Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T22:45:01.090Z Has data issue: false hasContentIssue false
  • English
  • Français

Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding

Published online by Cambridge University Press:  23 June 2014

Kiran Ramesh ,
Ashok Gopalarathnam ,
Kenneth Granlund ,
Michael V. Ol  and
Jack R. Edwards
Kiran Ramesh*
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA
Ashok Gopalarathnam
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA
Kenneth Granlund
Affiliation:
US Air Force Research Laboratory, Air Vehicles Directorate, AFRL/RBAL, Building 45, 2130 8th Street, WPAFB, OH 45433, USA
Michael V. Ol
Affiliation:
US Air Force Research Laboratory, Air Vehicles Directorate, AFRL/RBAL, Building 45, 2130 8th Street, WPAFB, OH 45433, USA
Jack R. Edwards
Affiliation:
Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Unsteady aerofoil flows are often characterized by leading-edge vortex (LEV) shedding. While experiments and high-order computations have contributed to our understanding of these flows, fast low-order methods are needed for engineering tasks. Classical unsteady aerofoil theories are limited to small amplitudes and attached leading-edge flows. Discrete-vortex methods that model vortex shedding from leading edges assume continuous shedding, valid only for sharp leading edges, or shedding governed by ad-hoc criteria such as a critical angle of attack, valid only for a restricted set of kinematics. We present a criterion for intermittent vortex shedding from rounded leading edges that is governed by a maximum allowable leading-edge suction. We show that, when using unsteady thin aerofoil theory, this leading-edge suction parameter (LESP) is related to the $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}A_0$ term in the Fourier series representing the chordwise variation of bound vorticity. Furthermore, for any aerofoil and Reynolds number, there is a critical value of the LESP, which is independent of the motion kinematics. When the instantaneous LESP value exceeds the critical value, vortex shedding occurs at the leading edge. We have augmented a discrete-time, arbitrary-motion, unsteady thin aerofoil theory with discrete-vortex shedding from the leading edge governed by the instantaneous LESP. Thus, the use of a single empirical parameter, the critical-LESP value, allows us to determine the onset, growth, and termination of LEVs. We show, by comparison with experimental and computational results for several aerofoils, motions and Reynolds numbers, that this computationally inexpensive method is successful in predicting the complex flows and forces resulting from intermittent LEV shedding, thus validating the LESP concept.

JFM classification

Mathematical Foundations: Computational methods Wakes/Jets: Separated flows Vortex Flows: Vortex interactions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 751 , 25 July 2014 , pp. 500 - 538
Copyright
© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbott, I. H. & von Doenhoff, A. E. 1959 Theory of Wing Sections. Dover.Google Scholar
Acharya, M. & Metwally, M. H. 1992 Unsteady pressure field and vorticity production over a pitching airfoil. AIAA J. 30 (2), 403–411.CrossRefGoogle Scholar
Ansari, S. A., Żbikowski, R. & Knowles, K. 2006a Nonlinear unsteady aerodynamic model for insect-like flapping wings in the hover. Part 1: methodology and analysis. Proc. Inst. Mech. Engrs G 220 (2), 61–83.CrossRefGoogle Scholar
Ansari, S. A., Żbikowski, R. & Knowles, K. 2006b Nonlinear unsteady aerodynamic model for insect-like flapping wings in the hover. Part 2: implementation and validation. Proc. Inst. Mech. Engrs G 220 (3), 169–186.CrossRefGoogle Scholar
