Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-24T00:16:34.404Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscous extension of potential-flow unsteady aerodynamics: the lift frequency response problem

Published online by Cambridge University Press:  08 April 2019

Haithem Taha Open the ORCID record for Haithem Taha [Opens in a new window]  and
Amir S. Rezaei Open the ORCID record for Amir S. Rezaei [Opens in a new window]
Haithem Taha*
Affiliation:
Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697, USA
Amir S. Rezaei
Affiliation:
Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The application of the Kutta condition to unsteady flows has been controversial over the years, with increased research activities over the 1970s and 1980s. This dissatisfaction with the Kutta condition has been recently rejuvenated with the increased interest in low-Reynolds-number, high-frequency bio-inspired flight. However, there is no convincing alternative to the Kutta condition, even though it is not mathematically derived. Realizing that the lift generation and vorticity production are essentially viscous processes, we provide a viscous extension of the classical theory of unsteady aerodynamics by relaxing the Kutta condition. We introduce a trailing-edge singularity term in the pressure distribution and determine its strength by using the triple-deck viscous boundary layer theory. Based on the extended theory, we develop (for the first time) a theoretical viscous (Reynolds-number-dependent) extension of the Theodorsen lift frequency response function. It is found that viscosity induces more phase lag to the Theodorsen function particularly at high frequencies and low Reynolds numbers. The obtained theoretical results are validated against numerical laminar simulations of Navier–Stokes equations over a sinusoidally pitching NACA 0012 at low Reynolds numbers and using Reynolds-averaged Navier–Stokes equations at relatively high Reynolds numbers. The physics behind the observed viscosity-induced lag is discussed in relation to wake viscous damping, circulation development and the Kutta condition. Also, the viscous contribution to the lift is shown to significantly decrease the virtual mass, particularly at high frequencies and Reynolds numbers.

JFM classification

Boundary Layers: Boundary layer structure Wakes/Jets: Wakes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 868 , 10 June 2019 , pp. 141 - 175
Copyright
© 2019 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

Abramson, H. N. & Chu, H.-H. 1959 A discussion of the flutter of submerged hydrofoils. J. Ship Res. 3 (2), 5–13.Google Scholar
Abramson, H. N., Chu, W.-H & Irick, J. T.1967 Hydroelasticity with special reference to hydrofoil craft. Tech. Rep. 2557. NSRDC Hydromechanics Lab.Google Scholar
Abramson, H. N. & Ransleben, G. E. 1965 An experimental investigation of flutter of a fully submerged subcavitating hydrofoil. J. Aircraft 2 (5), 439–442.10.2514/3.43681Google Scholar
Alben, S. 2008 The flapping-flag instability as a nonlinear eigenvalue problem. Phys. Fluids 20 (10), 104106.Google Scholar
Ansari, S. A., Żbikowski, R. & Knowles, K. 2006a Non-linear 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.Google Scholar
Ansari, S. A., Żbikowski, R. & Knowles, K. 2006b Non-linear unsteady aerodynamic model for insect-like flapping wings in the hover. Part 2. Implementation and validation. J. Aerospace Engng 220, 169–186.Google Scholar
Bass, R. L., Johnson, J. E. & Unruh, J. F. 1982 Correlation of lift and boundary-layer activity on an oscillating lifting surface. AIAA J. 20 (8), 1051–1056.10.2514/3.7964Google Scholar
Basu, B. C. & Hancock, G. J. 1978 The unsteady motion of a two-dimensional aerofoil in incompressible inviscid flow. J. Fluid Mech. 87 (01), 159–178.Google Scholar
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Birkhoff, G. 1962 Helmholtz and Taylor Instability, (Proc. Symp. Applied Mathematics) , vol. 13, pp. 55–76. American Mathematical Society.Google Scholar
