energies

Article Automated Design Optimization of a Mono Tiltrotor in Hovering and Cruising States

Lifang Zeng 1, Jianxin Hu 2,*, Dingyi Pan 1,* and Xueming Shao 1 1 School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China; [email protected] (L.Z.); [email protected] (X.S.) 2 Faculty of Mechanical Engineering & Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China * Correspondence: [email protected] (J.H.); [email protected] (D.P.)



Received: 2 February 2020; Accepted: 2 March 2020; Published: 4 March 2020


Abstract: A mono tiltrotor (MTR) design which combines concepts of a tiltrotor and coaxial rotor is presented. The aerodynamic modeling of the MTR based on blade element momentum theory (BEMT) is conducted, and the method is fully validated with previous experimental data. An automated optimization approach integrating BEMT modeling and optimization algorithms is developed. Parameters such as inter-rotor spacing, blade twist, taper ratio and aspect ratio are chosen as design variables. Single-objective (in hovering or in cruising state) optimizations and multi-objective (both in hovering and cruising states) optimizations are studied at preset design points; i.e., hovering trim and cruising trim. Two single-objective optimizations result in different sets of parameter selections according to the different design objectives. The multi-objective optimization is applied to obtain an identical and compromised selection of design parameters. An optimal point is chosen from the Pareto front of the multi-objective optimization. The optimized design has a better performance in terms of the figure of merit (FM) and propulsive efficiency, which are improved by 7.3% for FM and 13.4% for propulsive efficiency from the prototype, respectively. Further aerodynamic analysis confirmed that the optimized rotor has a much more uniform load distribution along the blade span, and therefore a better aerodynamic performance in both hovering and cruising states is achieved.

Keywords: mono tiltrotor (MTR); multi-objective optimization; BEMT; figure of merit; propulsive efficiency

1. Introduction The speed limit of a conventional helicopter is normally below 350 km/h [1] due to its inherent aerodynamic characteristics; e.g., the tip stall of the retreating blade and compressibility effects of the advancing blade. New concepts of rotorcraft design compounded with fixed wings have been put forward to break this speed limit. These conceptual rotorcrafts combine both the capabilities of conventional helicopters, such as vertical takeoff and hovering, and fixed-wing airplanes, such as the long range of cruising and large payload performance. An example of these conceptual designs is the Bell/Boeing V-22 Osprey [2], the first production tiltrotor in the world. The V-22 Osprey is capable of vertical take-off and landing, and its maximum flight speed is up to 509 km/h. Later, the innovative design of a mono tiltrotor (MTR) was proposed [3], in which only one rotor is used, combining the concepts of a tiltrotor and coaxial rotor. The original design of an MTR, which innovatively combines the concepts of a tiltrotor and coaxial rotor, is presented in Figure1. In particular, a contra-rotating coaxial rotor system is employed in this aircraft to achieve high aerodynamic performance with a compact structure. Compared with traditional helicopters, a coaxial rotor is able to fulfill torque trim without a separate anti-torque tail system, and it has a relatively high aerodynamic efficiency [4]. Benefiting from these advantages, the coaxial rotor design has been widely applied in shipboard

Energies 2020, 13, 1155; doi:10.3390/en13051155 www.mdpi.com/journal/energies EnergiesEnergies 20202020, 13, 13, x, 1155FOR PEER REVIEW 2 2of of 20 20 has been widely applied in shipboard helicopters, micro unmanned helicopters [5], advanced high-speedhelicopters, compounded micro unmanned helicopters helicopters [6], etc. [5], The advanced present high-speed MTR integrates compounded a coaxial helicopters tiltrotor system, [6], etc. twoThe fixed present wings, MTR a integrates fuselage aand coaxial a twin-boom tiltrotor system, tail system. two fixed It distinguishably wings, a fuselage uses and a coaxial a twin-boom rotor tailto changesystem. from It distinguishably a lifter to a uses propeller a coaxial by rotor converti to changeng from from the a lifter vertical/hovering to a propeller by flight converting mode from to forward/cruisingthe vertical/hovering flight flight mode. mode When to forward converting/cruising between flight different mode. When modes, converting the coaxial between rotor di shaft,fferent wingmodes, and the fuselage coaxial rotorare fixed shaft, to wing each and other, fuselage tilting are through fixed to eachseveral other, hinges tilting with through the support several hinges of a twin-boomwith the support tail system. of a twin-boom This convertible tail system. wing This supplies convertible enough wing lift and supplies results enough in a high lift and lift-to-drag results in ratioa high in lift-to-dragcruising state. ratio The in cruising lift-to-drag state. ratio The lift-to-dragis expected ratio to be is expectedmore than to betwice more the than ratio twice of thea conventionalratio of a conventional helicopter helicopter[7]. [7].

Figure 1. The mono tiltrotor (MTR) and its three flight modes. Figure 1. The mono tiltrotor (MTR) and its three flight modes. The aerodynamic shape designs of a traditional rotor craft are usually based on parametric studies or physicalThe aerodynamic analysis [8 ],shape which designs rely to aof large a traditional extent on rotor experience. craft are However, usually forbased a new on concept parametric rotor studiescraft such or asphysical MTR, thereanalysis is little [8], relevant which experience.rely to a large The extent physical on analysis experience. method However, will be complex for a new and concepttime-consuming rotor craft with such such as MTR, little there experience. is little Inrele thevant preliminary experience. design The physical stage of aanalysis new layout method craft, will it is bedi complexfficult for and designers time-consuming to decide onwith the such appropriate little expe initialrience. values In the of preliminary parameters. design This paperstage of proposes a new layouta high-e craft,fficiency it is difficult automated for designers optimization to decide framework on the appropriate for MTR preliminary initial values design of parameters. by integrating This paperaerodynamic proposes modeling a high-efficiency and advanced automated optimization optimization algorithms. framework It will for explore MTR the preliminary relationship design of the bydesign integrating parameters aerodynamic and help modeling designers toand determine advanced trade-o optimizationffs in the algorithms. preliminary It design will explore stage. the relationshipIn the designof the design of MTR, parameters the optimization and help of ade coaxialsigners rotor to determine system plays trade-offs a crucial in the role preliminary in achieving designhigh e stage.fficiency in both cruising and hovering states. Single-objective optimization was applied in theIn design the design of the of traditional MTR, the coaxial-rotoroptimization systemof a coax [9ial]; however,rotor system it is plays not able a cr toucial cover role most in achieving working highconditions efficiency of MTR.in both In cruising other words, and hovering incompatible states. Single-objective sets of design parameters optimization are was possibly applied rendered in the designfor MTR of the in cruisingtraditional and coaxial-rotor hovering, respectively system [9]; [however,10]. A highly it is not effi cientable hoveringto cover most state working normally conditionsrequires low of MTR. disk loading,In other words, while aincompatible highly efficient sets cruising of design state parameters corresponds are possibly to a minimized rendered drag for MTRprofile in cruising of the blades and hovering, and compressibility respectively losses. [10]. A Therefore, highly efficient the design hovering targets state for no thermally two statesrequires are lowinconsistent. disk loading, Recently, while fewa highly works efficient in coaxial cruising rotor designstate corresponds have focused to ona minimized multi-objective drag designprofile (forof theboth blades hovering and and compressibility cruising states), losses. and mostTherefore, of them the are design based ontargets parametric for the studies two withoutstates are any inconsistent.integrated optimization Recently, few strategies works in [coaxial11,12]. rotor In view design of this,have it focused is necessary on multi-objective to develop and design validate (for bothcomprehensive hovering and multi-objective cruising states), optimization and most strategies. of them are The based goal ofon the parametric present work studies is to without construct any an integratedautomated optimization and efficient strategies design approach [11,12]. for In theview coaxial of this, rotor it designis necessary of MTR to coveringdevelop multipleand validate states comprehensivein its flight envelope. multi-objective optimization strategies. The goal of the present work is to construct an automatedTo ensure and the efficient efficiency design and approach accuracy for ofthe thecoaxial MTR’s rotor optimization design of MTR framework, covering multiple two key statesfactors—aerodynamic in its flight envelope. modeling and optimization algorithms—should be thoroughly explored. First,

Energies 2020, 13, 1155 3 of 20 the blade element momentum theory (BEMT) method for MTR has been applied in the current aerodynamic modeling, owing to its high efficiency and reasonable accuracy. In detail, several aerodynamic methods—i.e., momentum theory (MT), blade element momentum theory (BEMT), free vortex method (FVM), and computational fluid dynamics (CFD)—are compared and discussed. The BEMT method is much faster than FVM and CFD, and its accuracy has been fully validated through previous experiments. Second, two optimization algorithms—i.e., adaptive simulated annealing (ASA) and non-dominated sorting genetic algorithm II (NSGA-II)—are employed, and their feasibilities for the highly non-linear and discontinuous design spaces of current MTR design have been explored. Within the selected algorithms, both single-objective and multi-objective optimizations are put forward in the current design. A corresponding aerodynamic analysis of the optimized design is also presented.

