Review of vortex methods for rotor aerodynamics and wake dynamics

Hakjin Lee1, Bidesh Sengupta1, M. S. Araghizadeh1, R.S. Myong1
1School of Mechanical and Aerospace Engineering, Gyeongsang National University, Jinju, Gyeongnam, 52828, South Korea

Tóm tắt

AbstractElectric vertical take-off and landing (eVTOL) aircraft with multiple lifting rotors or prop-rotors have received significant attention in recent years due to their great potential for next-generation urban air mobility (UAM). Numerical models have been developed and validated as predictive tools to analyze rotor aerodynamics and wake dynamics. Among various numerical approaches, the vortex method is one of the most suitable because it can provide accurate solutions with an affordable computational cost and can represent vorticity fields downstream without numerical dissipation error. This paper presents a brief review of the progress of vortex methods, along with their principles, advantages, and shortcomings. Applications of the vortex methods for modeling the rotor aerodynamics and wake dynamics are also described. However, the vortex methods suffer from the problem that it cannot deal with the nonlinear aerodynamic characteristics associated with the viscous effects and the flow behaviors in the post-stall regime. To overcome the intrinsic drawbacks of the vortex methods, recent progress in a numerical method proposed by the authors is introduced, and model validation against experimental data is discussed in detail. The validation works show that nonlinear vortex lattice method (NVLM) coupled with vortex particle method (VPM) can predict the unsteady aerodynamic forces and complex evolution of the rotor wake.

Từ khóa


Tài liệu tham khảo

Castles W Jr, Gray RB (1951) Empirical relation between induced velocity, thrust, and rate of descent of a helicopter rotor as determined by wind-tunnel tests on four model rotors. NACA Technical Note NACA-TN-2474

Wilson JC, Mineck RE (1974) Wind tunnel investigation of helicopter rotor wake effects on three helicopter fuselage models. NASA Technical Memorandum NASA-TM-X-3185-SUPPL

Landgrebe AJ (1971) An analytical and experimental investigation of helicopter rotor hover performance and wake geometry characteristics. AD0728835

McCroskey WJ, Fisher RK (1972) Detailed aerodynamic measurements on a model rotor in the blade stall regime. J Am Helicopter Soc 17(1):20–30. https://doi.org/10.4050/JAHS.17.1.20

Johnson B, Leishman JG, Sydney A (2010) Investigation of sediment entrainment using dual-phase, high-speed particle image velocimetry. J Am Helicopter Soc 55(4):42003. https://doi.org/10.4050/JAHS.55.042003

Crozier P, Leconte P, Delrieux Y, Gimonet B, Pape AL, des Rochettes HM (2006) Wind-tunnel tests of a helicopter rotor with active flaps. In: 32nd European Rotorcraft Forum, Maastricht, the Netherlands, 12–14 September 2006.

van der Wall BG, Burley CL, Yu Y, Richard H, Pengel K, Beaumier P (2004) The HART II test – measurement of helicopter rotor wakes. Aerosp Sci Technol 8(4):273–284. https://doi.org/10.1016/j.ast.2004.01.001

Datta A, Yeo H, Norman TR (2013) Experimental investigation and fundamental understanding of a full-scale slowed rotor at high advance ratios. J Am Helicopter Soc 58(2):1–17. https://doi.org/10.4050/JAHS.58.022004

Komerath NM, Smith MJ, Tung C (2011) A review of rotor wake physics and modeling. J Am Helicopter Soc 56(2):22006. https://doi.org/10.4050/JAHS.56.022006

Beaumier P (2018) Rotorcraft experimental databases: future needs in the fields of aeromechanics and aeroacoustics. In: 7th Asian/Australian Rotorcraft Forum, Jeju Island, South Korea, 30 October - 1 November 2018

Desopper A, Lafon P, Ceroni P, Philippe JJ (1989) Ten years of rotor flow studies at ONERA. J Am helicopter Soc 34(1):34–41. https://doi.org/10.4050/JAHS.34.34

Strawn RC, Barth TJ (1993) A finite-volume Euler solver for computing rotary-wing aerodynamics on unstructured meshes. J Am Helicopter Soc 38(2):61–67. https://doi.org/10.4050/JAHS.38.61

Srinivasan GR, Baeder JD, Obayashi S, McCroskey WJ (1992) Flowfield of a lifting rotor in hover - A Navier-Stokes simulation. AIAA J 30(10):2371–2378. https://doi.org/10.2514/3.11236

Duque EPN (1992) A numerical analysis of the British experimental rotor program blade. J Am Helicopter Soc 37(1):46–54. https://doi.org/10.4050/JAHS.37.46

Chen H, Kandasamy S, Orszag S, Shock R, Succi S, Yakhot V (2003) Extended Boltzmann kinetic equation for turbulent flows. Science 301(5633):633–636. https://doi.org/10.1126/science.1085048

Narducci R (2015) Hover performance assessment of several tip shapes using OVERFLOW. In: 53rd AIAA Aerospace Sciences Meeting. Kissimmee, Florida, USA, 5–9 January 2015

Chaderjian NM (2017) Navier-Stokes simulation of UH-60A rotor/wake interaction using adaptive mesh refinement. In: Proceedings of the AHS International 73rd Annual Forum of the American Helicopter Society. Fort Worth, Texas, USA, 9–11 May 2017.

