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

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Global stability analysis of an idealized compressor blade row. I. Single-blade passage analysis

Anton Glazkov

Miguel Fosas de Pando

Peter J. Schmid*

Li He

  • Mechanical Engineering, Physical Science and Engineering Division, KAUST, Thuwal 23955, Saudi Arabia

  • Departamento Ingeniería Mecánica y Diseño Industrial, Escuela Superior de Ingeniería, Universidad de Cádiz, Puerto Real 11519, Spain

  • Mechanical Engineering, Physical Science and Engineering Division, KAUST, Thuwal 23955, Saudi Arabia

  • Department of Engineering Science, University of Oxford, Oxford OX1 3PJ, United Kingdom

  • *Corresponding author: peter.schmid@kaust.edu.sa
PDF

Phys. Rev. Fluids 8, 103903 – Published 11 October, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.103903

Abstract

A direct-adjoint mean flow global stability investigation of self-excited instabilities in an idealized, two-dimensional compressor blade row at off-design conditions is carried out, with a focus on acoustic feedback mechanisms underlying the observed instabilities. This paper is the first part of this work, where nonlinear flows, impulse responses and the global modes are computed for a single-passage system, with good agreement between the linear and nonlinear structures. Structural sensitivities and feedback loops are identified with the aid of wavemakers and show that dominant structures arise due to feedback mechanisms linking the pressure and suction sides of the aerofoil via acoustic waves emanating from the trailing edge. A separate, second part extends this analysis to multiple-blade passages per period window by exploiting the theory of block-circulant matrices and Bloch-wave theory.

View figure in article

Physics Subject Headings (PhySH)

See Also

Global stability analysis of an idealized compressor blade row. II. Multiple-blade interactions

Anton Glazkov, Miguel Fosas de Pando, Peter J. Schmid, and Li He
Phys. Rev. Fluids 8, 103904 (2023)

Article Text

References (60)

  1. N. Peake and A. B. Parry, Modern challenges facing turbomachinery aeroacoustics, Annu. Rev. Fluid Mech. 44, 227 (2012).
  2. P. J. Schmid, Nonmodal stability theory, Annu. Rev. Fluid Mech. 39, 129 (2007).
  3. K. Taira, S. L. Brunton, S. T. M. Dawson, C. W. Rowley, T. Colonius, B. J. McKeon, O. T. Schmidt, S. Gordeyev, V. Theofilis, and L. S. Ukeiley, Modal analysis of fluid flows: An overview, AIAA J. 55, 4013 (2017).
  4. J. L. Lumley, The Structure of Inhomogeneous Turbulent Flows, in Proceedings of the International Colloquium on the Fine Scale Structure of the Atmosphere and Its Influence on Radio Wave Propagation, edited by A. M. Yaglam and V. I. Tatarsky (Doklady Akademii Nauk SSSR, Nauka, Moscow, 1967).
  5. P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech. 656, 5 (2010).
  6. K. Taira, M. S. Hemati, S. L. Brunton, Y. Sun, K. Duraisamy, S. Bagheri, S. T. M. Dawson, and C.-A. Yeh, Modal analysis of fluid flows: Applications and outlook, AIAA J. 58, 998 (2020).
  7. P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows, Applied Mathematical Sciences Vol. 142 (Springer, New York, 2001).
  8. S. Beneddine, D. Sipp, A. Arnault, J. Dandois, and L. Lesshafft, Conditions for validity of mean flow stability analysis, J. Fluid Mech. 798, 485 (2016).
  9. M. P. Juniper, Absolute and Convective Instability in Gas Turbine Fuel Injectors, in Proceedings of the ASME Turbo Expo 2012: Turbine Technical Conference and Exposition. Volume 2: Combustion, Fuels and Emissions, Parts A and B. Copenhagen, Denmark (American Society of Mechanical Engineers, 2012), pp. 189–198.
  10. O. Tammisola and M. P. Juniper, Coherent structures in a swirl injector at Re=4800 by nonlinear simulations and linear global modes, J. Fluid Mech. 792, 620 (2016).
  11. K. Gudmundsson and T. Colonius, Instability wave models for the near-field fluctuations of turbulent jets, J. Fluid Mech. 689, 97 (2011).
  12. M. Fosas de Pando, P. J. Schmid, and D. Sipp, A global analysis of tonal noise in flows around aerofoils, J. Fluid Mech. 754, 5 (2014).
  13. M. Fosas de Pando, P. J. Schmid, and D. Sipp, On the receptivity of aerofoil tonal noise: An adjoint analysis, J. Fluid Mech. 812, 771 (2017).
  14. D. C. Hill, A theoretical approach for analyzing the restabilization of wakes, Tech. Rep., NASA Technical Memorandum 103858, Ames Research Center, 1992.
  15. F. Giannetti and P. Luchini, Structural sensitivity of the first instability of the cylinder wake, J. Fluid Mech. 581, 167 (2007).
  16. J. Chomaz, Global instabilities in spatially developing flows: Non-normality and nonlinearity, Annu. Rev. Fluid Mech. 37, 357 (2005).
  17. P. J. Schmid and L. Brandt, Analysis of fluid systems: Stability, receptivity, sensitivity, Appl. Mech. Rev. 66, 024803 (2014).
  18. D. X. Wang and L. He, Adjoint aerodynamic design optimization for blades in multistage turbomachines—Part I: Methodology and verification, J. Turbomach. 132, 021011 (2010).
  19. D. X. Wang, L. He, Y. S. Li, and R. G. Wells, Adjoint aerodynamic design optimization for blades in multistage turbomachines—Part II: Validation and application, J. Turbomach. 132, 021012 (2010).
  20. A. Jameson, Aerodynamic design via control theory, J. Sci. Comput. 3, 233 (1988).
  21. M. B. Giles and N. A. Pierce, An introduction to the adjoint approach to design, Flow, Turbul. Combust. 65, 393 (2000).
  22. L. N. Trefethen and D. Bau III, Numerical Linear Algebra (SIAM, Philadelphia, 1997).
  23. R. W. Paterson, P. G. Vogt, M. R. Fink, and C. Lee Munch, Vortex noise of isolated airfoils, J. Aircr. 10, (1973).
  24. C. K. W. Tam, Discrete tones of isolated airfoils, J. Acoust. Soc. Am. 55, 1173 (1974).
  25. M. R. Fink, Prediction of airfoil tone frequencies, J. Aircr. 12, (1975).
  26. E. C. Nash, M. V. Lowson, and A. McAlpine, Boundary-layer instability noise on aerofoils, J. Fluid Mech. 382, 27 (1999).
  27. S. Pröbsting, J. Serpieri, and F. Scarano, Experimental investigation of aerofoil tonal noise generation, J. Fluid Mech. 747, 656 (2014).
  28. G. Desquesnes, M. Terracol, and P. Sagaut, Numerical investigation of the tone noise mechanism over laminar airfoils, J. Fluid Mech. 591, 155 (2007).
  29. L. E. Jones and R. D. Sandberg, Numerical analysis of tonal airfoil self-noise and acoustic feedback-loops, J. Sound Vib. 330, 6137 (2011).
  30. M. Sanjose, A. Towne, P. Jaiswal, S. Moreau, S. Lele, and A. Mann, Modal analysis of the laminar boundary layer instability and tonal noise of an airfoil at Reynolds number 150,000, Intl. J. Aeroacoustics 18, 317 (2019).
