the algorithm was the main goal, completing thealgorithm runs in a specified amount of time was ofimportance. Data was stored after each generation so thequality of the solutions and the cumulative computation timeup to the specified generation could be easily found.E. Multiple ObjectivesThe multiple objective functions were approached indifferent ways for the separate objective problems. For thefirst objective problem with six separate objective functions,a Pareto Curve would not be a suitable choice. For this
reason, a simple sum of the normalized objective functionswas used to combine all six into a single objective. Eachobjective function value was normalized against the referencevalue for the NLF0416 airfoil to solve scaling issues betweenthe objectives. Weighted sums were attempted, but proved tohave no noticeable difference from the non-weighted results.Wickramasinghe’s paper used a Hyper-Volume approach toevaluate the six objectives at once, but this approach alsocame with a large computational cost.For the second objective problem a Pareto Front was usedto evaluate the two objective functions, which was the sameapproach used by D’Angelo. A fairly simple rank-basedPareto front was used for this algorithm, where the parents forthe next generation of offspring were randomly selected fromthe first rank of the Pareto front from the previous generation.The concept of ε-dominance [4] was used to ensure thesolutions were spread out across the front and did not clusterin certain areas. Archive sizes of 100, 300, and 500 were usedwith the larger archive sizes performing slightly better andwith no significant difference in computation time. Thisapproach to a Pareto front is fairly simple and straightforward,and leaves possible room for improvement such as biasingparent selection to areas less covered in previous generations,but is sufficient for the purposes of this paper.F. Objective Function EvaluationAs stated earlier, the open source XFoil programdeveloped by Dr. Drela was used as a subsonic flow solver forthis algorithm. By calculating the airfoil upper and lowersurface coordinates from the Bezier Control Points andinputting them with the flight conditions into XFoil, the flowsolver will produce aerodynamic data such as the coefficientof lift, coefficient of drag, and point of flow separation. TheseXFoil outputs could then simply be read by the algorithm andtranslated into the appropriate objective function values.Although XFoil is a fairly robust program, supplyingmutated airfoils for evaluation sometimes resulted inproblems. Reducing the restrictions on the Bezier ControlPoint movements exacerbated this problem, and for thisreason several error control measures were implemented toprotect the algorithm from XFoil convergence failures. Theerror control measures used in this algorithm were:1.If the upper and lower surfaces of the airfoil crossed,the airfoil was discarded and another mutated airfoilwould take its place.Because XFoil uses a panel method, the anglebetween two adjacent panels must not be too great. Ifpanel angles were too large, the algorithm wouldattempt three times to spline points to reduce the panelangles before discarding the airfoil and mutating anew one.As a blanket error control measure, a timer was startedeach time XFoil was called upon for an evaluation. Ifthe timer expired prior to XFoil outputting results,XFoil was terminated and a new airfoil was mutatedand evaluated.2.3.If future work uses an Euler or RANS solver instead ofXfoil, these control measures could be removed from thealgorithm as long as adequate mesh generation and boundarycondition techniques are used.V. PROBLEM ONE RESULTSThe ES algorithm performed well in finding near-optimalsolutions and performed extremely well in computational timecompared to Wickramasinghe’s PSO algorithm. Eight runswere performed to evaluate the algorithm while changingcertain parameters. The specifics of the different runs aresummarized below in Table 2.A. Objective Function ValuesThe results for both the ES algorithm and the benchmarkalgorithm are summarized in Figure 5. Figure 7 shows anexample of the converged XFoil results for the best solution.Each of the eight runs produced roughly equivalent results,with the exception of run 6, differing only slightly in objectivefunction values. Run 6 produced inferior results simplybecause without sigma restart the 1/5th rule intensified thesearch prior to reaching the area of the optimum solution.Without sigma restart the moves reduced in size until theywere approximately zero, effectively stopping the searchBest Airfoil zier CPsBezier CPsy/c0.20.10-0.1-0.200.10.2Fig. 8. Airfoil GeometryFig. 7. XFoil converged solution example to airfoil in Figure 4b.14550.30.40.5x/c0.60.70.80.91