Pang C, Yang H, Gao Z, Chen S (2021) Enhanced adaptive mesh refinement method using advanced vortex identification sensors in wake flow. Aerosp Sci Technol 115:106796. https://doi.org/10.1016/j.ast.2021.106796

Wilbur IC, Moushegian A, Smith MJ, Whitehouse GR (2020) UH-60A rotor analysis with an accurate dual-formulation hybrid aeroelastic methodology. J Aircr 57(1):113–127. https://doi.org/10.2514/1.C035467

Yoon S, Diaz PV, Boyd Jr DD, Chan WM, Theodore CR (2017) Computational aerodynamic modeling of small quadcopter vehicles. In: Proceedings of the AHS International 73rd Annual Forum of the American Helicopter Society. Fort Worth, Texas, USA, 9–11 May 2017.

Leishman JG, Ananthan S (2006) Aerodynamic optimization of a coaxial proprotor. In: Proceedings of the AHS International 62nd Annual Forum and Technology Display, Phoenix, AZ, USA, 9–11 May 2006

Wie SY, Lee S, Lee DJ (2009) Potential panel and time-marching free-wake coupling analysis for helicopter rotor. J Aircr 46(3):1030–1041. https://doi.org/10.2514/1.40001

Gennaretti M, Bernardini G, Serafini J, Romani G (2018) Rotorcraft comprehensive code assessment for blade–vortex interaction conditions. Aerosp Sci Technol 80:232–246. https://doi.org/10.1016/j.ast.2018.07.013

Tugnoli M, Montagnani D, Syal M, Droandi G, Zanotti A (2021) Mid-fidelity approach to aerodynamic simulations of unconventional VTOL aircraft configurations. Aerosp Sci Technol 115:106804. https://doi.org/10.1016/j.ast.2021.106804

Leishman JG (2006) Principles of helicopter aerodynamics. Cambridge University Press, Cambridge

Johnson W (2013) Rotorcraft aeromechanics. Cambridge University Press, Cambridge

Landgrebe AJ, Moffitt RC, Clark DR (1977) Aerodynamic technology for advanced rotorcraft-part I. J Am Helicopter Soc 22(2):21–27. https://doi.org/10.4050/JAHS.22.21

Landgrebe AJ, Moffitt RC, Clark DR (1977) Aerodynamic technology for advanced rotorcraft-part II. J Am Helicopter Soc 22(3):2–9. https://doi.org/10.4050/JAHS.22.3.2

DeYoung J (1976) Historical evolution of vortex-lattice methods. NASA Langley Res Cent Vor Util N76–28164

Katz J, Maskew B (1988) Unsteady low-speed aerodynamic model for complete aircraft configurations. J Aircr 25(4):302–310. https://doi.org/10.2514/3.45564

Wachspress DA, Yu MK (2015) Lifting surface blade model for comprehensive rotorcraft analysis. In: Proceedings of the AHS International 71st Annual Forum and Technology Display, Virginia Beach, VA, USA, 5–7 May 2015

Rubbert PE, Saaris GR (1968) A general three-dimensional potential-flow method applied to V/STOL aerodynamics. SAE Trans 77:945–957. https://doi.org/10.4271/680304

Hess JL (1972) Calculation of potential flow about arbitrary three-dimensional lifting bodies. AD0755480.

Crispin Y (1982) Unsteady rotor aerodynamics using a vortex panel method. In: 9th Atmospheric Flight Mechanics Conference, San Diego, CA, USA, 9–11 August 1982.

Wachspress DA, Quackenbush TR, Boschitsch AH (2003) Rotorcraft interactional aerodynamics with fast vortex/fast panel methods. J Am Helicopter Soc 48(4):223–235. https://doi.org/10.4050/JAHS.48.223

Rajmohan N, He C (2016) A VPM/CFD coupling methodology to study rotor/ship aerodynamic interaction. In: AIAA Modeling and Simulation Technologies Conference, San Diego, California, USA, 4–8 January 2016

Zhao J, He C, Zhang L, Zhao H, Hu P (2011) Coupled viscous vortex particle method and unstructured computational fluid dynamics solver for rotorcraft aerodynamic interaction analysis. In: 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Orlando, Florida, USA, 4–7 January 2011.

