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Hindawi International Journal of Aerospace Engineering Volume 2020, Article ID 6650375, 18 pages https://doi.org/10.1155/2020/6650375 Research Article Geometry Design of Coaxial Rigid Rotor in High-Speed Forward Flight Bo Wang , Xin Yuan , Qi-jun Zhao , and Zheng Zhu National Key Laboratory of Rotorcraft Aeromechanics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 210016, China Correspondence should be addressed to Bo Wang; wangbo@nuaa.edu.cn Received 13 October 2020; Revised 16 November 2020; Accepted 24 November 2020; Published 7 December 2020 Academic Editor: Jacopo Seraﬁni Copyright © 2020 Bo Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The aerodynamic performance analysis and blade planform design of a coaxial rigid rotor in forward ﬂight were carried out utilizing CFD solver CLORNS. Firstly, the forward ﬂow ﬁeld characteristics of the coaxial rotor were analyzed. Shock-induced separation occurs at the advancing side blade tip and severe reverse ﬂow occurs at the retreating side blade root. Then, the inﬂuence of geometrical parameters of the coaxial rigid rotor on forward performance was investigated. Results show that swept-back tip could reduce the advancing side compressibility drag and elliptic shape of blade planform could optimize the airload distribution at high advance ratio ﬂights. A kind of blade planform combining swept-back tapered tip and nonlinear chord distribution was optimized to improve the rotor eﬃciency for a given high-speed level ﬂight based on geometric parameter studies. The optimized coaxial rotor increases lift-to-drag ratio by 30% under the design conditions. 1. Introduction performance compared with single rotor conﬁguration, as shown in the following. Coleman [1] summarizes experimental and theoretical Coaxial rigid rotor helicopters with auxiliary propulsion are research on coaxial rotor aerodynamics up to 1997, among able to attain better performance at high forward speed com- pared with conventional single-rotor helicopters. The appli- which hover measurement data from Harrington [2] and for- ward measurement data from Dingeldein [3] are commonly cation of lift oﬀset on coaxial rigid rotors plays an important role in the improvement of forward rotor eﬃ- used for analysis validation. Barbely et al. [4] provided a com- prehensive summary of computational investigations of ciency. For the single-rotor conﬁguration, the forward ﬂight coaxial rotors in hover and forward ﬂight. Bagai and Leish- speed is restricted by dynamic stall on the retreating side and compressibility on the advancing side of the disk. Lift man [5] developed a free vortex wake methodology (FVM) for multirotor conﬁgurations including coaxial rotors and capability of advancing blades is also limited to maintain lat- eral equilibrium. Lift oﬀset concept of coaxial rotor oﬀers a investigated the wake structure of coaxial rotors. Brown and Line [6, 7] developed and extended Vorticity Transport solution to the speed limitation problem. It allows the lateral Model (VTM) which performs well in retaining the structure asymmetry of the lift distribution. The advancing blades gen- erate most lift of the disk, and the retreating blades are oﬀ- and forming of the rotor wake. Kim and Brown [8, 9] ana- lyzed the performance of coaxial rotors in steady and man- loaded. The maintenance of lateral and directional equilibrium is achieved by balancing rolling moments and oeuvring ﬂight using VTM. Leishman and Ananthan [10, 11] established simple momentum theory and blade element torque produced by the upper and lower rotors. However, momentum theory (BEMT) for coaxial rotor system in hover the counterrotation of two rotors generates severe aerody- namic interaction, and operations are much more compli- and axial ﬂight. The viscous vortex particle method (VVPM) [12] is also applied into coaxial rotor studies. Tan et al. [13] cated. Here are limited analytical studies on coaxial rotor 2 International Journal of Aerospace Engineering and Singh et al. [14] used VVPM with vortex panel loads for Table 1: Computation method. coaxial rotor simulations. Computational ﬂuid dynamics Background Blade (CFD) is increasingly powerful with the development of Governing equations Euler equations Compressible RANS numerical simulation techniques and computer hardware. CFD could provide an accurate prediction on performance. Spatial discretization Cell-centered FVM Rajmohan et al. [15] developed a hybrid VPM/CFD method- Time integration LU-SGS ology to study coaxial rotor aerodynamics, which maintains Inviscid ﬂux Roe-MUSCL high accuracy and eﬃciency. CFD and hybrid-CFD ﬂow Viscous ﬂux 2nd-order central diﬀerence method solvers such as OVERFLOW, Helios, and RotCFD are used Turbulence model SA model to study the performance and ﬂow physics of coaxial rotors [4, 16–18], and performance predictions are in good agree- ment with the experimental data. There are more considerations in blade geometry design ∭ WdΩ + ∬ F − F dS =0, ð1Þ ðÞ c v compared with conventional single rotor. Eﬀorts on coaxial Ω ∂Ω ∂t rotor design are made based on various above-mentioned methodologies. Leishman and Ananthan [11] gave the opti- where Ω is the control volume and ∂Ω is its closed surface; dS mum blade twist of hovering Harrington coaxial rotor through BEMT method. Johnson et al. [19] designed a com- is the surface element, W is the vector of conservative vari- ables, F is the vector of convective ﬂuxes, and F is the vector pound helicopter utilizing lift oﬀset rotors using CAMRAD c v II, and the tapering planform and blade twist were optimized of viscous ﬂuxes. under the design conditions. Yeo and Johnson [20] carried out rotor planform and twist optimization for hover and 2 3 cruise performance of high blade loading coaxial rotors by 6 7 CAMRAD II. Bagai [21] described the aerodynamic design 6 7 TD ρu 6 7 of X2 main rotor in detail including blade planform, twist 6 7 6 7 distribution and airfoil conﬁgurations, and performance 6 7 W = , ρv 6 7 improvements it contributes to. 6 7 6 7 In previous work, a coaxial rotor solver has been devel- 6 ρw7 4 5 oped based on CLORNS [22], a high-ﬁdelity CFD solver. Hover performance of coaxial rotor has been investigated ρE 2 3 using the established CFD tool [23] by the present authors. ρV These studies have proved that our CFD solver could provide 6 7 6 7 a reliable performance prediction of coaxial rotor. An opti- ρuV + n p 6 7 r x 6 7 mization method combining the surrogate-based approach 6 7 and genetic algorithm was implemented in the blade geome- 6 7 F = ρvV + n p , ð2Þ r y 6 7 try shape design [24, 25]. In the present paper, the study of 6 7 6 7 impacts of diﬀerent geometry parameters on forward aerody- ρwV + n p 6 7 r z 4 5 namic performance is added to explain why a rotor planform ρHV + V p combining elliptic shape and swept-back tapered tip is cho- r m 2 3 sen. The design operating condition was level cruise at 210 knots. At high advance ratio ﬂights, compressibility on the 6 7 6 7 advancing blade tip is one of the main causes of rotor perfor- n τ + n τ + n τ 6 7 x xx y xy z xz 6 7 mance deﬁciency. The geometry combination of elliptic 6 7 6 7 F = n τ + n τ + n τ , shape and swept-back tapered tip could improve the rotor x yx y yy z yz 6 7 6 7 eﬃciency by providing a better airload distribution over the 6 7 n τ + n τ + n τ 6 7 x zx y zy z zz disk. The optimization was carried out based on the baseline 4 5 geometry, and the optimum blade has a higher lift-to-drag n Θ + n Θ + n Θ x x y y z z ratio. 2. Performance Prediction of Coaxial Rotors in where V = ðV − V Þ ⋅ n and V = V ⋅ n, in which V = ð r b m b b Forward Flight u , v , w Þ is the velocity of the moving blade grid, V = ð b b b u, v, wÞ is the velocity of the ﬂow, and n = ðn , n , n Þ is x y z 2.1. Methodology Descriptions. Rotor performance is pre- the normal vector to the surface pointing outward from dicted using a coaxial rotor solver based on CLORNS [22]. the control volume. ρ, p, E, and H represent density, pres- The governing equations for rotor are the 3D compressible sure, the total energy, and the total enthalpy, respectively. Reynolds Navier-Stokes equations (RANS) and for back- τ and Θ represent the viscous stress and the work per- ground are the Euler equations. Finite volume method ð·Þ ð·Þ (FVM) is used for spatial discretization. The governing equa- formed by the viscous stress and heat conduction, respec- tively. tions for rotor ﬂowﬁeld are as follows: International Journal of Aerospace Engineering 3 X X (a) Structured blade grid (b) Dual rotor grids of blade crossing (c) Moving-embedded grid system Figure 1: Computational grid for coaxial rotor. Table 2: Convergence of grids. Background Blade Total/10 Error of thrust coeﬃcient 175 × 135 × 159 179 × 39 × 61 Coarse 5.46 4.73% 201 × 155 × 187 199 × 51 × 91 Medium 9.52 2.61% 241 × 201 × 231 221 × 89 × 101 Fine 1 19.14 0.44% 361 × 243 × 351 241 × 101 × 159 Fine 2 46.27 Benchmark ∂u 2 ∂u ∂v time-steps correspond to 1.0 of azimuth. The inviscid spatial τ =2μ − μ∇⋅ V, τ = μ + , > xx xy terms are computed using a third-order MUSCL scheme with ∂x 3 ∂y ∂x Roe’s ﬂux diﬀerence splitting, and the viscous terms are com- ∂v 2 ∂v ∂w puted using second-order central diﬀerencing. The Spalart- τ =2μ − μ∇⋅ V, τ = μ + , yy yz > ∂y 3 ∂z ∂y Allmaras turbulence model is employed for the RANS closure. > Calculation methods are summarized in Table 1. > ∂w 2 ∂u ∂w : τ =2μ − μ∇⋅ V, τ = μ + , To allow for the counterrotation between the upper rotor zz xz ∂z 3 ∂z ∂x and lower rotor, moving-embedded grid system is used. Figure 1 shows the structured grid of moving-embedded grid ∂T Θ = uτ + vτ + wτ + κ , x xx xy xz system. Figure 1(a) is the structured blade grid. Compared ∂x > with the single rotor, the outer boundary of the blade grid ∂T should be set carefully to avoid intersections with another Θ = uτ + vτ + wτ + κ , y yx yy yz > ∂y rotor blade. Figure 1(b) shows the relative position of the > upper and lower blades when blade crossing occurs. It can ∂T Θ = uτ + vτ + wτ + κ , be seen that there is some spacing between the outer bound- z zx zy zz ∂z ary of upper blade grid and the lower blade surface; the same ð3Þ is between the outer boundary of lower blade grid and upper blade surface. Therefore, the transmission of ﬂowﬁeld infor- where μ denotes the dynamic viscosity coeﬃcient, κ is the mation between blade grids and background grid at the end of each subiteration could work correctly. thermal conductivity coeﬃcient, and T is the absolute The blade collective, lateral cyclic, and longitudinal cyclic static temperature. Time integration is performed using the Lower-Upper pitch angles for each rotor under speciﬁc conditions are determined using the Newton-Raphson method. By Symmetric Gauss-Siedel (LU-SGS) method. The chosen 4 International Journal of Aerospace Engineering trimming solution, the upper and lower rotors are torque bal- anced and achieve the target thrust coeﬃcient, rolling, and pitching moments. The blade