2. Aerodynamic Modeling of MTR

2.1. Numerical Methods The aerodynamic characteristics of a coaxial rotor are complex due to the interference between the upper and lower rotors. The lower rotor operates in the slipstream of the upper rotor, while the upper rotor is affected by the suction effect from the lower rotor. Approaches to investigate the coaxial rotor aerodynamics include experimental research [13], momentum theory (MT) [14], blade element momentum theory (BEMT) [7,15], free vortex method (FVM) [9,16] and also unsteady computational fluid dynamics (CFD) based on the turbulence model [17]. In particular, MT is based on the classic momentum theory of Glauert [18], in which viscous and compressibility effects are neglected. Leishman and Syal [14] used MT to analyze a hovering contra-rotating coaxial rotor system. In FVM, a vorticity transport equation is formed under the assumption that the flow is irrotational and incompressible [9]. By directly solving unsteady Reynolds-averaged Navier–Stokes (RANS) equations, CFD is most accurate among these numerical methods. It possesses the ability to provide insight into the unsteady aerodynamic characteristics of a coaxial rotor. From the viewpoint of computational cost, CFD is most expensive, which is unrealistic for optimization design [19]. Although the computational cost of FVM is less than that of CFD, an optimization process based on FVM with a large number of cases is still intensive. BEMT combines the blade element method with momentum theory to solve the induced velocity. It achieves a better modeling accuracy than MT by considering the spanwise distribution of the induced velocity. BEMT is applied to the investigation of certain parameters, such as the inter-rotor spacing, blades platform shapes, and blade twist. It can expeditiously and accurately predict different aerodynamic results of MTR by changing the rotor parameters. Some work was reported which has used BEMT for the aerodynamic optimization of a coaxial rotor in hovering or axial flight [9–12]. When carrying out an optimization design, the computational cost does not only come from the aerodynamic method, but also from other factors. For instance, the thrust trim and torque trim between upper and lower rotors will also require large computational resources, especially in FVM and CFD modeling. Regardless of the trim, on a desktop computer (a four-core Intel processor with CPU speeds of 3.2 GHz), a single case requires almost one day for unsteady CFD simulation; the BEMT model is more efficient, in that one individual case can be solved within several minutes. Multi-fidelity optimization techniques have always been applied to incorporate high-fidelity models into the rotor optimization loop while maintaining a moderate computational cost [20,21]. However, when considering trim, the computational cost will be increased dramatically. In view of these factors, among all the modeling approaches mentioned above, BEMT might be the most suitable and practical method for the optimization design of a coaxial rotor with a large number of samples, and it was chosen as the aerodynamic modeling tool in the current study. The BEMT method for coaxial rotor modeling, as reported in [15], can be applied in the modeling of the current MTR. Lock [22] assumed that the wake flow of the upper rotor is affected the flow field of the lower rotor, while neglecting the lower rotor effect on the upper one. The aerodynamic Energies 2020, 13, x FOR PEER REVIEW 4 of 20

the flow field of the lower rotor, while neglecting the lower rotor effect on the upper one. The aerodynamic interference between the upper and lower rotors, and both the upper-to-lower and Energieslower-to-upper2020, 13, 1155 effects, is considered in the current work. 4 of 20 The BEMT modeling approach is introduced in the following section; readers can refer to [15] for more detail. As mentioned above, the BEMT modeling can be divided into two separate steps: interferencei.e., the momentum between theory the upper step and and lower the boundary rotors, and element both thestep. upper-to-lower and lower-to-upper effects,In is the considered momentum in the theory current step, work. the flow around the rotor is assumed to be axisymmetric when the Therotor BEMT is hovering, modeling climbing approach vertically is introduced and crui insing the forward. following A section; simplified readers two-dimensional can refer to [15 theory] for morecan be detail. used As to mentionedpredict the above,nonuniform the BEMT inflow modeling distributions. can be Each divided rotor into disk two is separatedivided into steps: several i.e., the momentum theory step and the boundary element step. concentric rings, and the segment thrust dCT can be obtained by the non-dimensioned advanced In the momentum theory step, the flow+ around the rotor is assumed to be axisymmetric when V0 Vv0 i Ω ratio, λ∞ = , and the inflow ratio, λ= , where V0 is the inflow velocity, is the rotating the rotor is hovering,ΩR climbing vertically andΩR cruising forward. A simplified two-dimensional theory can be used to predict the nonuniform inflow distributions. Each rotor disk is divided into several speed, R is the radius of the rotor and vi is the induced velocity. concentric rings, and the segment thrust dCT can be obtained by the non-dimensioned advanced ratio, V0 V0+vi =−λλ() λ λ = , and the inflow ratio, λ = ,dC whereT 4V0 is the∞ inflow xdx velocity, Ω is the rotating speed, R(1) is ∞ ΩR ΩR the radius of the rotor and vi is the induced velocity. where x is the non-dimensional radial location. Then, the non-dimensional power coefficient dCP of each ring is dCT = 4λ(λ λ )xdx (1) − ∞ dC=−4λλλ2 () xdx where x is the non-dimensional radial location.P Then, the∞ non-dimensional power coefficient dCP (2)of each ring is In the boundary element step, for a single 2blade element, as shown in Figure 2, the incremental dCP = 4λ (λ λ )xdx (2) thrust of the rotor disk ring, dT , can be derived −by ∞the given lift Cl and drag coefficients Cd of the Inairfoil. the boundary element step, for a single blade element, as shown in Figure2, the incremental thrust of the rotor disk ring, dT, can be derived by the given lift Cl and drag coefficients Cd of the airfoil. 1 cos()φγ+ =+=Ω()φγ ρ 2 dT bcos dR b ( r ) Cl c2 dr (3) 12 2 coscos(φφγ cos+ γ) dT = b cos(φ + γ)dR = bρ(Ωr) Clc dr (3) 2 cos2 φ cos γ where b is the blade number, r is the radial location, ρ is the air density, c is the chord length,

+ Cd whereC bd is the blade number, r is the radial−1 Vv location,0 i ρ is the air density, c is the chord length, γ = γ = and the inflow angle φ = tan . The non-dimensional blade element thrust canC lbe  +  Ω Cl 1 V0 vi r and the inflow angle φ = tan− Ωr . The non-dimensional blade element thrust can be written as written as ( + ) bc cos φ()φγ+γ 2 dC = bcC cos x dx2 (4) T = l 2 dCTl2π Ccos φ2cos γ x dx (4) 2coscosπφγ wherewherec =ccRc=/R/. Then,. Then, the the non-dimensional non-dimensional power power coe coefficientfficient is is sinsin(φ()+φγ+γ) =bcbc 33 dCPdC=PlCl Cx xdx dx (5)(5) 2π2coscosπφγcos2 φ2cos γ

Figure 2. Force analysis of a blade element. Figure 2. Force analysis of a blade element. BEMT constructs thrust equivalence between blade element theory and MT to solve the induced velocity. For a single rotor disk, from Equations (1) and (4), it can be obtained that

cos(φ + γ) bc 4λ(λ λ ) = Cl x (6) − ∞ 2π cos2 φ cos γ Energies 2020, 13, 1155 5 of 20

By solving Equation (6), the radial velocity distribution can be obtained. For the MTR, the induced velocity cannot be directly solved by the separate BEMT equivalent equation of the upper and lower rotors because of the interference. An interaction model between the upper and lower rotors can be constructed based on BEMT, and the mode schematic is shown in Figure3. The Biot–Savart law shows that the induced velocity vi evolves with the axial spacing d as follows [7]: ! d vi0(d) = 1 + vi ε(d)vi (7) √R2 + d2 · ≡ where ε is the interference factor. The subscripts “u” and “l” are used to identify the relative parameters of the upper rotors and lower rotors, respectively. From the mass conservation law, the streamlines passing at radius ru on the upper rotors to the lower rotors at radius rsu are given by s V0 + vu(ru) + ε( d)vl(rsu) rsu = − ru a(ru) ru (8) V0 + ε(d)vu(ru) + vl(rsu) · ≡ · where a is the interference function. In the same way, the streamlines passing at radius rl on the lower rotors to the upper rotors at radius rsl are given by s V0 + vl(rl) + ε(d)vu(rsl) rsl = rl a(rl) rl (9) V + ε( d)v (r ) + vu(r ) · ≡ · 0 − l l sl Then, the interference-induced velocity for the upper and lower rotors is expressed as  v0(ru) = ε( d) vl(a(ru)ru) f or upper rotor  l − · (10)  v (r ) = ε(d) vu(a(r )r ) f or lower rotor u0 l · l l

The induced velocity at each rotor disk compromises two parts: i.e., its own induced velocity vi and the interaction induced velocity v . For the MTR, the BEMT thrust equivalence of Equation (6) 0i can be written as  ( + )  bcu cos φu γu  4λu(λu λ ε( d) λl(xsu)) = 2π Cl 2 xu,  − ∞ − − · cos φu cos γu (11)  bcl cos(φl+γl)  4λl(λl λ ε(d) λu(xsl)) = Cl 2 xl. − ∞ − · 2π cos φl cos γl From Equations (4) and (5) of blade element theory, we obtain the radial distribution of the thrust and power coefficients. The generated total thrust and consumed power by the MTR in axial flight are obtained by the summation of segment loads on upper and lower rotors, respectively:

Z 1 Z 1 Z 1 Z 1 = + = + CT dCTu dCTl , CP dCpu dCPl . (12) e e e e where e is the non-dimensional root cutout. Energies 2020, 13, 1155 6 of 20 Energies 2020, 13, x FOR PEER REVIEW 6 of 20

Figure 3. Interaction model in axial flight. Figure 3. Interaction model in axial flight. In addition, blade tip losses should be also considered properly, which will deteriorate the In addition, blade tip losses should be also considered properly, which will deteriorate the aerodynamic performance, since trailing vortices appear near the blade tip, and the strengths greatly aerodynamic performance, since trailing vortices appear near the blade tip, and the strengths greatly increase, causing a sudden drop of lift. The Prandtl correction factor will be used [1], and it can be increase, causing a sudden drop of lift. The Prandtl correction factor will be used [1], and it can be included in the BEMT modeling of the MTR. included in the BEMT modeling of the MTR. The goal function in the hovering state, the figure of merit (FM), is defined as [7] The goal function in the hovering state, the figure of merit (FM), is defined as [7] 3/2 CCT 3/2 FMFM== T .. (13) √2C (13) 2CP

The goal function in axial flight,flight, thethe propulsivepropulsive eefficiencyfficiency ((ηη),), is is defined defined as as [7 [7]]

CTλ η η== T ∞∞. . (14) CP (14) CP 2.2. Validation of the BEMT Modeling of MTR in Hovering State 2.2. Validation of the BEMT Modeling of MTR in Hovering State A coaxial rotor in hovering state is simulated based on the BEMT modeling in Section 2.1. The resultsA coaxial are rotor validated in hovering by comparing state is themsimulated the experiments based on the of BEMT Harrington modeling [4], inin whichSection full-scale 2.1. The windresults tunnel are validated investigations by comparing of both single them and the coaxial experiments rotors were of Harrington conducted. [4], The inmajor which rotor full-scale parameters wind aretunnel listed investigations in Table1. of both single and coaxial rotors were conducted. The major rotor parameters are listed in Table 1. Table 1. Major information of rotors in the experiments by Harrington [4].

Table 1. Major information of rotors in the experimentsHarrington Rotorby Harrington 2 [4]. Diameter (m), D Harrington 7.62 Rotor 2 DiameterBlade number, (m), D b 2 for single / 7.624 for coaxial Inter rotor spacing (m), d 0.6096 Blade number, b 2 for single /4 for coaxial Blade tip velocity (m/s), ΩR 120 InterSingle rotor/coaxial spacing rotor solidity, (m), dσ 0.076 0.6096/0.152 Blade tipRoot velocity chord (m), (m/s), cr ΩR 0.4572120 Single/coaxialTaper rotor ratio, TRsolidity, σ 0.076/0.152 1 Root cut ratio, e 0.2 Root chord (m), cr 0.4572 Taper ratio, TR 1 The power polars are presentedRoot cut ratio, in Figure e 4a, which gives the0.2 variation of the thrust coe fficient versus the power coefficient. The figure of merit (FM) is demonstrated in Figure4b, which is the ratio of theThe eff ectivepower thrust polars power are presented to the total in consumedFigure 4a, whic power.h gives the variation of the thrust coefficient versus the power coefficient. The figure of merit (FM) is demonstrated in Figure 4b, which is the ratio of the effective thrust power to the total consumed power.

Energies 2020, 13, 1155 7 of 20 Energies 2020, 13, x FOR PEER REVIEW 7 of 20

FigureFigure 4. 4.Performance Performance of rotor rotor 2: 2: a acomparison comparison of ofresults results by bythe thecurrent current blade blade element element momentum momentum theorytheory (BEMT), (BEMT), computational computational fluid fluid dynamics dynamics (CFD) (CFD)[19], [19], and experiments[4] experiments [4 (marked] (marked as asEXP). EXP).

InIn FigureFigure 4,4, thethe currentcurrent BEMTBEMT results are are al alsoso compared compared with with the the CFD CFD simulation simulation of of LakshminarayanLakshminarayan [19], [19], which which was was based based on on compressible compressible unsteady unsteady RANS RANS equations equations with with the the Spalart–AllmarasSpalart–Allmaras turbulence turbulence model. model. The The BEMT BEMT result resultss agrees agrees well with well both with the both experimental the experimental data dataand andCFD CFDsimulation. simulation. This Thisconfirms confirms that thatBEMT BEMT is a isreliable a reliable aerodynamic aerodynamic modeling modeling tool toolfor the for the investigationinvestigation of the the rotor rotor in in hovering hovering performance. performance. Although Although the the FMs FMs predicted predicted by BEMT by BEMT are a are little a little optimisticoptimistic comparedcompared toto thethe measurement, measurement, they they are are still still within within the the tolerance tolerance of theof the preliminarily preliminarily design. design. 2.3. Validation of the BEMT Modeling of MTR in Axial Flight 2.3. Validation of the BEMT Modeling of MTR in Axial Flight A coaxial rotor under different axial inflow is simulated based on the BEMT modeling in Section 2.1. A coaxial rotor under different axial inflow is simulated based on the BEMT modeling in The results are validated with previous experiments in [23]. Detailed model parameters of this coaxial Section 2.1. The results are validated with previous experiments in [23]. Detailed model parameters rotor are presented in Table2. Figure5 presents the results of the thrust coe fficient and propulsive of this coaxial rotor are presented in Table 2. Figure 5 presents the results of the thrust coefficient and efficiency. The solid lines represent the predictions of BEMT, and the circular symbols are the propulsive efficiency. The solid lines represent the predictions of BEMT, and the circular symbols experimental data. With the blade pitch angle increasing from 20 to 50 with an increment of 5 , are the experimental data. With the blade pitch angle increasing from 20°◦ to 50°◦ with an increment of ◦ the thrust coefficients (Figure5a) increase accordingly. With a fixed pitch angle, CT reduces rapidly 5°, the thrust coefficients (Figure 5a) increase accordingly. With a fixed pitch angle, CT reduces along with the inflow ratios. The BEMT results simulated in this paper agree very well with the rapidly along with the inflow ratios. The BEMT results simulated in this paper agree very well with measurements at small pitch angles (20 ~50 ). As pitch angle increases (40 ~50 ), the predicted C the measurements at small pitch angles (20°~50°).◦ ◦ As pitch angle increases (40°~50°),◦ ◦ the predicted T does not match the experimental trend perfectly at a low inflow ratio. This is because the angles of CT does not match the experimental trend perfectly at a low inflow ratio. This is because the angles attack for some blade sections exceed the linear aerodynamic range, where non-linear effects become of attack for some blade sections exceed the linear aerodynamic range, where non-linear effects more dominant. The nonlinear measurement results caused by deep blade stall at very low inflow become more dominant. The nonlinear measurement results caused by deep blade stall at very low ratios are not presented in Figure5. inflow ratios are not presented in Figure 5.

Table 2. Major parameters of coaxial-rotor 3155-6-1.5. Table 2. Major parameters of coaxial-rotor 3155-6-1.5.

RotorRotor 3155-6-1.5 3155-6-1.5 RadiusRadius (m), (m), R R 1.524 1.524 BladeBlade number, number, b b 6 6 InterInter rotor rotor spacing(m), spacing(m), d d 0.381 0.381 Blade tip velocity (m/s), ΩR 87.5 Blade tip velocity (m/s) , ΩR 87.5 Rotor solidity, σ 0.156 RotorRoot solidity, cut ratio, σe 0.156 0.15 Root cutAirfoil ratio, e Clark-Y0.15 Airfoil Clark-Y

Energies 2020, 13, 1155 8 of 20 Energies 2020, 13, x FOR PEER REVIEW 8 of 20

FigureFigure 5. Comparison of of the the BEMT-predicted BEMT-predicted result result and and experimental experimental data data for axial for axial flight flight at various at various inflowinflow ratios.ratios.

TheThe propulsive efficiencies efficiencies are are demonstrated demonstrated in inFigure Figure 5b,5b, with with blade blade pitches pitches from from 20° 20to ◦50°to in 50 ◦ in η λ aa stepstep of 10°. 10◦. A A linear linear dependence dependence of of ηandand λ∞ isis predicted predicted by by BEMT BEMT at atlow low inflow inflow ratios, ratios, which which is is ∞ consistentconsistent withwith thetheexperiments. experiments. Overall,Overall, thethe BEMT BEMT is is also also a reliablea reliable tool tool for for the the modeling modeling of aof coaxial a rotorcoaxial in rotor axial in flight. axial flight.