Baik, Y. S., Bernal, L. P., Granlund, K. & Ol, M. V. 2012 Unsteady force generation and vortex dynamics of pitching and plunging airfoils. J. Fluid Mech. 709, 37–68.CrossRefGoogle Scholar
Barnes, J. & Hut, P. 1986 A hierarchical $O(N \log N)$ force-calculation algorithm. Nature 324, 446–449.CrossRefGoogle Scholar
Beddoes, T. S. 1978 Onset of leading-edge separation effects under dynamic conditions and low Mach number. In 34th Annual Forum of the American Helicopter Society, vol. 17.Google Scholar
Brunton, S. L., Rowley, C. W. & Williams, D. R. 2013 Reduced-order unsteady aerodynamic models at low Reynolds numbers. J. Fluid Mech. 724, 203–233.CrossRefGoogle Scholar
Bryant, M., Gomez, J. C. & Garcia, E. 2013 Reduced-order aerodynamic modelling of flapping wing energy harvesting at low Reynolds number. AIAA J. 51 (12), 2771–2782.CrossRefGoogle Scholar
Carr, L. W. 1988 Progress in analysis and prediction of dynamic stall. J. Aircraft 25 (1), 6–17.CrossRefGoogle Scholar
Carr, L. W., Platzer, M. F., Chandrasekhara, M. S. & Ekaterinaris, J. 1990 Experimental and computational studies of dynamic stall. In Numerical and Physical Aspects of Aerodynamic Flows IV (ed. Cebeci, T.), pp. 239–256. Springer.CrossRefGoogle Scholar
Carrier, J., Greengard, L. & Rokhlin, V. 1988 A fast adaptive multipole algorithm for particle simulations. SIAM J. Sci. Stat. Comput. 9 (4), 669–686.CrossRefGoogle Scholar
Cassidy, D. A., Edwards, J. R. & Tian, M. 2009 An investigation of interface-sharpening schemes for multi-phase mixture flows. J. Comput. Phys. 228 (16), 5628–5649.CrossRefGoogle Scholar
Chandrasekhara, M. S., Ahmed, S. & Carr, L. W.1990 Schlieren studies of compressibility effects on dynamic stall of aerofoils in transient motion. AIAA Paper 90-3038.CrossRefGoogle Scholar
Chandrasekhara, M. S., Ahmed, S. & Carr, L. W. 1993 Schlieren studies of compressibility effects on dynamic stall of transiently pitching aerofoils. J. Aircraft 30 (2), 213–220.CrossRefGoogle Scholar
Choi, J.-I. & Edwards, J. R. 2008 Large eddy simulation and zonal modelling of human-induced contaminant transport. Indoor Air 18 (3), 233–249.CrossRefGoogle ScholarPubMed
Choi, J.-I. & Edwards, J. R. 2012 Large-eddy simulation of human-induced contaminant transport in room compartments. Indoor Air 22 (1), 77–87.CrossRefGoogle ScholarPubMed
Choi, J.-I., Oberoi, R. C., Edwards, J. R. & Rosati, J. A. 2007 An immersed boundary method for complex incompressible flows. J. Comput. Phys. 224 (2), 757–784.CrossRefGoogle Scholar
Chorin, A. J. 1973 Numerical study of slightly viscous flow. J. Fluid Mech. 57 (4), 785–796.CrossRefGoogle Scholar
Clements, R. R. 1973 An inviscid model of two-dimensional vortex shedding. J. Fluid Mech. 57 (2), 321–336.CrossRefGoogle Scholar
Clements, R. R. & Maull, D. J. 1975 The representation of sheets of vorticity by discrete vortices. Prog. Aerosp. Sci. 16 (2), 129–146.CrossRefGoogle Scholar
Doligalski, T. L., Smith, C. R. & Walker, J. D. A. 1994 Vortex interactions with walls. Annu. Rev. Fluid Mech. 26 (1), 573–616.CrossRefGoogle Scholar
Edwards, J. R. & Chandra, S. 1996 Comparison of eddy viscosity – transport turbulence models for three-dimensional, shock-separated flowfields. AIAA J. 34 (4), 756–763.CrossRefGoogle Scholar
Ekaterinaris, J. A. & Platzer, M. F. 1998 Computational prediction of aerofoil dynamic stall. Prog. Aerosp. Sci. 33 (11–12), 759–846.CrossRefGoogle Scholar
Eldredge, J. D. 2007 Numerical simulation of the fluid dynamics of 2D rigid body motion with the vortex particle method. J. Comput. Phys. 221 (2), 626–648.CrossRefGoogle Scholar
Eldredge, J. D., Wang, C. J. & Ol, M. V.2009 A computational study of a canonical pitch-up, pitch-down wing maneuver. AIAA Paper 2009-3687.Google Scholar