pitch θ of each rotor at azimuth ψ is as follows: θ = θ + θ cos ψ + θ cos ψ , U 0U 1cU U 1sU U ð4Þ θ = θ + θ cos ψ + θ cos ψ : L 0L 1cL 1sL L L In the trimming procedure, the rotor control input vector and the response vector are, respectively, given by X =fg θ , θ , θ , θ , θ , θ , 1 0U 1sU 1cU 0L 1sL 1cL no ð5Þ Y = 〠C ,〠C , LOS, C ,〠C , C , T Q MzU Mx MzL 0.12 0.14 0.16 0.18 0.20 0.22 where C , C , LOS, C , and C represent the thrust coeﬃ- T Q Mz Mx cient, torque coeﬃcient, lift oﬀset, pitching moment coeﬃ- EXP-HC1 EXP-HS1 cient, and rolling moment coeﬃcient, respectively. CAL-HC1 CAL-HS1 Subscripts U and L represent the upper rotor and lower rotor, respectively. ∑C = C + C , ∑C = C + C , Figure 2: CFD calculations compared with measured performance T TU TL Q QU QL ∑C = C + C , and LOS = jC j + jC j/∑C · R. of HC1 and HS1 in forward ﬂight. Mx MxU MxL MxU MxL T Then, the equation to solve is target number and its thrust coeﬃcient comparisons are shown in YX − Y =0, ðÞ Table 2. Fine 1 grid was ﬁnally chosen into calculation. The ð6Þ n+1 n −1 n target ðÞ ðÞ ðÞ size of Fine 1 background grid surrounding the rotor gird is X = X − J Y − Y , reﬁned to 0.05c (c is the blade tip chord length), under which size the mesh convergence is achieved. The subiteration of target target target where Y = f∑C , 0, LOS ,0,0,0g, and J is the the solver is set as 50, and then, the residual of density will Jacobian matrix. be reduced by at least 1 order. Figure 2 shows the measured forward ﬂight performance 2 3 ∂∑C ∂∑C ∂∑C ∂∑C ∂∑C ∂∑C T T T T T T of HC1 and HS1 compared to calculations. The calculations 6 7 ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ 6 7 0U 1sU 1cU 0L 1sL 1cL are in agreement with the measurements. The established 6 7 6 7 CFD solver overpredicted the forward performance, and ∂∑C ∂∑C ∂∑C ∂∑C ∂∑C ∂∑C Q Q Q Q Q Q 6 7 6 7 the calculations are closer to the experimental data with the ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ 6 7 0U 1sU 1cU 0L 1sL 1cL 6 7 increase of advance ratio. It also shows that coaxial rotor ∂LOS ∂LOS ∂LOS ∂LOS ∂LOS ∂LOS 6 7 costs more power than twice of single-rotor consumptions, 6 7 6 ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ 7 0U 1sU 1cU 0L 1sL 1cL 6 7 which could be mainly attributed to the severe interference J = : 6 7 ∂C ∂C ∂C ∂C ∂C ∂C 6 7 MzU MzU MzU MzU MzU MzU in the ﬂowﬁeld of coaxial rotor caused by the counterrotation 6 7 6 7 ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ of upper rotor and lower rotor. Figure 3 gives the conver- 0U 1sU 1cU 0L 1sL 1cL 6 7 gence of thrust coeﬃcient of coaxial rotor at μ =0:2. It shows 6 ∂∑C ∂∑C ∂∑C ∂∑C ∂∑C ∂∑C 7 Mx Mx Mx Mx Mx Mx 6 7 that the force came to convergence after 2 revolutions under 6 7 ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ 0U 1sU 1cU 0L 1sL 1cL 6 7 6 7 a given trimming operations. 4 ∂C ∂C ∂C ∂C ∂C ∂C 5 MzL MzL MzL MzL MzL MzL ∂θ ∂θ ∂θ ∂θ ∂θ ∂θ 0U 1sU 1cU 0L 1sL 1cL 2.3. Aerodynamic Features of Coaxial Rigid Rotors in Forward ð7Þ Flight. The model coaxial rotor used for analysis consists of two 2-bladed rotors with rectangular planform. The blades 2.2. Validations for Coaxial Rotor. The experimental data of have a NACA0012 proﬁle with a radius of 2.0 m and feature Harrington Rotor-1 [2, 3] are used to validate the computa- a linear aerodynamic twist rate of -5 deg/m. The rotor aspect ratio is 10. The separation distance between the upper rotor tional predictions. It consists of one 2-bladed rotors arranged to form a single system (HS1) and the same two 2-bladed and lower rotor is 0.15R. The setting of initial azimuthal loca- rotors arranged to form a coaxial system (HC1). The blade tions is shown in Figure 4: U1 (upper rotor blade 1) is at 90 , uses a NACA airfoil with a nonlinearly varying thickness U2 (upper rotor blade 2) is at 270 , L1 (lower rotor blade 1) is ° ° and a linearly varying chord length. The tip velocity is at 0 , and L2 (lower rotor blade 2) is at 180 . 142.95 m/s in forward ﬂight. The tip mach speed is 0.528 for calculations in the present Four groups of moving-embedded grids are generated to paper. All calculations are trimmed. The torque is balanced investigate the mesh convergence. Fine 2 gird is the ﬁnest, with a thrust coeﬃcient of 0.013 and a lift oﬀset of 0.35. Rotor whose thrust coeﬃcient is chosen as benchmark. The grid shaft angle is 0 . The equivalent lift to drag ratio of coaxial C ) Q Forward flight direction International Journal of Aerospace Engineering 5 0.005 0.004 0.003 0.002 0.001 0.000 0 180 360 540 720 900 1080 1260 1440 1620 Iteration Upper rotor Lower rotor Target C /2 Figure 3: The convergence of thrust coeﬃcients of the upper and lower rotors. Good insight into blade ﬂow features is made by looking the upper surface streamlines during one revolution in Figure 6. In the current operating state, there are little diﬀer- ences between the ﬂow topologies of the upper rotor and lower rotor. Therefore, here only gives the detailed ﬂow topology on the upper surface of the upper rotor. It can be 0.15R seen that the ﬂow feature varies with the azimuth angle and the radical position. There is an obvious deviation of the ﬂow over a large area (radius above 0.95R) at ψ =90 . Shock wave occurs in this area, which produces shock-induced separa- tion. And reverse ﬂow occurs in the root regions on the retreating side, generating large proﬁle drag and negative lift. Figure 4: Blade azimuthal locations at the beginning. The area of reverse ﬂow depends on the ﬂight speed and rotor