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Unsteady aerodynamics of porous aerofoils

Published online by Cambridge University Press:  24 February 2021

Peter J. Baddoo Open the ORCID record for Peter J. Baddoo [Opens in a new window] ,
Rozhin Hajian Open the ORCID record for Rozhin Hajian [Opens in a new window]  and
Justin W. Jaworski Open the ORCID record for Justin W. Jaworski [Opens in a new window]
Peter J. Baddoo*
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK
Rozhin Hajian
Affiliation:
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA02138, USA Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA02115, USA
Justin W. Jaworski
Affiliation:
Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA18015, USA
*
†Email address for correspondence: p.baddoo@imperial.ac.uk
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Abstract

We extend unsteady thin aerofoil theory to aerofoils with generalised chordwise porosity distributions by embedding the material characteristics of the porous medium into the linearised boundary condition. Application of the Plemelj formulae to the resulting boundary value problem yields a singular Fredholm–Volterra integral equation which does not admit an analytical solution. We develop a numerical solution scheme by expanding the bound vorticity distribution in terms of appropriate basis functions. Asymptotic analysis at the leading and trailing edges reveals that the appropriate basis functions are weighted Jacobi polynomials whose parameters are related to the porosity distribution. The Jacobi polynomial basis enables the construction of a numerical scheme that is accurate and rapid, in contrast to the standard choice of Chebyshev basis functions that are shown to be unsuitable for porous aerofoils. Applications of the numerical solution scheme to discontinuous porosity profiles, quasi-static problems and the separation of circulatory and non-circulatory contributions are presented. Further asymptotic analysis of the singular Fredholm–Volterra integral equation corroborates the numerical scheme and elucidates the behaviour of the unsteady solution for small or large reduced frequency in the form of scaling laws. At low frequencies, the porous resistance dominates, whereas at high frequencies, an asymptotic inner region develops near the trailing edge and the effective mass of the porous medium dominates. Analogues to the classical Theodorsen and Sears functions are computed numerically, and Fourier transform inversion of these frequency-domain functions produces porous extensions to the Wagner and Küssner functions for transient aerofoil motions or gust encounters, respectively. Results from the present analysis and its underpinning numerical framework aim to enable the unsteady aerodynamic assessment of design strategies using porosity, with implications for unsteady gust rejection, noise-reducing aerofoil design and biologically inspired flight.

JFM classification

Aerodynamics: Flow-structure interactions Acoustics: Aeroacoustics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , A16
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Abdou, M.A. 2003 Fredholm-Volterra integral equation with singular kernel. Appl. Maths Comput. 137 (2–3), 231–243.CrossRefGoogle Scholar
Ablowitz, M.J. & Fokas, A.S. 2003 Complex Variables: Introduction and Applications. Cambridge University Press.CrossRefGoogle Scholar
Abramowitz, M. & Stegun, I.A. 1964 Handbook of Mathematical Functions. Dover Publications.Google Scholar
Andreu-Angulo, I., Babinsky, H., Biler, H., Sedky, G. & Jones, A.R. 2020 Wing-gust interactions: the effect of transverse velocity profile. AIAA Paper 2020-0079.CrossRefGoogle Scholar
Atassi, H.M. 1984 The Sears problem for a lifting airfoil revisited – new results. J.Fluid Mech. 141, 109–122.CrossRefGoogle Scholar
Atassi, H.M., Dusey, M. & Davis, C.M. 1993 Acoustic radiation from a thin airfoil in nonuniform subsonic flows. AIAA J. 31 (1), 12–19.CrossRefGoogle Scholar