Bae ES, He C (2017) On high fidelity modeling of aerodynamic interaction between ship and rotor. In: 35th AIAA Applied Aerodynamics Conference, Denver, Colorado, USA, 5–9 June 2017

Bhagwat MJ, Leishman JG (2001) Stability, consistency and convergence of time-marching free-vortex rotor wake algorithms. J Am Helicopter Soc 46(1):59–71. https://doi.org/10.4050/JAHS.46.59

Lee D-J, Na SU (1994) Predictions of helicopter wake geometry and air loadings by using a time marching free wake method. In: Proceedings of the 1st Forum Russian Helicopter Society, Moscow

Jang JS, Park SH, Lee DJ (2014) Prediction of fuselage surface pressures in rotor–fuselage interactions using an integral solution of Poisson equation. J Am Helicopter Soc 59(4):1–11. https://doi.org/10.4050/JAHS.59.042001

He C, Zhao J (2009) Modeling rotor wake dynamics with viscous vortex particle method. AIAA J 47(4):902–915. https://doi.org/10.2514/1.36466

Winckelmans GS, Leonard A (1993) Contributions to vortex particle methods for the computation of three-dimensional incompressible unsteady flows. J Comput Phys 109(2):247–273. https://doi.org/10.1006/jcph.1993.1216

Singh P, Friedmann PP (2021) Dynamic stall modeling using viscous vortex particle method for coaxial rotors. J Am Helicopter Soc 66(1):1–16. https://doi.org/10.4050/JAHS.66.012010

Cao Y, Lv S, Li G (2014) A coupled free-wake/panel method for rotor/fuselage/empennage aerodynamic interaction and helicopter trims. Proc Inst Mech Eng Part G J Aerosp Eng 229(3):435–444. https://doi.org/10.1177/0954410014534203

Tan JF, Sun YM, Barakos GN (2018) Unsteady loads for coaxial rotors in forward flight computed using a vortex particle method. Aeronaut J 122(1251):693–714. https://doi.org/10.1017/aer.2018.8

Tan JF, Cai JG, Barakos GN, Wang C, Huang MQ (2020) Computational study on the aerodynamic interference between tandem rotors and nearby obstacles. J Aircr 57(3):456–468. https://doi.org/10.2514/1.C035629

Tan JF, Sun YM, Barakos GN (2018) Vortex approach for downwash and outwash of tandem rotors in ground effect. J Aircr 55(6):2491–2509. https://doi.org/10.2514/1.C034740

Tan JF, Zhou TY, Sun YM, Barakos GN (2019) Numerical investigation of the aerodynamic interaction between a tiltrotor and a tandem rotor during shipboard operations. Aerosp Sci Technol 87:62–72. https://doi.org/10.1016/j.ast.2019.02.005

Lee J, Chae S, Oh S, Yee K (2010) Parametric study for hovering performance of a coaxial rotor unmanned aerial vehicle. J Aircr 47(5):1517–1530. https://doi.org/10.2514/1.46460

Brocklehurst A, Barakos GN (2013) A review of helicopter rotor blade tip shapes. Prog Aerosp Sci 56:35–74. https://doi.org/10.1016/j.paerosci.2012.06.003

Colmenares JD, López OD, Preidikman S (2015) Computational study of a transverse rotor aircraft in hover using the unsteady vortex lattice method. Math Probl Eng 2015:478457. https://doi.org/10.1155/2015/478457

Chung KH, Kim JW, Ryu KW et al (2006) Sound generation and radiation from rotor tip-vortex pairing phenomenon. AIAA J 44(6):1181–1187. https://doi.org/10.2514/1.22548

Saetti U, Horn JF, Brentner KS, Villafana W, Wachspress D (2016) Rotorcraft simulations with coupled flight dynamics, free wake, and acoustics. In: Proceedings of the AHS International 72nd Annual Forum and Technology Display, West Palm Beach, Florida, USA, 17–19 May 2016.

Lee H, Lee D-J (2020) Rotor interactional effects on aerodynamic and noise characteristics of a small multirotor unmanned aerial vehicle. Phys Fluids 32:47107. https://doi.org/10.1063/5.0003992

Kwon OJ, Hodges DH, Sankar LN (1991) Stability of hingeless rotors in hover using three-dimensional unsteady aerodynamics. J Am Helicopter Soc 36(2):21–31. https://doi.org/10.4050/JAHS.36.21

Yoo KM, Hodges DH, Peters DA (1992) An interactive numerical procedure for rotor aeroelastic stability analysis using elastic lifting surface. In: 18th ICAS Conference, Beijing, China, 20–25 September 1992.