frequency. Radical ﬂow is more signiﬁcant on the retreating side. Figure 7 shows the temporal variation of CT over one rotor is deﬁned as revolution. From the ﬁgure, the unsteadiness is clearly seen with a dominant 2/rev frequency (blade number of one ∑C rotor). L/D = : ð8Þ For a coaxial rotor with two 2-bladed counterrotating ∑C +∑C /μ D Q rotors, there are 4 overlaps for each blade in one rotor revo- lution as shown in Figure 8. According to the research of The blade mesh has 251 × 131 × 80 points in the stream- Lakshminarayan and Baeder [18], blade overlaps generate wise, spanwise, and normal directions, respectively, and the signiﬁcant impulses in the instantaneous thrust and power in hover, which could be explained by the blade thickness background mesh has 241 × 281 × 201 points in the lateral, longitudinal, and vertical directions, respectively. Simulation (a venturi eﬀect) and loading (an upwash-downwash eﬀect). case used for analysis is set at advance ratio of 0.6. To under- Similarly, blade overlaps exist in forward ﬂight and have eﬀects on aerodynamic loading. The blades of the upper rotor stand the performance characteristics of coaxial rotor in high-speed forward ﬂight, the lift distribution over the rotor overlap with blades of the lower rotor in the following order for one revolution: 1/8 Rev, 3/8 Rev, 5/8 Rev, and 7/8 Rev disk is given in Figure 5 in the form of C ∙Ma contour. It (Rev means one revolution). Figure 9 shows the pressure dis- shows that the lift is mainly generated by the advancing tribution of 0.95R blade section in 1/8Rev and 3/8Rev when blades, and the retreating side provides almost nonlift. This the two sections begin to across. loading distribution is a consequence of the application of It can be seen that sectional pressure distributions of the lift oﬀset. The tip regions at ψ =90 are negatively loaded 1/8Rev and 3/8Rev are diﬀerent. This is because U1/L1 is at this condition, and its area is smaller than the single-rotor advancing blades at 1/8Rev time and U2/L1 is advancing conﬁguration due to the application of lift oﬀset. T 6 International Journal of Aerospace Engineering 2 2 C C L L 0.32 0.32 0.28 0.28 0.24 0.24 0.2 0.2 0.16 0.16 0.12 0.12 0.08 0.08 0.04 0.04 0 0 (a) Upper rotor (b) Lower rotor Figure 5: C ∙Ma contours over the rotor disk. ab c d Figure 6: Streamlines on the upper surface of the upper rotor during one revolution. 0.024 blades at 3/8Rev time. The investigated case has a lift oﬀ- 0.021 set of 0.35 after trimming. Therefore, the blades are unloaded when they are on the retreating side through 0.018 the adjustment of cyclic pitch. Figure 9 reveals a highly 0.015 asymmetric pressure ﬁeld below and above each airfoil section of the advancing blades, namely, lift is produced. 0.012 And for the sections of retreating blades, U2/L2 in 0.009 Figure 9(a)) and U1/L2 in Figure 9(b), the pressure ﬁeld 0.006 is close to symmetric, which means they produce little lift. 0.003 In Figure 9(a), the airfoil section of L1 is aﬀected by the presentation of U2 airfoil section, and in Figure 9(d), the 0.000 airfoil section of L1 is aﬀected by U1 airfoil section. It also –0.003 shows that when the lower rotor blades are on the retreat- 0.00 0.25 0.50 0.75 1.00 ing side, there is only small interference between their ﬂow Rev ﬁelds. This phenomenon is mainly brought by the lift oﬀ- set concept of ABC rotor, and the eﬀect of the blade thick- Upper rotor Lower rotor ness is weaken compared with hover state. Coaxial rotor The pressure contours on the upper surface of two rotors are shown in Figure 10, which represent typical positions Figure 7: Temporal variation of C of the upper and lower rotors during one revolution including crossing, and overlap. It over one revolution. shows that the low pressure area outside the blade appears when the upper rotor or lower rotor is on the advancing side. When on the retreating side, the blade surface has an even pressure distribution. T International Journal of Aerospace Engineering 7 0Rev 1/8Rev 1/6Rev 3/8Rev Upper rotor Lower rotor Figure 8: Counterrotation of coaxial rotors. C 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 C 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 P P U-2 U-1 U-2 U-1 L-1 L-2 L-2 L-1 (a) 1/8Rev (b) 3/8Rev Figure 9: Nondimensional pressure distribution of blade section at r/R =0:95. 3. Geometrical Parameter Influence Analysis in the tip region moves outwards and is weakened with the increase of swept-back angle. The gain of swept-back 3.1. Swept-Back Tip tip is brought by the reduction in Mach number normal to the leading edge of blade tip sections, which is signiﬁ- 3.1.1. Swept-Back Angle. Three types of blades with swept cant on the advancing side. Figure 13 shows the span back tip are chosen to simulate the eﬀect of swept-back thrust distribution at 90 azimuthal position. Swept-back angle on aerodynamic characteristics. All blades begin to tips have higher thrust than the rectangular tip, and the sweep back at 0.9R radius, and the swept-back angles are ° ° ° thrust increases with swept-back angle. This is in confor- 10 ,20 , and 30 , respectively. Figure 11 shows the lift- mity with the pressure distribution analysis in Figure 12. to-drag ratio of three blades compared with the rectangu- In the middle section, the thrust of blades with swept- lar blade (the rotor in Figure 4). In trimmed conditions, back tip is lower than the rectangular blade. swept-back blades have better rotor eﬃciency than the Figure 14 shows the blade section