Use of Global Drag Rise Boundaries to Investigate Ill-Posed Transonic Airfoil Optimization

John J. Doherty*, Handing Wang†, Yaochu Jin‡

University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom

This paper presents a series of transonic airfoils, designed using differing optimization approaches, which are evaluated over a wide range of operating conditions using global aerodynamic performance maps. Global drag rise boundaries, which are identified, modelled and directly optimized during design, include drag divergence and onset of wave drag. The AIAA ADODG Case 2 airfoil optimization case is used to compare the results of the new global performance design approach with conventional multi-point optimization. The impact of alternative design formulations is presented in terms of both global performance maps and selected drag rise characteristics around the Case 2 design condition. In particular, the trade-off between drag divergence and the preceding onset of wave drag is discussed. The new approach addresses the issue of early excessive drag creep, which is typically encountered for optimization focused on a narrow range of operating conditions. The study provides some further insights into how a well posed optimization formulation for transonic airfoil design can potentially be established.

NomenclatureCp = pressure coefficient CL = lift coefficientCD = total drag coefficient CDw = wave drag coefficient = angle of attack M = freestream Mach numberMcrit = critical Mach numberMDD = drag divergence Mach numberMshk = local Mach number just upstream of a shockMshk13 = boundary at which local Mshk = 1.3t/cmax = airfoil maximum thickness ratioxTR_u = upper surface transition location relative to chord lengthxTR_l = lower surface transition location relative to chord lengthRe = Reynolds number based on chord lengthK = Korn technology factor ( K = MDD + CL/10 + t/cmax )K94 = notional variation in MDD and CL for K=0.94 when t/cmax=0.121

I. IntroductionAs a result of significant effort by the research and development community over many decades, the use of CFD

based optimization tools is now widespread. However, in spite of this growing usage and the very extensive range of alternative approaches proposed, for some applications at least, identifying a consistent means to achieve a well-posed optimization problem formulation, which leads to a practical design result, is still challenging. Difficulties associated with ill-posed problem formulations are still apparent in transonic design1, leading to designs that may have undesirable features or may be impractical for real-world use. A particular challenge can result if optimization is focused upon specific operating conditions2, without the problem formulation also fully considering the performance at off-design conditions. Various studies have been published3-5, which demonstrate alternative * Reader, Department of Mechanical Engineering Sciences, Associate Fellow AIAA.† Research Fellow, Department of Computer Science.‡ Professor in Computational Intelligence, Department of Computer Science.

American Institute of Aeronautics and Astronautics1

approaches for directly including off-design considerations or smoothing off-design behaviour6. Generally, these off-design regions must be specified as part of the problem formulation and the end results can still be highly dependent upon the initial choices made.

In recent years results reported for the AIAA ADODG Case 2 transonic airfoil test case1 highlight that the original formulation may be ill-posed, with resulting designs only achieving improved performance over a small local range of operating conditions, while off-design performance can deteriorate rapidly and in an unexpected manner.

This paper describes an initial study, which uses a new problem formulation for transonic airfoil optimization, based upon global behaviour rather than specific operating conditions. Specific operating conditions, as would be needed for single or multi-point based optimization, are not defined and instead the performance is evaluated over a complete envelope of operating conditions, allowing direct optimization of the location of key boundaries such as drag divergence and onset of wave drag. The new optimization formulation is informed by previous work7, which considered global behavior of drag rise boundaries resulting from multi-point optimization compared with a sonic plateau design approach.

Details of the new optimization problem formulation are discussed and results from application of the method to a standard airfoil case are presented. Comparison is made with results from a multi-point optimization approach to highlight the potential benefits of the new problem formulation. The global performance of each of the differing optimization formulations are evaluated using an existing approach for generation of detailed aerodynamic performance maps and critical performance boundaries. The results are used to discuss issues associated with formulating an optimization problem to achieve desirable off-design characteristics.

II. Global Performance Maps

A. AeroMap detailed performance mapsReference 7 describes the AeroMap tool, which can be used to evaluate airfoil designs through generation of

detailed aerodynamic performance maps over a wide region of operating conditions. In this study AeroMap is used in combination with a rapid airfoil prediction method (VGK), which was originally developed at RAE8 and is available from IHS ESDU9. VGK solves the full-potential equations for compressible flow, coupled with integral methods to represent laminar and turbulent boundary layer effects. The method is applicable for subsonic and transonic flow over a single element airfoil, with a sharp or moderately blunt trailing edge, including conditions with weak shock waves (local Mach number just upstream of shock Mshk taking values up to 1.3). The method is applicable for attached flows, but can also provide useful predictions for cases which would have limited regions of separation, allowing onset of separation to also be predicted. Within VGK wave drag (CDw) is calculated using Lock’s method, which post-processes the flow solution to identify individual shock waves around the airfoil. The viscous drag coefficient (CDv) is obtained from a far-field wake momentum thickness. The total drag coefficient (CD) is simply the sum of the wave and viscous drag components.

AeroMap generates a large database of VGK calculations, over a wide region of operating conditions, from which global performance boundaries are then identified. A drag divergence boundary is identified, for variation of drag divergence Mach number (MDD) with CL. The criterion defines MDD as the lowest freestream Mach number, at each value of CL, for which the gradient of the total drag coefficient, with respect to freestream Mach number, is equal to 0.110:

∂CD

∂ M |M=M DD

=0.1 (1)

Many other performance boundaries are extracted by AeroMap, including onset of wave drag, onset of separation, trailing edge pressure divergence, critical Mach number, peak Mach number, crest Mach number, local Mach number at the top of a shock (Mshk).

To highlight the output from the AeroMap process, which will be used throughout the rest of the paper to evaluate optimized airfoil designs, it is first applied to the RAE2822 airfoil. The RAE2822 is used as the starting point for the Case 2 optimization test case available from the AIAA ADODG web site §. The Reynolds number is set at 6.5 x 106 for all subsequent prediction results. Case 2 is focused on airfoil optimization at M = 0.734, CL = 0.824 (DP1) so this operating condition will also be included in presented results, although it is not used directly in the new optimization problem formulation.

§ https://info.aiaa.org/tac/ASG/APATC/AeroDesignOpt-DG/default.aspx [retrieved September 2017]

American Institute of Aeronautics and Astronautics2

AeroMap is applied over the region of Mach number and CL operating conditions listed in Table 1. Each VGK calculation was run in fixed lift mode. Transition is fixed at 3% chord for the upper and lower surfaces. A total of 495 VGK calculations are generated during the AeroMap analysis of an airfoil and the equivalent elapse time for a single processor, including all data collation and post-processing, is approximately 500 seconds.

Figure 1 shows a selection of the aerodynamic performance maps generated for RAE2822. Individual contour maps are shown for total drag (CD), wave drag (CDw), aerodynamic efficiency metric ML/D and pitching moment (CM). The extracted sonic flow boundary (Mcrit) indicates the operating condition where sonic flow is first detected on the airfoil surface and the drag divergence boundary is also presented.

a) b)

c) d)

Fig. 1 Aerodynamic performance maps and drag rise boundaries for RAE2822 for Re = 6.5 x 106 a) wave drag b) pitching moment c) total drag and d) ML/D.

A boundary Mshk13 at which the local flow speed at the top of a shock reaches a value of M=1.3 is also plotted for

information. As discussed previously VGK is based upon solution of the full potential equations and has been shown to provide satisfactory modelling of “weak” shocks, but should not be used for “strong” shocks. In particular, the Mshk13 boundary is used to highlight where results from VGK would likely become less trustworthy and also to potentially indicate where shock induced separation may become more likely.

American Institute of Aeronautics and Astronautics3

Table 1 Range of operating conditions for AeroMap global performance analysisMinimum Maximum Resolution

M 0.60 0.76 0.005CL 0.50 0.85 0.025Re 6.5 x 106

XTR_u Fixed transition (x/c = 0.03)XTR_l Fixed transition (x/c = 0.03)

A boundary (K94) corresponding to a notional design achieving a Korn technology factor of K=0.94 is also plotted for reference in Fig. 1. K is defined by Equation 2, with the baseline RAE2822 airfoil having a thickness ratio t/cmax = 12.1%. The DP1 design condition would be just ahead of drag divergence if an optimized airfoil could achieve this level for the technology factor K.

K=M DD+CL /10+ t /cmax (2)

In Fig. 1d it can be seen that the maximum value for ML/D occurs at approximately M=0.69, CL=0.75, corresponding to the highest values of M and CL for which a low level of wave drag is present. The ADODG Case 2 design condition (see symbol plotted at DP1: M = 0.734, CL = 0.824) is located significantly away from this region of maximum ML/D and beyond the initial drag divergence boundary. This design condition is at the edge of applicability of VGK for the baseline airfoil, with clear signs of unsmooth behaviour in the data presented beyond this boundary.

B. Approximate global performance mapsSince the full AeroMap process uses many hundreds of individual VGK calculations to build detailed

performance maps for each individual airfoil, direct use within optimization is excessively expensive. Contour maps from the AeroMap analysis of RAE2822, as shown in Fig. 1, highlight that each of the aerodynamic performance metrics are non-linear, but generally smooth, over the range of operating conditions investigated. Hence an efficient process for providing approximated performance maps, based upon a significantly reduced number of VGK calculations, is used for the current optimization study. The process uses a 2D regression approach, by combining the k-nearest neighbors algorithm (KNN)11, together with a Kriging model12, to build an approximate global model using only 23 VGK calculations. The resulting KNN-Kriging model is used to predict performance over the same extent of M and CL as defined earlier in Table 1.

Additional knowledge of the specific design problem can be built into the generation of the KNN-Kriging model. For example, it can be anticipated that there will be regions for which wave drag is not present (or is too small to be detected by the VGK wave drag post-processor). In this case the KNN-Kriging iterative process can be trained to identify zero and non-zero CDw regions as further VGK calculations are added. The final approximate models for the aerodynamic performance maps will hence use enriched training data sets, which include 23 VGK calculations, together with any artificial samples added, for example, to emphasize regions where zero wave drag might be expected.

Figure 2 shows the KNN-Kriging wave drag map for the RAE2822 airfoil based upon 23 VGK calculations. The result from the full AeroMap analysis, based upon 495 VGK calculations, is also redrawn from Fig. 1 for comparison. The approximate map captures the general features of the wave drag contours reasonably well in a global sense, but there will clearly be more significant local errors. The onset of wave drag boundary, corresponding to the contour for which CDw is 1 drag count, is also modelled adequately in terms of general shape and location.

American Institute of Aeronautics and Astronautics4

a) b)

Fig. 2 Wave drag map for RAE2822 predicted by a) KNN-Kriging approximate map based upon 23 VGK calculations and b) AeroMap using 495 VGK calculations.

The approximate map is used to also predict a drag divergence boundary, using the same criteria in Equation 1. A discretized boundary is first predicted using the approximate map, as highlighted in Fig. 2a. An analytic expression for the MDD boundary is then regressed by a linear least square fit as

(3)

where k and b are the two estimated parameters determining the predicted location and gradient of the boundary. The regression error ER from the linear fit is used to ‘punish’ any non-linearity in the predicted drag divergence boundary. The resulting penalized drag divergence boundary is given by Equation 4 and is plotted in Fig. 2a.

(4)

The optimization process uses the size of the area SDD between this penalized drag divergence boundary and the upper right corner of the global map. Minimization of SDD during optimization will generally lead to delayed drag divergence (MDD boundary moves towards higher values of M and CL) and will also prefer a linear (smooth) MDD

boundary. The onset of wave drag boundary, shown in Fig. 2a, is used to derive a further area function Sw which captures

the area between the lower left corner of the space and the onset of wave drag boundary. In this case an optimization case, which aims to maximize this area Sw, will tend to delay the onset of wave drag to higher values of M and CL.

A constraint on the absolute value of the pitching moment, at a single fixed operating condition, is discussed within the optimization test cases presented subsequently. An approximate global map for the pitching moment has been investigated during current studies, using the data available from the same 23 VGK calculations. It was hence possible to predict an approximate value for the pitching moment at any required operating condition. However, the resulting error in the local absolute value of pitching moment was considered too large for current use. Hence, for the case where a pitching moment constraint is required, a further VGK calculation is added at the specific design operating condition DP1, in order to return an accurate value for pitching moment for use within the optimization constraint.

III. Optimization MethodUnlike single or multi point optimization for airfoil shape design, the proposed optimization method has access

to a full, but approximate, map of performance over a wide range of M and CL together with boundaries for drag

American Institute of Aeronautics and Astronautics

predictionDDM

VGK calculation = 0 predictionDwC

5

divergence and onset of wave drag. As mentioned previously, the drag divergence boundary (MDD) is desired to move towards the right top corner of the global space. Therefore, as shown in Fig. 3, the area SDD between the predicted drag divergence boundary and the upper right corner of the global space will be minimized. Similarly, the boundary for onset of wave drag may also be desired to move towards the upper right corner of the global map, so the area Sw will aim to be maximized. Where both boundaries are optimized together, the optimization objective function is defined as:

maximise f =Sw−SDD (5)

The pitching moment CM constraint, where applied, is associated with the DP1 operating condition and it should be larger than a threshold -0.092.

Fig. 3 Illustration of optimization problem formulation

To efficiently optimize the above objective function, the covariance matrix adaptation evolution strategy (CMA-ES)13 is employed, which is an evolutionary algorithm for difficult non-linear non-convex continuous black-box optimization problems. CMA-ES uses (5+⌊3lnN⌋) as the population size, where N is the number of geometry parameters and N=6 for the results reported here. Each airfoil design will required 23 VGK calculations to generate an approximate aerodynamic map, together with 1 additional VGK calculation when the CM constaint is applied. In the current study, 20 iterations are used during the evolutionary search. The full global optimization process requires (5+⌊3lnN⌋)*24*20 = 4800 VGK calculations, which runs in approximately 6 hours.

IV. Airfoil Design Study Baseline multi-point design results have been generated for the RAE2822 test case for subsequent comparison

with the new optimization approach. In order to provide a fair comparison with the subsequent results based upon global optimization, the same number of geometry parameters is used. In particular, 6 variables are used to modify the camber of the RAE2822 airfoil, while the thickness distribution is fixed.

A wide range of multi-point problem formulations have been investigated, many of which have generated extremely unsmooth off-design results. As highlighted earlier, the quality of the output design is highly dependent upon the choice of the design operating conditions chosen. A multi-point design case having relatively smoother off-design behaviour is presented here, referred to as MPT. The multi-point optimization is based on 5 operating conditions, as shown in Table 2, including the design condition (DP1) defined for the ADODG Case 2 for which a constraint for pitching moment is included. The objective function for the multi-point design is based upon the average of the total drag across all five design conditions.

For optimization using CMA-ES in combination with the approximated global maps, 6 Hicks-Henne shape functions are used to modify the camber of the RAE2822 airfoil, while the thickness distribution remains unchanged throughout.

American Institute of Aeronautics and Astronautics6

Table 2 Operating conditions formulti-point optimization (MPT)

DP1: M = 0.734, CL = 0.824, CM ≥ -0.092DP2: M = 0.719, CL = 0.824DP3: M = 0.704, CL = 0.824DP4: M = 0.734, CL = 0.804DP5: M = 0.734, CL = 0.704

Four different problem formulations are used for the global optimization cases. DD refers to the result of minimizing SDD to delay drag divergence. CDW refers to maximizing Sw so trying to delay the onset of wave drag. DD+CDW refers to maximization of the combined function (Sw - SDD) to both delay drag divergence while also avoiding early onset of wave drag. The fourth case also uses the combined function (Sw - SDD) together with a constraint for CM ≥ -0.092 at DP1.

A. Multi-point optimizationThe MPT airfoil design is compared with the RAE2822 starting point in Fig. 4. Negative camber has been

introduced over the forward chord, similar to the design results for this test case shown in reference 1.

Fig. 4 MPT airfoil. Results from AeroMap full analysis of the MPT airfoil design are presented in Fig. 5. The location of the 5

operating conditions used within the multi-point optimization formulation, have also been plotted for clarity. The resulting design has a relatively small localized region of drag reduction at the center of 5 specified design conditions, as can be seen in both the total drag and ML/D contours. The drag divergence boundary is non-smooth, such that for values of CL above 0.75, drag divergence has been delayed by a value of approximately 0.02 in Mach number, while at lower values of CL drag divergence is similar to the RAE2822 airfoil. The pitching moment contours are non-smooth within the localized region of improvement, which may make the design highly undesirable or indeed impractical for use.

It is noticeable that the multi-point design has significantly earlier onset of wave drag over the full extent of the performance map. In practice, this will be seen as drag creep at off-design conditions for lower M and CL as shown later. It is also apparent that there is now supercritical flow on the airfoil over the full performance map, hence there is no Mcrit boundary present.

Figure 6 shows a further comparison between the multi-point design case and RAE2822, by showing the variation of both total drag and wave drag around DP1 at constant CL = 0.824 and for constant M = 0.734. The multi-point design has improved the level of drag around the DP1 condition, however deep drag buckets and extensive drag creep at lower values of both M and CL is apparent. It can be seen this results primarily from wave drag, together with the associated smaller impact for viscous drag.

American Institute of Aeronautics and Astronautics7

a) b)

c) d)

Fig. 5 Optimization result from multi-point drag optimization at 5 operating conditions (case MPT3).

The pressure distribution for the MPT design is compared with the RAE2822 airfoil for the DP1 condition in Fig. 7. The designs are also compared at M = 0.7 and CL = 0.5, which corresponds approximately to a condition for which the RAE2822 airfoil has a near sonic rooftop. The origins of the wave drag at the lower levels of M and CL for MPT is associated with the formation of a leading edge suction peak, which is not present for the RAE2822 airfoil. At high CL conditions this suction peak will lead to a forward shock. As M increases this shock slowly moves aft on the chord and there may also be a double shock feature established. Below the DP1 design region there is a localized near shock free region, resulting from beneficial interaction from leading edge expansion waves, but the feature disappears quickly as M increases further. Final drag divergence is associated with a strengthening single aft shock.

Results for MPT at DP1 is given in Table 3, together with results for RAE2822. It can be seen in Fig. 5b, together with the results in Table 3, that the pitching moment constraint (CM ≥ -0.092) has been satisfied by a small margin.

American Institute of Aeronautics and Astronautics8

a) b)Fig. 6 Total drag and wave drag variation for multi-point optimization compared to baseline

a) constant CL = 0.824 and b) constant M = 0.734.

Fig. 7 Pressure coefficient for MPT airfoil.

B. Global drag rise based optimizationThe optimized airfoil geometries for each of the DD, CDW and DD+CDW optimization cases are compared with

the RAE2822 starting point in Fig. 8. All of the designs are quite different to the MPT result shown in Fig. 4, particularly in terms of the introduction of larger aft-camber. It should be noted that the pitching moment constraint is not applied for these three optimization cases.

The AeroMap analysis of the resulting design for the DD global optimization case is shown in Fig. 9. It can be seen that the optimization process has apparently been very successful in delaying the drag divergence boundary by about 0.038 in Mach number (compare to Fig. 1 and see Table 3). However the delay in the drag divergence boundary is accompanied by earlier onset of wave drag, similar to that seen for MPT although it is not as extreme. It is also noticeable that the resulting design has smoother overall ‘off-design’ behaviour, including for the pitching moment, which has increased in magnitude from -0.093 to -0.109 at DP1.

American Institute of Aeronautics and Astronautics9

Fig. 8 Global drag rise optimization airfoils.

a) b)

c) d)Fig. 9 Optimization objective based on delayed drag divergence (case DD).

American Institute of Aeronautics and Astronautics10

The AeroMap results for the CDW global optimization case is shown in Fig. 10. The optimization process has successfully delayed the onset of wave drag, compared to RAE2822 in Fig. 1. The Mcrit critical Mach number boundary has been shifted up to higher values of CL and the drag divergence boundary has also been delayed by about 0.01 in Mach number (see Table 3). The resulting design has very smooth overall ‘off-design’ behaviour. However the pitching moment, as measured at DP1, has significantly increased in magnitude from -0.092 to -0.1486 as it is again not constrained.

a) b)

c) d)Fig. 10 Optimization objective based on delayed onset of wave drag (case CDW).

The result of the combined global optimization objective DD+CDW case is shown in Fig. 11. In this case the optimization process has been successful in delaying the drag divergence boundary by approximately 0.018 in Mach number compared to RAE2822 (see Table 3), while also delaying the onset of wave drag and critical Mach number. Again the resulting design has smooth overall ‘off-design’ behaviour. As for the previous CDW case, the pitching moment, as measured at DP1, has significantly increased in magnitude from an initial value of -0.092 to -0.1499 due to the fact that it is not constrained.

American Institute of Aeronautics and Astronautics11

a) b)

c) d)Fig. 11 Optimization objective based on combined delay of both drag divergence and onset

of wave drag (case DD+CDW).

A fourth global drag rise optimization problem formulation was attempted, with the addition of the pitching moment constraint to the DD+CDW objective. In this case no improvement could be obtained over RAE2822. This case was run for further generations within the optimization process, but without any improvement. It seemed the addition of the pitching moment constraint completely opposed the aim to delay the drag rise and onset of wave drag, so the optimization process was unable to find a path for improvement.

Figure 12 shows a further comparison between the three successful global optimization cases and RAE2822, for variation of both total drag and wave drag around DP1 at constant CL = 0.824 and for constant M = 0.734. All designs have improved performance at DP1, although this condition was not directly specified in the problem formulation. There is also significantly improved smoothness at lower M and CL compared to the MPT design in Fig. 6, with no indication of drag creep for lower values of M and CL. The CDW and DD+CDW designs have each improved the level of drag at all M for CL = 0.824 compared to RAE2822, while all three cases have lower drag over nearly the full range of CL at M = 0.734. It can be seen these improvements result primarily from wave drag being controlled, together with the associated smaller impact for viscous drag.

American Institute of Aeronautics and Astronautics12

a) b)Fig. 12 Total drag and wave drag variation for global drag rise optimization cases compared to RAE2822

a) constant CL = 0.824 and b) constant M = 0.734.

The pressure distributions for the three global drag rise designs are each compared with the RAE2822 airfoil for the DP1 condition in Fig. 13. The designs are also compared to RAE2822 at M = 0.7, at a value of CL chosen to indicate whether the designs possess a near sonic rooftop feature, which was discussed in more detail in reference 7.

The DD design has a small leading edge suction peak, which is similar to but less extreme than that seen for the MPT design. As for the MPT case, the earlier onset of wave drag seen for the DD design is associated with this leading edge suction peak and the presence of a forward shock. This leading edge suction peak is not present for the CDW design and is very limited for the DD+CDW design.

The CDW design has a similar, but extended, sonic rooftop to RAE2822 at M=0.7, which occurs at a higher CL

for CDW through the introduction of aft-camber. It is notable that the CDW problem formulation results in the presence of this sonic rooftop feature. The ‘sonic plateau’ type design approach, as described by Harris 14 in reference to early supercritical airfoil studies by Whitcomb, was originally employed to delay drag divergence. This sonic plateau design approach was also investigated previously using AeroMap in reference 7. The DD+CDW design has a near sonic rooftop feature, but also has a small leading edge suction. As seen in Fig. 12, the DD+CDW design achieves a similar delay in the onset of wave drag to the CDW case, but additionally achieves later drag divergence.

Fig. 13 Pressure coefficient for global drag rise optimization designs.

Results for all of the optimization cases are listed in Table 3, compared against the RAE2822 starting case. Results corresponding to the ADODG Case 2 condition (M = 0.734, CL = 0.824) are provided, although there is no specific emphasis given to this condition during the global optimization process. The drag divergence Mach number for CL = 0.824 and the equivalent Korn technology factor K is also presented in Table 3.

American Institute of Aeronautics and Astronautics13

The MPT design achieves the lowest level of drag at DP1 and satisfies the pitching moment coefficient constraint. However, this design has excessive drag creep with significant deterioration in performance at off-design conditions. In addition, the pitching moment was seen to vary non-linearly.

The global drag rise optimization cases each achieve reasonable improvements in drag at the DP1 condition, although this specific operating condition was not prescribed in the optimization problem formulation. The DD design is closest to satisfying the pitching moment requirement, while the CDW and DD+CDW designs would violate the pitching moment requirement significantly. An attempt to include the pitching moment constraint suggested it was completely in opposition to the combined aim of delaying both drag divergence and the onset of wave drag.

