DSpace at VNU: Aerodynamic design optimization of helicopter rotor blades including airfoil shape for forward flight

DSpace at VNU: Aerodynamic design optimization of helicopter rotor blades including airfoil shape for forward flight tài...

JID:AESCTE AID:3209 /FLA [m5G; v1.145; Prn:13/01/2015; 11:17] P.1 (1-12) Aerospace Science and Technology ••• (••••) •••–••• Contents lists available at ScienceDirect 67 68 Aerospace Science and Technology 69 70 71 72 www.elsevier.com/locate/aescte 73 74 75 10 76 11 12 13 Aerodynamic design optimization of helicopter rotor blades including airfoil shape for forward flight 14 15 16 17 18 N.A Vu , J.W Lee b,2 23 24 25 26 27 28 29 30 79 81 82 a 83 Ho Chi Minh City University of Technology, Ho Chi Minh City, Viet Nam b Konkuk University, Seoul 143-701, Republic of Korea 84 85 20 22 78 80 a, 19 21 77 86 a r t i c l e i n f o Article history: Received 19 September 2013 Received in revised form 19 May 2014 Accepted 25 October 2014 Available online xxxx Keywords: Rotor blades design Airfoil Design optimization 31 32 33 34 35 a b s t r a c t 87 88 This study proposes a process to obtain an optimal helicopter rotor blade shape including both planform and airfoil shape for helicopter aerodynamic performance in forward flight An advanced geometry representation algorithm which uses the Class Function/Shape Function Transformation (CST) is employed to generate airfoil coordinates With this approach, airfoil shape was considered in terms of design variables The optimization process was constructed by integrating several programs developed by the author Airfoil characteristics are automatically generated by an analysis tool where lift, drag, and moment coefficients of airfoil are predicted for subsonic to transonic flow and a wide range of attack angles The design variables include twist, taper ratio, point of taper initiation, blade root chord, and coefficients of the airfoil distribution function Aerodynamic constraints consist of limits on power available in hover and forward flight, aerodynamic requirements (lift, drag and moment coefficients) for critical flow condition occurring on rotor blades The trim condition must be attainable in any flight condition Objective function is chosen as a combination expression of non-dimensional required power in hover and forward flight © 2015 Published by Elsevier Masson SAS 89 90 91 92 93 94 95 96 97 98 99 100 101 36 102 37 103 38 39 104 Introduction 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 In contrast to fixed wing design, most rotorcraft research focuses on the design of the rotor blade to optimize performance, vibration, noise, and so on because the rotor blade performance plays an essential role in most of the disciplines in helicopter design The aerodynamics of helicopter rotor blades is a complex discipline Diverse regimes of flow occur on blades, such as reverse flow, subsonic flow, transonic flow, and even supersonic flow In forward flight, a component of the free stream adds to the rotational velocity at the advancing side and subtracts from the rotational velocity at the retreating side The blade pitch angle and blade flapping as well as the distribution of induced inflow through the rotor will all affect the blade section angle of attack (AoA) [16] The non-uniformity of AoA over the rotor disk in conjunction with the inconstant distribution of velocity along the helicopter rotor blade makes aerodynamic analysis difficult There are two common approaches to blade aerodynamic performance design First, some researchers now focus on blade shape 59 60 61 62 63 64 65 66 E-mail addresses: vna2006@hotmail.com (N.A Vu), jwlee@konkuk.ac.kr (J.W Lee) Lecturer, Department of Aerospace Engineering Professor, Department of Aerospace Information Engineering, Member AIAA http://dx.doi.org/10.1016/j.ast.2014.10.020 1270-9638/© 2015 Published by Elsevier Masson SAS design by selecting the point of taper initiation, root chord, taper ratio, and maximum twist which minimize hover power without degrading forward flight performance [31] This approach usually deals with integration of several programs to build an optimization process Michael and Francis investigated the influence of tip shape, chord, blade number, and airfoil on rotor performance Their wind tunnel test demonstrates significant improvements that can be gained from planform tailoring and further development of airfoils, specifically for high speed rotor operation [19] Second, some works tried to solve this problem using numerical methods Joncheray used the vortex method, which schematizes the blade and rotational flow areas on the basis of a distribution of vortices, to calculate the air flow around a rotor in hover [13] Pape and Beaunier created an aerodynamic optimization for helicopter rotor blade shape in hover based on the coupling of an optimizer with a three-dimensional Navier–Stokes solver [22] Morris and Allen developed a generic computational fluid