pressure coeﬃcient at rectangular blade, and this improvement increases with 0.9R, where the swept-back begins. It reveals that swept- swept-back angle. back tips lower the negative pressure amplitude on the Case of μ =0:7 is taken into analysis. Figure 12 shows upper surface before the shock occurs (near 0.65c) in this the pressure distribution on the upper surface of blades position. With the increase of swept-back angle, the adverse with diﬀerent swept-back angle at 90 azimuthal position. pressure gradient is reduced. The shock wave moves toward There is an obvious area reduction of low pressure at the trailing edge, and its strength is weakened. Figure 15 0.9R spanwise position of swept-back blades compared shows the blade section pressure coeﬃcient at 0.95R. The with the rectangular blade tip. And the low pressure center = 0° Forward flight direction Forward flight direction Forward flight direction = 0° = 0° Forward flight direction = 0° 8 International Journal of Aerospace Engineering C 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 Time = 0 Time = 1/8Rev U-1 L-2 U-1 L-2 U-2 U-2 L-1 L-1 Time = 1/4Rev Time = 3/8Rev U-1 L-2 U-1 U-2 L-1 L-1 L-2 U-2 𝛹 Figure 10: Pressure distribution on the upper surface of two rotors at 0Rev, 1/8Rev, 1/4Rev, and 3/8Rev. upper surface pressure coeﬃcients of three swept-back blades and the rectangular blade are nearly the same before the shock occurs. On the upper surface, air compressibility makes the position of pressure jump moves toward the leading edge. On the lower surface, the pressure jump decreases with the swept-back angle, and the shock strength is weakened. 3.1.2. Swept-Back Position. Three types of blades with swept back tip are chosen to simulate the eﬀect of swept-back posi- tion on aerodynamic characteristics. All blades have a swept- back angle of 20 , and the swept-back positions begin at 0.85R, 0.9R, and 0.95R, respectively. Figure 16 shows the lift-to-drag ratio of three rotors compared with the rectangu- 0.4 0.5 0.6 0.7 lar blade (the rotor in Figure 4). In trimmed conditions, swept-back blades have better rotor eﬃciency than the rect- Rectangular blade 0.90R-20deg angular blade, and this improvement increases with swept- 0.90R-10deg 0.90R-30deg back area. Case of μ =0:7 is taken into analysis. Figure 17 shows the Figure 11: Lift-to-drag ratio of blades with diﬀerent swept-back pressure distribution on the upper surface of blades with dif- angle. ferent swept-back area at 90 azimuthal position. Low L/D International Journal of Aerospace Engineering 9 C 1 1.19 1.38 1.57 1.76 1.95 2.14 2.33 2.52 2.71 2.9 Figure 12: Pressure distribution on the upper surface of the upper rotor (ψ =90 , μ =0:7). 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 r/R r/R Rectangular blade 0.90R-20deg Rectangular blade 0.90R-20deg 0.90R-10deg 0.90R-30deg 0.90R-10deg 0.90R-30deg (a) Upper rotor (b) Lower rotor Figure 13: Span thrust distribution (ψ =90 , μ =0:7). –1.5 –1.5 –1.0 –1.0 –0.5 –0.5 0.0 0.0 0.5 0.5 1.0 1.0 1.5 1.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x/c x/c Rectangular blade 0.90R-20deg Rectangular blade 0.90R-20deg 0.90R-10deg 0.90R-30deg 0.90R-10deg 0.90R-30deg (a) Upper rotor (b) Lower rotor Figure 14: Blade section pressure coeﬃcient (ψ =90 , μ =0:7, r/R =0:90). dC /dr dC /dr P 10 International Journal of Aerospace Engineering –1.5 –1.5 –1.0 –1.0 –0.5 –0.5 0.0 0.0 0.5 0.5 1.0 1.0 1.5 1.5 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x/c x/c Rectangular blade 0.90R-20deg Rectangular blade 0.90R-20deg 0.90R-10deg 0.90R-30deg 0.90R-10deg 0.90R-30deg (a) Upper rotor (b) Lower rotor Figure 15: Blade section pressure coeﬃcient (ψ =90 , μ =0:7, r/R =0:95). tered at 0.7R, 0.7R, and 0.6R. The length of the minor axes are 1.2c, 1.35c, and 1.25c (c is the length of the root chord), respectively. Figure 19 shows the lift-to-drag ratio variation of three rotors compared with the rectangular blade (the rotor in Figure 4). It reveals that the elliptic chord distribu- tion has better forward performance than the rectangular planform. Among three elliptic blades, the blade “0.6R- 1.35c” has the highest L/D. Considering the higher L/D of the blade “0.7R-1.35c” compared with the blade “0.7R-1.20c ” may be brought by the change of rotor solidity, it implies that the rotor eﬃciency of elliptic blades is sensitive to the position of minor axes. Case of μ =0:7 is taken into analysis. Figure 20 shows the pressure distribution on the upper surface of blades 0.4 0.5 0.6 0.7 𝜇 with diﬀerent chord distribution at 90 azimuthal position. Figure 21 shows the spanwise thrust distribution at 90 Rectangular blade 0.90R-20deg azimuthal position. There are obvious diﬀerences in the 0.95R-20deg 0.85R-20deg pressure distributions between the rectangular blade and elliptic blades. The strength of low pressure area of elliptic Figure 16: Lift-to-drag ratio of rotors with diﬀerent swept-back blades is weakened. This phenomenon may be brought by position. the leading edge proﬁle, which is gently swept forward to the center of the ellipse and then gently swept back with the chord length tapering in the blade tip. Elliptic blade pressure region in the midblade (deep blue band in generates more lift in the middle part compared with rect- Figure 17) is narrowed with the increase of swept-back area. angular blade as shown in Figure 21. The position of the The swept-back tip reduces the shock wave drag, so the rotor lift peak moves towards the blade root with the center of eﬃciency is improved as shown in Figure 