Advertisement

Artificial eigenmodes in truncated flow domains

  • Lutz Lesshafft
Original Article

Abstract

Whenever linear eigenmodes of open flows are computed on a numerical domain that is truncated in the streamwise direction, artificial boundary conditions may give rise to spurious pressure signals that are capable of providing unwanted perturbation feedback to upstream locations. The manifestation of such feedback in the eigenmode spectrum is analysed here for two simple configurations. First, explicitly prescribed feedback in a Ginzburg–Landau model is shown to produce a spurious eigenmode branch, named the ‘arc branch’, that strongly resembles a characteristic family of eigenmodes typically present in open shear flow calculations. Second, corresponding mode branches in the global spectrum of an incompressible parallel jet in a truncated domain are examined. It is demonstrated that these eigenmodes of the numerical model depend on the presence of spurious forcing of a local \(k^+\) instability wave at the inflow, caused by pressure signals that appear to be generated at the outflow. Multiple local \(k^+\) branches result in multiple global eigenmode branches. For the particular boundary treatment chosen here, the strength of the pressure feedback from the outflow towards the inflow boundary is found to decay with the cube of the numerical domain length. It is concluded that arc branch eigenmodes are artefacts of domain truncation, with limited value for physical analysis. It is demonstrated, for the example of a non-parallel jet, how spurious feedback may be reduced by an absorbing layer near the outflow boundary.

Keywords

Global instability Non-reflecting boundary conditions Jets 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

Discussions with Xavier Garnaud and Wilfried Coenen greatly helped shape the ideas presented in this paper. The study was supported by the Agence Nationale de la Recherche under the Cool Jazz project, Grant Number ANR-12-BS09-0024.

References

  1. 1.
    Åkervik, E., Ehrenstein, U., Gallaire, F., Henningson, D.: Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. B/Fluids 27, 1–13 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Buell, J.C., Huerre, P.: Inflow/outflow boundary conditions and global dynamics of spatial mixing layers. In: Proceedings of 2nd Summer Program, Stanford University. Center Turbulence Research, pp. 19–27 (1988)Google Scholar
  3. 3.
    Cerqueira, S., Sipp, D.: Eigenvalue sensitivity, singular values and discrete frequency selection mechanism in noise amplifiers: the case of flow induced by radial wall injection. J. Fluid Mech. 757, 770–799 (2014)CrossRefGoogle Scholar
  4. 4.
    Chakravarthy, R.V.K., Lesshafft, L., Huerre, P.: Global stability of buoyant jets and plumes. J. Fluid Mech. 835, 654–673 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chomaz, J.-M.: Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chomaz, J.-M., Huerre, P., Redekopp, L.G.: Bifurcations to local and global modes in spatially developing flows. Phys. Rev. Lett. 60, 25–28 (1988)CrossRefGoogle Scholar
  7. 7.
    Coenen, W., Lesshafft, L., Garnaud, X., Sevilla, A.: Global instability in low-density jets. J. Fluid Mech. 820, 187–207 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Colonius, T.: Modeling artificial boundary conditions for compressible flow. Annu. Rev. Fluid Mech. 36, 315–345 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Couairon, A., Chomaz, J.-M.: Absolute and convective instabilities, front velocities and global modes in nonlinear systems. Physica D 108, 236–276 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ehrenstein, U., Gallaire, F.: On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536, 209–218 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ehrenstein, U., Gallaire, F.: Two-dimensional global low-frequency oscillations in a separating boundary-layer flow. J. Fluid Mech. 614, 315–327 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Garnaud, X., Lesshafft, L., Schmid, P.J., Huerre, P.: Modal and transient dynamics of jet flows. Phys. Fluids 25, 044103 (2013)CrossRefzbMATHGoogle Scholar
  13. 13.
    Heaton, C.J., Nichols, J.W., Schmid, P.J.: Global linear stability of the non-parallel Batchelor vortex. J. Fluid Mech. 629, 139–160 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Huerre, P.: Open shear flow instabilities. In: Batchelor, G.K., Moffatt, H.K., Worster, M.G. (eds.) Perspectives in Fluid Dynamics, pp. 159–229. Cambridge University Press, Cambridge (2000)Google Scholar
  15. 15.
    Huerre, P., Monkewitz, P.A.: Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Khorrami, M.R.: A Chebyshev spectral collocation method using a staggered grid for the stability of cylindrical flows. Int. J. Numer. Meth. Fluids 12, 825–833 (1991)CrossRefzbMATHGoogle Scholar
  17. 17.
    Kurz, H., Kloker, M.: Mechanisms of flow tripping by discrete roughness elements in a swept-wing boundary layer. J. Fluid Mech. 796, 158–194 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lesshafft, L., Huerre, P.: Linear impulse response in hot round jets. Phys. Fluids 19(2), 024102 (2007)CrossRefzbMATHGoogle Scholar
  19. 19.
    Loiseau, J.-C., Robinet, J.-C., Cherubini, S., Leriche, E.: Investigation of the roughness-induced transition: global stability analyses and direct numerical simulations. J. Fluid Mech. 760, 175–211 (2014)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Marquet, O., Sipp, D., Jacquin, L.: Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Michalke, A.: Survey on jet instability theory. Prog. Aerosp. Sci. 21, 159–199 (1984)CrossRefGoogle Scholar
  22. 22.
    Nichols, J.W., Lele, S.K.: Global modes and transient response of a cold supersonic jet. J. Fluid Mech. 669, 225–241 (2011)CrossRefzbMATHGoogle Scholar
  23. 23.
    Rodríguez, D., Cavalieri, A.V.G., Colonius, T., Jordan, P.: A study of linear wavepacket models for subsonic turbulent jets using local eigenmode decomposition of PIV data. Eur. J. Mech. B/Fluids 49, 308–321 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sipp, D., Lebedev, A.: Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358 (2007)CrossRefzbMATHGoogle Scholar
  25. 25.
    Trefethen, L. N., Embree, M.: Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press (2005). See also http://www.cs.ox.ac.uk/projects/pseudospectra/

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Laboratoire d’HydrodynamiqueCNRS/École polytechniquePalaiseauFrance

Personalised recommendations