  31. H. Wu, R. D. Sandberg, and S. Moreau, Stability characteristics of different aerofoil flows at Rec=150,000 and the implications for aerofoil self-noise, J. Sound Vib. 506, 116152 (2021).
  32. A. Glazkov, M. F. de Pando, P. J. Schmid, and L. He, following paper, Global stability analysis of an idealized compressor blade row. II. Multiple-blade interactions, Phys. Rev. Fluids 8, 103904 (2023).
  33. P. J. Schmid, M. F. de Pando, and N. Peake, Stability analysis for n-periodic arrays of fluid systems, Phys. Rev. Fluids 2, 113902 (2017).
  34. N. L. Sanger and R. P. Shreeve, Comparison of calculated and experimental cascade performance for controlled-diffusion compressor stator blading, J. Turbomach. 108, 42 (1986).
  35. T. Sasaki and F. Breugelmans, Comparison of sweep and dihedral effects on compressor cascade performance, J. Turbomach. 120, 454 (1998).
  36. H. Yang and L. He, Experimental study on linear compressor cascade with three-dimensional blade oscillation, J. Propul. Power 20, (2004).
  37. L. He and J. Yi, Two-scale methodology for URANS/Large Eddy Simulation solutions of unsteady turbomachinery flows, J. Turbomach. 139, 101012 (2017).
  38. L. He, Averaging for high fidelity modeling—Toward large eddy simulations in multi-passage multi-row configurations, J. Turbomach. 143, 021002 (2021).
  39. A. Glazkov, A direct-adjoint framework for stability and sensitivity analyses of turbomachinery aeroacoustics, Ph.D. thesis, University of Oxford, 2021.
  40. J. Sesterhenn, A characteristic-type formulation of the Navier-Stokes equations for high order upwind schemes, Comput. Fluids 30, 37 (2000).
  41. M. Fosas de Pando, Tonal noise generation in flows around aerofoils: A global stability analysis, Ph.D. thesis, École Polytechnique, 2012.
  42. N. A. Adams and K. Shariff, A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems, J. Comput. Phys. 127, 27 (1996).
  43. S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103, 16 (1992).
  44. C. A. Kennedy, M. H. Carpenter, and R. M. Lewis, Low-storage, explicit Runge-Kutta schemes for the compressible Navier-Stokes equations, Appl. Numer. Math. 35, 177 (2000).
  45. MPI Forum, https://www.mpi-forum.org/ (2021).
  46. 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).
  47. S. Abhyankar, J. Brown, E. M. Constantinescu, D. Ghosh, B. F. Smith, and H. Zhang, PETSc/TS: A modern scalable ODE/DAE solver library, arXiv:1806.01437.
  48. S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. D. Gropp et al., PETSc user's manual, Tech. Rep. ANL-95/11, Revision 3.13, Argonne National Laboratory, 2020.
  49. S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, A. Dener, V. Eijkhout, W. D. Gropp et al., PETSc Web page, https://www.mcs.anl.gov/petsc (2019).
  50. V. Hernandez, J. E. Roman, and V. Vidal, SLEPc, ACM Trans. Math. Softw. 31, 351 (2005).
  51. E. Åkervik, L. Brandt, D. S. Henningson, J. Hœpffner, O. Marxen, and P. Schlatter, Steady solutions of the Navier-Stokes equations by selective frequency damping, Phys. Fluids 18, 068102 (2006).
  52. J. Crouch, A. Garbaruk, and D. Magidov, Predicting the onset of flow unsteadiness based on global instability, J. Comput. Phys. 224, 924 (2007).
  53. B. T. Chu, On the energy transfer to small disturbances in fluid flow (Part I), Acta Mech. 1, 215 (1965).
  54. J. C. Schulze, P. J. Schmid, and J. L. Sesterhenn, Exponential time integration using Krylov subspaces, Int. J. Numer. Methods Fluids 60, 591 (2009).
  55. Lord Rayleigh, On the stability, or instability, of certain fluid motions, Proc. London Math. Soc. s1-11, 57 (1879).
  56. R. Sandberg and A. Wheeler, Effect of trailing-edge boundary conditions on acoustic feedback loops in high-pressure turbines, J. Sound Vib. 461, 114917 (2019).
  57. R. Parker, Resonance effects in wake shedding from compressor blading, J. Sound Vib. 6, 302 (1967).
  58. B. Hellmich and J. R. Seume, Causes of acoustic resonance in a high-speed axial compressor, J. Turbomach. 130, 031003 (2008).
  59. https://docs.scipy.org/doc/scipy/reference/interpolate.html.
  60. M. Hahsler, M. Piekenbrock, and D. Doran, dbscan: Fast density-based clustering with R, J. Stat. Softw. 91, 1 (2019).

Outline

Information

Phys. Rev. Fluids 8, 103903– Published 11 October, 2023
  • Received 13 December 2022
  • Accepted 25 May 2023
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by King Abdullah University of Science and Technology.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation