Figures 10.22 and 10.23 show the integrated loads of pitching moment and torque over the blade during one revolution for varying design parameters. The peak-to-peak moment value reduces with further outboard notch positions and the average moment tends to be more centred around zero when sweep is higher. The torque seems to be mostly affected by sweep and on the advancing and retreating side. The torque is reduced more on the advancing side than the increase on the retreating side. This is because it alleviates the compressibility effects. On the retreating side, the differences are more subtle. This data suggests that a highly swept blade would be optimal. Hav- ing a higher notch gradient would also improve the moments and having a notch more inboards would amplify the effects of the sweep. The quantities of the design parameters that make up the optimum design are obtained in the next section.
The vibratory pitching moment (defined as the pitching moment less the mean and 1/rev oscil- lation) in Figure 10.24 shows sensitivity to sweep mostly. However, its effect is captured in the peak-to-peak moment and hence it is not used in the objective evaluation.
Figure 10.22: M2Cm integrated over the full blade at each azimuth for the BERP-like rotors.
Figure 10.23: M2Cq integrated over the full blade at each azimuth for the BERP-like rotors.
10.7
Planform Optimisation
The original population obtained from the CFD results contained 27 points. The design param- eters selected were the average pitching moment (Cmavg), peak-to-peak pitching moment (∆Cm) and the torque coefficient (Cq). The objective function weights were determined such that on average, each of these parameters had the same influence. This was determined using the data from the original population which was obtained using the high-fidelity CFD solver. First each design was scaled with the baseline design case. The baseline case chosen was the BERP most similar to the typical fast flying rotor as shown in Figure 10.25. The parameters for it are NE = 12, NG = 25, SW = 0.25. The average ratio of Cmavg to ∆Cm was found to be 2.7893:1 and the average ratio of ∆Cm to Cq was found to be 0.9548:1. Therefore the ratio of Cmavg to ∆Cm to Cq is obtained as: 2.6634 : 0.9548 : 1.0000. The weight for Cq was then calculated as:
wCq =
2.6634
2.6634 + 0.9548 + 1 = 0.5767 (10.3)
Hence the weight of Cmavg and ∆Cm are given as:
wCmavg = 0.5767/2.6634 = 0.2165 (10.4)
w∆Cm = 0.5767/0.9548 = 0.6040 (10.5)
Cqwas also used as a constraint. Since the rotors were trimmed to a CT/σ = 0.09, CT/σ did not need to be constrained.
From this data, it was determined that the most influential design parameter was the sweep, followed by the notch position and then the gradient. ANNs were trained for each of the perfor- mance parameters as shown in Figure 10.26. These parameters were used to find the optimum blade using a GA which was compare with the Pareto front shown in Figures 10.27 to 10.29. The ANNs accuracy was also estimated relative to the change in the performance of the design obtained using the CFD data. The maximum error in the objective function obtained was found to be 2.7%.
The comparison of the optimum with the original baseline blade is shown in Table 10.5 for the trimmed rotors. A much better avg M2Cm was obtained and also a slightly better ∆M2Cm. The performance of the resulting optimum relative to the baseline design is shown in Figures 10.30, 10.31 and 10.32. The black line indicates the contour line of the baseline design and the red is the optimum blade design. On the moment plot, it can be seen that the optimised blade has larger areas of lower moment especially on the retreating side but also on the outboard region of the advancing side. A similar trend can be seen for the torque plot. Figure 10.32 shows the difference in the objective function components between the optimum and the baseline design. For the regions of higher OFV, the areas enclosed by red are larger showing that the optimsed blade increases the area where performance is good and vice versa for areas of poorer performance. The hover performance of the optimum blade was measured and compared to the baseline in Table 10.5. The results were obtained at a collective of 13 degrees. The CT/σ is slightly less than the baseline design mostly due to the added solidity, but the FM obtained was higher. Figure 10.33 compares the CP distribution for the BERP reference and optimised blade. It can be seen that the optimised one spreads the loading at the tip over more of the span. Therefore, overall, the optimised blade has better performance than the baseline blade especially in terms of moment where the average pitching moment was reduced to approximately a fifth of the baseline designs.
