With the rapid advancement of green aviation technologies, electrically driven ducted fans have seen widespread adoption in electric vertical takeoff and landing aircraft due to their high hover efficiency and low noise. However, uncertainties in manufacturing process and operation parameters can significantly affect thrust performance and introduce operational risks. Stochastic uncertain parameter and interval uncertain parameter are considered in this paper. The Polynomial Chaos Expansion integrated with the Legendre inclusion function is employed to analyze the aerodynamic performance of a ducted fan under hybrid uncertainty. The tip clearance is treated as a stochastic uncertain parameter, modeled as a normal distribution N (3.5, 0.122). The rotational speed is considered an interval uncertain parameter within the range of [2350, 2450] rpm. The hybrid uncertainty analysis shows that the lower and upper bound mean of the total thrust is [92.71 N, 100.96 N], and the lower and upper bound standard deviation is [0.666 N, 0.745 N], demonstrating noticeable performance deviation under hybrid uncertainty. In addition, statistical characteristics of the lower and upper bounds of total thrust are presented. The bound relative variance (BRV) of the overall performance is 5%, whereas the BRV of the ducted thrust increased to about 8% due to pressure variations near the lip region. Additionally, a hybrid uncertainty analysis for cruise condition has also been conducted.
RostamiMFarajollahiAH. Aerodynamic performance of mutual interaction tandem propellers with ducted UAV. Aero Sci Technol2021; 108: 106399. https://doi.org/10.1016/j.ast.2020.106399
3.
RaeisiB. Aerodynamic study of tilting asymmetrical ducted fans mounted at the wing tips of a vtol uav[D]. Ryerson University, 2016.
4.
ZengCChenHYuZ, et al.Study on aerodynamic characteristics of a vertical take-off and landing UAV with tilting ducted propeller[C]. In: International conference on advanced unmanned aerial systems. Springer Nature Singapore, 2024, pp. 310–321.
5.
LiuLWangTGaoZ, et al.Deep learning based adaptive deformation of aerodynamic shape for ducted propellers. Aero Sci Technol2023; 142: 108607. https://doi.org/10.1016/j.ast.2023.108607
6.
Meher-HomjiCBChakerMA. Motiwala HM. Gas turbine performance deterioration. In: College station: turbomachinery laboratories. Texas A&M University, 2001, pp. 139–175.
7.
GuoZChuWZhangH, et al.Statistical evaluation of stability margin of a multi-stage compressor with geometric variability using adaptive polynomial chaos-kriging model. Phys Fluids2023; 35(7): 076114. https://doi.org/10.1063/5.0158821
8.
Anusonti-InthraPCorleESmithB, et al.Sensitivity analysis and uncertainty quantification of a coaxial rotor system[C]. AIAA Aviation 2019 Forum, 2019, p. 3476.
WenYChenXLuoW, et al.Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles. Prog Aero Sci2011; 47(6): 450–479.
11.
BabaeiARSetayandehMRFarrokhfalH. Aircraft robust multidisciplinary design optimization methodology based on fuzzy preference function. Chin J Aeronaut2018; 31(12): 2248–2259. https://doi.org/10.1016/j.cja.2018.04.018
12.
HurtadoJEBarbatAH. Monte carlo techniques in computational stochastic mechanics. Arch Comput Methods Eng1998; 5: 3–29. https://doi.org/10.1007/bf02736747
13.
LangAVoigtMVogelerK, et al.Impact of ManuFacturing variability on multi-stage high-pressure com⁃ pressor performance. J Eng Gas Turbines Power2012; 134(11): 1.
HeXZhaoFVahdatiM. Uncertainty quantification of spalart–allmaras turbulence model coefficients for compressor stall. J Turbomach2021; 143(8): 081007. https://doi.org/10.1115/1.4050438
16.
SongZWuYLiuX, et al.Uncertainty quantification based on active subspace dimensionality-reduction method for high-dimensional geometric deviations of compressors. Phys Fluids2024; 36(10): 106104. https://doi.org/10.1063/5.0221789
17.
WunschDHirschCNigroR, et al.Quantification of combined operational and geometrical uncertainties in turbo-machinery design[C]//Turbo expo: power for land, sea and Air. American Soc Mech Eng2015; 56659: V02CT45A018. https://doi.org/10.1115/gt2015-43399
18.
MooreRE. Methods and applications of interval analysis[M]. Society for Industrial and Applied Mathematics, 1979.
19.
FengXZhangYWuJ. Interval analysis method based on legendre polynomial approximation for uncertain multibody systems. Adv Eng Software2018; 121: 223–234. https://doi.org/10.1016/j.advengsoft.2018.04.002
WeiTLiFMengG. A bivariate chebyshev polynomials method for nonlinear dynamic systems with interval uncertainties. Nonlinear Dyn2022; 107: 793–811. https://doi.org/10.1007/s11071-021-07020-y
22.
QiuZPWangPB. Parameter vertex method and its parallel solution for evaluating the dynamic response bounds of structures with interval parameters, science China physics. Mech & Astron2018; 61: 1–13. https://doi.org/10.1007/s11433-017-9164-6
23.
HanWWangZShenY, et al.Interval estimation for uncertain systems via polynomial chaos expansions. IEEE Trans Automat Control2020; 66(1): 468–475. https://doi.org/10.1109/TAC.2020.2982907
24.
WuJLuoZZhangY, et al.Interval uncertain method for multibody mechanical systems using chebyshev inclusion functions. Int J Numer Methods Eng2013; 95(7): 608–630. https://doi.org/10.1002/nme.4525
25.
