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Nonlinear dynamic responses of a perimeter-reinforced membrane wing in laminar flows

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Abstract

Dynamic responses of a two-dimensional (2D) perimeter-reinforced (PR) membrane wing in laminar viscous flows are investigated numerically. The 2D Navier–Stokes equations and a one-dimensional nonlinear equation for membrane vibration are coupled to describe the flow-induced vibrations of the membrane wing in laminar flows. The modified characteristic-based split scheme, Galerkin finite element method, spring analogy technique and loosely coupled partitioned approach are employed, respectively, for the flow simulation, computation of the membrane response, mesh movement of the flow domain and fluid–structure interaction. The accuracy and stability of the proposed numerical method and corresponding codes are validated using a benchmark model of fluid–membrane interaction. Finally, the bifurcation characteristics of the membrane dynamic response and vortex structure near the membrane wing with respect to the angle of attack, Reynolds number, rigidity and pre-strain are analysed in detail. This paper could give people more information about the dynamic behaviours of the PR membrane wing in the laminar flow regime.

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Abbreviations

\(\alpha \) :

Angle of attack

c :

Chord length

\(C_\mathrm{d} \) :

Structural damping normalized by \(u_\infty \)

\(\bar{C}_{L}\) :

Mean lift coefficient

\(\bar{C}_{D}\) :

Mean drag coefficient

\(\bar{C}_{p}\) :

Mean pressure coefficient, \(\bar{C}_{p} =2\bar{p}\)

\(\delta _0 \) :

Membrane pre-strain

\(\Delta p\) :

Pressure difference between the lower and upper surfaces of the flexible membrane normalized by \(\rho _\infty u_\infty ^2 \)

\(\xi \) :

Local coordinate on the flexible membrane normalized by c

E :

Elastic modulus of the membrane normalized by \(\rho _\infty u_\infty ^2 \)

\(f_\mathrm{D}\) :

Structural damping force per unit area normalized by \(\rho _\infty u_\infty ^2 \)

h :

Membrane thickness normalized by c

L :

Length of the flexible membrane before deforming

\({L}'\) :

Length of the flexible membrane before deforming normalized by c

\(L_\mathrm{S}\) :

Length of the flexible membrane after deforming normalized by c

\(\overline{L/D}\) :

Mean lift-to-drag ratio

\(n_\mathrm{P}\) :

Total number of grid nodes in the flow domain

\(n_\mathrm{E}\) :

Total number of grid elements in the flow domain

\(n_\mathrm{M}\) :

Total number of grid elements on the flexible membrane

p :

Pressure normalized by \(\rho _\infty u_\infty ^2 \)

\(\bar{p}\) :

Mean pressure normalized by \(\rho _\infty u_\infty ^2 \)

\(p^{+},p^{-}\) :

Pressures on the upper and lower surfaces of the flexible membrane normalized by \(\rho _\infty u_\infty ^2 \)

Re :

Reynolds number with respect to c and \(u_\infty \)

\(\rho _\infty \) :

Density of the incompressible flow

\(\rho _\mathrm{S} \) :

Density of the flexible membrane normalized by \(\rho _\infty \)

t :

Real time normalized by \(c/{u_\infty }\)

T :

Tension of the flexible membrane normalized by \({\rho _\infty u_\infty ^2 }/c\)

\(u_\infty \) :

Velocity component of the free stream in x direction

\(u_i\) :

Velocity components of the flow field by \(u_\infty \)

v :

Membrane velocity normalized by \(u_\infty \)

x, y :

Coordinate components of the flow domain normalized by c

z :

Membrane displacement normalized by c

\(z_\mathrm{mean}\) :

Time-averaged displacement of the membrane normalized by c

References

  1. Albertani, R., Stanford, B., Hubner, J.P., Ifju, P.G.: Aerodynamic coefficients and deformation measurements on flexible Micro Air Vehicle wings. Exp. Mech. 47, 625–635 (2007)

    Article  Google Scholar 

  2. Stanford, B., Ifju, P., Albertani, R., Shyy, W.: Fixed membrane wings for micro air vehicles: experimental characterisation, numerical modelling and tailoring. Prog. Aerosp. Sci. 46, 258–294 (2008)

    Article  Google Scholar 

  3. Newman, B.G.: Aerodynamic theory for membranes and sails. Prog. Aerosp. Sci. 24, 1–19 (1987)

    Article  MATH  Google Scholar 

  4. Lian, Y.S., Shyy, W., Viieru, D., Zhang, B.N.: Membrane wing aerodynamics for micro air vehicles. Prog. Aerosp. Sci. 39, 425–465 (2003)

    Article  Google Scholar 

  5. Shyy, W., Ifju, P., Viieru, D.: Membrane wing-based micro air vehicles. Appl. Mech. Rev. 58, 203–301 (2005)

    Google Scholar 

  6. Gursul, I., Cleaver, D.J., Wang, Z.: Control of low Reynolds number flows by means of fluid–structure interactions. Prog. Aerosp. Sci. 64, 17–55 (2014)

    Article  Google Scholar 

  7. Waszak, R.M., Jenkins, N.L., Ifju, P.: Stability and control properties of an aeroelastic fixed wing micro aerial vehicle. In: AIAA paper 2001-4005 (2001)

  8. Hubner, J.P., Hicks, T.: Trailing-edge scalloping effect on flat-plate membrane wing performance. Aerosp. Sci. Technol. 15, 670–680 (2011)

    Article  Google Scholar 

  9. Zhang, Z., Martin, N., Wrist, A., Hubner, J.P.: Geometry and prestrain effects on the aerodynamic characteristics of battern-reinforced membrane wings. J. Aircr. 53, 530–544 (2016)

    Article  Google Scholar 

  10. Song, A., Tian, X.D., Israeli, E., Galvao, R., Bishop, K., Swartz, S., Breuer, K.: Aeromechanics of membrane wings with implications for animal flight. AIAA J. 46(8), 2096–2106 (2008)

    Article  Google Scholar 

  11. Gordnier, R.E.: High fidelity computational simulations of a membrane wing airfoil. J. Fluid Struct. 25, 897–917 (2009)

    Article  Google Scholar 

  12. Visbal, M.R., Gordnier, R.E., Galbraith, M.C.: High-fidelity simulations of moving and flexible airfoils at low Reynolds numbers. Exp. Fluids 46, 903–922 (2009)

    Article  Google Scholar 

  13. Rojratsirikul, P., Genc, M., Wang, Z., Gursul, I.: Flow-induced vibrations of low aspect ratio rectangular membrane wings. J. Fluid Struct. 19, 1296–1309 (2011)

    Article  Google Scholar 

  14. Arbós-Torrent, S., Ganapathisubramani, B., Palacios, R.: Leading- and trailing-edge effects on the aeromechanics of membrane aerofoils. J. Fluid Struct. 38, 107–118 (2013)

    Article  Google Scholar 

  15. Genç, M.S.: Unsteady aerodynamics and flow-induced vibrations of a low aspect ratio rectangular membrane wing with excess length. Exp. Therm. Fluid Sci. 44, 749–759 (2013)

    Article  Google Scholar 

  16. Gordnier, R.E., Attar, P.J.: Impact of flexibility on the aerodynamics of an aspect ratio two membrane wing. J. Fluid Struct. 45, 138–152 (2014)

    Article  Google Scholar 

  17. Bleischwitz, R., de Kat, R., Ganapathisubramani, B.: Aspect-ratio effects on aeromechanics of membrane wings at moderate Reynolds numbers. AIAA J. 53(3), 780–788 (2015)

    Article  Google Scholar 

  18. Johnston, J.W., Romberg, W., Attar, P.J., Parthasarathy, R.: Experimental characterization of limit cycle oscillations in membrane wing Micro Air Vehicles. J. Aircraft 47(4), 1300–11308 (2010)

    Article  Google Scholar 

  19. Attar, P.J., Gordnier, R.E., Johnston, J.W., Romberg, W.A., Parthasarathy, R.N.: Aeroelastic analysis of membrane micro air vehicles—part I: flutter and limit cycle analysis for fixed-wing configurations. J. Vib. Acoust. 133, 021008 (2011)

    Article  Google Scholar 

  20. Scott, K.D., Hubner, J.P., Ukeiley, L.: Cell geometry and material property effects on membrane and flow response. AIAA J. 50(3), 755–761 (2012)

    Article  Google Scholar 

  21. Attar, P.J., Brian, J.M., Romberg, W.A., Johnston, J.W., Parthasarathy, R.N.: Experimental characterization of aerodynamic behavior of membrane wings in low-Reynolds-number flow. AIAA J. 50(7), 1525–1537 (2012)

    Article  Google Scholar 

  22. Galvao, R., Israeli, E., Song, A., Tian, X.D., Bishop, K., Swartz, S., Breuer, K.: The aerodynamics of compliant membrane wings modeled on mammalian flight mechanics. AIAA paper 2006-2066 (2006)

  23. Rojratsirikul, P., Wang, Z., Gursul, I.: Unsteady aerodynamics of membrane airfoils. AIAA paper 2008-0613(2008)

  24. Rojratsirikul, P., Wang, Z., Gursul, I.: Unsteady fluid–structure interactions of membrane airfoils at low Reynolds numbers. Exp. Fluids 46(5), 859–872 (2009)

    Article  Google Scholar 

  25. Rojratsirikul, P., Wang, Z., Gursul, I.: Effect of pre-strain and excess length on unsteady fluid–structure interactions of membrane airfoils. J. Fluid Struct. 18(3), 359–376 (2010)

    Article  Google Scholar 

  26. Gordnier, R.E., Attar, P.J.: Implicit LES simulations of a low Reynolds numbers flexible membrane wing airfoil. AIAA paper 2009-579 (2009)

  27. Curet, O.M., Carrere, A., Waldman, R., Breuer, K.S.: Aerodynamic characterization of a wing membrane with variable complicance. AIAA J. 52(8), 1749–1756 (2014)

    Article  Google Scholar 

  28. Sun, X., Zhang, J.Z.: Effect of the reinforced leading or trailing edge on the aerodynamic performance of a perimeter-reinforced membrane wing. J. Fluid Struct. 68, 90–112 (2017)

    Article  Google Scholar 

  29. Sun, X., Zhang, J.Z.: Nonlinear vibrations of a flexible membrane under periodic load. Nonlinear Dyn. 85(4), 2467–2486 (2016)

    Article  MathSciNet  Google Scholar 

  30. Smith, R., Shyy, W.: Computation of unsteady laminar flow over a flexible two-dimensional membrane wing. Phys. Fluids 7, 2175 (1995)

    Article  MATH  Google Scholar 

  31. Smith, R., Shyy, W.: Computation of aerodynamic coefficients for a flexible membrane airfoil in turbulent flow: a comparison with classical theory. Phys. Fluids 8, 3346 (1996)

    Article  MATH  Google Scholar 

  32. Jaworski, J.W., Gordnier, R.E.: High-order simulations of low Reynolds number membrane. AIAA paper 2011-1116 (2011)

  33. Sun, X., Zhang, J.Z., Ren, X.L.: Characteristic-based split (CBS) finite element method for incompressible viscous flow with moving boundaries. Eng. Appl. Comp. Fluid 6(3), 461–474 (2012)

    Google Scholar 

  34. Blom, F.J.: Considerations on the spring analogy. Int. J. Numer. Meth. Fluids 32(6), 647–668 (2000)

    Article  MATH  Google Scholar 

  35. Jameson, J.: Time dependent calculations using multigrid with application to unsteady flows past airfoils and wings. AIAA paper 91-1596 (1991)

  36. Zienkiewicz, O.C., Taylor, R.L., Nithiarasu, P.: The Finite Element Method for Fluid Dynamics, six edn. Butterworth-Heinemann, Elsevier (2005)

  37. Chung, J., Hulbert, G.: A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\(\alpha \) method. J. Appl. Mech. Trans. ASME 60(2), 371–375 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by Science Foundation of China University of Petroleum-Beijing (No. 01JB0303) and the National Natural Science Foundation of China (No. 51506224). The author would like to thank for the kindly support of these foundations.

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Authors and Affiliations

  1. National Engineering Laboratory for Pipeline Safety/MOE Key Laboratory of Petroleum Engineering, China University of Petroleum-Beijing, Beijing, 102249, China

    Xu Sun

  2. Petrochina Pipline R&D Center, Langfang, 065000, China

    Xiao-Long Ren

  3. School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an, 710049, China

    Jia-Zhong Zhang

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  1. Xu Sun
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Correspondence to Xu Sun.

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Sun, X., Ren, XL. & Zhang, JZ. Nonlinear dynamic responses of a perimeter-reinforced membrane wing in laminar flows. Nonlinear Dyn 88, 749–776 (2017). https://doi.org/10.1007/s11071-016-3274-3

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