NE NG SWEEP CT/σ CQ avg M2CM ∆M2Cm 11.75 35 0.21 0.0831761 0.000171 -0.000373 0.010782
After trim 0.0898057 0.000187 -0.000105 0.011019
Baseline 0.0904548 0.000186 -0.000517 0.010832
Hover Performance Comparison
NE NG SWEEP CT/σ FM Collective (deg)
11.75 35 0.21 0.28957 0.6873 13
Baseline 0.30390 0.6543 13
Table 10.5: BERP and baseline case (NE = 12, NG = 25, SW = 0.25) performance comparison related to Figure 10.30. NE is the notch position parameter and NG is the notch gradient parameter. avg M2CM is over one revolution and∆M2CMis the peak-to-peak amplitude over one revolution
Figure 10.25: The baseline BERP-like rotor in comparison to a swept tip design. The parameters for this rotor are NE = 12, NG = 25, SW = 0.25.
Cq scaled average Cm scaled ∆Cm
Figure 10.26: ANN predictions with training data and GA selection shown for each of the performance parameters. The white dots are the GA optimal selection and the black dots are the CFD training data for the ANNs. The dashed line is where the contour level = 1 i.e. the value for the baseline design.
Figure 10.27: Pareto front points compared with GA selection; red is NE = 11.5, green is NE = 11.75, blue is NE = 12. The white dots are the GA optimal selection and the cyan dots are the Pareto front solutions.
Figure 10.28: Pareto front for the BERP-like design.
(a) (b)
Figure 10.29: (a) Pareto front points compared with GA selection; red is NE = 11.5, green is NE = 11.75, blue is NE = 12, (b) OFV contour colour map in the design space. The white dots are the GA optimal selection, the cyan dots are the Pareto selection and the black dots are the CFD training data for the ANNs.
M2Cm M2Cq
Figure 10.30: Optimum (red contour lines) compared to reference (black contour lines) for M2Cm and M2Cq.
M2C
m M2Cq
Figure 10.32: OFV for the baseline and BERP-like rotor where the black contour lines represent the reference rotor with parameters: NE = 12, NG = 25, SW = 0.25, and the red lines represent the optimised rotor with parameters: NE = 11.75, NG = 35 and SW = 0.21 where OFV=−0.2×M2C
m−0.6×M2Cq.
CP distribution comparison
Planform comparison
Figure 10.33: Cp and planform distribution of the reference (blue) and optimised (red) BERP variant at
Chapter 11
Summary and Conclusions
11.1
Summary of the Optimisation Method
This thesis documents an optimisation method for helicopter rotor blades using Computational Fluid Dynamics (CFD). The objective was to “tweak” an existing good design to obtain even more performance out of it. The optimiser allows for high-fidelity CFD computations to be used to obtain accurate aerodynamic data for the objective, but at the same time, allows for this to be carried out efficiently by use of a lower order model, or metamodel. The information obtained from the metamodel is based on interpolation data from the high-fidelity model. This maintains accuracy and efficiency and makes the method usable for helicopter rotor aerodynamics. The main reason for this is that the optimisation method is decoupled from the high-fidelity CFD data. The two are linked through a database. The high-fidelity CFD data is used to populate the database to create a design space and the optimiser accesses this design space through a metamodel. In most cases, a parameterisation technique existed, but for some cases, specific parameterisation techniques had to be developed to create the design space.
The optimiser used was a Genetic Algorithm (GA). The reason for this is that the design space for rotor aerodynamics especially, is expected to be uneven and hence the likelihood of the optimiser getting trapped in local optima is high. Evolutionary methods avoid this problem and one of the most effective of these methods are GAs. GAs for all the cases were coupled with and relied on metamodels. A number of metamodels were tested, including polynomial and POD based methods, and the most promising ones were the Artificial Neural Networks (ANNs) and kriging methods.
Once all of this was put together, the method was demonstrated using a number of cases: • transonic aerofoil test case in steady flow
• wing planform optimisation in steady flow
• optimal selection of rotor aerofoil sections in unsteady flow • optimal selection of twist for hovering rotors in steady flow
• forward flying rotor optimisation in unsteady flow in conjunction with hover optimisation of the same
• optimisation of simplified fuselage bodies for drag reduction
• optimisation of a BERP-like rotor in forward flight constraining hover performance
For the transonic aerofoil case (Chapter 4), the RAE 2822 aerofoil was used and a parame- terisation technique was created using the Chebyshev polynomial method (described in Section 3.2.1). Three parameters were optimised for high lift to drag ratio while constraining drag and
drag divergence Mach number. The results showed an 8% increase in L/D with negligible change in pitching moment and drag divergence Mach number. A version of the Latin Hypercube Sam- pling method (LHS) was also tested against the full factorial method.
For the wing planform optimisation in Chapter 6, the aim was to obtain elliptic loading by varying the chord and sweep while maintaining lift and drag. The wing was split into three sections and their position and width were defined as the five design parameters. Due to the relatively larger number of parameters, the sampling method used was a version of adaptive fractional factorial method, where points were added to the database where there was the most promise of finding the optimum. The results showed a 13% improvement in the elliptic loading and a small improvement in lift-to-drag while constraining maximum drag.
In Chapter 5 for the rotor section optimisation, dM/dt calculations were performed to simu- late a rotor section in forward flight. NACA aerofoils were used and the design parameters were the thickness and camber as defined using the NACA numbering system. The objective was to reduce the effects of compressibility on the advancing side and of high angle of attack on the retreating side. This was captured using the moment curve over a full revolution of the rotor and the average drag. This case was also used to test the effect of the ANN parameters on its prediction accuracy and to compare the predictions of the kriging and ANN metamodels. The final design had resulted in over 50% reduction in the average pitching moment and at most, a 37% reduction in average drag both, inboard and outboard, over one revolution.
The optimisation method was then applied to full rotors in hover and in forward flight. Chapter 7 describes a simple rectangular rotor optimisation in hover for high Figure of Merit (FM) over a range of thrust coefficients. This was obtained by modifying the linear twist of the blade and measuring the highest FM and the gradient of the FM over a range of collective angles. The LHS method was tested here and its limitations for small sets of data was shown. The kriging method was also compared to the ANN for this case. The twist selected was slightly less than the maximum in the design space. The increase in the maximum FM from the original was 3%, but the maximum FM changed less with thrust than the baseline which is quantified by the gradient of the FM against the thrust. This was improved by 7.5%.
Chapter 8 describes the optimisation of the UH60-A rotor sweep and anhedral of the blade tip in both hover and forward flight. The parameterisation technique ensured that the area of the tip remained constant and that a smooth surface was maintained where the anhedral occurred. The objectives were to reduce stall effects on the retreating side and compressibility effects on the advancing side and the pitching moment was used to capture this via two parameters viz. the peak-to-peak and the average value over a full revolution. The vibratory pitching moment and the torque were constrained. The resulting blade had a lot more anhedral and slightly less sweep resulting in a 6.7% decrease in torque, a 17.6% decrease in the peak-to-peak pitching moment over a revolution and a 24% decrease in the average pitching moment in one revolution. The hover performance with the new planform was maintained.
The final rotor case was the BERP-like rotor tip (Chapter 10). Here a parameterisation tech- nique was defined using a set of equations that captured three design parameters viz. the notch position, notch gradient and sweep of the tip. Due to the significant change in area, the rotor was trimmed to ensure the same thrust/solidity of the rotor. The objective was the same for the forward flight optimisation of the UH60-A rotor. The forward flight conditions were taken from a typical fast-flying, moderate lift rotor. The hover performance, in terms of FM, with a BERP modification was maintained by modifying the twist and anhedral. This rotor was then used to optimise the BERP-like rotor in forward flight and the final design was then analysed in hover to check for any performance compromise. The objectives were captured using the pitching moment curves as before, but torque was also included in the objective as well as a constraint. Here, the
Harmonic Balance method was used to obtain all the forward flight data for the initial design space as opposed to the time marching method used in all the other cases. This significantly improved the efficiency of the method. The final blade had a steep notch at about 86% of the span with high sweep resulting in an improvement in the average pitching moment over a revolution of 80%, although the peak-to-peak moment was increased by 1.7%. The torque did not change much and hover performance was maintained.
The method was also applied to fuselage optimisation in Chapter 9. As an example, the JM- RTS fuselage from JAXA was selected as a suitable test case. The parameterisation used was a modification of the super-elliptic equation technique used to define the ROBIN fuselage. The objective was to reduce the drag by changing the angle of the wind screen. This test case was mainly carried out for development of the parameterisation method. The generation of geometries is automated in ICEMCFD. Also a very simplified uniform actuator disk was included to simulate the downwash from the rotor on to the fuselage. A 3.3% reduction in drag was achieved for a shape that reduced the gradient of the windscreen area.
The conclusions drawn about the method and each of these test cases is explained further in the next section.