Roura M, Cuerva A, Sanz-Andrés A, Barrero-Gil A (2010) A panel method free-wake code for aeroelastic rotor predictions. Wind Energy 13(4):357–371. https://doi.org/10.1002/we.358

Prandtl L (1921) Applications of modern hydrodynamics to aeronautics. NACA Technical Report NACA-TR-116

Katz J, Plotkin A (2001) Low-speed aerodynamics. Cambridge University Press, Cambridge

Glauert H (1947) The elements of aerofoil theory. Cambridge University Press, Cambridge

Goldstein S (1929) On the vortex theory of screw propellers. Proc R Soc Lond A 123(792):440–465. https://doi.org/10.1098/rspa.1929.0078

Lerbs H (1952) Moderately loaded propellers with a finite number of blades and an arbitrary distribution of circulations. Trans SNAME 60:73–123

Kawada S (1933) On the induced velocity and characteristics of a propeller. J Eng 20:147–162

Conlisk AT (2001) Modern helicopter rotor aerodynamics. Prog Aerosp Sci 37:419–476. https://doi.org/10.1016/S0376-0421(01)00011-2

Melo DB, Baltazar J, de Campos JACF (2018) A numerical wake alignment method for horizontal axis wind turbines with the lifting line theory. J Wind Eng Ind Aerodyn 174:382–390. https://doi.org/10.1016/j.jweia.2018.01.028

Kerwin JE, Lee C-S (1978) Prediction of steady and unsteady marine propeller performance by numerical lifting-surface theory. In: SNAME Annual Meeting. Society of Naval Architects and Marine Engineers, Jersey City, NJ, USA, 16–18 November 1978.

José AC, de Campos F (2007) Hydrodynamic power optimization of a horizontal axis marine current turbine with lifting line theory. In: 17th International Offshore and Polar Engineering Conference, Lisbon, Portugal, 1–6 July 2007.

Miller RH (1985) Methods for rotor aerodynamic and dynamic analysis. Prog Aerosp Sci 22:113–160. https://doi.org/10.1016/0376-0421(85)90008-9

Jones HE, Kunz DL (2001) Comprehensive modeling of the Apache with CAMRAD II. In: American Helicopter Society Structure Specialists Meeting, Williamsburg, VA, USA, 1 January 2001

Kunz DL, Jones HE (2001) Modeling and simulation of the Apache rotor system in CAMRAD II. In: American Helicopter Society Structure Specialists Meeting, Williamsburg, VA, USA, 1 January 2001

Yeo H, Saberi H (2021) Tiltrotor conversion maneuver analysis with RCAS. J Am Helicopter Soc 66(4):1–14. https://doi.org/10.4050/JAHS.66.042010

Yeo H, Bosworth J, Acree CW Jr, Kreshock AR (2018) Comparison of CAMRAD II and RCAS predictions of tiltrotor aeroelastic stability. J Am Helicopter Soc 63(2):1–13. https://doi.org/10.4050/JAHS.63.022001

Jain RK, Yeo H, Ho JC, Bhagwat M (2016) An assessment of RCAS performance prediction for conventional and advanced rotor configurations. J Am Helicopter Soc 61(4):1–12. https://doi.org/10.4050/JAHS.61.042005

Ho JC, Yeo H, Bhagwat M (2017) Validation of rotorcraft comprehensive analysis performance predictions for coaxial rotors in hover. J Am Helicopter Soc 62(2):1–13. https://doi.org/10.4050/JAHS.62.022005

Wachspress DA, Quackenbush TR (2006) Impact of rotor design on coaxial rotor performance, wake geometry and noise. In: Proceedings of the AHS International 62nd Annual Forum and Technology Display, Phoenix, AZ, USA, 9–11 May 2006

Moodie AM, Yeo H (2012) Design of a cruise-efficient compound helicopter. J Am Helicopter Soc 57(3):1–11. https://doi.org/10.4050/JAHS.57.032004

Guermond J-L (1990) A generalized lifting-line theory for curved and swept wings. J Fluid Mech 211:497–513. https://doi.org/10.1017/S0022112090001665

Phillips WF, Snyder DO (2000) Modern adaptation of Prandtl’s classic lifting-line theory. J Aircr 37(4):662–670. https://doi.org/10.2514/2.2649

Phlips PJ, East RA, Pratt NH (1981) An unsteady lifting line theory of flapping wings with application to the forward flight of birds. J Fluid Mech 112:97–125. https://doi.org/10.1017/S0022112081000311