WangLChenZYangG. A polynomial chaos expansion approach for nonlinear dynamic systems with interval uncertainty. Nonlinear Dyn2020; 101(4): 2489–2508. https://doi.org/10.1007/s11071-020-05895-x
26.
JiangCZhengJHanX. Probability-interval hybrid uncertainty analysis for structures with both aleatory and epistemic uncertainties: a review. Struct Multidiscip Optim2018; 57(6): 2485–2502. https://doi.org/10.1007/s00158-017-1864-4
27.
CacciolaPMuscolinoGVersaciC. Deterministic and stochastic seismic analysis of buildings with uncertain-but-bounded mass distribution. Comput Struct2011; 89: 2028–2036. https://doi.org/10.1016/j.compstruc.2011.05.017
28.
DoDMGaoKYangW, et al.Hybrid uncertainty analysis of functionally graded plates via multiple-impreciserandom-field modelling of uncertain material properties. Comput Methods Appl Mech Eng2020; 368: 113116. https://doi.org/10.1016/j.cma.2020.113116
29.
ZhuHXiaoMZhangJ, et al.Uncertainty design and optimization of a hybrid rocket motor with mixed random-interval uncertainties. Aero Sci Technol2022; 128: 107791. https://doi.org/10.1016/j.ast.2022.107791
30.
XiaBYuDLiuJ. Hybrid uncertain analysis of acoustic field with interval random parameters. Comput Methods Appl Mech Eng2013; 256: 56–69. https://doi.org/10.1016/j.cma.2012.12.016
31.
XiaBYuDLiuJ. Hybrid uncertain analysis for structural–acoustic problem with random and interval parameters. J Sound Vib2013; 332: 2701–2720. https://doi.org/10.1016/j.jsv.2012.12.028
32.
YinSYuDYinH, et al.Interval and random analysis for structure-acoustic systems with large uncertain-but-bounded parameters. Comput Methods Appl Mech Eng2016; 305: 910–935. https://doi.org/10.1016/j.cma.2016.03.034
33.
WuJLuoZZhangN, et al.A new uncertain analysis method and its application in vehicle dynamics. Mech Syst Signal Process2015; 50: 659–675. https://doi.org/10.1016/j.ymssp.2014.05.036
34.
WangLGuoCXuF, et al.Hybrid uncertainty propagation for mechanical dynamics problems via polynomial chaos expansion and legendre interval inclusion function. Mech Syst Signal Process2025; 223: 111826. https://doi.org/10.1016/j.ymssp.2024.111826
35.
MaBZhuXDuZ. Hybrid uncertainty analysis of the nozzle guide vane surface temperature in hydrogen-fueled gas turbine. Int J Engine Res2025; 14: 14680874251378611. https://doi.org/10.1177/14680874251378611
36.
QuXZYuJHYaoQS. Random-interval hybrid reliability analysis for fan performance in the presence of epistemic uncertainty. Scientia Sinica Technologica2020; 50(3): 299–311.
37.
JavedAPecnikRBuijtenenJP. Optimization of a centrifugal compressor impeller design for robustness to manufacturing uncertainties. J Eng Gas Turbines Power2016; 138(11): 1–11.
38.
Manohara SelvanKKowalczykL. Exploring the impact of manufacturing geometric uncertainties on the aerodynamic performance of a small scale compressor[C]. In: Turbo expo: power for land, sea, and air. American Society of Mechanical Engineers, 2018, p. 51029.
39.
JiTYChuWLGuoZT. Uncertainty quantification and sensitivity analysis of blade geometric deviation on compressor performance. Aeronaut J2024; 128: 1–19. https://doi.org/10.1017/aer.2024.65
40.
MaYLiangZChenM, et al.Interval analysis of rotor dynamic response with uncertain parameters. J Sound Vib2013; 332(16): 3869–3880. https://doi.org/10.1016/j.jsv.2013.03.001
41.
DaróczyLJanigaGThéveninD. Analysis of the performance of a H-Darrieus rotor under uncertainty using polynomial chaos expansion. Energy2016; 113: 399–412. https://doi.org/10.1016/j.energy.2016.07.001
42.
AkturkACamciC. Tip clearance investigation of a ducted fan used in vtol unmanned aerial vehicles—part I: baseline experiments and computational validation. J Turbomach2014; 136(2): 021004. https://doi.org/10.1115/1.4023468
SeoHSKwakBM. Efficient statistical tolerance analysis for general distributions using three-point information. Int J Prod Res2002; 40: 931–944. https://doi.org/10.1080/00207540110095709
45.
IsukapalliSS. Uncertainty analysis of transport-transformation models. The State University of New Jersey, 1999.
46.
ChuWLHeXDYangJB, et al.Effects of blade single and coupling errors on axial flow compressor performance. J Aero Power2023; 1: 1–14.
47.
LeishmanGJ. Principles of helicopter aerodynamics[M]. Cambridge University Press, 2006.
48.
TangXYuanKGuN, et al.An interval quantification-based optimization approach for wind turbine airfoil under uncertainties. Energy2022; 244: 122623. https://doi.org/10.1016/j.energy.2021.122623
49.
JiangCHanXLiuGR. Optimization of structures with uncertain constraints based on convex model and satisfaction degree of interval. Comput Methods Appl Mech Eng2007; 196(49–52): 4791–4800. https://doi.org/10.1016/j.cma.2007.03.024
