Preliminary airfoil design of an innovative adaptive variable camber compliant wing

2.1. Airfoil description

One of the most important things in design and optimization of an airfoil is the parameterization of airfoil, which determines not only the accuracy of airfoil description but also the speed. There are many ways in the parametric description of the airfoil geometry, such as free from deformation (FFD) method based on NURBS spline curve, Bezier Curve method, etc. M. Kulfan and E. Bussoletti [16, 17] proposed a “shape function” and a “class function” to describe the geometry of an airfoil, in which the “shape function” provides the ability to directly control key geometry parameters such as leading edge radius, trailing edge boat tail angel and closure to a specified aft thickness, while the “class function” generalizes the method for a wide variety of geometries; the “shape function” and “class function” methodology, which is also called the Class-Shape-Transformation (CST)methodology, can describe a 2D airfoil with a handful of control points. Lane et al. [18] pointed out that, compared to Bezier curves, the CST methodology describes airfoil with less parameters due to appropriate class functions that are similar to airfoil profiles. Guan et al. [19] studied the description accuracy of several methodologies such as Bezier Curve method, Hicks-Henne method and PARSEC, finding that the CST methodology is featured with less control parameters and higher accuracy, which is the reason that it is chosen in this study. In the CST methodology, the “class function” is used to define the basic geometry of an airfoil, from it all geometries optimized are derived; the “shape function” is used to modify the basic geometry to obtain needed geometry. In this study, as XFOIL was used to investigate the aerodynamic performance of optimized airfoils, the chord of airfoil investigated is set to be 1. For both upper and lower surfaces, the airfoil with chord of 1 can be stated as:

where, $\sqrt{x}\left(1-x\right)$ is the “class function” that defines an airfoil with round nose and sharp trailing edge, in which $\sqrt{x}$ is for providing a round nose and $\left(1-x\right)$ is required to ensure a sharp trailing edge; $∆{th}_{TE}$ is the thickness of trailing edge, and $x∆{th}_{TE}/2$ is used to control the trailing edge thickness; $\sum _{i=0}^n{A}_i{S\left(x\right)}^i$ is the “shape function”, representing a general function that describes the middle section shape of airfoil.

Usually, Bernstein Polynomial is used as basic function of the “shape function” to describe an airfoil, because it needs less parameter in describing airfoil compared with Bezier Curve method and other methods. The Bernstein polynomial of order $n$ (${BP}_n$) can be defined as:

where, the coefficients factors $K_{r,n}$ are binominal coefficients defined as:

Thus, an airfoil that uses Bernstein Polynomial of order n to describe can be stated as:

The order n in the Bernstein Polynomial determines the accuracy of the description, but it does not say the higher the better. It was discovered that a Bernstein Polynomial of order 5 is sufficient to describe the airfoil NACA2410 accurately, while Bernstein Polynomial of order larger than 12 starts to compromise accuracy of the description. To reduce computational cost and keep the basic shape of the airfoil from being greatly changed, in which case the improvement of aerodynamic performance is resulted by change of angle of attack rather than the optimized airfoil itself, middle points of the control points of both upper and lower surfaces were used as control points instead, which halve the number of control points and hence reduce much computational cost. The parametric geometry of NACA2410 with Bernstein Polynomial of order 5 and the control points are shown in Fig. 5.

Fig. 5Comparison between original geometry and parametric geometry of NACA2410

In the Fig. 5, the green dote line is the original geometry and the red dote line is the parametric geometry; the blue diamond-shaped dots in the middle are control points to be used in airfoil design. The parametric geometry shows a very good accuracy in the description of the original airfoil. To simulate the morphing process, only the 1st, 5th and 6th control points (defined as C1, C5, C6) in the Fig. 3 were used in the design process. When reviewing factors that affect the endurance of aircraft, the relative thickness, leading edge (controlled by C1), trailing edge (controlled by C5 and C6) and trailing edge thickness of airfoil should be taken into consideration, thus, relative thickness coefficient $k$ for controlling thickness at corresponding control points, trailing edge thickness $∆{th}_{TE}$ and position of trailing edge of upper surface $z_{TE}$ were also taken in to account as parameters. Therefore, in the design, the geometry of the whole optimized airfoil is a function of C1, C5, C6, $k$, $∆{th}_{TE}$ and $z_{TE}$. As the endurance parameter $E_p$ is determined by the geometry of airfoil, the function of $E_p$ can be presented as:

From Eq. (5) we can see that, taking endurance of aircraft as objective, only 6 parameters are needed for the airfoil design, which may significantly reduce computation cost. The reason why the thicknesses at corresponding control points were not manipulated individually lays to the fact that by doing so the computational cost will be dramatically increased, while the aerodynamic performance of design airfoils shows no commensurate improvement. Compared with other airfoil design or optimization methods that take hours, even days and weeks, this method only take about 60 minutes to obtain a satisfactory result.