3.3. OptimizationOptimization Design Design of of Current Current MT MTRR in in Hovering Hovering and and Cruising Cruising Flights Flights

3.1.3.1. OptimizationOptimization Procedures Procedures AA designdesign process process coupling coupling BEMT BEMT modeling modeling and and optimization optimization algorithms algorithms was was constructed, constructed, as shownas inshown Figure in6 .Figure The goal 6. The was goal to findwas theto find optimal the optima bladel parameters blade parameters with the with highest the highest e fficiency efficiency for the for MTR system.the MTR In system. this process, In this a design process, point—e.g., a design thepoint—e.g., thrust coe thefficient, thrustCT ,coefficient, with a certain CT magnitude—should, with a certain bemagnitude—should decided first. The be trim decided of both first. thrust The and trim inter-rotor of both torque thrust (i.e., and the inter-rotor non-dimensional torque (i.e., total the power coefficient of upper rotor and lower rotors, Cp and Cp ) should be fulfilled.C In particular,C thrust trim non-dimensional total power coefficient of upperu rotor andl lower rotors, pu and pl ) should be was achieved by adjusting the collective pitch of the upper rotor, and torque trim was satisfied by fulfilled. In particular, thrust trim was achieved by adjusting the collective pitch of the upper rotor, adjustingand torque the trim corresponding was satisfied lower by adjusting rotor’s collective the corresponding pitch. The lower classic rotor’s dichotomy collective method pitch. in [The24] was utilizedclassic dichotomy to solve the method trim procedure, in [24] was andutilized the convergenceto solve the trim was procedure, determined and by the the convergence tolerances E wasand E1, as shown in Figure7. determined by the tolerances E and E1 , as shown in Figure 7. TwoTwo single-objectivesingle-objective optimization optimization cases cases (for (for MT MTRR in inhovering hovering or orin cruising in cruising states) states) and anda a multi-objectivemulti-objective optimization optimization case case (for (for MTR MTR in inboth both hovering hovering and and cruising cruising states) states) are areconducted conducted at at certaincertain designdesign points. The The inter inter rotor rotor spacing, spacing, blade blade twist, twist, taper taper ratio ratio (TR ()TR and) and aspect aspect ratio ratio (AR) ( ARare) are chosenchosen asas designdesign variables,variables, whilewhile the blade soliditysolidity and blade number number are are fixed. fixed. The The definitions definitions of of TR andTR andAR areAR are = = TR==ct/cr, AR R/cm, (15) TR ctr/, c AR R / c m, (15) where ct, cr and cm are the lengths of the tip chord, root chord and mean aerodynamic chord, respectively c c c (aswhere shown t , inr Figure and 8m). are the lengths of the tip chord, root chord and mean aerodynamic chord, respectivelyA practical (as shown design in of Figure the MTR 8). has to allow certain constraints, such as the overall rotor weight, limitationA practical of the design blade of tip the speed, MTR has and to maximumallow certain allowable constraints, blade such twist. as the overall According rotor toweight, previous experiencelimitation of [8 ,the9], constraintsblade tip speed, of certain and parametersmaximum allowable are listed blade in Table twist.3, where Accordingdθ is to the previous blade twist experience [8,9], constraints of certain parameters are listed in Table 3, where dθ is the blade twist from root to tip. Coaxial rotor 2 in [4] is chosen as the baseline design of MTR in this optimization. from root to tip. Coaxial rotor 2 in [4] is chosen as the baseline design of MTR in this optimization. The best design of a coaxial rotor may require different blade twists and platforms for each rotor [9]. The best design of a coaxial rotor may require different blade twists and platforms for each rotor [9]. However, considering the structural symmetry and design consistency of the MTR system, the blade However, considering the structural symmetry and design consistency of the MTR system, the blade twist and platform of each rotor are kept the same in the current design. In addition, the ideal blade twist and platform of each rotor are kept the same in the current design. In addition, the ideal blade twisttwist is hyperbolic hyperbolic or or double double hyperbolic hyperbolic [25], [25 while], while only only linear linear blade blade twists twistsare considered are considered here for here forsimplification. simplification.

Energies 2020, 13, x FOR PEER REVIEW 9 of 20 Energies 2020, 13, 1155x FOR PEER REVIEW 9 of 20 Energies 2020, 13, x FOR PEER REVIEW 9 of 20

Figure 6. Framework of the optimization procedure for the mono tiltrotor (MTR) design. ASA: Figure 6. Framework of the optimization procedure for the mono tiltrotor (MTR) design. ASA: Figure 6.adaptiveFramework simulated of the optimization annealing; NSGA-II: procedure no forn-dominated the mono tiltrotor sorting (MTR) genetic design. algorithm ASA: II. adaptive Figureadaptive 6. Framework simulated of theannealing; optimization NSGA-II: procedure non-dominated for the mono sorting tiltrotor genetic (MTR) algorithm design. II. ASA: simulatedadaptive annealing; simulated NSGA-II: annealing; non-dominated NSGA-II: sorting non-dominated genetic algorithm sorting genetic II. algorithm II.

Figure 7. Coaxial trim in BEMT modeling ( CT is the total thrust coefficient). Figure 7. Coaxial trim in BEMT modeling (CT is the total thrust coefficient). Figure 7. Coaxial trim in BEMT modeling ( CT is the total thrust coefficient). Figure 7. Coaxial trim in BEMT modeling ( CT is the total thrust coefficient).

FigureFigure 8. 8.Chord Chord ofof rotorrotor platform.platform. Figure 8. Chord of rotor platform. Figure 8. Chord of rotor platform.

Energies 2020, 13, 1155 10 of 20

Table 3. Constraint conditions of design variables.

Lower Limit Baseline Upper Limit d/R 0.14 0.16 0.2 dθ 20 0 0 − taper ratio (TR) 0.4 1 1 aspect ratio (AR) 8 (R = 3.3 m) 8.33 (R = 3.81 m) 8.6 (R = 4.4 m)

3.2. Optimization Algorithms For an optimization project, the key to success is the selection of an appropriate optimization algorithm. Two core factors, accuracy and efficiency,are considered here when selecting the optimization methods. The aerodynamic design of MTR is a highly non-linear and discontinuous design problem. The adaptive simulated annealing (ASA) algorithm with open source code is used for the single-objective optimization of the MTR in hovering or cruising states. The ASA algorithm is very well suited to solving non-linear and discontinuous problems. In the current work, the non-dominated sorting genetic algorithm II (NSGA-II) is applied for the multi-objective optimization design of the MTR in both hovering and cruising states. NSGA-II is chosen here because of its high efficiency, fast convergence, good fitness properties and uniformly distributed Pareto front. It is also well suited to the highly non-linear and discontinuous design spaces in the MTR design.

3.2.1. Adaptive Simulated Annealing Algorithm for Single-Objective Optimization Adaptive simulated annealing (ASA) is a global optimization algorithm that relies on a random importance-sampling parameter space, which can be sampled much more efficiently than by using traditional annealing algorithms. The ASA code released by Ingber [26] is used for the single-objective optimization of the MTR in hovering or cruising states. The ASA algorithm is very well suited to solving non-linear and discontinuous problems with short running analysis codes. It has been applied for aircraft research [27] and is chosen here for MTR design. The parameters of ASA are selected according to [26]. The maximum number of generated designs is set as 700. Other parameters, such as relative rate of parameter annealing, cost annealing, parameter quenching, cost quenching and so forth, use default values in the ASA code.

3.2.2. Non-Dominated Sorting Genetic Algorithm II for Multi-Objective Optimization Genetic algorithms have been widely used for the multidisciplinary optimization of rotor design for a long time [28], because of their distinct advantages of using discrete points to search the entire design space. A Pareto front is always obtained by multi-objective optimization algorithms to make trade-offs within the set of design parameterizations that are all Pareto-efficient. In the Pareto front, each design has the “optimal” combination of objective values, and improving one objective is impossible without sacrificing one or more of the other objectives. The non-dominated sorting genetic algorithm II (NSGA-II) [29] has been widely applied for rotor design [30–32]. In this project, NSGA-II is applied for the multi-objective optimization design of the MTR in both hovering and cruising states. In NSGA-II, each objective parameter is treated separately, and a Pareto front is constructed by selecting feasible non-dominated designs. The monitoring of convergence to the Pareto-optimal is set in the NSGA-II program. In the NSGA-II program, A performance metric, γ, has been defined to evaluate the convergence to the Pareto-optimal. NSGA-II utilizes fast non-dominated sorting technology to decrease the computational complexity and is based on an elitist mechanism to improve accuracy. In the current optimization process, the population size N is set to 40, and 20 generations are created to obtain the last Pareto front. The main parameters of NSGA-II include the population size N, number of generations, crossover probability, cross over distribution index, and mutation distribution index. The setting of the last Energies 2020, 13, x FOR PEER REVIEW 11 of 20

the last three parameters was based on those in [29,33] The crossover probability was 0.9, and the distribution indexes for crossover and mutation were 20 and 20, respectively.