Ahmadi AR, Widnall SE (1985) Unsteady lifting-line theory as a singular perturbation problem. J Fluid Mech 153:59–81. https://doi.org/10.1017/S0022112085001148

Sclavounos PD (1987) An unsteady lifting-line theory. J Eng Math 21:201–226. https://doi.org/10.1007/BF00127464

Lee H (2019) Development of nonlinear vortex lattice method for predicting wind turbine performance and wake structures. Korea Advanced Institute of Science and Technology

Tulinius J (1972) Unified subsonic, transonic, and supersonic NAR vortex lattice. TFD-72–523 North American Rockwell Los Angeles

Joseph C, Mohan R (2021) A parallel, object-oriented framework for unsteady free-wake analysis of multi-rotor/wing systems. Comput Fluids 215:104788. https://doi.org/10.1016/j.compfluid.2020.104788

Wachspress DA, Yu MK, Brentner KS (2019) Rotor/airframe aeroacoustic prediction for EVTOL UAM aircraft. In: Vertical Flight Society’s 75th Annual Forum and Technology Display, Philadelphia, PA, USA, 13–16 May 2019.

Govdeli Y, Muzaffar SMB, Raj R, Elhadidi B, Kayacan E (2019) Unsteady aerodynamic modeling and control of pusher and tilt-rotor quadplane configurations. Aerosp Sci Technol 94:105421. https://doi.org/10.1016/j.ast.2019.105421

Cho MH, Lee I (1995) Aeroelastic analysis of multibladed hingeless rotors in hover. AIAA J 33(12):2348–2353. https://doi.org/10.2514/3.12990

Lee J-W, Oh S-J, Yee K-J et al (2007) Loose coupling approach of CFD with a free-wake panel method for rotorcraft applications. Int J Aeronaut Sp Sci 8:1–9. https://doi.org/10.5139/IJASS.2007.8.1.001

Lee J, Yee K, Oh S (2009) Aerodynamic characteristic analysis of multi-rotors using a modified free-wake method. Trans Jpn Soc Aeronaut Space Sci 52(177):168–179. https://doi.org/10.2322/tjsass.52.168

Zhu W, Morandini M, Li S (2021) Viscous vortex particle method coupling with computational structural dynamics for rotor comprehensive analysis. Appl Sci 11(7):3149. https://doi.org/10.3390/app11073149

Ballmann J, Eppler R, Hackbusch W (1987) Panel methods in fluid mechanics with emphasis on aerodynamics. In: Proceedings of the 3rd GAMM-seminar, Kiel, 16–18 January 1987

Jun S, Yee K, Lee J, Lee D-H (2011) Robust design optimization of unmanned aerial vehicle coaxial rotor considering operational uncertainty. J Aircr 48(2):353–367. https://doi.org/10.2514/1.C001016

Morino L, Kuo C-C (1974) Subsonic potential aerodynamics for complex configurations: a general theory. AIAA J 12(2):191–197. https://doi.org/10.2514/3.49191

Quackenbush TR, Wachspress DA, Boschitsch AH, Curbishley TB (1999) A comprehensive hierarchical aeromechanics rotorcraft model (CHARM) for general rotor/surface interaction. Princeton, NJ Continuum Dynamics, Inc

Zhang C, Quackenbush TR, Saberi H, Sheng C, Gaffey T (2015) Aeromechanics of the coaxial compound helicopter. In: 56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Kissimmee, Florida, USA, 5–9 January 2015

Wachspress DA, Quackenbush TR (2001) BVI noise prediction using a comprehensive rotorcraft analysis. In: Proceedings of the American Helicopter Society 57th Annual Forum, Washington D.C., USA, 9–11 May 2001

Johnson W (2012) A history of rotorcraft comprehensive analysis. NASA/TP-2012–216012

Yang Z, Sankar LN, Smith MJ, Bauchau O (2002) Recent improvements to a hybrid method for rotors in forward flight. J Aircr 39(5):804–812. https://doi.org/10.2514/2.3000

Wie SY, Lee JH, Kwon JH, Lee DJ (2010) Far-field boundary condition effects of CFD and free-wake coupling analysis for helicopter rotor. J Fluids Eng 132(8):084501. https://doi.org/10.1115/1.4002110

Wie SY, Im DK, Kwon JH, Lee DJ (2010) Numerical simulation of rotor using coupled computational fluid dynamics and free wake. J Aircr 47(4):1167–1177. https://doi.org/10.2514/1.46797

Shi Y, Xu G, Wei P (2016) Rotor wake and flow analysis using a coupled Eulerian-Lagrangian method. Eng Appl Comput Fluid Mech 10(1):384–402. https://doi.org/10.1080/19942060.2016.1174887