Attenborough, K. 1983 Acoustical characteristics of rigid fibrous absorbents and granular materials. J.Acoust. Soc. Am. 73 (3), 785–799.CrossRefGoogle Scholar
Ayton, L.J. 2016 Acoustic scattering by a finite rigid plate with a poroelastic extension. J.Fluid Mech. 791, 414–438.CrossRefGoogle Scholar
Baddoo, P.J. & Ayton, L.J. 2018 Potential flow through a cascade of aerofoils: direct and inverse problems. Proc. R. Soc. A 474 (2217), 20180065.CrossRefGoogle ScholarPubMed
Baddoo, P.J., Hajian, R. & Jaworski, J.W. 2019 A Jacobi spectral collocation method for the steady aerodynamics of porous aerofoils. AIAA Paper 2019-2959.CrossRefGoogle Scholar
Beckwith, R.M.H. & Babinsky, H. 2009 Impulsively started flat plate flow. J.Aircraft 46 (6), 2186–2188.CrossRefGoogle Scholar
Biler, H., Badrya, C. & Jones, A.R. 2019 Experimental and computational investigation of transverse gust encounters. AIAA J. 57 (11), 4608–4622.CrossRefGoogle Scholar
Bird, H.J., Otomo, S., Ramesh, K.K. & Viola, I.M. 2019 A geometrically non-linear time-domain unsteady lifting-line theory. AIAA Paper 2019-1377.CrossRefGoogle Scholar
Bisplinghoff, R.L., Ashley, H. & Halfman, R.L. 1996 Aeroelasticity. Dover Publications.Google Scholar
Bliss, D.B. 1982 Study of bulk reacting porous sound absorbers and a new boundary condition for thin porous layers. J.Acoust. Soc. Am. 71 (3), 533–545.CrossRefGoogle Scholar
Cordes, U., Kampers, G., Meißner, T., Tropea, C., Peinke, J. & Hölling, M. 2017 Note on the limitations of the Theodorsen and Sears functions. J.Fluid Mech. 811, R1.CrossRefGoogle Scholar
Corkery, S.J., Babinsky, H. & Graham, W.R. 2019 Quantification of added-mass effects using particle image velocimetry data for a translating and rotating flat plate. J.Fluid Mech. 870, 492–518.CrossRefGoogle Scholar
Darwish, M.A. 1999 Fredholm-Volterra integral equation with singular kernel. Korean J. Comput. Appl. Maths 6 (1), 163–174.CrossRefGoogle Scholar
DLMF 2019 NIST Digital Library of Mathematical Functions. Available at: http://dlmf.nist.gov/. Release 1.0.24 of 2019-09-15.Google Scholar
Eldredge, J.D. 2010 A reconciliation of viscous and inviscid approaches to computing locomotion of deforming bodies. Exp. Mech. 50, 1349–1353.CrossRefGoogle Scholar
Erdogan, F. & Gupta, G.D. 1972 On the numerical solution of singular integral equations. Q. Appl. Maths 29 (4), 525–534.CrossRefGoogle Scholar
Erdogan, F., Gupta, G.D. & Cook, T.S. 1973 Numerical solution of singular integral equations. In Methods of Analysis and Solutions of Crack Problems. Mechanics of Fracture (ed. G.C. Sih), vol. 1, pp. 368–425. Springer.CrossRefGoogle Scholar
Ergun, S. 1952 Fluid flow through packed columns. Chem. Engng Prog. 48 (2), 89–94.Google Scholar
Ffowcs Williams, J.E. 1972 The acoustics of turbulence near sound-absorbent liners. J.Fluid Mech. 51 (4), 737–749.CrossRefGoogle Scholar
Floryan, D., Van Buren, T., Rowley, C.W. & Smits, A.J. 2017 Scaling the propulsive performance of heaving and pitching foils. J.Fluid Mech. 822, 386–397.CrossRefGoogle Scholar
Fung, Y.C. 1993 An Introduction to the Theory of Aeroelasticity. Dover.Google Scholar
Garrick, I.E. 1936 Propulsion of a flapping and oscillating airfoil. NACA Rep. 567.Google Scholar
Garrick, I.E. 1938 On some reciprocal relations in the theory of nonstationary flows. NACA Rep. 629.Google Scholar
Geyer, T. & Sarradj, E. 2014 Trailing edge noise of partially porous airfoils. AIAA Paper 2014-3039.CrossRefGoogle Scholar
Geyer, T., Sarradj, E. & Fritzsche, C. 2010 Measurement of the noise generation at the trailing edge of porous airfoils. Exp. Fluids 48 (2), 291–308.CrossRefGoogle Scholar
Glegg, S. & Devenport, W. 2017 Aeroacoustics of Low Mach Number Flows: Fundamentals, Analysis, and Measurement. Academic Press.Google Scholar
Goldstein, M.E. & Atassi, H. 1976 A complete second-order theory for the unsteady flow about an airfoil due to a periodic gust. J.Fluid Mech. 74 (4), 741–765.CrossRefGoogle Scholar
Graham, J.M.R. 1970 Similarity rules for thin aerofoils in non-stationary subsonic flows. J.Fluid Mech. 43 (4), 753–766.CrossRefGoogle Scholar
Grosjean, C.C. 1986 The weight functions, generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the Jacobi and the Gegenbauer polynomials. J.Comput. Appl. Maths 16 (3), 259–307.CrossRefGoogle Scholar
Hajian, R. & Jaworski, J.W. 2017 The steady aerodynamics of aerofoils with porosity gradients. Proc. R. Soc. A 473, 20170266.CrossRefGoogle ScholarPubMed
Hajian, R. & Jaworski, J.W. 2019 Noncirculatory fluid forces on panels with porosity gradients. AIAA J. 57 (2), 870–875.CrossRefGoogle Scholar
Hajian, R. & Jaworski, J.W. 2020 Porous panels with fixed ends lose stability by divergence. J.Fluids Struct. 93, 102823.CrossRefGoogle Scholar
Hajian, R., Jaworski, J.W. & Grace, S.M. 2018 Acoustic emission from one-dimensional vibrating porous panels in a single-sided flow. AIAA Paper 2018-2962.CrossRefGoogle Scholar
Hanna, Y.G. & Spedding, G.R. 2019 Aerodynamic performance improvements due to porosity in wings at moderate Re. AIAA Paper 2019-3584.CrossRefGoogle Scholar
Howe, M.S. 1979 On the theory of unsteady high Reynolds number flow through a circular aperture. Proc. R. Soc. Lond. A 366 (1725), 205–223.Google Scholar
Howe, M.S. 2002 Theory of Vortex Sound. Cambridge University Press.CrossRefGoogle Scholar
Iosilevskii, G. 2011 Aerodynamics of permeable membrane wings. Eur. J. Mech. (B/Fluids) 30, 534–542.CrossRefGoogle Scholar
Iosilevskii, G. 2013 Aerodynamics of permeable membrane wings. Part 2: seepage drag. Eur. J. Mech. (B/Fluids) 39, 32–41.CrossRefGoogle Scholar
Jaworski, J.W. 2012 Thrust and aerodynamic forces from an oscillating leading edge flap. AIAA J. 50 (12), 2928–2931.CrossRefGoogle Scholar
Jaworski, J.W. & Gordnier, R.E. 2012 High-order simulations of low Reynolds number membrane airfoils under prescribed motion. J.Fluids Struct. 31, 49–66.CrossRefGoogle Scholar
Jaworski, J.W. & Gordnier, R.E. 2015 Thrust augmentation of flapping airfoils in low Reynolds number flow using a flexible membrane. J.Fluids Struct. 52, 199–209.CrossRefGoogle Scholar
Jaworski, J.W. & Peake, N. 2013 Aerodynamic noise from a poroelastic edge with implications for the silent flight of owls. J.Fluid Mech. 723, 456–479.CrossRefGoogle Scholar
Jaworski, J.W. & Peake, N. 2020 Aeroacoustics of silent owl flight. Annu. Rev. Fluid Mech. 52, 395–420.CrossRefGoogle Scholar
Kerschen, E.J., Tsai, C.T. & Myers, M.R. 1993 Influence of airfoil shape and incidence angle on high-frequency gust interaction noise. In Unsteady Aerodynamics, Aeroacoustics, and Aeroelasticity of Turbomachines and Propellers, pp. 765–782. Springer-Verlag.CrossRefGoogle Scholar
Kisil, A. & Ayton, L.J. 2018 Aerodynamic noise from rigid trailing edges with finite porous extensions. J.Fluid Mech. 836, 117–144.CrossRefGoogle Scholar
Krenk, S. 1975 On quadrature formulas for singular integral equations of the first and the second kind. Q.Appl. Maths 33 (3), 225–232.CrossRefGoogle Scholar
Kuijlaars, A.B.J., Martinez-Finkelshtein, A. & Orive, R. 2005 Orthogonality of Jacobi polynomials with general parameters. Elec. Trans. Numer. Anal. 19, 1–17.Google Scholar
Küssner, H.G. 1936 Zusammenfassender Bericht über den instationären Auftrieb von Flügeln. Luftfahrtforschung 13 (12), 410–424.Google Scholar
Leishman, J.G. 1997 Unsteady aerodynamics of airfoils encountering traveling gusts and vortices. J.Aircraft 34 (6), 719–729.CrossRefGoogle Scholar
Leonard, A. & Roshko, A. 2001 Aspects of flow-induced vibration. J.Fluids Struct. 15, 415–425.CrossRefGoogle 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.CrossRefGoogle Scholar
Loewy, R.G. 1957 A two-dimensional approximation to the unsteady aerodynamics of rotary wings. J.Aeronaut. Sci. 24 (2), 81–92.CrossRefGoogle Scholar
Moore, M.N.J. 2014 Analytical results on the role of flexibility in flapping propulsion. J.Fluid Mech. 757, 599–612.CrossRefGoogle Scholar
Moore, M.N.J. 2015 Torsional spring is the optimal flexibility arrangement for thrust production of a flapping wing. Phys. Fluids 27 (9), 091701.CrossRefGoogle Scholar
Morse, P.M. & Ingard, K.U. 1986 Theoretical Acoustics. McGraw-Hil.Google Scholar
Muskhelishvili, N.I. 1946 Singular Integral Equations. Noordhoff.Google Scholar
Nelson, P.A. 1982 Noise generated by flow over perforated surfaces. J.Sound Vib. 83 (1), 11–26.CrossRefGoogle Scholar
Perrotta, G. & Jones, A.R. 2017 Unsteady forcing on a flat-plate wing in large transverse gusts. Exp. Fluids 58 (8), 101.CrossRefGoogle Scholar
Peters, D.A. 2008 Two-dimensional incompressible unsteady airfoil theory–an overview. J.Fluids Struct. 24 (3), 295–312.CrossRefGoogle Scholar
Pitt Ford, C.W. & Babinsky, H. 2014 Impulsively started flat plate circulation. AIAA J. 52 (8), 1800–1802.CrossRefGoogle Scholar
Polyanin, A. 1998 Handbook of Integral Equations. CRC Press.CrossRefGoogle 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.CrossRefGoogle Scholar
Rienstra, S.W. 1992 A note on the Kutta condition in Glauert's solution of the thin aerofoil problem. J.Engng Maths 26 (1), 61–69.CrossRefGoogle Scholar
Schwarz, L. 1940 Berechnung der Druckverteilung einer harmonisch sich verformenden Tragfläche in ebener Strömung. Luftfahrtforschung 17, 379–386.Google Scholar
Scott, J.R. & Atassi, H.M. 1993 High frequency gust interaction with single loaded airfoils in subsonic flows. In Unsteady Aerodynamics, Aeroacoustics, and Aeroelasticity of Turbomachines and Propellers, pp. 743–764. Springer-Verlag.CrossRefGoogle Scholar
Sears, W.R. 1941 Some aspects of non-stationary airfoil theory and its practical application. AIAA J. 8 (3), 104–108.Google Scholar
Sedky, G., Lagor, F.D. & Jones, A. 2020 Unsteady aerodynamics of lift regulation during a transverse gust encounter. Phys. Rev. Fluids 5 (7), 074701.CrossRefGoogle Scholar
Söhngen, H. 1939 Die Lösungen der Integralgleichung und deren Anwendung in der Tragflügeltheorie. Math. Z. 45, 245–264.CrossRefGoogle Scholar
Stevens, P.R.R.J. & Babinsky, H. 2017 Experiments to investigate lift production mechanisms on pitching flat plates. Exp. Fluids 58 (1), 7.CrossRefGoogle Scholar
SureshBabu, A., Ramesh, K. & Gopalarathnam, A. 2019 Model reduction in discrete-vortex methods for unsteady airfoil flows. AIAA J. 57 (4), 1409–1422.CrossRefGoogle Scholar
Szegö, G. 1939 Orthogonal Polynomials. American Mathematical Society.Google Scholar
Taha, H. & Rezaei, A.S. 2019 Viscous extension of potential-flow unsteady aerodynamics: the lift frequency response problem. J.Fluid Mech. 868, 141–175.CrossRefGoogle Scholar
Theodorsen, T. 1935 General theory of aerodynamic instability and the mechanism of flutter. NACA Report 496.Google Scholar
Trefethen, L.N. 2000 Spectral Methods in MATLAB. Society for Industrial and Applied Mathematics (SIAM).CrossRefGoogle Scholar
Tzezana, G.A. & Breuer, K.S. 2019 Thrust, drag and wake structure in flapping compliant membrane wings. J.Fluid Mech. 862, 871–888.CrossRefGoogle Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic Press.Google Scholar
Wagner, H. 1925 Über die Entstehung des dynamischen Auftriebes von Tragflügeln. Z. Angew. Math. Mech. 5 (1), 17–35.CrossRefGoogle Scholar
Wegert, E. 1990 An iterative method for solving nonlinear Riemann-Hilbert problems. J.Comput. Appl. Maths 29 (3), 311–327.CrossRefGoogle Scholar
Wegert, E., Wolfersdorf, L.V. & Meister, E. 1987 Plane potential flow past a cylinder with porous surface. Math. Meth. Appl. Sci. 9 (1), 587–605.CrossRefGoogle Scholar
Wei, N.J., Kissing, J., Wester, T.T.B., Wegt, S., Schiffmann, K., Jakirlic, S., Hölling, M., Peinke, J. & Tropea, C. 2019 Insights into the periodic gust response of airfoils. J.Fluid Mech. 876, 237–263.CrossRefGoogle Scholar
Weidenfeld, M. & Arad, E. 2018 Mitigating the sound of a flapping airfoil using optimal structural properties distributions. J.Sound Vib. 432, 235–248.CrossRefGoogle Scholar
Weidenfeld, M. & Manela, A. 2016 On the attenuating effect of permeability on the low frequency sound of an airfoil. J.Sound Vib. 375, 275–288.CrossRefGoogle Scholar
Yang, Y., Li, M. & Liao, H. 2019 Three-dimensional effects on the transfer function of a rectangular-section body in turbulent flow. J.Fluid Mech. 872, 348–366.CrossRefGoogle Scholar