dynamics (CFD) based aerodynamic optimization tool for helicopter rotor blades in hover [21] Gunther Wilke performed a methodological setup of variable fidelity framework for the aerodynamic optimization of helicopter rotor blades and demonstrated its capabilities for a single and multi-objective test case [32] M Imiela and G Wilke investigated an optimization using a multi-fidelity approach with multiple design parameters on twist, chord, sweep, and anhedral 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 JID:AESCTE AID:3209 /FLA [m5G; v1.145; Prn:13/01/2015; 11:17] P.2 (1-12) N.A Vu, J.W Lee / Aerospace Science and Technology ••• (••••) •••–••• 2 67 Nomenclature 68 69 A , , A CST coefficients C d , C l , C m drag, lift, moment coefficient M Mach number M DD0 drag–divergence Mach number at zero lift Pf Ph P f ref P href required powers in forward flight required power in hover flight reference values in forward flight reference values in hover flight 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 70 71 72 73 74 [12] M Imiela created an optimization framework for helicopter rotors based on high-fidelity coupled CFD/CSM analysis [11] The optimization framework was first applied to various optimization problems in hover starting with the easy task of optimizing the twist rate for the 7A model rotor The last optimization in hover involved all design parameters, namely twist, chord, sweep, anhedral, transition point of two different airfoil, starting point of the blade tip showing its superiority over simpler optimization problems with respect to the achieved improvement [11] These CFD methods are reasonable for the hover case but very time consuming Moreover, application of the CFD method to the flow field passing the blade in forward flight is very complex Therefore, the CFD method is not suitable for the preliminary design phase where the need for quick estimation and considering of all factors including airfoil are required The airfoil shape which significantly affects the performance of helicopter rotor blades is usually considered as a separate problem Hassan et al developed a procedure based on the coupled threedimensional direct solutions to the full potential equation and two-dimensional inverse solution to an auxiliary equation for the design of airfoil sections for helicopter rotor blades [9] Bousman examined the relationship between global performance of a typical helicopter and the airfoil environment [4] McCroskey attempted to extract as much useful quantitative information as possible from critical examination and correlations of existing data obtained from over 40 wind tunnel tests [18] Therefore, this method is not applicable to a large number of new generations of airfoil shapes Marilyn J Smith [24] evaluated computational fluid dynamics (CFD) codes such as OVERFLOW [6], FUN2D [1], CFL3D [23], Cobalt LLC [25], and TURNS [27] to determine 2D airfoil characteristics With the advancement of computer technology, E.A Mayda and C.P van Dam developed a CFD-based methodology that automates the generation of 2D airfoil performance tables [17] The method employs ARC2D code, which controls a 2D Reynolds-Averaged Navier–Stokes (RANS) flow solver The method was shown to perform well for the largely “hands-off” generation of C81 tables, for use mainly in comprehensive rotorcraft analysis codes Nevertheless, the state of the art of rotorcraft studies is not only for analysis but also for design The method is a very expensive approach for rotorcraft analysis and design purposes where designers aim to compromise on many factors (design variables) to construct a certain objective The lack of less expensive analysis methods has been blocking multi-variable consideration of rotor blade design optimization Therefore, rotor blade airfoil shapes and planforms are usually examined in isolated design optimizations An effectively automated approach that is less expensive could contribute greatly to the rapid generation of C81 tables, to provide the ability to consider all aerodynamic aspects in rotor blade design optimization Vu et al have developed a tool that can rapidly and accurately compute airfoil data that are needed for rotorcraft design and analysis purposes [29] With the aim of allowing quick estimation in the preliminary design phase, this study proposes a process to obtain an optimal helicopter rotor blade shape including both planform and airfoil shape for helicopter aerodynamic performance In this study, a new geometry representation algorithm which uses the Class Function/Shape Function Transformation (CST) method was applied to consider airfoil shape The advantages of this CST method are high accuracy and the use of few variables in geometry representation [15] The effective tool for the automated generation of airfoil characteristics tables is employed in the design process The process associates a number of commercial software packages and in-house codes that employ diverse methodologies including the Navier– Stokes equation-solving method, the high-order panel method and Euler equations solved with the fully coupled viscous–inviscid interaction (VII) method The design process is represented in Fig This process also includes a sizing module After setting the size of the helicopter, the helicopter rotor blade shape optimization process is performed as the next step of the design process Following this process, a set of initial values for design variables is chosen from the sizing module The airfoil baseline, which is airfoil NACA0012, was chosen for the first step of the design process Then, blade shape variables such as chord distribution, twist distribution, and airfoil point coordinates are generated The required power for hover and forward flight is computed by the Konkuk Helicopter Design Program (KHDP), and the trim condition is checked Airfoil analysis is performed by the automated process program The airfoil aerodynamic characteristics are represented in C81 table format Some other additional codes to generate airfoil coordinates, chord distribution, and twist distribution are implemented in order to build a full framework for the optimization process in ModelCenter software ModelCenter is a powerful tool for automating and integrating design codes Once a model has been constructed, trade studies such as parametric studies, optimization studies, and Design of Experiment (DOE) studies may be performed [20] 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 Design process 105 2.1 Design considerations 107 106 108 The power required to drive the main rotor is formed by two components: induced power and profile power (to overcome viscous losses at the rotor) The induced power and the profile power primarily influence the blade aerodynamics performance design [16] Helicopter hover performance is expressed in terms of power loading or figure of merit (FM) A helicopter having good hover performance may have inferior performance in forward flight The compromise between hover and forward flight leads us to express the target design value in terms of the required power in hover and forward flight The conventional approach to blade aerodynamics performance design fixed the airfoil shape In general, the choice of airfoils is controlled by the need to avoid exceeding the section drag divergence Mach number on the advancing side of the rotor disk, the maximum section lift coefficients on the retreating side of the rotor disk and the zero-lift pitching moments The present work considers the effect of blade airfoil shape on required power Therefore, a baseline airfoil NACA0012 was chosen as a unique airfoil for the blade to simplify the process of optimum design The airfoil shape is represented by CST function coefficients These coefficients are also the design variables of the examined optimization problem The above discussion shows that the induced and profile power can be represented as functions of twist, taper ratio, point of taper 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 JID:AESCTE AID:3209 /FLA [m5G; v1.145; Prn:13/01/2015; 11:17] P.3 (1-12) N.A Vu, J.W Lee / Aerospace Science and Technology ••• (••••) •••–••• 67 68 69 70 71 72 73 74 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 Fig Design synthesis process 27 28 29 30 93 initiation, blade root chord, and coefficients of airfoil distribution function Aerodynamics performance is defined by the following requirements: 31 32 33 34 35 36 37 38 + The required power must be less than the power available + The helicopter must be able to trim at hover and forward flight condition + The airfoil should have the following characteristics: low zerolift pitching moment at low speed M = 0.3 approximately, high maximum lift between M = 0.3 and M = 0.5, high drag divergence Mach number at zero lift 39 40 2.2 Design synthesis process 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 The design synthesis process is shown in Fig The dashed-line rectangle represents a module which is integrated in ModelCenter software Each module is connected with the other modules by data input/output flows, which are the mutual part Four modules are implemented in this optimization framework: the chord, twist, and radius distribution generation module; the airfoil point coordinates generation module; the airfoil characteristics library with C81 format module; and the sizing, trim, and performance analysis module The chord, twist, and radius distributions are generated by a code in which the geometry representation can be changed; for example, it can be a linear or nonlinear function In this study, chord distribution is generated based on the root chord, the point of taper initiation, and the taper ratio Twist distribution is assumed to vary as a linear function along the blade Radius distribution was divided by the equal annulus area of the rotor disk These distributions are the input data for the trim code in