Sum of Obj Values Over Generations4.53.5Sum of Obj ValuesTABLE III.Best Airfoil200 Gen Run150 Gen Run100 Gen Run50 Gen Run43RUNS AND PARAMETERSGenerations1/5th RuleSigma RestartInitial PtsRun 150YesYesSeededRun 2100YesYesSeededRun 3150YesYesSeededRun 4200YesYesSeededGenerations1/5th RuleSigma RestartInitial PtsRun 5100NoN/ASeededRun 6100YesNoSeededRun 750YesYesRandomRun 40160180200Fig. 6. Objective Function vs. GenerationsTABLE II.before reaching optimum. The algorithm converged on asolution by about the 50th generation, so any subsequentgenerations generally added computation time withoutimproving results. Figure 6 shows the sum of the objectivefunctions over generations.Clearly by approximatelygeneration 50 the algorithm converged to a near-optimalsolution. Sigma restart improved the consistency of thealgorithm by ensuring it did not easily get stuck in localoptima, but did not significantly improve the final results orcomputation time. Using random initial solutions did notsignificantly impact the final results, but did slightly increasethe computation time because the algorithm requiredapproximately 10 generations to move from the random shapesto an airfoil shape equivalent to the NLF0416 airfoil. Eventhough the objective function results were very similar betweenthe runs, the airfoil geometry was noticeably different in someof the runs. An example airfoil geometry from another run isshown above in Figure 8.B. Computation TimeThe computation time required for the algorithm was veryreasonable considering the large computational cost of the flowsolver. The fastest run consisting of 50 generations, which alsoproduced the best objective function values, had a run time of1.11 hours. All computations were performed in Matlab on a2.0GHz dual-core processor. The PSO algorithm requiredapproximately 36 hours to run on a 2.3GHz dual-coreAirfoilNLF0416PSOESOBJECTIVE FUNCTION RESULTSObj. 1Obj. 2Obj. 3Obj. 4Obj. 5Obj. 1200.20870.22690.7560processor. Because the XFoil flow solver accounted for over80% of the computational time per run, the main difference incomputation time between the ES and PSO algorithms camefrom the number of function evaluations required. The PSOalgorithm required 10,000 function evaluations while the ESalgorithm required about 2,000. Using a Hyper-Volumemethod for evaluating multiple objectives rather than a linearcombination of the objective functions also increased run timeof the PSO algorithm.VI.PROBLEM TWO RESULTSFor problem two, the algorithm was generally unchangedwith the exception of adding a Pareto archive instead of aparent population, and the parents were selected randomlyfrom the first rank of this Pareto archive to create offspring.The algorithm still used a micro-population of eight membersand made no changes to the solution encoding, move operator,or any other fundamental aspects of the algorithm. Fifteendifferent runs were completed in order to gain a sufficientamount of data to evaluate the performance. Three runs foreach of the five scenarios below were completed:1.75 generations, 300 member archive2.150 generations, 300 member archive3.250 generations, 300 member archive4.150 generations, 100 member archive5.150 generations, 500 member archiveThe data from these runs is too much to display in onefigure, but in general the larger archives performed better withno change in computation time. Obviously, more generationsled to better results, but in general returns diminishedsubstantially after about 150 generations. There was no singlerun that dominated the other runs when comparing Paretofronts, but the run with 150 generations and a 500 memberFig. 9. Computational Time1456
archive performed well and will be used as the primarycomparison against D’Angelo’s results.Pareto Front3A. Pareto Front ComparisonFigure 11 shows the results from D’Angelo, both using afive parameter airfoil representation and a seven parameterrepresentation.D’Angelo’s best results using the fiveparameter representation was selected for comparison. Theseven parameter result utilized Bayesian learning, andperformed significantly better. The ES results for 150generations and a 300 member archive achieved much betterresults in terms of the Pareto Front. Although not shown, eachof the remaining 15 runs dominated at least a portion of theseven parameter Pareto front, and all but three of the runscompletely dominated the five parameter Pareto front fromD’Angelo.B. Computation TimeThe five parameter model from D’Angelo requiredapproximately two hours to complete using just XFoil for aflow solver on a Pentium IV 2.8 GHz. The computational timefor the seven parameter model was not stated, but it can beassumed with the higher degrees of freedom that the algorithmrequired at least as long as the five parameter model. The ESalgorithm required approximately one hour to complete 50generations, which resulted in a total computation time of 2.98hours for the 150 generation, 300 member archive run.However, even though the entire algorithm required more timeto complete, the Pareto front bettered the five parameter frontin approximately 18 minutes and bettered the seven parameterfront in approximately 25 minutes, as shown in Figure 11.2.5Cl21.5DAngelo 5 ParDAngelo 7 ParES 2Fig. 10. Pareto Front ComparisonVII. CONCLUSIONSThe goals of this paper included developing and evaluatinga less restrictive solution encoding for airfoil optimization,determining the appropriate optimization meta-heuristic andmove operator for the solution encoding, comparing resultsobtained against previously published results in the subject,and determining the impact solution encoding can have onoverall performance. The Bezier Control Point encoding wasselected because of its ability to define almost any airfoilshape, and when coupled with an ES move operator many ofthe previous restrictions placed on the control point movements(when utilizing Genetic Algorithms or Particle SwarmPareto Front vs. Computation Time32.5Cl2DAngelo 5 Par.DAngelo 7 Par.Initial Seeds8.73 min17.6 min24.7 min1.06 hrs1.95 hrs2.96 hrs1.510.0040.0060.0080.010.012CdFig. 11. Pareto Front vs. Computation Time14570.0140.0160.0180.02