Zhao Y, Shi Y, Xu G (2017) Helicopter blade-vortex interaction airload and noise prediction using coupling CFD/VWM method. Appl Sci 7(4):381. https://doi.org/10.3390/app7040381

Bae ES, Rand P, He C (2019) Hybrid Lagrangian-Eulerian approach for modeling aerodynamic interactions. In: AIAA Aviation 2019 Forum, Dallas, Texas, USA, 17–21 June 2019

Kelly ME, Duraisamy K, Brown R (2008) Predicting blade vortex interaction, airloads and acoustics using the vorticity transport model. In: Proceedings of the AHS Specialists’ Conference on Aeromechanics, San Francisco, CA, USA, 23–25 January 2008.

Renaud T, Le Pape A, Péron S (2013) Numerical analysis of hub and fuselage drag breakdown of a helicopter configuration. CEAS Aeronaut J 4:409–419. https://doi.org/10.1007/s13272-013-0081-0

Taylor MK (1950) A balsa-dust technique for air-flow visualization and its application to flow through model helicopter rotors in static thrust. NACA Technical Note NACA-TN-2220

Gray RB (1957) An aerodynamic analysis of a single-bladed rotor in hovering and low-speed forward flight as determined from smoke studies of the vorticity distribution in the wake. Dissertation, Princeton University

Landgrebe AJ (1972) The wake geometry of a hovering helicopter rotor and its influence on rotor performance. J Am Helicopter Soc 17(4):3–15. https://doi.org/10.4050/JAHS.17.4.3

Sullivan JP (1973) Experimental investigation of vortex rings and helicopter rotor wakes using a laser Doppler velocimeter. Massachusetts Institute of Technology Aerophysics Laboratory, Technical Report 183

Tangler JL (1977) Schlieren and noise studies of rotors in forward flight. In: Proceedings of the American Helicopter Society 33rd Annual Forum, Washington D.C., USA, May 1977.

Leishman JG, Bhagwat MJ, Bagai A (2002) Free-vortex filament methods for the analysis of helicopter rotor wakes. J Aircr 39(5):759–775. https://doi.org/10.2514/2.3022

Strawn RC, Djomehri MJ (2002) Computational modeling of hovering rotor and wake aerodynamics. J Aircr 39(5):786–793. https://doi.org/10.2514/2.3024

Lee DJ (2000) Numerical prediction of rotor tip-vortex roll-up in axial flights by using a time-marching free-wake method. In: Kamemoto K, Tsutahara M (eds) Vortex Methods. 1st International Conference on Vortex Methods, Kobe, 4 – 5 November 1999. World Scientific Publishing, Tokyo, pp 177–187. https://doi.org/10.1142/9789812793232_0021

Wie SY, Im DK, Kim E, Kwon JH, Lee DJ (2008) An analysis on the helicopter rotor aerodynamics in hover and forward flight using CFD/time-marching-free-wake coupling method. In: Proceedings of the ICCFD 5 - International Conference on Computational Fluid Dynamics. Springer

Hariharan N, Sankar L (2000) A review of computational techniques for rotor wake modeling. In: 38th Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, 10–13 January 2000.

Rosen A, Graber A (1988) Free wake model of hovering rotors having straight or curved blades. J Am helicopter Soc 33(3):11–19. https://doi.org/10.4050/JAHS.33.11

Wie S-Y, Lee J-H, Kwon J-H et al (2007) A study on the far-field boundary condition effects of CFD/time-marching-free-wake coupled method. J Korean Soc Aeronaut Space Sci 35(11):957–963

Morino L, Kaprielian Z, Sipcic SR (1983) Free wake analysis of helicopter rotors. Paper presented in the 9th European Rotorcraft Forum, Stresa, September 1983

Lee DJ, Na SU (1999) Numerical simulations of wake structure generated by rotating blades using a time marching, free vortex blob method. Eur J Mech - B/Fluids 18:147–159. https://doi.org/10.1016/S0997-7546(99)80011-9

Bhagwat MJ, Leishman JG (2002) Generalized viscous vortex model for application to free-vortex wake and aeroacoustic calculations. In: Proceedings of the American Helicopter Society 58th Annual Forum and Technology Display, Montreal, Canada, 11–13 June 2002.

