- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Stability analysis for n-periodic arrays of fluid systems

Peter J. Schmid*

Miguel Fosas de Pando†

N. Peake‡

  • Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom

  • Department of Mechanical Engineering and Industrial Design, University of Cádiz, 11519 Puerto Real, Spain

  • Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, United Kingdom

  • *pjschmid@imperial.ac.uk
  • †miguel.fosas@uca.es
  • ‡n.peake@damtp.cam.ac.uk
PDF

Phys. Rev. Fluids 2, 113902 – Published 13 November, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.113902

Abstract

A computational framework is proposed for the linear modal and nonmodal analysis of fluid systems consisting of a periodic array of n identical units. A formulation in either time or frequency domain is sought and the resulting block-circulant global system matrix is analyzed using roots-of-unity techniques, which reduce the computational effort to only one unit while still accounting for the coupling to linked components. Modal characteristics as well as nonmodal features are treated within the same framework, as are initial-value problems and direct-adjoint looping. The simple and efficient formalism is demonstrated on selected applications, ranging from a Ginzburg-Landau equation with an n-periodic growth function to interacting wakes to incompressible flow through a linear cascade consisting of 54 blades. The techniques showcased here are readily applicable to large-scale flow configurations consisting of n-periodic arrays of identical and coupled fluid components, as can be found, for example, in turbomachinery, ring flame holders, or nozzle exit corrugations. Only minor corrections to existing solvers have to be implemented to allow this present type of analysis.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (46)