Optimizers) could be removed.The remainder of thealgorithm was kept very simple and close to the canonical ESalgorithm in order to focus on the impact of solution encodingin algorithm performance.compared to a wide variety of more recent airfoil optimizationarticles which use CFD to evaluate the objective function.The results on both optimization problems showed that byusing a more appropriate solution encoding and move operator,the performance of the optimizer could be greatly enhanced,both in objective function value and computation time. This isdue to the ability to represent more optimal airfoil geometriesthan the other discussed encoding options, and a more efficientmove operator for the specified problem. The oscillations inthe airfoil surface and issues with weak representation of theleading edge identified in [5] were shown to not be asignificant factor using the updated airfoil representation.NOMENCLATUREThere are improvements that could be made to thisalgorithm, such as updating the many-objective evaluationmethod for problem 1 rather than using a sum of the sixobjective functions, improving the Pareto front selection codefor problem 2, or adding some of the improvements that arecommon to ES algorithms over the canonical method.However, the results in this paper have shown that selecting theappropriate solution encoding and move operator are crucialand essential prior to adding more complex elements to thecode. Bezier control points and an ES algorithm proved fittingfor this specific airfoil optimization problem, and future workin this area would benefit aerodynamic optimization greatly asa whole.Promising future work in this area includes updating theflow solver to an Euler or RANS solver that is able to tacklemore complex flow problems, such as transonic flow. Thisupdate would eliminate a majority of the restrictions placed onthe airfoil representation due to XFoil. More importantly, thiswould allow the algorithm and airfoil representation to beCD coefficient of drag coefficient of liftCL moment coefficient at zero liftCm0Re Reynold’s number normalized distance along the chord whereXtrturbulent flow starts.α angle of attackREFERENCES[1][2][3][4][5][6]1458D’Angelo, S., Minisci, E. “Multi-objective evolutionary optimization ofsubsonic airfoils by meta-modeling and evolution control.” Proceedingsof the Institution of Mechanical Engineers, Part G: Journal ofAerospace Engineering Vol. 221, 2007, pp. 805-814Feinkos, G., Giannakoglou, K. “Inverse design of airfoils based on anovel formulation of the ant colony optimization method.” InverseProblems in Engineering Vol. 11, 2003, pp. 21-38.Jahangirian, A., Shahrokhi, A. “Inverse design of rransonic airfoilsusing genetic algorithm and
Bezier; multi-objective optimization, aerodynamic optimization. I. INTRODUCTION Airfoil optimization has been attempted in a variety of ways for a wide range of objectives. Typically, an airfoil optimization problem tries to maximize the performance of an airfoil with respect to a specific set of performance parameters at a specified flight regime.
Figure 3.1.3: Selig S1223 Airfoil (10) Figure 3.1.4: Clark Y Airfoil (11) Lastly, the Clark Y airfoil was analyzed. This airfoil was chosen specifically for its flat bottom plate after researching the difficulties associa
NACA 0010 α 6 deg. -3.00 -2.00 CP Conformal Transformation 0.7072)(CL Thin Airfoil Theory (CL 0.658) -1.00 0.00 1.000.0 0.2 0.4 0.6 0.8 1.0 x/c Fig. 8. Comparison of the pressure distributions for thin airfoil theory with conformal transformation results for an uncambered NACA 0010 airfoil at 6 deg. angle of attack.
No suction was used in the design. The resulting airfoil is almost identical in every respect to the GLAS II airfoil, which verifies that Glauert's airfoil did not employ suction, but a step drop in the velocity as expected. A design method for slot-suction airfoils should include the effects of true suction at the slot location; that is, the .
simulated results of OA209 airfoil under coupled freestream velocity/pitching oscillation conditions, it is indicated that the dynamic stall characteristics of airfoil associate with the critical value of Cp peaks (i.e., the dynamic stall characteristics of OA209 airfoil would be enhanced when the maximum
NACA 4412 Airfoil 4 digit code used to describe airfoil shapes 1st digit - maximum camber in percent chord 2nd digit - location of maximum camber along chord line (from leading edge) in tenths of chord 3rd and 4th digits - maximum thickness in percent chord NACA 4412 with a chord of 6” Max camber: 0.24” (4% x 6”) Location of max camber: 2.4” aft of leading edge (0.4 x 6”)
waves. The objectives of the APEX experiment are To increase the understanding of airfoil performance in the high-altitude, low-Reynolds-number, and high-subsonic-Mach-number flight regime. To obtain flight test data of airfoil performance parameters that can be used for validation of airfoil design codes. Figure 2. Laminar separation .
volAIR - revolutionary Aerodynamics Innovation and Research. Our Solution: Revolutionary Airfoil Design Slotted, natural-laminar-flow (SNLF) airfoil . Somers S204, SNLF airfoil. 3 Low-speed tests show simultaneous decrease in cruise drag coefficient and increase in static maximum lift coefficien
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