the trimming process Ten coefficients of the airfoil distribution function were defined as the initial input data of the design process after obtaining the fitting curve of the airfoil baseline NACA0012 Then, airfoil coordinate points were generated by using the CST function The automated process generates an airfoil characteristics library with C81 format comprising the airfoil lift, drag, and moment coefficients with respect to the angle of attack for different Mach numbers (from 0.05 to 1.0) The airfoil characteristics in C81 format and rotor blades planform configuration are then used for performance and trim analysis It should be noted that the baseline rotor blades configuration can be obtained from the sizing process It is assumed that the sizing process generates rotor blades configuration similar to that of the Bo 105 helicopter This assumption is for comparison purposes of design optimization The KHDP program with the performance analysis module provides many options for the objective function The objective function of this study is chosen as a combination expression of nondimensional required power in hover and forward flight Helicopter data are analyzed by the performance code obtained from either the sizing module or user inputs After achieving the trim condition, meaning that the trim condition is attainable, the required power is evaluated in order to proceed to the next loop of the optimization process So, a new set of initial data (root chord, the point of taper initiation, taper ratio, pre-twist, and A to A coefficients of the airfoil distribution function) are generated depending on the optimization algorithm This loop continues until the convergence condition is satisfied 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 2.2.1 Geometry representation CST method [2] The CST method is based on analytical expressions to represent and modify the various shapes [15] The components of this function are “shape function” and “class function” Using the CST method, the curve coordinates are distributed by the following equation: y (x/c ) = N1 C N2 (x/c ) · S (x/c ) (1) For the formulation of the CST method, Bernstein polynomials are used as a shape function 115 116 117 118 119 120 121 122 123 124 125 126 n −i (2) 127 Fig shows the airfoil geometry represented using the CST method and non-uniform rational basis B-spline (NURBS) In this case, the control variables are the coordinates of the control points (five variables for the upper curve and five for the lower curve) 129 i S i (x) = K i x (1 − x) 128 130 131 132 JID:AESCTE AID:3209 /FLA [m5G; v1.145; Prn:13/01/2015; 11:17] P.4 (1-12) N.A Vu, J.W Lee / Aerospace Science and Technology ••• (••••) •••–••• 67 68 69 70 71 72 73 74 75 10 76 11 77 12 13 78 79 Fig RAE 2822 airfoil representation [2] 14 80 15 81 16 82 17 83 18 Fig Automated process of 2D airfoil characteristics estimation [29] 19 85 20 21 22 23 24 25 26 27 28 Fig Absolute errors in airfoil generation [2] 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 The CST method with four control variables fits the existing airfoil better than NURBS, which uses ten control variables [2] Fig shows the absolute errors of airfoil generation using CST and NURBS (five control points for each curve, fourth order blending functions) Generation by NURBS gives bigger errors at the tail part of the airfoil The advantage of the CST method in comparison with other methods such as Spline, B-Splines, or NURBS is that it can represent curves and shapes very accurately using few scalar control parameters In this study, the airfoil baseline was chosen as NACA0012 With the given data coordinate points in Cartesian coordinate space, a curve fitting was generated using fourth order Bernstein polynomials The class function for the airfoil was: C (x) = x0.5 (1 − x) (3) The airfoil distribution functions defined as upper and lower curves are presented sequentially as below 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 84 yl (x) = C (x) A l0 (1 − x)4 + A l1 4x(1 − x)3 + A l2 6x2 (1 − x)2 + Al3 4x (1 − x) + Al4 x y u (x) = C (x) A u0 (1 − x)4 + A u1 4x(1 − x)3 + A u2 6x2 (1 − x)2 + A u3 4x (1 − x) + A u4 x (4) where A u0 = 0.1718; A u1 = 0.15; A u2 = 0.1624; A u3 = 0.1211; A u4 = 0.1671; A l0 = −0.1718; A l1 = −0.15; Al2 = −0.1624; A l3 = −0.1211; Al4 = −0.1671 Changes in the coefficients A and A in the CST method are sufficient for airfoil shape modification [31] These coefficients were also the design variables of the examined optimization problem Five coefficients of the airfoil distribution function were defined as the initial input data of the design process after obtaining the fitting curve of the airfoil baseline NACA0012 Then, airfoil coordinate points were generated by using the CST function 2.2.2 An effective tool for the automated generation of airfoil characteristics tables [29] The aerodynamics of helicopter rotor blades is a complex discipline Diverse regimes of flow occur on blades, such as reverse flow, subsonic flow, transonic flow, and even supersonic flow An effectively automated approach that is less expensive could contribute greatly to the rapid generation of C81 tables, to provide the ability to consider all aerodynamic aspects in rotor blade design optimization This section describes the development of a methodology that integrates a number of commercial software components and inhouse codes that employ diverse methods including the 2D RANS equation-solving method, a high-order panel method, and Euler equations solved with the fully coupled viscous–inviscid interaction method The sequent applications of each method are as follows: 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 • A high-order panel with the fully coupled viscous–inviscid interaction method for M ∞ ≤ 0.4 • The Euler equations solved with the fully-coupled viscous– inviscid interaction method for 0.4 < M ∞ ≤ 0.7 • The 2D RANS equation-solving method for M ∞ > 0.7 103 104 105 106 107 108 The 2D RANS method is only used for M ∞ > 0.7 where the two less expensive methods (Euler equations and the high-order Panel solved with the fully coupled viscous–inviscid interaction method) are less suitable By integrating commercial software and in-house codes, a fully automated process has been developed for generating C81 tables quickly and accurately for arbitrary airfoil shapes Moreover, the commercial software including Gridgen V15 and Fluent 6.3.26, used for mesh generation and CFD modeling, are very common in the CFD research community Therefore, the proposed method could be applicable to any automation process employing Gridgen and Fluent in particular, as well as CFD tools in general The SC1095 that is used in the UH-60A main rotor was chosen for validation purposes because of the wealth of data available from the UH-60A Airloads flight test program [5], as well as the current evaluation of the UH-60A rotor loads by a number of researchers Fig shows the total automated process for airfoil characteristic estimation An airfoil analysis program, 2KFoil, was developed for subsonic isolated airfoils The code was adapted from the well known XFOIL code so as to be suitable for the present study The code employs a simplified envelope version of the en method for predicting transition locations The user-specified parameter “Ncrit” is set to 9.0 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 JID:AESCTE AID:3209 /FLA [m5G; v1.145; Prn:13/01/2015; 11:17] P.5 (1-12) N.A Vu, J.W Lee / Aerospace Science and Technology ••• (••••) •••–••• 67 68 69 70 71 72 73 74 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 34 99 Fig The automatic process of MSES execution [29] 100 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 101 (the ambient disturbance level of an average wind tunnel) for all of the predictions [8] MSES, a coupled viscous/inviscid Euler method for a single airfoil section and multiple sections design and analysis, was employed to predict airfoil characteristics from M ∞ = 0.4 to M ∞ = 0.7 The in-house code shown in Fig was developed to manage the MSES run Fluent 6.3.26, comprehensive software for CFD modeling, was employed to analyze 2D airfoil characteristics in the transonic region The software is widely utilized by CFD research and industries, thereby ensuring that the development is applicable to the community Moreover, it would be straightforward to support for other solvers An in-house code shown in Fig has been developed to manage the Fluent run A library of journal files that are utilized for the run of the case setting AoA = deg is created For instance, the journal files are created for the following M ∞ and AoA pairs: M ∞ = 0.75, AoA = deg; M ∞ = 0.80, AoA = deg; M ∞ = 0.85, AoA = deg; etc A journal file contains a sequence of Fluent commands, arranged as they would be typed interactively into the program or entered through a GUI The GUI commands are recorded as scheme code lines in journal files Figs and show the validation of the automated process for airfoil characteristics tables at M = 0.4 and M = 0.8 The lift, drag and pitching moment coefficients of the automated process calculation at M ∞ = 0.4 for AoA from −20 deg to 20 deg are shown in Fig The automated process results are very close to the ARC2D results Stall behavior still remains difficult for CFD researchers The current study and Mayda’s study have the same problem for this 102 Fig Automatic process of Fluent execution [29] 103 104 region For other regions, the automated process results and existing C81 table data are in good agreement The drag coefficient calculated by the automated process agrees very well with the C81 data as ARC2D The existing C81 data and the moment coefficient calculated by the automated process are also in good agreement The lift, drag and pitching moment coefficients of the automated process calculation at M ∞ = 0.8 for AoA from −20 deg to 20 deg are shown in Fig At this M ∞ , Fluent is employed to calculate the 2D airfoil characteristics In general, the ARC2D and automated process results have the same data trend due to using the same SA turbulence model The pitching moment varies non-linearly near AoA = deg because of the shock commencing on the airfoil The zero-lift drag coefficient data of the experiment and automated process are shown in Fig There is fairly good agreement between the experimental data and the calculated data It is seen that the calculated results represent the lower boundary of the experimental data Different Re and boundary layer transition locations cause scatter in the experimental data The automated process results show good agreement with the experiment in the drag–divergence zone where the drag coefficient sharply increases 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 2.2.3 Konkuk helicopter design program (KHDP) KHDP is a helicopter sizing, performance analysis, and trim analysis program that was developed at Konkuk University These codes were developed for use in the conceptual design phase and 129 130 131 132 JID:AESCTE AID:3209 /FLA [m5G; v1.145; Prn:13/01/2015; 11:17] P.6 (1-12) N.A Vu, J.W Lee / Aerospace Science and Technology ••• (••••) •••–••• 67 68 69 70 71 72 73 74 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 Fig Lift, drag and moment coefficients at M ∞ = 0.8 for the SC1095 airfoil [29] 45 46 Fig Lift, drag and moment coefficients at M ∞ = 0.4 for the SC1095 airfoil [29] 112 47 48 49 50 51 52 53 54 55 56 57 113 114 hence they used empirical formulas to reduce computing times [14] Blade element theory was implemented to calculate the required power in different helicopter operations, namely hover, climb, cruise, descent, and autorotation [26,10] Helicopter data are analyzed by the performance code obtained from either the sizing module or user inputs The differences between the calculated results and existing data are within 5% in general, hence acceptable for the preliminary design phase [28] 115 116 117 118 119 120 121 122 123 58 59 124 125 Optimization formulation and method 60 61 126 127 3.1 Design variables 62 63 64 65 66 111 128 The blade shape including maximum pre-twist, taper ratio, point of taper initiation, blade root chord are design variables Additionally, the A to A coefficients of the airfoil distribution function are design variables for airfoil shape The blade is assumed to 129 130 131 Fig Drag coefficients at zero lift as a function of M ∞ for the SC1095 aerofoil [29] 132 JID:AESCTE AID:3209 /FLA [m5G; v1.145; Prn:13/01/2015; 11:17] P.7 (1-12) N.A Vu, J.W Lee / Aerospace Science and Technology ••• (••••) •••–••• be rectangular until the station of the point of taper initiation and then tapered linearly to the tip The twist varies linearly from the root to the tip NACA0012 was chosen as the baseline airfoil 3.2 Constraints 10 11 12 13 14 15 16 17 18 19 20 21 The requirements are as follows: the airfoil sections should not stall in forward flight; the Mach number at the blades tip should avoid the drag divergence Mach number The drag–divergence Mach number at zero lift is a measure of the usefulness of a section near the tip of a helicopter rotor blade in forward flight It is a parameter to quantify the drag penalty associated with strong compressibility effects [7] The desirable Mach number in this case is M DD0 ≥ 0.81 However, estimation of the drag divergence Mach number ( M DD ) is not available in this process The purpose of these constraints is to avoid a very high drag at blades tip on the advancing side Therefore, the requirements are changed to constraints on the airfoil section drag coefficient The transonic data are estimated by solving the Navier–Stokes equation using Fluent software Therefore, the sectional drag coefficient constraint can be defined as below: 67 Table Constraints of optimization at 120 kts forward speed flight 68 Constraints max optimum Iteration/20 Figure of merit |C m0,M =0.3 | C lmax,M =0.4 C d0,M =M +0.02 0.0 0.7 0.0 1.4 0.0 1 0.01 2.0 0.04 0.5 0.73 0.0075 1.67 0.03 DD0 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 88 C d0 ≤ 0.01 at M = M DD0 + 0.02 (5) This constraint is constructed because a portion of the advancing blade generally operates beyond M DD Low drag rise beyond drag divergence is desirable The high M DD property requires a thin and less cambered airfoil, while the high C lmax requires a thick and more cambered airfoil These constraints are conflicting and difficult to achieve in one design Therefore, these constraints are compromised and built up in Table The maximum lift (0.3 ≤ M ≤ 0.5) is critical in delaying retreating blade stall Separation at high lift levels depends on both the free stream Mach number and airfoil shape For the typical airfoil employed on helicopter rotor blades, the maximum lift required is greater than 1.5 38 39 C lmax ≥ 1.5 at M = 0.4 (6) 40 41 42 43 44 45 46 47 48 49 50 Benson et al indicated that small nose-up