Ploumhans P, Winckelmans GS (2000) Vortex methods for high-resolution simulations of viscous flow past bluff bodies of general geometry. J Comput Phys 165:354–406. https://doi.org/10.1006/jcph.2000.6614

Cottet G-H, Koumoutsakos PD (2000) Vortex methods: theory and practice. Cambridge university press Cambridge

Ploumhans P, Winckelmans GS, Salmon JK et al (2002) Vortex methods for direct numerical simulation of three-dimensional bluff body flows: application to the sphere at Re = 300, 500, and 1000. J Comput Phys 178(2):427–463. https://doi.org/10.1006/jcph.2002.7035

Winckelmans GS (2004) Vortex methods. Encycl Comput Mech. https://doi.org/10.1002/0470091355.ecm055

Winckelmans G, Cocle R, Dufresne L, Capart R (2005) Vortex methods and their application to trailing wake vortex simulations. Comptes Rendus Phys 6(4–5):467–486. https://doi.org/10.1016/j.crhy.2005.05.001

Leonard A (1985) Computing three-dimensional incompressible flows with vortex elements. Annu Rev Fluid Mech 17:523–559. https://doi.org/10.1146/annurev.fl.17.010185.002515

Russo G (1990) Deterministic diffusion of particles. Commun Pure Appl Math 43(6):697–733. https://doi.org/10.1002/cpa.3160430602

Dehnen W (2002) A hierarchical O(N) force calculation algorithm. J Comput Phys 179:27–42. https://doi.org/10.1006/jcph.2002.7026

Warren MS, Salmon JK (1994) A parallel, portable and versatile treecode. In: 7th Society for Industrial and Applied Mathematics (SIAM) conference on parallel processing for scientific computing. San Francisco, CA, USA, 15–17 February 1995.

Salmon JK, Warren MS (1994) Fast parallel tree codes for gravitational and fluid dynamical N-body problems. Int J Supercomput Appl High Perform Comput 8(2):129–142. https://doi.org/10.1177/109434209400800205

Koumoutsakos P, Leonard A, Pépin F (1994) Boundary conditions for viscous vortex methods. J Comput Phys 113(1):52–61. https://doi.org/10.1006/jcph.1994.1117

Singh P, Friedmann PP (2018) A computational fluid dynamics–based viscous vortex particle method for coaxial rotor interaction calculations in hover. J Am Helicopter Soc 63(4):1–13. https://doi.org/10.4050/JAHS.63.042002

Su T, Lu Y, Ma J, Guan S (2020) Aerodynamic characteristics analysis of electrically controlled rotor based on viscous vortex particle method. Aerosp Sci Technol 97:105645. https://doi.org/10.1016/j.ast.2019.105645

Greengard L, Rokhlin V (1987) A fast algorithm for particle simulations. J Comput Phys 73(2):325–348. https://doi.org/10.1016/0021-9991(87)90140-9

Cheng H, Greengard L, Rokhlin V (1999) A fast adaptive multipole algorithm in three dimensions. J Comput Phys 155(2):468–498. https://doi.org/10.1006/jcph.1999.6355

Lee H, Lee D-J (2019) Wake impact on aerodynamic characteristics of horizontal axis wind turbine under yawed flow conditions. Renew Energy 136:383–392. https://doi.org/10.1016/j.renene.2018.12.126

Lee H, Lee D-J (2019) Numerical investigation of the aerodynamics and wake structures of horizontal axis wind turbines by using nonlinear vortex lattice method. Renew Energy 132:1121–1133. https://doi.org/10.1016/j.renene.2018.08.087

Raj NV (2000) An improved semi-empirical model for 3-D post-stall effects in horizontal axis wind turbines. Master Thesis, University of Illinois, Urbana-Champaign

Du Z, Selig M (1998) A 3-D stall-delay model for horizontal axis wind turbine performance prediction. In: 1998 ASME Wind Energy Symposium, Reno, NV, USA, 12–15 January 1998.

Peters DA (1985) Toward a unified lift model for use in rotor blade stability analyses. J Am Helicopter Soc 30(3):32–42. https://doi.org/10.4050/JAHS.30.3.32

Leishman JG, Beddoes TS (1986) A generalized method for airfoil unsteady aerodynamic behavior and dynamic stall using the indicial method. In: Proceedings of the 42nd Annual Forum of the American Helicopter Society, Washington D.C., USA, June 1986

Leishman JG, Beddoes TS (1989) A semi-empirical model for dynamic stall. J Am Helicopter Soc 34(3):3–17. https://doi.org/10.4050/JAHS.34.3.3

Øye S (1991) Dynamic stall simulated as time lag of separation. Technical report, Department of Fluid Mechanics,Technical University of Denmark, 1991.

Lee H, Lee DJ (2019) Prediction of aerodynamic noise radiated from a small multicopter unmanned aerial vehicle using acoustic analogy. Trans Korean Soc Noise Vib Eng 29(4):518–526. https://doi.org/10.5050/KSNVE.2019.29.4.518

Lee H, Lee D-J (2019) Noise prediction of multi-rotor unmanned aerial vehicle considering wake interaction effects. In: Proceedings of the Vertical Flight Society’s 75th Annual Forum and Technology Display, Philadelphia, PA, USA, May 13–16, 2019

Lee H, Lee D-J (2019) Numerical prediction of aerodynamic noise radiated from quadcopter unmanned aerial vehicles. In: Proceedings of INTER-NOISE and NOISE-CON Congress, InterNoise19, Madrid, Spain, 16–19 June 2019.

Lee HJ, Lee DJ (2018) Computational study of wake interaction in quadcopter unmanned aerial vehicle. In: 7th Asian/Australian Rotorcraft Forum, Jeju Island, South Korea, 30 October - 1 November 2018.

Lee H, Lee D-J (2020) Low Reynolds number effects on aerodynamic loads of a small scale wind turbine. Renew Energy 154:1283–1293. https://doi.org/10.1016/j.renene.2020.03.097

Lee H, Lee D-J (2019) Effects of platform motions on aerodynamic performance and unsteady wake evolution of a floating offshore wind turbine. Renew Energy 143:9–23. https://doi.org/10.1016/j.renene.2019.04.134

Caradonna FX, Tung C (1981) Experimental and analytical studies of a model helicopter rotor in hover. NASA Technical Memorandum NASA-TM-81232

Hand MM, Simms DA, Fingersh LJ, et al (2001) Unsteady aerodynamics experiment Phase VI: wind tunnel test configurations and available data campaigns. NREL Technical Report NREL/TP-500–29955

Gupta S, Leishman JG (2006) Performance predictions of NREL Phase VI combined experiment rotor using a free-vortex wake model. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, USA, 9–12 January 2006

Gupta S (2006) Development of a time-accurate viscous Lagrangian vortex wake model for wind turbine applications. Dissertation, University of Maryland

Bhagwat MJ, Leishman JG (2000) Stability analysis of rotor wakes in axial flight. J Am Helicopter Soc 45(3):165–178. https://doi.org/10.4050/JAHS.45.165

Li P, Chen R (2012) Rotor unsteady aerodynamics model using an efficient free-vortex method. Aircr Eng Aerosp Technol 84(5):311–320. https://doi.org/10.1108/00022661211255494

Kini S, Conlisk AT (2002) Nature of locally steady rotor wakes. J Aircr 39(5):750–758. https://doi.org/10.2514/2.3021

Gupta S, Leishman JG (2004) Stability of methods in the free-vortex wake analysis of wind turbines. In: 42nd AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, USA, 5–8 January 2004.

Bagai A, Leishman JG (1995) Rotor free-wake modeling using a pseudo implicit technique-including comparisons with experimental data. J Am Helicopter Soc 40(3):29–41. https://doi.org/10.4050/JAHS.40.29

Abedi H, Davidson L, Voutsinas S (2017) Enhancement of free vortex filament method for aerodynamic loads on rotor blades. J Sol Energy Eng 139(3):031007. https://doi.org/10.1115/1.4035887

Yeo H (2019) Design and aeromechanics investigation of compound helicopters. Aerosp Sci Technol 88:158–173. https://doi.org/10.1016/j.ast.2019.03.010

Alvarez EJ, Ning A (2018) Development of a vortex particle code for the modeling of wake interaction in distributed propulsion. In: AIAA AVIATION Forum and Applied Aerodynamics Conference, Atlanta, Georgia, USA, 25–29 June 2018

Alvarez EJ, Ning A (2019) Modeling multirotor aerodynamic interactions through the vortex particle method. In: AIAA Aviation Forum, Dallas, Texas, USA, 17–21 June 2018

Tan JF, Gao J, Barakos GN, Lin CL, Zhang WG, Huang MQ (2021) Novel approach to helicopter brownout based on vortex and discrete element methods. Aerosp Sci Technol 116:106839. https://doi.org/10.1016/j.ast.2021.10683

Huberson S, Rivoalen E, Voutsinas S (2008) Vortex particle methods in aeroacoustic calculations. J Comput Phys 227:9216–9240. https://doi.org/10.1016/j.jcp.2008.06.011

Berdowski T, Ferreira C, Walther J (2016) 3D Lagrangian VPM: simulations of the near-wake of an actuator disc and horizontal axis wind turbine. J Phys: Conf Ser 753:032004. https://doi.org/10.1088/1742-6596/753/3/032004

Willis DJ, Peraire J, White JK (2007) A combined pFFT-multipole tree code, unsteady panel method with vortex particle wakes. Int J Numer Meth Fluids 53(8):1399–1422. https://doi.org/10.1002/fld.1240