Evans, W. T. & Mort, K. W.1959 Analysis of computed flow parameters for a set of sudden stalls in low speed two-dimensional flow. NACA TN D-85.Google Scholar
Garmann, D. J. & Visbal, M. R. 2011 Numerical investigation of transitional flow over a rapidly pitching plate. Phys. Fluids 23, 094106.CrossRefGoogle Scholar
Garrick, I.1937 Propulsion of a flapping and oscillating aerofoil. NACA Rep. 567.Google Scholar
Ghosh Choudhuri, P., Knight, D. & Visbal, M. R. 1994 Two-dimensional unsteady leading-edge separation on a pitching aerofoil. AIAA J. 32 (4), 673–681.CrossRefGoogle Scholar
Granlund, K., Ol, M. V. & Bernal, L.2011 Experiments on pitching plates: force and flowfield measurements at low Reynolds numbers. AIAA Paper 2011-0872.Google Scholar
Granlund, K., Ol, M. V. & Bernal, L. P. 2013 Unsteady pitching flat plates. J. Fluid Mech. 733, R5.CrossRefGoogle Scholar
Hald, O. H. 1979 Convergence of vortex methods for Euler’s equations, II. SIAM J. Numer. Anal. 16 (5), 726–755.CrossRefGoogle Scholar
Hammer, P., Altman, A. & Eastep, F. 2014 Validation of a discrete vortex method for low Reynolds number unsteady flows. AIAA J. 52 (3), 643–649.CrossRefGoogle Scholar
Jones, K. D. & Platzer, M. F.1997 A fast method for the prediction of dynamic stall onset on turbomachinery blades. ASME Paper 97-GT-101.CrossRefGoogle Scholar
von Kármán, T. & Burgers, J. M. 1963 General Aerodynamic Theory – Perfect Fluids (ed. Durand, W. F.), Aerodynamic Theory: A General Review of Progress, vol. 2. Dover.Google Scholar
von Kármán, T. & Sears, W. 1938 Aerofoil theory for non-uniform motion. J. Aeronaut. Sci. 5 (10), 379–390.CrossRefGoogle Scholar
Katz, J. 1981 Discrete vortex method for the non-steady separated flow over an aerofoil. J. Fluid Mech. 102, 315–328.CrossRefGoogle Scholar
Katz, J. & Plotkin, A. 2000 Low-Speed Aerodynamics. Cambridge University Press.Google Scholar
Kinsey, T. & Dumas, G. 2008 Parametric study of an oscillating aerofoil in a power-extraction regime. AIAA J. 46 (6), 1318–1330.CrossRefGoogle Scholar
Kiya, M. & Arie, M. 1977 A contribution to an inviscid vortex-shedding model for an inclined flat plate in uniform flow. J. Fluid Mech. 82 (2), 241–253.CrossRefGoogle Scholar
Krist, S. L., Biedron, R. T. & Rumsey, C. L.1998 CFL3D user’s manual. NASA TM 208444.Google Scholar
Leishman, J. G. 2002 Principles of Helicopter Aerodynamics. Cambridge University Press.Google Scholar
Leonard, A. 1980 Vortex methods for flow simulation. J. Comput. Phys. 37 (3), 289–335.CrossRefGoogle Scholar
McAvoy, C. W. & Gopalarathnam, A. 2002 Automated cruise flap for aerofoil drag reduction over a large lift range. J. Aircraft 39 (6), 981–988.CrossRefGoogle Scholar
McCroskey, W. J.1981 The phenomenon of dynamic stall. NASA TM 81264.Google Scholar
McCroskey, W. J. 1982 Unsteady aerofoils. Annu. Rev. Fluid Mech. 14, 285–311.CrossRefGoogle Scholar
McCune, J. E., Lam, C. G. & Scott, M. T. 1990 Nonlinear aerodynamics of two-dimensional airfoils in severe maneuver. AIAA J. 28 (3), 385–393.CrossRefGoogle Scholar
McGowan, G. Z., Granlund, K., Ol, M. V., Gopalarathnam, A. & Edwards, J. R. 2011 Investigations of lift-based pitch-plunge equivalence for airfoils at low Reynolds numbers. AIAA J. 49 (7), 1511–1524.CrossRefGoogle Scholar
Morris, W. J. & Rusak, Z. 2013 Stall onset on aerofoils a low to moderately high Reynolds number flows. J. Fluid Mech. 733, 439–472.CrossRefGoogle Scholar
Mukherjee, R. & Gopalarathnam, A. 2006 Poststall prediction of multiple-lifting-surface configurations using a decambering approach. J. Aircraft 43 (3), 660–668.CrossRefGoogle Scholar
Ol, M. V., Bernal, L., Kang, C. K. & Shyy, W. 2009a Shallow and deep dynamic stall for flapping low Reynolds number airfoils. Exp. Fluids 46 (5), 883–901.CrossRefGoogle Scholar
Ol, M. V., McAuliffe, B. R., Hanff, E. S., Scholz, U. & Kaehler, C.2005 Comparison of laminar separation bubble measurements on a low Reynolds number aerofoil in three facilities. AIAA Paper 2005-5149.CrossRefGoogle Scholar
Ol, M. V., Reeder, M., Fredberg, D., McGowan, G. Z., Gopalarathnam, A. & Edwards, J. R. 2009b Computation vs. experiment for high-frequency low-Reynolds number aerofoil plunge. Intl J. Micro Air Veh. 1 (2), 99–119.CrossRefGoogle Scholar
Pitt Ford, C. W. & Babinsky, H. 2013 Lift and the leading-edge vortex. J. Fluid Mech. 720, 280–313.CrossRefGoogle Scholar
Ramesh, K.2013 Theory and low-order modelling of unsteady aerofoil flows. PhD thesis, North Carolina State University, Raleigh, NC.Google Scholar
Ramesh, K., Gopalarathnam, A., Edwards, J. R., Granlund, K. & Ol, M. V.2013textita Theoretical analysis of perching and hovering maneuvers. AIAA Paper 2013-3194.CrossRefGoogle Scholar
Ramesh, K., Gopalarathnam, A., Edwards, J. R., Ol, M. V. & Granlund, K. 2013b An unsteady aerofoil theory applied to pitching motions validated against experiment and computation. Theor. Comput. Fluid Dyn. 27 (6), 843–864.CrossRefGoogle Scholar
Ramesh, K., Gopalarathnam, A., Ol, M. V., Granlund, K. & Edwards, J. R.2011 Augmentation of inviscid aerofoil theory to predict and model 2D unsteady vortex dominated flows. AIAA Paper 2011-3578.CrossRefGoogle Scholar
Rival, D. E., Kriegseis, J., Schaub, P., Widmann, A. & Tropea, C. 2014 Characteristic length scales for vortex detachment on plunging profiles with varying leading-edge geometry. Exp. Fluids 55 (1), 1–8.CrossRefGoogle Scholar
Rosenhead, L. 1932 The point vortex approximation of a vortex sheet. Proc. R. Soc. Lond. A 134, 170–192.Google Scholar
Saffman, P. G. & Baker, G. R. 1979 Vortex interactions. Annu. Rev. Fluid Mech. 11 (1), 95–121.CrossRefGoogle Scholar
Sarpkaya, T. 1975 An inviscid model of two-dimensional vortex shedding for transient and asymptotically steady separated flow over an inclined plate. J. Fluid Mech. 68 (1), 109–128.CrossRefGoogle Scholar
Selig, M. S., Donovan, J. F. & Fraser, D. B. 1989 Airfoils at Low Speeds, Soartech, vol. 8. SoarTech Publications.Google Scholar
Spalart, P. R. & Allmaras, S. R.1992 A one-equation turbulence model for aerodynamic flows. AIAA Paper 92-0439.CrossRefGoogle Scholar
Theodorsen, T.1931 On the theory of wing sections with particular reference to the lift distribution. NASA Tech. Rep. 383.Google Scholar
Theodorsen, T.1935 General theory of aerodynamic instability and the mechanism of flutter. NACA Rep. 496.Google Scholar
Vatistas, G. H., Kozel, V. & Mih, W. C. 1991 A simpler model for concentrated vortices. Exp. Fluids 11 (1), 73–76.CrossRefGoogle Scholar
Visbal, M. R. & Shang, J. S. 1989 Investigation of the flow structure around a rapidly pitching aerofoil. AIAA J. 27 (8), 1044–1051.CrossRefGoogle Scholar
Wagner, H. 1925 Über die Entstehung des dynamischen Auftriebes von Tragflügeln. Z. Angew. Math. Mech. 5 (1), 17–35.CrossRefGoogle Scholar
Wang, C. & Eldredge, J. D. 2013 Low-order phenomenological modelling of leading-edge vortex formation. Theor. Comput. Fluid Dyn. 27 (5), 577–598.CrossRefGoogle Scholar
Xia, X. & Mohseni, K. 2013 Lift evaluation of a two-dimensional pitching flat plate. Phys. Fluids 25 (091901), 1–26.CrossRefGoogle Scholar
Young, J., Ashraf, M. A., Lai, J. C. S. & Platzer, M. F. 2013 Numerical simulation of fully passive flapping foil power generation. AIAA J. 51 (11), 2727–2739.CrossRefGoogle Scholar
Young, J., Lai, J. C. S. & Platzer, M. F. 2014 A review of progress and challenges in flapping foil power generation. Prog. Aerosp. Sci. 67, 2–28.CrossRefGoogle Scholar
Zhu, Q. 2011 Optimal frequency for flow energy harvesting of a flapping foil. J. Fluid Mech. 675, 495–517.CrossRefGoogle Scholar