Birnbaum, W. 1924 Der Schlagflugelpropeller und die kleinen Schwingungen elastisch befestigter Tragfluegel. Z. Flugtech. Motorluftschiffahrt 15, 128–134.Google Scholar
Birnbaum, W. & Ackermann, W. 1923 Die tragende Wirbelfläche als Hilfsmittel zur Behandlung des ebenen Problems der Tragflügeltheorie. Z. Angew. Math. Mech. 3 (4), 290–297.10.1002/zamm.19230030408Google Scholar
Bisplinghoff, R. L., Ashley, H. & Halfman, R. L. 1996 Aeroelasticity. Dover Publications.Google Scholar
Blasius, H. 1908 Grenzschichten in Flüssigkeiten mit kleiner Reibung. B.G. Teubner.Google Scholar
Brown, S. N. & Cheng, H. K. 1981 Correlated unsteady and steady laminar trailing-edge flows. J. Fluid Mech. 108, 171–183.Google Scholar
Brown, S. N. & Daniels, P. G. 1975 On the viscous flow about the trailing edge of a rapidly oscillating plate. J. Fluid Mech. 67 (04), 743–761.Google Scholar
Brown, S. N. & Stewartson, K. 1970 Trailing-edge stall. J. Fluid Mech. 42 (03), 561–584.Google Scholar
Chow, R. & Melnik, R. E. 1976 Numerical solutions of the triple-deck equations for laminar trailing-edge stall. In Proceedings of the 5th International Conference on Numerical Methods in Fluid Dynamics June 28–July 2 1976, Twente University, Enschede, pp. 135–144. Springer.Google Scholar
Chu, W.-H. 1961 An aerodynamic analysis for flutter in Oseen-type viscous flow. J. Aero. Sci. 29, 781–789.Google Scholar
Chu, W.-H. & Abramson, H. N. 1959 An alternative formulation of the problem of flutter in real fluids. J. Aero. Sci. 26 (10), 683–684.Google Scholar
Crighton, D. G. 1985 The Kutta condition in unsteady flow. Annu. Rev. Fluid Mech. 17 (1), 411–445.Google Scholar
Daniels, P. G. 1978 On the unsteady Kutta condition. Q. J. Mech. Appl. Maths 31 (1), 49–75.Google Scholar
Ding, Q. N. & Wang, D.-L. 2006 The flutter of an airfoil with cubic structural and aerodynamic non-linearities. Aerosp. Sci. Technol. 10 (5), 427–434.10.1016/j.ast.2006.03.005Google Scholar
Garrick, I. E.1937 Propulsion of a flapping and oscillating airfoil. Tech. Rep. NACA-TR-567.Google Scholar
Garrick, I. E.1938 On some reciprocal relations in the theory of nonstationary flows. NACA Tech. Rep. 629.Google Scholar
Glauert, H. 1926 The Elements of Aerofoil and Airscrew Theory. Cambridge University Press.Google Scholar
Goldstein, S. 1930 Concerning some solutions of the boundary layer equations in hydrodynamics. Math. Proc. Camb. Phil. Soc. 26 (1), 1–30.Google Scholar
Goldstein, S. 1938 Modern Developments in Fluid Dynamics: An Account of Theory and Experiment Relating to Boundary Layers, Turbulent Motion and Wakes. Clarendon Press.Google Scholar
Graftieaux, L., Michard, M. & Grosjean, N. 2001 Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows. Meas. Sci. Technol. 12 (9), 1422–1429.Google Scholar
Hemati, M. S., Eldredge, J. D. & Speyer, J. L. 2014 Improving vortex models via optimal control theory. J. Fluids Struct. 49, 91–111.Google Scholar
Henry, C. J.1961 Hydrofoil flutter phenomenon and airfoil flutter theory. Tech. Rep. 856. Davidson Laboratory.10.21236/AD0273328Google Scholar
Howarth, L. 1935 The theoretical determination of the lift coefficient for a thin elliptic cylinder. Proc. R. Soc. Lond. A 149 (868), 558–586.Google Scholar
Hussein, A. A. & Canfield, R. A. 2017 Unsteady aerodynamic stabilization of the dynamics of hingeless rotor blades in hover. AIAA J. 56 (3), 1298–1303.Google Scholar
Hussein, A. A., Taha, H., Ragab, S. & Hajj, M. R. 2018 A variational approach for the dynamics of unsteady point vortices. Aerosp. Sci. Technol. 78, 559–568.Google Scholar
Jobe, C. E. & Burggraf, O. R. 1974 The numerical solution of the asymptotic equations of trailing edge flow. Proc. R. Soc. Lond. A 340 (1620), 91–111.Google Scholar
Jones, M. A. 2003 The separated flow of an inviscid fluid around a moving flat plate. J. Fluid Mech. 496, 405–441.Google Scholar
Joukowsky, N. 1910 Über die konturen der Tragflächen der Drachenflieger. Z. Flugtech. Motorluftschiffahrt 1, 281–284.Google Scholar
Katz, J. & Plotkin, A. 2001 Low-Speed Aerodynamics. Cambridge University Press.Google Scholar