Table 3 Airfoil design results, Re = 6.5 x 106

CL = 0.824 M = 0.734, CL = 0.824MDD K () CDw CD CM ML/D

RAE2822 0.704 0.908

2.88 0.0081 0.0195

-0.0932 31.1

MPT 0.727 0.931

2.90 0.0021 0.0127

-0.0912 47.9

DD 0.742 0.946

2.69 0.0041 0.0142

-0.1093 42.6

CDW 0.714 0.917

1.73 0.0043 0.0149

-0.1486 40.8

DD+CDW 0.722 0.926

1.84 0.0027 0.0132

-0.1499 45.9

V. Conclusions

Aerodynamic performance maps have been used to investigate the global performance of various optimization formulations, based loosely upon the AIAA ADODG RAE2822 test case. A multi-point design has been used to highlight the ill-posed nature of transonic airfoil design, leading to localized improvements and undesirable off-design drag creep behaviour. A new optimization problem formulation for direct design of global drag divergence and onset of wave drag boundaries has been demonstrated. The separate and combined effects of delaying drag divergence and the onset of wave drag have each been investigated, highlighting that each objective leads to a differing design type. It has been possible to achieve designs having improved global performance, while avoiding undesirable off-design characteristics. However, studies so far suggest there may be a direct conflict between constraining pitching moment while attempting to delay both drag divergence and the onset of wave drag.

References1LeDoux, S. T., Vassberg, J. C., Young, D. P., Fugal, S., Kamenetskiy, D., Huffman, W. P., Melvin, R. G., and Smith, M. F.,

“Study Based on the AIAA Aerodynamic Design Optimization Discussion Group Test Cases,” AIAA Journal, Vol. 53, No. 7, July 2015, pp. 1910-1935.

2Drela, M., “Pros & Cons of Airfoil Optimization,” Frontiers of Computational Fluid Dynamics, edited by D. A. Caughey and M. M. Hafez, World Scientific, Singapore, 1998, pp. 363–381.

3Zingg, D.W., and Elias, S., “Aerodynamic Optimization Under a Range of Operating Conditions,” AIAA Journal, Vol. 44, No. 11, Nov. 2006, pp. 2787–2792.

4Buckley, H. P., Zhou, B. Y., and Zingg, D. W., “Airfoil Optimization Using Practical Aerodynamic Design Requirements,” Journal of Aircraft, Vol. 47, No. 5, 2010, pp. 1707–1719.

5Li, W., Huyse, L., and Padula, S., “Robust Airfoil Optimization to Achieve Consistent Drag Reduction Over a Mach Range,” NASA/CR-2001-211042, Aug. 2001.

6Li, W., Krist, S., and Campbell, R., “Transonic Airfoil Shape Optimization in Preliminary Design Environment,” Journal of Aircraft, Vol. 43, No. 3, May–June 2006, pp. 639–651.

7Doherty, J., “Transonic Airfoil Study Using Sonic Plateau, Optimization and Off-Design Performance Maps,” Paper AIAA-2017-3056, 35th AIAA Applied Aerodynamics Conference, 5-9 Jun 2017, Denver, Colorado.

8Ashill, P.R., Wood, R.F., and Weeks, D.J., “An improved, semi-inverse version of the viscous Garabedian and Korn method (VGK),” RAE Technical Report 87002, 1987.

American Institute of Aeronautics and Astronautics14

9ESDU 96028, “VGK method for two-dimensional aerofoil sections. Part I: Principles and results,” April 2004.10Gur, O., Mason, W. H., and Schetz, J. A., “Full-configuration drag estimation,” Journal of Aircraft, Vol. 47, No. 4, July

2010, pp. 1356-1367.11Hansen, N., and Ostermeier, A., “Completely derandomized self-adaptation in evolution strategies”, Evolutionary

Computation, Vol. 9, No. 2, 2001, pp. 159-95.12Altman, N.S., “An introduction to kernel and nearest-neighbor nonparametric regression,” The American Statistician, Vol.

46, No. 3, 1992, pp.175-185.13Williams, C.K., “Prediction with Gaussian processes: From linear regression to linear prediction and beyond,” in Jordan, M

(ed), Learning in Graphical Models, Nato Science Series D, Vol. 89, 1998, pp.599-621.14Harris, C. D., “NASA Transonic Airfoils: A Matrix of Family-Related Airfoils,” NASA, TP 2969, 1990.

American Institute of Aeronautics and Astronautics15