16. Figure 18 shows the ellipse moving inwards. And the lift peak value the spanwise thrust distribution at 90 azimuthal position. increases with the length of the minor axis (the largest Swept-back tips have lower thrust than the rectangular tip chord length of the blade). When performed in high in the middle blade, and the thrust in the middle blade advance ratio conditions, the blade tip of coaxial rotor decreases with swept-back angle. needs to be unloaded. Therefore, the elliptic shape is in favor of high forward rotor eﬃciency through concentrat- 3.2. Nonlinear Chord Distribution. Three types of blades with TD ing more lift in the middle part of blade. elliptic shape resembling X2 rotor [5] are chosen to simu- Figure 22 shows the streamline on the upper surface of late the eﬀects of chord distribution. The main part of the the upper rotor. In the current state (μ =0:7), severe shock- planform is an ellipse with speciﬁc length and position of induced separation occurs on the blade surface on the the minor axis. The elliptic sections of three blades are cen- L/D P International Journal of Aerospace Engineering 11 C 1 1.19 1.38 1.57 1.76 1.95 2.14 2.33 2.52 2.71 2.9 0.95R 0.90R 0.85R Figure 17: Pressure distribution on the upper surface of the upper rotor (ψ =90 , μ =0:7). 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 r/R r/R Rectangular blade 0.90R-20deg Rectangular blade 0.90R-20deg 0.95R-20deg 0.85R-20deg 0.95R-20deg 0.85R-20deg (a) Upper rotor (b) Lower rotor Figure 18: Span thrust distribution (ψ =90 , μ =0:7). advancing side. Compared with the rectangular blade, elliptic blades narrow the separation area and could reduce shock- induced drag. 4. Blade Geometry Design of Coaxial Rotor 4.1. Optimization Method. Blade geometry optimization is a complex multivariable problem, in which the CFD simula- tion with trimming of coaxial rotors is time consuming and computationally expensive. Therefore, an optimization method combining the surrogate-based approach and genetic algorithm is implemented in blade geometry shape design in this paper. This method is developed on previous work [25]. Firstly, a surrogate-based approach is established 0.4 0.5 0.6 0.7 based on LHS (Latin Hypercube Sampling) method and RBF (Radial Basis Function) technique. Then, it is trans- Rectangular blade 0.7R-1.35c planted in the process of genetic algorithm to evaluate the ﬁt- 0.7R-1.20c 0.6R-1.35c ness (objective function value). Figure 23 shows the full optimization ﬂowchart of the Figure 19: Lift-to-drag ratio of rotors with diﬀerent chord coaxial rigid rotor blade geometry. In this ﬂowchart, the distributions. surrogate-based approach reduces the calculation amount L/D dC /dr dC /dr T 12 International Journal of Aerospace Engineering C 1 1.19 1.38 1.57 1.76 1.95 2.14 2.33 2.52 2.71 2.9 1.20c 0.7R 1.35c 1.35c 0.6R Figure 20: Pressure distribution on the upper surface of the upper rotor(ψ =90 , μ =0:7). 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 r/R r/R Rectangular blade 0.7R-1.35c Rectangular blade 0.7R-1.35c 0.7R-1.20c 0.6R-1.35c 0.7R-1.20c 0.6R-1.35c (a) Upper rotor (b) Lower rotor Figure 21: Span thrust distribution (ψ =90 , μ =0:7). in ﬁtness function evaluation. Selected individuals with higher ﬁtness are simulated using CFD solver and then added to the initial population as new parents. The accuracy of the surrogate-based approach is improved with the addition of new parents predicted by high-accuracy CFD solver. 4.2. Parameterization of Blade Planform. Based on the former geometrical parameter analysis, a baseline planform combin- ing elliptic shape feature and swept-back tapered tip is parameterized as shown in Figure 24. This type of planform could concentrate the blade area on the middle part which is in favor of optimizing the airload distribution, and the swept-back tapered tip could reduce the compressibility of Figure 22: Streamline on the upper surface of the upper rotor the advancing blade tip. There are 8 control points, P1~P8, (ψ =90 , μ =0:7). that deﬁne the blade planform. P7 and P8 are ﬁxed at 0.25R spanwise position. The spline segments between P7 and P1, P8 and P4, and P4 and P5 are deﬁned by cubic functions. dC /dr dC /dr T International Journal of Aerospace Engineering 13 Start Initial points Initial generation population (Number: N) generation Fitness function Construction of approximation model based on RBF with N Selection of sample points parents Crossover CFD numerical & mutation simulation of coaxial rotors Objective and constraint functions NO Converged? YES Excellent sample points Error selection analyses (Number: m) Add m pointsselected NO into initial N points Converged? N = N + m YES Optimized blade Results analyses geometry End Figure 23: Flowchart of coaxial rotor blade geometry optimization procedure. 