Krylov, N. M. & Bogoliubov, N. N. 1943 Introduction to Non-Linear Mechanics. (AM-11). Princeton University Press.Google Scholar
Küssner, H. G. 1929 Schwingungen von Flugzeugflügeln. Jahrbuch der deutscher Versuchsanstalt für Luftfahrt especially Section E3 Einfluss der Baustoff-Dämpfung, pp. 319–320.Google Scholar
Kutta, W. M. 1902 Auftriebskräfte in strömenden Flüssigkeiten. Illustrierte Aeronautische Mitteilungen 6 (133), 133–135.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Langlois, W. E. & Deville, M. O. 2014 Slow Viscous Flow. Springer.Google Scholar
Li, J. & Wu, Z.-N. 2015 Unsteady lift for the Wagner problem in the presence of additional leading/trailing edge vortices. J. Fluid Mech. 769, 182–217.Google Scholar
Librescu, L., Chiocchia, G. & Marzocca, P. 2003 Implications of cubic physical/aerodynamic non-linearities on the character of the flutter instability boundary. Intl J. Non-Linear Mech. 38 (2), 173–199.Google Scholar
Lighthill, M. J. 1953 On boundary layers and upstream influence. II. Supersonic flows without separation. Proc. R. Soc. Lond. A 217 (1131), 478–507.Google Scholar
Loewy, R. G. 1957 A two-dimensional approximation to unsteady aerodynamics in rotary wings. J. Aero. Sci. 24, 81–92.Google Scholar
Mandre, S. & Mahadevan, L. 2010 A generalized theory of viscous and inviscid flutter. Proc. R. Soc. Lond. A 466 (2113), 141–156.Google Scholar
Menter, F. R. 1994 Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 32 (8), 1598–1605.Google Scholar
Messiter, A. F. 1970 Boundary-layer flow near the trailing edge of a flat plate. SIAM J. Appl. Maths 18 (1), 241–257.Google Scholar
Messiter, A. F. 1983 Boundary-layer interaction theory. Trans. ASME J. Appl. Mech. 50 (4b), 1104–1113.Google Scholar
Michelin, S. & Smith, S. G. L. 2009 An unsteady point vortex method for coupled fluid–solid problems. Theor. Comput. Fluid Dyn. 23 (2), 127–153.Google Scholar
Multhopp, H. 1950 Methods for Calculating the Lift Distribution of Wings (Subsonic Lifting-Surface Theory). Aeronautical Research Council.Google Scholar
Ogata, K. & Yang, Y. 1970 Modern Control Engineering. Prentice-Hall.Google Scholar
Orszag, S. A. & Crow, S. C. 1970 Instability of a vortex sheet leaving a semi-infinite plate. Stud. Appl. Maths 49 (2), 167–181.Google Scholar
Peters, D. A. 2008 Two-dimensional incompressible unsteady airfoil theory – an overview. J. Fluids Struct. 24, 295–312.Google Scholar
Pitt Ford, C. W. & Babinsky, H. 2013 Lift and the leading-edge vortex. J. Fluid Mech. 720, 280–313.Google Scholar
Prandtl, L. 1918 Gesammelte Abhandlungen zur angewandten Mechanik, Hydro-und Aerodynamik. Springer.Google Scholar
Prandtl, L. 1924 Über die Entstehung von Wirbeln in der idealen Flüssigkeit, mit Anwendung auf die Tragflügeltheorie und andere Aufgaben. In Vorträge aus dem Gebiete der Hydro-und Aerodynamik (Innsbruck 1922), pp. 18–33. Springer.Google Scholar
Preston, J. H. 1943 The Approximate Calculation of the Lift of Symmetrical Aerofoils taking Account of the Boundary Layer, with Application to Control Problems. HM Stationery Office.Google Scholar
Pullin, D. I. & Wang, Z. 2004 Unsteady forces on an accelerating plate and application to hovering insect flight. J. Fluid Mech. 509, 1–21.Google Scholar
Ramesh, K., Gopalarathnam, A., Edwards, J. R., Ol, M. V. & Granlund, K. 2013 An unsteady airfoil theory applied to pitching motions validated against experiment and computation. Theor. Comput. Fluid Dyn. 27, 1–22.Google Scholar
Ramesh, K., Gopalarathnam, A., Granlund, K., Ol, M. V. & Edwards, J. R. 2014 Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding. J. Fluid Mech. 751, 500–538.Google Scholar
Rezaei, A. S. & Taha, H.2017 Computational study of lift frequency responses of pitching airfoils at low Reynolds numbers. AIAA-Paper 2017-0716.Google Scholar
Robinson, A. & Laurmann, J. A. 1956 Wing Theory. Cambridge University Press.Google Scholar
Rott, N. 1956 Diffraction of a weak shock with vortex generation. J. Fluid Mech. 1 (01), 111–128.10.1017/S0022112056000081Google Scholar
Rott, N. & George, M. B. T.1955 An approach to the flutter problem in real fluids. Tech. Rep. 509.Google Scholar
Satyanarayana, B. & Davis, S. 1978 Experimental studies of unsteady trailing-edge conditions. AIAA J. 16 (2), 125–129.Google Scholar
Savage, S. B., Newman, B. G. & Wong, D. T.-M. 1979 The role of vortices and unsteady effects during the hovering flight of dragonflies. J. Expl Biol. 83 (1), 59–77.Google Scholar
Schlichting, H. & Truckenbrodt, E. 1979 Aerodynamics of the Airplane. McGraw-Hill.Google Scholar
Sears, W. R. 1956 Some recent developments in airfoil theory. J. Aero. Sci. 23, 490–499.Google Scholar
Sears, W. R. 1976 Unsteady motion of airfoils with boundary-layer separation. AIAA J. 14 (2), 216–220.Google Scholar
Shen, S. F. & Crimi, P. 1965 The theory for an oscillating thin airfoil as derived from the Oseen equations. J. Fluid Mech. 23 (03), 585–609.Google Scholar
Smith, F. T. 1983 Interacting flow theory and trailing edge separation – no stall. J. Fluid Mech. 131, 219–249.Google Scholar
Spence, D. A. 1954 Prediction of the characteristics of two-dimensional airfoils. J. Aero. Sci. 21, 577–587.Google Scholar
Stewartson, K. 1968 On the flow near the trailing edge of a flat plate. Proc. R. Soc. Lond. A 306 (1486), 275–290.Google Scholar
Stewartson, K. 1974 Multistructured boundary layers on flat plates and related bodies. Adv. Appl. Mech. 14 (145–239), 136.Google Scholar
Stewartson, K. 1981 D’Alembert’s paradox. SIAM Rev. 23 (3), 308–343.Google Scholar
Taha, H., Hajj, M. R. & Beran, P. S. 2014 State space representation of the unsteady aerodynamics of flapping flight. Aerosp. Sci. Technol. 34, 1–11.Google Scholar
Tchieu, A. A. & Leonard, A. 2011 A discrete-vortex model for the arbitrary motion of a thin airfoil with fluidic control. J. Fluids Struct. 27 (5), 680–693.Google Scholar
Theodorsen, T.1935 General theory of aerodynamic instability and the mechanism of flutter NACA Tech. Rep. 496.Google Scholar
Tietjens, O. K. G. & Prandtl, L. 1934 Applied Hydro- and Aeromechanics: Based on Lectures of L. Prandtl. Courier Corporation.Google Scholar
Truckenbrodt, E. 1953 Tragflächentheorie bei inkompressibler Strömung. Jahrbuch, pp. 40–65.Google Scholar
Veldmann, A. E. P. & Van de Vooren, A. I. 1975 Drag of a finite plate. In Proceedings of the Fourth International Conference on Numerical Methods in Fluid Dynamics, pp. 423–430. Springer.Google Scholar
Von Karman, T. & Sears, W. R. 1938 Airfoil theory for non-uniform motion. J. Aero. Sci. 5 (10), 379–390.Google Scholar
Wagner, H. 1925 Uber die Entstehung des dynamischen Auftriebs von Tragflugeln. Z. Angew. Math. Mech. 5, 17–35.Google Scholar
Wang, C. & Eldredge, J. D. 2013 Low-order phenomenological modeling of leading-edge vortex formation. Theor. Comput. Fluid Dyn. 27 (5), 577–598.Google Scholar
Weissinger, J. 1949 Über eine erweiterung der prandtlschen theorie der tragenden linie. Math. Nachrichten 2 (1–2), 45–106.Google Scholar
Wilcox, D. C. 1998 Turbulence Modeling for CFD. DCW Industries.Google Scholar
Woolston, D. S. & Castile, G. E.1951 Some effects of variations in several parameters including fluid density on the flutter speed of light uniform cantilever wings. NACA Tech. Rep. 2558.Google Scholar
Xia, X. & Mohseni, K. 2017 Unsteady aerodynamics and vortex-sheet formation of a two-dimensional airfoil. J. Fluid Mech. 830, 439–478.Google Scholar
Yan, Z., Taha, H. & Hajj, M. R. 2014 Geometrically-exact unsteady model for airfoils undergoing large amplitude maneuvers. Aerosp. Sci. Technol. 39, 293–306.Google Scholar
Yongliang, Y., Binggang, T. & Huiyang, M. 2003 An analytic approach to theoretical modeling of highly unsteady viscous flow excited by wing flapping in small insects. Acta Mechanica Sin. 19 (6), 508–516.Google Scholar
Zakaria, M. Y., Al-Haik, M. Y. & Hajj, M. R. 2015 Experimental analysis of energy harvesting from self-induced flutter of a composite beam. Appl. Phys. Lett. 107 (2), 023901.Google Scholar
Zakaria, M. Y., Taha, H. & Hajj, M. R. 2017 Measurement and modeling of lift enhancement on plunging airfoils: a frequency response approach. J. Fluids Struct. 69, 187–208.Google Scholar