In the demonstrative design stated later in this paper, the leading edge morphing is restrained as wind tunnel testing shows that XFOIL is unable to accurately estimate the morphing leading edge part [7]. To achieve a subtler control of both leading edge and trailing edge, Bernstein Polynomial of larger order may be applicable in the design, which will not be discussed in this paper.

2.2. Selection of solver and optimizer

As development of this method is driven by the needs of airfoils for a variable camber compliant wing for low speed high altitude long endurance application, XFOIL was chosen as the solver to evaluate the aerodynamic performance of optimized airfoils because of its fast calculation and accuracy for low speed applications [20].

Multi-Objective Particle Swarm optimization (MOPS)was chosen as optimizer in the airfoil design, it mimics the social behavior of animal groups such as flocks of birds or fish shoals. The process of finding an optimal design point is likened to the food-foraging activity of these organisms. Particle swarm optimization is a population-based search procedure where individuals (called particles) continuously change position (called state) within the search area. In other words, these particles “fly” around in the design space looking for the best position. The detailed explanation of this optimizer is shown in the reference [21].

Compared to the genetic algorithm adopted by many other researchers in airfoil design or optimization, the MOPS optimizer showed a much better performance in both searching speed and final output for optimum objectives in this method, which may provide some inspiration for other to use it in their research.

2.3. Introduction of a new selector

During the design process, as the optimizer changes variable parameters for thousands or even tens of thousand times to find an optimum result, a large number of deformed airfoils, for which XFOIL is unable to calculate or fails to converge at some of the off-design points, may be generated, resulting in a larger computation cost and poor off-design performance of the airfoil.

In the investigation, it was discovered that there were many airfoils generated with very good values at design point but poor geometry and poor off-design point performance, which means solely taking design objective as selector will compromise the robustness of the airfoil obtained. Therefore, in addition to the objective, as which maximum endurance parameter was taken in this investigation, optimizer should be “informed” what is a good airfoil, to help it identify acceptable airfoils.

To solve this problem and improve the robustness of this method, a new selector is introduced. It was found in the investigation that airfoils generated with good geometry and good off-design performance are prone to achieve convergence in XFOIL, therefore, after the new airfoil is generated, performance estimation through XFOIL will be carried out at many design points, such as at different angle-of-attacks, different Mach numbers or different Reynolds numbers. Then, MATLAB codes will automatically obtain the number of design points that has been converged and compare it with total number of design points, the ratio of number of converged design points to total number of design points (hereinafter refers to the Convergence Ratio) will be taken as a performance robustness indicator of the airfoil, making it a very good objective and selector for the optimizer. As it can be seen, the convergence ratio ranges from 0 to 1; 0 represents totally unacceptable airfoil, which will consume a very large portion of computation time, while 1 represents airfoil a very good airfoil that will be quickly evaluated by XFOIL; the larger the ratio is, the better performance robustness the airfoil will have. In this investigation, 61 design points in total were evaluated, with a convergence ratio of 1 was taken as objective and 0.90 as lower limit. This selector has shown great improvement in the selection of airfoils with robust off-design performance.

As for this method, endurance parameter, lift coefficient, drag coefficient, lift-to-drag ratio, maximum lift coefficient and momentum coefficient can be taken as both objectives and constraints at different Mach numbers, different Reynolds numbers and given lift coefficients. Control parameters of baseline airfoil can be automatically calculated and output. After input the control parameters into the optimizer and set the upper and lower limits of control parameters as well as objectives and constraints, the codes will automatically perform the searching process and output the best result at the end of search. This method can be possibly used to optimize natural laminar flow (NLF) airfoils and identify the transition location, which will not be discussed in this paper but may be studied in future research as it is not of a great concern right now.

The Fig. 6 shows the procedure of airfoil design. For each iteration after the original airfoil, the control parameters listed in Eq. (5) will be changed and correspondingly a new airfoil will be obtained. The performance of the new airfoil will be calculated at and off design point, totally 61 cases, with results output automatically. If the Convergence Ratio is larger than 0.9 in this iteration, the endurance parameter of the new airfoil will be compared with that of last optimum airfoil. If the new one is superior in the comparison, then the old one will hence be replaced by the new one. At the end of the iteration, the most optimum airfoil will be taken as final result.

Fig. 6Procedure of preliminary design of airfoils

3.1. Design objective

The original purpose of this study was to develop a wing for high altitude long endurance UAV, so the airfoil for cruise flight conditions was taken as primary target. Many efforts have been put into this work by other researchers, most of them taking lift-to-drag (L/D) ratio as objective in their airfoil design or optimization efforts. But in this study another objective was taken as follows.

When reviewing the laws of flight physics, for battery powered UAVs with fixed weight, it is clear that the power required for constant level flight depends on flight velocity $v_{\infty }$ and drag force $F_{drag}$:

As a function of flight velocity, air density $\rho$ (a function of flight altitude), area of the wing planform $S_{plan}$, and the coefficient of drag $c_D$, the drag force of aircraft can be stated as:

The minimum flight velocity is a strong requirement to ensure the minimum lift force for maintaining flight altitude. As a function of air density, flight velocity, coefficient of lift $c_L$ and area of the wing planform, the lift force of aircraft $F_{lift}$ can be stated as:

where, $m$ is the weight of aircraft and $g$ is the acceleration due to gravity. Since the lift force must be equal to the gravity force of aircraft in cruise flight, the flight velocity of aircraft in cruise flight $v_{cruise}$ can be deduced:

Inserting Eq. (8) and Eq. (6) into Eq. (5) yields the power required for cruise flight of aircraft:

As limited by payload of aircraft, the battery or fuel of propeller-driven long endurance aircraft is strictly limited in weight. Assuming that the energy output of aircraft by batteries or fuel for cruise flight is a constant value of $W$, the endurance of an aircraft $E_{cruise}$ in cruise flight can be deduced:

where, $W$, $\rho$, $S_{plan}$ and $mg$ are fixed values for a certain aircraft in cruise flight, the ${c_L}^{3/2}/c_D$, which is called endurance parameter $E_p$, offers the best aircraft endurance, and it was taken as design objective in this investigation. Increasing the endurance parameter improves endurance of the aircraft, though in most times the maximum $E_p$ occur at the same design point with the lift-to-drag ratio, sometimes it is not, and so $E_p$ is a more accurate indicator of aircrafts’ endurance.

3.2. Airfoil designed for cruise flight

In this study, the lift coefficient was taken as $c_l=\text{1.2}$ for the high altitude long endurance application, noting that the airfoils designed in this study is only for a uniform wing, and the wing twist and thickness distribution were not taken into account, which needs to be further studied in the future.

Usually, normal long endurance aircrafts are optimized for operating at one flight state due to the fact that a wing with fixed profile achieves optimum aerodynamic performance under certain conditions. However, this wing can operate at multiple flight states by changing the profile of the wing, so airfoils for three cruise flight states, assuming the in cruise flight operates at $c_l=\text{1.2}$, were designed in this study to determine the basic profile. The flight conditions, objectives and constraints taken in this study are shown in the following.

Flight conditions: $C_l=\text{1.2}$, $Ma=\text{0.15}$ and (1) $Re=\mathrm{}$1×10⁶; (2) $Re=\mathrm{}$1.5×10⁶; (3) $Re=\mathrm{}$2×10⁶.

Objectives: $E_{c_l=1.2}$: maximize; convergence ratio: 1.

Constraints: convergence ratio $>$ 0.95.

The airfoils designed and their aerodynamic performances are shown in the following figures, noting that the airfoils preliminarily designed take no consideration of structure and weight issues.

Fig. 7Airfoils designed at cl=1.2 and Ma=0.15 under different Reynolds numbers

From the Fig. 7 we can see that the trailing edges start to bend at approximately 75 % of chord, cambering the airfoil to provide better aerodynamic performances. From Fig. 3 we know that in the design process the trailing edge was freed at about 70 % of chord according to C5 stated above, so an initial bending at approximately 60 % of chord was expectable. However, the three airfoils all start to bend at 75 % of chord and tell us that this a better option for increasing endurance parameter, which is also supported by an investigation stated in Fig. 8, verifying this method’s effectiveness in designing airfoil. Through this preliminary design, the thickness distribution and internal structure of the adaptive variable camber compliant wing can be preliminary determined.

3.3. Aerodynamic performance of the airfoils and discussion

As the main objective of this preliminary airfoil design is to find an airfoil with superior performance that satisfies structural requirements for internal mechanisms, namely the thickness distribution along the chord line, the aerodynamic performance of the obtained 2-D airfoils were only numerically studied at this stage. More complex aerodynamic performance study of 3-D wings, numerically or experimentally, may be carried out in future study.

Fig. 8Aerodynamic performance of the three airfoils designed

a)

b)

c)

d)

The above figures show the basic aerodynamic performance of the three airfoils. From Fig. 8(a) we can see that, the maximum endurance parameters of the three airfoils designed all occur at approximately $c_l=\text{1.2}$, which is exactly the operating condition specified in the design process, then the endurance parameters come into a sharp decrease, indicating that the airfoils designed are the optimum when operate at this given condition. From Fig. 8(b) we can see that in addition to a maximum endurance parameter at design point, the three airfoils all have very good off-design performance in lift coefficient and post stall, with two of them have $C_{lmax}$ up to approximately 2, showing a good resistance to changing flight conditions and good post stall performance even without changing profile. The Fig. 8(c) shows that even lift coefficients of the three airfoils continue to increase after a small drop at $C_l=\text{1.3}$, the drag coefficients begin to increase significantly, and that is the reason why the endurance parameters encounter a sharp decrease. The pressure coefficient distribution in Fig. 8(d) shows that the airflow velocities at the lower surface decrease slowly and then increase at the trailing edge part. The pressure of airfoils at upper surface maintains at the leading half part, slowly drops due to acceleration of airflow velocity, and then encounters a sharp increase at approximately 85 % of chord, which is due to the transition of airflow from laminar flow to turbulent flow. It means that the airfoils designed have laminar flows attached on more than 80 % of the upper surface at operation conditions, which is a phenomenon seen in natural laminar flow (NLF) airfoils. The pressure differentiation between upper surface and lower surface also indicates that the higher lift performance of the airfoil designed is mainly due to the cambered trailing edge, which also demonstrates that the variable camber compliant wing has great potential in aviation application for long endurance, because larger lift coefficient can also mean smaller wet surface of aircraft of certain weight, thus parasite drag can be reduced.

3.4. Further application of the method

The performance of the three airfoils designed above shows not only the aerodynamic gaining of the adaptive variable camber wing, which adapts itself according to different flight conditions to achieve optimum performance, but also the effectiveness of this airfoil design method. From Fig. 6 we can see that the airfoils start to morph at about 75 % of chord, another investigation was carried out to demonstrate that this bending is in compliance with optimum performance requirement rather than constraints of control points.

Under same constraints on the control points and thickness ratio, another airfoil that also took endurance parameter as objective but without specifying lift coefficient was designed. The airfoil designed operates at $Ma=$0.15 and $Re=$ 1.5×10⁶, and its shape and aerodynamic performance (refers to as $Cl$ free in the figures), together with that of the airfoil above (refers to as $Cl=$ 1.2) and baseline airfoil NACA2410, is show in Fig. 8 and Fig. 9, all operate at the same flight conditions.

Fig. 9Airfoils design at Cl=1.2 and free of lift constraint

Fig. 10Aerodynamic performance comparison of the three airfoils

From Figs. 9-10 we can clearly see that the airfoil designed without giving specific lift coefficient starts to bend at approximately 55 % of chord, which is in compliance with the expectation; and its higher maximum endurance parameter occurs at approximately $Cl=$ 1.5. Both airfoils designed have much better aerodynamic performance compared with the baseline airfoil. This study demonstrates that this method is capable of designing airfoils with optimum performance operates at specific given flight conditions, which is very helpful in the preliminary airfoil design of an adaptive variable camber compliant wing aircraft.

In application of this method, it was found that the method is very convenient and efficient for airfoil design and optimization of not only variable camber compliant wing aircraft, but other aircrafts with fixed profiles. A subtler control of trailing edge or leading edge morphing is achievable by increasing the order of Bernstein Polynomial, but it is strongly recommended that the order should not exceed 12. Increasing order of Bernstein Polynomial to gain a more accurate control of airfoil profile should take into consideration the performance compliant skin and driving skin developed in changing the profile.