0.6 P1(x , y ) 1 1 P7 0.0 P2(x , y ) 2 2 –0.6 P3(1.0, y ) P8 P4(x , y ) P5(x , y ) 4 4 2 5 –1.2 P6(1.0, y ) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 x/R Figure 24: Parameterized blade geometry of the coaxial rotor. The spline segment between P1 and P2 is deﬁned by a (1) In the leading edge: parabolic function. The spline segments between P3 and P6 and P2 and P3 are deﬁned by linear functions. Except point P5, every connection point is also the tan- 3 2 > aðÞ x − 0:25 + bðÞ x − 0:25 , 0:25 < x ≤ x , 1 1 1 gent point of two segments it connects. For simpliﬁca- < y = tion, x4 is set as equal to x2, so there are 8 design kðÞ x − x + y , x < x ≤ x , 1 1 1 2 variables, fx1, x2, y1, y2, y3, y4, y5, y6g. kðÞ x − x + y , x < x ≤ 1:0, 2 2 2 Then, the blade geometry is determined by functions as ð9Þ follows: y/c 14 International Journal of Aerospace Engineering Table 3: Airfoil distribution. Spanwise position (/R) Starting point End point 0.2~0.25 DBLN526 OA209 0.25~(x -0.1) OA209 OA209 x ~x OA209 OA206 1 2 x ~1.0 OA206 OA206 where −y a =2 ⋅ , > 8 > x − 0:25 ðÞ > 1 > 0 20 40 60 80 100 > y Sample point > 1 b =3 ⋅ , ðÞ x − 0:25 1st 3rd ð10Þ 2nd 4th > y − y > 2 1 k = , > 1 > ðÞ x − x Figure 25: Objective value distribution of four generations arranged 2 1 from lowest to highest. > y − y > 3 2 k = : 1:0 − x 0:65 < x <0:7, 0:1< y <0:3, 1 1 −0:8< y <0:2, 0:85 < x <0:95, 2 2 ð13Þ −0:8< y < y , − 1:3< y < y − 1:0, 3 2 4 1 (2) In the trailing edge y < y , 0:3< y − y < y − y : 4 5 3 6 2 5 3 2 Coaxial rigid rotor conﬁguration chosen in the optimi- > aðÞ x − 0:25 + bðÞ x − 0:25 − 1:0, 0:25 < x ≤ x , 2 2 4 zation has two 4-bladed rotors, and the rectangular blade 3 2 y = aðÞ x − x + bðÞ x − x + y , x < x ≤ x , 3 2 3 2 4 2 is used for comparison of rotor aerodynamic performance. Both rectangular blade and coaxial rotor blade have a kðÞ x − x + y , x < x ≤ 1:0, 2 2 2 diameter of 5.2 m, and the root chord length is 0.2 m. The initial azimuthal angles of four blades of the upper ð11Þ ° ° ° ° rotor are set at 45 , 135 , 225 , and 315 . For the lower ° ° ° ° rotor, they are set at 0 ,90 , 180 , and 270 . The clearance between two rotors is 0.15R. Airfoil distribution of coaxial where rotor is shown in Table 3. There are three kinds of airfoils employed in the blade as shown in the table. Twist distri- bution has 2 parts, linear twist rate = 14 /m inboard (from −ðÞ y +1:0 > 4 ° a =2 ⋅ , 0.2R to 0.4R) and –8 /m outboard (from 0.4R to blade ðÞ x − 0:25 > 4 tip). Positive twist rate inboard could reduce the adverse eﬀect caused by reverse-ﬂow region. > ðÞ y +1:0 b =3 ⋅ , 2 The design condition is at μ =0:6 (forward velocity is ðÞ x − 0:25 > 4 210 kts), and blade-tip Mach number of rotation is 0.528. −ðÞ y − y Total thrust coeﬃcient is set at 0.013, and the lift oﬀset is 4 5 ð12Þ a =2 ⋅ , > set at 0.35 by the trimming procedure during the optimiza- x − x > ðÞ 4 2 > tion. All 8 design variables fx1, x2, y1, y2, y3, y4, y5, y6g are y − y > ðÞ 4 5 > normalized into v ~v (v ∈ ½0, 1, n =1,2, ⋯8). The objec- 1 8 n > b =3 ⋅ , > 3 x − x tive function is the equivalent lift-drag ratio; therefore, it is > ðÞ 4 2 an optimization problem with eight dimensions and one > y − y > 6 5 : k = : objective. 1:0 − x Design variables : v , v , v , v , v , v , v , v , 1 2 3 4 5 6 7 8 The design variables are constrained as follows to main- ð14Þ tain the rotor solidity and geometrical features of the Objective functions : Maximum : swept-back tapered tip and nonlinear chord distribution: L/D International Journal of Aerospace Engineering 15 0.8 0.4 0.0 –0.4 –0.8 –1.2 –1.6 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x/R Figure 26: Optimized blade geometry. 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.4 0.5 0.6 0.7 0.2 0.4 0.6 0.8 1.0 x/R Figure 27: Chord distribution of the optimized blade (normalized Rectangular (baseline) by the root chord length). Optimized planform Figure 28: The lift-drag ratio of optimized blade and baseline blade. 4.3. Results and Analyses. There are 100 sample points in the initial sampling which have 8 normalized variables. After dynamic characteristics of the upper and lower rotors have optimization, a converged result is obtained. Figure 25 shows little diﬀerence at high advance ratios, so that only the the value of the objective function L/D of each generation numerical simulated results of the upper rotor are pre- that is arranged from the lowest to the highest. The response sented here. It exhibits that the intense shock wave occurs value of the sample point is higher and higher with the devel- opment of optimization, which means the process moves over the outboard rectangular blade in high-speed forward ﬂight. The shock on the optimized blade is clearly weak- along a favorable direction. In the 4th generation, the optimi- ened, and the airload distribution along the spanwise of zation is regarded as converged according to error analyses. The optimized blade shape with a tapered swept-back tip the blade is more reasonable. Figure 30 shows the blade surface pressure contours in 3 and nonlinear chord distributions is shown in Figure 26, typical azimuthal locations, 0, 1/16 Rev, and 1/8 Rev. The sec- and its chord distribution is shown in Figure 27. ond location is where the upper rotor meets the lower rotor. Aerodynamic performance analyses of the optimized There are obvious negative pressure regions on two blades on rotor are carried out. Figure 28 gives lift-drag ratio of the optimized blade compared with the baseline rectangular the advancing side, and the negative pressure region on the surface of the optimized blade has been reduced compared blade. Under the current operating condition (μ =0:6), the with the baseline. This improvement is mainly due to the lift-drag ratio of the optimized rotor increases about 30%, reduction of the velocity component normal to the blade from 8.06 of the baseline rotor to 10.48. Although the optimi- leading edge brought by the tapered swept-back distribution. zation is performed at an advance ratio of 0.6, the forward The large area in the middle blade segment provides most of ﬂight aerodynamic performance is improved in a wide range the total lift, and the blade tip is oﬄoaded. This also explains of advance ratios. Moreover, the eﬀective-equivalent lift-drag the weakness phenomenon of shock wave in the outboard ratio of the coaxial rotor with optimized shape and single part of the blade as shown in Figure 29. NACA0012 airfoil distribution is 9.53. This result manifests Although the noise properties are not considered in the the advantage of the optimized planform. optimization design, the acoustic characteristics of the base- Figure 29 gives the streamline distribution of the upper line rotor and optimized rotor are also calculated. Figure 31 rotor over the blade surface on the advancing side. Aero- y/c y/c L/D 16 International Journal of Aerospace Engineering (a) Baseline rotor blade (b) Optimized rotor blade Figure 29: Streamline distributions of the baseline upper rotor and the optimized upper rotor (ψ =90 ). C 1.2 1.37 1.54 1.71 1.88 2.05 2.22 2.39 2.56 2.73 2.9 Time = 0 Time = 1/16Rev Time = 1/8Rev (a) Baseline rotor blade C 1.2 1.37 1.54 1.71 1.88 2.05 2.22 2.39 2.56 2.73 2.9 Time = 0 Time = 1/16Rev Time = 1/8Rev 𝛹 = 90° (b) Optimized rotor blade Figure 30: Blade surface pressure contour distribution of baseline rotor and optimized rotor. shows the sound pressure history ahead of the rotor plane 5. Conclusions and noise directivity patterns in the rotor plane. Severe HSI (high-speed impulsive) noise occurs on the coaxial rotor in A coaxial rotor solver developed on CLORNS was applied to high-speed forward ﬂight, and the rotor noise has a strong simulate the aerodynamics of the coaxial rotor in forward radiation directivity pattern mostly forward the rotor plane. ﬂight. The ﬂow-ﬁeld features of the lift oﬀset coaxial rotor It can be seen from the ﬁgure that the value of negative at high advance ratio were analyzed. Inﬂuences of two geo- sound pressure peak is decreased about 50%, and the metrical features—swept-back tip and elliptic chord distribu- sound pressure level is about 7 dB lower. Therefore, the tion—on rotor performance were studied. Then, a baseline optimized blade geometry also contributes to improving planform combining swept-back tapered tip and nonlinear noise characteristics. chord distribution was put forward and used in blade International Journal of Aerospace Engineering 17 300 90 120 60 150 30 –100 180 0 –200 –300 210 330 –400 0 90 180 270 360 240 300 Azimuthal angle (deg) Baseline Baseline Optimized Optimized (a) Time history forward the rotor plane (b) Directivity patterns in the rotor disc Figure 31: Acoustic characteristics of baseline rectangular blade and optimized blade. ﬁeld of hover and low advance ratio are more severe planform optimization. The optimization method combines surrogate-based approach and GA with acceptable computa- than high-speed forward ﬂights. Further geometry tion cost and accuracy. Performance analyses of the opti- design of coaxial rotor should take hover and low- mized coaxial rotor show an obvious improvement on rotor speed performance into consideration eﬃciency in forward ﬂight. There are some conclusions that The objective of this investigation was to understand the can be drawn from the present researches: geometrical parameter inﬂuences on the forward perfor- mance of the coaxial rotor. And the optimization focused (1) The coaxial rotor solver developed on CLORNS pro- on planform design. In the further research, the 2D airfoil vides reasonable aerodynamic performance predic- section design will be coupled with the 3D blade geometry tions. At high advance ratio, the advancing blade tip design in order to obtain a better coaxial rotor. Airfoil conﬁg- suﬀers strong compressibility and the retreating uration and twist distribution also will be taken into consid- blade root suﬀers severe reverse ﬂow, which brings erations when designing the blade geometry of coaxial rotors. adverse eﬀects on rotor eﬃciency (2) Swept-back tip mainly inﬂuences the aerodynamic Nomenclature characteristics of the blade tip region on the advanc- ing side by reducing the compressibility. Compared c: Blade chord with the rectangular shape, the elliptic chord distri- C : Drag coeﬃcient bution of the blade planform has smaller area in the C : Pressure coeﬃcient root and tip, which decreases the drag generated by C , C : Rolling/pitching moment coeﬃcient Mx Mz the blade root in reverse ﬂow and blade tip on the C : Torque coeﬃcient advancing side. Larger area is placed in the middle C : Lift coeﬃcient blade, which beneﬁts a favorable airload distribution L/D: Lift-to-drag ratio LOS: Lift oﬀset (%) (3) The combination of surrogate-based approach and GA Ma: Mach number makes the optimization cost less computation than only R: Blade radius GA and has a higher accuracy than only surrogate- V : Blade tip speed tip based approach. The eﬃciency of the optimized rotor θ: Pitch angle (deg) is signiﬁcantly increased compared with the rectangular θ , θ , θ : Collective/longitudinal cyclic/lateral cyclic pitch blade. The optimized geometry also contributes to 0 1c 1s angle (deg) improving the noise characteristics, and high-speed μ: Advance ratio impulsive noise in high-speed forward ﬂight is ψ: Azimuth angel decreased compared with the rectangular rotor U: Upper rotor (4) The optimized blade is the optimal solution under L: Lower rotor. current constraints, which is focused on high-speed performance. 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International Journal of Aerospace Engineering – Hindawi Publishing Corporation
Published: Dec 7, 2020
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