3.3. Realization of Optimization The realization of the current method can be described by the following flow chart in Figure 9. It includes two major modules: i.e., the trim modeling by BEMT and the optimization design. For the module of trim modeling, the authors developed the BEMT code based on the theory as illustrated in Section 2.1 and Figure 7 in the manuscript. For the module of optimization design, both algorithms (ASA and NSGA-II) were employed from the well-developed theories [24,26] and Energies 2020, 13, 1155 11 of 20 open source codes. The detailed procedure is as follows: first, the initial values of shape parameters, such as d/R, dθ, TR, AR, are input by the designers, which are shared to the design variables space threeof the parameters optimization was design based module. on those Second, in [29,33 the] The ho crossoververing performance probability FM was an 0.9,d cruising and the distributionefficiency η indexescan be solved for crossover by BEMT and trim mutation modeling, were and 20 andthe result 20, respectively.s are then sent to the design objective space of the optimization algorithm module. Third, the optimization algorithm will decide the design 3.3.parameters Realization of the of Optimization next group/generation and feed back to the BEMT trim modeling module. This circleThe is actuated realization by ofoptimization the current algorithms method can and be describeddoes not stop by theuntil following the preset flow generation chart in Figurenumber9. Itis includesfinished. twoThis major optimization modules: project i.e., the can trim be modelingimplemented by BEMT in existing and the design optimization optimization design. platforms such as Isight software and Artap [34] to validate the results.

Figure 9. Two main modules of MTR optimization system. FM: figurefigure of merit.

ForThe thetime module consumption of trim modeling,of a case varies the authors within developed a certain therange, BEMT which code relies based on on each the theoryset of asparameters. illustrated One in Section individual 2.1 and case Figure requires7 in almost the manuscript. 1~2 minutes, For and the one module optimization of optimization process design, can be bothfinished algorithms within one (ASA day and (with NSGA-II) 800 samples, were employed on a computer from theof a well-developedfour-core Intel processor theories [ 24with,26 ]CPU and openspeeds source of 3.2 codes. GHz). The detailed procedure is as follows: first, the initial values of shape parameters, such as d/R, dθ, TR, AR, are input by the designers, which are shared to the design variables space of3.4. the Analysis optimization of 0ptimization design Results module. Second, the hovering performance FM and cruising efficiency η can be solved by BEMT trim modeling, and the results are then sent to the design objective space of3.4.1. the Single-Objective optimization algorithm Optimization module. in Hovering Third, the or in optimization Cruising States algorithm will decide the design parameters of the next group/generation and feed back to the BEMT trim modeling module. This circle Two single-objective optimization cases for the current MTR in hovering or in cruising states is actuated by optimization algorithms and does not stop until the preset generation number is finished. C = 0.008 Thisare conducted. optimization For project the target can bethrust implemented coefficient, in the existing hovering design trim optimization is T 0 platforms and cruising such as trim Isight is = = −5 = −4 softwareCT 0 0.004 and. ArtapThe [34trim] to validatetolerances the are results. set as E 10 and E1 10 , respectively. Both single-objectiveThe time consumption optimization of acases case are varies based within on athe certain ASA range,optimization which relies algorithm on each with set of700 parameters. samples. One individualFigure 10 gives case requiresthe hove almostring FM 1~2 and minutes, cruising and η in one each optimization optimization process process. can beUnder finished the design within oneconstraints day (with (as 800 shown samples, in Table on a3), computer both optimization of a four-core objectives Intel processor converge with to the CPU optimal speeds goals of 3.2 after GHz). a certain number of samples. The best FM for hovering optimization is 0.6846, and the optimal η for 3.4.cruising Analysis optimization of 0ptimization is 0.6514. Results Figure 11 and Figure 12 illustrate the design parameter distribution versus objectives of all the 3.4.1. Single-Objective Optimization in Hovering or in Cruising States samples. All the design parameters gather to certain values (marked with a red circle in each figure), whichTwo are single-objective given in the third optimization and fourth cases columns for thein Table current 4. MTRSeveral in hoveringpoints can or be in observed cruising in states the areoptimization conducted. results: For the 1) targetthe optimal thrust taper coeffi ratioscient, for the both hovering hovering trim and is C Tcruising0 = 0.008 casesand reach cruising thetrim lower is 5 4 CT0 = 0.004. The trim tolerances are set as E = 10− and E1 = 10− , respectively. Both single-objective optimization cases are based on the ASA optimization algorithm with 700 samples. Figure 10 gives the hovering FM and cruising η in each optimization process. Under the design constraints (as shown in Table3), both optimization objectives converge to the optimal goals after a certain number of samples. The best FM for hovering optimization is 0.6846, and the optimal η for cruising optimization is 0.6514. Figures 11 and 12 illustrate the design parameter distribution versus objectives of all the samples. All the design parameters gather to certain values (marked with a red circle in each figure), which are given in the third and fourth columns in Table4. Several points can be observed in the optimization results: 1) the optimal taper ratios for both hovering and cruising cases reach the lower constraint limit, Energies 2020, 13, 1155 12 of 20 Energies 2020, 13, x FOR PEER REVIEW 12 of 20 sinceconstraint smaller limit,TRs sincecontribute smaller to TRs more contribute uniformly to induced more uniformly velocity distributionsinduced velocity along distributions each blade; 2)along the optimaleach blade; radii 2) for the both optimal cases radii are closefor both to the cases upper are constraintclose to the limit—obviously, upper constraint longer limit—obviously, and thinner bladeslonger haveand thinner less inter-blade blades have interference; less inter-blade 3) the best inte linearrference; blade 3) twiststhe best for linear the hovering blade twists case (for11.4) the − andhovering cruising case case (-11.4) ( 18.2) and are cruising divergent, case which (-18.2) results are divergent, from their wh diichfferent results reset from trim their thrusts—a different suitable reset − bladetrim thrusts—a twist design suitable can decrease blade twist the induced design lossescan decrease on the coaxialthe induced rotor, losses and it on can the distinctly coaxialdelay rotor, flow and separationit can distinctly at the bladedelay rootflow and separation stall at the at bladethe blade tip, which root and will contributestall at the to blade a more tip, uniform which loadwill distributioncontribute to on a more each bladeuniform [10 ].load However, distribution a large on bladeeach blade twist will[10]. beHowever, unproductive, a large sinceblade it twist induces will morebe unproductive, serious flow stallsince along it induces the blade; more 4) serious for inter-rotor flow stall spacing, along the distinct blade; gathering 4) for inter-rotor can be seen spacing, in the cruisingdistinct gathering case, which can is atbe theseen lower in the limit cruising 0.14, butcase, no which clear trendis at the is foundlower inlimit hovering 0.14, but case. no Comparedclear trend withis found the previousin hovering three case. design Compared parameters, with thethe inter-rotor previous three spacing design has aparameters, relatively weak the inter-rotor influence. Accordingspacing has to a the relatively intrinsic weak mechanism, influence. a larger According spacing to maythe intrinsic contribute mechanism, to less interference a larger spacing losses, but may it alsocontribute leads to to higher less interference profile drag losses, with a but higher it also exposed leads shaft.to higher profile drag with a higher exposed shaft.It can be found that the performance in both cases is obviously affected by the design constraints. The single-objectiveIt can be found optimization that the performance results give clear in both indications cases ofis theobviously effectsof affected each design by the parameter design andconstraints. their correlations. The single-objective The optimizations optimization of the results hovering give case clear and indi cruisingcationscase of the result effects in di offferent each selectionsdesign parameter of design and parameters. their correlations. Trade-offs The of design optimi parameterszations of shouldthe hovering be carefully case and considered cruising when case bothresult hovering in different and cruisingselections performances of design areparameters. concerned. Trade-offs Therefore, of a multi-objectivedesign parameters optimization should be is possiblycarefully required. considered when both hovering and cruising performances are concerned. Therefore, a multi-objective optimization is possibly required.

FigureFigure 10.10. OptimizationOptimization samplessamples inin hoveringhovering andand cruisingcruising statestate basedbased onon ASAASA (N(N isis thethe populationpopulation size).size).

Table 4. Parameters of original and single/multi-objective optimized rotors. Table 4. Parameters of original and single/multi-objective optimized rotors. Original Optimized Hover Optimized Cruise Optimized Hover and Cruise Optimized Optimized Optimized Hover and Original R (/m) 3.81Hover 4.29 Cruise 4.40 Cruise 4.40 d/R 0.16 0.18 0.14 0.18 R (/m) 3.81 4.29 4.40 4.40 dθ (/ ) 0 11.4 18.2 17.6 ◦ − − − cr (d//m)R 0.45720.16 0.7131 0.18 0.140.7298 0.72960.18 dTRθ (/°) 01 ₋11.4 0.40 ₋18.2 0.40 ₋17.6 0.40 ARcr 8.33 6.01 6.03 8.6 0.4572 0.7131 0.7298 0.7296 FM(/m) 0.6343 0.6846 0.6760 0.6805 η 0.5745 0.6451 0.6514 0.6512 TR 1 0.40 0.40 0.40 AR 8.33 6.01 6.03 8.6 FM 0.6343 0.6846 0.6760 0.6805 η 0.5745 0.6451 0.6514 0.6512

Energies 2020, 13, x FOR PEER REVIEW 13 of 20 EnergiesEnergies2020 2020, 13,, 115513, x FOR PEER REVIEW 13 of13 20 of 20

Figure 11. Parameters of optimization samples in hovering state. FigureFigure 11. 11.Parameters Parameters of optimizationof optimization samples samples in hoveringin hovering state. state.

FigureFigure 12. 12.Parameters Parameters of optimizationof optimization samples samples in cruisingin cruising state. state. Figure 12. Parameters of optimization samples in cruising state.