pitching moment is necessary to minimize rotor loads in forward flight [3] The pitching moment at zero lift should satisfy the criteria below |C m0 | ≤ 0.01 at M = 0.3 53 54 55 56 57 58 59 (7) The trim constraint in hover and forward flight is implemented by expressing the constraint in terms of the number of trim iterations ITER, and the maximum number of trim iterations allowed ITERmax 51 52 0< ITER ITERmax ≤1 (8) Another constraint used to ensure that the blade tip chord does not become too small All constraints are normalized The normalizing factors are chosen as a possible maximum value based on the experience of the designers This study performs optimization of the blade of the BO 105 helicopter 60 61 3.3 Objective function and optimization tool 62 63 64 65 66 69 The performance module allows for the objective function of the optimization problem to be very varied In this study, a linear combination of required power in hover and forward flight was performed as the objective function 89 Fig 10 The process of using a surrogate model in the Design Explorer option of ModelCenter [20] F = 0.75 Ph P href + 0.25 Pf P f ref ˆf x n = ∗ wi x f ( xi ) 91 92 (9) 93 94 Weight factors are 0.75 and 0.25 chosen by the designer’s experience Reference values P href , P f ref are used to normalize the objective function components All modules were wrapped in the ModelCenter program, which is a powerful tool for automating and integrating design codes Genetic algorithm is widely used to perform a global optimization problem However, this method requires a large number of runs Therefore, the Design Explorer tool was used to perform the optimization search using ModelCenter Design Explorer’s key technologies are the systematic and efficient sampling of the design space using Design of Experiments (DOE) methods and the intelligent use of “surrogate” models for problem analysis and optimization The smooth surrogate models serve as substitutes for potentially expensive and “noisy” computer simulations and make global analysis and optimization of complex systems practical The surrogate models used by Design Explorer are Kriging interpolation models [23] To create a surrogate model, Design Explorer executes the analysis code (ModelCenter model) multiple times and stores the results of each run in a table The input variable values for this series of runs are chosen to efficiently canvas the design space (using an orthogonal array) Initial one hundred forty samples (ten times of the number of design variables) are used to generate surrogate model The aim of Kriging interpolation is to estimate the value of an unknown function, f , at a point x∗ using weighted linear combinations of the values of the function at some other points, x1 , x2 , , xn The predicted value ˆf (x∗ ) is expressed as: ∗ 90 (10) i =1 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 The weights w i are solutions of a system linear equation which is obtained by calculating the partial first derivatives of the error variance and setting the results to zero The error of prediction ε(x) is expressed as: w i (x) f (xi ) i =1 127 128 129 130 n ε(x) = f (x) − 126 (11) 131 132 JID:AESCTE AID:3209 /FLA [m5G; v1.145; Prn:13/01/2015; 11:17] P.8 (1-12) N.A Vu, J.W Lee / Aerospace Science and Technology ••• (••••) •••–••• 67 68 69 70 71 72 73 74 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 63 64 65 66 128 Where: TAPR: Taper ratio; POTAP: Position of taper initiation; CHOR: Chord length; POWER_HOVER: required power in hover flight; AU0, AU4, AL0, and AL4: Coefficients of airfoil shape distribution function: TWIST: Twist of the rotor blades Fig 11 Sensitivity analysis of design variables [30] 129 130 131 132 JID:AESCTE AID:3209 /FLA [m5G; v1.145; Prn:13/01/2015; 11:17] P.9 (1-12) N.A Vu, J.W Lee / Aerospace Science and Technology ••• (••••) •••–••• 10 11 12 13 14 15 Table Design variables of optimization at 120 kts forward speed flight Design variables max optimum AU AU AU AU AU A L0 A L1 A L2 A L3 A L4 Chord/0.35 Twist/16 Taper ratio Taper position 0.1 0.1 0.05 0.05 0.05 − 0.3 − 0.3 − 0.3 − 0.3 − 0.3 0.588 0.5 0.35 0.35 0.35 0.35 0.35 −0.05 −0.05 −0.05 −0.05 −0.05 1.0 1.0 1.0 1.0 0.2724 0.1005 0.055 0.201 0.1227 −0.1149 −0.051 −0.2492 −0.0603 −0.0793 0.82 0.77 0.76 16 17 18 19 The process of using a surrogate model in the Design Explorer tool is shown in Fig 10 The surrogate models are selectively up- dated and refined as the optimization process progresses Global search mechanisms are implemented to avoid local minima A final pattern search guarantees that the best design found is at least a local minimum 67 68 69 70 71 Results 72 73 In this study, the convergence history of the objective function shows that the objective function is reduced to 0.956, so it reduces by 4.4% after the optimization process The figure of merit increases by 4.3% (from 0.7 to 0.73) From Eqs (9), we can easily obtain 5.3% reduction on the required power in 120 kts forward flight The study assumed that the drag divergence Mach number is 0.83 A portion of the advancing blade may operate beyond M DD in higher forward speed or maneuver flight In these flight conditions, the zero-lift drag could rise to 0.03 However, the objective function was considered for 120 kts forward speed where the Mach number at the tip of rotor blades could approach 0.81 74 75 76 77 78 79 80 81 82 83 84 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 44 109 Fig 12 Optimal rotor blade shape and airfoil for 120 kts forward speed 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 130 65 66 131 Fig 13 Convergence history of objective function 132 JID:AESCTE 10 AID:3209 /FLA [m5G; v1.145; Prn:13/01/2015; 11:17] P.10 (1-12) N.A Vu, J.W Lee / Aerospace Science and Technology ••• (••••) •••–••• 67 68 69 70 71 72 73 74 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 65 66 130 Fig 14 Convergence history of planform design variables 131 132 JID:AESCTE AID:3209 /FLA [m5G; v1.145; Prn:13/01/2015; 11:17] P.11 (1-12) N.A Vu, J.W Lee / Aerospace Science and Technology ••• (••••) •••–••• 11 67 68 69 70 71 72 73 74 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 65 66 130 Fig 15 Convergence history of constraints 131 132 JID:AESCTE AID:3209 /FLA Table The number of analysis of each module [m5G; v1.145; Prn:13/01/2015; 11:17] P.12 (1-12) N.A Vu, J.W Lee / Aerospace Science and Technology ••• (••••) •••–••• 12 Number of analysis Whole design process Airfoil characteristics module Performance analysis Trim analysis Blades planform configuration module 220 259 160 220 440 220 10 11 12 13 14 15 The constraints were built up not only for a 120 kts forward speed flight condition, but also for other critical flight conditions such as maneuver (see Fig 11, Table 2, Figs 12–15, Table 3) The sensitivity analysis of design variables to required hover power shows that all the design variables significantly affect the objective function [30] 16 17 Conclusion 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 This study developed an automated process for rotor blades design optimization including both blade shape and airfoil shape The airfoil analysis tool effectively automates the generation of airfoil characteristics tables where lift, drag, and moment coefficients of airfoil are predicted for subsonic to transonic flow and a wide range of attack angles Diverse methodologies (Navier–Stokes equation-solving method, the high-order panel method and Euler equations solved with the fully coupled viscous–inviscid interaction method) are employed This tool made it possible to consider both rotor blades planform and airfoil shape in one design optimization The maximization of the drag divergence Mach number leads to a reduction of the thickness and camber of the airfoil In contrast, the reduction of the thickness and camber of the airfoil can reduce the maximum lift characteristics and cannot avoid premature trailing edge separation The optimal airfoil in 120 kts forward flight design optimization has smaller thickness and camber This could be because the drag could increase significantly at the advancing side in 120 kts flight, so the camber and thickness should be reduced The compromise between drag and lift coefficients leads to a reduction in camber and thickness as well The rotor blade solidity in 120 kts flight does not change much in order to provide enough thrust The small taper at the tip could reduce the drag on the advancing side The objective function, reduced by 4.4% after the design optimization process However, the airfoil characteristics improved to the desired range of lift, drag, and moment coefficients These coefficients have a very important function for helicopter rotor blade performance Using this process, an integrated configuration of blade shape and airfoil shape can be quickly sized according to the requirements of helicopter rotor blades Conflict of interest statement None declared Acknowledgements 56 57 58 59 This work was supported by National Foundation for Science and Technology Development (NAFOSTED) of Vietnam (Project No 107.04-2012.25) 60 61 References 62 63 64 [1] W.K Anderson, D.L Bonhaus, An implicit upwind algorithm for computing turbulent flows on 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121 122 123 124 125 126 127 128 129 130 65 131 66 132 ... developed an automated process for rotor blades design optimization including both blade shape and airfoil shape The airfoil analysis tool effectively automates the generation of airfoil characteristics... Automated generation of aerofoil characteristics for rotorcraft application, Aircr Eng 84 (4) (2012) 221–230 [30] N.A Vu, J.W Lee, J.I Shu, Aerodynamic design optimization of helicopter rotor blades. .. rotor blade performance Using this process, an integrated configuration of blade shape and airfoil shape can be quickly sized according to the requirements of helicopter rotor blades Conflict of