  1. F. M. Esquivelzeta-Rabell, B. Figueroa-Espinoza, D. Legendre, and P. Salles, Shearing flow over an idealized wavy surface: Comparison between linear theory and DNS, Proceedings of the Third Brazilian Conference on Boiling, Condensation and Multiphase Flow, Curitiba, 2012 (unpublished).
  2. N. K. Ghaddar, K. Z. Korczak, B. B. Mikic, and A. T. Patera, Numerical investigation of incompressible flow in grooved channels. Part 1. Stability and self-sustained oscillations, J. Fluid Mech. 163, 99 (1986).
  3. N. K. Ghaddar, M. Magen, B. B. Mikic, and A. T. Patera, Numerical investigation of incompressible flow in grooved channels. Part 2. Resonance and oscillatory heat-transfer enhancement, J. Fluid Mech. 168, 541 (1986).
  4. J. Szumbarski, Instability of viscous incompressible flow in a channel with transversely corrugated walls, J. Theor. Appl. Mech. 45, 659 (2007).
  5. M. Choudhari, F. Li, and J. Edwards, Stability analysis of roughness array wake in a high-speed boundary layer, Proceedings of the 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition (AIAA, Reston, 2009), paper 2009-0170.
  6. S. D. M. Adams, R. V. Craster, and S. Guenneau, Bloch waves in periodic multi-layered acoustic waveguides, Proc. R. Soc. A 464, 2669 (2008).
  7. X. Garnaud and C. C. Mei, Bragg scattering and wave-power extraction by an array of small buoys, Proc. R. Soc. A 466, 79 (2010).
  8. J. I. Erdos, E. Alznert, and W. McNally, Numerical solution periodic transonic flow through a fan stage, AIAA J. 15, 1559 (1977).
  9. G. A. Gerolymos and V. C. Chapin, Generalized expression of chorochronic periodicity in turbomachinery blade-row interaction, Rech. Aérospat. 5, 69 (1991).
  10. G. A. Gerolymos, G. J. Michon, and J. Neubauer, Analysis and application of chorochronic periodicity in turbomachinery rotor/stator interaction computations, J. Propul. Power 18, 1139 (2002).
  11. L. He, Fourier methods for turbomachinery applications, Prog. Aero. Sci. 46, 329 (2010).
  12. H. D. Li and L. He, Single-passage solution of three-dimensional unsteady flows in a transonic fan rotor, Proc. Inst. Mech. Eng. A 215, 653 (2001).
  13. H. D. Li and L. He, Single-passage analysis of unsteady flows around vibrating blades of a transonic fan under inlet distortion, ASME J. Turbomach. 124, 285 (2002).
  14. M. Olaussona, L.-E. Eriksson, and S. Baralon, Evaluation of nonlinear rotor wake/stator interaction by using time domain chorochronic solver, Proceedings of the 8th International Symposium on Experimental and Computational Aerothermodynamics of Internal Flows, Lyon, 2007 (unpublished), paper ISAIF8-0064.
  15. G. Campa and S. M. Camporeale, Prediction of the thermoacoustic combustion instabilities in practical annular combustors, J. Eng. Gas Turbines Power 136, 091504 (2014).
  16. G. A. Mensah, G. Campa, and J. P. Moeck, Efficient computation of thermoacoustic modes in industrial annular combustion chambers based on Bloch-wave theory, J. Eng. Gas Turbines Power 138, 081502 (2016).
  17. J. P. Moeck, M. Paul, and C. O. Paschereit, Thermoacoustic instabilities in an annular Rijke tube, ASME Turbo Expo 2010: Power for Land, Sea, and Air (ASME, New York, 2010), paper GT2010-23577.
  18. S. Patil, T. Sedalor, D. Tafti, S. Ekkad, Y. Kim, P. Dutta, H.-K. Moon, and R. Srinivasan, Study of flow and convective heat transfer in a simulated scaled up low emission annular combustor, J. Thermal Sci. Eng. Appl. 3, 031010 (2011).
  19. G. Staffelbach, L. Y. M. Gicquel, G. Boudier, and T. Poinsot, Large eddy simulation of self excited azimuthal modes in annular combustors, Proc. Comb. Inst. 32, 2909 (2009).
  20. S. R. Stow and A. P. Dowling, Modelling of circumferential modal coupling due to Helmholtz resonators, ASME Turbo Expo 2003, collocated with the 2003 International Joint Power Generation Conference (ASME, New York, 2003), paper GT2003-38168.
  21. P. Wolf, G. Staffelbach, A. Roux, L. Y. M. Gicquel, T. Poinsot, and V. Moureau, Massively parallel LES of azimuthal thermo-acoustic instabilities in annular gas turbines, C. R. Mec. 337, 385 (2009).
  22. K. Ekici and K. C. Hall, Nonlinear analysis of unsteady flows in multistage turbomachines using the harmonic balance technique, Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit (AIAA, Reston, 2006), paper 2006-0422.
  23. A. K. Gopinath, E. van der Weide, J. J. Alonso, and A. Jameson, Three-dimensional unsteady multi-stage turbomachinery simulations using the harmonic balance technique, Proceedings of the 45th AIAA Aerospace Sciences Meeting and Exhibit (AIAA, Reston, 2007), paper 2007-0892.
  24. F. Bloch, Über die Quantenmechanik der Elektronen in Kristallgittern, Z. Phys. 52, 555 (1929).
  25. L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Non-Normal Matrices and Operators (Princeton University Press, Princeton, 2005).
  26. R. M. Gray, Toeplitz and circulant matrices: A review, Found. Trends Commun. Inf. Theory 2, 155 (2006).
  27. D. S. G. Pollock, Circulant matrices and time-series analysis, Int. J. Math. Edu. Sci. Technol. 33, 213 (2010).
  28. G. J. Tee, Eigenvectors of block circulant and alternating circulant matrices, Res. Lett. Inf. Math. Sci. 8, 123 (2005).
  29. P. J. Schmid, Nonmodal stability theory, Annu. Rev. Fluid Mech. 39, 129 (2007).
  30. P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows (Springer, New York, 2001).
  31. M. R. Jovanovic and B. Bamieh, Componentwise energy amplification in channel flows, J. Fluid Mech. 534, 145 (2005).
  32. P. Luchini and A. Bottaro, Adjoint equations in stability theory, Annu. Rev. Fluid Mech. 46, 493 (2014).
  33. V. Mehrmann and H. Voss, Nonlinear eigenvalue problems: A challenge for modern eigenvalue methods, GAMM Mitteil. 27, 121 (2004).
  34. F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev. 43, 235 (2001).
  35. J. Martins, P. Sturdza, and J. Alonso, The complex-step derivative approximation, ACM Trans. Math. Softw. 29, 245 (2003).
  36. J. Peter and R. P. Dwight, Numerical sensitivity analysis for aerodynamic optimization: A survey of approaches, Comput. Fluids 39, 373 (2010).
  37. L. Hascoët and V. Pascual, The Tapenade automatic differentiation tool: Principles, model, and specification, ACM Trans. Math. Softw. 39, 20 (2013).
  38. M. Fosas de Pando, D. Sipp, and P. J. Schmid, Efficient evaluation of the direct and adjoint linearized dynamics from compressible flow solvers, J. Comput. Phys. 231, 7739 (2012).
  39. The latest version of the code will be available in doi: 10.5281/zenodo.1040159.
  40. K. Taira and T. Colonius, The immersed boundary method: A projection approach, J. Comput. Phys. 225, 2118 (2007).
  41. C. S. Peskin, The immersed boundary method, Acta Numer. 11, 1 (2002).
  42. A. M. Roma, C. S. Peskin, and M. J. Berger, An adaptive version of the immersed boundary method, J. Comput. Phys. 153, 509 (1999).
  43. C. L. Ball, Advanced technology impact on compressor design and development: A perspective, NASA Report No. TM-102341, 1989 (unpublished).
  44. W. B. Gardner, Energy efficient engine (E3) technology status, Proceedings of the 20th Aerospace Sciences Meeting (AIAA, Reston, 1982), paper 1982-1052.
  45. V. Hernandez, J. E. Roman, and V. Vidal, SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems, ACM Trans. Math. Softw. 31, 351 (2005).
  46. P. R. Amestoy, I. S. Duff, and J.-Y. L'Excellent, Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. Methods Appl. Mech. Eng. 184, 501 (2000).

Outline

Information

Phys. Rev. Fluids 2, 113902– Published 13 November, 2017
  • Received 17 February 2017

©2017 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation