Skip to main content
SpringerLink
Log in

On the application of the asymptotic method of global instability in aeroelasticity problems

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

The asymptotic method of global instability developed by A.G. Kulikovskii is an effective tool for determining the eigenfrequencies and stability boundary of one-dimensional or multidimensional systems of sufficiently large finite length. The effectiveness of the method was demonstrated on a number of one-dimensional problems; and since the mid-2000s, this method has been used in aeroelasticity problems, which are not strictly one-dimensional: such is only the elastic part of the problem, while the gas flow occupies an unbounded domain. In the present study, the eigenfrequencies and stability boundaries predicted by the method of global instability are compared with the results of direct calculation of the spectra of the corresponding problems. The size of systems is determined starting from which the method makes a quantitatively correct prediction for the stability boundary.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. L. B. Aizin and V. P. Maksimov, “On the stability of flow of weakly compressible gas in a pipe of model roughness,” Prikl. Mat. Mekh. 42(4), 650–655 (1978) [J. Appl. Math. Mech. 42, 691–697 (1978)].

    Google Scholar 

  2. V. V. Bolotin, “Dynamical edge effect under elastic vibrations of plates,” Inzh. Sb. 31, 3–14 (1961).

    Google Scholar 

  3. O. Doaré. and E. de Langre, “Local and global instability of fluid-conveying pipes on elastic foundations,” J. Fluids Struct. 16(1), 1–14 (2002).

    Article  Google Scholar 

  4. O. Doaré and E. de Langre, “The role of boundary conditions in the instability of one-dimensional systems,” Eur._J. Mech. B/Fluids 25(6), 948–959 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  5. B. Echebarria, V. Hakim, and H. Henry, “Nonequilibrium ribbon model of twisted scroll waves,” Phys. Rev. Lett. 96(9), 098301 (2006).

    Article  Google Scholar 

  6. Ia. A. Kameniarzh, “On some properties of equations of a model of coupled termoplasticity,” Prikl. Mat. Mekh. 36(6), 1100–1107 (1972) [J. Appl. Math. Mech. 36, 1031–1038 (1972)].

    Google Scholar 

  7. A. G. Kulikovskii, “On the stability of homogeneous states,” Prikl. Mat. Mekh. 30(1), 148–153 (1966) [J. Appl. Math. Mech. 30, 180–187 (1966)].

    Google Scholar 

  8. A. G. Kulikovskii, “On the stability of Poiseuille flow and certain other plane-parallel flows in a flat pipe of large but finite length for large Reynolds’ numbers,” Prikl. Mat. Mekh. 30(5), 822–835 (1966) [J. Appl. Math. Mech. 30, 975–989 (1966)].

    Google Scholar 

  9. A. G. Kulikovskii, “Stability of flows of a weakly compressible fluid in a plane pipe of large, but finite length,” Prikl. Mat. Mekh. 32(1), 112–114 (1968) [J. Appl. Math. Mech. 32, 100–102 (1968)].

    Google Scholar 

  10. A. G. Kulikovskii, “The global instability of uniform flows in non-one-dimensional regions,” Prikl. Mat. Mekh. 70(2), 257–263 (2006) [J. Appl. Math. Mech. 70, 229–234 (2006)].

    MathSciNet  Google Scholar 

  11. A. G. Kulikovskii and I. S. Shikina, “On bending vibrations of a long tube with moving fluid,” Izv. Akad. Nauk Arm._SSR, Mekh. 41(1), 31–39 (1988).

    MATH  Google Scholar 

  12. E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Nauka, Moscow, 1979; Pergamon, Oxford, 1981), Course of Theoretical Physics10.

    Google Scholar 

  13. J. W. Miles, The Potential Theory of Unsteady Supersonic Flow (Univ. Press, Cambridge, 1959).

    MATH  Google Scholar 

  14. J. W. Nichols, J.-M. Chomaz, and P. J. Schmid, “Twisted absolute instability in lifted flames,” Phys. Fluids 21(1), 015110 (2009).

    Article  MATH  Google Scholar 

  15. N. Peake, “On the unsteady motion of a long fluid-loaded elastic plate with mean flow,” J. Fluid Mech. 507, 335–366 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Priede and G. Gerbeth, “Convective, absolute, and global instabilities of thermocapillary–buoyancy convection in extended layers,” Phys. Rev. E 56(4), 4187–4199 (1997).

    Article  Google Scholar 

  17. J. Priede and G. Gerbeth, “Absolute versus convective helical magnetorotational instability in a Taylor–Couette flow,” Phys. Rev. E 79(4), 046310 (2009).

    Article  MathSciNet  Google Scholar 

  18. A. Shishaeva, V. Vedeneev, and A. Aksenov, “Nonlinear single-mode and multi-mode panel flutter oscillations at low supersonic speeds,” J. Fluids Struct. 56, 205–223 (2015).

    Article  Google Scholar 

  19. G. A. Shugai and P. A. Yakubenko, “Convective and absolute instability of a liquid jet in a longitudinal magnetic field,” Phys. Fluids 9(7), 1928–1932 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Tuerke, D. Sciamarella, L. R. Pastur, F. Lusseyran, and G. Artana, “Frequency-selection mechanism in incompressible open-cavity flows via reflected instability waves,” Phys. Rev. E 91(1), 013005 (2015).

    Article  Google Scholar 

  21. V. V. Vedeneev, “Flutter of a wide strip plate in a supersonic gas flow,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 5, 155–169 (2005) [Fluid Dyn. 40, 805–817 (2005)].

    MathSciNet  MATH  Google Scholar 

  22. V. V. Vedeneev, “High-frequency plate flutter,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 2, 163–172 (2006) [Fluid Dyn. 41, 313–321 (2006)].

    MathSciNet  MATH  Google Scholar 

  23. V. V. Vedeneev, “High-frequency flutter of a rectangular plate,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 4, 173–181 (2006) [Fluid Dyn. 41, 641–648 (2006)].

    MathSciNet  MATH  Google Scholar 

  24. V. V. Vedeneev, “Nonlinear high-frequency flutter of a plate,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 5, 197–208 (2007) [Fluid Dyn. 42, 858–868 (2007)].

    MathSciNet  MATH  Google Scholar 

  25. V. V. Vedeneev, “Numerical investigation of supersonic plate flutter using the exact aerodynamic theory,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 2, 169–178 (2009) [Fluid Dyn. 44, 314–321 (2009)].

    MATH  Google Scholar 

  26. V. V. Vedeneev, “Study of the single-mode flutter of a rectangular plate in the case of variable amplification of the eigenmode along the plate,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 4, 163–174 (2010) [Fluid Dyn. 45, 656–666 (2010)].

    MathSciNet  MATH  Google Scholar 

  27. V. V. Vedeneev, “Panel flutter at low supersonic speeds,” J. Fluids Struct. 29, 79–96 (2012).

    Article  Google Scholar 

  28. V. V. Vedeneev, “Limit oscillatory cycles in the single mode flutter of a plate,” Prikl. Mat. Mekh. 77(3), 355–370 (2013) [J. Appl. Math. Mech. 77, 257–267 (2013)].

    MathSciNet  MATH  Google Scholar 

  29. V. V. Vedeneev, S. V. Guvernyuk, A. F. Zubkov, and M. E. Kolotnikov, “Experimental observation of single mode panel flutter in supersonic gas flow,” J. Fluids Struct. 26(5), 764–779 (2010).

    Article  Google Scholar 

  30. V. V. Vedeneev, S. V. Guvernyuk, A. F. Zubkov, and M. E. Kolotnikov, “Experimental investigation of singlemode panel flutter in supersonic gas flow,” Izv. Ross. Akad. Nauk, Mekh. Zhidk. Gaza, No. 2, 161–175 (2010) [Fluid Dyn. 45, 312–324 (2010)].

    Google Scholar 

  31. V. V. Vedeneev and S. V. Shitov, “Flutter of a periodically supported elastic strip in a gas flow with a small supersonic velocity,” Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela, No. 3, 105–126 (2015) [Mech. Solids 50, 318–336 (2015)].

    Google Scholar 

  32. P. A. Yakubenko, “Global capillary instability of an inclined jet,” J. Fluid Mech. 346, 181–200 (1997).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    V. V. Vedeneev

Authors
  1. V. V. Vedeneev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Vedeneev.

Additional information

Original Russian Text © V.V. Vedeneev, 2016, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 295, pp. 292–320.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vedeneev, V.V. On the application of the asymptotic method of global instability in aeroelasticity problems. Proc. Steklov Inst. Math. 295, 274–301 (2016). https://doi.org/10.1134/S0081543816080174

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543816080174

Search

Navigation