Energies 2020, 13, 1155 14 of 20

Energies 2020, 13, x FOR PEER REVIEW 14 of 20 3.4.2. Multi-Objective Optimization for the MTR in Both Hovering and Cruising States 3.4.2. Multi-Objective Optimization for the MTR in Both Hovering and Cruising States In the multi-objective optimization process, the figure of merit (FM) in the hovering sate and the propulsiveIn the multi-objective efficiency (η) in optimization the cruising process, state are the chosen figure as of objective merit (FM) functions. in the hovering Sensitivity sate analysisand the propulsiveunder different efficiency sets of (η generation) in the cruising numbers state was are implementedchosen as objective for the functions. multi-objective Sensitivity optimization. analysis underTwo optimization different sets cases of generation with 20 and numbers 30 generations was implemented (both with for population the multi-objective size N = 40)optimization. have been Twocarried optimization out, respectively. cases with The results20 and in30 Figure generations 13 show (both that with the 30-generationspopulation size case N = shares 40) have a similar been carriedPareto Frontout, respectively. to the 20-generation The results results, in Figure which 13 show indicates that thatthe 30-generations 20 generations case are susharesfficient a similar in the Paretocurrent Front optimization. to the 20-generation The following results, analysis which will focusindicates on the that case 20 withgenerations 20 generations. are sufficient in the current optimization. The following analysis will focus on the case with 20 generations.

Figure 13. SensitivitySensitivity analysis analysis results results under under different different generation generation numbers in multi-objective optimization (the Pareto frontfront in in the the red red circle circle is from is from the 20the generations 20 generations case.). case.). Figure 14a shows the detailed multi-objective optimization results which are obtained by the Figure 14a shows the detailed multi-objective optimization results which are obtained by the NSGA-II algorithm, where a total number of 800 samples is utilized to obtain the last Pareto Front NSGA-II algorithm, where a total number of 800 samples is utilized to obtain the last Pareto Front (Figure 14b). Among all these samples, the maximum FM for the hovering case is 0.6845 and the (Figure 14b). Among all these samples, the maximum FM for the hovering case is 0.6845 and the maximum η for the cruising case is 0.6513. They appear in different design samples, and both of maximum η for the cruising case is 0.6513. They appear in different design samples, and both of them are smaller than those in the single-objective optimization cases. Designers can make trade-offs them are smaller than those in the single-objective optimization cases. Designers can make trade-offs among the design points on the Pareto front. For example, if the cruising performance is investigated, among the design points on the Pareto front. For example, if the cruising performance is an optimal design from the optimized cruising side of the Pareto front can be selected, as marked in investigated, an optimal design from the optimized cruising side of the Pareto front can be selected, Figure 14b. This optimal design has almost the highest cruising efficiency with a moderate hovering as marked in Figure 14b. This optimal design has almost the highest cruising efficiency with a FM. This specific example design has been used as a case study and in the parameter analysis in the moderate hovering FM. This specific example design has been used as a case study and in the following sections. The parameter comparison of the original and optimized rotor (optimal choice in parameter analysis in the following sections. The parameter comparison of the original and Figure 14b) is listed in the second and fifth columns of Table4. optimized rotor (optimal choice in Figure 14b) is listed in the second and fifth columns of Table 4. Figure 15 demonstrates the design parameters along the Pareto front in the current multi-objective optimization. TR is around 0.4 along the entire Pareto front (Figure 15a), which is consistent with the single-objective optimization results. In Figure 15b, when the Pareto front inclines to the cruising-optimized side, the distribution of the inter-rotor spacing is relatively scattered, and it converges to 0.18 when it moves towards the optimized hovering side. Figure 15c indicates that larger blade twists (a larger absolute value of dθ ) are more suitable for cruising, whereas relatively smaller twists are better for hovering. The last sub-figure shows that most of the optimal blade radii along the Pareto front are close to 4.40. Besides, radii within the range of 4.25 to 4.40 are also suitable for hovering. The aerodynamic mechanism analysis and discussion of results will focus on the multi-objective optimization design (the optimal choice in Figure 14b). As shown in the last column of Table 4, the optimized blades of MTR are thinner and longer than the original one under the same blade solidity and blade number. Unlike the original rectangular blade platform, the optimized rotor platform is trapezoidal. Its AR reaches the upper limit of 8.6, while its TR reaches the lower limit of 0.4. The other two design variables—i.e., the inter rotor spacing and blade twist—are determined by

Energies 2020, 13, x FOR PEER REVIEW 15 of 20 the trade-offs between the hovering FM and cruising η. For the chosen optimized rotor, the inter rotor spacing is 0.18 and the linear blade twist is -17.6°. = Under the presupposed design point—i.e., hovering trim CT 0 0.008 and cruising trim = CT 0 0.004 —the optimized MTR has a performance of FM = 0.6805 and η = 0.6512. Compared with the original rotor (FM = 0.6343, η = 0.5745, for coaxial rotor 2 in [4]), FM and η have been improved by 7.3% and 13.4%, respectively. It is also worth mentioning that a compromise design is directly obtained from the multi-objective optimization, while the single-objective optimizations always Energies 2020, 13, 1155 15 of 20 presentEnergies 2020 biased, 13, x designs FOR PEER (as REVIEW shown in the last two rows in Table 4) with either a worse FM or a worse15 of 20 efficiency. the trade-offs between the hovering FM and cruising η. For the chosen optimized rotor, the inter rotor spacing is 0.18 and the linear blade twist is -17.6°. = Under the presupposed design point—i.e., hovering trim CT 0 0.008 and cruising trim = CT 0 0.004 —the optimized MTR has a performance of FM = 0.6805 and η = 0.6512. Compared with the original rotor (FM = 0.6343, η = 0.5745, for coaxial rotor 2 in [4]), FM and η have been improved by 7.3% and 13.4%, respectively. It is also worth mentioning that a compromise design is directly obtained from the multi-objective optimization, while the single-objective optimizations always present biased designs (as shown in the last two rows in Table 4) with either a worse FM or a worse efficiency.

Figure 14. Multi-objective optimization results based on NSGA-II (20 generations). Figure 14. Multi-objective optimization results based on NSGA-II (20 generations). Figure 15 demonstrates the design parameters along the Pareto front in the current multi-objective optimization. TR is around 0.4 along the entire Pareto front (Figure 15a), which is consistent with the single-objective optimization results. In Figure 15b, when the Pareto front inclines to the cruising-optimized side, the distribution of the inter-rotor spacing is relatively scattered, and it converges to 0.18 when it moves towards the optimized hovering side. Figure 15c indicates that larger blade twists (a larger absolute value of dθ) are more suitable for cruising, whereas relatively smaller twists are better for hovering. The last sub-figure shows that most of the optimal blade radii along the Pareto front are close to 4.40. Besides, radii within the range of 4.25 to 4.40 are also suitable for hovering.Figure 14. Multi-objective optimization results based on NSGA-II (20 generations).

Figure 15. Parameters along the Pareto front in multi-objective optimization.

FigureFigure 15. 15. ParametersParameters along along the the Pareto Pareto front front in in multi-objective multi-objective optimization. optimization.

Energies 2020, 13, 1155 16 of 20 Energies 2020, 13, x FOR PEER REVIEW 16 of 20

TheThe aerodynamichovering FM mechanism and cruising analysis η under and a discussionseries of trim of results thrust will coefficients focus on theare multi-objectiveinvestigated to optimizationcompare the global design performance (the optimal of choice the optimized in Figure MTR 14b). with As the shown original in theone. last As shown column in of Figure Table 416,, thethe optimized effects of bladesoptimization of MTR areare thinnerobvious and in longer both thanhovering the original and crui onesing under states, the sameDuring blade the solidity entire andcalculated blade number.range, the Unlike maximum the original performance rectangular increments blade platform,for hovering the are optimized ΔFM = rotor14.1% platform and those is trapezoidal.for cruising ItsareARΔηreaches= 25.0%. the upper limit of 8.6, while its TR reaches the lower limit of 0.4. The other two design variables—i.e., the inter rotor spacing and blade twist—are determined by the trade-offs Figure 17 and Figure 18 illustrate the span-wise load distributions—i.e., the thrust coefficient, between the hovering FM and cruising η. For the chosen optimized rotor, the inter rotor spacing is 0.18 dCT , and torque coefficient, dCQ —on the upper and lower rotors of the original and the and the linear blade twist is 17.6◦. multi-objective optimized MTR,− respectively. In these figures, solid lines with solid symbols are the Under the presupposed design point—i.e., hovering trim CT0 = 0.008 and cruising trim results of the optimized MTR, and dashed lines with hollow symbols are those of the original rotor. CT0 = 0.004—the optimized MTR has a performance of FM = 0.6805 and η = 0.6512. Compared with theIn addition, original rotor the triangle (FM = 0.6343, symbolsη = correspond0.5745, for coaxialto the upper rotor 2 rotor in [4 ]),and FM circular and η have symbols been correspond improved by to 7.3%the lower and 13.4%, rotor. respectively.Figure 17a shows It is also that worth most mentioningof the upper that rotor a compromise thrusts are larger design than is directly the lower obtained ones, fromwhich the results multi-objective from the effects optimization, of induced while speed the single-objective in the hovering optimizations state, while in always the cruising present state, biased as designsshown in (as Figure shown 18a, in the the last upper two rowsrotor inthrust Table distribu4) withtion either is avery worse close FM to or the a worse lower e onefficiency. because, at that Thetime, hovering the inflow FM caused and cruising by forwardη under speed a seriesplays a of dominant trim thrust role. coe Forfficients the torque are investigated distribution toin comparethe hovering the global state, performance as shown in of Figure the optimized 17b, distributions MTR with theon the original upper one. and As lower shown rotors in Figure remain 16, theconsistent effects of with optimization each other are due obvious to the intorque both trim hovering in the and optimization cruising states, proces Durings. The the same entire phenomenon calculated range,is seen the in maximum Figure 18b performance for cruising increments flight. In for conclusion, hovering are compared∆FM = 14.1% with and the those original for cruising MTR, arethe ∆optimizedη = 25.0%. MTR has a much more uniform load distribution along the blade span, which contributes to much better performance both in hovering and cruising states.

