This case was also used to test out the Latin Hypercube Sampling (LHS) method in comparison to the full factorial method. The validation data were additional points in the design space obtained using the HMB solver, but these results were not included in the training data for both the LHS and full factorial sampling techniques. Using the LHS method (with additional database boundary points) obtains the same trends but not as accurately as using the full factorial method as can be seen in Figure 4.8 for pitching moment, which had the worst prediction accuracy with the LHS. However, the trend was still captured and this was sufficient to obtain similar optimum design parameters.
α3 = 0.00044
α3 = 0.00344
Chapter 5
Rotor Aerofoils Optimisation
In this case, the aerofoil sections for a rotor blade are optimised for forward flight. The simulated conditions are shown in Figure 5.1. Typically rotors tend to have up to three different sections
Figure 5.1: Conditions of forward flight for optimisation.
on a rotor blade. Usually these sections are selected to improve the efficiency of the rotors or to counter the disadvantages of varying other design aspects of the rotor such as twist, planform, etc. For this case, the assumption is that the rotor is a rigid blade with twist and a cyclic pitch amplitude of 8 degrees. Three stations along the blade are selected to be optimised. The design parameters to be optimised for were thickness and camber and to do this, the NACA 5-digit aerofoils were used as the parameterisation technique. Table 5 shows the parameters to be used in creating the initial population.
The CFD calculations were carried out by simulating the motion of the section as a dM/dt unsteady calculation, the details of which can be found in Section 2.4. Therefore for each section, the Mach number variation changes depending on the station. The details for each station are also included in Table 5. Each CFD calculation was carried out over three cycles and the results from the final revolution were used to obtain the performance of the section.
5.1
Performance Parameters
The objective here is to minimise the adverse effects of compressibility on the advancing side and high angles of attack at low Mach number on the retreating side of the rotor disk. Both
Radii (r/R) 0.5 0.75 0.9
Mach number range 0.15 - 0.55 0.325 - 0.725 0.43 - 0.83
AoA 10±8 7±8 5±8
NACA Thickness (t/c) 9, 12, 15 and 18 NACA Camber (5-digit) 0, 130, 230, 330, 430
Table 5.1: Table showing test cases used for 2D optimisation.
these aerodynamic phenomena not only cause increase in torque (and hence power requirements) and drag, but they also cause high pitching moments that put high demands on the pitch links. Therefore a good performance parameter to include in the objective function would be the pitch- ing moment over a full cycle as well as the lift and drag coefficients. Figures 5.2 to Figures 5.7 show the variation in these three performance parameters for a full cycle (360o azimuth) at r/R = 0.5 and 0.9.
Looking at the coefficient of drag first, it can be seen that stall is easily observable at the in- board station in Figure 5.2. For the NACA 0009, on the retreating side (close to ψ = 270o, its average drag coefficient (Cdavg) is much higher. Also the oscillations caused by the shedding of vortices are large and with lower frequency which is also an indication of stall. With increasing camber and thickness, the average drag drops and the oscillations reduce. On the advancing side, an increase in the drag can be observed with increasing camber due to compressibility effects. Also at high thickness values, there is a higher increase in drag with increased camber on the advancing side than with thinner aerofoils.
The effect of compressibility at high Mach number on the advancing side at the outboard station is also evident from the drag curves in Figure 5.3. The NACA 0009 does not have the steep drag rise and the NACA 0012 also behaves the same to some extent. All the other aerofoils have a steep drag rise on the advancing side. At higher thickness values, camber has little effect on the drag rise. Not much change in drag occurs on the retreating side with change in camber and thickness although the NACA0009 seems to stall. The blip in the curves, especially visible at thicknesses of 15% and 18% are due to the sudden form of a shock on the surface of the aerofoil as shown in Figure 5.8.
Overall, taking the average drag value will distinguish the designs that are vastly poorly perform- ing sections from better ones. However, further performance parameters are necessary to fine-tune the optimisation for smaller changes in the thickness and camber.
Figures 5.4 and 5.5 show the pitching moment variation with azimuth at the inboard and outboard stations. For the moments, the aim is to try and keep it as close to zero as possible to reduce the load required on the pitch links. There are two parameters that can be used to do this viz. the average value,Cmavg and the peak-to-peak value, ∆Cm, over a full cycle. ∆Cm is a good indicator of the power required to maintain the blade’s pitching moment, but it does not account for all the oscillations in the curve. Together with the average moment, it then gives a more complete picture. However,Cmavg also reflects the power required to maintain pitching moment. Therefore, using both parameters will increase its weighting in the objective function. Therefore, the decision was to use the Cmavg since it gives a more complete picture than ∆Cm. This is more visible in the 9% thickness plot at r/R = 0.5 in Figure 5.4, where the symmetrical aerofoil has one of the lowest ∆Cm but it has the most oscillations due to stall which is better reflected in its Cmavg. At the outboard station, theCmavg reflects the same objective as ∆Cm. This confirms the use of the Cmavg over ∆Cm.
The Cl curves in Figures 5.6 and 5.7 can also be used to show the effects of stall and com- pressibility, but this has been adequately dealt with using the drag and moment curves. However, the average Cl is used as a constraint.
It is accepted that a more fair comparison would be to compare the performance at the same lifting capability of the design, for example by considering the zero-lift angles as opposed to the geometric angle of attack. However, for an unsteady case such as this, it is difficult to match the lift load throughout the cycle or to even obtain the same average lift value, because of the many aerodynamic phenomena that occur based on shape. For the cases shown here, a maximum of 50% difference in average Cl was contained within the data for the very extreme cases where the un-cambered thin sections and the highly cambered thick sections were compared. This shows the complexity of the optimisation task though it could be addressed with further computations. Once the performance parameters were established, the metamodel was employed. For the meta- model, ANNs were trained with the scaled data i.e. as a ratio to the reference section (NACA 0012) at the same conditions. The ANN parameters were 2 input, 2 hidden layers with 15 neurons each and 1 output for each performance parameter. The error convergence was set to 0.01 and convergence occurred at the end of the run. The trained ANN was then used to predict points every 10th of the range. The results agreed well with the data obtained from the solver - most points overlapping exactly. The trends in all the performance parameters (i.e. the average and peak-to-peak values for all the coefficients as well as the gradient of the moment and drag curves) as predicted using ANNs can be seen in Figure 5.9. It can be seen that drag has more variation with thickness than with camber and moment seems to have similar sensitivity to both thickness and camber.
Figure 5.2: Cd for varying NACA aerofoil camber - thickness 9, 12, 15 and 18% chord at the inboard station, r/R = 0.5. Mloc = 0.35, M∞= 0.2 in forward flight.
Figure 5.3: Cd for varying NACA aerofoil camber - thickness 9, 12, 15 and 18% chord at the outboard station, R = 90%. Mloc = 0.63, M∞= 0.2 in forward flight.
Figure 5.4: Cm for varying NACA aerofoil camber - thickness 9, 12, 15 and 18% chord at the inboard
Figure 5.5: Cm for varying NACA aerofoil camber - thickness 9, 12, 15 and 18% chord at the outboard
station, R = 90%. Mloc = 0.63, M∞= 0.2 in forward flight.
Figure 5.6: Cl for varying NACA aerofoil camber - thickness 9, 12, 15 and 18% chord at the inboard station, R = 50%. Mloc = 0.35, M∞= 0.2 in forward flight.
Figure 5.7: Cl for varying NACA aerofoil camber - thickness 9, 12, 15 and 18% chord at the outboard station, R = 90%. Mloc = 0.63, M∞= 0.2 in forward flight.
Figure 5.8: Pressure contours on the lower and upper surface of NACA 0015 for increasing azimuth angle. R = 90%, Mr/Rin hover would be 0.63 and M∞= 0.2 in forward flight.
5.2
Objective Function
To create the objective function, the reference design to which all designs will be scaled is the NACA 0012 section so that the weightings are truly proportional to the requirements and are not affected by the actual values of the performance parameters. To capture the objectives, Cdavg andCmavg over a complete revolution of the rotor was used to create the objective function while keeping Clavg ≥1. The drag component was the primary component in providing the selection pressure and the moment component, secondary. The guiding weights therefore, were determined by averaging the ratio of averageCdavg toCmavg:
1 n n ∑ i=1 Cdavg Cmavg = 1.2129 (5.1)
wheren= 20 is the number of sample points and the limiting weight for the loads were w(Cdavg) = 1 1 + 1.2129 = 0.452 (5.2) w(Cmavg) = 1.2129 1 + 1.2129 = 0.548 (5.3)
As there was large variance in this ratio and sinceCdavg is the primary component, the objective function used was
OF V =−0.5Cdavg −0.3Cmavg + 0.8 (5.4)
The ANN predictions of these two components at 50%R and 90%R and the effect camber and thickness have on them can be seen in Figure 5.9. Both drag and moment show that thick aerofoils are more suitable inboards and thinner ones, outboard, as expected. Moment also refines the optimisation of camber showing that more symmetrical aerofoils are suitable outboard and vice versa.
5.3
Optimisation
The GA employs these trained ANNs to obtain the optima. Figure 5.10 shows that the optima also fall on the Pareto front at all three stations. The objective function forces the global solution to a small region on the Pareto front. The combined Pareto and objective function method used the objective function to create the selection pressure for the parents for crossover, but does not determine the fitness. Table 5.2 shows the results of the GA for the inboard and outboard stations. Thinner more symmetrical sections are good for outboard stations and thicker, more cambered sections perform better more inboards, which is in agreement with most real-life helicopter rotors.
r/R = 0.5 r/R = 0.9
Camber Thickness OFV Camber Thickness OFV
33 15 0.44 9 9 0.30
43 16 0.42 7 11 0.14
32 14 0.36 5 10 0.12
Table 5.2: Aerofoils selected by the GA for the inboard and outboard stations. The * values are scaled with the reference section, the NACA 0012.
The kriging metamodel was also employed to compare its performance with the ANN predic- tions. A comparison between the optimum surface created by both is shown in Figure 5.11. It can be seen that kriging favours a smoother surface, although this did not significantly affect the outcome of the optimisation (a 1% difference in camber).
Cdavg ∆Cd
Cmavg ∆Cm
Clavg ∆Cl
∂Cm ∂Cd
Figure 5.9: Variation of loads for r/R = 50% and 90%; Red represents the upper extreme and blue, the lower extreme.
Inboard station
Midboard station
Outboard station
Figure 5.10: Comparison for the dM/dt cases inboard, midboard and outboard stations of the use of pareto optimisation, objective functions and both i.e. objective function used for selection pressure on designs.
Inboard station, 50% r/R
Outboard station, 90% r/R
Figure 5.11: Comparison of the optimum surface created by the ANN (blue) and Kriging (red) metamodels. The GA selection is shown as white dots.