FigureFigure 16.16. GlobalGlobal performanceperformance comparisoncomparison of thethe originaloriginal andand optimizedoptimized MTRMTR inin hoveringhovering andand cruisingcruising states.states.

Figures 17 and 18 illustrate the span-wise load distributions—i.e., the thrust coefficient, dCT, and torque coefficient, dCQ—on the upper and lower rotors of the original and the multi-objective optimized MTR, respectively. In these figures, solid lines with solid symbols are the results of the optimized MTR, and dashed lines with hollow symbols are those of the original rotor. In addition, the triangle symbols correspond to the upper rotor and circular symbols correspond to the lower rotor. Figure 17a shows that most of the upper rotor thrusts are larger than the lower ones, which results from the effects of induced speed in the hovering state, while in the cruising state, as shown in Figure 18a, the upper rotor thrust distribution is very close to the lower one because, at that time, the inflow caused by forward speed plays a dominant role. For the torque distribution in the hovering state, as shown in Figure 17b, distributions on the upper and lower rotors remain consistent with each other due to the torque trim in the optimization process. The same phenomenon is seen in Figure 18b for cruising flight. In conclusion, compared with the original MTR, the optimized MTR has a much

Figure 17. Spanwise load distribution on the upper and lower rotors of the optimized MTR in the hovering state.

Energies 2020, 13, x FOR PEER REVIEW 16 of 20

The hovering FM and cruising η under a series of trim thrust coefficients are investigated to compare the global performance of the optimized MTR with the original one. As shown in Figure 16, the effects of optimization are obvious in both hovering and cruising states, During the entire calculated range, the maximum performance increments for hovering are ΔFM = 14.1% and those for cruising are Δη = 25.0%. Figure 17 and Figure 18 illustrate the span-wise load distributions—i.e., the thrust coefficient, dCT , and torque coefficient, dCQ —on the upper and lower rotors of the original and the multi-objective optimized MTR, respectively. In these figures, solid lines with solid symbols are the results of the optimized MTR, and dashed lines with hollow symbols are those of the original rotor. In addition, the triangle symbols correspond to the upper rotor and circular symbols correspond to the lower rotor. Figure 17a shows that most of the upper rotor thrusts are larger than the lower ones, which results from the effects of induced speed in the hovering state, while in the cruising state, as shown in Figure 18a, the upper rotor thrust distribution is very close to the lower one because, at that time, the inflow caused by forward speed plays a dominant role. For the torque distribution in the hovering state, as shown in Figure 17b, distributions on the upper and lower rotors remain consistent with each other due to the torque trim in the optimization process. The same phenomenon is seen in Figure 18b for cruising flight. In conclusion, compared with the original MTR, the optimized MTR has a much more uniform load distribution along the blade span, which contributes to much better performance both in hovering and cruising states.

Energies 2020, 13, 1155 17 of 20

moreFigure uniform 16. Global load performance distribution comparison along the blade of the span, original which and contributes optimized MTR to much in hovering better performance and bothcruising in hovering states. and cruising states.

Energies 2020, 13, x FOR PEER REVIEW 17 of 20 Figure 17. Spanwise load distribution on the upper and lower rotors of the optimized MTR in the Figure 17. Spanwise load distribution on the upper and lower rotors of the optimized MTR in the hovering state. hovering state.

FigureFigure 18.18. Spanwise load distribution on the upper andand lower rotors ofof thethe optimizedoptimized MTRMTR inin thethe cruisingcruising state.state. 4. Conclusions 4. Conclusion In this paper, the BEMT method was applied in the aerodynamic modeling of an MTR coaxial In this paper, the BEMT method was applied in the aerodynamic modeling of an MTR coaxial rotor system. As part of this analysis, the interference between the upper and lower rotors was rotor system. As part of this analysis, the interference between the upper and lower rotors was accounted for. The accuracy of the current BEMT modeling is fully validated by a comparison accounted for. The accuracy of the current BEMT modeling is fully validated by a comparison with with previous experimental data and CFD simulation, confirming that BEMT is a reliable tool for previous experimental data and CFD simulation, confirming that BEMT is a reliable tool for aerodynamic modeling and the optimization design of MTR. A high-efficiency automated optimization aerodynamic modeling and the optimization design of MTR. A high-efficiency automated framework was then built by integrating the BEMT modeling and advanced optimization algorithms; optimization framework was then built by integrating the BEMT modeling and advanced i.e., single-objective optimization and multi-objective optimization. The trim of both thrust and torque optimization algorithms; i.e., single-objective optimization and multi-objective optimization. The was fulfilled during the optimization process. trim of both thrust and torque was fulfilled during the optimization process. The single-objective optimizations of the rotor in either hovering or cruising states were The single-objective optimizations of the rotor in either hovering or cruising states were independently studied. Both cases led to better aerodynamic performances of the rotor with their independently studied. Both cases led to better aerodynamic performances of the rotor with their specific working conditions; i.e., the figure of merit (FM) was improved from 0.6343 to 0.6846 in the specific working conditions; i.e., the figure of merit (FM) was improved from 0.6343 to 0.6846 in the hovering state, or the propulsive efficiency (η) was improved from 0.5745 to 0.6514 in the cruising hovering state, or the propulsive efficiency (η) was improved from 0.5745 to 0.6514 in the cruising state. The single-objective optimization results clearly show the effects of each parameter and their state. The single-objective optimization results clearly show the effects of each parameter and their inter-relationships on design objectives. It was found that the optimal design parameters of these two cases deviate from each other, which is not applicable in the realization of aircraft design. The multi-objective optimization, considering both hovering and cruising performance, is studied. A Pareto front is obtained from this project. The design parameter analysis along the Pareto front shows that AR reaches the upper limit of 8.6, while TR reaches the lower limit of 0.4. The other two design variables—i.e., the inter-rotor spacing and blade twist—are determined by the trade-offs between the hovering FM and cruising η. When the Pareto front inclines to the cruising-optimized side, the distribution of the inter-rotor spacing is relatively scattered, and it converges to 0.18 when moving towards the hovering-optimized side. Larger blade twists are more suitable for cruising, whereas relatively smaller twists are better for hovering. An example design point has been chosen from the Pareto front samples. Further analysis showed that the optimized design possessed a performance of FM = 0.6805 and η = 0.6512, which was improved by 7.3% and 13.4% from that of the prototype, respectively. A comprehensive performance in both hovering and cruising states is achieved in multi-objective optimization, in which the single-objective optimization cases only render a biased optimization design. The multi-objective optimized rotor platform is trapezoidal; its AR arrives to the upper limit, while its TR reaches to the lower limit. The optimized rotor has a uniform load distribution along the blade span, and therefore it contributes to a better performance in both hovering and cruising states.

Energies 2020, 13, 1155 18 of 20 inter-relationships on design objectives. It was found that the optimal design parameters of these two cases deviate from each other, which is not applicable in the realization of aircraft design. The multi-objective optimization, considering both hovering and cruising performance, is studied. A Pareto front is obtained from this project. The design parameter analysis along the Pareto front shows that AR reaches the upper limit of 8.6, while TR reaches the lower limit of 0.4. The other two design variables—i.e., the inter-rotor spacing and blade twist—are determined by the trade-offs between the hovering FM and cruising η. When the Pareto front inclines to the cruising-optimized side, the distribution of the inter-rotor spacing is relatively scattered, and it converges to 0.18 when moving towards the hovering-optimized side. Larger blade twists are more suitable for cruising, whereas relatively smaller twists are better for hovering. An example design point has been chosen from the Pareto front samples. Further analysis showed that the optimized design possessed a performance of FM = 0.6805 and η = 0.6512, which was improved by 7.3% and 13.4% from that of the prototype, respectively. A comprehensive performance in both hovering and cruising states is achieved in multi-objective optimization, in which the single-objective optimization cases only render a biased optimization design. The multi-objective optimized rotor platform is trapezoidal; its AR arrives to the upper limit, while its TR reaches to the lower limit. The optimized rotor has a uniform load distribution along the blade span, and therefore it contributes to a better performance in both hovering and cruising states. The optimized design has increased the hovering performance, FM, by 7.3% and the cruising performance η by 13.4% when compared with the prototype, respectively. The payload capability is in direct proportion to the hovering FM. On the other hand, the rate of climb will also be increased with the FM. Cruising η is the ratio of effective power and total power; a higher η means more fuel savings. The endurance of cruising will be increased by almost 10% after optimization. In addition, the maximum fight speed will be distinctly improved by the increase of η. In summary, the optimization project in the current work can achieve the original goal of MTR, which is to expand the overall flight envelope for a rotor craft. In general, there are three flight modes of the MTR: i.e., forward flight, vertical flight and converting flight. The current work mainly focuses on the BEMT aerodynamic modeling in the hovering state and axial flight, corresponding to the vertical flight and forward flight modes. The flow around the coaxial rotor is axisymmetric in the hovering state and axial flight, while for converting flight, an oblique flow acts on the coaxial rotor. The flow around the coaxial rotor is not axisymmetric. Several changes should be made to apply the BEMT model to inclined flow. First of all, three-dimensional rather than two-dimensional theory should be applied in the momentum theory of BEMT. Secondly, the aerodynamic interference areas both in the upper rotor and lower rotor are not circular, as with those in Figure3. Both shapes of interference from the upper rotor and the lower rotor are irregular. Therefore, the calculation method of interference areas in axial flight should be changed when applied to converting flight. Improved aerodynamic modeling should be explored for the converting flight mode in future work.

Author Contributions: Conceptualization, L.Z., J.H., D.P. and X.S.; Data curation, L.Z.; Formal analysis, L.Z., D.P. and X.S.; Funding acquisition, J.H. and D.P.; Investigation, L.Z. and J.H.; Methodology, L.Z.; Supervision, D.P. and X.S.; Writing—original draft, L.Z.; Writing—review & editing, J.H., D.P. and X.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Natural Science Foundation of China, grant number 51906224, and Zhejiang Provincial Natural Science Foundation, China, grant numbers LY18A020002. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Wayne, J. Helicopter Theory; Princeton University Press: Princeton, NJ, USA, 1980. 2. Bolkcom, C. V-22 Osprey Tilt-Rotor Aircraft. 2005. Available online: https://www.everycrsreport.com/files/ 20050804_RL31384_31272dfc23e2ff48a9e152b7caa0adcf3a100e5c.pdf (accessed on 18 January 2020). Energies 2020, 13, 1155 19 of 20

3. Baldwin, G.D.; Washington, P. Preliminary Design Studies of a Mono Tiltrotor (MTR) with Demonstrations of Aerodynamic Wing Deployment. In Proceedings of the AHS International Specialists Meeting, Chandler, AZ, USA, 23–25 January 2007. 4. Harrington, D. Full-scale-tunnel investigation of the static-thrust performance of a coaxial helicopter rotor. In NACA TN-2318; Langley Field: Hampton, VA, USA, 1951. 5. Bohorquez, F. Rotor Hover Performance and System Design of An Efficient Coaxial Rotary Wing Micro Air Vehicle. Ph. D. Thesis, University of Maryland, College Park, MD, USA, 2007. 6. Bagai, A. Aerodynamic design of the x2 technology demonstrator main rotor blade. In Proceedings of the 64th American Helicopter Society Annual Forum, America Helicopter Society, Montreal, QC, Canada, 29 April–1 May 2008; pp. 29–44. 7. Leishman, J.G. Aerodynamic performance considerations in the design of a coaxial proprotor. J. Am. Helicopter Soc. 2009, 54, 12005. [CrossRef] 8. Leishman, J.G. Principles of Helicopter Aerodynamics, 2nd ed.; Cambridge University Press: New York, NY, USA, 2006; pp. 125–140. 9. Syal, M.; Leishman, J.G. Aerodynamic optimization study of a coaxial rotor in hovering flight. J. Am. Helicopter Soc. 2012, 57, 1–15. [CrossRef] 10. Leishman, J.G.; Rosen, K.M. Challenges in the aerodynamic optimization of high-efficiency proprotors. J. Am. Helicopter Soc. 2011, 56, 12004. [CrossRef] 11. Rand, O.; Khromov, V. Aerodynamic optimization of coaxial rotor in hover and axial flight. In Proceedings of the 27th International Congress of the Aeronautical Sciences, Nice, France, 19–24 September 2010; pp. 1–13. 12. Leishman, J.G.; Ananthan, S. Aerodynamic optimization of a coaxial proprotor. In Proceedings of the 62th American Helicopter Society Annual Forum, American Helicopter Society, Phoenix, AZ, USA, 9–11 May 2006; Volume 1, p. 64. 13. Coleman, C.P. A Survey of Theoretical and Experimental Coaxial Rotor Aerodynamic Research. 1997. Available online: https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19970015550.pdf (accessed on 15 October 2019). 14. Leishman, J.G.; Syal, M. Figure of merit definition for coaxial rotors. J. Am. Helicopter Soc. 2008, 53, 290–300. [CrossRef] 15. Valkov, T.Y. Aerodynamic load computation on coaxial hingeless helicopter rotor. In Proceedings of the 28th Aerospace Sciences Meeting, Reno, NV, USA, 8–11 January 1990. 16. Yana, J.; Rand, O. Performance analysis of a coaxial rotor system in hover: Three points of view. In Proceedings of the 28th International Congress of the Aeronautical Sciences, Brisbane, Australia, 23–28 September 2012. 17. Pérez Gordillo, A.M.; Villegas Santos, J.S.; Lopez Mejia, O.D.; Suárez Collazos, L.J.; Escobar, J.A. Numerical and Experimental Estimation of the Efficiency of a Quadcopter Rotor Operating at Hover. Energies 2019, 12, 261. 18. Glauert, H. Airplane propellers. In Division L in Aero-Dynamic Theory Vol.4; Durand, W.F., Ed.; Springer: Berlin, Germany, 1935; pp. 169–360. 19. Lakshminarayan, V.; Baeder, J. Computational investigation of small scale coaxial rotor aerodynamics in hover. In Proceedings of the 47th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Orlando, FL, USA, 5–8 January 2009; p. 1069. 20. Leusink, D.; Alfano, D.; Cinnella, P. Multi-fidelity optimization strategy for the industrial aerodynamic design of helicopter rotor blades. Aerosp. Sci. Technol. 2015, 42, 136–147. [CrossRef] 21. Collins, K.B. Multi-Fidelity Framework for Physics Based Rotor Blade Simulation and Optimization. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, USA, 2008. 22. Lock, C. Interference Velocity for A Close Pair of Contra-Rotating Airscrews; British HM Stationery Office: London, UK, 1941. 23. Maynard, J.D. Wind-tunnel tests of single-and dual-rotating tractor propellers of large blade width. Tech. Rep. Arch. Image Libr. 1942, 727, 319–347. 24. Knuth, D.E. The Art of Computer Programming: Sorting and Searching; Addison-Wesley Professional: Englewood Cliffs, NJ, USA, 1998. 25. Leishman, J.G.; Ananthan, S. An optimum coaxial rotor system for axial flight. J. Am. Helicopter Soc. 2008, 53, 366–381. [CrossRef] 26. Ingber, L. Adaptive simulated annealing (ASA): Lessons learned. J. Control. Cybern. 1995, 25, 33–54. Energies 2020, 13, 1155 20 of 20

27. Bouttier, C.; Babando, O.; Gadat, S.; Gerchinovitz, S.; Laporte, S.; Nicol, F. Adaptive Simulated Annealing with Homogenization for Aircraft Trajectory Optimization. J. Oper. Res. Soc. Am. 2017, 569–574. [CrossRef] 28. Hajela, P.; Lee, J. Genetic algorithms in multidisciplinary rotor blade design. In Proceedings of the 36th Structures, Structural Dynamics and Materials Conference, New Orleans, LA, USA, 10–13 April 1995; pp. 2187–2197. 29. Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. A fast and elitist multi-objective genetic algorithm: Nsga-ii. IEEE Trans. Evol. Comput. 2002, 6, 182–197. [CrossRef] 30. Liu, Z.; Dong, L.; Moschetta, J.M.; Zhao, J.; Yan, G. Optimization of Nano-Rotor Blade Airfoil Using Controlled Elitist NSGA-II. Int. J. Micro Air Veh. 2014, 6, 29–42. [CrossRef] 31. Sessarego, M.; Dixon, K.R.; Rival, D.E.; Wood, D.H. A hybrid multi-objective evolutionary algorithm for wind-turbine blade optimization. Eng. Optim. 2015, 47, 1043–1062. [CrossRef] 32. Jiang, H.; Zhou, Z.; Jia, Y.; Zhang, Y.; Zhang, F. Coordinated Optimization of DFIG Rotor Crowbar and DC-chopper Resistances Based on NSGA-II. Energy Procedia 2019, 18, 589–594. [CrossRef] 33. Deb, K.; Agrawal, R.B. Simulated binary crossover for continuous search space. Complex Syst. 1995, 9, 115–148. 34. Panek, D.; Tamas, O.; Pavel, K. Artap: Robust design optimization framework for engineering applications. arXiv 2019, arXiv:1912.11550.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).